development of a local subgrid diffusivity model for large-eddy

DEVELOPMENT OF A LOCAL SUBGRID DIFFUSIVITYMODEL FOR LARGE-EDDY SIMULATION OF BUOYANCY-DRIVEN FLOWS: APPLICATION TO A SQUAREDIFFERENTIALLY HEATED CAVITY

A. SergentLIMSI-CNRS, France and LEPTAB, Universityof La Rochelle, La Rochelle, France

P. JoubertLEPTAB, University of La Rochelle,La Rochelle, France

P. Le QuereLIMSI-CNRS, Orsay, France

We present a local subgrid diffusivity model for the large–eddy simulation of natural-

convection 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 config-uration for a variety of practical situations such as cooling of electronic devices or airflow 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 Quere, LIMSI-CNRS, BP 133, 91403 Orsay Cedex, France.

E-mail: [email protected]

Numerical Heat Transfer, Part A, 44: 789–810, 2003

Copyright # Taylor & Francis Inc.

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/0407780390220511

789

configurations characterized by small Rayleigh numbers corresponding to relativelysmall temperature differences and typical dimensions. After the pioneering work ofBatchelor [1], a decisive step was the identification of the boundary-layer regime byElder [2] and Gill [3], which allowed for a heat transfer correlation that was laterimproved by Bejan [4]. Simultaneously, this configuration served as a prototypeconfiguration for the development of numerical algorithms aiming at solving theincompressible Navier-Stokes equations, and benchmark solutions for the squarecavity with adiabatic walls are now available up to the end of the steady laminarregime, either in two dimensions or in three dimensions (see G. de Vahl Davis [5], LeQuere [6], and Tric et al. [7]). However, in many configurations of practical interestthe temperature differences and dimensions are such that the corresponding flows areeither transitional or turbulent. The prediction of corresponding flow structure andrelated heat transfer in such configurations is thus of great interest but still remains aformidable challenge despite the numerous efforts that have already taken place.This is due mostly to the fact that these flows display complex behavior, partlybecause of their high sensitivity to the thermal boundary conditions on the walls ofthe cavity, which makes them very difficult to investigate. Even if most of the 2-Dmechanisms of instability are now reasonably understood [8], these flows are still achallenging field of research for 3-D effects and for the turbulent domain.

Many experimental studies have already been carried out for transitional orturbulent conditions [9–12], which provide valuable results for numerical simulationsand the tuning of turbulence models, but there still remains to fully characterize the3-D structure of the flows and to explain the differences observed between experi-ments and numerical results. In particular, among the major discrepancies that re-main unexplained in the turbulent regime is the value of the vertical stratification inthe core region of the cavity.

As said above, from a computational point of view, numerous studies havebeen devoted to this problem, and accurate solutions are now available in two and

NOMENCLATURE

Ck subgrid scale (SGS) diffusivity model

coefficient

Cu SGS viscosity model coefficient

g gravitational acceleration

hi subgrid heat flux

H cavity height

k turbulent kinetic energy

Nu Nusselt number

p pressure

Pr molecular Prandtl number

PrSGS SGS Prandtl number

qc kinetic energy at the cutoff

Ra Rayleigh number�SS strain rate tensor of resolved scales

t time

ui nondimensional velocities

U nondimensional horizontal velocity

W nondimensional vertical velocity

X horizontal coordinate

xi Cartesian space coordinate

Z vertical coordinate

a molecular thermal diffusivity

aSGS SGS diffusivity�DD grid filtered size~DD test filtered size, ~DD > �DDy nondimensional temperature

