Modeling Shallow Water Flows on General Terrains

A formulation of the shallow water equations adapted to general complex terrains is proposed. Its derivation starts from the observation that the typical approach of depth integrating the Navier-Stokes equations along the direction of gravity forces is not exact in the general case of a tilted curved bottom. We claim that an integration path that better adapts to the shallow water hypotheses follows the "cross-flow" surface, i.e., a surface that is normal to the velocity field at any point of the domain. Because of the implicitness of this definition, we approximate this "cross-flow" path by performing depth integration along a local direction normal to the bottom surface, and propose a rigorous derivation of this approximation and its numerical solution as an essential step for the future development of the full "cross-flow" integration procedure. We start by defining a local coordinate system, anchored on the bottom surface to derive a covariant form of the Navier-Stokes equations. Depth integration along the local normals yields a covariant version of the shallow water equations, which is characterized by flux functions and source terms that vary in space because of the surface metric coefficients and related derivatives. The proposed model is discretized with a first order FORCE-type Godunov Finite Volume scheme that allows implementation of spatially variable fluxes. We investigate the validity of our SW model and the effects of the bottom geometry by means of three synthetic test cases that exhibit non negligible slopes and surface curvatures. The results show the importance of taking into consideration bottom geometry even for relatively mild and slowly varying curvatures

The rise of fully turbulent flow

Over a century of research into the origin of turbulence in wallbounded shear flows has resulted in a puzzling picture in which turbulence appears in a variety of different states competing with laminar background flow. At slightly higher speeds the situation changes distinctly and the entire flow is turbulent. Neither the origin of the different states encountered during transition, nor their front dynamics, let alone the transformation to full turbulence could be explained to date. Combining experiments, theory and computer simulations here we uncover the bifurcation scenario organising the route to fully turbulent pipe flow and explain the front dynamics of the different states encountered in the process. Key to resolving this problem is the interpretation of the flow as a bistable system with nonlinear propagation (advection) of turbulent fronts. These findings bridge the gap between our understanding of the onset of turbulence and fully turbulent flows.Comment: 31 pages, 9 figure

Optimal control for studying wave energy in hydraulic systems

A class of novel models for water waves induced by elastic deformation in the topography is developed and analyzed. The depth-averaged shallow water equations including friction terms for the water free-surface and the well-known second-order elastostatic formulation for the bed deformation have been implemented. Friction forces and water hydrostatic pressure distribution are also accounted for in this model. At the interface between the water ow and the bed topography, transfer conditions are implemented. Furthermore, a hybrid nite element/nite volume method for solving free-surface run-up ow problems over deformable beds has been proposed. The deformations in the topography have been generated by a localized force which causes propagations of the water waves with dierent amplitudes and frequencies. Two dierent methods have been proposed for the transfer of informations through the interface. The rst one is the two-mesh procedure; in this method a proper interpolation has been implemented to transfer the data between the surface nodes and the control volumes using uniform nite volume meshes. In the second method, and to avoid the interpolation at the interface, a nite volume method using non-uniform meshes has been implemented. When the shallow water waves approach the coastline they begin to transform as they enter shallow water regime. As each wave begins to experience the seabed, both run-up and overtopping occur. To solve for this, a class of stable, accurate and simple numerical model for moving wet/dry fronts in shallow water equations using the parametrization concept and the point-wise Riemann solver has been proposed. Many parameters of shallow water equations are subject to uncertainties to the inherit randomness of natural processes. To incorporate uncertain parameters into the stochastic shallow water equations, the stochastic properties of dierent parameters that are considered uncertain, namely in ow boundary condition, the bed friction coecients and the domain topography are added to the system. Development of accurate and ecient tools for uncertainty quantication in shallow water ows has been proposed and carefully examined for single-layer, two-layer - nite volume models. To further quantify the uncertainty in shallow water ows the proposed methods have been extended to multi-layer shallow water ows with mass exchange terms subject to stochastic topography, uncertain friction and viscosity coecients. Several test examples and well-established benchmark problems have been used to assess the numerical performance of the proposed models and methods. Comparisons to experimental measurements have also been carried out in this thesis. Finally, an optimal control technique for bed reconstruction has been presented as in many engineering applications this information is not entirely provided

Robust finite volume schemes for 2D shallow water models. Application to flood plain dynamics

