A Sharp Interface Cartesian Grid Method for Simulating Flows with ...

370 UDAYKUMAR ET AL.
where locations x 1 to x 6 are as shown in Fig. 8. In Ralph and Pedley [38], the streamfunction
vorticity method has been used along with a body-fitted moving mesh to solve the
problem. In the present method, primitive variables are used and the mesh remains fixed.
We will compare our results for this case with experiments [34] and numerical results [38].
For low Strouhal numbers, the flow behaves in a quasi-steady fashion and the sequence of
events in time as the indentation grows, corresponds to those encountered when steadystate
solutions are obtained at the different geometries and juxtaposed. The physics of the
problem is interesting for high enough Strouhal numbers, when the quasi-steady assumption
no longer holds and the full unsteady, viscous dynamics needs to be captured to reproduce
the experimental results.
For the Strouhal number chosen here, i.e., for the frequency of oscillation of the indentation,
the flow in the channel downstream of the indentation falls in the unsteady regime.
The squeezing of the flow in the channel propagates waves downstream of the constriction
leading to the formation and advection of vortices on both the top and the bottom surfaces
of the channel. The streamlines are shown in Fig. 9 at various instants in one cycle of motion
of the indentation, for the case where St = 0.037, ε = 0.38, and β = 4.14. Our results
reproduce almost exactly the sequence of events occurring in the experiments and agree
with the numerical results of Ralph and Pedley [38]. In Fig. 9 we show the development
of a series of eddies downstream of the moving indentation. These eddies are formed successively
in time as the disturbance propagates through the channel. In agreement with the
numerical [38] and experimental [34] results, we find that the eddy B at the top wall splits
at a nondimensional time between t = 0.6 and t = 0.65 into three eddies. This behavior of
eddy B is shown in detail in Fig. 10. In Fig. 11, we compare the wave propagation characteristics
quantitatively with numerical and experimental results of Pedley and coworkers.
We show in that figure the location of the crests and troughs associated with each of eddies
B, C, and D with time. The slope of this curve gives the phase velocity of the disturbances
downstream of the indentation. Note that for eddies B and D, we measure trough position,
FIG. 9. Wave formation and propagation downstream of a moving indentation computed with the present
numerical method. The formation of the various eddies downstream of the indentation is indicated. The nondimensional
time instants at which the streamlines are shown are also indicated.