An accurate momentum advection scheme for a z-level coordinate models (original) (raw)

On accurate momentum advection scheme for a z-level coordinate models

2010

In this paper, we focus on a conservative momentum advection discretisation in the presence of zlayers. While in the 2D case conservation of momentum is achieved automatically for an Eulerian advection scheme, special attention is required in the multi-layer case. We show here that an artificial vertical structure of the flow can be introduced solely by the presence of the z-layers, which we refer to as the staircase problem. To avoid this staircase problem, the z-layers have to be remapped in a specific way. The remapping procedure also deals with the case of an uneven number of layers adjacent to a column side, thus allowing one to simulate flooding and drying phenomena in a 3D model.

On a momentum conservative z-layer unstructured C-grid ocean model with flooding

Ocean Modelling, 2012

We present a z-layer unstructured C-grid finite volume hydrostatic model. An efficient and highly scalable implicit technique for solution of the free surface equation is combined with an Eulerian approach for the advection of momentum. We show that an accurate velocity reconstruction procedure is of crucial importance not only for discretization of the Coriolis term, but also for the correct advection of momentum, especially in the multilayer case. Unlike other z-layer models the method presented here ensures that the staircase representation of bathymetry and free surface has no influence on the vertical structure of the flow. The method is therefore guaranteed to be strictly momentum conservative, also in the layers containing the free surface and bed. A number of test cases are presented to show that the model is able to accurately simulate Coriolis dominated flows and flooding and drying processes both in the depthaveraged case and in the presence of multiple z-layers. We use a simulation of 2004 Indian Ocean tsunami to evaluate the ability of the method to simulate fast propagating tsunami waves and detailed inundation processes. Results obtained using two different rupture models are compared to the tide gauge arrival times, satellite altimetry data and the inundation observations in the Banda Aceh area. The comparison is used not only to assess the quality of the underlying rupture models but also to determine the value of the available data sources for such an assessment.

Accurate vertical profiles of turbulent flow in z-layer models

Water Resources Research, 2014

Three-dimensional hydrodynamic z-layer models, which are used for simulating the flow in rivers, estuaries, and oceans, suffer from an inaccurate and often discontinuous bottom shear stress representation, due to the staircase bottom. We analyze the governing equations and clearly show the cause of the inaccuracies. Based on the analysis, we present a new method that significantly reduces the errors and the grid dependency of the results. The method consists of a near-bed layer-remapping and a modified nearbed discretization of the k 2 e turbulence model. We demonstrate the applicability of the approach for uniform channel flow, using a schematized two-dimensional vertical model and for the flow over a bottom sill using the Delft3D modeling system. Conversely, the z-layer discretization allows simple horizontal discretizations for pressure, advection, and diffusion and it efficiently handles shallow areas. However, the bottom and free-surface boundaries are represented as ''staircases,'' see Figure 1. Even using a partial-cell or shaved-cell approach [see e.g.,

The Momentum Conserving Scheme for Two-Layer Shallow Flows

Fluids, 2021

This paper confronts the numerical simulation of steady flows of fluid layers through channels of varying bed and width. The fluid consists of two immiscible fluid layers with constant density, and it is assumed to be of a one-dimensional shallow flow. The governing equation is a coupled system of two-layer shallow water models. In this paper, we apply a direct extension of the momentum conserving scheme previously used for solving the one layer shallow water equations. Computations of various steady-state solutions are used to demonstrate the performance of the proposed numerical scheme. Under the influence of a given flow rate, the numerical steady interface is generated in a channel topography with a hump. The results obtained confirm the analytic steady interface of the two-layer rigid-lid model. Furthermore, the same scheme was used with an additional artificial damping to simulate the maximal exchange flow in channels of varying width. The numerical steady interface agreed wel...

