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<html>
<head>
<title>
NAVIER_STOKES_2D_EXACT - Exact solutions to the 2D Incompressible Time-Dependent Navier Stokes Equations
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
NAVIER_STOKES_2D_EXACT<br>
Exact solutions to the <br>
2D Incompressible Time-Dependent Navier Stokes Equations
</h1>
<hr>
<p>
<b>NAVIER_STOKES_2D_EXACT</b>,
a C++ library which
evaluates exact solutions to the incompressible time-dependent
Navier-Stokes equations over an arbitrary domain in 2D.
</p>
<p>
<ul>
<li>
<b>Lucas</b>: steady flow, pressure is zero everywhere;
</li>
<li>
<b>Spiral</b>: velocity is zero on the boundary of the unit square;
</li>
<li>
<b>Taylor</b>: source term is zero everywhere.
</li>
</ul>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>NAVIER_STOKES_2D_EXACT</b> is available in
<a href = "../../c_src/navier_stokes_2d_exact/navier_stokes_2d_exact.html">a C version</a> and
<a href = "../../cpp_src/navier_stokes_2d_exact/navier_stokes_2d_exact.html">a C++ version</a> and
<a href = "../../f77_src/navier_stokes_2d_exact/navier_stokes_2d_exact.html">a FORTRAN77 version</a> and
<a href = "../../f_src/navier_stokes_2d_exact/navier_stokes_2d_exact.html">a FORTRAN90 version</a> and
<a href = "../../m_src/navier_stokes_2d_exact/navier_stokes_2d_exact.html">a MATLAB version</a> and
<a href = "../../py_src/navier_stokes_2d_exact/navier_stokes_2d_exact.html">a Python version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/navier_stokes_3d_exact/navier_stokes_3d_exact.html">
NAVIER_STOKES_3D_EXACT</a>,
a C++ library which
evaluates an exact solution to the incompressible time-dependent
Navier-Stokes equations over an arbitrary domain in 3D.
</p>
<p>
<a href = "../../m_src/navier_stokes_mesh2d/navier_stokes_mesh2d.html">
NAVIER_STOKES_MESH2D</a>,
MATLAB data files which
define triangular meshes for several
2D test problems involving the Navier Stokes equations
for fluid flow, provided by Leo Rebholz.
</p>
<p>
<a href = "../../cpp_src/spiral_data/spiral_data.html">
SPIRAL_DATA</a>,
a C++ library which
computes a velocity vector field that satisfies the continuity
equation, writing the data to a file that can be plotted
by gnuplot.
</p>
<p>
<a href = "../../cpp_src/stokes_2d_exact/stokes_2d_exact.html">
STOKES_2D_EXACT</a>,
a C++ library which
evaluates exact solutions to the incompressible steady
Stokes equations over the unit square in 2D.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Maxim Olshanskii, Leo Rebholz,<br>
Application of barycenter refined meshes in linear elasticity
and incompressible fluid dynamics,<br>
ETNA: Electronic Transactions in Numerical Analysis,<br>
Volume 38, pages 258-274, 2011.
</li>
<li>
Geoffrey Taylor,<br>
On the decay of vortices in a viscous fluid,<br>
Philosophical Magazine,<br>
Volume 46, 1923, pages 671-674.
</li>
<li>
Geoffrey Taylor, A E Green,<br>
Mechanism for the production of small eddies from large ones,<br>
Proceedings of the Royal Society of London, <br>
Series A, Volume 158, 1937, pages 499-521.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "ns2de.cpp">ns2de.cpp</a>, the source code.
</li>
<li>
<a href = "ns2de.hpp">ns2de.hpp</a>, the include file.
</li>
<li>
<a href = "ns2de.sh">ns2de.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "ns2de_prb.cpp">ns2de_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "ns2de_prb.sh">ns2de_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "ns2de_prb_output.txt">ns2de_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
Lucas Bystricky flow:
<ul>
<li>
<a href = "lucas_commands.txt">lucas_commands.txt</a>,
commands to draw the velocity field.
</li>
<li>
<a href = "lucas_data.txt">lucas_data.txt</a>,
velocity field data.
</li>
<li>
<a href = "lucas.png">lucas.png</a>,
a PNG plot of the velocity field.
</li>
</ul>
</p>
<p>
Spiral flow:
<ul>
<li>
<a href = "spiral_commands.txt">spiral_commands.txt</a>,
commands to draw the velocity field.
</li>
<li>
<a href = "spiral_data.txt">spiral_data.txt</a>,
velocity field data.
</li>
<li>
<a href = "spiral.png">spiral.png</a>,
a PNG plot of the velocity field.
</li>
</ul>
</p>
<p>
Taylor flow:
<ul>
<li>
<a href = "taylor_commands.txt">taylor_commands.txt</a>,
commands to draw the velocity field.
</li>
<li>
<a href = "taylor_data.txt">taylor_data.txt</a>,
velocity field data.
</li>
<li>
<a href = "taylor.png">taylor.png</a>,
a PNG plot of the velocity field.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>GRID_2D</b> returns a regular 2D grid.
</li>
<li>
<b>NS2DE_GNUPLOT</b> writes the Navier-Stokes velocity field to files for GNUPLOT.
</li>
<li>
<b>R8VEC_AMAX</b> returns the maximum absolute value in an R8VEC.
</li>
<li>
<b>R8VEC_AMIN</b> returns the minimum absolute value in an R8VEC.
</li>
<li>
<b>R8VEC_LINSPACE</b> creates a vector of linearly spaced values.
</li>
<li>
<b>R8VEC_MAX</b> returns the value of the maximum element in an R8VEC.
</li>
<li>
<b>R8VEC_MIN</b> returns the value of the minimum element in an R8VEC.
</li>
<li>
<b>R8VEC_NORM_L2</b> returns the L2 norm of an R8VEC.
</li>
<li>
<b>R8VEC_UNIFORM_AB_NEW</b> returns a scaled pseudorandom R8VEC.
</li>
<li>
<b>RESID_LUCAS</b> returns Lucas Bystricky residuals.
</li>
<li>
<b>RESID_SPIRAL</b> evaluates the residuals of the spiral flow problem.
</li>
<li>
<b>RESID_TAYLOR</b> returns residuals of the Taylor exact Navier Stokes solution.
</li>
<li>
<b>RHS_LUCAS</b> evaluates the right hand side of Lucas Bystricky's problem.
</li>
<li>
<b>RHS_SPIRAL</b> evaluates the right hand side of the spiral flow problem.
</li>
<li>
<b>RHS_TAYLOR</b> returns right hand sides of the Taylor vortex equation.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>UVP_LUCAS</b> evaluates Lucas Bystricky's exact Navier Stokes solution.
</li>
<li>
<b>UVP_SPIRAL</b> returns velocity and pressure for the spiral flow.
</li>
<li>
<b>UVP_TAYLOR</b> evaluates the Taylor exact Navier Stokes solution.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 06 March 2015.
</i>
<!-- John Burkardt -->
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</html>