A combination of Crouzeix-Raviart, Discontinuous Galerkin and MPFA methods for buoyancy-driven flows

A combination ofCrouzeix-Raviart, DiscontinuousGalerkin and MPFA methods for

buoyancy-driven flowsAnis Younes

Laboratoire d’Hydrologie et de Geochimie, University of Strasbourg, Strasbourg,France

Ahmed MakradiCentre Henry Tudor, Luxembourg-Kirchberg, Luxembourg

Ali ZidaneLHYGES, Strasbourg, France

Qian ShaoPublic Research Centre Henry Tudor, Esch-sur Alzette, Luxembourg, and

Lyazid BouhalaCentre Henry Tudor, Esch-sur-Alzette, Luxembourg

Abstract

Purpose – The purpose of this paper is to develop an efficient non-iterative model combiningadvanced numerical methods for solving buoyancy-driven flow problems.Design/methodology/approach – The solution strategy is based on two independent numericalprocedures. The Navier-Stokes equation is solved using the non-conforming Crouzeix-Raviart (CR)finite element method with an upstream approach for the non-linear convective term. The advection-diffusion heat equation is solved using a combination of Discontinuous Galerkin (DG) and Multi-PointFlux Approximation (MPFA) methods. To reduce the computational time due to the coupling,the authors use a non-iterative time stepping scheme where the time step length is controlled by thetemporal truncation error.Findings – Advanced numerical methods have been successfully combined to solve buoyancy-drivenflow problems on unstructured triangular meshes. The accuracy of the results has been verified usingthree test problems: first, a synthetic problem for which the authors developed a semi-analyticalsolution; second, natural convection of air in a square cavity with different Rayleigh numbers(103-108); and third, a transient natural convection problem of low Prandtl fluid with horizontaltemperature gradient in a rectangular cavity.Originality/value – The proposed model is the first to combine advanced numerical methods(CR, DG, MPFA) for buoyancy-driven flow problems. It is also the first to use a non-iterative timestepping scheme based on local truncation error control for such coupled problems. The developedsemi analytical solution based on Fourier series is also novel.

Keywords Buoyancy-driven flows, Crouzeix-Raviart finite elements, Discontinuous Galerkin,Incompressible Navier-Stokes equation, MPFA, Time stepping

Paper type Research paper

The current issue and full text archive of this journal is available atwww.emeraldinsight.com/0961-5539.htm

Received 17 July 2012Revised 4 February 2013

Accepted 25 April 2013

International Journal of NumericalMethods for Heat & Fluid Flow

Vol. 24 No. 3, 2014pp. 735-759

r Emerald Group Publishing Limited0961-5539

DOI 10.1108/HFF-07-2012-0156

This work was done during a stay of the first author at the Public Research Center Henri Tudorat Luxembourg and has been supported by the National Research Funding of Luxembourg(FNR) via Accompanying Measure AM2c program, FNR/11/AM2c/16. Part of this research workhas been achieved in the framework of the FNR CORE OMIDEF project (FNR/786643).

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Methods forbuoyancy-driven

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1. IntroductionBuoyancy-driven flow where the fluid motion is generated by density changes causedby temperature and/or concentration variations can be encountered in many fields(e.g. salt water intrusion, weather systems, ocean dynamics, solidifications processes,etc.). The modeling of these problems is an active research subject (Bejan, 2004) dueto the high non-linear coupling between the flow and energy (or transport) equations.In this work, we consider a viscous, incompressible, Newtonian and Boussinesqapproximated fluid. The buoyancy-driven flow problem is governed by momentum,mass and energy conservation equations. The system of Partial Differential Equations(PDEs) is as following:

rquqtþ r u:rð Þu� mr: ruð Þ þ rP ¼ �rgrz ð1Þ

r:u ¼ 0 ð2Þ

rCpqT

qtþ rCp urT �r: krTð Þ ¼ 0 ð3Þ

where r is the fluid density, u is the velocity vector, P is the pressure, g is the gravityacceleration, m is the dynamic viscosity, C p is the specific heat capacity and k isthe thermal conductivity. The right hand term of Equation (1) corresponds to thebody force.

In the case of a uniform temperature distribution, the flow system (1)-(2) andthe advection-diffusion heat transport Equation (3) are uncoupled and can be solvedseparately. However, in the case of Buoyancy-driven flow, these equations are linked.Indeed, the gravity acting on the density variations provokes buoyancy forces whichdrive the fluid flow. The state equation linking the density to the temperature isgenerally considered linear:

r ¼ r0 � r0b T � T0ð Þ ð4Þ

where r0 is the density of the flow at the reference temperature T0 and b is thecoefficient of expansion for the fluid.

In this work, efficient and accurate numerical methods are developed to solve thecoupled system of PDEs (1)-(3) on unstructured triangular meshes, which are adaptedfor general domains with complex geometry.

The Navier-Stokes Equation (1) is solved using the Crouzeix-Raviart (CR) method.This mixed finite element method employs different basis functions for the pressureand the velocity approximations. This method leads to a final non-singular linear systemwithout resorting to staggered grids or stabilization techniques (such as pressurestabilization techniques, penalty methods or artificial compressibility methods).

Generally, mixed finite elements (as for instance, Taylor-Hood finite elements), leadto a large final system where the unknowns are the degrees of freedom of both pressureand velocity. To avoid this drawback, the CR finite element method uses the lowest-order P1-P0 method (linear velocity, constant pressure). The method works for bothsteady state and transient flows and satisfies the local mass conservation property(Gresho and Sani, 1998). This property is essential when solving the transport equationin order to avoid artificial sources and sinks. The CR method is combined here with a

736

HFF24,3

special upstream discretization of the non-linear convective term in order to ensure astable solution in the case of high Reynolds number.

