Direct numerical simulation of a three-dimensional

Direct numerical simulation of a three-dimensional natural convection ow in a dierentially heated cavity of aspect ratio 4 M.Soria, X.Trias, C.D.Perez-Segarra, A.Oliva

Centre Tecnologic de Transferencia de Calor Universitat Politecnica de Catalunya C/Colom, 11 08222 Terrassa (Barcelona) Tlf: +34 93 739 80 20 Fax: +34 93 739 81 01 e-mail: [email protected] Abstract

The majority of the direct numerical simulations of turbulent and transition natural convection ows in cavities assume a two-dimensional behaviour. To investigate the eect of the three-dimensional uctuations, a complete direct numerical simulation has been carried out, in a cavity with aspect ratio 4, Raz = 6:4 108 and P r = 0:71, using a low cost PC cluster. A description of the parallel algorithm and the methodology used to verify the code and the accuracy of the statistics obtained is presented. The main features of the two- and three-dimensional ows are described and compared. Several rst and second order statistic distributions have been evaluated, including the Reynolds stress tensor. Signi cant dierences are observed between the second-order statistics of the two- and three-dimensional simulations. Nomenclature Ax , Az e E Eg Ed f L N

height and depth aspect ratios local kinetic energy volume integrated kinetic energy averaged kinetic energy generation rate averaged kinetic energy dissipation rate body force length number of nodes 1

t0 T Tc Th V x,y,z

beginning of integration period temperature cold wall temperature hot wall temperature volume coordinates thermal diusivity thermal expansion coeÆcient

Nu p ph ,pt Pr g u Ray Raz t

Nusselt number dynamic pressure spatial and temporal order of accuracy Prandtl number gravitational acceleration velocity Rayleigh number based on cavity width Rayleigh number based on cavity height time

t ta T = x, y, z (u) =

time step time averaging integration period Th Tc

mesh size ru + rut : ru grid concentration parameter kinematic viscosity generic unknown

1 INTRODUCTION

Natural convection in two-dimensional (2D) and three-dimensional (3D) parallelepipedic enclousures has been the subject of numerous studies. The majority of them can be classi ed in three groups: (i) cavities where the ow is due to internal heat generation, (ii) cavities heated from below (Rayleigh-Benard con guration), (iii) dierentially heated cavities (DHC), the situation that here is under consideration. A schema of a general DHC problem is presented in Fig. 1. Two opposite vertical walls of the cavity, y = 0 and y = Ly are kept isothermal at temperatures Tc and Th > Tc. The horizontal walls z = 0 and z = Lz can either be adiabatic ( @T@x = 0) or perfectly conducting (i.e., with a linear temperature pro le between the hot and the cold temperatures). Here, these variants will be respectively denominated AHW and PCHW. If Boussinesq approximation is used, DHC problem is governed by Raz = T Lz g and P r = numbers and the geometric aspect ratios Az = LLyz (height aspect ratio) and Ax = LLxy (depth aspect ratio). 3

In the case of 3D cavities, the vertical planes that close the cavity in the depth direction, orthogonal to the main ow (x = 0 and x = Lx) can be considered adiabatic solid walls (u = 0, @T@x = 0) or a periodic behaviour can be assumed, (x; y; z; t) = (x + Lx; y; z; t). Here, these variants will be respectively denominated solid vertical walls (SVW) and periodic vertical boundary conditions (PVBC). The SVW variant forces the ow to be 3D, independently of the Ra number. Non-slip boundary condition (u = 0) is always used for the other four planes. 2

A summary of previous numerical studies of DHC ows of air (P r = 0:71) that are relevant in our context is presented in the next paragraphs. All of them solve directly the incompressible Navier-Stokes equations (without turbulence models) and use air as working uid and, except where mentioned, the Boussinesq approximation. Steady state ows

The square cavity with AHW is a very well known con guration that often is used to test CFD codes. The original benchmark formulation [1] was for 103 Raz 106 . A multigrid method was used in [2] to solve the problem with meshes up to 640 640. Latter, solutions for the full range of steady-state solutions (Raz 108 ) have been obtained using dierent methods [3, 4, 5]. The 3D cubic cavity (Ax = Az = 1), with AHW and SVW is also a well known con guration, but has received comparatively less attention [6, 7]. For large height aspect ratio cavities, in a certain range of Ra numbers [8, 9, 10], a steady-state multicellular ow is obtained. Time-dependent two-dimensional ows

