# TCELL Generate Navier Stokes solutions in TCELL region.

TCELL is a MATLAB program which solves the time-dependent Navier Stokes equations in a two dimensional region called the "T Cell", by Hyung-Chun Lee.

As time increases, other flow parameters may be varied as well. For this code, the flow parameter that can vary is the magnitude of the inflow specified at the left boundary. The program stores the results of each timestep in a file. Each solution may be thought of as a "snapshot" of the gradual evolution of the flow field over time. In a typical run, 500 such files might be created.

A collection of snapshot files may be analyzed to seek a small underlying basis of fundamental flows. This basis may be used to approximate the entire set, or indeed to generate a reduced basis program that can quickly and cheaply compute new flow solutions whose flow parameters vary in small ways from those of the original set.

The "T-Cell" region is a T-shaped region contained in a subset of the unit square [01]x[01]. The top half of the region is a channel open to the left and right, with a rectangular recess in the bottom. The left channel opening extends from [0.0,0.5] to [0.0,1.0], and the right channel from [1.0,0.5] to [1.0,1.0]. The recess is a square, whose bottom extends from [0.25,0.0] to [0.75,0.0], and which extends upward to meet the channel.

```             +--------------------+
+--------------------+
->                      ->
Inlet --->      Channel       --->  Outlet
->                      ->
+-----+          +---+
+----+|  Recess  |+--+
||          ||
Void ||          || Void
|+----------+|
+------------+
```
The grid can be thought of as being generated by a uniform 81 by 81 uniformly spaced grid of nodes, from which the nodes corresponding to the two voids have been deleted. This leaves 4,961 nodes. These nodes are organized into 2400 6-node triangular elements which are quadratic in velocity and linear in pressure.

### Author:

Original FORTRAN77 version by Hyung-Chun Lee. MATLAB version by John Burkardt.

### Languages:

TCELL is available in a FORTRAN77 version and a MATLAB version.

### Related Data and Programs:

TCELL_FLOW, a dataset which contains 500 successive flow solutions of the T-Cell problem.

TCELL_FLOW_MOVIE, an animation of a time simulation of the flow computed by TCELL.

TCELL_MASS, a FORTRAN77 program which computes the mass matrix for the T-Cell problem.

### Reference:

1. Max Gunzburger,
Finite Element Methods for Viscous Incompressible Flows,
A Guide to Theory, Practice, and Algorithms,
ISBN: 0-12-307350-2,
LC: TA357.G86.
2. John Burkardt, Max Gunzburger, Hyung-Chun Lee,
Centroidal Voronoi Tessellation-Based Reduced-Order Modelling of Complex Systems,
SIAM Journal on Scientific Computing,
Volume 28, Number 2, 2006, pages 459-484.

### Source Code:

• tcell.m, the source code.
• fuini.txt, an input file of initial solution coefficient values.

### Examples and Tests:

RUN1 is a run of the program used to generated "snapshots", which are flow solutions at equally spaced time intervals. The program expects an input file, containing coefficients that define the solution at the initial time. During this run, the boundary conditions are varied in a way that is intended to make the system exhibit many varied behaviors. (Right now, to change the boundary conditions requires altering the source code of the program.)

• fuini_run1.txt, an input file of initial solution coefficient values, for the run that generates snapshots.
• tcell_run1_output.txt, the output file.
• elenode.txt, an output file containing the number of nodes and elements, the node coordinates, and the node indices that form each element.
• The 500 snapshot files created by this run are available in the TCELL snapshot dataset.

RUN2 is a run of the program used to generate a solution for comparison with one generated by a reduced order model. The program expects an input file, containing coefficients that define the solution at the initial time. During this run, the boundary conditions are varied in a new way. (Right now, to change the boundary conditions requires altering the source code of the program.)

• fuini_run2.txt, an input file of initial solution coefficient values, for the run that generates snapshots.

You can go up one level to the MATLAB source codes.

Last revised on 05 March 2016.