2 A NEW FORMULATION OF THE v " f MODEL USING ELLIPTIC BLENDING AND ITS APPLICATION TO HEAT TRANSFER PREDICTION
F. Billard1, J.C. Uribe1 and D. Laurence2
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1
School of Mechanical, Aerospace and Civil Engineering, the University of Manchester, Manchester M16 1QD, UK
2
EDF R&D, MFEE dpt. 6 quai Watier, 78400 CHATOU, FRANCE
[email protected]
Abstract
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The present paper proposes a new uncoupled formulation of the v 2 " f model that addresses numerical issues usually arising with such models without any loss in prediction accuracy. The model is based on elliptic blending, and unlike its “codefriendly” !predecessors, it correctly predicts nearwall asymptotic behaviours without impairing its overall robustness, which brings it to the same level of stability as most standard eddy viscosity models used in industrial applications. The model is tested in cases featuring heat transfer when buoyant forces strongly influence the turbulence regime, and its anisotropy. The case of air flowing upwards inside a heated infinite pipe and a heated channel are studied. The different formulations of the v 2 " f model are compared with the direct numerical simulations given in You et al. (2003) and Kasagi and Nishimura (1997), respectively for the pipe and the channel. With the new ! formulation, not only the overall predictions are improved, but also turbulence damping due to buoyancy is remarkably well predicted. Moreover, in case of relaminarisation induced by mixed convection regime, numerical robustness of the computation strongly depends on the formulation used for the v 2 and f equations, and the way near-wall balance of the leading order terms is handled. The new formulation presented herein proved robust and more capable of reproducing the relaminarisation.
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1
Introduction
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nents. The v 2 " f model was successfully used in a wide range of applications (impingement, separation, buoyancy), where near-wall phenomena are important (friction, heat transfer…). In many flows of! moderate complexity, it showed to be an economic yet accurate alternative to the more complex Reynolds Stress Models. Good predictions were reported not only in aerospace industry application, with the case of a subsonic and transonic flow around an airfoil (Kalitzin (1997)), but also in aero acoustics (Moreau et al. (2001)) or with heat transfer in a gas turbine (Kalitzin and Iaccarino (1999)). However elliptic blending models have not been tested thoroughly for cases where buoyant forces strongly affect turbulence and low values for turbulent quantities appear in large portions of the domain. In spite of good results reported, researchers have constantly been facing numerical issues in segregated solvers mainly due to the stiffness of the boundary conditions, this problem usually being addressed by a modification of the equations or by neglecting a term, at the expense of the prediction accuracy.
2
Mathematical analysis and rationale
In the original version of the model, proposed in Durbin (1991), the equations for the variables v 2 and f read:
Dv 2 v2 $ -' % !* $ v 2 0 = kf " # + /)% + t , 2 Dt k $x j /.( & k + $x j 21
(1)
The v 2 " f model, first introduced by Durbin C1 #1 $ v 2 2 ' P 2 (2) && # )) # C2 L "f # f = (1991) is an appealing near-wall eddy viscosity T % k 3( k model, as it does not require any damping function ! or any explicit use of a geometrical parameter, such The elliptic blending has as its first objective to !as the wall distance. The elliptic equation used to reproduce wall-blocking effects characterized by a represent the pressure-strain redistribution between !strong damping of the ratio v 2 /k , leading to the twoReynolds stresses accounts for non-local effects, dimensional trend of the near-wall turbulence. The specifically the wall-induced reduction of energy transfer from longitudinal to wall-normal compowall-normal Reynolds stress v 2 behaves as y 4 near !
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the wall and this can hardly be reproduced by boundary conditions given to the transport equation ruling
same requirement of predicting a wall-normal Reynolds stress two orders below the kinetic energy, the use of " instead of v 2 reduces the order of leading terms. The boundary limit given to f becomes:
v 2 . This is achieved by the wall limit given to the
source term k " f , where f represents the redistribution due to pressure-strain correlation. This term is 2
required to balance leading order terms "#v /k and 2 "#v ! which are !quadratic near the wall, hence the limit value of f that reads:
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Lien and Durbin (1996) proposed a change of variable f = f + g , and solved the elliptic equation for f with homogeneous boundary conditions. The function g is chosen such that it has the same wall-
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2
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!