For the advection-diffusion heat Equation (3), a combination of DiscontinuousGalerkin (DG) and Multi-Point Flux Approximation (MPFA) methods are used inthis work. Indeed, in the case of advection dominated heat transport, standardnumerical methods, such as Finite Elements (FE) or Finite Volumes (FV), generatesolution with numerical diffusion and/or non-physical oscillations. To avoid theseproblems, the DG method is used for the advection part of the heat equation. It leadsto a high-resolution scheme for advection that has been proven to be clearly superior tothe already existing finite element methods (Arnold et al., 2002). The DG methodmaintains the local conservation of finite volume methods but allows high-orderapproximations to enter through a variational formulation rather than by somehybridized difference or functional reconstruction (Kirby, 2000). With DG, theadvective fluxes are uniquely defined by solving a Riemann problem at the interfaceof two elements and the solution has been shown to be total variation diminishing inGowda and Jaffre (1993).

In the literature, Konz et al. (2009) compared DG and FV methods for the simulationof density-driven flow experiments in porous media. They obtained a very goodmatching between the DG and the experimental results for all concentration isolines.On the other hand, the results of FVs with a very fine mesh (containing threetimes more elements than used for the DG simulation) accurately reproduce the 50per cent isoline, but failed to reproduce the 2 per cent concentration isoline(Konz et al., 2009).

Note that, contrarily to hyperbolic systems, elliptic and parabolic systems, whensolved with DG do not lead to numerical results that are clearly superior to the alreadyexisting finite element methods (Arnold et al., 2002). Therefore, in this work, thediffusion part of the heat transport equation is solved using the MPFA method(Aavatsmark et al., 1996). This method uses more than two elements to compute fluxesacross edges and is well adapted for unstructured triangular meshes. The MPFAmethod has similar properties than the mixed finite element method. Indeed, both arelocally conservative and handle general irregular grids on anisotropic heterogeneousdomains. The link between MPFA and mixed finite elements of Raviart Thomas orBrezzi-Douglas-Marini has been studied by many authors (for instance, Edwards andRogers, 1998; Klausen and Russell, 2004; Vohralık, 2006; Younes and Fontaine, 2008a, b).In contrast to mixed finite elements, the MPFA method provides fluxes at elementinterfaces explicitly by weighted sums of discrete element values. Moreover, the MPFAmethod uses the same type of unknowns (average value per element) than the DG methodand therefore, both discretizations can be gathered into one system matrix (Younes andAckerer, 2008) which avoids operator splitting errors.

Generally, the Picard method is used for the linearization of the coupled system(1)-(3) at each time step. Flow and heat equations are solved sequentially which allowsthe use of different temporal discretization and therefore, allows achieving highaccuracy for each equation. In this work, the flow system (1)-(2) is solved using animplicit time discretization whereas an explicit time discretization is preferred for theadvective part of the heat Equation (3).

Time stepping schemes can be used to reduce the computational cost by optimizingthe time step size. Time stepping schemes with embedded error control havebeen developed in many non-linear problems like density-driven flow in porousmedia (Diersch and Kolditz, 1998), incompressible Navier-Stokes problems (Turek, 1996),

737

Methods forbuoyancy-driven

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unsaturated flow (Williams and Miller, 1999; Kavetski et al., 2001), elastoplasticconsolidation in geomechanics (Sloan and Abbo, 1999), reactive transport problems(Saaltink et al., 2004).

In the present study, the non-iterative time stepping scheme of Younes and Ackerer(2010), based on local truncation error control, is used for the buoyancy-driven flowsystem (1)-(3) and compared to the empirical time stepping where the time step size isadapted according to the performance of the iterative non-linear solver. To validatethe implemented methods, three test problems are studied. We develop in the first onea semi-analytical solution for coupled Stokes flow and heat equations by expandingthe temperature and the stream function in double Fourier series. The obtained semi-analytical solution is then compared against the numerical one. The second test problemis the de Vahl Davis benchmark (de Vahl, 1983) for natural convection of air in a squarecavity. This problem was largely studied in the literature and the obtained results arecompared to the published ones for a wide range of laminar steady flow regimes(Rayleigh number between 103 and 108). The last test problem is the one suggested byZaloznik et al. (2008) and Kosec and Sarler (2013) to study transient natural convectionwith horizontal temperature gradient of low Prandtl fluid in a rectangular cavity. Theresults of this test problem show an oscillatory phenomenon which can be studied byanalyzing the time evolution of the average Nusselt number.

The paper is organized as follows: Sections 2 and 3 are devoted to the numericaldiscretization of the flow and heat equations. The time stepping procedure isdetailed in Section 4. Validation of the numerical model is performed in Section 5by comparing the obtained results to the semi-analytical solution of the first testproblem and the numerical results from the literature for the second and thirdtest problems.

2. Flow discretizationThe flow system (1)-(2) is discretized using the non-conforming CR finite elementsfor the velocity approximation in combination with constant pressure per element.This combination satisfies the Babuska-Brezzi stability condition (Brezzi and Fortin,1991; Girault and Raviart, 1986; Gresho and Sani, 1998). The CR method has been usedin many problems such as the Darcy-Stokes problem (Bruman and Hansbo, 2005), theStokes problem (Crouzeix and Raviart, 1973) and the elasticity problem (Hansboand Larson, 2003). It results in a simple, stable and optimal order approximation of theStokes equations (Arnold, 1993).

In the following, we recall the main stages for the discretization of the Navier-Stokesequation (1) with the CR triangular elements. The non-linear convective term isdiscretized using an upstream approach with weight as in (see for instance, Schieweckand Tobiska, 1989; Djadel and Nicaise, 2008).

2.1 Velocity approximationWith the non-conforming finite element method, the degrees of freedom for the velocityvector u are its two components (ui, vi) at the mid-edge i that is located facing the node i(see Figure 1).

