A

Application of the two-point correlation technique for interpreting the € equation1

We shall now analyse the advantages of (3.16) over the corresponding equation that defines the trace to:

of the dissipation tensor t:ij.

~Ui ~Ui t:=v-­

OXk OXk

To derive the equation for the dissipation rate, to, we first differentiate (2.26) for the instantaneous fluctuation with O/OXl to obtain

Multiplying the above equation by 2voudoXl and using the mass conservation law results in the following equation:

a aUi aUi 2 aUi aUi aU" -u a aUi aUi 2 aUi au" aUi -v--+ v----+ ,,-v--+ v---at aXj aXj ax" aXj aXj ax" aXj aXj aXj aXj ax"

2- 2 2 2 aUi a U i 2 aUi aUk aUi 2 aUi a Ui 2 aUi a + VUk----+ v---+ VUk----- v----

aXj aX"aXj ax! ax! aXk aXj aXkaX! aXj aXkaX!

__ 2VaUi a 2p 2 aUi aUi XUkUi = --~~ + 2v ~6",~. (A.2)

(} vX! VXiVXj vXj vX!

Using the following transformations:

(A.3)

1 This chapter is reproduced from Jovanovic, Ye and Durst (1995) Statistical interpretation of the turbulent dissipation rate in wall-bounded flows, Journal Fluid Mech 293, by permission of the Cambridge University Press.

122 A Application of the two-point correlation technique for interpreting ...

2 28ui A 8Ui A 8Ui 8Ui 2 2 82Ui 82Ui V - L.l - - V L.l v- - - V -=----::-- -::----:::--

8Xl x 8 Xl - x 8 Xl 8 Xl 8 Xl8xn 8Xl8xn ' (A.4)

and averaging (A.2), we obtain the equation for determining the dissipation rate €:

_2V8ui 8Uk 8Ui _ V~[Uk 8Ui 8Ui]_ 2v 8Ui ~ 8Xl 8Xl 8Xk 8Xk 8Xl 8 Xl (J 8 Xl 8xi8 x l

[6] [7] [8]

To gain insight into the significance of the various terms and into the structure of the dissipation equation, we shall apply the two-point correlation technique which was already discussed in Sect. 3.1. The individual terms of (A.5) read as follows:

Term 1

(A.6)

Term 2

Term 3

(A.8)

Term 4

A Application of the two-point correlation technique for interpreting... 123

Term 5

Terms 6 and 7

4 OUi OUk OUi 0 OUi OUi v 0 __ P +T, = -2v--- - V-Uk-- = ---llxUiUkUi

, OXI OXI OXk OXk OXIOXI 40Xk

Term 8

Term 9

Term 10

+~ O~k [(lleUiUkUDo + (lleUiU~U~)ol

+V[(O~k lleUiU~uDo - (O~k lleUiUkUDo],

(A.ll)

(A.14)

By adding all of the derived terms defined by (A.6)-(A.14), we obtain the following form of the dissipation equation:

124 A Application of the two-point correlation technique for interpreting ...

v al a -- v- a 2 - a ---6. - - v-(6.<u·u~)o + -Uk-6. q - VUk-(6.<U·U')o = 4 x at at'" 4 aXk x aXk' , ,

V __ aUi --, aUi --, aUi v a 2q2 aUk --26.xUiUk,,--- +V(6.eUkU;)O"--- +v(6.eUiUk)O,,--- - -~...,,-

UXk UXk UXk 2 UXkUXZ uXz

a 2 --, aUk a __ a 2Ui a --, a 2Ui +2v( "'c "'c UiUi)O...,,- - V~UiUk ~ + v( "'c UiUk)O~

Uo"kUo"Z uXz UXz UXkUXZ uo"z UXkUXZ

a --, a 2Ui v a __ v a --, --, -, -v( "'c UkU;)O~ - -4 ,,---6.xUiu kui + -2 ,,---[(6.euiUkUi)O + (6.euiukui)O]

uo"z uXkUXZ UXk UXk

+v[( a~k 6.euiu~ uDo - (a~k 6.eu iu kUDo] - ;0 a~i 6. xUiP + ;0 a~i [(6. epuDo

_ v a _ a - v 2

+(6.eUiP')O] + Q[(a~i 6.eUiP')O - (a~i 6.epuDo]- 86.x6. X UiUi

2 -- 2 -- v 2 2--+v 6.x(6.eUiU;)O - 2v (6.e6.euiU;)O + 46.x6.xUiUi - v 6.x(6.euiuDo.

