Stability and receptivity analysis of flat-plate boundary layer with suction and blowing

Abstract

The effect of suction and blowing on the local stability of the flat-plate boundary layer is presented using non-normal theory. The 3D governing stability equations are derived using standard procedure in the form of normal velocity and vorticity. The governing stability equations are discretized using the Chebyshev spectral collocation method. The discretized governing equations with grid stretching form an eigenvalue problem, and it is solved using the QZ algorithm with an appropriate boundary conditions. The transient energy growth is computed by the linear superposition of the non-orthogonal eigenvectors. The energy curve is obtained by singular value decomposition (SVD) of the matrix exponential. The receptivity analysis is also considered based on the input–output framework to quantify a fluid system’s response with external forcing frequencies. The optimal fluid system response corresponding to the optimal initial condition is computed for non-modal and receptivity analysis. The flow is modally stable for suction even at a higher Reynolds number $(\hbox {Re}_{\delta ^*})$, while a reverse trend is observed for blowing. In a case of suction, peak response in energy of the fluid system is detected at resonant frequency $\omega = 0.14$ and 0.102 for $\alpha = 0.15$, $\beta = 1$ and $\alpha = 0$, $\beta = 1$, respectively. Similarly, for blowing, maximum flow response is detected at $\omega = 0.1$ and 0.102 for $\alpha = 0.15$, $\beta = 1$ and $\alpha = 0$, $\beta = 1$. The temporal growth rate $\omega _i$, energy growth, and resolvent norm are increased with increasing the Reynolds number or blowing intensity.

Appendix

1.1 Appendix A: Spatial grid convergence analysis

Table 3 Spatial grid convergence analysis for uniform suction with intensity is $0.5\%$ of $U_\infty (= 0.4\hbox { m/s})$ at a location of $x = 1.5\hbox { m}$ and $y = 1\hbox { mm}$. The $k+1$ and k represent fine and coarse grids, respectively

Table 4 Spatial grid convergence analysis for no suction/injection case with inlet velocity, $U_\infty =0.4\hbox { m/s}$ at a location of $x = 1.5\hbox { m}$ and $y = 1\hbox { mm}$. The $k+1$ and k represent fine and coarse grids, respectively

The spatial grid convergence study is a numerical technique used to check the precision and accuracy of the CFD simulation with regard to the grid resolution. In other words, it is a technique used to determine whether the simulation result is trustworthy and independent of the grid size or not. The fundamental principle of the spatial grid convergence analysis is to run the same simulation at several grid resolutions and compare the outcomes. The grid is refined until the changes in the output parameters (i.e. velocity and pressure) become negligible, indicating that the solution has converged. Thus, the grid resolution $2001\times {251}$ is enough for all base flow configurations with relative errors of less than $0.1\%$ (see Tables 1, 3, and 4).