2000, manceau, reproducing the blocking effect of the wall in one-point turbulence models2

8/3/2019 2000, Manceau, Reproducing the Blocking Effect of the Wall in One-Point Turbulence Models2

1/20

European Congress on Computational Methods in Applied Sciences and EngineeringECCOMAS 2000

Barcelona, 11-14 September 2000cECCOMAS

REPRODUCING THE BLOCKING EFFECT OF THE WALLIN ONE-POINT TURBULENCE MODELS

Remi Manceau

Thermal and Fluids Sciences Section, Department of Applied PhysicsDelft University of Technology

Lorentzweg 1, PO Box 5046, 2600 GA, Delft, The Netherlands

Email: [email protected]

Key words: Near-wall models, Low-Reynolds-number Models, Non-local Effects, Asymp-totic Behaviours, Elliptic Relaxation, Industrial Applications.

Abstract. The presence of a wall induces different effects on turbulence, which aredifficult to account for in one-point closures, in particular the non-local blocking effect.The balance of the dominant terms of the budget of the Reynolds stresses in the vicinity ofthe wall must be reproduced, which requires a careful modelling of the redistribution term.

The elliptic relaxation method enables the reproduction of this balance and, conse-quently, of the anisotropy in the near-wall region. However, a side effect of this method isthe amplification of the redistribution in the log layer, which can be corrected by improvingthe basis assumptions of the model, leading to a modified elliptic relaxation model.

Two simplified models, derived in order to reduce the number of equations, are pre-sented: the v2f model, whose interest for industrial applications is shown through resultsin the backstep and the ribbed-channel flow, and a new Reynolds-stress model, the ellipticblending model, which gives predictions comparable to the elliptic relaxation model, butwith 8 equations instead of 13.

1


2/20

Remi Manceau

1 INTRODUCTION

Accounting for the wall-induced effects is one of the most difficult problems turbulencemodellers faced during 20th century. The few existing theories are generally valid only inhomogeneous turbulence, e.g., [10], or not to close to solid boundaries [3]. All existing

models derive from these theories, and making them comply with the near-wall behaviourof turbulence is still nowadays one of the most active fields of research.This issue is particularly important for real life applications, since flows are always

somehow influenced by the presence of walls, either as boundaries (internal flows) or asobstacles (external flows). From an industrial point of view, an accurate prediction ofnear-wall turbulence is indispensable, since the quantities of primary interest are usuallythose evaluated at the wall, such as the friction and heat transfer coefficients Cf and Nu.

Moreover, very intense energy interactions that influence the rest of the flow takeplace very close to the wall : the production of turbulent kinetic energy that reaches itsmaximum in the buffer layer, and its dissipation rate into internal energy that reaches itsmaximum at the wall itself. The main difficulty is that this region is generally very small.

For instance, in external flows, at usual Reynolds numbers, the boundary layer is severalorders of magnitude smaller than the characteristic size of the flow; the viscous sublayeris also very small compared to the boundary layer, the size ratio behaving as Re

7/8 at

moderate Reynolds number, and as Re9/10 at high Reynolds number, Re being based

on the boundary layer thickness and the free-flow velocity.Therefore, measurements are very difficult in the boundary layer and particularly in

the viscous sublayer. Our knowledge of the phenomena inside the latter remained partialuntil the increasing power of super-computers made direct numerical simulation possi-ble in the middle of the 80s. However, the available data are still limited to very lowReynolds numbers and very simple geometries (Re 11, 000 for the most recent DNS ina channel [23]).

The aim of this paper is to present the main properties of near-wall turbulence to beaccounted for in turbulence models and in particular to describe how the blocking effectcan be reproduced in one-point models. First, the influence of the wall on turbulence, theasymptotic behaviour of the redistribution term and the weaknesses of standard modelsare presented. Secondly, the elliptic relaxation approach is described in the frame ofsecond moment closure. Attention is then focused on simplified models for industrialapplications: the v2f model and the elliptic blending model.

2 MODELLING WALL EFFECTS

2.1 Influence of the wall on turbulence

The wall exerts a strong influence on the instantaneous velocity and pressure fields, bycausing the appearance of streaks, hairpin vortices, ejections... However, our concern inRANS is ensemble means: the difficulty is thus to represent the mean effects of all theseinstantaneous phenomena on the averaged quantities. In this section, the importance ofdistinguishing among dynamic and kinematic effects is emphasized.

2


3/20

Remi Manceau

2.1.1 Dynamic effects

Mean shear The viscosity of the fluid imposes a no-slip boundary condition on meanvelocities. This induces strong gradients from which the turbulence productionoriginates.

Viscous damping The no-slip boundary condition at instantaneous level implies thatthe fluctuating velocity components tangent to the wall u and w behave as y. Con-sequently, the Reynolds stresses u2, w2 and uw behave as y2. The stronger dampingof components involving the wall-normal velocity is due to another phenomenon,the blocking effect.

Overlapping of scales When the turbulent Reynolds number ReT = k2/ diminishes,the scale separation between energetic eddies and dissipative eddies progressivelydisappears. Energetic scales are then influenced by molecular diffusion and influencein turn the dissipative eddies, which can no longer be considered as isotropic.

2.1.2 Kinematic effects

Wall echo effect The wall echo effect originates from the fact that an image term ap-pears in the Green function of the domain when it is bounded by a wall. This islinked to the pressure reflection on the wall. Contrary to what is usually claimed,this effect increases the pressure, and, accordingly, the redistribution term [16, 20],but is small compared to the blocking effect.

