https://www.cfd-online.com/W/index.php?title=Special:Contributions/Slffea&feed=atom&limit=50&target=Slffea&year=&month=CFD-Wiki - User contributions [en]2017-06-27T05:58:03ZFrom CFD-WikiMediaWiki 1.16.5https://www.cfd-online.com/Wiki/Cfd_simulation_of_vortex_sheddingCfd simulation of vortex shedding2008-10-29T23:32:55Z<p>Slffea: typo</p>
<hr />
<div>To numerically simulate [[Vortex shedding|vortex shedding]], [[CFD]] is used to calculate the unsteady flow that arises from a fluid moving past an obstruction. [[Downwind]] of the obstruction are regions of lower pressure which causes the fluid which initially was deflected around the obstruction to get sucked into these regions, begin to circulate, and form [[vortices]].<br />
<br />
For a [[Newtonian fluid]], it is sufficient to use the [[Navier-Stokes equations]] as the governing equations, and choose either the [[Finite element|finite element method]], '''FEM''', or [[Finite difference|finite differencing]], '''FD''', to do the simulation. Vortex shedding is more likely in [[Incompressible flow|incompressible fluids]] than [[Compressible flow|compressible fluids]], so the former is used when capturing this phenomenon. Although both types of fluids are represented by the [[Navier-Stokes equations]], their methods of solution differ significantly. For instance, when using '''FD''' for incompressible fluids, it is common to have an explicit expression for the velocity as you integrate the equations in time. In [[Compressible flow|compressible fluids]], the governing equations are often written in terms of the [[conservative form]] of [[Euler's equation]], with velocity and pressure extracted from the conserved quantities(e.g. mass, momentum, energy). Also, compressible flows usually involve the calculation of additional quantities such as density and temperature.<br />
<br />
==Example of 2 Dimensional Problem==<br />
The images below were created for the case of 2-D flow past a square obstruction with a [[Reynolds number]] of 1000 and freestream boundary conditions on the top, bottom, and right boundaries. The quantities of the Navier-Stokes equations were non-dimensionalized and the [[Schemes_by_Leonard_-_structured_grids|QUICKEST]] finite differencing algorithm by B.P. Leonard was used. The grid was uniform with either a 204 x 90 node mesh or a 102 x 45 node mesh.<br />
<br />
<br><br />
[[Image:vortex_prs.jpg|thumb|200px|left|pressure at 95.4s and 3180 timesteps.]]<br />
<br><br />
[[Image:vortex_vx.jpg|thumb|200px|left|velocity x at 95.4s and 3180 timesteps.]]<br />
<br><br />
[[Image:vortex_vy.jpg|thumb|200px|left|velocity y at 95.4s and 3180 timesteps.]]<br />
[[Image:Vortex_streaklines-0ang.jpg|400px|thumb|center|[[streaklines]] for square obstruction at 96s and 2400 timesteps for 0 angle of attack flow.]]<br />
<br><br />
<br><br />
[[Image:Vortex_pathlines-0ang.jpg|400px|thumb|center|[[pathlines]] for triangle obstruction at 96s and 2400 timesteps for 0 angle of attack flow.]]<br />
<br><br />
<br><br />
[[Image:Vortex_streaklines-15ang.jpg|400px|thumb|center|[[streaklines]] for square obstruction at 96s and 2400 timesteps for 15 angle of attack flow.]]<br />
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<br><br />
<br><br />
<br><br />
<br><br />
<br><br />
<br><br />
<br><br />
<br><br />
<br />
==Pressure Corrector==<br />
The incompressibility of the fluid is enforced through the [[Continuity equation|continuity equation]] using pressure as a corrective quantity whose gradient in either the x or y direction is added to the velocity in the respective direction. Calculating the pressure correction involves solving the [[Discrete Poisson equation|discrete Poisson equation]] for pressure.<br />
<br />
==Staggered grid==<br />
Use of a [[staggered grid]], where pressure and velocities in the x and y directions are calculated at alternating grid points, is commonly used with [[upwinding]] to avoid oscillatory behavior.<br />
<br />
== See also ==<br />
*[[Flow across a square cylinder]]<br />
<br />
==References==<br />
*Davis, R. W., and E. F. Moore, <I>A numerical study of vortex shedding from rectangles</I>, Journal of Fluid Mechanics, 116, 475 (1982).<br />
<br />
*Davis, R. W., E. F. Moore, and L. P. Purtell, <I>A numerical-experimental study of confined flow around rectangular cylinders</I>, Physics of Fluids, 27, 46 (1984).<br />
<br />
*Hoffman, Joe D., <I> Numerical Methods for Engineers and Scientist, fourth edition </I>, McGraw-Hill Inc., New York, 1992.<br />
<br />
*Leonard, B. P., <I>A stable and accurate convective modelling procedure</I>, Computer Methods in Applied Mechanics and Engineering, 19, pages 59-98 (1979).</div>Slffeahttps://www.cfd-online.com/Wiki/Talk:CodesTalk:Codes2008-06-13T16:47:18Z<p>Slffea: /* Femwater link broken */</p>
