A CFD-Compatible Amplification Factor Transport Equation for Oblique Tollmien-Schlichting Waves in Supersonic Boundary Layers

Abstract

Boundary layer transition is a hot research topic in fluid mechanics and aerospace engineering. In low-speed flows, two-dimensional Tollmien-Schlichting (T-S) waves always dominate the flow instability, which has been modeled by Coder and Maughmer from 2013. However, in supersonic flows, three-dimensional oblique Tollmien-Schlichting waves become dominant in flow instability. Inspired by Coder and Maughmer’s NTS amplification factor transport equation for two-dimensional Tollmien-Schlichting waves in low-speed flows and Kroo and Sturdza’s linear stability theory (LST) analysis results for oblique Tollmien-Schlichting waves in supersonic flows, a new amplification factor transport equation for oblique Tollmien-Schlichting waves has been developed based on LST. The compressible Falkner-Skan similarity equations are introduced to build the relationships between nonlocal variables and local variables so that all the variables used in the present model can be calculated using local variables. Applications of this new transport equation to the flows over supersonic flat plate, 3% thick biconvex airfoil, and one modified supersonic laminar airfoil show promising results compared with the standard LST analysis results.

Comparative Assessment of Modified γ - Re θ t

“…Although the RANS simulations are useful as an engineering tool, the accuracy of the results depends on the turbulence model used, especially for problems wherein the boundary layer transition is important and governs all the flow features. Accordingly, different turbulence models, such as the algebraic [20][21][22], laminar kinetic energy (k L ) [23,24], turbulence intermittency [25][26][27], stability theory-based [28][29][30][31], and correlation-based [32,33] models, have been developed to account for the effect of transition in RANS simulations. Several models have exhibited satisfactory prediction capability when applied to the simulation of transitional shear flows.…”

Linear amplification factor transport equation for stationary crossflow instabilities in supersonic boundary layers

“…However, in this paper, the authors mainly focus on the local model and try to establish a transport equation for stationary crossflow instabilities in compressible boundary layers using local variables. About local transition models, such as Langrtry and Menter's c À Re ht correlation-based transition model, 14 Walters et al's laminar kinetic energy transition model, 15 Fu and Wang's k-x-c transition model 16 and Xu et al's physical-mode-based models, [17][18][19] are getting more and more popular. Then, many researchers extended these models for crossflow instabilities [20][21][22][23][24] because these models are easy to be programmed and perform well in transition prediction.…”