Explicit algebraic models for turbulent flows with buoyancy effects

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1 Explicit algebraic models for turbulent flows with buoyancy effects W.M.J. Lazeroms, G. Brethouwer,. Wallin and A.V. Johansson Linné FLOW Centre, KTH Mechanics E-1 44 tocholm, weden Abstract An explicit algebraic model for the Reynolds stresses and turbulent heat flux is presented for turbulent parallel shear flows in which buoyancy forces are present. The derivation is based on a model framewor for two-dimensional mean flows that was already presented in a previous wor Lazeroms et al., 213, and new test cases are investigated here. The model is formulated in terms of the Reynolds-stress anisotropy and a normalized heat flux, which are expanded as a linear combination of appropriate basis tensors. The coefficients in this expansion can be found from a system of 18 linear equations. This formulation is complemented with a nonlinear equation for a quantity related to the total production-to-dissipation ratio, the solution of which is modelled with an appropriate expression. Fully explicit model expressions are found for two fundamentally different flow geometries: a horizontal channel for which the temperature gradient is aligned with gravity, and a vertical channel where the temperature gradient and gravity are perpendicular. In the first case, we consider both stable and unstable stratification, while in the second case, both mixed and natural convection flows are investigated. Comparison with DN data shows that the model gives predictions that are substantially better than standard eddy-viscosity/eddy-diffusivity models. 1 Introduction Explicit algebraic Reynolds-stress models EARM constitute a class of turbulence models that have proven to be a useful compromise between simple eddy-viscosity/eddy-diffusivity models and the more physical differential Reynolds-stress models DRM, for which the full transport equations for the Reynolds stresses and scalar flux are solved. uch explicit algebraic models are based on an equilibrium assumption that turns the transport equations into algebraic equations. A fully explicit algebraic solution to these equations considerably reduces the complexity of numerical simulations, while eeping most of the physics present in the DRM. An example of such a model that has proven to be succesful is the EARM of Wallin & Johansson 2, which was later extended to an explicit algebraic scalar-flux model EAFM by Wiström et al. 2 for describing passive scalars. Recently, the current authors have proposed a framewor for deriving new explicit algebraic models for active scalarslazeroms et al., 213, in which case the Reynolds stresses and turbulent heat flux are mutually coupled. Active scalars are mostly found in flows where buoyancy forces are active, e.g. atmospheric flows where the potential temperature influences the flow field. In these situations, the coupling between the Reynolds stresses and the heat flux maes the derivation of the model more complicated because they need to be solved for simultaneously in order to obtain fully explicit expressions. The framewor presented in Lazeroms et al. 213 consists of a system of 18 linear equations, which was derived specifically for two-dimensional mean flows. A fully explicit model was then constructed for the specific case of stably stratified parallel flows, which was shown to give very good results when compared with DN. Furthermore, some fundamental Also at wedish Defence Research Agency FOI, E tocholm, weden. 1

2 advantages of the model over other models of a similar type e.g. Mellor & Yamada 1982; Violeau 29; o et al. 22, 24 were discussed. The results for stably stratified parallel flows presented in Lazeroms et al. 213 are promising for atmospheric flows, in particular climate models for which the correct prediction of stably stratified flows occuring in the Arctic is crucial. Active-scalar transport is also important in many industrial applications with large temperature gradients, for which the influence of the temperature on the flow is no longer negligible. However, all these applications require a model that is also capable of predicting unstable stratification. In the current paper, we shall investigate a case with unstable stratification, for which some modifications to the model are needed. Moreover, the cases considered in Lazeroms et al. 213 focussed on a particular geometry for which the temperature gradient is aligned with gravity. In some cases, the temperature gradient is directed perpendicular to gravity, which needs to be accounted for by the model. This will also be considered in the present wor. Examples of such a geometry are mainly found in industrial applications e.g. convection in a chimney, heat exchangers, but it can also be seen as a next step towards a general two-dimensional model. The following section contains a summary of the derivation found in Lazeroms et al. 213, the final result of which is a linear system of 18 equations and two additional nonlinear equations from which an explicit algebraic model can be obtained. ection 3 focusses on the case of a horizontal channel in which the temperature gradient is aligned with gravity. While most of the model expressions for this case have already been derived in Lazeroms et al. 213, we discuss modifications that mae the model suitable for both stable and unstable stratification. Also some comparisons with direct numerical simulations DN are shown. ection 4 investigates the new case of a vertical channel in which the temperature gradient and gravity are perpendicular. Again, we derive a fully explicit model for this case and compare it with DN of vertical channel flow, both with mixed and natural convection. 2 Model framewor The starting point of the derivation is the Navier-toes equations, representing conservation of mass, momentum and energy. Applying the so-called Boussinesq approximation, effects of varying density or temperature are retained only in the gravity term. The Reynolds-averaged version of the resulting equations will then be as follows: U j x j =, DU i Dt = 1 ρ P x i +ν 2 U i x j x j u iu j x j β T Θg i, DΘ Dt = κ 2 Θ x j x j u jθ x j, where U i, P, Θ are the mean velocity, mean pressure, and mean temperature, respectively, and u i, p, θ arethe correspondingturbulent fluctuations. Furthermore, ν is the inematic viscosity, κ is the molecular heat diffusivity, β T is the thermal expansion coefficient, and g i is the gravitational acceleration. The materialderivative alongthe mean flow is indicated by D/Dt = / t+u / x. Complementing the set of equations1 for the mean quantities is a similar set for the turbulent fluctuations, fromwhichoneobtainsthetransportequationsforthe Reynolds-stressesu i u j andthe turbulent heat flux u i θ. The goal of the current modelling approach is to find explicit expressions for u i u j and u i θ in order to close the system of equations 1, starting from their transport equations, which have the following form: 1a 1b 1c Du i u j D ij = P ij +Π ij ε ij +G ij, Dt Du i θ D θi = P θi +Π θi ε θi +G θi. Dt 2a 2b 2

