Boundary element methods for convective heat transfer

Computer Methods in Applied Mechanics and Engineering 105 (1993) 261-284 North-Holland

CMA 355

Boundary element methods for convective heat transfer

Y. Shi and P.K. Banerjee Department of Civil Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA

Received 20 March 1992

A boundary element method is developed for the linearized convective heat transfer problem using a newly derived integral formulation based on a convective fundamental solution. Details of the numerical implementation are provided for this linearized problem, which can be solved with only surface discretization. Significant attention is given to the algorithm so that the method could be applied to problems involving heat transfer at high P~clet numbers. Several numerical examples of steady state flows in two-dimensions are included to demonstrate the accuracy of the present methodology, and to highlight its practical utility and limitations.

1. Introduction

The boundary element method~ for solving the heat equation have been studied by several authors (for example [1-4]). General purpose numerical implementations for two-dimensional, axisymmetric and three-dimensional problems have been detailed more recently by Dargush and Banerjee [5, 6].

The convective diffusion equation is one of the most basic governing equations describing macroscopic phenomena in classical physics [7]. However, it is still very difficult to numerically solve the convective diffusion equation, when the convective term is dominant (see for example [8]). A number of advances have been made by Dargush and Banerjee [9, 10] as well as Kanoh et al. [11] for low to moderate P~clet number, steady and unsteady convective heat transfer problems. These range from the use of advanced geometry and functional modelling to the development of accurate numerical integrations of the singular kernel function involved. However, all of the above boundary element references employ the stationary heat flow kernel, and involve a volume integral which takes care of the convective effects and thus limiting their range of applicability. For high P~clet numbers, the fundamental solution must incorporate more of the physics of the problem and the numerical implementation must include the strong front and back effects that result from it.

Recently, Tanaka et al. [12, 13] have presented a boundary element formulation of the three-dimensional steady linearized convective diffusion equation. Unfortunately, the behavior of the convective kernel function, which was provided by Carslaw and Jaeger [14], was not properly taken into consideration in the boundary integration and therefore the solution

Correspondence to: Dr. Prasanta K. Banerjee, University at Buffalo, State University of New York, Department of Civil Engineering, 212 Ketter Hall, Buffalo, NY 14260, USA.

0045-7825/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved

262 Y. Shi, P.K. Banerjee, Boundary element methods for convective heat transfer

obtained for a simple test problem showed significant error when compared to the exact solution for the upstream regions. Furthermore, as the P~clet number became larger, the solution even in the downstream portion also deviated gradually from the exact solution. These undesirable results may have been brought about by an unsatisfactory numerical evaluation of the singular boundary integrals. In their calculations, the standard Gaussian quadrature rule was used for all boundary integrals which is clearly neither efficient nor accurate enough for these problems.

In the present paper, the boundary element method is developed for heat transfer problems based upon convective fundamental solutions. The unsteady fundamental solution is derived here for the first time and presented in explicit form. The complete boundary integral equation is also developed from the governing convective diffusion equation. The numerical implementation, however, focuses entirely on the linear steady problem, since even for this case significant advances in numerical integration methodology were required to achiev~ accurate solutions. These include analytical integration, developments in the use of admissible solutions to indirectly obtain the diagonal terms of the potential and flux kernels and parametric representations for functions and geometry. The implementation used here overcomes many of the major difficulties faced by earlier workers [12, 13] by taking care of the physical behavior portrayed by the convective fundamental solutions. Several detailed numerical examples are included to highlight the robustness of the method.

2. Governing equations

We consider first a medium moving with a constant velocity U~. When calculating the rate at which heat crosses any plane, a convective term of components pcU~ must be added to the part due to conduction. Thus the components of the heat-flux vector are now

00 q , - - k Ox, "I~ pcOU~, (1)

where k is the thermal conductivity, c is the specific heat and 0 is the temperature. Substituting (1) into the equation of continuity,

O0 Oqj pc ~ + Ox~ = pc~, , (2)

yields the linearized governing equation for the convective heat transfer problem:

D00 020 K (3)

Dt Ox~Ox~

where K = k/cp is the diffusivity, ~ is the heat source rate per unit mass and

Do 0 0

denotes the linear material time derivative.

Y. Shi, P.K. Banerjee, Boundary element methods for convective heat transfer 263

The derivation of the fundamental solutions and integral formulation based on (3) are detailed in the following sections.

