Viscous Surface Waves Research Articles

On the decay of viscous surface waves in 3D

We consider the incompressible viscous surface wave problem in the setting that the fluid domain is a horizontal infinite layer in 3D. The fluid dynamics is governed by the gravity-driven incompressible Navier–Stokes equations, and the effect of surface tension is ignored on the upper free surface. We prove the optimal time-decay rate of the low-order energy of the solution with minimal derivative count 3, which implies that the Lipschitz norm of the velocity decays at the rate (1+t)−1. This together with a time-weighted estimate for the highest order spatial derivatives of the free surface function leads to the boundedness of the high-order energy, which improves the result of Wang [9].

Bifurcation Analysis and Propagation Conditions of Free-Surface Waves in Incompressible Viscous Fluids of Finite Depth

Viscous linear surface waves are studied at arbitrary wavelength, layer thickness, viscosity, and surface tension. We find that in shallow enough fluids no surface waves can propagate. This layer thickness is determined for some fluids, water, glycerin, and mercury. Even in any thicker fluid layers, propagation of very short and very long waves is forbidden. When wave propagation is possible, only a single propagating mode exists for a given horizontal wave number. In contrast, there are two types of non-propagating modes. One kind of them exists at all wavelength and material parameters, and there are infinitely many such modes for a given wave number, distinguished by their decay rates. The other kind of non-propagating mode that is less attenuated may appear in zero, one, or two specimens. We notice the presence of two length scales as material parameters, one related to viscosity and the other to surface tension. We consider possible modes for a given material on the parameter plane layer thickness versus wave number and discuss bifurcations among different mode types. Motion of surface particles and time evolution of surface elevation is also studied at various parameters in glycerin, and a great variety of behaviour is found, including counterclockwise surface particle motion and negative group velocity in wave propagation.

Compressible viscous heat-conducting surface wave without surface tension

In this paper, we consider a three-dimensional compressible viscous heat-conducting fluid in a horizontally periodic domain, bounded above by a free surface and below by a rigid bottom. The motion of the fluid is governed by the full compressible, gravity-driven Navier–Stokes equations with appropriate boundary conditions. On the free surface, the effect of surface tension is neglected, and the temperature is assumed to satisfy the Robin boundary condition. Motivated by Y. Guo and I. Tice [Anal. PDE 6, 287–369 (2013), Arch. Ration. Mech. Anal. 207, 459–531 (2013), and Anal. PDE 6, 1429–1533 (2013)], we establish the global well-posedness of this free boundary problem, provided that the initial data are close to a nontrivial equilibrium state.

Vanishing Viscosity and Surface Tension Limits of Incompressible Viscous Surface Waves

Consider the dynamics of a layer of viscous incompressible fluid under the influence of gravity. The upper boundary is a free boundary with the effect of surface tension taken into account, and the...

Lagrangian Approach to Global Well-Posedness of the Viscous Surface Wave Equations Without Surface Tension

In this paper, we revisit the global well-posedness of the classical viscous surface waves in the absence of surface tension effect with the reference domain being the horizontal infinite slab, for which the first complete proof was given in Guo–Tice [Anal. PDE 6,1429–1533 (2013)] via a hybrid of Eulerian and Lagrangian schemes. The fluid dynamics are governed by the gravity-driven incompressible Navier–Stokes equations. Even though Lagrangian formulation is most natural to study free boundary value problems for incompressible flows, few mathematical works for global existence are based on such an approach in the absence of surface tension effect, due to breakdown of Beale’s transformation. We develop a mathematical approach to establish global well-posedness based on the Lagrangian framework by analyzing suitable “good unknowns” associated with the problem, which requires no nonlinear compatibility conditions on the initial data.

Anisotropic decay and global well-posedness of viscous surface waves without surface tension

We consider a viscous incompressible fluid below the air and above a fixed bottom. The fluid dynamics is governed by the gravity-driven incompressible Navier-Stokes equations, and the effect of surface tension is neglected on the free surface. The global well-posedness and long-time behavior of solutions near equilibrium have been intriguing questions since Beale (1981) [2]. It was proved by Guo and Tice (2013) [6] that with certain additional low horizontal frequency assumption of the initial data in 3D an integrable decay rate of the velocity is obtained so that the global unique solution can be constructed, while the global well-posedness in 2D was left open. In this paper, by exploiting the anisotropic decay rates of the velocity, which are even not integrable, we prove the global well-posedness in both 2D and 3D, without any low frequency assumption of the initial data. One of key observations here is a cancellation in nonlinear estimates of the viscous stress tensor term in the bulk by using Alinhac good unknowns, when estimating the energy evolution of the highest order horizontal spatial derivatives of the solution.