Beyond a critical Rayleigh number, the DHC ows become time-dependent (periodic, chaotic and eventually fully turbulent) even with steady boundary conditions. The PCHW con guration is more unstable than the AHW, due to the presence of high temperature areas at the bottom of the cavity. Its transition to non-steadiness was studied in [11], obtaining a critical number of Raz = 2:109 106 , later con rmed by [12]. For the square AHW cavity, [13] determinated the critical number as Ra = 1:82 0:01 108 and studied the time-dependent chaotic

ows up to Ra = 1010 . For the case of cavities with AHW and Az = 4 [14], there is a Hopf bifurcation at Raz = 1:03 108 and chaotic behaviour is rst observed at Raz = 2:3 108 . Two-dimensional \turbulent" ows have been studied by [15], who carried out a direct simulation for Raz = 6:4 1010 and [14], who studied the situations with Raz = 6:4 108 , 2 109 and 1010 , recording statistics of the ow for Raz = 2 109 and 1010 . Recently, the AHW cavity with Az = 8, with Ray = 3:4 105 (unsteady), has been choosen as a test problem [16]. For this con guration, the critical Ray number for the transition to unsteadiness is Ray = 3:0619 105 . Time-dependent 2D ows have also been studied without using the Boussinesq approximation [17, 18], for Rayleigh numbers up to 1010 and dierent aspect ratios. 3

Time-dependent three-dimensional ows

The transition from 3D steady to 3D time-dependent ows was considered, for the case of a PCHW cubic cavity with SVW in [19], assuming symmetry. The critical Ra number was estimated to be between 2:25 106 and 2:35 106 , larger than the equivalent critical number of a 2D cavity. An oscilatory ow was studied [20] in the same conditions, without assuming symmetry, for Ra = 8:5 106 . For the case of the AHW cubic cavity (Az = Ax = 1, AHW, SVW), the transition to unsteadiness was studied in [21], assuming symmetry too. The critical Ra number obtained was between 2:5 108 and 3 108 . However, the same con guration was later studied without assuming symmetry [22], obtaining a non-symmetric transition for Ra = 3:19 107 , signi cantly lower than in the equivalent 2D case. The direct simulation of a turbulent ow with Ra = 1010 was carried out by [23], using a 62 122 62 mesh. Transition from two-dimensional to three-dimensional ows

In a general DHC problem with PVBC, where the boundary conditions do not force the ow to be 3D, there are three possible ow con gurations: 2D steady, 2D unsteady and 3D unsteady. A question relevant in our context is if there is a range of Rayleigh numbers where the ow is 2D but unsteady, this is, if the transitions to unsteadiness and three dimensionality are simultaneous. This was considered in detail in [24], for square cavities (Az = 1) with AHW and PCHW, using PVBC boundary conditions. In both cases, it was found that 3D perturbations are less stable than 2D perturbations, concluding that the assumption of 2D is not correct (in time-dependent square DHC). A 3D simulation was carried out for Ra = 108 , with PCHW boundary conditions, using Ax = 0:1 with four Fourier modes in the x direction. Statistics of the ow were recorded and compared with the statistics of the 2D ow. The most signi cant dierence found was an increase of the heat transfer coeÆcient in the 3D ow. For cavities of height aspect ratio 4, the same question was considered in [25]. Experimentally, it was found that in a cavity of Ax = 1:33 there is a transition to unsteadiness at Raz 108 (in good agreement with results reported in [14]). Numerical simulations (assuming 2D) con rmed this result. However, in a 3D PVBC simulation with a lower Rayleigh number, Raz = 9:6 107 , with Ax = 1, the ow 4

was found to be 3D. These results [25, 24] seem to indicate that for Az = 1 and Az = 4 (and large enough Ax), the

ows would never be unsteady and 2D. But this conclusion is not valid for other aspect ratios: for Az = 8, the critical number for the transition to unsteadiness is Ray = 306192 10, while the 2D to 3D transition is observed at higher Ray numbers, at least 3:84 105 [26]. Motivation and summary of the present work

The majority of the previous numerical simulations of chaotic and turbulent ows have been carried out assuming 2D ows. To the knowledge of the authors, there are only two previous 3D studies: a DHC cavity with AHW and SVW, at Ra = 1010 using a relatively low spatial resolution (62 122 62 with QUICK) [23] and a cavity with Ax = 0:1, Az = 1 at Ra = 108 , assuming PCHW and PVBC, using only four Fourier modes in x axis [24]. More work seems necesary to evaluate the eect of the uctuations in the direction orthogonal to the main ow over the averaged two-dimensional ows. The main goal of this paper is to use direct numerical simulations of DHC to compare the 2D and 3D ows in a situation with Az = 4, Ax = 1, P r = 0:71, Raz = 6:4 108 (i.e., higher than the critical Raz number for transition to chaotic regime [14] by a factor of 2.78), using periodic boundary conditions for the 3D ow (PVBC). The ow con guration choosen, according to the numerical and experimental results of [25] (that are con rmed by our own simulations), is 3D and time-dependent. It has been selected as an extension to 3D of one of the 2D problems studied in [14]. Statistics (u, T , u0iu0j , u0iT 0, T 0 T 0) of the 2D and 3D ows are compared. As expected, the 2D results are in good agreement with the data provided by [14] for Raz = 6:4 108 . 2 GOVERNING EQUATIONS AND NUMERICAL METHOD Governing equations