3. Fundamental solutions

Consider the response at x due to a unit point source at ~: in an infinite space. Define

y, = x ~ - s~, (4a)

r2 = YiYi , (4b)

and let

~, = 8(x - ~ ) 8 ( t - ~ ) , (4c)

where 8 ( x - ~:) and 8 ( t - ¢) are the generalized Dirac delta functions. The corresponding fundamental solution g must satisfy

Dog tCg - 8(x- !~ )8(t- ¢)=0 . (5) pc Dt k ax~ ax~

Green's function is then given by Carslaw and Jaeger [14] who used a Galelian transformation to obtain

g ( x - ~, t - ~') =

1 e 8 p c ( ~ K t ' ) 3/2

4~rkt' e

- r 2 / 4 K I ' , for three-dimension ;

for two-dimension, (6)

in which

2 r . = ( y~ - U : ' ) ( y d - U : ' ) ,

t ' = t - ¢ . (7)

Next, the infinite space, the unit step point heat source solution of (5) is required. Consider the effect of a unit step heat source acting at the point ~:. Thus, the source can be written as

,~ = 8(x - ~ ) H ( t ) , ¢8)

where 8 ( x - ~) and H ( t ) are standard Dirac delta and Heaviside step functions, respectively. By integrating (6) with respect to time ¢, it is possible to derive the unit step solution. In the past, several authors have resorted to numerical integration with respect to time ~.nce the


analytic integration is difficult. Carslaw and Jaeger [14] provided the solution for the steady state at time t--~oo by integrating (6) over time ~-from 0 to oo:

1 e(UkYk_Ur)/2g , = ~ 4"akr for 3D i

G~(x- ~) I 1 oUkYk/2K K ( Ur~ for2D [2~rk " °\21¢/ '

(9)

But for an arbitrary time t, authors have been able to complete the required integration by a suitable choice of substitution variables. Thus, in a moving medium, this source produces the following temperature distribution at t:

G(x - ~, t) = fo g(x - ~, t - T) d~

I l k [ r+Ut e_U,/2 ~ r -U t ] = 8 r e UkYt/2~ e Ur/2K erfc ~ , - ~ + erfc 2~v'-'~ J ' for 3D,

L4-'~k eV'y"2"~'(r' t) , for 2D,

where U 2 ffi U~U~ and a new function ~,(r, t) is introduced,

~(r, t) ffi t' dr' = Ko 21¢ / + K° In r ' 2 K / '

(lo)

(11)

Ko(a) is the modified Bessel function of the second kind of order zero and Ko(a,/J) is the so-called incomplete MacDonald function defined by [15]

K,~(a,/3) ffi ~' ~ i'jl c°~h~ A~ e-a~(x 2 - 1) ~-1/2 dx

ffi ~ ~0 ~ f ; e-a ~o,h~ sinh2~x dx Av (t2)

where the coefficient

A~ - 2~F(v + 1/2)F(1/2). (13)

The above solutions defined in (10) do not exist in the published literature and are presented here for the first time.

The two-dimensional transient convective diffusion kernel can also be formed by expressing the incomplete MacDonald function in series form. Thus,

Y. Shi, P.K. Baner]ee, Boundary element methods for convective heat transfer 265

G(x- ~, t) =

;t~rk e Ukyk/21¢ X rl! En+l " ~ " n=O

r 2 ~n 1 (U r) 1 eOkyd2 . ~ 4"KKt' (U2t~

eUkyd2'CK° "~K 4~rk nl En+l 4 r / ' n=O

for small t ,

for large t ,

(14)

where E,+ t is the exponential integral of order n + 1. It is useful to examine the nature of this fundamental solution more closely. First, consider

the limiting case of zero free-stream velocity. By introducing series expansions for the exponential and modified Bessel functions in (9) and then performing the limiting process, one obtains

lim G s = U-'~0

1 41r kr '

1

2~rk l n r + A,

for 3D,

for 2D. (15)

where A represents a constant independent of x and ~:. Thus, with zero free-stream velocity, the steady state convective fundamental solutions reduce to the well-known static forms. A contour plot of the temperature under a convective velocity of (-U1) in the xt-direction for the solution provided by (9) due to a steady source at the origin is depicted in Fig. 1. Notice that, in general, the temperature diminishes as one travels away from the applied source. Although, the response is radially symmetric in the static situation (UI - 0 ) as shown in Fig. l(a), with non-zero velocity, the character of the flow field changes dramatically. Figures l(b)-l(d) show the response at increasing values of U. The convective fundamental solution presented in (9) embodies both conductive and convective processes. At low velocity, conduction dominates and the response is nearly symmetric about the x t-coordinate of the source point. On the other hand, in a high speed medium, the effects are convected downstream with little influence from the conduction of the medium. The response becomes narrowly banded and clearly asymmetric. It should be emphasized that the single convective fundamental solution (9) captures the entire range of this behavior and therefore the numerical treatment must be done carefully.

