A thermodynamic property of gases called the fundamental derivative was first proposed by Bethe(1942) and later defined as the dimensionless quantity $\Gamma=\dfrac{c^4}{2v^3}\left(\dfrac{\partial^2 v}{\partial p^2}\right)_s$. The sign of $\Gamma$ reflects the sign of the curvature of the isentrope in the pressure-specific volume plane. The value of $\Gamma$ significantly affects the gas behavior and flow properties. Gases at relatively low pressure away from the critical pressure levels usually have values of $\Gamma$ above 1.0. For an ideal gas, $\Gamma = \dfrac{\gamma +1}{2}$, where $\gamma$ is the ratio of specific heats. Previous studies identified flow behaviors of gases with $\Gamma <0$ that are qualitatively opposite to classical gas dynamic theories based on perfect gas laws. For example, a divergent channel accelerates a subsonic flow and expansion shocks exist for gases with negative $\Gamma$. Although no experimental evidence has yet been found to confirm such non-classical gas flow behaviors, present interests in the use of super-critical heavy gases as well as pure academic curiosity call for more in-depth and definitive studies of such gas flows. A dense gas called $MDM$ is selected as the working fluid in the present work. A region of negative fundamental derivative is found near the critical point using the Van der Waals real gas Equation of State (EoS) for this heavy gas. Contrary to previous studies, the present work considers $\Gamma$ as a local thermodynamic variable instead of a constant in an isentropic flow or across a shock wave. Formulas of the relation of the fundamental derivative to other thermodynamic variables are given. To compare with the ideal gas model, the thermodynamic properties of this dense gas and the gas dynamic behaviors near its critical point are investigated. The conservation laws have been applied to develop the ordinary differential equation system for the quasi-one-dimensional isentropic flow. Since analytical solutions as in the classical theory are no longer possible for the non-ideal gas, numerical simulations are obtained for different upstream conditions. Various seemingly counter-classical gas dynamics flow behaviors are demonstrated. For example, a divergent-convergent nozzle is needed for transonic flow when the gas is within the negative fundamental derivative range. These unconventional gas behaviors are vitally interrelated in a flow of such non-ideal gas as it expands from high pressure to low pressure going through regions of $\Gamma >1$, $0<\Gamma <1$, and $\Gamma <0$ due to changes of its thermodynamic properties in the isentropic expansion process. Specific counter-classical behaviors are identified and discussed in this thesis.