This study proposes original combinations of higher order Godunov type finite volume schemes and time discretization schemes for the 2d shallow water equations, leading to fully second-order accuracy with well-balanced property. Also accuracy, positiveness and stability properties in presence of dynamic wet/dry fronts is demonstrated. The test cases are the classical ones plus extra new ones representing the geophysical flow features and difficulties

River antidunes and bars: new models and nonmodal analysis

The first part of this thesis deals with river alternate bars. Fluvial bars are regular widespread bed forms that are characterized by vertical and transversal scales which are comparable with the stream depth and width, respectively. Although well-established linear and weakly nonlinear stability analysis have already been performed, no non modal analysis has been proposed yet. We here demonstrate the remarkable non normality of the operator that governs bar dynamics in large regions of the parameter space in fair agreement with our tests in flume experiments. This entails the occurrence of dramatic transient growths in the evolution of bed perturbations. Such algebraic growths suggest a novel explanation, through a purely linear process, of the progressive increase in the dominant bar wavelength that is observed in flume experiments and real rivers during bar inception. The second part of this thesis deals with river antidunes. A supercritical free-surface turbulent stream flowing over an erodible bottom can generate a characteristic pattern of upstream migrating bedforms known as antidunes. This morphological instability, which is quite common in fluvial environments, has attracted speculative and applicative interests, and has always been modelled in 2D or 3D mathematical frameworks. However, in this work we demonstrate that antidune instability can be described by means of a suitable one-dimensional model that couples the Dressler equations to a mechanistic model of the sediment particle deposition/entrainment. The results of the linear stability analysis match the experimental data very well, both for the instability region and the dominant wavelength. The analytical tractability of the 1D modeling allows us (1) to elucidate the key physical processes which drive antidune instability, (2) to show the secondary role played by sediment inertia, (3) to obtain the dispersion relation in explicit form, and (4) to demonstrate the absolute nature of antidune instabilit

River antidunes and bars: new models and nonmodal analysis

The first part of this thesis deals with river alternate bars. Fluvial bars are regular widespread bed forms that are characterized by vertical and transversal scales which are comparable with the stream depth and width, respectively. Although well-established linear and weakly nonlinear stability analysis have already been performed, no non modal analysis has been proposed yet. We here demonstrate the remarkable non normality of the operator that governs bar dynamics in large regions of the parameter space in fair agreement with our tests in flume experiments. This entails the occurrence of dramatic transient growths in the evolution of bed perturbations. Such algebraic growths suggest a novel explanation, through a purely linear process, of the progressive increase in the dominant bar wavelength that is observed in flume experiments and real rivers during bar inception. The second part of this thesis deals with river antidunes. A supercritical free-surface turbulent stream flowing over an erodible bottom can generate a characteristic pattern of upstream migrating bedforms known as antidunes. This morphological instability, which is quite common in fluvial environments, has attracted speculative and applicative interests, and has always been modelled in 2D or 3D mathematical frameworks. However, in this work we demonstrate that antidune instability can be described by means of a suitable one-dimensional model that couples the Dressler equations to a mechanistic model of the sediment particle deposition/entrainment. The results of the linear stability analysis match the experimental data very well, both for the instability region and the dominant wavelength. The analytical tractability of the 1D modeling allows us (1) to elucidate the key physical processes which drive antidune instability, (2) to show the secondary role played by sediment inertia, (3) to obtain the dispersion relation in explicit form, and (4) to demonstrate the absolute nature of antidune instability