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! limit as f , g = "5 v 2 # . Doing so makes the term
k 2 appear in the f equation, which is neglected. L "g ! This version (denoted herein LDM) proved numeri! cally more stable than the original formulation, and ! the near-wall balance of leading order terms in the ! is correctly predicted thanks to the g v 2 equation term that is handled implicitly. However the term neglected in the f equation was shown in Laurence et al. (2004) to have a significant incidence on the results as this term was as large as the ! other source terms of the f equation. ! Laurence et al. (2004), Hanjalić et al. (2004) and Hanjalić et al. (2005) alternatively proposed to ! solve a transport equation for the variable " = v 2 /k
! instead of v 2 to reduce the stiffness of the near-wall problem. The equation for " , derived from those for !
!
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k and v 2 , reads:
! & D" " 2 $ ) ,k ," = f # P + ($ + t + Dt k k' % k * ,x j ,x j
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(4)
, -& $ ) ," 0 /(($ + t ++ 2 + ,x j /.' %" * ,x j 21
Near the wall, the balance between leading order terms of equation (4) (in a steady state) becomes:
& 2 $k $% $ 2% ) lim( f + # +# 2 +=0 y "0' k $y $y $y *
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(5)
Using the Taylor series expansion k = O(y 2 ) and " = O(y 2 ) as y " 0 , terms of (5) are: ! 2! #k #$ "$ # 2$ "$ " % 8 2 and " 2 % 2 2 . To meet the! k #y #y y #y! y
! !
! !
f w = "10#
$ y2
(6)
! This is now only a balance between quadratic terms. !
"20# 2! v2 (3) fw = $w y 4 This!is evaluated at the first cell off-wall and involves a balance between terms whose order is O(y 4 ) , unavoidably leading to numerical instabilities ! when used with a segregated solver. This numerical issue has been addressed in two different ways.
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The model version proposed in Hanjalić et al. (2004) uses! a boundary condition for f similar to (6), but the cross-diffusive term is neglected to enhance the numerical robustness of the resolution. In fact, the limit of the cross-diffusive term may be difficult to reproduce, because ! this term is taken explicitly. In Hanjalić et al. (2004) this term is not considered in equation (4) and in this case f only has to bavlance the molecular diffusion. However the budget of the " equation in a channel flow shows that the cross-diffusive term is of the same order as the turbulent transport of!" (Hanjalić et al. (2005)), and another strategy should be found, which would ! allow this term to be considered. Moreover, the stiffness of the boundary condition, yet reduced, still ! strongly impairs the numerical robustness when a segregated solver is used. Even if this boundary condition is of the same nature as the one given to " , the ellipticity of the operator leads to a strong sensitivity of the results to any ill prediction of the ratio (6). In practice, the time step had to be severely reduced, compared to the one used ! with industrial purpose eddy viscosity models. In Laurence et al. (2004), the reduced variable is used and the leading order term balance is handled by a change of variable, using the same methodology as in Lien and Durbin (1996). The elliptic operator solves the variable f defined as:
"
2 #k #$ # 2$ f = f+ " +" 2 k #y #y #y
(7)
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As before, a term is neglected in the f equation, but Laurence et al. (2004) showed it is about two orders of magnitude smaller than the one neglected ! in Lien and Durbin (1996). The ! main drawback of this version (denoted herein as " # f ) is that in this attempt of taking off all the leading order terms, the molecular diffusion is not solved in an implicit way anymore. Doing so then uncouples the variable and ! its boundary condition when the turbulent viscosity tends to zero. This leads to an erroneous behaviour of " near the wall, hence an over prediction of v 2 and " t in that region, especially in meshes with fine resolution near the walls. ! In light of the problems encountered in the three versions previously detailed, one can draw the conclusion that a zero boundary condition would be
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preferred for the elliptically relaxed variable. There is then no other option but to use a change of variable: f = f + g which ensures the leading order balance to be handled implicitly by the extra source term g in the v 2 (or " ) equation. This strongly improves the numerical robustness but comes un! avoidably with the necessity to neglect the term L2"g in ! the f equation, increasingly departing from ! ! the original and ! theoretically founded pressure-strain equation and reducing the quality of the predictions. Therefore, numerical solutions brought to make the ! model of Durbin “code-friendly” are always done at the expense of prediction accuracy.
