-
Notifications
You must be signed in to change notification settings - Fork 57
/
navier_stokes_2d_exact.html
359 lines (327 loc) · 10.7 KB
/
navier_stokes_2d_exact.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
<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 MATLAB 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 = "../../m_src/navier_stokes_3d_exact/navier_stokes_3d_exact.html">
NAVIER_STOKES_3D_EXACT</a>,
a MATLAB library which
evaluates exact solutions 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 defining meshes for several
2D test problems involving the Navier Stokes equations
for fluid flow, provided by Leo Rebholz.
</p>
<p>
<a href = "../../m_src/spiral_data/spiral_data.html">
SPIRAL_DATA</a>,
a MATLAB 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 = "../../m_src/stokes_2d_exact/stokes_2d_exact.html">
STOKES_2D_EXACT</a>,
a MATLAB 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 = "grid_2d.m">grid_2d.m</a>,
returns a regular 2D grid.
</li>
<li>
<a href = "ns2de_gnuplot.m">ns2de_gnuplot.m</a>,
writes the velocity vector field to files for GNUPLOT.
</li>
<li>
<a href = "r8vec_uniform_ab.m">r8vec_uniform_ab.m</a>,
returns a scaled pseudorandom R8VEC.
</li>
<li>
<a href = "resid_lucas.m">resid_lucas.m</a>,
evaluates the residuals at any point (x,y) and
time t.
</li>
<li>
<a href = "resid_taylor.m">resid_taylor.m</a>,
evaluates the residuals at any point (x,y) and
time t.
</li>
<li>
<a href = "rhs_lucas.m">rhs_lucas.m</a>,
evaluates the right hand sides of the Lucas Bystricky problem
at any point (x,y) and time t.
</li>
<li>
<a href = "rhs_taylor.m">rhs_taylor.m</a>,
evaluates the right hand sides of the Taylor vortex equations
at any point (x,y) and time t.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>
prints the YMDHMS date as a timestamp.
</li>
<li>
<a href = "uvp_lucas.m">uvp_lucas.m</a>,
evaluates the Lucas Bystricky flow field at any point (x,y) and
time t.
</li>
<li>
<a href = "uvp_spiral.m">uvp_spiral.m</a>,
evaluates the spiral flow field at any point (x,y) and
time t.
</li>
<li>
<a href = "uvp_taylor.m">uvp_taylor.m</a>,
evaluates the Taylor vortex flow field at any point (x,y) and
time t.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "ns2de_test.m">ns2de_test.m</a>,
calls all the tests.
</li>
<li>
<a href = "ns2de_test_output.txt">ns2de_test_output.txt</a>,
the output file.
</li>
<li>
<a href = "gnuplot_taylor_test.m">gnuplot_taylor_test.m</a>,
samples the solution at the initial time on a regular grid,
and stores GNUPLOT graphics files of the velocity field.
</li>
<li>
<a href = "gnuplot_taylor_test.m">gnuplot_taylor_test.m</a>,
samples the Spiral Flow solution at the initial time on a regular grid,
and stores GNUPLOT graphics files of the velocity field.
</li>
<li>
<a href = "parameter_spiral_test.m">parameter_spiral_test.m</a>,
samples the Spiral Flow solution to show the effect
of the parameters NU and RHO.
</li>
<li>
<a href = "parameter_taylor_test.m">parameter_taylor_test.m</a>,
samples the solution to show the effect of the parameters NU and RHO.
</li>
<li>
<a href = "resid_lucas_test.m">resid_lucas_test.m</a>,
samples the Lucas Bystricky residual at the initial time,
to estimate the range.
</li>
<li>
<a href = "resid_spiral_test.m">resid_spiral_test.m</a>,
samples the Spiral Flow residual at the initial time, to estimate the
range.
</li>
<li>
<a href = "resid_taylor_test.m">resid_taylor_test.m</a>,
samples the residual at the initial time, to estimate the
range.
</li>
<li>
<a href = "rhs_lucas_test.m">rhs_lucas_test.m</a>,
samples the Lucas Bystricky right hand side at the initial time,
to estimate the range.
</li>
<li>
<a href = "rhs_spiral_test.m">rhs_spiral_test.m</a>,
samples the Spiral Flow right hand side at the initial time, to estimate the
range.
</li>
<li>
<a href = "rhs_taylor_test.m">rhs_taylor_test.m</a>,
samples the right hand side at the initial time, to estimate the
range.
</li>
<li>
<a href = "uvp_lucas_test.m">uvp_lucas_test.m</a>,
samples the Lucas Bystricky flow field at the initial time, to estimate the
data range.
</li>
<li>
<a href = "uvp_lucas_test2.m">uvp_lucas_test2.m</a>,
samples the Lucas Bystricky flow field on the boundary
at the initial time, to estimate the
data range.
</li>
<li>
<a href = "uvp_spiral_test.m">uvp_spiral_test.m</a>,
samples the Spiral Flow solution at the initial time, to estimate the
data range.
</li>
<li>
<a href = "uvp_spiral_test2.m">uvp_spiral_test2.m</a>,
samples the Spiral Flow solution on the boundary
at the initial time, to estimate the
data range.
</li>
<li>
<a href = "uvp_taylor_test.m">uvp_taylor_test.m</a>,
samples the solution at the initial time, to estimate the
data range.
</li>
<li>
<a href = "uvp_taylor_test2.m">uvp_taylor_test2.m</a>,
samples the solution on the boundary at the initial time, to estimate the
data range.
</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>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 31 January 2015.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>