Transient Free Surface Research Articles

Studying macro- and mesoscopic wetting dynamics of a spreading oil droplet using multiple wavelength interferometry

In this study we present an interferometric technique based on multiple wavelengths to capture the transient free surface contour of nanolitre drops spreading on a wettable surface, in particular close to the three-phase contact line. Various data analysis procedures are evaluated in terms of error and noise sensitivity. The technique allows an unambiguous determination of the local liquid film thickness for optical path differences up to Δs≈3.19μm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta s \\approx 3.19\\,\\upmu \ ext {m}$$\\end{document} without the need of a known reference height. Film thicknesses as low as 0.1μm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0.1\\,\\upmu \ ext {m}$$\\end{document} can be measured with the present optical configuration. The entire three-dimensional droplet shape is investigated for different capillary numbers, allowing also reliable measurements of the time-resolved contact angle.

A Dimension-Reduced Line Element Method for 3D Transient Free Surface Flow in Porous Media

In order to reduce the numerical difficulty of the 3D transient free surface flow problems in porous media, a line element method is proposed by dimension reduction. Different from the classical continuum-based methods, homogeneous permeable pores in the control volume are conceptualized by a 3D orthogonal network of tubes. To obtain the same hydraulic solution with the continuum model, the equivalent formulas of flow velocity, continuity equation and transient free surface boundary are derivable from the principle of flow balance. In the solution space of transient free surface flow, the 3D problem is transformed into 1D condition, and then a finite element algorithm is simply deduced. The greatest advantage of the line element method is line integration instead of volume/surface integration, which has dramatically decreased the integration difficulty across the jump free surface. Through the analysis of transient free surface flow in the unconfined aquifer, trapezoidal dam, sand flume and wells, the transient free surface locations predicted from the proposed line element method generally agree well with the analytical, experimental and other numerical data in the available literatures, the numerical efficiency can also be well guaranteed. Furthermore, the hydraulic anisotropy has significant effect on the evolution of free surface locations and the shape of depression cones in spatial. The line element method can be expanded to model the 3D unsaturated seepage flow, two-phase flow and thermos problems in porous media because of the similarity between the similarity of Darcy’s law, Buckingham Law and Fourier’s law.

Sigma mapping for drainage problems with a time-dependent water table

Regional groundwater flow forecasting is frequently accomplished using model equations based on Darcy's law, the continuity equation, and kinematic and dynamic boundary conditions for a time-dependent water table, to predict hydraulic heads and fluxes. For homogeneous and isotropic aquifers, the model reduces to Laplace's equation for the hydraulic head and a transient non-linear free surface boundary condition. These equations are solved in the literature and in computer packages by a variety of numerical methods, including finite difference, finite element, and boundary element methods. The solution of this model is not simple and subject to instability problems. Computations in a vertical plane using finite-difference methods are not easy due to the irregular mesh needed near a highly curved water table, frequently involving smoothing of the numerical results to ensure stability. A technique to avoid the complications related to mesh generation near a curved boundary and its tracking entails using mappings to produce transformed domains. However, this is not so frequent to compute unsteady seepages. In particular, the so-called sigma mapping is widely used to model irrotational water waves; it applies to any flow involving a time-dependent free surface boundary. This mapping transforms the domain into a rectangle, so that the free surface becomes static in the transformed plane. There is close similarity between the mathematical model describing irrotational water waves and that of unsteady seepage flow, e.g., both are governed by Laplace's equation involving relevant boundary conditions at the moving boundary. Both moving free surfaces are different in character, e.g., the seepage is dissipative whereas the water wave is dispersive. However, it is possible to transfer mathematical approximations between both fields of research. This transfer is exploited in this work and the sigma mapping is newly and successfully applied to seepage flows involving both a steady and unsteady water table either receiving, or not, a recharge from rainfall or an artificial source.