Blocking effect Conjugate effects of the impermeability of the wall and the incompress-ibility of the fluid lead to the damping of the wall-normal instantaneous velocitycomponent v, which consequently behaves as y2 in the vicinity of the wall. Thiseffect is instantaneously felt far from the wall, through the pressure that adjusts toensure the incompressibility condition: a velocity directed toward the wall instan-taneously generates an increase of the pressure that contradicts it.After using Reynolds decomposition, this effect is inherited by both mean velocityV and fluctuating velocity v. In computations, the blocking of the mean velocityis well reproduced, since the incompressibility condition is explicitly imposed. Thisleads, for instance, to the fact that the mean velocity field feels obstacles at somedistance upstream. But the continuity equation is not resolved at fluctuating level,and the blocking effect must be accounted for by other means, through the onlyterm in Reynolds-stress transport equations that contains the pressure, namely theredistribution term. This issue is the main concern of this paper.

2.1.3 Consequences for modelling

The dynamic and kinematic effects are very different by nature and can appear sepa-rately. In particular, they do not act the same way when the flow reaches the leading edgeof a flat plate or when a wall is suddenly introduced into the flow. Dynamic effects act

3


4/20

Remi Manceau

progressively, the layer affected by viscosity developing as a function of the streamwiselocation [1] or of time [9, 24]. On the contrary, the kinematic effects instantaneously actfar from the wall [9].

Moreover, if all the effects previously described play a role close to a wall, they canappear separately in other situations. For instance, at the interface between two fluids of

different density, only kinematic effects are significant. On the other hand, in a free flow,when the Reynolds number diminishes, only the overlapping of scales appears. Therefore,it is impossible, and even dangerous, to model these effects using functions of ReT only,as some models do. They must better be accounted for by different mechanisms, and theblocking effect, which is non-local, needs a very careful modelling, as presented in thefollowing of this paper.

2.2 Asymptotic behaviours

The aim of this section is to emphasize the importance of the balance between dominantterms in the Reynolds-stress transport equations in the vicinity of the wall.

In a channel flow, the viscous damping and the blocking effect presented above implythe following expansions of the mean velocity, the fluctuating velocities, the fluctuatingpressure, the Reynolds stresses, the turbulent kinetic energy and its dissipation rate:

U = A1 y + A2 y2 + O(y3)

u = a1 y + a2 y2 + O(y3)

v = b2 y2 + O(y3)

w = c1 y + c2 y2 + O(y3)

p = p0 + p1 y + p2 y2 + O(y3)

(1)

u2 = a21 y

2 + 2a1a2 y3 + O(y4)

v2 = b22 y4 + 2b2b3 y

5 + O(y6)

w2 = c21 y2 + 2c1c2 y

3 + O(y4)uv = a1b2 y

3 + (a2b2 + a1b3) y4 + O(y4)

(2)

k = 12uiui =12 (a

21 + c

21) y

2 + O(y3) (3) = (a12 + c1

2) + O(y) (4)

The transport equations of the Reynolds stresses are

uiujt

+ Ukuiujxk

Cij

= 2uiujxkxk

Dij

uiujuk

xk DTij

1

ui

p

xj

1

uj

p

xi ij

uiuk Ujxk

ujuk Uixk

Pij

2uixk

ujxk

ij

(5)

where Cij , Dij , D

Tij,

ij, Pij et ij are convection, molecular diffusion, turbulent diffu-

4


5/20

Remi Manceau

Dij DTij

ij Pij ij

u2 2a21 + 12a1a2y 4a21b2y

3 4a1a2y 2A1a1b2y3 2a21 8a1a2y

v2 12b22y2 6b32y

5 4b22y2 0 8b22y

2

w2 2c21 + 12c1c2y 4b2c21y

3 4c1c2y 0 2c21 8c1c2y

uv 6a1b2y 5a1b22y4 2a1b2y A1b22y

4 4a1b2y

Table 1: Asymptotic behaviours of the different terms of the budgets of the Reynolds stresses.

sion, velocitypressure gradient correlation (or redistribution), production and dissipationterms, respectively. Note that ij is called herein redistribution term because it mainlyinduces an energy redistribution among components of the Reynolds stress, but is notonly redistributive, since it is not traceless.

The asymptotic behaviours of these different terms in a channel flow are given in ta-ble 1. It can be seen that, for all the components, production and turbulent diffusion

are negligible. The significant terms are Dij , ij and

ij. Reproducing the balance be-tween these terms as accurately as possible is thus necessary to predict the anisotropy: insimulations, the asymptotic behaviours of the Reynolds stresses are related to the termsinvolving their second derivatives, in particular to molecular diffusion. Therefore, atten-tion must be focused on the behaviour of ij , which is of the same order as D

ij, but also

on ij , which balances the difference between Dij and ij :

ij = ij Dij + O(y

n) (6)

ij is at a higher order than ij and Dij in u

2 and w2 budgets, but at the dominant order

for v2 and uv. In v2 budgets, ij balances ij Dij up to 4

th order (n = 5).