The uid is considered to be incompressible, Newtonian and with constant physical properties. To account for the density variations, the Boussinesq approximation is used. Thermal radiation 5

is neglected. Under these assumptions, the governing equations in dimensionless form are:

ru = 0 @u @t @T @t

= =

u ru + P r r 2 u

rp + f

u rT + r2 T

(1)

Where the body force vector is f = [0; 0; RaP rT ]. The reference length, time, velocity, temperature and dynamic pressure used for the dimensionless form are respectively Lz , L2z =, =Lz , T and 2 =L2z . The boundary conditions are: Isothermal at the vertical walls (T (x; 0; z) = 1, T (x; 14 ; z ) = 0). Adiabatic at the horizontal walls (AHW). Periodic boundary conditions are imposed at the vertical planes x = 0 and x = 14 (PVBC). At the four planes y = 0; y = 41 ; z = 0; z = 1, null velocity is imposed. The initial conditions used are not relevant because the statistics of the ow are recorded when statistically steady-state behaviour is reached. The dimensionless governing numbers are P r = 0:71, Raz = 6:4 108 , Az = 4 and Ax = 11 . Periodic vertical boundary conditions are used because they allow to study the 3D eects due to intrinsic instability of the main ow and not to the boundary conditions. Also, it has to be pointed out that this variant is more convenient from a computational point of view since the resulting ow has no boundary layers in the x axis and therefore, the mesh can be coarser and uniform in x. This is an important computational advantadge, because Fourier-based methods can be used to solve the Poisson equation. Numerical method

The spatial discretization is carried out using a staggered mesh [27], and a spectro-consistent second-order scheme [28]. In this way the global kinetic-energy balance is exactly satis ed even for coarse meshes. For the temporal discretization, a central dierence scheme is used for the time derivative term and a fully explicit second-order Adams-Bashforth scheme for 1

Except a situation where Ax = 2 has been considered.

6

convection, diusion and body force terms. An implicit rst-order Euler scheme is employed for the pressure gradient term and the mass conservation equation. According to the current state of computing technology, parallel computers are needed for the direct integration of Navier-Stokes equations in turbulent regimes. All the simulations described in this paper have been carried out using a low cost PC cluster. Based on a domain decomposition strategy, the parallelization of the explicit parts of the code is straightforward. However, the eÆcient solution of the Poisson equations, that must be solved almost to machine accuracy, is a critical aspect. This issue is specially relevant if a low cost PC cluster, with a high latency and low bandwidth network, is used. Here, a Direct Schur Fourier Decomposition (DSFD) method is used [29, 30]. DSFD allows to solve each Poisson equation almost to machine accuracy (mass residual is reduced by a factor of 10 12 ) using a single all-to-all communication episode. With 36 computing nodes and for a mesh of N = 64 154 310 control volumes, the integration of each time step takes roughly 5 seconds. This is 30 times less than the time needed by a single node (using Fourier decomposition combined with Additive Correction Multigrid [31] to solve the Poisson equations). Averaging operator

The main interest of the direct simulation is not the instantaenous elds but their statistics. In our con guration, after a long enough time from the initial conditions, the ow has the following properties: (i) It is statistically stationary, i.e., all the statistics are invariant under a shift in time, (ii) It is statistically 2D, i.e., its statistics are identic in each yz plane and n o (iii) It is statistically skew-symmetric, i.e., (y; z ) = ( 14 y; 1 z ) for = u2 ; u3 ; T 21 . Averages on the three statistically invariant transformations are carried out. For the case of vertical velocity, the averaging operator is:

u3 (y; z ) =

1 Z Lx Z t +ta u3 (x; y; z; t) 1 + ju3 (x; Ly y; Lz z; t) j dt dx 2Lxta 0 t ju3 (x; y; z; t) j 0

0

(2)

The statistic properties of the ow could be estimated only with time (or even space) averaging. 7

However, the triple averaging indicated in Eq. 2 is used in this paper with the aim of reducing the averaging period ta, and therefore the computational cost. In [24], time and depth averaging but not symmetry averaging are performed. In practice, the evaluation of averaging operator presents several diÆculties. The data to be averaged has to be de ned before the simulation since the storage of the instantaneous elds (about 100 Tbytes) is not possible. The initial instant for the evaluation of the averaging t0 is not known a priori. The averaging of products of uctuations such as a0b0 can not be directly evaluated and must be post-processed as ab ab. Since the computing time is long, the computation is usually interrupted (e.g., by hardware failures in one of the computing nodes), so partial averages of all the products of interest should be saved and then assembled. Symmetry and space averaging can not be evaluated during the parallel simulation since this would requiere massive data exchange between the processors. 3 Code and simulation veri cations 3.1 Code veri cation

