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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2. Conservation Equations for Turbulent Flows
Coverage of this section:
I Review of Tensor Notation
I Review of Navier-Stokes Equations for Incompressible
andCompressible Flows
I Reynolds & Favre Averaging and RANS & FANS
Equations
I Turbulent Kinetic Energy and Reynolds Stresses
I Closure Problem and Turbulence Modelling
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.1 Review of Tensor Notation
Tensor notation is used extensively throughout the textbook
andthis course and is therefore briefly reviewed and compared to
vectornotation before moving to a discussion of the
conservationequations for turbulent flows.
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.1 Review of Tensor NotationExpression Vector Notation Tensor
Notation
scalars pi, c pi, c(zeroth-order tensor)
operations
(+, , , /) e.g., pi c , pic
pi c , pic
vectors ~a, ~x ai , xi(3D space) (first-order tensor,
it is taken that i {1, 2, 3})
addition ~b = ~a + ~x bi = ai + xi = aj + xk
vector productsinner product ~a ~x = i aixi = c aixi = c
(scalar result)
i aixi = a1x1 + a2x2 + a3x3 (Einstein notation: sum implied)
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.1 Review of Tensor Notation
2.1.1 Einstein Summation ConventionEinstein summation
convention: repetition of an index in any termdenotes a summation
of the term with respect to that index overthe full range of the
index (i.e., 1, 2, 3).Thus, for the inner product
aixi =3
i=1
aixi = a1x1 + a2x2 + a3x3
the sum is implied and need not be explicitly expressed. Note
thatusing matrix-vector mathematical notation, the inner product
oftwo 3 1 column vectors, a and x, can be experssed as
aTx = [a1 a2 a3]
x1x2x3
= a1x1 + a2x2 + a3x34
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.1 Review of Tensor Notation
Expression Vector Notation Tensor Notation
cross product ~a ~x = ~r = ~i ~j ~ka1 a2 a3x1 x2 x3
ijkajxk = ri(vector result) ~r =
(a2x3 a3x2)~i(a1x3 a3x1)~j+(a1x2 a2x1)~k
ijk = permutation tensor
(sum over j & k implied)
outer product ~a~x = ~a~x =
~~J ax = J(dyadic result, (second-order tensor,
vector of vectors) 9 elements,
6 elements for symmetric tensor)
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.1 Review of Tensor Notation
2.1.2 Dyadic Quantity: A Vector of Vectors
In vector notation, a dyadic quantity,~~d is essentially a
vector of
vectors as defined by the outer product:
~~d = ~u~v
It is equivalent to the second-order tensor, dij ,
dij = uiuj
using tensor notation. In this case using matrix-vector
notation,the outer product of two 3 1 column vectors, u and v, can
beexperssed as
uvT =
u1u2u3
[v1 v2 v3] = u1v1 u1v2 u1v3u2v1 u2v2 u2v3
u3v1 u3v2 u3v3
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.1 Review of Tensor NotationExpression Vector Notation Tensor
Notation
dyads~~d = ~u~v dij = uiuj
dyad-vector products~~A ~x = ~b Ax = b
(vector result) equivalent to Ax = b
high-order tensors~~~Q Qijk
(third-order tensor,
27 elements, 10 symmetric)
~~~~R Rijkl(fourth-order tensor,
81 elements, 15 symmetric)
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.1 Review of Tensor Notation
Expression Vector Notation Tensor Notation
contracted quantities ~h hi = qijj(contacted 3rd-order
tensor,
vector)
~~P Pij = Rijkk(contacted 4th-order tensor,
second-order tensor, dyad)
p p = Riikk(double contacted tensor,
scalar quantity)
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.1 Review of Tensor Notation
2.1.3 Permutation Tensor
The permuation tensor, ijk , is a third-order tensor that
isintroduced for defining cross products with the following
propertiesfor its elements:
123 = 231 = 312 = 1 , even permutations
213 = 321 = 132 = 1 , odd permutations111 = 222 = 333 = 0 ,
repeated indices
112 = 113 = 221 = 223 = 331 = 322 = 0 , repeated indices
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.1 Review of Tensor Notation
2.1.4 Kronecker Delta Tensor
The Kronecker delta tensor, ij , is a second-order tensor that
isdefined as follows:
ij =
{1 , for i = j0 , for i 6= j
The Kronecker delta tensor is equivalent ot the identity
dyad,~~I
and the 3 3 indentity matrix, I, in matrix-vector
mathematicalnotation given by
I =
1 0 00 1 00 0 1
Note also that
ii = trace(I) = 310
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.1 Review of Tensor Notation
2.1.5 Indentity
The following identity relates the permutation and Kronecker
deltatensors:
ijkist = jskt jtks
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.1 Review of Tensor NotationExpression Vector Notation Tensor
Notation
differential operators
gradient ~V = ~ Vi = xi
divergence c = ~ ~a c = aixi
~u ~ ui xi
curl ~g = ~~a gi = ijk akxj
vector derivative~~P = ~~B Pij = Bi
xj
Laplacian c = 2 = ~ ~ c = 2
xii
~a = 2~A = ~ ~~A ai = 2Ai
xjj12
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.1 Review of Tensor Notation
2.1.6 Other Notation
In the course textbook and elsewhere you will some time see
theuse of the shorthand tensor notation:
~p = pxi
= p,i
and~ ~u = ui
xi= ui ,i
This notation will not be used by this instructor as it can
bedifficult to follow and is more prone to errors.
