Subsonic Indicial Aerodynamics for Unsteady Loads Calculation via Numerical and Analytical Methods: a Preliminary Assessment

This study deals with generating aerodynamic indicial-admittance functions for the unsteady loads of aircraft wings via CFD and approximate analytical formulations. The novel aspects concern both a critical analysis of the CFD simulations process and an effective analytical synthesis of the CFD outcome based on sound physical grounds, where the main factors affecting the results’ accuracy are separately considered for both impulsive and circulatory parts of the flow response. Considering both thin aerofoils and elliptical wings in compressible subsonic flow, the first part of this work aims at understanding the best practice of generating aerodynamic indicial functions via CFD and numerical results of the lift build-up are obtained for several Mach numbers and aspect ratios. The second part of this work aims at investigating convenient semi-analytical means of approximating such aerodynamic indicial functions by modifying those available for incompressible flow, taking advantage of Prandtl-Glauert scalability for the circulatory part and piston theory for the non-circulatory part. Suitable tuning of the analytical expressions is also derived in order to mimic the CFD results and make proper comparisons between the two approaches. Results are finally shown and critically addressed with respect to the physical and mathematical assumptions employed for the aerodynamic load calculations, within a rigorous and consistent framework. Especially in the case of thin aerofoils, the correct limit behaviour and excellent agreement found between numerical and (tuned) analytical results (i) validate both, (ii) identify the best practice in setting up the CFD simulations and (iii) demonstrate the accuracy and robustness of the proposed analytical technique for approximating aerodynamic indicial-admittance functions efficiently.

[1]  Gianluigi Rozza,et al.  Reduced Order Methods for Modeling and Computational Reduction , 2013 .

[2]  J. Gordon Leishman,et al.  Principles of Helicopter Aerodynamics , 2000 .

[3]  David A. Peters,et al.  A State-Space Airloads Theory for Flexible Airfoils , 2006 .

[4]  Harvard Lomax,et al.  Two-and three-dimensional unsteady lift problems in high-speed flight , 1952 .

[5]  J. E. Cooper,et al.  Curve fitting approach for transonic flutter prediction , 2003, The Aeronautical Journal (1968).

[6]  W. M. Adams,et al.  Nonlinear programming extensions to rational function approximation methods for unsteady aerodynamic forces , 1988 .

[7]  D. Williams.  Some mathematical methods in three dimensional subsonic flutter derivative theory , 1961 .

[8]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[9]  Peretz P. Friedmann,et al.  New approach to finite-state modeling of unsteady aerodynamics , 1986 .

[10]  A. Jameson,et al.  Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes , 1981 .

[11]  Garret N. Vanderplaats,et al.  Numerical Optimization Techniques for Engineering Design: With Applications , 1984 .

[12]  Joseph A Drischler,et al.  Approximate indicial lift functions for several wings of finite span in incompressible flow as obtained from oscillatory lift coefficients , 1956 .

[13]  Marin D. Guenov,et al.  Advances in Collaborative Civil Aeronautical Multidisciplinary Design Optimization , 2009 .

[14]  Ruxandra Botez,et al.  Method of Unsteady Aerodynamic Forces Approximation for Aeroservoelastic Interactions , 2002 .

[15]  J. G. Leishman,et al.  Subsonic unsteady aerodynamics caused by gusts using the indicial method , 1996 .

[16]  F. A. Woodward.  An improved method for the aerodynamic analysis of wing-body-tail configurations in subsonic and supersonic flow. Part 1: Theory and application , 1973 .

[17]  Paul G. A. Cizmas,et al.  Mesh Generation and Deformation Algorithm for Aeroelasticity Simulations , 2007 .

[18]  Jean-Luc Guermond,et al.  A unified unsteady lifting-line theory , 1991, Journal of Fluid Mechanics.

[19]  Leonardo Bergami,et al.  Indicial response function for finite-thickness airfoils, a semi-empirical approach , 2011 .

[20]  W. Eversman,et al.  Modified Exponential Series Approximation for the Theodorsen Function , 1991 .

[21]  J. Leishman,et al.  State-space representation of unsteady airfoil behavior , 1990 .

[22]  Ludwig Prandtl,et al.  Applications of Modern Hydrodynamics to Aeronautics , 1923 .

[23]  Harvard Lomax,et al.  The Indicial Lift and Pitching Moment for a Sinking or Pitching Two-Dimensional Wing Flying at Subsonic or Supersonic Speeds , 1951 .

[24]  I. E. Garrick.  On some reciprocal relations in the theory of nonstationary flows , 1938 .

[25]  W. Rodden,et al.  A doublet-lattice method for calculating lift distributions on oscillating surfaces in subsonic flows. , 1969 .

