Free-surface Equations Research Articles

Implicit Bidiagonal Scheme for Depth-Averaged Free-Surface Flow Equations

A general fast implicit bidiagonal numerical scheme, based on the MacCormack's predictor-corrector technique requiring the inversion of only block bidiagonal matrices, has been developed and subsequently applied for subcritical and supercritical free-surface flow calculations. The model has been applied to depth-averaged steady flows. There are two main advantages of the proposed method: the technique has fast convergence and utilizes a body fitted nonorthogonal local coordinate system to simulate irregular geometry flows. The model is used to analyze a wide variety of hydraulic engineering problems including flows in a converging-diverging subcritical flume, supercritical expansions at various Froude numbers, and supercritical converging chutes. For each of these test cases, the calculated results are compared with experimental data. The comparisons with measurements as well as with other numerical solutions show that the proposed method is comparatively accurate, fast, and reliable.

Special cylindrical free surface jets of liquid crystalline polymers and their stability

We examine two special solution families of the governing equations for free surface, axisymmetric, cylindrical liquid crystalline polymer (LCP) jet flows, in which only the uniaxial orientation pattern is accommodated. From these ideal transient solutions, we analyze the linearized stability of the coupled flow and microstructure in the LCP slender jet approximation. Our analysis reveals a window of uniaxial order parameters for both jets in which the LCP microstructure yields a negative first normal stress difference that may completely stabilize the jets provided the negative first normal stress difference dominates the isotropic surface tension and viscous stress during some period of time. It is shown that the negative first normal stress difference in the stable jets is the result of the low degree of orientation in the microstructural material.

An unstructured grid-based, parallel free surface solver

An unstructured grid-based, parallel-free surface solver is presented. The overall scheme combines a finite-element, equal-order, projection-type 3-D incompressible flow solver with a finite element, 2-D advection equation solver for the free surface equation. For steady-state applications, the mesh is not moved every timestep, in order to reduce the cost of geometry recalculations and surface repositioning. A number of modifications required for efficient processing on shared-memory, cache-based parallel machines are discussed, and timings are shown that indicate scalability to a modest number of processors. The results show good quantitative comparison with experiments and the results of other techniques. The present combination of unstructured grids (enhanced geometrical flexibility) and good parallel performance (rapid turnaround) should make the present approach attractive to hydrodynamic design simulations.

Finite element solution of free-surface ship-wave problems

An unstructured finite element solver to evaluate the ship-wave problem is presented. The scheme uses a non-structured finite element algorithm for the Euler or Navier–Stokes flow as for the free-surface boundary problem. The incompressible flow equations are solved via a fractional step method whereas the non-linear free-surface equation is solved via a reference surface which allows fixed and moving meshes. A new non-structured stabilized approximation is used to eliminate spurious numerical oscillations of the free surface. Copyright © 1999 John Wiley & Sons, Ltd.

Two-Dimensional Numerical Model of Spillway Flow

A numerical model using the finite-element and finite-volume methods is developed for the resolution of two-dimensional free-surface flow equations including air entrainment and applied to calculation of the flow in a spillway. The model is implemented on an unstructured triangular mesh where the standard Galerkin scheme and an upwind finite-volume scheme are developed to solve the continuity and the conservative momentum equations, respectively. The time integration is performed using the fourth-order-accurate Runge-Kutta method with time steps that satisfy the Courant-Friedrichs-Lewy condition. An artificial dispersion term is introduced to eliminate spurious oscillations. A test problem in a spillway is solved to verify the applicability of the model to practical design. A physically realistic solution is obtained that represents a series of flow state alternations from supercritical to subcritical and vice versa, as well as the surface level increase due to the entrained air. The temporal mean of the calculated solution is compared with experimental temporal mean data and examined by posteriorly evaluating the residual term due to vertical nonhomogeneity of velocity. The investigations prove that the model is valid as a primary analysis tool for hydraulic design of spillways.