The other limit that is of considerable interest involves the case in which the response point (x i) approaches the Mad point (fi). Once again series expansions are introduced to finally produce

lim G' = r--~0

1

4~rkr '

1

2~rk l n r + B,

for3D,

for 2D,

(16)


(a) u

PE:O.O

CONVECTIVE HEAT TRANSFER KERNEL (PE - 0.0)

.540 - A

.450,, B

.360 : C

3 .270 m O

.180 - E

.0900 -. F

0 . - G

(b) PE - 10.0

CONVEOTNE HEAT TRANSFER KERNEL (PE - 10.0)

.800 - A

.2SO • B

.200" 0

. 160 , , 0

• 100 • E

,0600 m F

O . o G

Fig. 1. Convective heat transfer fundamental solution.

Y. Shi, P. K. Banerjee, Boundary element methods for convective heat transfer

PE - 100.0

‘CONVECTIVE HEAT TRANSFER KERNEL (PE - 100.0)

PE - 1000.0

(d)

.120- A

.llO- 8

.OxJO- c

.0700 - 0

.OSOO- E

.0300-F

.OlOO = 0

.omo~ A

.0200 - E

.0120 - F

.00400-0

26-i

CONVECTIVE HEAT TRANSFER KERNEL (PE - 1000.0)

Fig. 1. Continued.


where B represents a constant which is spatially independent. Consequently, the singularities in G s are identical to those for static heat flow. This property will prove useful later when developing numerical integration algorithms.

4. Boundary integral representations

We now multiply both sides of (3) by ~ and integrate by parts [2] once over the time t and twice over the volume V, respectively. This will generate an equation expression 0 in terms of the boundary information on S and derivatives of g". We therefore start from

fo f [ DoG ] i - p c D t + k O~'~x, + ck d V d t= O. (17)

As a result, (17) is transformed into

fo f [ +

(18)

Dt 0xj Ox~ - ;

n~ is defined as the unit normal to the surface S at x. To complete the derivation of the integral equation for any point £ interior to S at time ~', the last volume integral appearing in (18) must be reduced to 0(£, ¢). This is accomplished, if

pc Dtt + ~ Ox, a-~ + 8(x- ~)8(t- , ) - O. (19)

Green's function ~ defined by (19) is the adjoint of the original Green's function. This adjoint Green's function can be obtained simply by suitably transposing the fundamental solution presented in an earlier section. That is,

~(x - 6, t - ¢) - g(~ - x, ¢ - t) . (2o)

Substituting (19) into (18) produces the desired integral equation,

c(6)0(6, ¢) - [f(6 - x , ¢ - t)O(x, t) - g( 6 - x , T - t ) q . ( x , t)] dS(x) dt

forf + [g(6 - x, ~" - t)ck(x, t)] dV(x) dt

+ fv [g( ~ - x , "r)pcO(x, 0)] dV(x), (21)


in which

ao(x, t) q . ( x , t ) = - k Ox i ni(x ) + pcUiO(x, t )ni(x) (22a)

f ( ~ - x, • - t) = - k Og( [~- x, • - t) ni(x) , Ox~

(22b)

and c(~:) is constant. When ~: is inside S, c(~) = 1. If ~ is on the boundary, then the values are determined by the rOative smoothness of S at ~. For ~: outside the region V, c(~) is zero. The last integral in (21) re~resents the effect of the initial temperature distribution throughout the region of interest.

For simplicity, (21) can be rewritten as

co = i f * o - g , q.] dS + tg * 4' + gpcO°l d e , (23)

in which 0 ° presents the initial condition. The • in (23) symbolizes a Riemann convolution integral, for example

g * q" = fo g( ~ - x, T - t ) q . ( x , t) d t . (24)

$. Numerical implementation

In this section, state-of-the-art boundary element technology is applied to the linear steady convective heat transfer problem. Recent boundary element developments for stationary medium heat transfer [5, 6] with some modifications are directly applied to these problems. The presentation below concentrates on those aspects of the numerical implementation which differ from that detailed in those references. The current implementation is limited to the two-dimensional steady state case, although certainly all of the integral formulations presented in the pre~,ious section are equally valid in two and three dimensions involving non-steady cases.