u molecular viscosity

uSGS SGS viscosity

v grid filtered variable

v test filtered variable

v subgrid variable

v time-averaged variable

v time fluctuation

tij subgrid stresses tensor

790 A. SERGENT ET AL.

three dimensions up to the end of the steady laminar regime. Concerning the tur-bulent regime, the state of the art is far from being satisfactory. An overview of theresults obtained for an early comparison exercise between different k–E type modelscan be found in Henkes and Hoogendoorn [13], which showed large differences withcorresponding results from 2-D direct numerical simulation (DNS) performed byPaolucci [14] using finite differences, or Xin and Le Quere [15] in a cavity of aspectratio 4 using spectral methods. The trends observed in [15] were later confirmed byLe Quere in a square cavity [16] and by Nobile [17]. These trends from 2-D simu-lations require, of course, confirmation in three dimensions, but for cavities withactive lateral walls, 3-D direct numerical simulations are presently restricted to va-lues of Rayleigh numbers corresponding the end of the laminar regime [7, 18] or tothe beginning of the transitional regime. As 3-D direct numerical simulations of fullyturbulent regimes seem presently out of reach and will long remain prohibitivelyexpensive, it is of great interest to develop alternative numerical approaches. Onepossible way is to develop specific Reynolds Averaged Navier–Stokes equations(RANS) models [19], but, due to the fact that these flows are characterized by largescale fluctuations in low-speed regions, the large-eddy simulation (LES) techniqueseems a very appealing approach and is therefore drawing increasing attention. SomeLES results have recently been reported in the literature, but essentially for mixed-convection flows in enclosures [20, 21]. Although in these articles the pure naturalconvection case of the differentially heated cavity is also considered, very little resultsare given for this configuration. In another recent article, Bastiaans et al. [22] per-formed 2-D and 3-D LES as well as 3-D DNS for turbulent thermal plumes inconfined enclosures. We nevertheless must note that this configuration gives acompletely different structure of the flow as compared with the precedent case andleads to very different numerical difficulties.

In all of these articles, different subgrid models are used and compared, butmost of them assume that the thermal subgrid diffusivity can be deduced from thesubgrid viscosity through a Reynolds analogy with a constant subgrid Prandtlnumber. Those that alleviate the assumption of constant subgrid Prandtl number bythe use of a dynamic approach, for instance, Peng and Davidson [20], still assumethat the subgrid diffusivity depends mainly on a time scale which is that of the re-solved dynamic scales. We believe that this assumption cannot hold for the wholerange of Prandtl numbers, due to the very different characteristics of the dynamicand thermal fields for either very small or very large Prandtl numbers.

We present in this article a LES approach for natural-convection flows basedon an original local diffusivity model based on its own time scale independent of thatof the subgrid viscosity. This model is tested in the configuration of a square dif-ferentially heated cavity at a Rayleigh number of 5� 1010, which is the same con-figuration as that used for the comparison exercise detailed by Henkes andHoogendoorn [13]. Note that this value is more than two orders of magnitude abovethe onset of unsteadiness.

This article is devoted mainly to presenting the local subgrid diffusivity modeland performing validation tests for the overall numerical procedure and in particularto understanding the interplay between the numerics and the modeling by comparingdifferent types of models and convective discretization schemes. The article isorganized as follows: after presenting the local diffusivity model, we study the

LARGE-EDDY SIMULATION OF BUOYANCY-DRIVEN FLOWS 791

respective contribution of the viscosity introduced by the subgrid diffusive termand by the QUICK scheme used to discretize the convective fluxes. This is done byan ‘‘a priori’’ test with a DNS approach, taken as reference. Then, we present theimprovement introduced by the thermal diffusivity model compared to a Reynoldsassumption using a constant subgrid Prandtl number.

As final introductory remark, let us mention that we are fully aware that 2-Dsimulations, no matter how chaotic they might be, cannot reproduce real turbulentflows, and the present developments and tests will have to be carried out in threedimensions 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 incompressiblefluid flow under Boussinesq assumption are classically derived by applying a con-volution filter to the unsteady momentum and energy equations. The resulting set ofnondimensional equations reads:

quiqxi

¼ 0

quiqt

þ quiujqxj

¼ � q�ppqxi

þ qqxj

Pr Ra�1=2 quiqxj

þ qujqxi

� �� qtij

qxjþ Pr �yydiZ

qyqt

þ qujyqxj

¼ qqxj

Ra�1=2 qyqxj

� �� qhjqxj

8>>>>>>>><>>>>>>>>:

ð1Þ

These equations were made dimensionless by using the height of the cavity, H, asreference length, the temperature difference between the two vertical isothermalwalls of the cavity, Dy, and the natural-convection characteristic velocity,UON ¼ aRe1=2H:

In the above equations, the subgrid-scale (SGS) stress tensor t and the subgridheat flux vector h, defined as

tij ¼ uiuj � �uui�uuj

hj ¼ ujy� �uuj�yy

(ð2Þ

need to be modeled. Because only eddy-viscosity type models are considered, theisotropic part of the SGS tensor is added to the filtered pressure, leading to thedefinition of a modified pressure,

�pp? ¼ �ppþ 1

3tii

The trace of the SGS tensor t defines a subgrid kinetic energy kSGS ¼ 12tkk,

which involves unresolved scales and interaction between unresolved and resolvedscales. Its budget equation reads (see, e.g., [23]):

qkSGSqt

þ q�uujkSGSqxj

¼ Dissþ Diff � gb yuz � �yy�uuz� �

� tij �SSij ð3Þ


where Diss and Diff are the dissipative and diffusive terms, respectively, while the lastterm appearing in this equation represents the production of subgrid kinetic energydue to the strain rate of resolved scales.

Positive values of �tij �SSij indicate energy transfer to the smallest scales of theflow, whereas negative values correspond to a backtransfer of energy from small tolarge scales. It is known that the behavior of this term constitutes one of severalpossible 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 presentwork belong to the eddy-viscosity family, and therefore assume a linear relationshipbetween the deviatoric part of the subgrid tensor td and the resolved strain ratetensor �SS:

tdij ¼ �uSGS �SSij ð4Þ

where uSGS is the subgrid viscosity and

�SSij ¼1

2

quiqxj

þ qujqxi

� �

In order to simulate flows which are not fully turbulent or with both laminarand transitional regions such as large Rayleigh number air flow in cavities, thesubgrid model should be able to adjust itself to locally inhomogeneous flow, which isusually done by using a Germano-Lilly dynamic procedure [26, 27]. However, asmentioned by Zhang [21], who used it to compute indoor air flow, this approach isnot reliable, as the computations become very unstable when there is no homo-geneous 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 cubical3-D cavity with noslip and adiabatic walls, whereas they retained the dynamicprocedure 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 conditionsby using the mixed-scale model, developed by Sagaut and Ta Phuoc [29, 30]. Thismodel stems from a Smagorinsky model in which this local adaptation is achieved bytaking explicitly into account the kinetic energy at the cutoff, qc. The subgridviscosity is then evaluated as

uSGS ¼ Cu�DD3=2 �SS

�� 1=2q1=4c ð5Þ

where �SS�� ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi

2 �SSij�SSij

q, qc stands for the kinetic energy at the cutoff, qc ¼ 1

2�uu0i�uu

0i, and Cu

is a constant. Following Bardina’s similarity hypothesis [31], the velocity field at thecutoff, �uu0i, can be estimated by filtering the resolved velocity field with a testfilter coarser than the implicit one, ~�DD�DD > �DD, and so �uu0i ¼ �uui � ~�uu�uui. The small-scaledependency 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.


2.2.2. Subgrid-diffusivity model. Concerning the heat transfer equation[Eq. (1)], most subgrid models currently used are eddy-diffusivity models and relatethe subgrid heat flux components, hj, to the large-scale temperature gradient bymeans of a SGS diffusivity, aSGS, as

hj ¼ �aSGSq�yyqxj

ð6Þ

As said above, aSGS is usually computed from uSGS assuming a Reynoldsanalogy and a constant subgrid Prandtl number: aSGS ¼ uSGS= PrSGS. This assump-tion amounts to considering that the small thermal scales depend solely, through thesubgrid viscosity model, on the resolved dynamic scales. Therefore it is suspected notto hold in the case of turbulent natural convection, where the flow is produced bybuoyant forces and not through a dynamic forcing. Furthermore, it is known that,even under turbulent conditions, the thermal and velocity scale distributions dependvery strongly on the molecular Prandtl number, and the assumption of constantPrSGS is likely not to hold over the whole range of molecular Prandtl number. Forinstance, in the case of a passive scalar diffusion at a Prandtl number of the order of1 in isotropic turbulence, Lesieur and Rogallo [32] present spectrums evolutions ofeddy viscosity and diffusivity, which confirm the fact that there is not a simplerelationship between uSGS and aSGS over the whole range of scales. Even though thisargument does not boldly carry over to natural convection, where temperaturecannot be considered as a passive scalar, the mere introduction of a subgrid Prandtlnumber is very surprising since the ratio uSGS=aSGS never appears as an independentparameter in the governing equations.

