Unsteady Transonic Small Disturbance Equation Research Articles

The unsteady transonic small disturbance equation: Data on oblique curves

We propose and solve a new problem for the unsteady transonic small disturbance equation. Data are given for the self-similar equation in a fixed, bounded region of similarity space, where on a part of the boundary the equation has degenerate type (a `sonic line') and on the remainder it is elliptic. Previous results on this problem have chosen data so that the solution is constant on the sonic line, but we set up a situation where the solution is not constant on the sonic part of the boundary. The solution we find is Lipschitz up to the boundary. Our solution sets the stage for resolution of some interesting Riemann problems for this equation and for other multidimensional conservation laws.

Self-similar solutions for the diffraction of weak shocks

We numerically solve a problem for the unsteady transonic small disturbance equations that describes the diffraction of a weak shock into an expansion wave. In the context of a shock moving into a semi-infinite wall, this problem describes the interaction between the reflected part of the shock and the part that is transmitted beyond the wall. We formulate the equations in self-similar variables, and obtain numerical solutions using high resolution finite difference schemes. Our solutions appear to show that the shock dies out at the sonic line, rather than forms at an interior point of the supersonic region.

On the Self-similar Diffraction of a Weak Shock into an Expansion Wavefront

We study an asymptotic problem that describes the diffraction of a weak, self-similar shock near a point where its shock strength approaches zero and the shock turns continuously into an expansion wavefront. An example arises in the reflection of a weak shock off a semi-infinite screen. The asymptotic problem consists of the unsteady transonic small disturbance equation with suitable matching conditions. We obtain numerical solutions of this problem, which show that the shock diffracts nonlinearly into the expansion region. We also solve numerically a related half-space problem with a “soft” boundary, which shows a complex reflection pattern similar to one that occurs in the Guderley Mach reflection of weak shocks.

HIGH RESOLUTION SOLUTIONS FOR THE SUPERSONIC FORMATION OF SHOCKS IN TRANSONIC FLOW

We present numerical solutions of two problems for the unsteady transonic small disturbance equations whose solutions contain shocks. The first problem is a two-dimensional Riemann problem with initial data corresponding to a slightly supersonic flow hitting the corner of an expanding duct at t = 0. The second problem is a boundary value problem that describes steady transonic flow over an airfoil. In both cases, the solutions contain regions of supersonic and subsonic flow, and an expansion wave interacts with a sonic line to produce a shock. We use high resolution methods, together with local grid refinement, to investigate the nature of the solution in the neighborhood of the point where the shock forms. We find that the shock originates in the supersonic region as originally proposed by Guderley, and very close to, but not at, the sonic line.

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.

The Triple Point Paradox for the Nonlinear Wave System

We present numerical solutions of a two-dimensional Riemann problem for the non- linear wave system which is used to describe the Mach reflection of weak shock waves. Robust low order as well as high resolution finite volume schemes are employed to solve this equation formulated in self-similar variables. These, together with extreme local grid refinement, are used to resolve the solution in the neighborhood of an apparent but mathematically inadmissible shock triple point. Rather than observing three shocks meeting in a single standard triple point, we are able to detect a primary triple point containing an additional wave, a centered expansion fan, together with a se- quence of secondary triple points and tiny supersonic patches embedded within the subsonic region directly behind the Mach stem. An expansion fan originates at each triple point. It is our opinion that the structure observed here resolves the von Neumann triple point paradox for the nonlinear wave system. These solutions closely resemble the solutions obtained in (A. M. Tesdall and J. K. Hunter, SIAM J. Appl. Math., 63 (2002), pp. 42-61) for the unsteady transonic small disturbance equation.

Subsonic solutions for compressible transonic potential flows

We establish an existence theorem for transonic isentropic potential flows where the subsonic region is bounded by the sonic line and thus the governing equation may become degenerate on the boundary partly or entirely. It has been conjectured by experiments and numerical studies that the self-similar multidimensional flow changes its type, namely, hyperbolic far from the origin (supersonic region) and elliptic near the origin (subsonic region). Furthermore, the potential equation has a different nonlinearity compared to other transonic problems such as the unsteady transonic small disturbance equation, the nonlinear wave equation, and the pressure gradient equation. Namely, the coefficients of the potential equation depend on the gradients while others are independent of the gradients. We provide techniques to handle the gradients, establish interior and boundary gradient estimates for the potential flow in a convex region, and answer the conjecture, that is, the flow is strictly elliptic and the region is subsonic.

