Small Disturbance Equation Research Articles

A low-loss vector exhaust guide vanes and its application

The exhaust device of gas-driven fan propulsion system (for a VTOL aircraft) adopts vector exhaust guide vane (VEGV) to achieve vector thrust in the range of 0-90°, which requires that the deflection angle of the VEGV reach about ±45∘ and the total pressure recovery coefficient above 0.985. Therefore, the exhaust device needs a wide-range and low-loss VEGV to ensure efficient operation of the propulsion system. And the key of this VEGV is to eliminate the separation of the suction side under large deflection angle. So, this paper designed a new type of VEGV that can meet the demands by reducing the inverse pressure gradient of the suction surface (one of the necessary conditions for two-dimensional separation). In order to realize this design process, the spectral method of small disturbance equation (SMSDE) for two-dimensional subsonic flow was developed. Then, two VEGVs were designed by the SMSDE and verified by numerical simulation with Reynolds number 106 and blade solidity 1.18. The results show that the two VEGVs eliminate the separation of suction side at inlet Mach number of 0.25 and blade stagger angle of 45°. And the VEGV with lateral blade guarantees this characteristic up to Mach 0.3. Besides, when the inlet Mach number is below 0.3, the VEGV with lateral blade can ensure that the total pressure recovery coefficient is greater than 0.985, the outlet area ratio is greater than 0.95 and the exhaust angle error is less than 3.5°. Finally, Experimental verification of the vector exhaust device using the VEGV with lateral blade was carried out. The results show that the new VEGV has a higher total pressure recovery coefficient and outlet area ratio than the traditional VIGV at large deflection angles.

Parametric Study on Lateral–Directional Stability of Hypersonic Waverider

Lateral–directional stability is a critical issue for the design of any kind of aircraft. For the hypersonic waverider, the nonaxisymmetric, flat, and slender geometric feature makes it susceptible to the problem of lateral–directional instability. Therefore, the influence of geometric feature on the lateral–directional static and dynamic stability of hypersonic waverider is studied so as to enable the designer to consider this problem in the preliminary aircraft design stage. A series of power-law waveriders is generated at a given design space of the leading-edge parameters. Then, the variation of the static and dynamic derivatives with the design parameter is obtained by use of a surrogate model. It is found that increasing the dihedral angle of the lower surface can improve both the lateral and directional static stability of the total waverider. Furthermore, the dynamic instability mechanisms are analyzed in detail based on the approximate formulas derived from the linearized small-disturbance equations of motion. Especially, a concept of the Dutch-roll dynamic derivative is proposed, which plays an important part in the damping of the Dutch roll. Finally, it is found that increasing the dihedral angle can also improve the damping of the dynamic modes, including the roll, spiral, and Dutch roll.

Linear stability of transversely modulated finite‐amplitude capillary waves on deep water

AbstractWe investigate the three‐dimensional linear stability of the periodic motion of pure capillary waves progressing in permanent form on water of infinite depth for the whole range of wave amplitudes. After introducing a coordinate transformation based on a conformal map for two‐dimensional steady capillary waves, we perform linear stability analysis of finite‐amplitude capillary waves in the transformed space. To solve the linearized equations for small amplitude disturbances, it is assumed that the wavelengths of the disturbances in the transverse direction are much longer than those in the propagation direction and, therefore, the disturbances are weakly three‐dimensional. This assumption along with the periodicity of solutions allows us to write the linearized equations as an eigenvalue problem in matrix form. Following a perturbation theory for matrices, we expand the solutions of this eigenvalue problem in terms of a small parameter measuring the weak three‐dimensionality, and numerically obtain approximate eigenvalues. For weakly three‐dimensional superharmonic disturbances, the numerical results demonstrate that the pure capillary waves are two‐dimensionally stable, but three‐dimensionally unstable for almost all wave amplitudes. On the other hand, for subharmonic disturbances that are known to be two‐dimensionally unstable, it is found that the long‐wavelength disturbances in the transverse direction reduce the two‐dimensional growth rate near the critical amplitude beyond which the pure capillary waves are unstable.

