We present direct numerical simulation (DNS) of a stationary turbulent hydraulic jump with inflow Froude number of 2, Weber number of 1820 and density ratio of 831, consistent with ambient water–air systems, all based on the inlet height and inlet velocity. A non-dissipative geometric volume of fluid (VOF) method is used to track the detailed interactions between turbulent flow structures and the nonlinear interface dynamics. Level set equations are also solved concurrent with VOF in order to calculate the interface curvature and surface tension forces. The mesh resolution is set to resolve a wide range of interfacial scales including the Hinze scale. Calculations are compared against experimental data of void fraction and interfacial scales indicating, reasonable agreement despite a Reynolds number mismatch. Multiple calculations are performed confirming weak sensitivity of low-order statistics and void fraction on the Reynolds number. The presented results provide, for the first time, a comprehensive quantitative data for a wide range of phenomena in a turbulent breaking wave using DNS. These include mean velocity fields, Reynolds stresses, turbulence production and dissipation, velocity spectra and air entrainment data. In addition, we present the energy budget as a function of streamwise location by keeping track of various energy exchange processes in the wake of the jump. The kinetic energy is mostly transferred to pressure work, potential energy and dissipation while surface energy plays a less significant role. Our results indicate that the rate associated with various energy exchange processes peak at different streamwise locations, with exchange to pressure work flux peaking first, followed by potential energy flux and then dissipation. The energy exchange process spans a streamwise length of order ${\sim}10$ jump heights. Furthermore, we report statistics associated with bubble transport downstream of the jump. The bubble formation is found to have a periodic nature. Meaning that the bubbles are generated in patches with a specific frequency associated with the roll-up frequency of the roller at the toe of the jump, with its footprint apparent in the velocity energy spectrum. Our study also provides the ensemble-averaged statistics of the flow which we present in this paper. These results are useful for the development and validation of reduced-order models such as dissipation models in wave dynamics simulations, Reynolds-averaged Navier–Stokes models and air entrainment models.

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This study focuses on the Rijke tube problem, which includes features relevant to the modelling of thermoacoustic coupling in reactive flows: a compact acoustic source, an empirical model for the heat source and nonlinearities. This thermoacoustic system features both linear and nonlinear flow regimes with complex dynamical behaviour. In order to synthesize accurate time series, we tackle this problem from a numerical point of view, and start by proposing a dedicated solver designed for dealing with the underlying stiffness -in particular, the retarded time and the discontinuity at the location of the heat source. Stability analysis is performed on the limit of low-amplitude disturbances by using the projection method proposed by Jarlebring (PhD thesis, Technische Universität Braunschweig, 2008), which alleviates the problems arising from linearization with respect to the retarded time. The results are then compared with the analytical solution of the undamped system and with the results obtained from Galerkin projection methods commonly used in this setting. This analysis provides insight into the consequences of the various assumptions and simplifications that justify the use of Galerkin expansions based on the eigenmodes of the unheated resonator. We demonstrate that due to the presence of a discontinuity in the spatial domain, the eigenmodes in the heated case predicted by using Galerkin expansion show spurious oscillations resulting from the Gibbs phenomenon. Finally, time series in the fully nonlinear regime, where a limit cycle is established, are analysed and dominant modes are extracted. By comparing the modes of the linear regime to those of the nonlinear regime, we are able to illustrate the mean-flow modulation and frequency switching, which appear as the nonlinearities become significant and ultimately affect the form of the limit cycle. Analysis of the saturated limit cycles shows the presence of higher-frequency modes, which are linearly stable but become significant through nonlinear growth of the signal. This bimodal effect is not exhibited when the coupling between different frequencies is not accounted for. In conclusion, a dedicated solver for capturing thermoacoustic instability is proposed and methods for analysing linear and nonlinear regions of the resulting time series are introduced.