The two components of the velocity vector u(u, v) are approximated as following,

u ¼X

l

uljl and v ¼X

l

vljl ð5Þ

738

HFF24,3

where the interpolation function ji is non-zero only inside the elements E and itsadjacent element E0 that share the edge i. A linear variation is assumed for the velocityuE(uE, uE) inside the element E:

uE ¼ uijEi þ ujjE

j þ ukjEk ; vE ¼ vijE

i þ vjjEj þ vkjE

k ð6Þ

For an edge i, the linear interpolation function ji is given by:

jEi ¼

1

Ej j x zk � zj

� �þ z xj � xk

� ��þ 1

2zjxk þ zjxi � zkxi � zkxj � zixj þ zixk

� ��ð7Þ

where |E| is the area of the element E, and (xi, zi) are coordinates of the vertex i ofthe element E. The interpolation function ji

E equals 1 on the mid-edge i and zero on themid-edges j and k of E (see Figure 1).

From (7), we obtain:

rjEi ¼

1

Ej jzk � zj

xj � xk

� �¼ 1

Ej jDzi

Dxi

� �ð8Þ

2.2 Discretization of the momentum transport equationThe variational formulation of the flow Equation (1) using the test function ji over thedomain O writes:Z

O

rquqtþ r u:rð Þu� mr: ruð Þ þ rP

� �ji ¼

ZO

�rgrzji ð9Þ

Discretization of the mass term. The mass term writes:

ZO

rquqt

� �ji ¼ r

qqt

Pl

ul

ROjljiP

l

vl

ROjlji

0@

1A ¼ r

Xl

qqt

ul

vl

� �ZO

jlji

0@

1A ð10Þ

ji

i

iE�

iE′�

Figure 1.The linear interpolation

function for velocity field

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Methods forbuoyancy-driven

flows

Using (7), we obtain: Xl

ZE

jlji ¼Ej j3

ð11Þ

For an interior edge i sharing two adjacent elements E and E 0, the mass term can thenbe written as follows:

ZO

rquqtji ¼

r3

Ej j þ E 0j jð Þqui

qtqvi

qt

0B@

1CA ð12Þ

Discretization of the convective term. This term is non-linear and can lead to an unstablescheme for problems with high Reynolds number, that is why we applyhere the upwind technique used in Schieweck and Tobiska (1989). The domain isdecomposed into lumping regions Ri (hatched area in Figure 2) assigned to the edge i.It is formed by the two simplex regions Si

E and SiE 0 for an inner edge i and by the sole

simplex region SiE for a boundary edge. The simplex region Si

E is defined by joiningthe barycentre of E with the nodes j and k forming the edge i.

We use the lumping operator which transforms a given function into a piecewiseconstant function. Denoting -¼u, the variational formulation leads to:Z

O

r u:rð Þuji ¼ZO

r $ :rð Þuji � rZRi

$ :rð Þu ð13Þ

Green’s formula leads to:

rZRi

$ :rð Þu ¼ rZRi

$ :ru$ :rv

� �¼ r

RRi

r: $uð Þ �RRi

ur:$RRi

r: $vð Þ �RRi

vr:$

0B@

1CA ð14Þ

Using the following upwind scheme:ZRi

r: $uð Þ ¼X

l

uil

ZGil

$ :Zil ð15Þ

i

j

k

j

i

k

GE’

E

SiE

Figure 2.The lumped region Ri

(hatched area) formed bythe two simplex regionsSi

E and SiE 0

740

HFF24,3

where uil is the upwind value:

uil ¼ uupw ¼ lilui þ 1� lilð Þul ð16Þ

Then, the lumping operator, used for the second integral of (14) gives:ZRi

ur:$ � ui

ZRi

r:$ ¼ ui

Xl

ZGil

$ :Zil ð17Þ

Hence, denoting Qil ¼RGil$ :Zil , we obtain:Z

Ri

r: $uð Þ �ZRi

ur:$ ¼X

l

Qil 1� lilð Þ ul � uið Þ ð18Þ

Let’s note s ¼ Qil

n , two functions can be used for lil¼f(s) (see Schieweck and Tobiska,1989):

f1 sð Þ ¼ 1 if sX00 else

�or f2 sð Þ ¼

1=2þ s

1þ sif sX0

1

2� 2selse

8<: ð19Þ

Therefore, denoting Dzi¼ zk�z j and Dxi¼ x j�xk, Qij (see Figure 2) is given by:

Qij ¼ZGij

$ :Zij ¼1

92ui þ 2uj � uk

� �Dzj � Dzi� ��

þ 2vi þ 2vj � vk

� �Dxj � Dxi� � ð20Þ

Finally, the convective term writes:

rZRi

$ :rð Þu ¼r

Pl

Qil 1� lilð Þ ul � uið ÞPl

Qil 1� lilð Þ vl � við Þ

0B@

1CA

8><>:

9>=>;

E

þ

Pl

Qil 1� lilð Þ ul � uið ÞPl

Qil 1� lilð Þ vl � við Þ

0B@

1CA

8><>:

9>=>;

E 0

ð21Þ

Discretization of the viscous and pressure terms. These terms are written in thefollowing form:Z

O

�mr: ruð Þ þ rPð Þji ¼ZO

r: �mruþ PIdð Þji ¼ZO

�r:rji ð22Þ

with r¼ mru�PId and Id the 2 � 2 identity matrixThe variational formulation leads to:

ZO

�r:rji ¼ �ZqO

ji r:g þZO

rrji ð23Þ

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Methods forbuoyancy-driven

flows

For an interior edge i sharing two elements E and E 0, the previous integral reduces to:ZO

�r:rji ¼ZE

rrji þZE 0

rrji ð24Þ

using Equation (8), we obtain:

ZE

rrji ¼mEj j

P3l¼1

DxiDxl þ DziDzl� �

ul

P3l¼1

DxiDxl þ DziDzl� �

vl

0BBB@

1CCCA� Dzi

Dxi

� �PE ð25Þ

Therefore, Equation (22), writes:

ZO

�mr: ruð Þ þ rPð Þji ¼mEj j

P3j¼1

DxiDxj þ DziDzj� �

uj

P3j¼1

DxiDxj þ DziDzj� �

vj

0BBBB@

1CCCCA�

Dzi

Dxi

!PE

8>>>><>>>>:

9>>>>=>>>>;

E

þ mE 0j j

P3j¼1

DxiDxj þ DziDzj� �

uj

P3j¼1

DxiDxj þ DziDzj� �

vj

0BBBB@

1CCCCA�

Dzi

Dxi

!PE 0

8>>>><>>>>:

9>>>>=>>>>;

E 0

ð26Þ

Discretization of the buoyancy force. The variational formulation for the right hand sideterm of Equation (9) writes:

�ZO

rgrzji ¼� rEg

ZE

rzji � rE 0g

ZE 0

rzji

¼ �rEg

ZE

r zjið Þ þ rEg

ZE

zrji � rE 0g

ZE 0

r zjið Þ þ rE 0g

ZE 0

zrji

ð27Þ

Green’s formula gives:

�rEg

ZE

r zjið Þ ¼ �rEgX

j

ZEj

zji:ZEj ð28Þ

Using a numerical integration based on the mid-edge value leads to:

6X

j

ZEj

zji:ZEj � � Eij jzEiZEi ¼ �zEiDzi

Dxi

� �ð29Þ

742

HFF24,3

The integral rEgR

E zrji writes:

rEg

ZE

zrji ¼ rEgzEDzi

Dxi

� �ð30Þ

where zEi (respectively zE ) is the altitude of the centre of the edge (respectively theelement). Therefore combining Equations (29) and (30) leads to:

�rEg

ZE

rzji ¼ rEg zE � zEið Þ Dzi

Dxi

� �ð31Þ

and the right hand side term writes:

�ZO

rgrzji ¼ rEg zE � zEið Þ Dzi

Dxi

!( )E

þ rEg zE 0 � zE 0 ið Þ Dzi

Dxi

!( )E 0

ð32Þ

Boundary conditions. Three kinds of boundary conditions are used in the followingmanner:

(1) A boundary edge with a prescribed velocity u. In this case, the Equation (9) isnot written since the components of the velocity ui and vi are prescribed.

(2) A free boundary edge. In this case we set r¼ 0 in the boundary integral inEquation (23).

(3) A boundary edge with a prescribed pressure P. In this case we set also thevelocity component in the tangential direction to zero. This condition is namedNormal flow/Pressure or straight-out boundary condition (Conca et al., 1994;Gresho and Sani, 1987; Jager and Mikelic, 2001).

2.3 Discretization of the mass conservation equationThe mass conservation equation is discretized for each element E as following:Z

E

qu

qxþ qv

qz¼0 ð33Þ

using (5), we obtain: ZE

Xl

ul

qjl

qxþ vl

qjl

qz¼0 ð34Þ

which writes: Xl

Dzlul þ Dxlvl ¼ 0 ð35Þ

The final flow system is obtained by substituting Equations (12), (21), (26), (32) and (35)into Equation (9).

3. Heat transport discretizationFor the heat transport Equation (3), the explicit DG method, where fluxes are upwindedusing a Riemann solver is used for the advection equation and combined with thesymmetric MPFA method for the diffusion equation.

743

Methods forbuoyancy-driven

flows

In the following, we summarize the main developments for the discretization ofEquation (3).

The equation is written in the following mixed form:

qTqtþ u:rT þ 1

rCpr: qTð Þ ¼ 0

qT ¼ �krT

�ð36Þ

The heat flux qT is assumed to vary linearly inside the element E, therefore:

r:qT ¼1

Ej jX

i

QET; qEi ð37Þ

where QET;qEi ¼

RqEi qT :gqEi is the heat flux across the edge qEi of E:

We use the P1 DG method where the approximate solution Th(x, t) is expressed withlinear basis functions fi

E on each element E:

Th x; tð ÞjE ¼X3

i¼1

TEi tð Þ fE

i xð Þ ð38Þ

where TiE(t) are the three unknown coefficients corresponding to the degrees of freedom

which are the average value of the temperature defined at the triangle centroid�xE ; �zEð Þand its deviations in each space direction (Cockburn et al., 1989) with the

corresponding interpolation functions:

fE1 x; zð Þ ¼ 1; fE

2 x; zð Þ ¼ x� �xE ; fE3 x; zð Þ ¼ z� �zE ð39Þ

The variational formulation of Equation (36) over the element E using fiE as test

functions leads to (see Younes et al. (2010) for details):

A½ �

dTE1

dtdTE

2

dtdTE

3

dt

0BBBBB@

1CCCCCA ¼ B�M 0

� TE1

TE2

TE3

24

35�X3

‘¼1

M ‘� TE‘

1

TE‘2

TE‘3

24

35þ

PqEj

QET;j

00

264

375 ð40Þ

with,

Aij ¼ZE

fEj f

Ei Bij ¼

ZE

fEj u:rfE

i

M 0ij ¼

XNE

‘¼1

lEqE‘

QEqE‘

qE‘j j

ZqE‘

fEi fE

j ; M ‘ij ¼ 1� lE

qE‘

� QEqE‘

qE‘j j

ZqE‘

fEi fE‘

j ‘ ¼ 1; ::; 3ð Þ

where Ej is the adjacent element to E such that qEj is the common edge of E and Ej andQEqE‘ ¼

RqE‘ u:gqEj the fluid flux across qEj. The upwind parameter lqEj

E is defined by:

lEqEj ¼

1 if u:gqEjX00 if u:gqEjo0

�ð41Þ

744

HFF24,3

To reduce numerical diffusion, an explicit scheme is used for the temporaldiscretization of advection in Equation (40). Unphysical oscillations appearing nearsharp fronts are avoided using an efficient geometric slope limiter (Younes et al., 2010).

The heat fluxes QT, jE across edges are approximated at the new time level using

the MPFA method. The basic idea of this method is to divide each triangle into threesub-cells as in Figure 3.

Inside the sub-cell (O, F1, G, F2) formed by the corner O, the centre G and themidpoint edges F1 and F2, we assume linear variation of the temperature between T1

E,Tf1 and Tf2, which are, respectively, the temperature at G and at the two continuitypoints f1 and f2. The symmetry of the MPFA is achieved when the continuity points arelocalized at Of1=OF1 ¼ Of2=OF2 ¼ 2=3. In this case (O, f1, G, f2) is a parallelogram.

