K. Tesch; Fluid Mechanics – Applications and Numerical Methods 1 Fluid Mechanics – Applications and Numerical Methods Krzysztof Tesch Fluid Mechanics Department Gda´ nsk University of Technology
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 1
Fluid Mechanics – Applications and
Numerical Methods
Krzysztof Tesch
Fluid Mechanics DepartmentGdansk University of Technology
Contents
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 2
Description of fluid at different scales
Turbulence modelling
Finite difference method
Finite element method
Finite volume method
Monte Carlo method
Lattice Boltzmann method
Other methods
References
Description of fluid at different
scales
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 3
Descriptions
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 4
Fluid motion may be described by three types ofmathematical models according to the observed scales:
Microscopic description (molecular mechanics anddynamics)
Mesoscopic description (kinetic theory, DPD) Macroscopic description (classical fluid mechanics)
Microscopic description
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 5
Molecular mechanics takes advantage of classicalmechanics equations to model molecular systemswhereas molecular dynamics simulates movements ofatoms in the context of N-body simulation. Themotion of molecules is determined by solving theNewtons’s equation of motion
md2ri
dt2= Gi +
N∑
j=1 6=i
fij. (1)
The force exerted on a molecule consists of theexternal force such as gravity Gi and theintermolecular force fij = −∇V usually described bymans of the Lennard-Jones potential
Microscopic description
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 6
V = 4ǫ
[
(
σ
‖r‖
)12
−(
σ
‖r‖
)6]
. (2)
In the above equations ‖r‖ is the distance betweenparticles, ǫ – the depth of the potential well thatcharacterises the interaction strength and σ – thefinite distance describing the interaction range.Further, the ensemble average makes it possible toobtain a macroscopic quantity from the correspondingmicroscopic variable. The disadvantage of moleculardynamics method is that the total number ofmolecules even in small volume is too large –proportional to 1023.
Molecular dynamics pseudocode
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 7
t← 0;Calculate initial molecule position r;while not the end of calculations do
fij ← −∇V ;a← m−1fij;r← r+ v∆t+ 1
2a∆t2;
t← t+∆t;
Molecular dynamics - example
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 8
Mesoscopic description
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 9
The key concept is the probability distributionfunction f (N) in the phase space. The phase space isconstituted of 3N spatial coordinates q1, . . . ,qN and3N momenta p1, . . . ,pN . The probability distributionfunction f (N) allows to express the probability to finda particle within the infinitesimal phase space
(q1,q1 + dq)× . . .× (qN ,qN + dq)×(p1,p1 + dp)× . . .× (pN ,pN + dp). (3)
The total number of molecules within the infinitesimalphase space is then
f (N) (q1, . . . ,qN ,p1, . . . ,pN ) dqN dpN . (4)
Mesoscopic description
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 10
The time evolution of the probability distributionfunction f (N) follows the Liouville equation
df (N)
dt=∂f (N)
∂t+
N∑
i=1
(
∂f (N)
∂pi· dpidt
+∂f (N)
∂qi· dqidt
)
= 0.
This means that the distribution function is constantalong any trajectory in phase space.The reduced probability distribution function isdefined as
Fs (q1, . . . ,qs,p1, . . . ,ps) =∫
R3(N−s)
∫
R3(N−s)
f (N) (q1, . . . ,qN ,p1, . . . ,pN ) dqN−s dpN−s.
Mesoscopic description
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 11
The above function is called the s-particle probabilitydistribution function. A chain of evolution equationsfor Fs for 1 ≤ s ≤ N is derived and called BBGKYhierarchy. This means that the sth equation for thes-particle distributions contains s+ 1 distribution.That hierarchy may be truncated. Truncating it at thefirst order results in Boltzmann equation
∂f
∂t+ v · ∇f = Ω(f) (5)
for the probability distribution function
f(r,v, t) = mNF1(q1,p1, t) (6)
for binary collisions with uncorrelated velocities beforethat collision.
Mesoscopic description – DPD
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 12
The DPD (Dissipative Particle Dynamics) methodsimulate only a reduced number of degrees of freedom(coarse-grained models). The motion of particles isdetermined by solving the Newtons’s equation ofmotion
md2ri
dt2= Gi +
N∑
j=1 6=i
(
fCij + fDij + fRij)
where the interaction forces are the sum of
fCij conservative or repulsion forces fDij dissipative forces fRij random force
Macroscopic description – conservation
equations
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 13
Conservation of mass
dρ
dt+ ρ∇ ·U = 0 (7)
Conservation of linear momentum
ρdU
dt= ρf +∇ · σ (8)
Decomposition of stress tensor σ = −ptδ+ τ.Another form of conservation of linear momentum
ρdU
dt= ρf −∇pt +∇ · τ (9)
Macroscopic description – energy equations
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 14
Energy Equation
Kinetic ρdekdt
= ρf ·U+∇ · (σ ·U)−∇ · qTotal ρdec
dt= ∂pt
∂t+∇ · (τ ·U)−∇ · q
Mechanical ρdemdt
= ∇ · (σ ·U)− σ : DInternal ρde
dt= σ : D−∇ · q
Enthalpy ρdhdt
= τ : D−∇ · q+ dptdt
Macroscopic description – general transport
equations
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 15
∂(ρf)
∂t+∇ · (ρUf) = Sf −∇ · k (10)
Left hand side represents transient and convectioneffects. It expresses rate of changeρdf
dt≡ ∂(ρf)
∂t+∇ · (ρUf). Right hand side represents
sources (positive and negative) and fluxes (transportdue to other mechanism than convection).
mass conservation equationf := 1, Sf := 0, k := 0
linear momentum conservation equationf ← U, Sf ← ρf , k← −σenergy conservation equationf := ek, Sf := ρf ·U, k := q− σ ·U
Macroscopic description – laws of
thermodynamics
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 16
Second law of thermodynamics
ρds
dt≥ −∇ · q
T. (11)
Entropy balance
ρds
dt=φ
T−∇ · q
T. (12)
First law of thermodynamics
ρde
dt= τ : D− pt∇ ·U−∇ · q. (13)
Macroscopic description – constitutive
equations
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 17
Mechanical (rheological) constitutive equations Equations of state Fluxes
Macroscopic description – mechanical
constitutive equations
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 18
Newtonian fluids τ = 2µD Non-Newtonian fluids
Generalised Newtonian fluids τ = 2µ(γ)D Differential type fluid τ = f(A1,A2, . . .)
Ai+1 =dAi
dt+Ai·
∂U
∂r+∇U·Ai, i = 1, 2, . . .
Integral type fluids
τ =t∫
−∞
f(t− τ) (δ−Ct(τ)) dτ
Rate type fluids τ = f(
τ,D, D)
τ+ λ1τ = 2µ(
D+ λ2D)
Generalised Newtonian fluids
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 19
τ1n = τ
1n
0+ (kγ)
1m
µ1n =(
τ0|γ|
)1n
+ k1m |γ| 1m− 1n
Szulman
τ = τ0 + kγn + µ∞γ
µ = τ0|γ| + k|γ|n−1 + µ∞
Generalised Herschel
τ1n = τ
1n
0+ (kγ)
1n
µ1n =(
τ0|γ|
)1n
+ k1n
Generalised Casson
m := n
τ = τ0 + kγn
µ = τ0|γ| + k|γ|n−1
Herschel-Bulkley
n := 1,
m := 1n
µ∞ := 0
τ = τ0 + k√γ + µ∞γ
µ = τ0|γ| +k√|γ|+ µ∞
Luo-Kuang
n := 12
√τ =√τ0 +√kγ
√µ =√
τ0|γ| +
√k
Casson
n := 2
τ = τ0 + kγµ = τ0|γ| + k
Bingham
n := 1
τ = kγn
µ = k|γ|n−1
Ostwald-de Waele
n := 1 τ0 := 0
τ = kγµ = k
Newton
τ0 := 0 n := 1
τ0 := 0
k := 0,
µ∞ := k,
τ0 := 0
Macroscopic description – equations of state
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 20
Fundamental equation of state e = f(s, ρ−1)
de
dt= T
ds
dt+ptρ2
dρ
dt(14)
Thermal equation of state pt = f(T, ρ−1)
pt = ρRT (15)
Caloric equation of state e = f(T, ρ−1)
de = cv dT +
(
T∂pt∂T− pt
)
dρ−1 (16)
Macroscopic description – fluxes
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 21
General form w = −T · ∇ϕ due to assumptionw = f(∇ϕ). More precisely w depends only on ϕ and∇ϕ. Fourier’s law
q = −λ · ∇T (17) Fick’s law
ji = −ρDij · ∇gi (18) Darcy’s law
U = −µ−1K · ∇p (19)
In the case of isotropy T = α δ andw = −α δ · ∇ϕ = −α∇ϕ.
Macroscopic description – applications 1
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 22
General form of the Navier-Stokes equation forNewtonian fluids
ρdU
dt= ρf −∇p+∇ ·
(
2µDD)
(20)
incompressible flow creeping flow inviscid flow Boussinesq approximation Oseen approximation filtration one-dimensional flows heat transfer
Macroscopic description – applications 2
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 23
– Incompressible flow (ρ = const)
ρdU
dt= ρf −∇p+∇ · (2µD) (21)
µ = const
ρdU
dt= ρf −∇p+ µ∇2U (22)
– creeping flow
ρ∂U
∂t= ρf −∇p+∇ · (2µD) (23)
2D creeping flow
∇4ψ = 0 (24)
Macroscopic description – applications 3
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 24
– inviscid flow (µ = 0)
ρdU
dt= ρf −∇p (25)
potential flows (∇×U = 0)
∇2ϕ = ∇2ψ = 0 (26)
– Boussinesq approximation– Oseen approximation U · ∇U ≈ U∞ · ∇U
ρ∂U
∂t+ ρU∞ · ∇U = ρf −∇p+∇ ·
(
2µDD)
(27)
Macroscopic description – applications 4
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 25
– filtration
ρdU
dt= ρf −∇p+ µ∇2U− R1U (28)
– one-dimensional flows
∇2U = a (29)
Macroscopic description – applications 5
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 26
– heat transfer The ‘fluid’ Fourier equation describesthe temperature field in the fluid.
cv
(
∂(ρT )
∂t+∇ · (ρTU)
)
= φµ +∇ · (λ∇T ) . (30)
For solids where U = 0 the above equation simplifiesto the ‘solid’ Fourier-Kirchhoff equation
c∂(ρT )
∂t= ∇ · (λ∇T ) + SE (31)
where internal energy sources are given by SE .
Comments
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 27
Generally, the ‘fluid’ equation should be solvedtogether with ‘solid’ equation. This is called conjugateheat transfer. Not having to know the heat transfercoefficient is an advantage of this approach. Thedisadvantage is the necessity of increasing the totalnumber of mesh elements due to the additional solidvolume.It is not always possible because of storagelimitations. Then either the temperature or heat fluxmust be specified at the wall. Alternatives, throughboundary conditions, are discussed further such asspecified temperature, specified heat flux, specifiedtemperature and heat flux, adiabatic or specified heattransfer coefficient.
Macroscopic description – dimensionless form
of equations
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 28
Mass conservation equation
∇+ ·U+ = 0 (32)Linear momentum conservation equation
ρ+(
Sh∂U+
∂t++U+ · ∇+U+
)
=
=ρ+f+
Fr− Eu∇+p+ +
µ+
Re∇2+U+ (33)
Fourier-Kirchhoff (internal energy)
ρ+c+v
(
Sh∂T+
∂t++U+ · ∇+T+
)
=
=Ec
Reφ+µ +
λ+
PrRe∇2+T+ (34)
Macroscopic description – dimensionless
numbers
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 29
Sh := LUt0
= tcht0
= LfU
=Ut0U2
L
Fr := U2
f0L=
ρ0U2
L
ρ0f0
Re := LUρ0µ0
= LUν0
=ρ0U
2
Lµ0U
L2
Eu := p0ρ0U2 =
p0L
ρ0U2
L
Ec := U2
cv0T0Pr := cv0µ0
λ0= ν0
λ0cv0ρ0
Sc := µ0ρ0D0
= ν0D0
Da := K0
L2
De := λ0t0
Wi := λ0γ0
Macroscopic description – compatibility
conditions
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 30
General transport equation
∂(ρf)
∂t+∇ · (ρUf) = Sf −∇ · k. (35)
From Reynolds’ transport theorem arises generalcompatibility condition n · [ρUf + k] = 0.
– Mass conservation: f := 1, Sf := 0, k := 0. C.C.takes form n · [ρU] = 0 or n · [U] ≡ [Un] = 0.– Linear momentum: f ← U, Sf ← ρf , k← −σ andC.C. n · [ρUU− σ] = 0 or n · [σ] = [σn] = 0.– Energy conservation: f := ek, Sf := ρf ·U,k := q− σ ·U and C.C. n · [ρUek − σ ·U+ q] = 0or n · [q] or [λ∂T
∂n] = 0.
Macroscopic description – boundary
conditions 1
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 31
Compatibility conditions are insufficient! Furtherconditions are needed:
adhesion l ·U = Ul = 0 thermal equilibrium on surfaces [T ] = 0
Boundary condition related to heat transfer (arisefrom C.C.)
Dirichlet: T = f1(x, y, z, t) Neumann: qn := n · q or qn = f2(x, y, z, t) mixed: αT − λ∂T
∂n= f3(P, t)
Macroscopic description – boundary
conditions 2
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 32
Other surfaces than walls
inlet: n− 1 conditions where n stands for thenumber of equations
outlet: Generally, σn = n · σ plus T distribution.Usually p distribution due to σn ≈ −pn plus∂T∂n
= 0
symmetry: ∂ϕ∂n
= 0 for all scalar variables ϕ periodicity (translation and rotation):
ϕ(P1) = ϕ(P2) where P1 and P2 arecorresponding points on periodic surfaces
Mathematical classification
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 33
The Navier-Stokes equations are second ordernonlinear partial differential equations. In general,they cannot be classified. However they possesproperties of quasi-linear, semi-linear and linear secondorder partial differential equations. Sometimes theycan be simplified to those and can be divided into:
hyperbolic, parabolic, elliptic.