Following Lai & So [11], it must be emphasized that a turbulence model must reproducethis balance in order to be valid down to the wall. In many cases, the correct behaviour ofij is not well reproduced. It is then preferable to built a model for

ij that compensatesfor this shortcoming, by balancing the difference ij D

ij, even if this does not correspond

to the true behaviour of ij.The redistribution term ij is generally split into a traceless part, the pressurestrain

term ij, and the pressure diffusion Dpij:

ij =1

p

uixj

+ujxi

ij

1

xk(uipjk + ujpik)

Dpij(7)

in order to distinguish the purely redistributive effect and the diffusion effect. Beyondthe non-unicity problem it induces [15], it leads to artificial asymptotic behaviours thatare difficult to reproduce. For instance, in v2 transport equation, the behaviours of the

5


6/20


7/20

Remi Manceau

thus allowing to take it outside the integral, which leads to the general form of standardmodels for the rapid term

2ij =Ulxm

(aijml + ajiml) (11)

Now, Bradshaw et al. [2] have shown, by using a DNS database of a channel flow [22], thatthe quasi-homogeneous approximation is valid beyond y+ = 40, but totally wrong below.

Secondly, the locality assumption consists in forgetting that the pressure-strain iswritten as an integral depending on all the points of the domain, and in modelling it byalgebraic expressions, such as (11), depending only on the point x. This is not questionablefrom a mathematical point of view (the value of the integral is only a function ofx), butrather from a physical point of view: this leads to the loss of the non-local character ofthe Reynolds-stress transport equations and thus makes very difficult the reproduction ofkinematic effects, whose importance has been emphasized in 2.1.

These two assumptions are then only applicable in free flows, and the presence of a wallrequires the introduction of strong corrections to the models: wall echo terms, damping

functions, complex non-linear terms. The wall echo terms [8] have the advantage ofreintroducing non-locality into the equations, since they explicitely depend on the distanceto the wall, but they lead to many problems in complex flows, and, above all, they arebased on an erroneous rationalization (cf. [16, 20]). Damping functions are mainly derivedin order to fit particular experimental or DNS data, and have no reason to be valid ingeneral situations. Moreover, many of them are functions of ReT only, which contradictsthe fact, emphasized in 2.1, that a dependence on ReT cannot account for both dynamicand kinematic effects, and they are not sufficient for reproducing the correct anisotropy.Complex non-linear terms also allow to impose elaborate physical constraints, such asrealizability in extreme conditions, but, following Speziale [26], it must be noted thatit should be preferable to avoid the use of quasi-homogeneous and locality assumptions

rather than correcting their consequences by introducing complex terms. Therefore, in thefollowing of this paper, attention is focused on the elliptic relaxation method, introducedby Durbin [6, 7], which is based on a totally different approach.

3 THE ELLIPTIC RELAXATION MODEL

3.1 Derivation of the model

From the solution of the Poisson equation for the fluctuating pressure, the followingintegral equation of the redistribution term ij can be derived:

ij(x) = ui(x)2p

xj(x) uj(x)2

p

xi(x)

ij(x, x)

G(x, x)dV(x) (12)

7


8/20

Remi Manceau

where G(x, x) is the Green function of the domain. Durbin [6] proposed to model the

two-point correlation ij by

ij(x, x) = ij(x

, x)exp

r

L

(13)

where r denotes the distance x x. However, using (13) in (12) leads to the loss ofthe correct asymptotic behaviour of ij. Indeed, it can be seen in (12) that

ij goes tozero when x approaches the wall because of ui and uj, which are expressed in x. Usingthe model (13), ui and uj are turned into functions of x

, and ij has no reason to go tozero anymore. A way of overcoming this problem is to use the model [16]

ij(x, x) = k(x)

ij(x, x)

k(x)exp

r

L

(14)

since the factor k(x) forces ij(x) to go to zero at the wall.Introducing (14) into (12), approximating G by its free-space value GIR3 = 1/4r

and using the fact that GIR3 (x, x

) = exp(r/L)/4r is the Green function associated to theoperator 1/L2 2, the following elliptic relaxation equation is obtained:

ijk

L22ijk

=hijk

(15)

In this equation, a quasi-homogeneous model hij (Rotta+IP, QI, SSG, ...) is used in theright hand side, noting that in homogenous situations, the second term of the left handside vanishes.

The main interest of this approach is that the redistribution term is given by a differ-ential equation, rather than by an algebraic expression as in the standard approach. Theelliptic character of (15) enables the reproduction of the non-locality ofij, which derives

from the model (14) for the correlation function. Moreover, the quasi-homogeneous as-sumption is only used for the modelling of the source term of (15), which is the form themodel tends to far from solid boundaries, at least sufficiently far for being in a region ofthe flow where this assumption is valid. In the vicinity of the wall, the solution of (15) isconstrained by its boundary conditions, as shown in the following section.

3.2 Near-wall behaviour

It has been noticed in 2.2 that the main point a near-wall turbulence model mustsatisfy is the balance between molecular diffusion, dissipation and redistribution. First, itis necessary to pay attention to the behaviour of ij, which must take an anisotropic form

ij in the vicinity of the wall and tend to

2

3ij far from it. A possible way of imposing thisis to decompose ij into

ij and (ij

ij), and to solve for the latter an elliptic relaxationequation similar to (15):

ij

ij

k L22

ij

ij

k=

2

3ij

ij

k(16)

8


9/20

Remi Manceau

By imposing the boundary condition (ij

ij)/k = 0 at solid boundaries, the total dissi-pation ij + (ij

ij) approaches the value

ij in the near-wall region. Far from the wall,the second term in the left hand side of (16) vanishes and the total dissipation tends to2

3ij .

