numerical simulation of a low-mach-number flow with a large temperature variation

Computers Fluids Vol. 21, No. 2, pp. 185-200, 1992 0045-7930/92 $5.00 + 0.00 Printed in Great Britain. All fights reserved Copyright © 1992 Pergamon Press plc

N U M E R I C A L S I M U L A T I O N O F A L O W - M A C H - N U M B E R

F L O W W I T H A L A R G E T E M P E R A T U R E V A R I A T I O N t

YASUYOSHI HORIBATA

Systems & Software Engineering Laboratory, Toshiba Corporation, 70, Yanagl-cho, Saiwai-ku, Kawasaki 210, Japan

(Received 8 April 1991; received for publication 8 November 1991)

Abstract---Compressibility is important in a low-Mach-number flow with a large temperature variation. However, it is well-known that time-dependent compressible flow schemes become ineffective at low Mach numbers. This ineffectiveness occurs because a wide disparity exists between the time scales associated with convection and the propagation of acoustic waves. For this reason, scale analysis modifies the compressible equations in order to remove acoustic waves from them. The pressure is divided into the thermodynamic and dynamic parts. The density variation caused by the variation in the dynamic part is neglected. The scale analysis shows the conditions under which the modified equations are applicable. Examination of small-amplitude waves shows that the modified equations contain internal gravity waves (buoyant effects), while they exclude acoustic waves. Similarly to the modified equations, the Boussinesq equations are derived under a further condition. A finite difference scheme integrates the modified equations. The scheme is essentially the MAC method. First, thermal convection of a Boussinesq fluid in a square cavity is simulated in order to validate the calculation method. The result is in good agreement with a benchmark solution. Second, thermal convection with a large temperature variation is simulated in a vertical pipe furnace used for the heat treatment of semiconductor wafers. An axisymmetric steady flow is obtained.

1. I N T R O D U C T I O N

As is well-known, time-dependent compressible flow schemes become ineffective at low Mach numbers. For an implicit scheme of the Beam-Warming, Briley-McDonald type, the convergence rate slows dramatically in a two-dimensional nozzle flow as the Mach number is lowered below 0.3; the scheme becomes virtually useless [1]. The rational Runge-Kutta method is an explicit scheme and possesses an excellent stability characteristic. However, its convergence rate slows in a flow in a square cavity as the Mach number is lowered [2]. This ineffectiveness occurs because a wide disparity exists between the time scales associated with convection and the propagation of acoustic waves. Implicit schemes permit a larger time step. However, the maximum value is normally less than 5-10 times that given by the CFL condition in explicit schemes because truncation errors become unacceptable [1].

Several methods have been reported for circumventing the ineffectiveness of time-dependent compressible flow schemes. The first method is a rescaling of the compressible equations. Briley et al. [3] rescaled the Euler equations to improve the convergence rate and calculated a two-dimensional flow for Mach numbers down to 0.05. Ramshaw et al. [4] rescaled the pressure gradient term in the compressible equations for a flow with nearly uniform pressure. They calculated the flame velocity of a one-dimensional steady flame and the two-dimensional unsteady burning in a closed, spherical volume. The second method is a preconditioning procedure for schemes. Turkel [5] utilized preconditioning for an explicit scheme, but did not report a calculation result. Choi and Merkle [1] used preconditioning for an implicit scheme of the Beam-Warming, Briley-McDonald type, and calculated two-dimensional flows for Mach numbers down to 0.05. The preconditioning method is applicable only to a steady flow. The third method employs a perturbation expansion of the compressible equations in powers of the Mach number. Guerra and Gustafsson [6] modified the Euler equations from a power series expansion of them in terms of the Mach number, and calculated a two-dimensional flow for a Mach number of 0.01. Paolucci [7, 8] expanded the Navier-Stokes equations in powers of the Mach number squared, and calculated the discharge of a pressurized gas from a two-dimensional container. Merkle and Choi

$Presented at the 2nd Japan-Soviet Union Joint Syrup. on Computational Fluid Dynamics, Tsukuba, Japan, 27-31 August 1990.

185

186 YASUYOSHI HORIBATA

[9, 10] modified the Euler equations by expanding them in powers of the Mach number squared, and constructed a scheme similar to the artificial compression method. They calculated two- dimensional steady flows with heat addition for Mach numbers down to 10 -5. Dwyer and Yam [11] utilized their result, and constructed a scheme similar to the SIMPLE method. They calculated the flow around a burning droplet, and forced and free convection over a heated ellipsoid.

