a review on numerical simulation of turbulent flow · 2010-12-23 · simulating turbulent flows. 1....

International Journal on Architectural Science, Volume 3, Number 2, p.77-102, 2002

77

A REVIEW ON NUMERICAL SIMULATION OF TURBULENT FLOW S.L. Liu College of Power and Nuclear Engineering, Harbin Engineering University, Harbin, China W.K. Chow Department of Building Services Engineering, The Hong Kong Polytechnic University, Hong Kong, China (Received 14 January 2002; Accepted 24 May 2002) ABSTRACT Methods for numerical simulation of turbulent flow were reviewed. The physical concept of each method is outlined. This would be useful for those trying to get a general view on the common methods available for simulating turbulent flows. 1. INTRODUCTION Most of the practical problems on fluid flow involve turbulent flows. Earlier studies on turbulent flow included empirical studies with experiments. Later, momentum integral equations were proposed by Karman and Pohlhausen [1] with the equations concerned simplified into ordinary differential equations. This method applied to boundary layer flow with pressure gradient being not too large; and separate flow on smooth surfaces. Doing this would give a much faster computing rate, making it a practical engineering tool. However, information on turbulence structure would be lost while integrating the momentum equation. This method cannot give an accurate description of the turbulence of the boundary layer flow such as the turbulent shear stresses distribution. An expression for the velocity profile in the boundary layer has to be assumed in an approximate method, it is not too suitable for studying flow problems with sudden changes in geometry, large pressure gradient and separate recirculating flow. To better describe the complicated flow, the internal turbulent flow structure has to be more clearly understood. The method of solving the differential equations was developed for that purpose. Rapid development of computer hardware in the last century allowed the integration of complicated ordinary differential equations encountered in boundary layer flow. In the mid 1960s, methods for calculating turbulent energy based on mean momentum conservation equations were developed actively. Partial differential equations for turbulent energy could be solved by the second generation of numerical methods. Various numerical methods were discussed in a conference on turbulent boundary layer held at

Stanford in 1968 [e.g. 2]. The most important conclusion drawn from this conference is the confirmation of developing methods for solving partial differential equations. This approach is far more superior than the best integral method available. More complicated but practical turbulence models on partial differential equations were then developed. Firstly, partial differential equations for turbulent energy were worked out with the partial differential equations for mean flow variables. Secondly, partial differential equations for turbulent length scale were introduced to reduce the number of empirical parameters required. Further, partial differential equations for turbulent stress tensor were developed. Although those models are not only applicable to boundary layer, studies on the complicated condition have triggered the development of this kind of model. Those models describing the turbulent fluid flow can help in understanding the turbulent phenomenon. Parallel to the rapid development in computing turbulent flow, experimental study was also developed. Coherent structure was proved to exist in the mixing layer by Brown and Roshko [3]. Numerical results were compared with experiments by Blackwelder [4]. Extensive studies on turbulent structure illustrated that large eddies, fast and slow spots, and dark spots existed between the shear layer and turbulent boundary layer. Up to the present moment, experimental data cannot yet be applied extensively to the differential equation model. Perhaps, large-scale transport was not truly made use of ‘localization’ concept in the model. Intermittent turbulent flow should be focused as pointed out by Libby [5].

International Journal on Architectural Science

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A fundamental concept in modeling turbulence is to solve the instantaneous Navier-Stokes equation in a certain computing domain. In this approach, no modeling is required. However, the computer capability cannot satisfy the requirement for solving the equations exactly for a complex problem, at least, not in the near future. There are new concepts such as taking the turbulent flow as a nonlinear mechanical system. The Navier-Stokes equation concerned is simplified into a set of ordinary differential equations. With them, nonlinear structure and stabilities are analyzed to investigate bifurcation and chaos. A better understanding on turbulence is achieved. But such studies are still at the preliminary stage, far from applying them to solve practical engineering problems. In fact, only temporary mean flow variables, including space, temperature and turbulent parameters, are of interest in engineering application. The turbulent generation and development course are of less interest. Therefore, temporary (Reynolds) averaging equation is still the most practical modeling technique in solving practical engineering problems. Particularly, for large-scale turbulent flows, a much more practical approach is to solve three-dimensional transient problems in partial differential equation form; and to combine with empirical turbulent model for smaller-scale turbulence flow problems using large-scale computers. The turbulent structure can be modeled to satisfy the engineering requirements. This is still in active development, though not too mature. But simple partial differential equations show the superiority of the method. For example, constants used in modeling simple two-dimensional partial differential equations can be derived. This new method is still updating and would become the most useful method for engineering application. The complexity of turbulent flow and multi-stage application in solving engineering problems have led to many types of turbulence models, with different regions of application. A brief review of turbulence models described by partial differential equations is presented as follows: Zero-equation model

There is only a partial differential equation for average speed, and no partial differential equation for turbulence parameters. This kind of model is commonly used in solving engineering problems.

One-equation model

In addition to the above, there is another partial differential equation for describing the turbulent velocity scale. This type of model is only applicable to some problems.

Two-equation model

There is one equation for turbulence parameters related to velocity scale and another equation related to turbulence length scale.

Stress-equation model

This model includes equation with all turbulent shear stress tensors, and is mainly applicable to more complicated flows. The approach has the potential to be widely used as a practical tool.

Algebraic-stress model

The Reynolds stress equations are simplified into algebraic equation, combined together with the two-equation models. This is an extended two-equation turbulent model, probably true that it has certain applications.

Large-eddy simulation

The three-dimensional Navier-Stokes equations were solved to confirm behaviour of the large-eddy. The smaller eddies are modeled with common equations. This approach is still under development and used for comparing with other simpler turbulence models. It has the potential to become a practical engineering tool.

Development of turbulence models and comparison with experiments were clearly reviewed by Bradshaw [6]. One-equation and two-equation turbulence models were analyzed by Mellor and Herring [7]. Zero-equation model for turbulent boundary layer was presented by Cebeci and Smith [8]. Development of computing turbulent flow was reviewed by Reynolds [9]. In this paper, turbulence models for incompressible flows were reviewed. Extension to compressible flows and heat transfer was outlined. 2. ZERO-EQUATION MODEL Expressing the instantaneous velocity ui in the Navier-Stokes equation in terms of the mean Reynolds value Ui and fluctuation derivation iu′ :

iii uUu ′+= Time averaging gives the incompressible mean flow equation [10] as:


79

0xU

i

i =∂∂

(2.1a)

)RS2(xx

P1xU

Ut

Uijij

jij

ij

i −ν∂∂

−∂∂

ρ−=

∂∂

+∂∂

(2.1b) Stress tensors in Cartesian co-ordinate were used in the above, with ν, P, ρ being the fluid kinematic viscosity, mean flow pressure and density respectively, ρRij being the Reynolds stress tensor, and Sij being the strain rate tensor.

jiij uuR ′′= and

∂

∂+

∂∂

=i

j

j

iij x

UxU

21S

For ‘closing’ the system of equations given by (2.1), additional equation is to be provided for Rij. Boussinesq approximation [11] was introduced:

ijtij2

ij S2q31R ν−δ= (2.2)

where νt is the turbulent eddy viscosity to be expressed in a suitable form. Turbulent fluctuation kinetic energy k can be expressed in terms of q:

ii2 Rq =

and

2qk

2=

In zero-equation model, νt is correlated with the average velocity Ui. In turbulent surface layer flow, transport term in the internal layer is very small, and can be taken as satisfying the equilibrium approximation. Analogous to molecular motion, νt at the surface wall can be expressed as:

y u *t κ=ν (2.3) In the above equation, κ is the Karman constant (close to 0.4); *u is the friction velocity expressed in terms of the wall shear stress τw:

2/1w

*u

ρτ

=

and y is the distance away from the wall: y = x2 For near-wall region, viscous effect is very important. The surface would restrict the turbulent fluctuation, reducing the rate of turbulent transport. Increasing the velocity gradient would increase the viscous stress, increasing dissipation of the turbulent kinetic energy. Equation (2.3) can be modified by making the following correction:

2

*t Ayexp1y u

−−κ=ν +

+

(2.4)

where y+ is the turbulent Reynolds Number, and A+ is an empirical constant of value 26.

ν=+ *uyy

Mixing-length modeling gives the general form of νt in terms of the mixing-length l: νt = l2 (2Snm Snm)1/2 (2.5a) For surface layer flow, it can be simplified into:

νt = l2

yU∂∂ (2.5b)

where U is the velocity parallel to the wall surface. In the logarithmic region, mean velocity gradient is not related to viscosity, i.e.

yu

yU *

κ=

∂∂ (2.6)

For internal flow layer, l = κy Therefore, equations (2.3) and (2.6) are equivalent. In the outer layer, l = λδ with δ being the width of the boundary layer. The proportionality constant λ lies between 0.075 to 0.09, depending on the definition of δ. For δ0.995, λ is 0.085. Outside the boundary layer, intermittency of turbulence would affect the turbulent viscosity. Intermittency factor has to be used for correcting the expression. Similarity gives:


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−δ

−= 78.0y5erf121r (2.7)

In the above equation, erf is the error function:

∫ −

π=

x

0 t dte2)x(erf

2

As discussed in above, different length scales have to be used for the inner or outer layer. Basic rules for those length scales were briefly outlined. More complicated formulae for engineering application were reported by PS: Patankar and Spalding [12], and CS: Cebeci and Smith [8]. Works by Van Driest [13] were extended with air flowing towards the wall surfaces included in the model of CS. This model was widely used in the following form: For inner layer (0 ≤ y ≤ yc):

νt = l2 yU∂∂ γtr (2.8)

where l is given by:

l =

−−κ

Ayexp1y

with κ equals to 0.4; and

*NuAA ν

=+

where A+ equals to 26; and N is given by:

