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.

Computational fluid dynamics (CFD) methods and schemes have been evolving at a rate that significantly outpaces the equipment needed to readily utilize them at scale. This lack of computational resources has resulted in an increased reliance on turbulence models and the need to know where turbulence models do well, where they do poorly, and where/how they can be improved upon. The BeVERLI Hill experiments aim to address this issue by providing experimental data that achieves a completeness level of three, which has never been done for this type of project. The experimental data collected is studied along side computational results from CFD solvers in order to help address and answer these questions. This paper provides an overview of the current computational status of the BeVERLI Hill project at Virginia Tech. The computational grids used for the analyses are presented such that the reader can gain an appreciation for the modeling techniques and methods being implemented. An analysis of the numerical error associated with the computational results is presented to provide confidence in the results obtained. An in-depth analysis will be presented that shows the results for the various grid levels that are being utilized to determine any grid based effects that are occurring within the solutions. Then, an analysis of the influence of the Reynolds numbers being run is shown. An investigation into the differences between the two different solvers being utilized, SENSEI and Fluent, is shown. An analysis of the effects on the solutions due to numerical limiters is presented to assist in increasing the computational efficiency of the workflow while not adversely affecting the results. Finally, an analysis of the differences between the two turbulence models being utilized is presented. Computational results are compared to available experimentally obtained data to further motivate and identify flow features. / Master of Science / An analysis has been done with high-fidelity computational fluid dynamic solvers that are utilized in order to solve for the flow over a three-dimensional bump called BeVERLI. An analysis is provided that discusses the use of different computational meshes, solvers, turbulence models, and numerical limiters within the computational tools to characterize the flow over the bump. An analysis of the estimated amount of numerical error within the solutions is provided along with a comparison to experimentally obtained data.

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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