Contemporary Engineering Sciences, Vol. 9, 2016, no. 2, 47 - 70
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ces.2016.511306
Geometric Multigrid Method for Steady
Buoyancy Convection in Vertical Cylinders
Fedir Pletnyov and Ayodeji A. Jeje
Department of Chemical and Petroleum Engineering, Schulich School of
Engineering, University of Calgary, Calgary, Alberta, T2N 1N4 Canada
Copyright © 2015 Fedir Pletnyov and Ayodeji A. Jeje. This article is distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
Abstract
An implementation of the Geometric Multi-Grid (GMG) method, with full
approximation scheme/storage (FAS) algorithm, in a numerical study of steady
Buoyancy-driven convection with axis-symmetric flows in vertical cylinders is
presented. The particular system examined is a cylinder with an aspect ratio a
(radius divided by height) of 4. The fluid is heated from below and cooled from
above, and the circular wall of the vessel is insulated. The Rayleigh (Ra) and fluid
Prandtl (Pr) numbers are respectively 56.4 10 and 7. A non-linear system of
equations was formulated in stream function-vorticity-temperature variables and
discretized using a monotonic conservative finite difference scheme of second order
accuracy. The steady state condition was solved for purposes of comparing two
numerical methods: the GMG FAS and the Gauss- Seidel method with
lexicographic ordering (GS-LEX). The GMG FAS method has significantly higher
efficiency in CPU performance compared to the pure GS-LEX method for fine grids
only if the tolerance value for stopping iteration process is chosen not too small. A
procedure for the selection of adjustment parameters for GMG FAS algorithm is
proposed and tested for different grid sizes. Details regarding convergence criteria
are addressed.
Keywords: multigrid method, natural convection, optimization parameters,
convergence criteria, vertical cylinder
Nomenclature
aspect ratio, thermal diffusivity, k/ρCp a /R H
48 Fedir Pletnyov and Ayodeji A. Jeje
acceleration of gravity relative accuracy
Grashof number dependent variable in the
general convection-
diffusion equation ( or )
mesh step in radial direction surface of the wall
mesh step in axial direction part of the boundary
( ) where heat enters
the fluid
height of cylinder part of the boundary
( ) where heat exits
the fluid
radial component of convection-
diffusion operator dimensionless temperature
axial component of convection-
diffusion operator
vorticity
rN number of radial mesh points stream function
zN number of axial mesh points
Nusselt number Subscripts
heat input from the bottom wall cold
heat output through the top wall hot
Prandtl number (= ) mesh point in direction
radial coordinate j mesh point in direction
radius of cylinder iteration number
Rayleigh number (=
)
radial component of vector
Rayleigh number by Leong [7]
(= )
axial component of vector
temperature vorticity transport equation
ΔT temperature difference across bottom
and top surfaces of domain energy transport equation
g
Gr
rh
zh 0Nu
H 0Nu
rL
zL
Nu
Nu c
Nu h
Pr / i r
r z
R n
Ra
4 / , PrT
g R Ra GrH
r
LeongRa3 /g TH
z
T
Geometric multigrid method for steady Buoyancy convection 49
k thermal conductivity of fluid ν kinematic viscosity, μ/ρ
Cp heat capacity of fluid μ viscosity of fluid
dimensionless radial component of
velocity
ρ density of fluid
dimensionless axial component of
velocity
Superscripts
axial coordinate + , - any
variable
coefficient of volumetric expansion _
1. Introduction
Natural convection of fluids within confined spaces routinely occurs in
many engineering systems such as storage tanks for fuels, water and industrial
chemicals. Vertical cylinders are most commonly used for the storage, as reactors
for chemical transformations, and as vessels for precipitation and mixing processes.
Descriptions of thermal convection inside vessels of different geometries and a
comprehensive review of numerical methods for modelling the circulation has been
presented by Lappa [6]. Implementation of the numerical schemes is of current
interest and steady state solutions for the circulation are important for exploring
issues of convergence, stability, uniqueness and accuracy. Steady state solutions are
also the starting point for linear stability analysis. The choice of a schemes for
simulating steady Buoyancy-driven convection is important and the criteria are
primarily the efficiency of the computer processing unit (CPU) and the accuracy of
the solution. Kuzmin [5] has recently undertaken a comprehensive review of
various numerical methods for the equations of transport phenomena. The relative
efficiency and accuracy of this numerical methods are often in conflict, a trade-off
is required. Leong [7] solved the transport equations for steady Rayleigh-Bernard
convection in cylinders by the central difference scheme in three dimensions (3D).