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1$ P '$ 2' f h = " & C1 "1+ C2 )&* " ) T% # (% 3( fw = "
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(14)
#$ k
(15)
$k $ # '1/ 2 ' T = max&& ,CT & ) )) %"( ( %"
(16)
% 3/2 % $ 3 (1/ 4 ( k ' L = CL max' ,C# ' * ** &") ) & "
(17)
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Proposal for a new version
The present formulation uses the simplified approach of the elliptic relaxation, namely elliptic blending, introduced in Manceau and Hanjalic (2002) and Manceau (2004) applied to Reynolds Stress Modelling. This is adapted here in order to strengthen the coupling between " and its source term, using a blending coefficient " , ranging from 0 at the wall to 1 far from it, which is ruled by the same modified Helmholtz equation. The source term f is defined as a blending between two different forms as follows: ! ! (8) f = (1" # P ) f w + # p f h The functions f w and f h are defined, respectively, as the model for f at the wall and away from the!wall. The part of the cross diffusion term associated ! ! to the molecular viscosity has been split as follows:
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D" " = (1# $ p ) f w + $ p f h # P Dt k (13) . 1 ' * ' * 2 % -k -" % -" 0)% + t ,, 3 + )$ p% + t , + k( & k + -x j -x j -x j 0/)( &" + -x j 32
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' 1* C#"1 = C# 1)1+ A1 (1$ % p ) , &+ (
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(18)
" t = Cµ k#T (19) ! With the ! following set of constants:
C"1
C" 2
A1
Cµ
C1
C2
1.44
1.83
0.04
0.22
1.7
1.2
2 #k #$ ! p $( 2 #k #$ ! (9) ! "# ! " k! " = (1% & ) ' 4 + & p " k #x j #x j k k #x j #x j 1.22 1
p 3
"#!
C!L
C !"
CT
1
0.161
90
6
The near-wall part of this term, 4"# /k , is included into
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! the ! f w , which becomes: f w = " #$ . As for k
model used in f h , Laurence et al. (2004) used a ! LRR-IP model (as in the original v 2 " f ), whereas Hanjalić et al. (2004) ! preferred a SSG model since it was shown to give a better prediction of the redis! tribution in the logarithmic layer. The latter choice is adopted here, using the!same model as in Hanjalić et al. (2004). The complete formulation is given subsequently.
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Equations of the
" #$
model:
D" C"#1P $ C" 2" % .( & + %" 1 = + 0*& + t 3 Dt T %x j /) ' " , %x j 2
! !
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!
(10)
Dk $ -' % * $k 0 = P "# + /)% + t , 2 Dt $x j .( & k + $x j 1
(11)
L2"# $ # = $1
(12)
2 This ! ! formulation !of the !v " f is a simple eddy viscosity version of the Reynolds Stress Model described in Manceau (2004), but constants have been modified due to the use of the reduced variable " . In ! particular, the power p , originally set to 2 in Manceau (2004), needed to be changed in order to comply with the requirement of leading order terms balance in the " equation. As already!pointed out in ! if the length scale (17) were supManceau (2004), posed to be a constant, the simple solution would be found for a boundary layer: " (y) = 1# exp(#y /L) , ! is linear at the wall. Choosing an exponent p which equal to 2 would lead the term " p f h to be involved in the near-wall equilibrium, which is not its purpose. ! Thus requiring the exponent p to be chosen at least ! equal to 3. ! As a preliminary test case, the flow between two infinite parallel walls is studied, for which DNS re! (1999) are available, at Reynolds sults of Moser et al. number 395 (based on channel half width and friction velocity). As shown in figure (1) the near-wall as-
ymptotic behaviour of v 2 is correctly reproduced whereas there is an over prediction done by the " # f , for which its near-wall values can be ten times as large as ! the DNS. !