[ ]2/1

www

)v8.11exp()v8.11exp(1vPN

+−= +++

+

where P+ is the differential pressure coefficient with respective to the co-ordinate x = x1 of the wall surface given in terms of the outer boundary velocity ue by:

dxdu

uu

P e3*

eν=+

and +

wv is the wall flow coefficient given in terms of the wall flow velocity vw by:

*

ww u

vv =+

For no mass transfer through the surface,

vw = 0 and N = (1 − 11.8 P+)1/2 For outer layer (yc ≤ y ≤ δ):

≤δ

≥γ−α

=νθ

θ∫∞

5000Re,u0168.0

5000Re,dy)Uu(

*e

tr

0 e

t (2.9)

In the above equation, δ* is the transformed boundary layer thickness, Reθ is given in terms of the thickness of the momentum lost in the boundary layer θ by:

Reθ = νθ eu

where

ξ+

=α1

55.10168.0

with

)]Z298.0Z243.0exp(1[55.0 12/1

1 −−−=ξ

−=

θ 1425ReZ1

The separation points of the inner and outer layer yc can be determined by equating νt given by equations (2.8) and (2.9). γtr is the intermittent factor in the region of infection. This is given by the following equation for two-dimensional flow:

−−−=γ ∫

x

x etrtr

rt udx)xx(Gexp1 (2.10)

In the above equation, xtr is the position where infection starts. This can be determined from Reθ

and Rex (Rex = ν

eux ) given by the CS model:

46.0xtrRe

xtrRe224001174.1trRe

+=θ (2.11)

G is an empirical coefficient:

34.12

3e xtrRe

u1200

1G −

ν=


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Viscous effects in CS and other similar models were applied successfully in simulating two-dimensional and axisymmetric incompressible flows in boundary layer. The concept was extended to compressible boundary layer flows with heat and mass transfer, and some three-dimensional boundary layer flows after the conference held at Stanford. For free shear flows including jets, wakes and plumes, the wall effects need not be considered. The mixing length l can be assumed to be constant at directions perpendicular to flow, with value proportional to the width of the mixing layer b(x). In general, b(x) is a function along the flow direction x, i.e. l = λ b(x) (2.12) For planar jet, l = 0.09 b ≅ 0.018 x For circular jet, l = 0.075b ≈ 0.015 x For developed pipe flow, mixing length l can be determined by Nikuradse formulae:

−−

−−=

42

Ry106.0

Ry108.014.0Rl (2.13)

In the above equation, R is the radius of the pipe, and y is the distance from the pipe wall. The advantages of the zero-equation are: direct, simple and needs not to solve another partial differential equation for turbulence parameters. It is good for simpler flows such as jet, boundary layer, pipe flow and tube flow. There are some obvious problems in this kind of turbulence models. The turbulent viscosity is a function of the positions. Partial balancing of turbulence energy would give dissipation, giving convective and diffusive components to be zero. For places with zero mean velocity gradient

=

∂∂ 0

yU , the turbulent viscosity (shear or heat

flow or diffusion also) will also be zero. This will not agree with practical cases. Therefore, taking turbulent viscosity to be proportional to the mean velocity gradient in mixing length model is not reasonable. Further, formula for calculating the mixing length are very complicated in many flow problems. Examples are the complicated boundary layer flow (such as concentric pipe flow), curvature

flow or flow over step and obstacles. There, the mixing length model cannot be applied. 3. ONE-EQUATION MODEL By multiplying iu ′ to the instantaneous Navier-Stokes equation in that particular direction, and then applying Reynolds resolution and time averaging, the transport equation of turbulent kinetic energy can be obtained [10]:

j

j

j

2

j

2

xJ

)P(2xqU

tq

∂

∂−ε−=

∂∂

+∂∂ (3.1a)

where P is the production rate of turbulent kinetic energy per unit mass; ε is the dissipation rate of turbulent kinetic energy per unit mass; and Jj is the diffusive flux of turbulent kinetic energy per unit mass.

j

iij x

UR

∂∂

−=P

ijijSS 2 ′′ν=ε and

′′ν−

ρ

′′+′′′= iji

jjiij Su4

up2uuuJ

where

∂′∂

+∂

′∂=′

j

i

i

jij x

uxu

21S

If ε is substituted by “isotropic dissipation rate” D given by:

j

i

j

i

xu

xu

∂′∂

∂′∂

ν=D

and Jj is substituted by *

jJ as:

j

2j

jii*j x

qup2uuuJ

∂∂

ν−ρ

′′+′′′=

Then, equation (3.1a) can be rewritten as:

j

*j

j

2

j

2

xJ

)DP(2xqU

tq

∂

∂−−=

∂∂

+∂∂ (3.1b)


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Note that the molecular viscosity coefficient ν of *jJ reflects the diffusion gradient of q2 and D is not

appropriate to be regarded as the dissipation rate. It can be proved that D = ε for homogeneous isotropic small-scale turbulent flows only at high Reynolds numbers, but not at low Reynolds numbers or at near-wall regions. Boussinesq approximation given by equation (2.2) and the eddy viscosity concept are still assumed in the one-equation model. The eddy viscosity νt is expressed as the product of the characteristic velocity q and the characteristic length l: νt = C2 q l (3.2) The turbulent velocity scale q is determined by the turbulent kinetic energy equation (3.1), and the length scale l is given by algebraic relation analogous to the zero-equation model. The cascading process (non-linear) of turbulent kinetic energy enables the transfer of turbulent energy from large eddies to the small eddies, making the small eddies follow the basic mechanism of homogeneous isotropy. At high Reynolds numbers, the dissipation rate of turbulent kinetic energy is independent of viscosity, but viscosity can lead to energy dissipation of the smallest eddies. Dimensional analysis gives:

lD

3

3qC= (3.3)

The diffusion gradient model is usually adopted for diffusive flux:

( )j

2

t4*j x

qCJ∂∂

ν+ν−= (3.4)

Selection of constants C2, C3 and C4 can be based on the relation of the surface layer logarithmic region. Taking l = κy, from equations (2.5b), (2.6) and (3.2):

C2 = qu*

According to the assumption of partial balancing, i.e. the production term is balanced with the dissipation term P = D, C3 can be deduced as:

C3 = 3

*

qu

It is found from experiments that at near-wall region of the surface layer:

2*

qu

= 0.15

Taking C2 to be 0.387 and C3 to be 0.0506, comparing the results from one-equation model and zero-equation model yields C4 of value 0.5. At the surface layer: when y+ ≤ 2A+ for viscous bottom layer, zero-

equation model is applied; when y+ > 2A+, one-equation model is applied,

the boundary conditions at q2 are found using the turbulent kinetic energy equation and determined by the results from zero-equation model.

This model was used by Kays and coworkers to study the effects of free stream turbulence on the heat transfer of the surface layer [14], and further examine the effects of dramatic change (non-balancing surface layer characteristics) in free stream conditions. A one-equation model applicable to wall of the surface layer was suggested by Norris and Reynolds [15]. In this model, it is not required to use the empirical decay factor A+ and its correction.

Having noted that at low Reynolds range, 2

2ql

ν is

used as the dissipation scale, and so D is given by:

ν

+=/q

C1qC 5

3

3 llD (3.5)

Considering also that the length scale of the viscous bottom layer would not have any special changes, it is still valid to take l = κy. At near-wall region, q ~ y, equation (3.5) is simplified to:

2

2

53qCCl

D ν=

When y approaches 0, D gradually becomes a constant. This is a necessary fixed physical characteristic in dissipation. Diffusive flux *

jJ still uses equation (3.4), and since the turbulence transport is restrained by the wall, the eddy viscosity is corrected as:

ν−−=ν

qyCexp1 qC 62t l (3.6)

Considering q ~ y at near-wall region, then on the wall where y = 0, equation (3.1) is simplified to:


83

0yq2 2

22=

∂

∂ν+− D

C3C5 = κ2 can be deduced, where C5 can be obtained from the given C3. The value of the constant C6 can be deduced from the known plain surface layer characteristics, the value of C6 used here is 0.014. This model was applied in channel flow by Norris and Reynolds [15]. The turbulent length scale l is taken as 0.4y at near-wall region, smoothly transited to the channel central line l = 0.13δ, where δ refers to the channel half-width. When there is no diffusion through the wall, the computed results of the uniform velocity flow field near the wall and the shear distribution agree well with the corresponding data on flat surface layer, demonstrating the wall function approach is verified. To solve for the channel flow equation, the wall dissipation rate was corrected by some assumptions. The equations were integrated outward starting from the wall to positions where the conditions at the channel central line are satisfied. Results indicated that dissipation rate of wall is very sensitive to the computing procedure. Higher precision computational technique is required and more accurate assumed value is needed. Otherwise, diverged or negative values of q2 might be predicted. One-equation model supplements the inadequacy of the zero-equation model by including the effects due to turbulence energy (convective flow) and mixing (diffusion), to avoid having zero turbulent

viscosity at positions where 0yU

=∂∂ .

