(υ 2 /k) f TURBULENCE MODEL AND ITS APPLICATION TO FORCED AND NATURAL CONVECTION

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1 (υ /k) f TURBULENCE MODEL AND ITS APPLICATION TO FORCED AND NATURAL CONVECTION K. Hanjalić, D. R. Laurence,, M. Popovac and J.C. Uribe Delft University of Technology, Lorentzweg, 68 CJ Delft, Nl UMIST, PO Box 88, Manchester M6 QD, UK EDF-DER-LNH, 6 quai Watier, 784 Chatou, France ABSTRACT We present the rationale and some validation of a version of Durbin s elliptic relaxation eddy-viscosity model, which solves a transport equation for the velocity scale ratio υ /k instead of υ. The new model, developed independently at TU Delft ([]) and UMIST ([]) in two variants, shows improved robustness, faster convergence and less sensitivity to grid nonuniformities. The two variants differ insignificantly in the formulation of the υ /k and f equations: the UMIST model endeavours to remain close to Durbin s original one, while TUD variant introduces quasi-linear pressure strain formulation but with some further numerically beneficial simplifications. The model validation in a range of attached, separating and impinging flows with heat transfer, as well as in natural convection in a tall cavity, showed satisfactory predictions in all cases considered. INTRODUCTION Since its appearance, the υ f model of Durbin (99) has attracted substantial interest both among academia and industry because of its simple formulation, plausible physical rationale and a range of successful applications. The introduction of an additional ( wall-normal ) velocity scale υ and of elliptic relaxation to sensitize υ to the inviscid wall blocking effect, are the main features of the model. Whilst the υ f model is far from providing a panacea for all situations (highly anisotropic three-dimensional flows may still need a full second-moment or higher-order nonlinear eddy-viscosity approach), it offers an interesting model option, superior to most common eddy-viscosity models. It seems especially suited for predicting near-wall phenomena - friction and heat transfer in not-so-complex flows. This has been demonstrated in successful predictions of several wall-bounded flows with heat transfer, featuring impingement, separation and buoyancy. However, the υ f model poses some numerical difficulties arising from the stiffness of the boundary condition, which make it less attractive for industrial applications. The source of stiffness is associated with the wall boundary condition f w = lim y υ ν /(εy 4 ), which makes the computations sensitive to the near-wall grid clustering especially when the first grid point is placed at too small y +. The problem Authors names are listed alphabetically

2 can be obviated by solving simultaneously the υ and f equation, but most commercial as well as inhouse codes use more convenient segregated solvers. Alternative formulations of the υ and f equations have been proposed by Lien and Durbin [] (hereafter LDM), which permit f w =, but these perform less satisfactorily than the original model [], [7]. In order to obviate the numerical stiffness whilst still retaining the major features of the Durbin s elliptic relaxation concept, a new model formulation was recently proposed independently by Hanjalic et al. [] and Laurence et al. [], which solves a transport equation for the velocity scale ratio υ /k instead of the equation for υ. Two model versions differ slightly in some details and coefficients, but their performances are very similar: both versions predict a range of generic flows with a quality comparable with - and in some cases better than - the original Durbin s model. Because of a more convenient formulation of the equation for υ /k and especially of the wall boundary condition for the elliptic function f, the model is less sensitive to small values of y +, what makes it more robust and less sensitive to nonuniformities and clustering of the computational grid. Other advantages of this formulation is the absence of ε from the transport equation for v /k. This makes this equation uncoupled from the usually troubling ε equation, what further contributes to the model robustness. It is also noted that the υ /k is nondimensional and bounded between the zero value (at a solid wall) and its isotropic value of /. The new model has so far been validated in a boundary layer on a flat plate, fully developed channel flow at several Re numbers, pulsating channel flow, impinging jets with heat transfer at different Re numbers, flow and heat transfer behind a backward-facing step, flow in an asymmetric diffusor and over a periodic hill and natural convection in a tall cavity. We present here the rationale and a summary of the two model variants, and illustrate their performance with a series of new validations. THE RATIONALE In Durbin s υ f model, the eddy viscosity is defined as ν t = C µ υ τ (where τ is the turbulence time scale), and evaluated by solving the conventional transport equations for the turbulence kinetic energy k and its dissipation rate ε in the form Dk Dt = P ε + [( ν + ν ) ] t k () x j Dε Dt = C εp C ε ε + τ x j σ k x j [( ν + ν t σ ε ) ] ε plus the transport equations for υ and a Helmholz-type elliptic equation for the relaxation function f, which introduces the effects of pressure-strain correlations as in a second-moment closure. A transport equation for the ratio (υ /k) can be derived directly from the υ and k equations of Durbin (99). The direct transformation yields: [( ) ] D(υ /k) = f (υ /k) P + ν + ν t (υ /k) + X () Dt k x k x k σ (υ /k) where the cross diffusion X is a consequence of transformation ( ) X = ν + ν t (υ /k) k (4) k x k x k σ (υ /k) The solution of the (υ /k) equation () instead of υ should produce the same results. However, from the computational point of view, the following advantages can be identified: x j ()