The Dielectric Barrier Discharge (DBD) plasma actuator is a commonly used flow control device. This device is capable of ionizing air using a high-frequency and high-voltage AC power source, creating positive ions and negative electrons. The ions travel in the direction of the external electric field, inducing a jet along the wall. This jet can delay the boundary layer separation, increase the lift-to-drag ratio, and suppress the fluctuations in the wake. The DBD plasma actuator has many advantages, such as ease of installation, fast-acting capabilities, and high energy efficiency. Additionally, it requires no moving mechanical parts, making it one of the most efficient active control methods. While numerous papers have studied the application of the plasma actuator, only a few have explored its fundamental physics and the detailed mechanisms responsible for its flow control effectiveness. This research is divided into two parts. The first part examines the fundamental physics behind the ionization process, investigating how plasma is created by the high-power and high-frequency AC power source. The second part delves into the fundamental physics of plasma control on flat plates and conical bodies. In this research, the plasma effect is simulated by a mathematical model and treated as a source term when coupled with the 2D Navier-Stokes equations. This model has some parameters that require calibration based on experimental results. The first case examines the plasma-induced flow field on a flat plate under duty-cycle control. In this scenario, a single DBD plasma actuator is installed on the surface of a flat plate. The air is initially quiescent and the flow field is solely induced by the plasma actuator. In this case, essential parameters are obtained for the plasma body force model through calibration with available experimental data, thereby providing a fully-developed model. The simplicity of this case makes it perfect for the calibration process. The second case examines the flow field around a square cylinder under plasma control, where a pair of DBD plasma actuators are installed on different parts of the square cylinder. The fully developed plasma model is integrated into the SIMPLE algorithm as a source term. During the simulation, the incoming flow velocity is always set to $1m/s$, and the flow fields at $Re=100$ and $Re=200$ are simulated under three different installation configurations. In the first configuration, a pair of plasma actuators is installed on the front surface of the cylinder, generating two induced jets moving away from each other. In the second configuration, one actuator is placed on the top while another one is installed on the bottom of the cylinder, inducing two jets in the stream wise direction. In the third configuration, a pair of plasma actuators is installed on the rear surface of the cylinder, with two jets moving toward each other. After detailed analysis, we found that the third configuration produces the best flow control results, completely suppressing lift fluctuations and significantly reducing the time-averaged drag coefficient when the plasma body force is strong enough. In the third case, the plasma effect is simulated using a self developed plasma body force model, while the flow field is simulated by solving the Reynolds-Averaged Navier-Stokes (RANS) Equations with the Spalart-Allmaras (SA) one-equation turbulence model. This case is divided into three steps. In the first step, a new plasma body force model tailored for circular surfaces is developed to replicate the experimental flow field around a circular cylinder. In the second step, the flow fields under different steady-state actuation signals are simulated and the results are analyzed in detail. In the third step, the flow fields under unsteady actuations with varying duty-cycle frequencies and duty-cycle ratios are simulated. For both steady and unsteady actuation, the simulation results are compared with available experimental data. In addition to the findings reported in Hui's 2022 experiment, the simulation results unveil three significant new discoveries. First, the friction force on the surface of the circular cylinder responds instantly to the plasma actuation signal, whereas the momentum of the flow within the measurement window exhibits a time delay. Second, the momentum in the cross-stream direction forms an arc-like shape during one duty-cycle period, while the momentum in the streamwise direction remains relatively constant. Third, the time-accurate momentum exhibits only one peak within a duty-cycle period, while the pressure and friction forces exhibit multiple peaks. Furthermore, the magnitude of the pressure force greatly surpasses that of the friction force.

In recent decades, the escalating global energy consumption has increased interest in thermodynamic power cycles, such as the Organic Rankine Cycle (ORC), which are known for their high waste heat recovery efficacy. The fluids in such power cycles have complicated molecular structures and are typically operated near the saturation vapor line and critical point. These gases exhibit similar behaviors to those investigated by Bethe (1942), who established a thermodynamic property known as the fundamental derivative (Γ), which measures the variation of the sound speed of a gas with pressure change during an isentropic process. Gases with Γ values less than 1 are referred to as Bethe-Zel’dovich-Thompson (BZT) gasesand are commonly used as working fluids in the ORC. BZT gases demonstrate a range of qualitatively distinct phenomena compared to conventional gases, especially when Γ < 0. These include expansion shock waves, double sonic shock waves, compression fans, and expansion shock fans. Despite the lack of experimental evidence to confirm such non-classical gas flow behaviors, the increasing interest in using supercritical heavy gases and academic curiosity warrant further in-depth study. A comprehensive literature review of past studies on the fundamental derivative and corresponding non-classical gas behaviors is conducted.

In order to enhance the prediction accuracy of the thermodynamic properties of dense gases and address the existing gap in the availability of precise and accurate equation of state within the Bethe-Zel’dovich-Thompson (BZT) region, this research investigates the potential of machine learning algorithms’ application in the thermodynamics field. By meticulouslyanalyzing the influence of various hyperparameters, an optimized artificial neural network (ANN) model has been developed and subsequently incorporated into our in-house code, which solves the Navier-Stokes equations using a finite-volume method. The successful implementation of the optimized ANN model to calculate the thermodynamics properties demonstrates the potential to significantly decrease the reliance on traditional real gas equations of state for simulation results. This advancement offers a more robust and reliable approach to predicting thermodynamic properties, thereby contributing to the broader field of thermodynamics research.

Besides the state-of-art technology in the thermodynamics part, this study also employs the Van der Waals model to illustrate the existence of negative fundamental derivative regions for dense gases and numerically investigates the unique phenomena of dense gas flow. A numerical solver based on the Jameson-Schmidt-Turkel scheme for dense gas flow is developed, and various counter-classical gas dynamics flow behaviors of real gases in different Γ regimes are emonstrated through selected cases. The simulation results of dense gas flow across various geometries are presented and analyzed, revealing intricate wave fields.