(A.15)

Equation (A.15) is composed of the terms that are included in (3.16) and higher order terms. The higher order terms are directly related to the equation for q2. To illustrate this point further, we first contract (2.29) to obtain

2 2 - --- ---oq - oq __ oUi OUiUiUk 2 op OUi OUi 2 £l + Uk~ + 2UiUk~ + 0 + -Uin- + 2v~~ - v!:l.xq = O. ut UXk UXk Xk {! UXi UXk UXk

(A.16) By applying the Laplace operator !:l.x to (A.16) and multiplying the resulting equation by v/4, we obtain

Introducing the transformation

v 2 ou· ou· v2 v2 __

2!:l.x ox: ox: = 8!:l.x!:l.xuiUi - 2!:l.X(!:l.~UiUDo,

(A.17) reduces to the form

v A aq2 vaUk a 2q2 v-u a A 2 vaq2 A -U aUiUk -Ux- + ------ + - k-Uxq + --Ux k +v--4 at 2 axz axZaxk 4 aXk 4 aXk axz

a 2Ui v __ aUi v __ a - v a __ x~ + -(6.xuiUk)"--- + -2Uiuk~6.xUi + -4~6.xUiUiuk

UXkUXZ 2 UXk UXk UXk

v a v 2 v 2 -- v 2 + 20 aXi 6. xuiP + 86.x6.xUiUi - 26.x (6.e UiuDo - 46.x6. X UiUi = O.

(A.17)

(A.18)

(A.19)

A Application of the two-point correlation technique for interpreting... 125

Subtracting (A.19) from (A.15), we obtain

a (A -') -u a (A -') (A -') aUi (A -') aUi -V-a i..l.~UiUi 0 - v k-a i..l.~UiUi 0 - V i..l.~UkUi O-a - v i..l.~UiUk O-a t Xk Xk Xk

V__ aUi v aq2 - a --, a2Ui a --, -2UiUk~X aXk - 4" aXk ~xUk + V(ael UkUi)O aXkaXI - V(ael UiUk)O

a2Ui a2 --, aUk v a --, --,-, X-a a - 2v(ac ac UiU;)O-a - -2 -a [(~~UiUkUi)O + (~~UiUkU;)ol

XI Xk "I "k UI Xk

-V[(a~k ~~UiU;UUO - (a~k ~~UiUkUDol- ;{! a~i [(~~puDo + (~~uiP')ol va - a - 12 -- 2 --

-"Q[(aei ~~UiP')O - (aei ~~puDol + 2V ~x(~~UiuDo + 2v (~~~~UiuDo = o. (A.20)

The deduced result is identical with (3.16), which defines the homogeneous part of the turbulent dissipation rate. It is obvious that there is no reason why we should keep the higher order terms that appear in (A.15). The treatment of the dissipation correlations utilizing the two-point correlation technique is an elegant way of eliminating the shortcomings mentioned above.

B

Approximate form of the two-point velocity correlation uiuj tensor

Here we consider the possibility of constructing an approximate form of the two-point correlation tensor uiuj as a basis for analysis of the partition of the stress dissipation and closure of the production terms:

(B.1)

(B.2)

in (3.47). In accordance with the assumption (of local homogeneity) regard­ing the small-scale structure of turbulence, we will require that the correlation tensor satisfies all the constraints that can be defined for homogeneous tur­bulence.

