SPEC CPU2000 Benchmark Description File
Benchmark Program General Category
Computational Fluid Dynamics
This problem is a particular case of the GAMM (Gesellschaft fuer Angewandte
Mathematik und Mechanik) benchmark devoted to numerical analysis of
oscillatory instability of convection in low-Prandtl-number fluids .
The physical problem is the following. There is a rectangular box filled by
a liquid whose Prandtl number is Pr=0.015. The aspect ratio of the cavity
length/height is 4. The left and right vertical walls are maintained at at
higher and lower temperatures respectively. This causes a convective motion
in the liquid. When the temperature difference is relatively small the
convective flow is steady. The flow looses its stability and become
oscillatory when the temperature difference exceeds a certain value.
The buoyancy force, which causes the convective flow, is characterized by a
parameter called Grashof number. Besides all, the Grashof number (Gr) is
proportional to the characteristic temperature difference (difference of
the temperatures at the vertical walls in this case).
The task of the GAMM benchmark is to calculate the critical value of the
Grashof number which corresponds to a bifurcation from steady to
oscillatory state of the flow. Together with the critical Gr it is
necessary to calculate the critical frequency (the frequency of the
resulting oscillations when Gr is equal to its critical value).
The critical values (critical Grashof number and critical frequency) depend
on all parameters of the problem and the boundary conditions. The GAMM
benchmark considers fixed values of the Prandtl number and the aspect ratio
(0.015 and 4 respectively), and varies the boundary conditions. The
boundary conditions used here correspond to the Rigid/adiabatic -
Free/adiabatic case defined in .
The numerical method used here is the spectral Galerkin method with the
basis functions defined globally in the whole region of the flow. Detail
description of the method may be found in . Some test calculations
illustrating the advantages of this method may be found in [2,3].
The Galerkin method requires large computer memory required to keep all
coefficients of the resulting dynamic system. To avoid this some
coefficients are recalculated each time when a calculation of rhs of the
dynamic system is necessary, leading to a rapid increase of the required
memory and cpu time when the number of the Galerkin basis functions is
A relatively small number of degrees of freedom makes it possible to study
linear stability of steady solutions, requiring solution of an eigenvalue
problem, which is usually impossible for an arbitrary CFD code. It becomes
possible with the use of the global Galerkin method, and it was
successfully done for convective flows described here [2,5] and for swirling flow in a closed
cylindrical container [3,4].
After linear stability analysis is completed and the bifurcation point is
calculated, we calculate an asymptotic approximation of the supercritical
flow. The asymptotic approach used is described in .
The details on its numerical application may be found in .
A variety of data may be provided via namelist input. However, only the
number of basis functions in horizontal and vertical directions is provided
in the SPEC input, the other data taking default values.
The output consists of data relating to
Calculation of steady state flow. Steady states are calculated using
Solution of eigenvalue problem corresponding to analysis of linear
stability of the calculated steady flow.
Repeat steps 1 and 2 until the critical value of a governing parameter
(Reynolds number, Grashof number, etc.) is found.
Calculate an asymptotic approximation of the oscillatory state of the
flow. Without going into details (see [3,6]), the first term of this
asymptotic expansion is defined by two scalar numbers which are called
here "Mu" and "Tau". Stability of the asymptotic
oscillatory state is defined by the non-zero Floquet exponent, which is
also calculated. Negative Floquet exponent means stability, and the
positive means instability.
After all stages of calculations are completed, the code reports the
following five numbers:
critical Grashof number
critical circular frequency
Known portability issues
Fixed format fortran 90 source format is used in galgel, usually requiring
the use of a compiler flag, such as "-fixed", for example.
Roux B. (ed.) Numerical simulation of oscillatory
convection in low-Pr fluids: A GAMM workshop. Notes on Numerical Fluid
Mechanics, Vieweg, Braunschweig, vol.27, 1990.
Gelfgat A.Yu. and Tanasawa I. Numerical analysis of
oscillatory instability of buoyancy convection with the Galerkin spectral
method. Numerical Heat Transfer, Part A, vol. 25, pp.627-648, 1994.
Gelfgat A.Yu., Bar-Yoseph P.Z. and Solan A.
Stability of confined swirling flow with and without vortex breakdown.
Journal of Fluid Mechanics, vol.311, pp.1-36, 1996.
Gelfgat A.Yu., Bar-Yoseph P.Z. and Solan A. Steady
states and oscillatory instability of swirling flow in a cylinder with
rotating top and bottom. Physics of Fluids, vol.8, pp.2614-2625,
Gelfgat A.Yu., Bar-Yoseph P.Z. and Yarin A. On
oscillatory instability of convective flows at low Prandtl number.
Transactions of ASME, Journal of Fluids Engineering, December volume of
1997 (to appear).
Hassard B.D., Kazarinoff N.D., Wan Y.-H. Theory and
Applications of Hopf bifurcation. Mathematical Society Lecture Notes
Series, vo.41. London, 1981.
Last updated: 12 October 1999