EXPLICIT ALGEBRAIC MODELS FOR STRATIFIED FLOWS

Transcription

1 EXPLICIT ALGEBRAIC MODELS FOR STRATIFIED FLOWS W.M.J. Lazeroms, G. Brethouwer, S. Wallin,2, and A.V. Johansson Linné FLOW Centre, KTH Mechanics, SE- 44 Stockholm, Sweden 2 Swedish Defence Research Agency FOI), SE-64 9 Stockholm, Sweden werner@mech.kth.se Abstract An explicit algebraic Reynolds stress and scalar flux model for stratified turbulent flows is formed from a coupled set of algebraic equations for the Reynolds stress anisotropy and the scaled scalar flux. The complete model is derived for two-dimensional mean flows, and evaluated for two test cases: turbulent channel flow and homogeneous shear flow, both with stable stratification. The channel flow results are compared with DNS data, and show a good agreement in the mean profiles, the Reynolds stress and the wallnormal scalar flux. The homogeneous shear flow case results in a critical gradient Richardson number close to.25. The derived model can thus correctly capture the effect of stable stratification on a turbulent shear flow. Introduction In recent years, explicit algebraic turbulence models have proven to be successful in many engineering applications. This class of models incorporates more physics than the simpler eddy-viscosity-based models, while requiring less computational efforts than solving the full Reynolds stress transport equations. Recent advances are the explicit algebraic Reynolds stress model EARSM) by Wallin & Johansson 2) and the explicit algebraic scalar flux model EASFM) by Wikström et al. 2) for the case of a passive scalar. For atmospheric flows, a treatment of the effects caused by density stratification is essential. In such cases, the temperature acts as an active scalar that influences the flow. This introduces a coupling between the transport equations for the Reynolds stresses and the turbulent heat flux, which gives more difficulties in deriving an explicit algebraic turbulence model in this case. Many attempts have nevertheless been made, e.g. by Mellor & Yamada 982), Cheng et al. 22), and most recently Violeau 29). Some authors use an inappropriate form of the weak equilibrium assumption, or very simple expressions for the pressure redistribution terms, causing the final model to be overly simplified. Others, such as So et al. 24), perform a complete and correct derivation of the model, but their equations need to be solved iteratively. The aim of this paper is to present an explicit algebraic model for the Reynolds stresses and turbulent heat flux for stably stratified flows. The approach is an extension of the EARSM by Wallin & Johansson 2) and the EASFM by Wikström et al. 2). The goal is an algebraic model that is fully explicit, robust i.e. no singular behaviour should occur), easy to implement, and approximately) self-consistent. An outline of the derivation of this model is given in Section 2. Thereafter, the model is applied to stably stratified turbulent channel flow, as well as homogeneous shear flow. Some practical issues associated with the implementation of the model, including some additional corrections, are discussed in Section 3. Section 4 shows the results of the simulations and compares them to DNS data. 2 Derivation of the model Explicit algebraic turbulence models can be derived by first considering the full transport equations for the Reynolds stresses u i u j and the turbulent heat fluxu i θ. These equations are rewritten in terms of the Reynolds stress anisotropy a ij and a normalized heat fluxξ i, defined by: a ij = u iu j K 2 3 δ ij, ξ i = u iθ KKθ, ) wherek = u i u j /2 is the turbulent kinetic energy and K θ = θ 2 /2 half the temperature variance. Algebraic equations for a ij and ξ i are then found by applying the weak equilibrium assumption Rodi 972)), which states that the advection and diffusion terms of these two dimensionless fluxes can be neglected. The result in the case of an active scalar is the following system of equations fora ij andξ i : Na ij = 8 5 S ij +C aik S kj +S ik a kj 2 3 a kms km δ ij ) +C 2 a ik Ω kj Ω ik a kj ) 2a) C 3 Γi ξ j +ξ i Γ j 2 3 Γ kξ k δ ij ), N θ ξ i = c S S ij +c Ω Ω ij )ξ j c Θ aij δ ij) Θj 2b) c Γ Γ i,

