Numerical Heat Transfer, Part A, 44: 789–810, 2003 Copyright # Taylor & Francis Inc. ISSN: 1040-7782 print=1521-0634 online DOI: 10.1080/0407780390220511
DEVELOPMENT OF A LOCAL SUBGRID DIFFUSIVITY MODEL FOR LARGE-EDDY SIMULATION OF BUOYANCYDRIVEN FLOWS: APPLICATION TO A SQUARE DIFFERENTIALLY HEATED CAVITY A. Sergent LIMSI-CNRS, France and LEPTAB, University of La Rochelle, La Rochelle, France
P. Joubert LEPTAB, University of La Rochelle, La Rochelle, France
P. Le Que´re´ LIMSI-CNRS, Orsay, France
We present a local subgrid diffusivity model for the large–eddy simulation of naturalconvection flows in cavities. This model, which does not make use of the Reynolds analogy with a constant subgrid Prandtl number, computes the subgrid diffusivity independently from the subgrid viscosity along the lines of the mixed scale model for eddy viscosity. First, an a-priori test is performed from a direct numerical simulation (DNS) approach in order to compare the respective effects of the subgrid viscosity model and that introduced by the QUICK scheme used to discretize the convective terms in the momentum equations. Then the subgrid diffusivity model is applied to the case of a two-dimensional square cavity filled with air for a Rayleigh number of 5 1010. Comparisons with DNS reference results demonstrate significant improvements in capturing the general pattern of the flow and particularly in predicting the transition to turbulence in the boundary layers as compared with Reynolds analogy results. The influence of subgrid diffusivity on the local heat transfer is also examined.
1.
INTRODUCTION
Natural convection in cavities with active lateral walls is a prototype configuration for a variety of practical situations such as cooling of electronic devices or air flow in buildings and as such has received a lot of attention over the past 40 years. The early studies concentrated on elucidating the flow structure and heat transfer in
Received 5 February 2002; accepted 21 January 2003. The computations were performed on the NEC SX5 of the CNRS-IDRIS computing center. We thank Dr. C. Tenaud and Dr. L. Ta Phuoc for many useful discussions. Address correspondence to Patrick Le Que´re´, LIMSI-CNRS, BP 133, 91403 Orsay Cedex, France. E-mail:
[email protected] 789
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NOMENCLATURE Ck Cu g hi H k Nu p Pr PrSGS qc Ra S t ui U
subgrid scale (SGS) diffusivity model coefficient SGS viscosity model coefficient gravitational acceleration subgrid heat flux cavity height turbulent kinetic energy Nusselt number pressure molecular Prandtl number SGS Prandtl number kinetic energy at the cutoff Rayleigh number strain rate tensor of resolved scales time nondimensional velocities nondimensional horizontal velocity
W X xi Z a aSGS D ~ D y u uSGS v v v v v tij
nondimensional vertical velocity horizontal coordinate Cartesian space coordinate vertical coordinate molecular thermal diffusivity SGS diffusivity grid filtered size ~ >D test filtered size, D nondimensional temperature molecular viscosity SGS viscosity grid filtered variable test filtered variable subgrid variable time-averaged variable time fluctuation subgrid stresses tensor
configurations characterized by small Rayleigh numbers corresponding to relatively small temperature differences and typical dimensions. After the pioneering work of Batchelor [1], a decisive step was the identification of the boundary-layer regime by Elder [2] and Gill [3], which allowed for a heat transfer correlation that was later improved by Bejan [4]. Simultaneously, this configuration served as a prototype configuration for the development of numerical algorithms aiming at solving the incompressible Navier-Stokes equations, and benchmark solutions for the square cavity with adiabatic walls are now available up to the end of the steady laminar regime, either in two dimensions or in three dimensions (see G. de Vahl Davis [5], Le Que´re´ [6], and Tric et al. [7]). However, in many configurations of practical interest the temperature differences and dimensions are such that the corresponding flows are either transitional or turbulent. The prediction of corresponding flow structure and related heat transfer in such configurations is thus of great interest but still remains a formidable challenge despite the numerous efforts that have already taken place. This is due mostly to the fact that these flows display complex behavior, partly because of their high sensitivity to the thermal boundary conditions on the walls of the cavity, which makes them very difficult to investigate. Even if most of the 2-D mechanisms of instability are now reasonably understood [8], these flows are still a challenging field of research for 3-D effects and for the turbulent domain. Many experimental studies have already been carried out for transitional or turbulent conditions [9–12], which provide valuable results for numerical simulations and the tuning of turbulence models, but there still remains to fully characterize the 3-D structure of the flows and to explain the differences observed between experiments and numerical results. In particular, among the major discrepancies that remain unexplained in the turbulent regime is the value of the vertical stratification in the core region of the cavity. As said above, from a computational point of view, numerous studies have been devoted to this problem, and accurate solutions are now available in two and