3 - In the original υ equation, the sink is represented by ευ /k, which is difficult to reproduce correctly in the near-wall layer as ε becomes large and υ /k tends to zero as the wall is approached; instead, in the (υ /k) the sink term contains the turbulence energy production P, which goes smoothly to zero at the wall leaving only the stable viscous diffusion as dominant near wall mechanism. - Because (υ /k) y when y, the wall boundary condition for (υ /k) deduced from the budged of (υ /k) equation in the limit when the wall is approached, reduces to the balance of only two terms with a finite value at the wall, the elliptic relaxation function f and the viscous diffusion D ν (υ /k), whereas P(υ /k)/k varying with y (in fact with y 4 when eddy viscosity is used) goes to zero at the wall: ν(υ f w = lim /k) (5) y y - The above boundary condition is more convenient and easier reproducible as compared with f w in the υ f model. In fact, the boundary condition for f w (eqn. 5) has the identical form as that for ε w and can be treated in analogous manner in the computational procedure. The mere fact that both the nominator and the denominator of f w are proportional to y instead of y 4 as in the original υ f model (with y = being a singular point in both cases), brings improved stability of the computational scheme. THE ϕ f MODEL (UMIST) The first model variant, introduced in [], with the new variable denoted as ϕ = υ /k - hence hereafter labelled as the ϕ f model, uses a reformulation of the elliptic function f in order to make it tend to zero at the wall: f = f + ν( ϕ k) + ν ϕ (6) k The corresponding ϕ and f equations, to be solved together with the k and ε equations (, ) are: Dϕ Dt = f P ϕ k + k L f f = T (C ) ν t σ k ϕ [ ϕ k + x j x j x j ] C P k ν k [ νt ϕ σ k x j ] (7) ϕ k ν ϕ (8) x j x j Where ν t, τ and L are defined as: ν t = C µ ϕkτ, τ = max [ k ε,c τ ] [ ν k /, L = C L max ε ε,c η ] ν /4 ε /4 (9) This modification ensures the correct behavior of f far from the wall in contrast to the LDM formulation of []. The coefficients used in this model are summarized in the table below: In Figure the budgets C µ C ε ) C ε σ ε σ k C C C τ C L C η a..4 ( + a ϕ of the terms in equation (8) are presented. The production of ϕ comes from f, while the destruction is represented by Pϕ/k. The diffusion term on the right hand side of (8) compensates for the misalignment of the maximum between the production and destruction by transporting ϕ this into the near-wall region. The cross-gradient term is mainly a sink term only positive in the viscous sub-region and it could be