3 P ij = u i u U j x u j u U i x P θi = u i u j Θ x j u j θ U i x j production G ij = β T gi u j θ +g j u i θ G θi = 2β T K θ g i buoyancy ε ij = 2 3 εδ ij Π ij = Π Rotta ij +Π LRR ij +Π B ij Π θi ε θi = Π WWJ θi +Π B θi dissipation pressure redistribution Table 1: Expressions used for the terms in equations 2. The pressure-redistribution terms are modelled using the slow term proposed by Rotta 1951, the rapid terms by Launder et al LRR, the non-linear model used by Wiström et al. 2 WWJ, and the buoyancy contributions B proposed by Launder The various terms in these equations are shown in more detail in table 1. The production and buoyancy terms are exact, while the other terms need to be modelled, as indicated in the table. The complete expressions for the modelled pressure-redistribution terms can be found in Lazeroms et al The left-hand sides of 2 contain the advection and diffusion terms, which will be treated in such a way that the equations become algebraic, as explained below. In order to close the system, the transport equations in 2 need to be complemented by suitableequationsormodelsfortheturbulentineticenergyk = 1 2 u u, thetemperaturevariance 2K θ = θ 2, andtheirrespectivedissipationratesεandε θ. Thefirstquantitycanbe found bytaing half the trace of 2a, leading to the transport equation for turbulent inetic energy: DK Dt D = P ε+g, 3 with D = 1 2 D, P = 1 2 P, and G = 1 2 G. A similar equation can be found for K θ : DK θ Dt D θ = P θ ε θ, with P θ = u j θ Θ x j. 4 In this wor, closure of the system is obtained by means of the K-ω model proposed by Wilcox 1993 together with a transport equation for K θ and a model expression for the scalar dissipation term ε θ. More details are given in appendix A. 2.1 Explicit algebraic models for 2-D mean flows The next step is to recast the transport equations 2 into a form that is purely algebraic with respect to u i u j and u i θ. Alternatively, we consider these equations in terms of the Reynoldsstress anisotropy a ij and the normalized heat flux ξ i, which are dimensionless quantities defined as follows: a ij = u iu j K 2 3 δ ij, ξ i = u iθ KKθ. 5 The main assumption in the current modelling approach is the so-called wea equilibrium assumption, first stated by Rodi 1972, 1976, through which both the advection and diffusion terms in the equations for a ij and ξ i can be neglected. As shown in Lazeroms et al. 213, this leads to the following set of algebraic equations for a ij and ξ i : Na ij = 8 15 ij +C 1 ai j + i a j 2 3 a m m δ ij +C2 a i Ω j Ω i a j C 3 Γi ξ j +ξ i Γ j 2 3 Γ ξ δ ij, N θ ξ i = c ij +c Ω Ω ij ξ j c Θ aij δ ij Θj c Γ Γ i, 6b 6a 3

4 c 1 C 1 C 2 C 3 c θ1 c θ5 c c Ω c Θ c Γ r 1.8 4/ Table 2: Values of model constants used in the current model. These values have been tested successfully in Lazeroms et al. 213 in the case of stably stratified channel flow and homogeneous shear flow. where the coefficients N and N θ are defined by: N = c 1 1+ P +G = c 1 1 a m m Γ ξ, 7a ε N θ = c θ1 + 1 P +G ε r 2 c P θ θ5 = c θ1 + 1 N c c θ5 1 rε θ 2 r 2 ξj Θ j, 7b where r = τ θ /τ is the ratio of turbulence timescales for the temperature τ θ = K θ /ε θ and the velocity τ = K/ε, which is assumed to be constant appendix A. Furthermore, the system of equations 6 contains the following dimensionless tensors and vectors: ij = τ Ui + U j 2 x j x i K Θ Θ i = τ, K θ x i, Ω ij = τ 2 Γ i = τ Ui U j x j x i Kθ K β Tg i. The coefficients C 1, C 2, C 3, c, c Ω, c Θ, c Γ, c 1, c θ1 and c θ5 are model constants to which we assign the calibrated values shown in table 2. The rationale behind this choice of parameters can be found in Lazeroms et al A solution to equations 6a and 6b for a ij and ξ i that is explicit in terms of ij, Ω ij, Θ i and Γ i canbe found by firstconsideringn and N θ asnown coefficientsand solvingthe resulting linear equations through an appropriate choice of basis tensors and vectors. This basis has to consist of combinations of ij, Ω ij, Θ i and Γ i with the property that all products of tensors and vectors encountered in the derivation should be reducible to the tensors and vectors contained in the basis. As previously shown by the authors Lazeroms et al., 213, in the case of two-dimensional mean flows with buoyancy forces, it is sufficient to consider a basis of ten symmetric and traceless tensors and eight vectors. We can therefore write the solution of 6 in the following way using matrix notation:, a = ξ = 4 4 β T + β θ T θ + =1 =1 4 λ θ V θ + =1 =1 4 λ g V g. 2 β g T g, =1 9a 9b The tensors T and vectors V are given in appendix B. olving for a ij and ξ i now boils down to solving a system of 18 linear equations for the coefficients β and λ, which is found by inserting 9 in 6, reducing all terms to the 18 basis tensors, and putting the coefficients in front of each basis tensor to zero. The resulting linear system for two-dimensional mean flows is presented in Lazeroms et al. 213 with the additional assumption that the constant C 1 in 6a is zero following Wallin & Johansson 2. The system can be solved by a computer algebra pacage to obtain the β- and λ-coefficients in 9. For the sae of robustness and simplicity, we also assumed c = c Ω =. 4

