Multiple pole rational-function approximations for unsteady aerodynamics

Subsonic Indicial Aerodynamics for Aerofoil's Unsteady Loads via Numerical and Analytical Methods

“…The same optimal results were efficiently obtained using either the unconstrained Nelder-Mead Simplex Method [131] (NMSM, where the absolute value of the poles was considered for a meaningful implementation of the optimization problem) or a Genetic Algorithm [132] (GA, where the initial population was taken as the SQP solution in order to boost convergence) as well, confirming consistency in the results and robustness of the proposed approach. No clustering of the optimal poles [84] was found and the global optimum was likely reached in all cases.…”

“…Finally, it is worth stressing that any error in approximating Theodorsen's and Sears' functions translates into an error in approximating Wagner's and Kussner's functions, respectively; in particular, a good approximation of the formers at high reduced frequencies (i.e., highly-unsteady flow) translates into a good approximation of the latter at low reduced time (i.e., transient response), while a good accuracy at low reduced frequencies (i.e., quasi-steady flow) translates into a good accuracy at high reduced time (i.e., asymptotic response) [90]. Many approximations were readily available in the literature [74][75][76][77][78][79][80][81][82][83][84][85][86][87][88][89] but the present ones grant the best agreement along the entire exact curves with the least number of poles and possess the correct limit values.…”

“…Ranging from thin symmetric to thick cambered aerofoils, the effects of domain size, spatial resolution, time stepping and integration scheme [73] on the Euler/RANS CFD solutions are first investigated and an effective analytical synthesis is then attempted on solid physical grounds [10]. A convenient methodology is proposed for approximating aerodynamic indicial functions [74][75][76][77][78][79][80][81][82][83][84][85][86][87][88][89][90][91] of compressible subsonic flow by modifying those of incompressible flow directly, based on acoustic wave theory [8,45] for the non-circulatory airload and Prandtl-Glauert's scalability rule [22,46] for the circulatory airload. In particular, an explicit parametric formula is newly proposed for modelling the latter as function of the Mach number in the absence of shock waves, while damped harmonic terms are effectively introduced alongside exponential terms for better approximating the former as well as preventing the nonlinear curve-fitting problem from being ill-conditioned, as less terms of the same analytical form become necessary.…”

Semi-Analytical Reduced-Order Models for the Unsteady Aerodynamic Loads of Subsonic Wings

“…Note that any error in approximating these functions translates into an error in approximating Wagner and Kussner functions, respectively, and vice-versa; in particular, a good accuracy in approximating the formers at high reduced frequencies (i.e., highly-unsteady flow) translates into a good accuracy in approximating the latter at low reduced time (i.e., transient response), while a good accuracy at low reduced frequencies (i.e., quasi-steady flow) translates into a good accuracy at high reduced time (i.e., asymptotic response). Jones' approximations [39] deviating quite significantly from the exact curves, many other approximations are readily available in the literature [109][110][111][112][113][114][115][116] but the present ones grant the best agreement along the entire exact curves with the least number of poles [96] (additional poles have been investigated but no significant improvement found) and possess the correct limit values (which is an essential feature for the present applications).…”

Semi-Analytical Reduced-Order Models for the Aeroelastic Behavior of Subsonic Flexible Wings via Generalised Plate and Beam Formulations

“…For three-dimensional flow, all these fundamental functions [44][45][46][47] are modified so to include the unsteady downwash of the trailed wing-tip vortices [48][49][50][51][52] and the results are then approximated for computational convenience [53][54][55][56][57][58][59][60][61][62][63][64][65][66]; Appendix B reports the applications to elliptical and trapezoidal wings [20][21]. Due to rigorous analytical continuation [41], with s the Laplace variable, a rational approximation [22,67] is suitably adopted for the equivalent of Theodorsen function in the complex reduced frequency p domain, namely:…”