Before using the code developed for direct numerical simulations, it is necessary to verify that it is correct (i.e., that it solves Eq. 1 with the prescribed boundary conditions and the expected order of accuracy). To do so, the Method of Manufactured Solutions (MMS) [32] was used. In MMS, an arbitrary analytic function ua (that will be the solution of the PDE system) is choosen. Then, the source term fa that matches with the arbitrary solution, is calculated analytically from the PDE to be solved. The initial and boundary conditions are obtained evaluating ua. This procedure ensures the obtention of an analytic solution even for complex equations such as the Navier-Stokes system. This analytic source term fa is evaluated at the discretization nodes and then used as input data for the numerical code. The time-dependent numerical solution un is compared against ua . The norm of the error is evaluated as: kek1 = kua unk1 = Max j ua (xi; yj ; zk ; tn) un (xi; yj ; zk ; tn) j

8

(3)

where the indices xi; yj ; zk ; tn sweep all the spatial and temporal domain. If this measure is repeated for systematically re ned grids, kek1 must tend to zero with the expected order of accuracy. In general, it can be expressed as: kek1 = Ct tpt + Chhph + H:O:T

(4)

where Ct and Ch are constants, t is the time step, h is a measure of the spatial discretization, and pt and ph are the orders of accuracy of the temporal and spatial discretizations. The values of pt and ph obtained with the code are evaluated separately and compared with the theoretical results. With the discretization schemes implemented in our case, pt = ph = 2 should to be obtained for uniform temporal and spatial discretizations. For too coarse meshes or too dense meshes, the higher order terms and round-o errors are respectively dominant in Eq. 4 and a second-order behaviour is not observed. However, within a wide range of h and t values, the expected orders of accuracy should be obtained. The results have been represented in Fig. 2, together with the ph and pt that t better with the numerical values. They are in good agreement with the theoretically orders of accuracy (that, using a spectro-consistent scheme, should be below 2 for non-uniform meshes [28]). To look for all possible errors, this procedure has been repeated for dierent numbers of discretization nodes, domain sizes in the three axis, and mesh concentration parameters, obtaining analogous results. For an explicit parallel algorithm based on spatial decomposition, it is also important to ensure that the results are strictly independent of the number of processors, since bugs in the halo-update routines can be otherwise undetected. The maximum number of processors used has to be large enough so that all their possible relative positions are exercised. That is, for a 2D domain decomposition at least nine processors (3 3) must be used to ensure that one of them has no external boundary conditions. For implicit parallel algorithms (even using direct linear solvers as in our case) slight variations in the results are possible.

9

3.2 Simulation veri cation

After the absence of errors in the code has been ensured, it is necessary to determinate if the simulation parameters used allow to obtain numerical results reasonably close to the asymptotic solution of the problem. The parameters are: mesh size, mesh concentration, time step, domain length in the direction orthogonal to the main ow (Lx), beginning of averaging period t0 and integration time to evaluate the ow statistics ta. For all of them, a compromise between accuracy and computing time must be accepted. As the ow under consideration is chaotic, even very small changes on the parameters produce totally dierent instantaneous elds after a certain integration time. However, the statistic properties of the ow are independent of them. A total number of eleven 2D and 3D simulations have been carried out and compared in order to estimate the accuracy of the results. Mesh size and concentration

The selection of an appropiated spatial resolution (number of nodes and concentration) is a critical aspect of direct numerical simulations. In [17] and [15], the spatial resolution was selected using as a guideline the Kolmogorov length scales and the size of the conductive sublayer. In [14], no a priori spatial resolution analysis was needed because, as explained by their authors, the spectral method is unstable if the smallest scales of the ow are not resolved. In our work, a strategy based on succesive mesh re nements has been used to obtain a qualitative estimation of the accuracy of the time-averaged 2D and 3D solutions. A summary of the eleven simulations carried out is presented in Table 1, where is a concentration factor for the hyperbolic tangent function used to generate the mesh (densi ed near the walls in y and z axis and uniform in x axis) and t is the averaged time step. Beginning with the coarsest mesh, referred as C, the number of control volumes in each axis has been multiplied by two and four in meshes B and A respectively. The concentration parameter is the same, y = z = 2, for meshes B and C. However, due to the CFL condition, it was necessary to decrease it to y = z = 1:5 for mesh A. Mesh A is only slightly ner than mesh B near the boundary layers, but its density is more than double in other areas of the domain. The simulation B2X, analogous to B but with Lx = 2 and double number of control volumes in 10

x axis, has been carried out to investigate the in uence of the domain size in the homogeneous

direction. The meshes A2D, B2D and C2D are a 2D projection of the respective 3D meshes A, B and C.

