A fully hydrodynamic model for three‐dimensional,free‐surface flows

The hydrostatic pressure assumption has been widely used in studying water movements in rivers, lakes, estuaries, and oceans. While this assumption is valid in many cases and has been successfully used in numerous studies, there are many cases where this assumption is questionable. This paper presents a three‐dimensional, hydrodynamic model for free‐surface flows without using the hydrostatic pressure assumption. The model includes two predictor–corrector steps. In the first predictor–corrector step, the model uses hydrostatic pressure at the previous time step as an initial estimate of the total pressure field at the new time step. Based on the estimated pressure field, an intermediate velocity field is calculated, which is then corrected by adding the non‐hydrostatic component of the pressure to the estimated pressure field. A Poisson equation for non‐hydrostatic pressure is solved before the second intermediate velocity field is calculated. The final velocity field is found after the free surface at the new time step is computed by solving a free‐surface correction equation. The numerical method was validated with several analytical solutions and laboratory experiments. Model results agree reasonably well with analytical solutions and laboratory results. Model simulations suggest that the numerical method presented is suitable for fully hydrodynamic simulations of three‐dimensional, free‐surface flows. Copyright © 2003 John Wiley & Sons, Ltd.     

Shallow-water flow solver with non-hydrostatic pressure: 2D vertical plane problems

A numerical solution for shallow-water flow is developed based on the unsteady Reynolds-averaged Navier–Stokes equations without the conventional assumption of hydrostatic pressure. Instead, the non-hydrostatic pressure component may be added in regions where its influence is significant, notably where bed slope is not small and separation in a vertical plane may occur or where the free-surface slope is not small. The equations are solved in the σ-co-ordinate system with semi-implicit time stepping and the eddy viscosity is calculated using the standard k–ϵ turbulence model. Conventionally, boundary conditions at the bed for shallow-water models only include vertical diffusion terms using wall functions, but here they are extended to include horizontal diffusion terms which can be significant when bed slope is not small. This is consistent with the inclusion of non-hydrostatic pressure. The model is applied to the 2D vertical plane flow of a current over a trench for which experimental data and other numerical results are available for comparison. Computations with and without non-hydrostatic pressure are compared for the same trench and for trenches with smaller side slopes, to test the range of validity of the conventional hydrostatic pressure assumption. The model is then applied to flow over a 2D mound and again the slope of the mound is reduced to assess the validity of the hydrostatic pressure assumption. © 1998 John Wiley & Sons, Ltd.     

A two‐dimensional vertical non‐hydrostatic σ model with an implicit method for free‐surface flows

An implicit finite difference model in the σ co‐ordinate system is developed for non‐hydrostatic, two‐dimensional vertical plane free‐surface flows. To accurately simulate interaction of free‐surface flows with uneven bottoms, the unsteady Navier–Stokes equations and the free‐surface boundary condition are solved simultaneously in a regular transformed σ domain using a fully implicit method in two steps. First, the vertical velocity and pressure are expressed as functions of horizontal velocity. Second, substituting these relationship into the horizontal momentum equation provides a block tri‐diagonal matrix system with the unknown of horizontal velocity, which can be solved by a direct matrix solver without iteration. A new treatment of non‐hydrostatic pressure condition at the top‐layer cell is developed and found to be important for resolving the phase of wave propagation. Additional terms introduced by the σ co‐ordinate transformation are discretized appropriately in order to obtain accurate and stable numerical results. The developed model has been validated by several tests involving free‐surface flows with strong vertical accelerations and non‐linear waves interacting with uneven bottoms. Comparisons among numerical results, analytical solutions and experimental data show the capability of the model to simulate free‐surface flow problems. Copyright © 2004 John Wiley & Sons, Ltd.     

Three‐dimensional modeling of non‐hydrostatic free‐surface flows on unstructured grids

A three‐dimensional numerical model is presented for the simulation of unsteady non‐hydrostatic shallow water flows on unstructured grids using the finite volume method. The free surface variations are modeled by a characteristics‐based scheme, which simulates sub‐critical and super‐critical flows. Three‐dimensional velocity components are considered in a collocated arrangement with a σ‐coordinate system. A special treatment of the pressure term is developed to avoid the water surface oscillations. Convective and diffusive terms are approximated explicitly, and an implicit discretization is used for the pressure term to ensure exact mass conservation. The unstructured grid in the horizontal direction and the σ coordinate in the vertical direction facilitate the use of the model in complicated geometries. Solution of the non‐hydrostatic equations enables the model to simulate short‐period waves and vertically circulating flows. Copyright © 2015 John Wiley & Sons, Ltd.     

