SPEC CPU2000 Benchmark Description File
Benchmark Name: 178.galgel
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.
Reference:
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.