 |
tupdated documentation to correct interaction force term - sphere - GPU-based 3… |
 |
git clone git://src.adamsgaard.dk/sphere |
 |
Log |
|
Files |
|
Refs |
|
LICENSE |
|
--- |
|
commit 60f837c2139f0dd664e1e009a6dec97be806cbcf |
|
parent 2d996a75c458c8904aded32488f73e5f37dc9135 |
|
Author: Anders Damsgaard <[email protected]> |
|
Date: Thu, 3 Apr 2014 10:58:30 +0200 |
|
|
|
updated documentation to correct interaction force term |
|
|
|
Diffstat: |
|
A doc/html/_images/math/0e86ca4660ab… | 0 |
|
A doc/html/_images/math/3357706feda9… | 0 |
|
A doc/html/_images/math/399743cbf7c4… | 0 |
|
A doc/html/_images/math/4fd7ec1b618d… | 0 |
|
A doc/html/_images/math/7f7495bf6b7e… | 0 |
|
A doc/html/_images/math/8d0831e0e18a… | 0 |
|
A doc/html/_images/math/9ec27d87740d… | 0 |
|
A doc/html/_images/math/a00f5eb30a7a… | 0 |
|
A doc/html/_images/math/a1e91a45b485… | 0 |
|
A doc/html/_images/math/b13f21416d84… | 0 |
|
A doc/html/_images/math/b70dd9c116fc… | 0 |
|
A doc/html/_images/math/dbb95aa092c1… | 0 |
|
A doc/html/_images/math/f94ccd83304b… | 0 |
|
A doc/html/_images/math/faee932adbe0… | 0 |
|
A doc/html/_images/math/ff29b7920019… | 0 |
|
M doc/html/_sources/cfd.txt | 36 ++++++++++++++++++++---------… |
|
M doc/html/cfd.html | 52 ++++++++++++++++++-----------… |
|
M doc/html/genindex.html | 4 ++++ |
|
M doc/html/objects.inv | 0 |
|
M doc/html/python_api.html | 6 ++++++ |
|
M doc/html/searchindex.js | 4 ++-- |
|
M doc/pdf/sphere.pdf | 0 |
|
M doc/sphinx/cfd.rst | 36 ++++++++++++++++++++---------… |
|
|
|
23 files changed, 89 insertions(+), 49 deletions(-) |
|
--- |
|
diff --git a/doc/html/_images/math/0e86ca4660ab213ac57b980a736e32499978d2dc.png… |
|
Binary files differ. |
|
diff --git a/doc/html/_images/math/3357706feda99b545e69da682825a94b64782cb6.png… |
|
Binary files differ. |
|
diff --git a/doc/html/_images/math/399743cbf7c481198eba2ac4794bf67ef9d5c294.png… |
|
Binary files differ. |
|
diff --git a/doc/html/_images/math/4fd7ec1b618d0e036da7606d6876e79d81480584.png… |
|
Binary files differ. |
|
diff --git a/doc/html/_images/math/7f7495bf6b7e8e4c468863b6fd083f72f3a844ac.png… |
|
Binary files differ. |
|
diff --git a/doc/html/_images/math/8d0831e0e18af6fd3f3f1060516faab8016dc054.png… |
|
Binary files differ. |
|
diff --git a/doc/html/_images/math/9ec27d87740d654f43e3238d5bfe718e521368ce.png… |
|
Binary files differ. |
|
diff --git a/doc/html/_images/math/a00f5eb30a7a379b737fd4fafa61160bc0fce4a8.png… |
|
Binary files differ. |
|
diff --git a/doc/html/_images/math/a1e91a45b4858dfcbacc9b0d3b28418f1a990df1.png… |
|
Binary files differ. |
|
diff --git a/doc/html/_images/math/b13f21416d84e13708696f34dea81026cda583c9.png… |
|
Binary files differ. |
|
diff --git a/doc/html/_images/math/b70dd9c116fc5a16341030868952b58cd10afa88.png… |
|
Binary files differ. |
|
diff --git a/doc/html/_images/math/dbb95aa092c199cb518b2fdf22908d217988c251.png… |
|
Binary files differ. |
|