Gravity-driven film flow inside an inclined corrugated pipe: An experimental investigation of corrugation shape and tip width

Fluorescence images were acquired in gravity-driven film flow through inclined corrugated pipes representing a range of corrugation shapes and tip widths. The film flow developed an identifiable statically deformed free surface with a wavelength similar to the substrate for most cases of corrugation shape and tip width. The amplitude and phase shift of the statically deformed free surface, as well as the steady-state film thickness, varied more with tip width than with corrugation shape. Transient fluctuations in the free surface elevation were examined for evidence of periodic traveling waves. In general, the film flow produced transient free surface fluctuations, and in many cases, periodic traveling waves with parameters that varied similarly with corrugation shape as with tip width. For flow conditions that produced positive phase shift, low amplitude, or minimal curvature of the statically deformed free surface, transient and periodic behavior were suppressed, supporting previous findings on the importance of the shape and position of the statically deformed free surface. An increase in corrugation tip width also reduced the transient and periodic response. These two findings implicate flow dynamics in the substrate trough as a leading factor in the development of transient and periodic behavior. Steady-state response and the existence of time-dependent behavior are influenced more by tip width than corrugation shape, in agreement with two-dimensional film flow over topography; however, transient fluctuation and periodic traveling wave parameters are similarly influenced by corrugation shape and tip width, which contrasts two-dimensional findings.

SPH simulations of transient non-isothermal viscoelastic flows with free surfaces

• Transient non-isothermal viscoelastic free surface flows are simulated by SPH. • SPH discretization of the temperature equation is derived. • The impacting droplet and the filling process of circular disc with core are studied. • Both Oldroyd-B and UCM fluid models are considered. • The work provides references to handle non-isothermal viscoelastic flows in an entirely meshfree framework. This study presents a smoothed particle hydrodynamics (SPH) method for simulating transient non-isothermal viscoelastic flows with free surfaces. The basic equations governing the non-isothermal free surface flow of an Oldroyd-B fluid are considered and approximated by SPH. To model non-isothermal fluid flows, a working SPH discretization of the temperature equation is derived, and the temperature-dependent viscosity and relaxation time are represented by the time–temperature superposition principle. The proposed SPH method is first validated by solving the viscoelastic Poiseuille flow problem under isothermal and non-isothermal conditions and comparing the SPH solutions with the analytical solutions or those obtained by other methods. Then, the SPH method is extended to deal with the impact of a non-isothermal viscoelastic droplet with a cold wall and the non-isothermal filling process of a circular disc with a core. The convergence of the method is verified by four particle spacings of different levels of refinement. The effects of some important parameters, including the temperature dependency coefficient, reference temperature of the fluid, and Péclet number, on the flow behavior are analyzed in detail. All the results demonstrate that the proposed SPH method is capable of accurately and stably simulating transient non-isothermal viscoelastic free surface flows in an entirely meshfree framework.

A Fully Coupled Time-Domain BEM-FEM Method for the Prediction of Symmetric Hydroelastic Responses of Ships with Forward Speed

This paper presents a direct time-domain method for the prediction of symmetric hydroelastic responses of ships progressing with forward speed in small amplitude waves. A transient time-domain free surface Green function is used for the idealisation of the seakeeping problem using an Earth fixed coordinate system. Free surface ship hydrodynamics are idealised in the time domain by a Green function, and forward speed effects are idealised by a space-state model. Modal actions are accounted for by Timoshenko beam structural dynamics. Flexible fluid structure interaction (FFSI) coupling is enabled by a body boundary condition, and a direct integration Newmark- β scheme is used to obtain symmetric dynamic responses. The method is validated against available published numerical and experimental results. A parametric study for different container ship hull forms confirms that (i) forward speed effects should be taken under consideration as far as practically possible and (ii) hull flexibility effects accounting for hull shear deformation and rotary inertia are more notable for slender hull forms.