Low regularity well-posedness for the viscous surface wave equation

In this paper, we prove the local well-posedness of the viscous surface wave equation in low regularity Sobolev spaces. The key points are to establish several new Stokes estimates depending only on the optimal boundary regularity and to construct a new iteration scheme on a known moving domain. Our method could be applied to some other fluid models with free boundaries.

The Viscous Surface Wave Problem with Generalized Surface Energies

We study a three-dimensional incompressible viscous fluid in a horizontally periodic domain with finite depth whose free boundary is the graph of a function. The fluid is subject to gravity and gen...

Decay of the 2D Periodic Stationary Viscous Surface Waves

We consider the evolution of viscous fluids in a 2D horizontally periodic slab bounded above by a free top surface and below by a fixed flat bottom. This is a free boundary problem. The dynamics of the fluid are governed by the incompressible stationary Navier–Stokes equations under the influence of gravity and the effect of surface tension. We develop the global theory of solutions in low regularity Sobolev spaces for small data by nonlinear energy estimates.

Numerical Solution of the Viscous Surface Wave with Discontinuous Galerkin Method

We consider an incompressible viscous flow without surface tension in a finite-depth domain of two dimensions, with free top boundary and fixed bottom boundary. This system is governed by the Navier–Stokes equations in this moving domain and the transport equation on the moving boundary. In this paper, we construct a stable numerical scheme to simulate the evolution of this system by discontinuous Galerkin method, and discuss the error analysis of the fluid under certain assumptions. Our formulation is mainly based on the geometric structure introduced in [Y. Guo and Ian Tice, Anal. PDE 6 (2013) 287–369; Y. Guo and Ian Tice, Arch. Ration. Mech. Anal. 207 (2013) 459–531; L. Wu, SIAM J. Math. Anal. 46 (2014) 2084–2135], and the natural energy estimate, which is rarely used in the numerical study of this system before.

Well-Posedness and Decay of the Viscous Surface Wave

In this paper, we consider an incompressible viscous flow without surface tension in a finite-depth domain of three dimensions, with a free top boundary and a fixed bottom boundary. The system is governed by the Navier--Stokes equations in this moving domain. Traditionally, this problem can be analyzed in the Lagrangian coordinates as a perturbation of linear equations in a fixed domain. In the series of papers [Anal. PDE, 6 (2013), pp. 287--369; Anal. PDE, 6 (2013), pp. 1429--1533; Arch. Ration. Mech. Anal., 207 (2013), pp. 459--531], Tice and Guo introduced a new framework using the geometric structure in the Eulerian coordinates to study both local and global well-posedness and decay of this system. Following this path, we extend their result in local well-posedness from the small data case to the general data case. Also, we give a simpler proof of global well-posedness in the small data case with horizontally infinite cross section. Other than the geometric energy estimates, the time-dependent Galerkin method, and the interpolation estimates with Riesz potential and minimal counts, which are introduced in these papers, we utilize three new techniques: (1) using the $\epsilon$-Poisson integral to construct a diffeomorphism between the fixed domain and the moving domain; (2) using a bootstrapping argument to prove the comparison estimates in the steady Navier--Stokes equations for general data of free surface; and (3) redefining the energy and dissipation to simplify the original complicated bootstrapping argument to show the interpolation estimates.

Decay of viscous surface waves without surface tension in horizontally infinite domains

We consider a viscous fluid of finite depth below the air, occupying a three-dimensional domain bounded below by a fixed solid boundary and above by a free moving boundary. The fluid dynamics are governed by the gravity-driven incompressible Navier–Stokes equations, and the effect of surface tension is neglected on the free surface. The long-time behavior of solutions near equilibrium has been an intriguing question since the work of Beale (1981). This is the second in a series of three papers by the authors that answers the question. Here we consider the case in which the free interface is horizontally infinite; we prove that the problem is globally well-posed and that solutions decay to equilibrium at an algebraic rate. In particular, the free interface decays to a flat surface. Our framework utilizes several techniques, which include a priori estimates that utilize a “geometric” reformulation of the equations; a two-tier energy method that couples the boundedness of high-order energy to the decay of low-order energy, the latter of which is necessary to balance out the growth of the highest derivatives of the free interface; control of both negative and positive Sobolev norms, which enhances interpolation estimates and allows for the decay of infinite surface waves. Our decay estimates lead to the construction of global-in-time solutions to the surface wave problem.