In conjunction with the traditional density-based solvers, this study also proposes an innovative pressure-based solution methodology for compressible flow, specifically designed to address the stiffness issue encountered near the critical point due to the heightened sensitivity of pressure with respect to density. A comprehensive derivation of the novel approach is presented, emphasizing its theoretical underpinnings and practical applications.

Due to the dramatic increase in global energy consumption in the past decade, thermodynamic power cycles such as Oraganic Rakine Cycle and Supercritical Carbon Dioxide Cycle have drawn lots of attention recently because of their high efficiency in waste heat recovery. The fluids in these power cycles usually have complex molecular structures. They operate near the saturation vapor line and critical point, where the gas behavior significantly deviates from the ideal gas behavior. Objective of this dissertation is to numerically investigate and understand the peculiar gas behavior in this region.The fundamental derivative of gas dynamics is a measurement of the variation of the speed of sound of a gas with respect to pressure in an isentropic process. The van der Waals model is used to confirm the existence of negative fundamental derivative regions for dense gas. Regions of Γ < 0, 0 < Γ < 1 and 1 < Γ are identified and mapped out in the p- v diagram. Jump relations are investigated in three regions of Γ, i, e.Γ > 1, 0 < Γ < 1 and Γ < 0 respectively. The non-monotone dependence of the speed of sound along a shock adiabat is demonstrated to increase the sound speed across normal shock waves in the region where 1 > Γ > 0. Expansion shock solution is obtained in the region where Γ < 0. Double sonic shock wave are confirmed admissible where fluid can be expanded and accelerated through a discontinuity with Mach number being unity before and after the discontinuity. Unconventional gas behaviors of the isentropic quasi-1D flow have been investigated systematically and in depth for dense gas. Analysis indicates that dense gas behavior in one dimensional nozzle is directly related to the value of Γ. For example, when Γ < 0, the divergent-convergent nozzle is needed to accelerate flow from subsonic to supersonic. A real gas numerical solver using Jameson-Schmidt-Turkel scheme is developed. Simulation result of dense gas over compression ramp and expansion ramp shows agreement with the analytical solution. It confirms the existence of complicated wave field such as expansion shock and expansion shock-fan. Results of unsteady shock tube simulation show that shock tube can produce different wave fields depending on the initial conditions. More results of dense gas over corners and arc surface are presented and discussed in details.

A numerical simulation was performed for a 2D plasma flow control of flow around a circular cylinder at Reynolds number 6.6e3.

The plasma actuator model proposed by Suzen et al. was used. Poisson equations for the electric potential and the charge density were solved to compute the body force field. The body force generated by the plasma actuator model was incorporated into the flow solver as source terms for the momentum equations.

The simulation with plasma actuators demonstrated a smaller wake and reduced vortex shedding behind the cylinder compared to the reference case without the actuators. The influence of the locations of the plasma actuators was investigated with a parametric study. The results indicate that the effectiveness of the plasma actuation is sensitive to the actuator locations.

This dissertation proposes numerical methods for the Euler and Navier-Stokes equations with spectral discretization in time and a fast space-time coupled LU-SGS (ST-LU-SGS) method for solving the resultant implicit equations. Firstly, the Fourier time spectral method is studied for periodic problems with test cases. The problem of non-symmetric solutions for symmetric periodic flow problems, caused by odd numbers of intervals in a period, is discovered and discussed in detail. The requirement of ensuring symmetric solution is proposed. In problems where frequency is not known a priori, a new frequency search approach based on Fourier analysis of the lift coefficient is proposed to work with the time spectral method. Computational results show that initial guesses of the frequency far away from the exact value can be used if the new approach is applied before employing a gradient based method. A new Chebyshev time spectral method is proposed to solve non-periodic unsteady problems and is validated by test cases. Computational results show that this method is very efficient in simulating both periodic and non-periodic unsteady flows, especially the non-periodic problems.

