EISEYIER Comput. Methods Appl. Mech. Engrg. 151 (1998) 43-62

Computer methods in applied

mechanics and enginearlng

Development of BEM for thermoviscous flow l?K. Banerjee*, K.A. Honkala

Department of Civil Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA

Abstract

The boundary element method (BEM) is applied to natural convection problems involving steady, two-dimensional, buoyancy- driven flow of an incompressible fluid. The buoyancy effect is incorporated in two distinct BEM formulations, involving coupled and uncoupled kernel functions. The integral equation is derived, the appropriate fundamental solutions are explicitly presented, and the implementation details are discussed. Numerical examples are provided to demonstrate the validity of the formulations, and for comparison with the more established finite element method (PBM). Various computational aspects are discussed, such as the convergence, accuracy and efficiency of the various approaches.

1. Introduction

Problems involving natural convection phenomena have received considerable attention in recent years. Since few practical problems lend themselves to analytical solutions, much effort has been ex- pended in finding numerical solutions. Specifically, the finite difference method, finite element method and boundary element method have all been used to obtain numerical solutions to natural convection problems.

The finite difference method has been used extensively to solve a wide range of computational fluid dynamics (CFD) problems (e.g. [l]). In the specific area of natural convection problems, Wilkes and Churchill [2], Elder [3] and Gill [4] p rovided a solid foundation upon which subsequent finite difference formulations were based.

More recently, the finite element method (FEM) has been applied to natural convection problems with similar success, as illustrated by Gartling [5], Marshall et al. [6] and Heinrich and Strada [7].

The boundary element method has been applied to a wide range of engineering problems (e.g. [S]), but is still in its infancy with regard to CFD. Dargush and Banerjee [9,10] have presented promising directions in the application of the BEM to CFD problems. Onishi et al. 1111 presented a BEM formulation for buoyancy-driven flow whereby the buoyancy effect was considered as a pseudo-body force. More recently, Tosaka and Fukushima [12] incorporated the buoyancy effect in the fundamental solution. However, the kernel functions were not detailed and no quantitative assessment of the formulation was provided. Furthermore, the present authors have not found a thorough discussion and comparison between the two BEM formulations, nor between the BEM and more established methods, such as FEM.

In the present work, an integral representation is derived directly from the governing differential equa- tions of fluid mechanics. This new representation is written exclusively in terms of velocity, temperature, surface traction and surface heat flux. There is no need to calculate velocity and temperature gradients.

*Corresponding author.

00457825/97/$17.00 @ 1997 Published by Elsevier Science S.A. All rights reserved PII SOO45-7825(97)00112-6

44 P.K. Banerjee, K.A. Honkala / Comput. Methods Appl. Mech. Engrg. I51 (1998) 43-61

A systematic procedure for deriving the appropriate fundamental solution is then summarized, and the explicit form of the kernel functions for coupled buoyancy is presented for the first time. Both the cou- pled and uncoupled BEM formulations are presented simultaneously, allowing meaningful comparisons to be made between the two formulations. In both cases, since the kernel functions automatically en- force incompressibility, penalty methods are unnecessary. Implementation details are also discussed to document the procedure adopted by the authors. Both Direct iteration and Newton iteration are used to solve the nonlinear equations.

Numerical examples are presented which allow a critical comparison to be made between the two BEM formulations and the FEM. Unexpectedly, similarities between the BEM and FEM iteration histories are observed. However, the coupled BEM formulation along with Newton iteration provides superior convergence characteristics in solving buoyancy problems, and permits a very accurate computation of the convective heat transfer coefficient.

2. Governing equations

In order to analyze fluid flow and heat transfer in natural convection problems, one must consider the conservation laws of mass, momentum and energy. In addition, a constitutive law and appropriate boundary conditions must be specified. The governing Navier-Stokes equations for steady, incompressible flow of a Newtonian fluid are

Vj,j = 0 (1)

/_b(Vj,ij + Vi,jj) - Pr,i - PVjvij + Pgi = O (2)

kT~,j-pcVjT,j+~~+PqT=O. (3)

In the above equations, Vj is the velocity vector, pr is the absolute pressure, T is the temperature, Al. is the absolute viscosity, p is the mass density, gi is the gravity vector, k is the thermal conductivity, c is the specific heat, @ is the viscous dissipation and qT is the volumetric heat generation. In the following analysis, 6, and qT will be neglected. Note that the gravity vector gi acts in the positive coordinate directions in (2).

Since it is the density variation that gives rise to the buoyancy phenomenon, it is useful to express the momentum equation (2) in terms of density difference, p - po, where po is the mass density at some predefined reference state. To be specific, if the fluid is brought to rest, and the temperature of the fluid is uniform, To, then the momentum equation (2) reduces to

- P0.i + POgi = O (4)

where p. is the pressure distribution at hydrostatic conditions. Similarly, it is convenient in BEM CFD to refer each of the primary variables to reference conditions.

For instance, the quantity (T - To) represents the temperature above the reference temperature, or the ‘perturbed temperature’. Thus, let

Vj = VfJj + Z4j (5)

PT =PO+P (6)

T=To+O (7)

where Voj is called the ‘free stream’ reference velocity vector, while Uj is the perturbed velocity, p is the perturbed pressure and 0 is the perturbed temperature.

