Flow Over Cylinders -Numerical Simulation (demo)

1. Flow Over Cylinders-Numerical Simulation (demo) Prepared by: Wang Xiangqi November 16th 2005

2. Introduction • Relative motion between an object and a fluid is common occurrence; obstacles disturb the flow and create particular shapes in their wakes. • The shape of the obstacle plays an important role. • To understand this phenomena, studies are done on a simple case: a two-dimensional flow past a circular/square cylinder for various Reynolds number.

3. Background Knowledge • Re<4: • Re<<1: • 4<Re<40: • 40<Re<400: (Von Karmann vortex street)

4. Governing Equations • Laminar: • Continuity: • Momentum: • Energy: • Turbulent model (k-ε) • k: • ε:

5. Physical Models Cylinder: wall Upper and lower sides: periodic Left side: velocity inlet Right side: Pressure outlet P: Monitored point Upper side Inlet P (0.06,0.06) Outlet Cylinder Lower sides • Inlet: velocity inlet, U=Reμ/ρD, T=298.15 K • Outlet: P=0 (atomistic pressure) • Periodic walls: ΔP=0 (zero pressure gradient) • Cylinder: T=473.15 K BCs:

6. Typical Grid Structure • Meshes: • The central square with a side length of six times the radius of the cylinder was used to facilitate meshing; • Boundary layer with 10 rows was used at the cylinder walls; • Total meshes: Circular (31,400); Square (34,600).

7. Numerical Procedure • Obtain the steady solution; • Kelvin-Helmoltz perturbation (10%*U) for a while (10s) if the Reynolds number is too low to active the instability of flow; (Numerical errors also can, but need a long time) • Compute the unsteady solution then. • Criteria: • Steady solution: convergence of average velocity of monitored point; • Unsteady solution: 1e-4 for all variables except 1e-6 for energy; time step size: 1s

8. Temperature and Pressure Patterns (Re=1) Circular Square • At Re=1, the flow is symmetrical upstream and downstream. • Thermal dissipation; • For square cylinder, the thickness of thermal layer is higher; • For square cylinder, the pressure pattern is elongated horizontally; • Global pattern is similar for both cylinders.

9. Temperature and Pressure Patterns (Re=100, t=1200s) Circular Square • At Re=100, phenomena of instability; • Convection is important; • Von Karmann vortex streets dominate; • The solution is unsteady and complicated; • Square cylinder eddies are stronger and show larger amplitudes

10. Temperature and Pressure Patterns (Re=1000) Circular Square • At Re=1000, turbulent flow; • The thermal layer is confined near the cylinder; • Boundary layer separation; • Von Karmann vortex street disappears;

11. Velocity Patterns Circular Re=1 Re=100 (t=1200s) Re=1000 Square

12. Vector Distribution (Circular, Re=1)

13. Vector Distribution (Circular, Re=100)

14. Vector Distribution (Circular, Re=1000)

15. Vector Distribution (Square, Re=1)

16. Vector Distribution (Square, Re=100)

17. Vector Distribution (Square, Re=1000)

18. Streamlines (Circular, Re=1)

19. Streamlines (Circular, Re=100)

20. Streamlines (Circular, Re=1000)

21. Streamlines (Square, Re=1)

22. Streamlines (Square, Re=100)

23. Streamlines (Square, Re=1000)

24. Occurrence of Instability (Re=100) Circular Square

25. Occurrence of Instability (Re=100) • Circular cylinder: • Unsteady calculation from steady solution. • Instability occurs due to the perturbation of numerical errors (?) • Square cylinder: • Steady solution cannot be achieved (maybe the mesh is not fine enough for square case to catch the boundary layer?) • Unsteady solution from zero-value field (t=0) • Instability occurs at around t=200s and results in von Karmann vortex street. • As compared to circular cylinder, the amplitude of square cylinder is larger.

26. Vector Distribution Re=100, t=1000s Line-6 Line-1 Line-2 Line-5 Line-3 Line-4

27. Nu Distribution Along the Wall Re=1 Re=100 Re=1000 Unsteady (t=1200s)

28. Pressure Coefficient Along the Wall Re=1 Re=100 Re=1000 Unsteady (t=1200s)

29. Conclusion • The difference flow patterns depending on Reynolds number are observed; • The most important results is the visualisation of the hydrodynamic instability called “Strouhal Instability” or “Von Karmann Vortex Street”; • The results are in good agreement with the experimental values; • The instability can be explained: • Mathematically, by the theory of the instability; • Physically, by the boundary layer separation.