diff --git a/doc/html/_images/math/f94ccd83304b6a8e5d454843c2d462d9bb6cba57.png… |
|
Binary files differ. |
|
diff --git a/doc/html/_images/math/faee932adbe0b3f8663c9be6fa88d65f456385a7.png… |
|
Binary files differ. |
|
diff --git a/doc/html/_images/math/ff29b7920019a21a45545381f042a08acfba3530.png… |
|
Binary files differ. |
|
diff --git a/doc/html/_sources/cfd.txt b/doc/html/_sources/cfd.txt |
|
t@@ -23,11 +23,13 @@ and the momentum equation: |
|
\rho \frac{\partial \boldsymbol{v}}{\partial t} |
|
+ \rho (\boldsymbol{v} \cdot \nabla \boldsymbol{v}) |
|
= \nabla \cdot \boldsymbol{\sigma} |
|
- + \rho \boldsymbol{f} |
|
+ - \boldsymbol{f}^i |
|
+ + \rho \boldsymbol{g} |
|
|
|
Here, :math:`\boldsymbol{v}` is the fluid velocity, :math:`\rho` is the |
|
-fluid density, :math:`\boldsymbol{\sigma}` is the `Cauchy stress tensor`_, and |
|
-:math:`\boldsymbol{f}` is a body force (e.g. gravity). For incompressible |
|
+fluid density, :math:`\boldsymbol{\sigma}` is the `Cauchy stress tensor`_, |
|
+:math:`\boldsymbol{f}^i` is the particle-fluid interaction vector and |
|
+:math:`\boldsymbol{g}` is the gravitational acceleration. For incompressible |
|
Newtonian fluids, the Cauchy stress is given by: |
|
|
|
.. math:: |
|
t@@ -78,7 +80,8 @@ with a body force :math:`\boldsymbol{f}` becomes: |
|
.. math:: |
|
\frac{D (\phi v_x)}{D t} |
|
= \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\sigma}) \right]_x |
|
- + \phi f_x |
|
+ - \frac{1}{\rho} f^i_x |
|
+ + \phi g |
|
|
|
In the Eulerian formulation, an advection term is added, and the Cauchy stress |
|
tensor is represented as isotropic and deviatoric components individually: |
|
t@@ -88,7 +91,8 @@ tensor is represented as isotropic and deviatoric components… |
|
+ \boldsymbol{v} \cdot \nabla (\phi v_x) |
|
= \frac{1}{\rho} \left[ \nabla \cdot (-\phi p \boldsymbol{I}) |
|
+ \phi \boldsymbol{\tau}) \right]_x |
|
- + \phi f_x |
|
+ - \frac{1}{\rho} f^i_x |
|
+ + \phi g_x |
|
|
|
Using vector identities to rewrite the advection term, and expanding the fluid |
|
stress tensor term: |
|
t@@ -98,9 +102,9 @@ stress tensor term: |
|
+ \nabla \cdot (\phi v_x \boldsymbol{v}) |
|
- \phi v_x (\nabla \cdot \boldsymbol{v}) |
|
= \frac{1}{\rho} \left[ -\nabla \phi p \right]_x |
|
- + \frac{1}{\rho} \left[ -\phi \nabla p \right]_x |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x |
|
- + \phi f_x |
|
+ - \frac{1}{\rho} f^i_x |
|
+ + \phi g_x |
|
|
|
Spatial variations in the porosity are neglected, |
|
|
|
t@@ -121,7 +125,8 @@ With these assumptions, the momentum equation simplifies t… |
|
+ \nabla \cdot (\phi v_x \boldsymbol{v}) |
|
= -\frac{1}{\rho} \frac{\partial p}{\partial x} |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x |
|
- + \phi f_x |
|
+ - \frac{1}{\rho} f^i_x |
|
+ + \phi g_x |
|
|
|
The remaining part of the advection term is for the :math:`x` component |
|
found as: |
|
t@@ -338,7 +343,8 @@ presented by Langtangen et al. (2002), the predicted velo… |
|
