Steady Supersonic Flow Research Articles

Global existence of shock for the supersonic Euler flow past a curved 2-D wedge

In this paper, we study the global existence of the supersonic shock for the steady supersonic Euler flow past a curved 2-D wedge. By using the method of characteristic, we show that the shock exists globally and the flow between the shock and wedge is continuous provided the wedge is a small perturbation of a straight wedge under a weighted global Sobolev norm and the vertex angle is less than the extreme angle.

Existence of multidimensional non-isothermal phase transitions in a steady van der Waals flow

This paper is concerned with the existence of multi-dimensional non-isothermal subsonicphase transitions in a steady supersonic flow with the van der Waals type statefunction. Due to the subsonic property, the Lax entropy inequality [15] is no longer validfor subsonic phase transitions. Hence, physical admissible planar waves are chosen bythe viscosity capillarity criterion [24]. Based on the uniform stability result in [28], weperform the iteration scheme [20] and establish the existence.

Stability of non-isothermal phase transitions in a steady van der Waals flow

This paper is concerned with the multidimensional stability of non-isothermal subsonic phase transitions in a steady supersonic flow of van der Waals type. For the sake of seeking physical admissible planar waves, the viscosity–capillarity criterion (Slemrod in Arch Ration Mech Anal 81(4):301–315, 1983) is chosen to be the admissible criterion. By showing the Lopatinski determinant being non-zero, we prove that subsonic phase transitions are uniformly stable in the sense of Majda (Mem Am Math Soc 41(275):1–95, 1983) under both one-dimensional and multidimensional perturbations.

Numerical investigation of the onset of instability of triple shock configurations in steady supersonic gas flows

Triple configurations of shock waves that arise in a steady supersonic flow of a real gas have been numerically simulated. Previously, we have theoretically predicted that a new triple shock configuration with a negative reflection angle can appear in a gas flow at large Mach numbers and small adiabatic indices. This configuration is now obtained for the first time in numerical experiments. It is shown that the formation of this triple shock configuration leads to instability of the entire flow pattern.

Nonexistence of global weak solution with only one stable supersonic conic shock wave for the steady supersonic Euler flow past a perturbed cone

Nonexistence of global weak solution with only one stable supersonic conic shock wave for the steady supersonic Euler flow past a perturbed cone

Global Conic Shock Wave for the Steady Supersonic Flow Past a Large Curved Cone at Infinity

In this paper, we establish the global existence and stability of a steady symmetric shock wave for the constant supersonic flow past an infinitely long and large curved conic body. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Through looking for the suitable "dissipative" boundary conditions on the shock and the conic surface together with the special form of shock equation, we show that the conic shock attached at the vertex of the cone exists globally in the whole space when the speed of the supersonic incoming flow is appropriately large.

Supersonic viscous corner flows

Results of numerical and experimental investigations of steady state supersonic viscous corner flows in simplified geometries that resemble typical wing body–fuselage intersections are presented. These flows are numerically calculated using the computational fluid dynamics code of Fluent and include the use of standard turbulence models. Experimental work includes the use of the oil film flow visualization technique to visualize the surface flow patterns. The flows considered are symmetric about the corner bisector. This article focuses on the development of the three shock structure along the various geometries, specifically focusing on the formation and growth of the shear layers that result from the Mach reflection. The resultant effects of this shear layer on pressure distributions, shear stress, and changes in Mach numbers are discussed. The shock wave boundary layer interaction is also presented.

Nozzle startup in an oncoming flow

Three variants of the startup of an axisymmetric convergent-divergent nozzle are considered with the static pressures at the entry and exit of the nozzle being the same at the beginning of the process. The subsonic startup corresponds to open nozzle acceleration in air. The supersonic startup simulates the sudden opening of a cover at the nozzle inlet under supersonic flight conditions. A successful nozzle startup with the formation of steady supersonic flow along the whole channel is realized in the third variant of supersonic startup with gas injection through a small region of the wall of the divergent nozzle section. The investigation is performed numerically, on the basis of the Euler equations for axisymmetric gas flows.