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three dimensions up to the end of the steady laminar regime. Concerning the turbulent regime, the state of the art is far from being satisfactory. An overview of the results obtained for an early comparison exercise between different k–E type models can be found in Henkes and Hoogendoorn [13], which showed large differences with corresponding results from 2-D direct numerical simulation (DNS) performed by Paolucci [14] using finite differences, or Xin and Le Que´re´ [15] in a cavity of aspect ratio 4 using spectral methods. The trends observed in [15] were later confirmed by Le Que´re´ in a square cavity [16] and by Nobile [17]. These trends from 2-D simulations require, of course, confirmation in three dimensions, but for cavities with active lateral walls, 3-D direct numerical simulations are presently restricted to values of Rayleigh numbers corresponding the end of the laminar regime [7, 18] or to the beginning of the transitional regime. As 3-D direct numerical simulations of fully turbulent regimes seem presently out of reach and will long remain prohibitively expensive, it is of great interest to develop alternative numerical approaches. One possible way is to develop specific Reynolds Averaged Navier–Stokes equations (RANS) models [19], but, due to the fact that these flows are characterized by large scale fluctuations in low-speed regions, the large-eddy simulation (LES) technique seems a very appealing approach and is therefore drawing increasing attention. Some LES results have recently been reported in the literature, but essentially for mixedconvection flows in enclosures [20, 21]. Although in these articles the pure natural convection case of the differentially heated cavity is also considered, very little results are given for this configuration. In another recent article, Bastiaans et al. [22] performed 2-D and 3-D LES as well as 3-D DNS for turbulent thermal plumes in confined enclosures. We nevertheless must note that this configuration gives a completely different structure of the flow as compared with the precedent case and leads to very different numerical difficulties. In all of these articles, different subgrid models are used and compared, but most of them assume that the thermal subgrid diffusivity can be deduced from the subgrid viscosity through a Reynolds analogy with a constant subgrid Prandtl number. Those that alleviate the assumption of constant subgrid Prandtl number by the use of a dynamic approach, for instance, Peng and Davidson [20], still assume that the subgrid diffusivity depends mainly on a time scale which is that of the resolved dynamic scales. We believe that this assumption cannot hold for the whole range of Prandtl numbers, due to the very different characteristics of the dynamic and thermal fields for either very small or very large Prandtl numbers. We present in this article a LES approach for natural-convection flows based on an original local diffusivity model based on its own time scale independent of that of the subgrid viscosity. This model is tested in the configuration of a square differentially heated cavity at a Rayleigh number of 5 1010, which is the same configuration as that used for the comparison exercise detailed by Henkes and Hoogendoorn [13]. Note that this value is more than two orders of magnitude above the onset of unsteadiness. This article is devoted mainly to presenting the local subgrid diffusivity model and performing validation tests for the overall numerical procedure and in particular to understanding the interplay between the numerics and the modeling by comparing different types of models and convective discretization schemes. The article is organized as follows: after presenting the local diffusivity model, we study the
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respective contribution of the viscosity introduced by the subgrid diffusive term and by the QUICK scheme used to discretize the convective fluxes. This is done by an ‘‘a priori’’ test with a DNS approach, taken as reference. Then, we present the improvement introduced by the thermal diffusivity model compared to a Reynolds assumption using a constant subgrid Prandtl number. As final introductory remark, let us mention that we are fully aware that 2-D simulations, no matter how chaotic they might be, cannot reproduce real turbulent flows, and the present developments and tests will have to be carried out in three dimensions to check for their degree of generality. 2. GOVERNING EQUATIONS AND SGS MODELS 2.1. Filtered Navier–Stokes Equations The governing equations for the large–eddy simulation of an incompressible fluid flow under Boussinesq assumption are classically derived by applying a convolution filter to the unsteady momentum and energy equations. The resulting set of nondimensional equations reads: 8 qui > > ¼0 > > > qx i > > > < qui qui uj q p q quj qtij 1=2 qui þ ¼ þ Pr Ra þ þ Pr ydiZ ð1Þ qx qx qt qx qx qx qx > j i j j i j > > > > > qy quj y q qhj > 1=2 qy > þ ¼ Ra : qt qxj qxj qxj qxj These equations were made dimensionless by using the height of the cavity, H, as reference length, the temperature difference between the two vertical isothermal walls of the cavity, Dy, and the natural-convection characteristic velocity, UON ¼ a Re1=2 H: In the above equations, the subgrid-scale (SGS) stress tensor t and the subgrid heat flux vector h, defined as ( tij ¼ ui uj ui uj ð2Þ hj ¼ uj y uj y need to be modeled. Because only eddy-viscosity type models are considered, the isotropic part of the SGS tensor is added to the filtered pressure, leading to the definition of a modified pressure, 1 p? ¼ p þ tii 3 The trace of the SGS tensor t defines a subgrid kinetic energy kSGS ¼ 12tkk , which involves unresolved scales and interaction between unresolved and resolved scales. Its budget equation reads (see, e.g., [23]): qkSGS q uj kSGS þ ¼ Diss þ Diff gb yuz yuz tij Sij qt qxj
ð3Þ
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where Diss and Diff are the dissipative and diffusive terms, respectively, while the last term appearing in this equation represents the production of subgrid kinetic energy due to the strain rate of resolved scales. Positive values of tij Sij indicate energy transfer to the smallest scales of the flow, whereas negative values correspond to a backtransfer of energy from small to large scales. It is known that the behavior of this term constitutes one of several possible criteria for validation of a SGS model [24, 25].
