Surface Flow Model Research Articles

Development of a 3D non-hydrostatic pressure model for free surface flows

A three-dimensional, non-hydrostatic pressure, numerical model for free surface flows is presented. By decomposing the pressure term into hydrostatic and non-hydrostatic parts, the numerical model uses an integrated time step with two fractional steps. In the first fractional step, the momentum equations are solved without the hydrostatic pressure term using Newton's method in conjunction with the generalised minimal residual (GMRES) method. This combined method does not require the determination of a Jacobian matrix explicitly but simply the product of the Jacobian and a vector, thereby reducing the amount of storage required and significantly decreasing the overall computational time required. By using Newton's method, the numerical model can handle implicitly almost all variables, unlike many other numerical models. Hence numerical stability is achieved effectively. In the second fractional step, the pressure-Poisson equation is solved iteratively with a preconditioned linear GMRES method. It is shown that preconditioning reduces the processing time dramatically. After the new pressure field is obtained the intermediate velocities, which are calculated from the previous fractional step, are updated and then these updated velocities preserve the local mass conservation. The newly developed model is verified against analytical solutions, with good agreement.

Asymptotic model for free surface flow of an electrically conducting fluid in a high-frequency magnetic field

We study the behaviour of a layer of an electrically conducting inviscid incompressible fluid in a high-frequency alternating magnetic field. We derive nonlinear asymptotic equations governing the evolution of the fluid layer in the high-frequency limit. As a test for the model, we consider the linearised stability problem for an infinite planar free surface of a layer of finite depth.

Coupled Surface-Subsurface Flow Model for Improved Basin Irrigation Management

The availability of a process-based coupled surface-subsurface model can lead to improved surface irrigation/fertigation management practices. In this study, a one-dimensional zero-inertia model is coupled with a one-dimensional unsaturated zone water-flow model: HYDRUS-1D. A driver program is used to effect internal iterative coupling of the surface and subsurface flow models. Flow depths calculated using the surface-flow model are used as Dirichlet boundary conditions for the subsurface-flow model, and infiltration amounts calculated by the subsurface model are in turn used in surface-flow mass balance calculations. The model was tested by using field data collected at the University of Arizona, Yuma Mesa, research farm. The maximum mean absolute difference between field-observed and model-predicted advance is 2 min. Applications of the coupled model in improved irrigation management are highlighted. In addition, the significance of the effects of soil moisture redistribution on irrigation water availability to crops and the capability of the coupled model in tracking those changes in soil water status over time are discussed using examples.

Analysis of a Local Hydrodynamic Model with Marangoni Effect

We study a mathematical model treating of the dynamic contact line problem, supposed to describe the main features of the advancing triple line (rolling motion and variable contact angle) and to remove the singularity. The model is composed by a macroscopic hydrodynamic free surface flow model (HFSM) (Navier–Stokes) coupled with a mesoscopic local surface model (LSM). Detailed mathematical and numerical analysis of the 1D steady-state local surface model are done existence and uniqueness of the exact and numerical solutions, extra properties of the derivatives, and convergence of finite element schemes. Some numerical results of the two models treated separately are presented for a 2D plunging tape configuration.

A Numerical Technique for Solving the Maxwell Model for Free Surface Flows

This work is concerned with the development of a numerical technique for solving free surface flows of a Maxwell fluid. The governing equations for the flow of a Maxwell type fluid together with appropriate boundary conditions are given. The free surface stress conditions are treated in details. A novel formulation for calculating the extra stress components on rigid boundaries is given. The numerical technique presented in this work employs the finite difference method on a staggered grid and employs the ideas of the MAC (Marker-and-Cell) method. Numerical results demonstrating that this numerical technique can solve viscoelastic flows governed by the Maxwell model are presented. Moreover, validation results are presented.