Therefore, half-edge heat fluxes Q1O ¼

R F1

O �krT and Q2O ¼

R F2

O �krT �

, taken

positive for outflow simplifies to (Younes and Ackerer, 2008):

Q1O

Q2O

� �¼ bE

�OF1��!

:OF1��!

OF1��!

:OF2��!

OF1��!

:OF2��! �OF2

��!:OF2��!

!Tf1 � TE

1

Tf2 � TE1

� �ð42Þ

with bE¼ 3k/|E|.This system is written for all sub-cells sharing the vertex O which create an interaction

region (Figure 4).Then by writing continuity of heat fluxes across half-edges and continuity of

temperature at continuity points, we obtain a local system [A](Tf)¼ [B](T). This localsystem is solved to obtain the temperature at the continuity points (Tfi) as function ofthe temperature at all elements sharing the vertex O. The obtained relation is thensubstituted into Equation (42) to obtain half-edge heat fluxes explicitly as a weightedsum of the temperature at cells forming the interaction volume. Finally, the summationof these heat fluxes is written using an implicit time discretization and substituted intothe Equation (40).

O

G f2

f1

QO1

Q 2O

F2

F1

R

P

Figure 3.Triangle splitting

into three sub-cells andlinear temperatureapproximation on

the sub-cell

745

Methods forbuoyancy-driven

flows

4. Coupling Navier-Stokes flow and heat transportWith the Boussinesq approximation, the density differences are confined to the buoyancyterm. Due to non-linearities, the simulations can require an excessive computational timeand/or heavy equipments. To reduce the computational cost while maintaining accuracyof the model, a non-iterative time stepping scheme based on local truncation errorcontrol (Younes and Ackerer, 2010) is used. The main stages of the scheme are recalled inthe following.

The local time truncation error of the temperature is evaluated using two approximationsof adjacent order of accuracy:

enþ1 � 1

2Tnþ1 � Tn þ Dtnþ1

DtnTn � Tn�1� �� � �

ð43Þ

The time step is accepted if the absolute error criterion is verified:

enþ1�� ��og ð44Þ

If this criterion is met, the following time step is controlled by the temporal truncationerror tolerance g using:

Dtnþ1 ¼ Dtn�min s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig

max enþ1j j

r; rmax

� �ð45Þ

If the error criterion is not satisfied, the current time step is repeated using the latesterror estimate:

Dtnþ1jþ1 ¼ Dtnþ1

j �max s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig

max enþ1j j

r; rmin

� �ð46Þ

where j indexes the recursive step size reduction, rmax and rmin are used to limitmultiplication and reduction factors and often set equal to 2.0 and 0.1, respectively, ands¼ 0.9 a safety factor (Sloan and Abbo, 1999). All the simulations in this work areperformed with a temporal truncation error tolerance g¼ 0.01.

OQ1

O

E1

E5

E4

E3

E2

F1

F5

F4

F3

F2

Tf1

Tf5

Tf4

Tf3Tf2

Figure 4.The interaction regionsharing the vertex O

746

HFF24,3

5. Validation of the coupled flow-heat transport problem5.1 The synthetic problem with semi-analytical solutionTo validate the numerical code, a synthetic problem is studied for coupled flow-heattransport. This problem is inspired from the salt water intrusion problem describedin (Henry, 1964; Segol, 1994). The test problem considers an idealized rectangulardomain in which a fluid with a temperature T1 enters from the left boundary of thedomain (Figure 5). A hydrostatic pressure is prescribed along the right boundarywhere the temperature is fixed to T0oT1. The top and the bottom of the domain areadiabatic no slip boundaries. Due to density effects, the coldest fluid intrudes fromthe right until equilibrium with the hot injected fluid is reached.

For this test case, we consider the inertial forces in the flow field (the first non-linearterm in Equation (1)) negligibly small compared with the viscous and pressure forces.Therefore, the steady state flow equations reduce to:

� qP

qxþ m

q2u

qx2þ q2u

qz2

!¼ 0

� qP

qzþ m

q2v

qx2þ q2v

qz2

!¼ f

qu

qxþ qv

qz¼ 0

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

ð47Þ

Using the change of variables (T¼ 0 corresponds to r0 and T¼ 1 corresponds to r1),the body forces f can be written in the following form:

f ¼ r0 þ ðr1 � r0ÞT½ �g ð48ÞThe semi-analytical solution. Using the change of variables defined in (Henry, 1964;Segol, 1994), the previous system can be written in terms of non-dimensionaltemperature T and stream function c as (see Segol, 1994):

aq4cqx4þ 2

q4cqx2qz2

þ q4cqz4

!¼ � qT

qxþ 1

x

� �ð49Þ

bq2T

qx2þ q2T

qz2

!¼ qc

qz

qT

qx� qc

qx

qT

qzþ 1

xqcqzþ qT

qxþ 1

xð50Þ

0

1

20

T0

Hydrostatic pressure

u = 0, v = 0, adiabatic

u = 0, v = 0, adiabatic

u = u0,v = 0T1

Figure 5.Domain and boundaryconditions for the semi

analytical problem

747

Methods forbuoyancy-driven

flows

with a ¼ ðmQ0Þ=ðr1 � r0Þgd3, b ¼ ðkÞ=ðrCpQ0Þ and the aspect ratio of the domainx ¼ L=d, where L is the length, d is the depth of the domain and Q0 is the injection fluximposed at the left boundary.

