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 potential-flow BEM for large-displacement FSI

This work concerns the interaction of light membrane structures enclosing incompressible fluids. Large displacements and collapsed boundaries (initially slender subdomains) are characteristic of this class of problems. A finite-element/boundary-element (FE/BE) coupled discretization is discussed in this work which addresses these challenges. Most common linear fluid models have a boundary-integral representation, restricting the problem to the boundary and making them amenable to a boundary element method (BEM) discretization. A marked advantage of this representation with respect to the conventional partial-differential-equation-based view of finite-element methods for fluids is the bypassing of volumetric meshing. The FE/BE approach is therefore an enabling method for fluid-structure interaction (FSI) problems involving membranes, and the present work serves as a proof of concept for this approach. Numerical experiments show the capabilities of the proposed scheme

Finite element/boundary element coupling for airbag deployment

Fluid–structure interaction of inflatables comprizes a family of applications, of which airbags are one. Challenges in this domain are complex geometries requiring relatively high resolution; large displacements possibly entailing severe mesh distortion; and strong coupling. A promising approach seems to be coupling a classical finite element formulation for the airbag fabric to a boundary element formulation for the enclosed fluid. Together with an appropriate time-integration scheme this method answers aforementioned challenges, as demonstrated by numerical simulation

Model comparison for inflatables using boundary element techniques

This paper focuses on numerical techniques for inflatable structures where, typically, the structure is a light membrane enclosing an incompressible fluid. Large displacements and subdomains with high aspect ratio are often characteristic of this class of problems, to which the finite-element / boundary-element (FE/BE) paradigm is an auspicious approach for numerical approximation. The FE/BE approach constitutes discretizing the structure with the finite-element method and employing a boundary-integral representation for the fluid problem for discretization with the boundary-element method. A marked advantage of this representation with respect to the conventional partial-differential-equation-based view of finite-element methods for fluids is the bypassing of volumetric meshing, rendering the FE/BE approach an enabling method for simulation of inflatables. The present work compares different structure and fluid models and judges their applicability based on numerical results

Finite element/boundary element coupling for airbag deployment

Fluid–structure interaction of inflatables comprizes a family of applications, of which airbags are one. Challenges in this domain are complex geometries requiring relatively high resolution; large displacements possibly entailing severe mesh distortion; and strong coupling. A promising approach seems to be coupling a classical finite element formulation for the airbag fabric to a boundary element formulation for the enclosed fluid. Together with an appropriate time-integration scheme this method answers aforementioned challenges, as demonstrated by numerical simulation

Numerical approximation of the Boltzmann equation : moment closure

This work applies the moment method onto a generic form of kinetic equations to simplify kinetic models of particle systems. This leads to the moment closure problem which is addressed using entropy-based moment closure techniques utilizing entropy minimization. The resulting moment closure system forms a system of partial differential equations that retain structural features of the kinetic system in question. This system of partial differential equations generate balance laws for velocity moments of a kinetic density that are symmetric hyperbolic, implying well-posedness in finite time. In addition, the resulting moment closure system satisfies an analog of Boltzmann's H-Theorem, i.e. solutions of the moment closure system are entropy dissipative. Such a model provides a promising alternative to particle methods, such as the Monte Carlo approaches, which can be prohibitive with regard to computational costs and inefficient with regard to error decay. However, several challenges pertaining to the analytical formulation and computational implementation of the moment closure system arise that are addressed in this work. The entropy minimization problem is studied in light of the classical results by Junk [18] showing that this technique suffers from a realizability problem, i.e. there exists realizable moments such that the entropy minimizer does not exist. Recent results by Hauck [17], Schneider [31] and Pavan [29] are used to investigate and circumvent this issue. The resulting moment closure system involves moments of exponentials of polynomials of, in principle, arbitrary order. In the context of numerical approximations, this is regarded as a complication. A novel and mathematically tractable moment system is developed that is based on approximating the entropy minimizing distribution. It will be shown that the resulting system retains the same structural features of the kinetic system in question. This system can be seen as a refinement of Grad's original moment system [15, 32]. Finally, a numerical approximation of the resulting moment systems is devised using discontinuous Galerkin (DG) finite elements method. Energy analysis based on the work of Barth [1, 2] is employed to investigate energy stable numerical flux functions to be used for the DG discretization of the moment systems. In contrast to the work of Barth [1], the numerical flux function suggested for the tractable system does not require a simplified construction since it is computable. In addition, higher order (approximated) moment systems, beyond the 10-moment system investigated by Barth [1], can be considered

Numerical approximation of the Boltzmann equation : moment closure

This work applies the moment method onto a generic form of kinetic equations to simplify kinetic models of particle systems. This leads to the moment closure problem which is addressed using entropy-based moment closure techniques utilizing entropy minimization. The resulting moment closure system forms a system of partial differential equations that retain structural features of the kinetic system in question. This system of partial differential equations generate balance laws for velocity moments of a kinetic density that are symmetric hyperbolic, implying well-posedness in finite time. In addition, the resulting moment closure system satisfies an analog of Boltzmann's H-Theorem, i.e. solutions of the moment closure system are entropy dissipative. Such a model provides a promising alternative to particle methods, such as the Monte Carlo approaches, which can be prohibitive with regard to computational costs and inefficient with regard to error decay. However, several challenges pertaining to the analytical formulation and computational implementation of the moment closure system arise that are addressed in this work. The entropy minimization problem is studied in light of the classical results by Junk [18] showing that this technique suffers from a realizability problem, i.e. there exists realizable moments such that the entropy minimizer does not exist. Recent results by Hauck [17], Schneider [31] and Pavan [29] are used to investigate and circumvent this issue. The resulting moment closure system involves moments of exponentials of polynomials of, in principle, arbitrary order. In the context of numerical approximations, this is regarded as a complication. A novel and mathematically tractable moment system is developed that is based on approximating the entropy minimizing distribution. It will be shown that the resulting system retains the same structural features of the kinetic system in question. This system can be seen as a refinement of Grad's original moment system [15, 32]. Finally, a numerical approximation of the resulting moment systems is devised using discontinuous Galerkin (DG) finite elements method. Energy analysis based on the work of Barth [1, 2] is employed to investigate energy stable numerical flux functions to be used for the DG discretization of the moment systems. In contrast to the work of Barth [1], the numerical flux function suggested for the tractable system does not require a simplified construction since it is computable. In addition, higher order (approximated) moment systems, beyond the 10-moment system investigated by Barth [1], can be considered