Transonic Shock Research Articles

Global Uniqueness of Transonic Shocks in Two-Dimensional Steady Compressible Euler Flows

We prove that for the two-dimensional steady complete compressible Euler system, with given uniform upcoming supersonic flows, the following three fundamental flow patterns (special solutions) in gas dynamics involving transonic shocks are all unique in the class of piecewise C 1 smooth functions, under appropriate conditions on the downstream subsonic flows: (i) the normal transonic shocks in a straight duct with finite or infinite length, after fixing a point the shock-front passing through; (ii) the oblique transonic shocks attached to an infinite wedge; (iii) a flat Mach configuration containing one supersonic shock, two transonic shocks, and a contact discontinuity, after fixing a point where the four discontinuities intersect. These special solutions are constructed traditionally under the assumption that they are piecewise constant, and they have played important roles in the studies of mathematical gas dynamics. Our results show that the assumption of a piecewise constant can be replaced by some weaker assumptions on the downstream subsonic flows, which are sufficient to uniquely determine these special solutions. Mathematically, these are uniqueness results on solutions of free boundary problems of a quasi-linear system of elliptic-hyperbolic composite-mixed type in bounded or unbounded planar domains, without any assumptions on smallness. The proof relies on an elliptic system of pressure p and the tangent of the flow angle w = v/u obtained by decomposition of the Euler system in Lagrangian coordinates, and a newly developed method for the L ∞ estimate that is independent of the free boundaries, by combining the maximum principles of elliptic equations, and careful analysis of the shock polar applied on the (maybe curved) shock-fronts.

Transonic Shocks for the Full Compressible Euler System in a General Two-Dimensional De Laval Nozzle

In this paper, we study the transonic shock problem for the full compressible Euler system in a general two-dimensional de Laval nozzle as proposed in Courant and Friedrichs (Supersonic flow and shock waves, Interscience, New York, 1948): given the appropriately large exit pressure p e(x), if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes p e(x). We solve this problem completely for a general class of de Laval nozzles whose divergent parts are small and arbitrary perturbations of divergent angular domains for the full steady compressible Euler system. The problem can be reduced to solve a nonlinear free boundary value problem for a mixed hyperbolic–elliptic system. One of the key ingredients in the analysis is to solve a nonlinear free boundary value problem in a weighted Holder space with low regularities for a second order quasilinear elliptic equation with a free parameter (the position of the shock curve at one wall of the nozzle) and non-local terms involving the trace on the shock of the first order derivatives of the unknown function.

Effects of aspect ratio in a transonic shock tube airfoil flow

In the present study, the flow visualizations were performed around the NACA 0012 models which differ in aspect ratios. We discussed the effects of the aspect ratio in the test models. Additionally the unsteady, two-dimensional, compressible Euler equations were solved for the NACA 0012 airfoil. Experiments were performed utilizing the conventional gas driven shock tube as the intermittent transonic wind tunnel. The aspect ratios of the models are about 0.86 and 1.5, respectively. The Mach numbers M 2 are about 0.84. The Reynolds numbers of the present experimental conditions were constant that Re based on chord length is about 4.0×105. The results are as follows: in different aspect ratios, the difference of the shock wave location is confirmed though the Mach number and Reynolds number are same. It indicates the different correction Mach number by the effects of the side wall boundary layer though the nominal Mach number measured the same value. Also, on the difference of shock wave location for the effects of the aspect ratio, the tend of CFD shows the qualitative agreement with the result of an experiment.

Global stability of steady transonic Euler shocks in quasi-one-dimensional nozzles

We prove global in time dynamical stability of steady transonic shock solutions in divergent quasi-one-dimensional nozzles. We assume neither the smallness of the relative slope of the nozzle nor the weakness of the shock. Key ingredients of the proof are an exponentially decaying energy estimate for a linearized problem together with methods from Luo et al. (2011) [12].

Experimental Study into the Flow Physics of Three-Dimensional Shock Control Bumps

This paper describes a fundamental experimental study of the flow structure around a single three-dimensional (3D) transonic shock control bump (SCB) mounted on a flat surface in a wind tunnel. Tests have been carried out with a Mach 1.3 normal shock wave located at a number of streamwise positions relative to the SCB. Details of the flow have been studied using the experimental techniques of schlieren photography, surface oil flow visualization, pressure sensitive paint, and laser Doppler anemometry. The results of the work build on the findings of previous researchers and shed new light on the flow physics of 3D SCBs. It is found that spanwise pressure gradients across the SCB ramp and the shape of the SCB sides affect the magnitude and uniformity of flow turning generated by the bump, which can impact on the spanwise propagation of the quasi-two-dimensional (2D) shock structure produced by a 3DSCB. At the bump crest, vortices can form if the pressure on the crest is significantly lower than at either side of the bump. The trajectories of these vortices, which are relatively weak, are strongly influenced by any spanwise pressure gradients across the bump tail. Asignificant difference between 2D and 3D SCBs highlighted by the study is the impact of spanwise pressure gradients on 3D SCB performance. The magnitude of these spanwise pressure gradients is determined largely by SCB geometry and shock position. Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc.

