Hello, I am developing a simple 3D fortran solver in finite difference using Modified SMAC scheme. My domain is 2pi*2pi*1 as taken by S. Chandrasekhar. My equations are having non dimensional parameters as Reynolds Number, Rayleigh Number and Prandtl No.
The code is running well but w- velocity contours(heating at bottom and cold at top) are coming only positive and negative alternatively and i am not getting rolls as expected. Can someone explain it.? Thanks in advance
It sounds like you may be having a problem with the boundary conditions -I would look into how the BCs of the problem are being incorporated into the FD scheme.
The boundary conditions are pretty simple. I am taking periodic Boundaries in X, Y and no penetrating and no slip in z direction(both top and bottom). Temperature BCs are t=1 @ bottom and t=0 at top boundary.
Okay - so in the X and Y direction you are taking the values of the values of the function as given (i.e., Dirichlet conditions) and then in the Z direction you are taking some Neumann condition. The implementation of the Dirichlet conditions should be straightforward - you simply replace the equations involving the boundary nodes by the exact values. For the Neumann condition, there is some approximation of the normal derivative involved - this is where I would look for a mistake first.
Hello Madhup,
The problem with boundary conditions described in your question is quite well known. I will start from the simple scheme in which the derivatives inside of the domain are calculated using the central differences. On the lower and upper boundaries I will apply forward and backward differences respectively. This solution is usually acceptable for most of the problems including convection in the Cartesian and spherical geometry.
If you are looking for more accurate solution I would like to suggest to read
Strict Stability of High-Order Compact Implicit
Finite-Difference Schemes: The Role of
Boundary Conditions for Hyperbolic PDEs, I
Saul S. Abarbanel and Alina E. Chertock
Journal of Computational Physics
160, 42–66 (2000)
Based on my recent experience it looks that the both solutions can be used to solve the boundary value problems.
In the past I was using often the book:
Fundamentals of Computational Fluid Dynamics
by Patrick J. Roache
where all problems of computational fluid dynamics are discussed very clearly and in accessible manner.
All the best in your calculations,
Janusz
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