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Turbulent kinetic energy (TKE) is a quantity of primary importance in shock–turbulence interaction (STI). Linear interaction analysis (LIA) serves as a theoretical method to predict the amplification of TKE in STI. LIA is constrained by low-amplitude fluctuations and neglects nonlinear effects. In this paper, we explore the nonlinear amplification of the kinetic energy of fluctuations in the interaction between a vorticity wave and a shock wave that serves as a building block flow for STI. A weakly nonlinear framework (WNLF) is introduced to analyze nonlinear effects in fluctuation kinetic energy (FKE) and identify the dominant physical mechanisms driving its amplification. The theoretical framework is validated through high-accuracy numerical simulations of shock–vorticity wave interactions. The simulation results are compared with the predictions derived from the WNLF for a range of intensities and inclinations of shock-upstream vorticity fluctuations at different Mach numbers, and WNLF is found to successfully scale the numerical data for FKE, thus confirming the validity and applicability of the framework. According to WNLF findings, at lower supersonic Mach numbers, the intermodal interaction between vorticity–vorticity modes is important, whereas the interaction between vorticity and acoustic modes becomes dominant at higher Mach numbers. Using the intermodal interactions, a model based on WNLF is proposed to predict TKE amplification in STI. In comparison to direct numerical simulation, the WNLF based model predicts the TKE amplification for moderate turbulent Mach number and lower supersonic flow Mach numbers. This is a significant improvement over the LIA results available in the literature. [ABSTRACT FROM AUTHOR]

Interactions of shear fluctuations with shock waves are ubiquitous in high-speed flow applications from scramjet propulsion to cosmic events like supernova explosions. They also serve as fundamental building blocks for the study of shock–turbulence interaction. In this work, we study the nonlinear effects in pressure arising due to the interaction of a normal shock with a two-dimensional shear wave. It employs the weakly nonlinear framework (WNLF) developed recently for vorticity amplification by Thakare et al. ["A weakly nonlinear framework to study shock–vorticity interaction," J. Fluid Mech. 933, A48 (2022)]. The analysis includes the effect of intermodal interactions that is neglected in the widely used linear interaction analysis (LIA) of shock–turbulence interaction. It is found that the deformation of the shock wave and the fluctuation mass flux normal to the shock contribute to the dominant physical mechanisms responsible for the observed nonlinearities. Interestingly, the WNLF predicts a drop in mean pressure behind the shock due to a second-order intermodal interaction, which is consistent with the well-established results by Lele ["Shock-jump relations in a turbulent flow," Phys. Fluids A 4, 2900–2905 (1992)] at low Mach numbers and brings out additional effects of shock deformation that are important at higher Mach numbers. We extend the WNLF to three-dimensional interaction of homogeneous isotropic turbulence with a normal shock. Comparison with existing direct numerical simulation data shows good agreement for low turbulent Mach numbers, which is a significant improvement over the prediction capability of LIA. We also compute the dilatation fields from WNLF and use them to distinguish between the acoustic and non-acoustic components of the second-order pressure fluctuations generated by the shock wave. [ABSTRACT FROM AUTHOR]

Linear interaction analysis (LIA) is routinely used to study the shock-turbulence interaction in supersonic and hypersonic flows. It is based on the inviscid interaction of elementary Kovásznay modes with a shock discontinuity. LIA neglects nonlinear effects, and hence it is limited to small-amplitude disturbances. In this work, we extend the LIA framework to study the fundamental interaction of a two-dimensional vorticity wave with a normal shock. The predictions from a weakly nonlinear framework are compared with high-order accurate numerical simulations over a range of wave amplitudes (∊), incidence angles (α) and shock-upstream Mach numbers (M¹). It is found that the nonlinear generation of vorticity at the shock has a significant contribution from the intermodal interaction between vorticity and acoustic waves. Vorticity generation is also strongly influenced by the curvature of the normal shock wave, especially for high incidence angles. Further, the weakly nonlinear analysis is able to predict the correct scaling of the nonlinear effects observed in the numerical simulations. The analysis also predicts a Mach number dependent limit for the validity of LIA in terms of the maximum possible amplitude of the upstream vorticity wave. [ABSTRACT FROM AUTHOR]

Shock waves in high-speed flows can drastically alter the nature of Reynolds stresses in a turbulent flow. We study the canonical interaction of homogeneous isotropic turbulence passing through a normal shock, where the shock wave generates significant anisotropy of Reynolds stresses. Existing Reynolds stress models are applied to this canonical problem to predict the amplification of the stream-wise and transverse normal Reynolds stresses across the shock wave. In particular, the efficacy of the different models for the rapid pressure–strain correlation is evaluated by comparing the results with available direct numerical simulation (DNS) data. The model predictions are found to be grossly inaccurate, especially at high-Mach numbers. We propose physics-based improvement to the Reynolds stress-transport equation in the form of shock-unsteadiness effect and enstrophy amplification for turbulent dissipation rate . The resulting model is found to capture the essential physics of Reynolds stress amplification, and match DNS data for a range of Mach numbers. Numerical error encountered at shock waves are also analysed and the model equations are cast in conservative form to obtain physically consistent results with successive grid refinement. Finally, the proposed model for canonical shock-turbulence interaction is generalised to multi-dimensional flows with shock of arbitrary orientation. [ABSTRACT FROM AUTHOR]

