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: stsby@mail.sysu.edu.cn

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

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.

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.

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

. .

1 , , . . , 2006, 35(4), 265-268 (Xi Hengdong, Sun Chao, Xia Keqing.Flow dynamics and heat transport in turbulent thermal convection.Physics, 2006, 35(4): 265-268 (in Chinese))

2 , . Rayleigh--Benard . , 2012, 42(3): 231-251 (Zhou Quan, Xia Ke-qing. Advances and outlook in turbulent Rayleigh--Benard convec-tion. Advances in Mechanics, 2012, 42(3): 231-251 (in Chinese))

3 Zhou Q, Stevens RJAM, Sugiyama K, et al. Prandtl--Blasius tem-perature and velocity boundary-layer profiles in turbulent Rayleigh--Benard convection. J Fluid Mech, 2010, 664: 297-312

4 Sun C, Cheung YH, Xia KQ. Experimental studies of the viscousboundary layer properties in turbulent Rayleigh--Benard convection.J Fluid Mech, 2008, 605: 79-113

5 Zhou Q, Xia KQ. Measured instantaneous viscous boundary layerin turbulent Rayleigh--Benard convection. Phys Rev Lett ,2010, 104:104301

6 Puits RD, Resagk C, Thess A. Thickness of the diffusive sublayer in

4 FFT-- 671

turbulent convection. Phys Rev E, 2010, 81: 0163077 Ahlers G, Grossmann S, Lohse D. Heat transfer and large scale dy-

namics in turbulent Rayleigh--Benard convection. Rev Mod Phys,2009, 81: 503-537

8 Grossmann S, Lohse D. Fluctuations in turbulent Rayleigh--Benardconvection: The role of plumes. Phys Fluids, 2004, 16(12): 4462-4472

9 Shishkina O, Wagner C. Local heat fluxes in turbulent Rayleigh--Benard Convection. Phys Fluids, 2007, 19(8): 085107

10 Stevens RJAM, Verzicco RLohse D. Radial boundary layer struc-ture and Nusselt number in Rayleigh--Benard convection. J FluidMech, 2010, 643: 495-507

11 Kadanoff LP. Turbulent heat flow: Structure and scaling. PhysicsToday, 2001, 54(8): 34-39

12 , , . . : , 2007 (Jiang Chunbo, Zhang Yongliang, Ding Zeping. Compu-tational Fluid Dynamics. Bejing: China Electric Power Press, 2007(in Chinese))

13 ,,. . : , 1999 (Lu Tao, Shi Jimin, Lin Zhenbao.Region Solving MethodNew Simulation Technology for PartialDifferential Equations. Bejing: Science Press, 1999 (in Chinese))

14 Matteo F, Steven GJ. The design and implementation of FFTW3.Proceedings of the IEEE, 2005, 93(2): 216-231

(: )

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: stsby@mail.sysu.edu.cn