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.2 Navier-Stokes Equations for a Compressible Gas
The Navier-Stokes equations describing the flow of
compressiblegases are a non-linear set of partial-differential
equations (PDEs)governing the conservation of mass, momentum, and
energy of thegaseous motion. They consist of two scalar equations
and onevector equation for five unknowns (dependent variables) in
termsof three independent variables, the position vector, ~x or xi
, andtime, t.
We will here review briefly the Navier-Stokes equations for
apolytropic (calorically perfect) gas in both tensor and
vectornotation. Integral forms of the equations will also be
discussed.
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.2 Navier-Stokes Equations for a Compressible Gas
2.2.1 Continuity Equation
The continuity equation is a scaler equation reflecting
theconservation of mass for a moving fluid. Using vector notation,
ithas the form
t+ ~ (~u) = 0
where and ~u are the gas density and flow velocity,
respectively.In tensor notation, the continuity equation can be
written as
t+
xi(ui ) = 0
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.2.1 Continuity Equation
For the control volume and control surface above, the
integralform of the continuity equation can be obtained by
integrating theoriginal PDE over the control volume and making
using of thedivergence theorem. The following integral equation is
obtained:
d
dt
V dV =
A~u ~n dA
which relates the time rate of change of the total mass within
thecontrol volume to the mass flux through the control surface.
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.2 Navier-Stokes Equations for a Compressible Gas
2.2.2 Momentum Equation
The momentum equation is a vector equation that represents
theapplication of Newtons 2nd Law of Motion to the motion of a
gas.It relates the time rate of change of the gas momentum to
theforces which act on the gas. Using vector notation, it has the
form
t(~u) + ~
(~u~u + p
~~I ~~)
= ~f
where p and ~~ are the gas pressure and fluid stress dyad or
tensor,respectively, and ~f is the acceleration of the gas due to
body forces(i.e., gravitation, electro-magnetic forces). In tensor
notation, themomentum equation can be written as
t(ui ) +
xj(uiuj + pij ij) = fi
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.2.2 Momentum Equation
For the control volume, the integral form of the
momentumequation is given by
d
dt
V~u dV =
A
(~u~u + p
~~I ~~) ~n dA+
V~f dV
which relates the time rate of change of the total
momentumwithin the control volume to the surface and body forces
that acton the gas.
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.2 Navier-Stokes Equations for a Compressible Gas
2.2.3 Energy Equation
The energy equation is a scalar equation that represents
theapplication of 1st Law of Thermodynamics to the gaseous
motion.It describes the time rate of change of the total energy of
the gas(the sum of kinetic energy of bulk motion and internal
kinetic orthermal energy). Using vector notation, it has the
form
t(E ) + ~
[~u
(E +
p
) ~~ ~u + ~q
]= ~f ~u
where E is the total specific energy of the gas given byE =e +
~u ~u/2 and ~q is the heat flux vector representing the fluxof heat
out of the gas. In tensor notation, it has the form
t(E ) +
xi
[ui
(E +
p
) ijuj + qi
]= fiui
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.2.3 Energy Equation
For the control volume, the integral form of the energy equation
isgiven by
d
dt
VE dV =
A
[~u
(E +
p
) ~~ ~u + ~q
]~n dA+
V~f ~u dV
which relates the time rate of change of the total energy within
thecontrol volume to transport of energy, heat transfer, and
workdone by the gas.
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.2 Navier-Stokes Equations for a Compressible Gas
The Navier-Stokes equations as given above are not
complete(closed). Additional information is required to relate
pressure,density, temperature, and energy, and the fluid stress
tensor, ijand heat flux vector, qi must be specified. The equation
set iscompleted by
thermodynamic relationships; constitutive relations; and
expressions for transport coefficients.
