2. Conservation Equations for Turbulent...

AER1310: TURBULENCE MODELLING 2. Conservation Equations for Turbulent Flows C. P. T. Groth c©2018

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

1


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.

2


2.1 Review of Tensor NotationExpression Vector Notation Tensor Notation

scalars π, c π, c(zeroth-order tensor)

operations

(+, −, ×, /) e.g., π − c ,π

cπ − c ,

π

c

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)

3



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]

x1

x2

x3

= a1x1 + a2x2 + a3x3

4



Expression Vector Notation Tensor Notation

cross product ~a× ~x = ~r =

∣∣∣∣∣ ~i ~j ~ka1 a2 a3

x1 x2 x3

∣∣∣∣∣ εijkajxk = ri

(vector result) ~r =(a2x3 − a3x2)~i

−(a1x3 − a3x1)~j

+(a1x2 − a2x1)~k

εijk = permutation tensor

(sum over j & k implied)

outer product ~a~x = ~a⊗~x =

~~J aαxβ = Jαβ(dyadic result, (second-order tensor,

vector of vectors) 9 elements,

6 elements for symmetric tensor)

5



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 =

u1

u2

u3

[v1 v2 v3] =

u1v1 u1v2 u1v3

u2v1 u2v2 u2v3

u3v1 u3v2 u3v3

6



dyads~~d = ~u~v dij = uiuj

dyad-vector products~~A · ~x = ~b Aαβxβ = bα

(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)

7



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)

8



2.1.3 Permutation Tensor

The permuation tensor, εijk , is a third-order tensor that isintroduced for defining cross products with the following propertiesfor its elements:

ε123 = ε231 = ε312 = 1 , even permutations

ε213 = ε321 = ε132 = −1 , odd permutations

ε111 = ε222 = ε333 = 0 , repeated indices

ε112 = ε113 = ε221 = ε223 = ε331 = ε322 = 0 , repeated indices

9



2.1.4 Kronecker Delta Tensor

The Kronecker delta tensor, δij , is a second-order tensor that isdefined as follows:

δij =

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

δii = trace(I) = 3

10



2.1.5 ε− δ Indentity

The following identity relates the permutation and Kronecker deltatensors:

εijkεist = δjsδkt − δjtδks

11



differential operators

gradient ~V = ~∇φ Vi =∂φ

∂xi

divergence c = ~∇ ·~a c =∂ai∂xi

~u · ~∇φ ui∂φ

∂xi

curl ~g = ~∇×~a gi = εijk∂ak∂xj

vector derivative~~P = ~∇~B Pij =

∂Bi

∂xj

Laplacian c = ∇2φ = ~∇ · ~∇φ c =∂2φ

∂xi∂xi

~a = ∇2~A = ~∇ · ~∇~A ai =∂2Ai

∂xj∂xj12



2.1.6 Other Notation

In the course textbook and elsewhere you will some time see theuse of the shorthand tensor notation:

~∇p =∂p

∂xi= 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.

13


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 four independent variables, the three-component position vector,~x or xi , and the scalar time, 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.

14



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

∂ρ

∂t+ ~∇ · (ρ~u) = 0

where ρ and ~u are the gas density and flow velocity, respectively.In tensor notation, the continuity equation can be written as

∂ρ

∂t+

∂

∂xi(ρui ) = 0

15



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

ρ dV = −∮A

ρ~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.

16



2.2.2 Momentum Equation

The momentum equation is a vector equation that represents theapplication of Newton’s 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(ρ~u) + ~∇ ·

(ρ~u~u + p

~~I − ~~τ)

= ρ~f

where p and ~~τ 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(ρui ) +

∂

∂xj(ρuiuj + pδij − τij) = ρfi

17



For the control volume, the integral form of the momentumequation is given by

d

dt

∫V

ρ~u dV = −∮A

(ρ~u~u + p

~~I − ~~τ)· ~n dA+

∫V

ρ~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.

