Free Surface Boundary Conditions Research Articles

A Dimension‐reduced Pressure Solver for Liquid Simulations

AbstractThis work presents a method for efficiently simplifying the pressure projection step in a liquid simulation. We first devise a straightforward dimension reduction technique that dramatically reduces the cost of solving the pressure projection. Next, we introduce a novel change of basis that satisfies free‐surface boundary conditions exactly, regardless of the accuracy of the pressure solve. When combined, these ideas greatly reduce the computational complexity of the pressure solve without compromising free surface boundary conditions at the highest level of detail. Our techniques are easy to parallelize, and they effectively eliminate the computational bottleneck for large liquid simulations.

Fully nonlinear analysis of second-order sloshing resonance in a three-dimensional tank

The 3D boundary element method (BEM) based on the potential flow theory is developed to study the second-order resonance problems in rectangular sloshing tanks. Fully-nonlinear free-surface boundary conditions are considered. A local surface fitting method is used to calculate tangential derivatives. A shared-memory parallel procedure based on OpenMP is adopted for the efficiency improvement. Both 2D and 3D sloshing cases are studied. It is found that the second-order resonance may occur due to the interaction among natural frequencies and excitations frequencies. The second-order resonance becomes obvious after a sufficient long time length. After increasing the excitation amplitude, the second-order resonance emerges much earlier. For 3D situations, the second-order resonance in rectangular tanks with either equal or unequal length and width sides are investigated. The second-order resonances due to the interaction of two orthogonal plane waves in length and width directions are observed.

A Continuous Desingularized Source Distribution Method Describing Wave–Body Interactions of a Large Amplitude Oscillatory Body

A Rankine source method with a continuous desingularized free surface source panel distribution is developed to solve numerically a wave–body interaction problem with nonlinear boundary conditions. A body undergoes forced oscillatory motion in a free water surface and the variation of wetted body surface is captured by a regridding process. Free surface sources are placed in continuous panels, rather than points in isolation, over the calm water surface, with free surface collocation points placed on the calm water surface. Nonlinear kinematic and dynamic free surface boundary conditions along the collocation points on the calm water surface are solved in a time domain simulation based on a Lagrange time dependent formulation. Compared with isolated desingularized source points distribution methods, a significantly reduced number of free surface collocation points with sparse distribution are utilized in the present numerical computation. The numerical scheme of study is shown to be computationally efficient and the accuracy of numerical solutions is compared with traditional numerical methods as well as measurements.

Numerical simulations using conserved wave absorption applied to Navier–Stokes equation model

A relaxation method was adopted by Mayer et al. (1998) to absorb waves under a time-dependent 2-D Euler model with full nonlinear free surface boundary conditions. In this paper, we propose a conserved wave absorption method based on the relaxation method. Instead of deciding the absorption coefficient empirically, the conserved absorption method dynamically determines the absorption coefficient by solving the mass conservation equation derived in this article. In order to validate the conserved absorption method, a 2-D numerical wave flume is established and a series of numerical tests are conducted. The numerical wave tank solves the Navier–Stokes equations and constructs free surface with a VOF (volume of fluid) scheme. In the first test, the conserved absorption is validated at the end of the channel by generating regular waves, 2-order and 5-order Stokes waves and solitary waves of various amplitudes and frequencies. In the second test, the conserved absorption is tested with reflected waves, by generating standing waves in a tank with a sponge layer adjacent to the wave-generating boundary and a vertical wall at the opposite end. The third test validates the conserved absorption with both incident waves and reflected waves, i.e., the sponge layers are at both ends of the tank. The fourth test investigates the conserved absorption with large-amplitude waves under a realistic sea scale. In these tests, measurements of real sea states (Mori et al., 2002) as well as theoretical and experimental results (Luth et al., 1994) are employed to make comparison against the simulated results. It is found that the conserved wave absorption is applicable to various waves, at both the head and the end of the wave channel. In addition, it can give acceptable results with a short absorption length.

