Viscous transonic spiral and radial flow

Similarity solutions of the viscous transonic equation describing source and source vortex flows have been found. These solutions contain shock-like transitions from the supersonic to the subsonic branch of the corresponding inviscid solutions, while the singularity near the sonic point of the inviscid solutions is shifted to a smaller radius. It is shown that this similarity solution is identical to the transonic viscous compressible source and sink flow solutions of Wu (1955) and Sakurai (1958).     

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–Poisson systems are investigated. First, a steady transonic shock solution with a supersonic background 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 a supersonic background charge is proved to be dynamically and exponentially stable with respect to small perturbations 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 on 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.     

The theory of transonic vortical gas flows

For cases of plane and axisymmetric transonic vortical gas flows, the approximate equations for the stream function are constructed directly on the physical plane in the vicinity of the sonic-line point at which the entropy is extremal. Certain particular solutions are found which are generalizations of the familiar integrals of transonic gasdynamics without vortices.The author wishes to thank Yu. D. Shmyglevskii for helpful discussions of this study.     

Asymptotic analysis of transonic flows

In this paper we derive the equations of the second and third approximations for the stream function of two-dimensional and axisymmetric potential transonic flow of an inviscid gas and find their particular solutions corresponding to certain transonic flows.A similar study concerning the second approximation of subsonic and supersonic flow was made by Van Dyke [1] and Hayes [2]. The second approximation for the velocity potential of transonic flow has been examined in detail by Hayes [3]. Euvrard [4, 5] has investigated the asymptotic behavior of transonic flow far from a body, while Fal'kovich, Chernov, and Gorskii [6] have studied the flow in a nozzle throat.The transonic asymptotic analysis for the stream function is presented in this paper.     

Finite volume method for three-dimensional transonic potential flow through turbomachinery blade rows

A finite-volume method has been developed for the calculation of transonic, potential flows through 3-D turbomachinery blades with complex geometries. The exact transonic potential flow equation is solved on a mesh constructed from small volume elements. A transformation is introduced through which cuboids of the physical plane are mapped into computational cubes. Two sets of overlapping volumes are used. While the thermodynamic properties are calculated at the primary volume centres, the flux balance is established on the secondary volumes. For transonic flows an artificial compressibility term (upwind density gradient) is added to density to produce the necessary directional bias in the hyperbolic region. The successive point over-relaxation Gauss-Seidel method has been used to solve the non-linear partial differential equations. Comparisons with experiments and/or other numerical solutions for various turbomachinery configurations show that the 3-D finite-volume approach is a relatively accurate, reliable and fast method for inviscid, transonic flow predictions through turbomachinery blade rows     

Euler flow solutions for transonic shock wave–boundary layer interaction

Steady 2D Euler flow computations have been performed for a wind tunnel section, designed for research on transonic shock wave–boundary layer interaction. For the discretization of the steady Euler equations, an upwind finite volume technique has been applied. The solution method used is collective, symmetric point Gauss–Seidel relaxation, accelerated by non-linear multigrid. Initial finest grid solutions have been obtained by nested iteration. Automatic grid adaptation has been applied for obtaining sharp shocks. An indication is given of the mathematical quality of four different boundary conditions for the outlet flow. Two transonic flow solutions with shock are presented: a choked and a non-choked flow. Both flow solutions show good shock capturing. A comparison is made with experimental results.     

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.     

Numerical boundary conditions for unsteady transonic flow calculations

In calculations of transonic flows it is necessary to limit the domain of computation to a size that is manageable by computers. At the boundary of the computational domain, boundary conditions are required to ensure a unique solution. Since wave solutions exist in the unsteady transonic flow field, incorrect boundary conditions may result in spurious reflections from the computational boundary. This may introduce errors into the solution. To prevent the spurious reflections, absorbing boundary conditions are often used on the computational boundary. In this paper we describe a method to derive absorbing boudary conditions for transonic calculations. We demonstrate both theoretically and numerically that the use of the absorbing boundary conditions will reduce the spurious reflections in the calculation.     

Self-similar solutions to Lin-Reissner-Tsien equation

The Lin-Reissner-Tsien equation describes unsteady transonic flows under the transonic approximation. In the present paper, the equation is reduced to an ordinary differential equation via a similarity transformation. The resulting equation is then solved analytically and even exactly in some cases. Numerical simulations are provided for the cases in which there is no exact solution. Travelling wave solutions are also obtained.     