2.2. Subgrid-Scale Models 2.2.1. Subgrid-viscosity model. All the SGS models studied in the present work belong to the eddy-viscosity family, and therefore assume a linear relationship between the deviatoric part of the subgrid tensor td and the resolved strain rate tensor S: td ij ¼ uSGS Sij
ð4Þ
where uSGS is the subgrid viscosity and 1 qui quj Sij ¼ þ 2 qxj qxi In order to simulate flows which are not fully turbulent or with both laminar and transitional regions such as large Rayleigh number air flow in cavities, the subgrid model should be able to adjust itself to locally inhomogeneous flow, which is usually done by using a Germano-Lilly dynamic procedure [26, 27]. However, as mentioned by Zhang [21], who used it to compute indoor air flow, this approach is not reliable, as the computations become very unstable when there is no homogeneous flow direction, as is the case in a 2-D cavity. This led Peng and Davidson [28] to use the Smagorinsky model associated to a damping function for a cubical 3-D cavity with noslip and adiabatic walls, whereas they retained the dynamic procedure in the case of a 3-D cavity with periodic flow in the depth direction. In this work we chose to adapt the subgrid model to local air flow conditions by using the mixed-scale model, developed by Sagaut and Ta Phuoc [29, 30]. This model stems from a Smagorinsky model in which this local adaptation is achieved by taking explicitly into account the kinetic energy at the cutoff, qc. The subgrid viscosity is then evaluated as 3=2 1=2 1=4 S qc
uSGS ¼ Cu D
ð5Þ ffiffiffiffiffiffiffiffiffiffiffiffi ffi q where S ¼ 2Sij Sij , qc stands for the kinetic energy at the cutoff, qc ¼ 12 u0i u0i , and Cu is a constant. Following Bardina’s similarity hypothesis [31], the velocity field at the cutoff, u0i , can be estimated by filtering the resolved velocity field with a test ~ >D , and so u0 ¼ ui u~i . The small-scale filter coarser than the implicit one, D i dependency of this model ensures that it will adapt itself to the local state of the flow, and vanish in fully resolved regions of the flow and near the walls.
