STABILITY OF TRANSONIC SHOCKS TO THE EULER-POISSON SYSTEM WITH VARYING BACKGROUND CHARGES

This paper is devoted to studying the stability of transonic shock solutions to the Euler-Poisson system in a one-dimensional nozzle of finite length.The background charge in the Poisson equation is a piecewise constant function.The structural stability of the steady transonic shock solution is obtained by the monotonicity argument.Furthermore,this transonic shock is proved to be dynamically and exponentially stable with respect to small perturbations of the initial data.One of the crucial ingredients of the analysis is to establish the global well-posedness of a free boundary problem for a quasilinear second order equation with nonlinear boundary conditions.

Two-Dimensional Riemann Problems:Transonic Shock Waves and Free Boundary Problems

We are concerned with global solutions of multidimensional(M-D)Riemann problems for nonlinear hyperbolic systems of conservation laws,focusing on their global configurations and structures.We present some recent developments in the rigorous analysis of two-dimensional(2-D)Riemann problems involving transonic shock waves through several prototypes of hyperbolic systems of conservation laws and discuss some further M-D Riemann problems and related problems for nonlinear partial differential equations.In particular,we present four different 2-D Riemann problems through these prototypes of hyperbolic systems and show how these Riemann problems can be reformulated/solved as free boundary problems with transonic shock waves as free boundaries for the corresponding nonlinear conservation laws of mixed elliptic-hyperbolic type and related nonlinear partial differential equations.

Numerical simulation of the dimensional transformation of atomization in a supersonic aerodynamic atomization dust-removing nozzle based on transonic speed compressible flow

To simulate the transonic atomization jet process in Laval nozzles,to test the law of droplet atomization and distribution,to find a method of supersonic atomization for dust-removing nozzles,and to improve nozzle efficiency,the finite element method has been used in this study based on the COMSOL computational fluid dynamics module.The study results showed that the process cannot be realized alone under the two-dimensional axisymmetric,three-dimensional and three-dimensional symmetric models,but it can be calculated with the transformation dimension method,which uses the parameter equations generated from the two-dimensional axisymmetric flow field data of the three-dimensional model.The visualization of this complex process,which is difficult to measure and analyze experimentally,was realized in this study.The physical process,macro phenomena and particle distribution of supersonic atomization are analyzed in combination with this simulation.The rationality of the simulation was verified by experiments.A new method for the study of the atomization process and the exploration of its mechanism in a compressible transonic speed flow field based on the Laval nozzle has been provided,and a numerical platform for the study of supersonic atomization dust removal has been established.

Transonic Rudder Buzz on Tailless Flying Wing UAV

Transonic rudder buzz responses based on the computational fluid dynamics or computational structural dynamics(CFD/CSD)loosely method are analyzed for a tailless flying wing unmanned aerial vehicle(UAV).The Reynolds-averaged Navier-Stokes(RANS)equations and finite element methods based on the detailed aerodynamic and structural model are established,in which the aerodynamic dynamic meshes adopt the unstructured dynamic meshes based on the combination of spring-based smoothing and local remeshing methods,and the lower-upper symmetric-Gauss-Seidel(LU-SGS)iteration and Harten-Lax-van Leer-Einfeldt-Wada(HLLEW)space discrete methods based on the shear stress transport(SST)turbulence model are used to calculate the aerodynamic force.The constraints of the rudder motions are fixed at the end of structural model of the flying wing UAV,and the structural geometric nonlinearities are also considered in the flying wing UAV with a high aspect ratio.The interfaces between structural and aerodynamic models are built with an exact match surface where load transferring is performed based on 3Dinterpolation.The flying wing UAV transonic buzz responses based on the aerodynamic structural coupling method are studied,and the rudder buzz responses and aileron,elevator and flap vibration responses caused by rudder motion are also investigated.The effects of attack,height,rotating angular frequency and Mach number under transonic conditions on the flying wing UAV rudder buzz responses are discussed.The results can be regarded as a reference for the flying wing UAV engineering vibration analysis.

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.

AN INVISCID MODEL FOR THE BASE PRESSURE OF TRANSONIC TURBINE CASCADE

An inviscid base pressure model for transonic turbine blade has been presented. It has been shown that for a given back pressure the base pressure at the trailing edge, and the profile loss of a turbine blade are fixed according to the model and the base pressure can be calculated with the help of an inviscid numerical scheme. A parameteric study on the model shows that a blade profile with positive curvature downstream of the throat is advantageous for generating less loss, whilst the worst situation is when the exit flow reaches the sonic condition.

