Christian S. Mayer

Direct numerical simulation of complete transition to turbulence via oblique breakdown at Mach 3

A pair of oblique waves at low amplitudes is introduced in a supersonic flat-plate boundary layer at Mach 3. Its downstream development and the concomitant process of laminar to turbulent transition is then investigated numerically using linear-stability theory, parabolized stability equations and direct numerical simulations (DNS). In the present paper, the linear regime is studied first in great detail. The focus of the second part is the early and late nonlinear regimes. It is shown how the disturbance wave spectrum is filled up by nonlinear interactions and which flow structures arise and how these structures locally break down to small scales. Finally, the study answers the question whether a fully developed turbulent boundary layer can be reached by oblique breakdown. It is shown that the skin friction develops such as is typical of transitional and turbulent boundary layers. Initially, the skin friction coefficient increases in the streamwise direction in the transitional region and finally decays when the early turbulent state is reached. Downstream of the maximum in the skin friction, the flow loses its periodicity in time and possesses characteristic mean-flow and spectral properties of a turbulent boundary layer. The DNS data clearly demonstrate that oblique breakdown can lead to a fully developed turbulent boundary layer and therefore it is a relevant mechanism for transition in two-dimensional supersonic boundary layers.

Numerical Investigation of Supersonic Transition for a Circular Cone at Mach 3.5

The Direct Numerical Simulation (DNS) of the transition process in a supersonic boundary layer from the laminar to the turbulent state signican tly challenges existing numerical codes. High{order accurate methods are commonly used to improve the accuracy of simulations and thus reduce the number of required grid points. In this paper the development of a high{order code which is tailored towards stability and nonlinear transition simulations over a circular cone is discussed. A thoroughly conducted validation is presented. In particular, small amplitude disturbances are introduced to study the linear wave amplication (eigenbehavior). The results are compared to Linear Stability Theory (LST). Moreover, a three{dimensional stability diagram { in the downstream{frequency{azimuthal mode domain (Rx F k) { is extracted from these calculations and analyzed. Finally, the possible occurrence of oblique breakdown is highlighted by performing simulations with continuously forced nite{amplitude disturbances.

Numerical investigation of the nonlinear transition regime in a Mach 2 boundary layer

The transition process in a supersonic flat-plate boundary layer at Mach 2 is investigated numerically using linear stability theory (LST) and direct numerical simulations (DNS). The experimental investigations by Kosinov and his co-workers serve as a reference and provide the physical conditions for the numerical set-up. In these experiments, the weakly nonlinear regime of transition was studied. This led to the discovery of asymmetric subharmonic resonance triads, which appear to be relevant for transition in a Mach 2 boundary layer. These triads were composed of one primary oblique wave of frequency 20 kHz and two oblique subharmonic waves of frequency 10 kHz. While the experimentalists have focused on this new breakdown mechanism, we have found that the experimental data also indicate the presence of another mechanism related to oblique breakdown. This might be the first experimental evidence of the oblique breakdown mechanism in a supersonic boundary layer. With the simulations presented here, the possible presence of oblique breakdown mechanisms in the experiments is explored by deliberately suppressing subharmonic resonances in the DNS and by comparing the numerical results with the experimental data. The DNS results show excellent agreement with the experimental measurements for both linear and nonlinear transition stages. Most importantly, the results clearly show the characteristic features of oblique breakdown. In addition, we also investigated the subharmonic transition route using LST and DNS. When forcing both the subharmonic and the fundamental frequencies in the DNS, a subharmonic resonance mechanism similar to that in the experiments can be observed.

Investigation of Asymmetric Subharmonic Resonance in a Supersonic Boundary Layer at Mach 2 Using DNS

The transition process in a supersonic at-plate boundary layer at Mach 2 is investigated numerically using linear stability theory (LST) and direct numerical simulations (DNS). Experimental investigations by Kosinov et al. serve as a reference and provide the physical conditions for the numerical setup. In these experiments, the weakly nonlinear regime of transition was studied resulting in the discovery of asymmetric subharmonic resonance triads composed of one primary oblique wave of frequency 20 kHz and two oblique subharmonic waves of frequency 10 kHz. The experimentalists concluded that during the transition process, specic subharmonic resonance triads are selected depending on the amplitude ratio between fundamental and subharmonic disturbances. With the simulations presented in this paper, the subharmonic transition route is studied in detail. A similar subharmonic resonance mechanism as observed in the experiments is also visible in our numerical simulations. Additionally, several other resonance triads can be identied using LST and DNS. In our simulations, the selection process for a specic triad is not inuenced by the amplitude ratio of fundamental and subharmonic disturbances. In contrast to the experiments, the DNS results show that the phase relation between disturbances of both frequency plays a more crucial role.

