Nonlinear Dynamic Boundary Conditions Research Articles

Existence and uniqueness for an elliptic problem with evolution arising in electrodynamics

We study the existence and uniqueness of solutions to an elliptic problem with a nonlinear dynamic boundary condition, relating the conormal derivative of the unknown to the time derivative of its jump across an internal interface. We firstly prove the well-posedness of a suitable linear version of this problem, by means of a classical result in abstract parabolic theory; then, we study the nonlinear case using a fixed point technique. Our mathematical scheme is of interest in the modelling of electrical conduction in biological tissues.

A Comparison of Dirichlet and Neumann Wavemakers for the Numerical Generation and Propagation of Nonlinear Long-Crested Surface Waves

We are studying numerically the problem of generation and propagation of long-crested gravity waves in a tank containing an incompressible inviscid homogeneous fluid initially at rest with a horizontal free surface of finite extent and of infinite depth. A nonorthogonal curvilinear coordinate system, which follows the free surface, is constructed and the full nonlinear kinematic and dynamic free surface boundary conditions are utilized in the algorithm. “Wavemakers” are modeled using both the Dirichlet and Neumann lateral boundary conditions and a full comparison is given. Overall, the Dirichlet model was more stable than the Neumann model, with an upper limit steepness S=2A/λ of 0.08 using good resolution compared with the Neumann’s maximum of 0.05.

A Numerical Wave Tank For The Generation And ProgagationOf Bichromatic, Nonlinear, Long-crested Surface Waves

We are studying numerically the problem of generation and propagation of bichromatic gravity long-crested waves in a tank containing an incompressible inviscid homogeneous fluid initially at rest with a horizontal free surface of finite extent and of infinite and finite depths. A non-othogonal curvilinear coordinate system, which follows the free surface is constructed and the full nonlinear kinematic and dynamic free surface boundary conditions are utilized in the algorithm.

Global solutions for quasilinear parabolic problems

Results on the global existence of classical solutions for quasilinear parabolic equations in bounded domains with homogeneous Dirichlet or Neumann boundary conditions are presented. Besides quasilinear parabolic equations, the method is also applicable to some weakly-coupled reaction-diffusion systems and to elliptic equations with nonlinear dynamic boundary conditions.

Optimal control problem for phase-field equations with nonlinear dynamic boundary conditions

Optimal control problem for phase-field equations with nonlinear dynamic boundary conditions

Parabolic problems with nonlinear dynamical boundary conditions and singular initial data

where Ω is a bounded domain in Rn of class C2, Γ = ∂Ω, ν denotes the outer normal on Γ, γ is the trace operator, and Au = −∆u + ωu, Bu = uν + ωu. Although we will consider this particular case, the techniques we use can also be applied to the case of systems in which, as in [10], Au = −∂j(ajk∂ku) + aj∂ju+a0, Bu = ajkνγ∂ku+b0γu, with smooth-enough coefficients. On the nonlinear terms, f and g, we assume that they are smooth functions with

On Simulation of Turbulent Nonlinear Free-Surface Flows

A method for numerical simulation of the unsteady, three-dimensional, viscous Navier–Stokes equations for turbulent nonlinear free-surface flows is presented and applied to simulations of a laminar standing wave and turbulent open-channel flow with a finite-amplitude surface wave. The solution domain is discretized with a boundary-orthogonal curvilinear grid that moves with the free surface, allowing surface deformations to be smoothly resolved down to the numerical grid scale. The nonlinear kinematic and dynamic boundary conditions for boundary-orthogonal curvilinear coordinates are developed and discussed with a novel approach for advancing the free surface in curvilinear space. Dynamic large-eddy-simulation techniques are used to model subgrid scale turbulence effects. The method is shown to correctly produce the shape of a nonlinear free-surface wave and its decay due to viscosity. Application to finite-amplitude waves moving over a turbulent channel flow allows demonstration of the clear differences between a channel flow with and without waves, particularly the instantaneous turbulence structure. An interesting sidelight is the appearance of short-crested cross-channel surface waves caused by natural resonance.

Wave Influences on Viscous Flow around a Ship in Steady Yaw Motion

This paper concerns an investigation of wave influences on viscous flow around a ship in steady yaw motion, which is based on detailed analysis of numerical solutions of Reynolds-averaged Navier-Stokes (RaNS) equations. The present numerical method solves the unsteady RaNS and continuity equations for mean-velocity and pressure and wave fields with exact nonlinear kinematic and approximate dynamic free-surface boundary conditions. In the following, an overview is given of the present numerical approach, and results are presented for the Series 60 CB=0.6 at yaw angle 10°for with/without-wave cases, including wherever possible comparison with experimental data. Then discussions are focused on identification of salient feature of wave-included effects on the mean-velocity and pressure fields, body surface pressure and frictional streamlines, and hydrodynamic forces acting on the hull. Lastly, some specific aspects of flow for wave breaking and wave-induced vortices are described.