A summary of the results is presented in Table 2 and Fig.3. In Table 2, by rows, from top to bottom, the magnitudes are: the total averaged kinetic energy generation ratio Eg , the total averaged turbulent kinetic energy generation ratio RaV R u03T 0d , the total averaged turbulent kinetic energy dissipation ratio V1 R (u0 )d , the overall averaged Nusselt number at the hot or cold wall, the maximum of the averaged Nusselt number at the hot wall and its position, the maxima of u3 and u03u03 at the central z plane and their respective y positions, the maxima of u3 and u03 u03 at the central z plane and their respective y positions. As can be seen in Table 2 and in Fig. 3, the dierences in the rst-order statistics in meshes A and B (in both 2D and 3D simulations) are relatively small. However, for the secondorder statistics, from the comparison of the results from meshes A and B we can not conclude that the results of mesh A are accurate enough. To clarify this question, two additional 2D simulations have been carried out: AB2D with a double density than B2D in all the domain and mesh AA2D, designed to be ner in the areas where the previous simulations indicated larger uctuations. The results of A2D, AA2D and AB2D are quite similar even for the second order statistics (despite their dierences in the minimum and maximum control volume sizes). This seems to indicate that the second order results of mesh A2D are reasonably accurate. As the distribution of the second order statistics is more complex (as discussed later) for the 2D simulations than for the 3D simulations, we can conclude that the second-order statistics of the 3D simulations are reasonably accurate too. Nevertheless, simulations with more resolution would be convenient in order to con rm the results here presented. Also, a methodology for quantitative estimations of the accuracy of the rst- and second-order statistics would be of high interest. Time step

As the time discretization of the transport equations is explicit, t is limited by CFL condition. The averaged time step used for each mesh can be found in Table 1. To determinate if t is 11

small enough as to obtain independent averaged results, the simulations with mesh B2D have been repeated for time steps two and four times smaller. The results (not included for brevity) indicate that the time steps used are correct. Domain length in the direction orthogonal to the ow

If PVBC are used, the aspect ratio in the periodic direction Ax must be chosen large enough to contain all relevant large-scale vortices. In order to elucidate if Ax = 1 is enough, the case B was repeated with Ax = 2 and the same mesh density. The results obtained are very similar for the rst-order statistics but somewhat dierent for the second-order statistics (Fig. 3). This indicates that a box larger in x axis would be necessary. Integration period ta

In order to estimate the averaging time ta needed to obtain asymptotic statistics, the evolution of several rst- and second-order statistics was studied as a function of ta. As expected, the convergence of the rst-order statistics is faster. A summary of the results is presented in Fig.4. A similar approach was used in [24], who compared the results obtained with three dierent integration periods. As an additional control parameter, the dimensionless imbalance of kinetic energy generation ratio, jEgEgEdj was also analyzed. For ta > 4:5 10 2 , this parameter is in the order of 10 5 . To select a ta, as for the other numerical parameters, a compromise between accuracy and computing time must be made. In our case, for all the simulations ta > 2 10 2 , which is in good agreement with the integration period used by [14] for the same Ra number, ta 1:56 10 2 (expressed in our dimensionless scale). The integration must begin at t = t0, when the eect of the initial conditions is lost and the

ow has reached a statistically stationary state. In order to ensure that a correct t0 has been selected, two veri cations are carried out. First, during the simulation, the evolution of the velocity and temperature at dierent positions and the overall Nusselt number are controlled. Second, the time integral in Eq. 2 is evaluated beginning at the nal instant, t0 + ta and going back until a certain initial instant t, when the statistic values obtained are considered stable. If stable statistics are reached, then we can conclude that the integration period used does not overlap with the transient period (i.e., t > t0). If stable statistics can not be obtained, 12

the simulation should proceed, to allow a longer ta. 4 RESULTS AND DISCUSSION Instantaneous elds

The temperature distribution maps of three dierent instants of the 2D and 3D2 simulations have been represented in Fig. 5. The main dierence is that in the 2D simulations the chaotic

uctuations are concentrated in the top and bottom areas of the cavity while the vertical boundary layers are almost totally stable. This can be clearly appreciated in animations of the velocity components and temperature distributions. The second-order statistics con rm this eect, as discussed below. Also, in the 2D results, the vortices at the end of the vertical boundary layers are more vigorous and stable, as can also be appreciated in the streamline maps of u (Fig. 6). Ocasionally, in the 3D simulation, there are large instability episodes where the three-dimensional structures generated at the top right and bottom left areas of the cavity propagate across all the boundary layers. However, this phenomenon (that has not been observed in the 2D results) is too infrequent to generate signi cant values of u01u01 and u02u02 at the vertical boundary layers. Averaged ow

The averaged ow maps have been represented in Fig. 6. A quantitative comparison with the results of [14] is diÆcult, but (as expected), the 2D results seem to be in good agreement3 . Despite the large dierences in the instantaneous maps, the 2D and 3D averaged ows are similar. In both cases there is a large core area with very low averaged velocity and a strati ed temperature distribution. The thermal vertical boundary layers are very similar. However, the main recirculation at the corners is stronger in the 2D simulation, that also presents two additional small vortices. Heat transfer 2 3

The results discussed in next paragraphs correspond to simulations A and A2D. In particular, three digits of the overall Nusselt number are equal.