The linearization principle for a free boundary problem for viscous, capillary incompressible fluids

The free boundary problem associated with a viscous incompressible surface wave subject to the capillary force on a free upper surface and the Dirichlet boundary condition on a fixed bottom surface is considered. In the spatially periodic case, a general linearization principle is proved, which gives, for sufficiently small perturbations from a linearly stable stationary solution, the existence of a global solution of the associated system and the exponential convergence of the latter to the stationary one. The convergence of the velocity, the pressure, and the free boundary is proved in anisotropic Sobolev--Slobodetskii spaces, and then a suitable change of variables is performed to state the problem in a fixed domain. This linearization principle is applied to the study of the rest state's stability in the case of general potential forces. Bibliography: 16 titles.

Local well-posedness of the viscous surface wave problem without surface tension

We consider a viscous fluid of finite depth below the air, occupying a three-dimensional domain bounded below by a fixed solid boundary and above by a free moving boundary. The domain is allowed to have a horizontal cross-section that is either periodic or infinite in extent. The fluid dynamics are governed by the gravity-driven incompressible Navier–Stokes equations, and the effect of surface tension is neglected on the free surface. This paper is the first in a series of three on the global well-posedness and decay of the viscous surface wave problem without surface tension. Here we develop a local well-posedness theory for the equations in the framework of the nonlinear energy method, which is based on the natural energy structure of the problem. Our proof involves several novel techniques, including: energy estimates in a “geometric” reformulation of the equations, a well-posedness theory of the linearized Navier–Stokes equations in moving domains, and a time-dependent functional framework, which couples to a Galerkin method with a time-dependent basis.

Almost Exponential Decay of Periodic Viscous Surface Waves without Surface Tension

We consider a viscous fluid of finite depth below the air, occupying a three-dimensional domain bounded below by a fixed solid boundary and above by a free moving boundary. The fluid dynamics are governed by the gravity-driven incompressible Navier–Stokes equations, and the effect of surface tension is neglected on the free surface. The long time behavior of solutions near equilibrium has been an intriguing question since the work of Beale (Commun Pure Appl Math 34(3):359–392, 1981). This paper is the third in a series of three (Guo in Local well-posedness of the viscous surface wave problem without surface tension, Anal PDE 2012, to appear; in Decay of viscous surface waves without surface tension in horizontally infinite domains, Preprint, 2011) that answers this question. Here we consider the case in which the free interface is horizontally periodic; we prove that the problem is globally well-posed and that solutions decay to equilibrium at an almost exponential rate. In particular, the free interface decays to a flat surface. Our framework contains several novel techniques, which include: (1) a priori estimates that utilize a “geometric” reformulation of the equations; (2) a two-tier energy method that couples the boundedness of high-order energy to the decay of low-order energy, the latter of which is necessary to balance out the growth of the highest derivatives of the free interface; (3) a localization procedure that is compatible with the energy method and allows for curved lower surface geometry. Our decay estimates lead to the construction of global-in-time solutions to the surface wave problem.

A lattice Boltzmann method for viscous free surface waves in two dimensions

SUMMARYWe propose a new method based on the combination of the lattice Boltzmann equation (LBE) and the kinematic boundary condition (KBC) method to simulate viscous free surface wave in two dimensions. In our method, the flow field is modeled by LBE, whereas the free surface is explicitly tracked by the local height function, which is calculated by the KBC method. The free surface boundary condition (FSBC) for LBE is revised from previous researches. Interpolation‐supplemented lattice Boltzmann (ISLB) method is introduced, which enables our approach to be applied on arbitrary, nonuniform mesh grids. Five cases are simulated respectively to validate the LBE–KBC method: the stationary flow and the solitary waves simulated by the revised‐FSBC are more accurate than the one obtained by the former‐FSBC; numerical results of standing waves show that our method is compatible to the existing two‐dimensional finite‐volume scheme; cases of small amplitude Stokes wave and waves traveling over a submerged bar show good agreement on wave celerity, wavelength, wave amplitude and wave period between numerical results and corresponding analytical solutions and/or experiment data.Copyright © 2012 John Wiley & Sons, Ltd.