[26]  J. G. Leishman,et al.  Indicial lift approximations for two-dimensional subsonic flow as obtained from oscillatory measurements , 1993 .

[27]  William R. Wells,et al.  Approximate indicial lift function for tapered, swept wings in incompressible flow , 1978 .

[28]  P Marzocca,et al.  Development of an indicial function approach for the two-dimensional incompressible/compressible aerodynamic load modelling , 2007 .

[29]  Jonathan E. Cooper,et al.  Introduction to Aircraft Aeroelasticity and Loads , 2007 .

[30]  G. Hancock,et al.  The unsteady motion of a two-dimensional aerofoil in incompressible inviscid flow , 1978, Journal of Fluid Mechanics.

[31]  R. T. Jones.  The unsteady lift of a wing of finite aspect ratio , 1940 .

[32]  Harvard Lomax,et al.  Lift Developed on Unrestrained Rectangular Wings Entering Gusts at Subsonic and Supersonic Speeds , 1954 .

[33]  Murray Tobak,et al.  On the use of the indicial-function concept in the analysis of unsteady motions of wings and wing-tail combinations , 1954 .

[34]  Arne V. Johansson,et al.  An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows , 2000, Journal of Fluid Mechanics.

[35]  Herbert Wagner.  Über die Entstehung des dynamischen Auftriebes von Tragflügeln , 1925 .

[36]  Paul D. Sclavounos,et al.  An unsteady lifting-line theory , 1987 .

[37]  A. Jirásek,et al.  Reduced order unsteady aerodynamic modeling for stability and control analysis using computational fluid dynamics , 2014 .

[38]  D. Peters,et al.  Finite state induced flow models. I - Two-dimensional thin airfoil , 1995 .

[39]  Steven L. Brunton,et al.  Empirical state-space representations for Theodorsen's lift model , 2013 .

[40]  James D. Baeder,et al.  Indicial Aerodynamics in Compressible Flow-Direct Computational Fluid Dynamic Calculations , 1997 .

[41]  T. Chung.  Computational Fluid Dynamics: FOUR. AUTOMATIC GRID GENERATION, ADAPTIVE METHODS, AND COMPUTING TECHNIQUES , 2002 .

[42]  Robert T. Jones,et al.  Classical aerodynamic theory , 1979 .

[43]  Eric Reissner,et al.  Effect of Finite Span on the Airload Distributions for Oscillating Wings. II - Methods of Calculation and Examples of Application , 1947 .

[44]  H. Matthies,et al.  State-space representation of instationary two-dimensional airfoil aerodynamics , 2004 .

[45]  Massimo Gennaretti,et al.  Study of Reduced-Order Models for Gust-Respnse Analysis of Flexible Fixed Wings , 2004 .

[46]  K. L. Roger,et al.  Airplane Math Modeling Methods for Active Control Design , 1977 .

[47]  Gertjan Looye,et al.  UNIFYING MANOEUVRE AND GUST LOADS ANALYSIS MODELS , 2009 .

[48]  H. Glauert,et al.  The Effect of Compressibility on the Lift of an Aerofoil , 1928 .

[49]  R. Vepa.  Finite state modeling of aeroelastic systems , 1977 .

[50]  A. Hauptman,et al.  Aerodynamic coefficients of a thin elliptic wing in unsteady motion , 1987 .

[51]  Dan Mateescu,et al.  Theoretical solutions for unsteady compressible subsonic flows past oscillating rigid and flexible airfoils , 2011 .

[52]  B. Peirce.  A short table of integrals , 1898 .

[53]  John R. Spreiter,et al.  Reciprocity relations in aerodynamics , 1952 .

[54]  Th. Van Holten,et al.  Some notes on unsteady lifting-line theory , 1976, Journal of Fluid Mechanics.

[55]  Daniella E. Raveh,et al.  Reduced-Order Models for Nonlinear Unsteady Aerodynamics , 2001 .

[56]  Luigi Morino,et al.  A General Theory of Unsteady Compressible Potential Aerodynamics , 1974 .

[57]  David Eller,et al.  Efficient Laplace-Domain Aerodynamics for Load Analyses , 2013 .

[58]  Paolo Mantegazza,et al.  A \free" approach to Computational Aeroelasticity , 2010 .

[59]  Gary M. Graham.  Aeroelastic Reciprocity: An Indicial Response Formulation. , 1995 .

[60]  Peter Eliasson,et al.  High-order implicit time integration for unsteady turbulent flow simulations , 2015 .

[61]  T. Kármán,et al.  Airfoil Theory for Non-Uniform Motion , 1938 .