In his approach, unsteady equations for vorticity- vector potential-temperature were
solved until steady state was attained. Neither computational efficiency nor the
effect of variations in grid sizes were explored. The alternating-direction implicit
(ADI) scheme was used along with Fast Fourier Transform (FFT). Preliminary
calculations indicate that methods involving transitioning from unsteady to steady
states are inefficient. The ADI+FFT method with transitioning required much more
CPU time compared to the non-transitioning Gauss-Seidel (GS) method. Liang et
al [8] had earlier successfully applied the GS method for natural convection in cylindrical geometries. The method is efficient for coarse grids. It is slow to converge
r
z
z 1( )
2x x x x
1( )
2x x x
50 Fedir Pletnyov and Ayodeji A. Jeje
for fine grid meshes and may yield inaccurate results if the convergence factor is
relatively large. An alternate method is the multigrid (MG) approach (Trottenberg
et al. [13], Wesseling [17], Wesseling and Oosterlee [18]) that yields accurate
solutions but it may be slow when small residuals determine convergence. More
commonly applied for fluid dynamic problems is the geometric rather than the
algebraic multigrid method that Muratova and Andreeva [9] have examined.
Competing numerical methods include the Jacobian-free Newton-Krylov scheme
that Wang et al. [15, 16] used with primitive variables to examine linear stability of
natural convection for axially and laterally heated cylinders.
. The geometric multigrid method (GMG) is applied for this study and the
motivation is to optimize the techniques for the non-linear equations that describe
Buoyancy-driven motion in cylindrical coordinates, with the convergence criteria
based on residuals rather than on relative errors for consecutive iterations. Results
are compared with those from using the GS algorithm. The dependent variables are
vorticity, stream function and temperature, and the equations are discretized with a
monotonic conservative finite difference scheme of second order.
2. The multigrid method
Numerical methods involve transforming differential equations, valid for
continuous domains, into algebraic equations for discrete points or elements of
specified networks within a space bounded by surfaces. Algebraic equations
obtained by a finite difference scheme, for example, may be solved by the Jacobi,
Gauss-Seidel or other iterative algorithms. When problems involve spatially
complex regions and a large number of algebraic equations are to be solved,
convergence may be slow or not readily achievable. Typically, solution by
algorithms such as the Gauss-Seidel solver involve considering errors in the
dependent variables and in the residuals for the calculations. In sequential iterations,
errors and residuals associated with large eigenvalues or high frequency are reduced
quickly but the errors at lower frequency slowly or hardly decrease, thus
engendering long computational times.
The multigrid method accelerates the rates of convergence of solutions by
sequentially manipulating the residuals from iterations in two or more grid meshes
of different resolutions. In the simplest application, a differential equation is
discretized for two networks, one fine and the other coarse, for the same domain.
The solution is started from one of the networks, say the one with the fine grid.
After a few iterations, the residual of the equation is projected onto the coarse grid
network with a restriction operator. The residual equation is solved on the coarse
grid for the correction term to the value of the variable in every single point of the
domain. This correction term is interpolated into the finer grid with the prolongation
operator, and is added to the solution earlier obtained for the fine mesh (V-cycle).
The coarse grid is effectively a temporary, computational adjunct [10]. Low
frequency components of errors are quickly removed. Post-smoothing iterations are
usually performed for each of the calculation cycles. The process can be applied on
a recursive basis for coarser grids to accelerate convergence.