than for the aiding flow, for which the logarithmic region no longer exists. This overall trend is well reproduced by all models, both in the k and the mean velocity profiles, as it can be seen from figure (2). It should be noted however that the SST model strongly under predicts the turbulent kinetic energy in the opposing flow.
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Figure 1: Channel flow, Re* = 395 . Comparisons of the different formulations of the v 2 " f . Profiles and near-wall zoom of v 2 .
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4 Application to forced and natural ! ! convection in a vertical channel
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The second case studied is a flow through a heated channel, the two walls being kept at different temperatures, and a constant pressure gradient drives the flow upwards. The gravity force induced by the density gradients acts upwards (buoyancy aiding flow) near hot wall, and downwards (buoyancy opposing flow) at the other side. The channel is infinitely long, enabling to use periodic boundary conditions. The Reynolds number considered, based on the channel halfwidth and the friction velocity averaged on the two walls, is 150 and the Grashof number, defined with channel width and the temperature difference, is 9.6 "10 5 . A direct numerical simulation was performed in Kasagi and Nishimura (1997). The industrial finite volume Code Saturne (Archambeau et al. (2004)) was used to perform a comparative study of the different v 2 /k based models previously presented, as well as the LDM and the SST model of Menter (1994). With buoyancy effects, the turbulence level is higher for the opposing flow ! !
Figure 2: Mean velocity profile, turbulent kinetic energy, and tur2
bulent anisotropy v /k , in wall coordinates. Near-wall close-ups are also shown for the last graph.
The main difference between v 2 " f based ! models lays in the prediction of the near-wall “wallnormal” anisotropy represented by the quantity v 2 /k already mentioned in the previous section, ! whose prediction is the aim of v 2 " f models. As
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pointed out in Kasagi and Nishimura (1997), this anisotropy is enhanced in the aiding flow,. At both walls, the " # f expectedly over predicts this quantity (linear instead of quadratic), due to the removal of the molecular diffusion in the " equation. With!this model, the velocity profile in the opposing flow shows a logarithmic layer that starts far too early, due to the resulting over prediction of the turbulent viscosity. ! In the aiding flow, the LDM strongly over predicts the quantity v 2 /k . This could be due to the fact that the model solves the two quantities v 2 and k separately, making more difficult to correctly predict their ratio. ! However this over prediction does not seem to have an effect on the velocity profile. The ! of the " # $ gives the correct asymptotic behaviour ! anisotropy at both sides and the velocity profile is satisfactorily predicted.
the same buoyancy parameter as the DNS, whereas the two other models predict it somewhat later (as many other near-wall models).
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5 Mixed and natural convection in a heated pipe The upward flow of air inside a heated pipe has been studied for different values of the heat flux. The Reynolds number based on the friction velocity and the pipe radius is 180, and the flow is studied for different values of the parameter Gr /Re 2 varying from 0.063 (mixed convection but with dominant forced convection) to 0.4 (mixed convection with dominant natural convection), corresponding to different values ! of the heat flux imposed at the wall. Re and Gr re! spectively stand for the Reynolds and the Grashof number defined with the bulk velocity and the heat flux. A DNS of this case was performed in You et al. (2003), and recently the case was extensively studied in Keshmiri et al. (2008) and Kim et al. (2008), where RANS models were compared. As noticed in You et al. (2003), when the heat flux increases, the heat transfer, characterized by the Nusselt number Nu first decreases, when a relaminarisation state is reached (for Gr /Re 2 =0.087), then Nu starts to increase again.