4 .6.4 P ϕ / k f ν t k ϕ k ((ν t /σ k ) ϕ).5.4 Exp ϕ model LDM SST. Cf Figure : Budgets of ϕ in a channel flow e+6 4e+6 6e+6 8e+6 Re x Figure : Friction coefficient on a flat plate neglected if f was decreased by altering the coefficients in the equations, as discussed below. The model has been implemented in an unstructured finite-volume code [] and tested in different configurations as reported in []. As an illustration of the basic model verification, we show in Fig. the wall friction factor in a boundary layer on a flat plate computed with the ϕ f model, LDM version of the υ f and Menter s SST model [], compared with the experiments of Wieghardt and Tillman [7]. It can be seen that the ϕ f model outperforms both the LDM and SST model. The computation was done with the same CFL number for all models on a mesh clustered towards the wall, with 8 cells in the wall normal direction. Other tests lead to similar conclusions: in the flow over periodic hills the results for all the formulations of the model were similar, whereas in an asymmetric diffuser the predictions were improved, as illustrated by profiles of the mean velocity and the streamwise turbulent stress at two locations in Fig.. Here below we present recent application of the ϕ f model to natural convection in a tall cavity. Profiles at x/h = 5.98 Exp ϕ - f LDM y/h U/Ub Profiles at x/h = uu/ub^ y/h..4.6 U/Ub uu/ub^ Figure : Mean velocity and streamwise turbulent stress at two locations in an asymmetric diffusor. Symbols: experiments, Buice & Eaton 997); lines: computations 4

5 THE ζ f MODEL (TUD) This model variant proposed in [] denotes the new variable as ζ = υ /k - hence hereafter denoted as ζ f model - differs in two features from the ϕ f model. First, in order to reduce the ζ equation to a simple source-sink-diffusion form, the term X (equation 4) is omitted. As shown in Fig., this term is not significant, though close to the wall it has some influence. In order to compensate for the omission of X one can re-tune some of the coefficients. Another novelty is the application of a quasilinear pressure-strain model in the f-equation, based on the formulation of Speziale, Sarkar and Gatski (99) (SSG), which brings additional improvements for non-equilibrium wall flows. It was shown by Wizman et al. [8] that the elliptic relaxation model based on SSG requires significantly less reduction of pressure strain than the original model based on IP. This, together with Fig. explains why the cross diffusion term X could be omitted in the ζ f model. The computations of flow and heat transfer in a plane channel, behind a backward facing step and in a round impinging jet show in all cases satisfactory agreement with experiments and direct numerical simulations. The incorporation of the quasi-linear SSG model for the pressure-scrambling term Π ij, = C Pa ij + C ks ij + C 4 k(a ik S jk + a jk S ik δ ija kl S kl ) + C 5 k(a ik Ω jk + a jk Ω ik ) () into the wall normal stress component, with the neglect of P ( ), yields the following form of the f equations in conjunction with the ζ equation () (with X = ): L f f = [ ] ( (C ) + C P ζ ) ( ) C4 P τ ε C 5 () k Adopting the coefficients for the SSG pressure-strain model, with C =.65 and C 5 =. (for arguments see []) and noting that the last term in equation () can be neglected as compared with the first term because (C 4 / C 5 ).8, we arrive to the following set of model equation constituting - together with the the k and ε equations (, )- the ζ f model: ν t = C µ ζkτ () L f f = [ ]( (C.) + C P ζ ) () τ ε Dζ Dt = f ζ k P + [( ν + ν ) ] t ζ (4) x k σ ζ x k with wall boundary conditions ζ = and f w = lim y ( ζ/y ) (eqn. 5 ). The equations set is completed by imposing the Kolmogorov time and length scale as the lower bounds, combined with Durbin s (996) realizability constraints: τ = max [ min ( k ε, ) a ( ν ) ] /,C τ 6Cµ S ζ ε [ ( k / L = C L max min ε, k / 6Cµ S ζ ),C η ( ν ε ) /4 ] where a (recommended a =.6). The following coefficient are recommended: (5) (6) C µ C ε C ε C C σ k σ ε σ ζ C τ C L C η..4( +./ζ)

