Influence of orientation on the evolution of small perturbations in compressible shear layers with inflection points

We investigate the influence of orientation on the evolution of small perturbations in compressible shear layers with inflection points. By using linear analysis, we demonstrate that perturbations along the shear plane are most affected by compressibility. The influence of compressibility gradually diminishes with increasing obliqueness of the perturbations with respect to the shear plane. It is demonstrated that the effective gradient Mach number is an appropriate compressibility parameter. We establish that spanwise perturbations, orthogonal to the shear plane, are impervious to compressibility effects. Direct numerical simulations of compressible mixing layers subject to the perturbations at various obliqueness angles verify the analytical findings.

Figure 1

A typical perturbation wave vector of obliqueness angle $\beta$. Here $x_1, x_2$, and $x_3$ are the streamwise, normal, and spanwise directions.

Figure 2

Classification of oblique modes on the ${\kappa }_1-{\kappa }_3$ plane. ${\beta }_c$ is the critical obliqueness angle where $M_g^e=1$.

Figure 3

Classification of perturbation modes based on their initial obliqueness angles $\beta$: (a) streamwise ($\beta =0$), (b) spanwise ($\beta =\pi /2$), and (c) oblique ($0<\beta <\pi /2$).

Figure 4

Temporal evolution of (a) the normalized momentum thickness, (b) vorticity thickness, (c) the perturbation kinetic energy, and (d) $R_{22}=\overline{u_2^{\prime }{u}_2^{\prime }}/2$. Effect of obliqueness angle at $M_c=0.3$.

Figure 5

Temporal evolution of (a) the normalized momentum thickness, (b) vorticity thickness, (c) the perturbation kinetic energy, and (d) $R_{22}=\overline{u_2^{\prime }{u}_2^{\prime }}/2$. Effect of convective Mach number at $\beta =0$.

Figure 6

Temporal evolution of (a) the normalized momentum thickness, (b) vorticity thickness, (c) the perturbation kinetic energy, and (d) $R_{22}=\overline{u_2^{\prime }{u}_2^{\prime }}/2$. Effect of convective Mach number at $\beta =\pi /2$.

Figure 7

Temporal evolution of (a) the normalized momentum thickness, (b) vorticity thickness, (c) the perturbation kinetic energy, and (d) $R_{22}=\overline{u_2^{\prime }{u}_2^{\prime }}/2$. Effect of convective Mach number at $\beta =\pi /6$.

Figure 8

The temporal evolution of (a) the normalized momentum thickness, (b) vorticity thickness, (c) the perturbation kinetic energy, and (d) $R_{22}=\overline{u_2^{\prime }{u}_2^{\prime }}/2$. Effect of convective Mach number at $\beta =\pi /3$.