18



2.2.3 Energy Equation

The energy equation is a scalar equation that represents theapplication of the 1st Law of Thermodynamics to the gaseousmotion. It describes the time rate of change of the total energy ofthe gas (the sum of kinetic energy of bulk motion and internalkinetic or thermal energy). Using vector notation, it has the form

∂

∂t(ρE ) + ~∇ ·

[ρ~u

(E +

p

ρ

)− ~~τ · ~u + ~q

]= ρ~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(ρE ) +

∂

∂xi

[ρui

(E +

p

ρ

)− τijuj + qi

]= ρfiui

19


2.2.3 Energy Equation

For the control volume, the integral form of the energy equation isgiven by

d

dt

∫V

ρE dV = −∮A

[ρ~u

(E +

p

ρ

)− ~~τ · ~u + ~q

]·~n dA+

∫V

ρ~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.

20



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, τijand 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.

21



2.2.4 Thermodynamic Relationships

In this course, we will assume that the gas satisfies the ideal gasequation of state relating ρ, p, and T , given by

p = ρRT

and behaves as a calorically perfect gas (polytropic gas) withconstant specific heats, cv and cp, and specific heat ratio, γ, suchthat

e = cvT =p

(γ − 1)ρand h = e +

p

ρ= cpT =

γp

(γ − 1)ρ

where R is the gas constant, cv is the specific heat at constantvolume, cp is the specific heat at constant pressure, and γ=cp/cv .

22



2.2.5 Mach Number and Sound Speed

For a polytropic gas, the sound speed, a, can be determined using

a =

√γp

ρ=√γRT

and thus the flow Mach number, M, is given by

M =u

a=

u√γRT

23



2.2.6 Constitutive Relationships

The constitutive relations provide expressions for the fluid stresstensor, τij , 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

τij = µ

[(∂ui∂xj

+∂uj∂xi

)− 2

3δij∂uk∂xk

](τii = 0, traceless)

where µ is the dynamic viscosity of the gas. Fourier’s Law can beused to relate the heat flux to the temperature gradient as follows:

qi = −κ∂T∂xi

or ~q = −κ~∇T

where κ is the coefficient of thermal conductivity for the gas.

24



2.2.7 Transport Coefficients

In general, the transport coefficients, µ and κ, are functions ofboth pressure and temperature:

µ = µ(p,T ) and κ = κ(p,T )

Expressions, such as Sutherland’s Law can be used to determinethe dynamics viscosity as a function of temperature (i.e.,µ=µ(T )). The Prandtl number can also be used to relate µ andκ. The non-dimensional Prandtl number is defined as follows:

Pr =µcpκ

and is typically 0.70-0.72 for many gases. Given µ, the thermalconductivity can be related to viscosity using the precedingexpression for the Prandtl number.

25



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.

26


2.3 Navier-Stokes Equations for an Incompressible Gas

For low flow Mach numbers (i.e., low subsonic flow, M<1/4), theassumption that the gas behaves as an incompressible fluid isgenerally a good approximation. By assuming that

• the density, ρ, is constant;

• temperature variations are small and unimportant such thatthe energy equation can be neglected; and

• the viscosity, µ, is constant;

one can arrive at the Navier-Stokes equations describing the flowof incompressible fluids.

27




Using vector notation, the continuity equation for incompressibleflow reduces to

~∇ · ~u = 0

In other words, the velocity vector, ~u, is a solenoidal vector fieldand is divergence free. In tensor notation, the solenoidal conditioncan be expressed as

∂ui∂xi

= 0

28




Using vector notation, the momentum equation for anincompressible fluid can be written as

∂~u

∂t+ ~u · ~∇~u +

1

ρ~∇p =

1

ρ~∇ · ~~τ

In tensor notation, the incompressible form of the momentumequation is given by

∂ui∂t

+ uj∂ui∂xj

+1

ρ

∂p

∂xi=

1

ρ

∂τij∂xj

29



2.3.3 Constitutive Relationships

For incompressible flows, the Navier-Stokes constitutive relationrelating the fluid stresses and fluid strain rate can be written as

τij = µ

(∂ui∂xj

+∂uj∂xi

)= ρν

(∂ui∂xj

+∂uj∂xi

)= 2ρνSij

where ν=µ/ρ is the kinematic viscosity and the strain rate tensor(dyadic quantity) is given by

Sij =1

2

(∂ui∂xj

+∂uj∂xi

)As in the compressible case, the fluid stress tensor forincompressible flow is still traceless and τjj =0.