Time history of regular to Mach reflection transition in steady supersonic flow

In this paper, we study the transition from regular to Mach reflection (RR → MR) in the dual solution domain due to the influence of an upstream disturbance, by considering the transition as an evolutionary rather than an abrupt process. From numerical simulation, we observe for the early stage of transition a multiple interaction structure, composed of a triple-shock structure, a type VI shock interaction and a shock/slipline interaction. In the end, we observe a pure unsteady MR structure. Under self-similar assumption of the triple point for the first stage and including additional Mach waves over the slipline for the last stage, we develop an idealized unsteady model to obtain the evolution of the Mach stem height and the time taken for the Mach stem to stabilize. The triple point is found to move at a nearly constant speed in the multiple interaction stage which occupies about one quarter of the transition time. In the pure unsteady MR stage, which occupies the rest of transition, the speed of the triple point drops nonlinearly until the Mach stem stabilizes.

Dynamic effects on the transition between two-dimensional regular and Mach reflection of shock waves in an ideal, steady supersonic free stream

There have been numerous studies on the steady-state transition criteria between regular and Mach reflection of shock waves generated by a stationary, two-dimensional wedge in a steady supersonic flow, since the original shock-wave reflection research by Ernst Mach in 1878. The steady, two-dimensional transition criteria between regular and Mach reflection are well established. There has been little done to consider the dynamic effect of a rapidly rotating wedge on the transition between regular and Mach reflection. This paper presents the results of an investigation on the effect of rapid wedge rotation on regular to Mach reflection transition in the weak- and strong-reflection ranges with the aid of experiment and computational fluid dynamics. The experimental set-up includes a novel facility to rotate a pair of large aspect ratio wedges in a 450 mm × 450 mm supersonic wind tunnel at wedge rotation speeds up to 11000 deg s−1. High-speed images and measurements are presented. A numerical solution of the inviscid governing flow equations was used to mimic the experimental motion and to extend the investigation beyond the limits of the current facility to explore the influence of variables in the parameter space. There is good agreement between experimental measurements and numerical simulation. This paper includes the first experimental evidence of the regular to Mach reflection transition beyond the steady-state detachment condition in the weak- and strong-reflection ranges. It also presents results of simulations for the dynamic regular to the Mach reflection transition which show a difference between the sonic, length-scale and detachment conditions. This paper includes experimental evidence of the Mach to regular reflection transition below the steady-state von Neumann condition.

Dynamic effects on the transition between two-dimensional regular and Mach reflection of shock waves in an ideal, steady supersonic free stream

There have been numerous studies on the steady-state transition criteria between regular and Mach reflection of shock waves generated by a stationary, two-dimensional wedge in a steady supersonic flow, since the original shock-wave reflection research by Ernst Mach in 1878. The steady, two-dimensional transition criteria between regular and Mach reflection are well established. There has been little done to consider the dynamic effect of a rapidly rotating wedge on the transition between regular and Mach reflection. This paper presents the results of an investigation on the effect of rapid wedge rotation on regular to Mach reflection transition in the weak- and strong-reflection ranges with the aid of experiment and computational fluid dynamics. The experimental set-up includes a novel facility to rotate a pair of large aspect ratio wedges in a 450 mm × 450 mm supersonic wind tunnel at wedge rotation speeds up to 11000 deg s−1. High-speed images and measurements are presented. A numerical solution of the inviscid governing flow equations was used to mimic the experimental motion and to extend the investigation beyond the limits of the current facility to explore the influence of variables in the parameter space. There is good agreement between experimental measurements and numerical simulation. This paper includes the first experimental evidence of the regular to Mach reflection transition beyond the steady-state detachment condition in the weak- and strong-reflection ranges. It also presents results of simulations for the dynamic regular to the Mach reflection transition which show a difference between the sonic, length-scale and detachment conditions. This paper includes experimental evidence of the Mach to regular reflection transition below the steady-state von Neumann condition.

