The gas dynamics of shock-induced gas filtration through densely packed granular columns with vastly varying shock intensity and the structural parameters are numerically investigated using a coupled Eulerian–Lagrangian approach. The results shed fundamental light on the thermal effects of the shock-induced gas filtration manifested by a distinctive self-heating hot gas layer traversing the medium. The characteristics of the thermal effects in terms of the thermal intensity and uniformity are found to vary with the shock Mach number, Ms, and the filtration coefficient of the granular media, Π. As the incident shock transitions from weak to strong, and (or) the filtration coefficient increases from O(10−5) to O(104), the heating mechanisms transition between three distinct heating modes. A phase diagram of heating modes is established on the parameter space (Ms, Π), which enables us to predict the characteristics of the thermal effect in different shock-induced gas filtrations. The thermal effects markedly accelerate the pressure diffusion due to the additional heat influx when the time scale of the former is smaller than or comparable to the latter. Based on the contour map displaying the coupling degree of the thermal effects and the pressure diffusion, we identify a decoupling criterion whereby the isothermal assumption holds if only the pressure diffusion is concerned. The thermal effects may well bring about considerable thermal shocks which pose a great threat to the integrity of the solid skeleton and further reduce the overall shock resistance performance of the porous media.

The dynamics of a shock-induced separation unit generated by a 20$^\circ$ sharp fin placed on a cylindrical surface in a Mach 2.5 flow was investigated. Specifically, the present work investigated the mechanisms that govern the mid-frequency range of separation shock unsteadiness in the fin shock wave–boundary layer interaction (SBLI) unit. Two-dimensional pressure fields were obtained over the cylinder surface spanning the entire fin SBLI unit using high-bandwidth pressure-sensitive paint at 40 kHz imaging rate that allowed probing the low- through mid-frequency ranges of the separation shock unsteadiness. The mean pressure field showed a progressive weakening of the separation shock with downstream distance, which is an artifact of the three-dimensional relief offered by the curved mounting surface. The root-mean-square (r.m.s.) pressure field exhibited a banded structure with elevated $p_{r.m.s.}$ levels beneath the intermittent region, separation vortex and adjacent to the fin root. The power spectral density (PSD) of the surface pressure fluctuations obtained beneath the intermittent region revealed that the separation shock oscillations exhibited the mid-frequency content over the majority of its length. Interestingly, neither the PSD nor the length of the intermittent region varied noticeably with downstream distance, revealing a constant separation shock foot velocity along the entire SBLI. The pressure fluctuation PSD beneath the separation vortex also exhibited the broadband peak at the mid-frequency range of the separation shock motions over the majority of its length within the measurement domain. By contrast, the region adjacent to the fin root exhibited pressure oscillations at a substantially lower frequency compared with the separation shock and the separation vortex. Two-point coherence and cross-correlation analysis provided unique insights into the critical sources and mechanisms that drive the separation shock unsteadiness. The separation vortex and separation shock dynamics were found to be driven by a combination of convecting perturbations that originated from the vicinity of the fin leading edge and the local interactions of the separated flow with the incoming boundary layer. The boundary layer locally strengthened or weakened the convecting pressure perturbations depending on the local momentum fluctuations within the boundary layer.

Shock refraction in a gas–liquid interface is ubiquitous in nature and engineering. This study investigates the shock refraction phenomena in air–water interfaces for various inclination angles. The interface inclination angles are achieved using a tiltable vertical shock tube. The time-resolved schlieren images are compared with numerical simulations performed using the BlastFoam solver in the OpenFOAM software. The stiffened gas equation of state is used to model water in the simulations. The shock polar analysis using modified shock relations for a stiffened gas is used to elucidate the refraction patterns. A regular refraction pattern with a reflected shock wave and a bound precursor refraction with a regular reflection are observed experimentally for the first time in an air–water system. Further, a new free precursor refraction pattern with a Mach reflection is observed. The transition criteria and the corresponding boundaries for each refraction pattern are demarcated in the ($M_S, \theta _w^c$)-plane. The refraction sequence and the range for various incident shock strength regimes are also identified.