2 in which the right-hand sides contain the factors: N = c a km S km Γ k ξ k, 3a) N θ = c θ + N c ) + c θ5 2 r 2) ξj Θ j. 3b) These equations have been expressed in terms of the following dimensionless quantities involving the mean velocity gradient U i / x j, the mean temperature gradient Θ/ x i and the gravitational vectorg i : S ij = τ Ui + U ) j, 4a) 2 x j x i Ω ij = τ Ui U ) j, 4b) 2 x j x i K Θ Θ i = τ, 4c) K θ x i Kθ Γ i = τ K β Tg i, 4d) where β T is the thermal expansion coefficient and τ is the dynamical timescale. The quantity r = τ θ /τ is the ratio of turbulence timescales for the temperature and the velocity, which is assumed to be constant. Furthermore, the equations above contain model constants that appear through the modelling of the pressureredistribution and dissipation terms in the transport equations for u i u j and u i θ. For these we take the same expressions as Wallin & Johansson 2) and Wikström et al. 2) extended with the buoyancy terms discussed by Launder 975), It is important to note that equations 2) are mutually coupled, which is ultimately caused by the effects of buoyancy. In contrast to the case of the passive scalar, where one can first solve the EARSM and then obtain the heat flux directly from a ij, the current situation requires 2a) and 2b) to be solved simultaneously. The solution procedure is exemplified below for two-dimensional mean flows. Explicit algebraic model for 2D mean flows The implicit equations 2) are solved by first treating the N and N θ in the nonlinear terms on the righthand side as known quantities. The linear part of the equations can be solved by expressing a ij and ξ i as a linear combination of suitable tensor groups. For 2D mean flows in which U i, Θ i and Γ i are coplanar, a ij and ξ i have three and two independent components, respectively, and the following expansions can be used in matrix notation): a = β S +β 2 S 2 3 II SI)+2β 4 SΩ, ξ = λ I +λ S)Θ, 5a) 5b) where II S = tr{s 2 }. By inserting these expansions in 2) and expressing all products in terms of the 3+2 basis tensors and vectors in 5), one can derive a linear system for the five coefficients {β,β 2,β 4,λ,λ }. Solving this system is straightforward, and this has been done for the two testcases discussed below. At this point, the coefficients {β,β 2,β 4,λ,λ } still depend on the unknown factorsn andn θ. Equations for these quantities can be derived by inserting the expansions 5) into 3), including the newly found expressions for the five coefficients. These equations will be nonlinear, which reflects the fact that we disregarded the nonlinearity of 2) until now. In fact, in the testcases discussed below the equation for N turns out to be a sixth-order polynomial equation, for which the roots cannot be expressed in elementary functions. It is therefore not straightforward to find exact solutions forn andn θ in order to obtain a self-consistent explicit algebraic model fora ij andξ i. The nonlinearity of the implicit equations is a natural consequence of the current modelling approach. The nonlinear terms are not obtained when one neglects the advection and diffusion terms of the dimensional anisotropy and heat flux, as is done by some authors. In the models of Violeau 29) and So et al. 24) the correct form of the weak equilibrium assumption is applied, but the problems associated with the nonlinear terms are either avoided by assuming constant N and N θ, or left to be solved by numerical means. We choose to find explicit expressions for N andn θ that roughly exhibit the behaviour of the exact solutions. Together with the solutions for the coefficients {β,β 2,β 4,λ,λ } and suitable expressions for N and N θ, equations 5) form a fully explicit algebraic model for a ij and ξ i. The model should be used together with suitable equations fork,k θ, ε andε θ. 3 Application to stably stratified shear flows In the following, we apply the two-dimensional model to two stably stratified testcases: turbulent channel flow and homogeneous shear flow. In both cases the mean quantities only vary in a direction y aligned with the buoyancy force. This further simplifies the linear system for {β,β 2,β 4,λ,λ } compared to the general 2D-case. Another simplification is made by choosing the constantc in equation 2) to be zero, which was also done by Wallin & Johansson 2). The solution forβ will now be of the following form: β = D { β ) +β ) II SRi +β 2) II2 S Ri2 Θ 2}, 6) where D is the determinant of the linear system, and the coefficients β i) are polynomials in N, N θ and the invariant II S. The buoyancy-dependent terms are characterized by the product of II S and the gradient

