Entropy-based viscous regularization for the multi-dimensional Euler equations in low-Mach and transonic flows

Context. Many problems in stellar astrophysics feature flows at low Mach numbers. Conventional compressible hydrodynamics schemes frequently used in the field have been developed for the transonic regime and exhibit excessive numerical dissipation for these flows. Aims. While schemes were proposed that solve hydrodynamics strictly in the low Mach regime and thus restrict their applicability, we aim at developing a scheme that correctly operates in a wide range of Mach numbers. Methods. Based on an analysis of the asymptotic behavior of the Euler equations in the low Mach limit we propose a novel scheme that is able to maintain a low Mach number flow setup while retaining all effects of compressibility. This is achieved by a suitable modification of the well-known Roe solver. Results. Numerical tests demonstrate the capability of this new scheme to reproduce slow flow structures even in moderate numerical resolution. Conclusions. Our scheme provides a promising approach to a consistent multidimensional hydrodynamical treatment of astrophysical low Mach number problems such as convection, instabilities, and mixing in stellar evolution.

A low diffusion version of the Harten–Lax–Leer convective pressure split (HLL-CPS) scheme for resolving the shear layers and the flow features at low Mach numbers is presented here. The low diffusion HLL-CPS scheme is obtained by reconstructing the velocities at the cell interface with the face normal Mach number and a pressure function. Asymptotic analysis of the modified scheme shows a correct scaling of the pressure at low Mach numbers and a significant reduction in numerical dissipation. The robustness of the HLL-CPS scheme for strong shock is improved by reducing the contribution of the contact wave in the vicinity of the shock. The improvement in robustness for strong shock is demonstrated analytically through linear perturbation and matrix stability analyses. A set of numerical test cases are solved to demonstrate the efficacy of the proposed scheme over a wide range of Mach numbers.

This study comprehensively investigates the response of a combusting droplet during its interaction with a high-speed transient flow imposed by a coaxially propagating blast wave. The blast wave is generated using a specially designed miniature shock generator that produces blast waves using the wire-explosion technique, facilitating a wide range of Mach numbers (1.03 < Ms < 1.8). The experiments are performed in two configurations: open field and focused blast wave. The charging voltage and the configuration determine the Mach number (Ms) and flow characteristics. The flame is found to exhibit two major response patterns: partial extinction followed by reignition and full extinction. Increasing the Mach number (Ms > 1.1) makes the droplet flame more vulnerable to extinction. Additionally, the flame exhibits stretching and shedding, followed by reignition at lower Mach numbers (Ms < 1.06). In all cases, the flame base lifts off in response to the imposed flow, and the advection of the flame base interacting with the flame tip results in flame extinction. The entire interaction occurs in two stages: (i) interaction with the blast wave and the decaying velocity profile associated with it, and (ii) interaction with the induced flow behind the blast wave as a result of the entrainment (delayed response). Alongside the flame's response, the droplet also interacts with the flow imposed by the blast wave, exhibiting different response modes including pure deformation, Rayleigh–Taylor piercing bag breakup and shear-induced stripping.

Features of the simulation of low-velocity inviscid and viscous compressible gas flows are considered, and a finite-volume discretization of gas dynamics equations at low Mach numbers on unstructured meshes is discussed. Preconditioning based on the use of physical variables is used to speed up the convergence of time marching to a steady state and to improve the accuracy of the steady-state solution. The structure of the preconditioning matrix and the diagonalization of the Jacobian of the preconditioned system of equations are discussed. The capabilities of this approach are demonstrated using model gasdynamic simulations ing a wide range of Mach numbers.

This work focuses on the study of the asymptotic preserving property of a semi-implicit method. The semi-implicit method, which is a pressure-based method, has been successfully used to simulate two-phase flows in numerous industrial applications. This method is used in our studies due to the fact that pressure-based methods generally perform well at low Mach numbers. The semi-implicit method is applied to the homogeneous equilibrium model (HEM) in this work to simulate two-phase flows. We show that the semi-implicit method is asymptotic preserving, i.e. the discretization for a compressible model tends to a consistent discretization for the related incompressible model at the low Mach number limit. Finally, test cases are performed to show that the numerical method is able to deal with low Mach number flows, as well as flows with a wide range of Mach numbers.

