HCELL
Fluid Flow in an H-shaped Region

HCELL is a FORTRAN77 program, using double precision arithmetic, which solves the time-dependent Navier Stokes equations in a two-dimensional region called the "H Cell".

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. One set of such data is stored as the TCELL flow dataset.

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. Programs that can be used for such an analysis include CVT_BASIS and POD_BASIS.

The "H-Cell" region is an H-shaped region.

          +----------------------------+
          |              :     :       |
    row 3 |   (3,1)      :(3,2): (3,3) |
          |              :     :       |
          +--------------+.....+-------+
                         |     |        
    row 2     empty      |(2,2)|  empty
                         |     |  
          +--------------+.....+-------+
          |              :     :       |
    row 1 |   (1,1)      :(1,2): (1,3) |
          |              :     :       |
          +----------------------------+

              col 1       col 2  col 3
      

Usage:

hcell

runs the program, reading the file "up000.txt" for the initial values of the horizontal and vertical velocities, and the pressure, at each node. Output files include the series "up001.txt", "up002.txt" and so on, containing flow quantities at successive time steps.

Author:

Hyun-Chun Lee,
Ajou University, Korea,
in collaboration with Max Gunzburger, Janet Peterson, and John Burkardt.

Related Data and Programs:

BUMP is an executable FORTRAN90 program which solves a fluid flow problem in a channel including a bump which obstructs and redirects the flow.

CHANNEL is a program which computes the pressure and velocity for a Navier Stokes flow in a channel. It is available in a FORTRAN77 version and a FORTRAN90 version.

CVT_BASIS_FLOW extracts dominant solution modes from a set of flow solutions using CVT methods.

DIRECTION_ARROWS_GRID is a MATLAB program which reads files of node and velocity data, and, using interpolation, creates a velocity direction plot with arrows place on a uniform grid of the user's specification.

FEM1D, is an executable FORTRAN77 program which applies the finite element method, with piecewise linear basis functions, to a two point boundary value problem;

FFNS_SPARSE is a MATLAB program for solving the steady incompressible Navier Stokes equations on an arbitrary triangulated region, using the finite element method and MATLAB's sparse facility.

FFS_SPARSE is a MATLAB program for solving the steady incompressible Stokes equations on an arbitrary triangulated region, using the finite element method and MATLAB's sparse facility.

FREE_FEM_NAVIER_STOKES is a MATLAB program for solving the steady incompressible Navier Stokes equations on an arbitrary triangulated region, using the finite element method.

FREE_FEM_STOKES is a MATLAB program which computes the pressure and velocity for a Stokes flow.

HCELL_FLOW_DISPLAY is a MATLAB program which can plot the data produced by HCELL.

HCELL_STEADY is an executable FORTRAN77 program that solves the time-independent or "steady" Navier Stokes equations in an H-cell region.

INOUT is an executable FORTRAN77 program which solves the Navier Stokes equations in the "In-Out" region.

MHD_CONTROL is an executable FORTRAN90 program which tries to control the evolution of an MHD system so that a particular state is achieved.

MHD_FLOW is an executable FORTRAN90 program for the evolution of an MHD system.

NAST2D is a C++ program which computes the pressure and velocity for a Navier Stokes flow.

NAST2D_F90 is a FORTRAN90 program which computes the pressure and velocity for a Navier Stokes flow.

TCELL is an executable FORTRAN77 program which solves the Navier Stokes equations in the T-Cell region.

POD_BASIS_FLOW extracts dominant solution modes from a set of flow solutions using POD methods.

VECTOR_STREAM_GRID is a MATLAB program which reads node and vector data from a file, computes an interpolatory function, evaluates on a uniform grid of points specified by the user, and displays a streamline plot of the vector field.

VELOCITY_ARROWS_GRID is a MATLAB program which reads files of node and velocity data, and, using interpolation, creates a vector plot with arrows place on a uniform grid of the user's specification.

Reference:

1. Max Gunzburger,
   Finite Element Methods for Viscous Incompressible Flows,
   A Guide to Theory, Practice, and Algorithms,
   Academic Press, 1989,
   ISBN: 0-12-307350-2,
   LC: TA357.G86.