+ \nabla \cdot (\phi v_x \boldsymbol{v}) |
|
= - \frac{1}{\rho} \frac{\Delta p}{\Delta x} |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x |
|
- + \phi f_x |
|
+ - \frac{1}{\rho} f^i_x |
|
+ + \phi g_x |
|
|
|
\Downarrow |
|
|
|
t@@ -347,7 +353,8 @@ presented by Langtangen et al. (2002), the predicted velo… |
|
+ \nabla \cdot (\phi v_x \boldsymbol{v}) |
|
= - \frac{1}{\rho} \frac{\Delta p}{\Delta x} |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x |
|
- + \phi f_x |
|
+ - \frac{1}{\rho} f^i_x |
|
+ + \phi g_x |
|
|
|
We want to isolate :math:`\Delta v_x` in the above equation in order to project |
|
the new velocity. |
|
t@@ -356,7 +363,8 @@ the new velocity. |
|
\phi \frac{\Delta v_x}{\Delta t} |
|
= - \frac{1}{\rho} \frac{\Delta p}{\Delta x} |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x |
|
- + \phi f_x |
|
+ - \frac{1}{\rho} f^i_x |
|
+ + \phi g_x |
|
- v_x \frac{\Delta \phi}{\Delta t} |
|
- \nabla \cdot (\phi v_x \boldsymbol{v}) |
|
|
|
t@@ -364,7 +372,8 @@ the new velocity. |
|
= - \frac{1}{\rho} \frac{\Delta p}{\Delta x} \frac{\Delta t}{\phi} |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x |
|
\frac{\Delta t}{\phi} |
|
- + \Delta t f_x |
|
+ - \frac{\Delta t}{\rho\phi} f^i_x |
|
+ + \Delta t g_x |
|
- v_x \frac{\Delta \phi}{\phi} |
|
- \nabla \cdot (\phi v_x \boldsymbol{v}) \frac{\Delta t}{\phi} |
|
|
|
t@@ -381,7 +390,8 @@ in `Chorin (1968)`_. |
|
- \frac{\beta}{\rho} \frac{\Delta p^t}{\Delta x} \frac{\Delta t}{\phi^t} |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi^t \boldsymbol{\tau}^t) \right]_x |
|
\frac{\Delta t}{\phi} |
|
- + \Delta t f_x |
|
+ - \frac{\Delta t}{\rho\phi} f^i_x |
|
+ + \Delta t g_x |
|
- v^t_x \frac{\Delta \phi}{\phi^t} |
|
- \nabla \cdot (\phi^t v_x^t \boldsymbol{v}^t) \frac{\Delta t}{\phi^t} |
|
|
|
diff --git a/doc/html/cfd.html b/doc/html/cfd.html |
|
t@@ -72,13 +72,15 @@ continuity equation for an incompressible fluid material i… |
|
<p><img src="_images/math/b588eea9cec4513a3be72255d8d3df214546bfe7.png" alt="\… |
|
</div><p>and the momentum equation:</p> |
|
<div class="math"> |
|
-<p><img src="_images/math/5b46624e0dc3d79b64f388898e2dff17d232656c.png" alt="\… |
|
+<p><img src="_images/math/a00f5eb30a7a379b737fd4fafa61160bc0fce4a8.png" alt="\… |
|
+ \rho (\boldsymbol{v} \cdot \nabla \boldsymbol{v}) |
|
= \nabla \cdot \boldsymbol{\sigma} |
|
-+ \rho \boldsymbol{f}"/></p> |
|
+- \boldsymbol{f}^i |
|
++ \rho \boldsymbol{g}"/></p> |
|
</div><p>Here, <img class="math" src="_images/math/d0b4b390a4806bb739c6b4adbdf… |
|
-fluid density, <img class="math" src="_images/math/769bfdcb2a43bde2cd368d82a6f… |
|
-<img class="math" src="_images/math/69b1fdf87f9a78aaef8057a34aea7a6c17dad726.p… |
|
+fluid density, <img class="math" src="_images/math/769bfdcb2a43bde2cd368d82a6f… |
|
+<img class="math" src="_images/math/dbb95aa092c199cb518b2fdf22908d217988c251.p… |
|
+<img class="math" src="_images/math/a1e91a45b4858dfcbacc9b0d3b28418f1a990df1.p… |
