Linear stability of high-speed boundary layer flows at varying Prandtl numbers

Abstract

Accurate prediction of laminar to turbulent transition at hypersonic velocities is a challenging task. The high temperature and low pressure encountered in such applications can lead to large variations in physical and transport properties of the gas. This can result in significant changes in the effective Prandtl number. In this paper, we study the physical effects of Prandtl number on the stability of high Mach number boundary layer flows. Prandtl number values in the range of 0.3 to 1.2 are chosen and a temporal linear stability problem is formulated. The resulting eigenvalue problem is solved using Chebyshev spectral collocation method.We study the effect of Prandtl number on the eigenspectrum for a range of Mach number, Reynolds number and disturbance wave number. It is found that certain combinations of these parameters lead to synchronization between the fast and the slow modes, which can contribute to the peak growth rate. Two types of branching patterns are observed depending on the Prandtl number, where either the fast or the slow mode can be destabilized due to mode-synchronization. The stability diagrams plotted for a range of Reynolds number and wavenumber show a destabilizing role for increasing the Prandtl number, leading to larger regions of instability, and increased growth rates. This also leads to a significant reduction in the critical Reynolds number specifically at intermediate Mach numbers. High-speed flat plate boundary layer flow is thus found to be highly sensitive to slight changes in the Prandtl number.

Prandtl number effects on the hydrodynamic stability of compressible boundary layers: flow–thermodynamics interactions

“…The instability growth rates at high Mach numbers increases tenfold as the Prandtl number is increased from 0.5 to 1.3. This is consistent with the findings of Ramachandran et al (2015), wherein a similar destabilization of the streamwise first mode is observed for M = 4.…”

“…As shown in figure 10(b), the growth rate for all Mach numbers considered at Pr = 1.3 is more than double the growth rate at Pr = 0.5. A similar destabilization of the second mode was also observed by Ramachandran et al (2015). The main novelty of the present work is to examine the physics underlying the destabilization with increasing Prandtl number.…”

“…The loops corresponding to first and second modes fuse at M = 6 for Pr = 0.9 as destabilization increases with Prandtl number. Ramachandran et al (2015) also observed a similar merger of the loops of first and second modes. Figure 4 shows the effect of Prandtl number on the stability characteristics of 3-D disturbances.…”

“…On the other hand, for Pr ≥ 0.7, the slow mode after synchronization with the fast mode becomes the dominant instability. Fedorov & Tumin (2011) and Ramachandran et al (2015) also report a similar branching pattern of the eigenspectrum depending on the Mach and Prandtl numbers. The effect of Prandtl number on the eigenfunctions of the fast and slow modes at M = 6, α = 0.05 is shown in figure 7.…”

“…However, studies examining the effect of Prandtl number have been limited. Ramachandran et al (2015) investigate the effect of the Prandtl number on the eigenspectrum of hypersonic boundary layers. They observe destabilization of both first and second modes with increasing Prandtl number.…”