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Receptivity of Straight Blunt Cones to Broadband Freestream Pulse Disturbances for Transition Prediction in Hypersonic Flow


Traditional stability analysis for hypersonic flows has focused on the development and relative amplification of a dominant disturbance mode, namely Mack's second mode. However, conventional $e^N$ method transition estimates based purely on the relative amplification of the second mode ignore the receptivity mechanisms which govern the response of a flow to different environmental disturbances. These receptivity mechanisms are now known to play a much more significant role in the laminar-to-turbulent transition of hypersonic flows, and numerous theoretical, computational, and experimental studies have been made to characterize them. Computational studies in particular are useful due to the large amount of high precision data they can generate with a relatively low cost, while experimental receptivity studies face difficulties due to the high noise environments present in conventional hypersonic windtunnels, as well as the general expense of hypersonic experiments. While computational receptivity studies have been prominent in the literature, most of these prior studies have focused on discrete frequency disturbances and have neglected to consider the more continuous and broadband disturbance spectra that can be found in experiments and in flight.

This work aims to expand the prior body of receptivity research by studying the response to broadband disturbances that are more reflective of realistic environmental forcing. This process involves simulating the receptivity to axisymmetric disturbances as well as some preliminary investigations of the receptivity to 3-D pulse disturbances which excite azimuthal wave modes. The axisymmetric receptivity coefficients are also used to explore further applications of the amplitude method for transition prediction. This work uses perfect gas linear stability theory (LST) and direct numerical simulation (DNS) to simulate the Mach 10 flow over 9.525 mm (Case B), 5.080 mm (Case I), and 1 mm (Case S) nose radius axisymmetric straight cones with 7-degree half-angles. Case B and Case I utilized approximately the same freestream conditions, while the unit Reynolds number for Case S is 60\% lower. The 2-D receptivity of Case B and Case I are studied using finite spherical and planar pulse disturbances in the freestream. These include freestream fast acoustic, slow acoustic, temperature, and vorticity disturbances in order to generate receptivity coefficient spectra for a wide variety of possible freestream noise sources. LST analysis predicted significant second mode growth for both Case B and Case I, with mode F being unstable for Case B and mode S being unstable for Case I. Case I was found to be more destabilizing to the second mode, as expected from the reduced nose radius. These LST results are used to validate the unsteady DNS, and to extract receptivity coefficients for the dominant second mode disturbance. The sharpest Case S is also studied using both axisymmetric planar acoustic pulses as well as azimuthally varying, 3-D acoustic pulse disturbances.

Unsteady DNS for Case B and Case I show that all of the disturbances excite second mode growth and generally follow the amplification profile predicted by LST. The results for the blunter Case B show much stronger freestream noise effects than for Case I, while Case I seems to be more sensitive to extremely low frequency boundary layer modes associated with upstream disturbances originating in the entropy layer. Additionally, the blunter Case B is observed to have a much stronger supersonic mode, which is qualitiatively observed in acoustic radiation from the boundary layer disturbance wavepacket. This is attributed to the destabilized discrete mode F in Case B which is more capable than the discrete mode S of slowing and becoming supersonic relative to the meanflow. DNS and LST data are used to extract second mode receptivity coefficients and phase spectra for Case B and Case I for application to an iterative transition prediction method. The receptivity coefficients are decomposed from the total surface disturbance by renormalizing the unsteady DNS data with the LST-derived amplifcation rate. Most pulse cases saw the same spectral receptivity coefficient behavior, with peak amplitudes being at the most amplified downstream frequencies. However, the planar fast acoustic pulse in both Case I and Case B was found to excite a much more broadband disturbance profile associated with continuous mode instabilities. This likely necessitates more advanced modal decomposition methods, like the bi-orthogonal decomposition, to cleanly resolve the modes of interest. Additionally, Case I was found to be much more receptive to the temperature and vorticity pulses than Case B while the acoustic responses were fairly similar. This may be due to stronger low frequency upstream forcing that was found in Case I that is associated with entropy layer modes, and indicates differing receptivity mechanisms between the two nose bluntnesses.

The receptivity cofficients for Case B and Case I are applied to a simplified 2-D implementation of Mack's amplitude method in an attempt to better approximate the transition location for those cases. Unlike the conventional $e^N$ method, the amplitude method directly estimates disturbance amplitudes in the boundary layer. This requires the use of receptivity data and freestream noise profiles along with other correlations for threshold breakdown amplitudes. The correlations for freestream noise and breakdown amplitude are taken from Marineau(AIAA Journal, 2017). While some improvement is seen over the base accuracy of standard $e^N$ method predictions, the estimated transition locations using the receptivity data from this study are found to still be significantly overpredicted. The Case I results, however, match much more closely with experimental measurements likely due to the stronger second mode. The disagreement with experiment and Marineau's results can be attributed to imprecise correlations of freestream noise and oversimplifications of the original amplitude method relations.

True flight conditions will inevitably contain oblique disturbances, whether through local geometry or from the nature of freestream noise. Therefore, while the 2-D second mode disturbance has found to dominate in a large selection of hypersonic flows, consideration of 3-D disturbances is necessary to complete our view of the transition process. To accomplish this a preliminary analysis of both planar 2-D and 3-D azimuthally varying pulse simulations were also run on the small bluntness (1 mm nose radius) Case S. LST analysis of this new case shows an upstream shift in the destabilized second mode positions, as well as higher destabilized frequencies that match expectations for the smaller nose radius. While the peak growthrates were highest for Case S the reduced freestream unit Reynolds number compared to Case B and Case I led to a weaker overall amplification of the second mode in the streamwise direction, as measured by peak N-factor. Freestream fast and slow acoustic pulses were modelled using gaussian distributions in both the streamwise and azimuthal directions. Results for the 2-D pulses demonstrate strong similarities to prior results for Case B and Case I, with the planar fast acoustic pulse inducing strong broadband continuous mode variation while the slow acoustic pulse induced only second mode disturbances. The 3-D azimuthally varying pulses followed a similar pattern, as downstream excitations were primarily isolated to low wavenumber modes in the slow acoustic case. The 3-D fast acoustic pulse is found to also excite a wide range of disturbance frequencies, though excited frequency bands outside of the second mode are generally confined to lower wavenumbers. Disturbances at the second mode frequencies, however, saw little to no decay at higher wavenumbers compared to the 3-D slow acoustic case. This indicates that fast acoustic disturbances are highly capable of exciting both a broad range of disturbance wavenumbers as well as a broad range of disturbance frequencies.

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