Tracer pathline statistics have rarely, if at all, been used in the investigation of compressible subsonic, transonic and supersonic flows past a circular cylinder. Here, we report pathline histogram, conditional histogram and zonal histogram data, focusing on the characteristics of particle instantaneous flow reversal and their dependencies on Mach number and tracer release position. The far-upstream Mach number and the diameter-based Reynolds number used for these three flow regimes are ( 0.2 , 10000 ) , ( 0.9 , 3900 ) and ( 1.2 , 10000 ) , respectively. The subsonic surface-release histogram profiles display unexpected persistent oscillations at a frequency 50 times of the periodic vortex shedding frequency, reflecting the effect of a very small magnitude zig-zagging type of particle motion in a region very close to the wall and slightly behind the mean flow separation location. No similar high-frequency oscillation is observed in the corresponding transonic and supersonic histograms likely because, at Ma ≥ 0.9 , periodic vortex shedding ceases to exist and the near-wake becomes a quasi-laminar recirculation zone bounded by a pair of converging slip-layers with a neck opening to the far-wake. At the instant of 5 subsonic vortex-shedding periods after being released from the cylinder surface, there are 1.67 % , 35.2 % and 39.7 % particles that experience flow reversal in the subsonic, transonic and supersonic flows, respectively. Surprisingly, in the present transonic flow, growth of the turbulent far-wake behind the pair of lambda-shocks is constrained over a considerable streamwise distance.

This paper discusses the sound generated when an inhomogeneity in density is convected in a low Mach number steady flow through a contraction in a duct of infinite extent, and also when the inhomogeneity exhausts through a nozzle into free space. The analyses of Candel (1972) and Marble (1973) for the case of duct flow were based on a frequency decomposition of the incident inhomogeneity and cannot adequately deal with sharp-fronted inhomogeneities and entropy spots. However, the practical difficulties of this earlier work can be avoided at low flow Mach numbers by conducting the analysis in terms of an approximate expression for the acoustic Green's function in the manner described by Howe (1975). This method also permits a considerable extension of the range of the earlier investigations to the determination of the sound generated when the inhomogeneity is swept out of a nozzle orifice into free space. It is shown that the acoustic pressure perturbations developed in a duct at a contraction are in general proportional to the fractional difference between the density of the inhomogeneity and that of the mean flow times a typical mean flow pressure level, and are due principally to the fluctuation in thrust accompanying the passage of the inhomogeneity through the region of variable pressure gradient. The pressure waves generated at a nozzle orifice and radiated into free space are O(M0) smaller, where M0 is a mean flow Mach number based on the speed of sound in the jet.

Two-dimensional open-cavity flows for free stream Mach number of 0.1 to 1.6 are investigated with direct numerical simulations for cavities of aspect ratio 2 and 6 to characterize the flow stability. We determine the neutral stability curve over a wide range of Mach numbers and Reynolds numbers. In particular, we examine the effects of compressibility on the flow in transonic regime at low Reynolds numbers. Rossiter modes are identified by spectral analysis. The corresponding spatial structures associated with the dominant and subdominant Rossiter modes are extracted with dynamic mode decomposition. The nature of these structures provides us insight to their contribution to instability of the flow. The interaction between the shock waves and the shear layer modes for the transonic flows are investigated.

N for a wide range of Mach numbers and step heights, which can be incorporated into the e N method. In the present study both two-dimensional direct numerical simulation of disturbances as well as linear stability theory is used to gain detailed information on the ow. It is found that N is a function of streamwise position, step location, step height and Mach number. Subsonic and supersonic results show dierences in the ow topology and in the N-factor development, but eective N factors are in the same order of magnitude.

