High-Order Finite-Difference Schemes for Numerical Simulation of Hypersonic Boundary-Layer Transition

The prediction of the laminar-turbulent transition of boundary layers is critically important to the development of hypersonic vehicles because the transition has a first-order impact on aerodynamic heating, drag, and vehicle operation. The success of transition prediction relies on a fundamental understanding of the relevant physical mechanisms. In the 20 years since the review by Kleiser & Zang (1991) on the direct numerical simulation (DNS) of the boundary-layer transition, significant progress has been made on DNS in the hypersonic flow regime and in the spatial DNS approach. Many high-order shock-capturing and shock-fitting finite-difference methods have been developed and extensively applied to numerical simulations of the hypersonic boundary-layer transition. DNS has become a powerful research tool and has led to discoveries of new transition mechanisms. This article reviews the recent progress of DNS on hypersonic boundary-layer receptivity, instability, and transition. The current status and future directions are also presented.

http://cfdhost.seas.ucla.edu/papers/2012/Zhong-X-2012-annurev-fluid-120710-101208.pdf

Direct Numerical Simulation on the Receptivity, Instability, and Transition of Hypersonic Boundary Layers

This paper is the first part of a two-part study on the mechanisms of the receptivity to disturbances of a Mach 4.5 flow over a flat plate by using both direct numerical simulations (DNS) and linear stability theory (LST). The main objective of the current paper is to study the linear stability characteristics of the boundary-layer wave modes and their mutual resonant interactions. The numerical solutions of both steady base flow and unsteady flow induced by forcing disturbances are obtained by using a fifth-order shock-fitting method. Meanwhile, the LST results are used to study the supersonic boundary-layer stability characteristics relevant to the receptivity study. It is found that, in addition to the conventional first and second modes, there exist a family of stable wave modes in the supersonic boundary layer. These modes play a very important role in the receptivity process of excitation of the unstable Mack modes, especially the second mode. These stable modes are termed mode I, mode II, etc., in this paper. Though mode I and mode II waves are linearly stable, they can have resonant (synchronization) interactions with both acoustic waves and the Mack-mode waves. Therefore, the stable wave modes such as mode I and mode II are critical in transferring wave energy between the acoustic waves and the unstable second mode. The effects of frequencies and wall boundary conditions for the temperature perturbations on the boundary-layer stability and receptivity are also studied.

Receptivity of a supersonic boundary layer over a flat plate. Part 1. Wave structures and interactions

The principal objective of this paper is to study some unsteady characteristics of an interaction between an incident oblique shock wave impinging on a laminar boundary layer developing on a flat plate. More precisely, this paper shows that some unsteadiness, in particular the low-frequency unsteadiness, originates in a supercritical Hopf bifurcation related to the dynamics of the separated boundary layer. Various direct numerical simulations were carried out of a shock-wave/laminar-boundary-layer interaction (SWBLI). Three-dimensional unsteady Navier–Stokes equations are numerically solved with an implicit dual time stepping for the temporal algorithm and high-order AUSMPW+ scheme for the spatial discretization. A parametric study on the oblique shock-wave angle has been performed to characterize the unsteady behaviour onset. These numerical simulations have shown that starting from the incident shock angle and the spanwise extension, the flow becomes three-dimensional and unsteady. A linearized global stability analysis is carried out in order to specify and to find some characteristics observed in the direct numerical simulation. This stability analysis permits us to show that the physical origin generating the three-dimensional characters of the flow results from the existence of a three-dimensional stationary global instability.

http://jch.robinet.pagesperso-orange.fr/pdf/Robinet_JFM_2006.pdf

Bifurcations in shock-wave/laminar-boundary-layer interaction : global instability approach

Direct numerical simulations (DNS) were performed to investigate the laminar–turbulent transition in a boundary layer on a sharp cone with an isothermal wall at Mach 6 and at zero angle of attack. The motivation for this research is to make a contribution towards understanding the nonlinear stages of transition and the final breakdown to turbulence in hypersonic boundary layers. In particular, the role of second-mode fundamental resonance, or (K-type) breakdown, is investigated using high-resolution ‘controlled’ transition simulations. The simulations were carried out for the laboratory conditions of the hypersonic transition experiments conducted at Purdue University. First, several low-resolution simulations were carried out to explore the parameter space for fundamental resonance in order to identify the cases that result in strong nonlinear interactions. Subsequently, based on the results from this study, a set of highly resolved simulations that proceed deep into the turbulent breakdown region have been performed. The nonlinear interactions observed during the breakdown process are discussed in detail in this paper. A detailed description of the flow structures that arise due to these nonlinear interactions is provided and an analysis of the skin friction and heat transfer development during the breakdown is presented. The controlled transition simulations clearly demonstrate that fundamental breakdown may indeed be a viable path to complete breakdown to turbulence in hypersonic cone boundary layers at Mach 6.

Direct numerical simulation of transition in a sharp cone boundary layer at Mach 6: fundamental breakdown

The spatial development of disturbances with small and moderate amplitudes in a two-dimensional (2-D) supersonic flat-plate boundary layer at Mach 48 is investigated using direct numerical simulations based on the compressible 3-D Navier-Stokes equations Disturbances are introduced into the boundary layer by blowing and suction within a narrow disturbance strip at the wall In response to the timewise periodic forcing, two types of disturbance waves are generated, a “first-mode” wave and a “multiple-viscous-solution” The “multiple-viscous-solution” was described by Mack (1969, 1984) but was not seen before in a direct numerical simulation The results of the simulations are compared with results of linear stability theory, and the agreement is very good In simulations for larger amplitudes, fundamental resonance is observed, where both types of 3-D waves are nonlinearly amplified and synchronize their phase velocities with the 2-D disturbance waves Subharmonic resonance is found for 3-D waves with large wave numbers, where the phase velocities of the linear 2-D and 3-D waves are nearly the same

Spatial numerical simulations of linear and weakly nonlinear wave instabilities in supersonic boundary layers


 Deep surge is a time and system dependent phenomenon that might be triggered by a flow instability. This instability could be subsequently used as an indicator for the surge margin. Based on former numerical investigations concerning steady flow phenomena in a centrifugal compressor, published at “Turbo Expo 2021” by Hochschule RheinMain and TU Darmstadt the investigations have been continued. A speed line is divided into four sections based on its slope, which are examined for instabilities by means of transient simulations with a numerical reduced model. For this purpose, a detailed frequency analysis is carried out to detect unstable flow behaviour. It is shown that the characteristic of the speed line is in a good agreement with the analysed frequency spectrum and the associated unsteady flow phenomena. These are for instance a highly changing incidence triggered by vortex tubes at the impeller inlet. In addition, the computations show an instability in the vaneless space behind the trailing edge, the occurrence of which is in a relationship to the experimental stability limit. Regardless of the system volume, the phenomenon could be used as an indicator for the stability of the compressor wheel.
 A second speed line is then used to validate this relationship.

W. Eißler

Papers

Spatial numerical simulations of linear and weakly nonlinear wave instabilities in supersonic boundary layers

Numerical Investigation of Unsteady Flow Phenomena in a Centrifugal Compressor Operating Near Surge With a Geometrically Reduced Model