Geometric multigrid method for steady Buoyancy convection 51
The algorithm can be made more efficient by discretizing the governing
equations for the problem on as coarse a mesh as possible. The solution obtained is
used as the initial guess for the next finer mesh in a series, with values for the
unknown intermediate variables interpolated from results for the coarse mesh. The
governing equations are again re-discretized for the new mesh. This is the scheme
for the full multigrid algorithm (FMG) for linear problems and the full
approximation scheme/storage (FAS) for non-linear problems [10]. The approach
used for this study is the geometric multigrid (GMG) that has been well described
and contrasted with the Algebraic Multigrid (AMG) method by Trottenberg and
Oosterlee [13], Chang et al. [3], Shakira [11], St�̈�ben [12] and others.
The framework of the algorithms used are “Mglin” and “Mgfas” routines
[10] for the full multigrid method. They are also known respectively as the nested
iteration method and the full approximation scheme/storage method (FAS). Both
FMG and FAS algorithms were extended in this study from single elliptical partial
differential equations (PDE) to a group of three equations that describe convective
currents in a confined space. Of the three equations, two are non-linear, the vorticity
and energy transport equations, and the third is a linear elliptic equation for stream
function from the definition of vorticity. For smoothing, the Gauss-Seidel iteration
method with a lexicographic ordering of the grid points (GS-LEX) was applied. The
Gauss-Seidel method with red-black ordering was not found to be suitable for
second order boundary conditions, where values for points near boundaries are
estimated before derivatives are calculated. The efficiency of the multigrid method
depends significantly on how its parameters are adjusted - 1) the number of
GS_LEX relaxation iterations (sweeps) ( 1 ) before coarse-grid correction is
computed (pre-smoothing); 2) the number of GS_LEX relaxation iterations
(sweeps) ( 2 ) after coarse-grid correction is computed (post-smoothing; and 3) the
number of V-cycles ( ncycle ) that is used in each grid level. Preliminary estimates
indicated that the FMG algorithm, for any selection of parameters (𝜈1, 𝜈2, ncycle),
does not improve performance time compared to the pure GS-LEX method. A high
number of pre- and post-smoothing iterations are required for convergence of the
non-linear system that the FMG algorithm is inefficient. For the same system, the
GMG-FAS algorithm has lower CPU time than the GS-LEX method when an
optimal set of parameters (𝜈1, 𝜈2, ncycle) is used.
The procedure for calculating the multigrid parameters are demonstrated for
the specific case of thermal convection inside a vertical cylinder with an aspect
ratio a, ratio of container radius to its height, of 4. The fluid is heated from below,
cooled from above and the circular vertical wall is insulated. Results for the axis-
symmetric, 2-dimensional case are similar to those reported by Leong [7] for his
problem analyzed in 3-dimensions. Conditions common for both studies are
aspect ratio a = 4 and 56.4 10Ra . A Rayleigh number of 2,500, as defined by
Leong, has been multiplied by 4a to be consistent with the definition in this study.
The fluid Prandtl number (Pr) is chosen as 7. Conclusions arrived at from the
specific case have been extended to the general case for cylinders with arbitrary
aspect ratios elsewhere.
52 Fedir Pletnyov and Ayodeji A. Jeje
3. The governing equations
Equations for axisymmetric heat and momentum transfer, by natural
convection in a fluid-filled vertical container that is heated from below and cooled
from above, are written in cylindrical coordinates, in dimensionless and
conservative form as 2
1
0L u fx
, where is the dependent variable, x1
and x2 are spatial coordinates, L1 and L2 are known convection-diffusion
differential operators and is a source function. The Boussinesq approximation
is applied to the equations in primitive variables (Turner [14]), and the equations
transformed into the stream function-vorticity-temperature equations:
1 10, (1)
1 10, (2)
Pr Pr
1 1. (3)
r z
r z
r rr r Gr
r r r z r z r
r rr r z z
r r r z r z
The variables , are the radial and axial components of the velocity vector,
defined as:
, where is the stream function, and
with being the vorticity.
The following dimensional quantities were used as scaling factors:
the internal radius of cylinder, , for length; for velocity, where is the
kinematic viscosity of the fluid and dimensionless temperature , where
subscripts h and c refer to conditions at the lower (hot) and upper (cold) surfaces
of the pool respectively. The product of the dimensionless quantities in the
equations, Grashof number (Gr) and Prandtl number (Pr) is Rayleigh number (Ra).
