Boundary Layer on an Inclined Flat Plate

Abstract

This chapter presents a global stability analysis of the two-dimensional incompressible boundary layer with the effect of streamwise pressure gradients. A symmetric wedge flow with different non-dimensional pressure gradient parameters ($\beta _{H}$) has been considered. The pressure gradient (${\text {d}}p/{\text {d}}x$) in the flow direction is zero for $\beta _{H} = 0$, favourable (negative) for $\beta _H > 0$ and adverse (positive) for $\beta _H < 0$. The base flow is computed by the numerical solution of the Falkner-Skan equation. The displacement thickness ($\delta ^*$) at the inflow boundary is considered for computing the Reynolds number. The governing stability equations for perturbed flow quantities are derived in the body-fitted coordinates. The stability equations are discretized using Chebyshev spectral collocation method. The discretized equations and boundary conditions form a general eigenvalues problem and are solved using Arnoldi’s algorithm. The global temporal modes have been computed for $\beta _H=0.022$, 0.044 and 0.066 for favourable and adverse pressure gradients. The temporal growth rate ($\omega _i$) is negative for all the global modes. The $\omega _i$ is smaller for the favourable pressure gradient (FPG) than that of the adverse pressure gradient (APG) at the same Reynolds number (${\text {Re}} = 340$). Thus, FPG has a stabilization effect on the boundary layer. Comparing the spatial eigenmodes and spatial amplification rate for FPG and APG show that FPG has a stabilization effect while APG has a destabilization effect on the disturbances.