Compressibility is important in a low-Mach-number flow when the variation of temperature in it is large. The present paper modifies the compressible equations by scale analysis, and integrates them using a scheme similar to the MAC method. Nakamura et al. [12] used a similar method, and calculated the combustion of a thin solid fuel and the combustion of a gas from a Bunsen burner. However, they did not refer to the ineffectiveness of compressible schemes at low Mach numbers and the relationship between the method and acoustic waves. Furthermore, modification of the compressible equations was not fully discussed. The present paper clarifies the physical aspects of the method, and calculates thermal convection with a large temperature variation.

The pressure is divided into the representative pressure and the deviation from it. The scale of the density variation is analyzed, and the compressible equations are modified. The scale analysis shows the conditions under which the modified equations are applicable. The dispersion relation is obtained for small-amplitude waves contained in the modified equations. Similarly to the modified equations, the Boussinesq equations are derived, and the two equations are compared. A finite difference scheme integrates the modified equations. First, thermal convection of a Boussinesq fluid in a square cavity is simulated in order to validate the calculation method. Second, thermal convection with a large temperature variation is simulated in a vertical pipe furnace used for the heat treatment of semiconductor wafers.

2. THE COMPRESSIBLE EQUATIONS

Assume a laminar flow. Use Cartesian coordinates x, y, z, with z vertically upwards. The Navier-Stokes equations are

and

Du Op Oali = + - - ( 1 ) P Dt Ox Ox~'

Dv Op Oa2i - ~ ( 2 )

P DI Oy c~xi

Dw 0p &r3i P Dt = - a-z + ~ Pg' (3)

where p is the density; u, v, w are the x, y, z components of the velocity, respectively; p is the pressure; g is the gravitational acceleration; and a• is the viscous stress tensor. The coefficient of bulk viscosity is negligible except in the study of the structure of shock waves and in the absorption and attenuation of acoustic waves. Hence, %. is given by

[ Oui Ou, 2 Ou, ~, ox,)

where/z is the coefficient of viscosity. The dissipation function is negligible except in the study of a supersonic flow. Hence, the

equation for internal energy is De p D o Oq~

P D--t p Dt Ox, + Q' (4)

where E is the internal energy per unit mass; Q is the rate of heat produced per unit volume by external agencies; and q,, is the heat flux density. The flux q~ is def ined by

OT q~ = - x - - ,

0xt

where T and x are the temperature and thermal conductivity, respectively.

L o w - M a c h - n u m b e r f low wi th a large t empera tu re va r i a t ion 187

The compressible continuity equation is

Dp aui =

For a perfect gas, the equation of state is

p = (y -- 1)pE,

where y is the ratio of specific heats. Moreover, the following relationship exists:

E =cvT,

where Cv is the specific heat at constant volume.

(s)

(6)

3. MODIFICATION OF THE COMPRESSIBLE EQUATIONS

Let Pr, Tr and Pr denote the representative density, temperature and pressure of a flow, respectively. The equation of state relates these quantities: p~ = p~RT,; R is the gas constant. In general, p~, T~ and pr are functions of time. Denoting by Pd the deviation from p~, the pressure is expressed as

p(t, x, y, z) =pr(t) + pd(t, X, y, Z). (7)

Using equation (7), rewrite the Navier-Stokes equations (1)-(3) in terms of Pd as

and

Du apd a0.t, -~ (8) s P Dt c~x t~x,

Dv ¢~Pd 00"2, 4 (9) P Dt ay c~x,

Dw t~pd a0.3i P P Dt = - c3"-~ + ~x~ - g" (10)

Equations (8)-(10) show that Pd is the dynamic part of the pressure. By using the equation of state, the density variation caused by the variation in Pd is estimated

a s

T P r d"

The representative pressure p, = prRTr is, in order of magnitude, equal to p, c2; c is the acoustic speed in the fluid. In a steady flow, according to Bernouilli's equation, the pressure variation is estimated as Apd ~ prU~; Ur is the representative fluid velocity. Thus, Ap leads to

