20. Examples of solutions of Navier-Stokes equations for unsteady flows.

This lecture is about solving the Navier-Stokes equations for unsteady flows. There are not many practical examples of laminar unsteady flows in rectangular solution domains. Two examples are presented in Chapter 8, Section 8.4.2 of the book: lid-driven and buoyancy-driven cavity flows with oscillating boundary conditions. Here two additional examples are presented.
In the first test case, we are simulating the flow containing four Taylor-Green vortexes in two versions. The first version assumes an inviscid fluid; under this assumption, the vortexes should remain unchanged forever. An unsteady simulation leads at some stage to the break-up of symmetries and the four vortexes collapse into a single one. We are interested in how long the four vortexes persist and in the rate of decay of kinetic energy; this is especially important in simulations of turbulent flows using DNS and LES methods, which will be the subject of one of the future lectures. The grid- and time-step dependence of solutions and the performance of different algorithms (SIMPLE and IFSM) are analyzed.
If the fluid is viscous and the Reynolds number is not high, the flow is laminar and the four vortexes become weaker with time due to viscous dissipation of kinetic energy.
In the second test case, we are looking at a cavity whose lid is suddenly set to motion with a constant velocity, and then after 5 seconds suddenly held fixed. The fluid is initially at rest, and the moving lid accelerates it, creating a large vortex. When the lid stops moving , viscosity leads to the weakening of the vortex. Again, we are looking at the grid and time-step dependence of solutions, and the performance of different algorithms implemented in two-dimensional research codes, which are available in source form in supplementary materials, and in the commercial CFD software Simcenter STAR-CCM+ from Siemens.
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