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2.2.2. Subgrid-diffusivity model. Concerning the heat transfer equation [Eq. (1)], most subgrid models currently used are eddy-diffusivity models and relate the subgrid heat flux components, hj, to the large-scale temperature gradient by means of a SGS diffusivity, aSGS, as hj ¼ aSGS
qy qxj
ð6Þ
As said above, aSGS is usually computed from uSGS assuming a Reynolds analogy and a constant subgrid Prandtl number: aSGS ¼ uSGS = PrSGS . This assumption amounts to considering that the small thermal scales depend solely, through the subgrid viscosity model, on the resolved dynamic scales. Therefore it is suspected not to hold in the case of turbulent natural convection, where the flow is produced by buoyant forces and not through a dynamic forcing. Furthermore, it is known that, even under turbulent conditions, the thermal and velocity scale distributions depend very strongly on the molecular Prandtl number, and the assumption of constant PrSGS is likely not to hold over the whole range of molecular Prandtl number. For instance, in the case of a passive scalar diffusion at a Prandtl number of the order of 1 in isotropic turbulence, Lesieur and Rogallo [32] present spectrums evolutions of eddy viscosity and diffusivity, which confirm the fact that there is not a simple relationship between uSGS and aSGS over the whole range of scales. Even though this argument does not boldly carry over to natural convection, where temperature cannot be considered as a passive scalar, the mere introduction of a subgrid Prandtl number is very surprising since the ratio uSGS =aSGS never appears as an independent parameter in the governing equations. We have thus developed a local subgrid diffusivity model [33] along the lines of the mixed-scale viscosity model [Eq. (5)]. In this approach, the subgrid diffusivity is expressed as a weighted geometric average of two contributions. The first one comes from the Smagorinsky model using the resolved thermal scales [34], while the second comes from the turbulent kinetic energy model [31] based on the subgrid scales. As the functional approach of Smagorinsky assumes that the most important effect of the interaction between the resolved and the subgrid scales is the energy exchange, we focus on the transport equation of the subgrid-scale heat flux, qffiffiffiffiffiffiffiffiffiffiffi FSGS ¼ 12hk hk . This equation reads: qF2SGS q uj F2SGS q ui qy þ ¼ Diss þ Diff gb yy yy hi hj þ hi tij qxj qt qxj qxj
ð7Þ
where Diss and Diff are dissipative and diffusive terms and the last term in parentheses is the subgrid energy production term, DSGS. Comparing with Eq. (3) for the subgrid kinetic energy, kSGS, we note that the mechanisms of energy exchange between the resolved and the subgrid scales are analogous for kSGS and FSGS. This analogy allows us to derive an expression for the viscous dissipation of the SGS heat flux energy similar to the viscous dissipation of the SGS kinetic energy: D E D E 2 T2 hDi ¼ aSGS D
ð8Þ
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where T is a scalar quantity of dimension K=LT, which should come from a second-order tensor so designed as to preserve invariant and symmetry properties of the flow and temperature fields. The classical assumption of local energy equilibrium leads to
q ui qy þ hi tij hDi ¼ hDSGS i ¼ hi hj qxj qxj
ð9Þ
Substituting tij and hi by their definitions and nSGS by its functional expression (that is, the product of the square of a characteristic length with the norm of the resolved strain rate tensor) results in D E 2 hDi ¼ aSGS D 1 þ
1 nSGS =aSGS
qy qy Sij S qxi qxj
ð10Þ
Invariance considerations lead replace S by Sij in the above expression, us to pffiffiffiffiffiffiffiffiffiffiffi qy which then allows us to write T as Tij Tij , where Tij ¼ qx Sij (no summation on j j). The characteristic time scale of the SGS heat flux associated to the local equilibrium assumption is thus D 1 ¼ jTj to Dy which leads to the following expression for the eddy diffusivity as a function of the gradients of the resolved scales: 3
2
aSGS ¼ C0 S
D T Dy
ð11Þ
In the mixed-scale approach, the adaptation of the subgrid diffusivity to the local solution comes from a second contribution obtained from a dimensional analysis based on the heat flux at the cutoff, Fc , computed following Bardina’s similarity hypothesis [31]: aSGS ¼ C0T
1=2 D F c 2 Dy
ð12Þ
The mixed-scale model subgrid diffusivity is then expressed as a geometric mean of the two previous expressions and reads
aSGS ¼ Ca
2 1=2 1=4 D jTj Fc 2 Dy
ð13Þ
This model clearly retains the property of vanishing in fully resolved regions of the flow and at solid boundaries.
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3.
NUMERICAL PROCEDURE
3.1. Time Integration The governing equations are integrated in time using a prediction-projection fractional time-step algorithm. In the prediction step, the momentum equations are advanced in time using the known pressure at the previous time step. The time discretization is second-order-accurate and combines a second-order backward Euler scheme for the time derivative, an implicit formulation for the diffusion terms, and an explicit second-order Adams-Bashforth extrapolation for the nonlinear terms. The resulting semidiscretized problem reads: 3V 4Vn þ Vn1 Pr þ 2ðV:HVÞn ðV:HVÞn1 ¼ Hpn þ 1=2 H2 V 2 Dt Ra
ð14Þ
where the star refers to the intermediate field V*. This intermediate velocity field V* is then projected onto the subspace of divergence-free vector fields. This is done by means of the computation of an auxiliary potential j satisfying the following Poisson equation:
HV ¼ H2 j
ð15Þ
This equation, supplemented by homogeneous Neuman boundary conditions, is solved by a direct method. The final velocity and pressure fields are deduced from the auxiliary potential j through the relations: V
nþ1
¼ V Hj;
pnþ1 ¼ pn þ
3 Pr j 1=2 HV 2 Dt Ra
ð16Þ
3.2. Spatial Discretization The spatial discretization for the momentum and scalar equations relies on a classical finite-volume method on staggered grids. All spatial derivatives in the conservation equations are discretized using a second-order-accurate centered scheme, except a QUICK scheme [35] for the nonlinear terms of the momentum equations in the case of LES computations. This will be discussed in more detail in Section 5. The overall scheme is formally second-order-accurate in space and time. The CPU time is respectively 0.67 ms and 0.51 ms per node and per time step for a LES or a DNS computation when running on a NEC SX5 computer.
