The resolvent analysis reveals the worst-case disturbances and the most amplified response in a fluid flow that can develop around a stationary base state. The recent work by Padovan et al. (J. Fluid Mech., vol. 900, 2020, A14) extended the classical resolvent analysis to the harmonic resolvent analysis framework by incorporating the time-varying nature of the base flow. The harmonic resolvent analysis can capture the triadic interactions between perturbations at two different frequencies through a base flow at a particular frequency. The singular values of the harmonic resolvent operator act as a gain between the spatiotemporal forcing and the response provided by the singular vectors. In the current study, we formulate the harmonic resolvent analysis framework for compressible flows based on the linearized Navier–Stokes equation (i.e. operator-based formulation). We validate our approach by applying the technique to the low-Mach-number flow past an airfoil. We further illustrate the application of this method to compressible cavity flows at Mach numbers of 0.6 and 0.8 with a length-to-depth ratio of $2$. For the cavity flow at a Mach number of 0.6, the harmonic resolvent analysis reveals that the nonlinear cross-frequency interactions dominate the amplification of perturbations at frequencies that are harmonics of the leading Rossiter mode in the nonlinear flow. The findings demonstrate a physically consistent representation of an energy transfer from slow-evolving modes toward fast-evolving modes in the flow through cross-frequency interactions. For the cavity flow at a Mach number of 0.8, the analysis also sheds light on the nature of cross-frequency interaction in a cavity flow with two coexisting resonances.

This work presents an experimental investigation focused on the analysis of aerodynamic properties between two interacting spheres in a supersonic rarefied flow. Atmospheric re-entries of space debris, whether natural or man-made, begin at altitude 120 km, and observations of historical re-entries have shown that fragmentation occurs between 90 and 50 km. The resulting fragments interact with each other, altering their own trajectories while traversing the different flow regimes between the free molecular and continuum regimes. This study focuses on the intermediate slip regime, where viscous effects of varying magnitude can influence the nature of the interactions of the shocks and modify them from the already known behaviour in the continuum regime. Specifically, this study examines how two spheres interact with each other upon re-entry into the atmosphere, focusing particularly on the six types of shock/shock interactions identified by Edney. The experiments were performed in the MARHy wind tunnel, in a steady Mach 4 laminar flow with static pressure 2.67 Pa. To highlight the differences between the six types of interferences, a variety of set-ups and devices were used: flow-field visualization, aerodynamic forces (through two diagnoses, aerodynamic balance and the swinging sphere technique) and wall pressure measurements. Results demonstrate the identification of differences according to the type of interference observed, showing in particular the viscous effect of rarefied flows by making a comparison with the continuum regime.

With a growing interest in low Reynolds number compressible flows, compressibility effects on the secondary instabilities developing on the circular cylinder periodic wake are investigated. The unsteady and time-averaged two-dimensional flows are characterised for Reynolds numbers ${{Re}} \in [200;350]$ and Mach numbers up to ${{M}}_{\infty }=0.5$, revealing different flow structures which influence the characteristics of the secondary unstable modes. The two-dimensional time-periodic solution is used as the base state for a global linear stability analysis performed by means of Floquet theory coupled with a time-stepping finite-difference approach of the nonlinear operator. The influence of compressibility on mode A and mode B secondary instabilities which are responsible for the three-dimensionalisation of the two-dimensional periodic wake is analysed. A stabilising or a destabilising effect of compressibility is observed on mode A, depending on the Reynolds number and the spanwise wavelength of the mode, while mode B is stabilised by the increase of the Mach number. Compressibility is indeed found to decrease the mode kinetic energy production due to base flow shear conversion, which drives the growth of mode B. This results in a delay of the three-dimensionalisation process of the wake due to compressibility.