5 II = m m γ θ = Γ Θ ϑ = Θ m m Θ II Ω = Ω m Ω m γ θ = Γ Θ m m ϑ Ω = Θ m mj Ω j Θ γ = Γ Γ m m γ θ Ω = Γ Θ m Ω m Θ 2 = Θ Θ γ Ω = Γ Γ m mj Ω j γ θ Ω = Γ Θ m mj Ω j Γ 2 = Γ Γ Table 3: Invariant combinations of ij, Ω ij, Θ i and Γ i occurring in the present model. Up to this point in the derivation, the factors N and N θ have not been considered, and the β- and λ-coefficients are still expressed in terms of these unnown coefficients. Additional nonlinear equations for N and N θ are found by inserting the newly found expressions for a ij and ξ i in 7: N = c 1 1 II β 1 2γ θ β θ1 II γ θ β θ2 +2γ θ Ω β θ3 +II γ θ Ω β θ4 N θ = c θ γ β g1 +2γ Ω β g2 γ θ λ θ1 γ θ λ θ2 +γ θ Ω λ θ3 γ θ Ω λ θ4 N c 1 1 r Γ 2 λ g1 γ λ g2 γ Ω λ g4, 1a + c θ5 2 1 Θ 2 λ θ1 +ϑ λ θ2 +ϑ Ω λ θ4 +γ θ λ g1 +γ θ λ g2 +γ θ Ω λ g3 +γ θ Ω λ g4, 1b which contain the various invariants that are given in table 3. Unfortunately, the polynomial degree oftheseequationsturnsouttobegreaterthan4evenforthesimplifiedflowcasesconsideredbelow, so that no general analytical solution for N and N θ exists. Therefore, approximate expressions for N and N θ need to be devised that arenearlyself-consistent, meaning that they should nevertheless give results close to the exact solution of 7. In the following, we shall follow Wiström et al. 2 and use the assumption c θ5 = 1 2 to simplify this problem, since in that case N θ can be directly found from N equation 1b. In summary, an explicit algebraic model for a ij and ξ i can be found in the case of 2-D mean flows with buoyancy by solving the linear system for the β- and λ-coefficients presented in Lazeroms et al. 213 with C 1 = c = c Ω =. This model is completed by a suitable expression for N that approximates equation 1a, and from which N θ can be found through equation 1b with c θ5 = 1 2. In the next sections, this modelling approach is tested for a number of different parallel shear flows. 3 tratification in a horizontal channel First, we consider a class of parallel flows that was already investigated in Lazeroms et al. 213, for which the mean temperature gradient is aligned with gravity in the cross-stream direction. In particular, we shall examine the case of a horizontal channel with a temperature difference T = T u T d betweenthe wallsseefigure1. Theimportant invariantcontrollingthe buoyancyterms in this case is γ θ, which can also be written as γ θ = 2II Ri, where Ri = β T gdθ/dy/du/dy 2 is the gradient Richardson number. For γ θ < or T u > T d the stratified channel flow is stable, whereas the case γ θ > or T u < T d corresponds to unstable stratification. 5

6 T u U y g z x T d Figure 1: Geometry for the horizontal channel with upper-wall temperature T u and lower-wall temperature T d. For this particular case the invariants in table 3 can be written in the following way: II = 1 τ U 2, γ θ = τ 2 β T g Θ 2 y y, Θ 2 = K τ Θ 2, K θ y II Ω = II, γ θ Ω = 1 2 II γ θ, Γ 2 = K θ K τβ Tg 2, 11 γ Ω = 1 2 II Γ 2, ϑ Ω = 1 2 II Θ 2, γ = γ θ = γθ Ω = ϑ =, which leads to a simplified form of the linear system for the β- and λ-coefficients, the solution of which is documented in Lazeroms et al Here we note that the denominators of these coefficients contain the following factors: D u = N 2 +2C 2 2 II, Q 1 = D u N 2 θ 2c ΘC 3 NN θ γ θ +c 2 Θ C2 3 Θ 2 Γ 2, Q 2 = 3D u NN 2 θ c ΘC 3 N θ 6N 2 +D u γ θ +4c 2 Θ C2 3 N Θ 2 Γ 2. 12a 12b 12c For stably stratified and neutral flows γ θ, these expressions are guaranteed to be stricly positive as long as N >, which means that no singular behaviour of the model is expected in these cases. This fact is the most important reason for choosing c = c Ω =, without which additional terms would appear in 12 that might cause singularities. However, in the unstably stratified case γ θ > it is not clear if Q 1 and Q 2 change sign. This will probably impose a limit on the negative Richardson number below which singularities could occur. Nevertheless, no singularities were found in the test cases discussed below. 3.1 Determining N and N θ The equation for N can be simplied in this case by using the invariant relations shown in 11. Equations 1 then become as follows: N = c 1 1 II β 1 +γ θ β θ2 β θ3 Γ 2 β g2 γ θ λ θ II λ θ4 Γ 2 λ g1, 13 N θ = c θ1 + 1 N c 1 1, 14 2 r where we also used the fact that λ g4 = see Lazeroms et al. 213 and the assumption c θ5 = 1 2 mentioned previously. An analytical solution for N and N θ satisfying the equations above does not exist, as the equation for N turns out to be a sixth-order polynomial. One can often rely on perturbation methods to find approximate solution to high-order equations when some parameters are small. For example, we could assume that the buoyancydependent invariants in 13, namely γ θ and Γ 2, are small enough so that the equation for 6

7 N can be linearized with respect to these quantities. To zeroth order in γ θ and Γ 2, N can be approximated by the expression proposed by Wallin & Johansson 2, which is exact for non-stratified flows. We call this expression N and it is defined as follows: c N = P 1 + 1/3 P 2 +sign P1 P P1 2 P 2 1/3, P2, 9 c /6cos P1 2 1 P 2 3 arccos P 1, P P 2 2 <, 1 P 2 15 with P 1 = 1 27 c II II Ω c 1, P 2 = P1 2 9 c II II Ω, and c 1 = 9 4 c 1 1. A corresponding approximation for N θ, which we call N θ, can then be found by putting N = N in equation 14. The linearized version of 13 can then be written as follows: N c 1 1 II β 1 +γ θ β θ2 β θ3 Γ 2 β g2 γ θ λ θ II λ θ4 Γ 2 λ g1, 16 where the coefficients β and λ are the zeroth-order approximations of β and λ, i.e.: β = β N = N,N θ = N θ,γ θ = Γ 2 =, 17a λ = λ N = N,N θ = N θ,γ θ = Γ 2 =. 17b Unfortunately, this approach is not generally valid. For stably stratified channel flow, the invariants γ θ and Γ 2 are small near the wall, but they become very large in the centre of the channel, which is the effect of the damping of turbulence in this region. Lazeroms et al. 213 proposed a more complicated approximation for N specifically for these stably stratified cases. In this approximation, equation 13 is not linearized, but a modified version of N is used on the right-hand side of this equation. This approach can be seen as a one-step iteration where all higher-order terms in 13 are ept, and where the modified N is used as an appropriate initial guess. For unstably stratified channel flow, however, the invariants γ θ and Γ 2 do not become large in the centre of the channel, so that the approximation given by 16 might be valid. In this case, it turns out that for strong levels of stratification, turbulent mixing can cause II to be zero in a large region around the centre of the channel. From 15, it follows that N c 1 1 in this limit, corresponding to zero production. A better approximation can be obtained if we account for the positive buoyancy production in the expression for N in regions where II, which can be obtained through the following correction: Ñ = N γ θ λ θ1 Γ 2 λ g1, 18 and the corresponding Ñ θ obtained from 14. This yields the following approximation for N in the case of unstably stratified channel flow: β N c 1 1 II 1 +γ θ β θ2 β θ3 Γ 2 β g2 γ θ λ θ II λ θ4 Γ 2 λ g1, 19 with β = β N = Ñ,N θ = Ñ θ and λ = λ N = Ñ,N θ = Ñ θ. 7