The Boussinesq approximation provides a convenient and useful means of incorporating the buoyancy effect into the equations of motion. First of all, it neglects all property variation except for the density in the gravity term of the momentum equation (2). Secondly, the Boussinesq approximation expresses the density variation in this term by a first order approximation:

P-P~=-P~P(T-To) (8)

P.K. Banerjee, K.A. Honkala I Compui. Methods Appl. Mech. Engrg. 151 (1998) 43-62 45

B=-LL

P

where /3 is the coefficient of thermal expansion. Eqs. (2) through (8) can be combined to yield the governing equations for buoyancy-driven flow:

P(uj,ij + W,jj) -P,i - PO(vOj + uj)ui,j + POP@gi = O (9)

k Cl.jj - nlc( Voj + uj) 0,j = 0 (10)

Ujd = 0 (11)

In order to complete the specification of the boundary value problem, one must specify the appropriate boundary conditions. Frequently, certain conditions must be prescribed on the derivatives of the primary derivatives. In this case, the fluid tractions and thermal heat flux provide convenient quantities

t: = (-GijPr + /lz(Vi,j + Vj,i))nj

q’ = -kT?jnj

where t: is the fluid traction vector, 4’ is the heat flux, &j is Kronecker’s delta function and nj is the outward normal vector. In terms of perturbed quantities, these are

rl = (-&jP + /J(Ui,j + uj,i))nj

While boundary conditions can be specified on either the primary or perturbed quantities, the latter will be considered here with no loss in generality. Fluid flow boundary conditions involve the specification of velocities Ui = iii on SUi, and/or tractions ti = & on S,i. Thermal boundary conditions can take several forms: 0 = @ on ST; q = 4 on S,; and/or q = h(O + 7’0 - Ta,,,b) on Sh, where h is the convective heat transfer coefficient and Tamb is the ambient temperature of the fluid outside the region. Furthermore, S,i U S,, = Su2 U St2 = ST U S, U Sh = S while the corresponding intersections are null.

3. Integral representation

The differential equations (9), (10) and (11) can be expressed using linear differential operator notation as LIJCJJ - F, = 0, where UJ is the generalized velocity vector and FI is the generalized force vector. Since this paper concentrates on two-dimensional flow, the capitalized indices I, J, K, etc. span the 1,2,3 and 4 components, while the lower case indices i, i, k, etc. span the 1 and 2 components.

The terms in the operator equation are

LIJ

r/,

I 0 0 kV2 0

L a 8x1

Ul

u2 0 0 .

P

a 8x2

0 0

(12)

(13)

46 P.K. Banerjee, K.A. Honk& 1 Comput. Methods Appl. Mech. Engrg. 151 (1998) 43-61

(14)

In the above, V2 = ( ),ij is the Laplacian operator, /?c is an index for the coupled formulation and pU is an indicator for the uncoupled formulation. Thus, PC = 0 and pU = 1 for the uncoupled formulation, while PC = 1 and flu = 0 for a coupled approach. It is evident that the buoyancy effect is incorporated in the linear operator LIJ in the coupled formulation, whereas it is left as a pseudo body force in the uncoupled formulation. In addition, the convected velocity and temperature terms are also included as pseudo body forces. It is clear that velocity and temperature are coupled through the convective terms in both formulations.

The corresponding integral representation can be derived by multiplying the above operator equation by the matrix ziIK and integrating over the region of interest. That is,

J (LIJUJ - FI)UIK dV = 0. V

The explicit form of G IK is not known a priori, but will be determined in the process of deriving the integral equation.

Following the procedure summarized by Dargush and Banerjee [9,10], repeated application of the divergence theorem and integration by parts produces the appropriate integral equation:

c1,(47~1(~) = J

s(x) {&K(X, @iCx) - t;:KCx, 5>“i(x)} dS(X)

-t J {&KW, O@(x) - C~KW, G%(X)) dS(W SW)

- J &K(X, S)hui(X)(V~j + Uj(X))nj(X) dS(X) SW)

- J 123~ (X, S)~c@(X> (Voj + Uj(X)bj(X) dS(X) S(X)

+ J fiiK,j(Xy S)~ui(X)(l/oi + uj(X>> dV(X) V(X)

-I- S

fi3K,j(X, S)~c@(X)(Voj + uj(X)) dV(X) V(X)

- p”Jv(x) &K (x, &oPgi @@‘) dV(X) (15)

In (H), X and 6 represent the coordinates of two points in space. The nomenclature X, S(X), V(X), etc. emphasizes the fact that integration is carried out with respect to the X coordinates. The notation (X, 6) indicates that certain terms are functions of both points X and 5. CI~(~) are constants such that when 6 is inside S(X), C,, = S IK. When 5 is outside S(X), C,,(t) = 0. When 5 is on the boundary, the values of C,&) are determined by the relative smoothness of S(X) at 5. For a smooth boundary at 5, CIK(c) = l/2&~. The term tiIK(X, e) is the fundamental solution matrix, and &K(X, 5) is its associated traction matrix. Before considering the development of both EIK(X, 5) and &K(X, 5) in the next section, it should be noted that (15) involves velocities, tractions, temperatures and heat flux. All gradients have been eliminated from the formulation. and need not be calculated.

