@article{
 title = {Approximate Modeling of Unsteady Aerodynamics for Hypersonic Aeroelasticity},
 type = {article},
 year = {2010},
 volume = {47},
 websites = {http://arc.aiaa.org},
 id = {dcd65864-4a75-3ae3-ba92-e2893e3c11cb},
 created = {2021-10-25T22:53:06.660Z},
 accessed = {2021-10-25},
 file_attached = {true},
 profile_id = {6476e386-2170-33cc-8f65-4c12ee0052f0},
 group_id = {5a9f751c-3662-3c8e-b55d-a8b85890ce20},
 last_modified = {2021-10-25T22:56:23.805Z},
 read = {false},
 starred = {false},
 authored = {false},
 confirmed = {false},
 hidden = {false},
 citation_key = {mcnamara:ja:2010},
 private_publication = {false},
 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},
 bibtype = {article},
 author = {McNamara, Jack J and Crowell, Andrew R and Friedmann, Peretz P and Glaz, Bryan and Gogulapati, Abhijit},
 doi = {10.2514/1.C000190},
 journal = {Journal of Aircraft},
 number = {6}
}

{"_id":"NYDSxTmqpmxono4JT","bibbaseid":"mcnamara-crowell-friedmann-glaz-gogulapati-approximatemodelingofunsteadyaerodynamicsforhypersonicaeroelasticity-2010","author_short":["McNamara, J., J.","Crowell, A., R.","Friedmann, P., P.","Glaz, B.","Gogulapati, A."],"bibdata":{"title":"Approximate Modeling of Unsteady Aerodynamics for Hypersonic Aeroelasticity","type":"article","year":"2010","volume":"47","websites":"http://arc.aiaa.org","id":"dcd65864-4a75-3ae3-ba92-e2893e3c11cb","created":"2021-10-25T22:53:06.660Z","accessed":"2021-10-25","file_attached":"true","profile_id":"6476e386-2170-33cc-8f65-4c12ee0052f0","group_id":"5a9f751c-3662-3c8e-b55d-a8b85890ce20","last_modified":"2021-10-25T22:56:23.805Z","read":false,"starred":false,"authored":false,"confirmed":false,"hidden":false,"citation_key":"mcnamara:ja:2010","private_publication":false,"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","bibtype":"article","author":"McNamara, Jack J and Crowell, Andrew R and Friedmann, Peretz P and Glaz, Bryan and Gogulapati, Abhijit","doi":"10.2514/1.C000190","journal":"Journal of Aircraft","number":"6","bibtex":"@article{\n title = {Approximate Modeling of Unsteady Aerodynamics for Hypersonic Aeroelasticity},\n type = {article},\n year = {2010},\n volume = {47},\n websites = {http://arc.aiaa.org},\n id = {dcd65864-4a75-3ae3-ba92-e2893e3c11cb},\n created = {2021-10-25T22:53:06.660Z},\n accessed = {2021-10-25},\n file_attached = {true},\n profile_id = {6476e386-2170-33cc-8f65-4c12ee0052f0},\n group_id = {5a9f751c-3662-3c8e-b55d-a8b85890ce20},\n last_modified = {2021-10-25T22:56:23.805Z},\n read = {false},\n starred = {false},\n authored = {false},\n confirmed = {false},\n hidden = {false},\n citation_key = {mcnamara:ja:2010},\n private_publication = {false},\n 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},\n bibtype = {article},\n author = {McNamara, Jack J and Crowell, Andrew R and Friedmann, Peretz P and Glaz, Bryan and Gogulapati, Abhijit},\n doi = {10.2514/1.C000190},\n journal = {Journal of Aircraft},\n number = {6}\n}","author_short":["McNamara, J., J.","Crowell, A., R.","Friedmann, P., P.","Glaz, B.","Gogulapati, A."],"urls":{"Paper":"https://bibbase.org/service/mendeley/6476e386-2170-33cc-8f65-4c12ee0052f0/file/3f2f37fd-1489-1e33-6f7d-856734372e60/full_text.pdf.pdf","Website":"http://arc.aiaa.org"},"biburl":"https://bibbase.org/service/mendeley/6476e386-2170-33cc-8f65-4c12ee0052f0","bibbaseid":"mcnamara-crowell-friedmann-glaz-gogulapati-approximatemodelingofunsteadyaerodynamicsforhypersonicaeroelasticity-2010","role":"author","metadata":{"authorlinks":{}},"downloads":1},"bibtype":"article","biburl":"https://bibbase.org/service/mendeley/6476e386-2170-33cc-8f65-4c12ee0052f0","dataSources":["qwkM8ZucCwtxbnXfc","ya2CyA73rpZseyrZ8","2252seNhipfTmjEBQ"],"keywords":[],"search_terms":["approximate","modeling","unsteady","aerodynamics","hypersonic","aeroelasticity","mcnamara","crowell","friedmann","glaz","gogulapati"],"title":"Approximate Modeling of Unsteady Aerodynamics for Hypersonic Aeroelasticity","year":2010,"downloads":1}