|
Newtonian fluids, the Cauchy stress is given by:</p> |
|
<div class="math"> |
|
<p><img src="_images/math/c9264cc703654b5651cb89a1c9f5e178b5d15cd0.png" alt="\… |
|
t@@ -115,27 +117,29 @@ momentum equations. The continuity equation becomes:</p> |
|
</div><p>For the <img class="math" src="_images/math/26eeb5258ca5099acf8fe96b2… |
|
with a body force <img class="math" src="_images/math/69b1fdf87f9a78aaef8057a3… |
|
<div class="math"> |
|
-<p><img src="_images/math/900be94839e1ee03a8aa1134961912314905eb27.png" alt="\… |
|
+<p><img src="_images/math/4fd7ec1b618d0e036da7606d6876e79d81480584.png" alt="\… |
|
= \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\sigma}) \right]_x |
|
-+ \phi f_x"/></p> |
|
+- \frac{1}{\rho} f^i_x |
|
++ \phi g"/></p> |
|
</div><p>In the Eulerian formulation, an advection term is added, and the Cauc… |
|
tensor is represented as isotropic and deviatoric components individually:</p> |
|
<div class="math"> |
|
-<p><img src="_images/math/32e2ba09618e2d303d91d673a25ef66e29e94750.png" alt="\… |
|
+<p><img src="_images/math/ff29b7920019a21a45545381f042a08acfba3530.png" alt="\… |
|
+ \boldsymbol{v} \cdot \nabla (\phi v_x) |
|
= \frac{1}{\rho} \left[ \nabla \cdot (-\phi p \boldsymbol{I}) |
|
+ \phi \boldsymbol{\tau}) \right]_x |
|
-+ \phi f_x"/></p> |
|
+- \frac{1}{\rho} f^i_x |
|
++ \phi g_x"/></p> |
|
</div><p>Using vector identities to rewrite the advection term, and expanding … |
|
stress tensor term:</p> |
|
<div class="math"> |
|
-<p><img src="_images/math/dcf5762fe2b4b81adb93ee084951f42a2f1eadbc.png" alt="\… |
|
+<p><img src="_images/math/0e86ca4660ab213ac57b980a736e32499978d2dc.png" alt="\… |
|
+ \nabla \cdot (\phi v_x \boldsymbol{v}) |
|
- \phi v_x (\nabla \cdot \boldsymbol{v}) |
|
= \frac{1}{\rho} \left[ -\nabla \phi p \right]_x |
|
-+ \frac{1}{\rho} \left[ -\phi \nabla p \right]_x |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x |
|
-+ \phi f_x"/></p> |
|
+- \frac{1}{\rho} f^i_x |
|
++ \phi g_x"/></p> |
|
</div><p>Spatial variations in the porosity are neglected,</p> |
|
<div class="math"> |
|
<p><img src="_images/math/c42a32017c99646f19bb5807728595d4526c3b30.png" alt="\… |
|
t@@ -146,11 +150,12 @@ zero:</p> |
|
<p><img src="_images/math/44fafcf5a158459730d0dd7c293b93cdcf62f0a4.png" alt="\… |
|
</div><p>With these assumptions, the momentum equation simplifies to:</p> |
|
<div class="math"> |
|
-<p><img src="_images/math/6e62843666dcc0fe9a45a1c69c532c82a95a9451.png" alt="\… |
|
+<p><img src="_images/math/7f7495bf6b7e8e4c468863b6fd083f72f3a844ac.png" alt="\… |
|
+ \nabla \cdot (\phi v_x \boldsymbol{v}) |
|
= -\frac{1}{\rho} \frac{\partial p}{\partial x} |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x |
|
-+ \phi f_x"/></p> |
|
+- \frac{1}{\rho} f^i_x |
|
++ \phi g_x"/></p> |
|
</div><p>The remaining part of the advection term is for the <img class="math"… |
|
found as:</p> |
|
<div class="math"> |
|
