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 semi-implicit lagrangian scheme for 3D shallow water flow with a two-layer turbulence model

A semi-implicit Lagrangian finite difference scheme for 3D shallow water flow has been developed to include an eddy viscosity model for turbulent mixing in the vertical direction. The α-co-ordinate system for the vertical direction has been introduced to give accurate definition of bed and surface boundary conditions. The simple two-layer mixing length model for rough surfaces is used with the standard assumption that the shear stress across the wall region at a given horizontal location is constant. The bed condition is thus defined only by its roughness height (avoiding the need for a friction formula relating to depth-averaged flow, e.g. Chezy, used previously). The method is shown to be efficient and stable with an explicit Lagrangian formulation for convective terms and terms for surface elevation and vertical mixing handled implicitly. The method is applied to current flow around a circular island with gently sloping sides which produce periodic recirculation zones (vortex shedding). Comparisons are made with experimental measurements of velocity using laser Doppler anemometry (time histories at specific points) and surface particle-tracking velocimetry.     

A three-dimensional hydrodynamic model in curvilinear co-ordinates with collocated grid

A three-dimensional hydrodynamic model has been developed for turbulent flows with free surface. In the horizontal x–y-plane, a boundary-fitted curvilinear co-ordinate system is adopted, while in the vertical direction, a σ-co-ordinate transformation is used to represent the free surface and bed topography or lower boundary. Using the finite volume method, the convection terms are discretized using Roe's second-order-accurate scheme. The governing equations are solved in a collocated grid system by a fractional three-step implicit algorithm that has been developed to handle the velocity–pressure–depth coupling problem of free surface incompressible fluid flows. The present study is the extension of previous work to three-dimensional turbulent flows. The model has been applied to three test cases. Comparison with available data shows that the model developed is successful, and is valuable to engineering application. © 1998 John Wiley & Sons, Ltd.     

Hybridization between σ- and z-co-ordinates for improving the internal pressure gradient calculation in marine models with steep bottom slopes

It is discussed in this paper how the pressure gradient error in general vertical co-ordinate models (in which the σ-transformation is a special case) can be reduced by means of hybrid models. For a better understanding, the derivation of such a general vertical co-ordinate model from the Cartesian co-ordinate model is given. Two types of hybridization between σ- and z-co-ordinate models, each using one parameter specifying the degree of hydridization, are presented: (i) the mixed layer transformation with a constant number of layers which are refined near the surface and (ii) the z/σ-transformation which introduces steps near the bottom. In order to achieve good results with the models using other than σ-co-ordinates, a profile-conserving momentum advection discretization is developed. The different co-ordinate transformations are tested with 2D barotropic and baroclinic flows over a topographic bump. Those models with nearly horizontal co-ordinate surfaces in the stratified area give the best correspondence with an isopycnal reference solution. © 1997 John Wiley & Sons, Ltd.     

THE PRESSURE GRADIENT FORCE IN SIGMA-CO-ORDINATE OCEAN MODELS

The error in computing the horizontal pressure gradient force near steep topography is investigated in a primitive equation, σ-co-ordinate, numerical ocean model (Blumberg and Mellor, in Three -Dimensional Coastal Ocean Models, Vol. 4, American Geophysical Union, Washington D.C., 1987, pp. 1–16). By performing simple test experiments where the density field is allowed to vary in both the vertical and the horizontal direction, severe errors are detected in the areas where the isopycnals hit the sloping bottom. An alternative method of computing the pressure force (Stelling and van Kester), Int. j. numer. methods fluids, 18 , 915–935 (1994) is adopted, resulting in substantial reduction of the errors. However, a systematic underestimation of the calculated quantities is revealed, leading to erroneous depth-mean values of the pressure force. In this study a modification of the Stelling and van Kester method is proposed which seems to improve the overall performance of the method. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids 24: 987–1017, 1997.     