8 5 a.5 b 4.4 U Θ T Figure 2: Comparison of the explicit algebraic model dashed with an eddy-diffusivity model dasheddotted and the DN by García-Villalba & del Álamo 211 solid for the stably stratified horizontal channel, for Re τ = 55 and Ri τ = blue and Ri τ = 48 red. hown are: a mean velocity scaled with u τ and b mean temperature scaled with T. 3.2 Numerical results Next, we show some results of the model and compare them with available DN. The model has been evaluated using a FORTRAN code automatically generated in Maple. The numerical solver uses second-order central differences for the spatial discretization on a collocated grid with 21 gridpoints along the channel half-height, and a first-order implicit Euler scheme in time to search for the steady state tably stratified channel flow The case of stably stratified channel flow has been discussed extensively in Lazeroms et al DN for this case was obtained by García-Villalba & del Álamo 211 in the geometry of figure 1 with a constant streamwise pressure gradient and a positive temperature difference T = T u T d. The flow is governed by three dimensionless parameters: the friction Reynolds number Re τ = u τ h/ν, the friction Richardson number Ri τ = β T g Th/u 2 τ, and the Prandtl number Pr = ν/κ, where h is the channel half-height and u τ is the friction velocity. Themainresultsofthemodelcalculationsareshownin figure2, wherewepresentacomparison of the model with DN and a standard eddy-diffusivity model see appendix A. The calculations were performed for Re τ = 55, Pr =.71 and two different values of Ri τ. One can see that the explicit algebraic model gives very good predictions of the DN, especially for the stratified case with Ri τ = 48, which respresents a high level of stratification. For such a stably stratified flow, turbulence is damped to a large extent in the centre of the channel, which explains the high pea in the normalized mean velocity. The eddy-diffusivity model is not capable of predicting the stratified case correctly. Only in the nonstratified case, the eddy-diffusivity model does a better job in predicting the mean temperature than the explicit algebraic model. However, as already discussed in Lazeroms et al. 213, this issue can be solved with the diffusion correction proposed by Högström et al. 21 see section 4.3, but such a correction is clearly not necessary for the stably stratified case. One should also note that the results presented in figure 2 include a number of near-wall corrections that have been discussed in Lazeroms et al uch corrections consist of near-wall damping functions in the β- and λ-coefficients, which are needed because the wea-equilibrium assumption on which the model derivation is based section 2.1 is generally not valid in the near-wall region. Furthermore, a low-re version of the K-ω model Wilcox, 1993 was used. 8

9 2 a b 15.1 U Θ T c 1 d uv +.4 vθ Figure 3: Comparison of the explicit algebraic model dashed with an eddy-diffusivity model dasheddotted and the DN by Iida & Kasagi 1997 solid for the unstably stratified horizontal channel, for Re τ = 15 and Gr = hown are: a mean velocity scaled with u + τ, b mean temperature scaled with T, c Reynolds shear-stress scaled with u + τ, d wall-normal heat flux scaled with κ Θ/ y y= Unstably stratified channel flow For the case of unstably stratified channel flow, we turn to the DN performed by Iida & Kasagi imilar to the previous case, these simulations have been obtained in the geometry of figure 1 with a constant streamwise pressure gradient, but with a negative temperature difference T = T u T d. The flow can be described by the friction Reynolds number Re τ = u τ h/ν, the Prandtl number Pr = ν/κ and the Grashof number Gr = β T g T 2h 3 /ν 2. In the DN, we have Re τ = 15, Pr =.71 and Gr = A comparison of the model results with the DN is shown in figure 3, along with the outcome of the standard eddy-diffusivity model discussed in appendix A. It appears that the model gives reasonably good predictions of the DN, but the differences between the two model predictions are small. Only the prediction of the mean temperature and to some extent the wall-normal heat flux are significantly better with the new model. However, one should note that the Reynolds number is quite low in this case, which maes the use of the DN for model comparison rather questionable. For this reason, we did not use the near-wall corrections discussed in Lazeroms et al. 213, since they would be active in the entire domain for such a low Reynolds number. Moreover, we need to chec if the formulation for N presented in section 3.1 is self-consistent. From 7a we can see that the exact value for N is directly related to the total productionto-dissipation ratio P + G/ε. ince we use an approximate expression for N, its values will 9