Simulation of non‐hydrostatic,density‐stratified flow in irregular domains

A numerical model has been developed for simulating density‐stratified flow in domains with irregular but simple topography. The model was designed for simulating strong interactions between internal gravity waves and topography, e.g. exchange flows in contracting channels, tidally or convectively driven flow over two‐dimensional sills or waves propagating onto a shoaling bed. The model is based on the non‐hydrostatic, Boussinesq equations of motion for a continuously stratified fluid in a rotating frame, subject to user‐configurable boundary conditions. An orthogonal boundary fitting co‐ordinate system is used for the numerical computations, which rely on a fourth‐order compact differentiation scheme, a third‐order explicit time stepping and a multi‐grid based pressure projection algorithm. The numerical techniques are described and a suite of validation studies are presented. The validation studies include a pointwise comparison of numerical simulations with both analytical solutions and laboratory measurements of non‐linear solitary wave propagation. Simulation results for flows lacking analytical or laboratory data are analysed a posteriori to demonstrate satisfaction of the potential energy balance. Computational results are compared with two‐layer hydraulic predictions in the case of exchange flow through a contracting channel. Finally, a simulation of circulation driven by spatially non‐uniform surface buoyancy flux in an irregular basin is discussed. Copyright © 2000 John Wiley & Sons, Ltd.     

An implicit two‐dimensional non‐hydrostatic model for free‐surface flows

An implicit finite volume model in sigma coordinate system is developed to simulate two‐dimensional (2D) vertical free surface flows, deploying a non‐hydrostatic pressure distribution. The algorithm is based on a projection method which solves the complete 2D Navier–Stokes equations in two steps. First the pressure term in the momentum equations is excluded and the resultant advection–diffusion equations are solved. In the second step the continuity and the momentum equation with only the pressure terms are solved to give a block tri‐diagonal system of equation with pressure as the unknown. This system can be solved by a direct matrix solver without iteration. A new implicit treatment of non‐hydrostatic pressure, similar to the lower layers is applied to the top layer which makes the model free of any hydrostatic pressure assumption all through the water column. This treatment enables the model to evaluate both free surface elevation and wave celerity more accurately. A series of numerical tests including free‐surface flows with significant vertical accelerations and nonlinear behaviour in shoaling zone are performed. Comparison between numerical results, analytical solutions and experimental data demonstrates a satisfactory performance. Copyright © 2006 John Wiley & Sons, Ltd.     

Development of a generalized multi‐layer model for 3‐D simulation of free surface flows

A mathematical model was developed for three‐dimensional (3‐D) simulation of free surface flows. In this model, the flow depth is divided into a number of layers and shallow water equations are integrated in each layer to derive the hydrodynamic equations. To give a general form to this model, each layer is assumed to be non‐horizontal with varying thickness in the flow domain. A non‐orthogonal curvilinear coordinate system is employed in the model, to allow for flexibility in dealing with the irregular geometry of natural watercourses. Due to the similarity in governing equations, two‐dimensional (2‐D) depth averaged programs can be developed into a multi‐layer model. The development for a depth averaged program and its numerical scheme is described in this paper. Experimental data and semi‐analytical solutions are used to evaluate the performance of the model. Three different cases of open channel flow are tested: 1‐flow in a straight open channel, 2‐the flow development region in a channel, and 3‐flow in a meandering channel. It is shown that the model has the capability to predict velocity distribution and secondary flows in complex 3‐D flow conditions. Copyright © 2004 John Wiley & Sons, Ltd.     

A 3‐D non‐hydrostatic pressure model for small amplitude free surface flows

A three‐dimensional, non‐hydrostatic pressure, numerical model with k–ε equations for small amplitude free surface flows is presented. By decomposing the pressure 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 non‐hydrostatic pressure term, using Newton's method in conjunction with the generalized minimal residual (GMRES) method so that most terms can be solved implicitly. This method only needs the product of a Jacobian matrix and a vector rather than the Jacobian matrix itself, limiting the amount of storage and significantly decreasing the overall computational time required. In the second step the pressure–Poisson equation is solved iteratively with a preconditioned linear GMRES method. It is shown that preconditioning reduces the central processing unit (CPU) time dramatically. In order to prevent pressure oscillations which may arise in collocated grid arrangements, transformed velocities are defined at cell faces by interpolating velocities at grid nodes. After the new pressure field is obtained, the intermediate velocities, which are calculated from the previous fractional step, are updated. The newly developed model is verified against analytical solutions, published results, and experimental data, with excellent agreement. Copyright © 2005 John Wiley & Sons, Ltd.     