30



2.3.4 Vorticity Transport Equation

The vorticity vector, ~Ω, is related to the rotation of a fluid elementand is defined as follows:

~Ω = ~∇× ~u or Ωi = εijk∂uk∂xj

For incompressible flows, the momentum equation can be used toarrive at a transport equation for the flow vorticity given by

∂~Ω

∂t− ~∇× ~u × ~Ω = ν∇2~Ω

31


2.3.4 Vorticity Transport Equation

Using ~∇× ~u × ~Ω = ~Ω · ~∇~u − ~u · ~∇~Ω, the vorticity transportequation can be re-expressed as

∂~Ω

∂t+ ~u · ~∇~Ω− ~Ω · ~∇~u = ν∇2~Ω

Using tensor notation, this equation can be written as

∂Ωi

∂t+ uj

∂Ωi

∂xj− Ωj

∂ui∂xj

= ν∂2Ωi

∂xj∂xj

32


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, u′i , about a mean component, Ui , asfollows:

ui = Ui + u′i

One can then develop and solve conservation equations for themean quantities (i.e., the Reynolds-averaged Navier-Stokes(RANS) equations) and incorporate the influence of thefluctuations on the mean flow via turbulence modelling.

33



2.4.1 Forms of Reynolds Averaging

1. Time Averaging: appropriate for steady mean flows

FT (~x) = limT→∞

1

T

∫ t+T/2

t−T/2f (~x , t ′) dt ′

2. Spatial Averaging: suitable for homogeneous turbulent flows

FV(t) = limV→∞

1

V

∫Vf (~x , t) dV

3. Ensemble Averaging: most general form of averaging

FE(~x , t) = limN→∞

1

N

N∑n=1

fn(~x , t)

where fn(~x , t) is nth instance of flow solution with initial andboundary data differing by random infinitessimalperturbations.

34


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.”

35



2.4.2 Reynolds Time Averaging

In Reynolds time averaging, all instantaneous flow quantities,φ(xi , t) and a(xi , t), will be represented as a sum of mean andfluctuating components, Φ(xi ) and φ′(xi , t) and A(xi ) and a′(xi , t),respectively, such that

φ(xi , t) = Φ(xi ) + φ′(xi , t) or a(xi , t) = A(xi ) + a′(xi , t)

For the flow velocity, we have

ui (xα, t) = Ui (xα) + u′i (xα, t)

36



The time averaging procedure is defined as follows and yields thetime averaged quantities:

φ(xi , t) = Φ(xi ) = limT→∞

1

T

∫ t+T/2

t−T/2φ(xi , t

′) dt ′

a(xi , t) = A(xi ) = limT→∞

1

T

∫ t+T/2

t−T/2a(xi , t

′) dt ′

By definition, time averaging of mean quantities merely recoversthe mean quantity:

Ui (xα) = limT→∞

1

T

∫ t+T/2

t−T/2Ui (xα) dt ′ = Ui (xα)

37



Similarly by definition, time averaging of time-averaged quantitiesyields zero:

u′i (xα, t) = limT→∞

1

T

∫ t+T/2

t−T/2

[ui (xα, t

′)− Ui (xα)]dt ′ = 0

38



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 T2

where T2 is the time scale for the variation of the mean.

39




t

u(x,t)

T1

T2

T1 T T2

40



Provided there exists this separation of scales, the time averagingprocedure for time-varying mean flows can be defined as follows:

φ(xi , t) = Φ(xi , t) =1

T

∫ t+T/2

t−T/2φ(xi , t

′) dt ′

a(xi , t) = A(xi , t) =1

T

∫ t+T/2

t−T/2a(xi , t

′) dt ′

with T1 T T2.

41



2.4.4 Properties of Reynolds Time Averaging

Multiplication by a scalar:

c a(xi , t) =c

T

∫ t+T/2

t−T/2a(xi , t

′) dt ′ = cA

Spatial differentiation:

∂a

∂xi=

1

T

∫ t+T/2

t−T/2

∂a

∂xidt ′ =

∂

∂xi

(1

T

∫ t+T/2

t−T/2a dt ′

)=∂A

∂xi

42



2.4.4 Properties of Reynolds Time Averaging

Temporal differentiation:

∂ui∂t

=1

T

∫ t+T/2

t−T/2

∂ui∂t

dt′

=Ui (xi , t + T/2) − Ui (xi , t − T/2)

T+

u′i (xi , t + T/2) − u′i (xi , t − T/2)

T

≈ ∂Ui

∂t

The latter is obtained by assuming that |~u′| |~U| and T T2.