However, to close the one-equation model, an algebraic expression for the turbulence length scale l should be given. For simple flows such as boundary layer and jet flow, using zero-equation model would be good enough. While for more complicated flows, there are difficulties in applying this model. The one-equation model can be visualized as an intermediate product in the process of developing turbulence models. This approach has led to the emergence and application of two-equation model and advanced models. 4. TWO-EQUATION MODEL As described in the previous section, it is rather difficult to express the turbulent length scale l by an algebraic equation. Also, the effects of

convective flow and diffusion on l are not included. Therefore, one-equation model is not commonly applied to solve practical engineering problems. To cope with this limitation, the transport equation for l should be established. It was suggested by Harlow [16], and Launder and Spalding [17] to substitute l by the dissipation rate of isotropic turbulent kinetic energy D as the second variable to be solved. The transport equation of dissipation rate of turbulent kinetic energy so established was combined with equation (3.1) to form the widely used two-equation model. The eddy viscosity concept is still used in two-equation model. The expression for turbulent eddy viscosity can be obtained from the one-equation model expression (3.2) and the correlation equation (3.3) of D with q2 and l:

DD

4

7

4

32tqCqCC ==ν (4.1)

where the constant C7 is equal to C2C3 of value 0.0225. The transport equations for q2 and D are based on the Navier-Stokes equation. Taking xi as skewed derivative for the momentum equation in the

direction xj, multiplying both sides by j

i

xu

2∂′∂

ν , and

then using Reynolds resolution and averaging, the equation for D is:

j

j

jj x

HW

xU

t ∂

∂−−=

∂∂

+∂∂ DD (4.2a)

where

+∂∂

′∂∂∂

′∂ν+

∂

′∂∂

′∂∂

′∂ν=

kj

i2

kj

i2

2

j

k

k

i

j

i

xxu

xxu

2xu

xu

xu

2W

+

∂∂

∂

′∂+

∂

′∂∂

′∂∂

∂ν

i

k

i

j

k

i

j

i

k

j

xu

xu

xu

xu

xU

2

kj

i2

k

ij xx

Uxu

u2∂∂

∂∂

′∂′ν (4.2b)

jkk

j

k

i

k

ijj xx

pxu

2xu

xu

uH∂∂

ν−∂

′∂∂

′∂ν+

∂′∂

∂′∂′ν=

D (4.2c)

where W is the production and dissipation of D induced by the interaction of eddy stretching, molecular viscosity, uniform flow field and turbulent fluctuation; and Hj is the diffusive flux of D along the xj direction.


84

Production and dissipation of D are generally assumed to be proportional to the production and dissipation of q2, i.e. W~(C8D − C9P) From dimensional analysis:

2

2

q~W D

Therefore, the source term of the D equation can be modelled as:

2

2

98 qCCW D

DP

−= (4.3a)

Introducing the anisotropic part of the Reynolds stress tensor:

( )2

ij2

ijij q

3/qRb

δ−= 2

ijt

q

S2ν−= (4.4)

Since

j

iij x

UR

∂∂

−=P 2tS2ν=

Putting in equation (4.1) gives:

DP

74

22tijij

2 C2qS4bbb =ν==

Therefore, equation (4.3a) can be rewritten as:

( ) 2

22

108 qbCCW D

−= (4.3b)

where

C10 = 7

9

C2C

The strain rate of the turbulent flow field in the source term is described by b2 which can be taken as a predictable value for measuring anisotropy. Dimensional analysis indicated that the third term on the right side of equation (4.2b) is of a higher order than the fourth term. The third term can be

simplified to 3/xU

4i

i

∂∂

D under partial isotropic

condition, which can be neglected for incompressible flow. Therefore, W, the source term of D, does not come from the two terms

explicitly containing uniform flow velocity, but from the first and second terms. These two terms are of higher order and of opposite sign. Difference of these two terms has to be of the same order as other terms in equation (4.2a). Taking isotropic decay of each term into account, equations (3.1b), (4.2a) and (4.3) can be simplified as:

D2t

q 2−=

∂∂ (4.4a)

Wt

−=∂∂D (4.4b)

2

2

8q

CW D= (4.4c)

For easy validation, this set of equations is solved as:

n2o

2

at1qq

−

+= (4.5a)

)1n(

o at1

+−

+= DD (4.5b)

)2(nq

ao

2o

D= (4.5c)

)2C(2n

8 −= (4.5d)

where 2

oq and Do are the corresponding initial values. Earlier experiments gave the value of n as 1, then C8 is 4. Under very strict isotropic conditions, n was found lying between 1.1 to 1.3 by Comte-Bellot and Corrsin [18]. A certain degree of difference exists between the decay of slightly anisotropic and isotropic turbulence, especially, the structure of low wave number spectrum will affect significantly this difference. From spectral analysis, turbulent energy is transferred from low wave number to high wave number. The wave number kL corresponds to the maximum value of total energy spectrum E(k) in the energy-containing region. Assuming the dissipation rate D at low wave number region is unchanged, D at high wave number region will decrease. In the dissipation range of high enough wave number (k > kL), viscosity has an important role to play. The turbulent energy in this range is dissipated as heat


85

energy, the vortex is uniformly isotropic, and its spectrum structure is totally dependent on the dissipation rate D and unrelated to the eddy structure of the energy-containing region. In the inertial range where k > kL and with lower wave number than the dissipation range, both the conditions of uniform isotropy and the conditions unrelated to the eddy structure of energy-containing region are satisfied. Structure of its spectrum is related only to the wave number k and the dissipation rate D. Kolmogoff inertial range energy spectrum gives: E ~ k –5/3 Dimensional analysis gives: E(k) = α D2/3 k−5/3 (4.6a)

where the constant α is determined from experiments to be of value 1.5. The energy spectra for the inertial range and the dissipation range take the general form and are not related to the boundary conditions. However, a general form does not exist in the energy-containing region, its energy spectrum is directly related to the boundary conditions. Within this region, the vortex is obviously non-uniformly isotropic. At low wave number regions (k < kL), the turbulent flow absorbs energy from the main flow and transfers it to the high wave number region, its energy spectrum is: E(k) = A km (4.6b) Note that E ~ k4 when k tends to be zero. The above spectrum relationship would give q2:

∫= dR)k(E2

q 2

or

3/23/2L

2 k 23

1m1q D−

+

+α= (4.7)

Note that the form of the large eddy spectrum is totally reflected by m, not by introducing the intensity A. Equation (4.7) indicated that the

length scale of eddy-containing energy is D

3q , and

the time scale is D

2q . At the maximum value E(kL)

in the energy spectrum corresponding to kL, equations (4.6a) and (4.6b) give:

)5m3/(33/2

L Ak

+

α=

D (4.8)

Substituting into equation (4.7) gives: D = C [q2](3m+5)/(2m+2) (4.9) where C is a constant. From equations (4.9) and (4.4):

)2m2/()5m3(

2o

2

o qq

++

=

DD

and

n/)1n(

2o

2

o qq

+

=

DD

Therefore: n = (2m + 2)/(m + 3) Obviously, the spectrum structure of low wave number helps to determine n. Since it is not possible to represent this spectrum structure with q2 and D, this model cannot be used to predict accurately the decay of grid turbulence in wind tunnel; but it can be used for selecting the constant C8. It is expected that this model can be applied in the whole wave vector k space of large-scale uniform energy distribution, i.e. the energy spectrum density in the whole low wave number space φii(k) as a fixed value. Three-dimensional energy spectrum can be obtained from the integral of the energy spectrum density of a sphere of radius k:

)(k2dA)(21)(E ii

2

A ii kkk φπ=φ= ∫∫

For uniformly distributed large-scale turbulence:

2k~)(E k Based on the above, it is suggested that m is 2, then

n equals to 56 and C8 equals to

311 .

The constant C9 is usually determined through experiments. When comparing the results of pure strain flow and pure shear flow from experiments and by numerical simulation, the results agreed well when taking C9 to be 2. Therefore, when equation (4.3a) is used for the source term W of


86

equation D, it is suggested by Norris and Reynolds

[15] to take C8 as 3

11, and C9 as 2. When equation

(4.3b) is used, it is recommended to take C8 as 3.73, and C10 as 30. The diffusion gradient model for the diffusive flux was suggested by Jones and Launder [19,20]:

jt11j x)C(H∂∂

ν+ν−=D (4.10)

where the constant C11 is 0.77. Thermal elasticity and diffusion were studied using irreversible thermodynamics by Lumley [21]. The diffusion flux of dissipation rate was proved to be related to the gradient of turbulent kinetic energy, and vice versa. Therefore, the following model was proposed:

j12

j

2

11j xA

xqAJ

∂∂

−∂∂

−=D (4.11a)

j22

j

2

21j xA

xqAH

∂∂

−∂∂

−=D (4.11b)

This model was applied by Lumley and Khajeh-Nouri [21] in stress-equation model, but not yet in two-equation model. Equation (4.11) can be used in turbulent kinetic energy diffusion gradient of the wake central region, but not use the simple non-coincidence model. It is worthwhile to study on this further for developing the two-equation model. Another two-equation model was suggested by Saffman-Wilcox [22], which the transport equation of dissipation rate D is substituted by “virtual vorticity” Ω transport equation:

+Ω

Ωβ−

∂∂

∂∂

α=∂Ω∂

+∂Ω∂ 2

2/1

j

i

j

i

j

2

j

2

xU

xU

xU

t

( )

∂Ω∂

σν+ν∂∂

j

2

tj xx

(4.12)

Introducing

2q)S2(

22/12*α=P (4.13a)

2q 2* Ω

β=D (4.13b)

Combining with the turbulent kinetic energy q2 transport equation (3.1b) to form a two-equation model. Eddy viscosity is:

Ω=ν

2q 2

t (4.14)

Equation (2.2) is used as the momentum equation to provide Rij. Constants used in the model are: α = 0.1638, α* = 0.3, β = 0.15, β* = 0.09, σ = 0.5 Two-equation model can be applied in many areas. For flows with backflow, it is the turbulence model having the widest engineering application area. Comparison of vast amount of predictions with experimental results indicates that it can be successfully applied in non-buoyant jet, plain surface layer; duct flow, passage flow or ejection tube flow; and non-eddy or weak eddy backflow. It yields better results in computing burning in finite space than for pure flow. Two-equation model, one-equation model and zero-equation model all adopt the assumption of Boussinesq turbulent stress and assume that the turbulent transport can be expressed in terms of the turbulent kinetic energy and length scale. Turbulent stress equation correlates the turbulent stress with the uniform flow field. It cannot accurately describe flows where the uniform flow field velocity gradient is zero but the turbulent stress is not zero. The eddy viscosity form scalar functions cannot realize the anisotropy of turbulent transport. These are the demerits of the turbulent eddy viscosity model. Therefore, problems would be encountered in analyzing strong eddy flow, buoyant flow, gravitational layers flow, curved-wall surface layer or curved flow, low Reynolds number flow and circular jet flow. In order to get better agreement between the predicted results from two-equation model and the experimental results, improvements have been made. First, the source term of D equation was re-examined and corrected. Secondly, the assumption of isotropy and the concept of scalar eddy viscosity are given up. Coefficients such as C7, C8 and C9 are regarded as the functions following certain principle. If Boussinesq equation is used, then νt is the tensor, where C7 is the tensor related to the strain rate. Examples of the main correction methods are as follows: (1) Low Reynolds number Turbulent flow is regarded as low Reynolds number flow if:


87

600qtRet

4<

ν=

D

The downstream of free jet flow, turbulence near fixed wall region or fully decayed are regarded as low Reynolds number flow [23]. Its characteristics are that the effect of molecular viscosity on the flow will be increased, which affect the uniform flow transport and the turbulence process. In the later stage of decay of isotropic turbulent flow, the source term W in the model is q2 ~ t−5/2, but not t−6/5,

i.e. n is of value 25 , C8

* is of value 5

14 .