For isotropic turbulence, the two-point correlation tensor is given by the most general linear combination of the basic tensors I;il;j and 8ij ([59, pp. 181-189]):

(B.3)

or the equivalent form

We can construct in a similar way the approximate form of UiUj for ho­mogeneous turbulence starting from the three basic tensors of second rank:

(B.5)

where Rij is defined as

R .. - 3UiUj 2 --tJ - q = UsUs· q2 ' (B.6)

128 B Approximate form of the two-point velocity correlation UiU~ tensor

From (B.5) we can form the additional tensors of second rank:

(B.7)

Since

(B.8)

the general quasi-linear combination of the correlation tensors of second rank reads

UiUj = A~i~j + BJij + CRij + FRip~p~j + ERjp~p~i. (B.9)

The scalar coefficients A, B, ... , E are functions of ~. From the condition of invariance for homogeneous turbulence ([59]):

Uiuj = UjU~,

uiuj(~k) = Uiuj(-~k) = UjU~(~k)' (B.I0)

(B.ll)

it follows that F = E and that the scalar coefficients in (B.9) A, B, ... , E are even powers of ~:

A(~) = AD + ~Amn~m~n + ... J B(~) = Bo + ~Bmn~m~n + .. .

............... E(~) = Eo + -bEmn~m~n + ...

(B.12)

Utilizing the condition of coincidence by setting ~ -t 0 in (B.9):

(B.13)

we obtain the following values for the coefficients Bo and Co :

Bo =0, 1 2

Co = "3 q . (B.14)

Introducing the above relationships into (B.9), the two-point correlation ten­sor reduces to the following form:

--,_12 1 1 UiUj - "3 q {AO~i~j + 2! Bmn~m~nJij + (1 + 2! Cmn~m~n)Rij

+Eo(Rip~p~j + Rjp~p~i) + ... }. (B.15)

We shall now define the conditions imposed by the continuity equation. Using the two-point correlation technique and by combining the partial dif­ferential operators at points A and B:

B Approximate form of the two-point velocity correlation UiU~ tensor 129

(8=k) A = ~ CJ=k) AB - 8~k' (B.16)

(8=k) B = ~ (8=k) AB + 8~k' (B.17)

we obtain

(8=k)B = (8=JA +2~. (B.18)

Applying the differential operator (B.18) to (Ui)A(Uk)B and utilizing the con­dition of symmetry (B.1O) result in the following equation:

8-8~k UiU~ = O. (B.19)

Making use of the continuity equation (B.19) for the two-point correlation tensor (B.15) yields

(B.20)

Since (B.20) is symmetric for the permutation indices i and j, Cpj must there­fore have the following form:

(B.21)

From (B.20) and (B.21), one can evaluate the trace of Bij as follows:

Bss = -CSS - 24Eo - 12Ao. (B.22)

By applying the Laplace operator ~~ to (B.15) and setting ~ = 0, we obtain

(B.23)

Introducing the turbulence mica-scale tensor Pij in a similar manner as for isotropic turbulence:

1 -Pij = - 5q2 (~{UiUj)O' (B.24)

and with the help of (B.23), we may write

1 Pij = -15 {(2Ao + Bss)8ij + CssRij + 4EoRij}. (B.25)

Inserting the conditions valid for isotropic turbulence:

(B.26)

130 B Approximate form of the two-point velocity correlation uiuj tensor

and requiring that the two-point correlation tensor (B.I0) coincides with the appropriate isotropic form:

(B.27)

results in the following relationship between the trace of Pij and coefficients in (B.15):

Pss = 2Ao + 4Eo·

Substituting (B.22) and (B.28) into (B.25) yields

1 Pij = 15 {([5 Pss + 4Eo + Css )8ij + CssRij + 4EoRij}.

By normalizing the coefficients Eo and Css with PSS:

Eo = psse, Css = PssC,

we obtain the final form of the two-point correlation tensor:

--, 12 12 1 1 5 UiUj = 3q ~j + 3q Pss{(2" - 2e)~i~j - Rmn(Gc + 2"e)~m~n8ij

5 2 1 2 -(1 - 2"e)r 8ij + GeT Rij + e(~p~p~j + Rjp~p~i)

(B.28)

(B.29)

(B.30)

+ ... }, (B.31)

and the turbulence micro-scale tensor:

(B.32)

The trace Pss of Pij is related to the Taylor micro-scale A by Pss = 1/ A2 and from (B.24) we find that € in homogeneous turbulence may be expressed as follows:

(B.33)