The stream function and the temperature are represented by double Fourier seriesof the form:

c ¼X1m¼1

X1n¼0

Am; n sinðmpzÞ cos npx

x

� �ð51Þ

T ¼X1r¼0

X1s¼1

Br; s cosðrpzÞ sin spx

x

� �ð52Þ

Substituting these relations into Equations (49) and (50), then by multiplying Equation (49)with 4 sinðgpzÞ cosðhpx

xÞ and Equation (50) with 4 cosðgpzÞ sinðhpxxÞ, then integrating

over the rectangular domain gives an infinite set of algebraic equations for Ag, h andBg, h namely:

e2ap4Ag; h g2 þ h2

x2

� �2

x ¼X1r¼0

Br; hhNðg; rÞ þ 4

pWðg; hÞ ð53Þ

e1bp2Bg; h g2 þ h2

x2

� �x ¼

X1n¼0

Ag; n gNðh; nÞ

þ e1

X1s¼1

Bg; sSNðh; sÞ þ Quad þ 4

pWðh; gÞ

ð54Þ

Details about the parameters e1, e2, N, W, Quad, can be found in Appendix.The algebraic system of Equations (53)-(54) has the same form as the one

obtained by Henry (1964) and Segol (1994). To avoid convergence difficulties, we usethe Levenberg-Marquardt algorithm (Levenberg, 1944; Marquardt, 1963), which hasa quadratic rate of convergence (Yamashita and Fukushima, 2001). This method isconsidered as one of the most efficient algorithms for solving systems of non-linearequations. The non-linear algebraic system of Equations (53)-(54) is written in the formF(X)¼ 0 where X is a vector formed by the coefficients Ag, h and Bg, h. The Levenberg-Marquardt algorithm includes two minimization methods: the gradient descent methodand the Gauss-Newton method. Far from the optimum, the Levenberg-Marquardtmethod behaves like a gradient descent method, whereas, it acts like the Gauss-Newtonmethod nearby the optimum. The Jacobian J is approximated numerically using finitedifferences.

Validation of the numerical model. The domain of rectangular shape is discretizedwith a regular triangular mesh of 3,200 elements. The fluid enters from the left(inland) boundary with u0¼ 0.92 ms�1, v0¼ 0 and the thermal conductivity is setto k¼ 0.046 W m�1 K�1, the densities are, respectively, r0¼ 1,000 kg m�3 andr1¼ 985 kg m�3. To avoid very small values of the parameter a for which, wecannot obtain a converged semi-analytical solution, the viscosity is set to m¼ 1 Pasin this synthetic problem. The corresponding parameters for the analytical solution area ¼ mQ0=ðr0 � r1Þgd3 ¼ 0:006 and b ¼ k=rCpQ0 ¼ 0:05.

748

HFF24,3

Figure 6 shows the temperature distribution and the velocity field at t¼ 1 s obtainedwith the numerical code. The steady state solution is obtained after t¼ 10 s and iscompared to the semi analytical solution in Figure 7. We use a new truncation basedon 424 coefficients of the Fourier series: 214 terms (A1..7, 0.30) for the expansion of thestream function and 210 terms (B0..6, 1..30) for the expansion of the temperature.Three temperature isolines (0.25, 0.5 and 0.75) are used for the comparison in Figure 7.This Figure shows a very good agreement between the semi-analytical and thenumerical results which demonstrate the validity of the developed numerical model.

To study the efficiency of non-iterative time stepping scheme, the solution isrecalculated using an empirical time stepping where the time step size is adjustedaccording to the convergence behavior as following: the size of the new time step isincreased by 10 per cent if the convergence of the previous time step has been achievedin fewer than four iterations and decreased by 20 per cent if the convergence has beenreached in more than ten iterations. If the convergence is not reached within 20

1

0.8

0.6

0.4

Z(m

)

0.2

00 0.5 1

X (m)1.5 2

0.10.20.30.40.50.60.70.80.9

T

Figure 6.Temperature, velocity and

contours of streamfunction (Dc¼ 0.1) at

t¼ 1 s for the semianalytical problem

1

0.8

0.6

0.4

Z(m

)

0.2

00

Note: The semi-analytical solution is calculated using a truncation based on 424 coefficientsfor the double Fourier series

0.5 1X (m)

1.5

0.25

0.5

0.75T

2

0.250.50.75

Figure 7.Simulated temperature

and velocity distributions,contours of stream

function (Dc¼ 0.1) andsemi-analytical

temperature isolines(dashed lines)

749

Methods forbuoyancy-driven

flows

iterations, the computation is stopped for the corresponding time step and is repeatedusing a halved time step size.

The simulation of the test problem with this empirical scheme needed 394 modelevaluations which required 66 s of CPU time. Using this scheme, the time step sizeincreases linearly at the beginning of the simulation and the truncation error reaches amaximum value of 0.018 (Figure 8). On the other hand, with the non-iterative scheme, thetime step size keeps a small value at the beginning of the simulation. This scheme is,however, more efficient since only 197 model evaluations are needed which required 32 s ofCPU time. The truncation error remains below 0.005 (Figure 8) leading to higher accuracy.

Note that the results of Figure 8 show also that the use of a fixed time stepfor buoyancy flow simulations, as done in some numerical models, is clearly notan efficient strategy since the truncation error has strong variations which suggestthe adaptation of the time step size accordingly.

5.2 The De Vahl Davis benchmarkThe classical De Vahl (1983) benchmark deals with natural convection of air (Pr¼ 0.71)in a differentially heated closed square cavity, depicted in Figure 9. The left and rightwalls are maintained at fixed temperatures, while the horizontal walls are adiabatic.The only free parameter of the test is the thermal Rayleigh number Ra. The study isperformed for Ra up to 108. Indeed, it was reported ( Janssen and Henkes, 1993) thatthe problem becomes unsteady for Ra¼ 2 � 108. The governing equations, written innon-dimensional form, are:

quqtþ u:ru�r: ruð Þ þ rP ¼ 0

Ra y

� �ð55Þ

Empirical iterative scheme Non iterative scheme

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

Tim

e st

ep s

ize

Time

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

Tim

e st

ep s

ize

Time

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

0.020

Tru

ncat

ion

erro

r

Time

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.000

0.001

0.002

0.003

0.004

0.005

0.006

Tru

ncat

ion

erro

r

Time

Figure 8.Time step and truncationerror evolutions with theempirical iterative schemeand the non-iterative timestepping scheme

750

HFF24,3

r:u ¼ 0 ð56Þ

qyqtþ ury�r: ryð Þ ¼ 0 ð57Þ

where y is the dimensionless temperature varying between y¼ 1 at the hot left walland y¼ 0 at the right cold wall.The problem is simulated for different values of Ra with a triangular discretizationbased on 20,164 elements. The results in the literature show many flow featuresdepending on the value of Ra (Mayne et al., 2000). The obtained steady state flowpatterns and temperature distribution are depicted in Figure 10 for the followingRayleigh numbers Ra¼ 103, 104, 105, 106, 107 and 108. These results, obtained with amesh of 142 � 142, are in good agreements with the results of Kosec and Sarler (2013)using a mesh of (101 � 101), those of Mayne et al. (2000) using the h-adaptive finiteelement solution and those of Wan et al. (2001) using a mesh of 301 � 301.

