The solution to two dimensional Navier-Stokes equations are solved in a closed cavity, where the fluid initially at rest is set into motion by the upper plate moving in its plane. Two different cases for the motion of the driving plate - unidirectional and oscillating are considered to observe the effect of fluid viscosity on the flow pattern.

The vorticity stream function method is used to solve the problem with FTCS employed for the parabolic nonlinear vorticity equation and Point Gauss Seidel method for elliptic stream function equation. The flow patterns are compared for the driven cavity against the oscillating plate cases for different Reynolds numbers. Influence of frequency on the stream function is checked for the same Reynolds numbers. Finally steady state solutions for the driven cavity and unsteady solutions for the oscillating plate are plotted. A symmetrical driven cavity is also tested in which the top and bottom plates are set in motion at the same time.

The driven cavity problem is a two dimensional cavity enclosed within three stationary and one moving wall. The fluid in the cavity is initially at rest and is set in motion when the top plate has been given a steady velocity of U=5 m/s. The rectangular cavity has width of 40 cm and height of 30 cm. A grid resolution of 1 cm \times 1 cm is used. A convergence criterion of 0.001 for the point Gauss-Seidel formulation and 0.002 for steady state of the vorticity equation is used.

A. U=5 m/s, Re=600 | B. U=5 m/s, Re=2000 |

Figure 1 and 2 show the x and y component of the velocity along with the scaled velocity vectors. The two cases A and B corresponding to Re = 600 and 2000 are obtained with changing the kinematic viscosity of the fluid keeping the boundary conditions same. It is clear from the figures that at higher Re, the flow has higher circulation and sharper velocity gradients.

A. U=5 m/s, Re=600 | B. U=5 m/s, Re=2000 |

Figure 3 shows the velocity vectors indicating the flow direction and vortex structure. The vector length is not to scale with the velocity magnitude and is exaggerated for flow visulization. Comparing Fig. 3A and 3B, it's seen at higher Re, the two recirculation zones at bottom corners of cavity increase in size. One interesting point about the main vortex is that location of vortex center changes with the Re and lies always on the right side of the vertical symmetry line of the cavity. The computed results are compared with available literature in Table 1. The location of the vortex is measured visually in the present case.

Reynolds Number | Present Simulation (X_{0},Y_{0}) |
Reynolds Number | NASA Technical Memorandum 112874 (W.A.Wood)^{1}(X_{0},Y_{0}) |

50 (not shown) | (0.60,0.73) | 100 | (0.615,0.740) |

600 | (0.60,0.60) | 400 | (0.556,0.608) |

2000 | (0.58,0.57) | 2000 | (0.522,0.556) |

A. U=5 m/s, Re=600 | B. U=5 m/s, Re=2000 |

The literature reports presence of six vortices in the driven cavity at Re=2000. The present simulation with its grid resolution can capture only four vortices. There are two more vortices present at the extreme lower corners of the cavity.

^{1}NASA Technical Memorandum 112874 - Viscous Driven Cavity Solver User's Manual by W.A.Wood. *Langley Research Center,Hampton, Virginia*| Complete Reference

The driven cavity problem is solved with a variation in the boundary condition. The top boundary is forced to move with a velocity given as U = U_{0}sin(ft) m/s, where U_{0} is 5 m/s. Results are simulated for two different frequencies f = 0.5, 4 and two Reynolds numbers Re = 600, 2000. The results are shown in Table 2 as *.avi* animation files for one period of oscillation of the top boundary.

Reynolds Number | Frequency (Hz) | X component Velocity (m/s) | Y component Velocity (m/s) |

600 | 0.5 | U_Re600_f05 | V_Re600_f05 |

600 | 4 | U_Re600_f4 | V_Re600_f4 |

2000 | 0.5 | U_Re2000_f05 | V_Re2000_f05 |

2000 | 4 | U_Re2000_f4 | V_Re2000_f4 |

*Effect of Re*: High Reynolds number flow has low viscosity which causes the main vortex to form near the top boundary and not much of the bottom fluid is entrained into the vortex. At low Re, a large portion of the cavity fluid responds to the oscillatory behaviour of the flow, though peak velocities observed are less (in comparison to high Re) because of higher viscous effects.

*Effect of Frequency*: At same Re, higher frequency reduces the size of the main vortex and localizes it near the top boundary. At low Re, fluid respose to high frequency is very sluggish and can be seen from the file V_Re600_f4. Though, the results are presented for only first period of oscillation, where the fluid starts with a stagnation initial condition, subsequent periods of oscillation do not show considerable difference in the flow behaviour. Such effects of steady oscillatory state and initial oscillations are probably high for very low Re or very high frequencies where the flow would lag in phase with the forcing boundary condition considerably.

Kaustubh Shankar Kulkarni Last modified: Tue Apr 13 17:26:23 CDT 2004