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All “code friendly” formulations of the v 2 " f as well!as the SST model are!compared to the DNS results, clearly showing strong differences between models in correctly predicting the sudden decrease of ! turbulent kinetic energy due to relaminarisation. As 2 shown on the k /U B (where U B is the bulk velocity) profiles in figure (3), the " # $ is the only model able to capture the correct level of turbulence for the three regimes, models still giving a turbulent re! the other ! gime for Gr /Re 2 =0.087, characterized by the peak of ! k they erroneously predict. In fact, the three v 2 " f models predict relaminarisation for different values of the buoyancy parameter (defined with the Rey! nolds and the Grashof number as explained in Keshmiri (2008)). As it can be seen!in figure (4) the " # $ model is able to predict relaminarisation for
Figure 3: Turbulent kinetic energy profiles across the pipe, for
Gr /Re 2 = 0.087, 0.241 and 0.400 (from top to bottom). No DNS data is available for the last value.
Figure 4: The Nusselt number as a function of the Buoyancy parameter Bo . RANS models are compared to the DNS of You et al. (2003) (DNS data are available for Buoyancy parameter corresponding to Gr /Re 2 = 0.063, 0.087 and 0.241)
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Finally, the SST model did not capture any ! relaminarisation, the Nusselt number only slightly varying in the mixed convection (figure (4)). It is worth noting the different models were not equally stable as computations were carried out, especially when very low values of the turbulent variables were reached. With segregated solvers, such as Code Saturne, the different terms in the transport equations can become very small. Using a time step close to Kolmogorov timescale would enable to retain positivity of the variables even in the presence of
unbalanced source terms, but for industrial applications, the model must be able to run in transient mode with time steps preferably related to the Courant number, not the turbulence model. Numerical instabilities were experienced with the LDM even with much smaller time steps, because of the low magnitude of the v 2 variable. Although the " # f model does not solve variables of such near-wall high y + order, numerical issues were experienced due to the erroneous values of the anisotropy in transients. ! ! Moreover the cross-diffusive term could impede ! convergence in some instances. The " # $ model did not show any problem in converging, this being the result of the use of the reduced variable " along with a formulation of the !equations that ensures the correct prediction of the near-wall asymptotic behaviour.
! Conclusion In the present work a new formulation of the v " f model for a segregated solver has been presented. The formulation addresses both the numerical issues of the original v 2 " f model and inaccuracies of previous formulations that attempt to make it more “code-friendly” by redefining the boundary condition for f . The model is based on the notion of elliptic ! it proved to retain correct the asympblending and totic behaviour near the wall. It also proved to be less numerically sensitive to cases where the turbu! lent variables reach small values inducing instabilities. The new formulation was also capable of reproducing the relaminarisation in the case of the heated pipe at the correct Gr /Re 2 . The computational resources (time step, mesh) needed for the new formulation are comparable to those used by standard industrial models (e.g. k " # ! SST model) with an improvement of the predicting capabilities. 2
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References
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Archambeau, F., Mechitoua, N. and Sakiz, M. (2004), A finite volume method for the computation of turbulent incompressible flows – industrial applications, Int. J. Finite Volumes (electronic journal) Durbin, P.A. (1991), Near-wall turbulence closure modelling without damping functions, Theoretical and Computational Fluid Dynamics, Vol.3, pp.113 Hanjalić, K., Laurence, D., Popovac, M. and Uribe J.C. (2005), v 2 /k " f turbulence model and its application to forced and natural convection, Engineering Turbulence Modelling and Experiments, Vol.6, pp.67-86 ! K., Popovac, M. and Hadžiabdić, M. Hanjalić, (2004), A robust near-wall elliptic-relaxation eddyviscosity turbulence model for CFD, Int. J. Heat and Fluid Flow, Vol.25, pp.897-901
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