The spatial convection-diffusion differential operators in equations (1) and
(2) are generalized as
, (4)
u
f
r z
1 1,r z
r z r r
r z
z r
R / R
c
h c
T T
T T
1, ,L a b f r z
Geometric multigrid method for steady Buoyancy convection 53
For the vorticity transport equation (1), , 1, 1,r a b , f Grr
;
and, for the energy equation (2), , Pr, ,a r b , 0f .
4. The boundary conditions
The boundary conditions are specified, following Berkovskii and Nogotov [1],
as:
a) the stream function :
on the bounding solid surfaces
along the axis of the cylinder where, because of symmetry, both
and . The last term is equivalent to
. (5)
b) vorticity :
along the axis, and (6)
is defined by steam function values at internal points of the domain.
The condition for vorticity, , at the bounding surface of the domain can be
defined by a second order approximation using values of stream function and
vorticity in a single node nearest to the wall. This approach was first derived by
Woods [19] for rectangular coordinate system. The Woods’s formula should be
corrected for the cylindrical coordinate system in terms of a uniform mesh, which
is superimposed on the solution domain consisting of r zN N discrete points in
and directions respectively, as:
0, 0
00, 1, 1, 0 02
1, 2, 2,2
,0 ,1 ,12
, 1 , 2 , 22
0, 0,
3, 0,
2
3 11 , 1,
2
3 1, 0,
2
3 1. .
2
r r r
z z z
j
j j j r
r
N k N k N k r
r
i i i
i z
i N i N i N
i z
at r r
rr h at r r
h
h at rh
at zrh
at z Hrh
, (7)
where 0,..., 1zj N , 0,..., 1ri N ; rh , zh and rN , zN are the mesh sizes and
the number of the mesh points in r and z directions respectively.
0
00
r
00r r
0
0z
rr
0
10
rr r r
00
r
r
z
54 Fedir Pletnyov and Ayodeji A. Jeje
c) temperature :
1 0,
0 1,
0 0 1.
at z
at z
at r and rr
(8)
The initial guesses for the variables are:
0ij ;
0ij ; for temperature linear temperature profile between bottom and top
of the cylinder has been used:
1 / ( 1) ;ij zj N , /i j , where / refers to
the internal mesh points of the domain. Values were also chosen for the Grashof
number, Gr (= / PrRa ), in Eq. (1) and the aspect ratio a defined the domain
5. The numerical scheme
In designing finite difference schemes that satisfy the maximum principle
for any mesh size , first order derivatives are represented using the
asymmetrical difference expressions. The schemes take into account the sign of the
coefficients preceding these derivatives. The term is approximated with
backwards difference formula if the coefficient is positive, and a forward
difference formula if the coefficient is negative, as suggested by Courant,
Isaacson and Rees [4]. This is the upwind scheme.
The finite difference operators, approximating convection-diffusion
operators (eq. 4) in monotonic conservative form with second order approximation
, are shown below:
1, j , j
1, j
2
1, j
2
, j 1, j
1, j
2
1, j
2
1
12
1
12
i i
ri
rrr
i
i i
irr
ri
L bhh
h b
bhh
h b
(9)
r zh and h
i
i
fb
x
ib
ib
2( )O h max( , )r zh h h
Geometric multigrid method for steady Buoyancy convection 55
, j 1 , j
1, j
2
1, j
2
, j , j 1
1, j
2
1, j
2
1
12
1
12
i i
zi
zzz
i
i i
izz
zi
L bhh
h b
bhh
h b
(10)
where is coefficient in equation (4).
1 , j 1 , j 1 1, j 1 1, j 1, j
12
2
1 1, j 1, j 1, j 1 1, j 1, j
2
1,
4
1,
4
, 1,..., , 1,..., .2
r i i i ii
zi
z i i i ii
r i
z z
z r z
h r
h r
i N j N
6. Estimation of Nusselt number
Nusselt number (Nu) characterizes the intensity of heat transfer between a
fluid and a bounding surface. Berkovskii and Polevikov [2] defined it as:
, (11)
where θ is dimensionless temperature and variable is the dimensionless normal
to the surface of the wall at .