The methodology for temporal and spatial discretization of the integral equation (23) follows that described in [6]. Thus, the constant time step and quadratic shape functions can be utilized to portray the functional behavior of the field variables over three-noded surface elements. As a result of the discretizations, the boundary integral equation for convective heat transfer as exemplified by (23), can be written as

fs osll cO = O~ No, d S - q,,,,, - ,,, n = l 1 m m

l = l

where N is the total number of time steps, M is the total number of surface elements, No, represents the shape functions, and 0~ and q,~,0 are the nodal values of temperature and flux at

270 Y. Shi, P.K. Baner]ee, Boundary element methods for convective heat transfer

the nth time step, respectively. The time integration in (23) can be performed analytically over the time step (n - 1) A~- and n A~-. Thus (see [6]),

G" = G(~ - x, n A c), (26a)

F" = F( ~ - x , n A'r) (26b)

can be obtained from (10) with appropriate change of arguments. For the steady state conditions which is the primary focus of the numerical implementation described here, (25) reduces to

] cO = ?=, 0,. FN, o d S - q.,o GNu, dS , (27)

where G and F are defined in Appendix A. As noted earlier, the behavior of the convective fundamental solutions G and F can be

quite different from their static counterparts. Consequently, new integration algorithms must be devised to perform the required quadrature in an accurate and efficient manner. The following three categories are distinguished when integrating over an element m:

(I) the source point does not lie on element m; (II) the source point lies on element m, but only the weakly singular G kernel is involved;

(III) the source point lies on element m, and the strongly singular F kernel is involved. In general, for non-singular integration of both the G and F kernels of category (I), subsegmentation is employed along with standard Oaussian quadrature formulas. However, for near-singular cases when # is close to element m, very high order formulas are needed to capture the kernel behavior. For integration of elements far upstream of the source point, only a minimal number of Gauss points are required. At the other extreme are elements just downstream of the source. In that case, graded subsegmentation is utilized along with high order integration formulae.

Several alternatives are possible for evaluation of category (II) integrals. The first involves the uses of subsegmentation and high order Gaussian quadrature. This approach is quite inefficient, although for large problems, the cost of category (II) integration is typically insignificant. A second alternative combines analytical and numencal integration. The analytical pattern is established on the basis of the non-dimensional parameters a and/3, where

U~,Yk - # = .

2K ' 2K (28)

In this case, a small fiat subsegment tangential to the singular element is identified and integrated analytically; the remaining subsegments are handled numerically. Thus the length of this subsegment S* can be determined from

2K S*~ ~ a* , (29)

and the desired integral is expressible as


fs . e-ZKo(a)N,o( ¢) dS(x) = ~ fo" e-k~Ko(a)N, , ( ¢(a)) da , (30)

where k = UkYklUr is constant over the fiat segment. The integral then can be analytically evaluated for the three singular cases involving a 3-noded quadratic boundary element as described below (see Fig. 2).

CASE I: Node 1 is the singular point. The shape functions for the quadratic element can be stated as

where

and

N ~ ( ¢ ( a ) ) = 2 ( ¢ ½)(~" 1 ) = 2 a 2 2 - - la z. - 3a/aL + 1 ,

N2(~'(a)) = -4~'(~' - 1 ) - -4a2 /a 2 -[- 4a/aL , (31)

N3(~'(a)) = 2~'(~ - ½ ) = 2a 21a 2 - a laL ,

= S I L = a l a L , (32)

UL (33) aL = 2 K "

