Numerical Simulation of Leading Edge Receptivity of Stetson's Mach 8 Blunt Cone Stability Experiments

Laminar-turbulent transition in boundary layers depends on the receptivity process, which is the process of environmental disturbances initially entering the boundary layers and generating disturbance waves. The receptivity of hypersonic boundary layers to freestream disturbances is altered considerably by the presence of bow shocks in hypersonic flow fields and by the entropy layers created by the blunt noses. This paper is part I1 of a previous paper (AIAA paper 2002-2849) on the numerical simulation study of the receptivity to freestream acoustic disturbance waves for Mach 7.99 axisymmetric flow over a 7' half-angle blunt cone. The main focus of this study is on the excitation of the second Mack mode waves in the receptivity process in the presense of freestream acoustic waves for the case of T' = 0 wall boundary condition. The numerical solutions are compared with experimental results of Stetson et al. (1984) and with those obtained from a linear stability analysis. Both steady and unsteady flow solutions of the receptivity problem are obtained by computing the full Navier-Stokes equations using a high-order accurate shock-fitting finite difference scheme, which can accurately account for the effects of bow-shock/freestreamsound interactions on the receptivity process. In addition, a normal-mode linear stability analysis is also used to study the stability and receptivity properties of the boundary layer affected by the entropy layer. The most important finding of this study is that the synchronization location between mode I and the second mode plays an important role in the receptivity of the second Mack mode in the boundary layer. In the current flow over a blunt cone, the synchronization location is located downstream of the branch I neutral stability location of the second mode. As a result, there are no noticeable second mode components in the region before the synchronization location even though the second mode is linearly unstable there. Therefore mode I plays the most important role in the receptivity process, and leads to the delay of the excitation of the second mode in the current flow over a blunt cone. *Professor, Mechanical and Aerospace Engineering Departt Graduate Student, Student Member AIAA ment, e-mail: xiaolinOseas.ucla. edu, Associate Fellow AIAA. Copyright 02003 by American Institute of Aeronautics and Astronautics, Inc. All rights reserved. INTRODUCTION The prediction of laminar-turbulent transition in hypersonic boundary layers is a critical part of the aerodynamic design and control of hypersonic vehicles. Despite extensive studies over several decades, the mechanisms of hypersonic boundary layer stability and transition are still not well understood. Mack ['I was the first t o find that there are higher acoustic instability modes in addition to the first-mode instability waves in high-Mach-number boundary layers. Among them, the second Mack mode becomes the dominant instability for hypersonic boundary layers at Mach numbers larger than about 4. The existence and dominance of the second Mack mode has been observed by experimental studies [2131, especially in hypersonic boundary layers over axisymmetric cones [3-71. carried out boundary layer stability experiments for an axisymmetric blunt cone at Mach 7.99. The half angle of the cone is 7' and the freestream Reynolds number based on the nose radius is about 33,449. The total length of the cone is about 270 nose radii, corresponding to a Reynolds number of about 9 million. The experiments measured detailed frequency spectra of the disturbance waves along the body surface. The instability waves were found to be dominated by second Mack mode instability. There are also significant harmonic components of the second modes. They also found evidence of entropy layer instability in the region outside of the boundary layers for the case of a blunt cone with large nose radius. Stability experiments of hypersonic flow over similar geometries have also been done by other authors. Demetriades [4151 did extensive stability experiments on hypersonic boundary layers over axisymmetric cones. Maslov and his colleagues L 6 1 7 ] reported their recent stability experiments on high speed flow. Because the issue of hypersonic boundary layer stability involves many complex mechanisms, i t is necessary to conduct validation studies by comparing numerical simulation results with those of stability experiments. The CFD code validation on hypersonic boundary layer transition is one of the subjects of the NATO RTO Working Group 10 [9]. The Mach 7.99, 7' blunt cone of Stetson's experiment is identified by the NATO Working Group as one of the few best available stability experiments for CFD code validation. In the previous paper [lo], we studied the receptivity Stetson et al. 1 41st Aerospace Sciences Meeting and Exhibit 6-9 January 2003, Reno, Nevada AIAA 2003-1133 Copyright © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. to freestream acoustic waves of the 7' half-angle blunt cone in Mach 7.99 flow, corresponding to the stability experiment by Stetson et al. [81. The numerical simulation of the full nonlinear Navier-Stokes equations were able to take in to account the nonlinear wave interactions, the bow shock and entropy layer effects, the wall curvature effects, and the non-parallel mean flow effects on the stability of the boundary layer. The steady flow solutions of the previous paper [lo] agreed very well wit,h those computed by Esfahanian and Herbert [11] and compared well with experimental results on surface pressures and tangential velocities out side of the boundary layer. The receptivity of the Stetson's Mach 7.99 flow over the blunt cone to freestream fast acoustic waves was simulated by solving the full Navier-Stokes equations. The simulation results showed a complex development of wave modes induced by freestream acoustic waves. The main finding from the simulations was that the unstable second mode does not develop in the region where linear stability analysis (LST) predicts dominant unstable second modes. Instead, the second modes are excited at a later location than predicted by the LST analysis because the excitation of the second mode is governed by a different receptivity process. It was found that the synchronization location between mode I and the second mode plays an important role in the receptivity of the second Mack mode in the boundary layer. In the Mach 7.99 flow over a blunt cone, the synchronization location is located downstream of the branch I neutral stability location of the second mode. As a result, there are no noticeable second mode components in the region before the synchronization location even though the second mode is linearly unstable there. Therefore mode I and its resonant interaction with the second Mack mode plays a very important role in the receptivity process, and leads to the lack of excitation of the second mode in hypersonic flow over a blunt cone. This paper is part I1 of the study presented in [lo] on the numerical simulation of the receptivity of Stetson's 7' blunt cone in Mach 7.99 flow for two purposes: 1) to compare numerical results with the experimental results, and 2) more importantly, to study the detailed receptivity and stability mechanism by numerical simulations. In [lo], the steady base flow was obtained by imposing an adiabatic wall boundary condition. Although the mean flow has an adiabatic wall with a zero temperature gradient on the wall, it has been generally accepted that the temperature perturbations should be set to zero because of the relatively high frequencies of the second mode. In our simulations, however, a wide range of frequencies are simultaneously imposed in the freestream. It is likely that the actual temperature perturbations may be somewhere between the two extremes of i? T'/i? y,, = 0 and T' = 0. Therefore, we simulate both cases of different temperature perturbations conditions on the wall with the same mean flow. The results of the test case of dT ' /dy , = 0 boundary condition were mainly presented in [ lo] . The purpose of this paper is to present the second part of the results on the receptivity for the case of using zero temperature perturbation (7" = 0) boundary condition. The combined results of the current paper and the previous paper [lo] are expected to show a better picture of the receptivity process in hypersonic boundary layers over a blunt cone. In the following sections, the governing equations and the receptivity simulations are briefly described. More details, along with the review of related works, can be found in part I of this work in [IO]. GOVERNING EQUATIONS A N D NUMERICAL METHODS The stability and receptivity of axisymmetric laminar hypersonic flow over a blunt cone at zero angle of attack is computed using a three-dimensional grid. The governing equations are the unsteady full three-dimensional Navier-Stokes equations written for the computation in the conservation-law form: dU* d F * j dF*,j -+-+= o at* ax; ax; where superscript "*" represents dimensional variables, and U' = { p * , p'u;, p *u ; , p*u; , e*) . The gas is assumed to be thermally and calorically perfect. The viscosity and heat conductivity coefficients are calculated using Sutherland's law together with a constant Prandtl number, Pr . The equations are transformed into body-fitted curvilinear computational coordinates in a computational domain bounded by the bow shock and the body surface. The location and the movement of the bow shock is an unknown to be solved for along with the flow variables by a shock-fitting method described in [la]. The unsteady bow shock shape and shock oscillations are solved as part of the computational solution. The numerical methods for spatial discretization of the 3-D full Navier-Stokes equations are a fifth-order shock-fitting scheme in streamwise and wallnormal directions, and a Fourier collocation method