The use of Fourier or Chebyshev spectral discretization in time results in implicit equations in time marching. Explicit Runge-Kutta methods have often been used to solve such implicit system of equations through the use of the dual-time stepping algorithm. Such methods are, however, slow despite the use of acceleration schemes such as implicit residual smoothing and multigrid. We propose a new space-time LU-SGS (ST-LU-SGS) implicit scheme for both the Fourier and Chebyshev time spectral methods. In this scheme, the time domain is regarded as one additional dimension in space. Computational experiments show that this new scheme is faster than the explicit Runge-Kutta solver. For Navier-Stokes flow test cases, computations using the ST-LU-SGS implicit scheme is over ten times faster than the explicit Runge-Kutta solver. The ST-LU-SGS implicit scheme also works very well with the proposed frequency search approach. The ST-LU-SGS scheme is as efficient as the Block-Jacobi implicit algorithm and is more robust than the Block-Jacobi implicit algorithm. The proposed ST-LU-SGS scheme works for problems with either low frequency or high frequency while the Block-Jacobi implicit algorithm fails for high frequency flow problems.

A Navier-Stokes Computational Fluid Dynamics (CFD) code is coupled with a Computa-

tional Structural Dynamics (CSD) code to study the flutter boundary of the NACA64A010

airfoil using Isogai’s structural model in transonic conditions. This model simulates aeroelas-

tic conditions on a sweptback wing. A well-known feature, only present in the inviscid flutter

boundary of this airfoil, is the existence of multiple flutter points for a fixed freestream Mach

number. The fully-turbulent flutter boundary has not been studied by many researchers us-

ing a Reynolds-Averaged Navier-Stokes approach. In the present study, the fully-turbulent

flutter boundary reveals the existence of multiple equilibrium positions for a narrow range

of flight conditions. The system moves away from the initial equilibrium position, finding a

new set of equilibrium points and oscillating around it. This new set of equilibrium points

reveals as stable or unstable for different structural properties of the wing.

We then proceed to study the effect of turbulent transition on flutter boundary. A laminar-

to-turbulent transition model is implemented in the CFD code and validated. The effect of

using a free-transition CFD code vs. a fully-turbulent approach is evaluated on three airfoils

with different characteristics for subsonic and transonic conditions. While free-transition

does not affect the pressure distribution at subsonic conditions, the transonic simulations

reveal a change in the shock-wave position when laminar-turbulent effects are included. The

effect of transition on the flutter boundary of the NACA64A010 airfoil at transonic conditions

is then investigated. A comparison between the free-transition, inviscid and fully-turbulent

flutter boundaries reveals similarities between the inviscid and free-transition elastic re-

sponses. Those similarities are due to the shift in the fully-turbulent shock-wave position,

when accounting for free-transition effects, moving closer to the inviscid shock location.

This thesis explores linear and non-linear computational methods for solving unsteady flow. The eventual goal is to apply these methods to two-dimensional and three-dimensional flutter predictions. In this study the quasi-one-dimensional nozzle is used as a framework for understanding these methods and their limitations. Subsonic and transonic cases are explored as the back-pressure is forced to oscillate with known amplitude and frequency. A steady harmonic approach is used to solve this unsteady problem for which perturbations are said to be small in comparison to the mean flow. The use of a linearized Euler equations (LEE) scheme is good at capturing the flow characteristics but is limited by accuracy to relatively small amplitude perturbations. The introduction of time-averaged second-order terms in the Non-Linear Harmonic (NLH) method means that a better approximation of the mean-valued solution, upon which the linearization is based, can be made. The nonlinear time-accurate Euler solutions are used for comparison and to establish the regimes of unsteadiness for which these schemes fails. The usefulness of the LEE and NLH methods lie in the gains in computational efficiency over the full equations.

The first part of this dissertation explores harmonic methods for both one-dimensional and quasi-one-dimensional flows. The focus is placed on the flow response to oscillating back-pressure for subsonic and transonic conditions. This study includes insight as to the applicability of these methods when resonance of the flow field is expected and estimates of the errors introduced when applying a linearization about a steady-state non-linear flow. The second part explores the unsteady aerodynamics and performance of NASA Rotor 67 when time-varying inlet-distortions are used. A Detached-Eddy Simulation is used to model the flow physics. Within this study, the impact of the inlet distortions are presented based on the unsteady aerodynamics and performance metrics as well as the tonal content of pressure fluctuations within the rotor passage.