Torstens Skujins

A Flexible Hypersonic Vehicle Model Developed with Piston Theory

Abstract : For high Mach number flows, M>4, piston theory has been used to calculate the pressures on the surfaces of a vehicle. In a two-dimensional inviscid flow, a perpendicular column of fluid stays intact as it passes over a solid surface. Thus, the pressure at the surface can be calculated assuming the surface were a piston moving into a column of fluid. In this work, first-order piston theory is used to calculate the forces, moments, and stability derivatives for longitudinal motion of a hypersonic vehicle. Piston theory predicts a relationship between the local pressure on a surface and the normal component of fluid velocity produced by the surfaces motion. The advantage of piston theory over other techniques, such as Prandtl-Meyer flow, oblique shock, or Newtonian impact theory, is that unsteady aerodynamic effects can be included in the model. The unsteady effects, considered in this work, include perturbations in the linear velocities and angular rates, due to rigid body motion. A flexible vehicle model is developed to take into account the aeroelastic behavior of the vehicle. The vehicle forebody and aftbody are modeled as cantilever beams fixed at the center-of-gravity. Piston theory is used to account for the changes in the forces and moments due to the flexing of the vehicle. Piston theory yields an analytical model for the longitudinal motion of the vehicle, thus allowing design trade studies to be performed while still providing insight into the physics of the problem.

Reduced-Order Modeling of Unsteady Aerodynamics Across Multiple Mach Regimes

A reduced-order model for unsteady aerodynamic calculations across a range of Mach regimes based on linear convolution and a nonlinear correction factor is developed. Separate investigations are conducted for the sub-, trans-, and supersonic Mach regimes, and overall good results are seen when reduced-order model results are compared with full-order computational-fluid-dynamics solutions, though the reduced-order model errors tend to decrease as the Mach number increases. To assist reduced-order model construction, the method-of-segments simplified model has been created and tested throughout these same Mach regimes. Finally, a practical example of the reduced-order model’s applicability is presented by following a single test case from subsonic up through supersonic flight.

Six-Degree-of-Freedom Simulation of Hypersonic Vehicles

A three-dimensional, six-degree-of-freedom hypersonic vehicle model is being developed that more accurately characterizes vehicle dynamics for control studies. The main focus areas for this paper are the development of the rigid body dynamics equations of motion, the parametrization of the vehicle geometry, the formulation of the three-dimensional aerodynamic loads, and the adaptation of an existing 1D propulsion model. The aerodynamic loads are calculated by using or combining two-dimensional shock/expansion theory and the Taylor-Maccoll equations for conical flow. Comparisons with computational results show good agreement of resultant force and moment for the top and bottom vehicle surfaces and for most trials for the side surfaces. Using these models, the vehicle is trimmed for steady cruise conditions, and its flight dynamics are linearized about that state. This analysis provides information regarding the stability and controllability of a generic hypersonic vehicle in three-dimensional flight.

Reduced-Order Modeling of Hypersonic Vehicle Unsteady Aerodynamics

Accurate and computationally ecient models of unsteady aerodynamic loads are necessary for the development of hypersonic vehicle control algorithms. This work focuses on using convolution of modal step responses to construct a reduced-order model for these loads. In order to allow the model to be valid over a wide range of modal input amplitudes and ight conditions, a nonlinear correction factor is introduced. Not limited to a specic geometry, the correction factor methodology is general enough to be applied to many dierent two and three-dimensional vehicle congurations. Good correlation is seen between results obtained from the reduced-order model and computational results.

Toward an Unsteady Aerodynamic ROM for Multiple Mach Regimes

The accurate prediction of unsteady aerodynamic loads is of utmost importance in an aeroelastic simulation framework, which often times will be used to simulate a vehicle passing through multiple Mach regimes on a single ight path. In terms of computational e ciency, it is bene cial for an unsteady aerodynamic model to have a single mathematical form for the aerodynamic loads through the di erent Mach regimes for an entire simulation. An unsteady aerodynamic reduced-order modeling methodology based on linear convolution combined with a nonlinear correction factor is developed to ful ll this need. Though each Mach regime presents unique modeling challenges, the transonic regime is particularly challenging in this regard due to moving shocks and other nonlinear ow eld e ects. The purpose of this paper is to characterize both the error of the reduced-order model within the transonic regime as well the applicability of a simpli ed calculation procedure for the nonlinear correction which can replace, at times, full CFD simulations within the modeling framework. The reduced-order model results show generally good agreement with computational simulations of the AGARD 445.6 wing undergoing oscillation of multiple elastic modes. Also, at higher oscillation frequencies, a distinct phase shift which develops between the model and full-order simulation results has been quanti ed. Finally, one factor which appears to adversely a ect the model’s accuracy is the increasing memory length of the system being modeled.

Reduced-Order Modeling of Hypersonic Unsteady Aerodynamics Due to Multi-Modal Oscillations

The development of hypersonic vehicle control algorithms requires the accurate prediction of vehicle aerodynamic loads. This paper focuses on nding time-accurate unsteady loads due to the oscillation of multiple vehicle elastic modes. The reduced-order modeling methodology consists of using convolution of modal step responses along with a correction factor to take dierent oscillation amplitudes, ight conditions, and nonlinear aerodynamic eects due to multi-modal oscillations into account. Thus, the model is valid over a range of conditions rather than one specic set of conditions upon which it is constructed. When compared with CFD simulations, the errors are shown to be relatively small in most cases over a range of Mach numbers, modal frequencies, and modal oscillation amplitudes. Also, an error estimation methodology is developed to give a general sense beforehand as to the errors expected to be incurred through the use of the model.

On the Applicability of an Unsteady Aerodynamic ROM to the Transonic Flight Regime

Accurate prediction of aerodynamic loads throughout multiple ight regimes is vital for the development of control algorithms for vehicles which pass through di erent regimes during the course of a ight. As such, a reducedorder modeling methodology based on linear convolution combined with a nonlinear correction factor initially developed for hypersonic ight has been extended to the transonic ight regime. Errors mostly remained small when the unsteady lift, drag and moment coe cient results were compared with direct CFD simulations over a range of oscillation frequencies and amplitudes, though the drag coe cient did show larger errors at higher oscillation frequencies. A methodology to determine the optimal number and location in the parameter space of the ROM construction sampling points has also been developed. Results showed good agreement between the coe cients calculated by this method and those calculated by direct CFD simulation.

Applicability of an Analytical Shock/Expansion Solution to the Elevon Control Effectiveness for a 2-D Hypersonic Vehicle Configuration

Canard control surfaces placed on the forebody of a hypersonic vehicle provide advantageous characteristics for the vehicle’s controllability. This study looked at how these canards affect the flow seen by the elevon control surfaces on the aftbody of the vehicle, and, in turn the controllability of the vehicle in general. A 2-D analytical formulation was compared with CFD Euler solutions. For this analytical formulation, it was found that adding a thickness correction, as opposed to assuming that the airfoils were flat plates, actually decreased on average the accuracy of the model when compared with the computational data. The effect of the canard on the elevon, measured using the elevon effectiveness ratio, decreased as the distance between the control surfaces increased. Higher Mach numbers combined with higher canard deflection angles in general resulted in a greater effect on the elevon.