Standing acoustic waves have been known to generate Eulerian time-mean ‘streaming’ flows at least since the seminal investigation of Lord Rayleigh in the 1880s. Nevertheless, a recent body of numerical and experimental evidence has shown that inhomogeneities in the ambient density distribution lead to much faster flows than arise in classical Rayleigh streaming. The emergence of these unusually strong flows creates new opportunities to enhance heat transfer in systems in which convective cooling cannot otherwise be easily achieved. To assess this possibility, a theoretical study of acoustic streaming in an ideal gas confined in a rectangular channel with top and bottom walls maintained at fixed but differing temperatures is performed. A two time scale system of equations is utilized to efficiently capture the coupling between the fast acoustic waves and the slowly evolving streaming flow, enabling strongly nonlinear regimes to be accessed. A large suite of numerical simulations is carried out to probe the streaming dynamics, to highlight the critical role played by baroclinically generated wave vorticity and to quantify the additional heat flux induced by the standing acoustic wave. Proper treatment of the two-way coupling between the waves and mean flow is found to be essential for convergence to a self-consistent steady state, and the variation of the resulting acoustically enhanced steady-state heat flux with both the amplitude of the acoustic wave and the $O(1)$ aspect ratio of the channel is documented. For certain parameters, heat fluxes almost two orders of magnitude larger than those realizable by conduction alone can be attained.

Cavity evolution in granular media is crucial in explosion-driven gas–particle flows, particularly in many engineering applications. In this study, a theoretical model was first proposed to describe the cavity evolution in granular media by extending the classical Rayleigh–Plesset model. A closed equation set comprising the radius, pressure and gas leak-off velocity equations was built by considering the gas expansion and non-Darcy gas-penetration effects. Both centrally symmetric and non-centrally symmetric cases of gas injection into granular media were investigated. Especially for modelling the non-symmetric scenario, the radius and gas leak-off velocity equations were proposed in each radial direction with angle $\theta$, and then the pressure equation was built up based on the integral gas leak-off along the cavity outline. Through non-dimensionalizing the theoretical equations, four key dimensionless numbers $\varPi_1,\ \varPi_4$ were obtained to characterize the response time of cavity expansion and the intensity of non-Darcy effects for both cases. This allowed us to determine a scaling law of effective cavity radius $R_{eff}^*=\sqrt {2\varPi _2/(7{\rm \pi} )}t^{*1/2}$ and the critical time $t_{cr}^*=\sqrt {12.5/\varPi _1}$ for two-dimensional cavity evolution. Additionally, the necessity of incorporating non-Darcy effects was ascertained under conditions of $\varPi _4>400$. The findings demonstrate that the proposed theoretical equations effectively predict the cavity evolution results under various operational conditions ($0.7<\varPi _1<7\times 10^2, 3<\varPi _4<1.1\times 10^3$), as validated by refined Euler–Lagrange numerical simulations.

Shock waves are of great interest in many fields of science and engineering, but the mechanisms of their formation, maintenance and dissipation are still not well understood. While all transport processes existing in a shock wave contribute to its compression and irreversibility, they are not of equal importance. To figure out the roles of viscosity and heat conduction in shock transition, the existence of smooth shock solutions and the counter-intuitive entropy overshoot phenomenon (the specific entropy is not monotonically increasing and exhibits a peak inside the shock front) are theoretically and numerically investigated, with emphasis on the effects of viscosity and heat conduction. Instead of higher-order hydrodynamics, the Navier–Stokes formalism is employed for its stability and simplicity. Supplemented with nonlinear thermodynamically consistent constitutive relations, the Navier–Stokes equations are adequate to demonstrate the general nature of shock profiles. It is found that heat conduction cannot sustain strong shocks without the presence of viscosity, while viscosity can maintain smooth shock transition at all strengths, regardless of heat conduction. Hence, the critical role in shock compression is played by viscosity rather than heat conduction. Nevertheless, the dispensability of heat conduction would not compromise its essential role in the emergence of an entropy peak. It is the entropy flux resulting from heat conduction that neutralises the positive entropy production and thus prevents the decreasing entropy from violating the second law of thermodynamics. This mechanism of entropy overshoot has not been addressed previously in the literature and is revealed using the entropy balance equation.