P.K. Banerjee, K.A. Honkala I Compu!. Methods Appl. Mech. Engrg. 151 (1998) 43-6-7 41

4. Fundamental solution

En route to the derivation of the integral equation, the explicit form of z’i~~(x, 5) and ~;K(X, 5) was not stated. However, these terms must satisfy very specific conditions, and this is the subject of the present section.

In order for z&~(X, 5) to satisfy (15), the fundamental solution tensor must satisfy a related differential operator equation

M/K(X, 5) + 6&(X - 5) = 0 (16)

where L,, is the adjoint differential operator, and S(X - 5) is the Dirac delta function. The tensor Iz,,(X, 0 is commonly referred to as the ‘ad-joint Green’s tensor’, the ‘adjoint fundamental solution’, or merely the ‘fundamental solution’.

For the problem under consideration,

0 d --

ax1

a2 LlJ = pa~l ax2

-PcPoPs1 -Pcpo&2 kV* 0

a a ax, ax2

0 0

(17)

Comparing (12) and (17), it is evident that L IJ and LIJ differ only in the coupled formulation, when - PC = 1. In the uncoupled formulation, PC = 0 and L IJ = LIJ. In this particular case the two operators

are said to be ‘self-adjoint’. In general, finding the solution of (16) presents a formidable task. However, a systematic approach

has been suggested by Hormander [13] to condense the system of equations down to a single, scalar differential equation. Tosaka [14] has illustrated this method by determining the fundamental solution to viscous flow problems without heat transfer. The entire procedure is presented in detail in Honkala

P51. If Iz,,(X, 5) is sought in the form

iJK(X, 6) = LiJ$f’*(x, 6) (18)

where 4*(X, 6) is a scalar function and LkJ is the matrix of cofactors of LIJ, then (16) becomes

LIJL~J~*(X, 8 = -%KS(X - 6)

However, since from linear algebra &Lt = det(LIK) ([16]), one obtains the scalar differential equa-

tion det(LIK)$*(X, 6) = -6(X - 5) to be solved for 4*(X, 6). For the present operator defined in (17), det(LIK) = pkV6. The scalar function 4*(X, t) that satisfies pkV64*(X, 5) = -6(X - 5) is known to be

4*(X, 0 = &r’lnr (19)

where r is the euclidean distance between X and 5. The scalar 4*(X, 6) has the property that

s v(x) pkV64*(X, 5) dV(X) = -1 when ,$’ is inside V(X). Once 4*(X, 5) is known, the terms of the

fundamental solution cJJK(X, 5) may be determined from (18). The generalized traction matrix ~JK(X, 5)

can also be derived to satisfy (15). Interestingly, $* is represented by (19) for both coupled and uncoupled buoyancy. However, the form

of Lt in (18) is different for the two cases. The kernel functions ii IK (X, 5) and hK (X, 6) are presented in explicit form for the first time in Appendix A. It should be noted that previously, Tosaka and Fukushima

48 F!K. Banerjee, K.A. Ho&da I Comput. Methods Appl. Mech. Engrg. 151 (1998) 43-61

Table 1 Generalized comoonents

Component Generalized velocity Generalized force

1 Velocity xl Force x1 2 Velocity x2 Force x2 3 Temperature Heat source 4 Pressure Mass sink

[12] did present 4*(X, 5) for coupled buoyancy, but did not derive the actual terms in z&&X, 5) and &K(X, S), which are required for a numerical implementation.

Although it is not necessary to assign a physical meaning to the terms in ~~JK(X, t), it is sometimes instructive to do so. Simply stated, UJK(X, 6) is the Jth component of the adjoint generalized velocity at point X due to the Kth component of the generalized force at point 5. Table 1 summarizes the generalized terms. For example, z&r (X, 6) is the adjoint temperature at X due to a unit force applied in the x1 direction at point 5 (which happens to be zero for uncoupled buoyancy).

In recent years, much effort has been expended in adequately incorporating the incompressibility (continuity) requirement. A popular approach has been to approximately satisfy continuity through a penalty function. This has been utilized extensively in finite element (e.g. [17]) and boundary element (e.g. 1181) formulations. However, these approximate penalty methods are completely unnecessary in the present formulation since the fundamental solution incorporates incompressibility exactly.

5. Nmnerical implementation

The integral equation (15) is an exact statement; no approximations have been made, beyond those involved in formulating the governing differential equations. However, in order to make realistic use of this equation in solving practical problems, engineering approximations are needed. The details of these approximations for the~ovis~ous flow have been extensively documented in 19,101. For the sake of completeness a brief summary of the issues involved in the numerical implementation is covered here.

5.2. Spatial discretization

To provide a computationally efficient method for evaluating the integrals in (15), the boundary and volume are divided into elements and cells, respectively; each of relatively simple shape. This procedure is similar to that employed in the finite element method, but was first applied to BEM by Lachat and Watson [19]. On each element or cell, the geometry and primary variables are represented by

p=l

P.K. Banerjee, K.A. Honkala / Comput. Methods Appl. Mech. Engrg. 151 (1998) 43-62 49

where U; is the Zth component of generalized velocity on boundary element e, Tf is the Zth component of generalized traction on boundary element e, Xf is the ith component of the position vector on volume cell c, and so forth. The quantities & and r,!~~ are shape functions, while X6, UfP, Tfp,XFq and UFq signify nodal values.