Higher regularity for free boundary and tangential derivative problems and global solutions to regular reflections for the unsteady transonic small disturbance (UTSD) equations

We establish C 2 , α -estimates for solutions of a class of quasilinear elliptic equations with free boundary and tangential derivative boundary problems. Using this regularity result we show the existence of global solutions to regular shock reflections for the unsteady transonic small disturbance (UTSD) equation. We also present Lipschitz estimates near the degenerate Dirichlet boundary (the sonic boundary) for the UTSD equation.

Free boundary and mixed boundary value problems for quasilinear elliptic equations: tangential oblique derivative and degenerate Dirichlet boundary problems

We present Hölder estimates and Hölder gradient estimates for a class of free boundary problems with tangential oblique derivative boundary conditions provided the oblique vector β does not vanish at any point on the boundary. We also establish the existence result for a general class of quasilinear degenerate problems of this type including nonlinear wave systems and the unsteady transonic small disturbance equation.

A class of quasilinear degenerate elliptic problems

We establish the existence of solutions for a class of quasilinear degenerate elliptic equations. The equations in this class satisfy a structure condition which provides ellipticity in the interior of the domain, and degeneracy only on the boundary. Equations of transonic gas dynamics, for example, satisfy this property in the region of subsonic flow and are degenerate across the sonic surface. We prove that the solution is smooth in the interior of the domain but may exhibit singular behavior at the degenerate boundary. The maximal rate of blow-up at the degenerate boundary is bounded by the “degree of degeneracy” in the principal coefficients of the quasilinear elliptic operator. Our methods and results apply to the problems recently studied by several authors which include the unsteady transonic small disturbance equation, the pressure-gradient equations of the compressible Euler equations, and the singular quasilinear anisotropic elliptic problems, and extend to the class of equations which satisfy the structure condition, such as the shallow water equation, compressible isentropic two-dimensional Euler equations, and general two-dimensional nonlinear wave equations. Our study provides a general framework to analyze degenerate elliptic problems arising in the self-similar reduction of a broad class of two-dimensional Cauchy problems.

Self-Similar Solutions for Weak Shock Reflection

We present numerical solutions of a two-dimensional Riemann problem for the unsteady transonic small disturbance equations that provides an asymptotic description of the Mach reflection of weak shock waves. We develop a new numerical scheme to solve the equations in self-similar coordinates and use local grid refinement to resolve the solution in the reflection region. The solutions contain a remarkably complex structure: there is a sequence of triple points and tiny supersonic patches immediately behind the leading triple point that is formed by the reflection of weak shocks and expansion waves between the sonic line and the Mach shock. An expansion fan originates at each triple point, thus resolving the von Neumann paradox of weak shock reflection. These numerical solutions raise the question of whether there is an infinite sequence of triple points in an inviscid weak shock Mach reflection.

A free boundary problem for a quasi‐linear degenerate elliptic equation: Regular reflection of weak shocks

AbstractWe prove the existence of a solution to the weak regular reflection problem for the unsteady transonic small disturbance (UTSD) model for shock reflection by a wedge. In weak regular reflection, the state immediately behind the reflected shock is supersonic and constant. The flow becomes subsonic further downstream; the equation in self‐similar coordinates is degenerate at the sonic line. The reflected shock becomes transonic and begins to curve there; its position is the solution to a free boundary problem for the degenerate equation. Using the Rankine‐Hugoniot conditions along the reflected shock, we derive an evolution equation for the transonic shock, and an oblique derivative boundary condition at the unknown shock position. By regularizing the degenerate problem, we construct uniform bounds; we apply local compactness arguments to extract a limit that solves the problem. The solution is smooth in the interior and continuous up to the degenerate boundary.This work completes a stage in our program to construct self‐similar solutions of two‐dimensional Riemann problems. In a series of papers, we developed techniques for solving the degenerate elliptic equations that arise in self‐similar reductions of hyperbolic conservation laws. In other papers, especially in joint work with Gary Lieberman, we developed techniques for solving free boundary problems of the type that arise from Rankine‐Hugoniot relations. For the first time, in this paper, we combine these approaches and show that they are compatible. Although our construction is limited to a finite part of the unbounded subsonic region, it suggests that this approach has the potential to solve a variety of problems in weak shock reflection, including Mach and von Neumann reflection in the UTSD equation, and the analogous problems for the unsteady full potential equation. © 2002 John Wiley & Sons, Inc.

