A continuation method for the calculation of airfoil flutter boundaries

Application of proper orthogonal decomposition to the discrete Euler equations

“…The aerodynamic equations are placed in discrete form following the upwind total variation diminishing (TVD) scheme of Harten-Yee [24; 25], with a correction for grid-point motion. The discretization, of second-order or ÿrst-order spatial accuracy (depending on parameter selection) and ÿrst-order temporal accuracy, is expressed as [6] (20) where the arraysẼ andF are modiÿed numerical uxes that implement the TVD formulation.…”

“…Combining Equations (3) and (6) shows that T = V T V , where = diag( m ) is the diagonal matrix of eigenvalues. If the eigenvectors are normalized so that V T V = I; then this result leads to the deÿnition of orthonormal POD modes,˜ T˜ = I; where˜ = −1=2 : For ÿnite dimensional systems, KL theory and POD computation are united through the singular value decomposition [21] of the snapshot matrix,…”

“…In addition, the snapshot matrix is expanded into a block-diagonal form, where each block corresponds to a given ÿeld variable. It is this matrix that is used in calculating S T S for the eigenvalue problem (6). Denoting the retained set of eigenvectors by V r , the retained set of POD modes is r = SV r (9) and the reduced-order model of the full system is obtained by substituting…”

Summer Research Program - 1998 Summer Faculty Research Program Volume 5B Wright Laboratory

“…The direct computational procedure BIFAE has been successfully validated against other inviscid methods, including Euler and transonic small-disturbance methods and Beran and Morton (1997)). Validation of BIFAE was also the subject of work by Buxton and Beran (1997).…”

In‐Situ Residual Tracking in Reduced Order Modelling

“…To properly capture the effects of aerodynamic nonlinearities on flutter onset, time-integration methods based on the transonic small-disturbance [6], Euler [15], and Navier-Stokes [7,8,21] equations have been developed to simulate the behavior of aeroelastic systems. Direct methods based on Hopf bifurcation theory also have been developed to compute critical flutter onset speeds of the discrete aeroelastic equations without time integration [3,14]. The former class of techniques generally require large computation times due to the time-accurate nature of the calculations and the large integration times required to establish flow stability properties.…”