However, since the operator 1 L22 is linear, and since only the difference ij ijappears in the Reynolds-stress transport equations, it is not necessary to solve an ellipticrelaxation equation for ij/k and another for (ij

ij)/k, but only one for their differencefij = (

ij ij +

ij)/k, which yields

fij L22fij =

1

k

hij

2

3ij +

ij

(17)

In this equation, the near-wall model ij = uiuj /k is used. Thus, the modelled Reynolds-stress transport equations become

uiujt

+ Cij = Pij + Dij + D

Tij + kfij

uiujk

(18)

in which DTij is given by the Daly & Harlow [4] model.Thus, according to table 1, in the vicinity of the wall, the budgets of the Reynolds-stress

transport equations reduce to

2uiuj

y2

uiujk

= kfij (19)

Using /k = 2/y2 (cf. Eq. 3 and 4), the solution of this differential equation is

uiuj = A y2 +

B

y

20 2fij y

4 (20)

Since uiuj = 0 is imposed at the wall, B = 0, and the asymptotic behaviour of uiuj de-pends on the boundary condition used for fij .

For v2 and uv, the boundary conditions must be chosen in order to cancel the termA y2 in (20). Therefore, for v2, the condition fw22 = 20

2v2/y4 is used. Note that fw22 isnot singular, since this boundary condition implies that v2 behaves as y4.

For Reynolds stresses behaving as y2 (cf. Eq. 2), any boundary condition can be applied,since the leading term in (20) is in y2. Therefore, fw11 = f

w33 =

1

2fw22 are used, in order

to ensure that fwkk = 0. The reason for imposing fwkk = 0, whereas

ij is not traceless,is that it ensures that the Reynolds-stress transport equations contract to the standardk equation, thus avoiding the need to modify the coefficients of the quasi-homogeneous

model used as the source term.The case of the Reynolds shear stress uv is more problematic. Indeed, (2) shows that

uv behaves as y3, but there is no term containing y3 in (20). Therefore, the same boundarycondition as for f22 is used for f12, i.e, f

w12 = 20

2uv/y4, which induces a behaviour ofuv in y4.

9


10/20

Remi Manceau

fw11 fw22 f

w33 f

w12 f

w13 f

w23

12fw22

202

v2

y412f

w22

202

uvy4

0 202

vwy4

Table 2: Boundary conditions for the components of the tensor fij .

In general cases, the latter boundary condition is easily extended to the component fw23:fw23 = 20

2vw/y4. For fw13, as for fw11 and f

w33, any boundary condition can be applied:

fw13 = 0 is simply used. Table 2 summarizes the boundary conditions used for the differentcomponents of fij .

3.3 Advantages and shortcomings

The main quality of this model is that it reproduces the non-local blocking effect. In-deed, the model for ij ij +

ij approaches progressively its asymptotic form kfwij in the

near-wall region, the transition being ensured by the elliptic operator. In particular, this

enables the stronger damping ofv2, which leads to the two-component state of turbulencevery close to the wall, as shown in Fig. 1(b).

An easy way of understanding the role played by the model is to compare the equationsof v2 and w2. Suppose that the Rotta+IP model is used for hij . In a channel, v

2 and w2

have exactly the same transport equation. Moreover, the elliptic relaxation equations forf22 and f33 are also exactly identical. Thus, if the boundary conditions f

w22 and f

w33 were

the same, v2 and w2 would be exactly equal across the channel. Fortunately, differentboundary conditions are used, which distinguish v2 and w2 and lead to the prediction ofthe correct anisotropy near the wall, shown in Fig. 1(b). Thus, the reproduction of theblocking effect is due to the fact that a differential equation is solved for fij, which enables

the introduction of differences between the components through the boundary conditions,without violating the frame indifference principle. In algebraic models for ij, the onlyway of distinguishing between components is to identify the wall-normal direction, e.g.,by using the invariants of the Reynolds stress tensor, thus introducing non-linearities,which complicate the equations and lead to numerical difficulties.

Moreover, the elliptic relaxation model only uses the quasi-homogeneous assumptionin the source term hij , which becomes active sufficiently far from the wall. This featurecan be illustrated by an a priori test: Eq. (15) is resolved for 22, taking k, L,

hij and

the boundary conditions from the channel flow DNS [23] (for a priori tests, it is moreconvenient to handle separately ij and ij

ij, the interpretation of the result beingmuch easier). Fig. 2(a) shows the results obtained using two different models for hij.

It can be seen that the elliptic relaxation equation enables the correction of the near-wall behaviour of the redistribution, and that the results obtained below y+ = 40 do notdepend on the quasi-homogeneous model used in the source term: the solution is totallydetermined by the boundary conditions. Thus, the quasi-homogeneous model only plays a

10


11/20

Remi Manceau

22

0 100 200 300 400 500 600

0

0.02

0.04

0.06

y+

(a)

22

0 100 200 300 400 500 600

0

0.02

0.04

0.06

y+

(b)

Figure 2: A priori tests of the effect of the elliptic relaxation equation. Channel flow at Re = 590.(a) Original formulation (17); (b) Modified formulation (22).

22from the DNS [23]; h

22given

by the Rotta+IP model (without elliptic relaxation); 22 given by the elliptic relaxation equationwith the Rotta+IP model as the source term; h

22given by the SSG model (without elliptic

relaxation); 22

given by the elliptic relaxation equation with the SSG model as the source term.

significant role further from the wall, where the quasi-homogeneous assumption is valid [2].On the other hand, some shortcomings of the model must be noted. First, it is not,

contrary to what is sometimes believed, a low-Reynolds-number model, but rather anear-wall model. In the near-wall region, all the effects described in 2.1 are reproduced,because the correct asymptotic behaviours are imposed, but low-Reynolds-number effectsfar from a wall, in particular the effect of the overlapping of scales, are not accounted for,since there is no dependence on ReT in the model.