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Index

anisotropy 6, 18, 56, 72, 95, 103, 110 invariant map, 6, 19, 20, 21, 25, 27,

28, 29, 30, 56, 57, 77, 78, 85, 91, 92, 93, 98, 111

mapping, 27, 29, 56, 57, 107 tensor, 18, 22, 68, 91

Antonia RA 7, 83, 104, 114 assumption of local homogeneity 3, 39,

42, 44, 45, 46, 47, 115, 119, 127 asymptotic states of the turbulence 74 axisymmetric

contraction, 18, 20, 21, 22, 24, 72, 98 deformation, 92 expansion, 18, 20, 21, 22, 24, 92 strain, 58, 72, 73, 79 strained turbulence, 21, 22, 23, 91,

92,94, 103 tensor, 13 turbulence, 13, 18, 20, 24, 27, 28, 29,

58, 73, 75, 77, 97, 111

backward-facing step 30, 31, 56, 60, 62 Batchelor GK 61 Boersma BJ 7 Bradshaw P 41, 54, 60, 73, 77, 94, 121 buffer region 29, 41, 42, 58, 63

Capp SP 51 channel centreline 28, 58, 64, 65, 68, 83,

105, 110, 113 Chou PY 3, 4, 5, 33, 38, 39, 51, 64, 87 Chou's correlation integrals 100 closure

problem, 1, 13

of the production terms, 127 coincidence 67, 78, 128 Coleman GN 53 concept of local homogeneity 44 continuity equation 9, 12, 18, 22, 37,39,

56, 59, 90, 92, 100, 101, 128, 129 Comte-Bellot G 96, 97 convective transport 15 correlations 3

integrals, 91, 100 tensors, 5, 89

Corrsin S 51, 96, 97 Crow SC 90, 92

Davydov BI 4, 5, 51, 66 Deardorff JW 6 deformation rate 89 differencing errors 103 differential operators 33, 128, 129 diffusion hypothesis 66 dissipation 6

correlations, 3, 4, 15, 16, 24, 102 equation, 122, 123 rate, 2, 5, 23, 33, 37, 41, 51, 52, 57,

58, 102, 122 tensor, 67, 70, 121

direct simulation 6, 24, 35, 39, 42, 44, 53, 58, 60, 61, 63, 72, 104, 105, 106, 107, 109, 112, 114, 115, 116

Durst F 8, 33, 121

Eggels JGM 7, 105, 115 effective viscosity 2, 24, 25 equations

138 Index

for the mean flow, 12 turbulent stresses, 14

for the higher-order moments, 16 equilibrium constraint 2, 3, 83 ergodic hypothesis 17 essential peculiarity of axisymmetric

turbulence 25 energy transfer 91

Fatica M 7, 30, 32 fast part of llij 87, 89, 91, 103, 105 favourable pressure gradient 31, 107 flow domain 9, 89, 91, 113 fluctuations 1, 14, 101, 121 Frenkiel FN 109

Gauss theorem 15 Gaussian 66, 109 George WK 51 Gilbert N 7, 8, 28, 39, 42, 48, 50, 59, 64,

79,80 gradient diffusion approximation 111 Green's theorem 90, 100, 118 grid-generated turbulence 18, 54, 55,

95, 96 Groth J 8

Hallback M 8 Haworth DC 17 Hanjalic K 6, 53, 66 Harlow FH 5 Hinze JO 33 higher-order moments 6 homogeneous

axisymmetric turbulence, 95 directions, 99 fields, 11 flows, 104 isotropic turbulence, 51, 52, 55, 103 part of €ij, 35, 67

€, 36, 38, 39, 40, 41, 42, 46, 52, 64, 80,99, 125

sheared turbulence, 103 turbulence, 3, 6, 13, 26, 35, 38, 60, 62,

64, 72, 73, 76, 87, 90, 91, 97, 99, 100, 103, 109, 116, 127, 128, 130

Hussein HJ 51

Ievlev A 17

inhomogeneous contribution to llij, 99 part of €ij, 35, 36, 39, 40, 41

€,41 turbulence, 14, 26, 97

initially isotropic turbulence 89, 90, 92 inner variables 40 integral length scale 53, 55 intercomponent energy transfer 6, 95 interpolation equation 55, 97 isotropic

homogeneous turbulence, 5, 6, 13, 58, 69, 76, 127, 129

state, 21 tensors, 13, 90 two-component limit, 24, 30, 58

isotropy assumption 4 invariant

functions, 58, 62, 72, 102, 111 theory, 17, 18, 21, 24, 25, 27, 56, 72,

85, 91, 92, 97 invariance 64, 109, 128

Johanson AV 8 Jones W 3 Jovanovic J 8, 23, 25, 33, 54, 60, 73, 77,

94, 121

Kasagi N 7 Khajeh-Nouri B 80 Kim J 7, 8, 40 kinematic constraits 37, 91, 104 kinematic relations 21 kinematic viscosity 40 kinetic energy of turbulence 2, 13, 15,

