Coupled Modal-Nonmodal Interactions Due to Periodic, Infinite Train of Convecting Vortices (TCV)

TL;DR

The paper addresses turbulence encounters caused by a periodic train of convecting freestream vortices (TCV) and demonstrates that modal instability and nonmodal receptivity interact in a coupled, strongly nonlinear manner. Using 2D compressible Navier–Stokes simulations with image-vortex forcing and a high-order scheme, the authors show that TCV excites both modal and nonmodal disturbances, triggering spectacular disturbance growth beyond what either route would yield alone. Key findings include the coexistence and nonlinear interaction of TS-like modal components and STWF-like nonmodal structures, the emergence of multi-deck vorticity features near the wall, and significant unsteady loading on the surface, all of which inform plausible mechanisms for severe turbulence encounters on aircraft surfaces. The results highlight the necessity of nonlinear, compressible analysis to predict receptivity and transition in realistic, freestream-excited boundary layers, with implications for vehicle design and flow control.

Abstract

Events during transition to turbulence either follow modal or non-modal routes, or combinations of the two. Here, we report a computational investigation of strong freestream excitation caused by a train of convecting vortices. For this TCV excitation, we show a strong interaction of modal and non-modal components causing a spectacular growth of disturbances. We propose this as the mechanism for the severe encounters due to convective vortical disturbances on the underlying shear layer.

Figures (10)

• Figure 1: Streamwise disturbance velocity at a height, $y = 0.2781 \delta^*$, at the indicated times due to monochromatic frequency wall excitation for $Re=1000$ and $\omega_0 = 0.06$ shown in the left column. The corresponding spectrum are shown in the right column. The central wavenumber of TS wave ($\alpha_{TS} = 0.1840$) and STWF ($\alpha_{STWF} = 0.2608$) are indicated with lines. The indicated times are non-dimensional.
• Figure 2: Streamwise disturbance component $u_d$ at a height $y = 0.0028$ obtained from linearized (left) and nonlinear (right) incompressible Navier-Stokes equation simulations for a single freestream convecting vortex with parameters $\Gamma = 0.1$, $c=0.3$ and $H = 2$. Vertical dashed line indicates its instantaneous position.
• Figure 3: Fourier transform of streamwise disturbance component $u_d$ at a height $y = 0.0028$ for linearized (left) and nonlinear (right) incompressible Navier-Stokes equation simulations for a single freestream convecting with parameters $\Gamma = 0.1$, $c=0.3$ and $H = 2$. The wavenumber is normalized with the maximum resolved wavenumber limit ($\alpha_{max}$).
• Figure 4: Schematic of the free stream excitation due to an infinite, periodic train of convecting vortices.
• Figure 5: Evolution of disturbance vorticity $\omega_d$ for a case of infinite, periodic train of convecting vortices. The convecting vortices have strength $\Gamma = -0.005$ at non-dimensional time $t=45$.