Transonic Small Disturbance Theory Research Articles

Feedback control of limit cycle oscillations and transonic buzz, using the nonlinear transonic small disturbance aerodynamics

In this paper, a systematic method to suppress transonic buzz with feedback is presented. A trailing edge control surface in the form of part-span flap was used only to modify and control the unsteady aerodynamic loading on the wing. The flap rotation was used to provide feedback, which consisted of a weighted linear combination of the amplitudes of the principal modes of the structure, referred to as the control law. A linear, optimal feedback control law, that is synthesized systematically based on pseudo-spectral time domain analysis, may be used in principle, to assess its capacity to actively suppress the buzz in the transonic flow domain by using a servo-controlled control surface to modify the unsteady, nonlinear aerodynamic loads on the wing. Thus, it is essential that a set of feasible control laws are first constructed. In this paper, this is done by applying the doublet-lattice method. Restrictions, such as near-zero structural damping in the flap mode, were imposed on the aeroelastic model to facilitate the occurrence of transonic buzz. The feasible set of control laws were then assessed using the nonlinear transonic small disturbance theory and an optimum control is selected to suppress the buzz. The essential differences of the behaviour of the closed-loop system in nonlinear transonic flow, when compared to the applications of linear optimal control in linear potential flow, are presented and discussed.

Synthesis of an active flutter suppression system in the transonic domain using a computational model

ABSTRACTControl laws for implementing active flutter suppression are generally derived from linear aeroelastic models. In this paper, families of control laws for implementing an active flutter suppression system were initially designed using linearised aeroelastic models based on the doublet lattice method after ignoring the aerodynamic loads associated with relatively faster time scales. Using these preliminary sets of control laws and the nonlinear transonic small disturbance theory, near-optimum control laws were chosen in the transonic domain to maximally increase the flutter speed of a typical aircraft wing by at least 16% or more. Thus it is shown that it is feasible to systematically design near-optimal control laws for active flutter suppression using computational models in transonic flow. The doublet lattice method coupled with the zeroth-order matrix Padé approximant provided the fastest method for synthesising a large number of preliminary control laws. The methodology was successfully demonstrated by applying it to two benchmarking examples.

THE WING-BODY AEROELASTIC ANALYSES USING THE INVERSE DESIGN METHOD

Flutter phenomenon is one of the most dangerous problems in aeroelasticity. When it occurs, the aircraft structure can fail in a few second. In recent aeroelastic research, computational fluid dynamics (CFD) techniques become important means to predict the aeroelastic unstable responses accurately. Among various flow equations like Navier-Stokes, Euler, full potential and so forth, the transonic small disturbance (TSD) theory is widely recognized as one of the most efficient theories. However, the small disturbance assumption limits the applicable range of the TSD theory to the thin wings. For a missile which usually has small aspect ratio wings, the influence of body aerodynamics on the wing surface may be significant. Thus, the flutter stability including the body effect should be verified. In this research an inverse design method is used to complement the aerodynamic deficiency derived from the fuselage. MGM (modified Garabedian-McFadden) inverse design method is used to optimize the aerodynamic field of a full aircraft model. Furthermore, the present TSD aeroelastic analyses do not require the grid regeneration process. The MGM inverse design method converges faster than other conventional aerodynamic theories. Consequently, the inverse designed aeroelastic analyses show that the flutter stability has been lowered by the body effect.

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.

FLUID-STRUCTURE INTERACTION ANALYSIS OF A HIGH-ASPECT-RATIO WING CONSIDERING STRUCTURAL NONLINEARITY

In this research, fluid-structure interaction problem including geometric structural nonlinearity is studied for a high-aspect-ratio wing. When a high-aspect-ratio wing structure is interacted with external airload, geometric structural nonlinearity can be caused by large deflection of a wing. For the investigation of such a fluid-structure interaction problem, the transonic small disturbance theory for the aerodynamic analysis and the large deflection beam theory for the structural analysis are used, respectively. For the coupling between fluid and structure, the transformation of a displacement from the structural mesh to the aerodynamic grid is performed by a shape function which is used for the finite element and the inverse transformation of force by work equivalent load method. Static deformations in the vertical and twist deflections caused by gravity loading are compared with experimental results. Also, static aeroelastic analysis results are compared with experimental data. From the analysis results, effects of structural nonlinearity on static aeroelastic characteristics are investigated.

