1 Chapter 3: Turbulence and its modeling Ibrahim Sezai Department of Mechanical Engineering Eastern Mediterranean University Fall 2009-2010 I. Sezai – Eastern Mediterranean University ME555 : Computational Fluid Dynamics 2 Introduction For Re > Recritical => Flow becomes Turbulent. Chaotic and random state of motion develops. Velocity and pressure change continuously with time The velocity fluctuations give rise to additional stresses on the fluid => these are called Reynolds stresses We will try to model these extra stress terms
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IntroductionFor Re > Recritical => Flow becomes Turbulent.Chaotic and random state of motion develops.Velocity and pressure change continuously with timeThe velocity fluctuations give rise to additional stresses on the fluid
=> these are called Reynolds stressesWe will try to model these extra stress terms
Large Eddies:- have large eddy Reynolds number,- are dominated by inertia effects - viscous effects are negligible- are effectively inviscidSmall Eddies:- motion is dictated by viscosity- Re ≈ 1- length scales : 0.1 – 0.01 mm- frequencies : ≈ 10 kHzEnergy associated with eddy motions is dissipated and converted into thermal internal energy
increases energy losses.- Larges eddies anisotropic- Smallest eddies isotropic
Common features in the transition process:(i) The amplification of initially small disturbances(ii) The development of areas with concentrated rotational
structures(iii) The formation of intense small scale motions(iv) The growth and merging of these areas of small scale motions
into fully turbulent flowsTransition to turbulence is strongly affected by:- Pressure gradient- Disturbance levels- Wall roughness- Heat transferThe transition region often comprises only a very small fraction of
the flow domainCommercial CFD packages often ignore transition entirely (classify the flow as only laminar or turbulent)
3.3Effect of turbulence on time-averaged Navier-Stokes eqns
In turbulent flow there are eddying motions of a wide range of length scalesA domain of 0.1x 0.1m contains smallest eddies of 10-100 μm sizeWe need mesh pointsThe frequency of fastest events ≈ 10 kHz Δt≈100μs neededDNS of turbulent pipe flow of Re = 105 requires a computer which is 10 million times faster than CRAY supercomputerEngineers need only time-averaged properties of the flowLet’s see how turbulent fluctuations effect the mean flow properties
Since div and grad are both differentiations the above rules can be extended to a fluctuating vector quantity a = A + a´ and its combinations with a fluctuating scalar :
To illustrate the influence of turbulent fluctuations on mean flow we consider
Substitute
( ) ( ) ( ) ( ); ;div div div div div div
div grad div grad
ϕ ϕ ϕ
ϕ
′ ′= = = ΦΑ +
= Φ
a A a a a (3.7)
( )
( )
( )
01
1
1
divu pdiv u v div grad ut xv pdiv v v div grad vt yw pdiv w v div grad wt z
ρ
ρ
ρ
=∂ ∂
+ = − +∂ ∂∂ ∂
+ = − +∂ ∂∂ ∂
+ = − +∂ ∂
u
u
u
u
(3.9a)
(3.8)
(3.9b)
(3.9c)
; ; ; ;u U u v V v w W w p P p′ ′ ′ ′ ′= + = + = + = + = +u U u
Consider continuity equation: note that this yields the continuity equation
A similar process is now carried out on the x-momentum equation (3.9a). The time averages of the individual terms in this equation can be written as follows:
Substitution of these results gives the time-average x-momentum equation
The extra stress terms result from six additional stresses, three normal stresses and three shear stresses:
-These extra turbulent stresses are termed the Reynolds stresses.-The normal stresses are always non-zero-The shear stresses are also non-zeroIf, for instant, u´ and v´ were statistically independent fluctuations the time
average of their product would be zeroSimilar extra turbulent transport terms arise when we derive a transport
equation for an arbitrary scalar quantity. The time average transport equation for scalar φ is
In compressible flow density fluctuations are usually negligibleThe density-weighted averaged (or Favre-averaged) form of the compressible turbulent flow are:
u´ gives the largest of the normal stresses (v´and w´) .
Fluctuating velocities are not equal anisotropic structure of turbulenceAs mean velocity gradients tend to zero turbulence quantities tend to zero turbulence can’t be sustained in absence of shearThe mean velocity gradient is also zero at the centerline of jets and wakes. No turbulence there.The value of is zero at the centerline of a jet and wake since shear stress must change sign here.