Ap P~. 2~,,,,2 - - ~ - - u r xvx . Pr P~

Here M is the Mach number defined by M =udc. On the other hand, in an unsteady flow, Apo is estimated as Apo ~ lpr Ur/~; Z and i are a time and a length of the order of the times and distances over which the fluid velocity changes appreciably [13]. Thus, Ap leads to

Ap l M. Pr TC

If the propagation of acoustic waves in the fluid can be regarded as instantaneous, then ~ >> l/c. Hence, one has

AP,~M" Pr

If the flow is in a gravitational field, then the dynamic part of the pressure necessarily varies with height. The variation Apd is estimated as APd ~ p~gh. Here h is the vertical length of the flow. The order of magnitude of the density variation caused by this Ap d is

Ap gh p---~ ,,, c--- ~ .


Similarly, the density variation caused by the temperature variation is estimated as

p~(ap _o~o Pr \ 0 r ] p Pr Tr'

where O is the representative temperature difference in the flow. Hence, if the condition gh. 6) M2 ' l M, (11)

is fulfilled, then, in comparison with the density variation due to the temperature variation, the density variation due to the variation in Pd is negligible.

Taking the substantial derivative of the equation of state, one obtains

1 D p 1 D E 1 Dp - - - = + (12) p Dt E Dt p Dt

If condition (11) is fulfilled, equation (12) reduces to

1 Dp 1 DE 1 dpr p D---t= E n t + " (13) p dt

Moreover, if the condition _ gh M2 ' l M, ~-2 '~ 1 (14) TC

is fulfilled, then p~ >>Pa. Accordingly, we rewrite equation (13) as

From this, one obtains

1 Dp 1 DE 1 dpr - - - = + ( 1 5 ) p Dt E Dt Pr dt

P~ = (7 - l)pE. (16)

From equation (15), the continuity equation becomes

1 DE 1 dp, t~ui e Dt p~ dt = t3x---~" (17)

Substituting equation (15) into equation (4) for the term Dp/Dt, one obtains

DE l(__Oq, dpr'~ P - ~ = ~ \ 0xi + Q + d t ] " (18)

The modified equations (8)-(10), (18), (17) and (16) are used in place of the compressible equations (1)-(6) under conditions (11) and (14).

4. THE R E P R E S E N T A T I V E P R E S S U R E

Eliminating DE/Dt from equations (17) and (18), one obtains

dp__j~=dt - T P r ~ + ( 7 - 1 ) ( - ~ + Q) . (19)

Integrating equation (19) over a closed space, one has

V-~ +(~, f u, n, df)pr=6, - l)(- f q, n, df + f Q dv), (20)

where V is the volume of the space and ni is the unit vector normal to the surface element d f Hence the geometry of the space, the velocity and heat flux on the boundary and the rate of heat produced inside the space by external agencies determine p,. If the flow is closed inside the space and the rate of heat added to the flow is constant, then pr rises linearly with time. Moreover, if the rate of heat added to the flow is always zero, then pr is constant. When the flow is open to the atmosphere, Pr is constant. When one is interested in a steady flow only, take pr as a constant.

Low-Math-number flow with a large ternpcrature variation 189

5. S M A L L - A M P L I T U D E WAVES C O N T A I N E D IN THE M O D I F I E D EQUATIONS

This section considers small-amplitude waves contained in the modified equations. The viscous, heat diffusion and heat production terms are irrelevant to the essential character-

istics of small-amplitude waves; hence they are omitted. Assume the representative pressure Pr is constant. The basic state is a stationary stratified fluid and is a function of z only. A suffix 0 and a prime denote the values of quantities at the basic state and small deviations from these values, respectively. Then, to the first order of smallness, we write equations (8)-(10), (18), (17) and (16) as

and

~u* @~ O--t-= - - 0--~-' ( 2 1 )

Ov * Op "~ 0--7 = - O--y-' ( 2 2 )

Ow* @~ O----t- = - O--~ - p " g ' ( 2 3 )

36.____*_* = _ w* d6.____o (24) Ot dz "

Ou* Or* OW* = w , d Ox + -~y + ~ dz In Po (25)

with

p, 1 = - - 6. *, (26)

6.0

u* = po u ' , v* = pov' , w* = po w ' , 6.*=po6.' .