43



2.4.5 Single-Point Correlations

What about time-averaged products?

a(xi , t)b(xi , t) = (A + a′) (B + b′)

= AB + a′B + b′A + a′b′

= AB + Ba′ + Ab′ + a′b′

= AB + Ba′ + Ab′ + a′b′

= AB + a′b′

In general, a′ and b′ are said to be correlated if

a′b′ 6= 0

and uncorrelated ifa′b′ = 0

44



2.4.5 Single-Point Correlations

What about triple products? Can show that

a(xi , t)b(xi , t)c(xi , t) = ABC + a′b′C + a′c ′B + b′c ′A + a′b′c ′

45


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

∂ui∂xi

= 0

or∂Ui

∂xi= 0

46



2.5.1 Derivation

For the incompressible form of the momentum equation we have

∂ui∂t

+ uj∂ui∂xj

+1

ρ

∂p

∂xi=∂ui∂t

+ uj∂ui∂xj

+1

ρ

∂p

∂xi=

1

ρ

∂τij∂xj

Considering each term in the time-averaged equation above wehave:

∂ui∂t

=∂Ui

∂t

1

ρ

∂p

∂xi=

1

ρ

∂p

∂xi=

1

ρ

∂P

∂xi

47



2.5.1 Derivation

1

ρ

∂τij∂xj

=1

ρ

∂τij∂xj

=2

ρρν∂Sij∂xj

= 2ν∂Sij∂xj

where the mean strain, Sij , is defined as

Sij =1

2

[∂Ui

∂xj+∂Uj

∂xi

]

48



2.5.1 Derivation

uj∂ui∂xj

=∂

∂xj(uiuj)− ui

∂uj∂xj

=∂

∂xj

(UiUj + u′iu

′j

)=

∂

∂xj(UiUj) +

∂

∂xj

(u′iu′j

)= Uj

∂Ui

∂xj+ Ui

∂Uj

∂xj+

∂

∂xj

(u′iu′j

)= Uj

∂Ui

∂xj+

∂

∂xj

(u′iu′j

)Thus we have

∂Ui

∂t+ Uj

∂Ui

∂xj+

1

ρ

∂P

∂xi=

1

ρ

∂

∂xj

(2µSij − ρu′iu′j

)49



2.5.2 Summary

In summary, the RANS describing the time-evolution of the meanflow quantities Ui and P can be written as

∂Ui

∂xi= 0

∂Ui

∂t+ Uj

∂Ui

∂xj+

1

ρ

∂P

∂xi=

1

ρ

∂

∂xj(τij + λij)

where τij is the fluid stress tensor evaluated in terms of the meanflow quantities and λij is the Reynolds or turbulent stress tensorgiven by

λij = −ρu′iu′j

50


2.6 Reynolds Turbulent Stresses and Closure Problem

2.6.1 Closure or RANS Equations

The Reynolds stressesλij = −ρu′iu′j

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, λij , 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 λij interms of mean flow solution quantities.

51


2.6 Reynolds Turbulent Stresses and Closure Problem

2.6.2 Reynolds Stress Transport Equations

Transport equations for the Reynolds stresses, λij =−ρu′iu′j 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 ,

∂ui∂t

+ uj∂ui∂xj

+1

ρ

∂p

∂xi=

1

ρ

∂τij∂xj

and noting that

1

ρ

∂τij∂xj

=µ

ρ

∂

∂xj

(∂ui∂xj

+∂uj∂xi

)= ν

[∂2ui∂xj∂xj

+∂

∂xi

(∂uj∂xj

)]= ν

∂2ui∂xj∂xj

52



one can write

∂ui∂t

+ uk∂ui∂xk

+1

ρ

∂p

∂xi− ν ∂2ui

∂xk∂xk= 0 (1)

