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.

The use of Computational Fluid Dynamics (CFD) tools throughout the engineering industry has become standard. Simulations are used during nearly all steps throughout the life cycle of products including design, production, and testing. Due to their wide range of use, industrial CFD codes are becoming more flexible and easier to use. These commercial codes require robustness, reliability, and efficiency. Consequently, linear eddy viscosity models (LEVM) are used to model turbulence for an increasing number of flow types. LEVM such as k-ε and k-ω provide modeling with little loss of computational efficiency and have proven to be robust. The LEVM that are most common in CFD tools, however, are not adequate for accurate prediction of complex flows. This includes flows with high streamline curvature, strong rotation and separation regions. Unfortunately, due to their ease of use in the commercial CFD tools, the models are used frequently for complex flows. Modifications have been made to LEVM such as k-ε in order to improve modeling, but generally, the modifications have only improved modeling of less complex flows. More advanced LEVM models have been developed using elliptic relaxation equations to help resolve these issues. The ν2-f model was developed to better capture flow physics for complex flows while being applicable to general flows. It is generally considered one of the most accurate LEVMs. It does, however, have issues with stability and robustness. Several improvements have been proposed. One of the most notable is its reformulation into the ζ-f model which offers several improvements while maintaining accurate flow prediction. The model improvement is still limited by being a LEVM. While models, such as differential Reynolds stress models, do exist which are able to capture relevant flow physics in complex flows, modeling difficulties make them impractical for use in a commercial CFD code. Algebraic Reynolds stress models have attempted to bridge this gap with varying levels of success. The models express the Reynolds stress tensor as a function of different higher level tensors. This is the same process used to derive non-linear eddy-viscosity models which add extra high-order terms to the Boussinesq approximation. According to Kassinos and Reynolds, however, this technique is fundamentally flawed. These models fail to capture all relevant information about the turbulence structure. The Reynolds stresses capture information regarding the turbulent componentiality, i.e. velocity components of turbulence. The dimensionality, which carries information regarding the direction of turbulent eddies, is not modeled, however. Kassinos and Reynolds constructed a structure-based model which attempts to capture turbulent componentiality and dimensionality by expressing the Reynolds stress tensor as a function of one-point turbulence structure tensors. Their original model introduced hypothetical turbulence eddies which could be averaged and then used to relate the eddy-axis transport equation to the proper structure tensors. The ideas behind this model were adapted into several different models including the R-D model and the Q-model. These formulations were able to accurately capture the flow physics for many complex flow types especially those with mean rotation. These resulting models, however, were overly complicated for application in commercial CFD codes. These structure-based models later resulted in the development of the algebraic structure based model (ASBM). The ASBM was developed in order to ensure computational efficiency while capturing relevant turbulence physics. The ASBM uses an algebraic model for the eddy statistics which is constructed from the local mean deformation and two turbulent scales. The original turbulent scales used were the turbulent kinetic energy and the large scale vorticity. Although the model was calibrated specifically for use with the turbulent kinetic energy and large scale vorticity transport equations, the algebraic model can be used in conjunction with any scalar transport equations as long as the field distribution of turbulent kinetic energy and turbulence time scale can be obtained. Based on its formulation, the ASBM, used in combination with any scalar transport equations, should be applicable to most commercial CFD codes. The objective of this work was to implement the ζ-f model and ASBM, coupled with k-ε and v2-f, in the commercial CFD solver FLUENT and validate its performance for canonical turbulent flows including a subsonic turbulent flat-plate, S3H4 2D hill, and backward-facing step. Each turbulent flow was evaluated using various turbulence models including Spalart-Allmaras, k-ε, k-ω, k-ω-SST, v2-f, ζ-f and two ASBM formulations and compared against experimental results. The ζ-f model produced improved results for both the flat plate and backward facing step as compared to all two-equation or less turbulence models and showed similar predictive capabilities to the v2-f model. It had difficulties predicting attached flow past the S3H4 2D hill just as the v2-f model. This, however, was expected due to its basis on the v2-f model. The model was also more stable than the v2-f model during calculation of the turbulent flat plate but showed no improvement in robustness for the more complex backward facing step. The semicoupled (linear eddy viscosity model based) v2-f-ASBM’s predictive capabilities were comparable to the two equation models for the turbulent flat plate case. It performed surprisingly well for the backward facing step and matched the experimental data within experimental uncertainty. The model did, however, have problems predicting the S3H4 2D hill just as the with the v2-f model.

Prediction of the near wall region is crucial to the accuracy of turbulent flow computational fluid dynamics (CFD) simulation. However, sufficient near-wall resolution is often prohibitive for high Reynolds number flows with complex geometries, due to high memory and processing requirements. A common approach in these cases is to use wall functions to bridge the region from the first grid node to the wall. This thesis presents an alternative method that relaxes the near wall resolution requirement by solving one dimensional transport equations for velocity and turbulence across a locally defined subgrid contained within wall adjacent grid cells. The addition of the subgrid allows for wall adjacent primary grid sizes to vary arbitrarily from low-Re model sizing (y+ ~ 1) to wall function sizing without significant loss of accuracy or increase in computational cost.

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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.