Cancellation of Tollmien-Schlichting waves with surface heating (Q2061033)

This article investigates the stability of three dimensional boundary layer flow interacting with vibration and surface heating. Here it is known that in general disturbances may become unstable and develop so-called Tollmien-Schlichting waves. However, experiments show that in certain cases the heating can also be chosen in such a way as to stabilize the flow. As the main result of this article the authors show that in an asymptotic model based on the temperature-dependent Navier-Stokes equations \begin{align*} \partial_t \rho + \text{div}(\rho v)&=0, \\ \rho(\partial_t v+ v\cdot \nabla v) &=- \nabla p + \begin{pmatrix} \frac{1}{\text{Re}} \partial_y (\mu(\partial_y v_1 + \partial_x v_2))\\ 0 \\ \frac{1}{\text{Re}} \partial_y (\mu(\partial_z v_2 + \partial_y v_3)) \end{pmatrix} + \dots\\ \rho(\partial_t T+ v\cdot \nabla T) &= (\gamma-1) M_{\infty}^2 (\partial_t p + v \cdot \nabla p) + \frac{1}{\mathrm{Pr}\mathrm{Re}} \partial_y (\mu \partial_y T) + \dots \end{align*} in a triple-deck theory it is possible to choose a time-dependent heating to suppress the generation of Tollmien-Schlichting waves and hence cancel the instability due to vibration. These results are complemented by numerical computations confirming these findings.