An efficient finite difference scheme for free‐surface flows in narrow rivers and estuaries

This paper presents a free‐surface correction (FSC) method for solving laterally averaged, 2‐D momentum and continuity equations. The FSC method is a predictor–corrector scheme, in which an intermediate free surface elevation is first calculated from the vertically integrated continuity equation after an intermediate, longitudinal velocity distribution is determined from the momentum equation. In the finite difference equation for the intermediate velocity, the vertical eddy viscosity term and the bottom‐ and sidewall friction terms are discretized implicitly, while the pressure gradient term, convection terms, and the horizontal eddy viscosity term are discretized explicitly. The intermediate free surface elevation is then adjusted by solving a FSC equation before the intermediate velocity field is corrected. The finite difference scheme is simple and can be easily implemented in existing laterally averaged 2‐D models. It is unconditionally stable with respect to gravitational waves, shear stresses on the bottom and side walls, and the vertical eddy viscosity term. It has been tested and validated with analytical solutions and field data measured in a narrow, riverine estuary in southwest Florida. Model simulations show that this numerical scheme is very efficient and normally can be run with a Courant number larger than 10. It can be used for rivers where the upstream bed elevation is higher than the downstream water surface elevation without any problem. Copyright © 2003 John Wiley & Sons, Ltd.     

An implicit three‐dimensional numerical model to simulate transport processes in coastal water bodies

A three‐dimensional baroclinic numerical model has been developed to compute water levels and water particle velocity distributions in coastal waters. The numerical model consists of hydrodynamic, transport and turbulence model components. In the hydrodynamic model component, the Navier–Stokes equations are solved with the hydrostatic pressure distribution assumption and the Boussinesq approximation. The transport model component consists of the pollutant transport model and the water temperature and salinity transport models. In this component, the three‐dimensional convective diffusion equations are solved for each of the three quantities. In the turbulence model, a two‐equation k–ϵ formulation is solved to calculate the kinetic energy of the turbulence and its rate of dissipation, which provides the variable vertical turbulent eddy viscosity. Horizontal eddy viscosities can be simulated by the Smagorinsky algebraic sub grid scale turbulence model. The solution method is a composite finite difference–finite element method. In the horizontal plane, finite difference approximations, and in the vertical plane, finite element shape functions are used. The governing equations are solved implicitly in the Cartesian co‐ordinate system. The horizontal mesh sizes can be variable. To increase the vertical resolution, grid clustering can be applied. In the treatment of coastal land boundaries, the flooding and drying processes can be considered. The developed numerical model predictions are compared with the analytical solutions of the steady wind driven circulatory flow in a closed basin and of the uni‐nodal standing oscillation. Furthermore, model predictions are verified by the experiments performed on the wind driven turbulent flow of an homogeneous fluid and by the hydraulic model studies conducted on the forced flushing of marinas in enclosed seas. Copyright © 2000 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.     

On the approximation of horizontal gradients in sigma co-ordinates for bathymetry with steep bottom slopes

Nowadays the simulation of free surface flow and transport in rivers, estuaries and seas is often based upon three-dimensional modelling systems. Most of these three-dimensional modelling systems use sigma co-ordinates in the vertical. By the use of the sigma transformation the water column can be divided into the same number of layers independently of the water depth. Especially for steep bottom slopes combined with vertical stratification of the density, sigma-transformed grids impose numerical problems for the accurate approximation of horizontal gradients. This paper deals with algorithms for the approximation in sigma co-ordinates of the horizontal diffusive fluxes of temperature and salinity and for the approximation of the horizontal pressure gradients. The approximation of the horizontal diffusive fluxes is based upon a finite volume method. The approximation of the pressure gradients is directly related to the approximation of the diffusive fluxes. Artificial vertical diffusion and artificial flow due to truncation errors are minimized. The method described in this paper is not hampered by the so-called ‘hydrostatic consistency condition’. This will be illustrated by numerical experiments.     

