Fractional step in ALE form - possible?
 

hello,

i'm using the fractional step on a structured grid to solve flow past a cylinder. discretization is based on FVM, CN2 for viscous, AB2 for convective term. the non-dimensionalized eqns in integral form are:

f[ (u(*)-u(n))/dt ]dV + f[ 1.5*u(n)*U(n)-0.5*u(n-1))*U(n-1)) ]dS = 0.5*f[ L(u(n)+u(*)).n ]dS /Re

f[ dp/dn ]dS = f[ U(*).n ]dS /dt

u(n+1)=u(*)- dt*(G(p)) (cc)

U(n+1)=U(*)- dt*(dp/dn) (fc)

f[ ] is the integration summation

U is the face normal velocity, u is the cell center

L is the discrete laplace operator, G is the discrete gradient.

cc mean G(p) is evaluated at the cell center

fc mean dp/dn is evaluated at the face center to prevent pressure oscillation.

i intent to modified it to the ALE (Arbitrary Lagrangian-Eulerian) form to simulate oscillating cylinder.

searching the net shows that a lot of ppl use FEM or FVM with artificial compressibility. there are some who use SIMPLE/FVM with ALE. however i can't seem to find ppl using ALE in fractional step. i wonder why? is it just preference or...?

from my understanding, the main modification in the ALE form is the convective velocity, which in this case is the changing of U (face normal velocity) to U - Ub, where Ub is the face normal mesh velocity.

hence, the overall change is only in the momentum and poisson eqn, where all the U will become U - Ub.

also, the change in BC will be wall velocity from 0 to cylinder velocity, that's all.

with these modification, i'll be able to simulate oscillating cylinder.

Re: Fractional step in ALE form - possible?
 

I suppose you'll have to define some grid velocity associated with the intermediate position (*). Also, in the case of a deforming grid, don't forget that dV belongs inside the time derivative term (you may not need this if your cylinder grid is rigid).

Re: Fractional step in ALE form - possible?
 

hmm... i just tried moving the whole grid to simulate the oscillating cylinder but the answer's not correct.

BC for (*) is supposed to be at t=n+1, so the grid velocity is the one at t=n+1. other than that, i don't see any other changes required.

Btw, i found that the lift/drag coefficient computed using tecplot and my own extrapolation subroutine are different, both in value and frequency, although both are wrong anyway. strangely, for sll other cases (stationary,rotating,unsteady), the cl/cd are quite similar.

i believe there's no need to modify my cl/cd formula for the . that's what i was told in another post.