52,57 Kistler AL 51 Klebanoff PS 109 Kleiser L 7, 8, 28, 39, 42, 48, 50, 59, 64,

79,80 Kolmogorov AN 1, 2, 53, 66, 75, 110,

111 length scale, 60

Kolovandin BA 33, 35, 39, 51, 52, 117

Laplace operator 4, 35, 39, 124, 129 Launder BE 2, 3, 5, 8, 53, 66, 72, 102 Le H 7, 30, 31 Lee MJ 7, 23, 24, 25, 92, 94 length scale 111

ratio, 55 limiting states of turbulence 18, 20, 24,

25, 27, 62, 92, 93, 99 Lin CC 3 local

homogeneity, 38, 42, 44, 87, 119 isotropy, 2

locally homogeneous turbulence, 42, 43, 45,

46, 59, 64, 66, 113 isotropic turbulence, 2, 75, 76

logarithmic region 29, 41, 42, 58, 63, 83 Loitsianski's invariant 52, 54 Lumley JL 6, 8, 17, 18, 51, 56, 58, 65,

72, 73, 80, 100

magnitude of the mean strain rate 22 Mansour NN 7, 8, 39, 40, 47, 49, 53, 61,

64, 68, 76, 77, 78, 79, 81, 82, 101 marginal distribution 12 mass conservation 121 mean motion 1 Millionshchikov MD 11 7 Moin P 7, 8, 30, 31 momentum equations 101 Monin AS 17 Monin-Lundgren equations 17 Monte Carlo method 17 Moser RD 7

Navier-Stokes equations 9, 12 NewmanG 6, 8, 17, 18, 56, 72, 73 Nieuwstadt FTM 7 normalization

condition, 11, 17, 118 constraint, 90, 100, 101, 118

normalized distance from the wall 40, 42

non-linearity 1, 12 numerical databases 7, 8, 9, 11, 63, 64,

72, 81, 103, 112

one-component state, 24, 29, 56, 57, 58, 64, 74, 92,

107 one-dimensional turbulence 56 Orlandi P 7, 30, 32 Orszag SA 6 Otic I 23, 25, 54, 60, 73, 77, 94, 121

Index 139

partial differential operators 85 partition of

€ij, 67, 68, 69, 73, 127 Tij, 111

periods of the fluctuations 10 Perot JB 41 pipe

diameter, 106, 115 flow, 30

plain strain 72 planar averages 11 plane deformation 26 Poisson equation 87 Pope SB 17 Prandtl L 1 Prandtl-Kolmogorov hypothesis 23 Prandtl number 7 pressure

gradient fluctuations, 88 fluctuations, 87 transport, 15, 65, 110, 117

pressure-strain 15, 26 correlations, 5, 24, 86, 88, 89, 90, 91,

92,94 pressure-velocity correlations 38, 43,

100 pressure-velocity gradient correlations

86 probability

density distribution, 16 density function, 10, 17 distribution function, 11 mean, 11, 12

production terms 26, 76

quadruple correlations, 119 expansions, 91

random variable 11 rapid distorsion 89

limit, 93 realizability 26, 27, 74, 97, 109, 112, 113 Reece GJ 5 reflection 13, 67, 109 relaminarization 29, 56 relaxation to isotropy 72, 73, 103 re-meshing technique 72 return to isotropy function 105

140 Index

Reynolds 0 1, 9, 10, 11, 12 conditions, 11 number, 3, 8, 9, 27, 28, 29, 30, 39, 42,