When seeking solutions of the Navier-Stokes equations for
eithersteady-state boundary value problems or unsteady initial
boundaryvalue problems, boundary conditions will also be required
tocomplete the mathematical description.
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.2 Navier-Stokes Equations for a Compressible Gas
2.2.4 Thermodynamic Relationships
In this course, we will assume that the gas satisfies the ideal
gasequation of state relating , p, and T , given by
p = RT
and behaves as a calorically perfect gas (polytropic gas)
withconstant specific heats, cv and cp, and specific heat ratio, ,
suchthat
e = cvT =p
( 1) and h = e +p
= cpT =
p
( 1)where R is the gas constant, cv is the specific heat at
constantvolume, cp is the specific heat at constant pressure, and
=cp/cv .
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.2 Navier-Stokes Equations for a Compressible Gas
2.2.5 Mach Number and Sound Speed
For a polytropic gas, the sound speed, a, can be determined
using
a =
p
=RT
and thus the flow Mach number, M, is given by
M =u
a=
uRT
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.2 Navier-Stokes Equations for a Compressible Gas
2.2.6 Constitutive Relationships
The constitutive relations provide expressions for the fluid
stresstensor, ij , and heat flux vector, qi , in terms of the other
fluidquantities. Using the Navier-Stokes relation, the fluid stress
tensorcan be related to the fluid strain rate and given by
ij =
[(uixj
+ujxi
) 2
3ijukxk
](ii = 0, traceless)
where is the dynamic viscosity of the gas. Fouriers Law can
beused to relate the heat flux to the temperature gradient as
follows:
qi = Txi
or ~q = ~T
where is the coefficient of thermal conductivity for the
gas.
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.2 Navier-Stokes Equations for a Compressible Gas
2.2.7 Transport Coefficients
In general, the transport coefficients, and , are functions
ofboth pressure and temperature:
= (p,T ) and = (p,T )
Expressions, such as Sutherlands Law can be used to determinethe
dynamics viscosity as a function of temperature (i.e.,=(T )). The
Prandtl number can also be used to relate and. The non-dimensional
Prandtl number is defined as follows:
Pr =cp
and is typically 0.70-0.72 for many gases. Given , the
thermalconductivity can be related to viscosity using the
precedingexpression for the Prandtl number.
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.2 Navier-Stokes Equations for a Compressible Gas
2.2.8 Boundary Conditions
At a solid wall or bounday, the following boundary conditions
forthe flow velocity and temperature are appropriate:
~u = 0 , (No-Slip Boundary Condition)
and
T = Twall , (Fixed Temperature Wall Boundary Condition)
or~T ~n = 0 , (Adiabatic Wall Boundary Condition)
where Twall is the wall temperature and ~n is a unit vector in
thedirection normal to the wall or solid surface.
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.3 Navier-Stokes Equations for an Incompressible Gas
For low flow Mach numbers (i.e., low subsonic flow, M
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.3 Navier-Stokes Equations for an Incompressible Gas
2.3.2 Momentum Equation
Using vector notation, the momentum equation for
anincompressible fluid can be written as
~u
t+ ~u ~~u + 1
~p = 1
~ ~~
In tensor notation, the incompressible form of the
momentumequation is given by
uit
+ ujuixj
+1
p
xi=
1
ijxj
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.3 Navier-Stokes Equations for an Incompressible Gas
2.3.3 Constitutive Relationships
For incompressible flows, the Navier-Stokes constitutive
relationrelating the fluid stresses and fluid strain rate can be
written as
ij =
(uixj
+ujxi
)=
(uixj
+ujxi
)= 2Sij
where =/ is the kinematic viscosity and the strain rate
tensor(dyadic quantity) is given by
Sij =1
2
(uixj
+ujxi
)As in the compressible case, the fluid stress tensor
forincompressible flow is still traceless and ij =0.
