This dissertation presents a numerical simulation study of linear hypersonic boundary-layer receptivity and stability over blunt compression cones with freestream hotspot perturbations. This study is conducted for freestream disturbances with broad, continuous frequency spectra over cones that have nose radii of 1, 0.5 and 0.1 mm under freestream conditions of Mach 6, 10 and 15. The simulations are carried out using the high-order shock-fitting finite-difference scheme developed by Zhong (1998), the results of which are shown to agree well with linear stability theory (LST) and experiments. The general receptivity mechanism is then studied by the simulation-LST comparisons under two parametric effects: nose bluntness and freestream Mach number. Among the new findings of the current study, the mechanisms of the receptivity process are found to be mainly caused by the fast acoustic waves that are generated behind the bow shock from the hotspot/shock interaction in the nose region. It is these fast acoustic waves that substantially enter the boundary layer and generate mode F through the synchronization of fast acoustic waves and mode F in the upstream part of the cone. Subsequently, the synchronization of modes F and S generates mode S, or the second mode, which eventually grows into a dominant level at the downstream part of the cone. Additionally, we have obtained the receptivity coefficients of mode S along the Branch-I neutral stability curve using a method that combines LST predicted N-factors and simulated disturbance amplitudes. These receptivity coefficients agree well with those obtained from the theoretical modal decomposition method.

In addition to obtaining the general receptivity mechanism and receptivity coefficients, we have also studied the parametric effects of nose bluntness and freestream Mach number on boundary-layer receptivity and stability over cones. Specifically, our results have shown that nose bluntness reduces the boundary-layer receptivity to freestream entropy perturbations and stabilizes the perturbed boundary layer over a cone. The boundary layer is more receptive to freestream entropy perturbations at higher freestream Mach numbers, while the perturbed boundary layer is stabilized at higher freestream Mach numbers.

The current receptivity and stability study has not only shed new light on the receptivity mechanism to freestream entropy spots over blunt cones, but also advanced the understanding of nose bluntness and freestream Mach number effects on the receptivity and stability over blunt cones. Furthermore, the currently-obtained broad, continuous spectra of unstable-second-mode receptivity coefficients could potentially provide the initial amplitudes for future amplitude-based transition predicting methods.

The current understanding of the effects of thermochemical nonequilibrium on hypersonic boundary-layer instability still contains uncertainties, and there has been little research into the effects of surface ablation, or two-dimensional roughness, on hypersonic boundary-layer instability. The objective of this work is to study the effects of thermochemical nonequilibrium on hypersonic boundary-layer instability. More specifically, two separate nonequilibrium flow configurations are studied: 1) flows with graphite surface ablation, and 2) flows with isolated two-dimensional surface roughness. These two flow types are studied numerically and theoretically, using direct numerical simulation and linear stability theory, respectively.

To study surface ablation, a new high-order shock-fitting method with thermochemical nonequilibrium and finite-rate chemistry boundary conditions for graphite ablation is developed and validated. The method is suitable for direct numerical simulation of boundary-layer transition in a hypersonic real-gas flow with graphite ablation. The new method is validated by comparison with three computational data sets and one set of experimental data. Also, a thermochemical nonequilibrium linear stability theory solver with a gas phase model that includes multiple carbon species, as well as a linearized surface graphite ablation model, is developed and validated. It is validated with previously published linear stability analysis and direct numerical simulation results. A high-order method for discretizing the linear stability equations is used which can easily include high-order boundary conditions. The developed codes are then used to study hypersonic boundary-layer instability for a 7 deg half angle blunt cone at Mach 15.99 and the Reentry F experiment at 100~kft. Multiple simulations are run with the same geometry and freestream conditions to help separate real gas, blowing, and carbon species effects on hypersonic boundary-layer instability. For the case at Mach 15.99, a directly simulated 525~kHz second-mode wave was found to be significantly unstable for the real-gas simulation, while in the ideal-gas simulations, no significant flow instability is seen. An N factor comparison also shows that real-gas effects significantly destabilize the flow when compared to an ideal gas. Blowing is destabilizing for the real gas simulation and has a negligible effect for the ideal gas simulation due to the different locations of instability onset. Notably, carbon species resulting from ablation are shown to slightly stabilize the flow for both cases. For the Reentry F flow conditions, inclusion of the ablating nose cone was shown to increase the region of second mode growth near the nose cone. Away from the nose cone, the second mode was relatively unaffected.

