Experiments on Convective Flows in Geophysical Fluid Systems 



REVIEW OF NUMERICAL METHODS IN CONVECTIVE FLOWS 



The systems of partial differential equations governing two-dimensional or 

 three-dimensional axisymmetric flows in natural convection divide into two 

 groups: those involving the time-derivative called "forecast" equations, de- 

 scribing the transport of vorticity, temperature, and angular momentum; and a 

 Poisson equation relating the stream function to vorticity. The first group con- 

 tains equations that are of mixed parabolic -hyperbolic type, although, due to the 

 nonlinearities, these terms must be used in a loose sense only. It is possible 

 for the system to be parabolic in one region of the flow and not in another, or at 

 one time during its evolution and not at another. The Poisson equation is, of 

 course, elliptic. An extensive review of numerical methods used in geophysical 

 fluid dynamics up to 1962 was given by Miyakoda (1962), and a more recent sur- 

 vey was made by Lilly (1965). Since then, still more progress has been made in 

 this field, some in adapting existing methods to the Navier- Stokes equations and 

 some in developing new ones. 



We will first discuss the methods used to solve the Poisson equation v^-/' = 

 ^. The most common and easily programmed technique is the optimum over- 

 relaxation [Frankel (1950); Forsythe and Wasow (1960), p. 242; Varga (1962), 

 p. 105; Miyakoda (1962), p. 98j. The scheme for rectangular coordinates x - 

 i* A and z = j • A with centered space differencing is given by 



,n+l ,n r / ,n ,n+l ,n ,n+l ^ /n ^ ,o\ . ^^ 



^ij = ^ij + ^(^i + i.j + V^'i-i.j +0i.j,i+0i,j_i-40ij-^i3 -A^) , (29) 



where the superscript n denotes the iteration cycle, and r is the relaxation 

 coefficient or acceleration parameter. The mesh is swept in the order j = 1, 

 i = 1, 2, ... I; j =2, i = 1, 2, ... I, etc. so that the scheme is explicit. The 

 constant r depends on the mesh size, and its variation with grid size is given 

 by the authors cited above. Experience shows that, to decrease an initial error 

 by a factor of 10^, the matrix must be swept ^3/4 x/l3 times. The total number 

 of arithmetic operations would then be --9 x 3/4 x i/ll. The fastest iterative 

 technique for the solution of Poisson' s equation in a rectangular region is a 

 variant of the Peaceman-Rachford (1953) method, in which there is relaxation 

 alternatively on rows and columns of the mesh and changes in the acceleration 

 parameter from cycle to cycle. These methods are also known as the 

 alternating-direction implicit or simply ADI methods. If the number of itera- 

 tion cycles is chosen as a power of 2, say 2^, then the acceleration parameters 

 may be predetermined in advance and will obtain maximum convergence 

 [Varga (1962), p. 226; Gary (1967)]. The scheme may be written 



rr +9\,/,"+l/2_ ,n+l/2_ ,n+l/2 n n n ^0^^ 



r r -l-^^,/,"■^l jP*'^ ;"+1 - /^ i\ /n+1/2 . ,n+l/2 , ,n+l/2 /Qf\K\ 



(r^+2)^.. - '/'i.j + i - ^i, j-1 - (Tk- 2)0ij + 0j^^ . + ^._j . , (30b) 



where the sequence of parameters r^ is given by 



1/16 



,^ = ^S(17-2k) , k= 1, 2, ...8 ; /3 = m(^) ; S = (^) , (31) 



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