Three-dimensional evolution of structure and acoustic wave generation in a compressible plane wake

Abstract

The compressible Navier-Stokes equations are numerically solved to study three-dimensional structures of a compressible plane wake undergoing transition to turbulence. High-order compact ÿnite di erence schemes are used for spatial derivatives and a fourth-order Runge-Kutta scheme is employed for time advancement. NavierStokes characteristic boundary conditions are used in the vertical direction and periodic boundary conditions in the streamwise and spanwise directions. Unstable disturbances are obtained from linear stability theory using a mapped Fourier method for the viscous compressible equations. Three-dimensional structures of the wake are studied by means of temporally evolving plane wakes forced with random disturbances and a combination of unstable modes. Forcing with a pair of oblique unstable modes yields streamwise/vertical counter-rotating vortices in the saddle region. As the streamwise/vertical vortices evolve outside, their self-induction causes inclined braidlike structures to form in the wake. The e ect of the oblique unstable modes is also to cause the spanwise roller breakdown. Acoustic waves of the plane wake are generated when roughly two-dimensional rollup structures appear and rotate in the wake. The three-dimensional evolution of the wake reduces the near-ÿeld sound pressure.

Acoustic Wave Generation in a Compressible Wake

“…Note that the streamwise and transverse Q structure is stronger on the top half of the wake. This is quite different from the behavior of an oblique fundamental, where opposite sides of the wake evolve similarly (6) . Unlike the two-dimensional subharmonic, the oblique subharmonic instability does not lead to pairing.…”

“…The length of the computational domain in the streamwise and spanwise directions, L 1 and L 3 , is determined from the linear stability analysis (6) using a mapped Fourier method for the viscous compressible equations. Therefore, we have…”

“…Staggered lift-up structures were found to move in a large-scale motion in a subsonic wake in the far downstream region (15) (1988). The effect of a pair of oblique subharmonic modes on the large-scale vortical motion is examined with a comparison between three-and two-dimensional results (6) . Similarly to the two-dimensional subharmonic, the oblique subharmonic evolves differently on opposite sides of the wake.…”

“…The sound field was not presented in their paper. Watanabe and Maekawa (6) (2004) simulated a supersonic wake at relatively higher Reynolds numbers using high-order compact finite difference schemes. Their results indicate that the rollup elliptical structure plays an important role for sound wave generation in a compressible plane wake.…”

Direct Numerical Simulation of Impinging Shock Wave/Transitional Boundary Layer Interaction with Separation Flow

“…For positive values of ω i , the disturbances φ' are amplified in the streamwise direction whereas for negative the eigenfunctions are damped. We solved for the governing disturbance equation as an eigenvalue problem using spectral collocation methods (26) .…”