Experimental and numerical results have shown that two-dimensional surface roughness can stabilize a hypersonic boundary layer dominated by second-mode instability. It is sought to understand how this physical phenomenon extends from an airflow under a perfect gas assumption to that of a flow in thermochemical nonequilibrium. To these ends, a new high-order shock-fitting method that includes thermochemical nonequilibrium and a cut-cell method, to handle complex geometries unsuitable for structured body-fitted grids, is presented. The new method is designed specifically for direct numerical simulation of hypersonic boundary-layer transition in a hypersonic real-gas flow with arbitrary shaped surface roughness. The new method is validated and shown to perform comparably to a high-order method with a body-fitted grid. For a Mach 10 flow over a flat plate, a two-dimensional roughness element was found to stabilize the second mode when placed downstream of the synchronization location. This result is consistent with previous results for perfect-gas flows. For a Mach 15 flow over a flat plate, a two-dimensional surface roughness element stabilizes the second-mode instability more effectively in a thermochemical nonequilibrium flow, than in a corresponding perfect gas flow.

Numerical simulations of multi-scale flow problems such as hypersonic boundary layer transition, turbulent flows, computational aeroacoustics and other flow problems with complex physics require high-order methods with high spectral resolutions. For instance, the receptivity mechanisms in the hypersonic boundary layer are the resonant interactions between forcing waves and boundary-layer waves, and the complex wave interactions are difficult to be accurately predicted by conventional low-order numerical methods. High-order methods, which are robust and accurate in resolving a wide range of time and length scales, are required. The objective of this dissertation is to develop and analyze new very high-order numerical methods with spectral-like resolution for flow simulations on structured grids, with focus on smooth flow problems involving multiple scales. These numerical methods include: the multi-layer compact (MLC) scheme, the directional multi-layer compact (DMLC) scheme, and the least square multi-layer compact (LSMLC) scheme.

In the first place, a new upwind multi-layer compact (MLC) scheme up to seventh order is derived in a finite difference framework. By using the ‘multi-layer’ idea, which introduces first derivatives into the MLC schemes and approximates the second derivatives, the resolution of the MLC schemes can be significantly improved within a compact grid stencil. The auxiliary equations are introduced, and they are the only nontrivial equations. The original equation requires no approximation which contributes to good computational efficiency. In addition, the upwind MLC schemes are derived on centered stencils with adjustable parameters to control the dissipation. Fourier analysis is performed to show that the new MLC schemes have very small dissipation and dispersion in a wide range of wavenumbers in both one and two-dimensional cases, and the anisotropic error is much smaller than conventional finite difference methods in the two-dimensional case. Comparison with discontinuous-Galerkin methods is performed with Fourier analysis as well. Furthermore, stability analysis with matrix method shows that high-order boundary closure schemes are stable because of compactness of the stencils. The accuracies and rates of convergence of the new schemes are validated by numerical experiments of the linear advection equation, the nonlinear Euler equations, and the Navier-Stokes equations in both one and two-dimensional settings. The numerical results show that good computational efficiency, very high-order accuracies, and high spectral resolutions especially on coarse meshes can be attained with the MLC scheme.

On the other hand, even though the MLC scheme is promising in most test cases, it shows weak numerical instabilities for a small range of wavenumbers when it is applied to multi-dimensional flows, which are mainly triggered by the inconsistency between its one and two-dimensional formulations. The instability could lead to divergence in long-time multi-dimensional simulations. Moreover, the cross-derivative approximation in the MLC scheme requires an ad-hoc selection of supporting grid points, and the cross-derivative approximation is relatively inefficient for very high-order cases. To address the remaining challenges of the MLC scheme and achieve better performance for multi-dimensional flow simulations, another two new schemes are developed – the directional multi-layer compact (DMLC) scheme, and the least square multi-layer compact (LSMLC) scheme.