Hypersonic similarity for the two dimensional steady potential flow with large data

In this paper, we establish the first rigorous mathematical result on the validation of the hypersonic similarity globally, which is also called the Mach-number independence principle, for the two dimensional steady potential flow. The hypersonic similarity is equivalent to the Van Dyke's similarity theory, that is, if the hypersonic similarity parameter K is fixed, the shock solution structures (after scaling) are consistent, when the Mach number of the flow is sufficiently large. One of the difficulty is that after scaling, the solutions are usually of large data since the perturbation of the hypersonic flow is usually not small related to the sonic speed. In order to make it, we first develop a modified Glimm scheme to construct the approximate solutions with large data and find fine structure of the elementary wave curves to obtain the global existence of entropy solutions with large data, for fixed K and sufficiently large Mach number of the incoming flow M∞. Finally, we further show that for a fixed hypersonic similarity parameter K, if the Mach number M∞→∞, the solutions obtained above approach to the solution of the corresponding initial-boundary value problem of the hypersonic small-disturbance equations. Therefore, the Van Dyke's similarity theory is verified rigorously for the first time.

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.

Dihedral influence on lateral–directional dynamic stability on large aspect ratio tailless flying wing aircraft

The influence of dihedral layout on lateral–directional dynamic stability of the tailless flying wing aircraft is discussed in this paper. A tailless flying wing aircraft with a large aspect ratio is selected as the object of study, and the dihedral angle along the spanwise sections is divided into three segments. The influence of dihedral layouts is studied. Based on the stability derivatives calculated by the vortex lattice method code, the linearized small-disturbance equations of the lateral modes are used to determine the mode dynamic characteristics. By comparing 7056 configurations with different dihedral angle layouts, two groups of stability optimized dihedral layout concepts are created. Flight quality close to Level 2 requirements is achieved in these optimized concepts without any electric stability augmentation system.

The sonic line as a free boundary

We consider the steady transonic small disturbance equations on a domain and with data that lead to a solution that depends on a single variable. After writing down the solution, we show that it can also be found by using a hodograph transformation followed by a partial Fourier transform. This motivates considering perturbed problems that can be solved with the same technique. We identify a class of such problems.

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 Robust Solution Procedure for the Transonic Small-Disturbance Equation in Fluid-Structure Interactions

A simple and robust discretization of the Low Frequency Transonic Small Disturbance (LF-TSD) equation is presented, suitable for applications such as fluid-structure interaction (FSI). Choices in the derivation and discretization of the LF-TSD equation are justified and the performance of the scheme is demonstrated for a typical nonlinear FSI problem.

마찰 감쇠를 고려한 에어포일의 천음속 공탄석 해석

마찰 감쇠가 있는 공탄성 해석을 위하여, 연계 시간 적분법을 사용하여 아음속/천음속 영역에서 공탄성 응답을 구하였다. 양력면에 발생하는 충격파에 의한 공기역학적 비선형성을 고려하기 위하여 동위상 주기 경계 조건이 적용된 미소교란 방정식을 비정상 공기력 계산에 적용하였다. 변위 종속적인 마찰 감쇠기가 있는 2차원 에어포일 시스템에 대하여 플러터 경계에 대한 수직력의 기울기와 마하수의 영향을 살펴보았다. For the aeroelastic analysis of a wing with friction damping, coupled time integration method was used to obtain time responses in the subsonic and transonic regions. To take into account aerodynamic nonlinearity induced by shock wave on the lifting surface, transonic small disturbance equation with in-phase periodic boundary condition was used for unsteady aerodynamic calculation. For 2-DOF airfoil system with displace-dependent friction dampers, the effects of normal load slope and Mach number on flutter boundary were investigated.