4.
CHARACTERISTICS OF THE SIMULATIONS
We consider the classical square cavity filled with air (Pr ¼ 0.71) and differentially heated on its vertical walls, the horizontal ones being adiabatic. No-slip boundary conditions are imposed on the walls of the cavity (Figure 1). This configuration is generally referenced as the ‘‘adiabatic window problem.’’ The Rayleigh
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Figure 1. Geometric configuration of a thermally driven square cavity.
number is taken as 5 1010, which is more than two orders of magnitude above the onset of unsteadiness. This configuration is particularly demanding, because there are regions in which the flow is ‘‘fully’’ turbulent while there are still regions which are completely laminar. Let us again stress that the purpose of this computation, which is a priori not physically meaningful, is to better understand the interplay between the numerical approximation and the subgrid models and to study the influence of the subgrid models on the flow and heat transfer prediction. Because of computing limitations, this will be done, as a first step in two dimensions, and we leave open the question as to whether the conclusions and improvements observed in two dimensions will carry over in three dimensions. We used a 64 128 grid in the X and Z directions in the case of LES computations and a 512 1,024 grid for DNS computations. The horizontal grid was distributed following a cosine distribution for DNS, but hyperbolic tangent law was used for the LES because a cosine distribution with 64 nodes missed resolving the boundary layer near to the wall. In both cases the mesh was uniform in the vertical direction. The dimensionless time step was 0.004 for LES and 0.001 for DNS. Statistical values were obtained for a time interval longer than 200 time units once the statistically steady state had been reached, which might require 100–200 time units. 5.
RESULTS
Present LES simulations are compared with our DNS results, which were previously validated by comparison with spectral DNS data of Xin and Le Que´re´ [15]. In preliminary LES computations, we used a centered approximation to discretize the convection fluxes in momentum equations. This resulted in ‘‘numerical’’ instabilities in the vertical boundary layers, which led us to use a QUICK scheme [35] for discretizing these terms. When the grid spacing is uniform and equal to 1, the 1-D approximation reads, with the classical notations of staggered grids, ukþ1=2 jkþ1=2 ¼
1 ukþ1=2 jk1 þ 9jk þ 9jkþ1 jkþ2 16 ukþ1=2 jk1 3jk þ 3jkþ1 jkþ2
This discretization, although formally second-order-accurate, introduces an artificial diffusion term in the equations. Indeed, the convection term can be written as the sum of a centered approximation and of a diffusion-like term:
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ðjk þ jkþ1 Þ 1 þ ukþ1=2 ðDjk þ Djkþ1 Þ 2 2 þ ukþ1=2 ðDjk Djkþ1 Þ
ukþ1=2 jkþ1=2 ¼ ukþ1=2
ð14Þ
where Djk ¼ 18 jkþ1 2jk þ jk1 . The second term is thus analogous to a diffusion term. Its characteristic amplitude is u Dx=8 and competes with either the molecular diffusion (in DNS) or the subgrid viscosity model (in LES), showing that it requires extremely fine grids for the artificial diffusion to be smaller than actual molecular diffusion for instance. For this reason, we performed an ‘‘a-priori test’’ making use of DNS data in order to characterize the behavior of the SGS stress tensor and of its equivalent produced by the QUICK scheme, respectively, along the lines of what was proposed by Clark et al. [24]. It consists of using fully resolved velocity fields to compute the local instantaneous subgrid stresses and to compare them with the prediction of the subgrid-scale model. 5.1. A Priori Tests SGS models should reproduce in the equations of conservation the energy transfer between resolved and unresolved scales of the flow. This quantity can be quantified as the time-averaged production of subgrid kinetic energy, P ¼ td ij Sij . For DNS, the SGS stress tensor can be computed exactly from fully resolved : data filtered on the LES grid with a cutoff length scale equal to D tij ¼ ui uj ui uj
and
1 tdij ¼ tij tkk dij 2
For LES, tdij is computed with the SGS viscosity model [Eq. (5)]. As explained above, the QUICK scheme introduces in equations a term homogeneous to a stress tensor t, which reads, with the notations of (14), 1 tij QUICK ¼ Dui uj þ Duj ui 2 The subgrid kinetic energy production characterizing the three different approaches (DNS, LES, and QUICK) was computed from the same DNS realization (512 1,024 grid points) filtered on the LES grid (64 128 grid points). This allowed us to compare the characteristics of the two dissipative contributions (LES and QUICK) with the subgrid kinetic energy production computed from DNS data (PDNS). These different terms are plotted in Figure 2 (the X grid is blown up for easier reading; the data from the 64 horizontal grid points are plotted on a uniform grid). Surprisingly, these figures show a good spatial correlation between the QUICK dissipative production tensor (PQUICK) and PDNS, as both productions are essentially located in the recirculating region. Moreover, when looking at the horizontal profiles in the recirculating zone (Figure 3) we can see that PQUICK follows fairly well the shape of the DNS production, although it fails to represent the back-scatter energy transfer near the wall. Nevertheless as this backscatter is very weak compared with the total production, the general behavior of the QUICK contribution is very satisfactory.