Global solutions of the Laplace equation with a nonlinear dynamical boundary condition

We study the boundedness and a priori bounds of global solutions of the problem Δu=0 in Ω×(0, T), (∂u/∂t) + (∂u/∂ν) = h(u) on ∂Ω×(0, T), where Ω is a bounded domain in ℝN, ν is the outer normal on ∂Ω and h is a superlinear function. As an application of our results we show the existence of sign-changing stationary solutions. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

A Large-Domain Approach for Calculating Ship Boundary Layers and Wakes and Wave Fields for Nonzero Froude Number

A large-domain approach is developed for calculating ship boundary layers and wakes and wave fields for nonzero Froude number. The Reynolds-averaged Navier–Stokes and continuity equations are solved with the Baldwin–Lomax turbulence model, exact nonlinear kinematic and approximate dynamic free-surface boundary conditions, and a body/free-surface conforming grid. The results are validated through comparisons with data for the Series 60 CB= 0.6 ship model at low and high Froude numbers and results of a precursory interactive approach. Both approaches yield satisfactory results; however, the large-domain results indicate improved resolution of the flow close to the hull and wake centerplane and of the Froude number differences due to near-wall turbulence modeling and nonlinear free-surface boundary conditions. Additional evaluation is provided through discussion of the recent CFD Workshop Tokyo 1994, where both methods were among the best. Last, some concluding remarks are made.

Free-Surface Wave-Induced Separation

Free-surface wave-induced separation is studied for a surface-piercing NACA 0024 foil over a range of Froude numbers (0, .2, .37, .55) through computational fluid dynamics of the unsteady Reynolds-averaged Navier-Stokes and the continuity equations with the Baldwin-Lomax turbulence model, exact nonlinear kinematic and approximate dynamic free-surface boundary conditions, and a body/free-surface conforming grid. The flow conditions and uncertainty analysis are discussed. A topological rule for a surface-piercing body is derived and verified. Steady-flow results are presented and analyzed with regard to the wave and viscous flow and the nature of the separation.

Conditional averaging of a turbulent free surface.

A fully nonlinear dynamical boundary condition on a free surface in viscous turbulent flow is considered. Conditional averaging with a fixed normal to a free surface in a point revealed simple statistical structures for the first- and second-order moments of the deformation rate tensor. These structures persist in a free-surface boundary layer. The exact results presented suggest a different direction for experimental and numerical studies of free-surface turbulence.

Nonlinear response of membranes to ocean waves using boundary and finite elements

The behavior of a highly deformable membrane to ocean waves was studied by coupling a nonlinear boundary element model of the fluid domain to a nonlinear finite element model of the membrane. The hydrodynamic loadings induced by water waves are computed assuming large body hydrodynamics and ideal fluid flow and then solving the transient diffraction/radiation problem. Either linear waves or finite amplitude waves can be assumed in the model and thus the nonlinear kinematic and dynamic free surface boundary conditions are solved iteratively. The nonlinear nature of the boundary condition requires a time domain solution. To implicitly include time in the governing field equation, Volterra's method was used. The approach is the same as the typical boundary element method for a fluid domain where the governing field equation is the starting point. The difference is that in Volterra's method the time derivative of the governing field equation becomes the starting point. The boundary element model was then coupled through an iterative process to a finite element model of membrane structures. The coupled model predicts the nonlinear interaction of nonlinear water waves with highly deformable bodies. To verify the coupled model a large scale test was conducted in the OH Hinsdale wave Research Laboratory at Oregon State University on a 3-ft-diameter fabric cylinder submerged in the wave tank. The model data verified the numerical prediction of the structure displacements and of the changes in the wave field. The boundary element model is an ideal modeling technique for modeling the fluid domain when the governing field equations is the Laplace equation. In this case the nonlinear boundary element model was coupled with a finite element model of membrane structures, but the model could have been coupled with other finite element models of more rigid structures, such as a pontoon floating breakwater.

On the qualitative behaviour of some semilinear parabolic problems

This paper is concerned with semilinear reaction-diffusion systems under nonlinear dynamical boundary conditions. We will prove that such problems are well-posed on some Bessel potential spaces and that the corresponding solution defines a local semiflow on these spaces. This result will enable us to investigate the dynamical properties of the solutions under discussion. In particular, we will prove some conclusions concerning global existence and blow up phenomena as well as singular perturbation results.

