A new perspective suitable for understanding the details of nonlinear pumping (formation of traveling shocks) inside a pressurized cavity is constructed in this paper. Full compressible axisymmetric three-dimensional Navier-Stokes equations are used as the starting point to cover all complexities of the problem that exceedingly increase for particular ranges of Mach, Reynolds and Prandtl numbers. Then a very high-order numerical method is introduced to preserve the user-defined order of accuracy for practical simulations. For removal of spurious waves, higher-order compact filters are derived. All equations are marched in time using the classical Runge-Kutta algorithm which is appropriate for problems involving fine-scale temporal fluctuations. As the most important part of simulation, Navier-Stokes Characteristic Boundary Conditions are used for accurate calculation of wave reflection specially at singular points, i.e., corner points and points across the axis of symmetry. A simultaneous characteristic-decomposition is devised in this paper which completely resolves stability problems arising from problem-dependent treatment of corner points. Numerical experiments are performed for high-Reynolds laminar flows inside the shock region to determine the effect of frequency change on both shock formation (stationary flow) and transient solution. The current approach which favorably compares to the previous experimental data, may be used as a robust tool for understanding the less-understood problem of shock/Stokes-Layer interaction and its consequences on transition to turbulence in Oscillating Pipe Flow.

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