Mathematical Model UNTRIM2

Short Description

The numerical method UNTRIM2 is closely related with its predecessors UNTRIM and UNTRIM2007. UNTRIM2 is a semi-implicit finite difference (-volume) model, based on the three-dimensional shallow water equations as well as on the three-dimensional transport equation for salt, heat, dissolved matter and suspended sediments. The essential, main and revolutionary difference, is due to the sub grid technology used to discretise bathymetry with much finer resolution compared to the computational grid.

Sub Grid Technology

Computational Grid and Sub Grid

• Computational Grid:

  UNTRIM2 is able to work on unstructured orthogonal grids (UOG). The modelling domain is covered by a grid consisting of a set of non-overlapping convex polygons, usually either triangles or quadrilaterals. The grid is said to be an unstructured orthogonal grid if within each polygon a point (hereafter called a center) can be identified in such a way that the segment joining the center of two adjacent polygons and the side shared by the two polygons, have a non-empty intersection and are orthogonal to each other.

• Sub Grid:

  Model bathymetry is defined with sub grid resolution. Sub grid resolution can be much finer compared to the resolution of the computational grid.

• Non-Linearities:

  Water volume within a polygon may be non-linearily dependent on water level.

  Flow area along edges may also become non-linearily dependent on water level.

In slide computational grid and bathymetry on sub grid (approx. 0,5 MB) a classical discretisation of bathymetry is shown on the left side. On the right side, for a (coarse) structured computational grid, with sub grid technology we obtain a magnificant approximation of bathymetry. Many details of the bathymetry become visible. This is due to the fact, that depth is no longer a constant for polygons as well as for edges.

Computational Results

Water level as well as current velocity are computed on the (coarse) computational grid only, but not on the (much finer) sub grid. Typically water level corresponds to the mean value within a computational polygon. And current velocity is equivalent to the mean velocity for the respective actual flow area along an edge.

In slide current velocity for grid with sub grid (appox. 0,25 MB), on the left the classical result is shown, whereas on the right the sub grid result is displayed. One can easily see, that for the classical grid, a polygon can be either completely dry or completely wet. In the sub grid case instead, a polygon may be also partially wet. This feature facilitates quasi-natural flooding of tidal flat areas in a numerical model.

Benefits

1. Bathymetry can be prescribed independent of computational grid resolution.
2. Accuracy is, in principle, only limited due to measurement errors of bathymetric data.
3. For any water water level, modelled water volume is very close to the natural water volume. Similar is true for the flow area.
4. Dry and wet areas can be prescribed in great detail.
5. Comparable accuracy with respect to bathymetry can be achieved at much lower CPU cost using sub grid technology, compared to a full discretisation of bathymetry with the classical grid approach.

Physical Processes

• reynolds-averaged Navier-Stokes equations (RANS)
  • local acceleration (inertia)
  • advective acceleration
  • Coriolis acceleration
  • barotropic pressure gradient
  • baroclinic pressure gradient
  • hydrostatic or non-hydrostatic pressure
  • horizontal turbulent viscosity
  • vertical turbulent viscosity influenced by density stratification
  • bottom friction
  • wind friction
  • sources and sinks
  • horizontal acceleration due to wave effects (by means of radiation stress)
• transport of tracers
  • local rate of change of concentration
  • advective rate of change of concentration
  • optional flux limiter : Minmod, van Leer or Superbee
  • horizontal turbulent diffusivity
  • vertical turbulent diffusivity influenced by density stratification
  • settling of particles, deposition and erosion (for suspended sediments)
  • heat-transfer to/from the atmosphere and to/from the bottom
  • sources and sinks
  • sinks with immediate return inflow at a different location, with optional modification of inflow-temperature as well as -salinity

Computational Results

• water level elevation at the free surface
• current velocity
• tracer concentration (e.g. salinity, temperature, suspended sediments)
• hydrodynamic pressure

Note: when UNTRIM2 is used in two-dimensional (depth-integrated) mode, results correspond to the depth-averaged values for the above-mentioned quantities.

Publications

1. Casulli, V. (2008), A high-resolution wetting and drying algorithm for free-surface hydrodynamics, International Journal for Numerical Methods in Fluids, Volume 60, Issue 4, pages 391 - 408. See abstract.
2. Casulli, V. and Stelling, G. S. (2010), Semi-implicit sub grid modelling of three-dimensional free-surface flows, International Journal for Numerical Methods in Fluids, online available, advance of print. See abstract.

Presentations

• UNTRIM² (sub grid) at BAW Hamburg, slides, November 11, 2010, approx. 15 MB.

Validation Document

Not available.

User Interface Description

This document contains a detailed description of all interface functions and routines available to the user. The following topics are dealt with in this document:

1. set data (set-interfaces),
2. get data (get-interfaces),
3. check grid consistency as well as accuracy of iteratively computed results (check-routines),
4. external routines called by the computational core (user-interface-routines) which are required to,
   1. define paths and names of the standard input data files, to
   2. define (set) the inital state (initial data), to
   3. set the forcing terms (e.g. along open boundaries) for each time step, and to
   4. retrieve the computational results.
5. tables with short descriptions of all get- and set-interfaces available, and
6. example standard input data files.

A PDF-version of the user interface description document is freely available for download:

• UNTRIM² (version August 2009) user interface description

MPI-Parallelisation

Not realised.

BAW-Specific Informations

Grid Generation

An unstructured orthogonal grid for UNTRIM2 can be prepared using JANET grid generator software, made by SmileConsult. For further informations related to the integration of JANET into BAW's programming environment please visit JANET program description.

Simulation

The mathematical model UNTRIM2 is partially integrated into BAW's programming environment. More detailed information concerning it's integration can be found visiting UNTRIM2 program description.

Graphical Presentation of Computed Results

To display UnTRIM2 results currently several methods are used at BAW. The more important ones are,

• SGHVIEW2D, for data available throughout the computational domain,
• VVIEW2D and/or LQ2PRO, for data at longitudinal- and/or cross-sections, as well as
• GVIEW2D, for data at specific locations.

Analyses of Computational Results

A great variety of methods for analyses of computational results is available which enables the user to respond to many different questions.

Coupling to Independent Sub-Models

Not realised.

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Overview