13

As expected from the similarity between the 2D and 3D averaged thermal boundary layers, the overall averaged Nusselt numbers are almost equal, 49:23 for the 2D and 49:20 for the 3D (see Table 2). The averaged local Nusselt distribution is also very similar, as can be observed in Fig. 7. However, there are dierences in the instantaneous Nusselt numbers, that are clearly 3D (Fig. 7, bottom left), and the standard deviation of the local Nu number is signi cantly dierent in the 2D and 3D results (Fig. 7, top). Kinetic energy balances

The transport equation for kinetic energy, e = 12 u u is obtained from the scalar product of velocity vector and momentum equation, @e @t

h

= r (eu) + P r r u ru + rut

i

Pr

r (pu) + u f

(5)

Where (u) = ru + rut : ru. The term P r is the kinetic energy dissipation ratio that arises from the viscous forces term P r r2u. Integration of (5) in the domain yields: Z h Z i d t E= eu + P r u ru + ru + pu dS + [u f dt @

P r ] d

(6)

Where E = R ed is the total kinetic energy. The rst term, that accounts for the boundary interactions, is null in domains where all the boundaries are either periodic or with null velocity uj@ = 0 (or a combination of both as in our case). Considering these types of boundary conditions, the instantaneous global kinetic energy balance equation can be expressed as: Z d E = [u f dt

P r ] d =

Z

[RaP rT u3

P r] d

(7)

It is clear from this expression that the only terms of the momentum equation that contribute to the evolution of the total kinetic energy are P r, that necessarily dissipates kinetic energy to thermal energy (as 0 ) and the body force term u f that can either generate or dissipate kinetic energy. Pressure gradient and convective terms have a local eect but do not contribute 14

to the global kinetic energy balance. The spectro-consistent [28] numerical scheme used for the simulations allows the discrete velocity elds to verify exactly Eq. (7) even for coarse meshes. This has been tested as an additional veri cation of the code. Taking the average (as de ned in Eq. 2) of Eq. 7, for a long enough ta, a global kinetic energy balance is obtained, that expressed per volume unit is: Z h i i Ra Z h 1 0 0 (u) + (u0 ) d

u3 T + u3 T d = V {z V

} {z } | |

Eg

(8)

Ed

That is, for a statistically stationary ow, Eg , the averaged kinetic energy generation rate (only due to the bouyancy forces in our case) must be equal to Ed, the averaged kinetic energy dissipation rate due to viscous forces. The values of Eg obtained in the dierent simulations can be found in Table 2. Since the instantaneous kinetic energy balances are exactly satis ed, the expression jEgEgEdj can be used to control if t0 and ta used for the averaging are appropiated. As mentioned before, it has been represented versus ta in Fig. 4. The overall kinetic energy generation rate Eg is very similar in the 2D and 3D simulations, but the turbulent generation term, RaV R u03 T 0d , and the turbulent dissipation term, V1 R (u0 )d , dier in a factor of 32:6 and 1:73 respectively (see Table 2). Second order statistics

The distributions of turbulent kinetic energy, its dissipation rate (as de ned in Eq. 8), T 0T 0 and u03T 0 have been represented in Fig. 8, while the four non-null components of the Reynolds stress tensors have been represented in Fig. 9. Obviously, the component u01 u01 is null in the 2D simulation. The components u01u02 and u01u03 are null even in the 3D simulations (except statistical noise that decreases with ta) since the ow is statistically 2D. Their maxima and their spatial distributions are signi cantly dierent (i.e., larger than the possible numerical inaccuracies). For the 2D results, as can be seen in Figs. 8 and 9, all the second-order statistics are almost null at the vertical boundary layers. This behaviour is also observed in the 2D results presented in [14] for Raz = 2 109 and Raz = 1010 . When instantaneous maps are assumed to be 2D, the vertical boundary layers are almost steady, and the uctuations are concentrated at 15

the top and bottom regions, where the second order statistics have more complex distributions than in the 3D simulations (e.g., the map of u02u02 obtained in the 2D simulation has also a minimum at the corners, not present in the 3D case). The 2D and 3D distributions of u03 T 0 are totally dierent. 5 CONCLUSIONS

Results of 3D and 2D direct numerical simulations of a buoyancy driven ow in a dierentially heated cavity have been presented. The correctness of the code has been veri ed using the method of manufactured solutions (MMS), that ensures that the order of accuracy is in good agreement with the theoretical expectation in the whole domain. A low cost parallel computer (a PC cluster with a conventional 100 Mbits/s network) has been used for the simulations. An explicit scheme has been used for temporal integration and a spectro-consistent second-order scheme for spatial discretization, that preserves the global kinetic energy balances even for coarse meshes. The parallel algorithm is based on spatial domain decomposition. A direct algorithm (DSFD) has been used to solve the Poisson equations with only one communication episode. This method allows a high accuracy and eÆciency (with speed-ups of about 30), even using a conventional network with high latency and low bandwith. A detailed description of the ows has been presented, including distributions of the Reynolds stress tensor components, the variance of the temperature, local Nusselt numbers and their standard deviation and the generation and dissipation of turbulent kinetic energy. In order to obtain a qualitative estimation of the accuracy of the solutions, a total of eleven direct simulations have been carried out, with dierent meshes (from 4:8 104 to 3:1 106 nodes for the 3D simulations), dierent integration periods and time steps (up to 2 106 time steps) and two lengths in the homogeneous direction of the domain. The accuracy of the rst-order statistics seems to be quite good. However, more CPU resources or higher-order numerical schemes would be needed in order to ensure that the second-order statistics obtained for the 3D ows are accurate enough. With the available data, it seems reasonably to conclude that all the results are at least qualitatively correct. In spite of that, it seems that it would be 16

