Approximate Modeling of Unsteady Aerodynamics for Hypersonic Aeroelasticity

Abstract

Various approximations to unsteady aerodynamics are examined for the aeroelastic analysis of a thin doublewedge airfoil in hypersonic flow. Flutter boundaries are obtained using classical hypersonic unsteady aerodynamic theories: piston theory, Van Dyke's second-order theory, Newtonian impact theory, and unsteady shock-expansion theory. The theories are evaluated by comparing the flutter boundaries with those predicted using computational fluid dynamics solutions to the unsteady Navier-Stokes equations. In addition, several alternative approaches to the classical approximations are also evaluated: two different viscous approximations based on effective shapes and combined approximate computational approaches that use steady-state computational-fluid-dynamics-based surrogate models in conjunction with piston theory. The results indicate that, with the exception of first-order piston theory and Newtonian impact theory, the approximate theories yield predictions between 3 and 17% of normalized root-mean-square error and between 7 and 40% of normalized maximum error of the unsteady Navier-Stokes predictions. Furthermore, the demonstrated accuracy of the combined steady-state computational fluid dynamics and piston theory approaches suggest that important nonlinearities in hypersonic flow are primarily due to steadystate effects. This implies that steady-state flow analysis may be an alternative to time-accurate Navier-Stokes solutions for capturing complex flow effects.

Generalized Formulation and Review of Piston Theory for Airfoils

“…However, piston theory enjoyed renewed attention in the 1990s, with the early implementation of piston theory with Euler solutions in CFD [8] and with continued development of aeroelastic panel codes [9]. The 2000s saw a marked resurgence in the interest in piston theory as a computationally inexpensive method for modelling supersonic and hypersonic aeroelastic problems, with applications being found in aeroservo-elasticity and aero-thermo-elasticity, with the further integration of piston theory with CFD; recent literature highlights the continued application of piston theory in reducing the computational cost of CFD for hypersonic aeroelasticity [10][11][12][13][14][15].…”

Role of Higher-Order Terms in Local Piston Theory

“…The use of second-and third-order classical piston theory [1] (CPT) is commonplace, with the role of the higher-order terms being well understood [2]. The advantages of local piston theory (LPT) relative to CPT have been demonstrated previously [3].…”

Local Piston Theory as an Alternative to Mesh Deformation: Slender Wing/Body Configurations

“…This is because the theoretical basis of LPT has been established [35] as a special case of the perturbed Euler equations for slender bodies; Euler-based LPT is mathematically consistent. The application of LPT to a mean-steady solution of the N-S equations has not been shown to be mathematically consistent, and has seen varying [18,19,36] degrees of success. The change in the aerodynamic loading following structural deformation as obtained from the N-S equations would include not only the influence of the interference flowfield and the local surface inclination, but also the interactions of the viscous boundary-layer.…”

“…The method achieves this cost reduction through simplification of the underlying physics; the 3D partial differential equations for a field are replaced by point-wise algebraic equations at the boundary. Applications of LPT in literature have typically been restricted to simple geometries such as airfoils [5,18,19], panels, low aspect-ratio wings [5], or wave-riders [19], and have primarily been concerned with dynamic aeroelasticity. Research into the application of LPT in interfering flows has been sparse, focusing on shock impingement on deforming plates [20,21].…”