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Figure 2. Total subgrid kinetic energy production (the X grid is blown up).
Figure 3. Profiles of the subgrid kinetic energy production at z ¼ 0.75.
On the other hand, PLES is poorly correlated with the DNS results: the subgrid kinetic energy is produced essentially near the wall (Figure 2), although its amplitude is much too large (Figure 3). This result should have been expected, because
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numerous previous investigations have shown the nonalignment between the tensors t and S. In particular, it was confirmed for turbulence at high Reynolds number by the a-priori test realized by Liu et al. [36]. As a partial conclusion, the QUICK scheme seems to share the simultaneous properties of stabilizing the computations and of reproducing the desirable effect of a SGS model for viscosity. Therefore, we chose not to introduce a subgrid viscosity in the momentum equations, but to use the QUICK scheme to discretize the convective terms. Further work is needed to determine the generalization of these observations to other configurations.
5.2. Application to Natural Convection in a Square Cavity In order to investigate the behavior of thermal SGS models with either a Reynolds analogy or with the local diffusivity model, we have considered the four cases reported in Table 1. They consist of two LES computations, a DNS, and a coarse DNS (DNSc), that is, a DNS performed with a QUICK scheme on a coarse grid. Note that for case 3 (LOC), as mentioned above, only the QUICK scheme is present for the SGS modeling of the dynamic scales. The coarse DNS (DNSc) thus allows us to investigate directly the influence of the local subgrid diffusivity, as the only difference in the two computations lies in the introduction of the SGS diffusivity. For case 2, REY, the constant Cu of the subgrid viscosity model in Eq. (5) is taken equal to 0.04, as recommended by Sagaut [30]. The value of PrSGS was typically taken in the range of 0:3 PrSGS 0:6 [37]. Our best results, reported below, were obtained for PrSGS equal to 0.6, in agreement with Bastiaans et al. [22], who, in the case of plane plumes, observed an improvement for mean flow representation when the SGS Prandtl number was increased. For case 3, LOC, the constant Ca of the local subgrid diffusivity was determined empirically by comparison with the DNS and the best value was found equal to 0.5. A more detailed discussion is postponed to sections 5.2.2 and 5.2.3. 5.2.1. Mean fields. Isocontours of the averaged stream function for cases 0 (DNS), 2 (REY), and 3 (LOC) are displayed in Figure 4. As can be seen, the flow structure for DNS presents two large recirculation zones in the downstream part
Table 1. Description of cases
Simulation NX NZ Advective terms discretization uSGS model aSGS model
0, DNS
1, DNSc
2, REY
3, LOC
DNS 514 1,026 Centered
DNS 66 130
LES 66 130
LES 66 130
QUICK
QUICK
QUICK
—
—
—
—
—
Mixed-scale model Cu ¼ 0.04 Reynolds analogy PrSGS ¼ 0.6
Local model Ck ¼ 0.5
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Figure 4. Isotherms (left); stream function C (right).
of the boundary layers. These recirculation zones are located at three-fourths of the cavity size starting from their origin, and they correspond to the ejection of large eddies in the core of the cavity. This ejection of either hot or cold fluid creates rather thermally homogeneous zones in the upper and lower parts of the cavity core, which results in an increased stratification in the core near mid-height (Figure 4, left). These results agree very well with the spectral DNS computations of Xin and Le Que´re´ [15] in an air-filled cavity of aspect ratio 4 and those of Le Que´re´ [16] in a square cavity. It is clear that the mean fields for case 2, REY, differ notably from those produced by DNS. In particular, this solution does not display the characteristic ejection processes and the recirculation zones are squeezed in the downstream corners. Consequently, the temperature field does not exhibit the previously observed upper and lower thermally homogeneous zones, but instead a quasi-linear stratification is observed in the core all over the cavity height.