Free boundary problems for the unsteady transonic small disturbance equation: Transonic regular reflection

We formulate and solve a transonic regular reflection problem for the unsteady transonic small disturbance equation, using a free boundary problem approach. Our method applies to self-similar shock reflection when the incident shock angle is large enough to permit a regular reflection configuration with a subsonic state behind the reflected shock. For the small-disturbance approximation in weak shock reflection, this corresponds to relatively large wedge angles. One contribution of this paper is the development of an asymptotic formula for the reflected shock, far from the reflection point, and for the subsonic state far downstream. These asymptotic series are valid for the small-disturbance approximation, for any incident shock angles. The main result in the paper is an existence theorem for the nonuniform subsonic flow behind the reflected shock. The flow velocity satisfies a quasilinear elliptic equation which is coupled to the Rankine-Hugoniot equations for the reflected shock, forming a free boundary problem on part of the boundary. Because the equation is not uniformly elliptic in the entire domain, we introduce a cut off to give a bounded domain, and also cut offs to the coefficients. Our result is incomplete in the following sense: we have been unable to remove the cut offs entirely. However, we prove that the flow we have constructed solves the original problem in a domain of finite size around the reflection point.

Constraints on possible singularities for the unsteady transonic small disturbance (UTSD) equations

We discuss the singular behavior of solutions to two-dimensional, general secondorder, uniformly elliptic equations in divergence form, with bounded measurable coefficients and discontinuous Dirichlet data along a portion of a Lipschitz boundary. We show that the conjugate to the solution develops a singularity that is at least logarithmic along the boundary at the points of discontinuity in the boundary data. A problem like this arises in the study of self-similar solutions to the hyperbolic conservation laws in two space dimensions given by the unsteady transonic small disturbance (UTSD) flow equations. These solutions model the reflection of a weak shock wave upon a thin wedge in the regime where the von Neumann paradox applies. The present result provides a step in the direction of understanding the nature of the solutions to the UTSD equations near a triple point. It shows that the flow behind the triple point cannot be strictly subsonic under some mild assumptions on the solutions. c 1999 John Wiley & Sons, Inc.

A Numerical Study of Riemann Problems for the Two-Dimensional Unsteady Transonic Small Disturbance Equation

We study a two-parameter family of Riemann problems for the unsteady transonic small disturbance (UTSD) equation, also called the two-dimensional Burgers equation. The two parameters, a and b, which define oblique shock initial data, correspond to the slopes of the initial shock waves in the upper half-plane. For each a and b, the three constant states in the upper half-plane satisfy the Rankine--Hugoniot conditions across the shocks. This leads to a two-parameter family of oblique shock interaction problems.In this paper we present a numerical study of global solution behavior for the values of a and b in a previously obtained bifurcation diagram. Our study supplements the related theoretical results and conjectures recently obtained by S. Canic and B. L. Keyfitz. We employ a high resolution numerical method which reveals fine solution structures. Our findings confirm theoretical results and conjectures about the solution patterns and deepen the understanding of the structure of several intricate wave in...

Riemann Problems for the Two-Dimensional Unsteady Transonic Small Disturbance Equation