The effect of substrate amplitude and wavelength on gravity-driven film flow inside an inclined corrugated pipe

An experimental investigation was conducted to determine the effect of substrate amplitude and wavelength on gravity-driven film flow inside an inclined corrugated pipe. Nine different geometries were examined, with substrate amplitude and wavelength varied independently. A statically deformed free surface occurred for all conditions. The amplitude of the statically deformed free surface depended on substrate amplitude and wavelength, with phase shift unaffected by changes in substrate geometry for many conditions under investigation. Fluctuations in free surface elevation were enhanced at low substrate amplitude and intermediate substrate wavelength. Notable reductions in transient free surface behavior were observed for conditions that resulted in a positive phase shift. Transient free surface behavior developed into periodic traveling waves without applied external forcing. Frequency selection for traveling waves was strong, and traveling waves were detected for a majority of the conditions examined. The frequency, phase velocity, and wavelength of the traveling waves showed a potential dependence on substrate geometry; however, there were ranges of substrate amplitude and wavelength for which traveling wave characteristics remained unaffected by changes in substrate geometry. An examination of the amplitude of the statically deformed free surface and transient free surface fluctuations revealed that waviness is a potentially suitable method for combining the effects of substrate amplitude and wavelength on film flow in corrugated pipes. The comparison of amplitudes highlighted a possible link between the statically deformed free surface and the emergence of transient behavior and traveling waves. Length scales proposed in our original work showed promise for characterizing some results.

Influence of mixed free-surface and pressurized flow on transients process in a tailrace tunnel of a tailrace surge chamber

In the calculation of hydraulic transients in a hydropower station, the hydraulic problems caused by free-surface and pressurized flow are relatively complex. In consideration of this phenomenon, numerical simulation is used in this paper to study the effect of the transient free-surface, pressurized flow in a tailrace tunnel on the minimum pressure at inlet of the draft tube and the lining segment of the tailrace tunnel and the effect on the primary frequency regulation of the unit. The results show that the existence of tailrace surge tank can reduce the disturbance of the mixture flow on the units. Meanwhile, free-surface and pressurized flow can speed up the attenuation of the surge in the surge tank. Reasonable arrangement can mitigate transients in the tailrace system, which may provide a reference for similar projects.

Time domain TEBEM method of ship motion in waves with forward speed by using impulse response function formulation

A numerical method for solving 3D unsteady potential flow problem of ship advancing in waves is proposed. The flow field is divided into inner and outer domain by an artificial matching surface. The inner domain includes ship-wetted surface, matching surface and free surface inside. The free-surface condition is formulated by perturbation about the double-body flow (DB model) or uniform incoming flow assumption (NK model). The outer domain contains matching surface, free surface outside and infinite far-field boundary, where only NK model is applied. The simple Green function and transient free surface Green function are used to form the boundary integral equation (BIE) by the impulse response function method for the inner and outer domain. Taylor Expansion Boundary Element Method (TEBEM) is utilized to solve the BIEs. Matching conditions for the inner-domain and outer-domain flow are enforced by the continuity of velocity potential and normal velocity on the matching surface. In this paper, several ships are selected to predict the hydrodynamic coefficients and displacement at different forward speed. The experiment measurements in physical tank is also used to validate the accuracy of numerical results. Good agreement of 6DOF ship motions in various wave headings can be obtained using TEBEM method.

Numerical investigation on a container ship navigating in irregular waves by a fully nonlinear time domain method

A three dimensional fully nonlinear time domain method is introduced to simulate ship advancing in regular and irregular waves with forward speed. The mixed Eulerian-Lagrangian (MEL) method and boundary element method are adopted to solve the boundary value problem. Spring analogy method is utilized to guarantee the optimization mesh of the transient free surface and wetted hull body. A local coordinate system is introduced, and the fully nonlinear disturbed wave is separated from the total wave. Auxiliary functions are deduced to decouple the strongly nonlinear hydrodynamic forces and ship motion. Then the motion and velocity potential can be updated with the fourth-order Runge Kutta method. A 13500 TEU container ship is chosen to verify the numerical codes, the numerical results in regular waves and irregular waves are both in good agreement with experimental data. The instantaneous pressure distribution around wetted body surface and wave elevation on the free surface of dangerous moments are presented and the strongly nonlinear states in the harsh irregular waves are analyzed.