Numerical simulations of vortex-induced vibrations on vertical cylindrical structure with different aspect ratios

This paper presents a computational fluid dynamics (CFD) study of vortex-induced vibration (VIV) for different aspect ratio (L/D) cylinder. Of particular interest was to measure hydrodynamic forces and numerically investigate the wake behaviour of VIV while varying the aspect ratio. The simulation models represented the actual experimental conditions with idealised free-surface boundary condition to capture the responses from fluid-structure interaction phenomenon. The simulations were performed in the subcritical flow region (7.4×103 < Re < 2×105), corresponding to a range of reduced velocity (Ur) from 2 to 14. The results of the cases studied were discussed and compared with the experimental data to verify the accuracy and validity of the present simulation. The comparisons have shown a similar curved-shape drag coefficient plot, and however underestimated the value of the drag coefficients over the reduced velocity. Additionally, the simulations seemed to capture a higher lift force response compared with the experimental data for a low aspect ratio. The correlation length was observed to be longer for larger aspect ratio and proportionally decreases as the aspect ratio decreases.

Numerical study on three-dimensional waves produced by a bottom jet

The eruption of an underwater volcano can initiate the disturbances of the sea surface and subsequently generate a group of outward-propagating tsunamis. The theme of this study is to introduce a three-dimensional (3D) fully nonlinear wave model for the simulation of wave motions induced by a bottom jet. A boundary-fitted coordinate system is utilized to conveniently adjust grids according to the transient moving free surface. The governing Laplace equation of the velocity potential is solved by an implicit finite-difference scheme while a mixed explicit/implicit iteration procedure is applied to solve the free-surface boundary conditions. In addition, a set of generalized Boussinesq equations are solved for comparison with the fully nonlinear model. Good agreements in comparisons with the existing numerical and analytical solutions are achieved for cases investigated. Waves induced by three types of bottom jets: namely (1) sudden eruption, (2) initial transient, and (3) periodic transient are discussed in this paper. For the case of sudden erupted jet, a system of 3D outgoing waves as the cylindrical wave pattern are presented and discussed. For the initial transient types, it shows the transition in the incipient stage has a great influence on the initial rising of the water surface and the induced leading waves. Furthermore, an interesting up-down phenomenon in the center of disturbed free surface due to the type of periodic jet is revealed.

Study on Optimized Hull form of Basic Ships Using Optimization Algorithm

In order to achieve minimum resistance for a ship hull, we used an optimization algorithm and computational fluid dynamics to develop a hull-form design. An ITTC 1957 modelship correlation line formula was used to estimate the frictional resistance coefficient, and the wave-making resistance coefficient was evaluated by the Rankine source panel method with nonlinear free-surface boundary conditions. The geometry of the hull surface was represented and modified by a B-spline surface-modeling technique during the optimization process. Wigley and series 60 (C_B=0.60) hulls were selected to obtain an optimized hull that produces minimum resistance. Experimental tests were carried out for the modified series 60 (C_B=0.60) hull.

Wave resonances in a narrow gap between two barges using fully nonlinear numerical simulation

The traditional potential flow theory to describe fully nonlinear waves is reformulated by separating the contributions from incident and scattered waves, in order to improve the computational efficiency. The nonlinear incoming wave is specified explicitly and the modified nonlinear free surface boundary conditions for the scattered wave are expressed in the full Lagrangian description. At each time step only the scattered wave is solved using a mixed Eulerian–Lagrangian scheme by a higher-order boundary element method. The accuracy of the newly developed model is illustrated by comparisons with existing experimental and numerical data in the case of wave diffraction around an array of circular cylinders. Wave resonances in the gap between two side-by-side barges in beam seas, as in Molin et al. [1], are simulated with the barges subjected to regular waves. To clearly understand the gap resonant responses, long time simulations are performed to achieve final steady states, and the resonant mode shapes of the gap surface are presented. The gap free surface RAOs (Response Amplitude Operators) in the case of mild waves are found to agree well with linear calculations. The nonlinear effects on the resonant response due to the free surface conditions are then investigated. The first resonant frequency is found to shift but the peak value is not changed much with increasing incoming wave steepness, which is known as stiff/soft spring behavior of a nonlinear system. Through the investigation of barges with different drafts, the stiff and soft spring behaviors are identified.

Five-unknowns generalized hybrid-type quasi-3D HSDT for advanced composite plates