MODELLING OF BYPASS TRANSITION WITH CONDITIONED NAVIER–STOKES EQUATIONS COUPLED TO AN INTERMITTENCY TRANSPORT EQUATION

A differential method is proposed to simulate bypass transition. The intermittency in the transition zone is taken into account by conditioned averages. These are averages taken during the fraction of time the flow is turbulent or laminar respectively. Starting from the Navier–Stokes equations, conditioned continuity, momentum and energy equations are derived for the laminar and turbulent parts of the intermittent flow. The turbulence is described by a classical k−ϵ model. The supplementary parameter, the intermittency factor, is determined by a transport equation applicable for zero, favourable and adverse pressure gradients. Results for these pressure gradients are given.     

Numerical solutions of the shallow water equations with discontinuous bed topography

A simple scheme is developed for treatment of vertical bed topography in shallow water flows. The effect of the vertical step on flows is modelled with the shallow water equations including local energy loss terms. The bed elevation is denoted with z_b^‐ for the left and z_b⁺ for the right values at each grid point, hence exactly representing a discontinuity in the bed topography. The surface gradient method (SGM) is generalized to reconstruct water depths at cell interfaces involving a vertical step so that the fluxes at the cell interfaces can accurately be calculated with a Riemann solver. The scheme is verified by predicting a surge crossing a step, a tidal flow over a step and dam‐break flows on wet/dry beds. The results have shown good agreements compared with analytical solutions and available experimental data. The scheme is efficient, robust, and may be used for practical flow calculations. Copyright © 2002 John Wiley & Sons, Ltd.     

Near‐wall turbulence modelling with enhanced dissipation

A low‐Reynolds number k–ε turbulence model is proposed that incorporates diffusion terms and modified C_ε(1,2) coefficients to amplify the level of dissipation in non‐equilibrium flow regions, thus reducing the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. Unlike the conventional k–ε model, it requires no wall function/distance parameter that bridges the near‐wall integration. The model is validated against a few flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical simulation of turbulent free surface flow with two‐equation k–ε eddy‐viscosity models

This paper presents a finite difference technique for solving incompressible turbulent free surface fluid flow problems. The closure of the time‐averaged Navier–Stokes equations is achieved by using the two‐equation eddy‐viscosity model: the high‐Reynolds k–ε (standard) model, with a time scale proposed by Durbin; and a low‐Reynolds number form of the standard k–ε model, similar to that proposed by Yang and Shih. In order to achieve an accurate discretization of the non‐linear terms, a second/third‐order upwinding technique is adopted. The computational method is validated by applying it to the flat plate boundary layer problem and to impinging jet flows. The method is then applied to a turbulent planar jet flow beneath and parallel to a free surface. Computations show that the high‐Reynolds k–ε model yields favourable predictions both of the zero‐pressure‐gradient turbulent boundary layer on a flat plate and jet impingement flows. However, the results using the low‐Reynolds number form of the k–ε model are somewhat unsatisfactory. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical modeling of cohesive sediments dynamics in estuaries: Part I—Description of the model and simulations in the Po River Estuary

The present work contributes to the numerical modeling of complex turbulent multiphasic fluid flows occurring in estuarine channels. This research finds its motivation in the increasing need for efficient management of estuaries by taking into account the complex turbulent stratified flows encountered in estuaries and costal zones. A time‐dependent, 3D finite element model of suspended sediment transport taking into account the effects of cohesiveness between sediments is presented. The model estuary is the forced time‐dependent winds, time elevation at open boundaries and river discharge. To cope with the stiffness problems a decoupling method is employed to solve the shallow‐water equations of mass conservation, momentum and suspended sediment transport with the conventional hydrostatic pressure. The decoupling method partitions a time step into three subcycles according to the physical phenomena. In the first sub‐cycle the pure hydrodynamics including the k–ε turbulence model is solved, followed by the advection–diffusion equations for pollutants (salinity, temperature, suspended sediment concentration, (SSC)), and finally the bed evolution is solved. The model uses a mass‐preserving method based on the so‐called Raviart–Thomas finite element on the unstructured mesh in the horizontal plane, while the multi‐layers system is adopted in vertical with the conventional conforming finite element method, with the advantage that the lowermost and uppermost layers of variable height allow a faithful representation of the time‐varying bed and free surface, respectively. The model has been applied to investigate the SSC and seabed evolution in Po River Estuary (PRE) in Italy. The computed results mimic the field data well. Copyright © 2007 John Wiley & Sons, Ltd.     