[62]  Ken Badcock,et al.  On the generation of flight dynamics aerodynamic tables by computational fluid dynamics , 2011 .

[63]  E. H. Dowell,et al.  A simple method for converting frequency domain aerodynamics to the time domain , 1980 .

[64]  P. Tallec,et al.  Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: Momentum and energy conservation, optimal discretization and application to aeroelasticity , 1998 .

[65]  M. Karpel,et al.  Minimum-state unsteady aerodynamic approximations with flexible constraints , 1996 .

[66]  L. Peterson,et al.  Improved exponential time series approximation of unsteady aerodynamic operators , 1988 .

[67]  D. Peters.  Two-dimensional incompressible unsteady airfoil theory—An overview , 2008 .

[68]  Marco Berci,et al.  Dynamic Response of Typical Section Using Variable-Fidelity Fluid Dynamics and Gust-Modeling Approaches—With Correction Methods , 2014 .

[69]  H. Dunn.  An analytical technique for approximating unsteady aerodynamics in the time domain , 1980 .

[70]  J. A. Drischler,et al.  Numerical determination of indicial lift and moment functions for a two-dimensional sinking and pitching airfoil at Mach numbers 0.5 and 0.6 , 1952 .

[71]  Carol D. Wieseman.  Methodology for matching experimental and computational aerodynamic data , 1988 .

[72]  Reik Thormann,et al.  DLM-Correction Method for Aerodynamic Gust Response Prediction , 2013 .

[73]  Ülgen Gülçat,et al.  Fundamentals of Modern Unsteady Aerodynamics , 2010 .

[74]  Thiemo Kier,et al.  Comparison of Unsteady Aerodynamic Modelling Methodologies with Respect to Flight Loads Analysis , 2005 .

[75]  Charbel Farhat,et al.  An ALE formulation of embedded boundary methods for tracking boundary layers in turbulent fluid-structure interaction problems , 2014, J. Comput. Phys..

[76]  W. R. Sears,et al.  Operational methods in the theory of airfoils in non-uniform motion , 1940 .

[77]  A. Jameson.  Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings , 1991 .

[78]  J. Anderson,et al.  Fundamentals of Aerodynamics , 1984 .

[79]  Yuan-Cheng Fung,et al.  An introduction to the theory of aeroelasticity , 1955 .

[80]  Natalia Alexandrov,et al.  Multidisciplinary design optimization : state of the art , 1997 .

[81]  Paolo Mantegazza,et al.  State Space Reduced Order Models for Static Aeroelasticity and Flight Mechanics of Flexible Aircrafts , 2003 .

[82]  R. N. Desmarais.  A continued fraction representation for Theodorsen's circulation function , 1980 .

[83]  B Gothert.  Plane and Three-Dimensional Flow at High Subsonic Speeds , 2013 .

[84]  Russell M. Cummings,et al.  Computational Investigation into the Use of Response Functions for Aerodynamic-Load Modeling , 2012 .

[85]  Bernard Mazelsky,et al.  Determination of indicial lift and moment of a two-dimensional pitching airfoil at subsonic mach numbers from oscillatory coefficients with numerical calculations for a Mach number of 0.7 , 1952 .

[86]  M. Berci.  SEMI-ANALYTICAL REDUCED-ORDER MODELS FOR THE UNSTEADY AERODYNAMIC LOADS OF SUBSONIC WINGS , 2015 .

[87]  Frederic M. Hoblit,et al.  Gust Loads on Aircraft: Concepts and Applications , 1988 .

[88]  Ashish Tewari,et al.  Multiple pole rational-function approximations for unsteady aerodynamics , 1993 .

[89]  J. Katz,et al.  Low-Speed Aerodynamics , 1991 .

[90]  Altan Kayran.  Küssner's Function in the Sharp Edged Gust Problem-A Correction , 2006 .

[91]  E. Reissner,et al.  Effect of Finite Span on the Airload Distributions for Oscillating Wings I : Aerodynamic Theory of Oscillating Wings of Finite Span , 1947 .

[92]  Richard B. Holmes,et al.  A Course on Optimization and Best Approximation , 1972 .

[93]  Jan Nordström,et al.  Application of a line-implicit scheme on stretched unstructured grids , 2009 .

[94]  B. Mazelsky.  Numerical determination of indicial lift of a two-dimensional sinking airfoil at subsonic Mach numbers from oscillatory lift coefficients with calculations for Mach number 0.7 , 1951 .

[95]  R. Vepa,et al.  On the use of Pade approximants to represent unsteady aerodynamic loads for arbitrarily small motions of wings , 1976 .

[96]  H. Glauert.  The elements of aerofoil and airscrew theory , 1926 .