The goal of this dissertation is to consolidate the theoretical foundation of variable-resolution (VR) methods in general and the partially-averaged Navier-Stokes (PANS) approach in particular. The accurate simulation of complex turbulent flows remains an outstanding challenge in modern computational fluid dynamics. High- fidelity approaches such as direct numerical simulations (DNS) and large-eddy simulation (LES) are not typically feasible for complex engineering simulations with cur- rent computational technologies. Low-fidelity approaches such as Reynolds-averaged Navier-Stokes (RANS), although widely used, are inherently inadequate for turbulent flows with complex flow features. VR bridging methods fill the gap between DNS and RANS by allowing a tunable degree of resolution ranging from RANS to DNS. While the utility of VR methods is well established, the mathematical foundations and physical characterization require further development. This dissertation focuses on the physical attributes of fluctuations in partially-resolved simulations of turbulence. The specific objectives are to: (i) establish a framework for assessing the physical fidelity of VR methods to examine PANS fluctuations; (ii) investigate PANS simulations subject to multiple resolution changes; (iii) examine turbulent transport closure modeling for partially-resolved fields; (iv) examine the effect of filter control parameters in the limit of spectral cut-off in the dissipative region; and (v) validate low-Reynolds number corrections with RANS for eventual implementation with PANS. While the validation methods are carried out in the context of PANS, they are considered appropriate for all VR bridging methods. The key findings of this dissertation are summarized as follows. The Kolmogorov hypotheses are suitably adapted to describe fluctuations of partially-resolved turbulence fields, and the PANS partially-resolved field is physically consistent with the adapted Kolmogorov hypotheses. PANS adequately recovers the correct energetics in instances of multiple resolution changes. Scaling arguments are used to determine the correct transport closure model for a partially-resolved field in a boundary layer. The need to modify the fε filter control parameter for cut-off in the dissipation range is highlighted. A low-Reynolds number near-wall correction was evaluated on a RANS model with the intent of adapting to it VR methods. Overall, PANS shows promise as a theoretically sound modeling approach, and this work lays the foundation for future PANS investigations.

Over the past 25 years, advances in the field of turbulence modeling have been made in an effort to resolve more scales, preserving unsteadiness within a flow. In this research two hybrid models, Scale-Adaptive Simulation (SAS) and Stress-Blended Eddy Simulation (SBES) are implemented in solving the highly unsteady flow over a rotating arrangement of three cylinders. Results are compared to those from wind tunnel experiments carried out at North Dakota State University. Both models show close agreement with first and second order turbulence quantities, and SBES shows much greater flow structure detail due to its ability to resolve smaller scales. The Strouhal number for the flow is found to be a function of the rotational speed of the arrangement with von Karman-like structures resulting from each cylinder's wake over a full rotation. SAS shows a constant computational cost as Re increases while the SBES's computational cost increases relatively linearly.

Traditional methods of closing the Boussinesq-based Reynolds-averaged Navier-Stokes equations are considered, and suggestions for improving two-equation turbulence models are made. The traditional smooth-wall boundary conditions are shown to be incorrect, and the correct boundary conditions are provided along with sample solutions to traditional models. The correct boundary condition at a smooth wall for dissipation-based turbulence models is that which forces both the turbulent kinetic energy and its first derivative to zero. Foundations for an energy-vorticity model suggested by Phillips are presented along with the near-smooth-wall behavior of the model. These results show that at a perfectly smooth wall, the turbulent kinetic energy may approach the wall at a higher order than is generally accepted. The foundations of this model are used in the development of a k-λ model for fully rough pipe flow. Closure coefficients for the model are developed through gradient-based optimization techniques. Results of the model are compared to results from the Wilcox 1998 and 2006 k-ω models as well as four eddy-viscosity models. The results show that the Phillips k-λ model is much more accurate than other models for predicting the relationship between Reynolds number and friction factor for fully rough pipe flow. However, the velocity profiles resulting from the model deviate noticeably from the law of the wall.