Similarily,

∂uj∂t

+ uk∂uj∂xk

+1

ρ

∂p

∂xj− ν

∂2uj∂xk∂xk

= 0 (2)

Thus, u′j × (1) + u′i × (2) can be written as

0 = u′j

(∂ui∂t

+ uk∂ui∂xk

+1

ρ

∂p

∂xi− ν ∂2ui

∂xk∂xk

)+u′i

(∂uj∂t

+ uk∂uj∂xk

+1

ρ

∂p

∂xj− ν

∂2uj∂xk∂xk

)53



The various terms appearing in the preceding equation can beexpressed as follows:

u′j∂ui∂t

+ u′i∂uj∂t

= u′j∂

∂t

(Ui + u′i

)+ u′i

∂

∂t

(Uj + u′j

)=

∂Ui

∂tu′j + u′j

∂u′i∂t

+∂Uj

∂tu′i + u′i

∂u′j∂t

= u′j∂u′i∂t

+ u′i∂u′j∂t

=∂

∂t

(u′iu′j

)= −1

ρ

∂λij∂t

54



u′jρ

∂p

∂xi+

u′iρ

∂p

∂xj=

u′jρ

∂

∂xi(P + p′) +

u′iρ

∂

∂xj(P + p′)

=∂P

∂xiu′j +

1

ρu′j∂p′

∂xi+∂P

∂xju′i +

1

ρu′i∂p′

∂xj

=1

ρ

[u′j∂p′

∂xi+ u′i

∂p′

∂xj

]

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νu′j∂2ui∂xk∂xk

+ νu′i∂2uj∂xk∂xk

= νu′j∂2

∂xk∂xk(Ui + u′i ) + νu′i

∂2

∂xk∂xk

(Uj + u′j

)= ν

∂2Ui

∂xk∂xku′j + νu′j

∂2u′i∂xk∂xk

+ ν∂2Uj

∂xk∂xku′i + νu′i

∂2u′j∂xk∂xk

= νu′j∂2u′i∂xk∂xk

+ νu′i∂2u′j∂xk∂xk

= ν∂2

∂xk∂xk

(u′i u′j

)− 2ν

∂u′i∂xk

∂u′j∂xk

= −νρ

∂2λij

∂xk∂xk− 2ν

∂u′i∂xk

∂u′j∂xk

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u′juk∂ui∂xk

+ u′i uk∂uj∂xk

= u′j (Uk + u′k)∂

∂xk(Ui + u′i ) + u′i (Uk + u′k)

∂

∂xk

(Uj + u′j

)= Uk

∂

∂xk

(u′i u′j

)+ u′ju

′k

∂Ui

∂xk+ u′i u

′k

∂Uj

∂xk

+Uk∂Ui

∂xku′j + Uk

∂Uj

∂xku′i + u′k

∂

∂xk

(u′i u′j

)= −Uk

ρ

∂λij

∂xk− λjk

ρ

∂Ui

∂xk− λik

ρ

∂Uj

∂xk

+∂

∂xk

(u′i u′ju′k

)− u′i u

′j

∂u′k∂xk

= −Uk

ρ

∂λij

∂xk− λjk

ρ

∂Ui

∂xk− λik

ρ

∂Uj

∂xk+

∂

∂xk

(u′i u′ju′k

)

57



Combining all of these terms, can write

∂λij∂t

+ Uk∂λij∂xk

+ λjk∂Ui

∂xk+ λik

∂Uj

∂xk=

∂

∂xk

[ν∂λij∂xk

+ ρu′iu′ju′k

]+u′j

∂p′

∂xi+ u′i

∂p′

∂xj

+2µ∂u′i∂xk

∂u′j∂xk

The preceding is a transport equation describing the time evolutionof the Reynolds stresses, λij .

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While providing a description for the transport of λij , the Reynoldsstress equations introduce a number of other correlations offluctuating quantities:

u′j∂p′

∂xi: symmetric second-order tensor, 6 entries

ρu′iu′ju′k : symmetric third-order tensor, 10 entries

2µ∂u′i∂xk

∂u′j∂xk

: 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

2u′iu′i =

1

2

(u′2 + v ′2 + w ′2

)= −1

2

λiiρ

= − 1

2ρ(λxx + λyy + λzz)

where u′2 =−λxx/ρ, v ′2 =−λyy/ρ, and w ′2 =−λzz/ρ.