Semi-linear second order PDEs
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 34
For two independent variables x, y:
A(x, y)∂2U
∂x2+B(x, y)
∂2U
∂x∂y+ C(x, y)
∂2U
∂y2+
F
(
x, y, U,∂U
∂x,∂U
∂y
)
= 0 (36)
for all (x, y) over a domain Ω the above equation is:
hyperbolic if B2 − 4AC > 0, parabolic if B2 − 4AC = 0, elliptic if B2 − 4AC < 0.
Important elliptic second order PDEs
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 35
Laplace equation
∂2U
∂x2+∂2U
∂y2= 0. (37)
Poisson equation
∂2U
∂x2+∂2U
∂y2= f(x, y). (38)
Helmholtz equation
∂2U
∂x2+∂2U
∂y2+ k2U = 0. (39)
Important parabolic second order PDEs
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 36
Heat equation
∂U
∂t− α∂
2U
∂x2= 0, (40)
∂U
∂t− α∂
2U
∂x2= f(x, t). (41)
Important hyperbolic second order PDEs
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 37
Wave equation
∂2U
∂t2− a2∂
2U
∂x2= 0. (42)
Telegraph equations
∂2U
∂x2− a∂
2U
∂t2− b∂U
∂t− c U = 0. (43)
Important mixed type second order PDEs
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 38
Euler-Tricomi equation
∂2U
∂x2− x∂
2U
∂y2= 0. (44)
It is of hyperbolic type for x > 0, parabolic atx = 0 and elliptic for x < 0.
Generalised Euler-Tricomi equation
∂2U
∂x2− f(x)∂
2U
∂y2= 0. (45)
Important mixed type second order PDEs
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 39
Potential gas flow
(1−Ma2∞)∂2ϕ′
∂x2+∂2ϕ′
∂y2= 0. (46)
It is of hyperbolic type for Ma2∞ > 1, parabolic atMa2∞ = 1 and elliptic for Ma2∞ < 1.The velocity potential for the x axis dominatedflow is
ϕ(x, y) = U∞x+ ϕ′(x, y). (47)
Velocity components are then given as
Ux(x, y) =∂ϕ
∂x= U∞ + U ′x(x, y), (48a)
Uy(x, y) =∂ϕ
∂y= U ′y(x, y). (48b)
Turbulence modelling
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 40
Turbulence features
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 41
Irregularity Unsteadiness 3-D in terms of space and vortex structures Diffusivity Dissipation Energy cascade Need for constant energy supply
Kolmogorov scales 1
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 42
Kolmogorov scales are the smallest vorticity scaleswhere nearly the whole dissipation takes place. Thereare three scales, for velocity UK = (νε)1/4, length
LK = (ν3ε−1)1/4
and time tK = (νε−1)1/2
.The Reynolds number for these scales
ReK =UKLKν
=(νε)1/4 (ν3ε−1)
1/4
ν= 1. (49)
It means that at this level the inertial forces are of thesame order as the viscous forces.The dissipation intensity of the kinetic energy offluctuation can also be estimated in terms of a lengthscale for large scale motion (vorticity) as
Kolmogorov scales 2
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 43
ε ∼ U2
t=U2
L/U =U3
L . (50)
It means that the energy U2 of the large scales isdissipated proportionally to time L/U . Substitutingthe dissipation in equation for Lk with that for ε wehave
LK =
(
ν3LU3
)1/4
. (51)
Introducing a Reynolds number for large scalesReL = UL
νit is possible to find a relation for the ratio
of length scales by means of this Reynolds number inthe form of
LLK∼( U3
ν3L
)1/4
L =
(ULν
)3/4
= Re3/4L . (52)
Turbulence glossary
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 44
DNS – Direct Numerical Simulation LES – Large Eddy Simulation RANS – Raynolds Averaged Navier-Stokes URANS – Unsteady Raynolds Averaged
Navier-Stokes DES – Detached Eddy Simulation SST – Shear Stress Transport RNG – ReNormalisation Group EARSM – Explicit Algebraic Reynolds Stress
Models RST – Reynolds Stress Transport
Turbulence modelling
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 45
DNS LES RANS
Models based on the Boussinesq hypothesis(0-eq, 1-eq, 2-eq models)
Models which do not take advantage of theBoussinesq hypothesis
RST models EARSM
Decompositions and averages
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 46
Decomposition f(r, t) = f(r) + f ′(r, t)
Realisations f(r) := limN→∞
1N
N∑
i=1
fi(r, t)
Time fτ (r) := limτ→∞
1τ
τ∫
0
f(r, t) dt
Time ft(r, t) :=1∆t
t+∆t∫
t
f(r, t) dt
Spatial fV (r, t) :=1|V |
∫∫∫
Vf(r, t) dV
Comments
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 47
In practice it is usually enough to know what theaverage velocity is (not the fluctuation). The velocityvector field is be decomposed into average andfluctuation components U = U+U′. The time ofaveraging ∆t should be chosen to be greater than thefluctuation range and smaller than the function that isgoing to be averaged. The averaging process of theNavier-Stokes equation introduces a number of newunknown functions.
RANS 1
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 48
Averaged mass conservation equation
∇ · U = 0 (53)
Averaged Navier-Stokes equation
ρ∂U
∂t+ ρ∇·
(
UU)
= ρf −∇p+µ∇2U−ρ∇·U′U′ (54)
Reynolds stress tensor R := −ρU′U′ and thetotal stress tensor σ := −pδ+ 2µD+R makes itpossible to obtain the averaged momentumequation
ρdU
dt= ρf +∇ · σ (55)
RANS 2
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 49
Averaged Fourier-Kirchhoff equation
cv
(
∂(ρT )
∂t+∇ ·
(
ρT U)
)
=
= 2µD2 +∇ · (λ∇T )− cv∇ ·(
ρT ′U′)
+ ρε.(56)
The averaging process of the Navier-Stokes equationintroduces six unknown (because of the symmetry)components of the Reynolds stress tensor. Theaveraged Fourier-Kirchhoff equations gives a furtherthree of the vector T ′U′.
Comments
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 50
It is important to realise that the closure of system ofthe mass conservation and Navier-Stokes equationshas been lost. Further modelling is required.Formulating additional relationships for unknownfunctions to achieve closure of equations is calledturbulence modelling. Any additional closure equationmust fulfil a few basic criteria such as coordinateinvariance. This is fulfilled by proper tensorformulation of the exact and modelled equations.Another criterion is called realisability meaning that asolution must be physical.Practically, however, it is difficult to achieve all theserequirements. This is because some parts of the exacttransport equations are modelled or even dropped.
Closure
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 51
There are two main approaches to achieve closure.The models may be divided between those whichassume the eddy viscosity hypothesis and those whichdo not
Models not assuming the eddy viscosity hypothesis
Reynolds stress transport equation Algebraic stress tensor models
Boussinesq hypothesis assumed
Reynolds stress transport equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 52
∂R
∂t+∇ ·
(
UR)
= −∇U · (RT +R)+
+∇ ·((
Ck2ε−1 + ν)
∇R)
−Π+2
3ρεδ (57)
The left hand side represents unsteadiness andconvection. On the right hand side the two first termsrepresent production. The two terms under divergenceare responsible for diffusion.The right hand side fourth term is the secondunknown tensor Π need to be modelled. The lastright hand side term 2
3ρεδ is the so called dissipation
tensor for isotropic turbulence.
Algebraic stress tensor models
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 53
Without using the eddy viscosity hypothesis twotransport equations for k and a second variable areformulated. Instead of the linear Boussinesqhypothesis – algebraic, non-linear relationships areformulated between the stress anisotropy tensor a andthe average flow properties. The tensor a is related tothe Reynolds stress tensor by:
a :=R
k− 2
3δ. (58)
Typically, relationships depend on average strain rateand spin tensors
a = f(D, Ω). (59)
Boussinesq hypothesis
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 54
The turbulence stresses is related to the mean flowRxy = µt
∂Ux
∂y. This linear relationship is
R = a0δ+ 2µtD. (60)
The trace of this relations allows to find a constant−2ρk = 3a0 which gives
R = −2
3ρkδ+ 2µtD. (61)
The Reynolds equation becomes
∂(ρU)
∂t+∇·
(
ρUU)
= ρf −∇pe+∇·(
2µeD)
(62)
where µe := µt + µ, pe := p+ 23ρk.
Eddy viscosity and diffusivity hypothesis
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 55
The eddy diffusivity hypothesis is introduced by directanalogy with the eddy viscosity hypothesis. It reducesthe number of unknown functions in theFourier-Kirchhoff equation −cvρT ′U′ = λt∇T .The Fourier-Kirchhoff equation then becomes
cv
(
∂(ρT )
∂t+∇ ·
(
ρT U)
)
= 2µD2+∇·(
λe∇T)
+ρε,
(63)where λt can be estimated by means of the turbulentPrandtl number λt =
µtcvPrt
. Effective conductivity isintroduced by means of the definitionλe := λt + λ = µtcv
Prt+ λ.
Trace of RST equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 56
Calculating the trace of the Reynolds stress transportequation
∂R
∂t+∇ ·
(
UR)
= −∇U · (RT +R)+
+∇ ·((
Ck2ε−1 + ν)
∇R)
−Π+2
3ρεδ (64)
results in kinetic energy k transport equation which isused in the preceding one- and two-equationturbulence models trR = −2ρk for µt = Cµρk
2ε−1.The traces of Π by definition trΠ = 0 and thetransport equation for k takes the following form
∂(ρk)
∂t+∇·(ρkU) = ∇U : R+∇·
((
µtσ−1k + µ
)
∇k)
−ρε(65)
Zero-equation model
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 57
Zero k and ε are assumed. It allows the Boussinesqequation to be reduced to R = 2µtD.Eddy viscosity µt is modelled by means of thePrandtl-Kolmogorov hypothesis. This hypothesiscomes directly from dimensionless analysis µt = cρUL.The velocity scale U is often approximated by meansof the maximal velocity |U|max and length scale L bythe volume of the flow domain |V | by U ∼ |U|max,L ∼ 3
√
|V |.The Boussinesq hypothesis takes the following form
R = Cρ 3√
|V ||U|maxD (66)
No new unknown functions! However, zero-equationmodels are not as accurate but they are robust (firstapproximation for more complex models).
One-equation model
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 58
One transport equation for k is introduced
∂(ρk)
∂t+∇·(ρkU) = ∇U : R+∇·
((
µtσ−1k + µ
)
∇k)
−ρε
where ‘production’ ∇U : R = 2µtD2. According to
Prandl-Kolmogorov hypothesis U =√k and ε ∼ U3
Lso
ε = k3/2L−1. Finally, the k transport equation arrives
ρdk
dt= 2µtD
2 +∇ ·((
µtσ−1k + µ
)
∇k)
− ρk3/2L−1
(67)where eddy viscosity is estimated as µt = ρ
√kL by
means of another Prandtl hypothesis.
Two-equation k-ε model
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 59
Two additional equations have to be formulated. Thefirst, for the kinetic energy k, comes from theReynolds stress transport equation
ρdk
dt= 2µtD
2 +∇ ·((
µtσk
+ µ
)
∇k)
− ρε
and that for the dissipation ε is analogous to it
ρdε
dt= Cε1
ε
k2µtD
2 +∇ ·((
µtσε
+ µ
)
∇ε)
−Cε2ρε2
k.
(68)Both of them are transport equations for a scalarfunction. The eddy viscosity depends on both k and εand is postulated, as previously, to have the formµt = Cµρ
k2
ε.
Comments
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 60
The five constants in equations are empirical, that is,should be deduced from experiment for a specificgeometry. This ‘standard’ set is given by
σk = 1, σε = 1.3,
Cµ = 0.09, Cε1 = 1.44, Cε2 = 1.92. (69)
Two-equation k-ω model
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 61
The turbulent frequency ω is proportional to the ratioof dissipation and kinetic energy ω ∼ ε
kand using the
constant Cµ, they are then related by ε = Cµkω. Theeddy viscosity takes the form µt = ρ k
ω. The two
transport equations take the following form
ρdk
dt= 2µtD
2+∇·((
µtσk1
+ µ
)
∇k)
−Cµρkω (70)
ρdω
dt= α1
ω
k2µtD
2 +∇ ·((
µtσω1
+ µ
)
∇ω)
− β1ρω2
(71)This ‘standard’ set is constant is σk1 = 2, σω1 = 2,Cµ = 0.09, α1 =
59, β1 =
340.
Two equations SST model
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 62
The shear stress model combines the k−ω model nearthe wall with the k − ε far from it. Firstly, the k − εmodel has to be transformed to the k − ω formulationby means of relation ε = Cµkω. This results in
∂(ρk)
∂t+∇·
(
ρkU)
= 2µtD2+∇·
((
µtσk2
+ µ
)
∇k)
−Cµρkω,
∂(ρω)
∂t+∇ ·
(
ρωU)
= α2ω
k2µtD
2 +∇ ·((
µtσω2
+ µ
)
∇ω)
+
−β2ρω2 + 2ρ
ωσω2∇k · ∇ω.
Additional cross-diffusion terms now appear. The‘standard’ set of constants is different from that forthe original k − ε σk2 = 1, σω2 = 0.856, Cµ = 0.09,α2 = 0.44, β2 = 0.0828.
Two equations SST model
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 63
Secondly, the equations for the k − ω model aremultiplied by a blending function F1 and thetransformed k − ε equations by (1− F1). Theequations then are added. This results in
∂(ρk)
∂t+∇·
(
ρkU)
= 2µtD2+∇·
((
µtσk3
+ µ
)
∇k)
−Cµρkω,
∂(ρω)
∂t+∇ ·
(
ρωU)
= α3ω
k2µtD
2 +∇ ·((
µtσω3
+ µ
)
∇ω)
+
−β3ρω2 + (1− F1)2
ωρσω3∇k · ∇ω.