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2.3 Navier-Stokes Equations for an Incompressible Gas
2.3.4 Vorticity Transport Equation
The vorticity vector, ~, is related to the rotation of a fluid
elementand is defined as follows:
~ = ~ ~u or i = ijk ukxj
For incompressible flows, the momentum equation can be used
toarrive at a transport equation for the flow vorticity given
by
~
t ~ ~u ~ = 2~
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.3.4 Vorticity Transport Equation
Using ~ ~u ~ = ~ ~~u ~u ~~, the vorticity transportequation can
be re-expressed as
~
t+ ~u ~~ ~ ~~u = 2~
Using tensor notation, this equation can be written as
it
+ ujixj j ui
xj=
2ixjxj
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.4 Reynolds Averaging
As discussed previously, turbulent flow is characterized by
irregular,chaotic motion. The common approach to the modelling
ofturbulence is to assume that the motion is random and adopt
astatistical treatment. Reynolds (1895) introduced the idea that
theturbulent flow velocity vector, ui , can be decomposed
andrepresented as a fluctuation, ui , about a mean component, Ui ,
asfollows:
ui = Ui + ui
Develop and solve conservation equations for the mean
quantities(i.e., the Reynolds-averaged Navier-Stokes (RANS)
equations) andincorporate the influence of the fluctuations on the
mean flow viaturbulence modelling.
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.4 Reynolds Averaging
2.4.1 Forms of Reynolds Averaging
1. Time Averaging: appropriate for steady mean flows
FT (~x) = limT
1
T
t+T/2tT/2
f (~x , t ) dt
2. Spatial Averaging: suitable for homogeneous turbulent
flows
FV(t) = limV1
VVf (~x , t) dV
3. Ensemble Averaging: most general form of averaging
FE(~x , t) = limN
1
N
Nn=1
fn(~x , t)
where fn(~x , t) is nth instance of flow solution with initial
andboundary data differing by random
infinitessimalperturbations.
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2.4.1 Forms of Reynolds Averaging
For ergodic random processes, these three forms of
Reynoldsaveraging will yield the same averages. This would be the
case forstationary, homogeneous, turbulent flows.
In this course and indeed in most turbulence modelling
approaches,time averaging will be considered. Note that Wilcox
(2002) statesthat Reynolds time averaging is a brutal
simplification that losesmuch of the information contained in the
turbulence.
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.4 Reynolds Averaging
2.4.2 Reynolds Time Averaging
In Reynolds time averaging, all instantaneous flow
quantities,(xi , t) and a(xi , t), will be represented as a sum of
mean andfluctuating components, (xi ) and
(xi , t) and A(xi ) and a(xi , t),respectively, such that
(xi , t) = (xi ) + (xi , t) or a(xi , t) = A(xi ) + a(xi ,
t)
For the flow velocity, we have
ui (x, t) = Ui (x) + ui (x, t)
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.4.2 Reynolds Time Averaging
The time averaging procedure is defined as follows and yields
thetime averaged quantities:
(xi , t) = (xi ) = limT
1
T
t+T/2tT/2
(xi , t) dt
a(xi , t) = A(xi ) = limT
1
T
t+T/2tT/2
a(xi , t) dt
By definition, time averaging of mean quantities merely
recoversthe mean quantity:
Ui (x) = limT
1
T
t+T/2tT/2
Ui (x) dt = Ui (x)
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.4.2 Reynolds Time Averaging
Similarly by definition, time averaging of time-averaged
quantitiesyields zero:
ui (x, t) = limT1
T
t+T/2tT/2
[ui (x, t
) Ui (x)]dt = 0
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2.4 Reynolds Averaging
2.4.3 Separation of Time Scales
In practice, the time period for the averaging, T , is not
infinite butvery long relative to the time scales for the turbulent
fluctuations,T1 ( i.e., TT1).
This definition of time averaging and T works well for
stationary(steady) flows. However, for non-stationary (unsteady
flows), thevalidity of the Reynolds time averaging procedure
requires a strongseparation to time scales with
T1 T T2where T2 is the time scale for the variation of the
mean.
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.4 Reynolds Averaging
2.4.3 Separation of Time Scales
t
u(x,t)
T1
T2
T1 T T2
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2.4.3 Separation of Time Scales
Provided there exists this separation of scales, the time
averagingprocedure for time-varying mean flows can be defined as
follows:
(xi , t) = (xi , t) =1
T
t+T/2tT/2
(xi , t) dt
a(xi , t) = A(xi , t) =1
T
t+T/2tT/2
a(xi , t) dt
with T1 T T2.
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.4 Reynolds Averaging
2.4.4 Properties of Reynolds Time Averaging
Multiplication by a scalar:
c a(xi , t) =C
T
t+T/2tT/2
a(xi , t) dt = cA
Spatial differentiation:
a
xi=
1
T
t+T/2tT/2
a
xidt =
xi
(1
T
t+T/2tT/2
a dt )
=A
xi
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2.4 Reynolds Averaging
2.4.4 Properties of Reynolds Time Averaging
Temporal differentiation:
uit
=1
T
t+T/2tT/2
uit
dt
=Ui (xi , t + T/2) Ui (xi , t T/2)
T+
ui (xi , t + T/2) ui (xi , t T/2)T
Uit
The latter is obtained by assuming that |~u| |~U| and T T2.
