Direct method for the design of optimal three-dimensional aerodynamic shapes

A direct optimization method for a broad class of three-dimensional aerodynamic shapes based on the approximation of the desired geometry by Bernstein-Bézier surfaces is developed. The high efficiency of the method is demonstrated by applying it to the design of an optimal supersonic section of an axisymmetric maximum-thrust de Laval nozzle. The method is also tested as applied to the design of a three-dimensional supersonic nozzle section in a dense multi-nozzle setup. In addition to three-dimensional supersonic nozzle sections with a circular throat, nozzles with a varying throat shape are considered. The results suggest that the method can be applied to various problems of 3D shape optimization.     

Spatial eigenmodes of a boundary layer with a triple-deck velocity field

The beforehand unclear relation between the viscous-inviscid interaction and the instability of viscous gas flows is illustrated using three-dimensional boundary-layer perturbations in the case of sub- and supersonic outer flows. The assumptions are considered under which asymptotic boundary layer equations with self-induced pressure are derived and the excitation mechanisms of eigenmodes (i.e., Tollmien-Schlichting waves) are described. The resulting dispersion relations are analyzed. The boundary layer in a supersonic flow is found to be stable with respect to two-dimensional perturbations, whereas, in the three-dimensional case, the modes become unstable. The increment of growth is investigated as a function of the Mach number and the orientation of the front of a three-dimensional Tollmien-Schlichting wave.     

A Variational Approach Via Bipotentials for a Class of Frictional Contact Problems

In the reference (Cui and Yin, Pacific J. Math. 233:257–289, 2007), under the assumptions that the supersonic incoming flow is isothermal and symmetrically perturbed with respect to a uniform supersonic constant state, the authors have shown the global existence and stability of a symmetric supersonic conic shock for such a supersonic flow past a circular cone. In this paper, we will remove all the symmetric assumptions in the previous paper and study the global existence problem on a really multidimensional shock wave. More concretely, we establish the global existence and stability of a three-dimensional supersonic conic shock wave for a perturbed steady supersonic isothermal flow past an infinitely long conic body.     

Transonic shocks in 3-D compressible flow passing a duct with a general section for Euler systems

This paper is devoted to the study of a transonic shock in three-dimensional steady compressible flow passing a duct with a general section. The flow is described by the steady full Euler system, which is purely hyperbolic in the supersonic region and is of elliptic-hyperbolic type in the subsonic region. The upstream flow at the entrance of the duct is a uniform supersonic one adding a three-dimensional perturbation, while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference. The problem to determine the transonic shock and the flow behind the shock is reduced to a free boundary value problem of an elliptic-hyperbolic system. The new ingredients of our paper contain the decomposition of the elliptic-hyperbolic system, the determination of the shock front by a pair of partial differential equations coupled with the three-dimensional Euler system, and the regularity analysis of solutions to the boundary value problems introduced in our discussion.

     

Method for solving the equations describing the interaction of a three-dimensional boundary layer with an outer inviscid flow

A new method is developed for solving the three-dimensional time-independent equations describing the interaction of a laminar boundary layer with an outer inviscid flow. The method also applies to the interaction of plane flows. By applying the method, the problem of the three-dimensional viscous supersonic gas flow over a roughness element (a hump and a cavity) is solved for the first time within the framework of the classical triple-deck theory. The asymptotic height of the roughness element corresponding to the nonseparated flow is determined, and separated flow patterns are constructed.     

Global transonic conic shock wave for the symmetrically perturbed supersonic flow past a cone

In this paper, we are concerned with the global existence and stability of a steady transonic conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Theoretically, as indicated in [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, 1948], it follows from the Rankine-Hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the vertex of the sharp cone in terms of the different pressure states at infinity behind the shock surface, which correspond to the supersonic shock and the transonic shock respectively. In the references [Shuxing Chen, Zhouping Xin, Huicheng Yin, Global shock wave for the supersonic flow past a perturbed cone, Comm. Math. Phys. 228 (2002) 47-84; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Polytropic case, preprint, 2006; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Isothermal case, Pacific J. Math. 233 (2) (2007) 257-289] and [Zhouping Xin, Huicheng Yin, Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone, Anal. Appl. 4 (2) (2006) 101-132], the authors have established the global existence and stability of a supersonic shock for the perturbed hypersonic incoming flow past a sharp cone when the pressure at infinity is appropriately smaller than that of the incoming flow. At present, for the supersonic symmetric incoming flow, we will study the global transonic shock problem when the pressure at infinity is appropriately large.     