Eliminating u*, v*, 6.* and p ' from equations (21), (22), (25) and equations (23), (24) and (26), one obtains

~:p'~ O~P'd ~ w * N ~ w * - - + - - = - - + ( 2 7 ) t:3x 2 ay 2 0t 0z g 3t

and

where

02p~ ~2w* 3t Oz = Ot 2 N2w*' (28)

N2 = g d In E0. (29) dz

In general, N 2 depends on z. However, assume N 2 is a constant over distances of the order of a wavelength.

Assume the following forms for p~ and w*:

P'd = ~ ( z ) exp i (kx + ly - cot),

and

w* = #(z) exp i (kx + ly - ogt).

Substituting these forms into equations (27) and (28), one obtains the equations for/~(z) and ~(z):

(k 2 + 12)p = zo~ ~ z + ff (30)

and dp

--ico dzz = (c°2 - N2)w" (31)

Eliminating p from equations (30) and (31), one obtains

d2ff ~ N 2dff (k2+12) ( °92-N2) f f=0 . (32) dz 2 g dz co 2


Assume the following form for ~ ' N 2

~ ( z ) = W(z)exp(---~g Z).

Then, from equation (32), one obtains

d2W r(k2 + lZ)(afl - N 2) (NZX~2]W dz2 L o)2 j ~0~ (33)

Moreover, assume the form W(z)~ exp(inz); then equation (33) yields the following relation:

k 2 + 12 602 = N 2 (34) /N2\2 "

k2 +12 + n2 + t-~g )

As is well-known, the dispersion relation for internal gravity waves inside the incompressible fluid

is [13]: k 2 + 12 o)2 = 2 (35) N° k2 + l 5 + n 2'

with

N~=-g--(OP°] ds° (36) Po \ OSo/p dz '

where s is the entropy per unit mass. If the density variation caused by the pressure variation is negligible, one can suppose the pressure is constant in determining the derivatives of the thermodynamic quantities. Hence, since To(OSo/CgTo)p is cp, one has

dso (~So~ dTo=c d d--z = kaTolp dz P ~z In To (37)

and

OSoi,: \~-~so ). \~ol. r 0 ( 0 o 0 ] : o0 (38) cp k0To/p Cp

Substituting equations (37) and (38) into equation (36), one obtains

N2o=gd ln To dz

This No is equal to N. Hence equation (34) is the dispersion relation for internal gravity waves deformed by the density variation due to thermal expansion.

Therefore, the modified equations contain internal gravity waves (buoyant effects), while they exclude acoustic waves.

6. R E L A T I O N S H I P TO THE B O U S S I N E S Q E Q U A T I O N S

According to the arguments in Section 3, if the condition

APr <~ 0

p-? Tr is fulfilled together with condition (l 1), equation (15) reduces to

Dp = 0. Dt

This reduces equations (4) and (5) to

DE tgq~ P~ O---t = -0x--~ + Q

(39)

(40)

Low-Math-number flow with a large temperature variation 191

and

Define p* by

Ou--2 = 0. (41) Ox~

Pd = P* -- prgz. (42)

Under conditions (11) and (39), the temperature variation dominates the density variation. According to condition (39), the temperature variation is small. Hence the resulting density variation is also small, and, to the first order of smallness, one has

¢~P (T- Tr)= - -Or -~r P - - P v = - ~ p

Substituting equations (42) and (43) into the r.h.s.s of equations (8), (9) and (10), we rewrite them as

and

Du dp* Oali P = aX + OX ' (44)

Dv Op$ Oo2i 0~ D t = ~-y + Ox~ (45)

Dw cgp~ P~ D--t = - d--Z

+

Equations (44)-(46), (40) and (41) are usually the Boussinesq equations require condition (39),

Oa3i . T - T r "¢- Or ~ g" (46)

referred to as the Boussinesq equations. Hence in addition to conditions (I 1) and (14).

7. NUMERICAL INTEGRATION

A finite difference scheme integrates the modified equations. The grid system is a standard spatially staggered mesh. The system defines scalar variables such as the pressure and internal energy at a common point, and displaces the velocity components by one-half the grid interval.