We have thus developed a local subgrid diffusivity model [33] along the lines ofthe mixed-scale viscosity model [Eq. (5)]. In this approach, the subgrid diffusivity isexpressed as a weighted geometric average of two contributions. The first one comesfrom the Smagorinsky model using the resolved thermal scales [34], while the secondcomes from the turbulent kinetic energy model [31] based on the subgrid scales.

As the functional approach of Smagorinsky assumes that the most importanteffect of the interaction between the resolved and the subgrid scales is the energyexchange, we focus on the transport equation of the subgrid-scale heat flux,FSGS ¼

ffiffiffiffiffiffiffiffiffiffiffi12hkhk

q. This equation reads:

qF2SGS

qtþ q�uujF2

SGS

qxj¼ Dissþ Diff � gb yy� �yy�yy

� �� hihj

q�uuiqxj

þ hitijq�yyqxj

� �ð7Þ

where Diss and Diff are dissipative and diffusive terms and the last term in par-entheses is the subgrid energy production term, DSGS. Comparing with Eq. (3) forthe subgrid kinetic energy, kSGS, we note that the mechanisms of energy exchangebetween the resolved and the subgrid scales are analogous for kSGS and FSGS. Thisanalogy allows us to derive an expression for the viscous dissipation of the SGS heatflux energy similar to the viscous dissipation of the SGS kinetic energy:

Dh i ¼ aSGS�DD2

D E� T

�� 2D Eð8Þ


where T�� is a scalar quantity of dimension K=LT, which should come from a sec-

ond-order tensor so designed as to preserve invariant and symmetry properties of theflow and temperature fields. The classical assumption of local energy equilibriumleads to

Dh i ¼ DSGSh i ¼ � hihjq�uuiqxj

þ hitijq�yyqxj

ð9Þ

Substituting tij and hi by their definitions and nSGS by its functional expression (thatis, the product of the square of a characteristic length with the norm of the resolvedstrain rate tensor) results in

Dh i ¼ � aSGS � D2

D E� 1þ 1

nSGS=aSGS

� q�yy

qxi

q�yyqxj

�SSij�SS��

ð10Þ

Invariance considerations lead us to replace �SS�� by �SSij in the above expression,

which then allows us to write T�� as ffiffiffiffiffiffiffiffiffiffiffi

TijTij

p, where Tij ¼

�qyqxj

��SSij (no summation on

j). The characteristic time scale of the SGS heat flux associated to the local equili-brium assumption is thus

1

to¼

�DDDy

�TTj j

which leads to the following expression for the eddy diffusivity as a function of thegradients of the resolved scales:

aSGS ¼ C02S

D3

DyT�� ð11Þ

In the mixed-scale approach, the adaptation of the subgrid diffusivity to thelocal solution comes from a second contribution obtained from a dimensionalanalysis based on the heat flux at the cutoff, Fc, computed following Bardina’ssimilarity hypothesis [31]:

aSGS ¼ C0T

�DDDy

Fc2

�� 1=2 ð12Þ

The mixed-scale model subgrid diffusivity is then expressed as a geometric mean ofthe two previous expressions and reads

aSGS ¼ Ca

�DD2

DyTj j1=2 Fc

2�� 1=4 ð13Þ

This model clearly retains the property of vanishing in fully resolved regions of theflow and at solid boundaries.


3. NUMERICAL PROCEDURE

3.1. Time Integration

The governing equations are integrated in time using a prediction-projectionfractional time-step algorithm. In the prediction step, the momentum equations areadvanced in time using the known pressure at the previous time step. The timediscretization is second-order-accurate and combines a second-order backward Eulerscheme for the time derivative, an implicit formulation for the diffusion terms, andan explicit second-order Adams-Bashforth extrapolation for the nonlinear terms.The resulting semidiscretized problem reads:

3V� � 4Vn þ Vn�1

2Dtþ 2 V:HVð Þn� V:HVð Þn�1¼ �Hpn þ Pr

Ra1=2H2V� ð14Þ

where the star refers to the intermediate field V*. This intermediate velocity field V* isthen projected onto the subspace of divergence-free vector fields. This is done bymeans of the computation of an auxiliary potential j satisfying the followingPoisson equation:

HV� ¼ H2j ð15Þ

This equation, supplemented by homogeneous Neuman boundary conditions, issolved by a direct method. The final velocity and pressure fields are deduced from theauxiliary potential j through the relations:

Vnþ1 ¼ V

� � Hj; pnþ1 ¼ pn þ 3

2Dtj� Pr

Ra1=2HV

� ð16Þ

3.2. Spatial Discretization

The spatial discretization for the momentum and scalar equations relies on aclassical finite-volume method on staggered grids. All spatial derivatives in theconservation equations are discretized using a second-order-accurate centeredscheme, except a QUICK scheme [35] for the nonlinear terms of the momentumequations in the case of LES computations. This will be discussed in more detail inSection 5.

The overall scheme is formally second-order-accurate in space and time. TheCPU time is respectively 0.67 ms and 0.51 ms per node and per time step for a LES ora DNS computation when running on a NEC SX5 computer.

4. CHARACTERISTICS OF THE SIMULATIONS

We consider the classical square cavity filled with air (Pr¼ 0.71) and differ-entially heated on its vertical walls, the horizontal ones being adiabatic. No-slipboundary conditions are imposed on the walls of the cavity (Figure 1). This con-figuration is generally referenced as the ‘‘adiabatic window problem.’’ The Rayleigh


number is taken as 5� 1010, which is more than two orders of magnitude above theonset of unsteadiness. This configuration is particularly demanding, because thereare regions in which the flow is ‘‘fully’’ turbulent while there are still regions whichare completely laminar. Let us again stress that the purpose of this computation,which is a priori not physically meaningful, is to better understand the interplaybetween the numerical approximation and the subgrid models and to study the in-fluence of the subgrid models on the flow and heat transfer prediction. Because ofcomputing limitations, this will be done, as a first step in two dimensions, and weleave open the question as to whether the conclusions and improvements observed intwo dimensions will carry over in three dimensions.

We used a 64� 128 grid in the X and Z directions in the case of LES com-putations and a 512� 1,024 grid for DNS computations. The horizontal grid wasdistributed following a cosine distribution for DNS, but hyperbolic tangent law wasused for the LES because a cosine distribution with 64 nodes missed resolving theboundary layer near to the wall. In both cases the mesh was uniform in the verticaldirection. The dimensionless time step was 0.004 for LES and 0.001 for DNS. Sta-tistical values were obtained for a time interval longer than 200 time units once thestatistically steady state had been reached, which might require 100–200 time units.

5. RESULTS

Present LES simulations are compared with our DNS results, which werepreviously validated by comparison with spectral DNS data of Xin and Le Quere[15].

In preliminary LES computations, we used a centered approximation to dis-cretize 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-Dapproximation reads, with the classical notations of staggered grids,

ukþ1=2jkþ1=2 ¼1

16ukþ1=2 � �jk�1 þ 9jk þ 9jkþ1 � jkþ2

� �� ukþ1=2

�� jk�1 � 3jk þ 3jkþ1 � jkþ2

� ��This discretization, although formally second-order-accurate, introduces an artificialdiffusion term in the equations. Indeed, the convection term can be written as thesum of a centered approximation and of a diffusion-like term:

Figure 1. Geometric configuration of a thermally driven square cavity.


ukþ1=2 � jkþ1=2 ¼ ukþ1=2 �ðjk þ jkþ1Þ

2þ 1

2ukþ1=2 � ðDjk þDjkþ1Þ�

þ ukþ1=2

�� ðDjk �Djkþ1Þ� ð14Þ

where Djk ¼ � 18 jkþ1 � 2jk þ jk�1

� �. The second term is thus analogous to a dif-

fusion term. Its characteristic amplitude is uDx=8 and competes with either themolecular diffusion (in DNS) or the subgrid viscosity model (in LES), showing that itrequires extremely fine grids for the artificial diffusion to be smaller than actualmolecular 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 stresstensor and of its equivalent produced by the QUICK scheme, respectively, along thelines of what was proposed by Clark et al. [24]. It consists of using fully resolvedvelocity fields to compute the local instantaneous subgrid stresses and to comparethem with the prediction of the subgrid-scale model.