10 Figure 4: Comparison of the formulation for N section 3.1 with the total production-to-dissipation ratio for unstably stratified channel flow, for Re τ = 15 and Gr = hown are: î P +G/ε from DN by Iida & Kasagi 1997, solid P +G/ε resulting from the full model, dashed N c 1 +1, dashed-dotted Ñ c 1 +1, dotted N c not be equal to the final outcome of P + G/ε anymore. It is however important to have an approximation for N close to the exact, self-consistent formulation in order to have the correct asymptotic behaviour of the production for large strain rates see Wallin & Johansson 2. This can be checed by comparing the values of N to P+G/ε, which is done in figure 4. We can see that the expression for N as formulated in section 3.1 is close to the curve of P+G/ε, except near the wall where the Durbin timescale appendix A is active. Hence, we can conclude that our formulation for N is approximately self-consistent. In fact, the modified initial guess Ñ already appears to be a good approximation in itself. The non-buoyant expression N, however, is clearly not sufficient because it goes to zero in the centre of the channel. 4 Mixed and natural convection in a vertical channel In the second type of parallel flows considered in this wor, the mean temperature gradient points in the cross-stream direction and gravity points in the streamwise direction. This means that the mean temperature gradient is now perpendicular to gravity, so that γ θ =. Instead, the important invariant becomes γ θ, which was zero in the previous cases. An example of such a geometry is the vertical channel shown in figure 5, where one wall is heated and the other wall is cooled. The heating causes an upward buoyancy force on one side of the channel, while the cooling on the other side causes a downward buoyancy force. Due to this asymmetry, γ θ changes sign within the channel. We shall consider two different cases for the vertical channel: natural convection where buoyancy is the only driving force, and mixed convection where the main driving force is a pressure gradient and buoyancy causes an aiding and opposing effect on the respective sides of the channel. For the geometry shown in figure 5, the invariants of table 3 can be expressed as follows: II = 1 2 II Ω = II, τ U 2, γ θ y = 1 2 τ3 β T g Θ y γ θ Ω = γθ, U y, Θ 2 = K K θ τ Θ 2, y Γ 2 = K θ K τβ Tg 2, 2 γ Ω = 1 2 II Γ 2, ϑ Ω = 1 2 II Θ 2, γ = γ θ = γ θ Ω = ϑ =. Using these relations, simplified expressions for the β- and λ-coefficients can be found, which are 1

11 x z y U g T h T c Θ Figure 5: Geometry for the vertical channel with wall-temperature difference T = T h T c >. given in appendix C. Note that these coefficients contain denominators with the factor Q = 3D u N 2 θ +2c2 Θ C2 3 Γ 2 Θ 2 12C 2 C 3 c Θ N θ γ θ, in which all terms are guaranteed to be positive except for the last one. Thus, the sign of the invariant γ θ also determines the possibility of singularities in the model, as was the case for γθ in the horizontal channel. No singular behaviour is expected in regions where γ θ < i.e. the cooled side of the channel with mixed convection, or the interior part of the channel with natural convection. However, it is not trivial to draw conclusions about the possibility of singularities if γ θ >. In the test cases considered here, no problems with singularities have been encountered, but this is not guaranteed for cases with very strong convection. 4.1 Determining N and N θ By applying the relations in 2 to equations 1 with c θ5 = 1 2 we find the following: N = c 1 1 II β 1 +γ θ β θ4 + Γ 2 β g2 2γ θ β θ1 γ θ λ θ2 Γ 2 λ g1, N θ = c θ N c 1 1 r 21a, 21b where we also used the fact that λ θ3 = λ g4 = see appendix C. Finding the approximation for N turns out to be more straightforward for the vertical channel than for the horizontal channel. For this purpose, we tae equation 21a and approximate the coefficients β and λ occurring here by β and λ, defined as follows: β = β N = N,N θ = N θ,γ θ = Γ 2 =, 22a λ = λ N = N,N θ = N θ,γ θ = Γ 2 =, 22b where N and N θ are again given by 15. In other words, we neglect the buoyancy-related terms occuring in β and λ in 21a, which basically linearizes this equation with respect to γ θ and Γ 2. In contrast to the horizontal channel with stable stratification, turbulence is strong in the centre of the vertical channel, so that we do not encounter the same problem as for the stably stratified horizontal channel, where both N and the invariants become very large in the center, which justifies this method of approximating 21a. 4.2 Correction for neglected terms in the heat-flux equation As explained in Lazeromset al. 213, the assumptionc = c Ω = is needed to avoidsingularities in the case of stably stratified horizontal channel flow, with the additional advantage that the 11

12 system for the β- and λ-coefficients is greatly simplified. The drawbac of this assumption is the fact that the streamwise heat flux uθ is underpredicted by the model. For horizontal channel flow, this is only a minor issue, since this component does not appear in the transport equations 1 and is only a secondary quantity. However, in vertical channel flow, the streamwise component appears in the buoyancy term G in the transport equation for K, which maes its correct prediction more important. To improve the model in this respect, we propose the following ad hoc correction to the heat flux: ξ i ξ i + ξ i, ξi = 1 N θ c ij + c Ω Ω ij ξ j, 23 i.e. an approximation of the neglected term in 6b involving the uncorrected heat flux is added a posteriori. For the new model constants we choose the values c = c Ω = 1, which corresponds to the optimal values for c and c Ω found by Wiström et al. 2. ince the buoyancy term G also appears in the definition of N equation 7a, correction 23 also needs to be incorporated in the expression for this quantity to eep the formulation self-consistent. From 7a one can see that this correction should involve the term Γ ξ, which can be approximated in the way described in section 4.1, which leads to the following correction: N N + c Γ i ij + c Ω Γ i Ω ij λ N θ V j, 24 where λ is given by 22b and V j corresponds to the basis vectors in appendix B. In the present context of the vertical channel, this correction amounts to: N N + c + c Ω γ θ λ θ II λ θ4. 25 Equations 23 and 24 are, however, formulated in a coordinate-free way, and they can also be used in a more general context. Also in the case of horizontal channel flow considered in section 3, 23 gives the appropriate correction to the streamwise heat-flux, while the correction term in 24 is zero. 4.3 Diffusion correction ince the explicit algebraic model is based on neglecting the diffusion terms in the equations for a ij and ξ i, the model is expected to give problems in regions where diffusion is important, e.g. in the centre of channel flow. uch regions can generally be characterized by a small production-todissipation ratio. Wallin & Johansson 2 proposed a correction to the non-stratified EARM to tae into account diffusion in regions with P/ε < 1, which was later extended by Högström et al. 21 for the passive scalar flux. In section 3.2 it became clear that such corrections are not needed for horizontal channel flow with buoyancy forces. For the unstably stratified case, the total production-to-dissipation ratio is clearly not small in the centre of the channel, so that the role of diffusion is limited here. Although this ratio becomes small for the stably stratified case, the complete damping of turbulence again maes a diffusion correction unnecessary. In the case of a vertical channel with mixed convection, however, the total production-to-dissipation ratio again becomes zero within the channel, so that the diffusion correction might be needed. Below we discuss a method to extend the diffusion correction to cases with convection. Wallin & Johansson 2 give an analysis showing that diffusion can be incorporated in the model through the constant c 1 : N θ c 1 c 1 C D D ε, 26 12