Constants marked with the subscript ‘3’, namely σk3,σω3, α3, β3 are linear combinations of constants fromthe component models C3 = F1C1 + (1− F1)C2.
Additional passive transport equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 64
The additional variable transport equation may beadded to the closed system after averaging, as
∂(ρf)
∂t+∇·
(
ρfU)
= −∇·(
ρf ′U′)
−∇·k+Sf . (72)The first term of the right hand side can be modelledby means of the eddy diffusivity hypothesis and theturbulent diffusivity coefficient Γ , −ρf ′U′ = Γ∇fand the additional transport equation takes the form
∂(ρf)
∂t+∇ ·
(
ρfU)
= ∇ ·((
µtSct
+ ρD
)
∇f)
+ Sf
(73)where the turbulent diffusivity coefficient Γ may berepresented as a function of the eddy viscosity and theturbulent Schmidt number Γ := µt
Sctwhere Sc := µ
ρD.
Turbulent boundary layer
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 65
1 2 5 10 20 50 100 2000
5
10
15
20
25
y+
U+
1 2 5 10 20 50 100 200
0
5
10
15
20
25
Friction velocity Uτ :=√
τ0ρ
Characteristic length l := νUτ
dimensionless distance y+ := yl
dimensionless velocity U+ := Ux
Uτ
y+ < 11 laminar sub-layer
5 < y+ < 30 bufferregion
11 < y+ < 250 turbu-lent sublayer (log-lawlayer)
y+ < 250 inner turbu-lent boundary layer
y+ > 250 outer turbu-lent boundary layer
Filtration of N-S equations 1
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 66
Filtration of the N-S equations is associated with LESmethod. Small scales are removed by means offiltering
f(r, t) =
∫∫∫
R3
+∞∫
−∞
f(r ′′, t′′)G(r− r ′′, t− t′′) dt′′ dV ′′
where G is a filter. Typically it is a product
G(r− r ′′, t− t′′) := Gt(t− t′′)3∏
i=1
Gvi(xi−x′′i ). (74)
For Gt(t− t′′) := τ−1H(t′′) andGvi(xi − x′′i ) := δ(xi − x′′i ) we have time average
fτ (r) := limτ→∞
1τ
τ∫
0
f(r, t) dt.
Filtration of N-S equations 2
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 67
Filtration of the N-S equations results in
ρ∂U
∂t+ρ∇·
(
UU)
= ρf−∇p+∇·(
2µD− ρ (L+C+R))
(75)where Leonard’s decomposition
L := UU− UU, C := UU′ +U′U, R := U′U′
(76)represents the cross stress tensor C (interactionsbetween large and small scales), Reynolds subgridtensor R (interactions among subgrid scales) andLeonard tensor L (interactions among the largescales).For L = 0 and C = 0 we have Reynolds equations.
Finite difference method
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 68
The method
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 69
The finite difference method (introduced by Euler inXVIII century) replaces the region by a finite mesh ofpoints at which the dependent variable isapproximated.
All partial derivativesat each mesh pointare approximated fromneighbouring valuesby means of Taylor’stheorem. Thismeans that derivativesat each pointare approximated bydifference quotients.
Taylor’s theorem
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 70
Assuming that f has continuous derivatives overcertain interval the Taylor expansion is used
f(x0 +∆x) =m−1∑
n=0
dnf(x0)
n!+
dmf(c)
m!. (77)
where x := x0 +∆x, c = x0 + θ∆x and θ ∈]0; 1[.The above equation may also be written as
f(x0 +∆x) = f(x0) + f ′(x0)∆x+1
2f ′′(x0)∆x
2
+1
6f ′′′(x0)∆x
3 + . . . +1
m!f (m)(c)∆xm. (78)
Taylor’s theorem
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 71
Instead of f (m) at unknown point c it is rewritten interms of another unknown quantity of order ∆xm
f(x0 +∆x) = f(x0) + f ′(x0)∆x+ f ′′(x0)∆x2
2
+ . . . + f (m−1)(x0)∆xm−1
(m− 1)!+O(∆xm) (79)
Discarding (truncating) O(∆xm) one gets anapproximation to f . The error in this approximation isO(∆xm). Roughly speaking it says that knowing thevalue of f and the values of its derivatives at x0 it ispossible to write down the equation for its value atthe point x0 +∆x.
First order finite difference
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 72
Taking under consideration the Taylor expansion up tothe first derivative
f(x0 +∆x) = f(x0) + f ′(x0)∆x+O(∆x2) (80)
then neglecting O(∆x) and rearranging gives the firstorder finite difference approximation to f ′(x0)
f ′(x0) ≈f(x0 +∆x)− f(x0)
∆x. (81)
This approximation is called a forward approximation.Replacing ∆x by −∆x in Taylor expansion one getsbackward approximation
f ′(x0) ≈f(x0)− f(x0 −∆x)
∆x. (82)
Second order finite difference
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 73
Taking under consideration the Taylor expansion up tothe second derivative
f(x0+∆x) = f(x0)+f′(x0)∆x+f
′′(x0)∆x2
2+O(∆x3)
(83)then neglecting O(∆x2). Doing the same for −∆xand combining the two above we have
f ′(x0) ≈f(x0 +∆x)− f(x0 −∆x)
2∆x(84)
after neglecting O(∆x2). This is so called the secondorder central difference approximation to f ′(x0).
Second order finite difference
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 74
Higher order approximation to derivatives is alsopossible. This can be done by taking more terms inthe Taylor expansion. Doing so up to the third we get
f(x0 +∆x) = f(x0) + f ′(x0)∆x+1
2f ′′(x0)∆x
2
+1
6f ′′′(x0)∆x
3 +O(∆x4). (85)
Replacing ∆x for −∆x and combing the results thendropping O(∆x4) gives the second order symmetricdifference approximation to f ′′
f ′′(x0) ≈f(x0 +∆x)− 2f(x0) + f(x0 −∆x)
∆x2.
(86)
Differences
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 75
Selected finite differences approximation to first andsecond derivatives are given in the following table.These can be used to solve ordinary differentialequations by replacing derivatives by theirapproximations.
Approximation Type Order
f ′(x0)f(x0+∆x)−f(x0)
∆xforward 1st
f ′(x0)f(x0)−f(x0−∆x)
∆xbackward 1st
f ′(x0)f(x0+∆x)−f(x0−∆x)
2∆xcentral 2nd
f ′′(x0)f(x0+∆x)−2f(x0)+f(x0−∆x)
∆x2symmetric 2nd
Differences
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 76
Equations for f ′ approximate the slope of the tangentin x0 by means of chords (backward, forward andcentral finite difference).
Differences
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 77
The typical subscript notation is
f(x0 +mh, y0 + nh) ≡ fi+mj+n. (87)
Now it is possible to express selected finite differencesapproximations to derivatives in somewhat simplermanner
Approximation Type Order
f ′ifi+1−fi
hforward 1st
f ′ifi−fi−1
hbackward 1st
f ′ifi+1−fi−1
2hcentral 2nd
f ′′ifi+1−2fi+fi−1
h2symmetric 2nd
Poisson equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 78
Poisson equation is ∇2Uz = a. Two dimensionalversions of this equation is written as
∂2Uz∂x2
+∂2Uz∂y2
= a. (88)
The next step would be to replace second orderderivatives by symmetric finite differenceapproximation
Ui+1j − 2Uij + Ui−1jh2
+Uij+1 − 2Uij + Uij−1
h2= a.
(89)It can be rewritten to give Uij as a function ofsurrounding variables
Uij =Ui+1j + Ui−1j + Uij+1 + Uij−1 − ah2
4. (90)
Poisson equation - mesh and boundary
conditions
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 79
The domain is discretised in the x and y directions bymeans of constant mesh size h (figure on the left). Uzis unknown at black mesh points and known at whitepoints from the boundary condition.
For instance the Dirichletboundary conditionspecifies the values of Uzdirectly. In this case Uz = 0meaning no slip wall. If theboundary values are knownthen discrete Poissonequation gives a systemof linear equations for Uij.
Poisson equation - solution methods
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 80
The accuracy of results depends on the size of themesh represented here by h. Mesh size should bedecreased until there is no significant influence onnumerical results.The set of linear equations can be solved eitherdirectly by means of an appropriate method (Gausselimination for instance) or indirectly by means ofiterative solution methods or the relaxation method(point-Jacobi iteration)
Un+1ij =
Uni+1j + Un
i−1j + Unij+1 + Un
ij−1 − ah24
(91)
or point-Gauss-Seidel (faster than point-Jacobi)
Un+1ij =
Uni+1j + Un+1
i−1j + Unij+1 + Un+1
ij−1 − ah24
. (92)
Poisson equation - solution methods
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 81
Another indirect method is so called SuccessiveOver-Relaxation method
Un+1ij = (1− w)Un
ij+
wUni+1j + Un+1
i−1j + Unij+1 + Un+1
ij−1 − ah24
(93)
where w is a relaxation parameter. For w ∈]1, 2[ wehave over-relaxation and for w := 1 this methodcorresponds to the point-Gauss-Seidel method. Onecan also consider under-relaxation method forw ∈]0, 1[.The best choice of w value needs numericalexperiments. It also depends on specific problems.
Poisson FDM pseudocode
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 82
Data: Read input variables and BCsw ← 1; n← 1;repeat
R← 0;for i← 1 to imax do
for j ← 1 to jmax do
if not boundary(Unij) then
Un+1ij ← Un
i+1j+Un+1i−1j+U
nij+1+U
n+1ij−1−ah
2
4;
R← max(
|Un+1ij − Un
ij|, R)
;
Un+1ij ← (1− w)Un
ij + wUn+1ij ;
n← n+ 1;
until n ≤ nmax and R > Rmin;
Results - Poisson equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 83
24
68
102
4
6
810
0
3
6
2 4 6 8 10
2
4
6
8
10
x
y
Laplace equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 84
Laplace equation is ∇2ϕ = 0. Two dimensionalversions of this equation is written as
∂2ϕ
∂x2+∂2ϕ
∂y2= 0. (94)
Replacing second order derivatives by symmetric finitedifference approximation
ϕi+1j − 2ϕij + ϕi−1jh2
+ϕij+1 − 2ϕij + ϕij−1
h2= 0.
(95)It can be rewritten to give ϕij as a function ofsurrounding variables
ϕij =ϕi+1j + ϕi−1j + ϕij+1 + ϕij−1
4. (96)
Laplace eq. – Neumann boundary condition
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 85
Neumann boundary condition specifies values of thederivative ∂
∂nof a solution ϕ on boundary ∂Ω to fulfil
∂ϕ
∂n= N(x, y) (97)
where the normal derivatives is defined as
∂ϕ
∂n= n · ∇ϕ = nx
∂ϕ
∂x+ ny
∂ϕ
∂y(98)
and (x, y) ∈ ∂Ω. If U = ∇ϕ we get
∂ϕ
∂n= n ·U = nxUx + nyUy. (99)
Laplace eq. – Neumann boundary condition
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 86
We have two equation for apoint located on boundary ‘2’
∂2f
∂x2=fi+1j − 2fij + fi−1j
h2,
∂f
∂x=fi+1j − fi−1j
2h= Nij .
Point fi−1j is located outsidethe Ω area. Eliminating it weget
∂2f
∂x2=
2fi+1j − 2Nijh− 2fijh2
and
fij =2fi+1j + fi+1j + fij−1 − 2Nijh
4.
Laplace FDM pseudocode
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 87
Data: Read input variables and BCsw ← 1; n← 1;repeat
R← 0;for i← 1 to imax do
for j ← 1 to jmax do
if boundary(ϕnij) 6= 0 then
switch ϕ(Unij) do
case 1: ϕn+1ij ←
ϕni+1j+ϕ
n+1
i−1j+ϕn
ij+1+ϕn+1
ij−1
4;
;
case 2: ϕn+1ij ←
2ϕni+1j+ϕn
ij+1+ϕn+1
ij−1−2hNij
4;
;...
case 6: ϕn+1ij ←
2ϕni+1j+2ϕn
ij+1−2hNij
4;
;...
R← max(
|ϕn+1ij − ϕn
ij |, R)
;
ϕn+1ij ← (1− w)ϕn
ij + wϕn+1ij ;
n← n+ 1;
until n ≤ nmax and R > Rmin;
Results - Laplace equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 88
2 4 6 810 12 14 16 2
46810
−10−50
Biharmonic equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 89
The biharmonic equation is ∇4ψ ≡ ∇2 · ∇2ψ = 0.Two dimensional versions of this equation is written as
∂4ψ
∂x4+ 2
∂4ψ
∂x2∂y2+∂4ψ
∂y4= 0. (101)
It is a fourth-order elliptic partial differential equationthat describes creeping flows in terms of a streamfunction ψ where the velocity components areUx =
∂ψ∂y
and Uy = −∂ψ∂x.
The Dirichlet boundary condition specifies both: astream function ψ and its normal derivative ∂ψ
∂n. Two
conditions are needed due to the fourth order of thebiharmonic equation.
Biharmonic equation - approximation to
derivatives
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 90
The finite difference approximations to ∂4ψ∂x4
, ∂4ψ∂y4
are
∂4ψ
∂x4=ψi+2j + ψi−2j − 4ψi+1j − 4ψi−1j + 6ψij
h4,
(102a)
∂4ψ
∂y4=ψij+2 + ψij−2 − 4ψij+1 − 4ψij−1 + 6ψij
h4.