Nonlinear water-wave structure interaction

A numerical model of the interaction of highly deformable fluid-filled membranes and ocean waves has been developed and verified. The fluid domain is modeled using a boundary element model and the membrane is modeled using a finite element model. The finite element model was developed previously, and required little modification. However, the hydrodynamics in the finite element model was simplistic and the boundary element model of the fluid domain was developed to provide a better representation of the potentially nonconservative hydrodynamic loadings on the membrane. The finite element model predicts the nonlinear dynamic behavior of cable and membrane structures where the nonlinearities arise from large displacements, surface rotations, nonlinear stress-strain relationships and nonconservative loadings. The total Langrangian description and the principle of virtual work are used to formulate the equilibrium equations. Explicit stress-strain relationships are neither assumed nor required. The hydrodynamic loadings induced by water waves on structures are computed using a boundary element model. Large body hydrodynamics and ideal fluid flow are assumed and the diffraction/radiation problem solved. Either linear waves or finite amplitude waves are assumed in the model; thus the nonlinear kinematic and dynamic free surface boundary conditions are treated iteratively. To implicitly include time in the governing field equations Volterra's method was used. The models were coupled in an iterative procedure to model the wave structure interaction. Numerical results were compared to results of large scale test conducted on a 3-ft diameter membrane cylinder.

Nonlinear Local Analysis of Steady Flow About a Ship

A nonlinear local analysis of the steady potential flow at a ship bow and stern, and more generally at any point along a ship waterline, is presented. The hull boundary condition and the nonlinear kinematic and dynamic free-surface boundary conditions are satisfied exactly, at the actual position of the free surface, in this analysis. The bow-flow analysis shows that the free surface at a ship bow is tangent to the stem. This theoretical result appears to agree with existing experimental measurements of steady bow waves of the Wigley hull. Simple analytical expressions defining the fluid velocity at the bow and the stern, and more generally at any point along the wave profile, in terms of the elevation of the free surface at the corresponding point are also given. These analytical expressions and the available experimental measurements of wave profiles along the Wigley hull show that the velocity of the flow disturbance due to this hull is fairly small compared to the hull speed everywhere along the wave profile except in very small regions around the bow and the stern, where the total fluid velocity is nearly equal to the hull speed in magnitude but directed vertically. Nonlinearities therefore appear to be quite important, although only in very small regions surrounding a ship bow and stern. A genuine nonlinear method of calculation must then be able to represent the very rapid variation in the direction of the fluid velocity occurring within small regions around a ship bow and stern. In particular, a sufficiently fine discretization is required in these regions.

Non-linear dynamic analysis of anisotropic cylindrical shells containing a flowing fluid

An analytical model is presented to predict the influence of non-linearities associated with fluid flow on the dynamic behaviour of a structure consisting of shells and a surrounding fluid medium. The model requires the use of two linear operators to control the equilibrium of the shell and the velocity potential, a linear boundary condition of impermeability and a non-linear dynamic boundary condition. The method is based on thin shell theory and the non-rotational flow of non-viscous fluids, in combination with finite element analysis. It is applicable to non-uniform thin anisotropic cylinders subjected to different boundary conditions. The displacement functions for the wall and liquid column are derived from Sanders' equations and from the velocity field associated with the column, respectively. The set of matrices describing their relative contributions to equilibrium is determined by exact analytical integration. The coupled equations are solved for the non-flow problem. For cases of fluid flow, certain analytical modifications are proposed to restore the situation to conventional modal analysis. The non-linear equations of motion are solved by the fourth-order Runge-Kutta numerical method. The frequency variations are then determined with respect to the amplitude of the motion. The trends of the non-linearities are of a softening type.

A Two-Dimensional Potential Flow Model of the Wave Field Generated by a Semisubmerged Body in Heaving Motion

A two-dimensional potential flow model is formulated to predict the wave field and forces generated by a sere!submerged body in forced heaving motion. The potential flow problem is solved on a boundary fitted coordinate system that deforms in response to the motion of the free surface and the heaving body. The full nonlinear kinematic and dynamic boundary conditions are used at the free surface. The governing equations and associated boundary conditions are solved by a second-order finite-difference technique based on the modified Euler method for the time domain and a successive overrelaxation (SOR) procedure for the spatial domain. A series of sensitivity studies of grid size and resolution, time step, free surface and body grid redistribution schemes, convergence criteria, and free surface body boundary condition specification was performed to investigate the computational characteristics of the model. The model was applied to predict the forces generated by the forced oscillation of a U-shaped cylinder. Numerical model predictions are generally in good agreement with the available second-order theories for the first-order pressure and force coefficients, but clearly show that the third-order terms are larger than the second-order terms when nonlinearity becomes important in the dimensionless frequency range 1≤ Fr≤ 2. The model results are in good agreement with the available experimental data and confirm the importance of the third order terms.