A coupled surface‐subsurface model for hydrostatic flows under saturated and variably saturated conditions

In this paper, the governing differential equations for hydrostatic surface‐subsurface flows are derived from the Richards and from the Navier‐Stokes equations. A vertically integrated continuity equation is formulated to account for both surface and subsurface flows under saturated and variable saturated conditions. Numerically, the horizontal domain is covered by an unstructured orthogonal grid that may include subgrid specifications. Along the vertical direction, a simple z‐layer discretization is adopted. Semi‐implicit finite difference equations for velocities, and a finite volume approximation for the vertically integrated continuity equation, are derived in such a fashion that, after simple manipulation, the resulting discrete pressure equation can be assembled into a single, two‐dimensional, mildly nonlinear system. This system is solved by a nested Newton‐type method, which yields simultaneously the (hydrostatic) pressure and a nonnegative fluid volume throughout the computational grid. The resulting algorithm is relatively simple, extremely efficient, and very accurate. Stability, convergence, and exact mass conservation are assured throughout also in presence of wetting and drying, in variable saturated conditions, and during flow transition through the soil interface. A few examples illustrate the model applicability and demonstrate the effectiveness of the proposed algorithm.     

A conservative semi‐implicit method for coupled surface–subsurface flows in regional scale

A semi‐implicit method for coupled surface–subsurface flows in regional scale is proposed and analyzed. The flow domain is assumed to have a small vertical scale as compared with the horizontal extents. Thus, after hydrostatic approximation, the simplified governing equations are derived from the Reynolds averaged Navier–Stokes equations for the surface flow and from the Darcy's law for the subsurface flow. A conservative free‐surface equation is derived from a vertical integral of the incompressibility condition and extends to the whole water column including both, the surface and the subsurface, wet domains. Numerically, the horizontal domain is covered by an unstructured orthogonal grid that may include subgrid specifications. Along the vertical direction a simple z‐layer discretization is adopted. Semi‐implicit finite difference equations for velocities and a finite volume approximation for the free‐surface equation are derived in such a fashion that, after simple manipulation, the resulting discrete free‐surface equation yields a single, well‐posed, mildly nonlinear system. This system is efficiently solved by a nested Newton‐type iterative method that yields simultaneously the pressure and a non‐negative fluid volume throughout the computational grid. The time‐step size is not restricted by stability conditions dictated by friction or surface wave speed. The resulting algorithm is simple, extremely efficient, and very accurate. Exact mass conservation is assured also in presence of wetting and drying dynamics, in pressurized flow conditions, and during free‐surface transition through the interface. A few examples illustrate the model applicability and demonstrate the effectiveness of the proposed algorithm. Copyright © 2015 John Wiley & Sons, Ltd.     

An implicit three‐dimensional fully non‐hydrostatic model for free‐surface flows

An implicit method is developed for solving the complete three‐dimensional (3D) Navier–Stokes equations. The algorithm is based upon a staggered finite difference Crank‐Nicholson scheme on a Cartesian grid. A new top‐layer pressure treatment and a partial cell bottom treatment are introduced so that the 3D model is fully non‐hydrostatic and is free of any hydrostatic assumption. A domain decomposition method is used to segregate the resulting 3D matrix system into a series of two‐dimensional vertical plane problems, for each of which a block tri‐diagonal system can be directly solved for the unknown horizontal velocity. Numerical tests including linear standing waves, nonlinear sloshing motions, and progressive wave interactions with uneven bottoms are performed. It is found that the model is capable to simulate accurately a range of free‐surface flow problems using a very small number of vertical layers (e.g. two–four layers). The developed model is second‐order accuracy in time and space and is unconditionally stable; and it can be effectively used to model 3D surface wave motions. Copyright © 2004 John Wiley & Sons, Ltd.     

