an efficient solution for 2d rayleigh benard convection
DESCRIPTION
This paper describes a novel efficient method for solving the continuity, momentum and energy equations describing 2 dimensional Rayleigh Benard convection.TRANSCRIPT
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45 4 Vol. 45No. 42013 7 Chinese Journal of Theoretical and Applied Mechanics July2013
FFT-- 1)
( 510275)
--.. (fast Fourier transform, FFT). FFT. (Ra) (Nu). --
O357.5 A DOI10.6052/0459-1879-12-334
.
[1]. . -- (Rayleigh--Benard convec-tion, RBC) [2].
(Boussinesq) . . [3-6] [7-8]. [9-10]. . . ,.
.
. . . (fast Fourier trans-form, FFT).
1
-. --.. 1. .. ( plumes )
20121126 120130412.1). E-mail: [email protected]
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4 FFT-- 667
1 [11]
Fig. 1 Model of thermal convection[11]
.
L0 = H V0 =
RaPr
H
0 = t0 = 1Ra/Pr
H2
p0 = Ra Pr2
H2.
V = 0Vt
+ (V ) V = p + 1Ra/Pr
2V + j
t+ (V ) = 1
Ra Pr2
(1)
Vp. Ra = gH
3
Pr =
. gH. Pr 1 ( 1) Pr 1 Pr 6.
x = 0 , x = 1 : V = 0 , x
= 0
y = 0 : V = 0 , = 1
y = 1 : V = 0 , = 2
(2)
2 .
.
2.1 [12].
(1)
(1) 1.(2) 2.
.Vn+1 V
t+ pn+1 = 0 (3)
Vn+1 = 0 (4)
Vn+1. (3)
2 pn+1 = 1t V (5)
. pn+1.
(3) 3.(4) 4 .
. Vn+1 n+1.
2.2
. [12]. . 2. u v p t.
2 Fig. 2 Computational grid
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668 2013 45
2.3
.
2 p = d (6)
a1 j pi1, j + b1 j pi, j + c1 j pi+1, j+
a2 j pi, j1 + b2 j pi, j + c2 j pi, j+1 = di, j (7)
(7) (8)
1x2
(pi1, j + pi+1, j
)+ a j pi, j1+(
b j 2x2
)pi, j + c j pi, j+1 = di, j (8)
2.3.1
.(1) i j
pk+1i, j =1
b j 2x2
[di, j 1
x2(pki1, j + pki+1, j)
(a j pki, j1 + c j pki, j+1)]
(9)
(2) i j
pk+1i, j =1
b j 2x2
[di, j 1
x2(pk+1i1, j + pk+1i+1, j)
(a j pk+1i, j1 + c j pk+1i, j+1)]
(10)
.2.3.2 x FFT
x.
[13]
pk, j =
2M
Mi=0
i pi, j cospiikM
pi, j =
2M
Mk=0
k pk, j cospiikM
k =
12, k = 0,M
1, 1 6 k 6 M 1
(11)
2.M+1. k0 6 k 6 M(
b1 4x2
sin2 pik2M
)pk,0 + (a1 + c1) pk,1 = dk,0
a j pk, j1 +(b j 4
x2sin2 pik
2M
)pk, j + c j pk, j+1 = dk, j
(aN + cN) pk,N1 +(bN 4
x2sin2 pik
2M
)pk, j = dk,N
(12)
FFTW [14].
3 Ra = 1.0 109Pr = 4.3.
. 512560 t = 2.0104.
.. FFT.3.1
FFT .. 20 1.
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4 FFT-- 669
1 20Table 1 Comparison between the computation time
of 2105 time steps
MethodLeapfrog
overrelaxationiteration
Direct solutionusing FFT
time/min 418 376
2FFT.3.2
0 . 2 . 2
(1.0106). FFT.
2 Table 2 Comparison between the velocity divergences
MethodLeapfrog
overrelaxationiteration
Directsolution using
FFTmaximum mean
velocitydivergence
3.3104 3.5107 2.51014
3.3
2 . 2.. .
Nu =
RaPr vx xy (13)
3
100 2. 2 .
3 Table 3 Comparison between the Nu of mean fields
MethodLeapfrog
overrelaxationiteration
Directsolution
using FFTNu 51.33 51.49
3.4
[2]. 3 Pr = 4.3
110521010 17. 3 2 . .
3 Fig.3 Nu number versus Ra number
4(a) . . 4(b).. 4(c) 100.
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670 2013 45
4 Fig. 4 Formations and interactions of the plumes in RBC
.
3.5 5
.
FFT FFT . .
5 Fig. 5 Velocity and temperature fields in a RBC cell with partition plates
4
. . FFT FFT. FFT 2.
FFT
. .
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(: )
AN EFFICIENT SOLUTION FOR 2D RAYLEIGH--B ENARD CONVECTION USING FFT
Xu Wei Bao Yun1)(Department of Applied Mechanics and EngineeringSun Yat-sen UniversityGuangzhou 510275China)
Abstract The computation efficiency of direct numerical simulation (DNS) for Rayleigh--Benard (RB) convection inrectangular containers is studied. For unsteady flow of RB convection, the solution of pressure Poissons equation is akey issue affecting the computation efficiency in the whole solving process. Combined with fast Fourier transform (FFT)and chase method, a direct solution of pressure Poissons equation is presented. Comparing the precision and speed withhopscotch overrelaxation iteration, it is found that the direct solution of pressure poisson equation for RB convectionusing FFT is efficient. The results of temperature fields for the plume and the large scale circulation, and the change ofNu number for series Ra numbers are given.
Key words Rayleigh--Benard convection, FFT, direct solution of Poissons equation, DNS
Received 26 November 2012, revised 12 April 2013.1) Bao Yun, professor, research interests: computational fluid dynamics. E-mail: [email protected]