Another point to be brought out is the amplification of the redistribution in the loglayer. It can be seen in Fig. 2(a) that the main effect of the elliptic relaxation equation

is to correct the redistribution in the viscous and buffer layers. However, a side effectcan be noted: in the logarithmic layer, the redistribution ij given by the model is higherthan the one predicted by the quasi-homogeneous model hij. This is actually due tothe operator 1 L22, which has an amplification effect. Indeed, in the log layer, thefollowing behaviours can be assumed: hij = A/y, k = u

2/C1/2 and = u3/y, where

A is some constant value. Thus, the source term behaves as 1/y, and if a solution ofthe form ij =

hij is sought, an amplification factor 1.51 is obtained. With the

Rotta+IP model, which already overestimates the redistribution in the log layer, it wouldbe preferable to have an amplification factor smaller than 1. With the SSG model, whichreproduces correctly ij in this region, a neutral operator, i.e. corresponding to = 1,would lead to better predictions. This issue is examined in 3.4, where a modified elliptic

relaxation equation [20, 21] is presented.Finally, it must be emphasized that the elliptic relaxation approach leads to 6 additional

differential equations in the system, for the 6 independent components of the tensorfij. This makes the model rather unpopular among industry. Therefore, Durbin [6]

11


12/20

Remi Manceau

proposed an eddy-viscosity model that uses an elliptic relaxation equation to reproducethe behaviour of the eddy viscosity in the vicinity of the wall, the so-called v2f model,which is presented in 4.1. In 4.2, a new approach for second moment closures is proposed,based on the same general ideas than the elliptic relaxation, but using only 1 additionalequation.

3.4 Reformulation of the elliptic relaxation equation

Manceau et al. [20, 21] sought the reason for the amplification of the redistributiongiven by the elliptic relaxation operator in the log layer by investigating the validity ofthe basis assumptions of the model using a DNS database. They showed that this problemis due to the approximation (13) of the correlation function, which does not account for itsanisotropy. In particular, by using a model for the correlation function that is asymmetricin the main direction of inhomogeneity, identified by the gradient of the length scale

f(x, x) = exp

r

L + (x x) L

(21)

a new formulation of the elliptic relaxation equation can be derived:

(1 + 16(L)2)fij L22fij 8LL fij = f

hij (22)

This formulation is characterized by the amplification factor

=1

1 + 2(12 1)C2LC3/2

2(23)

Different values of the coefficient can be chosen, depending on the effect required inthe log layer. Fig. 2(a) shows that, when the Rotta+IP model is used for hij , which

overestimates the redistribution in the log layer, a coefficient that leads to a dampingof the redistribution is needed. In Fig. 2(b), the 22 profile obtained by an a priori testwith = 0.25 is shown. It can be seen that the use of (22) enables the correction of thebehaviour in the log layer, without degrading the predictions in the viscous and bufferlayers. With the SSG model, whose results are degraded in the log layer by the originalelliptic relaxation equation (cf. Fig. 2a), it can be seen in Fig. 2(b) that the formulation(22), with = 0.17, gives more satisfactory results than the original one.

Results obtained by the full integration of the Reynolds stress model equations (17)and (18) are plotted in Fig. 3. The SSG model is used for hij . As noted above, theoriginal formulation leads to difficulties in predicting the redistribution in the log layer,and the coefficients of the model (cf. [28]) have been chosen in order to compensate for

this problem. This leads to the underestimation of the peak of u2 and of the mean velocityin the buffer layer that can be observed in Fig. 3. Replacing (17) by the formulation (22),with = 0.083, allows the use of a wider range of coefficients, which enables a betterprediction of the peak of u2 and of the mean velocity in the buffer layer. More detailsabout these simulations can be found in [17].

12


13/20

Remi Manceau

U

+

1 10 100 10000

5

10

15

20

1 1000

5

10

15

20

y+

(a)

ui

uj

+

0 100 200 300 400 500 600

0

2

4

6

8

y+

(b)

Figure 3: Comparison of the results given by the original and modified formulations for the channel flowat Re = 590. The SSG model is used for

hij . (a) Mean velocity profiles. DNS [23]; original

formulation (22); modified formulation (22). (b) Reynolds stresses. Symbols: DNS ( u2, v2, w2, uv); original formulation (22); modified formulation (22).

4 SIMPLIFIED APPROACHES FOR INDUSTRIAL APPLICATIONS

As emphasized in 3.3, since it involves 6 transport equations for the Reynolds stressesand 6 elliptic relaxation equations, the full Reynolds-stress model proposed by Durbin isvery seldom used in industry. In the present section, two simplified models suitable forindustrial applications are presented: the v2f model and the elliptic blending model.

4.1 The v2f model

In industrial applications, k type models are widely used. However, high-Reynolds-number models also need corrections for reproducing the damping of the eddy viscosity inthe near-wall region. Generally, these corrections are based on the use of damping func-tions, which introduce strong non-linearities and thus numerical instabilities. In order toavoid such corrections, Durbin [6] proposed to use the fact, initially noted by Launder [12],that the eddy-viscosity can be well reproduced by the model T = Cv2k/ in a channelflow. Thus, in a channel, by using the same v2 and f22 equations as in the full Reynoldsstress model presented in 3, which gives an accurate prediction of the profile ofv2 in thenear-wall region (cf. Fig. 1b), the correct damping ofT is obtained. This leads to a modelconsisting of 3 transport equations, for k, and v2, and 1 elliptic relaxation equation.