Solving the Navier-Stokes equations using direct numerical simulation (DNS) is computationally impractical, especially at high Reynolds numbers. Recent technological advances in supercomputing have paved the way for Large Eddy Simulations (LES) to circumvent this problem by resolving large scale turbulence motions and modeling only the small (subgrid) scales. However, LES modeling still requires advanced knowledge of the turbulence and LES models are currently very simplistic. Because of this, there has been considerable interest in hybrid turbulence models, which can perform either Reynolds Averaged Navier-Stokes (RANS) modeling or Large Eddy Simulation (LES). The self-adapting model presented is fundamentally different from prior LES models and these current hybrid models in that it achieves a completely natural evolution from RANS to LES to (with enough mesh resolution) DNS. A modified k/e model and a Reynolds stress transport model is implemented in this manner and is compared to DNS data of isotropic decaying turbulence. The results indicate that this modeling approach is practical and efficient. In addition, this approach is extensible and not restricted to a particular (RANS) transport equation.

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The advent of advanced vehicle concepts involving Urban Air Mobility (UAM) and small Unmanned Aerial Systems (sUAS) has brought about a new class of rotorcraft technology which operate predominantly in low-Reynolds ($Re$) number regimes. In such regimes, the flow experiences complex boundary layer phenomena like laminar separation, flow transition and reattachment. These effects are known to greatly alter the flow at and near the rotor wall, influencing its aerodynamic performance as well as the noise generated. Capturing these effects in our computational models is necessary to further our understanding of rotor aerodynamics and acoustics. The current study has introduced a novel RANS transition turbulence model, SSG/LRR-$\omega$-$\gamma$, that is capable of modeling different modes of transition involving natural, bypass, separation-induced and crossflow transition. The model framework uses a Reynolds stress transport model, SSG/LRR-$\omega$, as the base turbulence formulation and is coupled with Menter's $\gamma$ transition model. It was validated using a number of canonical cases that exhibited different transition mechanisms and the model performed equivalently or better than existing state-of-the-art transition models. It is worthy to note that the proposed model was able to perform well in three-dimensional flows, demonstrated using the case of a prolate spheroid. This underscores the capability of Reynolds stress models to accurately capture complex flow curvatures, improving upon the capabilities of linear eddy viscosity models. The transition model, integrated into OpenFOAM, was then employed to analyze two different UAV propellers. The rotor flow was examined using a URANS simulation with an overset grid. The objective was twofold: firstly, to validate the predictions generated by the proposed model for low-Reynolds number (low-$Re$) rotors, and secondly, to evaluate its effectiveness across a range of operating conditions. Comparisons were drawn against established fully turbulent and transition models. The analysis showed that transition models in general tended to be consistent in their predictions and less sensitive to changing operating conditions when compared to fully turbulent models. They also demonstrated the ability to accurately predict the mechanisms leading to separation and transition. Further, the proposed transition model demonstrated superior capability in capturing detailed flow features, particularly in the wake, compared to other fully turbulent and transition models, which is attributed to its Galilean invariant framework. To leverage the boundary layer information obtained from the proposed model, a semi-empirical broadband noise prediction method was implemented. This method utilized boundary layer data predicted by URANS simulations to estimate blade self-noise. An evaluation of the fully turbulent $k$-$\omega$ SST model and the proposed transition model revealed that both exhibited reasonable accuracy at lower rotor advance ratios. However, the transition model performed better at higher advance ratios. It was also observed that CFD-based approaches provided superior prediction accuracy compared to lower-fidelity aerodynamic models in the context of blade self-noise prediction Finally, the proposed aerodynamic and acoustic computational framework was applied to a design case study of swept propellers to understand the advantages of blade sweep on rotor aerodynamics and noise. A qualitative analysis of the flow suggested that the swept rotor exhibited lower levels of blade wake interaction compared to the unswept geometry, in line with the experimental observations. / Doctor of Philosophy / Advanced vehicle concepts such as air taxis for Urban Air Mobility (UAM) and other multi-copter applications like drone delivery, reconnaissance, etc. are emerging sectors in aviation that have garnered great industrial as well as academic interest. However, since these vehicles are expected to fly at low altitudes within urban settings, noise mitigation is of particular interest to improve their public acceptance. The vehicle configurations in these applications predominantly comprise of rotorcraft which operate at low Reynolds ($Re$) numbers and tip speeds. These operating conditions introduce complex phenomena like flow transition and separation within the boundary layer that significantly alter their aerodynamic as well as aeroacoustic performance. The current work proposes a novel transition turbulence model that improves prediction of these complex boundary layer mechanisms in low-$Re$ propellers compared to the state-of-the-art. Furthermore, this work establishes a fast broadband noise prediction method by leveraging the detailed flow data from the transition model. The focus of this method is on modeling those propeller noise sources that are directly influenced by the aforementioned boundary layer phenomena (blade self-noise). The noise prediction study revealed that transition models yield consistent predictions across different operating conditions. Finally, a brief design study is conducted using the proposed aerodynamic and acoustic framework to assess the flow dynamics and possible noise mitigation capabilities of a swept propeller.