The discretization of the convective volume integrals in (15) requires some explanation. These terms can be represented by a momentum flux tensor,

fo~~l(VOl + Ul), POUl wo2 + u2)

a/j = PO”2(vOl + ul)~ Po~2(Vo2 + u2)

fOC@(VOl + Ul)r POcwo2 + u2) I

where now I = 1,2,3. However, the functional variation of these terms is given by UIj = C:;, +qcljq. The primary distinction

here is that the variation of Olj can be quadratic for an eight noded cell, however the variation of the individual components ul, u2, 0 may be less than quadratic. It should be recalled that these terms appeared in the derivation of (15) when the divergence theorem was applied.

5.3. Integration

The integrals involved in the BEM are efficiently evaluated on the computer using numerical quadra- ture. The details of the integration used here are covered in [9,10] and [15]. However, several comments can be made to summarize the techniques:

(i) Boundary integrals are evaluated using logarithmic Gaussian integration and/or coordinate trans- formations along with standard Gaussian integration, depending on the particular terms in the integrand.

(ii) The ‘rigid b o d y’ method of Cruse (see ref. [20]) . integrals.

IS used to evaluate the strongly singular boundary

(iii) Volume integration is much more costly than boundary integration. Consequently, efficient quadra- ture schemes must be utilized to evaluate the domain integrals.

5.4. Assembly

While the indices, Z and K, in (15) indicates that four equations could be written for each node, only the first three equations are necessary. Although the fourth equation could be used to determine pressure, it is not needed to solve for velocities and temperature. Therefore, to minimize the number of primary variables and integration effort, only the first three equations of (15) are utilized. This results in the rather pleasing set of primary variables: ul, u2 and 0.

The three equations can be written for each functional node, and the integrals subsequently evaluated. The resulting algebraic equations can then be rearranged in the following form:

(20)

where {x} is a vector of unknowns, {y} is a vector of known boundary quantities and {f} is the vector of nonlinear pseudo-body forces, including the convective terms. The {x} and {y} vectors include velocities, temperatures, fluid tractions and heat fluxes. For single region models, [A] is a fully populated, square matrix. However, computational advantages can often be gained by dividing the problem domain into subregions. Integration is then performed independently over each subregion, and [A] acquires a block- banded structure.

50 PK. Bane~ee, R.A. ~0~~ I Cornput. Mettwds Appi. Me&. Engrg. 151 (1998) 43-61

5.5. Solution

Two basic methods are used to solve the nonlinear system of equations, (20). The simpler method involves ‘Direct Iteration’, whereby nonlinear effects and forcing functions merely accumulate on the right-hand side:

The disadvantages of this approach are the inability to handle large nonlinear contributions, as well as slow convergence properties. However, it is simple to program and is a useful starting algorithm. One should note that the coefficient matrix can be computed and factored once, and stored for subsequent iterations since it does not change.

The second method used for solving the nonlinear equations is a Newton-Raphson algorithm, com- monly known as ‘Newton’s method’. The iteration scheme is represented by

at? k K )I ax {A-8+1} = -{$c} (22)

{xk+*} = (2) + (Axk+l}

The primary advantage with this algorithm is that the interior equations are brought inside the system matrix, similar to the approach employed initially by Banerjee and Raveendra [21] in elastoplasticity.

Whatever method is used to solve the nonlinear system, a convergence tolerance is required to ter- minate the algorithm. A criteria is used which checks for convergence of the velocity vector only:

Ek+l = i=l

i=l

(23)

where N is the total number of velocity ~mponents. Iteration is te~inated when &* is reduced below a small tolerance value.

6. Numerical examples

All of the fo~ulations discussed above have been implemented as a segment of GPBEST, a general purpose boundary element code. In this section, a few examples are included to demonstrate the validity of the boundary element formulations.

In order to compare the boundary element formulations with a more mature technique, a finite element program was also developed. The program uses a penalty function finite element formation, as outlined by Heir&h and Strada [7] and Gartling [5]. Since the elements, shape functions, boundary conditions and convergence criteria are similar between the BEM and FEM programs, some useful comparisons between the two methods can be made.

Tabie 2 classifies the various combinations of options used in these numerical examples. In particular, it is evident that, while buoyancy can be handled through coupled or uncoupled boundary element formulations, an analogous distinction cannot be drawn for the finite element formulations. In this instance, it is only useful to compare nonlinear iteration techniques between the two methods.

The numerical examples were run using properties for air at 25 “C. In this regard, it is important to realize that several limitations are required in order to analyze a compressible fluid, such as air, using an incompressible formulation. Spiegel and Veronis [22] have discussed this matter in detail. In particular, the specific heat c, is replaced by c,, and the density variation is restricted to small amplitudes.

P.K. Banerjee, K.A. Honk&a I Comput. Methods Appl. Mech. Engrg. 151 (1998) 43-62 51

Table 2 Comparison of solution options

Method Buoyancy Nonlinear coupling iteration

Boundary element Uncoupled Direct Boundary element Uncoupled Newton Boundary element Coupled Direct Boundary element Coupled Newton Finite element Uncoupled Direct Finite element Uncoupled Newton

It is useful to define the nondimensional numbers pertinent to natural convection problems (e.g. 1231). The Rayleigh number, Prandtl number and Average Nusselt number are given by

where L is a characteristic length for the particular problem and Q is the heat flux leaving a surface.

