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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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The wall shear is the shear force exerted on the wall by the ﬂuid. For a wall who’s normal vector points in the y direction, τw = µ ∂u . ∂y y=0 Wall shear is the main parameter that drives these sort of turbulent ﬂows. The velocity can be non-dimensionalized using wall units as well, u+ = u . uτ Often we will need to determine the friction velocity based on a mean ﬂow velocity. In order to do this we will need to know the shape of the velocity profile in order to calculate the wall shear. Closed forms for the velocity profile of turbulent ﬂows are generally not known. The one seventh power law, 1 y7 u = u0 y0 , that some people may be familiar with will not give reasonable results since it has infinite gradient at the wall. 1 17 ∂yu = u0 y6y0 It is typical to approximate the wall shear is using experimental observations. For example in assignment 3 you are given an approximate formula for ﬂow in a pipe. τw = 0.03325ρu˜7/4ν1/4R−1/4, where u˜ is the cross-stream average velocity. In general the wall shear is determined based on the skin friction coe cient, ﬃ τw = Cf · 1 ρu20 . 2 Thus the friction velocity is, uτ = u0 Cf . 2 The skin friction for various ﬂows can be found in reference books and papers, one common approximation that can be used in general cases is the Schlichting correlation,

Cf = (2 log10 (Re) − 0.65)−2.3 for Re < 109. For ﬂow in a duct or a pipe the Darcy friction factor can be used. ∆P = f · 1 ρu˜2 1 . L 2 DH 3

Here DH is the hydraulic diameter, DH = 4A . B Where A is the cross sectional area and B is the wetted perimeter. Equating the balance of forces, τw = ∆P A = f · 1 ρu˜2. BL 8 Values of f can be read form a Moody chart. Thus the friction velocity is, uτ = u˜ f . 8 Figure 1 shows the mean velocity profile near the wall. Turbulent motions die out close to the wall since the local Reynolds number is small. This laminar region near the wall is called the viscous sublayer or the laminar sublayer and is defined as 0 < y+ < 5. Further away from the wall, the velocity profile follows a ‘law of the wall’ or ‘log law’ profile. The ‘log law’ region is usually defined as y+ > 30. In between these two regions is the bu er region. In order to ensure the accuracy of cfd simulations, the viscous ff sublayer should be adequately captured. This means that you must have at least one ﬂuid cell within the viscous sublayer. 4

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25 20 U+=y+ 15 U+ 10 log law 5 0 100 101 102 103 10-1 y+ viscous sublayer bu er layer log-law region ff inner layer outer layer Figure 1: Mean velocity profiles for turbulent ﬂow near a wall. Image source: https://en.wikipedia.org/wiki/Law of the wall 5