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Unlike the Reynolds analogy, the use of the local subgrid diffusivity model (case 3, LOC) provides a solution where the recirculating structures are quite well located compared to DNS. As a consequence, the thermal field displays the corresponding structure of an increased stratified core region surrounded by more thermally homogeneous regions. The corresponding vertical temperature profiles at mid-width are reported in Figure 5. As mentioned above, case 3 (LOC) correctly represents the upper and lower zones of the cavity, but nevertheless fails to predict precisely the vertical temperature gradient at mid-height. The thermal stratification for case 3 (LOC) is larger (the average stratification is 1.3 in units of Dy=H for 0:4 Z 0:6) than that corresponding to DNS (1.15), whereas the stratification produced by LES-REY is somewhat smaller (1.07). The stratification in the core is closely related to the location of the recirculation areas. In case 2 (REY), they are pushed away in the edges of the cavity, which leads to a large region of uniform stratification in the core of the cavity similar to that found at the end of the steady laminar regime. On the other hand, the recirculation areas of case 3 (LOC) are located slightly more upstream than those for DNS (see the stream function in Figure 4), which explains the increased stratification at mid-height. Let us now focus on the vertical boundary layer. When looking at the temperature and vertical velocity profiles at mid-height (Figure 6), we can see that these profiles are superimposed for cases 0 (DNS) and 3 (LOC). This thus confirms that the effects of the SGS diffusivity model and of the QUICK scheme do act jointly to reproduce the laminar part of the boundary layer. We have checked that turning off separately the SGS diffusivity model or the QUICK scheme makes the solution deteriorate badly: replacing the QUICK scheme by a centered scheme in the momentum equations introduces large wiggles in the boundary layers from their starting corner. Turning off the SGS diffusivity model reproduces the structure of the solution obtained with the Reynolds analogy, and the corresponding solution displays a much too thick boundary layer. Another difference to be emphasized between both SGS approaches concerns the temperature and velocity profiles across the recirculation zone at Z ¼ 0.75 (Figure 7). It should be noted that these profiles depend strongly on Z location within the recirculating regions. With this remark in mind, the profiles corresponding to the local
Figure 5. Centerline thermal stratification.
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Figure 6. Profiles at Z ¼ 0.5: (a) mean vertical velocity; (b) mean temperature.
subgrid diffusivity model agree rather well with the DNS data. Their shapes are very similar, although quantitative differences are visible. On the contrary, those obtained with the Reynolds analogy display the characteristic laminar profiles, consistent with the absence of any recirculating structure, which is highlighted by the fact that the horizontal velocity profile remains always positive (Figure 7a).
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Figure 7. Profiles at Z ¼ 0.75: (a) mean horizontal velocity; (b) mean vertical velocity; (c) mean temperature.
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Figure 7. Continued.