This model can easily be generalised to general geometries, without changing the equa-tions, but only by changing the meaning of the variables: v2 is then no longer a componentof a tensor, but a scalar variable which is given by a transport equation formally identical

to the transport equation of the Reynolds stress component v2

in a channel (changingthe name of the variable would have made this distinction more clear); in a similar way,a new scalar variable called f is introduced, which satisfies the same elliptic relaxationequation as f22 in a channel.

More details about this model can be found in [6] or in [19]. The purpose of this section

13


14/20

Remi Manceau

is only to present some results that illustrate the interest of the model for wall-boundedflows.

In Fig. 4, mean velocity, k and v2 predictions in the case of a channel flow are plotted.Two different sets of results are compared to DNS data. The first one is given by theoriginal v2f model, using the elliptic operator [1 L22] in the f equation, the sec-

ond one by the reformulated model, using [(1 + 16(L)2) L22 8LL ], with = 0.083. It can be seen that the k and v2 profiles are correctly reproduced down to thewall by both formulation. On the other hand, the mean velocity profile is improved by themodified model, which is due to the suppression of the amplification of the redistributionin the log layer noted in 3.3, which enables the use of a wider range of coefficients. Moredetails can be found in [17].

Fig. 5(a) shows results obtained with the original v2f model in a backstep flow atRe = 5, 100, compared with the DNS results obtained by Le et al. [13]. It can be seenthat all the characteristics of the flow that models generally fail to predict are correctlyreproduced by the v2f model: first, the recirculation length, which is predicted with

only 3 % error (6.59h instead of 6.39h); secondly, the intensity of the backflow, andconsequently the friction coefficient (not shown here, cf. [18, 19]); finally, the recovery ofthe boundary layer downstream of the recirculation bubble. These results show the abilityof the model to reproduce separated flows as well as the near-wall turbulence, whichillustrates the interest of this model for industrial complex flows: the most importantquantities are often those which are evaluated at solid boundaries and which are stronglyinfluenced by separation and reattachment, namely the friction coefficient and the Nusseltnumber.

Heat transfer predictions are indeed very sensitive to the level of turbulence and thequality of a turbulence model can be partly evaluated through heat transfer cases. In1998, a comparison between different turbulence models, including low-Reynolds-number

Reynolds stress models, in the case of a periodic ribbed-channel flow, has been proposedfor the 7th ERCOFTAC/IAHR workshop on refined turbulence modelling [27]. Fig. 5(bc)show the velocity profiles and the streamlines obtained at Re = 37, 200. It can be ob-served in Fig. 5(b) that the backflow is correctly reproduced in the recirculation zone.No reattachment point is obtained, the two main recirculation bubbles being actuallyconnected, as can be seen in Fig. 5(c). Two secondary recirculation zones are also pre-dicted. The agreement between model and experiments is far from perfect in the regionabove the ribs (1 y/h 5), but no conclusion can be drawn because 3D effects aresuspected in the experiments (cf. [19]). The Nusselt number distribution obtained be-tween two consecutive ribs at Re = 12, 600 is plotted in Fig. 5(d). The heat fluxes are

simply modelled by a turbulent diffusivity hypothesis. It can be seen that, despite thisvery simple heat flux model, the Nusselt number distribution is well predicted, especiallybetween x/h = 0 and x/h = 3, which shows that, in forced convection, using a turbulencemodel that reproduces correctly the near-wall characteristics is more important than us-ing an elaborate heat flux model. It must be emphasized that the v2f model gave the

14


15/20

Remi Manceau

U+

1 10 100 10000

5

10

15

20

y+

(a)

k

,v2

0 100 200 300 400 500 6000

1

2

3

4

5

y+

(b)

Figure 4: Results given by the v2fmodel in a channel flow at Re = 590. (a) Mean velocity. DNS [23];Original formulation (17); Modified formulation (22). (b) Turbulent energy and v2. Sym-

bols: DNS ( k, v2); Original formulation (17); Modified formulation (22).

y/h

0 10 200

1

2

U/Ub

(a)

y/h

0 2 4 6 80

1

2

3

4

5

U/Ub

(b)

y/h

x/h

(c)

Nu/Nus

0 1 2 3 4 5 60

1

2

x/h

(d)

Figure 5: Results given by the v2

f model (original formulation) in two separated flows. (a) Backward-facing step at Re = 5, 100. Mean velocity profiles. DNS [13]. v2f. (b) Ribbed-channel flowat Re = 37, 200. Mean velocity profiles. Experiments [5]; v2f. (c) Ribbed-channel flow atRe = 37, 200. Streamlines. (d) Ribbed-channel flow at Re = 12, 600. Heat transfer enhancement Nu/Nus,Nus being the Nusselt number for a turbulent flow in a smooth circular pipe, given by the Dittus-Boeltercorrelation: Nus = 0.023Re

0.8Pr0.4. Experiments [14]; v2f.

15


16/20

Remi Manceau

best results among all the models used by different teams participating to the workshop(cf. [27]). The present computations with the v2f model have been detailed in [18, 19].

4.2 The elliptic blending model

The limitations of the turbulent viscosity approach make necessary the use of full

Reynolds-stress models in number of industrial cases. It has been shown in 3 thatDurbins Reynolds stress model is a very interesting approach for wall-bounded flows, butremains quite unpopular because of the 6 additional equations it implies.