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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 =

√u′2

U, v =

√v ′2

U, w =

√w ′2

U

where U is a reference velocity.

For isotropic turbulence, u′2 = 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

u′2 : v ′2 : w ′2 = 4 : 2 : 361


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

2u′iu′i = −1

2

λiiρ

The following equation for the transport of k can be obtained:

∂k

∂t+Ui

∂k

∂xi=λijρ

∂Ui

∂xj+∂

∂xi

(ν∂k

∂xi− 1

ρp′u′i −

1

2u′iu′ku′k

)−ν

∂u′i∂xj

∂u′i∂xj

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

Ui∂k

∂xi: convection transport of k

Production:

λijρ

∂Ui

∂xj: production of k by mean flow

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Diffusion:

ν∂k

∂xi: molecular diffusion of k

1

ρp′u′i : pressure diffusion of k

1

2u′iu′ku′k : turbulent transport of k

65



Dissipation:

ν∂u′i∂xj

∂u′i∂xj

= ε : dissipation of k at small scales

where ε is the dissipation rate of turbulent kinetic energy.

66


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 (κ)dκ

where E (κ) is the spectral distribution of turbulent energy,κ is the wave number of the Fourier-like energy mode, and ` is thewave length of the energy mode such that

E (κ)dκ = turbulent energy contained between κ and κ+ dκ

and where

` =1

κ

67



68



ηκDIκEI

Dissipationrange

Inertial subrangeEnergy-containing range

Slope 2

Slope -5/3

E(κ)

κ

69



2.9.2 Range of Turbulent Scales

The large-scale turbulent motion (κ→ 0) contains most of theturbulent kinetic energy, while most of the vorticity resides in thesmall-scale turbulent motion (κ→ 1/η), where η, 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 Kolmogorov’s universalequilibrium theory and his first similarity hypothesis that

dk

dt= −ε , and η =

(ν3

ε

)1/4

70



For high Reynolds number turbulence, dimensional analysis andexperimental measurements confirm that the dissipation rate, ε,turbulent kinetic energy, k, and largest scale representing the largescale motions (i.e., scale of the largest eddies), `0, are related asfollows:

ε ∝ k3/2

`0

When discussing features of turbulence, it was noted that itcontains a wide range of scales. This implies that

`0 η

71



Using the expression above for `0, an examination of the lengthscales reveals that

`0

η=

`0

(ν3/ε)1/4≈ `0

ν3/4

(k3/2

`0

)1/4

≈

(k1/2`0

ν

)3/4

≈ Re3/4t

where Ret is the turbulent Reynolds number. Thus `0 η for highturbulent Reynolds number flows (i.e., for Ret 1). The latter isa key assumption entering into Kolmogorov’s universal equilibriumtheory and his three hypotheses.

72



2.9.3 Kolmogorov -5/3 Law

Kolmogorov also hypothesized an intermediate range of turbulentscales lying between the largest scales and smallest scales whereinertial effects dominate (this is the basis for the second similarityhypothesis). He postulated that in this inertial sub-range, E (κ)only depends on κ and ε. Using dimensional analysis he arguedthat

E (κ) = Ckε2/3

κ5/3

orE (κ) ∝ κ−5/3

73



74


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.

75


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.

76


2.10.1 Two-Point Velocity Correlations

Two-Point Autocorrelation Tensor (In Time):

Rij(xi , t; t ′) = u′i (xi , t)u′j(xi , t + t ′)

Two-Point Velocity Correlation Tensor (In Space):

Rij(xi , t; ri ) = u′i (xi , t)u′j(xi + ri , t)

For both correlations,

k(xi , t) =1

2Rii (xi , t; 0)

77



2.10.2 Integral Length and Time Scales

The integral length and time scales, τ and `, can be defined asfollows:

`(xi , t) =3

16

∫ ∞0

Rii (xi , t; r)

k(xi , t)dr

τ(xi , t) =

∫ ∞0

Rii (xi , t; t ′)

2k(xi , t)dt ′

where r = |ri |=√ri ri and 3/16 is a scaling factor.