The role of microstructure in taming the Rayleigh capillary instability of cylindrical jets

It is observed both in nature and in technological processes that filaments with anisotropic molecular-scale structure are less susceptible to breakup due to capillary instability than homogeneous, isotropic fluids in similar filament flows. Here we provide rigorous evidence that the strong coupling of microstructure to the hydrodynamics of cylindrical axisymmetric free surface filaments, indeed fundamentally alters the linearized stability of cylindrical jets. We extend Rayleigh's classical inviscid analysis of cylindrical jets to the three-dimensional (3D), macroscopic flow-orientation equations derived from the Doi kinetic theory for liquid crystalline polymers (LCPs). These equations assume rigid rod-like molecules and incorporate LCP effects of molecular relaxation, anisotropic drag, polymer kinetic energy, LCP density, and an intermolecular potential which couple orientation dynamics to standard free surface fluid equations. Depending on the LCP density, there are between one and three flow-independent orientation equilibria which persist in a constant-velocity, cylindrical free surface flow: an isotropic phase exists at all concentrations, whereas two anisotropic phases exist at sufficiently high LCP density. These equilibrium LCP cylindrical jets have two independent sources of instability, hydrodynamic and orientational, each identified within the coupled flow/orientation free surface equations. For this paper we restrict to equilibria free of orientational instabilities. All streamwise perturbations of wavelength greater than the jet circumference are unstable to capillary instability; only the strength of the instability and most dominant wavelength are affected by LCP microstructure. The degree to which microstructure reduces the capillary instability depends on two critical scaling parameters: an LCP capillary number Ca lcp (a ratio of LCP-induced surface stress to interfacial capillary stress); and the anisotropic drag/friction parameter σ. The most striking result is: for sufficiently large Ca lcp and highly anisotropic drag ( σ ∼ 0) the capillary growthrate can be uniformly lowered, arbitrarily close to zero. For sufficiently small Ca lcp, all capillary-dominated growthrates are reduced, but are bounded below in terms of an explicit, sharp estimate and bounded above by the Rayleigh formula. The upshot is: inviscid LCP jets are predicted to yield bigger drops which form on longer timescales than an inviscid isotropic fluid with the same surface tension.

A semi‐implicit scheme for the barotropic primitive equations of atmospheric dynamics

The primitive equations of a barotropic atmosphere in isobaric co‐ordinates are reformulated, in order to express the geopotential gradient as a function of the pressure at the Earth’s surface. Furthermore, the free surface equation is written in conservative form. A finite difference, semi‐implicit, semi‐Lagrangian scheme in isobaric co‐ordinates is developed. The numerical scheme is mass conservative, is proven to be stable and requires the solution of a single five‐diagonal system. Numerical simulations show that the model captures the main dynamical features of large scale atmospheric motion.

Shape Optimization of Elliptical Road Containers Due to Liquid Load in Steady-State Turning

SUMMARY A numerical analysis of liquid load in elliptical road containers undergoing a steady-state turning manoeuvre is presented for containers of unrestricted size and for containers of fixed capacity. The liquid load is expressed in terms of the horizontal and vertical forces and the overturning moment created by these forces around the middle bottom point of the container. The moment is obtained by integrating the liquid free-surface equation and using numerical calculation of the resulting expressions. An optimization study with an objective to minimize the peak overturning moment is performed on containers of fixed capacity to identify the optimal height/width ratio of the container. The results of this investigation being of a general nature, i.e. with no restriction for an application, are particularly intended for road tankers under a steady turning manoeuvre, where they can be applied with high degree of accuracy.

Finite element simulation of combined buoyancy and thermocapillary driven convection in open cavities

AbstractThermocapillary‐induced and buoyancy‐driven convective flows that commonly occur in crystal growth are numerically simulated using Galerkin finite element method. The physical domain comprises of a open cavity with aspect ratio one and differentially heated vertical walls. The top gas–melt interface is free to deform subject to 90° contact angle boundary conditions at the two vertical walls. The unsteady two‐dimensional Navier–Stokes equations are discretized in time using Chorin‐type splitting scheme and pressure is determined from the Poisson's equation. The free surface is taken to be resting on vertical spines and its evolution in time is determined from the kinematic free surface equation. The governing equations for heat and momentum are solved in the Arbitrary Lagrangian Eulerian frame of reference to handle the moving boundary. The influence of Grashof number, Marangoni number, Bond number, Ohnesorge number and Prandtl number on the flow field and heat transfer is investigated.

Dynamics of Liquid Sloshing in Horizontal Cylindrical Road Containers

A study of the liquid behaviour in horizontal cylindrical road containers undergoing a steady turning manoeuvre is presented and discussed. The steady state solutions are derived analytically from the hydrostatic equations. The transient solutions are obtained by numerical integration of the Navier-Stokes, continuity and free-surface equations. The non-dimensional governing equations are solved in primitive variables by using a modified marker-and-cell technique which involves the interpolation-reflection type boundary conditions developed for this investigation. The mathematical model of the liquid motion includes all essential non-linear effects and allows the damped natural frequencies of liquid vibrations to be obtained as well as the magnitudes of the liquid slosh loads. This study also enables the coupled directional dynamics of the ‘vehicle-liquid tank’ system undergoing different road manoeuvres to be investigated by integrating the non-linear fluid slosh model and an appropriate vehicle model simultaneously.