High-order divergence-free velocity reconstruction for free surface flows on unstructured Voronoi meshes

In this paper, we present an efficient semi-implicit scheme for the solution of the Reynolds-averaged Navier-Stokes equations for the simulation of hydrostatic and nonhydrostatic free surface flow problems. A staggered unstructured mesh composed by Voronoi polygons is used to pave the horizontal domain, whereas parallel layers are adopted along the vertical direction. Pressure, velocity, and vertical viscosity terms are taken implicitly, whereas the nonlinear convective terms as well as the horizontal viscous terms are discretized explicitly by using a semi-Lagrangian approach, which requires an interpolation of the three-dimensional velocity field to integrate the flow trajectories backward in time. To this purpose, a high-order reconstruction technique is proposed, which is based on a constrained least squares operator that guarantees a globally and pointwise divergence-free velocity field. A comparison with an analogous reconstruction, which is not divergence-free preserving, is also presented to give evidence of the new strategy. This allows the continuity equation to be satisfied up to machine precision even for high-order spatial discretizations. The reconstructed velocity field is then used for evaluating high-order terms of a Taylor method that is here adopted as ODE integrator for the flow trajectories. The proposed semi-implicit scheme is validated against a set of academic test problems, and proof of convergence up to fourth-order of accuracy in space is shown.     

Low Reynolds number k–ε model for near‐wall flow

A wall‐distance free k–ε turbulence model is developed that accounts for the near‐wall and low Reynolds number effects emanating from the physical requirements. The model coefficients/functions depend non‐linearly on both the strain rate and vorticity invariants. Included diffusion terms and modified C_ε(1,2) coefficients amplify the level of dissipation in non‐equilibrium flow regions, thus reducing the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. The model is validated against a few flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Copyright © 2005 John Wiley & Sons, Ltd.     

An efficient and accurate non‐hydrostatic model with embedded Boussinesq‐type like equations for surface wave modeling

A novel approach that embeds the Boussinesq‐type like equations into an implicit non‐hydrostatic model (NHM) is developed. Instead of using an integration approach, Boussinesq‐type like equations with a reference velocity under a virtual grid system are introduced to analytically obtain an analytical form of pressure distribution at the top layer. To determine the size of vertical layers in the model, a top‐layer control technique is proposed and the reference location is employed to optimize linear wave dispersion property. The efficiency and accuracy of this NHM with Boussinesq‐type like equations (NHM‐BTE) are critically examined through four free‐surface wave examples. Overall model results show that NHM‐BTE using only two vertical layers is capable of accurately simulating highly dispersive wave motion and wave transformation over irregular bathymetry. Copyright © 2008 John Wiley & Sons, Ltd.     

OPERATOR–SPLITTING COMPUTATION OF TURBULENT FLOW IN AN AXISYMMETRIC 180° NARROWING BEND USING SEVERAL k—ϵ MODELS AND WALL FUNCTIONS

This paper describes a finite element implementation of an operator-splitting algorithm for solving transient/steady turbulent flows and presents solutions for the turbulent flow in an axisymmetric 180° narrowing bend, a benchmark problem dealt with at the 1994 WUA-CFD annual meeting. Three k–ϵ based models are used: the standard linear k–ϵ model, a non-linear k–ϵ model and an RNG k–ϵ model. Flow separation after the bend, as observed in the experiment, is predicted by the RNG model and by both the linear and non-linear k–ϵε models with van Driest mixing length wall functions. Good agreement with experimental data of pressure distribution on bending walls is obtained by the present numerical simulation. Results show that there is very little difference between the linear and non-linear k–ϵε models in terms of predicted velocity fields and that the non-linearities mainly affect the distribution of turbulent normal stress and pressure, in analogy to the effect of second-order viscoelastic fluid models on laminar flow. Both the linear and non-linear k–ϵε models fail to predict any flow separation if logarithmic wall functions are used.     

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.