In the second place, a new upwind directional multi-layer compact (DMLC) scheme is developed for multi-dimensional simulations. The main idea of the DMLC scheme is to introduce auxiliary equation for cross derivative in multi-dimensional cases. Consequently, the spatial discretization can be fulfilled along each dimension independently. With this directional discretization technique, the one-dimensional formulation of the MLC scheme can be applied to all spatial derivatives in a multi-dimensional governing equation. Therefore, the DMLC scheme overcomes the inconsistency between one and two-dimensional formulations of the MLC scheme, and it also avoids the ad-hoc cross-derivative approximations. Two-dimensional Fourier analysis demonstrates that all modes of the DMLC scheme are stable in the full range of wavenumbers, and it has better spectral resolution and smaller anisotropic error than the MLC scheme. Stability analysis with matrix method indicates that stable boundary closure schemes are much easier to be obtained in the DMLC scheme. Numerical tests in the linear advection equation and the nonlinear Euler equations validate that the DMLC scheme are more accurate and require less CPU time than the MLC scheme on the same mesh. In particular, the long-time simulation results reveal that the DMLC scheme is always stable for both periodic and non-periodic boundary conditions in two-dimensional cases.

In the third place, a new upwind least square multi-layer compact (LSMLC) scheme is developed for multi-dimensional simulations. The main idea of the LSMLC scheme is using the weighted least square approximation to redesign the two-dimensional formulation for cross derivatives. It avoids the ad-hoc selection of grid points in the MLC scheme. Meanwhile, the two-dimensional upwind scheme can be derived by introducing upwind correction into the weight function. The upwind factor β can adjust the dissipation and stability of the LSMLC scheme. Lagrange multiplier is used to ensure that the LSMLC scheme satisfies both the consistency constraint at the base point and the one-dimensional constraint from the MLC scheme. The LSMLC scheme does not increase computational cost on structured meshes, and can be implemented in the same way as the MLC scheme. A parametric study based on two-dimensional Fourier analysis shows that the truncated Gaussian distribution (TGD) weight function leads to better LSMLC scheme among other weight functions because it removes the numerical instability and maintain small dissipations. The LSMLC scheme has larger dissipation than the MLC scheme, and shows similar spectral resolution. Stability analysis with matrix method indicates that a combination of an interior LSMLC scheme and MLC boundary closure schemes can improve the boundary stability while maintaining small dissipation. Numerical tests in the linear advection equation and the nonlinear Euler equations validate that the LSMLC scheme produces slightly larger errors compared with MLC scheme. The long-time simulation results reveal that the LSMLC scheme is always stable for both periodic and non-periodic boundary conditions in two-dimensional cases.

Overall, the new very high-order multi-layer compact finite difference methods have the properties of simple formulations, high-order accuracies, spectral-like resolutions, and compact stencils, and they are suitable for accurate simulation of smooth multi-scale flows with complex physics. Among the three schemes developed in this dissertation, the DMLC scheme is always the best choice for multi-dimensional simulations because it shows comprehensive improvements from the MLC scheme with consistent stability, higher accuracy and spectral resolution, and better computational efficiency. The LSMLC scheme is also appropriate considering it has consistent stability and it is easy to be implemented.

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.

Hypersonic boundary layer research has studied surface features, such as isolated or distributed roughness, extensively for turbulence tripping. However, there are reports of a counterintuitive phenomenon within the literature whereby surface roughness can delay the onset of laminar-turbulent transition. The reports did not attract widespread attention, leaving the phenomenon's underlying mechanism uninvestigated for several decades. A renewed interest in boundary layer control strategies motivated Fong and Zhong in 2012 to conduct an extensive numerical study on what has been termed the ``roughness effect''. The research found that roughness elements immersed within the boundary layer and placed at the synchronization location for a particular unstable frequency can attenuate higher unstable frequencies while amplifying lower unstable frequencies. Thus, providing a passive means to delay laminar-turbulent transition with discrete surface roughness. However, these previous numerical investigations are limited to a flat plate geometry, 2-D spanwise roughness, limited in the scope of their freestream Mach number, and focus exclusively on Mack's second mode instability. In order to advance our knowledge of the roughness effect, the objectives of this dissertation are fourfold: (1) To investigate the roughness effect on a straight blunt cone geometry, (2) To investigate the long-term downstream consequences of the roughness effect, (3) Provide experimental evidence of second mode attenuation in a flow with a growing boundary layer containing a range of unstable frequencies, and the consequences of off-design flow conditions, and (4) To investigate the appearance of the supersonic mode in a low-enthalpy warm wall flow of the current study.