On the numerical simulation of compressible flows

In this thesis, numerical tools to simulate compressible flows in a wide range of situations are presented. It is intended to represent a step forward in the scientific research of the numerical simulation of compressible flows, with special emphasis on turbulent flows with shock wave-boundary-layer and vortex interactions. From an academic point of view, this thesis represents years of study and research by the author. It is intended to reflect the knowledge and skills acquired throughout the years that at the end demonstrate the author’s capability of conducting a scientific research, from the beginning to the end, present valuable genuine results, and potentially explore the possibility of real world applications with tangible social and economic benefits. Some of the applications that can take advantage of this thesis are: marine and offshore engineering, combustion in engines or weather forecast, aerodynamics (automotive and aerospace industry), biomedical applications and many others. Nevertheless, the present work is framed in the field of compressible aerodynamics and gas combustion with a clear target: aerial transportation and engine technology. The presented tools allow for studies on sonic boom, drag, noise and emissions reduction by means of geometrical design and flow control techniques on subsonic, transonic and supersonic aerodynamic elements such as wings, airframes or engines. Results of such studies can derive in new and ecologically more respectful, quieter vehicles with less fuel consumption and structural weight reduction. We start discussing the motivation for this thesis in chapter one, which is placed into the upcoming second generation of supersonic aircraft that surely will be flying the skies in no more than 20 years. Then, compressible flows are defined and the equations of motion and their mathematical properties are presented. Navier Stokes equations arise from conservation laws, and the hyperbolic properties of the Euler equations will be used to develop numerical schemes. Chapter two is focused on the numerical simulation with Finite Volumes techniques of the compressible Navier-Stokes equations. Numerical schemes commonly found in the literature are presented, and a unique hybrid-scheme is developed that is able to accurately predict turbulent flows in all the compressible regimens (subsonic, transonic and supersonic). The scheme is applied on the flow around a NACA0012 airfoil at several Mach numbers, showing its ability to be used as a design tool in order to reduce drag or sonic boom, for example. At subsonic regimens, results show excellent agreement with reference data, which allowed the study of the same case at transonic conditions. We were able to observe the buffet phenomenon on the airfoil, which consists of shock-waves forming and disappearing, causing a dramatic loss of aerodynamic performance in a highly unsteady process. To perform a numerical simulation, however, boundary conditions are also required in addition to numerical schemes. A new set of boundary conditions is introduced in chapter three. They are developed for three-dimensional turbulent flows with or without shocks. They are tested in order to assess its suitability. Results show good performance for three-dimensional turbulent flows with additional advantages with respect traditional boundary conditions formulations. Unfortunately, compressible flows usually require high amounts of computational power to its simulation. High speeds and low viscosity result in very thin boundary layers and small turbulent structures. The grid required in order to capture this flow structures accurately often results in unfeasible simulations. This fact motivates the use of turbulent models and wall models in order to overcome this restriction. Turbulent models are discussed in chapter four. The Reynolds-Averaged Navier Stokes (RANS) approach is compared with Large-Eddy Simulation (LES) with and without wall modeling (WMLES). A transonic diffuser is simulated in order to evaluate its performance. Results showed the ability of RANS methods to capture shock-wave positions accurately, but failing in the detached part of the flow. LES, on the other hand, was not able to reproduce shock-waves positions accurately due to the lack of precision on the shock wave-boundary-layer interaction (SBLI). The use of a wall model, nevertheless, allowed to overcome this issue, resulting in an accurate method to capture shock-waves and also flow separation. More research on WMLES is encouraged for future studies on SBLIs, since they allow three-dimensional unsteady studies with feasible levels of computational requirements. With all these tools, we are able to solve at this point any problem concerned with the aerodynamic design of high-speed vehicles which were identified in previous paragraphs. Finally, multi-component flows are discussed in chapter five. Our hybrid scheme is upgraded to deal with multi-component gases and tested in several cases. We demonstrate that with a redefinition of the discontinuity sensor multi-components flows can be solved with low levels of diffusion while being stable in the presence of high scalar gradients. Because of the work of this thesis, a complete numerical approach to the numerical simulation of compressible turbulent multi-component flows with or without discontinuities in a wide range of Reynolds and Mach numbers is proposed and validated. Direct applications can be found in civil aviation (subsonic and supersonic) and engine operation.En aquesta tesis es presenten tècniques numèriques per a la simulació de compressibles en una gran varietat de situacions. L’objectiu és el de donar un pas endavant en la investigació científica de la simulació numèrica de fluids compressibles, amb especial èmfasi en fluxos turbulents amb interaccions entre ones de xoc, capa límit y vòrtex. Algunes de les aplicacions que es poden beneficiar d’aquesta investigació són: enginyeria marítima, combustió en motors, predicció meteorològica, aerodinàmica en la industria automotriu y aeronàutica, aplicacions biomèdiques y moltes altres. Tot i així, aquest treball s’emmarca en el camp de l’aerodinàmica compressible y la combustió de gasos amb un clar objectiu: el transport aeri i la tecnologia de motors. Les ferramentes presentades permeten l’estudi del sònic boom, resistència aerodinàmica, soroll y reducció d’emissions mitjançant el disseny geomètric i tècniques de control de flux en elements aerodinàmics tals com ales o motors en règims subsònics, transsònics i supersònics. Els resultats de tals estudis poden donar lloc a nous vehicles més ecològics, respectuosos amb el medi ambient, més silenciosos, amb menor peso estructural i menys consum de combustible.Postprint (published version