3 Richardson number: β T g Θ y Ri = ) 2. 7) U y The other coefficients {β 2,β 4,λ,λ } have similar forms. DeterminingN andn θ To determinen andn θ, we first use an assumption proposed by Wikström et al. 2) and put the constant c θ5 in equation 3b) to /2. The third term then vanishes and a simple relation between N and N θ remains. By using the expansions in 5), equation 3a) becomes: N = c II S β +2II S Riλ. 8) Putting the expressions for β and λ in 8) leads to a polynomial equation which is quartic in N and quadratic in N θ. As mentioned before, a closed-form solution of this equation cannot be found. To avoid any numerical complications with finding the correct root of equation 8), we would like to have a fully explicit, yet approximate expression forn and N θ. For this purpose we start with the non-stratified case, in which equation 8) becomes cubic. An exact solution of this equation, which we calln, was found by Wallin & Johansson 2), and one can determine the correspondingn θ from 3b). One might obtain a better approximation for the stratified case by performing what is essentially a one-step iterative method, i.e. determining N directly from 8) after having inserted N and N θ in the right-hand side. This approximation might improve if one can modify N in such a way that it gets closer to the exact solution. Here we propose the following modification: Ñ = c +f N) N c +)+C r3 f N) 2, 9) where we have the following functions: f N) = exp ) C r II S Ri, a) f N) 2 = C r2 II S Ri) 2 /+C r2 II S Ri) 2 ), b) which aim to improve the overpredicted) value ofn whenii S Ri is large, such as in the centre of the channel. For the correct choice of constants C r - C r3, usingñ and the correspondingñθ in the right-hand side of 8) gives an approximation for N that is close to the exact solution. Equations fork,k θ and their dissipation rates Additional equations are needed to close the current turbulence model. Here we choose the framework of the K-ω-model proposed by Wilcox 993). In addition to a transport equation for the kinetic energy K, one can use a model equation for the inverse timescale ω, which is related to the dissipation rate ε of kinetic energy byε = β ωk. An important difference between the current model and the one by Wikström et al. 2) for the passive scalar is the fact that the latter can be solved without the need for an equation fork θ, as long as one chooses c θ5 = /2. In the current model, however, the coefficients {β,β 2,β 4,λ,λ } depend on K θ through Θ 2 as shown in equation 6). Therefore, we also need a suitable expression for K θ to close the model equations. By assuming production and dissipation of K θ to be equal, and the timescale ratio r to be constant, one can derive an explicit expression for K θ that is consistent with the current model formulation. We choose to use this expression for the channel flow to avoid numerical difficulties associated with solving the full transport equation fork θ. In the case of homogeneous shear flow, however, the full transport equation will be solved, since dk θ /dt is nonzero here. Near-wall treatment Wallin & Johansson 2) discuss some modifications to their EARSM for non-stratified wall-bounded flows in order to obtain the correct behaviour in the near-wall region. These corrections include a modified turbulence timescale and the use of damping functions in the coefficients {β,β 2,β 4 } to ensure the correct near-wall asymptotics for the anisotropy. Such corrections will also be used in the current model in the case of channel flow, both in a ij andξ i. The near-wall corrections should be used together with the low-re version of thek-ω-model, in which the coefficients in the ω-equation are modified to include a dependence on the turbulence Reynolds number see Wilcox 993)). Another issue in the near-wall region is the singular behaviour of equation for ω at the wall. This problem can be solved by writingω = ω w + ω whereω w y 2 is the near-wall asymptote. One can then solve a nonsingular) transport equation for ω and use the simple boundary condition ω = at the wall. Details can be found in Gullman-Strand 24). 4 Simulation results The model formulated above has been applied to stably stratified homogeneous shear flow and channel flow. All equations have been implemented in a computer-algebra package which generates the code for the numerical solver. Values for the model constants used here are shown in Table. These values are partly taken from Wallin & Johansson 2) and Wikström et al. 2), and others have been re)calibrated both to ensure robustness and to give satisfactory results. Homogeneous shear flow First we consider homogeneous shear flow, in which the mean gradients S U = du/dy and S T = dθ/dy are uniform. The near-wall corrections dis-