L is the length of the element. Then (30) becomes

node 1

~~~~__~e 3 Quadratic Element

¢ .---f ¢

(~) (a)

Case I

Node 1 is the Singular Point

Case II


Case III


Fig. 2. Two.dimensional boundary elements.


fs e_~Ko(ot)Nx(~) dS = 2~ (2C2/a2-3C,/a , . + Co) , L

fs- e-~K°(a)N2(~') dS = 2~ (_4C21a2L + 4C,/aL),

where s e - Z K o ( a ) N 3 ( ~ r ) dS = 2~ (2C2/a2 _ C,/O~L )

(7,. = fo a" e-k"Ko(a) d a , n = 0, 1, 2.

The expression of C. is detailed in Appendix B. The constant k for this case can be expressed as

(34a)

(34b)

(34c)

(35)

k - (U2n , - U, n 2 ) / U . (36)

CASE H: Node 3 is the singular point. The shape functions for the quadratic element can be stated as

N , ( g ' ( a ) ) = 2g'(g" - ½) = 2,~ ~/,~. - ,~/,~,. ,

N2 (~ ' ( a ) ) = - 4 ~ ' ( ~ ' - 1) = - 4 a 2 / a 2 + 4a/o~z. , ( 37 )

N3(~'(a)) = 2 ( ~ ' - ½)(~'- 1)ffi 2a2/a~- 3¢=/aL + 1.

It is not difficult to obtain the desired integrals following the same procedure as above,

fs 2~ . e="Ko(a)N,(~ ") dS = -U (2C2/a~ - C,/a,.), (3Sa)

fs. e=#'Ko(a)N=(~') dS ffi ~ (-4C=/a~ + 4C,/¢=L) , (38b)

fs.e -ZKo(oON3(()dS ffi ~ (2C=/a[ - 3C,/aL + Co). (380

But notice here the constant k is

k ffi ( U r n 2 - U2n,)/U. (39)

C A S E III: Node 2 is the singular point. The shape functions for the quadratic element can be stated as

N , ( ~ ( . ) ) = ½ ~(~'- ~)= ½,,'/~.- ½,,/=,.,

N2(~'(a)) = (1 - ~')(1 + ~')= 1 - a=/a~, (40)

2 2 N,(~'(a)) = ½ ~'(~" + 1)= ½,~ h~,. + ~ / ~ , . ,


where

and

Ol ~ m m

Ol 1

UL~

ai = 2K '

~2 = a , ( 4 1 ) if2

i = 1, 2. (42)

L i are the lengths of the element on both sides of the singular point (see Substituting (40) and (41) into (30), we obtain

fsO-"':o,O)"o,,)dS • = e Ko(a)N,o(~',(a)) da

Fig. 2(c)).

Thus the required integrals are given by

f, dS = [½ c[/,,,) + ½ ' ) ,,,.

. (C21al 2~ + 2 2 _ , (C2/a 2 C~/0~2)] (44a)

fs. e_aKo(oe)N2( ~,) dS = 2~ [(1-C~la~)+(1-C~/a~)], (44b)

. e - a K o ( a ) N a ( ~ ) d S = [2(C2/a 1 (C2/ot2 + (44c)

where s

.C~, = f f a" e-k"~Ko(a) d a , n = 0, 1, 2, i = 1, 2. (45)

The constants kj are

k, ffi (U,n 2 - U 2 n , ) / U , k2 --- (U2n , - U ~ n 2 ) / U . (46)

After the integration of category (I) and (II) type integrals, the ensuing nodal collocation process then produces the usual global set of equations

[F]{0) - [G]{ qn) = {0). (47)

The diagonal terms of F have not been completely determined due to the strongly singular nature of F. These diagonal contributions cannot be calculated by imposing a uniform 'equipotential' field on the body directly because the 'equipotential' and zero flux condition no longer exists. Instead, (47) can be written as

[F]{1} -[O]{pcU,,} = 0 .

Using (48), the desired diagonal terms, F d can be obtained from the summation of the off-diagonal terms of IF] and the vector { [ G ] { p c U , } } . This can be expressed as


F d= - ~ F + ~ pcGU,,. (49) p=l q=l p#d

The indirect calculation for diagonals of F kernels can also be extended to calculate the diagonals of G kernels which are also difficult to obtain by numerical integration for the strongly convective problem. This is possible, provided 0 and the corresponding qn are taken to be known quantities, as evaiuated from known simple temperature fields which satisfy the governing equation for an arbitrary body in the convective heat transfer problem. The required number of these temperature fields clearly depends on the numbers of quantities to be evaluated as the unknowns in (48). Besides the diagonal term of F kernel, there are two diagonal terms for extreme nodes F and one for the middle node for G kernel, that are difficult to obtain numerically. Thus, one or two more known temperature fields could be imposed besides the uniform temperature. The temperature field for any arbitrary body can be taken as: (i) O = X 2 ,

