Oscillatory flows of a choked underexpanded supersonic impinging jet issuing from a convergent nozzle have been computed using the axisymmetric unsteady Navier-Stokes system. The MLDFSS(Modified Low-Diffusion Flux-Splitting Scheme) is used to evaluate the inviscid fluxes on the cell surface. Central difference is used to calculate viscous terms. For time accurate calculations, the subiteration time advance scheme of Pulliam is adopted. In the present computation, it is assumed that the choked sonic flow condition is achieved at the nozzle exit. Thus the inlet boundary of this computation is the nozzle exit where the pressure, the velocity and the temperature are fixed from the isentropic relations. Tests are conducted to select a suitable grid and a time step, which can capture the salient features of flow oscillation. By scrutinizing the pressure oscillation patterns including the amplitude and the time period of the oscillation, it is found that grid $(Δx_{min} = Δr_{min} = 0.001D)$ and time step $(Δt = 0.003D/V_e)$ are sufficient for unsteady analysis.
This study focuses on the oscillatory flow features associated with the variation of the nozzle-to-plate distance (H/D) and nozzle pressure ratio $(P_0/P_a)$. Frequencies of the surface pressure oscillation and flow structural changes from computational results have been analyzed. First, as the distance from the nozzle to the plate increases within a specific range of H/D, the pressure oscillation frequency decreases. However, the frequency changes with a jump at a certain H/D at which the oscillation enters into a different stage. The staging behavior with the distance variation is in good agreement with that of impinging tone in the previous studies. In the oscillatory case, the plate shock moves up and down along the jet axis and the frequencies of the plate shock oscillations are identical with those of the surface pressure oscillations. Not only does the plate shock oscillate, but also the jet flow structure changes periodically. It is found that the pressure distribution of the expansion region in the first cell does not vary with time for all the cases. However, the pressure distribution over the compression region in the first cell and the region beyond are seen to undergo significant changes with time. It is also found that the pressure distribution along r=0.75D line exhibits a similar pattern with a standing wave in an open-ended pipe. The pressure distribution along r=0.75D line changes its pattern as H/D crosses the demarcation points in staging, while the pressure distribution along the centerline shows no significant variations. Further, the pressure distributions along r=0.75D line for H/D=2.6 and 3.6 are of the same pattern signifying that these two cases belong to the same stage, while the corresponding pressure distributions along the jet axis are entirely different each other as H/D values are widely apart.
Staging behavior of the frequency of the surface pressure oscillation has been observed for both cases of nozzle-to-plate distance variation and pressure ratio variation. The frequency does not vary smoothly with either the distance or the pressure ratio. The frequency jumps discontinuously at a specific value of H/D or $P_0/P_a$. The staging behavior with the pressure variation in which the oscillation frequency undergoes a step change when the pressure ratio crosses a specific value is different from that of the distance variation in the sense that the frequency of the surface pressure oscillation remains constant in a given stage. These two distinct staging behaviors of the oscillation frequency are found to correlate well if the frequency and the distance are normalized by the length of the first shock cell. It is further found that the staging behavior for both cases is strongly correlated with the change of the pressure wave pattern in the jet shear layer, but not with the jet shock cell structure.