Aerodynamic Computations Using a Finite Volume Method with an HLLC Numerical Flux Function

A finite volume method for the simulation of compressible aerodynamic flows is de- scribed. Stabilisation and shock capturing is achieved by the use of an HLLC consistent numerical flux function, with acoustic wave improvement. The method is implemented on an unstructured hybrid mesh in three dimensions. A solution of higher order accuracy is obtained by reconstruc- tion, using an iteratively corrected least squares process, and by a new limiting procedure. The numerical performance of the complete approach is demonstrated by considering its application to the simulation of steady turbulent transonic flow over an ONERA M6 wing and to a steady inviscid supersonic flow over a modern military aircraft configuration.

Shock wave standoff distance for a sphere slightly above Mach one

This paper describes a numerical simulation of bow shock formation ahead of a sphere at steady supersonic flow in the Mach number range of 1.025–1.20. Turbulent viscous flow results are presented using the Spalart–Allmaras turbulence model. The purpose of this study is to determine the shock standoff distance for a spherical projectile at slightly supersonic free flight speeds. Results are compared to experimental data, including double exposure holographic interferograms obtained from a 40 mm polycarbonate sphere launched by a light gas gun. The shock standoff distance was determined from the interferograms. The present numerical simulations were found to agree with previously published data, and reached down to M = 1.025—a range where almost no previously published data exists. The computed flow structure and shock wave locations agree well with recently obtained free-flight interferograms.

Instability of one global transonic shock wave for the steady supersonic Euler flow past a sharp cone

AbstractIn this paper, we are concerned with the instability problem of one global transonic conic shock wave for the supersonic Euler flow past an infinitely long conic body whose vertex angle is less than some critical value. This is motivated by the following descriptions in the bookSupersonic Flow and Shock Wavesby Courant and Friedrichs: if there is a supersonic steady flow which comes from minus infinity, and the flow hits a sharp cone along its axis direction, then it follows from the Rankine-Hugoniot conditions, the physical entropy condition, and the apple curve method that there will appear a weak shock or a strong shock attached at the vertex of the cone, which corresponds to the supersonic shock or the transonic shock, respectively. A long-standing open problem is that only the weak shock could occur, and the strong shock is unstable. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand this. In particular, under some suitable assumptions, because of the essential influence of the rotation of Euler flow, we show that a global transonic conic shock solution is unstable as long as the related sharp circular cone is perturbed.

Instability of one global transonic shock wave for the steady supersonic Euler flow past a sharp cone

AbstractIn this paper, we are concerned with the instability problem of one global transonic conic shock wave for the supersonic Euler flow past an infinitely long conic body whose vertex angle is less than some critical value. This is motivated by the following descriptions in the bookSupersonic Flow and Shock Wavesby Courant and Friedrichs: if there is a supersonic steady flow which comes from minus infinity, and the flow hits a sharp cone along its axis direction, then it follows from the Rankine-Hugoniot conditions, the physical entropy condition, and the apple curve method that there will appear a weak shock or a strong shock attached at the vertex of the cone, which corresponds to the supersonic shock or the transonic shock, respectively. A long-standing open problem is that only the weak shock could occur, and the strong shock is unstable. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand this. In particular, under some suitable assumptions, because of the essential influence of the rotation of Euler flow, we show that a global transonic conic shock solution is unstable as long as the related sharp circular cone is perturbed.

On the evolution of weak discontinuities in radiative magnetogasdynamics

A method of wavefront analysis is used to analyze the formation of shock waves in a two-dimensional steady supersonic flow past plane and axisymmetric bodies in radiative magnetogasdynamics. The gas is assumed to be perfectly conducting and to be permeated by a magnetic field orthogonal to the trajectories of the gas particles. The medium is taken to be sufficiently hot for the effects of thermal radiation to be significant, which is treated by the optically thin approximation to the radiative transfer equation. Transport equations, which lead to the determination of the distance at which the first characteristic could intersect with a successive one and also to conditions, which insure that no shock will ever evolve on the wave front, are derived. The effect of upstream flow Mach number and the magnetic field strength on the behavior of the wavefront are examined.