A combined approach of direct numerical simulation, body-fitted surface roughness, and linear stability theory are used to numerically investigate the roughness effect. Four cases are computed as part of the research objective. Case C.1 is a Mach 8 flow computed for the design of a passive transition-delaying roughness configuration, along with studying the roughness effect on a straight blunt cone. Case C.1-Ext is a longer cone simulation of C.1 and is computed to investigate the long-term downstream response of the roughness effect. C.2 is similar to C.1 except for a smaller nose radius and is computed for experimental validation. The last case, C.3, is a Mach 5 flow and is computed to study the roughness effect on a straight blunt cone in off-design flow conditions and for experimental validation.

The first objective to investigate the roughness effect on a straight blunt cone advances the research from a flat plate to more realistic test article geometries. Much of the experimental work done in hypersonic boundary layer stability research is done on straight cones due to the axisymmetric flows in hypersonic wind tunnels. The investigation found that the roughness effect behaves like a flat plate where unstable frequencies higher than the synchronization frequency are attenuated, and lower frequencies are amplified. The investigation also found that some flow features around the roughness elements, such as separation zones, are either smaller in size or absent in conical flow fields. The investigation also confirmed that the second mode's attenuation is a result of the element's proximity to the synchronization location and not due to its proximity with the branch I/II neutral points.

The long-term downstream effect of second mode attenuation is also investigated for a single roughness and roughness array. The numerical investigation found that the range of targeted frequencies is attenuated as expected, especially for the roughness array, which proves to be effective at attenuating unstable frequencies over a longer distance. However, the amplitudes of frequencies below the targeted range grow many times higher than they would have otherwise on a cone with no roughness. The passive transition-delaying control strategy, rather than dissipating the disturbance energy, acts to transfer the energy to lower unstable frequencies, guaranteeing eventual turbulent transition. The result demonstrates that roughness must be applied to the entire cone to have an effective control strategy.

The experimental results in this dissertation come from a joint numerical and experimental investigation of transition-delaying roughness with Dr. Katya Casper at Sandia National Laboratories. A numerical simulation is undertaken to design a surface roughness array that would attenuate Mack's second mode instability and maintain laminar flow over a Mach 8 hypersonic blunt cone. Multiple experimental runs at the Mach 8 condition with different Reynolds numbers are performed, as well as an off-design Mach 5 condition. The roughness array successfully delays transition in the Mach 8 case as intended but does not delay transition in the Mach 5 case. For validation and further analysis, numerical cases C.2 and C.3 are computed using the Mach 8 and Mach 5 experimental flow conditions. Stability analysis of case C.2 shows that the roughness array is adequately designed to attenuate the second mode. Analysis of case C.3 reveals the Mach 5 boundary layer is dominated by Mack's first mode instability and is not attenuated by the array. This investigation of multiple flow conditions combined with experimental results helps validate the numerical code and provides empirical evidence for the roughness effect.

While investigating transition delaying surface roughness, acoustic-like waves are observed emanating from the boundary layer of case C.1-Ext. The acoustic-like wave emissions are qualitatively similar to those attributed to the supersonic mode. However, the supersonic mode responsible for such emissions is often found in high-enthalpy flows with highly cooled walls, making its appearance in a flow with relatively low freestream enthalpy and a warm wall unexpected. Stability analysis on the steady-state solution reveals an unstable mode S with a subsonic phase velocity and a stable mode F whose mode F- branch takes on a supersonic phase velocity. The stable supersonic mode F is thought to be responsible for the acoustic-like wave emissions. Unsteady simulations are carried out using blowing-suction actuators at two different surface locations. Analysis of the temporal data and spectral data reveals constructive/destructive interference occurring between a primary and a satellite wave packet in the vicinity of the acoustic-like wave emissions, which has a damping effect on individual frequency growth. Based on this study's results, it is concluded that a supersonic discrete mode is not limited to high-enthalpy, cold wall flows and that it does appear in low-enthalpy, warm wall flows; however, the mode is stable.