6 It is noted that instead of using equation (5) for f w, one can make further simplifications to satisfy zero wall boundary condition for f (in analogy with the original Jones-Launder (97) formulation of the low-re-number dissipation equation) by solving equation () but for f with f w = and getting f from: ( ) ζ / f = f + ν (7) x n which is then used in ζ equation. The viscous terms in equations (7) and (6) have the same near-wall asymptotic behaviour. However, as also experienced with the ε equation, this is not necessary since the boundary condition f w = ζ/y is very robust. Basic validation and tuning of the ζ f model was performed with respect to DNS of a plane channel at several Re numbers (Re τ = 9, 59 and 8. In Fig. 4 profiles of velocity and turbulent quantities (including ζ and f) are shown, nondimensionalised with the inner-wall scales for a channel flow at Re τ = 8, and compared with the DNS of Tanahashi et al. (4). Note that results with ζ f model are obtained with the mesh with the wall-nearest cell-center at y + =., which corresponds to the first y + used for DNS. The agreement for U +, k + and uv + is excellent over the whole cross-section, and so is for ζ in the wall region where the elliptic effects are influential. A further DNS-based model validation is illustrated in Fig. 5 for a pulsating turbulent channel flow generated by an imposed sinusoidal pressure variation for the Stokes length scale l + S =7 (where l+ S = /ω +, ω + = ων/u τ and ω is the forcing frequency), for which Scotti and Piomelli reported recently direct numerical simulations. No special remedy was introduced to capture high-frequency unsteadiness, yet the reproduction of the cyclic variation of the wall shear stress and of velocity profiles at different phases is quite satisfactory, providing support of adequate response of the ζ model to the imposed unsteadiness and its applicability to unsteady flows. 5 5 U +, DNS Re τ =8 U +, 5 4 k +, DNS uv +, DNS f +, DNS y ε +, DNS P k +, DNS ζ +, DNS y + Figure 4: Velocity and turbulent quantities in channel flow Re τ = 8. Symbols: DNS data of Tanahashi et al. (4). Full line: ζ-f 6

7 τ w <U> 4 symbols: LES lines: τ/8 τ/8 τ/8 τ/8 4τ/8 5τ/8 6τ/8 7τ/8 DNS, Scotti&Piomelli, τ/t. y + Figure 5: Pulsating Channel, Re τ = 5. Cycle variation of the wall shear stress (left) and velocity profiles at different phase angles. Symbols: LES of Scotti and Piomelli (). Lines: SOME APPLICATIONS TO HEAT TRANSFER We present now some results for heat transfer obtained with the ϕ f and ζ f models. First, we consider forced convection and present some results of computation of velocity fields and heat transfer in two generic test cases: a separating flow behind a backward facing step and in a round impinging jet. The temperature field was obtained by solving the RANS energy equation with constant fluid properties, using the isotropic eddy diffusivity ν t /σ T where ν t is given by equation () and σ T =.9 Backward-facing step flow and heat transfer. Figure 6 presents the mean velocity profiles at several cross-sections, as well as the friction factor and Stanton number along the bottom wall behind a step in a backward facing step flow. Computations are performed with the standard Durbin s υ f and the ζ f models. Velocity profiles obtained with the two models at several locations within the separation bubble and around reattachment are practically indistinguishable and both in good agreement with experiments of Vogel and Eaton (985). The same can be concluded for the friction factor and Stanton number in the recirculation bubble and a few step height downstream from reattachment, but a more significant difference appear further downstream in the recovery region, where the ζ f model captures better the trend of the friction factor, and slightly worse for the Stanton number. The difference between the two models could be attributed to the difference in the pressure strain models. The agreement with the experiments in general can be regarded as fully satisfactory for both models. Round impinging jests. Figure 7 shows the mean velocity profiles and Nusselt numbers in a normally impinging round jet issuing from a fully developed pipe flow for two Reynolds numbers, and 7. The distance between the pipe exit and the plate is H/D =. The results are compared with the available experimental results of Baughn and Shimizu (989) and Baughn et al. (99), as well as the υ f computations using the coefficients of Behnia et al. (998). The velocity profiles (left) for both Re numbers are similar with some improvements returned by the ζ f model, especially for r/d =.. Similar quality of predictions has been obtained for the Nusselt number (right), especially for Re=, where the ζ f model shows some slight overprediction, but a more realistic shape of the curve. For Re=7, both models show visible overpredictions in the stagnation region, though much better than reported by most other turbulence models. Here to, the ζ f model shows some improvement over the υ f model. Natural convection in a tall cavity. The natural convection cavity studied by Betts and Bookhari [5] has been computed using the LDM, Launder-Sharma k-ε and the ϕ f model. The flow inside the cavity is driven by the temperature difference between the vertical walls creating boundary layers where the shear stress changes rapidly close to the wall, thus requiring a model capable of resolving correctly 7