t@@ -335,11 +340,12 @@ presented by Langtangen et al. (2002), the predicted ve… |
|
<img class="math" src="_images/math/90c8bfc206db2d9f4d0dd102507c9646a70755db.p… |
|
<img class="math" src="_images/math/a1ffc0a012620941fe660cedabff822ce7162eca.p… |
|
<div class="math"> |
|
-<p><img src="_images/math/7ee03c7bfc1e46255c9d2d47d9b733a068c9ec2b.png" alt="\… |
|
+<p><img src="_images/math/8d0831e0e18af6fd3f3f1060516faab8016dc054.png" alt="\… |
|
+ \nabla \cdot (\phi v_x \boldsymbol{v}) |
|
= - \frac{1}{\rho} \frac{\Delta p}{\Delta x} |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x |
|
-+ \phi f_x |
|
+- \frac{1}{\rho} f^i_x |
|
++ \phi g_x |
|
|
|
\Downarrow |
|
|
|
t@@ -348,14 +354,16 @@ presented by Langtangen et al. (2002), the predicted ve… |
|
+ \nabla \cdot (\phi v_x \boldsymbol{v}) |
|
= - \frac{1}{\rho} \frac{\Delta p}{\Delta x} |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x |
|
-+ \phi f_x"/></p> |
|
+- \frac{1}{\rho} f^i_x |
|
++ \phi g_x"/></p> |
|
</div><p>We want to isolate <img class="math" src="_images/math/b5e8dba2403c07… |
|
the new velocity.</p> |
|
<div class="math"> |
|
-<p><img src="_images/math/038474380a078000f31889a32a1dcf79ac38c223.png" alt="\… |
|
+<p><img src="_images/math/faee932adbe0b3f8663c9be6fa88d65f456385a7.png" alt="\… |
|
= - \frac{1}{\rho} \frac{\Delta p}{\Delta x} |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x |
|
-+ \phi f_x |
|
+- \frac{1}{\rho} f^i_x |
|
++ \phi g_x |
|
- v_x \frac{\Delta \phi}{\Delta t} |
|
- \nabla \cdot (\phi v_x \boldsymbol{v}) |
|
|
|
t@@ -363,7 +371,8 @@ the new velocity.</p> |
|
= - \frac{1}{\rho} \frac{\Delta p}{\Delta x} \frac{\Delta t}{\phi} |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x |
|
\frac{\Delta t}{\phi} |
|
-+ \Delta t f_x |
|
+- \frac{\Delta t}{\rho\phi} f^i_x |
|
++ \Delta t g_x |
|
- v_x \frac{\Delta \phi}{\phi} |
|
- \nabla \cdot (\phi v_x \boldsymbol{v}) \frac{\Delta t}{\phi}"/></p> |
|
</div><p>The term <img class="math" src="_images/math/fdb63b9e51abe6bbb16acfb5… |
|
t@@ -372,13 +381,14 @@ values in the solution procedure (Langtangen et al. 2002… |
|
corresponds to <a class="reference external" href="https://en.wikipedia.org/wi… |
|
in <a class="reference external" href="http://www.ams.org/journals/mcom/1968-2… |
|
<div class="math"> |
|
-<p><img src="_images/math/dbcdbe7c53fa70f8517907ca1b3c440b28512dfc.png" alt="v… |
|
+<p><img src="_images/math/9ec27d87740d654f43e3238d5bfe718e521368ce.png" alt="v… |
|
|
|
v_x^* = v_x^t |
|
- \frac{\beta}{\rho} \frac{\Delta p^t}{\Delta x} \frac{\Delta t}{\phi^t} |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi^t \boldsymbol{\tau}^t) \right]_x |
|
\frac{\Delta t}{\phi} |
|
-+ \Delta t f_x |
|
+- \frac{\Delta t}{\rho\phi} f^i_x |
|
++ \Delta t g_x |
|
- v^t_x \frac{\Delta \phi}{\phi^t} |
|
- \nabla \cdot (\phi^t v_x^t \boldsymbol{v}^t) \frac{\Delta t}{\phi^t}"/></p> |
|
</div><p>Here, <img class="math" src="_images/math/1eb29f9de3753a59530941141fc… |
|
