Computational Fluid Dynamics 5 Solution Behaviour

1. Computational Fluid Dynamics 5Solution Behaviour Professor William J Easson School of Engineering and Electronics The University of Edinburgh

2. Flow over a backward-facing step • Flow expands and leaves a recirculating vortex behind the step • Solve to 2nd order and maintain laminar flow • How long does the domain have to be to ensure that the solution is valid • Upstream? • Downstream? • Hint: Try x1=2H, 5H, 10H; x2=5H, 10H, 15H 2H x1 H x2 Re = 500, ν = 10-6, H=25mm, u=0.01ms-1

3. Vectors & shear stress x1=x2=5H

4. x1=10H, x2=10H & 15H

5. Shear Stressx1=10H, x2=10H & 15H

6. Shear stress for x2=10H,15H

7. Things you can do • Create simple geometries in Star-Design • Produce meshes of different densities and of varying density (by changing the parameters before meshing) • Solve for laminar flow in a 2D channel • Present the output in a variety of formats • Solve for 2D laminar jets • Solve for 2D flows with wall attachment • Solve to 1st & 2nd order simulations (check this) • Test the appropriateness of your mesh density (check) • Test the appropriateness of the extent of your domain

8. Solution behaviour Anderson Ch3 Versteeg & Malalasekera Ch2

9. Types of equation • Parabolas, Hyperbolas and Ellipses - reminder • Same class of curves • Can be ‘cut’ from a cone • General equation • Used in definition of equation ‘types’ Hyperbola Parabola Ellipse

10. General quasi-linear PDE(not the NS equations) Define a vector equation from the above simultaneous equations

11. Equation types Define The solution of this set of equations has been reduced to a ‘simple’ matrix equation that has a key matrix [N]. If the eigenvalues of [N] are real, the equation is hyperbolic complex – elliptic zero – parabolic

12. Why does this matter? • Each equation type has a different mathematical behaviour • Mathematical behaviour is related to physical behaviour • Physical behaviour should be taken into account when posing the problem • Hyperbolic and parabolic equations lend themselves to “marching” solutions and are generally more stable to solve • Marching can be in space or in time • Elliptic equations must be solved everywhere at once and are generally more difficult to solve

13. Classification of NS • General NS equations are of ‘mixed’ class

14. Example – 3D incompressible flow through a constriction at Re=1000 steady – elliptic unsteady - parabolic Not well-posed well-posed

15. Well-posed problem • If there exists a unique solution which depends continuously on the boundary conditions, the problem is well-posed • In the above example the steady-state problem was not well-posed, so an unsteady simulation was performed which was well-posed

16. Examples for this week • Flow through a 3D pipe at Re =107, 106 105 104 103 • Can you deduce the friction factors? • What is the effect of increasing surface roughness at 107? • Force on a cylinder in a steady turbulent flow (can be done in 2D) • What is the drag coefficient? • Consider the grid design and domain carefully • Allow the walls of your virtual water/wind tunnel to ‘slip’