A two‐scale low‐Reynolds number turbulence model

In this study, a two‐scale low‐Reynolds number turbulence model is proposed. The Kolmogorov turbulence time scale, based on fluid kinematic viscosity and the dissipation rate of turbulent kinetic energy (ν, ε), is adopted to address the viscous effects and the rapid increasing of dissipation rate in the near‐wall region. As a wall is approached, the turbulence time scale transits smoothly from a turbulent kinetic energy based (k, ε) scale to a (ν, ε) scale. The damping functions of the low‐Reynolds number models can thus be simplified and the near‐wall turbulence characteristics, such as the ε distribution, are correctly reproduced. The proposed two‐scale low‐Reynolds number turbulence model is first examined in detail by predicting a two‐dimensional channel flow, and then it is applied to predict a backward‐facing step flow. Numerical results are compared with the direct numerical simulation (DNS) budgets, experimental data and the model results of Chien, and Lam and Bremhorst respectively. It is proved that the proposed two‐scale model indeed improves the predictions of the turbulent flows considered. Copyright © 2000 John Wiley & Sons, Ltd.     

Finite‐volume‐type VOF method on dynamically adaptive quadtree grids

This paper describes a finite‐volume volume‐of‐fluid (VOF) method for simulating viscous free surface flows on dynamically adaptive quadtree grids. The scheme is computationally efficient in that it provides relatively fine grid resolution at the gas–liquid interface and coarse grid density in regions where flow variable gradients are small. Special interpolations are used to ensure volume flux conservation where differently sized neighbour cells occur. The numerical model is validated for advection of dyed fluid in unidirectional and rotating flows, and for two‐dimensional viscous sloshing in a rectangular tank. Copyright © 2004 John Wiley & Sons, Ltd.     

An implicit numerical algorithm for solving non‐hydrostatic free‐surface flow problems

Details are given of the development of a two‐dimensional vertical numerical model for simulating unsteady free‐surface flows, using a non‐hydrostatic pressure distribution. In this model, the Reynolds equations and the kinematic free‐surface boundary condition are solved simultaneously, so that the water surface elevation can be integrated into the solution and solved for, together with the velocity and pressure fields. An efficient numerical algorithm has been developed, deploying implicit parameters similar to those used in the Crank–Nicholson method, and generating a block tri‐diagonal algebraic system of equations. The model has been applied to simulate a range of unsteady flow problems involving relatively strong vertical accelerations. The results show that the numerical algorithm described is able to produce accurate predictions and is also easy to apply. Copyright © 2001 John Wiley & Sons, Ltd.     

High‐order compact numerical schemes for non‐hydrostatic free surface flows

The development of a numerical scheme for non‐hydrostatic free surface flows is described with the objective of improving the resolution characteristics of existing solution methods. The model uses a high‐order compact finite difference method for spatial discretization on a collocated grid and the standard, explicit, single step, four‐stage, fourth‐order Runge–Kutta method for temporal discretization. The Cartesian coordinate system was used. The model requires the solution of two Poisson equations at each time‐step and tridiagonal matrices for each derivative at each of the four stages in a time‐step. Third‐ and fourth‐order accurate boundaries for the flow variables have been developed including the top non‐hydrostatic pressure boundary. The results demonstrate that numerical dissipation which has been a problem with many similar models that are second‐order accurate is practically eliminated. A high accuracy is obtained for the flow variables including the non‐hydrostatic pressure. The accuracy of the model has been tested in numerical experiments. In all cases where analytical solutions are available, both phase errors and amplitude errors are very small. Copyright © 2006 John Wiley & Sons, Ltd.     

Depth‐integrated free‐surface flow with a two‐layer non‐hydrostatic formulation

This paper presents the derivation of a depth‐integrated wave propagation and runup model from a system of governing equations for two‐layer non‐hydrostatic flows. The governing equations are transformed into an equivalent, depth‐integrated system, which separately describes the flux‐dominated and dispersion‐dominated processes. The depth‐integrated system reproduces the linear dispersion relation within a 5 error for water depth parameter up to kd = 11, while allowing direct implementation of a momentum conservation scheme to model wave breaking and a moving‐waterline technique for runup calculation. A staggered finite‐difference scheme discretizes the governing equations in the horizontal dimension and the Keller box scheme reconstructs the non‐hydrostatic terms in the vertical direction. An semi‐implicit scheme integrates the depth‐integrated flow in time with the non‐hydrostatic pressure determined from a Poisson‐type equation. The model is verified with solitary wave propagation in a channel of uniform depth and validated with previous laboratory experiments for wave transformation over a submerged bar, a plane beach, and fringing reefs. Copyright © 2011 John Wiley & Sons, Ltd.     