Numerical solutions to shock reflection and shock interaction problems for the self-similar transonic two-dimensional nonlinear wave systems

We discuss numerical results for two dimensional Riemann problems for the nonlinear wave system. We consider four sectorial Riemann data and present the numerical results for two configurations. The first configuration is that incident shocks create reflected shocks, and the state behind the reflected shocks becomes subsonic. The second configuration is an interaction between two transonic shocks; one is created by an interaction with the sonic circle, and the other is created by the shock reflection. We implement Lax–Liu positive schemes and Strang splitting, and obtain linear correlations of the incident shock strength and the reflected shock strength for the first configuration. Furthermore we present intriguing numerical results on interactions of two transonic shocks for the second configuration.

An Interaction of a Rarefaction Wave and a Transonic Shock for the Self-Similar Two-Dimensional Nonlinear Wave System

We study a two dimensional Riemann problem for the self-similar nonlinear wave system which gives rise to an interaction of a transonic shock and a rarefaction wave. The interesting feature of this problem is that the governing equation changes its type from supersonic in the far field to subsonic near the origin. The subsonic region is then bounded above by the sonic line (degenerate) and below by the transonic shock (free boundary). Furthermore due to the rarefaction wave in the downstream, which interacts with the transonic shock, the problem becomes inhomogeneous and degenerate. We establish the existence result of the global solution to this configuration, and present analysis to understand the solution structure of this problem.

Numerical solutions of transonic two-dimensional flows at a ninety-degree wedge

We present numerical results on self-similar two-dimensional Riemann problems governed by the compressible Euler system and the nonlinear wave system, which give rise to a transonic shock. We consider a configuration for a vertical incident shock moving to the right above a rectangular object. The incident shock then interacts with a sonic circle soon after it moves beyond the object, and creates a transonic region. We implement Lax–Liu positive schemes and Strang splitting, and obtain linear correlations of the incident shock strength and the shock strength at the vertical wall. We further implement Roe average methods and finite volume methods on quadrilateral grids to capture a contact discontinuity of the Euler system near the corner of the object. The contact discontinuity creates a new supersonic state and a transonic shock inside the transonic region.

Degenerate Goursat-type boundary value problems arising from the study of two-dimensional isothermal Euler equations

As a ladder step to study transonic problems, we investigate two families of degenerate Goursat-type boundary value problems arising from the two-dimensional pseudo-steady isothermal Euler equations. The first family is about the genuinely two-dimensional full expansion of gas into a vacuum with a wedge; the other is a semi-hyperbolic patch that starts on sonic curves and ends at transonic shocks. Both the vacuum and the sonic sets cause parabolic degeneracy that results in substantial difficulties such as singularities of solutions and uniform a priori estimates. Main ingredients in this study are various characteristic decompositions for the pseudo-steady Euler equations in order to obtain necessary a priori estimates. Furthermore, we are able to verify the uniform Holder continuity of solutions with exponent 1/2 for the gas expansion problem and up to 2/7 for the semi-hyperbolic problem.

Monotonicity and uniqueness of a 3D transonic shock solution in a conic nozzle with variable end pressure

We focus on the uniqueness problem of a 3D transonic shock solution in a conic nozzle when the variable end pressure in the diverging part of the nozzle lies in an appropriate scope. By establishing the monotonicity of the position of shock surface relative to the end pressure, we remove the nonphysical assumptions on the transonic shock past a fixed point made in previous studies and further obtain uniqueness.

Experimental and Numerical Study of Oscillating Transonic Shock Waves in Ducts

AnexperimentalandcomputationalstudyofaM1 � 1:4transonicshockwaveinaparallel-walledductsubjectto downstream pressure perturbations in the frequency range of 16–90 Hz has been conducted. The dynamics of unsteady shock motion and aspects of the unsteady transonic shock and turbulent tunnel-floor boundary-layer interaction have been investigated. The numerical computations were performed using an unsteady Reynoldsaveraged Navier–Stokes scheme. It is found that the (experimentally measured) shock dynamics are generally well replicatedbythenumericalscheme,especiallyatrelativelylow(�40 Hz)frequencies.However,variationsinshock/ boundary-layer interaction structure during unsteady shock motion observed in experiments are not always well predicted by the simulation. Significantly, the computations predict variations in shock/boundary-layer interaction size due to shock motion that are much larger and in the opposite sense to the variations observed in experiments. Comparison of the unsteady results from the present study with steady (experimental) results from the literature suggeststhatunsteadyReynolds-averaged Navier–Stokescodeusedin thepresentstudy modelsthe unsteadyshock/ boundary-layer interaction behavior as quasi-steady, whereas experiments suggest that it is more genuinely unsteady. Further work developing numerical methods that demonstrate a more realistic sensitivity of shock/ boundary-layer interaction structure to unsteady shock motion is required.