A 2-D model of Rayleigh instability in capillary tubes??surfactant effects

The Rayleigh instability of stagnant liquid films lining the interior of capillary tubes is analyzed with the aid of a 2-D free surface flow model; this axisymmetric model is previously validated using already published theoretical and experimental results. The Galerkin-finite element method is used to transform the complete set of governing equations and boundary conditions into a discrete set, which is then simultaneously solved at each time step by Newton's method. Predictions of well known simplified models represented by nonlinear evolution equations derived on the one-dimensional flow assumption are compared with those obtained from the present one. The comparisons are made for pure liquids and also for liquids contaminated with insoluble surfactants; they show that the simpler models represent the free surface evolution reasonable well. However, the 1-D models generally underestimate the time needed to complete the unstable process that ends––if the film is thick enough––when the inner gas phase becomes disconnected due to the formation of liquid lenses regularly spaced; these discrepancies become larger when surface active agents are present. Surfactant effects and the wealth of information produced by the 2-D model are both evidenced through sample results presented at the end of the paper.

A 2-D model of Rayleigh instability in capillary tubes––surfactant effects

The Rayleigh instability of stagnant liquid films lining the interior of capillary tubes is analyzed with the aid of a 2-D free surface flow model; this axisymmetric model is previously validated using already published theoretical and experimental results. The Galerkin-finite element method is used to transform the complete set of governing equations and boundary conditions into a discrete set, which is then simultaneously solved at each time step by Newton's method. Predictions of well known simplified models represented by nonlinear evolution equations derived on the one-dimensional flow assumption are compared with those obtained from the present one. The comparisons are made for pure liquids and also for liquids contaminated with insoluble surfactants; they show that the simpler models represent the free surface evolution reasonable well. However, the 1-D models generally underestimate the time needed to complete the unstable process that ends––if the film is thick enough––when the inner gas phase becomes disconnected due to the formation of liquid lenses regularly spaced; these discrepancies become larger when surface active agents are present. Surfactant effects and the wealth of information produced by the 2-D model are both evidenced through sample results presented at the end of the paper.

Development and Application of a Methodology for Sediment Source Identification. I: Modified Unit Sedimentograph Approach

Identification of sediment source areas within a watershed is useful for designing appropriate management alternatives that reduce sediment losses. In most watersheds, data collection programs are present at the stream gauging station, and sediment source identification must be treated as an inverse problem. A physically based watershed-scale surface flow and erosion model, KINEROS was utilized as the modeling tool in this study. A modified unit sedimentograph approach was used for source identification based on hydrograph and sedimentograph data collected at the watershed outlet. Data from several events were examined over two small water- sheds, and a statistical procedure was employed to assess the erosion vulnerability of different regions in the two watersheds. These results were presented in the form of maps to obtain a spatial picture of potential sediment generating areas. A statistical analysis was also utilized to determine if erosion potentials of different regions of a watershed were significantly different. It appears that identification of source areas is possible when a limited number of regions are contributing sediments.

Arbitrary Lagrangian–Eulerian finite element analysis of free surface flow using a velocity–vorticity formulation

This paper describes the application of velocity–vorticity formulation of the Navier–Stokes equations for two-dimensional free surface flow using an arbitrary Lagrangian–Eulerian method. The velocity Poisson equations and the vorticity transport equations are solved using a finite element method to obtain the velocity and the vorticity fields in the interior region of the computational domain. The boundary-fitted coordinates system is adopted to solve the boundary equations for kinematic and dynamic conditions at the free surface using a finite difference method. The numerical model for the velocity–vorticity formulation is validated for a square cavity flow at Re=400 and 1000. The solitary wave reflected from a vertical wall is chosen as a test case for comparison and validation of the free surface flow model. Then the proposed numerical model is used to obtain flow results for the following free surface flow cases: (i) interaction between two opposite solitary waves, (ii) seiche phenomenon in a rectangular reservoir, and (iii) solitary wave through a submerged rectangular structure in a viscous fluid. The efficiency of the present numerical model for numerical treatment of free surface flows is discussed. Furthermore the advantage of this formulation with respect to primitive variables formulation is addressed from the computational point of view.