The main quality of this model, i.e., the reproduction of the blocking effect, is dueto the fact that the elliptic relaxation equations ensure a smooth transition between thefar-from-the-wall form hij of the redistribution term

ij, and its near-wall asymptoticvalues. Thus, it can be noted that each elliptic relaxation equation provides a transitiondepending only on the geometry and of the length scale L, which is the same for all thecomponents. Thus, it appears that these 6 equations are somewhat redundant. The sameeffect can be reproduced by a blending function , which ensures the transition betweenfar-from-the-wall and near-wall forms in the following manner:

ij = (1 k) wij + k

hij (24)

where k must be zero at the wall and must tend to 1 far from it. The geometrical effectcan be reproduced by using an elliptic relaxation equation for :

L22 =1

k(25)

with the boundary condition = 0 at the wall. The reason for using 1/k in the sourceterm of (25) and then multiplying by k in (24) is that it ensures that the factor kbehaves as y3 in the vicinity of the wall, which makes the second term in the right handside of (24) negligible in this region.

To obtain the same effect as with Durbins model, wij must be chosen such that thebalance between viscous diffusion, dissipation and redistribution in the near-wall regionis respected, as shown in 3.2. Thus, wij/k must tend to the values f

wij given in table 2.

This can be achieved in a channel by

w11 = 1

2w22 ;

w22 = 5

kv2 ; w33 =

1

2w22 ;

w12 = 5

kuv (26)

since /k tends to 2/y2 (cf. equations 3 and 4). In a similar way, for the dissipationtensor, the transition between uiuj/k and

2

3ij is ensured by the same blending method:

ij = (1 k)uiuj

k + k

2

3ij (27)

Fig. 6 shows the results obtained with this model in a channel flow, the Rotta+IP modelbeing used for hij. The results given by the full Reynolds stress elliptic relaxation model

16


17/20

Remi Manceau

U

+

1 10 1000

5

10

15

20

y+

(a)

uiu

j+

0 100 200 300 400 500 600

0

2

4

6

8

y+

(b)

Figure 6: Comparison in a channel flow at Re = 590 between the results given by the elliptic blendingmodel and by the elliptic relaxation model. (a) mean velocity. DNS [23]; elliptic relaxationmodel; elliptic blending model. (b) Reynolds stresses. Symbols: DNS [23] ( u2, v2, w2, uv); elliptic relaxation model; elliptic blending model.

(original formulation), already shown in Fig. 3, is also plotted for comparison. It can beseen that the results of the present model are almost as good as those given by the fullelliptic relaxation model. Indeed, even if the prediction of the Reynolds stresses is slightlyless accurate, the anisotropy is globally correctly reproduced, and it must be kept in mindthat this model contains only 8 equations (taking equation into account), instead of13 for Durbins model. This reduction of the number of equations is very interesting forindustrial applications.

The model needs to be generalised to complex geometries, since (26) is not frameindifferent. Therefore, some directional information must be introduced in a general

tensorial form of (26). In order to avoid the use of geometry-related quantities, such asthe wall-normal vector, which is ill-behaved in complex geometries, it can be noted thatthe gradient of is generally normal to the wall, and it is still defined inside the domain.Thus, a vector n can be defined by n = /. With this definition, (26) can begeneralized by

wij = 5

k

uiuk njnk + ujuk nink

1

2ukul nknl ninj

1

2ukul nknl ij

(28)

which is applicable to any geometry.

5 CONCLUSION

The importance of modelling the different effects of the wall on turbulence has beenemphasized. In particular, the blocking effect, which is non-local and which is felt bythe Reynolds stresses through the redistribution term, requires a particular attention.In the vicinity of a wall, this term balances the difference between molecular diffusionand dissipation, and respecting this balance is indispensable for predicting the correct

17


18/20

Remi Manceau

near-wall anisotropy.Standard models for the redistribution term, based on quasi-homogenous and local-

ity assumptions need strong corrections for being integrable down to solid boundaries,and the reproduction of the two-component limit of turbulence requires highly nonlinearterms. On the contrary, Durbins elliptic relaxation model, which is not based on the

same assumptions, does not need nonlinear corrections and predicts accurately the stronganisotropy in the near-wall region.

Its main shortcoming, the amplification of the redistribution in the log layer by theelliptic operator, has been corrected by accounting for the anisotropy of the two-pointcorrelations involved in the integral equation of the redistribution term. The modifiedelliptic relaxation equation which is then derived gives better predictions of the meanvelocity profile and of the peak of u2 in a channel flow.

However, Durbins full Reynolds-stress model is very seldom used in industrial com-putations, since it involves 6 transport equations for the Reynolds stresses and 6 ellipticrelaxation equations. Two simplified models, suitable to industrial applications, have

been presented.First, the v2f model, which is based on the turbulent viscosity concept, but reproducesthe damping of the latter by elliptic relaxation. Results presented in the cases of flows ina channels, over a backward-facing step and in a periodic ribbed-channel with one heatedwall, show the ability of the model to reproduce accurately the near-wall turbulencecharacteristics, and in particular the friction coefficient and the Nusselt number, whichare of primary interest for industry.

Secondly, the elliptic blending model has been proposed. This new model is a fullReynolds-stress model derived in order to reproduce the main features of the ellipticrelaxation model, but with only 1 additional equation, instead of 6. The results obtainedin a channel flow show the ability of this model to reproduce the anisotropy in the near-

wall region.

