Direct Numerical Simulation Of Turbulence With Scalar Transfer Around Complex Geometries Using The Immersed Boundary Method And Fully Conservative Higher Order Finite Difference Schemes

Direct Numerical Simulation of Turbulence with Scalar Transfer around Complex Geometries Using the Immersed Boundary Method and Fully Conservative Higher-Order Finite-Difference Schemes.

This book will interest researchers, scientists, engineers and graduate students in many disciplines, who make use of mathematical modeling and computer simulation. Although it represents only a small sample of the research activity on numerical simulations, the book will certainly serve as a valuable tool for researchers interested in getting involved in this multidisciplinary field. It will be useful to encourage further experimental and theoretical researches in the above mentioned areas of numerical simulation.

This book focuses on turbulent flows generated and/or influenced by multiscale/fractal structures. It consists of six chapters which demonstrate, each one in its own way, how such structures and objects can be used to design bespoke turbulence for particular applications and also how they can be used for fundamental studies of turbulent flows.

Direct and Large Eddy Simulation approaches are used to study internal tide generation and turbulence at a model topography. A three dimensional finite difference code using an immersed boundary method is developed to simulate stratified turbulent flows over complex geometry on a Cartesian grid. The thesis is comprised of two phases. In the first phase, a mixed spectral/finite difference code is used to solve the governing equations in generalized coordinates. Direct and large eddy simulations are performed using a body fitted grid (BFG) to study the internal waves generated by the oscillation of a barotropic tide over a model ridge of triangular shape. The objective is to assess the role of nonlinear interactions including turbulence in situations with low tidal excursion number. The criticality parameter, defined as the ratio of the topographic slope to the characteristic slope of the tidal rays, is varied from subcritical to supercritical values. The barotropic tidal forcing is also systematically increased. In laminar flow at low forcing, numerical results of the energy conversion agree well compared to linear theory in subcritical and supercritical cases but not at critical slope angle. In critical and supercritical cases with higher forcing, there are convective overturns, turbulence and significant reduction (as much as 25%) of the radiated wave flux with respect to laminar flow results. The phase dependence of turbulence within a tidal cycle is examined and found to differ substantially between the ridge slope and the ridge top where the beams from the two sides cross. In the second phase, a sharp-interface Immersed Boundary Method (IBM) is developed to simulate high Reynolds number density-stratified turbulent flows in complex geometry. The basic numerical scheme corresponds to a central second-order finite difference method, third-order Runge-Kutta integration for the advective terms and an alternating direction implicit (ADI) scheme for the viscous and diffusive terms. Both direct numerical simulation (DNS) and large eddy simulation (LES) approaches are considered. The focus is on accurate computation of the internal gravity wave field and turbulence near an underwater obstacle in a model problem where a tide oscillates over the obstacle. Methods to enhance the mass conservation and numerical stability of the solver to simulate high Reynolds number flows are discussed. The solver is validated using Direct Numerical Simulations (DNS) of channel flow with and without stratification, and tidal flow over a laboratory-scale (order of few meters) smoothed triangular ridge. The results including baroclinic energy flux, mean flow properties and turbulent kinetic energy agree reasonably well with our previous results obtained using a body-fitted grid (BFG). The deviation of IBM results from BFG results is found to increase with increasing steepness of the topography relative to the internal wave propagation angle. LES is performed on a large scale ridge, of the order of few kilometers in length, at significantly larger Reynolds number. A non-linear drag law is utilized to parameterize turbulent losses due to bottom friction. The large scale problem exhibits qualitatively similar behavior to the laboratory scale problem with some differences: slightly larger intensification of the boundary flow and somewhat higher nondimensional values for conversion, baroclinic wave flux and turbulent kinetic energy. The phasing of wave breaking and turbulence exhibits little difference between small and large scale obstacles. We conclude that IBM is a viable approach to the simulation of internal waves and turbulence in high Reynolds number stratified flows over topography.

This book offers an introduction to the key ideas, basic analysis, and efficient implementation of discontinuous Galerkin finite element methods (DG-FEM) for the solution of partial differential equations. It covers all key theoretical results, including an overview of relevant results from approximation theory, convergence theory for numerical PDE’s, and orthogonal polynomials. Through embedded Matlab codes, coverage discusses and implements the algorithms for a number of classic systems of PDE’s: Maxwell’s equations, Euler equations, incompressible Navier-Stokes equations, and Poisson- and Helmholtz equations.

With applications to climate, technology, and industry, the modeling and numerical simulation of turbulent flows are rich with history and modern relevance. The complexity of the problems that arise in the study of turbulence requires tools from various scientific disciplines, including mathematics, physics, engineering and computer science. Authored by two experts in the area with a long history of collaboration, this monograph provides a current, detailed look at several turbulence models from both the theoretical and numerical perspectives. The k-epsilon, large-eddy simulation and other models are rigorously derived and their performance is analyzed using benchmark simulations for real-world turbulent flows. Mathematical and Numerical Foundations of Turbulence Models and Applications is an ideal reference for students in applied mathematics and engineering, as well as researchers in mathematical and numerical fluid dynamics. It is also a valuable resource for advanced graduate students in fluid dynamics, engineers, physical oceanographers, meteorologists and climatologists.

The book aims to provide the reader with an updated general presentation of multiscale/multiresolution approaches in turbulent flow simulations. All modern approaches (LES, hybrid RANS/LES, DES, SAS) are discussed and recast in a global comprehensive framework. Both theoretical features and practical implementation details are addressed. Some full scale applications are described, to provide the reader with relevant guidelines to facilitate a future use of these methods.