Prediction of surface currents and drifter trajectories on the inner Scotian Shelf

The predictive skill of a simple model of surface flow on the Scotian Shelf is assessed using oceanographic data collected in February 1996. The model is forced by wind stress, water density, and sea level along the open boundaries of the model domain. The skill of the model with respect to subtidal variations of alongshore current and bottom pressure is quantified by the ratio γ2 = Var(O − P)/Var(O), where Var(O − P) and Var(O) are the variance of the prediction errors and observations, respectively. Skill is highest for bottom pressure (γ2 = 0.2) followed, in order, by cross‐shelf gradients in bottom pressure (γ2 = 0.5), horizontally averaged currents (0.4 < γ2 < 0.7), and currents at individual current meters (0.6 < γ2 < 1.4). The skill of the model with respect to drifter position is quantified by the search radius, centered on the predicted drifter position, that ensures a 50% chance of locating the drifter. Skill varies significantly with time but is generally highest when the drifter motion is strongest. We conclude with a comparison of the performance of the model against predictions based on the release point of the drifters and the operational scheme presently used to guide marine search and rescue in this region.

Three-dimensional hydrodynamic model for free surface flow

A three-dimensional mathematical model for free surface flow based on the Reynolds-averaged Navier-Stokes equations is developed in a sigma co-ordinate system. The time-splitting method is used to separate advection and diffusion terms from the pressure terms in the governing equations. The pressure variable is further separated into hydrostatic and hydrodynamic pressures so that the computer rounding errors can be largely avoided. The resulting hydrodynamic pressure equation is solved by a multigrid method, while the hydrostatic pressure equations are solved very efficiently by an Alternating Direction Implicit (ADI) scheme. The convection terms are discretized by the Roe's scheme of second-order accuracy. A staggered mesh is used. The model is tested against available analytical solutions and experimental data. Good agreement has been achieved.

REGIONAL SCALE MODELING OF SURFACE AND GROUND WATER INTERACTION IN THE SNAKE RIVER BASIN1

ABSTRACT: Changes in irrigation and land use may impact discharge of the Snake River Plain aquifer, which is a major contributor to flow of the Snake River in southern Idaho. The Snake River Basin planning and management model (SRBM) has been expanded to include the spatial distribution and temporal attenuation that occurs as aquifer stresses propagate through the aquifer to the river. The SRBM is a network flow model in which aquifer characteristics have been introduced through a matrix of response functions. The response functions were determined by independently simulating the effect of a unit stress in each cell of a finite difference groundwater flow model on six reaches of the Snake River. Cells were aggregated into 20 aquifer zones and average response functions for each river reach were included in the SRBM. This approach links many of the capabilities of surface and ground water flow models. Evaluation of an artificial recharge scenario approximately reproduced estimates made by direct simulation in a ground water flow model. The example demonstrated that the method can produce reasonable results but interpretation of the results can be biased if the simulation period is not of adequate duration.

Comments on the paper “Mathematical model for the free surface flow under a sluice gate” by Titus Petrila, published in Applied Mathematics and Computation 125 (2002) 49–58

Comments on the paper “Mathematical model for the free surface flow under a sluice gate” by Titus Petrila, published in Applied Mathematics and Computation 125 (2002) 49–58

Estimation of optimal parameters for a surface hydrology model

The estimation of infiltration, roughness, and rainfall intensity parameters in a surface flow model is studied. The problem is formulated as an optimal control problem in which the parameters to be estimated are viewed as controls in the surface flow model and a fit-to-data functional is viewed as a control functional. The control system is obtained as the approximation of the distributed model by means of finite elements in the spatial domain and finite differences in the time domain. The leap-frog and implicit differences are used for the time differencing approximations. The resulting minimization problem is solved by a descent method, The adjoint equations are introduced in the context of a general method to calculate efficiently the derivatives of the functional. The adjoint equations are developed for the continuous as well as the finite difference equations. A numerical example is presented. This example illustrates the effect of the accuracy of prior rainfall intensity information on the estimates of roughness and infiltration parameters.

A sigma coordinate 3D k– ε model for turbulent free surface flow over a submerged structure

A fully three-dimensional unsteady flow model is developed to simulate free surface flow over a submerged structure. A new sigma coordinate is used to map the physical domain containing the wavy free surface and uneven bottom to a rectangular prism, and to keep the size of the submerged block unchanged in the sigma coordinate system. The numerical difficulty encountered in the conventional sigma coordinate system in which the block changes dynamically due to the time varying free surface is thus eliminated. A split operator scheme is used in the numerical solution so that different numerical schemes can be purposely chosen to deal with the distinctive mathematical and physical characteristics of the phenomena at different steps. k– ε model is used in the parameterization of turbulence due to its efficiency and reasonable performance. The model is applied to simulate the propagation of a solitary wave with good results. It is subsequently used to simulate a free surface flow against a submerged cube with one face perpendicular (or 45° inclined) to the flow. The numerical results compare favorably with the experimental measurements. In particular, no excessive turbulent kinetic energy is accumulated at the impingement regions.