A mass-momentum consistent, Volume-of-Fluid method for incompressible flow on staggered grids

Computers & Fluids, 2021

The computation of flows with large density contrasts is notoriously difficult. To alleviate the difficulty we consider a discretization of the Navier-Stokes equation that advects mass and momentum in a consistent manner. Incompressible flow with capillary forces is modeled and the discretization is performed on a staggered grid of Marker and Cell type. The Volume-of-Fluid method is used to track the interface and a Height-Function method is used to compute surface tension. The advection of the volume fraction is performed using either the Lagrangian-Explicit / CIAM (Calcul d'Interface Affine par Morceaux) method or the Weymouth and Yue (WY) Eulerian-Implicit method. The WY method conserves fluid mass to machine accuracy provided incompressibility is satisfied. To improve the stability of these methods momentum fluxes are advected in a manner "consistent" with the volume-fraction fluxes, that is a discontinuity of the momentum is advected at the same speed as a discontinuity of the density. To find the density on the staggered cells on which the velocity is centered, an auxiliary reconstruction of the density is performed. The method is tested for a droplet without surface tension in uniform flow, for a droplet suddenly accelerated in a carrying gas at rest at very large density ratio without viscosity or surface tension, for the Kelvin-Helmholtz instability, for a 3mm-diameter falling raindrop and for an atomizing flow in air-water conditions.

A finite volume analogue of the finite element: With accurate flooding and drying

Ocean Modelling, 2010

A new unstructured grid, finite volume ocean model, H2Ocean, is described, that is suitable for flooding and drying problems. Here we derive the finite volume analogue of the P NC 1 À P 1 finite element, by interpreting the advective term in the continuity equation in a flux sense. A corresponding non-overlapping control volume is then selected for the momentum equation. The resulting model employs the median node-dual control volume for the water elevations, and edge-wise control volume for the velocities. In contrast to the P NC 1 À P 1 approach, the finite volume model not only guarantees mass conservation in a global sense, but it also explicitly guarantees mass conservation in each individual cell. Another feature of this new model is its ability to conserve momentum locally, and by applying a correction factor it can preserve constant energy head along a streamline in the case of rapidly varied flows. By using the upwind water depths in the flux computations, no special flooding and drying procedures need to be implemented. The new model is efficient, does not produce non-physical negative water depths and generates accurate results for a wide variety of flooding and drying problems. Compared with the results obtained from the P NC 1 À P 1 finite element model, the new model produces better solutions in the simulation of inundations and in capturing shock waves.

A conservative unstructured scheme for rapidly varied flows

International Journal for Numerical Methods in Fluids, 2008

This article introduces a new semi-implicit, staggered finite volume scheme on unstructured meshes for the modelling of rapidly varied shallow water flows. Rapidly varied flows occur in the inundation of dry land during flooding situations. They typically involve bores and hydraulic jumps after obstacles such as road banks. Near such sudden flow transitions, the grid resolution is often low compared with the gradients of the bathymetry. Locally the hydrostatic pressure assumption may become invalid. In these situations, it is crucial to apply the correct conservation properties to obtain accurate results. An important feature of this scheme is therefore its ability to conserve momentum locally or, by choice, preserve constant energy head along a streamline. This is achieved using a special interpolation method and control volumes for momentum.

Depth-integrated free-surface flow with parameterized non-hydrostatic pressure

International Journal for Numerical Methods in Fluids, 2013

Non-hydrostatic free-surface models can provide better descriptions of dispersive waves by increasing the number of layers at the expense of computational efficiency. This paper proposes a parameterized nonhydrostatic pressure distribution in a depth-integrated two-layer formulation to reduce computational costs and to maintain essential dispersion properties for modeling of coastal processes. The non-hydrostatic pressure at mid flow depth is expressed in terms of the bottom pressure with a free parameter, which is determined to match the exact linear dispersion relation for the water depth parameter up to kd D 3. This reduces the depth-integrated two-layer formulation to a hybrid system with a tridiagonal matrix in the pressure Poisson equation. Linear dispersion relations and shoaling gradients derived from the present model as well as conventional one-layer and two-layer models provide a baseline for performance evaluation. Results from these three models are compared with previous laboratory experiments for wave transformation over a submerged bar, a plane beach, and a fringing reef. The present model provides comparable results as the two-layer model but at the computational requirements of a one-layer model. non-hydrostatic pressure simultaneously . The splitting method consists of a hydrostatic and a non-hydrostatic step with two solution approaches. The fractional step method [5] utilizes the nonhydrostatic pressure only in the non-hydrostatic step, but may introduce splitting errors that affect wave propagation significantly . The projection method also known as the pressure correction technique [7] utilizes the non-hydrostatic pressure in both steps to eliminate the splitting errors in the numerical solution.