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In this work, we investigate the performance limitations characterizing certain common closure models for nonlinear models of fluid flow. The need for closures arises when for computational reasons first-principles models, such as the Navier-Stokes equations, are replaced with their simplified (filtered) versions such as the Large-Eddy Simulation (LES). In the present work, we focus on a simple model problem based on the 1D Kuramoto-Sivashinsky equation with a Smagorinsky-type eddy-viscosity closure model. The eddy viscosity is assumed to be a function of the state (flow) variable whose optimal functional form is determined in a very general form in the continuous setting. It is found by solving a PDE-constrained optimization problem in which the least-squares error between the output of the LES and the true flow evolution is minimized with respect to the functional form of the eddy viscosity. This problem is solved using a gradient-based technique utilizing a suitable adjoint-based variational data-assimilation approach implemented in the optimize-then-discretize setting using state-of-the-art techniques. The numerical computations are thoroughly validated. The obtained results indicate how the standard Smagorinsky closure model can be refined such that the corresponding LES evolution approximates more accurately the evolution of the original (unfiltered) flow. / Thesis / Master of Science (MSc)

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The accurate and efficient numerical simulation of geophysical flows is of great interest in numerical weather prediction and climate modeling as well as in numerous critical areas and industries, such as agriculture, construction, tourism, transportation, weather-related disaster management, and sustainable energy technologies. Oceanic and atmospheric flows display an enormous range of temporal and spatial scales, from seconds to decades and from centimeters to thousands of kilometers, respectively. Scale interactions, both spatial and temporal, are the dominant feature of all aspects of general circulation models in geophysical fluid dynamics. In this thesis, to decrease the cost for these geophysical flow computations, several types of multiscale methods were systematically developed and tested for a variety of physical settings including barotropic and stratified wind-driven large scale ocean circulation models, decaying and forced two-dimensional turbulence simulations, as well as several benchmark incompressible flow problems in two and three dimensions. The new models proposed here are based on two classes of modern multiscale methods: (i) interpolation based approaches in the context of the multigrid/multiresolution methodologies, and (ii) deconvolution based spatial filtering approaches in the context of large eddy simulation techniques. In the first case, we developed a coarse-grid projection method that uses simple interpolation schemes to go between the two components of the problem, in which the solution algorithms have different levels of complexity. In the second case, the use of approximate deconvolution closure modeling strategies was implemented for large eddy simulations of large-scale turbulent geophysical flows. The numerical assessment of these approaches showed that both the coarse-grid projection and approximate deconvolution methods could represent viable tools for computing more realistic turbulent geophysical flows that provide significant increases in accuracy and computational efficiency over conventional methods. / Ph. D.

Wakes of lifting bodies contain vortex sheets that roll up into strong streamwise vortices. The long time behavior of such vortices depends on the turbulence in the wake and the stability characteristics of the vortices themselves. In the near wake of a rectangular wing the flow field consists of a spiraling wake that winds around a pair of vortex cores. The study of the turbulence structure and life of wing tip vortices is of great importance to air traffic control in congested airports. In this dissertation a computer code is developed for the temporal as well as spatial simulations of trailing vortices. A sixth-order compact finite-difference method is used in the cross plane. The streamwise derivatives are represented either by a Fourier series for temporal simulations (periodic flow) or by a sixth-order compact scheme for spatial simulations. The time marching scheme is a third-order Runge-Kutta method. The code is used to study the nonlinear development of temporal helical instability waves in a trailing vortex. Contours of a passive scalar are used to study the entrainment process that redistributes angular and axial momenta between the core and its surroundings. Such a process leads to quenching of the instability waves in the vortex core. The code is also used to predict the spatial development of mean flow in the wake of a rectangular wing. New treatment of the outflow boundary condition on the pressure is formulated so that a strong streamwise vortex can exit the computational domain without distortion. Temporal large-eddy simulation (LES) is performed to study the development of large scale structures in the wake and their interaction with the tip vortex. A modified MacCormack scheme developed by Gottlieb and Turkel(1976) has been used to solve the LES equations. A model of the initial conditions in the near wake of a rectangular wing is devised to investigate mechanisms of turbulence production in the spiral wake around the core of a tip vortex. The model consists of a streamwise vortex sheet whose strength is found from Prandtl lifting line theory. A Gaussian streamwise velocity profile is superimposed on the field of the vortex sheet. This profile represents the spanwise vorticity. The integrated spanwise vorticity of this profile is zero. A novel feature of this study is that the mean flow contains both streamwise and spanwise vorticity. The model is then used to initialize the flow field for temporal LES of the instabilities of the spanwise vorticity during roll up. The results show that the sinuous mode prevails in the spiral wake around the core. The strength and streamwise length scale of the instability vary along the span because of the continuous variation of the wake thickness due to stretching by the tip vortex. The large scale structures produced by the instability of the spiral wake cause the formation of undulations on the core - consistent with the hypothesis of Devenport et al. (1996). / Ph. D.

Increasing the turbine inlet temperature of gas turbine engines significantly increases their power output and efficiency, but it also increases the likelihood of thermal failure. Internal passages with tiny ribs are typically cast into turbine blades to cool them, and the ability to accurately predict the flow and heat transfer within these channels leads to higher design reliability and prevention of blade failure resulting from local thermal loading. Prediction of the flow through these channels is challenging, however, because the flow is highly turbulent and anisotropic, and the presence of rotational body forces further complicates the flow. Large Eddy Simulations are used to study these flows because of their ability to predict the unsteady flow effects and anisotropic turbulence more reliably than traditional RANS closure models. Calculations in a stationary duct are validated with experiments in the developing flow, fully developed, and 180° bend regions to establish the accuracy and prediction capability of the LES calculations and to aid in understanding the major flow structures encountered in a ribbed duct. It is found that most flow and heat transfer calculations come to within 10-15% of the measurements, typically showing excellent agreement in all comparisons. In the developing flow region, Coriolis effects are found to destabilize turbulence and increase heat transfer along the trailing wall (pressure side), while decreasing leading wall heat transfer by stabilizing turbulence. Coriolis forces improve flow turning in the 180° bend by shifting the shape of the separated recirculation zone at the tip of the dividing wall and increasing the mainstream flow area. In addition, turbulence is attenuated near the leading wall throughout the bend, while Coriolis forces have little effect on trailing wall turbulence in the bend. Introducing and increasing centrifugal buoyancy in the developing flow region increases trailing wall heat transfer monotonically. Along the leading wall, buoyancy increases the size of the recirculation zones, shifting the peak heat transfer to a region upstream of the rib, which decreases heat transfer at low buoyancy parameters but increases it as the buoyancy parameter is increased beyond a value of 0.3. Centrifugal buoyancy in the 180° bend initially decreases the size of the recirculation zone at the tip of the dividing wall, increasing flow area and decreasing flow impingement. At high buoyancy, however, the recirculation zone shifts to the middle of the bend, increasing flow resistance and causing strong flow impingement on the back wall. The Boussinesq approximation is used in the buoyancy calculations, but the accuracy of the approximation comes into question in the presence of large temperature differences. A variable property algorithm is developed to calculate unsteady low speed flows with large density variations resulting from large temperature differences. The algorithm is validated against two test cases: Rayleigh-Bénard convection and Poiseuille-Bénard flow. Finally, design issues in rotating ribbed ducts are considered. The fully developed assumption is discussed with regard to the developing flow region, and controlling the recirculation zone in the 180° bend is considered as a way to determine the blade tip heat transfer and pressure drop across the bend. / Ph. D.