Triple-deck and Prandtl stages of marginal separation

Abstract

The method of matched asymptotic expansion is used to analyze the early phases of the laminar-turbulent transition triggered by the presence of a laminar separation bubble. From the perspective of high Reynolds number asymptotics, the early evolution of the bypass transition can be divided into four characteristic stages. The first (steady) stage terminates in a marginal or Goldstein singularity, while each further (unsteady) stage ends in a finite-time singularity of the numerical solution to the underlying model equations. The breakdown of the classical boundary layer (i) is initiated when an imposed adverse pressure gradient is present. The viscous-inviscid interaction (triple-deck theory) of the marginal separation stage (ii) allows the characterization of locally bounded recirculating flow regions. However, this stage fails to cover the (repeated) bursting of the bubble, which triggers the transition to turbulence. Experimental observations show that the bubble bursting is initiated by the formation of a coherent structure (’spike’) at the rear of the bubble. In the first part of the thesis, we concentrate on the so-called triple-deck stage (iii), which describes the spike formation. This stage is another interactive stage with shorter spatial and temporal scales. We analyze the governing equations for the leading order and the higher orders of the triple-deck stage. The finite-time blow-up of this stage is investigated in detail to obtain the initial condition for the subsequent Euler–Prandtl stage. Since the solutions are not unique, numerical results for the blow-up profiles are presented up to the third order for two specific cases. The main part of the current work is the study of the vortex formation, which is governed by the non-interactive Euler–Prandtl stage (iv). This stage consists of the inviscid Euler region and the viscous Prandtl layer. Here, the focus lies on the asymptotical description and the numerical approach to compute the dynamics of the Prandtl layer. The resulting equations of the Cauchy problem are solved based on a spectral collocation method using Chebyshev polynomials for both spatial directions. The time is discretized by a backward Euler finite differences scheme. The validity of the numerical results from both parts of the Euler–Prandtl stage depends on the absence of the Van Dommelen–Shen singularity within the Prandtl layer. The slip velocity at the wall is imposed from the Euler region and the initial condition follows from the triple-deck blow-up stage. Even though the occurrence of a Van Dommelen–Shen singularity is expected in the evolution of the Prandtl layer, numerical results reveal that a finite-time blow-up still needs to be reached with the present scheme. Improved spatial discretization and further time steps are likely required. However, numerical results in the Euler region, and, thus, the required slip velocity at the wall are only obtained up to a specific instant of time due to computational costs and the imposed accuracy requirements.