8 Exp. Vogel&Eaton, Re=8 v -f model y/h x/h.5 U/U b C f St. -. Exp. Vogel&Eaton, Re= v -f model. Exp. Vogel&Eaton, Re=8 v -f model x/h 5 5 x/h Figure 6: Backward-facing step, Re = 8. Velocity profiles, friction factor and Stanton number. Symbols: experiments of Vogel and Eaton (985). Full line: ζ-f; dotted line: υ -f model.4. Exp. Cooper et al., Re=7 Exp. Cooper et al., Re= v -f model r/d=. r/d=.5 r/d=. r/d=.5 r/d=. r/d=.5 r/d=. 5 Exp. Cooper et al., Re=7 Exp. Baughn&Shimizu, Re= Exp. Baughn et al., Re= v -f model z/d. Nu U /U b 4 5 r/d Figure 7: Impinging jets, Re= and 7. Velocity profiles and Nusselt number. Symbols: experiments of Baughn and Shimizu (989), Baughn et al. (99). Full line: ζ-f; dotted line: υ -f the near wall region. The experiments were done at two different Rayleigh numbers, Ra =.86x 6 and.4x 6 in a cavity with an aspect ratio of H/W = In Figure 8 the velocity profiles are plotted for two Rayleigh numbers at selected locations. The predictions of vertical velocity by the ϕ f model are in much better agreement with the experiment than other two models, especially in the middle and the top portion of the cavity for both Rayleigh numbers considered. The LDM model over predicts the velocity peak in the middle of the cavity, with an overprediction at higher Rayleigh number of 4% and %. Near the horizontal walls, the ϕ f model slightly 8

9 V(m/s) V (m/s) V(m/s) Exp y/h =.5 ϕ - f LDM k-ε L-S Exp y/h =.7 Exp y/h =.95 a) x (m) V (m/s) V (m/s) V (m/s) Exp y/h =.5 ϕ - f LDM k-ε L-S Exp y/h =.7 Exp y/h = x(m) Figure 8: Velocity profiles at selected cavity heights for Ra =.86x 6 (left) and Ra =.4x 6 (right). b) T (K) Exp y/h =.6 LDM ϕ - f k-ε L-S T (K) Exp y/h = x(m) x(m) Figure 9: Temperature profiles at two characteristic cavity heights for Ra =.4x 6. underpredicts the velocity on the decelerating side, but the slope of the velocity profile in the mid-width is closer to the experiment, compared with the LDM. The temperature profiles for the high Rayleigh number can be seen in Figure 9. In terms of temperature profiles there is not much difference between the two formulations, both give good predictions at the different stations of the cavity. CONCLUSIONS A new version of the Durbin s elliptic relaxation eddy-viscosity model, based on solving the time scale ratio υ /k instead of υ, has brought significant improvements in the model robustness and in quality of predictions. the major features of the new model are the new boundary conditions for the elliptic relaxation function, and fixed sign source terms. Two model variants, developed independently at UMIST and TU Delft, follow the same rationale, but differ in some minor details. Each model variant has been implemented in an in-house unstructured CFD codes, both using uncoupled solvers, and tested in a variety of attached, separating, impinging and buoyant flows and heat transfer with very different grids. In all cases considered, the model confirmed the already known advantages of the elliptic relaxation approach and yielded good agreement with the available experiments and DNS data. The new model is envisaged as a prospective model for industrial CFD, especially when wall friction and heat transfer are in focus. Acknowledgement. This work has been partially supported by the EU CEC Research Directorate-General through projects MinNOx, No. ENK6-CT--5 (TU Delft), and FLOMANIA, No. G4RD-CT-6 (UMIST). 9

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