Obviously, the constant in W model is related to the Reynolds number Ret. According to the corrections proposed by Launder-Sharma in 1974 [24]:

D

4

27tq

)200/tRe1(4.3expC

+−=ν (4.15a)

2

2

988q

PCfCW DD

−= (4.15b)

where

−−=

16tReexp3.01f

2

8 (4.15c)

(2) Circular jet flow and weak shear flow In circular jet flow, for the source term W in D equation, it was suggested by Rodi [25] to use:

=dx

dUfC m

9

i.e. the function relation of decrease of C9 along the main flow direction x. For weak shear flow:

=

DPfC7

i.e. C7 decreases as DP increases.

(3) Buoyant flow In flow systems affected by force field, non-uniform density will result in buoyancy. Buoyant fluctuations will affect the turbulence field and the production term of dissipation rate. The effect of

buoyant flow on the transport equation of q2 and the source term W in D equation is corrected as follows:

j

*j

bj

2

j

2

xJ

G)(2xqU

tq

∂

∂−+−=

∂∂

+∂∂ DP

A buoyant term Gb is added in the turbulent kinetic energy equation:

iT

tiiib x

Tg Tug G∂∂

σν

β=′′β−=

where β is the expansion coefficient, and the constant σT is 1.0.

pT1

∂ρ∂

ρ−=β

The source term of D equation is:

2

2fb

b98 q

)RC1()GP(CCW D

D

++−= (4.17)

where the flux Richardson number Rf is introduced. The two different situations of horizontal and vertical shear layer buoyant flow were considered by Rodi, and Rf was defined correspondingly. Based on that, the authors proposed:

−

+−=

uvtanh1

GPG

Rb

bf (4.18)

where u and v are the velocity rate in the horizontal and vertical direction respectively. It is not difficult to see that if Cb is of value 1, then in the horizontal shear layer buoyant flow (v = 0), equation (4.17) is simplified into equation (4.3a), the same as when there is no buoyant flow correction. Adjusting Cb can control the correction level of W, generally, values of Cb lie from 0.8 to 1. It can be seen readily that the effect of buoyancy in vertical shear layer is greater than in horizontal shear layer. After correction of buoyancy, abrupt change in velocity can be predicted effectively in the separate layer flow; but there is no significant improvement in the prediction of abrupt change in temperature. (4) Rotational flow For rotational flow, the standard two-equation model has overestimated the turbulent kinetic energy and eddy viscosity in the linear region near the axis, ignoring the central backflow region of


88

strong eddy flow or making that region more downstream. Therefore, the effect of centrifugal force on turbulent flow field has to be considered in order to improve this model in predicting strong eddy flow. The correction of the source term W in D equation was given by Morse [26]:

( ) 2

2

9f8 q CR6.1CW D

DP

−−= (4.19a)

D/rw

rwvR f

∂∂′′= (4.19b)

where v and w are radial and tangential velocity components in the cylindrical coordinates respectively. Equation (4.19) indicated that the

rotational velocity gradient

∂∂

rw

r would

increase the dissipation rate and reduce the turbulent kinetic energy. Another correction was given by Smith [26]:

2

2

9fb8 qC)RC1(CW D

DP

−−= (4.20a)

∂∂

=

r)wr(

rWq

41R 22

4

f D (4.20b)

To sum up, the above corrections can improve the predicted results to a certain extent. To achieve greater improvement in the prediction of buoyant flow and rotational flow, eddy viscosity model should be given up, instead, anisotropic turbulent stress-equation model or large eddy simulation should be used. 5. STRESS-EQUATION MODEL Most turbulent flows encountered in practice are anisotropic and fluctuated more along one particular direction. Therefore, eddy viscosity is better to be described as a tensor rather than a scalar quantity. Sometimes, the turbulent energy might change to uniform kinetic energy, the turbulence production rate might become negative, giving Reynolds stress having the opposite sign to the uniform kinetic strain rate (as in circular duct axial flow), eddy viscosity should then be negative. The concept of eddy viscosity is not applicable under such situations. Therefore, the stress tensor is computed using direct closure and the Reynolds stress transport equation. This is the stress-equation model which is also called the second-

order closure model. Though there are higher order closure models (e.g. third-order closure model) or even finer large eddy simulation, stress-equation model is regarded as the most complicated model in engineering applications for predicting complicated turbulence. Applying Reynolds resolution of the instantaneous Navier-Stokes equation, multiplying the momen-tum equation along the i and j direction by ju ′ and

iu ′ respectively, and then taking the average after adding the two equations yield the stress transport equation [10], the stress transport equation for incompressible fluid is:

k

ijkijijij

k

ijk

ij

xJ

DTPxR

Ut

R∂

∂−−+=

∂

∂+

∂

∂ (5.1a)

where Pij is the production tensor with Pii = P.

∂∂

+∂

∂−=

k

ijk

k

jikij x

UR

xU

RP (5.1b)

Tij is the transport tensor related to the pressure strain rate as:

∂

′∂+

∂′∂′

ρ=

i

j

j

iij x

uxu

p1T (5.1c)

Obviously, Tii = 0 That means Tij has no effect on the turbulent kinetic energy, but only changes the distribution of turbulent kinetic energy along the three directions, making the non-homogeneous isotropic turbulence becoming homogenous isotropic. This term is usually called the redistribution term. This clearly demonstrates the difference between stress-equation model and two-equation model, and is the key point in modelling stress-equation model. Dij is the dissipation rate tensor:

k

j

k

iij x

uxu

2D∂

′∂

∂′∂

ν= (5.1d)

and Dii = 2D Dij is the second-order related term of fluctuation velocity gradient. In small-scale vortices with the effect of molecular viscosity, the derivative of fluctuation velocity is much greater than the


89

fluctuation velocity itself, therefore, its modelling is very important. However, at the moment, the modelling of Dij is only at the preliminary stage, accurate modelling of dissipation tensor is the key to further development of stress-equation model. Jijk is the diffusive flux of Rij,

( )k

ijkjiikjjkiijk x

Ruuuupup1J

∂

∂ν−′′′+δ′′+δ′′

ρ=

(5.1e) and Jiik = Jk

* Generally, Jijk does not affect the average value of Reynolds stress of the whole system (except for some boundary conditions, e.g. fixed wall), but only adjust the distribution of Reynolds stress, making that spatially homogeneous. Apart from Pij, modelling is required for Tij, Dij and Jijk in the stress transport equation given by (5.1). Besides, the length scale l should be given or determined from the D equation. Experiments indicated that at high Reynolds numbers, small-scale eddies can be regarded as isotropic, i.e. “partial isotropic assumption”. Therefore,

)4(30x

uxu

jkijikijkj

k

i δδ−δδ−δδ=∂′

′∂

∂′∂

ν llll

D (5.2a)

Then, the dissipation rate tensor can be expressed as:

Dijij 32D δ= (5.2b)

The pressure strain term Tij is the key problem in experimental research and stress-equation modelling. The fluctuation momentum equation is obtained by subtracting the Reynolds equation (2.1b) from the instantaneous Navier-Stokes equation. Then, by taking its dispersion rate, the fluctuation pressure equation can be obtained:

)()Ruu(xxx

uxU

2p1ijji

ji

2

j

i

i

j2 xℜ=−′′∂∂∂

−∂′∂

∂

∂−=′∇

ρ

(5.3) This is a Poisson equation with p′ as the variable. This equation is elliptical, which shows that the fluctuation pressure at point x is dependent upon the whole flow field. The equation does not

contain the molecular viscosity coefficient, i.e. the transfer of p′ is not directly affected by the dissipation of molecular viscosity. In systems far from the fixed wall, Poisson equation (3.3) is solved as:

∫ττ

ℜπ

−=ρ′

r)(d)(

41p ξ

ξ (5.4)

Equation (5.4) gives a spherical integral. The centre of the sphere is at p′ . The radius of the integral sphere is ξ−= xr . Taking the gradient of the average rate in uniform flow to be a constant, then equation (5.1c) can be written as: Tij = Tij1 + Tij2 (5.5a) where

r)(d

xu

xuuu

41T

i

j

j

i

m

m2

1ijξτ

∂

′∂+

∂′∂

ξ∂ξ∂′′∂

π= ∫τ l

l (5.5b)

r)(d

xu

xuu

xU

21T

i

j

j

i

m

m2ij

ξτ

∂

′∂+

∂′∂

ξ∂′∂

∂∂

π= ∫τ

l

l (5.5c)

where the derivatives of ξl and ξm are the functions of ξ. Explicit forms of the integrals in equation (5.5) are difficult to obtain. Therefore, the modelling can only be based on the characteristics of each term and the function of the pressure strain term. Corrections and improvements can be made in comparing the computed results with the experimental data. There is only fluctuation in momentum, but no uniform flow rate in Tij1, which represents the contribution of turbulence fluctuation to the pressure strain Tij. A model of Tij1, (the “Linear Return to Isotropy” technique), was suggested by Rotta [27]: Tij1 = −Ct1 Dbij (5.6) Note that the constant Ct1 depends on the process of turbulence isotropy. It was suggested by Rotta that the value of Ct1 lies between 2 to 10, while the value currently used is from 3 to 6. If Tij2 is neglected in Tij expressed in equation (5.5a), Ct1 should be larger than 6. By realizing equation (5.6), characteristics of Tij1 are: There is only turbulence rate and no uniform

flow rate.