13 where C D = 2.2 and D is the diffusion term in the K-equation. For the latter term, a model expression is used that is only active in regions with P/ε < 1. This expression can be extended to the case of convection in a vertical channel in the following way: { D ε max 1 P +G }, ε max {1+II β eq 1 +γθ βeq θ4 + Γ 2 β eq g2 +2γ θ βeq θ1 +γθ λeq θ2 + Γ 2 λ eq g1 c + c Ω γ θ λ eq θ II λ eq } θ4 c θ1 1/2r,, 27 in which β eq and λeq are obtained from the β and λ by putting γ θ = Γ 2 = and N = c 1 and N θ = c θ1 1/2r. Hence, we have linearized the expression for P+G/ε with respect to γ θ and Γ 2, and for N and N θ we use the exact values in the case of P +G/ε = 1 see equation 7, similar to the approach of Wallin & Johansson 2. The last term in 27 corresponds to the heat-flux correction discussed in the previous subsection. The diffusion correction for the turbulent heat flux presented by Högström et al. 21 can be extended in a similar way: c θ1 c θ1 C θ D D ε, 28 with C θ D = 8.. In summary, corrections 26 and 28 with model expression 27 should be used throughout the models for N and N θ, except in 27 itself where the constant values c 1 = 1.8 and c θ1 = 4.51 table 2 are used. 4.4 Numerical results In the following we present a comparison of the model with available DN data. The model results were obtained using the same numerical solver as described in section 3.2. Furthermore, both the correction for the heat flux described in section 4.2 and the diffusion correction described in section 4.3 were used Mixed convection For the validation of the model in the case of mixed convection, we use the DN by Kasagi & Nishimura These simulations were done in the geometry shown in figure 5 where the flow is driven both by the temperature difference between the walls and by a pressure gradient. In contrast to pressure-driven channel flow without buoyancy, the friction velocity u τ is not the same for each wall. The flow is therefore best described using the friction velocity based on the average of the shear stress at each wall, defined as: u τ 2 = u2 τ,1 +u2 τ,2. 2 In the DN considered here, the friction Reynolds number Re τ = u τh/ν is equal to 15, where h is the channel half-width. The strength of the buoyancy forcing is described by the Grashof number Gr = β T g T2h 3 /ν 2 = Furthermore, the Prandtl number Pr = ν/κ was chosen to be.71. A comparison of the model and the DN for a number of turbulence statistics is shown in figure 6. It is clear that the model is able to capture an important qualitative effect, namely the asymmetry in the profiles caused by the buoyancy force. The heated wall on the left side of the channel enhances the flow in the direction of the pressure force, while the cooled wall on the other side causes an opposing flow. Apart from this qualitative result, the model appears to give a reasonably good quantitative prediction of the statistics shown, especially for the mean velocity and mean temperature profiles figures 6a,b. Also shown in figure 6 are the results for 13

14 2 a.5 b 15 U 1 Θ T 5 uv c vθ.4.2 d Figure 6: Comparison of the explicit algebraic model dashed with an eddy-diffusivity model dasheddotted and the DN by Kasagi & Nishimura 1997 solid for the vertical channel with mixed convection, for Re τ = 15 and Gr = hown are: a mean velocity scaled with u τ, b mean temperature scaled with T, c Reynolds shear-stress scaled with u τ, d wallnormal heat flux scaled with κ Θ/ y y=h. The heated wall is located on the left side of the figure. the K-ω model with standard expressions for the eddy viscosity and eddy diffusivity, as described in appendix A. The explicit algebraic model gives only a slightly better prediction than the standard model, but one should again note that the Reynolds number for this particular case is quite low, and no near-wall corrections other than the Durbin timescale appendix A have been used. Figure 6d also shows a dip in the heat flux profile around the position of the mean velocity maximum, which is probably caused by the diffusion correction. Figure 7 compares the results for the streamwise heat flux with the DN, both with and without the heat-flux correction shown in 23. We can see that this correction indeed has the desired effect of improving the prediction of uθ, and the prediction is reasonably good in the centre of the channel. However, the model is unable to correctly predict the near-wall peas. imilar problems in the near-wall region can be seen in figures 6c,d. Also note that a standard eddydiffusivity model would predict uθ =, which is a clear disadvantage compared to the explicit algebraic model. In order to chec to appriopriateness of the expressions for N presented in section 4.1, we again compare them to the total production-to-dissipation ratio P + G/ε resulting from the model. This is shown in figure 8 together with the DN. We can see that the approximation for N c 1 +1 agrees almost perfectly with P +G/ε, except very close to the wall where the Durbin timescale again causes an inconsistency. Furthermore, the non-buoyant expression N already appears to be close to the desired result, which shows that it can indeed be used as an initial guess without further corrections. 14