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.4 Reynolds Averaging
2.4.5 Single-Point Correlations
What about time-averaged products?
a(xi , t)b(xi , t) = (A + a) (B + b)= AB + aB + bA + ab
= AB + Ba + Ab + ab
= AB + Ba + Ab + ab
= AB + ab (1)
In general, a and b are said to be correlated if
ab 6= 0
and uncorrelated ifab = 0
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2.4 Reynolds Averaging
2.4.5 Single-Point Correlations
What about triple products? Can show that
a(xi , t)b(xi , t)c(xi , t) = ABC + abC + ac B + bc A + abc
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.5 Reynolds Averaged Navier-Stokes (RANS) Equations
2.5.1 Derivation
Applying Reynolds time-averaging to the incompressible form
ofthe Navier-Stokes equations leads to the Reynolds
AveragedNavier-Stokes (RANS) equations describing the time
variation ofmean flow quantities.
Application of time-averaging to the continuity equations
yields
uixi
= 0
orUixi
= 0
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2.5 Reynolds Averaged Navier-Stokes (RANS) Equations
2.5.1 Derivation
For the incompressible form of the momentum equation we have
uit
+ ujuixj
+1
p
xi=uit
+ ujuixj
+1
p
xi=
1
ijxj
Considering each term in the time-average equation above we
have:
uit
=Uit
1
p
xi=
1
p
xi=
1
P
xi
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.5 Reynolds Averaged Navier-Stokes (RANS) Equations
2.5.1 Derivation
1
ijxj
=1
ijxj
=2
Sijxj
= 2Sijxj
where the mean strain, Sij , is defined as
Sij =1
2
[Uixj
+Ujxi
]
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2.5 Reynolds Averaged Navier-Stokes (RANS) Equations
2.5.1 Derivation
ujuixj
=
xj(uiuj) ui uj
xj=
xj
(UiUj + uiu
j
)=
xj(UiUj) +
xj
(uiuj
)= Uj
Uixj
+ UiUjxj
+
xj
(uiuj
)= Uj
Uixj
+
xj
(uiuj
)Thus we have
Uit
+ UjUixj
+1
P
xi=
1
xj
(2Sij uiuj
)49
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2.5 Reynolds Averaged Navier-Stokes (RANS) Equations
2.5.2 Summary
In summary, the RANS describing the time-evolution of the
meanflow quantities Ui and P can be written as
Uixi
= 0
Uit
+ UjUixj
+1
P
xi=
1
xj(ij + ij)
where ij is the fluid stress tensor evaluated in terms of the
meanflow quantities and ij is the Reynolds or turbulent stress
tensorgiven by
ij = uiuj
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2.6 Reynolds Turbulent Stresses and Closure Problem
2.6.1 Closure or RANS Equations
The Reynolds stressesij = uiuj
incorporate the effects of the unresolved turbulent
fluctuations(i.e., unresolved by the mean flow equations and
description) onthe mean flow. These apparent turbulent stresses
significantlyenhance momentum transport in the mean flow.
The Reynolds stress tensor, ij , is a symmetric tensor
incorporatingsix (6) unknown or unspecified values. This leads to a
closureproblem for the RANS equation set. Turbulence modelling
providesthe necessary closure by allowing a means for specifying ij
interms of mean flow solution quantities.