(102b)
The fourth order mixed derivative is approximated as
∂4ψ
∂x2∂y2=ψi+1j+1 + ψi−1j−1 + ψi−1j+1 + ψi+1j−1
h4
+4ψij − 2ψi+1j − 2ψi−1j − 2ψij+1 − 2ψij−1
h4. (103)
Biharmonic equation - discrete equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 91
From the discrete biharmonic equations ψij can beexpressed as a function of surrounding variables
ψij =−ψi+2j − ψi−2j − ψij+2 − ψij−2 + 4ψij
20
+ 8ψi−1j + ψij+1 + ψij−1 + ψi+1j
20
− 2ψi+1j+1 + ψi−1j−1 + ψi−1j+1 + ψi+1j−1
20. (104)
Biharmonic equation - boundary conditions
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 92
From the below figure two purely geometricrelationships arise ∂ψ
∂n= n · ∇ψ = −Ul,
∂ψ∂l
= l · ∇ψ = Un. For an impermeable boundary one
gets Un = 0⇒ ∂ψ∂l
= 0. The general relationshipbetween volumetric flow rate and the stream functionsis
V =
∫
L
U · n dL =
∫
L
∂ψ
∂ldL =
∫
L
dψ = ψA − ψB.
(105)
Biharmonic FDM pseudocode
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 93
Data: Read input variables and BCsw ← 1; n← 1;repeat
R← 0;for i← 1 to imax do
for j ← 1 to jmax do
if not boundary(Unij) then
Un+1ij ← −Un
i+2j−Uni−2j−U
nij+2−U
nij−2+4Un
ij
20+
8Uni−1j+U
nij+1+U
nij−1+U
ni+1j
20−
2Uni+1j+1+U
ni−1j−1+U
ni−1j+1+U
ni+1j−1
20;
R← max(
|Un+1ij − Un
ij|, R)
;
Un+1ij ← (1− w)Un
ij + wUn+1ij ;
n← n+ 1;
until n ≤ nmax and R > Rmin;
Results - biharmonic equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 94
15
1015
20 14
812
0
0.5
1
Complex geometry
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 95
Point 1 is located insidethe Ω area
f1 =h f0 + d f2h+ d
(106)
Point 1 is located outsidethe Ω area
f1 =h f0 − d f2h− d (107)
Navier-Stokes equations
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 96
Typical numerical approaches for the incompressibleNavier-Stokes equations:
Ωz-ψ (vorticity-stream function) formulationmethod
Artificial compressibility method Pressure/velocity correction (operator splitting
methods)
Projection methods MAC (Marker-and-Cell) Fractional step method SIMPLE (Semi-Implicit Method for Pressure
Linked Equations), SIMPLER (SIMPLERevisited), SIMPLEC
Bad idea
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 97
The incompressible Navier-Stokes equations
∂U
∂t+U · ∇U = −1
ρ∇p+ ν∇2U. (108)
Explicit forward difference in time
Un+1 −Un
∆t+Un ·∇Un = −1
ρ∇pn+ν∇2Un. (109)
Problems:
∇ ·Un+1 6= 0,pn+1 =?
Better idea
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 98
The incompressible Navier-Stokes equations
∂U
∂t+U · ∇U = −1
ρ∇p+ ν∇2U, (110a)
∇ ·U = 0. (110b)
Un+1 −Un
∆t+Un · ∇Un = −1
ρ∇pn+1 + ν∇2Un, (111a)
∇ ·Un+1 = 0. (111b)
Problems:
∇2pn+1 = ρ∆t∇ · (Un −∆tUn · ∇Un +∆t ν∇2Un)
BCs?
Semi implicit
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 99
The incompressible Navier-Stokes equations
∂U
∂t+U · ∇U = −1
ρ∇p+ ν∇2U, (112a)
∇ ·U = 0. (112b)
Semi implicit approach
Un+1 −Un
∆t+Un · ∇Un+1 = −1
ρ∇pn+1 + ν∇2Un+1, (113a)
∇ ·Un+1 = 0. (113b)
Fully implicit
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 100
The incompressible Navier-Stokes equations
∂U
∂t+U · ∇U = −1
ρ∇p+ ν∇2U, (114a)
∇ ·U = 0. (114b)
Fully implicit approach
Un+1 −Un
∆t+Un+1 · ∇Un+1 = −1
ρ∇pn+1 + ν∇2Un+1,
(115a)
∇ ·Un+1 = 0. (115b)
Artificial compressibility method
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 101
The incompressible Navier-Stokes equations
∂U
∂t+U · ∇U = −1
ρ∇p+ ν∇2U, (116a)
∇ ·U = 0. (116b)
Explicit forward difference in time
Un+1 −Un
∆t+Un · ∇Un = − 1
ρ0∇pn + ν∇2Un, (117a)
βpn+1 − pn
∆t+∇ ·Un = 0. (117b)
Projection method
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 102
Ut −Un
∆t= −Un · ∇Un + ν∇2Un, (118)
∇ ·Ut 6= 0, BC
Un+1 −Ut
∆t= −1
ρ∇pn+1, (119)
∇ ·Un+1 = 0, ¬BC
∇2pn+1 =ρ
∆t∇ ·Ut (120)
Square cylinder
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 103
Square cylinder
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 104
Square cylinder
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 105
Square cylinder
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 106
Square cylinder
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 107
Square cylinder
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 108
Square cylinder
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 109
Vorticity-stream function formulation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 110
The incompressible 2D Navier-Stokes equations∂U
∂t+U · ∇U = −1
ρ∇p+ ν∇2U, (121a)
Ωz =∂Uy∂x− ∂Ux
∂y. (121b)
2D Helmholtz equation (Ux =∂ψ∂y, Uy = −∂ψ
∂x)
∂Ωz
∂t+U · ∇Ωz = ν∇2Ωz, (122a)
∇2ψ = −Ωz. (122b)
Ωn+1z − Ωn
z
∆t+Un · ∇Ωn
z = ν∇2Ωnz , (123a)
∇2ψn+1 = −Ωn+1z . (123b)
Finite element method
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 111
Method of weighted residuals
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 112
The mathematical foundation of the finite elementmethod is in the method of weighted residuals.Imagine ordinary differential equation
y′′(x) + 20x = 0 (124)
subjected to boundary conditions y(0) = y(1) = 0.The exact general solution of this equation is
y(x) := −10
3x3 + C1x+ C2 (125)
and a specific solution subjected to boundaryconditions
y(x) := −10
3(x3 − x). (126)
Method of weighted residuals
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 113
The method seeks an approximate solution y in thegeneral form
y(x) :=N∑
i=1
CiNi(x) (127)
where Ni are known trial functions which should becontinuous and fulfilled boundary conditions. Theconstants Ci are unknown and they will bedetermined. A residual R appears when substitutingapproximate solution y into the differential equations
R(x) := y′′(x) + 20x 6= 0. (128)
The unknown Ci constant are determined fori = 1, . . . N from
∫ 1
0
Wi(x)R(x) dx = 0. (129)
Method of weighted residuals
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 114
∫ 1
0
Wi(x)R(x) dx = 0 i = 1, . . . , N (130)
Choices for the weighting functions Wi
Collocation method Wi(x) := δ(x− xi) Subdomain method
Wi(x) := H(x− xi−1)−H(x− xi) Galerkin’s method Wi(x) := Ni(x)
∫ 1
0
Ni(x)R(x) dx = 0 i = 1, . . . , N (131)
Least Squares Method Wi(x) :=∂R∂Ci
∫ 1
0
∂R
∂CiR(x) dx = 0 i = 1, . . . , N (132)
Method of weighted residuals - example
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 115
A polynomial trial functions can be assumed
N(x) := xr(x− 1)s. (133)
It is continuous and fulfils boundary conditions. Justone trial function for r = s = 1 is the simplest case
N1(x) := x(x− 1). (134)
The approximate solution y(x) :=∑N
i=1CiNi(x)where N := 1 takes the following form
y(x) = C1N1(x) = C1(x2 − x). (135)
Residual may now be expressed as
R(x) = 2C1 + 20x 6= 0. (136)
Method of weighted residuals - example
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 116
The unknown constant C1 may be determined uponintegrating (Galerkin’s method of weighted residuals)
∫ 1
0
x(x− 1)(2C1 + 20x) dx = 0. (137)
This gives −13(5 + C1) = 0 and allows to determine
C1 = −5. The approximate solution is now
y(x) = −5x(x− 1) (138)
and can be compared with the exact solution
y(x) := −10
3(x3 − x). (139)
Method of weighted residuals - example
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 117
The simplest case with just one trial functionapproximates the exact solution more or lessacceptably. Better agreement is possible with morethan one trial functions.
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
x
y
y
y
Method of weighted residuals - example
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 118
The two polynomial trial functions can be assumed
N1(x) := x(x− 1), N2(x) := x2(x− 1). (140)
Both are continuous and fulfil boundary conditions.The approximate solution y(x) :=
∑Ni=1CiNi(x)
where N := 2 takes the following form
y(x) = C1N1(x)+C2N2(x) = C1(x2−x)+C2(x
3−x2).(141)
Residual may now be expressed as
R(x) = 2C1 + 2C2(3x− 1) + 20x 6= 0. (142)
Method of weighted residuals - example
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 119
The unknown constants C1, C2 may be determinedupon integrating
∫ 1
0
x(x− 1)(2C1 + 2C2(3x− 1) + 20x) dx = 0,
∫ 1
0
x2(x− 1)(2C1 + 2C2(3x− 1) + 20x) dx = 0.
This gives 10 + 2C1 + C2 = 0 and1 + 1
6C1 +
215C2 = 0 and allows to determine
C1 = C2 = −103. The approximate solution is now
y(x) = −10
3x(x− 1)(x+ 1). (143)
Method of weighted residuals - example
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 120
The case with two trial function approximates theexact solution very well. It is far better that theprevious case. There is no visible difference. In fact, itis even the exact solution
−10
3x(x− 1)(x+ 1) = −10
3(x3 − x). (144)
Method of weighted residuals - comments
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 121
The method of weighted residuals constitutesfoundation of the final element method
The method exploits an integral formulation tominimise residual errors
Trail functions of this method are global. It isusually difficult task to find a proper one thatsatisfies boundary conditions. The moredimensions the worse.
Finite Element method
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 122
The approximate solution is defined as
y(x) =N+1∑
i=1
yini(x) (145)
over the interval [x1;xN+1] which is divided into Nsubintervals (elements). The unknown values of yi atpoint xi correspond to constants Ci of the weightedresiduals method. However, the trial function is localand defined as
ni(x) =
x−xi−1
xi−xi−1xi−1 ≤ x ≤ xi,
xi+1−xxi+1−xi
xi ≤ x ≤ xi+1,
0 otherwise
(146)
Trial functions
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 123
Trial function
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 124
For instance, if x ∈ [x3;x4] we have a linearinterpolation
y(x) = y3n3(x) + y4n4(x) = y3x4 − xx4 − x3
+ y4x− x3x4 − x3
.
(147)The above definition arises from the equation of theline passing through two different points (x3, y3) and(x4, y4)
(y − y3)(x4 − x3) = (y4 − y3)(x− x3) (148)
or knowing that y = a+ bx we solve for a and b from
y3 = a+ bx3, (149a)
y4 = a+ bx4. (149b)
Method
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 125
Let us consider the same differential equationy′′ + 20x = 0 as previously. The approximate solutionis in the form y(x) =
∑N+1i=1 yini(x). The idea of
minimising the residual R(x) = y′′ + 20x 6= 0 isadapted from the weighted residuals to the finiteelement method∫ xN+1
x1
nj(x)R(x) dx = 0 j = 1, . . . , N + 1.
The unknown values yj may be determined uponintegrating (Galerkin’s finite element method)
∫ xN+1
x1
nj(x)
(
N+1∑
i=1
(yini(x))′′ + 20x
)
dx = 0
j = 1, . . . , N + 1. (150)
Method
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 126
For the three trial functions nj we have threeequations. However, in any sub interval only two trialfunctions are nonzero. If so∫ x2
x1
n1(x) ((y1n1(x) + y2n2(x))′′ + 20x) dx = 0
∫ x2
x1
n2(x) ((y1n1(x) + y2n2(x))′′ + 20x) dx+
∫ x3
x2
n2(x) ((y2n2(x) + y3n3(x))′′ + 20x) dx = 0
∫ x3
x2
n3(x) ((y2n2(x) + y3n3(x))′′ + 20x) dx = 0
Method
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 127
We can decouple the second equation for the timebeing and couple it latter at the global assemblyprocess. We have then
∫ xj+1
xj
nk(x) ((yjnj(x) + yj+1nj+1(x))′′ + 20x) dx = 0
j = 1, . . . , N ; k = j, j + 1. (151)
The above equation is valid for each subinterval(element) and suggest the so called ‘element’formulation. It is also well known that
∫ xN+1
x1
y(x) dx =N∑
j=1
∫ xj+1
xj
y(x) dx. (152)
‘Element’ formulation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 128
The approximate solution is expressed as
ye = yjN1(x) + yj+1N2(x) = N · ye (153)
where the known local trial functions N and theunknown nodal values ye are collected as vectors
N := (N1, N2), (154a)
ye := (yj, yj+1). (154b)
The local trial functions are simply a linearinterpolation
N1 :=xj+1 − xxj+1 − xj
xj ≤ x ≤ xj+1, (155a)
N2 :=x− xj
xj+1 − xjxj ≤ x ≤ xj+1. (155b)
‘Element’ formulation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 129
For each element we have the Galerkin residualcondition
∫ xj+1
xj
NR dx = 0 j = 1, . . . , N. (156)
Taking under consideration our differential equationy′′ + 20x = 0 and the approximate solution ye it isnow possible to express the residual as
∫ xj+1
xj
N
(
d2yedx2
+ 20x
)
dx = 0. (157)
The second derivative has to be replaced. This is dueto linear nature of the trial functions.
‘Element’ formulation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 130
Integration by parts makes it possible to replace thesecond derivative
Ndyedx
∣
∣
∣
∣
xj+1
xj
−∫ xj+1
xj
dN
dx
dyedx
dx+
∫ xj+1
xj
N20x dx = 0.
Finally, matrix form of the Galerkin residual conditionfor each element is now
∫ xj+1
xj
dN
dx
dN
dxdx·ye =
∫ xj+1
xj
N20x dx+Ndyedx
∣
∣
∣
∣
xj+1
xj
j = 1, . . . , N.
