A new parametric model for the growth rate of turbulent mixing layers is proposed. A database of experimental and numerical mixing layer studies was extracted from the literature to support this effort. The domain of the model was limited to planar, spatial, nonreacting, free shear layers that were not affected by artificial mixing enhancement techniques. The model is split into two parts which were each tuned to optimally fit the database; equations for an incompressible growth rate were derived from the error function velocity profile, and a function for a compressibility factor was generalized from existing theory on the convective Mach number. The compressible model is supported by a detailed evaluation of the currently accepted models and practices, including error analysis of the convective Mach number derivation and a critical analysis of Slessor’s re-normalization technique which affected his 1998 compressibility parameter. Analysis of the database suggested that a distinction should be made between thickness definitions that are based on the velocity profile and those based on the density profile. Additionally, the accumulation of different normalization approaches throughout the literature was shown to have introduced non-physical variance in the trends. Resolution of this issue through a consistent normalization process has greatly improved the normality and scatter of the data and the goodness-of-fit of the models, resulting in R2 = 0.9856 for the incompressible model and R2 = 0.9004 for the compressible model.

<p>Laminar or turbulent flows over porous surfaces have received extensive attention in the past few decades, due to their potential to achieve passive flow controls. These surfaces either in natural exhibit roughness or are engineered in purpose, and usually entail special features such as increasing/reducing surface drags. An increasing interest has arisen in the interaction between these surfaces and high-speed compressible flows, which could inform the next-level flow control studies at supersonic and hypersonic speeds for the designs of high-speed vehicles. In this dissertation, the interaction between high-speed compressible turbulent flows and acoustically permeable surface is investigated. The surface property is modeled via the Time-Domain Impedance Boundary Condition (TDIBC), which avoids the inclusion of the geometric details in the numerical simulations.</p> <p>We first perform Large-Eddy Simulations of compressible turbulent channel flows over one impedance wall for three bulk Mach numbers:Mb = 1.5, 3.5 and 6.0. The bulk Reynolds number Reb is tuned to achieve similar viscous Reynolds number Re∗τ ≈ 220 across all Mb to ensure a nearly common state of near-wall turbulence structures over impermeable walls. The TDIBC based on the auxiliary differential equations (ADE) method is applied to bottom wall of the channel. A three-parameter complex impedance model with a resonating frequency tuned to the large-eddy turn-over frequency of the flow is adopted. With a sufficiently high permeability, a streamwise traveling instability wave that is confined in nature and that increases the surface drag, is observed in the near-wall region and changes the local turbulent events. As a result, the first and second order mean flow statistics are found to deviate from that of a flow over impermeable walls. We then perform a linear stability analysis using a turbulent background base flow and confirm that the instability wave is triggered by a sufficiently high permeability and manifests a confined nature. The critical resistance Rcr (interpreted as the inverse of the permeability), above which the instability is suppressed, is found to be sub-linearly proportional to the bulk Mach number Mb, indicating less permeability required to trigger the instability in high Mach number flows.</p> <p>Due to the extremely high computational cost in high Mach number wall-bounded flow calculations, the next-phase optimization/flow control design using the porous surface becomes unaffordable. An ’economical’ flow setup that can server the purpose of rapid flow generation would greatly benefit the planned research. For such reason, we carry out a study about the effect of the domain size on the near-wall turbulence structures in compressible turbulent channel flows, to identify such type of flow setup. Apart from the concept of minimal flow units (MFU, as in the literature) entailing a minimal domain size required for near-wall turbulence to be sustained, efforts have also been made to identify a range of the domain size that can sustain both the inner and outer layer turbulence, and lead to only small deviations in mean flow statistics from the baseline data, which herein defined as minimal turbulent channel (MTC). The motivation of proposing the concept of MTC is to provide a computationally efficient setup for the rapid generation of near-wall turbulence with minimal compromise on the fidelity of the simulated field for investigations requiring numerous simulations, such as machine learning, flow control/optimization designs. It is found that the mean flow statistics from a computational domain spanning 700 − 1100 and 230 − 280 local viscous units in streamwise and spanwise directions, respectively, agree reasonably well with the reference calculations of all three Mach numbers under investigation, and are thus identified as the range in which the MTC stays. The large scale near-wall turbulence structures observed in full scale DNS simulations, and their spatially coherent connections, are roughly preserved in MTC, indicated by the existence of the grouped streamwise aligned hairpin vortices of various sizes and the resulted patterns of uniform momentum zones and thermal zones in the instantaneous flow field. In an MTC, the energy transfer paths among the kinetic energy of the mean field, turbulent kinetic energy and mean internal energy are slightly modified, with the most significant change observed in the viscous dissipation. The mean wall-shear stress and mean wall heat flux see less than 5% error as compared to the full scale simulations. Such reduced-order flow setup requires less than 3% of the computational resource as compared to the full scale simulations.</p>