In the quest to understand the basic universal features of compressible convection, one would like to disentangle genuine consequences of compression from spatial variations of transport properties. For instance, one may choose to consider a fluid with uniform dynamic viscosity, but, then, compressible effects will generate a density gradient and consequently the kinematic viscosity will not be uniform. In the present work, we consider a very peculiar equation of state, whereby entropy is solely dependent on density, so that a nearly isentropic fluid domain is nearly isochoric. Within this class of equations of state, there is a thermal adiabatic gradient and a key property of compressible convection is still present, namely its capacity to viscously dissipate a large fraction of the thermal energy involved, of the order of the well-named dissipation number. In the anelastic approximation, under the assumption of an infinite Prandtl number, the number of governing parameters can be brought down to two, the Rayleigh number and the dissipation number. This framework is proposed as a playground for compressible convection, an opportunity to extend the vast corpus of theoretical analyses on the Oberbeck–Boussinesq equations regarding stability, bifurcations or the determination of upper bounds for the turbulent heat transfer. Here, in a two-dimensional geometry, we concentrate on the structure of numerical solutions. For all Rayleigh numbers, a change in the vertical temperature profile is observed in the range of dissipation number between $0$ and less than $0.4$, associated with the weakening of ascending plumes. For larger dissipation numbers, the heat flux dependence on this number is found to be well predicted by Malkus's model of critical layers. For dissipation numbers of order unity, and large Rayleigh numbers, dissipation becomes related to the entropy heat flux at each depth, so that the vertical dissipation profile can be predicted, and so does the total ratio of dissipation to convective heat flux.

We present a detailed derivation of the Reynolds equation and its corresponding energy equation for three-dimensional, steady, laminar, compressible flows of single-phase Navier–Stokes fluids in thrust bearings. These equations are shown to be valid over most of the dense and supercritical gas regime except for the vicinity of the thermodynamic critical point. It is shown that the primary thermodynamic function governing the lubrication flow of high-pressure gases is the effective bulk modulus defined as the ratio of the bulk modulus to the shear viscosity. Numerical solutions to our Reynolds equation are obtained using a finite difference scheme for both moderate and high-speed flows. Approximate solutions to our Reynolds equation for high-speed flows are also derived through a perturbation analysis. It is found that boundary layers form on three out of four edges of the thrust pad. At the inner and outer radii of the pad, the flow is governed by a nonlinear heat equation. As the main flow leaves the pad, the flow is governed by a nonlinear relaxation equation. These three boundary layer solutions are rendered consistent by the construction of boundary layer solutions in the corner regions. A composite solution is developed which provides a single approximation and has the same accuracy as the individual approximations in their respective regions of validity.

Resonant gas oscillations in a closed straight tube contain shocks for frequencies near the linear resonant frequency. As the tube geometry changes from straight to a cone with slope $a,$ the shock strength decreases until, for large enough $a,$ the motion is continuous. The analytical result for small $a$ follows from a nonlinearization of the linear resonant response, while the result for large $a$ is a dominant single mode approximation. The connection between these two forms is analysed numerically. The shocked solutions change to multimode continuous solutions as $a$ increases to cross the curve $a_{s}\doteqdot 12.8M^{1/3}$, where $M\ll 1$ is the Mach number of the input. Then the amplitudes of the higher modes decrease as $a$ continues to increase until, having crossed $a_{m}\doteqdot 30M^{1/3}$, the single mode solution emerges.

In this paper, we propose a series expansion of the baroclinic torque in low-Mach-number flows, so that the accuracy and universality of any buoyancy term could be examined analytically, and new types of buoyancy terms could be constructed and validated. We first demonstrate that the purpose of introducing a buoyancy term is to approximate the baroclinic torque, and straightforwardly the error of any buoyancy term could be defined with the deviation of its curl from the corresponding baroclinic torque. Then a regular perturbation method is introduced for the elliptic equation of the hydrodynamic pressure in low-Mach-number flows, resulting in a sequence of Poisson equations, whose solutions lead to the series representation of the baroclinic torque and the new types of buoyancy terms. It is found that the frame invariance of the momentum equation is maintained with one of the new types of buoyancy terms. With the error definition of buoyancy terms and the series representation of the baroclinic torque, the validity and accuracy of previous and new buoyancy terms are examined. Finally, numerical simulations confirm that, with a decreasing density variation or an increasing order of our new buoyancy term, the simplified equations can converge to the original low-Mach-number equations.