6. I. Buoyancy Burgers flow

A useful problem of CFD involves the sim~taneous convection and diffusion occurring in a uniformly moving fluid, known as ‘Burgers flow’ [I]. Since the convective terms are retained in the solution, and an analytical solution can be effected, Burgers flow provides a useful way to check and test CFD formulations.

A simple extension of the conventional Burgers flow problem can be made to incorporate the effect of buoyancy. Fig. 1 illustrates the problem along with the appropriate bounda~ conditions. Consider a fluid moving with a uniform x-component of velocity, UO, such that the velocity and temperature are constant in the y-direction. The y-momentum and energy equations reduce to two ordinary differential equations:

SLAV’ - /eN’(x) c ~~g~(T - To) = 0

pc&T’(x) - kT”(x) = 0,

where primes represent derivatives with respect to X.

Fig. 1. Buoyant-enhan~d Burgers flow, problem geometry.

52 P.K. Banerjee, K.A. Honkala I Comput. Methods Appl. Mech. Engrg. 151 (1998) 43-61

The temperature distribution is readily determined:

T(x) = (24)

The differential equation for V(x) can be obtained by breaking down the solution into homogeneous and particular components, The particular solution incorporates the known temperature distribution (24). The final result is

PG - -x + CI exp

PUO where

AI=l-AZ-C1

The boundary element model consists of 12 boundary elements and 5 equally spaced volume cells. Since Burgers tIow is a numerical problem, the Rayleigh number was chosen to be Ra = 3.5 x lo-* in order to keep the buoyancy effect small but noticeable. The analytical and BEM temperature and velocity profiles are plotted in Fig. 2. Excellent results are obtained for both quantities. It should be noted that identical solutions are obtained for all four boundary element options, although convergence occurs fastest for the coupled buoyancy formulation with Newton iteration.

0.8

0.6 - Vel (Exact)

0.4

“--...- Temp (Exact) 0.2

r, Temp (BEM)

0.0 ! Y 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.a 2.0

X - Coordinate

Fig. 2. Buoyancy-enhan~d Burgers flow, velocity and temperature profiles.

P.K. Banerjee. K.A. Honkala I Comput. Methods Appl. Mech. Engrg. 151 (1998) 43-62 53

6.2. Buoyancy driven cavity

Buoyancy driven flow in the square cavity has been a popular problem for testing CFD algorithms. The problem to be solved is illustrated in Fig. 3. The temperature difference between the two vertical walls induces the density variation and results in a basic clockwise motion in the cavity. Unfortunately, an analytical solution is unavailable for general values of Rayleigh number. However, a comprehen- sive comparison of many numerical solutions to this problem was made by De Vahl Davis and Jones [24]. These authors pointed to a ‘probable’ solution using a finite difference method along with mesh refinement and Richardson extrapolation.

A number of boundary element researchers have also considered this problem. Tosaka and Fukushima [12] used a coupled buoyancy approach to solve the steady problem. Kitagawa et al. [18] employed an uncoupled formulation, while enforcing incompressibility with a penalty function. More recently, Onishi et al. [25] presented both steady and transient solutions for this problem. However, only Kitagawa et al. (181 provided anything more than just a qualitative assessment of the results.

For the sake of comparison, a relatively low Rayleigh number flow was run on a relatively coarse uniform mesh. A value of Ra = lo3 allowed all six comparison methods, listed in Table 2, to converge from a null initial velocity vector.

The resulting velocity vectors and temperature contours are shown in Fig. 4. A common comparison parameter is the average Nusselt number, which describes the amount of heat leaving the cold wall. While the known value of Nu is 1.118 from De Vahl Davis and Jones [24], the present BEM predicted 1.117 and the FEM predicted 1.134. The superior accuracy of the BEM prediction can be attributed to the fact that thermal fluxes are solved directly at boundary nodes, whereas in the FEM formulation, these quantities are determined indirectly from nodal temperatures [6]. As a result, the BEM is much less prone to error in predicting thermal fluxes at the boundaries, and is ideally suited for the calculation of heat transfer coefficients.

The convergence history for the uncoupled formulations are plotted in Fig. 5. The methods using Newton iteration proved superior to those employing Direct iteration. In fact, the choppy iteration history of Direct iteration indicates that trouble should be expected if larger Ra flow problems would be attempted using these methods.

The nearly identical behavior of the uncoupled BEM and FEM solutions for Direct iteration is strik- ing, and somewhat unexpected since the formulations are quite distinct. However, in both methods, the buoyancy effect enters the problem exclusively through a right-hand-side contribution. The system matrices are merely linear representations of Stokes flow and Fourier heat conduction. Thus, although the actual equations are different, the results in Fig. 5 suggest that there is a limit to the convergence rate for such a representation.

For the various BEM solutions to the driven cavity problem, the improved convergence of the cou- pled formulation over the uncoupled formulation is clearly evident in Fig. 6. However, it is even more beneficial to use Newton iteration. Convergence is faster when the system matrix embodies more of the physics of the problem. Thus, the coupled BEM formulation with Newton iteration proved to have superior convergence characteristics, requiring only four iterations.

However, it should be noted that overall computing time for the BEM solutions was still significantly higher than that required for FEM with the same mesh. Most of this additional time was spent evaluating the volume integrals of (15). Although, further work is clearly needed to reduce the computational effort associated with this phase of the analysis when comparisons are based upon achieving the same level of accuracy, the present boundary element approach does become competitive with FEM [15].