Element equations
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 131
The so called ‘stiffness’ matrix for each element e maybe introduced
Ke :=
∫ xj+1
xj
dN
dx
dN
dxdx. (158)
The above matrix is symmetric. The so called‘displacement’ vector is also introduced
Fe :=
∫ xj+1
xj
N20x dx+ Ndyedx
∣
∣
∣
∣
xj+1
xj
. (159)
The Galerkin residual condition for each element maynow be written as
Ke · ye = Fe. (160)
Element equations
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 132
It is possible to simplify the matrices even further. Forthe linear trial functions one gets the ‘stiffness’ matrix
Ke :=
∫ xj+1
xj
(
dN1
dxdN1
dxdN1
dxdN2
dxdN1
dxdN2
dxdN2
dxdN2
dx
)
dx =1
xj+1 − xj
(
1 −1−1 1
)
and the ‘displacement vector’
Fe :=
∫ xj+1
xj
(
N120xN220x
)
dx+
(
N1dyedx
∣
∣
xj+1
xj
N2dyedx
∣
∣
xj+1
xj
)
.
If the gradients are dropped, as discussed further, wehave
Fe = −10
3(xj − xj+1)
(
2xj + xj+1
xj + 2xj+1
)
.
Element equations - example
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 133
For the interval [0; 1] divided equally into 3 elementswe have the element matrices
K1 = K2 = K3 =
(
3 −3−3 3
)
.
The global assembly process (coupling):
K =
K111 K12
1 0 0K12
1 K221 +K11
2 K122 0
0 K122 K22
2 +K113 K12
3
0 0 K123 K22
3
results in
K =
3 −3 0 0−3 6 −3 00 −3 6 −30 0 −3 3
.
Element equations - example
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 134
The ‘displacement’ vector for equally divided interval[0; 1] take form
F1 =
(
1027− dy(0)
dx2027
+dy( 1
3)
dx
)
,F2 =
(
4027− dy( 1
3)
dx5027
+dy( 2
3)
dx
)
,F3 =
(
7027− dy( 2
3)
dx8027
+ dy(1)dx
)
.
After the global assembly process one finally gets
F =
F 11
F 21 + F 1
2
F 22 + F 1
3
F 23
=
1027− dy(0)
dx602712027
8027
+ dy(1)dx
.
System of equations - example
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 135
The global (assembled) system of linear equations is
3 −3 0 0−3 6 −3 00 −3 6 −30 0 −3 3
·
y1y2y3y4
=
1027− dy(0)
dx602712027
8027
+ dy(1)dx
.
It cannot, however, be solved yet. This is due tonecessity of applying the global boundary conditions.These are y1 = y4 = 0. Two typical methods ofapplying them are discussed further.
Boundary conditions
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 136
Extracting only these equations that are related tounknown functions y2 and y3 for y1 = y4 = 0 results in
· · · ·· 6 −3 ·· −3 6 ·· · · ·
·
·y2y3·
=
·602712027
·
or simpler in
(
6 −3−3 6
)
·(
y2y3
)
=
(
602712027
)
The above system may now be solved to obtain theunknown values y2, y3.
Boundary conditions
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 137
The second method does not change the layout of thematrices. However, it involves modification of specificelements by multiplying them by a ‘large’ number.These elements are located on the diagonal of the‘stiffness’ matrix and corresponding positions of the‘displacement’ vector (if non-zero)
3 · 107 −3 0 0−3 6 −3 00 −3 6 −30 0 −3 3 · 107
·
y1y2y3y3
=
0602712027
0
.
ODE FEM pseudocode
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 138
Data: Read N elements, nodes and BCsCreate global matrix K and vectors F, y;for e← 1 to N do
Ke ←∫
edNdx
dNdx
dx;Fe ←
∫
eN20x dx;
Add Ke to K;Add Fe to F;
Apply BCs;Solve linear system K · y = F;
Results - ODE
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 139
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
x
y
3 elements, 4 nodes
Results - ODE
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 140
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
x
y
6 elements, 7 nodes
Results - ODE
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 141
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
x
y
9 elements, 10 nodes
Results - ODE
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 142
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
x
y
15 elements, 16 nodes
Linear and quadratic interpolation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 143
Considering line equation ye = a+ bx and utilising itfor two different points (xj, yj) and (xj+1, yj+1) wecan get the following system of equations
(
yjyj+1
)
=
(
1 xj1 xj+1
)
·(
ab
)
. (161)
It can be easily solved for a and b(
ab
)
=
(
1 xj1 xj+1
)−1
·(
yjyj+1
)
. (162)
Keeping in mind that ye = N · ye whereN = (N1, N2) and ye = (yj, yj+1) we can utilise thesolution for a and b to get
ye = a+bx =xj+1 − xxj+1 − xj
yj+x− xj
xj+1 − xjyj+1 = N1yj+N2yj+1.
Linear and quadratic interpolation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 144
Introducing L1 and L2 for a one-dimensional element
L1 := N1 =xj+1 − xxj+1 − xj
, (163a)
L2 := N2 =x− xj
xj+1 − xj(163b)
one can formulate similar system of equation forquadratic interpolation ye = a+ bx+ cx2 through thepoints (xj, yj), (xj+ 1
2, yj+ 1
2) and (xj+1, yj+1)
yjyj+ 1
2
yj+1
=
1 xj x2j1 xj+ 1
2x2j+ 1
2
1 xj+1 x2j+1
·
abc
. (164)
Linear and quadratic interpolation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 145
0 0.2 0.4 0.6 0.8 1
0
0.5
1
L1 L2
0 0.2 0.4 0.6 0.8 1
0
0.5
1
N1 N2 N3
The quadratictrial functionscan be alsoexpressed interms of lineartrial functions
N1 := L1 (2L1 − 1) ,
N2 := 4L1L2,
N3 := L2 (2L2 − 1)
Quadratic interpolation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 146
The ‘stiffness’ matrix for each element e may now becalculated as
Ke :=
∫ xj+1
xj
dN
dx
dN
dxdx =
1
3(xj+1 − xj)
7 −8 1−8 16 −81 −8 7
.
The ‘displacement’ vector is now
Fe :=
∫ xj+1
xj
N20x dx =10
3(xj+1−xj)
xj2(xj + xj+1)
xj+1
.
The Galerkin residual condition for each element isthe same as previously
Ke · ye = Fe. (166)
ODE quadratic FEM pseudocode
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 147
Data: Read N linear elements, n nodes and BCsInsert midpoints xj+ 1
2← xj+xj+1
2;
n← 2n− 1 ;Create global matrix K and vectors F, y;for e← 1 to N do
Ke ←∫
edNdx
dNdx
dx;Fe ←
∫
eN20x dx;
Add Ke to K;Add Fe to F;
Apply BCs;Solve linear system K · y = F;
ODE - linear vs quadratic interpolation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 148
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
x
y
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
x
y
6 elements, 7 nodes. 3 elements, 7 nodes.
Mesh refinement
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 149
The generalised p-norm is given by
‖f‖p :=(∫
L
|f(x)|p dx) 1
p
(167)
where for p := 2 we have a special case
‖f‖2 :=√
∫
L
f2(x) dx. (168)
The error E := y − y of a finite element solution ymay now be defined by means of 2-norm. It may takethe following form
‖E ′‖22 ≤ CN∑
e=1
r2e . (169)
Mesh refinement
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 150
The element residue re is defined as
re := |Le| ‖f + y′′e‖2 (170)
but due to the linear form of trial functions N ′′i = 0 itis true that y′′e = 0. This means that the elementresidue is re := |Le| ‖f‖2 and solution error
‖E ′‖22 ≤ CN∑
e=1
|Le|2‖f‖22. (171)
Element’s length is |Le| = xj+1 − xj and utilising thetrapezoidal rule we can express the element residue as
re := |Le|√
∫
Le
f(x)2 dx ≈ (xj+1 − xj)32
√
f2j + f2
j+1
2.
The above approximation is used for mesh refinement.
Mesh refinement - results
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 151
0.00
0.50
1.00
0
0.5
1
1.5
x
y
0 0.2 0.4 0.6 0.8 10
1
2
3
x
Residue
Mesh refinement - results
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 152
0.00
0.50
0.75
1.00
0
0.5
1
1.5
x
y
0 0.2 0.4 0.6 0.8 10
1
2
3
x
Residue
Mesh refinement - results
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 153
0.00
0.25
0.50
0.75
1.00
0
0.5
1
1.5
x
y
0 0.2 0.4 0.6 0.8 10
1
2
3
x
Residue
Mesh refinement - results
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 154
0.00
0.25
0.50
0.75
0.87
1.00
0
0.5
1
1.5
x
y
0 0.2 0.4 0.6 0.8 10
1
2
3
x
Residue
Mesh refinement - results
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 155
0.00
0.25
0.50
0.63
0.75
0.87
1.00
0
0.5
1
1.5
x
y
0 0.2 0.4 0.6 0.8 10
1
2
3
x
Residue
Mesh refinement - results
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 156
0.00
0.25
0.38
0.50
0.63
0.75
0.87
1.00
0
0.5
1
1.5
x
y
0 0.2 0.4 0.6 0.8 10
1
2
3
x
Residue
FEM for Poisson equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 157
Letthe two dimensional formof Poisson equation on Ω
∂2U
∂x2+∂2U
∂y2= −a (172)
be subjected to the Dirichlet boundary conditionU(x, y) = 0 for every (x, y) ∈ ∂Ω. It is true that
∫∫
Ω
f(x, y) dx dy =∑
e
∫∫
Ωe
f(x, y) dx dy. (173)
‘Element’ formulation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 158
The approximate solution is expressed as
Ue = N ·Ue (174)
where the known local trial functions N and theunknown nodal values Ue are
N := (N1, N2, N3), (175a)
Ue := (U1, U2, U3). (175b)
For each element we have the Galerkin residualcondition
∫∫
Ωe
NR dx = 0. (176)
‘Element’ formulation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 159
Taking under consideration Poisson equation and theapproximate solution Ue it is now possible to expressthe residual as
∫∫
Ωe
N
(
∂2Ue∂x2
+∂2Ue∂y2
+ a
)
dx dy = 0. (177)
The second derivative has to be replaced (due tolinear nature of the trial functions). This can be doneby means of Green’s first identity∫∫
S
(
ψ∂2ϕ
∂x2+ ψ
∂2ϕ
∂y2
)
dx dy =
∫
∂S
ψ∂ϕ
∂ndL
−∫∫
S
(
∂ψ
∂x
∂ϕ
∂x+∂ψ
∂y
∂ϕ
∂y
)
dx dy. (178)
‘Element’ formulation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 160
Integration by means of Green’s identity makes itpossible to replace the second derivative
∫∫
Ωe
(
∂N
∂x
∂Ue∂x
+∂N
∂y
∂Ue∂y
)
dx dy
−∫
∂Ωe
N∂Ue∂n
dL−∫∫
Ωe
Na dx dy = 0. (179)
The matrix form of the Galerkin residual condition foreach element can now be expressed
∫∫
Ωe
(
∂N
∂x
∂N
∂x+∂N
∂y
∂N
∂y
)
dx dy ·Ue =
∫∫
Ωe
Na dx dy.
Element equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 161
Introducing the ‘stiffness’ matrix for each element e
Ke :=
∫∫
Ωe
(
∂N
∂x
∂N
∂x+∂N
∂y
∂N
∂y
)
dx dy (180)
and the ‘displacement’ vector
Fe :=
∫∫
Ωe
Na dx dy (181)
one may obtain the Galerkin residual condition foreach element in the form
Ke ·Ue = Fe. (182)
Element equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 162
The expanded version of the ‘stiffness’ matrix is
Ke :=
∫∫
Ωe
(
∂N1
∂x∂N1
∂x+∂N1
∂y∂N1
∂y∂N1
∂x∂N2
∂x+∂N1
∂y∂N2
∂y∂N1
∂x∂N3
∂x+∂N1
∂y∂N3
∂y∂N1
∂x∂N2
∂x+∂N1
∂y∂N2
∂y∂N2
∂x∂N2
∂x+∂N2
∂y∂N2
∂y∂N2
∂x∂N3
∂x+∂N2
∂y∂N3
∂y∂N1
∂x∂N3
∂x+∂N1
∂y∂N3
∂y∂N2
∂x∂N3
∂x+∂N2
∂y∂N3
∂y∂N3
∂x∂N3
∂x+∂N3
∂y∂N3
∂y
)
dx dy.
Similarly, the same for the ‘displacement’ vector
Fe := a
∫∫
Ωe
N1
N2
N3
dx dy. (183)
The actual form of matrices depends on the trialfunctions. Linear form of these are discussed further.
Linear trial function
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 163
Considering plane equation
Ue = a+ bx+ cy (184)
one can formulate the following system of equations
U1 = a+ bxi + cyi (185a)
U2 = a+ bxj + cyj (185b)
U3 = a+ bxk + cyk (185c)
for three different points (xi, yi), (xj, yj), (xk, yk).Solving these for a, b and c results in
Ue =U1
2Se(ai+bix+ciy)+
U2
2Se(aj+bjx+cjy)+
U3
2Se(ak+bkx+cky).
Linear trial function
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 164
The linear trial functions then are
N1 =1
2Se(ai + bix+ ciy),
N2 =1
2Se(aj + bjx+ cjy),
N3 =1
2Se(ak + bkx+ cky)
where
ai = xjyk − xkyj ; aj = xkyi − xiyk; ak = xiyj − xjyi;bi = yj − yk; bj = yk − yi; bk = yi − yj;ci = xk − xj; cj = xi − xk; ck = xj − xi;
Se =1
2|ckbj − cjbk|;
Linear interpolation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 165
00.5
1 0
0.5
1
0
0.5
1
x
y
z
00.5
1 0
0.5
1
0
0.5
1
x
y
z
00.5
1 0
0.5
1
0
0.5
1
x
y
z
00.5
1 0
0.5
1
0
0.5
1
x
y
z
Integrals and derivatives
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 166
Now it is possible to calculate necessary derivativesappearing in the ‘stiffness’ matrix
∂N1
∂x=
bi2Se
,∂N2
∂x=
bj2Se
,∂N3
∂x=
bk2Se
,
∂N1
∂y=
ci2Se
,∂N2
∂y=
cj2Se
,∂N3
∂y=
ck2Se
.