Wind-tunnel experiments have been conducted on cylindrical models with canted fins. The fins introduced a swirling motion into the wake downstream of a blunt-based afterbody aligned with a Mach 2 flow. Measurements of the velocity field downstream of the models and the pressure distribution at the model base show evidence of two wake flow patterns distinctively differing from the classical supersonic wake, depending on the degree of rotation introduced. For a fin-cant angle of 16$^\circ$, a rotating wake flow with a central, downstream-directed vortex tube and a concentric, counter-rotating, toric vortex pair forms. A higher fin-cant angle of 32$^\circ$, in turn, results in a swirling flow surrounding a region of low-momentum flow at the axis. Near the central axis of the flow field an upstream flow establishes, extending from the far wake up to the model base. Numerical simulations have been performed to explain the fluid-dynamic processes and the origins of the experimentally observed structural changes of the rotating wakes. The results of the large-scale-turbulence-resolving simulations agree qualitatively well with the measured flow fields. The numerical results show that the centrifugal forces decrease the base pressure and cause the experimentally observed structural changes in the wake.

We use resolvent analysis to develop a physics-based, open-loop, unsteady control strategy to attenuate pressure fluctuations in turbulent flow over a rectangular cavity with a length-to-depth ratio of $6$ at a Mach number of $1.4$ and a Reynolds number based on cavity depth of $10\,000$. Large-eddy simulations (LES) of the baseline uncontrolled flow reveal the dominance of Rossiter modes II and IV that generate high-amplitude unsteadiness via trailing-edge impingement and oblique shock waves that obstruct the free stream. To suppress the oscillations, we introduce three-dimensional unsteady blowing along the cavity leading edge. We leverage resolvent analysis as a linear model with respect to the baseline flow to guide the selections of the optimal spanwise wavenumber and frequency of the unsteady actuation input for a fixed momentum coefficient of 0.02. Instead of choosing the most amplified resolvent forcing modes, we seek a disturbance that yields sustained amplification of the primary response mode-based kinetic energy distribution over the entire cavity length. This necessary but not sufficient guideline for effective mean flow modification is evaluated using LES of the controlled cavity flows. The most effective control case reduces the pressure root mean square level up to $52\,\%$ along cavity walls relative to the baseline and is approximately twice that achievable by comparable steady blowing. Dynamic mode decomposition on the controlled flows confirms that the optimal actuation input indeed suppresses the formation of the large-scale Rossiter modes. It is expected that the present flow control guideline derived from resolvent analysis will also be applicable at higher Reynolds numbers with the aid of physical insights and further validation.

This paper describes a computational investigation of multimode instability growth and multimaterial mixing induced by multiple shock waves in a high-energy-density (HED) environment, where pressures exceed 1 Mbar. The simulations are based on a series of experiments performed at the National Ignition Facility (NIF) and designed as an HED analogue of non-HED shock-tube studies of the Richtmyer–Meshkov instability and turbulent mixing. A three-dimensional computational modelling framework is presented. It treats many complications absent from canonical non-HED shock-tube flows, including distinct ion and free-electron internal energies, non-ideal equations of state, radiation transport and plasma-state mass diffusivities, viscosities and thermal conductivities. The simulations are tuned to the available NIF data, and traditional statistical quantities of turbulence are analysed. Integrated measures of turbulent kinetic energy and enstrophy both increase by over an order of magnitude due to reshock. Large contributions to enstrophy production during reshock are seen from both the baroclinic source and enstrophy–dilatation terms, highlighting the significance of fluid compressibility in the HED regime. Dimensional analysis reveals that Reynolds numbers and diffusive Péclet numbers in the HED flow are similar to those in a canonical non-HED analogue, but conductive Péclet numbers are much smaller in the HED flow due to efficient thermal conduction by free electrons. It is shown that the mechanism of electron thermal conduction significantly softens local spanwise gradients of both temperature and density, which causes a minor but non-negligible decrease in enstrophy production and small-scale mixing relative to a flow without this mechanism.