y: distance away from the wallNear the wall (y small) is small viscous forces dominateAway from the wall (y large) is large inertia forces dominate.Near the wall U only depends on y, ρ, μ and τ (wall shear stress), so
The most useful form emerges if we view the wall shear stress as the cause of a velocity deficit Umax – U which decreases the closer we get to the edge of the boundary layer or the pipe centerline. Thus
Linear sub layer – the fluid layer in contact with smooth wall
Very near the wall there is no turbulent (Reynolds) shear stresses flow is dominated by viscous shear
For shear stress is approximately constant,
Integrating and using U = 0 at y = 0,
After some simple algebra and making use of the definitions of u+ and y+ this leads to
Because of the linear relationship between velocity and distancefrom the wall the fluid layer adjacent to the wall is often known as the linear sub-layer
- The inner region: 10 to 20% of the total thickness of the wall layer; the shear stress is (almost) constant and equal to the wall shear stress τw . Within this region there are three zones
- the linear sub-layer: viscous stresses dominate the flow adjacent to the surface
- the buffer layer: viscous and turbulent stresses are of similar magnitude- the log-law layer: turbulent (Reynolds) stresses dominate.
- The outer region or law-of-the-wake layer: inertia dominated core flow far from wall; free from direct viscous effects.
For y ⁄ δ > 0.8 fluctuating velocities become almost equalisotropic turbulence structure here. (far away the wall)For y ⁄ δ < 0.2 large mean velocity gradientshigh values of .(high turbulence production) .Turbulence is anisotropic near the wall.
Classical models: use the Reynolds eqn’s.(all commercial CFD codes)Large eddy simulation: the flow eqn’s are solved for the
- mean flow- and largest eddiesbut the effect of the smaller eddies are modeled- are at the research state- calculations are too costly for engineering use.
The mixing length and k-ε models are the most widely used and validated.They are based on the presumption that“there exists an analogy between the action of viscous stressesand Reynolds stresses on the mean flow”Viscous stresses are proportional to the rate of deformation. For incompressible flow:
Notation:i = 1 or j = 1 x-directioni = 2 or j = 2 y-directioni = 3 or j = 3 z-direction
Turbulent stresses are found to increase as the mean rate of deformation increases.It was proposed by Boussinesq in 1877 thatReynolds stress could be linked to mean rate of deformation
Similarly; Kinematic turbulent or eddy viscosity.Eqn.(3.23) shows that
Turbulent momentum transport is assumed to be proportional to mean gradients of velocity
By analogy turbulent transport of a scalar is taken to be proportional to the gradient of the mean value of the transported quantity. In suffix notation we get
Where Γt is the turbulent diffusivity.We introduce a turbulence Prandtl / Schmidt number as
Attempts to describe the turbulent stresses by means of simple algebraic formulae for μt as a function of positionThe k-ε models:
Two transport eqn’s (PDE’s) are solved:1. For the turbulent kinetic energy, k2. For the rate of dissipation of turbulent kinetic energy, ε.Both models assume that μt is isotropic(i.e. the ratio between Reynolds stresses and mean rate of deformation is
Approximate the Reynolds stresses in terms of algebraic equations instead of PDE type transport equations.Is an economical form of Reynolds stress model.Is able to introduce anisotropic turbulence effects into CFD simulations
where C is a dimensionless constant of proportionality.Turbulent viscosity is given byThis works well in simple 2-D turbulent flows where the only significant
Reynolds stress is and only significant velocity gradient is
To facilitate the subsequent calculations it is common to write the components of the rate of deformation eij and the stresses τij in tensor (matrix) form:
Governing equation for turbulent kinetic energy k1- Multiply x, y and z momentum eqns (3.9a-c) by u´, v´ and w´,respectively2- Multiply x, y and z Reynolds eqns (3.12a-c) by u´, v´ and w´, respectively3- subtract the two of the resulting equationsRearrange:
yields the equation for turbulent kinetic energy k:
( ) 1( ) 2 22
( ) ( ) ( ) ( ) ( ) ( ) ( )
ij i i j ij i j ijij
k div kU div p e u u u e e u u EtI II III IV V VI VII
Gives a negative contribution to (3.32) due to the appearance of the sum of squared fluctuating deformation rate e´ij
The dissipation of turbulent kinetic energy is caused by work done by the smallest eddies against viscous stresses.