The same concerns integrals appearing in the‘displacement’ vector
∫∫
Se
Nα1 N
β2N
γ3 dx dy = 2Se
α!β!γ!
(α+ β + γ + 2)!. (188a)
Element equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 167
Now it is possible to simplify the matrices evenfurther. For the linear trial functions one gets the‘stiffness’ matrix
Ke =1
4Se
b2i + c2i bibj + cicj bibk + cickbibj + cicj b2j + c2j bjbk + cjckbibk + cick bjbk + cjck b2k + c2k
(189)
and the ‘displacement vector’
Fe =Se3
aiajak
. (190)
Four element example
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 168
1(0, 0) 2(10, 0)
3(10, 10)4(0, 10)
5(5, 5)
1
2
3
4Ω
∂Ω
U = 0
U = 0
U=0
U=0
Global matrix
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 169
The global assembly process (coupling) for theconsidered four element case:
K=
K111 +K11
4 K121 0 K12
4 K131 +K13
4K12
1 K221 +K11
2 K122 0 K23
1 +K132
0 K122 K22
2 +K113 K12
3 K232 +K13
3K12
4 0 K123 K22
3 +K224 K23
3 +K234
K131 +K13
4 K231 +K13
2 K232 +K13
3 K233 +K23
4 K331 +K33
2 +K333 +K33
4
.
The element matrices are identical
K1 = K2 = K3 = K4 =1
2
1 0 −10 1 −1−1 −1 1
.
Finally, the global ‘stiffness’ matrix is
K =
1 0 0 0 −10 1 0 0 −10 0 1 0 −10 0 0 1 −1−1 −1 −1 −1 4
.
Global vector
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 170
The ‘displacement’ element vector for the consideredfour elements
F1 = F2 = F3 = F4 =25
3
111
.
After the global assembly process one finally gets
F =
F 11 + F 1
4
F 21 + F 1
2
F 22 + F 1
3
F 23 + F 2
4
F 31 + F 3
2 + F 33 + F 3
4
=50
3
11112
.
Poisson FEM pseudocode
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 171
Data: Read N elements, nodes and BCsCreate global matrix K and vectors F, y;for e← 1 to N do
Ke ←∫∫
Ωe
(
∂N∂x
∂N∂x
+ ∂N∂y
∂N∂y
)
dx dy;
Fe ←∫∫
ΩeNa dx dy;
Add Ke to K;Add Fe to F;
Apply BCs;Solve linear system K · y = F;
Results - PDE
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 172
24
68
10
24
6810
0
3
6
4 elements, 5 nodes
Results - PDE
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 173
24
68
10
24
6810
0
3
6
8 elements, 9 nodes
Results - PDE
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 174
24
68
10
24
6810
0
3
6
50 elements, 36 nodes
Results - PDE
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 175
24
68
10
24
6810
0
3
6
200 elements, 121 nodes
Results - PDE
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 176
24
68
10
24
6810
0
3
6
800 elements, 441 nodes
FEM for Laplace equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 177
Let the two dimensional form of Laplace equation
∂2ϕ
∂x2+∂2ϕ
∂y2= 0 (191)
or ∇2ϕ = 0 on Ω be subjected to both boundaryconditions on ∂Ω:
Neumann
∂ϕ
∂n= n·∇ϕ = n·(Ux, Uy) = nxUx+nyUy =: −fN ,
Dirichlet (as previously)
ϕ = const =: fD. (192)
‘Element’ formulation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 178
As previously, FEM formulation for Poisson equationsubjected to Dirichlet and Neumann BCs is∫∫
Ωe
(
∂N
∂x
∂Ue∂x
+∂N
∂y
∂Ue∂y
)
dx dy
−∫
∂Ωe
N∂Ue∂n
dL−∫∫
Ωe
Na dx dy = 0.
Poisson ∇2ϕ = −a with Dirichlet BC
∫∫
Ωe
(
∂N
∂x
∂N
∂x+∂N
∂y
∂N
∂y
)
dx dy·Ue =
∫∫
Ωe
Na dx dy,
Laplace ∇2ϕ = 0 with Dirichlet + Neumann BC∫∫
Ωe
(
∂N
∂x
∂N
∂x+∂N
∂y
∂N
∂y
)
dx dy·ϕe = −∫
∂Ωe
NfN dL.
Element equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 179
Introducing the ‘stiffness’ matrix as previously foreach element e
Ke :=
∫∫
Ωe
(
∂N
∂x
∂N
∂x+∂N
∂y
∂N
∂y
)
dx dy (193)
and the ‘displacement’ vector
Fe := −∫
∂Ωe
NfN dL (194)
one may obtain the Galerkin residual condition foreach element in the form
Ke ·ϕe = Fe. (195)
Element equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 180
The expanded version of the ‘stiffness’ matrix look thesame as previously but the ‘displacement’ vector isnow
Fe := fN
∫
∂Ωe
N dL = −1
2|L|fN1. (196)
The vector 1 may take of the three following forms
110
,
101
,
011
. (197)
|L| stands for element side length.
Laplace FEM pseudocode
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 181
Data: Read N elements, nodes and BCsCreate global matrix K and vectors F, y;for e← 1 to N do
Ke ←∫∫
Ωe
(
∂N∂x
∂N∂x
+ ∂N∂y
∂N∂y
)
dx dy;
Fe ← −∫
∂ΩeNfN dL;
Add Ke to K;Add Fe to F;
Apply BCs;Solve linear system K · y = F;
Geometry and mesh - Laplace equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 182
∂ϕ∂x=Ux
∂ϕ∂y
= 0
∂ϕ∂y
= 0
∂ϕ∂n
= 0
ϕ=const
456 nodes and 818 elements
Results - Laplace equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 183
−10
12 0
1
−2
1
−10
12 0
1
1
2
Geometry and mesh - Laplace equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 184
∂ϕ∂x=Ux
∂ϕ∂y
= 0
∂ϕ∂y
= 0
∂ϕ∂n
= 0∂ϕ∂y
= 0
ϕ=const
351 nodes and 607 elements
Results - Laplace equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 185
−10
12 0
1
−2
1
−10
12 0
1
0
1
2
Creeping flow – Stokes equations
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 186
Re≪ 1
0 = ρg −∇p+ µ∇2U,
∇ ·U = 0.
0 = ρgx −∂p
∂x+ µ
(
∂2Ux∂x2
+∂2Ux∂y2
)
,
0 = ρgy −∂p
∂y+ µ
(
∂2Uy∂x2
+∂2Uy∂y2
)
,
∂Ux∂x
+∂Uy∂y
= 0.
FEM formulation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 187
The approximate solution is expressed as
Ue = N ·Ue
where the quadratic trial functions N and theunknown nodal values Ue are
N := (N1, N2, N3, N4, N5, N6),
Ue := (U1, U2, U3, U4, U5, U6).
For each element we have the Galerkin residualcondition
∫∫
Ωe
NR dx = 0.
Quadratic interpolation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 188
The quadratictrial func-tions can beexpressed interms of lineartrial functions
N1 := L1 (2L1 − 1) ,
N2 := L2 (2L2 − 1) ,
N3 := L3 (2L3 − 1) ,
N4 := 4L1L2,
N5 := 4L2L3,
N6 := 4L1L3.
Momentum conservation equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 189
The Galerkin residual condition
∫∫
Ωe
N
(
ρgx −∂pe∂x
+ µ
(
∂2Uxe∂x2
+∂2Uxe∂y2
))
dx dy = 0
by means of Green’s first identity
∫∫
S
(
ψ∂2ϕ
∂x2+ ψ
∂2ϕ
∂y2
)
dx dy =
∫
∂S
ψ∂ϕ
∂ndL
−∫∫
S
(
∂ψ
∂x
∂ϕ
∂x+∂ψ
∂y
∂ϕ
∂y
)
dx dy
Momentum conservation equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 190
is
ρgx
∫∫
Ωe
N dx dy −∫∫
Ωe
N∂N
∂xdx dy · pe
− µ∫∫
Ωe
(
∂N
∂x
∂N
∂x+∂N
∂y
∂N
∂y
)
dx dy ·Uxe = 0
orKpxe · pe +Kxye ·Uxe = gxFe.
Similarly
Kpye · pe +Kxye ·Uye = gyFe.
Momentum conservation equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 191
Kpxe :=
∫∫
Ωe
N∂N
∂xdx dy
Kpye :=
∫∫
Ωe
N∂N
∂ydx dy
Kxye := µ
∫∫
Ωe
(
∂N
∂x
∂N
∂x+∂N
∂y
∂N
∂y
)
dx dy
Fe := ρ
∫∫
Ωe
N dx dy
Mass conservation equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 192
The Galerkin residual condition
∫∫
Ωe
N
(
∂Uxe∂x
+∂Uye∂y
)
dx dy = 0.
The matrix form of the Galerkin residual condition foreach element can now be expressed
∫∫
Ωe
N∂N
∂xdx dy ·Uxe +
∫∫
Ωe
N∂N
∂ydx dy ·Uye = 0
orKuxe ·Uxe +Kuye ·Uye = 0.
Mass conservation equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 193
Kuxe :=
∫∫
Ωe
N∂N
∂xdx dy
Kuye :=
∫∫
Ωe
N∂N
∂ydx dy
Element equations
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 194
Kpxe · pe +Kxye ·Uxe = gxFe,
Kpye · pe +Kxye ·Uye = gyFe,
Kuxe ·Uxe +Kuye ·Uye = 0
K6×6xye 06×6 K6×3
pxe
06×6 K6×6xye K6×3
pye
K3×6uxe K3×6
uye 03×3
·
U6×1xe
U6×1ye
p3×1e
=
gxF6×1e
gyF6×1e
03×1
K15×15e · u15×1
e = f15×1e
Finite volume method
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 195
The method
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 196
Equations are investigated in integral form incomparison with FDM. FVM is more flexible thanFDM in terms of spatial discretisation.Let us consider differential form of the generaltransport equation
∂(ρf)
∂t+∇ · (ρfU) = ∇ · (Γ∇f) + Sf . (203)
To obtain the integral form of this equation one needsGauss’s (divergence) theorem. Two dimensionalversion has the following form
∫∫
Ωi
∇ ·w dΩ =
∮
∂Ω+i
w · dL (204)
where dΩ ≡ dx dy and dL ≡ n dL ≡ ı dy − dx.
Integral form
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 197
Integrating over the two dimensional domain (finite‘volume’) Ωi and utilising Gauss’s theorem results in
d
dt
∫∫
Ωi
ρf dΩ+
∮
∂Ω+i
ρfU · dL =
∮
∂Ω+i
Γ∇f · dL+∫∫
Ωi
Sf dΩ.
For the sake of simplicity it is further assumed that wedeal with an incompressible case ρ = const.First and last integral in the above equation suggestthe following definition of an average fi value of fover Ωi
fi :=1
|Ωi|
∫∫
Ωi
f dΩ. (205)
The average value fi is typically located at the centreof the volume Ωi.
Spatial discretisation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 198
The next step would be the spatial discretisation overthe volume Ωi boundary ∂Ωi. The line integralrepresents the total flux out of volume Ωi and isreplaced by a sum
∮
∂Ω+i
w · dL ≈∑
k
wk ·∆Lk. (206)
Boundary ∂Ωi consists of lines indexed by subscript k.There are at least three lines (triangle). The vector wis either fU or Γ∇f . Because that vector w istypically not constant along each line it has to beapproximated by a single value wk at the centre ofeach line.
Time discretisation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 199
The last step would be time discretisation. Amongmany possibilities the simplest is the first orderforward finite difference approximation
dfidt≈ fn+1
i − fni∆t
. (207)
Time step of this approximation is denoted here as ∆t.Finally, one gets the following discretised version oftransport equation (i.e. Finite Volume Scheme)
ρfn+1i − fni
∆t|Ωi|+ ρ
∑
k
(fU)k ·∆Lk =
∑
k
(Γ∇f)k ·∆Lk + Sfi|Ωi|. (208)
|Ωi| stands for the area of control volume Ωi.
1D FVM diffusion problem
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 200
xi−1 xi xi+1
xi−12
xi+12
∆xi
∆xi−1 ∆xi+1
x0x1 x2 x3 = xN
xN+1
∆x1 ∆xN
∆x12
∆xN2
1D FVM diffusion problem
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 201
One dimensional and steady state diffusion equationof a f quantity arises from the general transportequation (convection-diffusion equation)
∂(ρf)
∂t+∇ · (ρUf) = ∇ · (Γ∇f) + Sf . (209)
If the diffusion coefficient Γ is constant then theabove equation simplifies to
∇ · (Γ∇f) + Sf = 0 (210)
or more precisely
d
dx
(
Γdf
dx
)
+ Sf = 0. (211)
1D FVM diffusion problem
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 202
The integral form of one dimensional diffusionequation takes the following form
∫ xi+1
2
xi− 1
2
d
dx
(
Γdf
dx
)
dx+
∫ xi+1
2
xi− 1
2
Sf dx = 0. (212)
There is no need to take advantage of Gauss’stheorem. This is because the first term can beintegrated directly
(
Γdf
dx
)
i+ 12
−(
Γdf
dx
)
i− 12
+ Sfi∆xi = 0 (213)
where ∆xi := xi+ 12− xi− 1
2.