(ii) 0 = e v:k/~, where the existence of only the xl-direction free-stream velocity is assumed. The corresponding flux fields qn on the boundary that could maintain these temperature fields, could then be computed easily. It should be noted here that only one set of these solutions is needed when b e is on the mid-node of an element or on an extreme node of an element that is smooth at b e , and both sets of solutions are needed when ~ is on an extreme node of an element that is not smooth at ~e. In the latter case, the solution of a system of simultaneous equations is required to indirectly evaluate the quantity G d over both neighboring elements. In the present implementation, both the partial analytical integration shown in (30) (together with Appendix B) and the imposition of the known solution outlined above gave essentially identical results which in turn was found to be better than a complete numerical integration of G.

6. Numerical examples

6.1. Burgers problem

Tne classic uniaxial linear Burgers problem provides an excellent test of the convective heat transfer formulations. The heat flows in the x-direction with uniform velocity U. Meanwhile, the temperature is specified as T O at the inlet and zero at the outlet. The length of the flow field is L. The analytical solution for the above boundary condition,

7"(0) -- To, T(t.) -- 0 , (50)

is given by Schlicting [16] as

T = To{ 1-exp[Pe(xlL-1)]} 1 - expf-Pe) ' (51)

where Pe = ULI~ is the P6clet number.


L . W '

.PS

%~. .50

.25 F ~ a l ) , t I c a l

o ~ (PlC - . 1 ~ 1 )

o BDI (FT = I . ) A B£)'I (PlC - 10. ) M BE)4 ( 1 ~ I . l i n g O . )

.ee L__ .leg . 2 5 .SO .?S I . g 8

×,,I.

Fig. 3. Burgers problem: temperature distribution•

The boundary element model employs 18 quadratic surface elements encompassing the rectangular domain. The elements are graded, provided a very fine discretizion near the exit, where T varies substantially for large Pe. The results are shown in Fig. 3. Excellent agreement with the analytical solution is obtained for this boundary-only analysis, even for the highly convective case of P e - 1000.

6.2. Flow around a cylinder

As the next convective heat transfer example, the case of heat flow around a unit diameter circular cylinder is considered, it should be emphasized that with the assumption of uniform convective velocity, the problem permits a boundary-only analysis. Thus, the only mesh that is needed is that displayed in Fig. 4. However, a number of interior points were added in the flow field for post-processing purposes.

The hot flow around a cooling cylinder is considered. The cylinder is heated externally by a hot flow, flowing from left to right at a unit temperature, and cooled to zero on the surface of the cylinder. The resulting steady-state temperature distributions at P~clet number 0 and 1000 are displayed in Fig. 5. The strong convective character is quite noticeable at larger Pe as the effect of the cold cylinder is swept downstream. Note also that the solution at the exit is significantly affected by the end condition.

I , , ! I

• • • • • -m • • • •

• |

Fig. 4. Flow around a cylinder: boundary element mesh.


(a) . s , . A

.750- B

P E - 0.0 .583 : C

.417 : O

.260- E

.0633. F

(b) .s17. ^

PE is 1000,0

.760- B

I . m - C

• 4t7 ,, D

.260- E

.0632. F

Fig. 5. Hot flow around a cooling cylinder: temperature contours.

6.3. Flow past airfoils

For illustrative purposes, a boundary-only heat transfer analysis was conducted for convective flow around a pair of NACA-0018 airfoils. The boundary element model of the blades is shown in Fig. 6. A hot fluid at unit temperature flows from left to right with a unit magnitude of the free stream velocity. Meanwhile, the airfoils are assumed to be stationary with their outer surface maintained at zero temperature.


NAOA-0018 AIRFOILS - BOUNDARY ELEMENT MODEL

""SHRINK ELEMENTS OPTION

I

". f

d , ' p o

~'~X Fig. 6. NACA-0018 airfoils: boundary element model.

It should be emphasized that this problem was run once again as a boundary-only analysis, however, a number of sampling points were included in the fluid surrounding the airfoils in order to graphically portray the response. Figure 7(a) depicts the temperature distribution in the fluid at a P6clet (Pe) number of 10, where Pe = U ! / K , with fluid velocity U, thermal

(a) TEMPERATURE .eTS = A

