Nonlinear Nonmodal Disturbances over Blunt Cones at Mach 6
Paredes∗, A. Scholten†, M. Choudhari‡, F. Li‡, M. H. Carpenter‡, and M. Bailey§
∗ National Institute of Aerospace, Hampton, VA 23666, USA
† North Carolina State University, Raleigh, NC 27695, USA
‡ NASA Langley Research Center, Hampton, VA 23681, USA
§ The University of Arizona, Tucson, AZ 85721, USA
The linear amplification of modal disturbances that lead to boundary-layer transition in two-dimensional/axisymmetric hypersonic configurations is strongly reduced by the presence of a blunt nosetip, and the mechanisms underlying the low N-factor values at the observed onset of transition over the cone frustum are currently unknown.
Linear nonmodal analysis has shown that both planar and oblique traveling disturbances that peak within the entropy layer experience appreciable energy amplification for moderate to large nosetip bluntness.
Previous linear analysis is extended by including the nonlinear effects via the solution of the perturbation form of the harmonic Navier-Stokes equations (HNSE).
Nonlinear nonmodal results are presented for planar traveling disturbances in a Mach 6 flow over a blunt circular cone. The analysis demonstrates that entropy-layer (EL) disturbances generated close to the nose tip can seed the amplification of higher frequency Mack’s second-mode (MM) instabilities further downstream, and hence, can lead to a reduction in the transition N-factor.
Keywords: Boundary Layer Stability, Numerical Algorithms, Nonmodal Analysis.
A thorough review concerning boundary-layer transition on sharp and blunt cones at hypersonic speeds is provided by Ref. . As discussed in this review, both experimental and numerical studies have shown that laminar-turbulent transition on sharp axisymmetric cones at zero degrees angle of attack is caused by the modal amplification of MM instabilities.
According to earlier research , increasing the nosetip Reynolds number ReRN leads to enhanced stabilization of the MM instabilities due to the effects of a stronger entropy layer, which agrees with the experimental findings indicating a downstream shift of the transition onset location.
However, this trend is also observed to eventually reverse beyond a critical nosetip bluntness. As summarized by Ref. , there is a drop in the MM N-factor at the transition location for ReRN > 40,000, but nonzero, low values of the transition N-factor are found up to approximately ReRN ≈ 500,000.
The physical mechanisms responsible for the low values of the MM N-factor at the transition location are unknown. To help shed further light on this topic, the present work investigates the nonlinear interactions involving the axisymmetric nonmodal EL disturbances and MM instabilities.
The analysis of disturbance growth is based on the decomposition of the flow variables, q(ξ,η,ζ,t) = (ρ,u,v,w,T)T, into a laminar basic state, q¯(ξ,η,ζ), and the unsteady perturbations, q˜(ξ,η,ζ,t)T. Here, (ξ,η,ζ) denote the streamwise direction, the wall-normal direction, and the azimuthal direction, respectively, while (u,v,w) represent the corresponding velocity components. The density and temperature are denoted by ρ and T, respectively.
For nonlinear analysis, the perturbation form of the HNSE is used. A fully implicit formulation is proposed by discretizing the periodic variation in the azimuthal direction and the time domain with Fourier collocation. The HNSE, expressed as F(q˜) = 0, are solved with the NewtonRaphson method. The linear systems are iteratively solved with the preconditioned GMRES method.
For the nonmodal analysis, a variational formulation based on the HNSE is used. The optimal initial disturbance, q˜0, is defined as the initial (i.e., inflow) condition at ξ0 that maximizes the objective function, J, which is defined as the outlet energy gain with respect to the energy norm at the inflow station, GE = E(ξ1)/E(ξ0). More details can be found in Ref. .
The flow conditions and geometry are based upon the tests conducted by Jewell et al.  in the U.S. Air Force Research Laboratory (AFRL) Mach 6 high-Reynolds-number facility, i.e., M∞ = 5.9, T¯∞ = 76.74 K, T¯w = 300 K, and Re∞ = 30.5×106 m-1, with a nosetip of 1.524 mm radius and a cone half angle of 7◦. The selection of this case is motivated by the low transition N-factor value of NT = N(ξT = 0.223 m) = 3.75 reached by the MM wave with f = 660 kHz.
The nonlinear nonmodal analysis is used to investigate the potential interactions between the nonmodal EL disturbances with f = 330 kHz and MM waves with f = 660 kHz. The inflow forcing is limited to the fundamental wave to avoid the introduction of the high frequency MM disturbance at the inflow.
The temperature contours of the fundamental mode (f1 = 330 kHz), and its first harmonic ( f2 = 660 kHz) are shown in Fig. 1 for an initial fundamental amplitude of A0 = E0 = 0.06. The MM wave with f = 660 kHz is generated by the nonlinear self-interaction of the initially seeded f = 330 kHz disturbance as it undergoes a nonmodal growth.
The highest amplitude is observed somewhat downstream of the optimization location at ξ ≈ 0.24 m, and it surpasses the peak of the fundamental wave which occurs around ξ = 0.125 m. Therefore, the EL disturbances excited near the nose tip can excite the high frequency MM disturbances and the subsequent amplification of these MM disturbances can lead to an overall increase in the disturbance energy beyond the linear evolution of the same EL disturbance.
Figure 1: Contours of temperature for the fundamental frequency (f = 330 kHz), and the first harmonic (f = 660 kHz) of the optimal disturbance computed by using the HNSE and (ξ0,ξ1) = (0.05,0.223) m. The solid and dashed black lines indicate the edges of the boundary layer and the entropy layer, respectively. Reproduced from Ref. 
This material is based upon research supported in part by the U. S. Office of Naval Research under award numbers N00014-20-1-2261 and N00014-23-1-2456, and by the Hypersonic Technology Project (HTP) under the NASA Aeronautics Research Mission Directorate (ARMD). The development of the iterative solver is supported by the NASA Langley Research Center under the CIF/IRAD Program.
 P. Schneider. Hypersonic laminar-turbulent transition on circular cones and scramjet forebodies. Progress in Aerospace Sciences, 40:1–50, 2004.
 F. Stetson. Nosetip bluntness effects on cone frustum boundary layer transition in hypersonic flow. AIAA Paper 83-1763, 1983.
 Paredes, M. Choudhari, F. Li, J. Jewell, R. Kimmel, E. Marineau, and G. Grossir. Nosetip bluntness effects on transition at hypersonic speeds: experimental and numerical analysis. Journal of Spacecraft Rockets, 56(2), 2019.
 Scholten, P. Paredes, M. Choudhari, F. Li, M. Carpenter, and M. Bailey. Nonlinear nonmodal analysis of hypersonic flow over blunt cones. AIAA Paper 2023-0000, 2023.
 S. Jewell, R.E. Kennedy, S.J. Laurence, and R.L. Kimmel. Transition on a variable bluntness 7-degree cone at high Reynolds number. AIAA Paper 2018-1822, 2018.