4 c C C 2 C 3 c θ c θ5 c S c Ω.8 4/ Ri = a) Ri =.5 c Θ c Γ r C r C r2 C r Table : Values of model constants used in the current work K K Ri =.2 cussed above are not needed in this case. Stably stratified homogeneous shear flow is suitable for drawing conclusions about the existence of a critical Richardson number, since Ri defined in 7) is constant in this case. Consider the transport equation for the turbulent kinetic energy: dk dt = P ε+g, ) in which P = uvs U is shear production and G = β T gvθ is buoyancy destruction. Through an analysis of the K-ω-equations one can show that P + G)/ε tends to a constant value for t. In this steady state, the kinetic energy K will either increase or decrease, depending on Ri. We will define the critical Richardson number as the value of Ri for which dk/dt for large times, which is equivalent to P +G)/ε. From Figure a), which shows the time evolution of K for five different Richardson numbers, it is clear that the increase of K is suppressed more and more when buoyancy effects are increased. Our investigation shows that the critical Richardson number of the current model is close to.25, which is the value dictated by the Miles-Howard stability criterion for this type of flows. Figures b-c) show the time evolution of three components of the anisotropy, which tend to constant values. In the case without stratification, a 2 attains the well-known value of.3 and tends towards zero for increasing Ri. The components a and a 22 have equal but opposite values for Ri =, whereas for increased stratification energy is added to the streamwise fluctuations and removed from the cross-stream fluctuations. The steady-state behaviour appears to be independent of any initial conditions as long as Ri does not change. Channel flow In the case of channel flow, all the features of the model discussed in Section 3 are implemented. We consider a channel with a constant streamwise pressure gradient and a constant temperature difference T between the two walls. The key parameters in this case arere τ = u τ h/ν,ri τ = β T g Th/u 2 τ, and Pr = ν/κ where h is the channel half-height, u τ the friction velocity, ν the kinematic viscosity, and κ the molecular heat diffusivity. Simulations are performed forre τ = 55,Ri τ = {,48,96} andpr =.7. Figure 2 shows the outcome of the simulations for a number of statistics. The model results are com- a 2 a αα.2 Ri =.25 Ri = S U t b) S U t c) Ri =.3 Ri =.25 Ri =.2 Ri =.5 Ri = Ri = r Ri = S U t Figure : Time evolution of a) the turbulent kinetic energy and b,c) the Reynolds stress anisotropy in homogeneous shear flow for different Richardson numbers. pared with DNS data obtained by García-Villalba & del Álamo 2). Generally speaking, the model appears to give good predictions. The mean velocity and temperature profiles, as well as the Reynolds stress and the wall-normal heat flux, are in good agreement with the DNS data. The overprediction of the mean temperature for Ri τ = is a known feature of the passivescalar EASFM, and this may be improved by adding a diffusion correction Högström et al. 2)). However, the model clearly fails to give a good prediction of the streamwise heat flux in the outer region. Although the result is better than the zero value predicted by a standard eddy-diffusivity model, the a a 22

5 6 a).5 b) 5 4 U Θ T uv + uθ c) e) vθ θ rms d) f) Figure 2: Comparison of the channel flow results dashed lines) with DNS solid lines), for Ri τ = {,48,96}. The arrow points in the direction of increasing Richardson number. Shown are a) the mean velocity profile, b) the mean temperature profile, c) the Reynolds shear stress, d) the wall-normal component and e) the streamwise component of the turbulent heat flux, and f) the rms of the temperature fluctuations. high peak values near the centre of the channel are not reached. This is caused by the recalibration of the constantsc S andc Ω compared to the EASFM of Wikström et al. 2). We choose to put both constants to zero to ensure that the denominator D appearing in the model see equation 6)) does not change sign. The flux component uθ is quite sensitive to these constants, but the robustness of the model has a higher priority. Also shown in Figure 2 is the rms value of the temperature fluctuations, which is determined from the model expression for K θ mentioned in the previous section. It appears that this simplified expression generally works well in predicting K θ, except