Numerical investigation of transonic flow over deformable airfoil with plunging motion

In this article, the transonic inviscid flow over a deformable airfoil with plunging motion is studied numerically. A finite volume method based on the Roe scheme developed in a generalized coordinate is used along with an arbitrary Lagrangian-Eulerian method and a dynamic mesh algorithm to track the instantaneous position of the airfoil. The effects of different governing parameters such as the phase angle, the deformation amplitude, the initial angle of attack, the flapping frequency, and the Mach number on the unsteady flow field and aerodynamic coefficients are investigated in detail. The results show that maneuverability of the airfoil under various flow conditions is improved by the deformation. In addition, as the oscillation frequency of the airfoil increases, its aerodynamic performance is significantly improved.

TRANSONIC SHOCK SOLUTIONS TO THE EULER SYSTEM IN DIVERGENT-CONVERGENT NOZZLES

In this paper,we study the transonic shock solutions to the steady Euler system in a quasi-one-dimensional divergent-convergent nozzle.For a given physical supersonic inflow at the entrance,we obtain exactly two non-isentropic transonic shock solutions for the exit pressure lying in a suitable range.In addition,we establish the monotonicity between the location of the transonic shock and the pressure downstream.

ANISOTROPIC MULTISTAGE FINITE ELEMENT METHOD FOR TWO DIMENSIONAL VISCOUS TRANSONIC FLOW IN TURBOMACHINERY

A new method based on the anisotropic tensor force finite element and Taylor-Galerkin finite element is presented in the present paper.Its application to two-dimensional viscous transonic flow in turbomachinery improves the conver- gence rate and stability of calculation,and the results obtained agree well with the experimental measurements.

Flow Diagnosis and Optimization Based on Vorticity Dynamics for Transonic Compressor/Fan Rotor

This paper presents two optimized rotors. The first rotor is as part of a 3-blade row optimization (IGV-rotor-stator) of a high-pressure compressor. It is based on modifying blade angles and advanced control of curvature of the airfoil camber line. The effects of these advanced blade techniques on the performance of the transonic 1.5-stage compressor were calculated using a 3D Navier-Stokes solver combined with a vortex/vorticity dynamics diagnosis method. The first optimized rotor produces a 3-blade row efficiency improvement over the baseline of 1.45% while also improving stall margin. The throttling range of the compressor is expanded largely because the shock in the rotor tip area is further downstream than that in the baseline case at the operating point. Additionally, optimizing the 3-blade row block while only adjusting the rotor geometry ensures good matching of flow angles allowing the compressor to have more range. The flow diagnostics of the rotor blade based on vortex/vorticity dynamics indicate that the boundary-layer separation behind the shock is verified by on-wall signatures of vorticity and skin-friction vector lines. In addition, azimuthal vorticity and boundary vorticity flux (BVF) are shown to be two vital flow parameters of compressor aerodynamic performance that directly relate to the improved performance of the optimized transonic compressor blade. A second rotor-only optimization is also presented for a 2.9 pressure ratio transonic fan. The objective function is the axial moment based on the BVF. An 88.5% efficiency rotor is produced.

Numerical Study on Transonic Flow with Local Occurrence of Non-Equilibrium Condensation

Characteristics of transonic flow over an airfoil are determined by a shock wave standing on the suction surface. In this case, the shock wave/boundary layer interaction becomes complex because an adverse pressure gradient is imposed by the shock wave on the boundary layer. Several types of passive control techniques have been applied to shock wave/boundary layer interaction in the transonic flow. Furthermore, possibilities for the control of flow fields due to non-equilibrium condensation have been shown so far and in this flow field, non-equilibrium condensation occurs across the passage of the nozzle and it causes the total pressure loss in the flow field. However, local occurrence of non-equilibrium condensation in the flow field may change the characteristics of total pressure loss compared with that by non-equilibrium condensation across the passage of flow field and there are few for researches of locally occurred non-equilibrium condensation in a transonic flow field. The purpose of this study is to clarify the effect of locally occurred non-equilibrium condensation on the shock strength and total pressure loss on a transonic internal flow field with circular bump. As a result, it was found that shock strength in case with local occurrence of non-equilibrium condensation is reduced compared with that of no condensation. Further, the amount of increase in the total pressure loss in case with local occurrence of non-equilibrium condensation was also reduced compared with that by non-equilibrium condensation across the passage of flow field.

Transonic Airfoil Design and Optimization for an Unmanned Air Vehicle Concept

Design,optimization and analysis of a new energy efficient 16%thick transonic airfoil were completed for a notional air vehicle concept.The airfoil specifications included a range of Reynolds number per foot from 1.7 million to 2.5 million and Mach number from 0.4 to 0.8.Shape optimization in geometry and inverse design modules of the airfoil analysis program MSES were used to design a new 16%thick reference airfoil.The performance of the reference airfoil was then optimized with an objective of minimizing drag coefficient for 7 design points with conflicting requirements in Reynolds and Mach number by using the MSES/LINDOP optimizer.The optimization results in an upward shift of drag bucket in the direction of higher lift coefficient.Both surface pressure distribution and Mach contour plots show that supersonic compression waves on airfoil surface are terminated at Mach 0.78 with a normal shock wave and associated flow separation,which causes not only a decrease of the maximum suction pressure,but also a decrease in lift and increase in drag coefficient.The new optimized airfoil shows robust performance when operating within the specified design constraints.