This dissertation focuses on the development and application of numerical algorithms for solving compressible multiphase fluid-material interaction problems. The first part of this dissertation is motivated by the extraordinary shock-resisting ability of elastomer coating materials (e.g., polyurea) under explosive loading conditions. Their performance, however, highly depends on their dynamic interaction with the substrate (e.g., metal) and ambient fluid (e.g., air or liquid); and the detailed interaction process is still unclear. Therefore, to certify the application of these materials, a fluid-structure coupled computational framework is needed. The first part of this dissertation developes such a framework. In particualr, the hyper-viscoelastic constitutive relation of polyurea is incorporated into a high-fidelity computational framework which couples a finite volume compressible multiphase fluid dynamics solver and a nonlinear finite element structural dynamics solver. Within this framework, the fluid-structure and liquid-gas interfaces are tracked using embedded boundary and level set methods. Then, the developed computational framework is applied to study the behavior a bilayer coating–substrate (i.e., polyurea-aluminum) system under various loading conditions. The observed two-way coupling between the structure and the bubble generated in a near-field underwater explosion motivates the next part of this dissertation. The second part of this dissertation investigates the yielding and collapse of an underwater thin-walled aluminum cylinder in near-field explosions. As the explosion intensity varies by two orders of magnitude, three different modes of collapse are discovered, including one that appears counterintuitive (i.e., one lobe extending towards the explosive charge), yet has been observed in previous laboratory experiments. Because of the transition of modes, the time it takes for the structure to reach self-contact does not decrease monotonically as the explosion intensity increases. Detailed analysis of the bubble-structure interaction suggests that, in addition to the incident shock wave, the second pressure pulse resulting from the contraction of the explosion bubble also has a significant effect on the structure's collapse. The phase difference between the structural vibration and the bubble's expansion and contraction strongly influences the structure's mode of collapse. The third part focuses on the development of efficient solution algorithms for compressible multi-material flow simulations. In these simulations, an unresolved challenge is the computation of advective fluxes across material interfaces that separate drastically different thermodynamic states and relations. A popular class of methods in this regard is to locally construct bimaterial Riemann problems, and to apply their exact solutions in flux computation, such as the one used in the preceding parts of the dissertation. For general equations of state, however, finding the exact solution of a Riemann problem is expensive as it requires nested loops. Multiplied by the large number of Riemann problems constructed during a simulation, the computational cost often becomes prohibitive. This dissertation accelerates the solution of bimaterial Riemann problems without introducing approximations or offline precomputation tasks. The basic idea is to exploit some special properties of the Riemann problem equations, and to recycle previous solutions as much as possible. Following this idea, four acceleration methods are developed. The performance of these acceleration methods is assessed using four example problems that exhibit strong shock waves, large interface deformation, contact of multiple (>2) interfaces, and interaction between gases and condensed matters. For all the problems, the solution of bimaterial Riemann problems is accelerated by 37 to 87 times. As a result, the total cost of advective flux computation, which includes the exact Riemann problem solution at material interfaces and the numerical flux calculation over the entire computational domain, is accelerated by 18 to 81 times. / Doctor of Philosophy / This dissertation focuses on the development and application of numerical methods for solving multiphase fluid-material interaction problems. The first part of this dissertation is motivated by the extraordinary shock-resisting ability of elastomer coating materials (e.g., polyurea) under explosive loading conditions. Their performance, however, highly depends on their dynamic interaction with the underlying structure and the ambient water or air; and the detailed interaction process is still unclear. Therefore, the first part of this dissertation developes a fluid-structure coupled computational framework to certify the application of these materials. In particular, the special material property of the coating material is incorparated into a state-of-the-art fluid-structure coupled computational framework that is able to model large deformation under extreme physical conditions. Then, the developed computational framework is applied to study how a thin-walled aluminum cylinder with polyurea coating responds to various loading conditions. The observed two-way coupling between the structure and the bubble generated in a near-field underwater explosion motivates the next part of this dissertation. The second part of this dissertation investigates the failure (i.e., yielding and collapse) of an underwater thin-walled aluminum cylinder in near-field explosions. As the explosion intensity varies by two orders of magnitude, three different modes of collapse are discovered, including one that appears counterintuitive (i.e., one lobe extending towards the explosive charge), yet has been observed in previous laboratory experiments. Via a detailed analysis of the interaction between the explosion gas bubble, the aluminum cylinder, and the ambient liquid water, this dissertation elucidated the role of bubble dynamics in the structure's different failure behaviors and revealed the transition mechanism between these behaviors. The third part of this dissertation presents efficient solution algorithms for the simulations of compressible multi-material flows. Many problems involving bubbles, droplets, phase transitions, and chemical reactions fall into this category. In these problems, discontinuities in fluid state variables (e.g., density) and material properties arise across the material interfaces, challenging numerical schemes' accuracy and robustness. In this regard, a promising class of methods that emerges in the recent decade is to resolve the exact wave structure at material interfaces, such as the one used in the preceding parts of the dissertation. However, the computational cost of these methods is prohibitive due to the nested loops invoked at every mesh edge along the material interface. To address this issue, the dissertation develops four efficient solution methods, following the idea of exploiting special properties of governing equations and recycling previous solutions. Then, the acceleration effect of these methods is assessed using various challenging multi-material flow problems. In different test cases, significant reduction in computational cost (acceleration of 18 to 81 times) is achieved, without sacrificing solver robustness and solution accuracy.