For this test problem, the Nusselt number is of particular interest since it measuresthe amount of heat transferred between the plates. For the studied problem, the non-dimensional Nusselt number is given by (Wan et al., 2001):

Nu ¼ �qyqx

����wall

ð58Þ

The distribution of the local Nusselt number for different Rayleigh numbers is plottedin Figure 11 for the hot and cold walls. The obtained distributions are similar to thoseof Wan et al. (2001).

An advanced comparison with published results is performed in Table I byanalyzing the obtained maximum mid-plane velocities umaxjx¼0:5 and vmaxjy¼0:5 and

average Nusselt number with the results of (Mayne et al., 2000; Wan et al., 2001; Kosecand Sarler, 2013). Table I shows a very good agreement between the present and thealready published results for a wide range of studied Rayleigh numbers (between 103

and 108).

g

��/�n = 0

��/�n = 0

no-slip boundary

no-slip boundary

no-s

lip b

ound

ary

no-s

lip b

ound

ary

� = 1� =

1

� =

0

Figure 9.Domain and boundary

conditions for thebuoyancy-driven cavity

751

Methods forbuoyancy-driven

flows

5.3 The periodic oscillatory flow for low Prandtl numberThis test problem was studied by Katsuyoshi and Hiroyuki (1996) and Zaloznik et al.(2008) and is similar to the De Vahl Davis benchmark. The problem considers transientnatural convection with horizontal temperature gradient of low Prandtl fluid in a

Ra = 103

0

0.2

0.4

0.6

0.8

1

X (m)

Z(m

)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

X (m)

Z(m

)

0 0.2 0.4 0.6 0.8 1

0.050.150.250.350.450.550.650.750.850.95

�

Ra = 104

Ra = 105 Ra = 106

Ra = 107 Ra = 108

0.050.150.250.350.450.550.650.750.850.95

�

0

0.2

0.4

0.6

0.8

1

X (m)

Z(m

)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

X (m)

Z(m

)

0 0.2 0.4 0.6 0.8 1

0.050.150.250.350.450.550.650.750.850.95

�

0.050.150.250.350.450.550.650.750.850.95

�

0

0.2

0.4

0.6

0.8

1

X (m)

Z(m

)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

X (m)

Z(m

)

0 0.2 0.4 0.6 0.8 1

0.050.150.250.350.450.550.650.750.850.95

�

0.050.150.250.350.450.550.650.750.850.95

�

Figure 10.Computed steady stateflow patterns andtemperature distributionfor different Rayleighnumbers

752

HFF24,3

rectangular cavity of height H and width L¼ 4H. We use a low Prandtl fluidPr¼ 0.0137 and a Rayleigh number Ra¼ 2.81 � 105 as in Zaloznik et al. (2008). Thistest case is interesting due to the oscillatory phenomena resulting from the balancebetween the buoyancy and the shear forces (Kosec and Sarler, 2013). The oscillationscan be shown from the time evolution of either the average Nusselt number or the flow

(a)

0 10 20 30 40 50 60 70 80 90 1000.0

0.2

0.4

0.6

0.8

1.0

Z(m

)

Nu

Ra =103

Ra =104

Ra =105

Ra =106

Ra =107

Ra =108

(b)

0 10 20 30 40 50 60 70 80 90 1000.0

0.2

0.4

0.6

0.8

1.0

Z(m

)

Nu

Ra =103

Ra =104

Ra =105

Ra =106

Ra =107

Ra =108

Figure 11.Distribution of Nusselt

number for the hot (a) andcold (b) walls for different

Rayleigh numbers

umax (x¼ 0.5) vmax (y¼ 0.5) Nuav

Ra¼ 103

Mayne et al. (2000) (h-adaptive) 3.649 3.696 1.115Wan et al. (2001) (301 � 301) 3.489 3.686 1.117Kosec and Sarler (2013) (101 � 101) 3.645 3.695 1.089Present study (142 � 142) 3.64 3.685 1.113Ra¼ 104

Mayne et al. (2000) (h-adaptive) 16.179 19.617 2.259Wan et al. (2001) (301 � 301) 16.122 19.79 2.254Kosec and Sarler (2013) (101 � 101) 16.45 20.03 2.258Present study (142 � 142) 16.0 19.66 2.201Ra¼ 105

Mayne et al. (2000) (h-adaptive) 34.774 68.692 4.483Wan et al. (2001) (301 � 301) 33.39 70.63 4.598Kosec and Sarler (2013) (101 � 101) 35.03 69.69 4.511Present study (142 � 142) 33.27 69.5 4.532Ra¼ 106

Mayne et al. (2000) (h-adaptive) 64.691 220.833 8.881Wan et al. (2001) (301 � 301) 65.4 227.11 8.976Kosec and Sarler (2013) (101 � 101) 65.91 221.37 8.97Present study (142 � 142) 64.5 220.1 8.90Ra¼ 107

Mayne et al. (2000) (h-adaptive) 145.266 703.253 16.39Wan et al. (2001) (301 � 301) 143.56 714.48 16.656Kosec and Sarler (2013) (101 � 101) 149.61 687.2 16.92Present study (142 � 142) 134.67 619.88 16.52Ra¼ 108