Heat enters the fluid through a segment of the boundary at ( ) and exits
through a different part of the boundary at ( ). Here describes the
distribution of the dimensionless heat transfer across the entire bounding surface.
The net heat transfer from the wall to the fluid, and from the fluid to the wall
are evaluated from the integrals:
,
2
b bb
b
Nun
n
0Nu 0Nu Nu
56 Fedir Pletnyov and Ayodeji A. Jeje
,Nu Nud Nu Nud
For the steady rate of energy transfer through a control volume, | |Nu Nu .
Since parts of the boundary and may be unknown in advance, it is
convenient, according to Berkovskii and Polevikov [2], to estimate
from the formulas:
1,
2
1.
2
Nu Nu Nu d
Nu Nu Nu d
(12)
7. Criteria for convergence
a) Relative errors for the dependent variables, from consecutive iterations, are
specified as:
max( , , ) , (13)
where ( ) ( ) 1
, j , j , j , j
, j
, j
1max ; ,
; , , ;
s s
i i i is
s
is
i r zN N
r zN N is the total number of the points in grid, n is the iteration number; is
tolerance for relative error, 3 810 10 .
a) Maximum value of the residual (defect) in the equations (1)-(3) from all
points of the domain.
( )
max, , j 1max , , ,id d
. (14)
Here 1 is tolerance for maximum value of the defect 3
1 0.1 10 .
b) Average value of the residual in the equations (1)-(3) from all points of the
domain.
c)
, j
, 2
, j
; , , ;i
aver
i r z
dd
N N
(15)
,Nu Nu
Geometric multigrid method for steady Buoyancy convection 57
Calculations show that values for maximum and average residuals for
vorticity transport equation (1) are several orders greater than for residuals in
equations (2) and (3). Thus, it is more important to keep track of the largest values
of the residuals max,d , ,averd in vorticity transport equation (1) for criteria (14) and
(15).
d) Convergence and balance of the heat transfer (Nusselt number) from the
bottom to the top of the cylinder
1
3
1
n n
n
Nu Nu
Nu
. (16)
8. Results and discussion
Results of steady convective patterns from the numerical simulation are
presented in the following with a review of the criteria for convergence and
selection of grid sizes.
8.1. Comparison of GS and GMG methods
The accuracy of the results from solving equations by iterative method such
as the Gauss-Seidel’s is tightly connected to the stopping criterion for the iteration
process. The convergence criteria for this study are equations (13) – (16). The
solution to the discretized form of the PDEs is correct if the values for a dependent
variable at a fixed point within the domain are the same as the mesh size tends to
zero ( 0h ). The relative error criterion, equation (13), with the GS method often
leads to solutions that are not the same for different mesh sizes when the tolerance
is fixed.
The number of steady ring rolls when a relative error of 310 is specified is
shown in the contour map of streamlines in Figure 1 for different mesh sizes (56.4 10Ra ; a= 4). Only the grids 31x17 and 65x65 exhibited three rolls. If the
relative error is reduced to 4 510 10 , all the networks with finer mesh sizes
show three rolls only, thus indicating that this is the correct results. Figure 1 also
includes contours of the residuals for vorticity. In Table 1, the highest values for
each of the residuals d , d and d for equations (1) – (3) are presented for
different grid sizes. These high residual values are at the boundary between adjacent
rings
58 Fedir Pletnyov and Ayodeji A. Jeje
rotating in opposite directions. Corresponding heat transfer rates entering at
the bottom and exiting at the top of the cylinder are also given in the Table.
Maximum residuals for all the three equations increased with refinement of the
mesh network. An important observation is that the tolerance that produces correct
solutions is not known a priori.
Table 1. Effect of mesh sizes on maximum values for residuals d , d and d ,
and the ultimate number of rings with the relative error 310 , Ra=
( =2500), Pr=7, a=4.
