Computational Fluid Dynamics 5 Gridding & time-dependance

1. Computational Fluid Dynamics 5Gridding & time-dependance Professor William J Easson School of Engineering and Electronics The University of Edinburgh

2. 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

3. Things you can do • Simulate steady, turbulent flow • Simulate flow past objects in a domain • Calculate the drag coefficient using the sum of forces on an object in a flow • Determine whether flow solution is dominated by hyperbolic, parabolic or elliptic behaviour • Utilise time-dependant equations to enhance convergence for elliptically-dominated solutions

4. Moody Diagram (Chart)

5. Gridding Anderson and Versteek & Malalasekera both weak on gridding

6. Structured grids (eg hex) Advantages: • Equations relatively simple • Cell shape is easily controlled • Numerical errors are smaller • Can be fitted to flow direction • Can be fitted to gradients in flow • Optimises memory use Disadvantages: • Fitting to complex geometries requires a lot of time and skill

7. Structured grid

8. Structured grid (2)

9. Un-structured grids (eg tet) Advantages: • Can be fitted to unusual shapes • ‘One-click’ creation Disadvantages: • Generally larger errors • Limited control over cell shape – ‘skewness’

10. Unstructured grid

11. Exercise 1 Simulate laminar flow through a straight pipe 8mm dia and 50mm long. Increase the cell density next to the wall • Using a structured grid by creating a prism layer • Using a triangular grid with a growth function Note the number of cells created in each case and the relative convergence time

12. Time-dependant flows Anderson Ch4 Versteeg & Malalasekera Ch8

13. Time dependant modelling • Many flows are inherently time-dependant and therefore require to be modelled as such • Some steady-state flow solutions are not well-behaved and therefore require to be modelled by time-dependent equations to achieve convergence

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

15. Explicit techniques • Solution at time-step t+Δt is obtained by marching forward from time-step t and obtaining gradients for estimating new values from those at previous time-step • Easy to program and quick to solve, but can be unstable if Δt too large (conditionally stable) • Δtmax is proportional to (Δx)2, so quickly becomes very small and limits usefulness of solution

16. Implicit techniques • Solution at time-step t+Δt is obtained by marching forward from time-step t and obtaining gradients for estimating new values from those at previous and current time-step • Crank-Nicolson is an early example • Requires solution of large number of simultaneous equations (also extra ‘loops’ inside iteration process) • Conditionally, or even unconditionally, stable • Large Δt results in errors – not instability • May be quicker to converge than explicit due to improved stability

17. Time-dependant example Vortex shedding past a cylinder • 2D example • Use laminar flow, 16mm cylinder at 7.5mm/s (Re~120 ) • Create unstructured grid using sizing function • Use an unsteady solver • Calculate the shedding frequency and check that it matches the strouhal number