Wing box transonic-flutter suppression using piezoelectric self-sensing actuators attached toskin

The main objective of this research is to study the capability of piezoelectric (PZT)self-sensing actuators to suppress the transonic wing box flutter, which is a flow–structureinteraction phenomenon. The unsteady general frequency modified transonic smalldisturbance (TSD) equation is used to model the transonic flow about the wing. The wingbox structure and piezoelectric actuators are modeled using the equivalent plate method,which is based on the first order shear deformation plate theory (FSDPT). Thepiezoelectric actuators are bonded to the skin. The optimal electromechanical couplingconditions between the piezoelectric actuators and the wing are collected from previouswork. Three main different control strategies, a linear quadratic Gaussian (LQG) whichcombines the linear quadratic regulator (LQR) with the Kalman filter estimator (KFE), anoptimal static output feedback (SOF), and a classic feedback controller (CFC), are studiedand compared. The optimum actuator and sensor locations are determined using thenorm of feedback control gains (NFCG) and norm of Kalman filter estimatorgains (NKFEG) respectively. A genetic algorithm (GA) optimization technique isused to calculate the controller and estimator parameters to achieve a targetresponse.

Wing box transonic-flutter suppression using piezoelectric self-sensing diagonal-link actuators

The main objective of this research is to study the capability of Piezoelectric (PE) self-sensing actuators to suppress the transonic wing-box flutter, which is a flow-structure interaction phenomenon. The unsteady general frequency modified Transonic Small Disturbance (TSD) equation is used to model the transonic flow about the wing. The wing-box structure and the piezoelectric actuators are modeled using the equivalent plate method, which is based on the first-order shear deformation plate theory (FSDPT). The piezoelectric actuators are used as diagonal-links. The optimal electromechanical-coupling conditions between the piezoelectric actuators and the wing are collected from previous work. Three main different control strategies; Linear Quadratic Gaussian (LQG) which combines the Linear Quadratic Regulator (LQR) with the Kalman Filter Estimator (KFE), Optimal Static Output Feedback (SOF), and Classic Feedback Controller (CFC); are studied and compared. The optimum actuators and sensors locations are determined using the Norm of Feedback Control Gains (NFCG) and Norm of Kalman Filter Estimator Gains (NKFEG), respectively. A genetic algorithm (GA) optimization technique is used to calculate the controller and estimator parameters to achieve a target response.

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.

Multi-fidelity design optimization of transonic airfoils using physics-based surrogate modeling and shape-preserving response prediction

A computationally efficient design methodology for transonic airfoil optimization has been developed. In the optimization process, a numerically cheap physics-based low-fidelity surrogate (the transonic small-disturbance equation) is used in lieu of an accurate, but computationally expensive, high-fidelity (the compressible Euler equations) simulation model. Correction of the low-fidelity model is achieved by aligning its corresponding airfoil surface pressure distribution with that of the high-fidelity model using a shape-preserving response prediction technique. The resulting method requires only a single high-fidelity simulation per iteration of the design process. The method is applied to airfoil lift maximization in two-dimensional inviscid transonic flow, subject to constraints on shock-induced pressure drag and airfoil cross-sectional area. The results showed that more than a 90% reduction in high-fidelity function calls was achieved when compared to direct high-fidelity model optimization using a pattern-search algorithm.

Simulation of transonic flows using quad-core OpenMP Euler, flux modified transonic small disturbance, and Fluent codes