Numerical simulation of viscous liquid sloshing in arbitrarily shaped reservoirs

ANOVEL approach for the application of the reflectiontype boundary conditions for a liquid with an interface in arbitrarily shaped containers, subjected to a step frame acceleration, is presented. The two-dimensional sloshing problem is solved numerically using a modified marker and cell method that has been altered to accommodate the new boundary conditions. Application of the method to predict the dynamic behavior of liquids in rectangular and cylindrical containers is also included. Contents The prediction of dynamic loading arising from liquid sloshing is crucial in many engineering applications. The linear theories of liquid sloshing14 currently in use give sufficiently accurate solutions for small amplitudes of liquid oscillations. However, in the majority of real situations these solutions become inaccurate. In this case a numerical solution based on the Navier-Stokes, continuity, and free-surface equations with correctly imposed boundary conditions is appropriate. The correct application of boundary conditions for the freesurface flow problem is a difficult task. Often the conventional reflection, Abbett's,5 Viecelli's,6 and Nichols and HirtV methods are used. The reflection method is the simplest, but it is not accurate on boundaries with high curvature. Abbett's and Viecelli's methods are more accurate, but are very complicated. Furthermore, these methods are applied to boundary points lying exactly on the boundary (i.e., an irregular mesh has to be used). The proposed which we call the interpolation-reflection method, combines simplicity and adequate accuracy and gives the boundary values directly on the boundary points of a regular staggered grid for arbitrarily shape boundaries. A brief description is presented in this paper, For more extensive treatment the reader should consult the accompanying paper. If the boundary grid point is lying inside the flow domain, the boundary value is computed by interpolation (in the direction normal to the boundary) of an appropriate order (i.e., first or second). If the point is located outside the flow (a fictitious grid point), then the boundary value is also interpolated on the point that is a mirror reflection of the boundary point with respect to the wall. Following this, the computed value is reflected to the boundary point using odd or even symmetry for the no-slip or free-slip condition, respectively. The derivation of the boundary V velocity for the freeslip condition and second-order interpolation is illustrated in Fig. 1. One can see that the line traced normal to the wall from the boundary point B to the center of the wall curvature O intersects the grid lines in nodes 1 and 2. The location of these nodes depends on the angle a. For second-order interpolation

On a model to propagate surface waves — A second order approach

Surface water waves can travel large distances with negligible energy dissipation. The description of such waves is therefore often done using the Laplace equation with suitable boundary conditions. In most cases linear models can give reasonable descriptions of the wave transformation and good values of design parameters. However there is a growing interest on non-linear models since the magnitude of non-linear effects should be considered for certain wave conditions. In this paper the development of a time dependent Boundary Integral Element Model, which includes significant non-linear effects at an affordable computational time, is reported. The savings in CPU time where achieved mostly because it was possible to set-up and invert the coefficient matrix only once. The free-surface equations are established using a Taylor series expansion and an analysis using dimensionless parameters is done to retain the most important terms. The formulation is presented in some detail and computational aspects are also referred. Results for different types of wave generation and propagation are presented. The applicability of the model to irregular wave propagation is also illustrated.

A systematic derivation of the leading-order equations for extensional flows in slender geometries

We consider the extensional flow and twist of a finite, slender, nearly straight, Newtonian viscous fibre when its ends are drawn apart at prescribed velocity. The initial cross-section of the fibre may be arbitrary and may vary graduallyin the axial direction. We derive the leading-order equations for the fibre's free surface and its flow velocity from a regular perturbation expansion of the full Stokes flow problem in powers of the aspect ratio. In order to obtain these equations systematically, it is necessary to consider terms beyond the leading order in the perturbation expansion

A finite difference technique for solving the Newtonian jet swell problem

A finite difference technique that incorporates a numerical mapping has been successfully applied to analyse both planar and axisymmetric Newtonian jets. A pressure gradient equation and a free-surface slope equation have been derived for free-surface iteration. The computation of pressure inside the jet surface using the pressure gradient equation is stable and accurate at high Reynolds numbers. The free-surface slope equation is needed for updating the free surface and is applicable for jets with strong surface tension effects. The present development can simulate the Newtonian jets for Reynolds numbers as high as 2000 and capillary number as low as 10−5. Numerical predictions by the present technique are close to the results of previous finite element simulations.