High Ra flows are characterized by large gradients in velocity and temperature, and an adequate mesh distribution is necessary to effect accurate solutions. In particular, the 16 cell model is not suitable much beyond Ra = 103. Consequently, refined 64 cell and 144 cell, two region models shown in Fig. 7 were developed to obtain higher Ra solutions. Table 3 provides a comparison of Nusselt numbers for the present BEM with those obtained by De Vahl Davis and Jones [24]. The deviation is only 0.6% at Ra = 10’ for the 144 cell model. This is quite remarkable in light of the coarseness of the BEM mesh, and again highlights the potential of a boundary element approach which includes surface tractions and heat flux as primary variables.

54

UP0

V=O

T=+T1

P.K. Banerjee, K.A. Honkala I Comput. Methods Appl. Mech. Engrg. I51 (1998) 43-61

Y A

ll=O V=O (InSUhtCd)

I L 0

lI=O

V=O

T s-T1

x I I - \

I

,

(cold wall) I 0 I 1

\ I

\ \ . / ,

Fig. 3. Buoyancy-driven cavity.

Fig. 4. Boundary element results for nonlinear buoyancy-driven cavity, Ra = 1000~ (a) velocity vectors; (b) temperature contours.

EpSllOn l.OE+Ol 3

.-a.. BEM.Unooupled.Dlreci

.+.. BEM,Uncoupled.Newton

+ FEM,Direct

+ FEM,Nwton

EPallon

nooupled.Nwton

- BEM.Cwpled,Direct

--g BEM.CoUplBd,NBwton

l.OE-064 , I I I I I , , 2 4 6 8 10 12 14 16 16 2” 22 2 4 6 8 10 12 14 16 18 20 22

Iteration # Iteration #

Fig. 5. Buoyancy-driven cavity iteration history, uncoupled formulations, Ra = 1COO.

Fig. 6. Buoyancy-driven cavity iteration history, boundary element formulations, Ra = 1000.

(a) (b)

Fig. 7. Buoyancy-driven cavity, refined boundary element models: (a) 64 cells; (b) 144 cells.

Table 3 Buoyancy-driven cavity

Nusselt number comparison

Ra 18 104 10s

De Vahl Davis and Jones [24] 1.118 2.238 4.505 BEM-64 cell model 1.118 2.240 4.619 BEM-144 cell model 1.118 2.243 4.532

P.K. Banerjee, K.A. Honk& I Comput. Methods Appl. Mech. Engrg. 151 (1998) 43-62 55

4.3. Natural convection in a double pane window

A natural convection problem with immense practical significance involves heat transfer between two closely spaced vertical surfaces. This problem arises in the design of double pane windows, building walls, boiler casings, and so forth. For the most part, work on this problem has been experimental in nature (e.g. [26-281). However, some numerical attempts have been made, such as Korpela et al. [29].

The problem geometry is depicted in Fig. 8. No-slip boundary conditions are applied around the entire cavity. Although radiation effects can be significant in the actual design problem, these are ignored here. The relevant parameters used to define the flow are

Temperature difference: AT = Tl - T2

Rayleigh number: Ra = cp2gL3/3 AT

b.k

Nusselt number: .4 Nu = y = - 4c

Aspect ratio: A = F

where q is the total heat transfer out of the cold surface and qC is the heat transfer due to conduction only. At small values at Rayleigh number, the heat transfer is due primarily to conduction, although small currents are set up between the vertical surfaces. In this case, the heat transfer is given by

kA AT qc = L

so that the heat transfer is inversely proportional to the window spacing L. The primary motivation for using double pane windows is to retard the heat transfer by forcing the heat through the air medium, which has a low value of thermal conductivity k. Hence, in the conduction regime, increasing the spacing L, increases the insulating effect.

As the Rayleigh number increases due to a greater temperature differential, convective currents trans- port significant amounts of heat, besides the conduction that is present. This phenomenon reduces the

Y

i H

T=T1

(hot)

?- 0 -L-I

no slip conditions

arolladbolmd=Y

T=TZ

(cold)

‘a x q=o

Fig. 8. Window buoyancy problem definition.

56 F.K. Banerjee, K.A. Honkala 1 Comput. Methods Appl. Mech. Engrg. 151 (1998) 43-61

(a)

Fig. 9. Window buoyancy, boundary element model: (a) four regions; (b) 240 interior cells.

(a) 0-Q Fig. 10. Window buoyancy velocity vectors: (a) Ra = 2958; (b) Ra =: 4437; (c) Ra = 5916; (d) Ra = 6656.

P.K. Banerjee, K.A. Honkala I Comput. Methods Appl. Mech. Engrg. 151 (1998) 43-62 51

insulating capability of the window. Therefore, it is desirable to understand the natural convection rela- tionships involved in designing adequate air insulation in windows and walls.

Fig. 9 shows the boundary element model of the air space between the two vertical surfaces. The aspect ratio under consideration is A = 20. Four regions are used to improve integration and solution efficiency. The coupled buoyancy formulation was used with Newton iteration.