In the present study, compressible low-Reynolds-number flow past a stationary isolated sphere was investigated by direct numerical simulations of the Navier–Stokes equations using a body-fitted grid with high-order schemes. The Reynolds number based on free-stream quantities and the diameter of the sphere was set to be between 250 and 1000, and the free-stream Mach number was set to be between 0.3 and 2.0. As a result, it was clarified that the wake of the sphere is significantly stabilized as the Mach number increases, particularly at the Mach number greater than or equal to 0.95, but turbulent kinetic energy at the higher Mach numbers conditions is higher than that at the lower Mach numbers conditions of similar flow regimes. A rapid extension of the length of the recirculation region was observed under the transitional condition between the steady and unsteady flows. The drag coefficient increases as the Mach number increases mainly in the transonic regime and its increment is almost due to the increment in the pressure component. In addition, the increment in the drag coefficient is approximately a function of the Mach number and independent of the Reynolds number in the continuum regime. Moreover, the effect of the Mach and Reynolds numbers on the flow properties such as the drag coefficient and flow regime can approximately be characterized by the position of the separation point.

Shock-wave reflection over concave surfaces poses a difficulty in its analysis due to the unsteady nature of the reflection process and the occurrence of various types of Mach reflections caused by it. In a pseudo-steady flow, the reflection's configuration is self-similar since the shock wave reflects over a surface with constant inclination. The unsteady Mach reflection introduces an additional complexity as it is affected by the changing inclination of the surface, forcing the reflection to continuously adjust itself to the varying boundary condition. In this study, validated simulations of Mach reflection (MR) over cylindrical concave surfaces with different radii were performed for three inviscid perfect gases with moderate incident shock Mach numbers (Ms) ranging from 1.3 to 1.5. The reflection was investigated up to the point of transition from MR to transitioned regular reflection. A similar behaviour of the configuration and evolution of the Mach stem was observed, one that is independent of the surface radius and type of gas. With regards to different gases, the speed of sound a0 is a dominant factor since it dictates the propagation of wall disturbances. A universal condition of the rate of surface change was found, accounting for different radii, different gases and Ms variation. Analysis based on shock dynamics is employed to explain how disturbances caused by surface variations play a significant role in the behaviour of the reflection. This method successfully supports the similarity that was demonstrated and facilitates a more informed perception of the MR process.

Fluid motion has two well-known fundamental processes: the vector transverse process characterized by vorticity, and the scalar longitudinal process consisting of a sound mode and an entropy mode, characterized by dilatation and thermodynamic variables. The existing theories for the sound mode involve the multi-variable issue and its associated difficulty of source identification. In this paper, we define the source of sound inside the fluid by the objective causality inherent in dynamic equations relevant to a longitudinal process, which naturally favours the material time-rate operator $D/Dt$ rather than the local time-rate operator $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t$, and describes the sound mode by inhomogeneous advective wave equations. The sources of sound physical production inside the fluid are then examined at two levels. For the conventional formulation in terms of thermodynamic variables at the first level, we show that the universal kinematic source can be condensed to a scalar invariant of the surface deformation tensor. Further, in the formulation in terms of dilatation at the second level, we find that the sound mode in viscous and heat-conducting flow has sources from rich nonlinear couplings of vorticity, entropy and surface deformation, which cannot be disclosed at the first level. Preliminary numerical demonstration of the theoretical findings is made for two typical compressible flows, i.e. the interaction of two corotating Gaussian vortices and the unsteady type IV shock/shock interaction. The results obtained in this study provide a new theoretical basis for, and physical insight into, understanding various nonlinear longitudinal processes and the interactions therein.

A focused laser can cause local optical breakdown of a gas, which leads to rapid deposition of energy into a high-temperature plasma kernel that expands and induces a complex flow. For some conditions, hot gas is rapidly ejected along the laser axis up to distances several times the kernel size, with a particularly curious feature: relatively small changes in, for example, initial pressure can cause the direction of this ejection to reverse. Detailed axisymmetric simulations of a model energy kernel in an inert gas provide a hydrodynamic description of this phenomenon, reproducing key observations in corresponding experiments, including the vortex-ring-like features that constitute the ejection. These simulations are analysed to show how changes in the early-time kernel can lead to ejection or its reversal via alteration in the relative strength and position of the vorticity produced. A corresponding semi-infinite geometry is used to isolate two mechanisms: vorticity production by the generated shock and by baroclinic torque at the kernel boundary. Dependence on the initial kernel asymmetry is quantified, as it ultimately determines whether the vorticity, upon its subsequent evolution, develops into the ring-like structure that ejects. Even simple elongation of the energy kernel alone can reverse the direction.