Liquid sloshing in rectangular road containers

Abstract A study of liquid behaviour in rectangular road containers undergoing a turning or braking manoeuvre is presented and discussed. The steady-state solution in terms of liquid heights, forces and overturning moments is derived analytically from the hydrostatic equations. The transient response of the liquid is obtained via numerical solution of the continuity, Navier-Stokes and free-surface equations. The governing equations are discretized in a Eulerian mesh and solved with respect to the nondimensional primitive variables together with the boundary conditions at rigid walls and the free surface using a modified marker-and-cell technique. Such an approach allows one to take into account all basic nonlinearities proper to the sloshing problem and to obtain the damped frequencies and magnitudes of the sloshing parameters. The present study is a contribution to the overall dynamics of coupled “vehicle-liquid” systems performing some road manoeuvres.

Vibration Damping Through Liquid Sloshing, Part I: A Nonlinear Analysis

Energy dissipation due to sloshing liquid in rigid torus shaped containers is studied using the potential flow model in conjunction with the boundary layer correction. The analysis is based on the perturbation method to solve for the nonlinear free surface equation. Special consideration is given to the case of resonant interactions which yield interesting damping characteristics. The results suggest that large diameter ratios and low liquid heights lead to efficient damper design. The concept is applicable to a wide spectrum of problems characterized by low frequency oscillations as encountered in wind, ocean and earthquake engineering.

An Investigation of Liquid Sloshing in Toroidal Containers

Liquid sloshing within torus-shaped partially filled containers is studied both analytically and experimentally, The potential flow model in conjunction with the boundary-layer correction is used and the perturbation method applied to solve for the nonlinear free surface equation. The pressure and damping forces exerted by the fluid on the solid walls are derived and compared with the results of the experimental program. Nonlinear phenomena involving the case of resonant coupling between the first and second liquid sloshing transverse modes at large container diameter ratios and low liquid heights are thoroughly investigated as they yield large damping ratios. The energy dissipation characteristics are particularly attractive at low frequencies generally encountered in a marine environment. The concept is likely to be useful for the design of stabilizers for buoys, ocean drilling platforms, and ships.

Developing and optimizing a 3-D-spectral/finite difference hydrodynamic model for the CRAY X-MP

Abstract This paper deals with the development of a vectorizable and numerically efficient algorithm for solving the 3-D hydrodynamic equations describing the sea's motion in any area of arbitrary geographical extent. The method employs a finite difference grid in the horizontal with a spectral approach through the vertical. By this means a decomposition into equations describing changes in the sea surface elevation, (solved using a vectorizable explicit algorithm) and equations describing vertical current structure is possible. This latter set can be solved implicitly using vectorizable methods. The implicit solution of these equations enables them to be solved with a significantly longer time step than the free surface equations, thereby achieving a computationally efficient and fully vectorizable code for the total solution. The numerical accuracy and computational requirements of the method on the CRAY X-MP are discussed in detail, together with some brief indication of the algorithms implementation on CYBER-205, or massively parallel machines,based on transputer arrays.

A Boundary Element Method for the Numerical Simulation of 3-D Nonlinear Water Waves Created by a Submerged Lifting Body

A boundary element method is developed for the simulation of 3-D non-linear water waves created by a submerged lifting body. Triangular panels are used to represent general 3-D shapes. All the variables are linearly approximated on each panel. The double mesh method is introduced to solve the non-linear free-surface equations and the Laplace equation with high efficiency and accuracy. On the down-stream boundary a new condition is used which is different from the commonly used Sommerfeld condition and proved effective for wave problems. A new implementation of the Kutta condition for the lifting body is proposed where the directions of wake are determined by an iterative procedure so as to satisfy the equal pressure condition at the trailing edge. The unbounded flow around a ship-like body with two wings is first calculated and then the free-surface flows and the hydrodynamic forces for the body are calculated. The computed results are compared with the measured ones. Discussions are made about the validity of the numerical method.

Numerical simulation of 3-D turbulent flows over dredged trenches

A 3-D free surface flow in open channels based on the Reynolds equations with thek-e turbulence closure model is presented in this paper. Insted of the “rigid lid” approximation, the solution of the free surface equation is implemented in the velocity—pressure iterative procedure on the basis of the conventional SIMPLE method. This model was used to compute the flow in rectangular channels with trenches dredged across the bottom. The velocity, eddy viscosity coefficient, turbulent shear stress, turbulent kinetic energy and elevation of the free surface can be obtained. The computed results are in good agreement with previous experimental data.