In this paper a 5-unknowns generalized hybrid-type quasi-3D HSDT for the static analysis of functionally graded single and sandwich plates is presented. Generalized hybrid-type modeling can adopted with any kind of shear strain shape functions for the inplane and transverse displacement, and therefore infinite hybrid-type (non-polynomial, polynomial, mixed type) displacement based shear deformation theory complying with the free surface boundary condition can be obtained. The key feature of this theory is that, in addition to including stretching, it has only 5 unknowns in the displacement field modeling as the first order shear deformation theory (FSDT). The generalized hybrid-type theory is also quasi-3D because the 3D Hooke’s law equation is utilized, i.e. σzz≠0. The generalized governing equations and boundary conditions are derived by employing the principle of virtual works. A generalized Navier-type closed-form solution is obtained for functionally graded single and sandwich plates subjected to transverse load for simply supported boundary conditions. Analytical results from the new generalized hybrid-type quasi-3D higher order shear deformation theory (HSDT) are compared with the FSDT, other quasi-3D HSDTs, and refined HSDTs. The fundamental conclusions that emerge from the present numerical results suggest that: (a) infinite shears strain shape function can be evaluated by using the present theory; (b) polynomial shear strain functions appear to be a good choice for the implementing of a quasi-3D HSDT based on this generalized quasi-3D hybrid type HSDT; (c) this generalized theory can be as accurate as the 6-unknown generalized hybrid-type quasi-3D HSDT; (d) the best HSDT with stretching effect and 5-unknows can be obtained from the present generalized theory, this can be done by optimizing a theory that for example has a given non-polynomial inplane and transverse shears strain shape functions.

Constant acceleration exit of two-dimensional free-surface-piercing bodies

The forced constant acceleration exit of two-dimensional bodies through a free-surface is computed for various 2D bodies (symmetric wedges, asymmetric wedges, truncated wedges and boxes). The calculations are based on the fully non-linear time-stepping complex-variable method of Vinje and Brevig (1981). The model was formulated as an initial boundary-value problem (IBVP) with boundary conditions specified on the boundaries (dynamic and kinematic free-surface boundary conditions) and initial conditions at time zero (initial velocity and position of the body and free-surface particles). The formulated problem was solved by means of a boundary-element method using collocation points on the boundary of the domain and stepped forward in time using Runge–Kutta and Hamming predictor–corrector methods. Numerical results for the deformed free-surface profile, pressure along the wetted region of the bodies and force experienced by the bodies are given for the exit. The analytical added-mass force is presented for the exit of symmetric wedges and boxes with constant acceleration using conformal mappings. To verify the numerical results, the added-mass force and the numerical force are compared and give good agreement for the exit of a symmetric wedge at a time zero (t=0) as expected but only moderate agreement for the box.

Space potential particles to enhance the stability of projection-based particle methods

Particle methods have been seldom verified by a Karman vortex simulation, which is commonly performed as a typical benchmark in computational fluid dynamics. This is mainly due to a difficulty in suppression of occurrence of unphysical voids manifested usually in a strong vortex on account of definition of free surface by the Lagrangian tracking framework with inconsistency in volume conservation. This paper presents a simple and effective scheme as a free-surface boundary condition of projection-based particle methods, namely the MPS (moving particle semi-implicit) and Incompressible SPH (ISPH) methods to handle the free surface with consistency in volume conservation. The new scheme is introduced into the Poisson pressure equation (PPE) with consideration of a potential in void space as space potential particle (SPP), to reproduce physical motions of particles around free surface through a particle–void interaction. The enhancing effect of the newly proposed SPP scheme is shown by simulating a few numerical tests, including a whirling water flow, a two-phase surfacing flow, and a set of Karman vortex simulations.

Optimization of linear and nonlinear sidewall storage units using coupled boundary element-finite element methods

In this research, a numerical model is developed for optimization of storage tanks based on sloshing phenomenon. The sloshing of the liquid can increase the dynamic pressure on the partially filled storage tank sidewalls and bottom to endanger its integrity. The coupled boundary element-finite element methods were used to solve the Laplace equation as governing equation and nonlinear free surface boundary conditions. The numerical model is validated against available data. Different sections for the liquid storage tanks such as rectangular tank with different aspect ratios, trapezoidal tanks with linear and nonlinear sidewalls, different sidewall angles and different widths were examined. Finally the best cross sections of storage tank based on minimization of the pressure and forces exerted on the perimeter of the tank due to sloshing phenomenon were suggested. The results showed that using nonlinear sidewalls in storage tank dramatically decreases the sloshing pressure on the perimeter of the tank.