5.1. A Priori Tests

SGS models should reproduce in the equations of conservation the energytransfer between resolved and unresolved scales of the flow. This quantity can bequantified as the time-averaged production of subgrid kinetic energy, P ¼ � tdij �SSij

�.

For DNS, the SGS stress tensor can be computed exactly from fully resolveddata filtered on the LES grid with a cutoff length scale equal to �DD:

tij ¼ uiuj � �uui�uuj and tdij ¼ tij �1

2tkkdij

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),

tij QUICK

�� ¼ 1

2Dui �uuj þDuj �uui� �

The subgrid kinetic energy production characterizing the three different app-roaches (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 allowedus to compare the characteristics of the two dissipative contributions (LES andQUICK) 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 easierreading; 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 dis-sipative production tensor (PQUICK) and PDNS, as both productions are essentiallylocated in the recirculating region. Moreover, when looking at the horizontal profilesin the recirculating zone (Figure 3) we can see thatPQUICK follows fairly well the shapeof the DNS production, although it fails to represent the back-scatter energy transfernear the wall. Nevertheless as this backscatter is very weak compared with the totalproduction, the general behavior of the QUICK contribution is very satisfactory.


On the other hand, PLES is poorly correlated with the DNS results: the subgridkinetic energy is produced essentially near the wall (Figure 2), although its amplitudeis much too large (Figure 3). This result should have been expected, because

Figure 3. Profiles of the subgrid kinetic energy production at z ¼ 0.75.

Figure 2. Total subgrid kinetic energy production (the X grid is blown up).


numerous previous investigations have shown the nonalignment between the tensorst and �SS. In particular, it was confirmed for turbulence at high Reynolds number bythe a-priori test realized by Liu et al. [36].

As a partial conclusion, the QUICK scheme seems to share the simultaneousproperties of stabilizing the computations and of reproducing the desirable effect of aSGS model for viscosity. Therefore, we chose not to introduce a subgrid viscosity inthe momentum equations, but to use the QUICK scheme to discretize the convectiveterms. Further work is needed to determine the generalization of these observationsto other configurations.

5.2. Application to Natural Convection in a Square Cavity

In order to investigate the behavior of thermal SGS models with either aReynolds analogy or with the local diffusivity model, we have considered the fourcases reported in Table 1. They consist of two LES computations, a DNS, and acoarse DNS (DNSc), that is, a DNS performed with a QUICK scheme on a coarsegrid. Note that for case 3 (LOC), as mentioned above, only the QUICK scheme ispresent for the SGS modeling of the dynamic scales. The coarse DNS (DNSc) thusallows us to investigate directly the influence of the local subgrid diffusivity, as theonly difference in the two computations lies in the introduction of the SGS diffu-sivity.

For case 2, REY, the constant Cu of the subgrid viscosity model in Eq. (5) istaken equal to 0.04, as recommended by Sagaut [30]. The value of PrSGS was typicallytaken in the range of 0:3 � PrSGS � 0:6 [37]. Our best results, reported below, wereobtained for PrSGS equal to 0.6, in agreement with Bastiaans et al. [22], who, in thecase of plane plumes, observed an improvement for mean flow representation whenthe SGS Prandtl number was increased.

For case 3, LOC, the constant Ca of the local subgrid diffusivity was de-termined empirically by comparison with the DNS and the best value was foundequal 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 flowstructure for DNS presents two large recirculation zones in the downstream part

Table 1. Description of cases

0, DNS 1, DNSc 2, REY 3, LOC

Simulation NX�NZ DNS

514� 1,026

DNS 66� 130 LES 66� 130 LES 66� 130

Advective terms

discretization

Centered QUICK QUICK QUICK

uSGS model — — Mixed-scale

model Cu¼ 0.04

—

aSGS model — — Reynolds ana-

logy PrSGS¼ 0.6

Local model

Ck¼ 0.5


of the boundary layers. These recirculation zones are located at three-fourths of thecavity size starting from their origin, and they correspond to the ejection of large ed-dies in the core of the cavity. This ejection of either hot or cold fluid creates ratherthermally homogeneous zones in the upper and lower parts of the cavity core, whichresults in an increased stratification in the core near mid-height (Figure 4, left). Theseresults agree very well with the spectral DNS computations of Xin and Le Quere [15]in an air-filled cavity of aspect ratio 4 and those of Le Quere [16] in a square cavity.