Source Code:

• hcell.f, an initial draft of the program, with no treatment of the boundary conditions.
• hcell.csh, commands to compile, link and load the source code.
• hregion_01.f, a first complete draft of the program.
• hregion_02.f, a second draft which includes setting one reference pressure, and a correction to the inflow/outflow boundary conditions.
• hregion_03.f, draft #3 includes a routine to write velocities to a TECPLOT file.
• hregion_03.csh, commands to compile, link and load the program.
• hregion_03.out, printed output from a run of the program.
• hregion_04.f, draft #4 included corrections to the UBDRY routine, which fixed an outlet that wasn't letting out, and dropped an extra pressure condition that was accidentally causing problems in the velocity equation in the extreme upper right node.
• hregion_04.csh, commands to compile, link and load the program.
• hregion_04.out, printed output from a run of the program.
• hregion_05.f, draft #5 implements the refinement scheme by setting NREFINE as a parameter, and computing all other variables directly from there.
• hregion_05.csh, commands to compile, link and load the program.

Examples and Tests:

• xy3.txt, the coordinates of the nodes associated with linear elements;
• xy6.txt, the coordinates of the nodes associated with quadratic elements;
• up000.txt, a text file containing data to be used as the initial value of the solution;
• p100.txt, a file of pressure information at time step 100, suitable for plotting with "xy3.txt" and HCELL_CONTOUR_PLOT;
• uv100.txt, a file of velocity information at time step 100, suitable for plotting with "xy6.txt" and HCELL_VECTOR_PLOT;
• uv100.tec, a TECPLOT file containing the velocities at time step 100;

There are six sets of inflow and outflow vents in the flow region. A basis flow profile is specified for each set of vents. However, for each vent group I, there is also a multiplier BETA(I) which can be used to modulate the flow strength. To perform parameter studies, it was decided to consider datasets, each of which contained seven different settings for all the BETA's. (Thus, such a dataset contains 42 numbers.) Datasets were generated in a variety of ways:

• canon_06_07.txt, the "canonical basis vectors", plus a vector which is entirely 1;
• cvt_06_07.txt, a set generated by CVT_DATASET;
• ham_06_07.txt, a set generated by HAMMERSLEY_DATASET;
• lcen_06_07.txt, a set generated by LATIN_CENTER_DATASET;
• lcvt_06_07.txt, a "latinized CVT" set generated by applying the LATINIZE program to the CVT_06_07.TXT file;
• uni_06_07.txt, a set generated by UNIFORM_DATASET;
• uni_06_10.txt, a set of 10 points, generated by UNIFORM_DATASET, and using a different seed;

List of Routines:

• MAIN is the main program for HCELL.
• CH_IS_DIGIT returns .TRUE. if a character is a decimal digit.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• DAXPY adds a multiple of one vector to another.
• DDOT forms the dot product of two vectors.
• DGBFA factors a double precision band matrix by elimination.
• DGBSL solves a double precision band linear system.
• DGEFA factors a double precision matrix by gaussian elimination.
• DGESL solves the double precision system
• DIGIT_INC increments a decimal digit.
• DIGIT_TO_CH returns the character representation of a decimal digit.
• DSCAL scales a vector by a constant.
• ELEMENT_NODE_BANDWIDTH determines the bandwidth associated with the grid.
• ELEMENT_NODE_WRITE writes the element node list to a file.
• FILE_NAME_INC generates the next file name in a series.
• HCELL_DOF_COUNT determines the number of degrees of freedom in the region.
• HCELL_DOF_SET assigns degrees of freedom to each node.
• HCELL_ELEMENT_COUNT determines the number of elements in the region.
• HCELL_ELEMENT_NODE determines the nodes that make up each element.
• HCELL_NODE_NUM determines the number of nodes in the region.
• HCELL_NODE_XY assigns coordinates to each node.
• IDAMAX finds the vector element of largest magnitude.
• NODE_EPS creates an EPS file containing an image of the nodes.
• NSTOKE solves the Navier-Stokes equations using Taylor-Hood elements.
• P_WRITE writes the pressures values for a given timestep to a file.
• QUAD_A_SET sets quadrature information for the assembly routine.
• R8_HUGE returns a "huge" double precision number.
• REFBSP evaluates a linear basis functions on the reference triangle.
• REFQBF evaluates quadratic basis functions on the reference triangle.
• SETGRD sets up the grid for the problem.
• TRANS transforms data between the reference and physical elements.
• UBDRY evaluates the boundary conditions.
• UV_READ returns an initial value for the solution coefficient vector.
• UV_WRITE writes the velocity values for a given timestep to a file.
• XY3_WRITE writes the coordinates of nodes of 3-node triangles to a file.
• XY6_WRITE writes the coordinates of nodes of 6-node triangles to a file.

You can go up one level to the FORTRAN77 source codes.