Time-domain TEBEM method for mean drift force and moment of ships with forward speed under the oblique seas

A numerical method for solving 3D unsteady potential flow problem of ship advancing in waves is put forward. The flow field is divided into an inner and an outer domain by introducing an artificial matching surface. The inner domain is surrounded by ship wetted surface and matching surface as well as part of the free surface. The free surface condition for the inner domain is formulated by perturbation about the double-body flow or uniform incoming flow assumption. The outer domain is surrounded by matching surface and the rest free surface as well as infinite far-field radiation boundary. The free surface condition for the outer domain is formulated by perturbation about uniform incoming flow. The simple Green function and transient free surface Green function are used to form the boundary integral equation (BIE) for the inner and outer domains, respectively. Taylor Expansion Boundary Element Method (TEBEM) is utilized to solve the double-body flow and inner domain and outer domain unsteady flow BIE. Matching conditions for the inner domain flow and outer domain flow are enforced by the continuity of velocity potential and normal velocity on the matching surface. Direct pressure integration on ship wetted surface is used to obtain the first-order and second-order wave forces (moments). The numerical predictions on the displacement, added resistance, sway mean drift force and yaw mean drift moment of the modified KVLCC2 ship at different forward speeds are investigated by the proposed TEBEM method. It is also compared with the other numerical results. The physical tank experiment results are also developed to validate the accuracy of numerical tank results. Compared with the experiment solutions, a good agreement can be obtained by TEBEM method.

Hydrodynamic Analysis and Motions of Ship with Forward Speed via a Three-Dimensional Time-Domain Panel Method

A new three-dimensional (3D) time-domain panel method is developed to solve the ship hydrodynamic problem and motions. For an advancing ship with a constant forward speed in regular waves, the ship’s hull can be discretized and processed into a number of quadrilateral panels. Based on Green’s theorem, an analytical expression for Froude–Krylov (F–K) forces evaluation on the quadrilateral panels is derived without accuracy loss. Within the linear potential theory, the transient free surface Green function (TFSGF) is applied to solve the boundary value problem. To improve the efficiency and numerical stability of TFSGF evaluation, a precise integration method with variable parameters setting for extended identity matrix is developed to compute the TFSGF in the computation domain. Then, radiation and diffraction forces can be evaluated by means of the impulse response function method. The Wigley I hull form is taken as a study case, and the computed hydrodynamic coefficients, wave exciting forces, and motions by the present method are compared with previous literature experimental data and prior published results. It manifests that the three-dimensional time-domain panel method proposed in this paper has good accuracy.

A one-dimensional line element model for transient free surface flow in porous media

A simple and physically sound model has been proposed to investigate the transient free surface flow behavior in porous media. Instead of the continuum-based model, the pore space between solid grains is conceptualized as a network of horizontal and vertical flow paths and then the flow paths are characterized by straight line elements. Based on the Darcy's law and flux equivalence between the continuum-based model and line element model, equivalent hydraulic parameters and continuity equation are derived through the control volume. By introducing a local coordinate system, unified governing equations and equivalent boundary conditions of the line element model are obtained in the form of one-dimensional formulations. The two-dimensional transient free surface flow problem is reduced to a one-dimensional problem of line element network and a finite-element algorithm is developed. The proposed one-dimensional line element model is verified by three illustrative examples, which indicate that the numerical results can satisfactorily match the analytical solutions and experimental observations from different sources. It is found that the solution accuracy has weak sensitivity of time-step size but sensitive to the spacing size, while numerical efficiency can be well guaranteed for different time-step sizes and spacing sizes. Based on the numerical results, the proposed model can give better performance for the evolution of free surfaces than the residual flow procedure, and is efficient to model transient surface flow with a complex drainage system.