№ Mesh
Size max
d
max
d
max
d
| | Remark
1 31x17 0.21 1.0 1355 29.7 29.8 3 rolls
2 65x65 2.4 12.4 18700 30.8 30.7 3 rolls
3 129x129 15.8 44.4 76860 33.2 33.1 5 rolls
4 169x169 47.0 135.0 106500 32.9 34.4 5 rolls
Application of the relative error criterion (Eq. 13) requires additional
analysis and it is not reliable. Values for the unknowns can be close in successive
iterations and convergence is indicated but the solution may be incorrect. It is
essential to continue to decrease the tolerance 3 4 510 ,10 ,10 ,... for a given
mesh size and only to stop calculation when solutions cease changing.
The criterion for which maximum value of residuals is prescribed (Eq. 14)
is more reliable but it requires a large number of iterations. In the calculations,
stream function contours and temperature isotherms already may be established and
nearly invariant for consecutive iterations, and the heat transfer rates as estimated
with the Nusselt number (Nu) constant, but the iteration continues until the residuals
are reduced below the imposed limit. Further iterations most significantly reduce
residuals at the boundary between rolls than elsewhere.
Instead of requiring that residuals at all points of a grid fall below a limiting
value as the condition for convergence, a criterion that the average residual for all
Nu
Nu
56.4 10
LeongRa
Nu Nu
Geometric multigrid method for steady Buoyancy convection 59
the grid points of the domain is prescribed (Eq. 15) appears to be more efficient for
stopping iteration. This is less stiff than the criterion that each point must have a
residual less than a value. To save time and effort, even with using an average value
for the residuals in a domain, care is required in selecting the limit. For example,
the same vorticity and temperature contours, as shown in Figure 2, are obtained for
both values of tolerance 2 1 and 2 0.01 , yet setting 2 0.01 consumes much
more CPU time without improving accuracy. The results in Table 2 are for a fine
mesh with 𝜀2 = 0.01. Corresponding values for criteria (13 to 16) at termination are
shown. Specification of average residuals appear to be most efficient as will be
further discussed later.
Table 2. Values of relative error, maximum and average residuals when
calculation was stopped. The inlet and outlet Nusselt numbers are also shown for
a grid mesh 513x129.
How fast convergence to a solution is achieved is compared for the three criteria,
using the GS-LEX method for a 275 x 65 mesh, in Figure 3. The spikes in the curves
corresponds to when flow patterns appear to transition between forms. Changes in
flow structures are reflected in Figure 4 at 1000, 2000, 7000, 15000 and 27000
iterations. The patterns (in section) evolved from stationary fluid to four rolls, five
rolls, four rolls, and finally, the stable three rolls. If iteration is terminated before
the last spike in Figure 3, e.g. at iteration number ~ 24000, the flow pattern obtained
is not the ultimate even though the Nusselt number has attained a steady values.
For this case, the three criteria appear to have worked equally well.
Relative
error Max d Aver d Nu
Nu
71.12 10 25.7 0.00998 30.10 30.10
60 Fedir Pletnyov and Ayodeji A. Jeje
Figure 1. Contour plots of stream function and residuals of vorticity transport
equation (1) at different mesh sizes. Relative error tolerance 310 (criterion
(13); Ra= , Pr=7, a=4.
56.4 10
Geometric multigrid method for steady Buoyancy convection 61
Figure 2. Contour plots of the stream function, vorticity, residual for vorticity and
isotherms. The grid is 513x129 and the average residual vorticity2 0.01 .
The effect of grid sizes and the shape of the cells were also examined. In the
classical multigrid approach, with finite difference discretization of two-
dimensional problems, the cell shape is square. This shape allows implementation
of standard restriction and interpolation operators [10, 13]. Since a radial section
through a cylinder with a radius-to-height ratio a equal 4, as considered in this study,
is not square, the grid has more subdivisions in the radial than in the axial directions
to obtain square cells. The coarsest mesh has 17 x 5 points or 16 x 4 cells. Other
grid sizes applied are 33x9, 65x17, 129x33, 257x65 and 513x129.
The results shown in Figure 2 are the same for the GS-LEX and FAS multigrid
methods for a 513x129 mesh size and for the average residual convergence criterion
0.01 . Effects of variations in grid size, number of iterations and the
convergence criteria are shown in Table 3 using pure GS-LEX smoother. For each
mesh, the average residuals 𝜀2 are maintained at 0.01 and 1, and the number of
iterations and convergence criteria ε and 𝜀1 determined. In these cases, 𝜀1 and 𝜀2
are residuals for vorticity.