78



2.10.3 Taylor’s Hypothesis

The two types of two-point correlations can be related by applyingTaylor’s hypothesis which assumes that

∂

∂t= −Ui

∂

∂xi

This relationship assumes that |u′i | |Ui | and predicts that theturbulence essentially passes through points in space as a whole,transported by the mean flow (i.e., assumption of frozenturbulence).

79


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

∂ρ

∂t+

∂

∂xi(ρui ) = 0

∂

∂t

(ρ+ ρ′

)+

∂

∂xi

[(ρ+ ρ′)

(Ui + u′i

)]= 0

80


2.11.1 Reynolds Time Averaging for Compressible Flows

The Reynolds time averaging yields

∂

∂t(ρ) +

∂

∂xi

[ρUi + ρ′u′i

]= 0

The introduction of high-order correlations involving the densityfluctuations, such as ρ′u′i , 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.

81



2.11.2 Definition

Favre time averaging can be defined as follows. The instantaneoussolution variable, φ, is decomposed into a mean quantity, φ, andfluctuating component, φ′′, as follows:

φ = φ+ φ′′

The Favre time-averaging is then

ρφ(xi , t) =1

T

∫ t+T/2

t−T/2ρ(xi , t

′)φ(xi , t′) dt ′ = ρφ+ ρφ′′ = ρφ

where

φ(xi , t) ≡ 1

ρT

∫ t+T/2

t−T/2ρ(xi , t

′)φ(xi , t′) dt ′ , ρφ′′ ≡ 0

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2.11.3 Comparison of Reynolds and Favre Averaging

I Decomposition

Reynolds : φ = φ+ φ′ , Favre : φ = φ+ φ′′

I Time Averaging

Reynolds : φ = φ+ φ′ = φ , Favre : ρφ = ρ(φ+ φ′′) = ρφ

I Fluctuations

Reynolds : φ′ = 0 , Favre : ρφ′′ = 0

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Further comparisons are possible. For Reynolds averaging we have

ρφ = ρφ+ ρ′φ′

and for Favre averaging we have

ρφ = ρφ

Thusρφ = ρφ+ ρ′φ′

or

φ = φ+ρ′φ′

ρ

84



We also note thatφ′′ 6= 0

To see this, start with

φ′′ = φ− φ = φ− φ− ρ′φ′

ρ

Now applying time averaging, we have

φ′′ = φ− φ− ρ′φ′

ρ= φ− φ− ρ′φ′

ρ= −ρ

′φ′

ρ6= 0

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Returning to the compressible form of the continuity equation, wecan write

ρui = ρUi + ρ′u′i = ρui

and therefore the Favre-averaged form of the continuity equation isgiven by

∂

∂t(ρ) +

∂

∂xi(ρui ) = 0

It is quite evident that the Favre-averaging procedure hasrecovered the original form of the continuity equation withoutintroducing additional high-order correlations.

86


2.12 Favre-Averaged Navier-Stokes (FANS) EquationsContinuity Equation:

∂

∂t(ρ) +

∂

∂xi(ρui ) = 0

Momentum Equation:

∂

∂t(ρui ) +

∂

∂xj(ρui uj + pδij) =

∂

∂xj

(τij − ρu′′i u′′j

)Favre-Averaged Reynolds Stress Tensor:

λ = −ρu′′i u′′j

Turbulent Kinetic Energy:

1

2ρu′′i u

′′i = −1

2λii = ρk

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2.12 Favre-Averaged Navier-Stokes (FANS) EquationsEnergy Equation:

∂

∂t

[ρ

(e +

1

2ui ui

)+

1

2ρu′′i u

′′i

]+

∂

∂xj

[ρuj

(h +

1

2ui ui

)+

uj2ρu′′i u

′′i

]=

∂

∂xj

[(τij − ρu′′i u′′j

)ui − qj

]+∂

∂xj

[−ρu′′j h′′ −

1

2ρu′′j u

′′i u′′i + ρu′′i τij

]Turbulent Transport of Heat and Molecular Diffusion of TurbulentEnergy:

qtj = ρu′′j h′′ , ρu′′i τij

Turbulent Transport of Kinetic Energy:

1

2ρu′′j u

′′i u′′i

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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

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