This article shows the importance of flow compressibility on the heat transfer in confined impinging jets, and how it is driven by both the Mach number and the wall heat flux. Hence, we present a collection of cases at several Mach numbers with different heat-flux values applied at the impingement wall. The wall temperature scales linearly with the imposed heat flux and the adiabatic wall temperature is found to be purely governed by the flow compression. Especially for high heat-flux values, the non-constant wall temperature induces considerable differences in the thermal conductivity of the fluid. This phenomenon has to date not been discussed and it strongly modulates the Nusselt number. In contrast, the heat transfer coefficient is independent of the varying thermal properties of the fluid and the wall heat flux. Furthermore, we introduce the impingement efficiency, which highlights the areas of the wall where the temperature is influenced by compressibility effects. This parameter shows how the contribution of the flow compression to raising the wall temperature becomes more dominant as the heat flux decreases. Thus, knowing the adiabatic wall temperature is indispensable for obtaining the correct heat transfer coefficient when low heat-flux values are used, even at low Mach numbers. Lastly, a detailed analysis of the dilatation field also shows how the compressibility effects only affect the heat transfer in the vicinity of the stagnation point. These compressibility effects decay rapidly further away from the flow impingement, and the density changes along the developing boundary layer are caused instead by variable inertia effects.

A small-disturbance model is presented for the complex dynamics of vortex flows of moist air in a straight, circular pipe with non-equilibrium and homogeneous condensation. The model explores the nonlinear interactions among the vortex near-critical swirl ratio and the small amount of water vapour in the air. The condensation rate is calculated according to classical nucleation and droplet growth models. The asymptotic analysis gives the similarity parameters that govern the flow problem. These are the flow inlet swirl ratio $\unicode[STIX]{x1D714}$, the inlet Mach number $Ma_{0}$, the initial humidity $\tilde{\unicode[STIX]{x1D714}}_{0}$, the number of water molecules in a characteristic fluid element $n_{C}$, the inlet centreline super-saturation ratio $S_{0}$ and the ratio of characteristic condensation and flow time scales $K$. Also, the flow field may be described by an ordinary first-order nonlinear differential equation for the flow evolution coupled with a set of four first-order ordinary differential equations along the pipe for the calculation of the condensate mass fraction. An iterative numerical scheme which combines the Runge–Kutta integration technique for the flow dynamics with Simpson’s integration rule for the calculation of the condensation variables is developed. Specifically, equilibrium states are determined, including the possibility of the appearance of multiple states under the same boundary conditions, and the stability characteristics of these states are described. The model is used to study the effects of humidity and of energy supply from nanoscale condensation processes on the large-scale dynamics of vortex flows as well as the effect of flow swirl on condensation processes in swirling flows.

A small-disturbance asymptotic model is derived to describe the complex nature of a pure water vapour flow with non-equilibrium and homogeneous condensation around a thin airfoil operating at transonic speed and small angle of attack. The van der Waals equation of state provides real-gas relationships among the thermodynamic properties of water vapour. Classical nucleation and droplet growth theory is used to model the condensation process. The similarity parameters which determine the flow and condensation physics are identified. The flow may be described by a nonlinear and non-homogeneous partial differential equation coupled with a set of four ordinary differential equations to model the condensation process. The model problem is used to study the effects of independent variation of the upstream flow and thermodynamic conditions, airfoil geometry and angle of attack on the pressure and condensate mass fraction distributions along the airfoil surfaces and the consequent effect on the wave drag and lift coefficients. Increasing the upstream temperature at fixed values of upstream supersaturation ratio and Mach number results in increased condensation and higher wave drag coefficient. Increasing the upstream supersaturation ratio at fixed values of upstream temperature and Mach number also results in increased condensation and the wave drag coefficient increases nonlinearly. In addition, the effects of varying airfoil geometry with a fixed thickness ratio and chord on flow properties and condensation region are studied. The computed results confirm the similarity law of Zierep & Lin (Forsch. Ing. Wes. A, vol. 33 (6), 1967, pp. 169–172), relating the onset condensation Mach number to upstream stagnation relative humidity, when an effective specific heat ratio is used. The small-disturbance model is a useful tool to analyse the physics of high-speed condensing steam flows around airfoils operating at high pressures and temperatures.