The number of iterations increased, the relative errors required for convergence
decreased and the tolerance for local maximum residual increased as the mesh
became finer.
62 Fedir Pletnyov and Ayodeji A. Jeje
Figure 3. The average and maximum residuals for vorticity, relative errors and the
heat transfer rates in relation to the number of iterations using the GS-LEX
method. Grid size is 257x65.
Values of the rates of heat transfer flows into the domain at the bottom ( Nu ) and
out at the top | |Nu ) are presented in Table 4 with respect to grid sizes. Two
considerations are important – closure of the heat balance at steady state and
convergence of the solution to a finite value. Closure is more readily satisfied (as
Nu is always approximately equal | |Nu) than convergence is achieved
especially for coarse grids.
Geometric multigrid method for steady Buoyancy convection 63
Figure 4. Contour plot of stream function and temperature isotherms at iteration
numbers (1000, 2000, 7000, 15000 and 27000) between spikes of error or residual
(max and average) values in Figure 3. Grid size is 257x65.
Table 3. Values of the relative error, maximum residual for vorticity at two values
of average residual tolerance 2 1 and 0.1.
64 Fedir Pletnyov and Ayodeji A. Jeje
Table 4. Heat transfer rates at the bottom (Nu+) and to the top (|Nu-|) of the
cylinder with respect to grid sizes.
8.2. Multigrid-FAS algorithm optimization
For the Multigrid method to be preferred over the GS method, optimal
adjustment parameters are to be found (to achieve minimum CPU time). These are
the number of V-cycles and number of pre- and post-smoothing iterations. The
procedure for selection is demonstrated for a grid with 65x17 points. The tolerance
for average residual of vorticity was specified as 2 0.01 . The steps are:
1. Select a small number for pre/post smoothing iterations: iter PRE/POST = 2.
2. Determine the number of V-cycles and the CPU time that give average residual
values nearest 2 0.01 .
3. Increase number of pre/post smoothing iterations and keep track for the required
CPU time. The number of V-cycles and CPU time would decrease.
4. Terminate the iteration when number of V-cycles ncycle = 2.
Results of calculations for the foregoing procedure are presented in Table 5
and Figure 5 (a, b). The GMG FAS algorithm parameters are V-cycles ncycle
equals 2 and pre/post smoothing iterations equal 1350. The GS-LEX method
required 7802 iterations.
N Grid size Nu | |Nu
1 17x5 23.9 23.6
2 33x9 29.7 29.4
3 65x17 30.4 30.0
4 129x33 30.3 30.0
5 257x65 30.2 30.0
6 513x129 30.1 30.1
Geometric multigrid method for steady Buoyancy convection 65
For finer grids (129x33, 257x65 and 513x129), the best number of V-
cycles ncycle is 2 and the number of PRE/POST smoothing iterations increases
with grid size as shown in Table 6 for 2 1 and 2 0.01 . Fewer PRE/POST
smoothing iterations are required for 2 1 compared to when 2 0.01 . The CPU
times and the ratios of CPU times for the GMG FAS and GS-LEX algorithms in
Table 7 illustrate the higher efficiency of the GMG-FAS method. The application
of the larger average residual 𝜀2 = 1, also yielded faster comparative times than for
the residual at 0.01, especially for the finer meshes. The specification of low
average residuals leads to sharp increases in PRE/POST smoothing iterations and
to decreased efficiency for the GMG FAS algorithm.
Table 5. Finding the optimal number of the iterations for PRE/POST smoothing of
the GMG FAS algorithm - grid size 65x17, 2 0.01 .