Fig. 10 shows the velocity profiles for various values of Rayleigh number, while the temperature contours are displayed in Fig. 11. Note that in these figures, the x-scale has been exaggerated to improve visualization. The transition from predominantly conduction heat transfer to natural convection heat transfer enhancement is clearly seen. Apparently, above Ra = 3 x 103, the flow becomes multicelled, a phenomenon observed by Elder [27] and Korpela et al. [29]. This is the reason that so many interior cells are used in the boundary element model.

To evaluate the accuracy of the boundary element solution, Nusselt number results are compared with ElSherbiny et al. [28], who gave the following equation based on experimental correlation for this geometry:

0.0fj4Ra’/3 h5 “h’5 ) 1

Although this equation is based on perfectly conducting horizontal surfaces, it is still valid for high aspect ratio, medium Rayleigh number flows with insulated horizontal surfaces. Table 4 shows the good correlation obtained between the boundary element solution and the experimental results. Once again the present BEM is capable of providing accurate heat transfer coefficients.

00 W Fig. 11. Window buoyancy temperature contours: (a) Ra = 2958; (b) Ra = 4437: (c) Ra = 5916; Cd) Ra = 66%.

58 PK. Banerjee, K.A. Honkala I Comput. Methods Appl. Mech. Engrg. 151 (1998) 43-61

Table 4 Two pane window heat transfer results

Nusselt number

Ra BEM

2958 1.06 4437 1.09 5916 1.18 6656 1.23

ElSherbiny et al. [28]

1.07 1.14 1.22 1.25

These results indicate that the natural convection phenomenon brings with it a 25% penalty in in- sulation performance for Ra = 6656, when compared with the heat transfer strictly due to conduction. Hence, numerical and experimental analyses such as these are invaluable in designing and optimizing the window spacing and other parameters present in insulation design problems.

7. Conclusions

Two distinct BEM formulations have been presented to analyze natural convection problems, i.e. coupled and uncoupled. Unlike previous boundary element work on this topic, the present formulation avoids the use of velocity and temperature gradients, satisfies incompressibility without introducing penalty methods, and provides the kernel functions in explicit form. A general multiregion numerical implementation is developed, which includes adaptive integration and two nonlinear solution algorithms. Furthermore the validity of the approach is firmly established based upon a quantitative assessment of the results of several numerical examples.

The following conclusions were reached as a result of this work:

(1)

(2)

(3)

(4)

(5)

(6)

The BEM is well suited to Nusselt number computation since heat fluxes are primary quantities on the boundary. Very accurate results can be obtained with the use of a relatively coarse mesh. In all cases, Newton iteration was more efficient than Direct iteration. The Newton iteration schemes required fewer iterations and approached the final solution in a more uniform manner. The coupled BEM was more efficient than the uncoupled BEM. The reason is that more of the physics of the problem is available at the kernel level in the coupled BEM. The coupled BEM along with Newton iteration proved to have the fastest convergence of all the methods evaluated here. The Direct iteration histories of uncoupled BEM and FEM were virtually identical, indicating a similarity in the way the two approaches incorporate nonlinear convective volumetric effects. Computational efficiencies are still needed in volume integration in order to make these new BEM formulations more attractive for general problems of natural convection. The BEM approach can be used to substantiate results obtained by other numerical methods for natural convection, particularly when experimental data is not readily available.

Acknowledgments

This work was supported in part by the NASA Lewis Center under Grant NAG3-712. The authors are indebted to Dr. C. Chamis for his support and encouragement.

Appendix A

The explicit form of the kernel functions for steady, 2D incompressible flow are presented below.

P.K. Banerjee, K.A. Honkala I Comput. Methods Appl. Mech. Engrg. 151 (1998) 43-62

yi = Xi - & r2 = yiyj

1

ull = -4w - (2&+lrd3

631 = PC pop [ (6 + 8 In r)ylylgr - r2(7 + 12 1n r)gI] 128npk

c32 = PC pop [(6 + 81nr)y2ylgl - r2(7 + 121nr)g2] 128mpk

1 ii33 = -- lnr

2nk

634 = -Pc $ (1 + In r)ylgl

1 Yl c41 = -- 27~ r2

1 Y2 c42 Ez -- 29~ r2

lid3 = 0

ii&l = -2/_&(X - .$)

fl 1 = _ 1 YlYlYPh

9-i r4

fl2 _ 1 YlY2YPl

7r r4

t13 = 0

59

60 P.K. Banerjee, K.A. Honkala I Comput. Methods Appl. Mech. Engrg. 151 (1998) 43-61

f 21

= J_YzYlYlnl ?r r4

f22 _ 1 Y2Y2Y@l

7F r4

tyj = 0

( n2 ~Y2Yr% t;4=2pS(X-[)n2+E --+----

P r2 r4 1

POP f3l = Pc-

[ -(6+8~nr)(nly~g~+ylnrg~) -8

YlYPlYmgm 128qu r2

+ (26 + 24 In r)gl ypq 1

832 = Pc- POP 128~p E - 16 + 8 In 4 hw + yma f - 8 Y2YPlYmgm

r2 + (26 + 24 In r)g2ytni

3

1 YPl t33 = --

2~ r2

f34 =2: pc Pop [

YlnlYmgm 4~ r2

+ (1 + In r)nlgl I

f41 = t142 = f43 = f+l = 0

References

[l] P.J. Roache, Computational Fluid Dynamics (Hermosa, Albuquerque, 1972). [2] J.O. Wilkes and S.W. Churchill, The finite-difference computation of natural convection in a rectangular enclosure, AIChE J.