Numerical Simulations of Controlled Transition for a Sharp Circular Cone at Mach 8

The linear and nonlinear development of disturbance waves in an axisymmetric boundary layer on a sharp circular cone at Mach 8 are investigated by numerical solution of the full 3D, time dependent, compressible Navier-Stokes equations. Disturbances are introduced by wall-normal blowing and suction near the upstream boundary of the computational domain. Small amplitude disturbances are introduced to study their linear stability behavior and the results are compared with Linear Stability Theory (LST) for validation. These results are also used to guide parameter selection for simulations of nonlinear transition phenomena. In particular, for simulations with large amplitude 2D disturbances, fundamental resonance can be observed. This resonance leads to nonlinear amplification of 3D modes and the classical aligned ⁄-vortex pattern as the boundary layer begins to transition to turbulence. Simulations of oblique breakdown are also performed and the two transition mechanisms are compared.

Detailed Comparison of DNS with PSE for Oblique Breakdown at Mach 3

Oblique breakdown in a supersonic flat-plate boundary layer is investigated using Direct Numerical Simulations (DNS) and Parabolized Stability Equations (PSE). This paper constitutes an extension to our previous studies of the complete transition regime of oblique breakdown. In these studies, the flow was assumed to be symmetric in the spanwise direction. A new DNS has been performed where the symmetry condition was removed. This simulation demonstrates that the “classical” oblique breakdown mechanism initialized by two symmetric instability waves with equal disturbance amplitudes loses its symmetry late in the turbulent stage for a low-noise environment. Hence, for the streamwise extent of the computational domain in our studies, the symmetry condition is justified. Furthermore, new data from a longer time average of the original symmetric simulation of oblique breakdown (CASE 3) are discussed. These data verify that a converged time average is reached. The final part of the paper focuses on a comparison of PSE results obtained from NASA’s LASTRAC code to the DNS results. This comparison corroborates that the nonlinear PSE approach can successfully predict transition onset and that despite the large amplitude forcing used to introduce the oblique mode disturbances in the DNS, the latter constitutes a generic reference case for oblique breakdown at Mach 3 and, therefore, can be used to validate reduced order models for the full transition zone.

The Late Nonlinear Stage of Oblique Breakdown to Turbulence in a Supersonic Boundary Layer

Oblique breakdown to turbulence was initiated by low amplitude forcing in a laminar flat-plate boundary layer at Mach three. The growth of the instability waves was investigated using spatial Direct Numerical Simulations (DNS). Excellent agreement with theory was obtained in the linear stage corroborating that the entire transition process from the linear regime to the final breakdown was captured. A fully turbulent flow was reached demonstrating that this transition scenario is a viable path to turbulence. Key events in the late nonlinear stage of breakdown are studied in detail.

Numerical investigation of transition initiated by a wave packet on a cone at Mach 3.5

Transition initiated by a wave packet in a cone boundary layer at Mach 3.5 has been investigated using LST and DNS. Disturbances have been introduced into the boundary layer by pulsing the wall-normal velocity through a hole on the cone surface. The computational setup is very close to experiments by Corke et al. and Matlis. The present study can be divided into three parts. In the first part, the linear development of a wave packet is studied in detail. Disturbance spectra in the frequency–azimuthal mode number plane based on wall-pressure amplitudes and time envelopes of the disturbance signal are discussed. The second part of the present study focuses on the identification of possible, asymmetric resonance triads for the most dominant oblique instability waves of the wave packet. New triads have been found that have not yet been reported for a supersonic boundary layer. These triads might explain some major findings in the third and final part of the present work, which focuses on the weakly nonlinear development of a wave packet that was generated by a large amplitude pulse. The initial disturbance development of this wave packet remains still linear, while farther downstream nonlinear wave interactions alter the shape and the disturbance spectrum of the packet. The disturbance spectrum from this study and the results of other investigations performed in parallel (Laible et al.) suggest that oblique breakdown might be the strongest nonlinear transition mechanism for a supersonic boundary layer.