Steady supersonic flow over a bending wall

This paper studies the steady supersonic flow moving over a bending wall that is the small perturbation of a convex one. Under the assumption that the total variation of the tangent of the perturbation is sufficiently small, the global weak solution is constructed via wave-front tracking scheme.

A study of the flow structure for Mach reflection in steady supersonic flow

In this paper we study the waves generated over the slipline and their interactions with other waves for Mach reflection in steady two-dimensional supersonic flow. We find that a series of expansion and compression waves exist over the slip line, even in the region immediately behind the leading part of the reflected shock wave, previously regarded as a uniform flow. These waves make the leading part of the slipline, previously regarded as straight, deviate nonlinearly towards the reflecting surface. When the transmitted expansion waves from the upper corner first intersect the slipline, an inflexion point is produced. Downstream of this inflexion point, compression waves are produced over the slipline. By considering the interaction between the various expansion or compression waves, we obtain a Mach stem height, the shape and position of the slipline and reflected shock wave, compared well to computational fluid dynamics (CFD) results. We also briefly consider the case with a subsonic portion behind the reflected shock wave. The global flow pattern is obtained through CFD and the starting point of the sonic line is identified through a simple analysis. The sonic line appears to coincide with the first Mach wave from the upper corner expansion fan after transmitted from the reflected shock wave.

Reduced-Order Modeling of Two-Dimensional Supersonic Flows with Applications to Scramjet Inlets

Control-oriented models of hypersonic vehicle propulsion systems require a reduced-order model of the scramjet inlet that is accurate to within 10% but requires less than a few seconds of computational time. To achieve this goal, a reduced-order model is presented, which predicts the solution of a steady two-dimensional supersonic flow through an inlet or around any other two-dimensional geometry. The model assumes that the flow is supersonic everywhere except in boundary layers and the regions near blunted leading edges. Expansion fans are modeled as a sequence of discrete waves instead of a continuous pressure change. Of critical importance to the model is the ability to predict the results of multiple wave interactions rapidly. The rounded detached shock near a blunt leading edge is discretized and replaced with three linear shocks. Boundary layers are approximated by displacing the flow by an empirical estimate of the displacement thickness. A scramjet inlet is considered as an example application. The predicted results are compared to two-dimensional CFD solutions and experimental results. Nomenclature a = local soundspeed [m/s] c = specific heat [J/kg K] h = specific enthalpy [J/kg] H = length normal to flow [m] M = Mach number n = number of a given quantity L = length tangent to flow [m] p = pressure [Pa] Pr = Prandtl number r = radius [m] R = normalized gas constant [J/kg K] R = 8314.47 J/kmol K T = temperature [K] u = velocity magnitude [m/s] W = molecular weight [kg/kmol] x = forward body-frame coordinate [m] Y = mass fraction z = vertical body-frame coordinate [m] = shock angle = ratio of specific heats = thickness of layer [m] = ratio = flowpath angle = dynamic viscosity [kg/m s] = ln p0=p = += 2

Evolution of weak discontinuities in a non-ideal radiating gas

A method of wavefront analysis is used to study the formation of shock waves in a two-dimensional steady supersonic flow of a non-ideal radiating gas past plane and axisymmetric bodies. The gas is taken to be sufficiently hot for the effect of thermal radiation to be significant, which is, of course, treated by the optically thin approximation to the radiative transfer equation. Transport equations, which lead to the determination of the shock formation distance and also to conditions which insure that no shock will ever evolve on the wavefront, is derived. The influence of the parameter of the non-idealness, upstream flow Mach number in the presence of thermal radiation on the behavior of the wavefront are examined.