Two-dimensional compressible flows in radial equilibrium are investigated in the ideal dilute-gas regime and the non-ideal single-phase regime close to the liquid–vapour saturation curve and the critical point. Radial equilibrium flows along constant-curvature streamlines are considered. All properties are therefore independent of the tangential streamwise coordinate. A differential relation for the Mach number dependency on the radius is derived for both ideal and non-ideal conditions. For ideal flows, the differential relation is integrated analytically. Assuming a constant specific heat ratio $\gamma$, the Mach number is a monotonically decreasing function of the radius of curvature for ideal flows, with $\gamma$ being the only fluid-dependent parameter. In non-ideal conditions, the Mach number profile also depends on the total thermodynamic conditions of the fluid. For high molecular complexity fluids, such as toluene or hexamethyldisiloxane, a non-monotone Mach number profile is admissible in single-phase supersonic conditions. For Bethe–Zel'dovich–Thompson fluids, non-monotone behaviour is observed in subsonic conditions. Numerical simulations of subsonic and supersonic turning flows are carried out using the streamline curvature method and the computational fluid dynamics software SU2, respectively, both confirming the flow evolution from uniform flow conditions to the radial equilibrium profile predicted by the theory.

The choking condition of a one-dimensional steady-state diabatic flow, with wall friction, of a perfect gas through a constant cross-section pipe has been analysed by applying a recent analytical solution. Such a choking condition can be achieved for an initially subsonic flow and a supersonic one, even when the heat flux goes from the fluid to the wall. Entropy–enthalpy diagrams have been analysed, and a universal choking condition for compressible diabatic flows with wall friction has been determined. The analytical solution of a supersonic flow, in which a normal shock occurs, has been obtained for a diabatic flow with wall friction and compared with the results of a numerical model.

Green hydrogen is expected to have a key role in the transition towards a carbon-neutral society, particularly in the transportation sector. Exploration of novel solutions for the direct injection of hydrogen in internal combustion engines (ICEs) is a main topic for both academia and industry. Here, the authors explore the gas-dynamic and mixing features of hydrogen under-expanded jets exiting from non-axisymmetric cross-sections, with the aim to provide guidelines for designing novel generations of ICE hydrogen injectors. Triangular and star shapes have been compared with elliptical and rectangular sections with different aspect ratios. Differences in the shock wave systems are reported, and explanations of the gas-dynamic mechanisms developed by each section are proposed. Non-uniformity in the radial expansion of the jet boundary has been noticed in all of the non-axisymmetric sections, leading to an axis switching of the jet in some cases. Differences in the radial mixing and axial jet penetration have been reported too, and a robust correlation with the vorticity distribution along the jet boundary has been observed.

Indirect noise is a significant contributor to aircraft engine noise, which needs to be minimized in the design of aircraft engines. Indirect noise is caused by the acceleration of flow inhomogeneities through a nozzle. High-fidelity simulations showed that some flow inhomogeneities can be chemically reacting when they leave the combustor and enter the nozzle (Giusti et al., Trans. ASME J. Engng Gas Turbines Power, vol. 141, issue 1, 2019). The state-of-the-art models, however, are limited to chemically non-reacting (frozen) flows. In this work, first, we propose a low-order model to predict indirect noise in nozzle flows with reacting inhomogeneities. Second, we identify the physical sources of sound, which generate indirect noise via two physical mechanisms: (i) chemical reaction generates compositional perturbations, thereby adding to compositional noise; and (ii) exothermic reaction generates entropy perturbations. Third, we numerically compute the nozzle transfer functions for different frequency ranges (Helmholtz numbers) and reaction rates (Damköhler numbers) in subsonic flows with hydrogen and methane inhomogeneities. Fourth, we extend the model to supersonic flows. We find that hydrogen inhomogeneities have a larger impact on indirect noise than methane inhomogeneities. Both the Damköhler number and the Helmholtz number markedly influence the phase and magnitude of the transmitted and reflected waves, which affect sound generation and thermoacoustic stability. This work provides a physics-based low-order model which can open new opportunities for predicting noise emissions and instabilities in aeronautical gas turbines with multi-physics flows.