Acknowledgments

A part of the work presented in this paper has been supported by Electricite de France,during the Ph.D. thesis of the author. In particular, the simulations of the backstep andribbed-channel flows have been performed with EDF finite element code N3S.

18


19/20

Remi Manceau

REFERENCES

[1] D. Aronson, A. V. Johansson, and L. Lofdahl. Shear-free turbulence near a wall. J.Fluid Mech., 338:363385, 1997.

[2] P. Bradshaw, N. N. Mansour, and U. Piomelli. On local approximations of the

pressurestrain term in turbulence models. In Proc. of the Summer Program, pages159164. Center for Turbulence Research, Stanford University, 1987.

[3] P. Y. Chou. On velocity correlations and the solutions of the equations of turbulentfluctuation. Quart. of Appl. Math., 3:3854, 1945.

[4] B. J. Daly and F. H. Harlow. Transport equations in turbulence. Phys. Fluids,13:26342649, 1970.

[5] L. E. Drain and S. Martin. Two-component velocity measurements of turbulentflow in a ribbed-wall flow channel. In Intl Conf. Laser Anemometry-Advances andApplications, pages 99112, 1985.

[6] P. A. Durbin. Near-wall turbulence closure modeling without damping functions.Theoret. Comput. Fluid Dynamics, 3:113, 1991.

[7] P. A. Durbin. A Reynolds stress model for near-wall turbulence. J. Fluid Mech.,249:465498, 1993.

[8] M. M. Gibson and B. E. Launder. Ground effects on pressure fluctuations in theatmospheric boundary layer. J. Fluid Mech., 86(3):491511, 1978.

[9] J. C. R. Hunt and J. M. R. Graham. Free-stream turbulence near plane boundaries.J. Fluid Mech., 84(2):209235, 1978.

[10] A. N. Kolmogorov. The local structure of turbulence in incompressible viscous fluidfor very large Reynolds numbers. Dokl. Aked. Nauk., URSS, 30:299303, 1941.

[11] Y. G. Lai and R. M. C. So. On near-wall turbulent flow modelling. J. Fluid Mech.,221:641673, 1990.

[12] B. E. Launder. On the computation of convective heat transfer in complex turbulentflows. J. Heat Transfer, 110:11121128, 1988.

[13] H. Le, P. Moin, and J. Kim. Direct numerical simulation of turbulent flow over abackward-facing step. In Proc. Ninth Symp. Turb. Shear Flows, 132, pages 16.Kyoto, 1993.

[14] T.-M. Liou, J.-J. Hwang, and S.-H. Chen. Simulation and measurement of enhancedturbulent heat transfer in a channel with periodic ribs on one principal wall. Intl J.Heat Mass Transfer, 36:507517, 1993.

19


20/20

Remi Manceau

[15] J. L. Lumley. Pressure-strain correlation. Phys. Fluids, 18(6):750, 1975.

[16] R. Manceau. Modelisation de la turbulence. Prise en compte de linfluence des paroispar relaxation elliptique. PhD thesis, Universite de Nantes, 1999.

[17] R. Manceau and K. Hanjalic. A new form of the elliptic relaxation equation toaccount for wall effects in RANS modelling. To be published in Phys. Fluids.

[18] R. Manceau and S. Parneix. Computations of turbulent flows using the v2f modelin a finite element code. In Proc. Fourth Intl Symp. on Engng Turbulence Modellingand Measurements, pages 319328, 1999.

[19] R. Manceau, S. Parneix, and D. Laurence. Turbulent heat transfer predictions usingthe v2f model in a finite element code. Intl J. Heat and Fluid Flow, in press.

[20] R. Manceau, M. Wang, and D. Laurence. Inhomogeneity and anisotropy effects onthe redistribution term in RANS modelling. To be published in J. Fluid Mech.

[21] R. Manceau, M. Wang, and D. Laurence. Assessment of inhomogeneity effects onthe pressure term using DNS database: implication for RANS models. In Proc. FirstIntl Symp. Turb. Shear Flow Phenomena, 8, pages 239244. Santa Barbara, 1999.

[22] N. N. Mansour, J. Kim, and P. Moin. Reynolds-stress and dissipation-rate budgetsin a turbulent channel flow. J. Fluid Mech., 194:1544, 1988.

[23] R. D. Moser, J. Kim, and N. N. Mansour. Direct numerical simulation of turbulentchannel flow up to Re = 590. Phys. Fluids, 11(4):943945, 1999.

[24] J. B. Perot and P. Moin. Shear-free turbulent boundary layers: physics and modeling.Technical report, Dept Mech. Engng, Stanford University, 1993.

[25] J. C. Rotta. Statistische Theorie nichthomogener Turbulenz. Zeitschrift fur Physik,129:547572, 1951.

[26] C. G. Speziale. A review of Reynolds stress models for turbulent shear flows. Tech-nical Report 95-15, ICASE, NASA, 1995.

[27] UMIST, Manchester. Proc. Seventh ERCOFTAC/IAHR workshop on refined turbulencemodelling, 1998.

[28] V. Wizman, D. Laurence, M. Kanniche, P. Durbin, and A. Demuren. Modeling near-wall effects in second-moment closures by elliptic relaxation. Intl J. Heat and FluidFlow, 17(3):255266, 1996.

20

2000, manceau, reproducing the blocking effect of the wall in one-point turbulence models2

Documents