Wall heat transfer effects on Klebanoff modes and Tollmien–Schlichting waves in a compressible boundary layer

Abstract

The influence of wall heat transfer on fluctuations generated by free-stream vortical disturbances in a compressible laminar boundary layer is investigated. These disturbances are thermal Klebanoff modes, namely low-frequency, streamwise-elongated laminar streaks of velocity and temperature, and oblique Tollmien-Schlichting waves, induced by a leading-edge adjustment receptivity mechanism. The flow is governed by the linearized unsteady boundary-region equations, which properly account for the nonparallel and spanwise diffusion effects, and for the continuous forcing of the free-stream convected gusts. Wall cooling stabilizes the laminar streaks when their spanwise wavelength is much larger than the boundary-layer thickness. For these conditions, the disturbances confine themselves in the outer edge layer further downstream, where the compressibility effects are marginal. Klebanoff modes for which the spanwise diffusion is comparable with the wall-normal diffusion possess an asymptotic solution similar to the incompressible case, and are stabilized by wall heating. The unstable waves, which appear in high-Mach-number subsonic and supersonic conditions, are stabilized by wall cooling and destabilized by wall heating. Removing heat from the surface significantly shifts downstream the starting location of instability, while the streamwise wavelength and the growth rate are less affected by the wall heat flux. Perturbation methods, such as the WKBJ technique and the triple-deck theory, are used effectively to validate the numerical results and to explain the flow physics.

Growth of boundary-layer streaks due to free-stream turbulence

“…A realistic streak profile along the whole wall-normal extent of the boundary layer was obtained by R9 and good agreement with the experimental data by Westin et al [56] was shown. Ricco & Dilib [43] employed R9's approach to show that the peak of the streak amplitude in the boundary-layer core may be completely suppressed if intense wall transpiration is applied and Ricco and co-authors [47,46,45] have studied the Klebanoff modes in compressible boundary layers.…”

Nonlinear unsteady streaks engendered by the interaction of free-stream vorticity with a compressible boundary layer

“…Ricco & Wu (2007) showed that the growing disturbances evolved from the so-called quasi-three-dimensional Lam-Rott eigensolutions, which are excited by the free-stream disturbance and may be identified as highly oblique low-frequency T-S waves in the so-called first-mode family (Mack 1975(Mack , 1984. Ricco et al (2009) found that wall cooling suppresses the streaks and enhances the growth of the instability waves.…”

“…Two types of different thermal boundary conditions at the wall may be imposed: the Dirichlet boundary condition,τ (0)=0, which was also employed by Ricco & Wu (2007), and the Neumann boundary condition, ∂τ /∂η| η=0 =0. The conditionτ (0)=0 is used herein as Ricco et al (2009) showed that there is no substantial difference in the development of the Klebanoff modes.…”

“…Ricco & Wu (2007) extended the incompressible analysis by Leib et al (1999) to the compressible case and explained the formation and growth of thermal streaks, which are thought to play a significant role in the secondary instability. Ricco et al (2009) and Ricco et al (2013) further studied the influence of wall heat transfer and wall suction, respectively, on the thermal streaks. The boundary-layer signature in the region relatively close to the leading edge corresponds to the inhomogeneous solution forced by the free-stream disturbance.…”

Laminar streak growth above a spanwise oscillating wall

“…The most commonly investigated strategy for controlling the boundary-layer instability growth is the use of wall suction, which has been shown to reduce the growth of Tollmien-Schlichting waves (Bodonyi & Duck 1990, 1992Fransson & Alfredsson 2003), laminar streaks (Byström et al 2007;Davidsson & Gustavsson 2008;Ricco & Dilib 2010) and Görtler vortices (Floryan & Saric 1983;Balakumar & Hall 1999). Other techniques for controlling the growth of boundary-layer disturbances include wall cooling and heating (El-Hady 1992;Hubbard & Riley 1995;Ricco et al 2009) and wall forcing . In the present study the evolution and attenuation of laminar streaks are investigated above a spanwise oscillating plate.…”