51, 52, 53, 56, 62, 68, 69, 71, 74, 80, 95, 9~ 99, 105, 106, 119

stress tensor, 4, 22 stresses, 3, 13, 89

Reynolds WC, 6,72 Rodi W 3, 5, 53 Rogers MM 7 Rogallo RS 7, 8, 23, 24, 25, 58, 60, 72,

73, 74, 75, 76, 77, 79, 92, 94 rotation 13, 21, 30, 32, 72, 73, 76

axis, 32, 103 rotating pipe flow 32 Rotta J 4, 55, 90,95

Saffman's invariant 54 scalar

coefficients, 67, 128 function, 21, 27, 68, 72, 77, 90, 91, 95 invariants, 6, 18

scailing argument 64 Schumann U 6, 8, 26 second-moment turbulence closure 4, 89 second-order moments 3 second-rank tensor 13 Seidl V 56 shear 72, 74 Shih TH 8 skewness factor 51 simulation databases 6, 42, 45, 64, 66,

77 single point triple correlations 65 sink-flow 56, 116 slow part of IIij 87, 89, 91, 103 small scales 72 Spalart PR 7, 29, 31, 106, 107, 113, 116 Spalding DB 2, 3 spatial

invariance, 99 resolution, 72

Speziale CG 32 Sreenivasan KR 53 stationary fields 11 statistically homogeneous turbulence 89 strain 21, 23

rate tensor, 21, 91 term, 86

stress tensor 91 surface

element, 88 integral, 89

symmetry 13, 90, 100, 109, 113, 117, 118, 129

requirements, 100

Taylor micro-scale 43, 48, 52, 130 Taylor-Proudman theorem 30 Taylor series expansion 35, 48, 69, 101 Tennekes H 51, 64 time average 10 time-dependent fluctuation 10 Tselepidakis DP 8 turbulence

closure, 26, 27, 103 decomposition, 1, 10, 12 micro-scale, 35, 129, 130 one-component, 20, 27, 56, 62 two-component, 18, 59, 62, 74, 101,

111 isotropic, 20, 29, 62, 74, 92

turbulent boundary layer, 28, 29, 31, 106, 107,

113 channel flow, 27, 28, 39, 48, 56, 58,

59, 64, 68, 69, 71, 78, 79, 103, 104, 114

diffusivity, 66 eddy viscosity, 21 motion in two dimensions, 61 pipe flow, 105, 115 production, 54 Reynolds number, 7, 52, 53 stresses, 2, 14, 17, 21, 26 transport, 3, 6, 15, 18, 64, 109, 113

trace 13, 37, 67, 110, 121, 130 of Tij 112

transport term 45, 46, 64, 81, 86, 111, 113, 114, 115, 116

triple correlations, 64, 109 products, 109 velocity correlations, 100, 113, 119

true rate of dissipation 41 two-component

limit, 27, 62, 71, 93, 97, 102 state, 19, 20, 21, 27, 59, 61

turbulence, 19, 20, 62 two-dimensional turbulence 56, 61, 62 two-point

correlations, 34, 38, 42, 51, 53, 56, 64, 65, 66, 67, 76, 78, 91, 116, 119, 127, 128, 129, 130

correlation technique, 3, 33, 37, 42, 44, 48, 63, 85, 87, 89, 117, 119, 121, 122, 125, 128

pressure-velocity correlations, 43, 44, 85,87, 115

velocity correlation, 35, 36, 44, 45, 46, 89, 90, 100, 115, 117, 127

Uberoi MS 97, 98 uniform mean flow 100, 117 unit vector 13, 21, 111

validation 103 Vatutin IA 33,35,39,51,117 velocity

derivative, 51 fluctuations, 101 pressure-gradient correlations 3, 15,

16, 37, 85, 87, 88, 99, 101, 103, 105, 106, 115, 119

Index 141

pressure gradient terms, 101, 104, 107 viscous

destruction, 51, 54 diffusion, 15, 101, 102 dissipation, 110 incompressible flow, 9 sublayer, 42, 44, 58

volume averages, 11 element, 87 integral, 87 integrands, 89

von Karman T 3, 53 vorticity 61

wall bounded flows, 6, 18, 27, 30, 83, 101,

112 shear velocity, 40, 106, 115 variables, 81

weakly inhomogeneous turbulence 109

Yaglom AM 17 Ye QY 7, 8, 33, 121