Mack’s second mode has been known to be the dominant disturbance leading to transition to turbulence in traditional hypersonic boundary layer flows at zero angle of attack. Physically, the second mode exists due to trapped acoustic waves within the boundary layer. The second mode has been widely studied and the conditions that stabilize or amplify the second mode are well documented. Predicting the second mode amplification is the basis of contemporary transition prediction techniques such as the eN method. There has been a renewed interest in studying hypersonic boundary layer stability in high-enthalpy flows with highly-cooled walls due to its applicability to experiments and some real flight conditions. One physical phenomenon that occurs in these flows is the creation of a supersonic mode, which is associated with an unstable mode F1 synchronizing with the slow acoustic spectrum. This causes the disturbance to travel upstream supersonically relative to the mean flow outside the boundary layer and radiate sound away from the boundary layer. The supersonic mode has been known to exist for decades, but has until recently been deemed negligible in comparison to the second mode. However, a resurgence in interest in the supersonic mode has shown the supersonic mode to exist in unexpected conditions with considerable peak growth rates compared to the second mode. Namely, recent research in the field has shown the supersonic mode in hot-wall flows, upending the notion that it is an artifact of highly-cooled walls. Additionally, a dominant supersonic mode with significantly larger growth rate than the second mode has been found on very blunt cones. Therefore, because the supersonic mode has not been systematically investigated, the mechanism of its creation and the conditions under which it exists are not yet clear.

The objective of this work is to systematically investigate the supersonic mode using numerical and theoretical tools to simulate hypersonic flow over blunt cones. Specifically, this work aims to (1) Determine the characteristics of the supersonic mode and under what conditions it exists, (2) Explore the effectiveness of Linear Stability Theory (LST) on predicting the supersonic mode, and (3) Examine the impact of the supersonic mode on transition to turbulence under realistic flight or experimental conditions. This work explores the supersonic mode on a 1 mm nose radius cone in various free stream flow configurations with a 5-species, two-temperature nonequilibrium gas model for air. A combined approach of Direct Numerical Simulation (DNS) and Linear Stability Theory (LST) are used to numerically investigate the supersonic mode. New LST equations with linearized Rankine-Hugoniot shock relation boundary conditions are derived and verified. In addition, a theoretical schematic has been developed to aid future experimentalists and those performing DNS in visualizing the supersonic mode. Mach numbers of 5 and 10 are considered with wall-temperature-to-free-stream-temperature ratios (Tw/T∞) between 0.2 and 1.43. Additionally, the impact of thermochemical nonequilibrium on the supersonic mode is assessed. Both LST and DNS results have confirmed the existence of the supersonic mode on a Mach 5 axisymmetric cold-wall (Tw/T∞ = 0.2) cone. On a warmer wall (Tw/T∞ = 0.667) under the same free stream conditions, LST indicated the supersonic mode was stabilized, although some weak sound radiation was still apparent in DNS.

For the Mach 10 case, LST predicted a stable supersonic mode for both wall temperature cases (Tw/T∞ = 1.43, Tw/T∞ = 0.43), however a prominent supersonic mode was observed in DNS. The supersonic mode was determined to be excited via a modal interaction that is ignored in LST due to the independent mode assumption. Furthermore, the supersonic mode in the Mach 10 case with Tw/T∞ = 0.43 exhibited a stronger peak growth rate for the supersonic mode compared to Mack’s traditional second mode. These findings illustrate the need for combined LST and DNS studies of the supersonic mode. Overall, this study has determined that the supersonic mode is destabilized by largely the same factors as Mack’s second mode. Namely, wall cooling is destabilizing, increasing Mach number/stagnation enthalpy is destabilizing, and vibrational nonequilibrium is stabilizing. The impact of chemical nonequilibrium is hypothesized to be slightly destabilizing, although was not able to be confirmed with the cases explored here. Based on the results presented here, transition prediction analyses relying on LST, such as the eN method, should be used with caution when applied to the supersonic mode, as it has been shown that LST may not fully capture the mechanism of the supersonic mode’s creation.

