NS Equations and convergence problems
 

Hi all

I am trying to solve Compressible Naiver-Stokes equations for steady flow through a pipe. I am using Finite volume method for solving these equations, since am solving flow through a pipe I have generated a grid in cylindrical coordinates (meaning my grid consists of radial lines and circles in a plane and stacked along the length of the pipe.) .
I am using Fourth order Range kutta Method for time integration and the fluxes are calculated using central difference method. Am using collocated variable arrangement and use surface integrals for finding the convective and viscous fluxes. This calculation is done using Gauss theorem which converts the volume integral to surface integral.

I am trying to solve a simple problem of flow through a pipe with l/d =10 and d = 19 mm. Boundary conditions specified using ghost cells , a layer of cells are added to the physical cells and at inlet I specify the total pressure and total temperature and extrapolate v3 from inside the physical cells also v1=v2=0.0 at exit I specify static pressure and extrapolate temperature and velocity. On wall I am using noslip and no penetrating condition also the pressure gradient on the wall is made zero and is done by copying the pressure from the neighbouring cell to the face corresponding to the wall also the normal temperature is made zero as I am assuming adiabatic wall and the temperature is also copied from the cell center to the wall face.

the fluxes on the face are calculated by taking the Qavg = (Qi + Qi+1)/2, and the gradients of velocity for calculating viscous stresses are first calculated at cell centers and then are averaged on to the face.


My actual problem is am not getting convergence, My code diverges after say 1500 time steps with time step being calculated by CFL =0.2

Since I am calculating fluxes as said above in order to nullify even-odd decoupling I am adding a fourth order dissipation which is calculated using FVM i.e by calculating 2nd , 3rd and 4th order gradients, and for fourth order gradients am using (1/4!) * average_face_area**4 * max_eigen_value on the face. This terms is multiplied to the fourth order derivative as the coefficient. since its a fourth order gradient its value is +ve.

can any body help me by suggesting whether am following correct path. am struggling for getting convergence.

any help in any manner is welcome and that would be of great..

thanks in advance

Dear Ramesh,

From your description of the convective and viscous flux computations, it seems that you are using a simple central differencing for the inviscid part. The simple averaging of the states in fact is essentially a zero artificial dissipation method, so you are relying entirely on the physical dissipation to stabilise your code. For a reasonable Re, this might not be sufficient enough. I presume this is the cause of your divergence. Also, though not of very immediate concern, the gradient computation by averaging the centroidal gradients to the face isn't a nice option either, they can lead to odd--even decoupling which is undesirable. Please take a look at the relevant portions in the book by J. Blazek, which I believe is an excellent book from viewpoint of writing CFD codes.

Hope this helps.

Regards,

Ganesh

Dear Rameshk;
I think your problem is mainly due to grid generation. this grid generation technique causes problems in regions near to pipe centerline. in other words "r" approaches zero at the vicinity of centrline and therefore some fluxes may become infinitively large and this leads in divergence.
try other techniques to control the mesh shape in pipe centerline regions, such as:

- Unstuctured mesh
- Multi block structured mesh (it is called butterfly mesh for pipeline meshing)

regards

Thanks Hamidzoka and Ganesh
 

though I aim using a cylindrical mesh My solver is in cartesian coordinates, so I have triangular prisms as volumes at the axis and quads in rest of the places.

@ Ganesh

I am adding artificial dissipation also in my code but since I have an dissipation term which is of second order I am not adding the second order dissipation term, If this second order term is added to the scheme the order of scheme gets reduced.

my doubt is even if I have physical mechanism of dissipation present in the equations do we need to add second order artificial dissipation terms.