The difficulties associated with the nonlinearity of the transonic small disturbance equation are avoided by considering the linear problem governing the rate of change of the flow velocity with respect to the airfoil thickness ratio parameter and integrating solutions of this problem over the thickness ratio. This procedure is equivalent to summing a series of perturbations in airfoil thickness and has the advantage of moving all the nonlinearity to a firstorder ordinary differential equation, where it causes little difficulty. Solutions for nonlifting airfoils exhibit all the commonly observed features of transonic flow, such as the formation and movement of a shock on the airfoil surface and the presence of a detached shock upstream of the airfoil for low supersonic Mach numbers. Numerical results are given for subsonic and supersonic flow and for flows at Mach number unity which are in general agreement with other theoretical results and with experiment. Preliminary results for mixed flows with subsonic freestream Mach numbers indicate a rearward movement of the shock impinging on the airfoil surface with increasing airfoil thickness and exhibit the commonly observed rapid acceleration behind the shock. Extensions to lifting and unsteady flows are also discussed, as well as possible applications of the method to other problems in fluid mechanics.

Compressible flow past thin wings and slender bodies are studied. First, subsonic flow past thin wings are analyzed by means of potential flow theory. The kernel function method is introduced for arbitrary planforms undergoing simple harmonic oscillations. The spanwise polynomial approximation and the chordwise trigonometric function approximations which automatically satisfy the Kutta condition and inherit the leading edge singularity are used. The Doublet-Lattice method as a more general numerical approach is given for the analysis of the flow past additional surfaces like tail or store surfaces, which are not necessarily in plane with the wing surface or surfaces having spanwise deflections. Airfoil response to the arbitrary unsteady motion is also given for subsonic flows. A brief review of shock waves and Mach waves in a supersonic flow is given. Afterwards, unsteady supersonic potential flow is studied for a simple harmonically oscillating point source. First, unsteady flow past an airfoil is considered. Then, supersonic flow about thin wings are analyzed using the Mach box technique. Introduction to supersonic kernel function method is briefly provided. Arbitrary unsteady motion of an airfoil in a supersonic flow is presented. Slender body theory is introduced to analyze the cross flow past bodies of considerable fineness where the cross flow is shown to be approximately incompressible under certain conditions. In connection with the slender body approximation, the Munk’s airship theory is utilized to predict the stability derivatives of missile like bodies.

We present a new approach – the entropy-viscosity method (EVM) – for numerical modelling of high Reynolds number flows and investigate its potential by simulating fully developed incompressible turbulent flow, first in a stationary pipe and subsequently in a flexible pipe. This method, which was first proposed by Guermondet al.(J. Comput. Phys., vol. 230 (11), 2011, pp. 4248–4267), introduces the concept of entropy viscosity, computed based on the nonlinear localized residual obtained from the energy equation. Specifically, this nonlinear viscosity based on the local size of entropy production is added to the spectral element discretization employed in our work for stabilization at insufficient resolution. Unlike its original formulation, which includes anad hoctuneable parameter$\unicode[STIX]{x1D6FC}$, here, we determine the value of$\unicode[STIX]{x1D6FC}$by assuming that the entropy viscosity is analogous to the eddy viscosity of the Smagorinsky model. However, the overall approach has the flavour of the implicit large eddy simulation (ILES) instead of the standard large eddy simulation (LES). Given the empiricism of our approach, we have performed systematic studies of homogeneous isotropic turbulence for validation (see appendix A). We have also carried out a more complete numerical simulation study to investigate incompressible turbulent flow in a stationary pipe at$Re_{D}=5300$and$Re_{D}=44\,000$, following the work of Wu & Moin (J. Fluid Mech., vol. 608, 2008, pp. 81–112) who performed very accurate direct numerical simulations (DNS) of these two cases. We found that the mean flow, turbulence fluctuations, and two-point correlations of the EVM-based LES are in good agreement with the DNS of Wu & Moin despite the fact that we employed grids with resolution two orders of magnitude smaller. If we instead use the standard Smagorinsky model in our simulations, the computations become unstable due to insufficient resolution of the smaller scales. Another important difference is that the entropy-viscosity model scales with the cube of the distance from the wall and approaches zero at the wall, which is theoretically correct, as shown by oura posterioritests. Based on the validated EVM approach, we then simulated fully developed turbulent flow at$Re_{D}=5300$in a flexible pipe subject to prescribed vibrations in the cross-flow plane corresponding to a standing wave of amplitude$A$and wavelength$\unicode[STIX]{x1D706}=3D$, where$D=2R$is the pipe diameter and$R$is the radius. We have simulated 11 cases corresponding to increasing values of wave steepness$s_{o}=2A/\unicode[STIX]{x1D706}$, with$s_{o}\in [0,0.067]$. We found a quadratic dependence of the friction factor on$s_{o}$with the minimum at approximately$s_{o}\approx 0.01$, so, surprisingly, we have a slight decrease in drag at first and then a substantial increase compared to the stationary pipe. To obtain the turbulence statistics, we averaged the simulated flow over twenty time periods at the nodes and anti-nodes separately. We found substantial changes in the mean velocity profile at distances$(1-r)^{+}>5$while the peaks of turbulent intensities were amplified by 50 % in the axial direction and by 200 % in the normal and azimuthal directions at$s_{o}=0.067$. The peak shear stress at the node increased by more than 200 % whereas at the anti-node it attained negative values. Turbulent budgets revealed large changes close to the wall at$(1-r)^{+}<50$while flow visualizations showed that many more strong worm-like vortices were generated in the near-wall regions compared to the stationary pipe. We have also computed various spatio-temporal correlations that show that the pressure fluctuations are very sensitive to the pipe vibration and scale quadratically with$s_{o}$. Both pressure and velocity correlations exhibit cellular patterns consistent with the standing-wave pipe motion.