This article investigates what performance can be achieved when executing an aerodynamic code on a quad-core based personal computer with an OpenMP compiler in a Windows environment. Euler equations are solved in two dimensions for simplicity, to predict flow field around an aerofoil on an O-type boundary fitted structured grid. The grid is generated by solving a system of Poisson's equations utilising an efficient method of false transients, coupled with an approximate factorisation technique and variable time steps cycling process. Comparison between the OpenMP Euler, flux modified transonic small disturbance and Fluent results for transonic flow fields, with shock waves, around a NACA0012 aerofoil is presented. References Bailey, D., Barszcz, E., Barton, J., Browning, D., Carter, R., Dagum, L., Fatoohi, R., Fineberg, S., Frederickson, P., Lasinski, T., Schreiber, R., Simon, H., Venkatakrishnan, V., and Weeratunga, S., The NAS Parallel Benchmarks, Contractor Report 203186, NASA, USA, 1994. Engquist, B., and Osher, S., Stable and Entropy Satisfying Approximations for Transonic Flow Calculations, Mathematics of Computation, 34, 149, January 1980, pp. 45--75. Gear, J., Ly, E., and Phillips, N. J. T., A Time Marching, Type-Dependent, Finite Difference Algorithm for the Modified Transonic Small Disturbance Equation, Proceedings of the 21st Congress of the International Council of the Aeronautical Sciences (ICAS98), ICAS and AIAA, Melbourne, Australia, 1998, Paper A98-31513. Hirsch, C., Numerical Computation of Internal and External Flows: Computational Methods for Inviscid and Viscous Flows, Volume 2, John Wiley and Sons, Great Britain, 1992. Jin, H., Frumkin, M., and Yan, J., The OpenMP Implementation of NAS Parallel Benchmarks and its Performance, Technical Report NAS-99-011, NASA, Moffett Field, USA, 1999. Liu, Y., and Vinokur, M., Nonequilibrium Flow Computations 1: An Analysis of Numerical Formulations of Conservation Laws, Contractor Report 177489, NASA, California, USA, 1988. Lock, R. C., Test Cases for Numerical Methods in Two-Dimensional Transonic Flows, Report Number 575, AGARD, 1970. Ly, E., Improved Approximate Factorisation Algorithm for the Steady Subsonic and Transonic Flow over an Aircraft Wing, Proceedings of the 21st Congress of the International Council of the Aeronautical Sciences (ICAS98), ICAS and AIAA, Melbourne, Australia, 1998, Paper Number A98-31699. Ly, E., and Nakamichi, J., Time-Linearised Transonic Computations Including Entropy, Vorticity and Shock Wave Motion Effects, The Aeronautical Journal, November 2003, pp. 687--695. Ly, E., and Norrison, D., Automatic Elliptic Grid Generation by an Approximate Factorisation Algorithm, ANZIAM Journal, 48 (CTAC2006), pp. C188--C202, July 2007. Murman, E., Analysis of Embedded Shock Waves Calculated by Relaxation Methods, AIAA Journal, 12, 5, May 1974, pp. 626--633. Pulliam, T., Kutler, P., and Rossow, V., Harvard Lomax: His Quiet Legacy to Computational Fluid Dynamics, 14th AIAA Computational Fluid Dynamics Conference, Norfolk, VA, June 1999. Steger, J.L., and Warming, R. F., Flux Vector Splitting of the Inviscid Gasdynamic Equations with Application to Finite Difference Methods, Technical Memorandum TM-78605, NASA, California, USA, 1979. Van Leer, B., Flux-Vector Splitting for the Euler Equations, Proceedings of the 8th International Conference on Numerical Methods in Fluid Dynamics, Aachen, West Germany, 1982.

Toward Applying Algebraic Multigrid to Transonic Flow Problem

This article presents a new approach to solve the transonic flow problem by the algebraic multigrid (AMG) method. The proposed algorithm is demonstrated to attain excellent convergence rates, independent of the problem size, through the entire range of flow regimes. The mathematical difficulties of the problem are associated with the fact that the governing equation changes its type from elliptic (subsonic flow) to hyperbolic (supersonic flow). A pointwise relaxation method when applied directly to the upwind discrete operator, in the supersonic flow regime, is unstable. Resolving this difficulty is the main achievement of this work. A variety of issues regarding the AMG coarsening and construction of transfer operators is addressed in order to achieve the required efficiency for the problems under consideration. We demonstrate the AMG performance on a variety of model problems involving the quasi-linear full potential equation in two dimensions for subsonic, sonic, and supersonic flow and various flow directions (with respect to the grid). In addition, we present a preliminary result verifying the capabilities of the constructed algorithm to deal with a nonlinear problem—transonic small-disturbance equation.