Transonic Shocks in Multidimensional Divergent Nozzles

We establish existence, uniqueness and stability of transonic shocks for steady compressible non-isentropic potential flow system in a multidimensional divergent nozzle with an arbitrary smooth cross-section, for a prescribed exit pressure. The proof is based on solving a free boundary problem for a system of partial differential equations consisting of an elliptic equation and a transport equation. In the process, we obtain unique solvability for a class of transport equations with velocity fields of weak regularity(non-Lipschitz), an infinite dimensional weak implicit mapping theorem which does not require continuous Frechet differentiability, and regularity theory for a class of elliptic partial differential equations with discontinuous oblique boundary conditions.

Stability of Transonic Shock Solutions for One-Dimensional Euler–Poisson Equations

In this paper, both structural and dynamical stabilities of steady transonic shock solutions for one-dimensional Euler-Poission system are investigated. First, a steady transonic shock solution with supersonic backgroumd charge is shown to be structurally stable with respect to small perturbations of the background charge, provided that the electric field is positive at the shock location. Second, any steady transonic shock solution with the supersonic background charge is proved to be dynamically and exponentially stable with respect to small perturbation of the initial data, provided the electric field is not too negative at the shock location. The proof of the first stability result relies on a monotonicity argument for the shock position and the downstream density, and a stability analysis for subsonic and supersonic solutions. The dynamical stability of the steady transonic shock for the Euler-Poisson equations can be transformed to the global well-posedness of a free boundary problem for a quasilinear second order equation with nonlinear boundary conditions. The analysis for the associated linearized problem plays an essential role.

Transonic shock in finitely long nozzle with porous medium boundary condition

In this paper, we investigate the 2-D Euler equations with complex boundary conditions. For the entrance and exit of finitely long nozzle, we use the supersonic incoming flow condition and the end pressure condition, respectively. In addition, for the nozzle wall, the lower one is solid and the upper one is permitted to have leakage. We establish the well-posedness if appropriate seepage discharge is given. We also obtain the shock position by solving a free boundary problem without assuming the shock must go through a fixed point.

Semi-hyperbolic Patches of Solutions to the Two-dimensional Euler Equations

We construct semi-hyperbolic patches of solutions, in which one family out of two nonlinear families of characteristics starts on sonic curves and ends on transonic shock waves, to the two-dimensional Euler equations. This type of solution appears in the transonic flow over an airfoil and Guderley reflection, and is common in the numerical solutions of Riemann problems.

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.

Numerical solutions to the self-similar transonic two-dimensional nonlinear wave system

We present numerical results on a two-dimensional Riemann problem governed by the self-similar nonlinear wave system that gives rise to a transonic shock. We consider a configuration for a vertical incident shock moving to the right above a rectangular object. The incident shock then interacts with a sonic circle soon after it moves beyond the object, and creates a transonic region. We implement Lax–Liu positive schemes and Strang splitting, and obtain several numerical solutions for the model system. With the numerical results that we have obtained, we present several analyses of the transonic shock strengths and the positions of the transonic shocks with various Riemann data. Moreover, due to the presence of the corner of the object, numerical oscillations are apparent. We discuss regularity results for the solution near the corner of the object. Copyright © 2010 John Wiley & Sons, Ltd.

The existence and monotonicity of a three-dimensional transonic shock in a finite nozzle with axisymmetric exit pressure

We establish the existence of a multidimensional transonic shock solution in a class of slowly varying nozzles for the three dimensional steady full Euler system with axially symmetric exit pressure in the diverging part lying in an appropriate scope. We also show that the shock position depends monotonically on the exit pressure.

A CONTINUOUS, TWO-WAY FREE BOUNDARY IN THE UNSTEADY TRANSONIC SMALL DISTURBANCE EQUATIONS

We formulate a problem for the unsteady transonic small disturbance equations which describes a situation analogous to the reflection of a weak shock off a wedge, with the incident shock replaced by an incident rarefaction. We linearize this problem and solve it exactly, and we compute a numerical solution of the full nonlinear problem. The solution of this problem has several features in common with the solution of the weak shock reflection problem, known as Guderley Mach reflection. In both cases, a rarefaction wave reflects off a sonic line and forms a transonic shock. There is transonic coupling between the supersonic and subsonic regions across the sonic line and shock. In both situations, this sonic line/shock can be considered a free boundary in the formulation of a new type of free boundary problem which has not previously been formulated or analyzed. The free boundary problem that arises in the context of the problem considered here is, however, simpler than the free boundary problem that arises in the weak shock reflection problem.