High-speed turbulent flows often encounter high heat loads due to the presence of shock waves. The turbulent energy flux correlation in the mean energy conservation equation is a key unclosed term that determines the heat transfer rate. In this work, we employ existing turbulence models to predict the turbulent energy flux in canonical shock-turbulence interaction. The shortcomings of these models are highlighted, and a new heat-flux limiter model is proposed with the aid of linear theory results. We also write the transport equation for the turbulent energy flux across a shock wave and use it to develop a physics-based model for the same. It is found to predict the peak energy flux at the shock wave and its variation in the acoustic-adjustment region behind the shock. Numerical error incurred while solving the model equations at a shock wave are analyzed and a numerically robust model is obtained by eliminating the nonconservative source terms. The model predictions are compared with available direct numerical simulation data and a good match is obtained for a range of Mach numbers. [ABSTRACT FROM AUTHOR]

High-speed turbulent flows with shock waves are characterized by high localized surface heat transfer rates. Computational predictions are often inaccurate due to the limitations in modelling of the unclosed turbulent energy flux in the highly non-equilibrium regions of shock interaction. In this paper, we investigate the turbulent energy flux generated when homogeneous isotropic turbulence passes through a nominally normal shock wave. We use linear interaction analysis where the incoming turbulence is idealized as being composed of a collection of two-dimensional planar vorticity waves, and the shock wave is taken to be a discontinuity. The nature of the postshock turbulent energy flux is predicted to be strongly dependent on the angle of incidence of the incoming waves. The energy flux correlation is also decomposed into its vortical, entropy and acoustic contributions to understand its rapid non-monotonic variation behind the shock. Three-dimensional statistics, calculated by integrating two-dimensional results over a prescribed upstream energy spectrum, are compared with available data from direct numerical simulations. A detailed budget of the governing equation is also considered in order to gain insight into the underlying physics. [ABSTRACT FROM AUTHOR]

Reynolds-averaged turbulence models can result in large numerical error at flow discontinuities like shock waves. This is due to the nonconservative nature of the source terms in the governing equations. In this paper, the k-∈ turbulence model is used to compute the canonical interaction of a normal shock with homogeneous isotropic turbulence. The characteristics of the nonconservative error is studied as a function of shock strength, grid resolution, and upstream turbulence level. The model equations are cast in an equivalent conservation form that gives physically consistent results at a shock wave. The predicted amplifications of turbulent kinetic energy and its dissipation rate match direct numerical simulation data and linear theory estimates over a range of Mach numbers. [ABSTRACT FROM AUTHOR]

Reynolds-averaged Navier-Stokes prediction of shock wave/turbulent boundary layer interactions can yield significant error in terms of the size of the separation bubble. In many applications, this can alter the shock structure and the resulting surface properties. Shock-unsteadiness modification of Sinha et al. (Physics of Fluids, Vol.15, No.8, 2003) has shown potential in improving separation bubble prediction in compression corner flows. In this article, the modification is applied to oblique shock wave interacting with a turbulent boundary layer. The challenges involved in the implementation of the shock-unsteadiness correction in the presence of multiple shock waves and expansion fans are addressed in detail. The results show that a robust implementation of the model yields appreciable improvement over standard k-ω turbulence model predictions. [ABSTRACT FROM AUTHOR]

The RANS (Reynolds averaged Navier-Stokes) equations can yield significant error when applied to practical flows involving shock waves. We use the interaction of homogeneous isotropic turbulence with a normal shock to suggest improvements in the k - e model applied to shock/ turbulence interaction. Mahesh et al. [J. Fluid Mech. 334, 353 (1997)] and Lee et al. [J. Fluid Mech. 340, 22 (1997)] present direct numerical simulation (DNS) and linear analysis of the flow of isotropic turbulence through a normal shock, where it is found that mean compression, shock unsteadiness, pressure-velocity correlation, and up-stream entropy fluctuations play an important role in the interaction. Current RANS models based on the eddy viscosity assumption yield very high amplification of the turbulent kinetic energy, k, across the shock. Suppressing the eddy viscosity in a shock improves the model predictions, but is inadequate to match theoretical results at high Mach numbers. We modify the k equation to include a term due to shock unsteadiness, and model it using linear analysis. The dissipation rate equation is similarly altered based on linear analysis results. These modifications improve the model predictions considerably, and the new model is found to match the linear theory and DNS data well. [ABSTRACT FROM AUTHOR]

Shock waves drastically alter the nature of Reynolds stresses in a turbulent flow, and conventional turbulence models cannot reproduce this effect. In the present study, we employ explicit algebraic Reynolds stress model (EARSM) to predict the Reynolds stress anisotropy generated by a shockwave. The model by Wallin and Johansson (2000) is used as the baseline model. It is found to over-predict the post-shock Reynolds stresses in canonical shock turbulence interaction. The budget of the transport equation of Reynolds stresses computed using linear interaction analysis shows that the unsteady shock distortion mechanism and the pressure–velocity correlations are important. We propose improvement to the baseline model using linear interaction analysis results and redistribute the turbulent kinetic energy between the principle Reynolds stresses. The new model matches DNS data for the amplification of Reynolds stresses across the shock and their post-shock evolution, for a range of Mach numbers. It is applied to oblique shock/boundary-layer interaction at Mach 5. Significant improvements are observed in predicting surface pressure and skin friction coefficient, with respect to experimental measurements. • A new EARSM to predict Reynolds stress anisotropy generated by shock wave. • Model coefficients represent physical mechanisms at shock wave. • Model results match DNS data over a range of Mach numbers. • Model shows significant improvement in matching with SBLI experiments. [ABSTRACT FROM AUTHOR]