We study a two-parameter family of Riemann problems for the unsteady transonic small disturbance (UTSD) equation, also called the two-dimensional Burgers equation, which is used to model the transition from regular to Mach reflection for weak shock waves. The related initial-value problem consists of oblique shock data in the upper half-plane, with two parameters a and b corresponding to the slopes of the initial shock waves. The study of quasi-steady solutions leads to a problem that changes type when written in self-similar coordinates. The problem is hyperbolic in the region where the flow is supersonic, and elliptic where the flow is subsonic. In this paper we give a complete description of the flow in the hyperbolic region by resolving the hyperbolic wave interactions in the form of quasi-one-dimensional Riemann problems. In the region of physical space where the flow is subsonic, we pose the related free-boundary problems and discuss the behavior of the subsonic solution using results from our previous work. Based on this approach we establish the existence of regions of different qualitative behavior in parameter (a, b) space. Our results reveal that the UTSD equation seems to be particularly suitable for the study of the so-called von Neumann paradox in which linearly degenerate waves can be ignored. We establish the region in the parameter space where a prototype of von Neumann reflection takes place. In other regions of parameter space we find prototypes for Mach reflection, regular reflection, and transitional Mach reflection. The lack of linearly degenerate waves in this model is resolved by the presence of a small rarefaction wave emerging from the triple point.

An Elliptic Problem Arising from the Unsteady Transonic Small Disturbance Equation

We prove a theorem on existence of a weak solution of the Dirichlet problem for a quasilinear elliptic equation with a degeneracy on one part of the boundary. The degeneracy is of a type (“Keldysh type”) associated with singular behavior—blow-up of a derivative—at the boundary. We define an associated operator which is continuous, pseudo-monotone and coercive and show that a weak solution displaying singular behavior at the boundary exists.

Calculation of steady and unsteady pressures on wings at supersonic speeds with a transonic small-disturbance code

A transonic unsteady aerodynamic and aeroelasticity code has been developed for application to realistic aircraft configurations. The new code is called CAP-TSD which is an acronym for Computational Aeroelasticity Program - Transonic Small Disturbance. The CAP-TSD code uses a time-accurate approximate factorization algorithm for solution of the unsteady transonic small-disturbance equation that is efficient for solution of steady and unsteady transonic flow problems including supersonic freestream flows. The new code can treat complete aircraft geometries with multiple lifting surfaces and bodies. Applications to wings in supersonic freestream flow are presented. Comparisons with selected exact solutions from linear theory are presented showing generally favorable results. Calculations for both steady and oscillatory cases for the F-5 and RAE tailplane models are compared with experimental data and also show good overall agreement. Selected steady calculations are further compared with a steady flow Euler code.

Transonic flutter analysis of aerodynamic surfaces in the presence of structural nonlinearities

The influence of structural nonlinearities on the transonic aeroelastic behavior of a two-degree-of-freedom system consisting of an airfoil which is plunging and pitching is considered. The preload and freeplay nonlinearity in the spring model are treated using the asymptotic expansion method which takes into account the influence of the higher harmonics. The aerodynamic load vector is obtained by solving the unsteady transonic small-disturbance (TSD) equation using an alternating-direction implicit (ADI) finite-difference algorithm. The frequency domain solution of the aeroelastic equation is obtained using the U-g method, and the aeroelastic response is carried out by applying Newmark-β and Wilson-ϑ methods. A comparison is made between the results obtained from these two solutions. The effects of preload and freeplay magnitudes on the stability of the system and the corresponding reduced frequency and mass density ratio at neutrally stable response are studied.

Unsteady transonic flow calculations for realistic aircraft configurations

A transonic unsteady aerodynamic and aeroelasticity code has been developed for application to realistic aircraft configurations. The new code is called CAP-TSD which is an acronym for Computational Aeroelasticity Program - Transonic Small Disturbance. The CAP-TSD code uses a time-accurate approximate factorization (AF) algorithm for solution of the unsteady transonic small-disturbance equation. The AF algorithm is very efficient for solution of steady and unsteady transonic flow problems. It can provide accurate solutions in only several hundred time steps yielding a significant computational cost savings when compared to alternative methods. The new code can treat complete aircraft geometries with multiple lifting surfaces and bodies including canard, wing, tail, control surfaces, launchers, pylons, fuselage, stores, and nacelles. Applications are presented for a series of five configurations of increasing complexity to demonstrate the wide range of geometrical applicability of CAP-TSD. These results are in good agreement with available experimental steady and unsteady pressure data. Calculations for the General Dynamics one-ninth scale F-16C aircraft model are presented to demonstrate application to a realistic configuration. Unsteady results for the entire F-16C aircraft undergoing a rigid pitching motion illustrated the capability required to perform transonic unsteady aerodynamic and aeroelastic analyses for such configurations.