diff --git a/doc/html/genindex.html b/doc/html/genindex.html |
|
t@@ -168,6 +168,10 @@ |
|
</dl></td> |
|
<td style="width: 33%" valign="top"><dl> |
|
|
|
+ <dt><a href="python_api.html#sphere.sim.deleteAllParticles">deleteAllParticl… |
|
+ </dt> |
|
+ |
|
+ |
|
<dt><a href="python_api.html#sphere.sim.disableFluidPressureModulation">disa… |
|
</dt> |
|
|
|
diff --git a/doc/html/objects.inv b/doc/html/objects.inv |
|
Binary files differ. |
|
diff --git a/doc/html/python_api.html b/doc/html/python_api.html |
|
t@@ -581,6 +581,12 @@ won’t work. Default = [0.0, 0.0, 0.0].</li> |
|
</dd></dl> |
|
|
|
<dl class="method"> |
|
+<dt id="sphere.sim.deleteAllParticles"> |
|
+<tt class="descname">deleteAllParticles</tt><big>(</big><big>)</big><a class="… |
|
+<dd><p>Deletes all particles in the simulation object.</p> |
|
+</dd></dl> |
|
+ |
|
+<dl class="method"> |
|
<dt id="sphere.sim.disableFluidPressureModulation"> |
|
<tt class="descname">disableFluidPressureModulation</tt><big>(</big><big>)</bi… |
|
<dd><p>Set the parameters for the sine wave modulating the fluid pressures |
|
diff --git a/doc/html/searchindex.js b/doc/html/searchindex.js |
|
t@@ -1 +1 @@ |
|
-Search.setIndex({objects:{"":{sphere:[5,0,1,""]},sphere:{status:[5,1,1,""],con… |
|
-\ No newline at end of file |
|
+Search.setIndex({objects:{"":{sphere:[5,0,1,""]},sphere:{status:[5,2,1,""],con… |
|
+\ No newline at end of file |
|
diff --git a/doc/pdf/sphere.pdf b/doc/pdf/sphere.pdf |
|
Binary files differ. |
|
diff --git a/doc/sphinx/cfd.rst b/doc/sphinx/cfd.rst |
|
t@@ -23,11 +23,13 @@ and the momentum equation: |
|
\rho \frac{\partial \boldsymbol{v}}{\partial t} |
|
+ \rho (\boldsymbol{v} \cdot \nabla \boldsymbol{v}) |
|
= \nabla \cdot \boldsymbol{\sigma} |
|
- + \rho \boldsymbol{f} |
|
+ - \boldsymbol{f}^i |
|
+ + \rho \boldsymbol{g} |
|
|
|
Here, :math:`\boldsymbol{v}` is the fluid velocity, :math:`\rho` is the |
|
-fluid density, :math:`\boldsymbol{\sigma}` is the `Cauchy stress tensor`_, and |
|
-:math:`\boldsymbol{f}` is a body force (e.g. gravity). For incompressible |
|
+fluid density, :math:`\boldsymbol{\sigma}` is the `Cauchy stress tensor`_, |
|
+:math:`\boldsymbol{f}^i` is the particle-fluid interaction vector and |
|
+:math:`\boldsymbol{g}` is the gravitational acceleration. For incompressible |
|
Newtonian fluids, the Cauchy stress is given by: |
|
|
|
.. math:: |
|
t@@ -78,7 +80,8 @@ with a body force :math:`\boldsymbol{f}` becomes: |
|
.. math:: |
|
\frac{D (\phi v_x)}{D t} |
|
= \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\sigma}) \right]_x |
|
- + \phi f_x |
|
+ - \frac{1}{\rho} f^i_x |
|
+ + \phi g |
|
|
|
In the Eulerian formulation, an advection term is added, and the Cauchy stress |
|
tensor is represented as isotropic and deviatoric components individually: |
|
t@@ -88,7 +91,8 @@ tensor is represented as isotropic and deviatoric components… |
|
+ \boldsymbol{v} \cdot \nabla (\phi v_x) |
|