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2.6 Reynolds Turbulent Stresses and Closure Problem
2.6.2 Reynolds Stress Transport Equations
Transport equations for the Reynolds stresses, ij =uiuj can
bederived by making use of the original and time-averaged forms
ofthe momentum equations.Starting with the momentum equation for
incompressible flowgoverning the time evolution of the
instantaneous velocity vector,ui ,
uit
+ ujuixj
+1
p
xi=
1
ijxj
and noting that
1
ijxj
=
xj
(uixj
+ujxi
)=
[2uixjxj
+
xi
(ujxj
)]=
2uixjxj
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2.6.2 Reynolds Stress Transport Equations
one can write
uit
+ ukuixk
+1
p
xi
2uixkxk
= 0 (1)
Similarily,
ujt
+ ukujxk
+1
p
xj
2ujxkxk
= 0 (2)
Thus, uj (1) + ui (2) can be written as
0 = uj
(uit
+ ukuixk
+1
p
xi
2uixkxk
)+ui
(ujt
+ ukujxk
+1
p
xj
2ujxkxk
)53
AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.6.2 Reynolds Stress Transport Equations
The various terms appearing in the preceding equation can
beexpressed as follows:
ujuit
+ uiujt
= uj
t
(Ui + ui
)+ ui
t
(Uj + uj
)=
Uit
uj + uj
uit
+Ujt
ui + ui
ujt
= ujuit
+ uiujt
=
t
(uiuj
)= 1
ijt
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2.6.2 Reynolds Stress Transport Equations
uj
p
xi+
ui
p
xj=
uj
xi(P + p) +
ui
xj(P + p)
=P
xiuj +
1
ujp
xi+P
xjui +
1
uip
xi
=1
[ujp
xi+ ui
p
xi
]
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2.6.2 Reynolds Stress Transport Equations
uj2uixkxk
+ ui2ujxkxk
= uj2
xkxk(Ui + ui ) + u
i
2
xkxk(Ui + ui )
= 2Uixkxk
uj + uj
2uixkxk
+ 2Ujxkxk
ui + ui
2ujxkxk
= uj2uixkxk
+ ui2ujxkxk
= 2
xkxk
(ui uj
) 2 u
i
xk
ujxk
=
2ijxkxk
2 ui
xk
ujxk
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2.6.2 Reynolds Stress Transport Equations
ujukuixk
+ ui ukujxk
= uj (Uk + uk)
xk(Ui + ui ) + u
i (Uk + u
k)
xk
(Uj + uj
)= Uk
xk
(ui uj
)+ uju
k
Uixk
+ ui uk
Ujxk
+UkUixk
uj + UkUjxk
ui + uk
xk
(ui uj
)= Uk
ijxk
jk
Uixk
ik
Ujxk
+
xk
(ui ujuk
) ui uj
ukxk
= Uk
ijxk
jk
Uixk
ik
Ujxk
+
xk
(ui ujuk
)
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2.6.2 Reynolds Stress Transport Equations
Combining all of these terms, can write
ijt
+ Ukijxk
+ jkUixk
+ ikUjxk
=
xk
[ijxk
+ uiujuk
]+uj
p
xi+ ui
p
xi
+2uixk
ujxk
The preceding is a transport equation describing the time
evolutionof the Reynolds stresses, ij .
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2.6.2 Reynolds Stress Transport Equations
While providing a description for the transport of ij , the
Reynoldsstress equations introduce a number of other correlations
offluctuating quantities:
ujp
xi: symmetric second-order tensor, 6 entries
uiujuk : symmetric third-order tensor, 10 entries
2uixk
ujxk
: symmetric second-order tensor, 6 entries
leading to 22 additional unknown quantities. This illustrates
wellthe closure problem for the RANS equations.
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2.7 Turbulence Intensity and Kinetic Energy
2.7.1 Turbulent Kinetic Energy
Turbulent kinetic energy contained in the
near-randomlyfluctuating velocity of the turbulent motion is
important incharacterizing the turbulence.
The turbulent kinetic energy, k , can be defined as follows:
k =1
2uiui =
1
2
(u2 + v 2 + w 2
)= 1
2
ii
= 12
(xx + yy + zz)
where u2 =xx/, v 2 =yy/, and w 2 =zz/.
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2.7 Turbulence Intensity and Kinetic Energy
2.7.2 Turbulence Intensity
Relative turbulence intensities can be defined as follows:
u =
u2
U, v =
v 2
U, w =
w 2
U
where U is a reference velocity.
For isotropic turbulence, u2 = v 2 =w 2 , and thus
u = v = w =
2
3
k
U2
For flat plate incompressible boundary layer flow,
U=U,u>0.10, and the turbulence is anisotropic such that
u2 : v 2 : w 2 = 4 : 2 : 361
AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.7.2 Turbulence Intensity
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2.8 Turbulent Kinetic Energy Transport Equation
2.8.1 Derivation
Can derive a transport equation for the turbulent kinetic
energythrough contraction of the Reynolds stress transport
equationsusing the relation that
k =1
2uiui =
1
2
ii
The following equation for the transport of k can be
obtained:
k
t+Ui
k
xi=ij
Uixj
+
xi
(k
xi 1pui
1
2uiukuk
) u
i
xj
uixj
As for the Reynolds stress equations, a number of
unknownhigher-order correlations appear in the equation for k
requiringclosure.