Transonic Flutter Suppression of the 2-D Flap Wing with External Store using CFD-based Aeroservoelasticity

An analysis procedure for the combined problem of control algorithm and aeroelastic system which is based on the computational fluid dynamics(CFD) technique has been developed. The aerodynamic forces in the transonic region are calculated from the transonic small disturbance(TSD) theory. An linear quadratic regulator(LQR) controller is designed to suppress the transonic flutter. The optimal control gain is estimated by solving the discrete-time Riccati equation. The system identification technique rebuilds the CFD-based aeroelstic system in order to form an adequate system matrix which involved in the discrete-time Riccati equation. Finally the controller, that is constructed on the basis of system identification technique, is used to suppress the flutter phenomenon of the airfoil with attached store. This approach, that is, the CFD-based aeroservoelasticity design, can be utilized for the development of effective flutter controller design in the transonic region.

VIBRATION AND TRANSONIC FLUTTER ANALYSIS FOR F-16 STORES CONFIGURATION CLEARANCE

This paper supports quick and accurate prediction of the flutter onset speed of an F-16 Block 40/50 configured with external stores in the transonic flight regime. Current flutter prediction methods are reviewed and hypothesized mechanisms for limit cycle oscillation (LCO) are summarized. New efforts to correlate transonic small disturbance (TSD) theory methods with flight tests are outlined. Vibration analysis and structural optimization of an F-16 finite element model were used to match ground vibration testing results. Frequency tuning was found to be critical for accurate flutter speed predictions. Sensitivity to nonlinear aerodynamic effects and store modeling was examined.

Nonlinear Aeroelastic Analysis of a High-Aspect-Ratio Wing with Large Deflection Effects

In this study, nonlinear static and dynamic aeroelastic analyses for a high-aspect-ratio wing have been performed. To achieve these aims, the transonic small disturbance (TSD) theory for the aerodynamic analysis and the large deflection beam theory considering a geometrical nonlinearity for the structural analysis are applied, respectively. For the coupling between fluid and structure, the transformation of a displacement from the structural mesh to the aerodynamic grid is performed by a shape function which is used for the finite element and the inverse transformation of force by work equivalent load method. To validate the current method, the present analysis results of a high-aspect-ratio wing are compared with the experimental results. Static deformations in the vertical and torsional directions caused by an angle of attack and gravity loading are compared with experimental results. Also, static and dynamic aeroelastic characteristics are investigated. The comparisons of the flutter speed and frequency between a linear and nonlinear analysis are presented.

Dynamic Aeroelastic Instabilities of an Aircraft Wing with Underwing Store in Transonic Regime

The research used the transonic small disturbance theory to better understand the dynamic aeroelastic phenomena and factors that affect the onset of flutter and store induced limit-cycle oscillations (LCO) in the transonic regime. Several parametric studies of the flutter and LCO of an aircraft wing with underwing store in the transonic regime were conducted, as well as an investigation of the effect of inclusion of store aerodynamics on the onset of flutter. The flutter sensitivity was analyzed for the following store parameters: (1) location of underwing store center of gravity with respect to aerodynamic root chord; (2) location of underwing store along the span of the wing; and (3) underwing clearance (pylon length). The parametric studies indicated that as the store center of gravity is moved fore of the elastic axis, the flutter velocity of the wing increased. Also, as the store is moved towards the aerodynamic tip chord, the flutter velocity of the wing decreased. The research work also concluded that as the underwing clearance is increased, the flutter velocity of the wing decreased. In addition to these results, it also helped in understanding that addition of store aerodynamics had no significant influence on the flutter velocity of the wing. Also, studies were conducted to identify the onset of LCO for different configurations of underwing store and flight regimes (unmatched analysis), thereby identifying the parameters that induce LCO. The sensitive parameters that affect flutter and LCO are identified.

Numerical Investigation on the Aeroelastic Instability of a Complete Aircraft Model

A nonlinear aeroelastic analysis system has been developed using the coupled numerical techniques of computational fluid dynamics (CFD) and computational structural dynamics (CSD) in the present study and the analysis system has been applied to an actual aircraft design procedure to examine aeroelastic stabilities. An aerodynamic analysis is performed using the transonic small disturbance (TSD) theory for computational efficiency and the results are compared with experimental data. An aeroelastic analysis in the time domain is performed to obtain the flutter boundaries of the aircraft model. Finally, the various effects of launchers and control surfaces are investigated.