The rate of dissipation per unit mass (m2/s3) is denoted by
The k – ε model equationsIt is possible to develop similar transport eqns. for all other turbulence quantitiesThe exact ε – equation, however, contains many unknown and unmeasurabletermsThe standard k – ε model(Launder and Spalding,1974) has two model eqns(a) one for k and (b) one for ε
( )2 2 2 2 2 211 22 33 12 13 232 2 2 2 2ij ije e e e e e e eμ μ′ ′ ′ ′ ′ ′ ′ ′− ⋅ = − + + + + +
The standard model uses the following transport equations used for k and ε
In words the equations are
The equations contain five adjustable constants . The standard k – ε model employs values for the constants that are arrived at by comprehensive data fitting for a wide range of turbulent flows:
κ = 0.41 (Von Karman’s constant)E = 9.8 (wall roughness parameter) for smooth wallsFor heat transfer we use the universal near wall temperature distribution
valid at high Reynolds numbers (Launder and Spalding, 1974):
Finally P is the “pee-function”, a correction function dependent on the ratio of laminar to turbulent Prandtl numbers (Launder and Spalding, 1974)
( ) ,,
,
,
,
with temperature at near wall point turbulent Prandtl number
wall temperature / Prandtl numberwall heat flux thermal conducti
The exact solution for the transport of takes the following form
Equation (3.45) describes six partial differential equations: one for the transport of each of the six independent Reynolds stresses( )Two new term appear compared with ke eqn (3.32)
2 2 21 2 3 1 2 1 3 2 3 2 1 1 2 3 1 1 3 3 2 2 3, , , , , since ,u u u u u u u and u u u u u u u u u u and u u u u′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′= = =
CFD computations with Reynolds stress transport equations retain the production term in its exact form
The diffusion term Dij can be modelled by the assumption that the rate of transport of Reynolds stresses by diffusion is proportional to gradient of Reynolds stress.
The dissipations rate εij is modeled by assuming isotropy of the small dissipative eddies. It is set so that it effects the normal Reynolds stresses (i = j) only and in equal measure. This can be achieved by
Where ε is the dissipation rate of turbulent kinetic energy defined by (3.33). The Kronecker delta, δij is given by δij = 1 if i = j and δij = 0 if i ≠ j
For Πij term:The pressure – strain interactions are the most difficult to modelTheir effect on the Reynolds stresses is caused by two distinct physical processes:
1. Pressure fluctuations due to two eddies interacting with each other
2. Pressure fluctuations due to the interactions of an eddy with a region of flow of different mean velocityIts effect is to make Reynolds stresses more isotropic and to reduce the Reynolds shear stressesMeasurements indicate that;The wall effect increases the isotropy of normal Reynolds stresses by damping out fluctuations in the directions normal tothe wall and decreases the magnitude of the Reynolds shear stresses.
A comprehensive model that accounts for all these effects is given in Launder et al (1975). They also give the following simpler form favoured by some commercial available CFD codes:
The rotational term
ωk : rotation vector
eijk = +1 if i, j and k are different and in cyclic order
eijk = –1 if i, j and k are different and in anticyclic order
eijk = 0 if any two indices are the sameTurbulent kinetic energy k is.
The six equations for Reynolds stress transport are solved along with a model equation for the scalar dissipation rate ε. Again a more exact form is found in Launder et al (1975), but the equation from the standard k – εmodel is used in commercial CFD for the sake of simplicity
The usual boundary conditions for elliptic flows are required for the solution of the Reynolds stress transport equations:
In the absence of any information inlet distributions are calculated from
For computations at high Reynolds numbers wall-function-type boundary conditions can be used which are very similar to those of the k-ε model .
Near wall Reynolds stress values are computed from formulae such as where cij are obtained from measurements.
Low Reynolds number modifications to the model can be incorporated to add the effects of molecular viscosity to the diffusion terms and to account for anisotropy in the dissipation rate in the Rij-equations.
Wall damping functions to adjust the constants of the ε-equation and a modified dissipation rate variable give more realistic modelling near solid walls.
3.5.4 Algebraic stress equation models (ASM)- ASM is an economical way of accounting for the
anisotropy of Reynolds stresses.- Avoid solving the Reynolds stress transport eqns- Instead, use algebraic eqn to model Reynolds stresses.The Simplest method is:
To neglect the convection and diffusion terms altogether
A more generally applicable method is:To assume that the sum of the convection and diffusion
terms of the Reynolds stresses is proportional to the sum of the convection and diffusion terms of turbulent kinetic energy
Introducing approximation (3.52) into the Reynolds stress transport equation (3.45) with production term (3.46), modeled dissipation rate term (3.50) and pressure – strain correction term (3.49) on the right hand side yields after some arrangement the following algebraic stress model:
- appear on both sides (on rhs within Pij ) - eqn(3.53) is a set of 6 simultaneous algebraic eqns. for 6 unknowns,- solved iteratively if k and ε are known.- The standard k-ε model eqns has to be solved also (3.34 - 3.37)
The nonlinear k - ε model accounts for anisotropy.Accounts for the secondary flow in non-circular duct flows.
The RNG k - ε model (Renormalization Group)The RNG procedure:
systematically removes the small scales of motion from the governing equations by expressing their effects in terms of large scale motions and a modified viscosity. (Yakhot et al 1992):
Only the constant β is adjustable; the above value is calculated from near wall turbulent data. All other constants are explicitly computed as part of the RNG process.This model can be applied with
The isotropic Reynolds stress formula (3.38) (k-ε model)The nonlinear form (3.57) (nonlinear k-ε model )