Analysis of the incompressible Navier-Stokes equations with a quasi free-surface condition

Numerical solution of free-surface flows with a free-surface that can be represented by a height-function, is of great practical importance. Dedicated methods have been developed for the efficient solution of steady free-surface potential flow. These methods solve a sequence of sub-problems, corresponding to the flow equations subject to a quasi free-surface condition. For steady free-surface Navier-Stokes flow, such dedicated methods do not exist. In the present report we propose an extension to Navier-Stokes flow of an iterative method which has been applied successfully to steady free-surface potential flow. We then examine the sub-problem corresponding to the incompressible Navier-Stokes equations subject to the quasi free-surface condition. We consider perturbations of a uniform, horizontal flow of finite depth and we show that the initial boundary value problem associated with the incompressible Navier-Stokes equations and the quasi free-surface condition allows stable wave solutions that exhibit a behavior that is typical of surface gravity waves. This indicates that the iterative method proposed is indeed suitable for solving steady free-surface Navier-Stokes flow. Implementation of the iterative method and numerical experiments are treated in a forthcoming report

Numerical solution of steady free-surface Navier-Stokes flow

Numerical solution of flows that are partially bounded by a freely moving boundary is of great practical importance, e.g., in ship hydrodynamics. The usual time integration approach for solving steady viscous free surface flow problems has several drawbacks. Instead, we propose an efficient iterative method, which relies on a different but equivalent formulation of the free surface flow problem, involving a so-called quasi free-surface condition. It is shown that the method converges if the solution is sufficiently smooth in the neighborhood of the free surface. Details are provided for the implementation of the method in {PARNAX. Furthermore, we present a method for analyzing properties of discretization schemes for the free-surface flow equations. Detailed numerical results are presented for flow over an obstacle in a channel. The results agree well with measurements as well as with the predictions of the analysis, and confirm that steady free-surface Navier-Stokes flow problems can indeed be solved efficiently with the new method

A Godunov-type scheme with applications in hydrodynamics

In spite of the absence of shock waves in most hydrodynamic applications, sufficient reason remains to employ Godunov-type schemes in this field. In the instance of two-phase flow, the shock capturing ability of these schemes may serve to maintain robustness and accuracy at the interface. Moreover, approximate Riemann solvers have greatly relieved the initial drawback of computational expensiveness of Godunov-type schemes. In the present work we develop an Osher-type flux-difference splitting approximate Riemann solver and we examine its application in hydrodynamics. Actual computations are left to future research

A pressure-invariant conservative Godunov-type method for barotropic two-fluid flows

Conservative discretizations of two-fluid flow problems generally exhibitpressure oscillations. In this work we show that these pressure oscillationsare induced by the loss of a pressure-invariance property under discretization,and we introduce a non-oscillatory conservative method for barotropic two-fluidflows. The conservative formulation renders the two-fluid flow problem suitableto treatment by a Godunov-type method. We present a modified Osher scheme forthe two-fluid flow problem. Numerical results are presented for atranslating-interface test case and a shock/interface-collision test case

Efficient numerical solution of steady free-surface Navier-Stokes flow

Numerical solution of flows that are partially bounded by a freely moving boundary is of great importance in practical applications such as ship hydrodynamics. The usual method for solving steady viscous free-surface flow subject to gravitation is alternating time integration of the kinematic condition, and the Navier-Stokes equations with the dynamic conditions imposed, until steady state is reached. This paper shows that at subcritical Froude numbers this time integration approach is necessarily inefficient and proposes an efficient iterative method for solving the steady free-surface flow problem. The new method relies on a different but equivalent formulation of the free-surface flow problem, involving a so-called quasi free-surface condition. The convergence behavior of the new method is shown to be asymptotically mesh width independent. Numerical results are presented for 2D flow over an obstacle in a channel. The results confirm the mesh width independence of the convergence behavior and comparison of the numerical results with measurements shows good agreement

On the adjoint solution of the quasi-1D Euler equations: the effect of boundary conditions and the numerical flux function

This work compares a numerical and analytical adjoint equation method with respect to boundary condition treatments applied to the quasi-1D Euler equations. The effect of strong and weak boundary conditions and the effect of flux evaluators on the numerical adjoint solution near the boundaries are discussed

Riemann-problem and level-set approaches for two-fluid flow computations II. Fixes for solution errors near interfaces.

Fixes are presented for the solution errors (`pressure oscillations') that may occur near two-fluid interfaces when applying a capturing method. The fixes are analyzed and tested. For two-fluid flows with arbitrarily large density ratios, a variant of the ghost-fluid method appears to be a perfect remedy. Results are presented for compressible water-air flows. The results are promising for a further elaboration of this important application area. The paper contributes to the state-of-the-art in computing two-fluid flows

Comparison of two adjoint equation approaches with respect to boundary-condition treatments for the quasi-1D Euler equations

For computation of nonlinear aeroelastic problems, an efficient error estimation and grid adaptation algorithm is highly desirable, but traditional error estimation or grid adaptation do not suffice, since they are insufficiently related to relevant engineering variables and are incapable of significantly reducing the computing time. The dual formulation however, can be used as an a-posteriori error estimation in the quantity of interest. However, derivation of the dual problem, especially the accompanying boundary conditions, is not a trivial task. This document compares a discrete and analytical ad joint equation method with respect to boundary-condition treatments applied on the quasi-1D Euler equations. Flux evaluation of the primal problem is do ne by a Linearised Godunov scheme. For our future goal, solving ftuid-structure problems, the discrete approach seems preferable