near the centre of the channel. One should note, however, that the DNS data also contain effects due to internal gravity waves, which is a feature that one cannot capture with the current model. For Ri τ = the underpredicted values of θ rms are caused by neglecting the diffusion of K θ, and a diffusion correction similar to the one mentioned for the mean temperature may be used to improve the prediction here. 5 Conclusions In this work we derived an explicit algebraic model for the Reynolds stresses and the turbulent heat flux in the case of stable stratification. Fully explicit expressions for the model coefficients were found in the case of 2D mean flows, and the model was applied to homogeneous shear flow and turbulent channel flow. In the first testcase, the current set of values for the model constants proved to give a critical gradient

6 Richardson number close to the theoretical value of.25 from linear stability theory. For Richardson numbers above this value, the turbulent kinetic energy predicted by the model decays to zero. Although there has been some debate about the existence of a critical Richardson number in e.g. real atmospheric flows, it is just a natural feature of the current model. Additional ad-hoc) corrections to the model would be needed to avoid the occurrence of a critical Richardson number. The model also predicts the correct behaviour of the anisotropy, which shows that for increased stratification the Reynolds shear stress tends towards zero, the vertical fluctuations are damped, and relatively more energy is contained in the streamwise component. In the case of channel flow, we compared the model results with DNS data for different values of the friction Richardson number Ri τ, shown in Figure 2. The agreement with the DNS is generally good for the mean velocity and temperature profiles, apart from the known issue of overprediction in the passivescalar case. As mentioned, this issue might be resolved by adding a diffusion correction, but it is currently unclear what such a correction will look like for active scalars. This may be the subject of further research. The Reynolds stresses and wall-normal heat flux also agree well with the DNS data. Only the streamwise heat flux is severely underpredicted in the outer region of the channel, but it is still an improvement compared to standard eddy-diffusivity models that yield uθ =. Presently, the model has only been applied to stably stratified testcases. It is desirable to also test the performance of the model for unstable stratification, and to apply it to more realistic atmospheric testcases. Acknowledgments The authors wish to thank Prof Manuel García- Villalba for providing the DNS data. The received travel scholarship from Fonden Erik Petersohns Minne is also gratefully acknowledged. References Rev. Geophys. Space Phys., Vol. 2, pp Rodi, W. 972), The prediction of free turbulent boundary layers by use of a two equation model of turbulence, PhD thesis, University of London. So, R.M.C., Jin, L.H. and Gatski, T.B. 24), An explicit algebraic Reynolds stress and heat flux model for incompressible turbulence: Part II Buoyant flow, Theoret. Comput. Fluid Dynamics, Vol. 7, pp Violeau, D. 29), Explicit algebraic Reynolds stresses and scalar fluxes for density-stratified shear flows, Phys. Fluids, Vol. 2, 353. Wallin, S. and Johansson, A.V. 2), An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows, J. Fluid Mech., Vol. 43, pp Wikström, P.M., Wallin, S. and Johansson, A.V. 2), Derivation and investigation of a new explicit algebraic model for the passive scalar flux, Phys. Fluids, Vol. 2, pp Wilcox, D.C. 993), Turbulence Modeling for CFD, DCW Industries, Inc. Cheng, Y., Canuto, V.M. and Howard, A.M. 22), An improved model for the turbulent PBL, J. Atmos. Sci., Vol. 59, pp García-Villalba, M., del Álamo, J.C. 2), Turbulence modification by stable stratification in channel flow, Phys. Fluids, Vol. 23, 454. Gullman-Strand, J. 24), Turbulence and scalar flux modelling applied to separated flows, PhD Thesis, KTH Royal Institute of Technology, Stockholm. Högström, C.-M., Wallin, S., Johansson, A.V. 2), Passive scalar flux modelling for CFD, In Second International Symposium on Turbulence and Shear Flow Phenomena, Stockholm, Vol. II, pp Launder, B.E. 975), On the effects of a gravitational field on the turbulent transport of heat and momentum, J. Fluid Mech., Vol. 67, pp Mellor, G.L. and Yamada, T. 982), Development of a turbulence closure model for geophysical fluid problems,