Nonlinear instability and convection in a vertically vibrated granular bed

AbstractThe nonlinear instability of the density-inverted granular Leidenfrost state and the resulting convective motion in strongly shaken granular matter are analysed via a weakly nonlinear analysis of the hydrodynamic equations. The base state is assumed to be quasi-steady and the effect of harmonic shaking is incorporated by specifying a constant granular temperature at the vibrating plate. Under these mean-field assumptions, the base-state temperature decreases with increasing height away from the vibrating plate, but the density profile consists of three distinct regions: (i) a collisional dilute layer at the bottom, (ii) a levitated dense layer at some intermediate height and (iii) a ballistic dilute layer at the top of the granular bed. For the nonlinear stability analysis (Shukla &amp; Alam, J. Fluid Mech., vol. 672, 2011b, pp. 147–195), the nonlinearities up to cubic order in the perturbation amplitude are retained, leading to the Landau equation, and the related adjoint stability problem is formulated taking into account appropriate boundary conditions. The first Landau coefficient and the related modal eigenfunctions (the fundamental mode and its adjoint, the second harmonic and the base-flow distortion, and the third harmonic and the cubic-order distortion to the fundamental mode) are calculated using a spectral-based numerical method. The genesis of granular convection is shown to be tied to a supercritical pitchfork bifurcation from the density-inverted Leidenfrost state. Near the bifurcation point the equilibrium amplitude ($A_{e}$) is found to follow a square-root scaling law, $A_{e}\sim \sqrt{{\it\Delta}}$, with the distance ${\it\Delta}$ from the bifurcation point. We show that the strength of convection (measured in terms of velocity circulation) is maximal at some intermediate value of the shaking strength, with weaker convection at both weaker and stronger shaking. Our theory predicts that at very strong shaking the convective motion remains concentrated only near the top surface, with the bulk of the expanded granular bed resembling the conduction state of a granular gas, dubbed as a floating-convection state. The linear and nonlinear patterns of the density and velocity fields are analysed and compared with experiments qualitatively. Evidence of 2:1 resonance is shown for certain parameter combinations. The influences of bulk viscosity, effective Prandtl number, shear work and free-surface boundary conditions on nonlinear equilibrium states are critically assessed.

Fully nonlinear solution of bi-chromatic deep-water waves

Fully nonlinear bi-chromatic unidirectional waves propagating in deep-water are investigated using the homotopy analysis method. The velocity potential of the waves is expressed by Fourier series and the nonlinear free surface boundary conditions are satisfied by continuous mapping. The bi-chromatic wave elevation and velocity profiles underneath the wave crest and trough are presented and compared with the available perturbation results. Unlike the perturbation method, the present approach is not dependent on small parameters; therefore solutions are possible for steep waves. The Fast Fourier Transform analysis is then applied to study the effect of higher order wave components. The fully nonlinear dispersion relation is established. Comparisons of the wave characteristics demonstrate that the present method is effective to study the strongly nonlinear wave–wave interactions.

Super-Grid Modeling of the Elastic Wave Equation in Semi-Bounded Domains

AbstractWe develop a super-grid modeling technique for solving the elastic wave equation in semi-bounded two- and three-dimensional spatial domains. In this method, waves are slowed down and dissipated in sponge layers near the far-field boundaries. Mathematically, this is equivalent to a coordinate mapping that transforms a very large physical domain to a significantly smaller computational domain, where the elastic wave equation is solved numerically on a regular grid. To damp out waves that become poorly resolved because of the coordinate mapping, a high order artificial dissipation operator is added in layers near the boundaries of the computational domain. We prove by energy estimates that the super-grid modeling leads to a stable numerical method with decreasing energy, which is valid for heterogeneous material properties and a free surface boundary condition on one side of the domain. Our spatial discretization is based on a fourth order accurate finite difference method, which satisfies the principle of summation by parts. We show that the discrete energy estimate holds also when a centered finite difference stencil is combined with homogeneous Dirichlet conditions at several ghost points outside of the far-field boundaries. Therefore, the coefficients in the finite difference stencils need only be boundary modified near the free surface. This allows for improved computational efficiency and significant simplifications of the implementation of the proposed method in multi-dimensional domains. Numerical experiments in three space dimensions show that the modeling error from truncating the domain can be made very small by choosing a sufficiently wide super-grid damping layer. The numerical accuracy is first evaluated against analytical solutions of Lamb’s problem, where fourth order accuracy is observed with a sixth order artificial dissipation. We then use successive grid refinements to study the numerical accuracy in the more complicated motion due to a point moment tensor source in a regularized layered material.

Large-Scale Liquid Simulation on Adaptive Hexahedral Grids.