.SiS - B

.37S - C

E X .12S - O

CONVECTIVE THERMOVlSCOUS FLOW ( RE,PE -10, N~IGLE ,. 0 }

Fig. 7. NACA-0018 air~'oils: temperature contours.


(b) TSUPSnATUM .roTS. A

.62S - B

.37S - C

~ ' ~ X .12S ,, D

CONVECTIVE THERMOVISCOU8 FLOW ( RE.PE -I00. ANGLE - 01

(c) TEMPERA11JRE • l f f 5 " A

.6211. B

• 8711 - C

• 1 8 - 0

CONVECTIVE THERMOVISCOU8 FLOW ( RE,PE -1000, ANGLE . 0 )

Fig. 7. Continued.

diffusivity K and airfoil chord length L. Meanwhile, Figs. 7(b)-7(d) show the response at progressively high P6clet number. At Pe = 10000, quartic surface elements were required in order to obtain an accurate solution. The strong convective character is quite noticeable at large Pe as the effect of the cold airfoils is swept downstream. Also, in Figs. 7(c) and 7(d) there is virtually no interaction between the airfoils. This type of behavior is expected from a physical standpoint. It occurs in the analysis because of the banded nature of the convective fundamental solutions illustrated previously (e.g., Fig. 1). However, interaction will take place if the angle of attack is altered. Figure 7(e) shows the response at a 30 ° angle for Pe = 1000. In these examples the problems were analyzed as exterior problems, hence no exit boundary


(d) TEMPERATURE .O t iS . A

- - i IL~ i i m i i

A A : " A A A A X ' -

. . . . ~ j

. . . . ~ ,~_~ . . . . . . ,, m , i a

1

11

.S2S- •

. 37S . C

[ X

.12S - D

CONVECTIVE THERMOViSCOUS FLOW ( RE,PE -10000. ANGLE - 0 )

(e) TEMPERATURE .67S - A

.62S - B

.37S - C

OONVEOllVE T~iERMOVIBOOUB FLOW ( RE,PE -1000, ANGLE - 30 )

Fig. 7. Continued.

. I H " 0

conditions were required since these are assumed to be at infinity. These solutions therefore are not influenced by any artificial boundary conditions at an assumed boundary.

7. Concluding remarks

Fundamental solutions and integral representations have been derived for application to convective heat transfer. The inclusion of linearized convective terms in the differential operator injects more of the physics of high speed flow into the fundamental solution than


does the usual stationary medium approach. Unfortunately, the typical numerical integration schemes currently in use for standard elliptic problems are not sufficiently robust to capture the behavior of the convective kernels. More sophisticated algorithms incorporating both analytical and numerical components are needed.

Once these are developed, however, very accurate results can be obtained for the linearized problems, as shown in the numerical examples. However, it is expected that the ultimate role for these solutions will be as a starting point for non-linear boundary element solutions of the complete convective thermoviscous fluid flow problems.

Acknowledgment

The work described in this paper was made possible by the NASA Grant No. NAG3-712 with supplemental funding from Pratt and Whitney Aircraft, Connecticut. The authors are indebted to Dr C. Chamis and Mr D. Hopkins of NASA and Drs E. Todd, R.B. Wilson and Mr H. Craig of Pratt and Whitney for their support and encouragement.

Appendix A. Kernels for steady heat transfer

where [ ] 1 1 Unr e_~Ko(a) + e_~rl (a)

G = 2~rk e - °K° (a ) ' F = 2+rr 2t¢ 2K '

_ r2 ffi U 2 Uky k Ur y, = x, ~ , . y , y , . = u , u , . /3 = ~ . ~ = 2 . "

Appendix B. The derivation of function Cn

The integral in (35) is rewritten as

Cn = fo an e-k~Ko(a)da ' n •0, 1,2. (B.1)

where series expansion of Ko(a) can be represented as (see [17] for details)

1 ) = ( n ! ) 2 -1 m - - 3t

m m

-ln(½a) ~ a~a2" + ~ b~a 2~- ln(½t~)~ a.a2" + ~ b.t~ 2" 0 < a ~ 2 n=O nffiO nffiO n=O

6 - ! / 2 - a

n = 0

2<~a<oo

(B.2)

Y. Shi, P~ K. Baner]ee, Boundary element methods for convective heat transfer 281

where

,,=o (n!) 2

, / = 0.57721 56649 01532 86060 6512.. •

and the coefficients are

- - - T ) , n > ~ l ,

an (n l )2 '

b o = - 7 ,

b " = a n m ffi!

c o = 1.25331 414,

c a - -0 .01062 446,

c s = -0.00251 540,

is Euler 's cons tant ,

c I = -0 .07832 358,

c 4 = 0.00587 872,

c a = 0.00053 208.

c 2 = 0.02189 568,