We present a novel analytical model for the evolution of hydroacoustic waves in weakly compressible fluids characterised by depth variations of the sound speed profile. Using a perturbation expansion in terms of the small vertical variation of the sound speed, we derive a novel expression for the second-order velocity potential and show that this solution does not exist in the case of homogeneous sound speed. At the third order, we derive a linear Schrödinger equation governing the evolution of the wave envelope for large length and time scales, which features new terms depending on the sound speed distribution. We show that for generalised sound speed vertical profiles the frequency of the hydroacoustic signal can increase or decrease with respect to the constant sound speed case, depending on the profile. This has substantial implications on the speed of the wavetrain envelope. Our findings suggest the need to extend existing models that neglect the sound speed vertical variation, especially in view of applications to tsunami early warning.

We developed a numerical method for the set of equations governing fully compressible convection in the limit of infinite Prandtl numbers. Reduced models have also been analysed, such as the anelastic approximation and the anelastic liquid approximation. The tests of our numerical schemes against self-consistent criteria have shown that our numerical simulations are consistent from the point of view of energy dissipation, heat transfer and entropy budget. The equation of state of an ideal gas has been considered in this work. Specific effects arising because of the compressibility of the fluid are studied, like the scaling of viscous dissipation and the scaling of the heat flux contribution due to the mechanical power exerted by viscous forces. We analysed the solutions obtained with each model (fully compressible model, anelastic and anelastic liquid approximations) in a wide range of dimensionless parameters and determined the errors induced by each approximation with respect to the fully compressible solutions. Based on a rationale on the development of the thermal boundary layers, we can explain reasonably well the differences between the fully compressible and anelastic models, in terms of both the heat transfer and viscous dissipation dependence on compressibility. This could be mostly an effect of density variations on thermal diffusivity. Based on the different forms of entropy balance between exact and anelastic models, we find that a necessary condition for convergence of the anelastic results to the exact solutions is that the product $\unicode[STIX]{x1D716}q$ must be small compared to unity, where $\unicode[STIX]{x1D716}$ is the ratio of the superadiabatic temperature difference to the adiabatic difference, and $q$ is the ratio of the superadiabatic heat flux to the heat flux conducted along the adiabat. The same condition seems also to be associated with a convergence of the computed heat fluxes. Concerning the anelastic liquid approximation, we confirm previous estimates by Anufriev et al. (Phys. Earth Planet. Inter., vol. 152, 2005, pp. 163–190) and find that its results become generally close to those of the fully compressible model when $\unicode[STIX]{x1D6FC}T{\mathcal{D}}$ is small compared to unity, where $\unicode[STIX]{x1D6FC}$ is the isobaric thermal expansion coefficient, $T$ is the temperature (here $\unicode[STIX]{x1D6FC}T=1$ for an ideal gas) and ${\mathcal{D}}$ is the dissipation number.

An acoustically forced fluid system is known to generate a time-averaged mean, or streaming, flow that evolves on a slow time scale compared to the acoustic-wave period. Classical acoustic streaming in a homogeneous fluid is typically associated with a one-way coupled system wherein the oscillatory acoustic fields inform the streaming mean flow, without any appreciable feedback. In contrast, Michel & Chini (J. Fluid Mech., vol. 858, 2019, pp. 536–564) investigate acoustic streaming in a stratified fluid and demonstrate that the streaming is sufficiently strong to induce significant rearrangements of the background temperature and density fields, resulting in a strong coupling between the acoustic waves and mean flow. This new class of streaming, referred to as baroclinic acoustic streaming, is shown to result in altered streaming patterns with enhanced heat transport that makes possible a range of new applications.