The design method for marine propulsor (propeller/stator) is presented for an autonomous underwater vehicle (AUV) that operates at a very high loading condition. The design method is applied to Virginia Tech Dragon AUV. It is based on the parametric geometry definition for the propulsor, use of high-fidelity CFD RANSE solver with the transition model, construction of the surrogate model, and multi-objective genetic optimization algorithm. The CFD model is validated using the paint pattern visualization on the surface of the propeller for an open propeller at model scale. The CFD model is then applied to study hydrodynamics of ducted propellers such as forces and moments, tip leakage vortex, leading-edge flow separation, and counter-rotating vortices formed at the duct trailing edge. The effect of variation of thickness for stator blades and different approaches for modeling the postswirl stator is presented. The field trials for Dragon AUV shows that there is a good correlation between expected and achieved design speed under tow condition with the designed base propulsor. The marine propulsor design is further improved with an objective to maximize the propulsive efficiency and minimize the rolling of AUV. The stator is found to eliminate the swirl component of velocity present in the wake of the propeller to the maximum extent. The propulsor designed using this method (surrogate-based optimization) is demonstrated to have an improved torque balance characteristic with a slight improvement in efficiency than the base propulsor design. / Master of Science / The propulsion system is the critical design element for an AUV, especially if it is towing a large payload. The propulsor for towing AUVs has to provide a very large thrust and hence the propulsor is highly loaded. The propeller has to rotate at very high speed to produce the required thrust and is likely to cavitate at this high speed. Also at this high loading condition, the maximum ideal efficiency of the propulsor is very less. Another challenge is the induced torque from the propeller on AUV that can cause the rolling of an AUV which is undesirable. This problem can be addressed by installing the stator behind the propeller that will produce torque in the opposite direction of the propeller torque. In this work, we present a design methodology for marine propulsor (propeller/stator) that can be used in AUV towing a large payload. The propulsor designed using this method has improved torque characteristics and has the efficiency close to 80 % of the ideal efficiency of ducted propeller at that loading condition.

Computational fluid dynamics methods have seen an increasing role in aerodynamic analysis since their first implementation. However, there are several major limitations is these methods of analysis, especially in the area of modeling separated flow. There exists a large demand for high-fidelity experimental data for turbulence modeling validation. Virginia Tech has joined NASA in a cooperative project to design and perform an experiment in the Virginia Tech Stability Wind Tunnel with the purpose of providing a benchmark set of data for the turbulence modeling community for the flow over a three-dimensional bump. This process requires thorough risk mitigation and analysis of potential flow sensitivities. The current study investigates several aspects of the experimental design through the use of several computational fluid dynamics codes. An emphasis is given to boundary condition matching and uncertainty quantification, as well as sensitivities of the flow features to Reynolds number and inflow conditions. Solutions are computed for two different RANS turbulence models, using two different finite-volume CFD codes. Boundary layer inflow parameters are studied as well as pressure and skin friction distribution on the bump surface. The shape and extent of separation are compared for the various solutions. Pressure distributions are compared to available experimental data for two different Reynolds numbers. / Master of Science / Computational fluid dynamics (CFD) methods have seen an increasing role in engineering analysis since their first implementation. However, there are several major limitations is these methods of analysis, especially in the area of modeling of several common aerodynamic phenomena such as flow separation. This motivates the need for high fidelity experimental data to be used for validating computational models. This study is meant to support the design of an experiment being cooperatively developed by NASA and Virginia Tech to provide validation data for turbulence modeling. Computational tools can be used in the experimental design process to mitigate potential experimental risks, investigate flow sensitivities, and inform decisions about instrumentation. Here, we will use CFD solutions to identify risks associated with the current experimental design and investigate their sensitivity to incoming flow conditions and Reynolds number. Numerical error estimation and uncertainty quantification is performed. A method for matching experimental inflow conditions is proposed, validated, and implemented. CFD data is also compared to experimental data. Comparisons are also made between different models and solvers.