N Iter PRE/POST
ncycle Resid, max Resid, aver
CPU, sec
1 2 1400 1.13 0.0076 76.134
2 10 280 1.13 0.0076 23.533
3 20 140 1.13 0.0076 16.692
4 30 95 0.96 0.0065 14.857
5 40 70 1.13 0.0076 13.564
6 50 55 1.31 0.0081 12.73
7 60 46 1.28 0.0086 12.322
8 65 42 1.4 0.0094 12.058
9 70 39 1.4 0.0094 11.9
10 75 37 1.22 0.0082 12.034
11 80 34 1.44 0.0097 11.673
12 85 32 1.44 0.0097 11.615
13 90 31 1.16 0.0077 11.779
14 95 29 1.3 0.0087 11.601
15 100 28 1.13 0.0076 11.697
16 105 26 1.4 0.0094 11.344
17 110 25 1.32 0.0088 11.409
18 115 24 1.27 0.0086 11.38
19 120 23 1.28 0.0086 11.299
20 125 22 1.32 0.0088 11.211
21 130 21 1.4 0.00949 11.129
22 140 20 1.13 0.0076 11.308
66 Fedir Pletnyov and Ayodeji A. Jeje
Table 5. (Continued): Finding the optimal number of the iterations for PRE/POST
smoothing of the GMG FAS algorithm - grid size 65x17, 2 0.01 .
N Iter PRE/POST ncycle Resid, max Resid, aver
CPU, sec
23 145 19 1.29 0.0087 11.105
24 155 18 1.16 0.0078 11.222
25 160 17 1.45 0.0097 10.878
26 170 16 1.45 0.0097 10.85
27 195 15 1.4 0.0094 10.782
28 210 13 1.4 0.0094 10.72
29 215 13 1.14 0.0077 10.941
30 230 12 1.27 0.0086 10.805
31 250 11 1.32 0.0088 10.719
32 275 10 1.32 0.0088 10.649
33 305 9 1.34 0.00897 10.618
34 340 8 1.45 0.0097 10.494
35 355 8 0.995 0.0067 10.892
36 390 7 1.4 0.0094 10.548
37 550 5 1.32 0.0088 10.477
38 700 4 1.13 0.0076 10.627
39 1350 2 1.48 0.0099 10.176
Table 6. Optimal number of PRE/POST smoothing iterations for lowest GMG
CPU for different grid sizes and average residual tolerances 2 1 and 0.01.
Mesh Size
Optimal number of iter PRE/POST
Tolerance 2 for average residual
1.0 0.01
65x17 615 1350
129x33 680 3110
257x65 1950 12000
513x129 4500 42000
Geometric multigrid method for steady Buoyancy convection 67
Figure 5. GMG CPU time (a) and number of PRE/POST smoothing iterations (b)
with respect to number of V-cycles. Grid size 65x17, tolerance 2 0.01 for
vorticity.
68 Fedir Pletnyov and Ayodeji A. Jeje
Table 7. Computer performance time (CPU, min) at different grid sizes for the
GS-LEX and GMG-FAS methods.
Mesh
Tolerance 2 for average residual
1.0 0.01
CPU GMG, min
CPU GS, min
Ratio CPU GS:GMG
CPU GMG, min
CPU GS, min
Ratio CPU GS:GMG
65x17 0.13 0.12 0.92 0.17 0.18 1.05
129x33 0.46 1.43 3.13 1.83 2.3 1.26
257x65 4.85 19.39 4.00 28.26 34.65 1.23
513x129 56 261 4.66 980.50 1066.00 1.09
9. Conclusion
The GMG-FAS algorithm has been used to obtain the solution for steady
axisymmetric natural convection inside a vertical cylinder heated from below,
cooled from above and insulated on the side. Three coupled equations for vorticity,
stream function and temperature were involved. The equations were discretized by
the monotonic conservative finite difference scheme of the second order accuracy.
The method has been shown to be more efficient in CPU performance time
compared to pure GS-LEX method for grid sizes 65x17 and finer only if not too
small tolerance values are selected. A procedure of the selection of the adjustment
parameters for the GMG-FAS algorithm has also been proposed and tested for
different grid sizes. Minimum CPU time is achieved for two V-cycles for all grid
sizes and the number of the PRE- and POST smoothing iterations was equal,
increasing as the grid became finer.
Application of average residual over the domain as the condition for
convergence was more efficient than the use of maximum residual for a point
within the domain. The choice of values for the tolerance for average residuals is
also important. The number does not have to be small for the results to be accurate.
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Received: December 6, 2015; Published: January 15, 2016