Turbulent kinetic energy is isotropic.


90

For example, when R11 > q2/3, then b11 > 0 and T111 < 0. It functions as a negative source term in the R11 equation, making the turbulent kinetic energy become isotropic. The reverse is also true. Tij2 contains the uniform flow velocity gradient. It represents the contribution to the pressure strain term Tij by the interaction of uniform flow field and turbulence fluctuation field. Modelling proposals for Tij2 were put forward individually by Launder and Spalding [28], Naot [29] and Lumley [30]. The Quasi-Isotropic Model (QIM) for Tij2 is:

)GG(x

UT mjimij

m2ij ll

l+

∂∂

= (5.7)

where Glmij = αRliδmj + β(Rlmδij + Rljδim + Rijδlm + Rimδlj)

+ Ct2Rmjδil + [ηδilδmj + ζ(δljδim + δlmδij)]q2 (5.8) Equation (5.7) can be rewritten as:

+βδ−+β−β+α−= kkijij2tij2ij PQ)C(P)(T

2

i

j

j

i qxU

xU)(

∂

∂+

∂∂

ζ+η (5.9)

where

∂∂

+∂∂

−=i

kjk

j

kikij x

UR

xU

RQ (5.10)

Obviously, Qkk = Pkk. The constant in equation (5.8) can be obtained from certain restrained conditions. According to continuous equation Glmii = 0, α + 5β + Ct2 = 0 (5.11a)

η + 4ζ + β = 0 (5.11b) From equations (5.5c) and (5.7),

∫ττ

∂

∂π

−= 2

m

i2

mim rd

rR

21G l

l

where dτ = 4πr2dr When r approaches ∞ , Ril tends to be 0, then Glmim = 2Ril. Using equation (5.8), 3α + 4β = 2 (5.11c) 3η + 2ζ + Ct2 = 0 (5.11d)

From equation (5.11), the constant in Glmij can be represented in the form of a single constant Ct2:

1110C4 2t +

=α ,

112C3 2t +

−=β ,

552C25 2t +

−=η ,

553C10 2t +

=ζ (5.12)

Substituting equation (5.12) into equation (5.9), and noting that:

( ) ( )[ ]kk2tkkijkkij QCP31P +β+β+αδ=βδ−

Therefore,

δ−

−−

δ−

+−= kkijij

2tkkijij

2t2ij Q

31Q

112C8

P31P

118C

T

∂

∂+

∂∂−

−i

j

j

i22t

xU

xU

q55

1C15 (5.13)

The characteristics of equation (5.13) are: the sum of all terms on the main diagonal line

is zero, that is identical to the characteristics of pressure strain term Tij;

for isotropic turbulence 3

qR2

ijij δ= , more

acceptable results are achieved, and it is not related to the value of Ct2, i.e.,

∂

∂+

∂∂

=i

j

j

i22ij x

UxU

q2.0T (5.14)

Comparing with experimental data, taking the constants in equations (5.6) and (5.13) as Ct1 to be 5.6, Ct2 to be 0.45 or Ct1 as 5, Ct2 as 0.4 agrees better with the experimental results. Since Ct2 is taken as smaller value, from the three coefficients in equation (5.13), the absolute value of the first coefficient is much larger than the second and the third ones. Neglecting the latter two terms, equation (5.13) can be simplified as:

δ−−= kkijij

*2t2ij P

31PCT (5.15)


91

This is generally called the Isotropization of Production Model (IPM), where the constant *

2tC is smaller than 0.6. Experiments indicated that there is a linear relationship between Ct1 and *

2tC :

1t*2t C115.01C −= (5.16)

Though IPM is a simplified form of QIM, a lot of calculations and experiments did not show that QIM is better than IPM. This demonstrates the complexity of turbulence modelling. The above discussion is on modelling the pressure strain term in free stream system. If there is fixed wall or free surface in the system or on the boundary, there will be reflections of pressure fluctuations on the surface, giving an additional effect on the pressure strain term. The modelling method currently used is similar to modelling the pressure strain term in free turbulence, by introducing correction of fixed wall in constrained flow; and then by linear addition of the pressure strain term to the fixed wall correction. The effect of fixed wall is related to the distance to the fixed wall. The following correction of Tij1 by fixed wall was suggested by Shir [31]:

−−δ=

nikkjjkkiijmkkm2W1W1ij x

f nnR23nnR

23nnR

qCT lD

(5.17) The correction of Tij2 by fixed wall was proposed by Gibson and Launder [32]:

−−δ=

nik2jkjk2ikijmk2kmW2W2ij x

f nnT23nnT

23nnTCT l

(5.18) where nk is the direction cosine of the unit vector n of the vertical fixed wall along the xk direction;, xn is the distance from the concerned location to the fixed wall; l is the turbulent length and the constants C1w and C2w are taken as 1 and 0.3 respectively.

Dl

3

3qC=

n

3

n xq14.0

xf

Dl

=

Considering the linear addition of fixed wall correction, equation (5.5a) can be rewritten as: Tij = Tij1 + Tij1W + Tij2 + Tij2W (5.19)

The inadequacy of fixed wall correction is that: equation (5.17) cannot accurately reflect that fixed wall will only have a greater effect on the positive stress in the main flow direction; and equation (5.18) cannot adequately reflect the non-linear characteristics of the process. For non-uniform turbulence, diffusive flux Jijk needs modelling. At sufficiently high Reynolds numbers, the viscosity term in the diffusive flux can be neglected. Research results of asymmetric

plain duct flow showed that dy

vpd ′′ is very small. It

is incorrect to say that the pressure diffusion term under general conditions is unimportant. However, with only limited research, the pressure diffusion term in Jijk is neglected. Differential equation can also be established for third-order related kji uuu ′′′ , but this requires higher-order enclosure. At present, the requirement for such enclosure has not been met. Based on the analysis of differential transport equation of third-order relation, the diffusion gradient model was proposed by Hanjalic and Launder [33]:

∂

∂+

∂∂

+∂

∂−=

n

ijkn

n

ikjn

n

jkin

2

Jijk xR

RxR

RxR

RqCJD

(5.20) where CJ is a constant. For two-dimensional surface layer, computed results showed that CJ is of value 0.04. Stress equation (5.1) and dissipation equation (4.2) supplement the Reynolds equation and the continuous equation. This is a complete stress-equation method which the dissipation equation takes into account the length scale, and calculates all non-zero Reynolds stress. The advantage of stress-equation model is that it takes into account the effect of anisotropy. In calculating the size of separated zones, eddy buoyant flow inside burning room and furnace, and systems with strongly anisotropic turbulence transport system, the results obtained are better than from two-equation model. Applying this to circular axial flow, the real situation of non-coincidence of average velocity gradient being zero and the shear stress being zero is calculated. Generally, stress-equation model has a wide applicability, at the moment, it can be used in micro-computers to perform calculations on engineering problems. The results for wall jet flow and circular free jet flow using stress-equation model are not satisfactory. The results for general backflow process are also not necessarily better than using two-equation model. The problems of stress-


92

equation model are: as in two-equation model, it cannot precisely reflect the changes in the length scale (e.g. in modelling the most inaccurate dissipation equation); and there is inadequate support for pressure strain term modeling. Moreover, though using the anisotropic concept, whether it can achieve higher precision than two-equation model needs further investigation; and it is too complicated for engineering applications. 6. ALGEBRAIC-STRESS MODEL The main concept of the algebraic-stress model is to simplify the differential equations into algebraic expressions to avoid the complexity of the model, but keeping the anisotropic characteristics of the turbulent flow. In the differential equation (5.1a), only the convective flow term Cij and diffusion term Bij contain the derivative of Rij, so proper handling of Cij and Bij can simplify equation (5.1a) into algebraic expression.

DtDR

C ijij =

k

ijkij x

JB

∂

∂−=

If Tij2 in the pressure strain term Tij uses IPM, and Pkk = 2P, then equation (5.1a) can be rewritten as:

+−

+

−δ=−

2ij

1t1t

ijijijq

RC1

2C

32BC DPD

δ−− Pijij

*2t 3

2P)C1( (6.1)

At present, there are four kinds of simplification methods for convective flow term Cij and diffusion term Bij: 1) Assuming that the transport rate of Reynolds

stress (Cij − Bij) has a linear relationship with the transport rate of turbulent kinetic energy 2(P − D), it was proposed by Rodi [34] that:

)(q

R2BC 2

ijijij DP −=− (6.2)

It was suggested by Launder [35] that:

)(32BC ijijij DP −δ=− (6.3)

2) Assuming partial balance of turbulent kinetic energy as in two-equation model gives:

Cij − Bij = 0 (6.4) 3) It was suggested by Launder [36] to use

different formula for Cij and Bij:

αδ−α+= ij2

ijij 3

1q

R)1(2C P (6.5a)

βδ−β+= ij2

ijij 3

1q

R)1(2B D (6.5b)

where the values of the constants α and β were suggested by Launder [36] to be 0.3 and -0.8 respectively. To make α = β = 0, equation (6.5) has to be transformed to equation (6.2). Based on the above assumptions (6.2) to (6.5), the corresponding algebraic-stress equation can be deduced:

[ ]

δ−

−+−

+δ= PDDP ijij

1t

*2t

ij2

ij 32P

)1/(2CC1

31qR

(6.6)

δ−

−+δ= P

D ijij1t

*2t

ij2

ij 32P

CC1

31qR (6.7)

[ ]

δ−

−α++β−α+−

+δ= PDDP ijij

1t

*2t

ij2

ij 32P

)1/)(1(2)(2CC1

31qR

(6.8) Principally speaking, q2 = Rii is not an independent variable in the algebraic-stress equation. However, q2 and D are still regarded as two parameters, the algebraic stress equation, turbulent kinetic energy equation and dissipation rate equation are combined to form the algebraic-stress model. Taking:

1t

*2t

C)C1( −

=λ

Assuming partial balance, with P = D, the algebraic-stress equation (6.7) can be written as:

∂∂

+∂

∂λ−δλ−=

k

ijk

k

jik

22

ijij xU

RxU

Rqq)21(31R

D(6.9)


93

Comparing this with the Boussinesq equation of isotropic turbulent flow:

∂∂

+∂

∂−δ=

j

i

i

j4

72

ijij xU

xUqCq

31R

D (6.10)

Other than the difference in constant, it can be seen that the scalar quantity, eddy viscosity is given by:

D

4

7tqC=ν

Under isotropic conditions, it is replaced by the

tensors ik

2Rq

λ

D, jk

2Rq

λ

D in equation (6.9).