Efficient Aeroelastic Analysis Method Including Inverse Design for Body Effects

EROELASTICITY is the research field of studying theinteractionbetweenthestructuraldynamicsandaerodynamics.It deals with the complex coupled phenomena of aircraft in extremeflightconditionsinthedesignstageofaircraftdevelopment.Amongthe aeroelastic problems, flutter phenomenon is the most dangerousone. Furthermore, if the aircraft flies in the transonic range, theaerodynamiccomplexityresultingfromshockwaveoscillation,flowseparation, and so forth, dramatically increases. Therefore, theinvestigation of nonlinear aerodynamics based on computationalfluid dynamics (CFD) has become important in recent aeroelasticanalysis. An aeroelastic examination using an efficient unsteadyaerodynamic analysis with the transonic small-disturbance (TSD)equation, which has been widely recognized as one of the mostefficient CFD-based approaches, has especially strong computa-tional advantages in many different types of parametric studies,because TSD code does not require CFD grid regeneration but onlychangesinwingsurfaceslopeduetostructuralmotiononafixed-gridsystem [1]. Based on these advantages, full-span aircraft have beenstudied.Inanactiveway,thefuselageisusuallyrepresentedbyaboxwithin the embedded multigrid. Instead of the former method, theCFD-basedaeroelasticanalysisofacomplexfull-spanaircraftmodelhasbeenperformedbyYooetal.[2]assumingthattheaerodynamicsof a wing are not influenced by the body shape. This assumption ismeaningful from an aeroelastic point of view, because the main partof lift and flexibility is contributed by the wing. Nevertheless, thefuselage can distort the aerodynamic small-disturbance pressure onthewingsurface, and it is necessary to investigate the bodyeffect influtter analysis [3].Tocomputeapressureperturbationduetothefuselage,aninversedesign method is applied in this research. A typical inverse designmethodismainlyusedtoautomaticallyoptimizetheairfoilgeometrytothelevelofanadequatetargetpressure.Thereareseveralkindsofinverse design methods, such as the MGM (modified Garabedian–McFadden) method [4], defection correction method [5], spring-analogy traction method [6], free-form deformation method, and soon. Among them, the MGM method is simple and easy to adoptbecauseitusesonlyoneauxiliaryequationtoobtainthetargetairfoilshape.ThissimplicitycanbeeffectivefortheCFD-basedaeroelasticanalysis problems, which need a tremendous computing time. TheMGM method assumes that the pressure difference between targetandbaselineairfoilsisafunctionofsurfacedisplacementanditsfirstandsecondderivatives.TheMGMauxiliaryequationregardstherateof change in surface slope as a pseudotime term and iterates until itconverges to zero.Figure1showstheflowchartofinversedesignprocedure.Aswiththe other CFD analysis process, the grid is generated first. Next, thesteadyaerodynamicsareanalyzed.Thenthepressuredistributiononawingsurfaceiscomparedwiththetargetpressuredistribution.Iftheresidualtermisnotzero,allthestepsarerepeated.However,themainadvantage is in the loop. Unlike other CFD codes, the TSD-basedaeroelastic code does not require grid regeneration for every step. Itproceeds with information on surface slope variation.In this study, an efficient aeroelastic analysis method includingbody aerodynamics is introduced. First, TSD theory and aeroelasticequation are explained briefly. Next, the MGM inverse designmethodisusedtomodifytheaerodynamicforcetoincludethebodyeffect.Insequence,thevalidationoftheMGMmethodisgiven.Thenthewing-bodymodelisselectedasanumericalexample.Finally,themodified aeroelastic analysis follows.

TRANSONIC AEROELASTIC ANALYSIS OF AIRCRAFT WINGS CONSIDERING THE BOUNDARY-LAYER EFFECTS

Aerodynamic solver using the transonic small-disturbance (TSD) equation has frequently been used to perform practical aeroelastic analysis for many aircraft models. In the present study, the more accurate aeroelastic analysis solver using the TSD equation was developed by considering the viscous effects of the boundary-layer. The viscous effects were considered using Green's lag-entrainment equations and an inverse boundary-layer method. Through aerodynamic analyses for several aircraft wings, the viscous-inviscid interaction approach could improve the accuracy of the aerodynamic computation using the TSD equation. Finally, the aeroelastic characteristics were investigated using comparisons of the time responses between the inviscid and viscous flows.