convenient to use an aspect ratio larger that Ax = 1 in the homogeneous direction. The eect of the integration period has been analyzed. The 2D results are in good agreement with the results previously published. Respect to the comparison of 2D and 3D results, their rst-order statistics are very similar, in particular the averaged local and overall Nusselt numbers. This is not the case for the second-order statistics, that are substantially dierent (by two orders of magnitude in certain cases), far beyond the possible numerical inaccuracies. The assumption of instantaneous 2D maps in uences the distribution of all the second order statistics. The main dierences are at the vertical boundary layers, where the 2D simulation incorrectly predicts very low values for all the second-order statistics. However, for the Ra number considered, the overall eect of the turbulent uctuations is relatively small and this incorrect prediction does not have an important impact on the averaged ow. If the direct numerical simulations of buoyancy driven ows are to be used to develop or enhance RANS turbulence models, the assumption of two-dimensionality is not valid, even at the relatively low Rayleigh number considered here. On the other hand, the 2D simulations may be enough to capture the main features of the natural convection ows, such as the local and overall Nu numbers. But it would be necessary to extend the present results to higher Ra numbers to con rm this conclusion. Acknowledgements

This work has been nancially supported by the \Comision Interministerial de Ciencia y Tecnologa", Spain (project TIC99-0770).

FIGURES AND TABLES

Figure 1: General DHC problem.

18

0.01 0.001

u (ph=1.98) v (ph=1.78) w (ph=1.88) t (ph=1.93) div (ph=1.94)

0.0001 1e-05 1e-06 1e-07 1e-08 1e-09 1e-10 1e-11 1e-12 0.01 1e-10

0.1 u (pt=2.00) v (pt=2.00) w (pt=2.00) t (pt=2.00)

1e-11

1e-12

1e-13

1e-14 0.0001

Figure 2: Maximum dierences between the analytical and numerical solutions (in the three components of momentum equation, the energy equation and the divergence operator). Top: errors versus mesh size h = Lx=Nx for meshes concentrated in axis y and z (for = 1:5), with Nx = Ny = Nz . Bottom: errors versus time step. In parentheses (see boxes) the spatial and temporal orders of accuracy (ph, pt).

3000

450000 A B B2X

A B B2X

400000

2000 350000 300000 avg(u3’u3’)

avg(u3)

1000

0

-1000

250000 200000 150000 100000

-2000 50000 -3000

0 0

0.05

0.1

0.15

0.2

0.25

0

0.05

0.1

y

0.15

0.2

0.25

y

3000

450000 A2D AA2D AB2D B2D

2000

A2D AA2D AB2D B2D

400000 350000 300000 avg(u3’u3’)

avg(u3)

1000

0

-1000

250000 200000 150000 100000

-2000 50000 -3000

0 0

0.05

0.1

0.15 y

0.2

0.25

0

0.05

0.1

0.15

0.2

0.25

y

Figure 3: Comparison of the rst- and second-order statistics in the section z = 1516 Lz (where the discrepancies are largest). Top: u3 and u03u03 for the 3D simulations. Bottom: Analogous results for the 2D simulations.

1404.5 1404 1403.5

max-avg(u3)

1403 1402.5 1402 1401.5 1401 1400.5 1400 0

0.005

0.01 0.015 0.02 Integration Period (∆ta)

0.025

0.03


0.025

0.03


0.025

0.03

20000 18000

max-avg(u3’u3’)

16000 14000 12000 10000 8000 6000 4000

Energy dissipation-generation imbalance

0

0.005

0.01

0.001

0.0001

1e-05

1e-06 0

0.005

Figure 4: Estimation of the integration period ta needed to evaluate rst- and second-order statistics (using simulation parameters B). Top: Maximum u3 at the central z plane versus ta. Center: Maximum u03u03 at the central z plane ta . Bottom: Accumulation of kinetic energy, jEgEgEdj versus ta.

Figure 5: Instantaneous temperature maps corresponding to the statistically steady state, with a vertical section at y = 0:95Ly . Top: Results of the 3D simulation. Bottom: Analogous results of the 2D simulation.

1 3D 2D

1000

500 0.6 avg(T)

avg(u2),avg(u3)

0.8

0 0.4 -500 0.2 -1000 0 0

0.2

0.4

0.6

0.8

1

z

Figure 6: Averaged ow eld. Top: From left to right, averaged results of 3D and 2D simulation, streamlines and isotherms. Bottom: Vertical pro les at y = 0:5Ly of u2 , u3 and T .