Regular grids are attractive for numerical fluid simulations because they give rise to efficient computational kernels. However, for simulating high resolution effects in complicated domains they are only of limited suitability due to memory constraints. In this paper we present a method for liquid simulation on an adaptive octree grid using a hexahedral finite element discretization, which reduces memory requirements by coarsening the elements in the interior of the liquid body. To impose free surface boundary conditions with second order accuracy, we incorporate a particular class of Nitsche methods enforcing the Dirichlet boundary conditions for the pressure in a variational sense. We then show how to construct a multigrid hierarchy from the adaptive octree grid, so that a time efficient geometric multigrid solver can be used. To improve solver convergence, we propose a special treatment of liquid boundaries via composite finite elements at coarser scales. We demonstrate the effectiveness of our method for liquid simulations that would require hundreds of millions of simulation elements in a non-adaptive regime.

Dynamic Response of Submerged Vertical Cylinder with Lumped Mass under Seismic Excitation

An analytical approach is presented to assess the response of offshore structures under seismic excitation. This paper evaluates the impacts of different fluid field models and the mass of equipment at the top of offshore structure which is simulated as lumped mass on the responses of offshore structures. To do this, two and three dimensional fluid field models are developed. In three dimensional models different approximation regarding the free surface boundary condition associated with high and low frequency excitations are adopted. Then the alternation of response of structure with changing in the value of lumped mass is calculated. Finally the impacts of different models on the value of maximum displacement for Kobe earthquake are evaluated. It is shown that different approximations regarding the fluid field could largely change the value of maximum displacement evaluated by the models.

An iterative Rankine boundary element method for wave diffraction of a ship with forward speed

A 3-D time-domain seakeeping analysis tool has been newly developed by using a higher-order boundary element method with the Rankine source as the kernel function. An iterative time-marching scheme for updating both kinematic and dynamic free-surface boundary conditions is adopted for achieving numerical accuracy and stability. A rectangular computational domain moving with the mean speed of ship is introduced. A damping beach at the outer portion of the truncated free surface is installed for satisfying the radiation condition. After numerical convergence checked, the diffraction unsteady problem of a Wigley hull traveling with a constant forward speed in waves is studied. Extensive results including wave exciting forces, wave patterns and pressure distributions on the hull are presented to validate the efficiency and accuracy of the proposed 3-D time-domain iterative Rankine BEM approach. Computed results are compared to be in good agreement with the corresponding experimental data and other published numerical solutions.

Viscous and potential forces on an advancing surface-piercing flat plate with a fixed drift angle

Transverse force and yaw moment acting on a surface-piercing flat plate with yaw angle and forward speed are studied. Following the Froude hypothesis, the problem is decomposed into a tail-separated forward flow and a bottom-tip-separated cross-flow parts. The objective is to quantify the role of different components in the resulted force and moment acting on the plate. A 3D potential flow code using distribution of Rankine sources and dipoles is used to solve the linear potential flow problem. The free-surface boundary condition is linearized based on the undisturbed flow velocity (Neumann–Kelvin linearization). The plate’s trailing-edge flow separation is modeled using a dipole distribution behind the plate on a linearized vortex sheet. The force and moment due to the cross-flow separation from the plate’s bottom tip are calculated by means of a slender body cross-flow method with a rigid free-surface boundary condition. Hence the free-surface effects are confined in the forward flow problem and neglected in the cross-flow problem. Simulations are carried out for different aspect ratios and drift angles. The plate’s thickness is taken into account. Convergence and sensitivity studies are performed carefully. Results are compared with previous experimental and numerical studies. The importance of the second-order forces and validity of the linear assumptions are touched upon. The overall agreement of the results is acceptable. The relative importance of the two viscous and potential components are studied. It is shown that the decomposed problem could follow the experimental results up to a relatively high Froude number.

A steady-state approach for evaluation of surge and swab pressures in flows with free surface boundary conditions

A new approach for predicting surge and swab pressures in open-ended pipes running in steady-state is proposed. In addition to the fluid flow through the annulus and drill pipe, the upper ends of drill pipe and annular space are assumed to be opened to the atmosphere, creating two free surfaces. The approach is based on the modified Bernoulli׳s equation applied to Bingham fluid flows. The pressure differences are not only due to viscous losses but also to fluid velocity differences, and to the fluid column weight differences. The model results are compared with measured data for a Newtonian fluid and the differences lie within an error range of 0% to +10%. A sensitivity analysis is conducted, depicting that the model depends on the Reynolds, Bingham and Froude numbers, and also on the pipe geometry (diameter ratios). • A new approach for evaluation of surge and swab pressures is proposed. • The pressure gradient is as a function of the Reynolds, Bingham and Froude numbers. • The differences between measured and computed values lie within a 10% error band.