Algebraic factorizations for 3D non-hydrostatic free surface flows

The high amount of computer resources required to simulate complex free surface flows has prompted for developing fractional step schemes capable of reducing the computational effort. These schemes are borrowed from a wider family of methods originally devised for the incompressible Navier–Stokes equations. An alternative approach is to perform an algebraic splitting on the coefficient matrix of the linear system resulting from the discretized problem, ending up with the successive solution of sub-problems of smaller size. The resulting schemes are shown in different cases to be the algebraic counterpart of the standard fractional step formulations. This algebraic procedure was again originally devised in the context of incompressible Navier–Stokes system, but we believe it is far more general: in this paper it is indeed extended to the more involved 3D free surface flow model. The inexact block factorization technique is applied to the coefficient matrix arising from the problem at hand and two significant choices for the approximation are discussed and numerically tested.

Pressure Surface Separations in Low-Pressure Turbines—Part 1: Midspan Behavior

This paper describes an investigation into the behavior of the pressure surface separation at midspan in a linear cascade. It is found that the pressure surface separation can be a significant contributor to the profile loss of a thin, solid, low-pressure turbine blade that is typical of current engine designs. Numerical predictions are first used to study the inviscid behavior of the blade. These show a strong incidence dependence around the leading edge of the profile. Experiments then show clearly that all characteristics of the pressure surface separation are controlled primarily by the incidence. It is also shown that the effects of wake passing, freestream turbulence and Reynolds number are of secondary importance. A simple two-part model of the pressure surface flow is then proposed. This model suggests that the pressure surface separation is highly dissipative through the action of its strong turbulent shear. As the incidence is reduced, the increasing blockage of the pressure surface separation then raises the velocity in the separated shear layer to levels at which the separation can create significant loss.

Modeling of Conjunctive Two-Dimensional Surface-Three-Dimensional Subsurface Flows

In the rainfall-runoff process, interaction between surface and subsurface flow components plays an important role, especially in rainwater abstraction and overland flow initiation at the early stage of rainfall events. Coupling of surface and subsurface flow submodels, therefore, is necessary for advanced comprehensive and sophisticated rainfall-runoff simulation. This article presents a conjunctive two-dimensional (2D) surface and three-dimensional (3D) subsurface flow model, which uses the noninertia approximation of the Saint-Venant equations for 2D unsteady surface flow and a modified version of the Richards equation for 3D unsteady unsaturated and saturated subsurface flows. The equations are written in the form of 2D and 3D heat diffusion equations, respectively, and solved numerically. The surface and subsurface flow components are coupled interactively using the common boundary condition of infiltration through the ground surface. The conjunctive model is verified with Smith and Woolhiser’s exper...

Surface flow model: inverse problems and predictions

The inverse problem to estimate the scaling coefficients for infiltration, rainfall, intensity, and roughness in a surface flow model is considered. A probability density function (pdf) is constructed from an observed hydrograph associated with a given rainfall event, the deterministic surface flow model, prior information on bounds on the scale coefficients, and prior information on the coefficients. From this pdf a basin pdf is obtained describing the likelihood of basin infiltration and roughness properties within a sample space of basin properties. With the occurrence of a new rainfall event, the basin pdf is utilized to describe the behavior of the basin in the presence of this event with a basin-event pdf. From this information predictions of flooding behavior may be obtained as well as the expected water levels along the stream. Numerical examples are presented that include different observational errors.

Mathematical model for the free surface flow under a sluice gate

A mathematical model with its analytical solution for the complete sluice gate problem is envisaged. The technique which uses a boundary value problem of Hilbert type formulation can be extended even to the cases of a free surface flow emerging from a vessel or of a “controlled” free surface stream under a sluice gate. The free surface two-dimensional flow of an inviscid incompressible heavy fluid, in the presence of a unlimited wall, is supposed all the time to be steady potential, subcritical upstream and supercritical downstream.