Approximate Modeling of Unsteady Aerodynamics for Hypersonic Aeroelasticity. McNamara, J. J, Crowell, A. R, Friedmann, P. P, Glaz, B., & Gogulapati, A. Journal of Aircraft, 2010. ![Approximate Modeling of Unsteady Aerodynamics for Hypersonic Aeroelasticity [link]](https://bibbase.org/img/filetypes/link.svg "link")
Paper doi abstract bibtex 1 download Various approximations to unsteady aerodynamics are examined for the aeroelastic analysis of a thin double-wedge 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 steady-state effects. This implies that steady-state flow analysis may be an alternative to time-accurate Navier-Stokes solutions for capturing complex flow effects. Nomenclature fA p g = estimated aeroelastic system matrix a = nondimensional offset between the elastic axis and the midchord, positive for elastic-axis locations behind midchord a o , a i , b i , A i = coefficients used for damping and frequency identification a 1 = speed of sound b = semichord, c=2 Cx = local deviations of kriging model C L;SS , C L;SUR SS = static component of lift coefficient computed using a computational fluid dynamics and a computational-fluid-dynamics-based surrogate C M;SS , C M;SUR SS = static component of moment coefficient about the midchord computed using a computational fluid dynamics and a computational-fluid-dynamics-based surrogate C p = pressure coefficient fC p g = estimated aeroelastic system matrix C p = component of piston theory pressure due to combined surface velocity and surface inclination C p;SS = component of piston theory pressure due strictly to surface inclination C p;vel = component of piston theory pressure due strictly to surface velocity c = chord length, reference length c l , c m = coefficients of lift and moment about the elastic axis F Z = flutter prediction parameter F j = intermediate function used to compute the flutter prediction parameter h = plunge degree of freedom of the airfoil h i = states in state-space representation of autoregressive model K = diagonal generalized stiffness matrix K h , K = spring constants in pitch and plunge k = discrete time L = sectional lift force L 1 = normalized maximum error M = diagonal generalized mass matrix M EA = sectional aerodynamic moment about the elastic axis M f = flutter Mach number M 1 = freestream Mach number m = Mass n m = number of modes p, p 1 = pressure and freestream pressure Q = vector of generalized forces q = vector of generalized degrees of freedom q i = generalized displacements q 1 , q f = dynamic pressure and dynamic pressure at flutter R = gas constant for air Rx = global approximation of kriging model r = real part of eigenvalue r = nondimensional radius of gyration of the airfoil S = sample sites of the parameter space S = airfoil static imbalance s = imaginary part of eigenvalue T e = sample time t = time = freestream velocity v n = normal velocity of airfoil surfaces W = snapshot matrix, computational fluid dynamics response data to s w d = displacement of the surface of the structure X j , Y j = flutter parameter matrices fX p g = state matrix x, y, z = spatial coordinates x rot = point about which airfoil angle of attack is measured x = nondimensional offset between the elastic axis and the cross-sectional center of gravity yx = kriging approximation Zx; t = position of structural surface Z str x = function describing surface geometry = pitch degree of freedom s = angle of attack = ratio of specific heats k1 = input for autoregressive moving-average model of aeroelastic system = damping ratio = estimated matrix eigenvalue m = airfoil mass ratio = air density = slope of the airfoil surface i = vector of displacements for mode i ! = frequency ! h = frequency corresponding to stiffness associated with the plunge degree of freedom of the airfoil ! = frequency corresponding to stiffness associated with the pitch degree of freedom of the airfoil
@article{mcnamara2010,
title = {Approximate {Modeling} of {Unsteady} {Aerodynamics} for {Hypersonic} {Aeroelasticity}},
volume = {47},
url = {http://arc.aiaa.org},
doi = {10.2514/1.C000190},
abstract = {Various approximations to unsteady aerodynamics are examined for the aeroelastic analysis of a thin double-wedge 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 steady-state effects. This implies that steady-state flow analysis may be an alternative to time-accurate Navier-Stokes solutions for capturing complex flow effects. Nomenclature fA p g = estimated aeroelastic system matrix a = nondimensional offset between the elastic axis and the midchord, positive for elastic-axis locations behind midchord a o , a i , b i , A i = coefficients used for damping and frequency identification a 1 = speed of sound b = semichord, c=2 Cx = local deviations of kriging model C L;SS , C L;SUR SS = static component of lift coefficient computed using a computational fluid dynamics and a computational-fluid-dynamics-based surrogate C M;SS , C M;SUR SS = static component of moment coefficient about the midchord computed using a computational fluid dynamics and a computational-fluid-dynamics-based surrogate C p = pressure coefficient fC p g = estimated aeroelastic system matrix C p = component of piston theory pressure due to combined surface velocity and surface inclination C p;SS = component of piston theory pressure due strictly to surface inclination C p;vel = component of piston theory pressure due strictly to surface velocity c = chord length, reference length c l , c m = coefficients of lift and moment about the elastic axis F Z = flutter prediction parameter F j = intermediate function used to compute the flutter prediction parameter h = plunge degree of freedom of the airfoil h i = states in state-space representation of autoregressive model K = diagonal generalized stiffness matrix K h , K = spring constants in pitch and plunge k = discrete time L = sectional lift force L 1 = normalized maximum error M = diagonal generalized mass matrix M EA = sectional aerodynamic moment about the elastic axis M f = flutter Mach number M 1 = freestream Mach number m = Mass n m = number of modes p, p 1 = pressure and freestream pressure Q = vector of generalized forces q = vector of generalized degrees