15 6 4 2 uθ Figure 7: The streamwise heat flux, scaled with κ Θ/ y y=h, in the vertical channel with mixed convection, for Re τ = 15 and Gr = hown are: the DN by Kasagi & Nishimura 1997 solid, the model with correction 23 dashed, the model without correction 23 dashed-dotted Figure 8: Comparison of the formulation for N section 4.1 with the total production-to-dissipation ratio for the vertical channel with mixed convection, for Re τ = 15 and Gr = hown are: î P +G/ε from DN by Kasagi & Nishimura 1997, solid P +G/ε resulting from the full model, dashed N c 1 +1, dashed-dotted N c Natural convection Next we turn to the case of natural convection, in which buoyancy is the only driving force. DN for this case in the geometry of figure 5 have been performed by Versteegh & Nieuwstadt 1998, 1999, which will be used for comparisons. This configuration results in an antisymmetric mean velocity profile, with upward flow on the heated side of the channel and downward flow on the cooledside. Theflowcanbe describedbytheprandtlnumberpr = ν/κandthe modifiedrayleigh number H = PrRa = β T g T2h 3 /κ 2, where h is again the channel half-width. In the DN we have Pr =.71 and H = ome model predictions for the natural-convection case are shown in figure 9 along with the DN and the corresponding results for a standard eddy-diffusivity modelappendix A. The model is able to give reasonably good predictions for the mean velocity, mean temperature, and Reynolds shear stress. The wall-normal heat flux appears to be somewhat underpredicted, but this is even worse for the standard eddy-diffusivity model. One interesting aspect of the natural-convection case is the slightly negative values of the Reynolds shear stress figure 9c close to the wall. 15

16 15 a.5 b.4 2h κ U 1 5 Θ T x 1 4 c 15 d 2h 2 κ 2 uv hvθ κ T Figure 9: Comparison of the explicit algebraic model dashed with an eddy-diffusivity model dasheddotted and the DN by Versteegh & Nieuwstadt 1998, 1999 solid for the vertical channel with natural convection and H = hown are: a mean velocity, b mean temperature, c Reynolds shear-stress scaled, d wall-normal heat flux. Only the heated half of the channel is shown. If this quantity is described with a nonnegative eddy viscosity uv = ν t U/ y with ν t, the position where uv changes sign should coincide with the maximum in the mean velocity. As mentioned by Versteegh & Nieuwstadt 1999, this is not the case in the DN, which means that it is impossible to describe uv with a nonnegative eddy viscosity. This fact reveals a clear advantage of the current model over any eddy-viscosity approach, since the explicit algebraic model gives contributions to uv that are not proportional to the mean strain rate 12. Again we would lie to stress that no near-wall corrections have been used in the presented results other than the Durbin timescale, and adding the near-wall corrections used in Lazeroms et al. 213 does not seem to give any clear improvement of the results. This might be caused by the fact that only buoyancy acts as a driving force here, which is a fundamental difference with previous cases. The need for near-wall corrections becomes apparent when considering the streamwise heat flux shown in figure 1. It appears that uθ becomes negative near the wall, while the DN data stay positive throughout the channel. This effect might even be worsened by the correction in 23. However, one should note once more that the parameters used here result in a quite low Reynolds number Re τ 17, which can be deduced from data given by Versteegh & Nieuwstadt For such a low Reynolds number the conventional near-wall corrections would be active in almost the entire domain. Finally, we again turn to the self-consistency of the formulation for N as shown in section 4.1. Figure11revealsthattheapproximationforN c 1 +1isveryclosetothecurveforP+G/ε,except 16

17 huθ κ T Figure 1: The streamwise heat fluxin the vertical channel with natural convection and H = hown are: the DN by Versteegh & Nieuwstadt 1999 solid and the result of the explicit algebraic model dashed Figure 11: Comparison of the formulation for N section 4.1 with the total production-to-dissipation ratio for the vertical channel with natural convection and H = hown are: solid P +G/ε resulting from the full model, dashed N c 1 +1, dashed-dotted N c close to the wall where the Durbin timescale correction is active. The non-buoyant expression N is not a good approximation in itself, but it does not deviate a lot from the desired behaviour, so that it can still be used as an initial guess without additional corrections. Figure 11 also shows that the production-to-dissipation ratio does not become small in the centre of the channel, which maes the use of the diffusion correction discussed in section 4.3 questionable. However, we encountered numerical difficulties when trying to evaluate the model without diffusion corrections, possibly due to the negative values attained by uθ near the wall. Ideally, the current case should therefore be investigated with appropriate near-wall corrections, for a higher Reynolds number, and without the diffusion correction. 5 Conclusions everal extensions of the explicit algebraic model proposed by Lazeroms et al. 213 have been discussed for different types of parallel shear flows that include buoyancy forces. In particular, we used the model framewor for two-dimensional mean flows given by 9 and 1 and derived new expressions for the β- and λ-coefficients for the vertical channel appendix C, while the horizontal 17

18 channel case was investigated using the β- and λ-coefficients already found by Lazeroms et al This shows that the linear system for these coefficients obtained by Lazeroms et al. 213 can readily be used to derived the linear part of the model for different flow cases. The biggest issue, however, appears to be the nonlinear part of the model, expressed by equations 1 for N and N θ. In sections 3.1 and 4.1, we discussed an approximate solution methods for these higher-order polynomial equations, which are based on a linearization with respect to the buoyancy-dependent invariants γ θ, γ θ and Γ 2. We also discussed that such a linearization is not always valid because there are cases in which the buoyancy-dependent invariants are not sufficiently small in the entire domain. Although a simple linearization suffices for the vertical channel section 4.1, the case of the horizontal channel is much more difficult. The particularly complicated case of stably stratified channel flow was already discussed in Lazeroms et al. 213, where a one-step iteration method with a modified initial guess was used. In section 3.1, we showed that a linearization method can still be used for unstably stratified channel flow with an additional correction for regions where the strain rate II tends to zero. To summarize, different expressions for N and N θ have been used for different flow cases, which is not desirable in practical applications. However, the discussion presented here can shed light on a possible general formulation for N, and since the different flow cases shown here are easily expressed in terms of the invariants in table 3, such a formulation might not be too far away. It is, however, beyond the scope of the current paper. Furthermore, some additional corrections were discussed for the vertical-channel case. In section 4.2, a method was proposed to correct the heat-flux model for discrepancies that might occur due to the terms that were neglected by the assumption c = c Ω =. For the vertical channel discussed here, this boils down to a correction for the streamwise heat flux uθ, which determines the buoyancy production G of turbulent inetic energy. However, correction 23 is expressed in a coordinate-free way and it can also be used in other cases. ection 4.3 presents an extension of the diffusion corrections already proposed by Wallin & Johansson 2 and Högström et al. 21, which should improve the model in regions where the neglected diffusion terms of a ij and ξ i are dominant, i.e. regions where the production-to-dissipation ratio P +G/ε is small. Again, this correction has only been used in specific cases, and a more general formulation is needed for practical applications. We would lie to note once more that the DN used for comparison in sections 3.2 and 4.4 all have a fairly low Reynolds number except for the DN by García-Villalba & del Álamo 211 for stably stratified channel flow already discussed in Lazeroms et al This maes it difficult to correctly evaluate the model in the near-wall region, where explicit algebraic models are nown to become less accurate and where near-wall correction are usually applied. Therefore, for a correct evaluation of the model, new DN data for higher Reynolds numbers is needed. It might also be necessary to modify the traditional near-wall corrections shown in Lazeroms et al. 213 in order to obtain a good near-wall behaviour for the vertical channel with natural convection. Nevertheless, we have shown in this wor that the explicit algebraic model is capable of giving reasonably good predictions even without a near-wall treatment, and that these predictions are often significantly better than for a standard eddy-diffusivity model. Although the test cases presented here are simplified parallel shear flows, the results are a useful first step towards a more general model for two-dimensional mean flows. The financial support from the Bert Bolin Centre for Climate Research, which in turn is funded by the wedish Research Council VR and wedish Research Council for Environment, Agricultural ciences and patial Planning FORMA, is gratefully acnowledged, as well as direct funding by the wedish Research Council through grant number

