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:

  Das Wasservolumen in einem Berechnungspolygon ist nichtlinear vom Wasserstand abhängig.

  Die durchströmte Fläche entlang einer Kante kann ebenfalls nichtlinear vom Wasserstand abhängen.

In dem Bild Berechnungsgitter und Bathymetrie auf SubGrid (ca. 0,5 MB) wird links die Topografie eines klassischen, aus Dreiecken aufgebauten unstrukturierten Gitters gezeigt. In einem Polygon sowie entlang jeder Kante variiert die Tiefe nicht. In der rechten Bildhälfte wird für ein (grobes) strukturiertes Berechnungsgitter die SubGrid-Technologie zur Diskretisierung der Bathymetrie genutzt. Man erkennt den Detailreichtum dieses Wattgebietes, da in jedem Polygon und entlang jeder Kante die Tiefe nunmehr veränderlich sein kann.

Berechnungsergebnisse

Die Ergebnisse für z. B. Wasserstand und Strömung liegen auf dem (groben) Berechnungsgitter und nicht auf dem (feineren) SubGrid vor. Typischer Weise entspricht der Wasserstand dem mittleren Wasserstand innerhalb eines Polygons, und die Strömungsgeschwindigkeit gibt für jede Kante jeweils den Mittelwert für die aktuell durchströmte Fläche wieder.

In dem Bild Strömung auf Gitter mit SubGrid (ca. 0,25 MB) wird links die Strömung für das klassische, und rechts für das Gitter mit SubGrid dargestellt. Man erkennt, dass im klassischen Fall jedes Polygon nur gänzlich von Wasser bedeckt oder vollkommen trockengefallen sein kann. In dem Gitter mit SubGrid kann ein Polygon (und eine Kante) hingegen auch teilweise von Wasser bedeckt sein. Damit gelingt eine naturähnlichere Simulation der Überflutung von Wattgebieten.

Vorteile

1. Die Topografie kann unabhängig von der Auflösung des Berechnungsgitters beschrieben werden.
2. Die Genauigkeit, mit der die Topografie wiedergegeben werden kann, wird prinzipiell nur durch die Genauigkeit der gemessenen Daten begrenzt.
3. Für jede Wasserspiegellage entspricht das Wasservolumen in dem Modell dem in der Natur vorhandenen Wasservolumen. Gleiches gilt für die durchströmten Querschnitte.
4. Überflutete und trocken gefallene Gebiete können präzise beschrieben werden.
5. Die gleiche Genauigkeit hinsichtlich der Diskretisierung der Bathymetrie wird mit SubGrid zu deutlich geringeren CPU-Kosten erzielt, als dies mit einem klassischen Berechnungsgitter, ohne SubGrid-Technologie, möglich wäre.

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

Publications

1. Casulli, V. and Zanolli, P. (1998), A Three-Dimensional Semi-Implicit Algorithm for Environmental Flows on Unstructured Grids, Proc. of Conf. on Num. Methods for Fluid Dynamics, University of Oxford.
2. Casulli, V. (1999), A Semi-Implicit Finite Difference Method for Non-Hydrostatic, Free-Surface Flows, International Journal for Numerical Methods in Fluids, 30: 425 - 440.
3. Casulli, V. and R.A. Walters (2000), An unstructured, three-dimensional model based on the shallow water equations, International Journal for Numerical Methods in Fluids 2000, 32: 331 - 348.
4. Casulli, V. and Zanolli, P. (2002), Semi-Implicit Numerical Modelling of Non-Hydrostatic Free-Surface Flows for Environmental Problems, Mathematical and Computer Modelling, 36: 1131 - 1149.
5. Casulli, V. and Zanolli, P. (2004), High Resolution Methods for Multidimensional Advection-Diffusion Problems in Free-Surface Hydrodynamics, Ocean Modelling, to appear.

Validation Document

In addition to a quite general introduction to UNTRIM the validation document contains also more detailed informations related to the following topics:

1. physical system,
2. model functionality,
3. conceptual model,
4. algorithmic implementation,
5. software implementation,
6. validation studies, and
7. literature.

A PDF-version of the validation document is freely available for download:

• UNTRIM standard validation document

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:

• UNTRIM user interface description
• UNTRIM² (version August 2009) user interface description

MPI-Parallelisation

The core of the mathematical model UnTRIM has been parallelized using MPI by Jacek Jankowski. A detailed description is available in the technical report MPI Version Manual. This version is actually applied internally mainly in the department of Hydraulic Engineering in Inland Areas.

BAW-Specific Informations

Grid Generation

An unstructured orthogonal grid for UNTRIM 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 UNTRIM is fully integrated into BAW's programming environment. More detailed information concerning it's integration can be found visiting UNTRIM program description.

Graphical Presentation of Computed Results

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

• HVIEW2D, 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

At BAW UNTRIM can be used together with the following independent sub-models:

1. spectral wave model k-modell (see unk.dat);
2. sedimentological model SediMorph (see sedimorph.dat).

The above mentioned sub-models can be used in direct coupling with the computational core of UNTRIM.

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