STABILIZATION EFFECT OF FRICTIONS FOR TRANSONIC SHOCKS IN STEADY COMPRESSIBLE EULER FLOWS PASSING THREE-DIMENSIONAL DUCTS

Transonic shocks play a pivotal role in designation of supersonic inlets and ramjets. For the three-dimensional steady non-isentropic compressible Euler system with frictions, we constructe a family of transonic shock solutions in rectilinear ducts with square cross-sections. In this article, we are devoted to proving rigorously that a large class of these transonic shock solutions are stable, under multidimensional small perturbations of the upcoming supersonic flows and back pressures at the exits of ducts in suitable function spaces.This manifests that frictions have a stabilization effect on transonic shocks in ducts, in consideration of previous works which shown that transonic shocks in purely steady Euler flows are not stable in such ducts. Except its implications to applications, because frictions lead to a stronger coupling between the elliptic and hyperbolic parts of the three-dimensional steady subsonic Euler system, we develop the framework established in previous works to study more complex and interesting Venttsel problems of nonlocal elliptic equations.     

Flight Control of Biomimetic Air Vehicles Using Vibrational Control and Averaging

A combination of vibrational inputs and state feedback is applied to control the flight of a biomimetic air vehicle. First, a control strategy is developed for longitudinal flight, using a quasi-steady aerodynamic model and neglecting wing inertial effects. Vertical and forward motion is controlled by modulating the wings’ stroke and feather angles, respectively. Stabilizing control parameter values are determined using the time-averaged dynamic model. Simulations of a system resembling a hawkmoth show that the proposed controller can overcome modeling error associated with the wing inertia and small parameter uncertainties when following a prescribed trajectory. After introducing the approach through an application to longitudinal flight, the control strategy is extended to address flight in three-dimensional space.     

Simulation of current layer dynamics in the magnetogasdynamic interaction with an argon flow

A nonstationary three-dimensional magnetohydrodynamic (MHD) model is used to numerically simulate the formation of a current layer interacting with a transverse magnetic field in a supersonic argon flow. The structural features of the current layer and the characteristics of the process are analyzed at various intensities of the MHD interaction. The problem is solved using the MacCormack method with splitting in spatial coordinates and flux-corrected transport.     

Analysis of supersonic flow in Oswatitsch inlet configuration

The aim of this work is a study of the flow parameters relevant for the Oswatitsch intake configuration. Geometry of the cone and inlet are equivalent to those on MiG-21bis fighter aircraft. Supersonic flow parameters for a channeled cone were studied at two different operational Mach numbers. Center body cone is movable along longitudinal axis and analysis is focused on specific positions for supersonic flight regimes. To study the problem, numerical model of axisymmetric two-dimensional Euler flow is applied. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global supersonic conic shock wave for the steady supersonic flow past a cone: Polytropic gas

In this paper, we establish the global existence and stability of a steady conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body as long as the vertex angle is less than a critical value. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Based on the delicate asymptotic expansion of the background solution, one can verify that the boundary conditions on the shock and the conic surface satisfy the “dissipative” property. From this property, by use of the reflected characteristics method and the special form of the 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 coming flow is appropriately large. On the other hand, we remove the smallness restriction on the sharp vertex angle in order to establish the global existence of a shock or a global weak solution, moreover, our proof approach is different from that in [Shuxing Chen, Zhouping Xin, Huicheng Yin, Global shock wave for the supersonic flow past a perturbed cone, Comm. Math. Phys. 228 (2002) 47-84] and [Zhouping Xin, Huicheng Yin, Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone, Anal. Appl. 4 (2) (2006) 101-132].     

超声速、高超声速粘性气体分离流动的显隐式差分解法

本文利用反扩散的两步显、隐式差分方法,求解了超声速、高超声速粘性气体绕二维、三维压缩拐角的层流和湍流分离运动.结果表明,它既能获得很好的精度,又能大大缩短计算机时.     

Modeling of flow separation of assist gas as applied to laser cutting of thick sheet metal

The present paper describes the results of mathematical modeling of supersonic flows of a viscous compressible gas, obtained by numerically solving three-dimensional full Navier–Stokes equations, and also the results of experiments with visualization of gas jet flows in channels geometrically similar to the laser cut. Separation of the gas flow from the cut front is predicted numerically and then validated by experiments on a model setup. The gas flow structure arising in a narrow channel behind a sonic (conical) or supersonic nozzle is described. Specific features of originating in the flow separation on a smooth surface in a narrow channel are examined, and mechanisms controlling the separation are proposed. Flow separation directly affects the changes in the shape and structure of striations and is the one of main reason for the worse quality of the laser cut surface. It is shown that the changes in the structures of striations over the thickness of the sheet being cut are closely related to aerodynamic features of jet flows of the assisting gas in the cut channel.     