of freedom q i = generalized displacements q 1 , q f = dynamic pressure and dynamic pressure at flutter R = gas constant for air Rx = global approximation of kriging model r = real part of eigenvalue r = nondimensional radius of gyration of the airfoil S = sample sites of the parameter space S = airfoil static imbalance s = imaginary part of eigenvalue T e = sample time t = time = freestream velocity v n = normal velocity of airfoil surfaces W = snapshot matrix, computational fluid dynamics response data to s w d = displacement of the surface of the structure X j , Y j = flutter parameter matrices fX p g = state matrix x, y, z = spatial coordinates x rot = point about which airfoil angle of attack is measured x = nondimensional offset between the elastic axis and the cross-sectional center of gravity yx = kriging approximation Zx; t = position of structural surface Z str x = function describing surface geometry = pitch degree of freedom s = angle of attack = ratio of specific heats k1 = input for autoregressive moving-average model of aeroelastic system = damping ratio = estimated matrix eigenvalue m = airfoil mass ratio = air density = slope of the airfoil surface i = vector of displacements for mode i ! = frequency ! h = frequency corresponding to stiffness associated with the plunge degree of freedom of the airfoil ! = frequency corresponding to stiffness associated with the pitch degree of freedom of the airfoil},
number = {6},
journal = {Journal of Aircraft},
author = {McNamara, Jack J and Crowell, Andrew R and Friedmann, Peretz P and Glaz, Bryan and Gogulapati, Abhijit},
year = {2010},
}
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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 steady-state effects. This implies that steady-state flow analysis may be an alternative to time-accurate Navier-Stokes solutions for capturing complex flow effects. Nomenclature fA p g = estimated aeroelastic system matrix a = nondimensional offset between the elastic axis and the midchord, positive for elastic-axis locations behind midchord a o , a i , b i , A i = coefficients used for damping and frequency identification a 1 = speed of sound b = semichord, c=2 Cx = local deviations of kriging model C L;SS , C L;SUR SS = static component of lift coefficient computed using a computational fluid dynamics and a computational-fluid-dynamics-based surrogate C M;SS , C M;SUR SS = static component of moment coefficient about the midchord computed using a computational fluid dynamics and a computational-fluid-dynamics-based surrogate C p = pressure coefficient fC p g = estimated aeroelastic system matrix C p = component of piston theory pressure due to combined surface velocity and surface inclination C p;SS = component of piston theory pressure due strictly to surface inclination C p;vel = component of piston theory pressure due strictly to surface velocity c = chord length, reference length c l , c m = coefficients of lift and moment about the elastic axis F Z = flutter prediction parameter F j = intermediate function used to compute the flutter prediction parameter h = plunge degree of freedom of the airfoil h i = states in state-space representation of autoregressive model K = diagonal generalized stiffness matrix K h , K = spring constants in pitch and plunge k = discrete time L = sectional lift force L 1 = normalized maximum error M = diagonal generalized mass matrix M EA = sectional aerodynamic moment about the elastic axis M f = flutter Mach number M 1 = freestream Mach number m = Mass n m = number of modes p, p 1 = pressure and freestream pressure Q = vector of generalized forces q = vector of generalized degrees of freedom q i = generalized displacements q 1 , q f = dynamic pressure and dynamic pressure at flutter R = gas constant for air Rx = global approximation of kriging model r = real part of eigenvalue r = nondimensional radius of gyration of the airfoil S = sample sites of the parameter space S = airfoil static imbalance s = imaginary part of eigenvalue T e = sample time t = time = freestream velocity v n = normal velocity of airfoil surfaces W = snapshot matrix, computational fluid dynamics response data to s w d = displacement of the surface of the structure X j , Y j = flutter parameter matrices fX p g = state matrix x, y, z = spatial coordinates x rot = point about which airfoil angle of attack is measured x = nondimensional offset between the elastic axis and the cross-sectional center of gravity yx = kriging approximation Zx; t = position of structural surface Z str x = function describing surface geometry = pitch degree of freedom s = angle of attack = ratio of specific heats k1 = input for autoregressive moving-average model of aeroelastic system = damping ratio = estimated matrix eigenvalue m = airfoil mass ratio = air density = slope of the airfoil surface i = vector of displacements for mode i ! = frequency ! h = frequency corresponding to stiffness associated with the plunge degree of freedom of the airfoil ! = frequency corresponding to stiffness associated with the pitch degree of freedom of the airfoil","number":"6","journal":"Journal of Aircraft","author":[{"propositions":[],"lastnames":["McNamara"],"firstnames":["Jack","J"],"suffixes":[]},{"propositions":[],"lastnames":["Crowell"],"firstnames":["Andrew","R"],"suffixes":[]},{"propositions":[],"lastnames":["Friedmann"],"firstnames":["Peretz","P"],"suffixes":[]},{"propositions":[],"lastnames":["Glaz"],"firstnames":["Bryan"],"suffixes":[]},{"propositions":[],"lastnames":["Gogulapati"],"firstnames":["Abhijit"],"suffixes":[]}],"year":"2010","bibtex":"@article{mcnamara2010,\n\ttitle = {Approximate {Modeling} of {Unsteady} {Aerodynamics} for {Hypersonic} {Aeroelasticity}},\n\tvolume = {47},\n\turl = {http://arc.aiaa.org},\n\tdoi = {10.2514/1.C000190},\n\tabstract = {Various approximations to unsteady aerodynamics are examined for the aeroelastic analysis of a thin double-wedge 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 steady-state effects. This implies that steady-state flow analysis may be an alternative to time-accurate Navier-Stokes solutions for capturing complex flow effects. 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