Baroclinic stability for a family of two‐level,semi‐implicit numerical methods for the 3D shallow water equations

The baroclinic stability of a family of two time‐level, semi‐implicit schemes for the 3D hydrostatic, Boussinesq Navier–Stokes equations (i.e. the shallow water equations), which originate from the TRIM model of Casulli and Cheng (Int. J. Numer. Methods Fluids 1992; 15 :629–648), is examined in a simple 2D horizontal–vertical domain. It is demonstrated that existing mass‐conservative low‐dissipation semi‐implicit methods, which are unconditionally stable in the inviscid limit for barotropic flows, are unstable in the same limit for baroclinic flows. Such methods can be made baroclinically stable when the integrated continuity equation is discretized with a barotropically dissipative backwards Euler scheme. A general family of two‐step predictor‐corrector schemes is proposed that have better theoretical characteristics than existing single‐step schemes. Copyright © 2006 John Wiley & Sons, Ltd.     

A method to discretize non‐planar fractures for 3D subsurface flow and transport simulations

A method is presented to discretize inclined non‐planar 2D fractures within a 3D finite element grid for subsurface flow and transport simulations. Each 2D fracture is represented as a triangulated surface. Each triangle is then discretized by 2D fracture elements that can be horizontal, vertical or inclined and that can be triangular or rectangular. The 3D grid representing a porous rock formation consists of hexahedra and can be irregular to allow grid refinement. An inclined fracture was discretized by (a) inclined triangles and (b) orthogonal rectangles and flow/transport simulations were run to compare the results. The comparison showed that (i) inclined fracture elements must be used to simulate 2D transient flow, (ii) results of 2D/3D steady‐state and 3D transient flow simulations are identical for both discretization methods, (iii) inclined fracture elements must be used to simulate 2D/3D transport because orthogonal fracture elements significantly underestimate concentrations, and (iv) orthogonal elements can be used to simulate 2D/3D transport if fracture permeability is corrected and multiplied by the ratio of fracture surface areas (orthogonal to inclined). Groundwater flow at a potential site for long‐term disposal of spent nuclear fuel was simulated where a complex 3D fracture network was discretized with this technique. The large‐scale simulation demonstrates that the proposed discretization procedure offers new possibilities to simulate flow and transport in complex 3D fracture networks. The new procedure has the further advantage that the same grid can be used for different realizations of a fracture network model with no need to regenerate the grid. Copyright © 2007 John Wiley & Sons, Ltd.     

A non‐linear k–ε model with realizability for prediction of flows around bluff bodies

The incompressible flow around bluff bodies (a square cylinder and a cube) is investigated numerically using turbulence models. A non‐linear k–ε model, which can take into account the anisotropy of turbulence with less CPU time and computer memory then RSM or LES, is adopted as a turbulence model. In tuning of the model coefficients of the non‐linear terms are adjusted through the examination of previous experimental studies in simple shear flows. For the tuning of the coefficient in the eddy viscosity (=C_μ), the realizability constraints are derived in three types of basic 2D flow patterns, namely, a simple shear flow, flow around a saddle and a focal point. C_μ is then determined as a function of the strain and rotation parameters to satisfy the realizability. The turbulence model is first applied to a 2D flow around a square cylinder and the model performance for unsteady flows is examined focussing on the period and the amplitude of the flow oscillation induced by Karman vortex shedding. The applicability of the model to 3D flows is examined through the computation of the flow around a surface‐mounted cubic obstacle. The numerical results show that the present model performs satisfactorily to reproduce complex turbulent flows around bluff bodies. Copyright © 2003 John Wiley & Sons, Ltd.     

A semi‐implicit finite difference method for non‐hydrostatic,free‐surface flows

In this paper a semi‐implicit finite difference model for non‐hydrostatic, free‐surface flows is analyzed and discussed. It is shown that the present algorithm is generally more accurate than recently developed models for quasi‐hydrostatic flows. The governing equations are the free‐surface Navier–Stokes equations defined on a general, irregular domain of arbitrary scale. The momentum equations, the incompressibility condition and the equation for the free‐surface are integrated by a semi‐implicit algorithm in such a fashion that the resulting numerical solution is mass conservative and unconditionally stable with respect to the gravity wave speed, wind stress, vertical viscosity and bottom friction. Copyright © 1999 John Wiley & Sons, Ltd.