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In this thesis, we study the entropy stability of the compressible Navier-Stokes model along with a modification of the model. We use the discretization of the inviscid terms with the Ismail-Roe entropy conservative flux. Then, we study entropy stability of the augmentation of viscous, heat and mass diffusion finite difference approximations to the entropy conservative flux. Additionally, we look at different choices of the diffusion coefficient that arise from combining the viscous, heat and mass diffusion terms. Lastly, we present numerical results of the discretizations comparing the effects of the viscous terms on the oscillations near the shock and show that they preserve entropy stability.

The Euler and Navier-Stokes-Fourier equations will be directly expressed as distribution evolution equations, where a new and proper continuum prescription will be derived. These equations of motion will be numerically solved with the development of a new and unique finite element formulation. Out of this framework, the 7D phasetime element has been born. To provide optimal stability, a new quantization procedure is established based on the principles of quantum theory. The entirety of this framework has been coined the "quantized velocity finite element method" (QVFEM). The work performed herein lays the foundational development of what is hoped to become a new paradigm shift in computational fluid dynamics. / Doctor of Philosophy / To model any of the four fundamental states of matter, for practical engineering applications, we must first recognize the complexity of such states. In consequence, a new and novel approach is presented on how to numerically simulate the dynamics of a gas using both the Euler and Navier-Stokes-Fourier equations of continuum mechanics and thermodynamics. In contrast to direct numerical simulation, a statistical mechanical prescription will be given where the equations of motion will be quantized using methods taken from the study of quantum mechanics. This newly developed discretization of the phase space and time, or phasetime, provides optimal stability for compressible flow simulations. From the newly proposed framework, the 7D phasetime element has been born.

We have examined three problems involving steady flows of Navier-Stokes fluids. In each problem non-classical effects are considered. In the first two problems, we consider fluids which have bulk viscosities which are much larger than their shear viscosities. In the last problem, we examine steady supersonic flows of a Bethe-Zel'dovich-Thompson (BZT) fluid over a thin airfoil or turbine blade. BZT fluids are fluids in which the fundamental derivative of gasdynamics changes sign during an isentropic expansion or compression. In the first problem we consider the effects of large bulk viscosity on the structure of the inviscid approximation using the method of matched asymptotic expansions. When the ratio of bulk to shear viscosity is of the order of the square root of the Reynolds number we find that the bulk viscosity effects are important in the first corrections to the conventional boundary layer and outer inviscid flow. At first order the outer flow is found to be frictional, rotational, and non-isentropic for large bulk viscosity fluids. The pressure is found to have first order variations across the boundary layer and the temperature equation is seen to have two additional source terms at first order when the bulk viscosity is large. In the second problem, we consider the reflection of an oblique shock from a laminar flat plate boundary layer. The flow is taken to be two-dimensional, steady, and the gas model is taken to be a perfect gas with constant Prandtl number. The plate is taken to be adiabatic. The full Navier-Stokes equations are solved using a weighted essentially non-oscillatory (WENO) numerical scheme. We show that shock-induced separation can be suppressed once the bulk viscosity is large enough. In the third problem, we solve a quartic Burgers equation to describe the steady, two-dimensional, inviscid supersonic flow field generated by thin airfoils. The Burgers equation is solved using the WENO technique. Phenomena of interest include the partial and complete disintegration of compression shocks, the formation of expansion shocks, and the collision of expansion and compression shocks. / Ph. D.