We present a theoretical and numerical stability analysis for a piston-driven planar shock against two-dimensional perturbations. The results agree with the well-established theory for isolated planar shocks: in the range of $h_c< h<1+2 {\mathcal {M}}_2$, where $h$ is the D'yakov–Kontorovich (DK) parameter related to the slope of the Rankine–Hugoniot curve, $h_c$ is its critical value corresponding to the onset of the spontaneous acoustic emission (SAE) and ${\mathcal {M}}_2$ is the downstream Mach number, non-decaying oscillations of shock-front ripples occur. The effect of the piston is manifested in the presence of additional frequencies occurring by the reflection of the sonic waves on the piston surface that can reach the shock. An unstable behaviour of the shock perturbation is found to be possible when there is an external excitation source affecting the shock, whose frequency coincides with the self-induced oscillation frequency in the SAE regime, thereby being limited to the range $h_c< h<1+2 {\mathcal {M}}_2$. An unstable evolution of the shock is also observed for planar shocks restricted to one-dimensional perturbations within the range $1< h<1+2 {\mathcal {M}}_2$. Both numerical integration of the Euler equations via the method of characteristics and theoretical analysis via Laplace transform are employed to cross-validate the results.

We propose a physical model to predict indirect noise generated by the acceleration of compositional inhomogeneities in nozzles with viscous dissipation (non-isentropic nozzles). First, we derive the quasi-one-dimensional equations from the conservation laws of multicomponent flows. Second, we validate the proposed model with the experimental data available in the literature for binary mixtures of four gases. Third, we calculate the nozzle transfer functions for different Helmholtz numbers and friction factors, in both subsonic and supersonic flows with/without shock waves. We show that friction and dissipation have a significant effect on the generation of indirect noise, for which the physical mechanism is identified and explained. Fourth, we find a semi-analytical solution with path integrals, which provides an asymptotic expansion with respect to the Helmholtz number. Fifth, we introduce the compositional-noise scaling factor, which is applied to quickly estimate compositional noise from the knowledge of only one single-component gas transfer function. The approximation error is less than $1\,\%$. The proposed low-order model provides accurate estimates of the transfer functions and physical insight into indirect noise for multicomponent gases. This opens up new possibilities to accurately predict, and understand, sound generation in gas turbines.

In this paper, the reflection of curved shock waves over a symmetry plane in planar supersonic flow is studied. This includes stable Mach reflection (MR) and the regular reflection (RR) to MR transition process. Curved shock theory (CST) is applied to derive the high-order parameters in front of and behind the shock wave. The method of curved shock characteristics is used to establish an analytical model to predict the wave configurations. The shock structures provided by the proposed model agree well with the numerical results. Flow structures, such as the height of the Mach stem and the shape of the shock wave and slip line, are studied by applying the analytical model. Isentropic waves generated from a curved wall are found to significantly influence the flow patterns. It appears that the compression waves obstruct the formation of the sonic throat and increase the Mach-stem height. The expansion waves have the opposite effect. The evolution mechanism of the Mach stem is found in conjunction with the RR-to-MR transition process. The CST is extended to a moving frame and used to model the transition. The time history of the moving triple point illustrates the effects of the incident shock angle and isentropic waves on the transition process.