Mathematical modeling of shock-wave processes under gas solid boundary interaction

The results of numerical simulations are presented for planar air flows in a bounded volume of square cross section diminishing due to a uniform motion of the walls, for a flow of a propane-air mixture under sinusoidal variation of the size of the square domain, and for three-dimensional supersonic air and propane-air flows in channels of variable square cross section. Specific features of shock-wave processes that are associated with the piston effect and cumulation are established. The hypersonic analogy between planar and spatial flows is confirmed, which allows one to use two-dimensional solutions in estimating three-dimensional flows. The equations of a multicomponent ideal perfect gas and one-stage kinetics of chemical reactions are used to describe the flows. The method of numerical simulations is based on S.K. Godunov’s scheme and implemented within an original software package.     

The study of three-dimensional gas flow in a nozzle using the completely implicit method

We study three-dimensional potential gas flow in a nozzle. The matrix of the system of linear equations obtained in approximating the equation for the velocity potential by symmetric differences has a strongly sparse form. This property is used for an approximate expansion of it as a product of triangular matrices. To guarantee stability in the supersonic region of flow additional terms of artificial viscosity type are introduced into the equation for the velocity potential. We study gas flows in axisymmetric and three-dimensional nozzles: elliptic, superelliptic, and nozzles with nonsymmetric subsonic part. Comparison of the results with the data of other authors has shown the high effectiveness of such an approach. Translated fromMetody Matematicheskogo Modelirovaniya, 1998, pp. 76–86.     

Numerical simulation of unsteady vortex structures in near wake of poorly streamlined bodies on multiprocessor computer system

On the basis of the conservative difference method, spatially unsteady flows near complexly shaped objects are studied. The mathematical model is based on the inviscid gas model. For subsonic, transonic, and supersonic regimes, the nonstationary aerodynamics of various aerospace objects is examined. The three-dimensional structure of the unsteady vortex near wake and its influence on the basic aerodynamic characteristics of aerial vehicles are visualized. The numerical simulation is performed using parallel algorithms on supercomputers of cluster architecture.     

Calculation of radial supersonic gas flow past sharp-nosed cones

The paper presents the results of numerical analysis of supersonic source flow of a gas past sharp-nosed cones. Axisymmetric and three-dimensional flows are considered. The flow geometry and the distributions of the gas-dynamic parameters in the shock layer are analyzed; their asymptotic properties are established. The numerical solutions are compared with solutions for uniform gas flow past a cone.Translated from Matematicheskie Modeli Estestvoznaniya, Published by Moscow University, Moscow, 1995, pp. 32–47.     

A remark on determination of transonic shocks in divergent nozzles for steady compressible Euler flows

In this paper we construct a class of transonic shock in a divergent nozzle which is a part of an angular sector (for two-dimensional case) or a cone (for three-dimensional case) which does not contain the vertex. The state of the compressible flow depends only on the distance from the vertex of the angular sector or the cone. It is supersonic at the entrance, while for appropriately given large pressure at the exit, a transonic shock front appears in the nozzle and the flow becomes subsonic after passing it. The position and strength of the shock is automatically adjusted according to the pressure given at the exit. We demonstrate these phenomena by using the two-dimensional and three-dimensional full steady compressible Euler systems. The idea involved is to solve discontinuous solutions of a class of two-point boundary value problems for systems of ordinary differential equations. Results established in this paper may be used to analyze transonic shocks in general nozzles.     

On the numerical integration of multi-dimensional,initial boundary value problems for the Euler equations in quasi-linear form

A matricial formalism to solve multi-dimensional initial boundary values problems for hyperbolic equations written in quasi-linear based on the λ scheme approach is presented. The derivation is carried out for nonorthogonal, moving systems of curvilinear coordinates. A uniform treatment of the integration at the boundaries, when the boundary conditions can be expressed in terms of combinations of time or space derivatives of the primitive variables, is also presented. The methodology is validated against a two-dimensional test case, the supercritical flow through the Hobson cascade n.2, and in three-dimensional test cases such as the supersonic flow about a sphere and the flow through a plug nozzle. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 781–814, 1998     

Long-range energy-state maneuvers for minimum time to specified terminal conditions

Some three-dimensional minimum-time paths to a specified final line (or point), heading, and energy are presented for an example supersonic aircraft. The optimum maneuvers have been determined using the calculus of variations and the energy-state approximation. These are compared with suboptimal solutions obtained with the additional assumption that only three discrete values of bank angle (?φ_max, 0, +φ_max) are available. Constraints on thrust, Mach number, angle-of-attack, dynamic pressure, and load factor are included. For ranges long enough that maximum velocity is attained en route, the initial and final ares can be determined separately, which greatly simplifies the solutions.