REVIEW OF THE GREEN＇S FUNCTION METHOD FOR STEADY ＆UNSTEADY，SUBSONIC，TRANSONIC ＆SUPERSONIC AERODYNAMICS AROUND COMPLEX CONFIGURATIONS

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HIGH ORDER ACCURACY SCHEME FOR 2-D TRANSONIC FLOWS

According to Taylor's expansion formula, a kind of derivatives computation method is presented, which can achieve arbitrary order of accuracy. Then it is combined with a flux difference splitting technique and a flux limiter to construct the desiredscheme that is suitable for non-oscillatory shock-capturing calculation. TVD typeRunge-Kutta method is used for temporal discretization. Several steady and unsteadynumerical experiments demonstrate that the scheme is a robust solver for transonicflows with high accuracy and high resolution of shock wave structures.

UNSTEADY TRANSONIC AERODYNAMIC LOADINGS ON THE AIRFOIL CAUSED BY HEAVING, PITCHING OSCILLATIONS AND CONTROL SURFACE

An implicit upwind finite volume solver for the Euler equations using the improved flux - splitting method is established and used to calculate the transonic flow past the airfoils with heaving, pitching oscillations and the control surface. Results are given for the NACA64A - 10 airfoil which is in harmonic heaving and pitching oscillation and with the control surface in the transonic flow field. Some computational results are compared with the experiment data and the good agreements are shown in the paper.

AN EFFICIENT METHOD FOR SOLVING THE MIXED DIRECT-INVERSE PROBLEM OF THE TRANSONIC ROTATIONAL FLOW IN PLANE CASCADES

The method in [1] has been extended to the case of rotational flow in this paper. A new method for dealing with the shock wave is presented. This method has the advantages of both the shock-fitting and the shock capturing methods. The direct problem and the mixed direct-inverse prob- lem of the rotational flow in a transonic plane cascade at both design and off design conditions are solved, and the results show that the present method has rapid convergence rate and high accuracy even for the flow with moderately strong shocks. The calculations have been carried out on the DPS-8 computer, and for the direct problem, only 50-80 iterations are needed, and 50-80 seconds of CPU time are required.

Unsteady Inverse Problem of Type B for 2-D Transonic Flow: A Variational Formulation

A family of variational principles (VP) has been developed for the unsteady inverse problem of the second type I B. It opens new ways for the inverse shape design of unsteady airfoils and can serve as key basis of multipoint inverse shape design of steady airfoils and cascades.

TIME MARCHING INTEGRAL EQUATION METHOD FOR UNSTEADY TRANSONIC FLOWS

In this paper,we have proposed a time marching intregral equation method which does not have the limitation of the time linearized integral equation method in that the latter method can not satisfac- torily simulate the shock-wave motions.Firstly,a model problem——one dimensional initial and boundary value wave problem is treated to clarify the basic idea of the new method.Then the method is implemented for 2-D and 3-D unsteady transonic flow problems.The introduction of the concept of a qua- si-velocity-potential simplifies the time marching integral equations and the treatment of trailing vortex sheet condition.The numerical calculations show that the method is reasonable and reliable.

Numerical Investigation into the Transient Behavior of the Spike-Type Rotating Stall for a Transonic Compressor Rotor

In this paper,a numerical investigation into a spike-type rotating stall process is carried out considering a transonic compressor rotor(the NASA Rotor 37).Through solution of the Unsteady Reynolds-Averaged Navier-Stokes(URANS)equations,the evolution process from an initially circumferentially-symmetric near-stall flow field to a stable stall condition is simulated without adding any artificial disturbance.At the near-stall operating point,periodic fluctuations are present in the overall flow of the rotor.Moreover,the blockage region in the channel periodically shifts from middle span to the tip.This fluctuating condition does not directly lead to stall,while the full-annulus calculation eventually evolves to stall.Interestingly,a kind of“early disturbance”feature appears in the dynamic signals,which propagates forward ahead of the rotor.

Aerodynamic design of transonic fan/compressor by 3D viscous RNS combined with genetic algorithms

This paper presents an aerodynamic design of a small transonic fan by 3D viscous RNS solver combined with genetic algorithms.The aerodynamic design system based on the 3D viscous RNS solver reduces the dependency on the design experience for designers.Furthermore the optimum with genetic algorithms is an effective method for improving the transonic fan performance as a part of the design system.The design result showed that the transonic fan designed by this method reaches the design requirement even with more efficiency value.