Oscillating Viscous Boundary Layer Adjacent to a Solid Wall
To demonstrate the effectiveness of the proposed numerical viscous boundary conditions for a solid wall, the case of an oscillating viscous boundary layer is simulated. The oscillating boundary layer is generated by a time periodic source in the energy equation at a very low frequency. Upon writing out in full, the energy equation, including the source term, is as follows:
d p д u д V
— +——– 1— = 0.01 exp
dt дх д y F
All the viscous stress terms of Eq. (6.16) are included in the computation. To ensure that there are at least 7 to 8 mesh points in the boundary layer (so that it can be resolved computationally) the mesh Reynolds number, R = р0а0Ах//л is taken to be 5.0, and the oscillation period set equal to 1250 time steps (at At = 0.07677). An exact solution of this problem is not available. However, far from the source, the boundary layer resembles the well-known Stokes oscillatory boundary layer. In the Stokes boundary layer the oscillatory velocity component u is given by the following formula:
In Eq. (6.25), the amplitude and phase factors є and 5 are constants of the Stokes solution. For the purpose of comparison with numerical solution, these two constants are determined by fitting solution (6.25) to the numerical solution at the point of maximum velocity fluctuation at two instants of time.
Figure 6.16 shows the computed velocity profile of the oscillating boundary layer along the wall at x = 0. In this figure, T is the period of oscillation. Shown also are the velocity profile of Eq. (6.25) at every quarter period. The agreement between the results of the numerical simulation and the approximate analytical solution appears to be good. On considering that only 7 to 8 mesh points are used to resolve the entire viscous boundary layer, the performance of the numerical viscous boundary conditions and the DRP scheme must be regarded as very good.