1D FVM diffusion problem - Dirichlet BC
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 203
The average value of Sf at the centre of finite volumecan be approximated by means of trapezoidal rule bymeans of known values of Sf at the boundary of finitevolume
Sfi =Sfi− 1
2+ Sfi+ 1
2
2. (214)
Let us consider ODE
y′′(x) + 20x = 0 (215)
subjected to the Dirichlet boundary conditionsy(0) = y(1) = 0. The specific solution is
y(x) := −10
3(x3 − x). (216)
1D FVM diffusion problem
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 204
In other word, the diffusion coefficient Γ := 1 andsource term Sf := 20x. If so, then discrete version ofone dimensional diffusion equation is now
dfi+ 12
dx−
dfi− 12
dx+ Sfi∆xi = 0. (217)
Derivatives or diffusive fluxes at the boundary of finitevolume are approximated by means of the secondorder scheme as
dfi+ 12
dx≈ fi+1 − fixi+1 − xi
, (218a)
dfi− 12
dx≈ fi − fi−1xi − xi−1
. (218b)
1D FVM diffusion problem
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 205
The specific form of a finite volume scheme is now
fi+1 − fixi+1 − xi
− fi − fi−1xi − xi−1
+ Sfi∆xi = 0. (219)
It can also be rewritten to give fi as a function ofsurrounding variables
fi =∆xi+1fi−1 +∆xi−1fi+1 + Sfi∆xi−1∆xi+1∆xi
∆xi−1 +∆xi+1
where ∆xi−1 := xi − xi−1 and ∆xi+1 := xi+1 − xi.For ∆xi−1 = ∆xi+1 = ∆xi = h (i.e. uniform mesh)the finite volume scheme is reduced to a finitedifference scheme
fi =fi−1 + fi+1 + Sfih
2
2. (220)
1D FVM pseudocode
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 206
Data: Read volumes data and BCsCreate nodes and ghost nodes;n← 1;repeat
R← 0;for i← 2 to imax − 1 do
fn+1i ← ∆xi+1f
ni−1+∆xi−1f
ni+1+Sfi∆xi−1∆xi+1∆xi
∆xi−1+∆xi+1;
R← max(
|fn+1i − fni |, R
)
;
Update ghost nodes;n← n+ 1;
until n ≤ nmax and R > Rmin;
Results for nonuniform and uniform grids
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 207
0.2 0.5 0.72 0.920
0.5
1
1.5
x
y
0.13 0.38 0.63 0.880
0.5
1
1.5
x
y
1D FVM diffusion problem - Neumann BC
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 208
Let us consider the same ordinary differential equation
y′′(x) + 20x = 0 (221)
subjected to both the Dirichlet y(0) = 0 andNeumann y′(1) = 0 boundary conditions. The specificsolution is now
y(x) := −10x(
x2
3− 1
)
. (222)
Results for nonuniform and uniform grids
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 209
0.2 0.5 0.72 0.920
2
4
6
x
y
0.13 0.38 0.63 0.880
2
4
6
x
y
2D FVM diffusion problem
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 210
Two dimensional and steady state diffusion equationof a f quantity arises, as previously, from the generaltransport equation (convection-diffusion equation)
∇ · (Γ∇f) + Sf = 0 (223)
If the diffusion coefficient is constant Γ := 1 and thesource term Sf := a then the above equationsimplifies to
∇ · ∇f + a = 0 (224)
or ∇2f = −a which is Poisson equation.
2D FVM diffusion problem
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 211
The discrete version of the diffusion equation(simplified version of general transport equation) is
∑
k
(Γ∇f)k ·∆Lk + Sfij|Ωij | = 0. (225)
For a structural and Cartesian mesh (next slide) thenormal ∆Lk vectors are
∆LAB = |AB |ı = ∆yi ı, (226a)
∆LBC = |BC| = ∆xi , (226b)
∆LCD = |CD| (−ı) = −∆yi ı, (226c)
∆LDA = |DA| (−) = −∆xi . (226d)
2D FVM diffusion problem
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 212
fi−1j−1
fij−1
fi+1j−1
fi−1j fij fi+1j
fi−1j+1 fij+1 fi+1j+1
A
BC
D∆~LAB
∆~LBC
∆~LCD
∆~LDA
2D FVM diffusion problem
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 213
The discrete version of two dimensional diffusionequation is now
Γ∂fi+ 1
2j
∂x∆yi + Γ
∂fij+ 12
∂y∆xi
− Γ∂fi− 1
2j
∂x∆yi − Γ
∂fij− 12
∂y∆xi + Sfij|Ωij | = 0
(227)where the area of volume Ωij is |Ωij| = ∆xi∆yi andthe diffusion coefficient is constant Γ := 1. If so, then
∂fi+ 12j
∂x∆yi +
∂fij+ 12
∂y∆xi −
∂fi− 12j
∂x∆yi
−∂fij− 1
2
∂y∆xi + Sfij∆xi∆yi = 0. (228)
2D FVM diffusion problem
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 214
The average value of Sf at the centre of finite volumeis approximated by means of known values of Sf atthe boundary of finite volume
Sfij =1
4
(
Sfi− 12j− 1
2+ Sfi+ 1
2j− 1
2Sfi− 1
2j+ 1
2+ Sfi+ 1
2j+ 1
2
)
.
Derivatives at the boundary of finite volume areapproximated by means of the second order scheme as
∂fi+ 12j
∂x≈ fi+1j − fijxi+1j − xij
=fi+1j − fij∆xi+1
, (229a)
∂fij+ 12
∂y≈ fij+1 − fijyij+1 − yij
=fij+1 − fij∆yj+1
, (229b)
2D FVM diffusion problem
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 215
∂fi− 12j
∂x≈ fij − fi−1jxij − xi−1j
=fij − fi−1j∆xi−1
, (230a)
∂fij− 12
∂y≈ fij − fij−1yij − yij−1
=fij − fij−1∆yj−1
. (230b)
The specific form of a finite volume scheme is now
Sfij∆xi∆yi+fi−1j∆yi∆xi−1
+fi+1j∆yi∆xi+1
+fij−1∆xi∆yj−1
+fij+1∆xi∆yj+1
−
fij
(
∆yi∆xi−1
+∆yi∆xi+1
+∆xi∆yj−1
+∆xi∆yj+1
)
= 0.
2D FVM diffusion problem
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 216
It can also be rewritten to give fij as a function ofsurrounding variables
fij =
fi−1j∆yi∆xi−1
+fi+1j∆yi∆xi+1
+fij−1∆xi∆yj−1
+fij+1∆xi∆yj+1
+ Sfij∆xi∆yi∆yi
∆xi−1+ ∆yi
∆xi+1+ ∆xi
∆yj−1+ ∆xi
∆yj+1
.
For∆xi = ∆yi = ∆xi−1 = ∆xi+1 = ∆yi−1 = ∆yi+1 = h(i.e. uniform mesh) the finite volume scheme reducedto a finite difference scheme for Poisson equation
fij =fi−1j + fi+1j + fij−1 + fij+1 + Sfijh
2
4. (231)
2D FVM pseudocode
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 217
Data: Read volumes data and BCs
Create nodes and ghost nodes;
n← 1;
repeat
R← 0;
for i← 2 to imax − 1 do
for j ← 2 to jmax − 1 do
fn+1ij ←fni−1j∆yi
∆xi−1+
fni+1j∆yi
∆xi+1+
fnij−1∆xi
∆yj−1+
fnij+1∆xi
∆yj+1+Sfij∆xi∆yi
∆yi∆xi−1
+∆yi
∆xi+1+
∆xi∆yi−1
+∆xi
∆yi+1
;
R← max
(
|fn+1ij − fnij |, R
)
;
Update ghost nodes;
n← n+ 1;
until n ≤ nmax and R > Rmin;
Nonuniform and uniform volumes and nodes
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 218
Results for nonuniform and uniform mesh
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 219
02
46
810 0
24
6810
0
3
6
02
46
810 0
24
6810
0
3
6
Results for nonuniform and uniform mesh
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 220
Monte Carlo method
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 221
Monte Carlo integration
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 222
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
x
y
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
xy
f(x) := xx
Monte Carlo integration
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 223
The simplest Monte Carlo integration is based onsampling uniformly distributed points (U ,U)
∫ 1
0
f(x) dx ≈ 1
n
n∑
i=1
F (U ,U) (232)
where
F (x, y) :=
1; if f(x) ≥ y
0; otherwise. (233)
Estimation of π
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 224
−1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
x
y
Estimation of π
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 225
The area |D| = π of an unit circle
D :=
(x, y) : x2 + y2 ≤ 1
(234)
is estimated as
π =
∫∫
D
F (x) dx ≈ 4
n
n∑
i=1
F (U ,U) (235)
where
F (x, y) :=
1; if x2 + y2 ≤ 1
0; otherwise. (236)
Wiener process and random walk 1D
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 226
0 20 40 60 80 1000
0.2
0.4
0.6
0.8
1
Poisson equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 227
Poisson equation subjected to the Dirichlet boundarycondition
∇2U(x) = −f(x); ∀x ∈ Ω, (237a)
U(x) = g(x); ∀x ∈ ∂Ω. (237b)
It can be solved as an expected value of random pathsof a stochastic process
U(x) = E
[
g(Wτ ) +12
∫ τ
0
f(Wt) dt
]
(238)
where t is a terminal time of a random walk
τ = inf t :Wt ∈ ∂Ω . (239)
Poisson equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 228
If the Dirichlet boundary condition is g(x) = 0 then
U(x) = 12E
[∫ τ
0
f(Wt) dt
]
. (240)
We can also estimate
∫ τ
0
f(Wt) dt ≈m∑
i=1
fi(Wτ )∆t =m∑
i=1
fi(Wτ )h
V (h).
(241)Let us consider an ordinary differential equationy′′(x) = −20x subjected to boundary conditionsy(0) = y(1) = 0 (i.e. U := y, f(x) := 20x,g(x) := 0).
Monte Carlo ODE pseudocode
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 229
Data: Read input variablesfor i← 1 to imax do
if not boundary(xi) thenS ← 0;for k ← 1 to n do
I ← 0;α← i;while not boundary(xα) do
α← α + 2⌊U(0, 1) + 12⌋ − 1;
I ← I + 20f(xα);
S ← S + I;
yi ← h2
2Sn;
Results – ODE
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 230
n = 10
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
x
y
Results – ODE
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 231
n = 100
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
x
y
Results – ODE
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 232
n = 1000
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
x
y
Monte Carlo Poisson pseudocode
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 233
Data: Read input variables and BCsfor i← 1 to imax do
for j ← 1 to jmax do
if not boundary(Uij) thenτn ← 0;for k ← 1 to n do
S ← 0;α← i; β ← j;while not boundary(Uαβ) do
α← α+ 2⌊U(0, 1) + 12⌋ − 1;
β ← β + 2⌊U(0, 1) + 12⌋ − 1;
S ← S + 1;τn ← τn + S;
Uij ← −a h2
2τnn;
Results – Poisson equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 234
24
68
102
4
6
810
0
3
6
n = 10
Results – Poisson equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 235
24
68
102
4
6
810
0
3
6
n = 50
Results – Poisson equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 236
24
68
102
4
6
810
0
3
6
n = 100
Results – Poisson equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 237
24
68
102
4
6
810
0
3
6
n = 500
Results – Poisson equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 238
24
68
102
4
6
810
0
3
6
n = 5000
Lattice Boltzmann method
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 239
Definitions and ideas
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 240
The kinetic theory of gases treats gas as a largenumber of small molecules. They are in constant (andrandom) motion and constantly collide with oneanother.Knowing the position and velocity of each particle atsome instant in time it would be possible to know theexact dynamical state of the whole system. Themotion of particles could then be described by meansof classical mechanics. This would allow for predictionof all future states of the system.Due to the large number of molecules a statisticaltreatment is possible and necessary.
Assumptions
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 241
The gas is compose of small molecules whichmeans that the average distance separatingparticles is large in comparison with their size.
Molecules are in constant and random motion The large number of molecules make it possible to
apply statistical treatment Molecules have the same mass and spherical
shape Molecules constantly and elastically collide The only interaction is due to collision (no other
forces on one another)
Phase space and the distribution function
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 242
For N molecules we can think of phase space in whichthe coordinates consist of the position xi, velocityvectors (vi) and the time t. For a three dimensionalcase we have 6N dimensional phase space (threecoordinates + three velocities times N molecules).The system can be described by a probabilitydistribution function f that depends on 6N variablesplus time t.For a single molecule this reduces to 6 dimensionalphase space (x1, x2, x3, v1, v2, v3). This can betreated as a statistical approach in which a system isrepresented by an ensemble of many copies.
Interpretation of the distribution function
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 243
The elementary volume and dV and product dv ofelementary velocities are defined as
dV :=D∏
i=1
dxi, dv :=D∏
i=1
dvi (242)
where D means the physical dimension size. Thedistribution f that depends on r,v, t represents theprobability of finding a particular molecule mass witha given position and velocity per unit phase space.
∫
RD
∫
RD
f(r,v, t) dV dv (243)
The above integrate represents the total mass ofmolecules.
Continuous Boltzmann equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 244
If no collisions occur then the probability of finding aparticular molecule mass with a given position andvelocity at (r,v, t) equals the probability at(r+ dr,v + dv, t + dt)
f(r+ dr,v+ dv, t+ dt) dV dv−f(r,v, t) dV dv = 0.
If, however, collisions take place then
f(r+ dr,v+ dv, t+ dt) dV dv−f(r,v, t) dV dv =
Ω(f) dV dv dt
where Ω is so called collision operator. It takes underconsideration collisions during dt interval.
Continuous Boltzmann equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 245
We can now expand the left hand side of the previousequation by means of Taylor’s theorem
f(r+ dr,v + dv, t + dt) ≈
f(r,v, t) + dr · ∇f + dv · ∇vf +∂f
∂tdt. (244)
The two above equations give the Boltzmann equation
∂f
∂t+ v · ∇f +
F
m· ∇vf = Ω(f) (245)
where v = drdt
and mdvdt
= F.