Report on a local in time solvability of free surface problems for the Navier–Stokes equations with surface tension

We consider the free boundary problem of the Navier–Stokes equation with surface tension. Our initial domain Ω is one of a bounded domain, an exterior domain, a perturbed half-space or a perturbed layer in ℝ n (n ≥ 2). We report a local in time unique existence theorem in the space with some T > 0, 2 < p < ∞ and n < q < ∞ for any initial data which satisfy compatibility condition. Our theorem can be proved by the standard fixed point argument based on the L p –L q maximal regularity theorem for the corresponding linearized equations. Our results cover the cases of a drop problem and an ocean problem that were studied by Solonnikov (Solvability of the evolution problem for an isolated mass of a viscous incompressible capillary liquid, Zap. Nauchn. Sem. (LOMI) 140 (1984) pp. 179–186 (in Russian) (English transl.: J. Soviet Math. 32 (1986), pp. 223–238)), Solonnikov (Unsteady motion of a finite mass of fluid, bounded by a free surface, Zap. Nauchn. Sem. (LOMI) 152 (1986), pp. 137–157 (in Russian) (English transl.: J. Soviet Math. 40 (1988), pp. 672–686)), Solonnikov (On nonstationary motion of a finite isolated mass of self-gravitating fluid, Algebra Anal. 1 (1989), pp. 207–249 (in Russian) (English transl.: Leningrad Math. J. 1 (1990), pp. 227–276)), Solonnikov (Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval, Algebra Anal. 3 (1991), pp. 222–257 (in Russian) (English transl.: St. Petersburg Math. J. 3 (1992) 189–220)), Beale (Large time regularity of viscous surface waves, Arch. Rat. Mech. Anal. 84 (1984), pp. 307–352) and Tani (Small-time existence for the three-dimensional incompressible Navier–Stokes equations with a free surface, Arch. Rat. Mech. Anal. 133 (1996), pp. 299–331).

Finite volume simulation of viscous free surface waves using the Cartesian cut cell approach

AbstractThe application of the Cartesian cut cell approach in the numerical simulation of two‐dimensional viscous free surface flows is described. The Arbitrary Lagrangian–Eulerian method is adopted to update the moving free water surface in a semi‐Lagrangian scheme, in which a finite volume method of second‐order accuracy in space is used for solving the flow field based on an Eulerian description at each time step. The cut cell approach is employed to track the free surface and solid boundaries across a stationary background Cartesian grid covering the whole fluid, air and solid regions. In this approach, the cells full of air and solid are not calculated explicitly, and apart from the fluid cells, cut cells and merged cells are treated separately in terms of corresponding boundary conditions. In order to validate the present numerical method, current flow past a circular cylinder at various low Reynolds numbers and wave sloshing in a rectangular container are tested first. Further numerical results are obtained for the propagation of regular waves and a wave passing over a submerged dike. The model is also applied to the simulation of radiation waves induced by a forced oscillating submerged circular cylinder. The results indicate that the present numerical model using the Cartesian cut cell approach is highly efficient for solving the wave fields, and fully automatic for generating boundary fitted meshes. These features are particularly useful for moving boundary problems in a larger computational domain and with a longer simulation time. Copyright © 2009 John Wiley &amp; Sons, Ltd.

Viscous waves and wave-structure interaction in a tank using adapting quadtree grids

Viscous waves and waves over a submerged cylinder in a stationary tank are simulated using a volume-of-fluid numerical scheme on adaptive hierarchical grids. A high resolution interface-capturing method is used to advect the free surface interface and the Navier–Stokes equations are discretised using finite volumes with collocated primitive variables and solved using a Pressure Implicit with Splitting of Operators (PISO) algorithm. The cylinder is modelled by using the technique of Cartesian cut cells. Results of flow of a single fluid past a cylinder at Reynolds number Re=100 are presented and found to agree well with experimental and other numerical data. Viscous free surface waves in a tank are simulated using uniform and quadtree grids for Reynolds numbers in the range from 2 to 2000, and the results compared against analytical solutions where available. The quadtree-based results are of the same accuracy as those on the equivalent uniform grids, and retain a sharp interface at the free surface while leading to considerable savings in both storage and CPU requirements. The nonlinearity in the wave is investigated for a selection of initial wave amplitudes. A submerged cylinder is positioned in the tank and its influence on the waves as well as the hydrodynamic loading on the cylinder is investigated.

BEM modeling of surface water wave motion with laminar boundary layers

This study is concerned with numerical modeling of viscous surface wave motion using boundary element method (BEM). The equations of motion for thin boundary layers at the solid surfaces are coupled with the potential flow in the bulk of the fluid, and a mixed BEM-finite difference technique is used to obtain the viscosity-related quantities such as wave damping rate, shear stress, and velocity distribution inside the boundary layer. The technique is presented for standing surface wave motion. An excellent agreement is obtained between the numerical predictions and the previous results. The extension to other free surface problems is straightforward.