Laminar-turbulent transition prediction in hypersonic boundary layer remains one of the most challenging topics in the design of hypervelocity vehicle. It requires thorough understanding of the physical mechanisms underlay freestream wave receptivity and nonlinear breakdown process. Freestream wave receptivity concerns the evolution of freestream disturbance passing through the shock and exciting the boundary layer normal modes that eventually become unstable. Nonlinear breakdown focuses on the study of the relevant mechanisms in the secondary instability region that leads to laminar-turbulent transition. These two topics have been extensively studied separately for decades. Significant progress has been made in terms of understanding how the instability waves form and develop in the early region as well as what are the viable paths from breakdown to turbulent. However, the linkage between receptivity and breakdown is still not well understood. The nature transition process commonly observed in hypersonic boundary layer consists of the following ingredients: freestream wave receptivity, linear growth, secondary instability and breakdown to turbulent. The transition location highly depends on the freestream wave disturbance profile. In order to attain a better understanding of the natural transition process, it is necessary to conduct a complete simulation from freestream wave receptivity all the way to nonlinear breakdown. This kind of simulation is considered beyond the capability of current computer power. The objective of current research is to devise a new three-step approach to simulate the flow from receptivity process to breakdown. In order to achieve the goal, direct numerical simulations (DNS) are performed over various freestream conditions and cone geometries to investigate the hypersonic boundary layer stability, freestream wave receptivity and nonlinear breakdown. In the study of nose bluntness effect on hypersonic boundary layer stability, three cone models with different nose radii are investigated by linear stability theory (LST). It is found that, if only considering the second-mode instabilities, the onset of instability is always delayed as the nose bluntness increases. In the effort to simulate the entire process from freestream wave receptivity to nonlinear breakdown, a new approach is applied to break the simulation into three steps: meanflow calculation, linear receptivity simulation and nonlinear breakdown simulation. Extensive case studies demonstrate that it is feasible to simulate the flow from receptivity to breakdown using our new simulation approach. From the breakdown simulations, it is found that the breakdown is the result of fundamental resonance that occurs between the two-dimensional second-mode wave and their three-dimensional modes. In the secondary instability growth region, the two-dimensional and three-dimensional modes need to attain the same amplitude level for the breakdown to take place.

The current understanding and research efforts on surface roughness effects in hypersonic boundary-layer flows focus, almost exclusively, on how roughness elements trip a hypersonic boundary layer to turbulence. However, there were a few reports in the literature suggesting that roughness elements in hypersonic boundary-layer flows could sometimes suppress the transition process and delay the formation of turbulent flow. These reports were not common and had not attracted much attention from the research community. Furthermore, the mechanisms of how the delay and stabilization happened were unknown. A recent study by Duan et al. showed that when 2-D roughness elements were placed downstream of the so-called synchronization point, the unstable second-mode wave in a hypersonic boundary layer was damped. Since the second-mode wave is typically the most dangerous and dominant unstable mode in a hypersonic boundary layer for sharp geometries at a zero angle of attack, this result has pointed to an explanation on how roughness elements delay transition in a hypersonic boundary layer. Such an understanding can potentially have significant practical applications for the development of passive flow control techniques to suppress hypersonic boundary-layer transition, for the purpose of aero-heating reduction. Nevertheless, the previous study was preliminary because only one particular flow condition with one fixed roughness parameter was considered. The study also lacked an examination on the mechanism of the damping effect of the second mode by roughness. Hence, the objective of the current research is to conduct an extensive investigation of the effects of 2-D roughness elements on the growth of instability waves in a hypersonic boundary layer. The goal is to provide a full physical picture of how and when 2-D roughness elements stabilize a hypersonic boundary layer. Rigorous parametric studies using numerical simulation, linear stability theory (LST), and parabolized stability equation (PSE) are performed to ensure the fidelity of the data and to study the relevant flow physics. All results unanimously confirm the conclusion that the relative location of the synchronization point with respect to the roughness element determines the roughness effect on the second mode. Namely, a roughness placed upstream of the synchronization point amplifies the unstable waves while placing a roughness downstream of the synchronization point damps the second-mode waves. The parametric study also shows that a tall roughness element within the local boundary-layer thickness results in a stronger damping effect, while the effect of the roughness width is relatively insignificant compared with the other roughness parameters. On the other hand, the fact that both LST and PSE successfully predict the damping effect only by analyzing the meanflow suggests the mechanism of the damping is by the meanflow alteration due to the existence of roughness elements, rather than new mode generation. In addition to studying the unstable waves, the drag force and heating with and without roughness have been investigated by comparing the numerical simulation data with experimental correlations. It is shown that the increase in drag force generated by the Mach wave around a roughness element in a hypersonic boundary layer is insignificant compared to the reduction of drag force by suppressing turbulent flow. The study also shows that, for a cold wall flow which is the case for practical flight applications, the Stanton number decreases as roughness elements smooth out the temperature gradient in the wall-normal direction. Based on the knowledge of roughness elements damping the second mode gained from the current study, a novel passive transition control method using judiciously placed roughness elements has been developed, and patented, during the course of this research. The main idea of the control method is that, with a given geometry and flow condition, it is possible to find the most unstable second-mode frequency that can lead to transition. And by doing a theoretical analysis such as LST, the synchronization location for the most unstable frequency can be found. Roughness elements are then strategically placed downstream of the synchronization point to damp out this dangerous second-mode wave, thus stabilizing the boundary layer and suppressing the transition process. This method is later experimentally validated in Purdue’s Mach 6 quiet wind tunnel. Overall, this research has not only provided details of when and how 2-D roughness stabilizes a hypersonic boundary layer, it also has led to a successful application of numerical simulation data to the development of a new roughness-based transition delay method, which could potentially have significant contributions to the design of future generation hypersonic vehicles.

