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SPEC CPU2000 Benchmark Description File

Benchmark Name


Benchmark Author

Alexander Gelfgat

Benchmark Program General Category

Computational Fluid Dynamics

Benchmark Description

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 [1].

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 [1].

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 [2]. 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 increased.

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 [6]. The details on its numerical application may be found in [3].

Input Description

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.

Output Description

The output consists of data relating to

  1. Calculation of steady state flow. Steady states are calculated using Newton iterations.
  2. Solution of eigenvalue problem corresponding to analysis of linear stability of the calculated steady flow.
  3. Repeat steps 1 and 2 until the critical value of a governing parameter (Reynolds number, Grashof number, etc.) is found.
  4. 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
  • parameter Mu
  • parameter Tau
  • Floquet exponent

Programming Language

Fortran 90

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.


  1. 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.
  2. 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.
  3. 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.
  4. 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, 1997.
  5. 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).
  6. 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