Substituting (B.2) into (B.1) , we obtain the expression of Cn for small a as

where

m

N m m

a m Ol 2m+n In (½a)e -k* da + b m Ol 2m+n e -k* da mffiO • m = O

~ amA2m+n + ~ bmB2,~4,,, m = 0 m = 0

O

A . = fo a " e -k" In( ½ a) d a

1 [nA,,_ 1 + B , , _ l - a * " l n ( ½ a * ) e - k " ' ] , n > 0 ,

A 0 -'-- f : e -ka In( ½ a) da

1 ,) = - ~ [e -k"" ln(½a + E~(ka*) + ~, + In21,

f-" 1 B n = a n e - k " dot = ~ (nBn-1 - a*" e - k " * ) ,

if " 1 • B o = e -k" d a = -~ ( 1 - e -k" ) .

n > 0 ,

(e.3)

For large a >~ 2, we have from (B.1),


P amA2m+ n + ~ b mB'2m+n

m--O m - O

where

2mCm ~n+ "~" mffi0 ('k -~- 1)(n_m+112) 7(n - m + ½, (k + 1)a*) + m= 1 2mCmLm_n, (B.4)

1 A: = -~ (nA._l + B._,) , n > O,

1 A ~ = - ~ l n 2 ,

n! B" = k . + l ,

and 7(a, x) is an incomplete gamma function defined as

7(a,x)= fo e-tt °-l d t - ( a - 1 ) 7 ( a - 1, x ) - x °-1 e -x , Rea l ( a )>0 ,

and "~( ½, X 2) ~--" 2 fo e-*2 dt = x/-6 err x ,

Ln ffi _ ,, I [a *-(n-t/2) e -(k+l)a" + (k + 1)L._t)] n >0 n - ½ ' '

X/ ~r erf~ L o = (k + 1)-~/27(½, (k + 1)a*) = k'+ I

Two special cases are now discussed. If k = O: From (B.1)

0

C. ffi fo a"Ko(a) da , (B.Sa)

where

~0 ~e O0 00 Co ffi Ko(a ) da ffi- Z a.A°. + Z b.B°., (B.Sb)

" n - O nmO

I

A°ff i fo a"ln(½a)da = B°[ln(½a*) - 11,

J

B° ffi fo a" da = 1 a,,,+l n + l

For large a ~> 2

6

, ffi o ( B . 6 a ) t, o ~ 2"c,.L., n ~ 0

e

CI ffi fo aK°(a) da ffi -a*Kl(a*) + 1, (B.6b)

Q

c 2 = f o a2Ko(a) da ffi -a*ZKl(a*) - a*Ko(a*) + Co, (B.6c)


where

L° _-- f a*-n-"2 e-"" da, = .1.1 n - /2 [a*-("-l 'z)e-" + L]_ l ] .

Lo=f e -a" ~ da* = v '~ erI V ' ~ .

I f k = -T-l:

C. = ~ a" e±'Ko(a) da . (B.7)

Taking into consideration the following relations:

fo e±~z~+t 2"F(v + 1) ÷ t v

e- t K~(t) dt = 2v + 1 [K~(z) "4- K~+l(z)] -w- ~-2~ rl '

d { x " r . ( x ) } = - x " K . _ ~ ( x ) dx

z - , O , r . ( z ) - - ½r(~)(½z) -~ ,

we obtain

where

t

Co = f o e ±~Ko(a ) da = e±°*[Ko(a *) - Kl(a*)] -w- 1, (B.8a)

C~ ffi fo a e'-"Ko(a) da

= ½a *~ e±°'[Ko(a *) -- Kt(a*)l-~ ~a .2 e±"'[K,(a *) -- Ks(a*)] + ~, (B.8b)

C a ffi f tx 2 e ±'Ko(a ) da

= - ~ * ~ e*-*'[K,(,~ *) - K,(,~*)I + I,~ *~ e-'"'[K,(,~*) + Kg,~*)l ~" 6 , (B.Sc)

r .+ , (x ) = r . _ , ( x ) + 2n r . ( x ) . x

References

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[2] P.K. Banerjee and R. Butterfield, Boundary Element Methods in Engineering Science (McGraw-Hill, London, 1981).

[3] P.K. Banerjee, R. Butterfield and G.R. Tomlin, Boundary element methods for two.dimensional problems of transient ground water flow, Internat. L Numer. Methods Geomech. 5 (1981) 15-31.


[4] L.C. Wrobei and C.A. Brebbia, Time-dependent potential problems, in: C.A. Brebbia, ed., Progress in Boundary Element Methods, Vol. 1 (Pentech Press, London, 1981) 192-212.

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[ll]

[121

[13]

G.F. Dargush and P.K. Banerjee, A time dependent incompressible viscous flow for moderate Reynolds numbers, Internat. J. Numer. Methods Engrg. 31 (1991) 1627-1648. M. Kanoh, G. Aramaki and T. Kuroki, Stability experiment of FDM, FEM and BEM in convective diffusion, in: C.A. Brebbia and G. Maier, eds., Boundary Elements VIII (Springer, Berlin, 1986) 725-733. Y. Tanaka, T. Honma and I. Kaji, On mixed boundary element solutions of convection-diffusion problems in three dimensions, Appl. Math. Modelling 10 (1986) 170-175. Y. Tanaka, T. Honma and I. Kaji, Boundary element solution of a three-dimensional convective diffusion equation for large P~clet numbers, in: M. Tanaka and T.A. Cruse, eds., Boundary Element Methods in Applied Mechanics (Pergamon, Oxford, 1988) 295-304.

[14] H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids (Clarendon Press, Oxford, 1947). [15] M.M. Agrest and M.S. Maksimov, Theory of Incomplete Cylindrical Functions and their Applications

(Springer, New York, 1971). [16] H. Schlicting, Boundary Layer Theory (McGraw-Hill, New York, 1955). [17] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972) 375-379.

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Boundary element methods for convective heat transfer