45

1.8

40

1.6

35

1.4

30

1.2

25

1

20

0.8

15

0.6

10

0.4

5

0.2

0

sdev(Nu)

avg(Nu)

3D 2D

0 0

0.2

0.4

0.6

0.8

1

z

Figure 7: Top: Averaged local Nusselt numbers (at the left) and standard deviation of local Nusselt numbers (at the right) for 2D and 3D simulations. Bottom: Instantaneous local Nusselt numbers at the hot wall (y = 0) for 3D (left) and 2D simulations (right).

Figure 8: Top: Results of the 3D simulation, from left to right, u0i u0i, (u0), T 0T 0, Bottom: Analogous results of the 2D simulation.

u03 T 0

.

Figure 9: Non-null components of u0iu0j . Top: Results of the three-dimensional simulation, from left to right, u01u01 , u02 u02, u03u03 , u02 u03. Bottom: Analogous results of the 2D simulation.

Nx

A B B2X C A2D B2D C2D AB2D AA2D

Ny

64 32 64 16 -

Nz

156 78 78 39 156 78 39 156 218

Lx

312 156 156 78 312 156 78 312 438

y

1 1 2 1 -

z

1.5 2.0 2.0 2.0 1.5 2.0 2.0 2.0 1.5

ymin 1.5 2.0 2.0 2.0 1.5 2.0 2.0 2.0 1.0

ymax 4:95 10 5:08 10 5:08 10 1:10 10 4:95 10 5:08 10 1:10 10 2:44 10 3:48 10

Table 1. Main parameters of the 3D and 2D simulations.

4 4 4 3 4 4 3 4 4

zmin 2:70 10 6:83 10 6:83 10 1:40 10 2:70 10 6:83 10 1:40 10 3:38 10 1:90 10

3 3 3 2 3 3 2 3 3

zmax 9:75 10 9:75 10 9:75 10 2:03 10 9:75 10 9:75 10 2:03 10 4:78 10 1:26 10

4 4 4 3 4 4 3 4 3

t 5:35 10 1:35 10 1:35 10 2:73 10 5:35 10 1:35 10 2:73 10 6:70 10 3:00 10

3 2 2 2 3 2 2 3 3

4:73 10 5:14 10 5:15 10 6:25 10 4:89 10 3:17 10 6:25 10 1:19 10 2:42 10

8 8 8 8 8 8 8 8 8

Eg 1 V Ra 1

V

R

R

u03 T 0

(u0 )

Nu Numax z u3 max y u03 u03 max y u3 max z u03 u03 max z

Eg 1 V Ra 1

V

R

R

u03 T 0

(u0 )

Nu Numax z u3 max y u03 u03 max y u3 max z u03 u03 max z

A 4:415 1010 1:341 108 1:542 109 4:920 101 4:303 101 4:525 10 3 5:599 103 7:508 10 3 3:004 105 1:326 10 2 1:075 103 6:106 10 2 3:541 105 1:061 10 1

A2D 4:418 1010 4:124 106 8:861 108 4:923 101 4:249 101 4:525 10 3 5:640 103 7:508 10 3 3:838 103 1:351 10 2 1:137 103 8:408 10 2 5:779 105 1:562 10 1

B 4:421 1010 1:232 108 1:330 109 4:915 101 4:305 101 3:647 10 3 5:607 103 7:508 10 3 2:701 105 1:351 10 2 1:096 103 6:406 10 2 3:862 105 1:161 10 1

AA2D 4:415 1010 1:855 107 9:161 108 4:921 101 4:237 101 4:462 10 3 5:648 103 7:508 10 3 3:852 103 1:401 10 2 1:179 103 8:709 10 2 5:131 105 1:602 10 1

Table 2. Summary of the 3D and 2D results.

B2X 4:432 1010 9:995 107 1:089 109 4:918 101 4:293 101 3:647 10 3 5:618 103 7:508 10 3 1:841 105 1:351 10 2 1:098 103 7:207 10 2 4:196 105 1:231 10 1

AB2D 4:410 1010 1:514 107 9:179 108 4:923 101 4:214 101 4:476 10 3 5:643 103 7:758 10 3 4:648 103 4:329 10 2 1:152 103 8:609 10 2 5:432 105 1:572 10 1

C 4:406 1010 1:842 108 1:338 109 4:918 101 4:538 101 3:152 10 3 5:579 103 7:508 10 3 3:555 105 1:351 10 2 8:420 102 3:303 10 2 3:632 105 6:206 10 2

B2D 4:419 1010 2:925 106 8:652 108 4:920 101 4:254 101 3:647 10 3 5:640 103 7:508 10 3 3:417 103 1:351 10 2 1:077 103 8:208 10 2 5:440 105 1:502 10 1

C2D 4:383 1010 2:259 107 8:605 108 4:949 101 4:428 101 3:152 10 3 5:658 103 7:508 10 3 4:343 104 1:351 10 2 7:784 102 7:407 10 2 4:511 105 8:759 10 1

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