5.2.2. Turbulent quantities. Figure 8 presents the resolved Reynolds stresses, the resolved turbulent kinetic energy k ¼ 12 h u00i u00i i, the resolved temperature variance 00 00 00 y i, and the resolved turbulent heat fluxes h h y u00i y i. Because an important part of the flow (the upstream part of the boundary layer and a large part of the core) remains laminar, turbulent fluctuations are only significant in the downstream part of the boundary layers and, consequently, we will again examine only the profiles at Z ¼ 0.75 along the hot wall. In general, we observe good agreement for the turbulent statistic patterns obtained either with DNS or in case 3 (LOC), even though the absolute levels are not always well respected. Note again that these absolute levels depend strongly on the location of the laminar–turbulent transition. On the other hand, the Reynolds analogy (case 2, REY) fails to predict the turbulent quantities correctly. The fluctuations are always too weak, particularly those for the velocities are too small by a factor of about 10. This is very likely due to the fact that LES-REY is too dissipative and, indeed, detailed examination of the eddy diffusivities shows that the maximum value for the ratio haSGS =ai is equal to 5.3 for the Reynolds model, while it is only 0.6 for the local model. It is clear that one of the main issues linked to the prediction of these flows which display both laminar and transitional zones lies in the ability of the different approaches to locate the transition to turbulence correctly. We have looked more precisely at this point by plotting the temperature fluctuations along the heated wall for the four different models (Figure 9): the fine DNS (case 0, Figure 9a), the coarse DNS (case 1, DNSC, Figure 9b), and the SGS diffusivity model for two values of Ca (0.5 for Figure 9c and 1.25 for Figure 9d, the coarse DNS corresponding to Ca ¼ 0). These curves indeed confirm the large sensitivity of the location of the laminar– turbulent transition, whereas the amplitudes of the fluctuation seem much less
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00 00 U 00 00 Figure 8. Turbulent statistics at Z ¼ 0.75: (a) hU
00 profiles
00 i;00 (b) hW W i; (c) turbulent kinetic energy k; 00 00 y00 . (d) temperature variance y y ; (e) turbulent heat flux U y ; (f) turbulent heat flux W
sensitive. It can be seen that increasing Ca shifts the transition upstream very substantially. This is somewhat paradoxical, since one could intuitively imagine that increasing the diffusivity would on the contrary delay the transition. The rationale for this behavior remains to be explained. 5.2.3. Heat flux. Figure 10 presents the time-averaged local Nusselt number along the hot wall. For DNS, the Nusselt distribution is of laminar type but presents a local increase (relative to the laminar decrease) in the downstream region of the boundary layer, indicating that expulsion of the large eddies from the boundary layer to the core of the cavity has a net effect on the local heat transfer. This is in fact the part of the boundary layer where the motion is turbulent (Figure 9a).
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Figure 9. Temperature fluctuation profile versus Z at the first grid point (x ¼ 2.6 10 74) for three different times; mean temperature profile at the first grid point (rescaled to fit in the figures).
Figure 10. Mean Nusselt number at hot wall.
For LES results, this relative increase is also observed, but at a more upstream location (about at z ¼ 0.6 instead of z ¼ 0.77 for DNS), in correspondence with the location of the large fluctuations already shown in Figure 9. Another difference is observed in the upstream part of the boundary layer, where the heat transfer is overpredicted by LES. Nevertheless, in this region the variation for case 3 (LOC) is
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still of laminar type, in contrast to the one of case 2 (REY), where a change in the shape is observed at nearly z ¼ 0.1, indicating a much too early transition toward turbulence. We must note that this is a typical behavior observed with k–e models [13] and can probably be attributed to the Reynolds analogy. It is noted that the distribution of Nusselt number (Figure 10) corresponding to the coarse DNS (case 1, DNSC) does not display the typical relative increase but instead decreases continuously along the hot wall in a laminar-shape way. The presence of a subgrid-scale diffusivity seems therefore to be indispensable to reproduce the turbulent-like behavior of the local heat transfer. This is consistent with the evolution along the hot wall of the temperature fluctuations profiles (Figure 9) according to the subgrid diffusivity model coefficient. Indeed, we note that there is a region where fluctuations grow and which moves upstream as the SGS diffusivity is increased.
6.
CONCLUSION
We have developed a local subgrid diffusivity model for the large-eddy simulation of natural-convection flows. This model does not rely on the Reynolds analogy with a constant subgrid Prandtl number, and the subgrid diffusivity is computed independently from the subgrid viscosity along the lines of the mixed-scale model. This model was applied to natural convection in a square cavity filled with air for Rayleigh numbers up to 5 1010. As it was found necessary to use a QUICK scheme for stability considerations, a priori tests were first performed from a DNS approach in order to compare the respective effects of the subgrid viscosity model and of the QUICK scheme used to discretize the convective terms in the momentum equations. An unexpected result was that the QUICK scheme seems to reproduce an energy transfer between scales in good agreement with DNS, unlike the subgrid viscosity model. The results computed with the local subgrid diffusivity were then compared with those obtained with a Reynolds analogy and DNS computations. The following major conclusions have emerged: 1. Generally speaking, the local model predicts the air flow structure correctly. Results are in good agreement with the DNS reference for the mean velocity and temperature as well as for the turbulent quantities and the localization of the transition to turbulence in the boundary layers. 2. The model based on Reynolds analogy fails to predict the localization of the recirculating structure, which affects the mean centerline thermal stratification. 3. The phenomenology of the transition to turbulence in the boundary layer is strongly dependent on the amplitude of the subgrid-scale diffusivity term. It is clear that much further work is needed to investigate the generality of these observations. It is also clear that 3-D simulations are needed to investigate the effects of allowing for extra degrees of freedom in the third direction on the structure of the mean flow and turbulence quantities.
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