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Oct 30, 2023

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Assignment 2 hints A few hints for the assignment. Partial convergence: The pseudo-code abandons the matrix solver after 100 iterations even if convergence has not been reached. This means that the time development is not accurate however you can still reach the correct steady state value. As the ﬂow reaches a steady state, the solver will begin to reach convergence within the 100 iteration limit. What if you allow more iterations before quitting? What if you do not place any limit? Is 100 the optimal value in any sense? Other matrix solvers: The greatest improvement in speed will come from expressing the pressure correction Poison equation as a matrix Ax = b then using one of the matrix solvers included in MATLAB such as bigstab (biconjugate gradient method, stabilized). Matrix code: Your code will execute significantly faster in matlab if you write it in terms of matrix operations rather than using loops. For example when we did advection of a scalar, the upwind scheme could be written as, for ix=2:nx -1 sigma_new ( ix ) = sigma_old ( ix ) ... - Cr*( sigma_old(ix)-sigma_old(ix -1)); end This could also be written as sigma_new (2: nx -1) = sigma_old (2: nx -1) ... - Cr*( sigma_old (2:nx -1)-sigma_old (1:nx -2)); The latter will be much faster. This technique can be extended to a two dimensional matrix as used in your assignment. You will need to be very careful (and clever) in order to correctly implement the Gauss-Seidel method using these sort of matrix expressions since you need to ensure that the updates are performed in the correct order. Crank-Nicolson implicit time stepping Approximate the time derivative using the trapezoidal rule, utα+ = utα − ∆t∂α (1/ρ) Pt+1/2 + ∆t*1/2*(ν∂β∂βutα+ + ν∂β∂βutα − utβ+∂βutα+ − utβ∂βutα)

This is an implicit equation. Unlike the implicit heat equation, we can not solve this using linear algebra because it is non-linear. We can use an iterative procedure, utα+,k+1 = utα − ∆t∂α 1 P t+ 1 ,k + 1 ν∂β∂βutα+,k + ν∂β∂βutα − utβ+,k∂βutα+,k − utβ∂βutα . ρ 2 ∆t 2 We make a similar substitution as used in projection methods with a new intermediate velocity at the kth sub-step, u α,k = utα + 1 ν∂β∂βutα+,k + ν∂β∂βutα − utβ+,k∂βutα+,k − ∗ utβ∂βutα . ∆t 2 With, utα+,k+1 = u α,k − ∆t∂α 1 P ∗ t+ 1 ,k ρ 2 The Poisson equation is again solved in terms of the divergence of an intermediate velocity, ∂αu α,k = ∆t 1 ∂α∂α P t+ ∗ 1 ,k . ρ 2 Initial guesses for the new velocity and pressure are usually taken to be, uαt+,0 = utα P t+ 1 ,0 = P t− 1 . 2 2 With these initial guesses, one step is equivalent to the projection method. Taking only two sub-steps of the above algorithm results is a method sometimes termed the piso algorithm. Since the approximate pressure calculated during the first sub- step is somewhat close to the correct value, few additional iterations of the pressure solver are required during the second sub-step for a reasonable improvement in accuracy. Further steps may be taken for comparatively little cost. So far we have not mentioned simple, the default scheme used by FLUENT which has been used in all tutorials so far. This scheme is similar overall to the pressure correction method from these notes however since it aims to solve the steady state equations

the time step is replaced by under relaxation factors which are related to the stability of the scheme. Wall bounded turbulent ﬂow - y+ Wall bounded turbulent ﬂows including channel ﬂows and boundary layers are charac- terised in terms of ‘wall units’, y+ = uτ y . ν Here uτ is the friction velocity which is based on the wall shear, uτ = τw . ρ 2

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