Piecewise linear interface-capturing volume-of-fluid method in axisymmetric cylindrical coordinates

The paper describes a novel implementation of the piecewise linear interface-capturing volume-of-fluid method (PLIC-VOF) in axisymmetric cylindrical coordinates. The principal innovative feature involved in this work is that both the forward and inverse reconstruction problems are solved analytically, resulting in an appreciable speed-up in computing time in comparison with an iterative approach. All reconstruction formulae are introduced explicitly, and an example illustrating their derivation is included for clarity. The numerical implementation of the PLIC-VOF interface tracking method developed here is described in detail, as well as its coupling with the 3D incompressible Navier-Stokes solver PSIBOIL, which features a finite-volume approach based on a fixed, rectangular grid. This coupling includes a method to calculate the surface tension force within an axisymmetric VOF framework by means of height functions. The method is first verified to ensure its correct implementation in the code, and to evaluate its performance, and several advection tests are employed to demonstrate successful solution of the basic transport problem. It is shown that convergence to equilibrium may be achieved for static problems, involving the control of parasitic currents, for a variety of grid arrangements, and for different material properties. Several axisymmetric dam-break and rising bubble problems, for which highquality measured data are available, are also presented to serve as validation tests. In all cases, very good agreement of simulation results with experimental data has been recorded, and the dynamics of the problem well reproduced. @& nbsp;2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Analysis and numerical simulation of free surface flows

Alexandre Caboussat

Mathematical and numerical aspects of free surface flows are investigated. On one hand, the mathematical analysis of some free surface flows is considered. A model problem in one space dimension is first investigated. The Burgers equation with diffusion has to be solved on a space interval with one free extremity. This extremity is unknown and moves in time. An ordinary differential equation for the position of the free extremity of the interval is added in order to close the mathematical problem. Local existence in time and uniqueness results are proved for the problem with given domain, then for the free surface problem. A priori and a posteriori error estimates are obtained for the semi-discretization in space. The stability and the convergence of an Eulerian time splitting scheme are investigated. The same methodology is then used to study free surface flows in two space dimensions. The incompressible unsteady Navier-Stokes equations with Neumann boundary conditions on the whole boundary are considered. The whole boundary is assumed to be the free surface. An additional equation is used to describe the moving domain. Local existence in time and uniqueness results are obtained. On the other hand, a model for free surface flows in two and three space dimensions is investigated. The liquid is assumed to be surrounded by a compressible gas. The incompressible unsteady Navier-Stokes equations are assumed to hold in the liquid region. A volume-of-fluid method is used to describe the motion of the liquid domain. The velocity in the gas is disregarded and the pressure is computed by the ideal gas law in each gas bubble trapped by the liquid. A numbering algorithm is presented to recognize the bubbles of gas. Gas pressure is applied as a normal force on the liquid-gas interface. Surface tension effects are also taken into account for the simulation of bubbles or droplets flows. A method for the computation of the curvature is presented. Convergence and accuracy of the approximation of the curvature are discussed. A time splitting scheme is used to decouple the various physical phenomena. Numerical simulations are made in the frame of mould filling to show that the influence of gas on the free surface cannot be neglected. Curvature-driven flows are also considered.