Numerical Analysis of Store-Induced Limit-Cycle Oscillation

Store-induced limit-cycle oscillation of a rectangular wing with tip store in transonic flow is simulated using a variety of mathematical models for the flowfield: transonic small-disturbance theory (with and without inclusion of store aerodynamics) and transonic small-disturbance theory with interactive boundary layer (without inclusion of store aerodynamics). For the conditions investigated, assuming inviscid flow, limit-cycle oscillations are observed to occur as a result of a weakly subcritical Hopf bifurcation and are obtained at speeds lower than those predicted 1) nonlinearly for clean-wing flutter and 2) linearly for wing/store flutter. The ability of transonic small-disturbance theory to predict the occurrence and strength of this type of limit-cycle oscillation is compared for the different models. Differences in unmatched and matched aeroelastic analysis are described. Solutions computed for the clean rectangular wing are compared to those computed with the Euler equations for a case of static aeroelastic behavior and for a case of forced, rigid-wing oscillation at Mach 0.92.

Studies of Store-Induced Limit-Cycle Oscillations Using a Model with Full System Nonlinearities

The authors investigate limit-cycle oscillations of a wing/store configuration. Unlike typical aeroelastic studies that are based upon a linearized form of the governing equations, herein full system nonlinearities are retained, and include transonic flow effects, coupled responses from the structure, and store-related kinematics and dynamics. Unsteady aerodynamic loads are modeled with the equations from transonic small disturbance theory. The structural dynamics for the cantilevered wing are modeled by the nonlinear equations of motion for a beam. The effects of general store-placement are modeled by the nonlinear equations of motion related to the position-induced nonlinear kinematics. Chordwise deformations of the wing surface, as well as pylon and store flexibility, are assumed negligible. Nonlinear responses are studied by examining bifurcation and related response characteristics using direct simulation. Particular attention is given to cases for which large-time, time-dependent behavior is dependent on initial conditions, as observed for some configurations in flight test. Comparisons of results in which selective nonlinearities are excluded indicate that the accurate prediction of nonlinear responses such as limit cycle oscillations (LCOs) may depend upon consideration of all nonlinearities related to the full system.

Frequency and time domain flutter computations of a wing with oscillating control surface including shock interference effects

In this study, the nonlinear aeroelastic characteristics of a wing with an oscillating control surface have been examined in transonic and supersonic regimes. The various effects of rotational stiffness on flutter have also been observed. A modified transonic small-disturbance (TSD) theory is used to more effectively analyze the unsteady aerodynamics of a wing with an oscillating control surface. In the flutter analysis, a coupled time integration method (CTIM) and a transient pulse method (TPM) were used to examine the effects of rotational stiffness reduction on the control surface. The present study shows that the severe decrease of flutter speed and the flutter mode transition can be induced by the reduction of rotational stiffness. In particular, it is shown that the aerodynamic effects of control surface oscillation play an important role in this flutter speed reduction.

Comparison of Static and Dynamic Neural Networks for Limit Cycle Oscillation Prediction

A dynamic artificial neural network in the form of a multilayer perceptron with a delayed recurrent feedback connection is investigated to determine its ability to predict linear and nonlinear flutter response characteristics. Flight-test results show that limit cycle oscillation (LCO) response characteristics are strongly dependent on Mach number in the transonic flight regime. This effect is also evident in the classical transonic small-disturbance theory governing equations. A dynamic network is considered in order to examine the effects of sequential Mach-number dependence on the network's predictive capability. The architecture of a dynamic network allows for modeling data dependent on a sequentially or linearly increasing parameter (usually time, but in this case Mach number). The predictive capabilities are compared to those of a static artificial neural network. The network is developed and trained using linear flutter analysis and flight-test results from a fighter test. Eleven external store carriage configurations are used as training data, and three configurations are used as test cases. The dynamic network was successful in predicting the aeroelastic oscillation frequency and amplitude responses over a range of Mach numbers for two of the test cases. The dynamic network showed slightly better correlation to flight-test results for the typical LCO test case but slightly worse correlation for the flutter case. Predictions for the nontypical LCO test case were not good for either network.

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.

Multidisciplinary Design of Vehicle Structures with Improved Roll Maneuverability-Transonic Regime

The objective of this research is the preliminary design of aircraft structures for improved steady-state roll performance in the transonic regime. The control surface effectiveness (a parameter that indicates the relative rolling power produced by an aileron of a flexible aircraft with respect to the rigid aircraft) is employed for quantifying the roll performance. Control surface effectiveness is evaluated by iteratively solving the antisymmetric trim equation on roll rate. Each iteration of the trim equation necessitates the computation of nonlinear airloads. To make the multidisciplinary design optimization computationally practical, the transonic small disturbance (TSD) theory is used for the aerodynamic analysis. Transonic loads including shock effects are captured by employing the TSD theory for which the CAP-TSD program is used. For design optimization, a nongradient-based method is adopted. Control surface effectiveness variation with respect to the statistically significant structural variables (skins, spars, ribs, and posts) is approximated using the response surface method. Multidisciplinary optimization for minimum weight, with control effectiveness, stress, displacement, and frequency constraints, is performed.

