Linear and nonlinear analysis of shallow wakes

Abstract

The bottom friction and the limited vertical extent of the water depth play a significant role in the dynamics of shallow wakes. These effects along with the effect of the strength of the shear layer define the wake parameter $S$. A nonlinear model, based on a second-order explicit finite volume solution of the depth-averaged shallow water equation in which the fluxes are obtained from the solution of the Bhatnagar–Gross–Krook (BGK) Boltzmann equation, is developed and applied to shallow wake flows for which laboratory data are available. The velocity profiles, size of the recirculating wake, oscillation frequency, and wake centreline velocity are studied. The computed and measured results are in reasonable agreement for the vortex street (VS) and unsteady bubble (UB) regimes, but not for the steady bubble (SB). The computed length of the recirculation region is about 60% shorter than the measured value when $S$ belongs to the SB regime. As a result, the stability investigation performed in this paper is restricted to $S$ values away from the transition between SB and UB. Linear analysis of the VS time-averaged velocity profiles reveals a region of absolute instability in the vicinity of the cylinder associated with large velocity deficit, followed by a region of convective instability, which is in turn followed by a stable region. The frequency obtained from Koch's criterion is in good agreement with the shedding frequency of the fully developed VS. However, this analysis does not reveal the mechanism that sets the global shedding frequency of the VS regime because the basic state is obtained from the VS regime itself. The mechanism responsible for VS shedding is sought by investigating the stability behaviour of velocity profiles in the UB regime as $S$ is decreased towards the critical value which defines the transition from the UB to the VS. The results show that the near wake consists of a region of absolute instability sandwiched between two convectively unstable regions. The frequency of the VS appears to be predicted well by the selection criteria given in Pier & Huerre (2001) and Pier (2002), suggesting that the ‘wave-maker’ mechanism proposed in Pier & Huerre (2001) in the context of deep wakes remains valid for shallow wakes. The amplitude spectra produced by the nonlinear model are characterized by a narrow band of large-amplitude frequencies and a wide band of small-amplitude frequencies. Weakly nonlinear analysis indicates that the small amplitude frequencies are due to secondary instabilities. Both the UB and VS regimes are found to be insensitive to random forcing at the inflow boundary. The insensitivity to random noise is consistent with the linear results which show that the UB and VS flows contain regions of absolute instabilities in the near wake where the velocity deficit is large.