We approximate equations (8), (9) and (10) as

u.+ I _ u" 1 = - - - - ~Sxp,] +t + M~, (47)

At p"

U n + t - - V n l n

A ~ = pn f i yp~+t "-F M y (48)

and W n + t __ W n 1

= - - - 6 "" ÷ z . ( 4 9 ) ~'d + M~. At p"

Here the finite difference operator 6 is defined by

1 6~(~ ) = S ¢ [ ¢ ( ~ + A ~ / 2 ) - ~ ( ~ - A ~ / 2 ) ] ,

where ~b denotes the dependent variable; ~ is the independent variable; and A~ is the interval over which the operation takes place. The superscripts n and n + 1 refer to the time levels n and n + 1, respectively. The terms M,, My and Mw denote the convection, diffusion and gravity terms in equations (8), (9) and (10), respectively. Applying the operators fix, ~y and ~z to equations (47), (48) and (49), respectively, and adding them together, one obtains:

1 - At [(6xu + 6yv + 6zw) "+ J - (6xu + 6yv + fi~w)"] + 6xM"~ + 6yM7 + 6~M"~. (50)


From equation (19), we evaluate the term (6xU + ~yl) -4-t~zW) n+ I on the r.h.s, of equation (50) as

+ + " + ' = - Qo+' +

The successive overrelaxation (SOR) method solves equation (50) for p~+l. Unless the value ofpd is specified at a point in the flow, p~+~+ C also becomes a solution; C

is an arbitrary constant. Only the gradient ofpa is significant in the modified equations; hence this arbitrariness occurs. The arbitrariness is similar to that for the pressure in the incompressible Navier-Stokes equations, and has no physical meaning.

We approximate equation (8) as

un--I __U n ~Un+l Oun+l OUn+l 1 0~.+1 p" = - p"u" - - - p"v" - - - p " w " - - Op ~___~_+ + v ~i

A t ¢3x ~y ~3z Ox Ox~

Using second-order upwind difference representations for the convection terms and central difference representations for the other space derivatives, the incomplete LU biconjugate gradient (ILUBCG) method solves the sparse system of linear equations for u "+ 1.

Similarly, we obtain the velocity components v "+l and w "+ ~, and the internal energy E "+~ from equations (9), (10) and (18), respectively; p~"+ ~ is obtained from equation (20).

8. VALIDATION OF THE METHOD

The thermal convection of a Boussinesq fluid has often been calculated in a square cavity [14-17]. de Vahl Davis extrapolated a benchmark solution of this problem from two results calculated using different grid intervals [14]. This section simulates the thermal convection in order to validate the calculation method.

Figure 1 depicts schematically the problem of thermal convection in a square cavity. The upper and lower boundaries are insulated. The left and right walls are perpendicular to the gravity direction. The left and right boundaries are at temperatures Tu = 305 K and T c = 295 K, respectively. The fluid is of Prandtl number 0.71.

Consider the case in which the Rayleigh number,

Ra = f l g ( T n - T c ) D 3 / z v ,

is 10 6. Here X, v, D and fl are the thermometric conductivity, kinematic viscosity, cavity width and thermal-expansion coefficient, respectively; fl is evaluated at 300 K.

I T ~

D

l

Z

T H

~ T 0 (az

T = T C

Y

> X

O /

O z

< D >

Fig. 1. Thermal convection in a square cavity.

Low-Mach-number flow with a large temperature variation

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Fig. 2. Grid system.

193

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, ~ l l / l l / / , . . . . W llltt . . . . " . . . . . ' ~ I I '

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. . . . . . . . . . . . "111111~' . . . . " " " " . . . . . . "////llll ' I1~'~ . . . . . . . . . .

• . . . . . . . . . - - - - ~ ~ _u ,_ , , ,

Fig. 3. Velocity vectors.


(a) (b]

Fig. 4. I so therms. The con tou r s range f rom - 0 . 5 to 0.5 wi th a con tou r interval o f 0.1. (a) Present work. (b) de Vahl Dav i s [14].

Figure 2 shows the grid system. The grid system is composed of 50 points in both direc- tions.

A steady state is obtained. The maximum value of the local Mach number is ,,- 1.4 x 10 -4 . From M ,-~ 1.4 x 10 -4 , O ,~ 10 K , T r ,~ 300 K and h ~ 0.1 m, one has

Mz,, ~ 10_8, gh,,, 9 x 10_6 , --,,~O 3 x 10 -2. c 2 Tr

Hence conditions (11) and (14) are fulfilled. Figure 3 shows the velocity vectors. Figure 4 shows the isotherms and the corresponding

isotherms for the benchmark solution [14]. The nondimensional temperature is defined by T = (T - (TH + Tc)/2)/(TH - Tc). These isotherms are similar to each other.