The influence of viscosity on the Mach reflection of shock waves in a steady flow of a monatomic gas is studied by solving the Navier–Stokes equations numerically. Based on the nested block grid refinement technique, the flow near the shock wave intersection is simulated, and its behaviour with increasing Reynolds number is studied. The computations are performed for the interaction of both strong (free-stream Mach number $M_\infty = 4$) and weak ($M_\infty = 1.7$) shock waves. In the strong reflection of shock waves at all Reynolds numbers in the examined range, it is found that there exists a small-size zone behind the shock wave intersection where the flow parameters differ from those predicted by the Rankine–Hugoniot relations and hence deviate from the predictions of the inviscid three-shock theory. The structure of this zone is self-similar: in coordinates normalised to the mean free path of molecules in the free stream. The structure is identical at all Reynolds numbers considered in the study. As the Reynolds number increases, the size of this zone in physical coordinates decreases, but the maximum difference between the viscous and inviscid solutions in this zone remains constant, reaching approximately $10\,\%$ for pressure. In the weak reflection of shock waves, the flow structure behind the shock wave intersection is not self-similar, i.e. the flow fields at different Reynolds numbers do not coincide in the normalised coordinates, but converge, as the Reynolds number increases, to the parameters predicted by the inviscid three-shock theory.

In the classical irreversible thermodynamics (CIT) framework, the Navier–Stokes–Fourier constitutive equations are obtained so as to satisfy the entropy inequality, by and large assuming that the entropy flux is equal to the heat flux over the temperature. This article is focused on the derivation of second-order constitutive equations for polyatomic gases; it takes the basis of CIT, but most importantly, allows up to quadratic nonlinearities in the entropy flux. Mathematical similarities between the proposed model and the classic Stokes–Laplace equations are exploited so as to construct analytic/semi-analytic solutions for the slow rarefied gas flow over different shapes. A set of second-order boundary conditions are formulated such that the model's prediction for the drag force is in excellent agreement with the experimental data over the whole range of Knudsen numbers. We have also computed the normal shock structure in nitrogen for Mach ${Ma} \lesssim 4$. A very good agreement was observed with the kinetic theory, as well as with the experimental data.

A closed-form solution is presented for internal compressible flows with an arbitrary combination of area change, heat addition, friction and non-uniform flow. The approach assumes steady flow of a calorically perfect ideal gas using the integral form of the governing equations. It is shown that the equations reduce to a simple, bi-quadratic equation in exit Mach number. This solution is compared to previously known solutions for Rayleigh flow, Fanno flow, isentropic flow and normal shock relations. Theoretical solutions for sudden expansion, sudden contraction, and supersonic–supersonic two-stream constant area mixing are also presented. These solutions were found to be in agreement with data available in the literature.

Viscous effects on an ideal gas flow in a supersonic convergent–divergent nozzle are a well-studied subject in classical gas dynamics. However, the ideal assumption fails on fluids that exhibit complex behaviours such as near-critical-region and non-ideal dense vapours. Under those conditions, a realistic equation of state (EOS) plays a vital role for a precise and realistic computation. This work examines the problem for solving the quasi-one-dimensional viscous compressible flow using a realistic EOS. The governing equations are discretised and solved using the fourth-order Runge–Kutta method coupled with a state-of-the-art EOS to calculate the thermodynamic properties. The role of the Grüneisen parameter along with viscous and real gas effects and their influence on the sonic point formation are discussed. The study shows that the flow may not achieve the supersonic regime for any pressure ratio depending on the combination of that parameter and the normalised friction factor. In addition, the analysis yields the discharge coefficient and the isentropic nozzle efficiency, which may achieve maximum values as a function of the stagnation conditions. Finally, the study also evaluates the formation and intensity of normal shock waves by using the Rankine–Hugoniot relations, which now depend on the real gas and viscous effects in opposition to the inviscid solution. Moreover, the methodology used captures the sonic point and shock wave position by a space marching algorithm using the Brent method for scalar minimisation. Experimental data available in the open literature corroborate the approach.

The classical Mott-Smith solution for one-dimensional normal shock wave structure is extended to the two-dimensional regular shock reflection problem. The solution for the non-equilibrium molecular velocity distribution function along the symmetry-plane streamline is obtained as a weighted sum of four Maxwellians. An analysis of applicability of the solution has been performed using the results of direct simulation Monte Carlo calculations for a range of incident shock wave intensities. Accuracy of the solution improves with increasing $Ma_n$, the Mach number normal to the shock front, so that the solution becomes rather accurate for strong shocks with $Ma_n>8$.