EPFL2004

Influence of macro-roughness of walls on steady and unsteady flow in a channel

Tobias Meile

High-head storage hydropower plants mainly operate their turbines during periods of high energy demand. The sudden starting and stopping of turbines (hydropeaking) lead to highly unsteady flow in channels and rivers. Besides hydropeaking, other anthropogenic actions and natural events such as sluice gate operations, flushing of reservoirs, debris jam and break up, ice jam and break up, sudden stopping and starting of turbines of runoff river hydropower plants, flashfloods or dambreaks can also cause highly unsteady flow. From an ecological point of view, hydropeaking consists in a non-natural disturbance of the flow regime. Possible mitigation measures aiming to reduce the effect of hydropeaking in a river downstream of the powerhouse can be divided into the installation of detention basins and the improvement of the river morphology. Morphological measures such as macro-roughness at banks might increase the flow resistance as well as the passive retention, which both increase the natural retention capacity of rivers and thus modify the form of the surge wave. In prismatic and nearly prismatic channels, highly unsteady flow conditions can be calculated using the elementary surge wave theory or numerical methods based on the Saint-Venant equations, respectively. For channels with large-scale roughness at the side walls, which may occur by the arrangement of particular morphological measures, no systematic experimental investigations have been done so far on the propagation of surge waves including downstream water-depth and flow velocity. The aim of this research project was to study how macro-roughness elements at the channel banks influence the unsteady flow conditions due to surge waves. 41 configurations of macro-rough banks and various discharges have been tested in a 40 m long flume with a bed slope of 1.14‰. A special experimental setup has been designed which is able to generate surge waves characterized by different discharge ratios. The first step of the experimental investigations focused on the determination of the flow resistance under steady flow conditions caused by large scale roughness elements at the channel banks, namely rectangular cavities (depressions). In a second step, positive and negative surge waves induced at the channel entrance have been tested in the same geometries. Five different discharge scenarios have been considered for each geometry. For comparison, experiments have also been performed in a prismatic reference channel without macro-roughness elements at the banks. The analysis of the experiments for steady flow conditions results in the following conclusions: The total head-loss is governed by different phenomena such as vertical mixing layers, wake-zones, recirculation gyres, coherent structures and skin friction. The flow resistance is significantly increased in the macro-rough configurations due to the disturbance of the bank geometry. Three different approaches have been considered in order to relate the additional, macro-rough flow resistance fMR to the forms of the banks by empirical formulas. The first approach is based on a powerlaw optimization, the second one on a model based on form drag and the third one uses the Evolutionary Polynomial Regression method. By separating the observed flow conditions in a square grooved, a reattachment and a normal recirculating flow type, the developed macro-rough flow resistance formulas are in good agreement with the laboratory experiments. Formulas of the second approach are suggested for practical applications in river engineering. Water body oscillations in cavities have been observed in axi-symmetric macrorough configurations. They lead to water-surface oscillations and transverse velocity components. Peaking excitations with heavy oscillations occur for Strouhal numbers values at 0.42 and at 0.84. The analysis of the experiments with unsteady flow, namely surge waves, shows: The elementary surge wave theory for the prismatic reference configuration including secondary waves and wave breaking could be verified in the laboratory flume. The absolute surge wave celerity Vw and the celerity c are lower for the macro-rough configurations. The decrease of the absolute surge wave celerity Vw is mainly due to the increased flow resistance and lies between 5% and 25% for both, positive and negative waves coming from upstream. Due to the dispersive character, the positive and negative surges from upstream are characterized by a sudden change (front), followed by a progressive change (body) of the water level. The sudden change can be clearly distinguished from the progressive change in the prismatic channel. In the channel with macro rough banks the separation is often hardly visible. The front height decreases between 5% to 25% along the laboratory flume for the prismatic configuration. In the macro-rough channel, the decrease can reach 70%. Behind the wave front of a positive surge, the water level continuous rising to a level which is higher than in the prismatic channel due to the channel bank macro-roughness. The decrease of the height of the surge wave front along the channel could be approached fairly good with an empirical relationship applicable for prismatic and macro-rough configurations. Numerical simulations based on the boundary conditions of the experiments have been done for steady flow tests (2-D simulations). They reveal that the integration of the turbulent stresses is required for the reproduction of the steady flow tests. Test cases with unsteady flow (1-D simulations) show that surge waves in rivers can be computed by integrating the macro-rough flow resistance in the source terms as well as by a modification of the continuity equation taking into account the passive retention.

EPFL2007

An Improved CFD Tool to Simulate Adiabatic and Diabatic Two-Phase Flows

Bogdan Alexandru Nichita

With increasing computer capabilities, numerical modeling of two-phase flows has developed significantly over the last few years. Although there are two main categories, namely 'one' fluid and 'two' fluid methods, the 'one' fluid methods are more commonly used for tracking or capturing the interface between two fluids. Level set (LS), volume-of-fluid (VOF), front tracking, marker-and-cell (MAC) and lattice Boltzmann (LB) methods are all 'one' fluid methods. It is clear that there is no perfect method; each method has advantages and disadvantages which make it more appropriate for one kind of problem than for others. For instance, a LS method will accurately compute the curvature and the normal to the interface, but tends to loss mass which is physically incorrect. On the other hand, a VOF method will conserve mass up to machine precision, but the computation of the curvature and normal to the interface is not as accurate. In order to minimize the disadvantages of these methods, several authors have used two or more methods together to model two-phase flows. This is the case for the CLSVOF (Couple Level Set Volume Of Fluid) method, where LS and VOF are coupled together in order to better capture the interface. In CLSVOF, the level set function is used to compute the interface curvature and normal to the interface, while the volume of fluid function is used to capture the interface. For two-phase flows in microchannels, surface tension forces play an important role in determining the dynamics of bubbles whereas gravitational forces are generally negligible. Also it is very important to consider the interaction between the boundaries and the fluids by prescribing or computing the correct contact angle between them. The commercial CFD code FLUENT allows the use of static constant angles, or the use of User Defined Functions (UDF) to compute the dynamic contact angles. It is inappropriate to use a static contact angle to model cases involving moving contact lines. For such cases a dynamic contact angle scheme should be implemented. In this study, FLUENT was used to model adiabatic and diabatic, time dependent two-phase flows. Since FLUENT already contains a VOF method, a LS method was implemented and coupled with VOF into FLUENT via UDFs. Furthermore, since the LS function, used to compute the surface tension force, ceases to be a signed distance to the interface even after one time step, a re-initialization equation was solved after each time step. This involved using a fifth order WENO (Weighted Essentially Non Oscillatory) scheme to discretize the space derivatives (otherwise oscillations of the interface occurred), and a first order Euler method for the time integration. In another part of the study, a 3D dynamic contact angle model based on volume fraction, interface reconstruction, and experimentally available advancing and receding static contact angles was also developed and implemented into FLUENT via UDFs. Several validations for the developed CLSVOF method and dynamic contact angle model are presented in this thesis, these includes a static bubble, a bubble rising in a stagnant liquid for Morton numbers ranging from 102 to 10-11, droplet deformation due to a vortex flow field, droplets spreading over a wall under the gravity effect and droplets sliding over a wall due to gravity. These validations demonstrated the high accuracy and the stability of our methods for modeling these phenomena. A heat and mass transfer model was also implemented into the commercial CFD code FLUENT for simulating of boiling (and condensation) heat transfer. Several simulations were presented with water and R134a as working fluids. The influence of the contact angle and the wall superheat was also studied.

EPFL2010