Prediction of Ship Motions Via a Three-Dimensional Time-Domain Method Following a Quad-Tree Adaptive Mesh Technique

Abstract A three-dimensional (3D) time-domain method is developed to predict ship motions in waves. To evaluate the Froude-Krylov (F-K) forces and hydrostatic forces under the instantaneous incident wave profile, an adaptive mesh technique based on a quad-tree subdivision is adopted to generate instantaneous wet meshes for ship. For quadrilateral panels under both mean free surface and instantaneous incident wave profiles, Froude-Krylov forces and hydrostatic forces are computed by analytical exact pressure integration expressions, allowing for considerably coarse meshes without loss of accuracy. And for quadrilateral panels interacting with the wave profile, F-K and hydrostatic forces are evaluated following a quad-tree subdivision. The transient free surface Green function (TFSGF) is essential to evaluate radiation and diffraction forces based on linear theory. To reduce the numerical error due to unclear partition, a precise integration method is applied to solve the TFSGF in the partition computation time domain. Computations are carried out for a Wigley hull form and S175 container ship, and the results show good agreement with both experimental results and published results.

An ISPH with k–ε closure for simulating turbulence under solitary waves

In this paper an Incompressible Smoothed Particle Hydrodynamics (ISPH) method solving the 2D RANS (Reynolds Averaged Navier-Stokes) equations with the k–ε turbulence closure is constructed. In the present model, the concept of “massless ISPH” utilizing the definition of “particle density” (number of computational particles within unit volume) is stressed. The skills of this numerical model are tested by applying to two laboratory experiments: (1) A non-breaking solitary wave propagating over a bottom-mounted barrier and (2) a solitary wave breaking on a 1 on 50 slope. In the former case flow separation occurs behind the barrier as the wave crest passes by and a vortex is generated, which later interacts with free surface causing breaking. In the latter case wave breaking and bottom friction both generate significant turbulence. For both cases, the effects of initial seeding of turbulent kinetic energy, required in the k–ε model, are studied and it is concluded that initial values of O (10−10) to O (10−8) m2/s2 should be used. An adaptive wall boundary condition for k–ε turbulence model is employed to avoid the unrealistic production of turbulence near the wall boundary. The numerical results, in terms of free surface profile, mean velocity field, vorticity field, turbulent kinetic energy and turbulent shear stress, are compared with experimental data. Very reasonable agreement is observed. This paper presents the first comprehensively validated 2D ISPH model with the k–ε turbulence closure, which can be applied to transient free surface wave problems.

Froude–Krylov nonlinear computations of three dimensional wave loads by a hybrid time domain boundary element method

Wave-induced loads on advancing ships are studied by using a three dimensional nonlinear hybrid high order boundary element method with the combination of the Rankine source Green function applied in the internal domain and the transient free surface Green function adopted in the external domain. A quadratic representation of both the boundary and the unknowns on the boundary is adopted. In the internal domain the forward speed problem is solved with the steady ship wave as the basis flow. In the external domain, an efficient algorithm combined with Chebyshev approximations and asymptotic expressions for the transient free surface Green function is adopted to the solution of radiation condition. The nonlinear Froude–Krylov and hydrostatic restoring forces resulted from large amplitude incident waves are taken into consideration over the instantaneous wetted hull surface. This paper presents an effective solution to the problems of excessive memory occupation and large time consuming in the long-duration simulation. A nonlinear numerical program is originally developed and the preliminary linear motions of Wigley-3 and S175 container hull at relatively high speed are analyzed. Furthermore, the nonlinear behavior of hydrodynamic forces, motions, and the wave loads of S175 hull are investigated in regular and irregular seas, especially the phenomenon of ‘super harmonics’ in severe regular waves are highlighted thoroughly. Comparisons with experiment results indicate the present method owns satisfactory numerical stability, accuracy, and efficiency for the practical application.