Evaluate the following quantities:

Nu0--the average Nusselt number on the vertical boundary of the cavity at g = 0; Numax--the maximum value of the local Nusselt number on the boundary at $ = 0

(together with its location); Numin--the minimum value of the local Nusselt number on the boundary at g = 0

(together with its location); Z~max--the maximum horizontal velocity on the vertical mid-plane of the cavity

(together with its location); V~max--the maximum vertical velocity on the horizontal mid-plane of the cavity

(together with its location);

where

. ~=x /D , 2 = z / D , f f=uD/z , 6 ,=wD/x .

Present results de Vahl Davis [14] Difference (%)

T a b l e 1. Differences (%) between the solution reported in this paper and a benchmark solution

Nuo Num~ @'~ = Numl. @1 = '~max @'~ = g'max @'f =

8.706 17.800 0.0344 0.952 1 67.74 0.847 224.05 0.0374 8.817 17.925 0.0378 0.989 1 64.63 0.850 219.36 0.0379

- 1.3 - 0 . 7 - 9 . 0 - 3 . 7 0 4 .8 - 0 . 4 2.1 - 1.3

Low-Mach-number flow with a large temperature variation 195

Height(ram)

Heater

Heat seal-

Insulator H2T.5

Top - " 1 0 3 9 . 5

s Outer tube

Center

3 0 4 . 5

Bottom 1 2 3 8 . 5

Bottom2 161.8

Brick [ - " 100

i : : 0

Insulator

i< ~'. 0 2 9 0 n

Fig. 5. Vertical pipe furnace.

The local Nusselt number on the vertical boundary of the cavity at x = 0 is defined by

Nu(z) = qwan D x r . - T c '

where

qwall = \ 026 ./,¢ = 0"

Evaluate the derivative in equation (51) using the first-order (two point) approximation. Evaluate the average Nusselt number,

ire° Nu0 = ~ Nu(z) dz,

by numerical integration using Simpson's rule. Compute the maximum and minimum values, and their locations by numerical differentiation using the fourth-order Legendre interpolation.

Table 1 compares the quantities with those for the benchmark solution [14]. The largest difference is - 9 . 0 % for the location of the maximum local Nusselt number. The second largest difference is 4.8% for tima x.

Table 2. Emissivity and temperature of the furnace wall

Emissivity Temperature (°C)

Upper insulator 0.18 Equation (52) Heater

Top 0.4 1016 Center 0.4 1003 Bottom I 0.4 1054 Bottom 2 0.4 1038

Brick 0.9 970 Heat seal 0.2 850 Lower insulator 0.18 Equation (52)

(a)

(b)

~O00~m

SO0

0

196 YASUYOSH1 HORIBATA

[ OOm~

0

- l O 0

-100 0 100,..,

- 1 0 0 0 11 O0 ,.~ Fig. 6. Grid system for the vertical pipe furnace. (a) Vertical cross section. (b) Horizontal cross section.

Many numerical solutions have been reported [15-17]. They show that the preceding result is in relatively good agreement wtih the benchmark solution.

9. T H E R M A L C O N V E C T I O N IN A V E R T I C A L PIPE F U R N A C E

Figure 5 is a schematic diagram of a vertical pipe furnace used for the heat treatment of semiconductor wafers. It consists of a heater, an insulator, a brick, a heat-seal and a cover. It contains outer and inner tubes made of quartz glass. The inner tube contains tools made of quartz glass; they sustain wafers. The pressure inside the furnace is almost equal to the atmospheric pressure.

Consider an air flow in the empty furnace. Assume that the coefficient of viscosity and thermal conductivity are constant.


• ] l O , O c m / s )

Co) [b)

197

I000~ i000~

600 500

I I ., ,,---:.:.=;...[ ; ,,. ::.:..'i=..--:-

-tOO 0 lOOm~ -100 0 100mm Fig. 7. Fields on the vertical cross section. (a) Velocity vectors. (b) Isotherms. The contours range from

835 to 1054°C with a contour interval of I°C.