= \frac{1}{\rho} \left[ \nabla \cdot (-\phi p \boldsymbol{I}) |
|
+ \phi \boldsymbol{\tau}) \right]_x |
|
- + \phi f_x |
|
+ - \frac{1}{\rho} f^i_x |
|
+ + \phi g_x |
|
|
|
Using vector identities to rewrite the advection term, and expanding the fluid |
|
stress tensor term: |
|
t@@ -98,9 +102,9 @@ stress tensor term: |
|
+ \nabla \cdot (\phi v_x \boldsymbol{v}) |
|
- \phi v_x (\nabla \cdot \boldsymbol{v}) |
|
= \frac{1}{\rho} \left[ -\nabla \phi p \right]_x |
|
- + \frac{1}{\rho} \left[ -\phi \nabla p \right]_x |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x |
|
- + \phi f_x |
|
+ - \frac{1}{\rho} f^i_x |
|
+ + \phi g_x |
|
|
|
Spatial variations in the porosity are neglected, |
|
|
|
t@@ -121,7 +125,8 @@ With these assumptions, the momentum equation simplifies t… |
|
+ \nabla \cdot (\phi v_x \boldsymbol{v}) |
|
= -\frac{1}{\rho} \frac{\partial p}{\partial x} |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x |
|
- + \phi f_x |
|
+ - \frac{1}{\rho} f^i_x |
|
+ + \phi g_x |
|
|
|
The remaining part of the advection term is for the :math:`x` component |
|
found as: |
|
t@@ -338,7 +343,8 @@ presented by Langtangen et al. (2002), the predicted velo… |
|
+ \nabla \cdot (\phi v_x \boldsymbol{v}) |
|
= - \frac{1}{\rho} \frac{\Delta p}{\Delta x} |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x |
|
- + \phi f_x |
|
+ - \frac{1}{\rho} f^i_x |
|
+ + \phi g_x |
|
|
|
\Downarrow |
|
|
|
t@@ -347,7 +353,8 @@ presented by Langtangen et al. (2002), the predicted velo… |
|
+ \nabla \cdot (\phi v_x \boldsymbol{v}) |
|
= - \frac{1}{\rho} \frac{\Delta p}{\Delta x} |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x |
|
- + \phi f_x |
|
+ - \frac{1}{\rho} f^i_x |
|
+ + \phi g_x |
|
|
|
We want to isolate :math:`\Delta v_x` in the above equation in order to project |
|
the new velocity. |
|
t@@ -356,7 +363,8 @@ the new velocity. |
|
\phi \frac{\Delta v_x}{\Delta t} |
|
= - \frac{1}{\rho} \frac{\Delta p}{\Delta x} |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x |
|
- + \phi f_x |
|
+ - \frac{1}{\rho} f^i_x |
|
+ + \phi g_x |
|
- v_x \frac{\Delta \phi}{\Delta t} |
|
- \nabla \cdot (\phi v_x \boldsymbol{v}) |
|
|
|
t@@ -364,7 +372,8 @@ the new velocity. |
|
= - \frac{1}{\rho} \frac{\Delta p}{\Delta x} \frac{\Delta t}{\phi} |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x |
|
\frac{\Delta t}{\phi} |
|
- + \Delta t f_x |
|
+ - \frac{\Delta t}{\rho\phi} f^i_x |
|
+ + \Delta t g_x |
|
- v_x \frac{\Delta \phi}{\phi} |
|
- \nabla \cdot (\phi v_x \boldsymbol{v}) \frac{\Delta t}{\phi} |
|
|
|
t@@ -381,7 +390,8 @@ in `Chorin (1968)`_. |
|
- \frac{\beta}{\rho} \frac{\Delta p^t}{\Delta x} \frac{\Delta t}{\phi^t} |
|
+ \frac{1}{\rho} \left[ \nabla \cdot (\phi^t \boldsymbol{\tau}^t) \right]_x |
|
\frac{\Delta t}{\phi} |
|
- + \Delta t f_x |
|
+ - \frac{\Delta t}{\rho\phi} f^i_x |
|
+ + \Delta t g_x |
|
- v^t_x \frac{\Delta \phi}{\phi^t} |
|
- \nabla \cdot (\phi^t v_x^t \boldsymbol{v}^t) \frac{\Delta t}{\phi^t} |
|