Comparative study on ship motions in waves based on two time domain boundary element methods

This paper presents a comparative study on the motions of a ship advancing in waves using two different three-dimensional time domain boundary element methods: a Transient Free surface Green's Function (TFGF) method and a Rankine Higher Order Boundary Element Method (HOBEM). Three models based on the HOBEM are also considered to identify the dominant factors affecting the accuracy of solutions: (i) the N–K model using Neumann–Kelvin (N–K) linearization, (ii) the N–K+DB model using the N–K linearization for the free surface boundary conditions and the double-body (DB) m-terms in the body boundary condition, (iii) the Rankine HOBEM based on DB linearization. The Wigley I and the Series 60 (CB = =0.7) are taken as study objects. The comparisons show that the Rankine HOBEM based on DB linearization is generally more accurate than the Rankine HOBEM with other two models and the TFGF method. The hydrodynamic coefficients are mainly affected by the m-terms, especially at low frequencies; while the influences of different linearizations of the free surface boundary conditions are negligible. Both the m-terms and the leading terms of steady velocity potential reserved in the free surface boundary conditions are important for motion responses, while the influence of the former is stronger than the latter.

Modified incompressible SPH method for simulating free surface problems using highly irregular multi-resolution particle configurations

In this study, the conventional incompressible smoothed particle hydrodynamics (I-SPH) formulations are modified to be applicable for highly irregular multi-resolution particle distributions for simulation of transient free surface flows. The modifications are based on distinguishing between particle density calculated by interpolation functions in SPH method and fluid density as a physical quantity. I-SPH is a fully Lagrangian method having great potential in dealing with large deformations of the free surface. Flow domain is discretized into a set of moving particles, and each particle carries mass, velocity and pressure during simulation time. Variations in these field parameters can be determined by temporal discretization of Navier–Stokes equation followed by a spatial discretization using I-SPH method. In addition to this, three pressure gradient models are employed for simulation of free surface flows to examine efficiency of the models coupled with the modifications. Furthermore, a simple and efficient method to find free surface particles which is compatible with highly irregular multi-resolution particle distributions is proposed. The modified approach is tested against different free surface problems using highly irregular particle configurations. The classic pressure gradient model employed in SPH method coupled with constant smoothing distance gives accurate results with lower computational times over highly irregular particle distributions. The method can be applied for adaptive refinement strategy to reduce computational times of transient free surface problems.

Prediction of added resistance of a ship in waves at low speed

This paper presents predictions of the added resistance of a ship in waves at a low speed according to the IMO minimum propulsion power requirement by a hybrid Taylor expansion boundary element method (TEBEM). The flow domain is divided into two parts: the inner domain and the outer domain. The first-order TEBEM with a simple Green function is used for the solution in the inner domain and the zero order TEBEM with a transient free surface Green function is used for the solution in the outer domain. The TEBEM is applied in the numerical prediction of the motions and the added resistance in waves for three new designed commercial ships. The numerical results are compared with those obtained from the seakeeping model tests. It is shown that the prediction of the ship motions and the added resistance in waves are in good agreement with the experimental results. The comparison also indicates that the accuracy of the motion estimation is crucial for the prediction of the wave added resistance. In general, the TEBEM enjoys a satisfactory accuracy and efficiency to predict the added resistance in waves at a low speed according to the IMO minimum propulsion power requirement.

Analysis of the hydroelastic effect on a container vessel using coupled BEM–FEM method in the time domain

ABSTRACTThis paper investigates numerically the effect of flexibility for a container ship in calculation vertical bending moment and shear force etc. For this purpose, the hydrodynamic forces are determined using a lower order panel method based on 3D transient free surface Green’s function. For the solution of the structural problem, 1D FEM is developed in the time domain. The Panel code and the FEM code are coupled in the time domain through the body boundary condition. The structural deformation and velocity at each section in each time step are calculated using the Newmark–Beta method. Structural deflection, vertical bending moment, shear force, dynamic pressure distribution along the hull for S175 Ship are computed for both rigid and flexible structure and compared to check the influence of the flexibility. From the results, it is seen that the structural flexibility plays a very important role for bending moment and shear force calculations.