At the top and bottom boundaries, the temperature is subject to the mixed boundary condition

--K d T = K(To~-- Tbdy ) - -q . an ~y

(52)

where t3T/an is the normal derivative of T; K is the heat transfer coefficient; T~ is the temperature at the outer surface of the wall; and qr is the radiative flux density on the surface. Calculate the flux q, from radiation exchange among the furnace surfaces [18].

The wall consists of the heater, brick and heat-seal parts. The heater part is divided into four zones: top, center, bottom 1 and bottom 2. The temperature of the wall is determined from


tO0~

- tO0

z= 96.0~

(e) (~ [ • 0 cm/s)

tOOm~

0

-100

-I00 0 ]O0~m

(b) z= 96 .0m~

- t O 0 0 100~m

Fig. 8. Fields on the horizontal cross section at z = 96 mm. (a) Velocity vectors. (b) Isotherms. The contours range from 850 to 979°C with a contour interval of I°C.

experimental data. See Table 2 for the values of the temperature and emissivity of the wall. Values of K = 2 W/m2°C, To = 200°C and K = 12 W/m2°C, To = 100°C are used at the top and bottom boundaries, respectively.

Figure 6 shows the grid system. The grid system is composed of 75 points along the axis, 20 points in the azimuthal direction, 18 points in the radial direction.

An axisymmetric steady flow is obtained. The maximum value of the local Mach number is ~3 .6 x 10 -4. From M ~ 3.6 x 10-% O ,-~ 220°C, Tr ~ 1000°C and h ~ 1 m, one has

M2~10_7 , qh~2x 10-% ~9,,~0.2. c 2 Tr

Hence conditions (11) and (14) are fulfilled. Figure 7 shows the velocity vectors and isotherms on the vertical cross section at the axis. Figures

8-10 show the velocity vectors and isotherms on the horizontal cross sections at z = 96, 340 and 1050 mm, respectively.

Z= 340.Omm

(a) (-- 1 • Otto/s)

lO0~ ' ' ~ "

-tO0 ,.~

t OOm,~

- iO0

(b) Z= 340.Omm

- 1 O 0 0 1 0 0 , ~ - 1 O 0 0 I O0,~m

Fig. 9. Fields on the horizontal cross section at z - -340 ram. (a) Velocity vectors. (b) Isotherms. T h e

contours range from 1001 to 1020°C with a contour interval of I°C.


Z= 1050.0mm

199

((l) ( - - 1 • O t t o / S ) (b) Z= 1050.0mm

_,oo _ ,oo

-100 0 1000~ -100 0 100m~.

Fig. 10. Fields on the horizontal cross section at z = 1050 ram. (a) Velocity vectors. (b) Isotherms. The contours range from 991 to 1015°C with a contour interval of I°C.

Air rises along the wall due to heating, and descends along the axis after hitting the top boundary. The maximum downdraft velocity is 26 cm/s. Then, the downdraft collides with the almost stagnant air in the low-temperature region near the brick and heat-seal parts, and is bent completely before it reaches z = 100 ram. The temperature varies rapidly here. In the heat-seal part, air falls along the wall, and forms a vortex. Part of the air rises along the axis after it reaches the bottom. Both flows have much lower velocity than the downdraft.

The updraft caused by the heating detaches itself from the wall after it passes the zone of bo t tom l, and comes into contact with the downdraft at the axis. Here a vortex develops. The updraft attaches itself to the wall in the higher-temperature zones of bo t tom 1, bot tom 2 and the top; whereas, it detaches itself from the wall, and a weak downdraft develops near the wall in the lower-temperature zone of the center.

10. C O N C L U S I O N S

Numerical simulation of a low-Mach-number flow with a large temperature variation is described, and simulation results are presented. The compressible equations are modified by scale analysis. The modified equations contain internal gravity waves, while they exclude acoustic waves. The scale analysis shows the conditions under which the modified equations are applicable. They require fewer conditions than the Boussinesq equations. The simulation result of thermal convection of a Boussinesq fluid in a square cavity is in good agreement with a benchmark solution. A simulation result is presented for thermal convection with a large temperature variation in a vertical pipe furnace used for the heat treatment of semiconductor wafers.

Acknowledgements--The author wishes to thank Dr H. Takeda of Recruit Co. Ltd and Professor K. Kuwahara of the Institute of Space and Astronautical Science for their valuable advice.

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numerical simulation of a low-mach-number flow with a large temperature variation

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