Mayne et al. (2000) (h-adaptive) 283.689 2,223.442 29.63Wan et al. (2001) (301 � 301) 296.71 2,259.08 31.486Kosec and Sarler (2013) (101 � 101) 278.49 2,095.23 32.12Present study (142 � 142) 291.47 1,753.73 29.497

Table I.Comparison of mid-plane

velocities umaxjx¼0:5 andvmaxjy¼0:5 and average

Nusselt number fordifferent Rayleigh

numbers

753

Methods forbuoyancy-driven

flows

velocity. This test case is highly sensitive (Kosec and Sarler, 2013) and the results canbe dependent on the spatial resolution. In our case, the problem is solved using veryfine meshes. Figure 12 shows the time evolution of the average Nusselt number at thehot wall using three meshes of, respectively, 250,000, 320,000 and 500,000 triangles.This figure shows complicated periodic oscillations of the Nusselt number as has beenobserved by Kosec and Sarler (2013). The three meshes lead to close final periods ofoscillations with small phase shifts. For to0.12, a very good matching is observedbetween the three mesh results. Note that the amplitudes and the periods of oscillationsare different from those obtained by Kosec and Sarler (2013).

Figure 13 depicts temperature and flow fields at different times. This figure shows alarge roll cell over the entire domain at the beginning of the simulation (t¼ 0.002).Then, at t¼ 0.06, two secondary roll cells are formed and lead to three developed rollcells later (at t¼ 0.09). Finally, two main stable cells are obtained at t¼ 1. These resultsare in very good agreement with the temperature and flow fields given in Zaloznik et al.(2008) and Kosec and Sarler (2013).

6. ConclusionAn efficient numerical model with rapid convergence was developed for buoyancy-driven flow problems. The model, which may seem complicated, is based on advancednumerical methods. On one hand, the Navier-Stokes equation was solved using thelocally mass conservative CR finite element method with an upstream approach forthe non-linear convective term. On the other hand, the advection-diffusion heatequation was solved using a combination of DG and MPFA methods. These methodsare well suited for transport equations since the DG method provides accurate resultsin the case of advection dominated problems and the MPFA method conserves masslocally and can handles general irregular grids. To reduce the computational time dueto the coupling caused by buoyancy forces, we use a non-iterative time steppingscheme where the time step length is controlled by the temporal truncation error.The developed model was validated against three different test problems. The first onewas a synthetic problem for which we developed a semi analytical solution using

0.0 0.1 0.2 0.3 0.4 0.5 0.64.0

4.5

5.0

5.5

6.0 mesh1 (250,000 triangles)

mesh2 (320,000 triangles)

mesh3 (500,000 triangles)

Nu

t

Figure 12.Results of three very finemeshes for the timeevolution of the Nusseltnumber on the hot wall

754

HFF24,3

Fourier series. The second was the de Vahl Davis benchmark (De Vahl, 1983) fornatural convection of air in a square cavity. The last test problem concerned transientnatural convection with horizontal temperature gradient of low Prandtl fluid in arectangular cavity.

Accurate results were obtained for the three test problems. A good agreement wasobserved between numerical and semi-analytical solutions for the first test problem

t = 0.02 t = 0.06 t = 0.09 t = 1.0

0.8

0.6

0.4

Z(m

)

0.2

00 0.1 0.2

X (m)

1

0.8

0.6

0.4

Z(m

)

0.2

00 0.1 0.2

X (m)

1

0.8

0.6

0.4

Z(m

)

0.2

00 0.1 0.2

X (m)

1

0.8

0.6

0.4

Z(m

)

0.2

00 0.1 0.2

X (m)

1

0.8

0.6

0.4

Z(m

)

0.2

00 0.1 0.2

X (m)

1

0.8

0.6

0.4

Z(m

)

0.2

00 0.1 0.2

X (m)

1

0.8

0.6

0.4

Z(m

)

0.2

00 0.1 0.2

X (m)

1

0.8

0.90.80.70.60.50.40.30.20.1

�

0.6

0.4

Z(m

)

0.2

00 0.1 0.2

X (m)

1

Figure 13.Flow patterns and

temperature distributionat different times

755

Methods forbuoyancy-driven

flows

and between the obtained and already published results for the second and thirdtest problems.

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AppendixThe non-linear algebraic equations using the Stokes equation for the flow are:

e2ap4Ag; h g2 þ h2

x2

� �2

x ¼X1r¼0

Br; hhNðg; rÞ þ 4

pWðg; hÞ

e1bp2Bg; h g2 þ h2

x2

� �x ¼

X1n¼0

Ag; ngNðh; nÞ

þ e1

X1s¼1

Bg; sSNðh; sÞ þ Quad þ 4

pW ðh; gÞ

where:

e1 ¼2 if g¼ 01 if g 6¼ 0

�

e2 ¼2 if h¼ 01 if h 6¼ 0

�

Nðh; nÞ ¼ ð�1Þhþn � 1

hþ nþ ð�1Þh�n � 1

h� n

758

HFF24,3

Wðh; gÞ ¼ð�1Þh�1

h; if g ¼ 0

0; if g 6¼ 0

(

Quad ¼ p4

X1m¼1

X1n¼0

X1r¼0

X1s¼1

Am; n Br; s ðmsLR � nrFGÞ

with:

F ¼ dðm�rÞ; g þ dðr�mÞ; g � dðmþrÞ; g

L ¼ dðm�rÞ; g þ dðr�mÞ; g þ dðmþrÞ; g

G ¼ ð�1Þhþn�s � 1

hþ n� sþ ð�1Þh�nþs � 1

h� nþ s� ð�1Þhþnþs � 1

hþ nþ s� ð�1Þh�n�s � 1

h� n� s

R ¼ ð�1Þhþn�s � 1

hþ n� sþ ð�1Þh�nþs � 1

h� nþ sþ ð�1Þhþnþs � 1

hþ nþ sþ ð�1Þh�n�s � 1

h� n� s

and di, j is the Kronecker delta such that

di; j ¼1 if i ¼ j0 if i 6¼ j

�

Corresponding authorProfessor Anis Younes can be contacted at: [email protected]

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759

Methods forbuoyancy-driven

flows