Direct solution of two-dimensional Navier-Stokes equations for static aeroelasticity problems

Abstract

A new method has been developed to calculate the steady flow and structural deformations for fluid/structure interaction problems. The discretized fluid dynamic and structural equations are regarded as a single set of coupled, nonlinear, algebraic equations. The equilibrium solution is directly obtained using Newton's method. The governing equations used for the fluid flow are the two-dimensional Navier-Stokes equations, and a finite element model is used to represent the structure. This paper describes the analytical method and presents sample calculations demonstrating the technique. The results show rapid convergence and good agreement with experimental data. Nomenclature BW = half bandwidth of tangent matrix D = set of structural displacement variables E - set of governing equations for the aeroelastic system F = flux vector in x direction G = flux vector in y direction h = distance from nozzle throat to centerline / = number of fluid dynamic control volumes in streamwise direction J = number of fluid dynamic control volumes in cross-stream direction L = reference length, total length of nozzle N = total number of variables in system p = pressure pe = exit static pressure pt - total pressure R = system residual R = magnitude of residual r - radius at nozzle throat T = tangent matrix (system Jacobian) t = wall thickness U = set of fluid dynamic variables X = set of variables for the aeroelastic system AX = change in system variables during a Newton iteration x = streamwise coordinate y = cross-stream coordinate