The structural instability of transonic flow associated with amalgamation/splitting of supersonic regions

We study inviscid transonic flow in a channel with a bump modeling an airfoil at zero incidence. A few simple bump configurations are considered. Numerical simulations show the existence of singular free-stream Mach numbers, which trigger the splitting/amalgamation of local supersonic regions. An analysis of this phenomenon contributes to the understanding of the nonuniqueness of transonic solutions at certain free-stream conditions.     

A transonic quasi-3D analysis for gas turbine engines including split-flow capability for turbofans

A numerical approximation is taken to the solution of the complex flows existing in gas turbine engines with transonic blading. The quasi-3D approach decouples the problem into through-flow and blade-to-blade solutions. An industrially practical finite element through-flow solution is developed and for blade-to-blade solutions a transonic finite areas method is utilized. The finite element code developed is capable of operating in an analysis or a design mode. In both modes a dynamic relaxation factor is employed and considerable reduction in solution time can be achieved. Comparisons to streamline curvature methods are carried out for simple analytical and complex industrial problems.     

A finite volume method for two-dimensional transonic potential flow through turbomachinery blade rows

To predict inviscid transonic flow through turbomachinery blade rows, the exact transonic potential flow equation is solved on a mesh constructed from small area elements. A transformation is introduced through which distorted squares of the physical plane are mapped into computational squares. Two sets of overlapping elements are used; while the thermodynamic properties are calculated at the primary element centres, the flux balance is established on the secondary elements. For transonic flows an artificial compressibility term (upwind density gradient) is added to density in order to produce the desired directional bias in the hyperbolic region. while the entropy does not increase across mass conservative shock jump regions. Comparisons withexperiments and with other numerical and analytical solutions for various turbomachinery configurations show that this approach is comparatively accurate, reliable, and fast.     

Invariance groups and reduction of the unsteady transonic small disturbance equation

A geometric approach is undertaken towards the solution of the unsteady transonic small disturbance equation describing the low frequency flow field about a thin airfoil. From group properties given in Anderson and Ibragimov, Lie-Bäcklund Transformations in Applications (1979) we derive three invariance groups for the equation. Based on these groups a reduction of the equation is performed. The reduction leads to a steady equation and to an ordinary differential equation from which group invariant solutions of the unsteady transonic small disturbance equation can be obtained.     

On a class of exact particular solutions of the equations of transonic gas flows

The paper presents exact particular solutions of the equations of transonic gas flows, analogous to the solutions derived in [1–3] for the case of short waves. These solutions are used to construct the flow around a body in a supersonic stream with an attached shock.     

Laval nozzle flow calculation

The inverse problem of the theory of the Laval nozzle is considered, which leads to the Cauchy problem for the gasdynamic equations; the streamlines and the flow parameters are found from the known velocity distribution on the axis of symmetry.The inverse problem of Laval nozzle theory was considered in 1908 by Meyer [1], who expanded the velocity potential into a series in powers of the Cartesian coordinates and constructed the subsonic and supersonic solutions in the vicinity of the center of the nozzle. Taylor [2] used a similar method to construct a flowfield which is subsonic but has local supersonic zones in the vicinity of the minimal section. Frankl [3] and Fal'kovich [4] studied the flow in the vicinity of the nozzle center in the hodograph plane. Their solution, just as the Meyer solution, made it possible to obtain an idea of the structure of the transonic flow in the vicinity of the center of the nozzle.A large number of studies on transonic flow in the vicinity of the center of the nozzle have been made using the method of small perturbations. The approximate equation for the transonic velocity potential in the physical plane, obtained in [3–6], has been studied in detail for the plane and axisymmetric cases. In [7] Ryzhov used this equation to study the question of the formation of shock waves in the vicinity of the center of the nozzle, and conditions were formulated for the plane and axisymmetric cases under which the flow will not contain shock waves. However, none of the solutions listed above for the inverse problem of Laval nozzle theory makes it possible to calculate the flow in the subsonic and transonic parts of the nozzles with large gradients of the gasdynamic parameters along the normal to the axis of symmetry.Among the studies devoted to the numerical calculation of the flow in the subsonic portion of the Laval nozzle we should note the study of Alikhashkin et al., and the work of Favorskii [9], in which the method of integral relations was used to solve the direct problem for the plane and axisymmetric cases.The present paper provides a numerical solution of the inverse problem of Laval nozzle theory. A stable difference scheme is presented which permits analysis with a high degree of accuracy of the subsonic, transonic, and supersonic flow regions. The result of the calculations is a series of nozzles with rectilinear and curvilinear transition surfaces in which the flow is significantly different from the one-dimensional flow. The flowfield in the subsonic and transonic portions of the nozzles is studied. Several asymptotic solutions are obtained and a comparison is made of these solutions with the numerical solution.The author wishes to thank G. D. Vladimirov for compiling the large number of programs and carrying out the calculations on the M-20 computer.     