In other words, C7 is a tensor function rather than a constant:

2ik

7qR

C ∝

or

2jk

7q

RC ∝

Anisotropic characteristics are reflected. Considering thin free shear flow, the uniform flow

velocity U1 ≠ 0, U2 = U3 = 0, 0xU

xU

3

1

1

1 =∂∂

=∂∂

.

From equation (6.9):

222 q)21(

31R λ−=

2

122

2

12 xU

RqR∂∂

λ−=

D

From equation (6.10):

2

14

712 xUqCR∂∂

−=

D

Therefore,

( )λ−λ=λ= 2131

qR

C 222

7

Taking Ct1 as 5 and Ct2

* to be 0.6, then λ is 0.08 and C7 is 0.0224. Therefore, it can be seen that for certain flows, two-equation model is an exception

of the simplest algebraic-stress model (6.7) under

the condition that DP is a constant value.

Algebraic-stress model is the simplified expression of stress-equation model under certain conditions, two parameters q2 and D are required to be solved. The effects of anisotropy are simplified as the correction of C7 in the two-equation model. Therefore, it can be grouped under two-equation model, i.e. extended two-equation model. Algebraic-stress model can take into account the anisotropic effect of turbulent transport to a certain extent without increasing the number of differential equations. However, algebraic-stress model is only suitable for flows not deviating too much from partial balancing conditions, it cannot calculate the effects of the reverse gradient diffusion. Also, it would have much difficulties in handling three-dimensional convergence problems. Therefore, at the moment, stress-equation model is re-commended for solving complicated anisotropic flows. 7. LARGE-EDDY SIMULATION There was rapid development in all the above turbulence modelling approaches, though empirical relations have to be used. Parameters appeared in the model depend on the flow problems, and no universal empirical data are found. The large eddy simulations (LES) approach might be a more universal method in dealing with turbulent flows. The basic idea of which is to solve the transient Navier-Strokes equations numerically in three-dimensional form to get large-scale eddy flows. Smaller scale turbulent motions are modelled by some pre-determined functions. Experiments illustrated that there are eddies of different scales in turbulent flows. Large-scale vortices would affect turbulent energy, Reynolds stresses and other dispersing flow parameters. Structures of large-scale turbulence depend strongly on the boundary conditions. This is due to flow characteristics and so cannot be modelled in a universal fashion. Dissipation would lead to smaller-scale eddies. For flows with high Reynolds number, these smaller eddies tend to give isotropic flow. The flow is not too affected by the boundary conditions, and with more similarities. Therefore, more general models can be worked out. It is still difficult to simulate flow down to the dissipation scale with the available computer hardware. However, it is possible to study the flow at the scale of those small inertial scales. Therefore, ‘compute the large-scale eddies, model the small-scale eddies’ would give better simulations on turbulent flows.


94

LES was applied by Deardorff [37] to simulate two-dimensional high Reynolds number channel flow. Predicted time-averaged velocities and turbulence intensities agreed well with experiments. Values of model constants were confirmed by Yeh and Van Atta [38] through experimental studies. Theoretical background on LES was discussed by Ferziger and Leslie [39]. LES is now applied to simulate different turbulent flows. Examples are the modelling of solid wall boundary by Moin and Kim [40] to explain the turbulence structure in different regions. Turbulence-induced non-equilibrium flow were studied by Kobayashi et al. [41]. By using wave-filtering and sub-grid model to get more accurate calculations, residual stress model was proposed by Piomell and co-workers [42,43]. Results were applied to study the average speeds and Reynolds stress across a sectional area in the transition region of channel flows. LES was applied by Tsai and Leslie [44] to study the development of turbulent boundary layer at low Reynolds number. From the predicted data, statistics on pressure distribution was obtained and effects of pressure on the Reynolds stresses equation were studied. This information cannot be achieved from experiments. Flow motions behind obstacle were studied by Kobayashi et al. [45] and Friedrich [46] to investigate how LES can be applied for modelling turbulence. Predicted length of attached flow was 15% to 30% higher than experimental results by Kim et al. [47]. In comparing with those results predicted by two-equation turbulence models, turbulence intensities along the flow direction were much lower with a peak value found near to the flow re-attachment point. Results from LES are much more reasonable. Further, distributions of average speeds and turbulence kinetic energy would be fluctuated easily as the number of grid points is insufficient. Turbulent mixing layer with LES was studied by Hamba [48]. Average speed, turbulence kinetic energy, coefficient of skewness, salient vales, balancing of turbulence energy and other functional relations were studied. From sub-grid model based on second-order closure, coherent structure of the convective boundary layer and more accurate frequency spectrum were analyzed by Schmidt and Schumann [49]. Turbulence dispersion due to chemical reactions were included to study the effects of reaction rate and mixing ratio on dynamics and eddy diffusions in turbulent reacting flows by Schumann [50]. Obviously, LES can give a better understanding on how turbulence can be modelled. 7.1 Fundamental Equations and Subgrid

Simulation In large eddy simulation of turbulent flows, the function f(x,t) can be expressed as the sum of filter

(large-scale) part f and the deviated (subgrid) part f ′′ :

fff ′′+= (7.1) The filter part f is defined as the integral of the whole flow field with the filter function as the central function:

=)t,x,x,x(f 321

∫ ∆Π=D 321321iiii

3

1idydydy)t,y,y,y(f);y,x(G (7.2)

where Gi is the filter function along the xi direction; ∆i is the filter width along the xi direction, which is nearly proportional to the grid mesh hi (∆i = nhi , n > 1). The best value of the characteristic filter width ∆A should be taken as: ∆A = 2 (h1 h2 h3)1/3 (7.3a) In subgrid simulation, the characteristic filter width is usually taken as: ∆ = (h1 h2 h3)1/3 (7.3b) The filter wave refers to the weighted average of function f of a volume with the filter width ∆i as the side length. The purpose of the filter function is to filter off wave of high wave numbers and maintain the kinetic energy cascading process, i.e. allow energy to transfer from large eddies to small eddies. The grid mesh width and the filter function have a great influence on the large-scale and subgrid flow field structure. The filter functions commonly used are the Gaussian filtering function, box filter and Fourier acute cut-off filter. Gaussian filtering function is defined as:

( ) ( )

∆

−−

∆π=∆ 2

i

2ii

2/1

2i

iiiiyx

6exp6;y,xG (7.4)

Box filter of variable width is defined as:

( )

∆≤−

∆=∆

elsewhere,0

2yx,1

;y,xG

iii

i

iiii (7.5)

Acute cut-off filter in physical space is defined as:

( ) ( )[ ]( )ii

iiiiiii yx

/yxsin2;y,xG

−π∆−π

=∆ (7.6a)


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In Fourier space, it is defined as:

<=

elsewhere,0

kk when,1)k(G

cii

ii (7.6b)

where )k(G ii is the Fourier coefficient of filter function Gi; kci is the cut-off wave number along the xi direction, which is related to the filter width

icik

∆π

= . The purpose of the acute cut-off filter is

similar to expand the function using Fourier grade with the given finite point function value. If box filter (7.5) is used in filter expression (7.2), and taking hi = h, ∆i = ∆ = h, then,

( ) 321321

2/hx

2/hx

2/hx

2/hx

2/hx

2/hx3 dxdxdxx,x,xf

h1f

1

1

2

2

3

3

∫ ∫ ∫+

−

+

−

+

−

=

(7.7) This is the expression using “volume balancing” to solve for the variable f [51], where f is the average value of the volume of grid cells, and it is the function of grid points (x1, x2, x3). The filter part f has a fundamental difference with the time-averaged value f . For statistically fixed flows, the time-averaged value is fixed usually, which ignores all the fluctuations; however the filter part after filtering is not fixed, but still contains lots of fluctuations. If the filter width is sufficiently small, all fluctuations can be maintained. The continuous equation for non-compressible liquid flow, momentum equation and energy equation after filtering can be written as:

0xu

i

i =∂∂

(7.8)

( )

∂∂

ν∂∂

+∂∂

ρ−=+

∂∂

+∂∂

j

i

jiijji

j

i

xu

xxp1Ruu

xtu

(7.9)

( )

∂∂

α∂∂

=+∂∂

+∂∂

jjTjj

j xT

xRuT

xtT (7.10)

where T,p,u i and ρ are the filter velocity, filter pressure, filter temperature and density respectively of the fluid flow; ν and α are the molecular kinetic

viscosity coefficient and heat diffusion coefficient. The subgrid terms Rij and RTj can be expressed as:

jijijiij uuuuuuR ′′+′′+′′′′= (7.11a)

jjjTj uTuTuTR ′′+′′+′′′′= (7.11b) where (−ρRij) is usually called the subgrid Reynolds stress, and (−ρRTj) is the subgrid heat flux. jiuu ′′′′ or juT ′′′′ is made from the subgrid part and does not contain large-scale activities, its importance and grade are much lower than Reynolds stress or heat flux. jjiji uT ,uu ,uu ′′′′′′ and

juT ′′ are changing very slowly, their values are very small after multiplying to high-frequency fluctuation and filtering. Taking into account the compromise during the decrease of tensor (i = j), introducing the new variable τij and corrected filter pressure P , which are defined as:

δ−−=τ kkijijij R

31R (7.12)

kkR31pP +

ρ= (7.13)

In equation (7.13), Rkk is included in the corrected filter pressure P that needs not to be calculated

individually. 2

R kk is the subgrid turbulent kinetic

energy; on the wall, ρ

=pP . ji uu and juT can

be obtained by the Taylor grade expansion of the product of the two filtered rate [52]. Obviously, the result is related to the filter function used, the relation between Gauss filter and box filter is as follows:

( ) ( )4

kk

ji22

jiji 0xxuu

24uuuu ∆+

∂∂

∂∆+= (7.14a)

( ) ( )4

kk

j22

jj 0xxuT

24uTuT ∆+

∂∂

∂∆+= (7.14b)

where the second term on the right side of equation (7.14) is called the Leonard term. They have the grade ∆2 as the subgrid scale. The contribution of Leonard term to subgrid kinetic energy transport is negligible and can be neglected. It has insignificant effect on the predicted results, especially in using staggered grid (volume balancing method), neglecting this term yields better results. Its function is to produce energy


96

dispersion from the higher to lower waver numbers to improve modelling the velocity frequency spectrum [52,53]. To improve the calculation precision, Fourier exchange was applied by Mansour et al. [54] to replace Taylor grade expansion equation (7.14), for example: ( ) ( ) ( )jiji uuFGFuuF = (7.15)

where F(y) is the Fourier exchange of y. Given the

iu field, the right side of the above equation can be

calculated, and ji uu can be determined using Fourier. Generally, equation (7.14) is more convenient and it can guarantee the precision of 0(∆4). In large eddy simulation, the following eddy viscosity model by Smagorinsky [55] is usually adopted for subgrid stress: τij = 2 νs Sij (7.16a)

∂∂

+∂

∂=

j

i

i

jij x

uxu

21S (7.16b)

νs = (CsDs∆)2 (2SijSij)1/2 (7.17) where νs is the subgrid eddy viscosity; Sij is the filter strain rate tensor; Cs is the Smagorinsky constant usually taken as 0.1 to 0.2; and Ds is the near-wall decay factor [56], similar equation of (2.4) is:

−−=

+

+

Ayexp1Ds (7.18)

The selection of Smagorinsky constant Cs is very important in the LES method. Though theoretically, the values of Cs lie between 0.2 to 0.3 [57], due to the physical effect of numerical diffusion on the subgrid viscosity, using smaller values of Cs can yield better results. Detailed discussion on Cs has been made by Mason and Callen [58]. The subgrid movement in turbulent flow is described by the eddy viscosity model (7.16). At near-wall regions, such modelling is similar to the Prandtl mixed length solution. Assuming lo to be the Smagorinsky mixed length,

then .C os ∆=

lIn a homogeneous isotropic

turbulent flow, Cs is taken as 0.23. The numerical modelling results of turbulent velocity field and kinetic energy decay agreed well with the experimental results and predictions [59]. For a larger Cs (≈ 0.2), turbulent modelling can give out spatial structure of smooth variations, but if Cs is

too large, large-scale decay would be resulted, and might even lead to calculation failure. When applying LES to turbulent flow with heat transfer, linear flow in building structure, channel flow and surface layer turbulence, taking Cs to be 0.1 would yield very good results, with very little deviation from continuous resolution. Cs has an important effect on velocity fluctuations, the modelling results for smaller Cs (≈ 0.1) would have fluctuations. It is found that in large eddy simulation of flat Poiseuille flow (e.g. turbulent channel flow of height 2δ), the scale of turbulent

kinetic energy basically depends on δol . When ol

≥ 0.05δ, scale eddy cannot be solved. To ensure the stable existence of vortex, ol < 0.02δ should be taken. When ol < 0.02δ is given, the variation of Cs with characteristic filter width ∆ is very small, taking Cs as 0.2 would yield satisfactory results [58]. That is to say, if ol is within the required scale, i.e. sufficiently small grid mesh width, Cs is not too sensitive to subgrid modelling at this time. In another model of subgrid eddy viscosity, eddy tensor ijΩ is used to replace the strain rate tensor Sij:

∂∂

−∂

∂=Ω

j

i

i

jij x

uxu

21

i.e. νs = (BsDs∆)2 (2ΩijΩij)1/2 (7.19) Comparing the LES results with the experimental results, Bs

2 = 1.5 Cs2. It was pointed out by

Tennekes and Lumley [10] that ijijijij~SS ΩΩ− .

But unexpectedly, Bs ≠ Cs; this requires further study. It was assumed by Bardina et al. [60] that the interaction between scale eddy and subgrid eddy would produce the smallest scale eddy and the largest subgrid scale eddy. This structure can be described by the difference between filter velocity and second filter velocity. The subgrid mixing model was then proposed:

( ) ijsjijiBij S2uuuuCR ν−−= (7.20) where CB is taken as 1 for keeping the Galilean exchange of kinetic equation unchanged [61]. The results of the LES method combining the mixed length modelling and Gauss filter agree well with those of the direct numerical modelling, especially when the grids are divided into very small grids. The smallest structure in one-dimensional energy spectrum can be predicted accurately. The


97

calculated results of subgrid stress are better than those from combined Smagorinsky model and Gauss filter (the values are small). The simple subgrid heat flux gradient diffusion model is used for RTj:

jsTj x

TR∂∂

α−= (7.21a)

s

ss σ

ν=α (7.21b)

where αs is the subgrid heat diffusion coefficient; and σs is the subgrid Prandtl number. Equation (7.21b) is similar to the usual turbulence modeling,

T

TT σ

ν=α

where Prandtl number σT ≈ 0.9. Subgrid Prandt number has an important impact on temperature prediction. Antonopoulos-Domis [59] found that σs is smaller than 0.5 for isotropic turbulence. In LES heat transfer simulation by Ciofalo and Collins [62], σs was taken as 0.25, 0.5 and 1 to calculate the average temperature field

T . The minimum value of σs of 0.25 agreed well with the known temperature profile. However, in LES heat transfer calculation of channel flow, even when such small value of σs is used, the prediction of temperature gradient at the central region where turbulent fluctuation is the smallest is still too large, it might be due to the inherent inadequacy of the subgrid heat flux gradient diffusion model. Using Smagorinsky subgrid eddy viscosity model (7.16) and subgrid heat flux gradient diffusion model (7.21), then the momentum equation (7.9) and energy equation (7.10) can be rewritten as:

( )j

jii

xuu

tu

∂

∂+

∂∂

( ) ( )

∂∂

∂∆−

∂

∂ν+

∂∂

ν+ν∂∂

+∂∂

−=kk

ji22

i

js

j

is

ji xxuu

24xu

xu

xxP

(7.22)

( ) ( ) ( )

∂∂

∂∆−

∂∂

α+α∂∂

=∂

∂+

∂∂

kk

j22

js

jj

j

xxuT

24xT

xxuT

tT

(7.23)

Equations (7.22), (7.23) and (7.8) can be used as a set of equations for numerical resolution of large eddy simulation. 7.2 Numerical Method Dividing the three-dimensional flow field into six parallel units, then the general pressure and temperature variables are defined at the central grid point (l,m,n), i.e. P(l,m,n), T(l,m,n). While the velocity weight is defined at the centre of the flux interface, i.e.:

+ n,m,

21u1 l ,

+ n,

21m,u 2 l ,

+

21n,m,u3 l

If the collocated scheme is used, all floating variables will be defined at grid points (l,m,n). Mixed finite differential method can be used in solving equations (7.8), (7.22) and (7.23). Time derivative uses forward difference, viscosity takes the Crank-Nicolson implicit form, and the convective term and other subgrid related parts take the second-order Adams-Bashforth explicit form. Applying this semi-implicit mixed difference, the momentum equation (7.22) and energy equation (7.23) can be rewritten as:

+−=∆− −

+1n

ini

ni

1ni H

21H

23

tuu

( )i

1n

jj

ni

2

jj

1ni

2

s xP

xxu

xxu

21

∂∂

−

∂∂∂

+∂∂

∂ν+ν

++

(7.24)

+−=∆− −

+1nn

n1nS

21S

23

tTT

( )

∂∂∂

+∂∂

∂α+α

+

jj

n2

jj

1n2

s xxT

xxT

21 (7.25)

where represents the average value of the whole computing range in a given period; Hi and S can be expressed as:

( ) ( ) +∂∂

∂ν−ν+

∂∂

−=jj

i2

sjij

i xxuuu

xH

∂∂

∂∆∂∂

−

∂

∂+

∂∂

∂ν∂

kk

ji22

ji

j

j

i

j

s

xxuu

24xxu

xu

x


98

( ) ( ) +∂∂

∂α−α+

∂∂

−=jj

2

sjj xx

TuTx

S

∂∂

∂∆∂∂

−∂∂

∂α∂

kk

j22

jjj

s

xxuT

24xxT

x

where the spatial derivatives of equations (7.24) and (7.25) use the central difference. Dispersion equation (7.24) can be divided for solving in two steps. First, calculate the central velocity field *

iu by ignoring the pressure gradient term in the momentum equation; then solve for the

pressure field 1nP + of the time mesh (n + 1). After that, correct the central velocity field to satisfy the continuous equation to obtain the velocity field

1niu + of (n + 1) time mesh, i.e.