12(l) (1966) 161-166. [3] J.W. Elder, Numerical experiments with free convection in a vertical slot, J. Fluid Mech. 24 (1966) 823-843. [4] A.E. Gill, The boundary-layer regime for convection in a rectangular cavity, J. Fluid Mech. 26 (1966) 515-536. [S] D.K. Gartling, Convective heat transfer analysis by the finite element method, Comput. Methods Appl. Mech. Engrg. I2

(1977) 365-382. [6] R.S. Marshall, J.C. Heinrich and O.C. Zienkiewicz, Natural convection in a square enclosure by a finite-element, penalty

function method using primitive fluid variables, Numer. Heat Trans. 1 (1978) 315-330. [7] J.C. Heinrich and M. Strada, Penalty finite element analysis of natural convection at high rayleigh numbers, in: R.H. Gallagher,

D.H. Norrie, J.T. Oden and O.C. Zienkiewicz, eds., Finite Elements in Fluids, Vol. 4 (Wiley, New York, 1982) 285-303. [X] PK. Banerjee and R. Butterfield, Boundary Element Methods in Engineering Science (McGraw-Hi& London, 1981). (91 G.F. Dargush and PK. Banejee, A boundary element method for steady incompressible thermoviscous flow, Int. J. Numer.

Methods Engrg. 31 (1991) 1605-1626. [lo] G.F. Dargush and PK. Banerjee, A time-dependent incompressible viscous bem for moderate reynolds numbers, Int. .I. Numer.

Methods Engrg. 31 (1991) 1627-1648. [ll] K. Onishi, T. Kuroki and M. Tanaka, An application of a boundary element method to natural convection, Appl. Math.

Modelling 8 (1984) 383-390. [12] N. Tosaka and N. Fukushima, Integral equation analysis of laminar natural convection problems, in: M. Tanaka and C.A.

Brebbia, eds., Boundary Elements VIII, Vol. 2 (Springer-Verlag, Berlin, 1986) 803-812. [13] L. Hormander, Linear Partial Differential Operators (Springer-Verlag, Berlin, 1964). /14] N. Tosaka, Integral equation formulation with the primitive variables for incompressible viscous fluid flow problems, Comput.

Mech. 4 (1989) 89-103. 151 K.A. Honkala, Boundary element methods for two-dimensional coupled, thermoviscous flow, Ph.D. Dissertation, State

University of New York at Buffalo, 1992. ‘161 G. Strang, Linear AIgebra and its Applications, 2nd edition (Academic Press, London, 1980). 171 T.J. Hughes, W.K. Liu and A. Brooks, Finite element analysis of incompressible viscous flows by the penalty function

formulation, J. Comput. Phys. 30 (1979) l-60. 1181 K. Kitagawa, C.A. Brebbia, M. Tanaka and L.C. Wrobel, A boundary element analysis of natural convection problems by

penalty function formulation, Boundary Elements X, Vol. 2 (Springer-Verlag, Berlin, 1988) 323-341. [19] J.C. Lachat and J.O. Watson, Effective numerical treatment of boundary integral equations: A formulation for three-

dimensional elasto-statics, Int. J. Numer. Methods Engrg. 10 (1976) 991-1005.

P.K. Banerjee, K.A. Honkala I Comput. Methods Appl. Mech. Engrg. 151 (1998) 43-62 61

[20] PK. Banerjee, The boundary element method in engineering, McGraw Hill, London and New York, 1994.

[21] PK. Banerjee and S.T. Raveendra, Advanced boundary element analysis of two- and three-dimensional problems of elasto-

plasticity, Int. J. Numer. Methods Engrg. 23 (1986) 985-1002.

[22] E.A. Spiegel and G. Veronis, On the Boussinesq approximation for a compressible fluid, Astrophysics J. 131 (1960) 442447.

[23] N.S. Land, A Compilation of Nondimensional Numbers (NASA SP-274, Washington, DC, 1972).

[24] G. De Vahl Davis and 1.P Jones, Natural convection in a square cavity: A comparison exercise, Numerical Methods in Thermal

Problems, Vol. 2 (Pineridge Press, Swansea, 1981) 552-572.

[25] K. Onishi. T. Kuroki and N. Tosaka, Further development of BEM in thermal fluid dynamics, in: PK. Banerjee and L.

Morino, eds., Boundary Element Methods in Nonlinear Fluid Dynamics, Developments in Boundary Element Methods, Vol.

6 (Elsevier Applied Science, New York, 1990) 319345.

[26] E.R.G. Eckert and W.O. Carlson, Natural convection in an air layer enclosed between two vertical plates with different

temperatures, Int. J. Heat Mass Trans. 2 (1961) l&120.

[27] J.W. Elder, Laminar Free convection in a vertical slot, J. Fluid Mech. 23, Part 1, (1965) 77-98.

[28] SM. ElSherbiny, G.D. Raithby and K.G.T. Hollands, Heat transfer by natural convection across vertical and inclined air

layers, J. Heat Trans. 104, Feb. (1982) 96-102.

[29] S.A. Korpela. Y. Lee and J.E. Drummond, Heat transfer through a double pane window, J. Heat Trans. 104 (1982) 539-544.