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2.8 Turbulent Kinetic Energy Transport Equation
2.8.2 Discussion of Terms
Terms in this transport equation can be identified as
follows:
k
t: time evolution of k
Uik
xi: convection transport of k
Production:
ij
Uixj
: production of k by mean flow
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2.8.2 Discussion of Terms
Diffusion:
k
xi: molecular diffusion of k
1
pui : pressure diffusion of k
1
2uiukuk : turbulent transport of k
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2.8.2 Discussion of Terms
Dissipation:
uixj
uixj
= : dissipation of k at small scales
where is the dissipation rate of turbulent kinetic energy.
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2.9 Kinetic Energy Spectrum and Kolmogorov -5/3 Law
2.9.1 DefinitionFurther insight into the energy contained in the
unresolvedturbulent motion can be gained by considering the
turbulent kineticenergy spectrum. The turbulent kinetic energy can
be expressed as
k =
0
E ()d
where E () is the spectral distribution of turbulent energy, is
the wave number of the Fourier-like energy mode, and ` is thewave
length of the energy mode such that
E ()d = turbulent energy contained between and + d
and where
` =1
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2.9 Kinetic Energy Spectrum and Kolmogorov -5/3 Law
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2.9 Kinetic Energy Spectrum and Kolmogorov -5/3 Law
DIEI
Dissipationrange
Inertial subrangeEnergy-containing range
Slope 2
Slope -5/3E()
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2.9 Kinetic Energy Spectrum and Kolmogorov -5/3 Law
2.9.2 Range of Turbulent Scales
The large-scale turbulent motion ( 0) contains most of
theturbulent kinetic energy, while most of the vorticity resides in
thesmall-scale turbulent motion ( 1/), where , the Kolmogorovscale,
is the smallest scale present in the turbulence.
The dissipation of the turbulence kinetic energy occurs at
theKolmogorov scale and it follows from Kolmogorovs
universalequilibrium theory that
dk
dt= , and =
(3
)1/4
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2.9.2 Range of Turbulent Scales
For high Reynolds number turbulence, dimensional analysis
andexperimental measurements confirm that the dissipation rate,
,turbulent kinetic energy, k, and largest scale representing the
largescale motions (i.e., scale of the largest eddies), `0, are
related asfollows:
k3/2
`0
When discussing features of turbulence, it was noted that
itcontains a wide range of scales. This implies that
`0
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2.9.2 Range of Turbulent Scales
Using the expression above for `0, an examination of the
lengthscales reveals that
`0
=`0
(3/)1/4 `03/4
(k3/2
`0
)1/4(k1/2`0
)3/4 Re3/4t
where Ret is the turbulent Reynolds number. Thus `0 for
highturbulent Reynolds number flows (i.e., for Ret 1). The latter
isa key assumption entering into Kolmogorovs universal
equilibriumtheory.
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2.9 Kinetic Energy Spectrum and Kolmogorov -5/3 Law
2.9.3 Kolmogorov -5/3 Law
Kolmogorov also hypothesize an intermediate range of
turbulentscales lying between the largest scales and smallest
scales whereinertial effects dominate. He postulated that in this
inertialsub-range, E () only depends on and . Using
dimensionalanalysis he argued that
E () = Ck2/3
5/3
orE () 5/3
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AER1310: TURBULENCE MODELLING 2. Conservation Equations for
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2.9 Kinetic Energy Spectrum and Kolmogorov -5/3 Law
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2.9.3 Kolmogorov -5/3 Law
Although the Kolmogorov -5/3 Law is not of prime importance
toRANS-based turbulence models, it is of central importance to
DNSand LES calculations. Such simulations should be regarded
withskeptism if they fail to reproduce this result.
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2.10 Two-Point Correlations
2.10.1 Two-Point Velocity Correlations
So far we have only considered single-point or
one-pointcorrelations of fluctuating quantities. Two-point
correlations areuseful for characterizing turbulence and, in
particular, the spatialand temporal scales and non-local behaviour.
They provide formaldefinitions of the integral length and time
scales characterizing thelarge scale turbulent motions.
There are two forms of two-point correlations:
I two-point correletions in time; and
I two-point correlations in space.
Both forms are based on Reynolds time averaging.
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2.10.1 Two-Point Velocity Correlations
Two-Point Autocorrelation Tensor (In Time):
Rij(xi , t; t) = ui (xi , t)u
j(xi , t + t
)
Two-Point Velocity Correlation Tensor (In Space):
Rij(xi , t; ri ) = ui (xi , t)uj(xi + ri , t)
For both correlations,
k(xi , t) =1
2Rii (xi , t; 0)
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2.10 Two-Point Correlations
2.10.2 Integral Length and Time Scales
The integral length and time scales, and `, can be defined
asfollows:
`(xi , t) =3
16
0
Rii (xi , t; r)
k(xi , t)dr
(xi , t) =
0
Rii (xi , t; t)
2k(xi , t)dt
where r = |ri |=ri ri and 3/16 is a scaling factor.