+−=∆− −1n

ini

ni

*i H

21H

23

tuu

( )

∂∂∂

+∂∂

∂ν+ν

jj

ni

2

jj

*i

2

s xxu

xxu

21 (7.26a)

i

1n*i

1ni

xtuu

∂φ∂

−=∆− ++

and i

*i1n2

xu

t1

∂∂

∆=φ∇ +

(7.26b) where

( ) 1n2s

1n1n

2tP +++ φ∇ν+ν

∆−φ= (7.27)

The velocity field, temperature field and pressure field of time mesh (n + 1) can be obtained by solving equations (7.25), (7.26) and (7.27). The conditions of diffusion stability for mixed difference forms (7.24) and (7.25) are [63]:

( )41

x1

x1

x1t 2

322

21

s <

∆+

∆+

∆∆ν−ν (7.28a)

( )41

x1

x1

x1t 2

322

21

s <

∆+

∆+

∆∆α−α (7.28b)

In large eddy simulation, diffusion coefficient is proportional to the square of grid scale, and it has the same level with the simulated diffusion coefficient. Therefore, a simulated diffusion term is added as an implicit term, and the corresponding term in explicit term is deleted. This will effectively increase the diffusion stability. The

Courant-Friedrichs-Lewy (CFL) stability condition for convective term is:

1x

umaxtCFL

i

i <

∆

∆= (7.29)

Generally, it is suitable to take CFL to be smaller than 0.6. Central difference has spatial second-order accuracy, while Adams-Bashforth form has temporal second-order accuracy; and Crank-Nicolson has the same accuracy too [64]. Therefore, dispersion equations (7.24) and (7.25) have second-order temporal and spatial accuracy. 7.3 Initial and Boundary Conditions The selection of initial conditions has no influence on fully-developed turbulent flow, and no effect on the accuracy of the results. It would only affect slightly the calculation time for achieving balance. Therefore, the existing similar flow field is usually taken as the initial value. Given the reference average velocity or the corresponding Reynolds number (water diameter as characteristic length), the shear coefficient and the corresponding uniform pressure gradient can be estimated to determine the wall stress and shear velocity, and the shear velocity can be used as the initial velocity fluctuation. Initial temperature can be taken as a constant. The turbulent structure at near-wall region is very small, and the velocity gradient is very steep. To calculate accurately the wall stress and velocity distribution, the grid size should be very small at the near-wall region, and reasonable boundary condition should be given. The wall boundary condition for momentum equation is no-slip condition, which accurately solve for the wall layer. Fine grids are required to give no-slip condition, the computing load would be very large because of the large number of grid points. This method is limited to turbulent flow at low Reynolds numbers. For example in turbulent channel flow, one-third of the total grid points are required at near wall surface of 7% of the height, and it is limited to Reynolds number Reδ = *u δ/ν within the range from 180 to 640. Therefore, using this boundary condition would limit the development of large eddy simulation in practical application. For engineering problems of turbulent flow, the whole wall layer can be modeled. The turbulent velocity can be determined by applying similar boundary condition that does not directly calculate


99

the wall stress. This method needs not solve for the wall layer structure. Near-wall region is very close to isotropy which is easier for modelling. Therefore, coarser grids can be used and the grids at first layer can be located at the logarithmic region (y+ ≈ 50 – 200). Obviously, this method can effectively save the computing expenses. Dynamics of wall layer is largely dependent on the burst of turbulent flow. The outer high-energy fluid flow toward the wall surface (sweep) would induce impingement and stretch of eddy, which would increase the wall velocity fluctuation, and in turn increase the wall stress. Low-energy fluid would be transported to the outer layer (ejection), which would reduce the wall stress. Near-boundary condition should simulate the interaction of the inner and outer layer to reflect the physics behind wall dynamics. Considering the relation between normal wall activity and wall stress, Piomelli et al. [65] had proposed the following approximative-boundary condition, i.e. ejection boundary condition:

)z,y,x(vCuR)z,x(R 2s*WW,12 ∆+−= (7.30a)

0)z,x(vW = (7.30b)

W2s

2sW,32 R

z,y,x(u)z,y,x(w

)z,x(R∆+∆+

= (7.30c)

where is the time-averaging value; y2 is the normal coordinate of the first layer grid points, the distance from the wall to grid points of this layer is

W2 yyy −=∆ ; u , v and w are the filter velocity part along the main flow direction (x), the vertical wall direction (y) and the horizontal direction (z) respectively; C is a constant at level 1 which can be taken as 1; the subscript w is the value on the wall; ∆s is the displacement between wall stress and velocity; the displacement vector in the surface layer makes an angle θ with the wall, i.e. ∆s = ∆y ctg θ (7.31)

when

ν

∆=< + *

2yu

y30 < 50 to 60, θ ≈ 8°, i.e. ∆s

≈ 7∆y; at 6° ≤ θ ≤ 13°, then 9 ≥ ∆s/∆y ≥ 4, ∆s is not sensitive to the calculated results. Using the logarithm of the surface layer,

( )2.5ylog5.2

uz,y,xu

)y(u 2*

2s2 +=

∆+= ++

(7.32)

The shear velocity *u can be calculated from the given ,)z,y,x(u 2s∆+ and then used for determining the wall stress τW and RW. The boundary condition corresponding to the energy equation is the given heat flux, i.e. equivalent to the given temperature gradient on the wall. For adiabatic wall surface, the heat flux is zero. 8. CONCLUSIONS Numerical calculation of turbulent flow is based on Direct Numerical Simulation (DNS) of Navier-Stokes equation. It is a very useful tool which does not need to carry out equation modelling, but requires sufficient number of grid points. For predicting all turbulent flow scales, or even solving for the smallest scale (Kolmogorov micro-scale), at least Re9/4 grid points with Re1/2 time interval are required. The storage capacity and computing time required has gone far beyond the capability of present computers. Therefore, DNS is limited to a few simple boundary condition flows of relatively lower Reynolds numbers. However, it helps in our understanding of turbulent flow and it is important in directing research in turbulence models. For simple free turbulence, boundary layer and simple non-buoyant channel flows without eddy or backflow, mixed length or one-equation model can be applied. For non-buoyant backflow with weak eddy or no eddy, two-equation model can be applied. These models are widely applied in engineering problems and academic research. However, turbulent viscosity model has the same characteristic that: it is assumed that the transport of turbulent eddy and transport of molecular heat activity have a certain degree of similarity, the turbulent viscosity is introduced and defined. Therefore, the anisotropy of turbulence transport cannot be realized, and the model cannot accurately describe flows where the uniform flow field gradient is zero and the turbulent stress is not zero. For strong eddies, buoyant flows and compressible flows, etc, turbulent viscosity coefficient model is not applicable. Comparing with the two-equation model, the anisotropic stress-equation model and algebraic-stress model can better predict rotational flows and buoyant flows, but there is only slight difference in predicting the uniform flow field. Both models are required to use the less precise turbulence kinetic energy dissipation (D) equation. Therefore, to increase the precision of such models, it is essential to improve the turbulent kinetic dissipation equation or the turbulent scale equation. Because


100

of the large increase in differential equations in stress-equation model, larger computing capacity is also required, and that is the main reason why this model is not widely used. Smaller computing capacity and less time are required in algebra-stress model. However, it is only suitable for flow processes not deviating too much from the partial balance conditions, so it cannot calculate the reverse diffusion gradient effect, and there are convergence problems associated with three-dimensional calculations. Therefore, its practicability is much doubted. Large eddy simulation uses large eddies to calculate small eddies, it not only takes into account the convective flow and diffusion effect, but also maintains the fluctuation characteristic of the filter part. LES can accurately calculate the turbulent flow field and statistical characteristic parameters, it is suitable for complicated anisotropic turbulent flows. At present, LES is widely applied in numerical modelling of turbulent flows with different flow characteristics, the results can provide directional guidance to research on simpler turbulence models, and improve the understanding of turbulence models and the analysis of experimental data. However, there are some problems associated with large eddy simulations. Firstly, the wavelength of the minimum eddy in LES is within the normal range, i.e. within the dissipation range or inertial range, where the structure of eddy is related only to the wavelength and the energy transferred from the large eddies, and not related to the boundary conditions. Therefore, it is feasible to model nearly isotropic small eddies, but the demand for computer hardware is difficult to meet. Secondly, although the effect of large eddy on statistical characteristics of small eddy is not very large, its effect on eddy with grid scale is still unknown. Further investigation on modeling such kind of eddy problems is needed. Thirdly, the selection of eddy viscosity of grid eddy has a great impact on the calculated results. It implies that not only the dissipation of grid eddy, but also the color dispersion should be studied. Large eddy simulation is now under active development and it might become a practical engineering application. Numerical modeling of turbulence and solving conditions are important factors affecting the accuracy of calculation. Boundary conditions have a strong effect on the calculation results. Numerical dissipation and frequent diffusion would also influence the computations. For instance, a simple difference would induce larger numerical diffusion. Therefore, improvement in turbulence

model should go along with improvement in numerical method. For example, using second-order accurate mixed difference form and division calculation is an effective numerical method. ACKNOWLEDGEMENT This project is funded by Area of Strategic Development in Advanced Buildings Technology in a Dense Urban Environment of The Hong Kong Polytechnic University. REFERENCES 1. T. Von Karman, “` Uber laminare und turbulente

Reibung”, Zeitschrift für Angewandte Mathematik und Mechanik, Akademie Verlag, Berlin, Vol. 1, pp. 233-252 (1921).

2. S.J. Kline, M.V. Morkovin, G. Sovran and D.J. Cockrell, Proceedings of 1968 AFOSR-IFP-Stanford Conference on Computation of Turbulent Boundary Layers, Department of Mechanical Engineering, Stanford University, USA (1969).

3. G.L. Brown and A. Roshko, “On density effects and large structures in turbulent mixing layers”, Journal of Fluid Mechanics, Vol. 64, Part 4, pp. 775-816 (1974).

4. R.F. Blackwelder, “Analogies between transitional and turbulent boundary layers”, Physics of Fluids, Vol. 26, No. 10, pp. 2807-2815 (1983).

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