Control Surface Reversal in the Transonic Regime Including Viscous Effects

This paper investigates control surface reversal in the transonic flight regime. Surface-pressure distributions on a rigid and deformable wing are presented using linear and nonlinear aerodynamic methods, which incorporate the effects of surface shocks and viscosity with an interactive boundary layer. Transonic small-disturbance (TSD) aerodynamics is used to determine reversal of a typical fighter wing to study the interactions among control surface deflections, structural flexibility, and embedded shocks in a flowfield where significant viscous effects can exist. Pressure distributions on the wing are examined, and control surface reversal calculations are presented. The results show that the reduction of the reversal dynamic pressure can be as much as 30% when an aerodynamic method capable of representing a shock is used. A limited number of Euler computational fluid dynamic (CFD) results are presented for the purpose of comparison with transonic small disturbance results. The adequacy of using TSD theory in reversal prediction as compared to that obtained from either Euler or Navier-Stokes CFD aerodynamics is discussed. It is suggested that the inviscid transonic small disturbance aerodynamic provides a capability of representing shocks with only a modest increase of computer time beyond that necessary with the linear surface panel method. It is recommended that the inviscid version of TSD be incorporated in the preliminary design of aircraft operating at transonic speeds.

Parametric investigation of nonadiabatic compressible flow around airfoils

The compressible flow of a mixture of dry air (an inert carrier gas) and water vapor with nonequilibrium and homogeneous condensation around thin airfoils is examined. The investigation is based on a recent transonic small-disturbance (TSD) theory. The theory provides the governing similarity parameters of the flow and condensation process. The parameters give the relationship between the flow properties, the amount of water vapor mass in the air, and the airfoil’s chord and thickness ratio. The results of varying the similarity parameters are demonstrated by numerical simulations. It is found that heat addition as a result of condensation causes significant changes in the compressible flow behavior and affects the aerodynamic performance of airfoils. Increasing the free-stream frozen Mach number or the airfoil’s chord (with fixed free-stream temperature, pressure, and amount of water vapor) augments the condensation region and the amount of heat transfer to the flow. The heat transfer to the flow also results in the appearance of a compression wave in addition to a regular shock wave. Also, decreasing the free-stream temperature with a fixed amount of water vapor in the air or increasing the free-stream temperature with a fixed free-stream supersaturation ratio enlarges the condensation region and reduces the shock wave strength. These results demonstrate the possible use of condensation phenomena to control aerodynamic characteristics of airfoils operating in internal flow systems.

Control-Surface Reversal in the Transonic Regime

This paper investigates control surface reversal in the transonic flight regime. Linear and nonlinear rigid and aeroelastic analyses are performed to examine the effects of including flow nonlinearities in the prediction of this aeroelastic phenomenon. Transonic small disturbance theory is employed in the analysis of a simple rectangular wing and a typical fighter type wing to study the interactions among control surface deflections, structural flexibility, and embedded shocks in the flow field. Pressure distributions on the wings are examined. Rigid rolling moment calculations are presented, followed by control surface effectiveness and control surface reversal trends as the Mach number is varied from a subsonic value, through the transonic regime, into the supersonic region. These results are discussed based on the predictions of the pressure coefficients generated by the solution of the transonic small-disturbance equation. Finally, generalizations are presented about the effects of including aerodynamic nonlinearities in the prediction% of steady aeroelastic phenomenon in transonic flow conditions.

On the Design of Lifting Airfoils with High Critical Mach Number Using Full Potential Theory

We wish to construct airfoils that have the highest free-stream Mach number for a given set of geometric constraints for which the flow is nowhere supersonic. Nonlifting airfoils that maximize the critical Mach number \(\) for a given cross-sectional area are known to possess long sonic segments at their critical speed. To construct lifting airfoils, we proceed under the conjecture that an airfoil with a high value of \(\) has the longest possible arc length of sonic velocity over its upper and lower surface. In Kropinski et al. (1995) the lifting problem was tackled in transonic small-disturbance theory. In this paper we numerically construct lifting airfoils with high \(\) using the full potential theory and we show that these airfoils have significantly higher \(\) than some standard airfoils. We also construct airfoils with higher values of the lift coefficient, by relaxing the speed constraint on the lower surface of the airfoil to have a value less than sonic.