Linear and Nonlinear PSE for Stability Analysis of the Blasius Boundary Layer Using Compact Scheme

Abstract

A highly accurate finite-difference PSE code has been developed to investigate the stability analysis of incompressible boundary layers over a flat plate. The PSE equations are derived in terms of primitive variables and are solved numerically by using compact method. In these formulations, both nonparallel as well as nonlinear effects are accounted for. The validity of present numerical scheme is demonstrated using spatial simulations of two cases; two-dimensional (linear and nonlinear) Tollmien-Schlichting wave propagation and three-dimensional subharmonic instability breakdown. The PSE solutions have been compared with previous numerical investigations and experimental results and show good agreement.

Verification of parabolized stability equations for its application to compressible boundary layers

“…The comparison [4] of resulting neutral curves showed better agreement between results from linear PSE (LPSE) method and experiment [5−8] , as well as with and Gaster's results [1] . The nonlinear PSE (NPSE) [4,9] method also yielded results indicated good agreement with those obtained by DNS, including the amplitudes and shapes of the profiles of fundamental disturbance, higher harmonics, as well as mean flow distortion. Consequently, it has been used for further study of the stability problems of incompressible [10] boundary layers.…”

“…(9) are expressed as follows, where τ = u /T is a function of temperature in basic flow, P r is Prandtle number, P r = c * p μ * e /k * e , and γ is the ratio of specific heats. …”

Accuracy analysis of super compact scheme in non‐uniform grid with application to parabolized stability equations

“…Amplitudes were measured at Á = 1:3. This ÿgure indicates that the results of the SCFDM agree well with those of the fourth-order compact [17] (not shown here) and experimental data, when k = 1:01 is used. For k = 1:15, a numerical error is generated from the ÿrst steps of computation as shown in the ÿgure.…”

“…Figure 16 presents the variations of non-dimensional skin friction coe cient versus Reynolds number for the subharmonic breakdown. The ÿgure also shows the comparison of the results with those obtained by the fourth-order compact method [17] to indicate the accuracy of the computation. As it can be seen for k = 1:15 used in the SCFDM, the numerical errors initiated from the ÿrst steps of computation, a ect the rise of the skin friction coe cient.…”

Applications of parabolized stability equation for predicting transition position in boundary layers

“…Joslin et al [6] showed that the PSE could exactly predict the evolution of two-dimensional (2D) TollmienSchlichting (T-S) waves and the transition of subharmonic instability waves and oblique waves. Esfahanian et al [7] used the PSE method to do the stability analysis and predict the transition position for the incompressible flat plate. The disturbances imposed at the entrance were consisted of one 2D wave and a pair of three-dimensional (3D) waves.…”