Class of differentially invariant solutions of the submodel of axisymmetric flows of an ideal gas

A class of differentially invariant solutions of a problem with the pressure independent of the radial coordinate is considered for a submodel of steady axisymmetric flows of a polytropic gas. The overdetermined system turns out to be compatible and is integrated. All solutions defining transonic and supersonic flows with a limiting surface are found. These solutions are compared with invariant solutions obtained previously.     

Multiple solutions and stability of the steady transonic small-disturbance equation

Numerical solutions of the steady transonic small-disturbance(TSD) potential equation are computed using the conservative Murman-Cole scheme. Multiple solutions are discovered and mapped out for the Mach number range at zero angle of attack and the angle of attack range at Mach number 0.85 for the NACA 0012 airfoil. We present a linear stability analysis method by directly assembling and evaluating the Jacobian matrix of the nonlinear finite-difference equation of the TSD equation. The stability of all the discovered multiple solutions are then determined by the proposed eigen analysis. The relation of stability to convergence of the iterative method for solving the TSD equation is discussed. Computations and the stability analysis demonstrate the possibility of eliminating the multiple solutions and stabilizing the remaining unique solution by adding a sufficiently long splitter plate downstream the airfoil trailing edge. Finally, instability of the solution of the TSD equation is shown to be closely connected to the onset of transonic buffet by comparing with experimental data.     

钝头翼型和机翼跨音速绕流的渐近展开分析

本文利用渐近展开匹配法分析钝头翼型的跨音速绕流,导出了描述前缘附近流动的一级近似、二级近似下的速位方程、边界条件及相应的近似解析解,并构成关于翼表面速度的一致有效合成解,消除了跨音速小扰动近似的前缘奇性,对于大展弦比后掠翼绕流,可利用翼型绕流分析结果,消除机翼前缘奇性。     

Isentropic Gas Flow for the Compressible Euler Equation in a Nozzle

We study the motion of isentropic gas in a nozzle. Nozzles are used to increase the thrust of engines or to accelerate a flow from subsonic to supersonic. Nozzles are essential parts for jet engines, rocket engines and supersonicwind tunnels. In the present paper, we consider unsteady flow, which is governed by the compressible Euler equation, and prove the existence of global solutions for the Cauchy problem. For this problem, the existence theorem has already been obtained for initial data away from the sonic state, (Liu in Commun Math Phys 68:141–172, 1979). Here, we are interested in the transonic flow, which is essential for engineering and physics. Although the transonic flow has recently been studied (Tsuge in J Math Kyoto Univ 46:457–524, 2006; Lu in Nonlinear Anal Real World Appl 12:2802–2810, 2011), these papers assume monotonicity of the cross section area. Here, we consider the transonic flow in a nozzle with a general cross section area. When we prove global existence, the most difficult point is obtaining a bounded estimate for approximate solutions. To overcome this, we employ a new invariant region that depends on the space variable. Moreover, we introduce a modified Godunov scheme. The corresponding approximate solutions consist of piecewise steady-state solutions of an auxiliary equation, which yield a desired bounded estimate. In order to prove their convergence, we use the compensated compactness framework.     

Theoretical and Numerical Studies of Transonic Flow of Moist Air Around a Thin Airfoil

Numerical studies of a two-dimensional and steady transonic flow of moist air around a thin airfoil with condensation are presented. The computations are guided by a recent transonic small-disturbance (TSD) theory of Rusak and Lee (2000) on this topic. The asymptotic model provides a simplified framework to investigate the changes in the flow field caused by the heat addition from a nonequilibrium process of condensation of water vapor in the air by homogeneous nucleation. An iterative method which is based on a type-sensitive difference scheme is applied to solve the governing equations. The results demonstrate the similarity rules for transonic flow of moist air and the effects of energy supply by condensation on the flow behavior. They provide a method to formulate various cases with different flow properties that have a sufficiently close behavior and that can be used in future computations, experiments, and design of flow systems operating with moist air. Also, the computations show that the TSD solutions of moist air flows represent the essence of the flow character computed from the inviscid fluid flow equations. Received 5 October 2000 and accepted 21 March 2002