Collision operator
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 246
The simplification of the complicated collisionoperator Ω is needed. It should, however, fulfil at leasttwo conditions:
conservation of collision invariants ϕ
∫
RD
ϕΩdv = 0 (246)
where collision invariants are: 1 (obvious), v and12‖v‖2.
tendency to the Maxwell-Boltzmann distribution(relaxation to local equilibrium)
BGK approximation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 247
The BGK (Bhatnagar-Gross-Krook) approximation isthe most popular simplification of the collisionoperator
Ω =1
τ(f eq − f) . (247)
It expresses relaxation to local equilibrium f eq withthe relaxation time τ . Both conditions are fulfilled.The Boltzmann equations without external forces F isnow
∂f
∂t+ v · ∇f =
1
τ(f eq − f) . (248)
Now the equation is linear! More precisely it is a linearpartial differential equation.
Maxwell-Boltzmann distribution
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 248
It is the basic law of the kinetic theory of gases. TheMaxwell-Boltzmann distribution is used for moleculesbeing not far from thermodynamic equilibrium. Othereffects like quantum effects and relativistic speeds areneglected. The distribution is
f eq = ρ (2πRT )−D2 e−
‖v−U‖2
2RT =
ρ(√
2πcs
)−D
e−
‖c‖2
2c2s . (249)
This distribution is valid for freely moving moleculeswithout interacting with one another. The exceptionsare only elastic collisions.
From Boltzmann eq. to conservation eqs
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 249
It is possible to derive the conservation laws from theBoltzmann equation. Firstly, from the interpretationof distribution function f it arises the definition ofmacroscopic density
ρ(r, t) =
∫
RD
f(r,v, t) dv. (250)
The average value of a quantity ϕ is defined as
〈ϕ〉 =
∫
RD
ϕf dv
∫
RD
f dv=
1
ρ
∫
RD
ϕf dv. (251)
The integration is carried out over velocity space.
Averaged Boltzmann equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 250
Multiplying the Boltzmann equation by ϕ and thenintegrating over velocity space results in
∫
RD
ϕ∂f
∂tdv +
∫
RD
ϕv · ∇f dv +∫
RD
ϕF
m· ∇vf dv =
∫
RD
ϕΩ(f) dv. (252)
Taking advantage of the average definition theaveraged Boltzmann equation may now be rewrittenas
∂
∂t(ρ〈ϕ〉) +∇ · (ρ〈ϕv〉)− ρf · 〈∇vϕ〉 = 0. (253)
Mass conservation equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 251
Substituting ϕ := 1 into the averaged Boltzmannequation we have
∂ρ
∂t+∇ · (ρ〈v〉) = 0. (254)
Comparing the above equation with the massconservation equation
∂ρ
∂t+∇ · (ρU) = 0 (255)
it becomes obvious that the macroscopic velocity U
must be
U = 〈v〉 = 1
ρ
∫
RD
v f dv. (256)
Momentum conservation equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 252
Substituting ϕ← v into the averaged Boltzmannequation we have
∂
∂t(ρU) +∇ · (ρ〈vv〉)− ρf = 0. (257)
Introducing the microscopic velocity c in the meanvelocity frame
c(r,v, t) := v −U(r, t) (258)
it is possible to define the stress tensor
σ := −ρ〈cc〉 = −∫
RD
ccf dv. (259)
Now, the averaged Boltzmann equations becomes themacroscopic momentum conservation equation
∂
∂t(ρU) +∇ · (ρUU) = ρf +∇ · σ. (260)
Energy conservation equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 253
Substituting ϕ := 12‖v‖2 ≡ 1
2v · v into the averaged
Boltzmann equation we have
e :=1
2〈‖c‖2〉 = 1
2ρ
∫
RD
‖c‖2f dv. (261)
Introducing the heat vector q definition
q :=1
2ρ〈c‖c‖2〉 = 1
2
∫
RD
c‖c‖2f dv (262)
we have the macroscopic energy conservation equation
∂
∂t
(
ρ
(
e+1
2‖U‖2
))
+∇·(
ρ
(
e+1
2‖U‖2
)
U
)
=
ρf ·U+∇ · (σ ·U− q) . (263)
Equation of state
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 254
From the definition of the stress tensor σ := −ρ〈cc〉and the internal energy e := 2−1〈c · c〉 we have
tr〈cc〉 = 2e. (264)
Additionally, by means of the stress tensor we havepressure definition p := −D−1 trσ. Combining theseresults in
2ρe = pD. (265)
The above equation together with the equipartition ofenergy for mono-atomic gases gives the equation ofstate
p = ρRT = ρc2s. (266)
Velocity space discretisation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 255
The velocity space v is discretised into a finite set ofQ velocities vn where Q = |vn|. Discretedistributions are defined by means of the discretisedvelocity space
fn(r, t) = Wn f(r,vn, t), (267a)
f eqn (r, t) = Wn feq(r,vn, t). (267b)
Wn are the weights of the Gaussian quadrature rule.The density may now be approximated as
∫
RD
f(r,v, t) dv ≈∑
n
Wnf(r,vn, t) =∑
n
fn(r, t).
Discrete Boltzmann equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 256
The discrete distribution function fn satisfies thediscrete Boltzmann equation (with BGKapproximation)
∂fn∂t
+ vn · ∇fn =1
τ(f eqn − fn) . (268)
The fluid density, velocity and internal energy are nowcalculated from the discrete distribution function:
Quantity Continuous discreteρ
∫
RD f dv∑
n fnρU
∫
RD v f dv∑
n vnfnρe 1
2
∫
RD ‖v −U‖2f dv 12
∑
n ‖vn −U‖2fn
Quadratures
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 257
To recover correct form of the Navier-Stokesequations the discrete velocity set has to be chosen sothe following quadratures hold exactly
∀0≤m≤3
∫
RD
f eqm∏
i=0
v dv =∑
n
f0n
m∏
i=0
vn. (269)
The above may be reduced to
I :=
∫
RD
e−
‖v‖2
2c2s ψ(v) dv ≈∑
n
Wne−
‖vn‖2
2c2s ψ(vn) (270)
where
Wn = e‖vn‖2
2c2s
(√2πcs
)D
wn (271)and
wn = π−D2
D∏
i=1
ωi. (272)
Space discretisation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 258
The space discretisation follows the velocity spacediscretisation. This means it is discretised into alattices (D1Q3, D2Q9, D3Q27 discussed further).From the quadratures it arises speed of the model
c =√3cs. (273)
The speed c is used for space discretisation in thefollowing manner ∆xi = c∆t where ∆t representstime step (time space discretisation).One may also introduce dimensionless lattice velocities
en :=vn
c. (274)
Lattice Boltzmann equation
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 259
Introducing the substantial derivative symboldndt
:= ∂∂t+ vn · ∇ makes it possible to rewrite the
discretised Boltzmann equation
dnfndt
=1
τ(f eqn − fn) . (275)
The substantial derivative is approximated by means of
dnfn(r, t)
dt=fn(r+ vn∆t, t+∆t)− fn(r, t)
∆t+O (∆t) .
From the two above one gets the Lattice Boltzmannequation
fn(r+vn∆t, t+∆t)−fn(r, t) =1
τ
(
f0n(r, t)− fn(r, t)
)
.
where dimensionless collision time is τ := τ∆t.
Equations
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 260
Continuous Boltzmann equation∂f
∂t+ v · ∇f = Ω(f) (276)
Continuous Boltzmann equation with BGKapproximation
∂f
∂t+ v · ∇f =
1
τ(f eq − f) (277)
Discrete Boltzmann equation
∂fn∂t
+ vn · ∇fn =1
τ(f eqn − fn) (278)
Lattice Boltzmann equation
fn(r+vn∆t, t+∆t)−fn(r, t) =1
τ
(
f0n(r, t)− fn(r, t)
)
(279)
Typical lattices
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 261
D1Q3 lattice
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 262
wn =
23, n = 0;
16, n ∈ 1, 2.
D2Q9 lattice
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 263
wn =
49, n = 0;
19, n ∈ 1, . . . , 4;
136, n ∈ 5, . . . , 8.
D3Q27 lattice
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 264
wn =
827, n = 0;
227, n ∈ 1, . . . , 6;
154, n ∈ 15, . . . , 26;
1216
, n ∈ 7, . . . , 14;
D3Q19 lattice
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 265
wn =
13, n = 0;
218, n ∈ 1, . . . , 6;
136, n ∈ 7, . . . , 18;
D3Q15 lattice
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 266
wn =
29, n = 0;
19, n ∈ 1, . . . , 6;
172, n ∈ 7, . . . , 14;
Expansion of the Maxwell-Boltzmann
distribution
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 267
The Maxwell-Boltzmann distribution can berearranged
f eq = ρ(√
2πcs
)−D
e−
‖v‖2
2c2s e−
(
‖U‖2
2c2s−v·U
c2s
)
. (280)
Now f eq can be expanded into a Taylor series in termsof the fluid velocity
f0 := ρ(√
2πcs
)−D
e−
‖v‖2
2c2s
(
1 +v ·Uc2s
+(v ·U)2
2c4s− ‖U‖
2
2c2s
)
.
This is valid for low Mach numbers
f eq = f0 +O(‖U‖3
c3s
)
= f0 +O(
Ma3)
. (281)
Discrete Maxwell-Boltzmann distribution
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 268
The discrete equilibrium distribution is defined bymeans of the discretised velocity space
f0n(r, t) = Wnf
0(r,vn, t) (282)
where weights are
Wn = e‖vn‖2
2c2s
(√2πcs
)D
wn. (283)
Together with the Taylor expansion for low Machnumbers we have
f0n := wnρ
(
1 +vn ·Uc2s
+(vn ·U)2
2c4s− ‖U‖
2
2c2s
)
.
Streaming and collision
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 269
The Lattice Boltzmann equation
fn(r+vn∆t, t+∆t) = fn(r, t)+1
τ
(
f0n(r, t)− fn(r, t)
)
.
The collision step
f tn(r, t+∆t) = fn(r, t) +1
τ
(
f0n(r, t)− fn(r, t)
)
.
The streaming step
fn(r+ vn∆t, t +∆t) = f tn(r, t+∆t).
Streaming
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 270
Streaming
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 271
Streaming
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 272
LBM pseudocode
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 273
k ← 0;repeat
R← 0;ρ←
∑
n fn;U← 1
ρ
∑
n vnfn;
Calculate residue;
f0n ← wnρ
(
1 + vn·Uc2s
+ (vn·U)2
2c4s− ‖U‖2
2c2s
)
;
f tn(r, t+∆t)← fn(r, t) +1τ(f0n(r, t)− fn(r, t));
Apply Bounceback;fn(r+ vn∆t, t +∆t)← f tn(r, t+∆t);Apply other BCs;k ← k + 1;
until k < kmax and R > Rmin;
Chapman-Enskog expansion
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 274
The Navier-Stokes equations can be recovered fromthe Lattice Boltzmann equation
∂
∂t(ρU) +∇ · (ρUU) = −∇p+ µ∇2U (284)
through the Chapman-Enskog expansion (multi-scaleanalysis). The expansion of the discreteMaxwell-Boltzmann distribution is used
f0n := wnρ
(
1 +vn ·Uc2s
+(vn ·U)2
2c4s− ‖U‖
2
2c2s
)
.
The first RHS term is responsible for ∇p, the secondfor ∇2U, the last two terms are related to ρUU.
Simplifications
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 275
Knowing the structure of the discreteMaxwell-Boltzmann distribution we can now drop thenonlinear terms
f0n := wnρ
(
1 +vn ·Uc2s
)
(285)
to recover Stokes equations
∂
∂t(ρU) = −∇p+ µ∇2U. (286)
For both cases the dynamic viscosity is defined as
µ := ρ
(
τ − 1
2
)
c2s∆t. (287)
Example - input
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 276
1 60 2023 1000000000122222222222222222222222222222222222222222222222224 1000000000122222222222222222222222222222222222222222222222225 1000000000122222222222222222222222222222222222222222222222226 1000000000122222222222222222222222222222222222222222222222227 1000000000122222222222222222222222222222222222222222222222228 1000000000111111111111111111111111111111111111111111111111119 100000000000000000000000000000000000000000000000000000000001
10 10000000000000000000000000000000000000000000000000000000000111 10000000000000000000000000000000000000000000000000000000000112 10000000000000000000000000000000000000000000000000000000000113 10000000000000000000000000000000000000000000000000000000000114 10000000000000000000000000000000000000000000000000000000000115 10000000000000000000000000000000000000000000000000000000000116 10000000000000000000000000000000000000000000000000000000000117 10000000000000000000000000000000000000000000000000000000000118 10000000000000000000000000000000000000000000000000000000000119 10000000000000000000000000000000000000000000000000000000000120 10000000000000000000000000000000000000000000000000000000000121 10000000000000000000000000000000000000000000000000000000000122 1111111111111111111111111111111111111111111111111100000000012324 22526 PT27 0 20 10 2028 0 .42930 VB31 50 0 60 032 0 .01
Advantages over conventional methods
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 277
Possibility of microscopic interactionsincorporation
Easier dealing with complex boundaries Parallelisation of the method (the collision and
streaming processes are local)
Other methods
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 278
Other methods
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 279
Boundary element method Smoothed-particle hydrodynamics
References
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 280
List of references
Contents
Description of fluidat different scales
Turbulencemodelling
Finite differencemethod
Finite elementmethod
Finite volumemethod
Monte Carlo method
Lattice Boltzmannmethod
Other methods
References
K. Tesch; Fluid Mechanics – Applications and Numerical Methods 281
[1] Ferziger J.H., Peric M., Computational Methods for Fluid Dynamics,Springer-Verlag, Berlin, 2002
[2] Pope S. B., Turbulent Flows, Cambridge University Press, Cambridge,2000
[3] Succi S., The Lattice Boltzmann Equation for Fluid Dynamics and
Beyond, Oxford University Press, Oxford, 2001
[4] Tesch K., Fluid Mechanics, Wyd. PG, 2008, 2013 (in Polish)
[5] Wilcox D. C., Turbulence Modeling for CFD, DCW Industries,California, 1994
[6] Zienkiewicz O. C., Taylor R. L., The Finite Element Method,Butterworth-Heinemann, Oxford, 2000