It is clear that the mean fields for case 2, REY, differ notably from thoseproduced by DNS. In particular, this solution does not display the characteristicejection processes and the recirculation zones are squeezed in the downstream cor-ners. Consequently, the temperature field does not exhibit the previously observedupper and lower thermally homogeneous zones, but instead a quasi-linear stratifi-cation is observed in the core all over the cavity height.

Figure 4. Isotherms (left); stream function C (right).


Unlike the Reynolds analogy, the use of the local subgrid diffusivity model(case 3, LOC) provides a solution where the recirculating structures are quite welllocated compared to DNS. As a consequence, the thermal field displays the corres-ponding structure of an increased stratified core region surrounded by more ther-mally homogeneous regions.

The corresponding vertical temperature profiles at mid-width are reported inFigure 5. As mentioned above, case 3 (LOC) correctly represents the upper andlower zones of the cavity, but nevertheless fails to predict precisely the verticaltemperature gradient at mid-height. The thermal stratification for case 3 (LOC) islarger (the average stratification is 1.3 in units of Dy=H for 0:4 � Z � 0:6) than thatcorresponding to DNS (1.15), whereas the stratification produced by LES-REY issomewhat smaller (1.07). The stratification in the core is closely related to thelocation of the recirculation areas. In case 2 (REY), they are pushed away in theedges of the cavity, which leads to a large region of uniform stratification in the coreof the cavity similar to that found at the end of the steady laminar regime. On theother hand, the recirculation areas of case 3 (LOC) are located slightly more up-stream than those for DNS (see the stream function in Figure 4), which explains theincreased stratification at mid-height.

Let us now focus on the vertical boundary layer. When looking at the tem-perature and vertical velocity profiles at mid-height (Figure 6), we can see that theseprofiles are superimposed for cases 0 (DNS) and 3 (LOC). This thus confirms thatthe effects of the SGS diffusivity model and of the QUICK scheme do act jointly toreproduce the laminar part of the boundary layer. We have checked that turning offseparately the SGS diffusivity model or the QUICK scheme makes the solutiondeteriorate badly: replacing the QUICK scheme by a centered scheme in the mo-mentum equations introduces large wiggles in the boundary layers from theirstarting corner. Turning off the SGS diffusivity model reproduces the structure of thesolution obtained with the Reynolds analogy, and the corresponding solution dis-plays a much too thick boundary layer.

Another difference to be emphasized between both SGS approaches concernsthe temperature and velocity profiles across the recirculation zone atZ¼ 0.75 (Figure 7).It should be noted that these profiles depend strongly on Z location within the re-circulating regions. With this remark in mind, the profiles corresponding to the local

Figure 5. Centerline thermal stratification.


subgrid diffusivity model agree rather well with the DNS data. Their shapes are verysimilar, although quantitative differences are visible. On the contrary, those obtainedwith the Reynolds analogy display the characteristic laminar profiles, consistent withthe absence of any recirculating structure, which is highlighted by the fact that thehorizontal velocity profile remains always positive (Figure 7a).

Figure 6. Profiles at Z ¼ 0.5: (a) mean vertical velocity; (b) mean temperature.


Figure 7. Profiles at Z ¼ 0.75: (a) mean horizontal velocity; (b) mean vertical velocity; (c) mean

temperature.


5.2.2. Turbulent quantities. Figure 8 presents the resolved Reynolds stresses,the resolved turbulent kinetic energy k ¼ 1

2 h�uu00i �uu00i i, the resolved temperature varianceh�yy00�yy00i, and the resolved turbulent heat fluxes h�uu00i �yy

00i. Because an important part ofthe flow (the upstream part of the boundary layer and a large part of the core) re-mains laminar, turbulent fluctuations are only significant in the downstream partof the boundary layers and, consequently, we will again examine only the profilesat Z¼ 0.75 along the hot wall. In general, we observe good agreement for the turbu-lent statistic patterns obtained either with DNS or in case 3 (LOC), even though theabsolute levels are not always well respected. Note again that these absolute levelsdepend strongly on the location of the laminar–turbulent transition.

