1 Discretization of Fluid Models (Navier Stokes) Dr. Farzad Ismail School of Aerospace and Mechanical Engineering Universiti Sains Malaysia Nibong Tebal Pulau Pinang Week 5 (Lecture 1 and 2)

2 Choice of Grid Discretization It is not critical for unsteady problems but it is crucial for steady-state problems –Elliptic problems have issues with certain errors not being removed even after many iterations. Collocated: all variables are stored at the same location –Worst choice, since there will be some errors that will never be removed! Compact: –Better choice, but there will be at least one type of errors that will never be removed! Staggered- best choice for elliptic problem, but…

3 Staggered Grid What about p, F,G, H’s?

4 Initial & Boundary Conditions For elliptic problems, IC is not so much a big deal –Will just determine how quick solution reach steady-state But the BC is very critical! –Not only determines the efficiency of the computations –But will also determine what is the final outcome –Unphysical BC may lead to huge problems?Why? Concept of Ghost Cells No slip condition along and no outflow/inflow through boundaries

5 Boundary Conditions (cont’d) How to compute ghost cells? –Combine BC and the equations of motions to extrapolate the values outside the computational domain –Take horizontal boundary, v=0 (obvious) but u may not be unless wall is not moving (no slip) –Can use mass equation

6 Boundary Conditions (cont’d) Nature of u,v at boundaries –v=0 at boundary but u may not be zero; v even function –v y = -u x = 0 (by mass), implies that u is constant along wall and that integrating u wrt x gives u(x,y) as a constant plus a linear Variation along y. –The ghost quantities (previous Fig) via the y-momentum:

7 Vo and H(y) are zero on the boundary, this simplifies to where v T are expanded from v 0 via Taylor series where v T are expanded from v 0 via Taylor series Since v 0 =0 and assume that the square of v and the high order terms are negligible, hence

8 Since v is an even function, The same can be done for other walls and similar approach for (u,v)

9 Algorithm for Incompressible NS Apply the appropriate BC Solve for the preliminary u*, v* based on (u,v,p) of IC and BC Compute the Pressure-Poisson equation to solve for p* –Iterate until reach required error tolerance Check the discrete velocity divergence, if criterion is met, proceed to the next time level. If not go back to step 1 and continue the iteration within the same time-level