[Rime]

Hi!
How to solve unsteady N-S equations without using projection method (solve the coupled equation)?

Thank you

[mprinkey]

You can use fully coupled methods that linearize the mass and momentum components together and put those entries into a single sparse matrix and solve that repeatedly. Construction of particularly the mass conservation and pressure terms needs to be handled carefully--either staggered mesh or Rhie-Chow. That linearized system may need to be iterated a few times to converge. This leads to a very large sparse badly conditioned linear system and it generally takes a lot more work to solve than the individual fields in segregated methods. This is because the pressure equation for mass conservation is VERY hard to solve while the momentum equations are fairly easy--so the full system ends up stagnating because of the twitchy pressure behavior. It can be very useful for some problems with complicated physics, but for single-phase incompressible flow, this is generally NOT recommended especially for transient flows.

You can also try to use Newton-Krylov methods with some underlying preconditioner (say an NS projection method!) to converge to the full-nonlinear system solution at each timestep. This may be recommended for some difficult problems or for very large timesteps, but again, probably not for transient incompressible NS.

Projection methods are popular because they are fast and accurate enough, especially if the timestep must be small for accuracy reasons (as in LES/DNS).. And by "projection methods" here, I mean general fractional-step non-iterative methods. Note that all SIMPLE/PISO-type schemes for incompressible flow involve a projection step of sorts that solves for a pressure(-like) variable and uses it to make the mass flux (aka velocity) field conservative. The projection process of rendering the mass flux field conservative iteration-by-iteration or timestep-by-timestep is a central portion of almost every primative-variable incompressible CFD solver that I am aware of.

[Jonas Holdeman]

If you want to try something really different that doesn't involve projection-based mixed methods, look at several FEM codes on the cfd wiki using a "pressureless" method.

cfd-online.com/Wiki/Source_code_archive_-_educational

[FMDenaro]

If you want to have some idea of the general coupled problem that can be then decomposed in fractional steps, I suggest


https://www.researchgate.net/profile...3fd6000000.pdf

http://faculty.nps.edu/fxgirald/Home...t_JCP_2002.pdf

[Rime]

thank you for the interesting answers and documents.

I have a question, what's the problem that i can have if I use for spatial discretization "collocation in the same node" (velocity and pressure)?

[mprinkey]

The problem is generally called "pressure-velocity coupling" or "even-odd decoupling." This is a problem where the pressure and velocity fields can decouple on collated grids and exhibit cell-to-cell oscillations. On 2D Cartesian grids, this results in a checkerboard-type oscillations. It occurs because the central differencing approximation of the pressure gradient is approximated as (p[i+1] - p[i-1])/ (2*dx). The p[i] value does not enter the computation and so the odd-values of i and the even values i can (and do) drift apart, often catastrophically.

The solution to this is either staggered mesh, whereby the pressure terms across a momentum cell can be computed as (p[east] - p[west])/dx where east and west of the pressures on the faces of the momentum cell. Or, by using Rhie-Chow to interpolate the discrete momentum relation (not including the pressure term) from the two neighboring cells to their shared face and evaluating the face velocity/massflux using that momentum interpolation and the pressure gradient term as (p[1] - p[0])/dn where 0 and 1 are the cells on the positive and negative side of the face and dn is the distance between the centroids of those two cells.