Noise induced hearing loss is a large concern for the Department of Defense. Personnel on aircraft carriers are exposed to dangerous noise levels of noise from tactical aircraft, causing hearing damage which results in significant costs for medical care and treatment. Additionally, NASA and the FAA have begun to investigate the viability of reintroducing supersonic commercial transport in the United States and one of the largest problems to address is reducing the noise impact of these aircraft on communities. The overarching goal of jet noise research is to optimize noise reduction techniques for supersonic jets. In order to achieve this, a more complete theoretical framework which links the jet boundary conditions to the turbulence production in the jet plume and the far-field radiated noise must be established. The research presented herein was conducted on the hypothesis that introducing thermal non-uniformities into a heated supersonic jet flow can favorably alter the turbulence structure in the jet shear layer, leading to reductions in radiated noise. To investigate the impact of temperature on the turbulence development in the jet, spatially resolved three-component velocity vectors were acquired using particle image velocimetry (PIV) performed on two small-scale perfectly expanded Mach 1.5 jet flows, one with a uniform temperature profile and another containing a geometrically offset temperature non-uniformity. Using the PIV data, the mean velocities, Reynolds stresses, and correlation coefficients were obtained from both jet flows and compared to analyze changes in the mean turbulence field. Small but significant reductions in the shear layer turbulence were observed in the near nozzle region of the thermally offset jet when compared to the uniform jet case. The changes result in a thickening of the shear layer nearest the location of the cold plume which alters the integral length scales of the coherent turbulent structures in the offset jet in a manner consistent with other techniques presented in the literature that reduce jet noise. Applying quadrant analysis, a conditional averaging technique, to the jet turbulence plume revealed changes in the statistical flow field of Reynolds shear stress structures. The changes provide strong evidence of the presence of intermittent stream-wise vortical structures which serve to reduce the spatial correlation levels of turbulence in the thermally offset jet flow when compared to the uniform baseline jet. / Doctor of Philosophy / Increasingly large and powerful engines are required as the mission requirements for tactical aircraft become more advanced. These demands come at the cost of an increased production of noise which is particularly hazardous to crewpersons operating on Navy aircraft carriers during take-off and landing. Noise-induced hearing loss from extended exposure to high noise levels has become a major medical expenditure for the Navy. To address this issue in tactical aircraft engines, the sources of jet plume noise must be reduced, but doing so requires improved understanding of the connections between nozzle boundary conditions, the jet turbulence plume, and the radiated noise while keeping in consideration system constraints and performance requirements. The current study introduces a novel method for controlling supersonic jet noise induced by turbulence through the introduction of an offset non-uniform temperature perturbation at the nozzle mouth. Non-invasive flow measurements were conducted using stereoscopic particle image velocimetry to obtain high-resolution velocity and turbulence data. Analysis of the flow data indicate that an offset reduced temperature plume introduced at the nozzle exit has a first-order effect on the turbulence evolution which result in small, but significant reductions in jet noise levels. The reductions observed are attributed to a disruption in the coherence of the primary noise generating turbulence structures in the jet plume which are associated with the formation of stream-wise vortical structures induced by the cold plume.

Supersonic hot jet noise causes significant hearing impairment to the military workforce and results in substantial cost for medical care and treatment. Detailed insight into the turbulence structure of high-speed jets is central to understanding and controlling jet noise. For this purpose a new instrument based on the Doppler global velocimetry technique has been developed. This instrument is capable of measuring three-component velocity vectors over ex-tended periods of time at mean data-rates of 100 kHz. As a demonstration of the applicability of the time-resolved Doppler global velocimetry (TR-DGV) measurement technique, statistics of three-component velocity measurements, full Reynolds stress tensors and spectra along the stream-wise direction in a cold, supersonic jet at exit Mach number Mj = 1.4 (design Mach number Md = 1.65) are presented. In pursuance of extending the instrument to planar op- eration, a rapid response photomultiplier tube, 64-channel camera is developed. Integrating field programmable gate array-based data acquisition with two-stage amplifiers enables high-speed flow velocimetry at up to 10 MHz. Incor- porating this camera technology into the TR-DGV instrument, an investigation of the perfectly expanded supersonic jet at two total temperature ratios (TTR = 1.6 and TTR = 2.0) was conducted. Fourth-order correlations which have direct impact on the intensity of the acoustic far-field noise as well as convective velocities on the lip line at several stream-wise locations were obtained. Comprehensive maps of the convective velocity and the acoustic Mach number were determined. The spatial and frequency scaling of the eddy convective velocities within the developing shear layer were also investigated. It was found that differences in the radial diffusion of the mean velocity field and the integral eddy convective velocity creates regions of locally high convective Mach numbers after the potential core. This, according to acoustic analogies, leads to high noise radiation efficiency. The spectral scaling of the eddy convec- tive velocity indicates intermittent presence of large-scale turbulence structures, which, coupled with the emergence of Mach wave radiation, may be one of the main driving factors of noise emission observed in heated supersonic jets. / Ph. D.