On the other hand, the Reynolds analogy (case 2, REY) fails to predict theturbulent quantities correctly. The fluctuations are always too weak, particularlythose for the velocities are too small by a factor of about 10. This is very likely due tothe fact that LES-REY is too dissipative and, indeed, detailed examination of theeddy diffusivities shows that the maximum value for the ratio aSGS=ah i is equal to 5.3for 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 flowswhich display both laminar and transitional zones lies in the ability of the differentapproaches to locate the transition to turbulence correctly. We have looked moreprecisely at this point by plotting the temperature fluctuations along the heated wallfor the four different models (Figure 9): the fine DNS (case 0, Figure 9a), the coarseDNS (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 Ca¼ 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

Figure 7. Continued.


sensitive. It can be seen that increasing Ca shifts the transition upstream very sub-stantially. This is somewhat paradoxical, since one could intuitively imagine thatincreasing the diffusivity would on the contrary delay the transition. The rationalefor this behavior remains to be explained.

5.2.3. Heat flux. Figure 10 presents the time-averaged local Nusselt num-ber along the hot wall. For DNS, the Nusselt distribution is of laminar typebut presents a local increase (relative to the laminar decrease) in the downstreamregion of the boundary layer, indicating that expulsion of the large eddies fromthe boundary layer to the core of the cavity has a net effect on the local heattransfer. This is in fact the part of the boundary layer where the motion is tur-bulent (Figure 9a).

Figure 8. Turbulent statistics profiles at Z ¼ 0.75: (a) �UU00 �UU00h i; (b) �WW00 �WW00h i; (c) turbulent kinetic energy k;

(d) temperature variance �yy00�yy00 �

; (e) turbulent heat flux �UU00�yy00 �

; (f) turbulent heat flux �WW00�yy00 �

.


For LES results, this relative increase is also observed, but at a more upstreamlocation (about at z¼ 0.6 instead of z¼ 0.77 for DNS), in correspondence with thelocation of the large fluctuations already shown in Figure 9. Another differenceis observed in the upstream part of the boundary layer, where the heat transfer isoverpredicted by LES. Nevertheless, in this region the variation for case 3 (LOC) is

Figure 9. Temperature fluctuation profile versus Z at the first grid point (x ¼ 2.6� 1074) for three dif-

ferent times; mean temperature profile at the first grid point (rescaled to fit in the figures).

Figure 10. Mean Nusselt number at hot wall.


still of laminar type, in contrast to the one of case 2 (REY), where a change in theshape is observed at nearly z¼ 0.1, indicating a much too early transition towardturbulence. 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 tothe coarse DNS (case 1, DNSC) does not display the typical relative increase butinstead decreases continuously along the hot wall in a laminar-shape way. Thepresence of a subgrid-scale diffusivity seems therefore to be indispensable to re-produce the turbulent-like behavior of the local heat transfer. This is consistent withthe 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 aregion where fluctuations grow and which moves upstream as the SGS diffusivity isincreased.

6. CONCLUSION

We have developed a local subgrid diffusivity model for the large-eddy simu-lation of natural-convection flows. This model does not rely on the Reynolds ana-logy with a constant subgrid Prandtl number, and the subgrid diffusivity is computedindependently 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 forRayleigh 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 therespective effects of the subgrid viscosity model and of the QUICK scheme used todiscretize the convective terms in the momentum equations. An unexpected resultwas that the QUICK scheme seems to reproduce an energy transfer between scales ingood agreement with DNS, unlike the subgrid viscosity model.

The results computed with the local subgrid diffusivity were then comparedwith those obtained with a Reynolds analogy and DNS computations. The followingmajor 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 velocityand temperature as well as for the turbulent quantities and the localizationof the transition to turbulence in the boundary layers.

2. The model based on Reynolds analogy fails to predict the localization ofthe recirculating structure, which affects the mean centerline thermalstratification.

3. The phenomenology of the transition to turbulence in the boundary layeris strongly dependent on the amplitude of the subgrid-scale diffusivityterm.

It is clear that much further work is needed to investigate the generality of theseobservations. It is also clear that 3-D simulations are needed to investigate the effectsof allowing for extra degrees of freedom in the third direction on the structure of themean flow and turbulence quantities.


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development of a local subgrid diffusivity model for large-eddy

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