Understanding non-equilibrium effects of hypersonic turbulent boundary layers is essential in order to build cost efficient and reliable hypersonic vehicles. It is well known that non-equilibrium effects on the boundary layers are notable, but our understanding of the effects are limited. The overall goal of this study is to improve the understanding of non-equilibrium effects on hypersonic turbulent boundary layers. A new code has been developed for direct numerical simulations of spatially developing hypersonic turbulent boundary layers over a flat plate with finite-rate reactions. A fifth-order hybrid weighted essentially non-oscillatory scheme with a low dissipation finite-difference scheme is utilized in order to capture stiff gradients while resolving small motions in turbulent boundary layers. The code has been validated by qualitative and quantitative comparisons of two different simulations of a non-equilibrium flow and a spatially developing turbulent boundary layer.

With the validated code, direct numerical simulations of four different hypersonic turbulent boundary layers, perfect gas and non-equilibrium flows of pure oxygen and nitrogen, have been performed. In order to rule out uncertainties in comparisons, the same inlet conditions are imposed for each species, and then mean and turbulence statistics as well as near-wall turbulence structures are compared at a downstream location. Based on those comparisons, it is shown that there is no direct energy exchanges between internal and turbulent kinetic energies due to thermal and chemical non-equilibrium processes in the flow field. Instead, these non-equilibria affect turbulent boundary layers by changing the temperature without changing the main characteristics of near-wall turbulence structures. This change in the temperature induces the changes in the density and viscosity and the mean flow fields are then adjusted to satisfy the conservation laws. The perturbation fields are modified according to the adjusted mean field and the conservation laws. From this, it can be concluded that Morkovin’s hypothesis is still valid with thermal and chemical non-equilibrium, and the effects of non-equilibrium can be compensated by taking the variations of mean density and viscosity into account. In the present study, it is shown that a semi-local scale is a proper scale that can account for the non-equilibrium effects.

The Trettel-Larsson (TL) transformation is extended to two dimensions using mass conservation and an interpretation of Morkovin's hypothesis. This extension reveals the previously unknown limitations of the TL transformation. The TL transformation only works in channel flows, in the inner layer of high Reynolds number boundary layers, and in boundary layers where the viscosity varies with the square root of the density. The error in the TL velocity transformation correlates with the second derivative of the transformed coordinate (the semi-local scaling). The second derivative of the transformed coordinate measures the amount that the transformed outer layer coordinate does not equal the untransformed outer layer coordinate. The extended TL theory now includes a streamwise coordinate transformation and production rate scalings. The limitations of the TL theory state that the analogy between compressible turbulent boundary layers and incompressible turbulent boundary layers is limited in theory but nonetheless useful in practice when possible.