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2.10 Two-Point Correlations
2.10.3 Taylors Hypothesis
The two types of two-point correlations can be related by
applyingTaylors hypothesis which assumes that
t= Ui
xi
This relationship assumes that |ui | |Ui | and predicts that
theturbulence essentially passes through points in space as a
whole,transported by the mean flow (i.e., assumption of
frozenturbulence).
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2.11 Favre Time Averaging
2.11.1 Reynolds Time Averaging for Compressible Flows
If Reynolds time averaging is applied to the compressible form
ofthe Navier-Stokes equations, some difficulties arise. In
particular,the original form of the equations is significantly
altered. To seethis, consider Reynolds averaging applied to the
continuityequation for compressible flow. Application of
time-averaging tothe continuity equations yields
t+
xi(ui ) = 0
t
(+
)+
xi
[(+ )
(Ui + ui
)]= 0
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2.11.1 Reynolds Time Averaging for Compressible Flows
The Reynolds time averaging yields
t() +
xi
[Ui + ui
]= 0
The introduction of high-order correlations involving the
densityfluctuations, such as ui , can complicate the turbulence
modellingand closure. Some of the complications can be circumvented
byintroducing an alternative time averaging procedure: Favre
timeaveraging, which is a mass weighted time averaging
procedure.
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2.11 Favre Time Averaging
2.11.2 Definition
Favre time averaging can be defined as follows. The
instantaneoussolution variable, , is decomposed into a mean
quantity, , andfluctuating component, , as follows:
= +
The Favre time-averaging is then
(xi , t) =1
T
t+T/2tT/2
(xi , t)(xi , t ) dt = + =
where
(xi , t) 1T
t+T/2tT/2
(xi , t)(xi , t ) dt , 0
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2.11 Favre Time Averaging
2.11.3 Comparison of Reynolds and Favre Averaging
I Decomposition
Reynolds : = + , Favre : = +
I Time Averaging
Reynolds : = + = , Favre : = (+ ) =
I Fluctuations
Reynolds : = 0 , Favre : = 0
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2.11.3 Comparison of Reynolds and Favre Averaging
Further comparisons are possible. For Reynolds averaging we
have
= +
and for Favre averaging we have
=
Thus = +
or
= +
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2.11.3 Comparison of Reynolds and Favre Averaging
We also note that 6= 0
To see this, start with
= =
Now applying time averaging, we have
=
=
=
6= 0
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2.11.3 Comparison of Reynolds and Favre Averaging
Returning to the compressible form of the continuity equation,
wecan write
ui = Ui + ui = ui
and therefore the Favre-averaged form of the continuity equation
isgiven by
t() +
xi(ui ) = 0
It is quite evident that the Favre-averaging procedure
hasrecovered the original form of the continuity equation
withoutintroducing additional high-order correlations.
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2.12 Favre-Averaged Navier-Stokes (FANS) EquationsContinuity
Equation:
t() +
xi(ui ) = 0
Momentum Equation:
t(ui ) +
xj(ui uj + pij) =
xj
(ij ui uj
)Favre-Averaged Reynolds Stress Tensor:
= ui ujTurbulent Kinetic Energy:
1
2ui u
i =
1
2ii = k
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2.12 Favre-Averaged Navier-Stokes (FANS) EquationsEnergy
Equation:
t
[
(e +
1
2ui ui
)+
1
2ui u
i
]+
xj
[uj
(h +
1
2ui ui
)+
uj2ui u
i
]=
xj
[(ij ui uj
)ui qj
]+
xj
[uj h
1
2uj u
i ui + u
i ij
]Turbulent Transport of Heat and Molecular Diffusion of
TurbulentEnergy:
qtj = uj h , ui ij
Turbulent Transport of Kinetic Energy:
1
2uj u
i ui
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2.13 Turbulence Modelling
Turbulence Modelling provides a mathematical framework
fordetermining the additional terms (i.e., correlations) that
appear inthe FANS and RANS equations.
Turbulence models may be classified as follows:I Eddy-Viscosity
Models (based on Boussinesq approxmiation)
I 0-Equation or Algebraic ModelsI 1-Equation ModelsI 2-Equation
Models
I Second-Moment Closure ModelsI Reynolds-Stress, 7-Equation
Models
89