The nozzle is a kind of equipment with a wide range of usage in the aerospace industry, of which its performance is vital for rocket engines by changing the geometry of the inner wall of the pipe section to accelerate the airflow. There are two types of nozzles commonly used: one is a tapered nozzle, and the other is a Laval nozzle. To analyze the flow phenomenon in nozzle is vital before transferring prototype design into industrial production, whether by laboratory experiment or computational simulation. In this paper, two different numerical methods are adopted to simulate gas behavior inside a simplified nozzle with upper and bottom symmetrical bumps. The first is to solve one dimensional Euler equations, and the other is to solve a scalar variable named velocity potential with small disturbance equations (SDE). The solutions under various inlet Mach numbers are compared by analysing the velocity fields and Mach number contours obtained by these two approaches. Similarities and differences between the Euler method and the potential SDE method for subsonic flow and supersonic flow are the key emphases in this work. For either subsonic or supersonic flow, the Mach number distribution along the nozzle’s center line shows a consistent trend for both methods. In contrast, the values of maximum or minimum Mach number have corresponding differences. Moreover, by using potential SDE simulation, several types of shock waves are successfully captured. All results show that the incoming airflow decelerates at the leading edge of the nozzle then accelerates when passing through the bumps, and finally decelerates back to the speed of inlet flow. The difference is that flows with varied Mach numbers has distinct velocity distributions in the nozzle.

We have modelled ultracompact HII regions (UCHIIR) in terms of steady subsonic ionized flows in a clumpy medium. Mass loss from neutral clumps allows the regions to be long-lived. We examine the form of global flows for different dependences of the volume mass injection rate, $\dot{q}$, on radius and Mach number, and describe the solutions in detail. We find that three observed UCHIIR morphologies are reproduced with these models. Mach number independent flows that include a radial variation can give centre- brightened core--halo morphologies. Mach number dependent flows reproduce naturally the uniform UCHIIR morphology. In a hybrid model, including subsonic and supersonic flows, we allow a supersonic wind to shock in the ionized region. The ionized subsonic gas has a high density and so dominates the emission. The shell produced has a velocity structure very different from that of fully supersonic models. Several morphologies of spherical UCHIIR can be understood in terms of these various models; however, kinematic data are crucial as a discriminant between them.

Study of the radiation of acoustic sources in a duct with flow