19 A The K-ω and K θ models The model presented in this wor should be used together with an appropriate model for K, ε, K θ and ε θ. Here we use the framewor of the K-ω model proposed by Wilcox 1993, where ω = ε/β K is an inverse timescale. The transport equations for K and ω are expressed as follows: DK Dt U = u i u j β ωk + [ ν + ν ] t K β T g u θ, x j x σ K x β ω 2 +2 ωω w + x j x + νt ω w γ ω +ω w x σ ω x K β Tg u θ, D ω Dt = α ω +ω w K u iu j U [ ν + ν ] t ω σ ω x 29a 29b in which ν t is the effective eddy viscosity, defined below. The equation for ω = ω+ω w has been written in a form that maes it suitable for calculations in wall-bounded flows. The quantity ω w represents the near-wall asymptote of ω: ω w = 6ν βy 2, while we solve for the remainder ω, which is zero at the wall. This form of the ω-equations was proposed by Gullman-trand 24 also shown in Gullman-trand et al. 24. Equations 29 contain the following constants: α = 5/9, β = 3/4, β = 9/1, σ K = σ ω = 2.. The parameter γ in front of the ω-buoyancy term was given the constant value.5 in Lazeroms et al. 213, noting that this value may depend on the orientation of the wall with respect to gravity. ince the value γ = is more appropriate for vertical wall-bounded flows see, e.g., Rodi 2, we choose the following expression for this parameter: } γ γ =.5tanh{ θ which gives γ = for γ θ = vertical channel and γ =.5 for γ θ = horizontal channel. The quantity K θ is determined through the following transport equation: DK θ = u j θ Θ ε θ + [ ν + ν ] t Kθ, 3 Dt x j x σ K x where the dissipation rate ε θ is modelled as: γ θ, ε θ = K θ ε, 31 rk with a constant timescale ratio r = K θ /ε θ /K/ε =.55. The effective eddy viscosity ν t can be expressed in the following ways: ν t = 1 2 Kτ β 1 +γ θ β θ2 β θ3 Γ 2 β g2 explicit algebraic model, horizontal channel = 1 2 Kτ β 1 +γ θ β θ4 + Γ 2 β g2 explicit algebraic model, vertical channel = K ω +ω w, standard eddy-viscosity model 19

20 The standard eddy-viscosity model is used for comparison in sections 3.2 and 4.4, together with a related eddy diffusivity κ t = ν t /.9. In the case of these standard models, the Reynolds shear stress and wall-normal heat flux are obtained from: uv = ν t U y, vθ = κ Θ t y. Note that various parts of the model have been formulated in terms of the turbulence timescale τ, for which we use the following expression proposed by Durbin 1993 that limits it to the Kolmogorov timescale: K τ = max ε,c τ ν, 32 ε with C τ = 6.. This expression is needed to improve the model in the near-wall region, where the turbulent inetic energy K tends towards zero. B Basis tensors and vectors in the explicit algebraic model for 2-D mean flows Below in table 4 we have listed the ten symmetric and traceless tensors and eight vectors that form the basis needed to obtain the explicit algebraic model for a ij and ξ i in two-dimensional mean flows with buoyancy equation 9. T 1 = T 2 = II I T 3 = Ω II ΩI = II Ω T 2 II T 4 = Ω Ω T θ1 = Γ Θ+Θ Γ 2 3 γθ I T θ2 = Γ Θ +Θ Γ 2 3 γθ I T θ3 = Γ ΘΩ ΩΘ Γ 2 3 γθ Ω I T θ4 = Γ ΘΩ ΩΘ Γ 2 3 γθ Ω I V θ1 = Θ V θ2 = Θ V θ3 = ΩΘ V θ4 = ΩΘ V g1 = Γ V g2 = Γ V g3 = ΩΓ V g4 = ΩΓ T g1 = Γ Γ 1 3 Γ 2 I T g2 = Γ ΓΩ ΩΓ Γ Table 4: Complete set of tensors and vectors needed for two-dimensional mean flows with stratification. The symbol refers to the dyadic product or tensor product of two vectors, i.e. A B ij = A ib j. Note that I = δ ij is the identity tensor in 3-D. C β- and λ-coefficients for vertical parallel flows The linear system for the coefficients in the expansion for a ij and ξ i equation 9, which is presented in Lazeroms et al. 213, can be solved for the specific case of vertical parallel flows with Θ Γ =. The result is shown below. Note that these expressions include the assumptions C 1 = c = c Ω =. β 1 = 8 N, β 2 = λ θ3 = λ g2 = λ g3 = λ g4 =, β 4 = 8 C 2, 15D u 15D u 2