<hr />
<div>I added a short note at the top of this page about limiting the length of the code descriptions - this is the result of a discussion in the wiki forum late last year - and I added a link to the copyright page to hopefully head off any more submissions of copyrighted material. I haven't started to look to hard at the individual pages yet - I'll start that later this week. Finally, I have removed all of the links to nonexistent pages in the free solvers section. I'm not sure how that looks. Any opinions? --[[User:Jasond|Jasond]] 15:48, 6 June 2007 (MDT)<br />
<br />
:It looks great I think. It is very good that someone works on the code section. It is one of the most frequented sections. About the description of the codes. What I think is most important is that we never accept any advertisements in these descriptions. Everything should be verifiable and objective truths. Hence, just writing that a code is accurate and uses state-of-the-art models and numerics is not good. Describing which models and methods that are used should be okay though, as long as adjectives like "good", "accurate", "best" etc. are used very sparingly. About the copyright issue. We must of course ensure that we don't break any copyrights. However, I don't think that companies will have any problems with using their descriptions of their codes, on the opposite, that is probably what they want. Some descriptions have also been written by the code companies themselves. For example, the [[Gridgen]] description was written by John Chawner and Rick Matus, two of the top people in Pointwise. It is really not that good that companies themselves write these decriptions since that has a tendency to always produce advertising material. Pointwise have been very good at avoiding advertisements and unverifiable adjectives though. --[[User:Jola|Jola]] 03:22, 7 June 2007 (MDT)<br />
<br />
::I guess that I generally agree. However, the webpage copying looks to be more widespread than I initially thought. I would rather have no description pages at all than have pages that are cut-and-paste jobs. On the [[Gridgen]] page, I guess my objection is that it is rather long - and they have their own web presence for that sort of thing. It does fit your requirements, though, and maybe we should add the part about "verifiable and objective truths" to the text on the code page. --[[User:Jasond|Jasond]] 13:55, 7 June 2007 (MDT)<br />
<br />
:::Yes, just having cut-and-paste copies of web-sites that can just as well be linked to directly is not good. We will just have old data that takes time to maintain and gives nothing extra. And just as you talked about if the original authors are not aware of it we will also break their copyrights. I think that you have started a good job of cleaning this section up. I will try to add something about the CFD-Wiki policies and the "verifiable and objective truths" requirement unless you or someone else have done it tomorrow. Now I have to go to bed, it is running late over here in Sweden --[[User:Jola|Jola]] 15:46, 7 June 2007 (MDT)<br />
<br />
::::As of right now, there are only a few issues left (as far as I know): [[Delaundo]]- which is a public domain code (so I left the text as is even though it is cut-and-pasted), [[vtk]],[[vtk.Net]] - Tony posted these but does give specific attribution, and there are a few pages that are basically nothing. I leave these as is unless there is opinion otherwise. I would still like to shorten the [[Gridgen]] page, but I won't do that unless you specifically agree that it needs to be shortened (and this is the last mention of it from me). I think I'll leave the policy modification to you, but I might add a little more text to my "note to contributors" - I want to keep things reasonable, but I don't want to discourage contribution. For the free codes, it might be a good idea to figure out a way to include license information on the main [[Codes]] page - maybe break the list into a "GPL" list and an "Other free license" list. Wouldn't it also be nice to have some sort of information on the language used? --[[User:Jasond|Jasond]] 10:54, 8 June 2007 (MDT)<br />
<br />
== What does a typical cfd software toolchain look like? ==<br />
<br />
Can anybody help me figure out the file formats and programs involved in completing cfd calculations etc.? Starting with blender, the open source 3D modeling program, I can get a mesh file (gmsh can read this), but then the majority of the cfd simulators do not seem to have a lot of documentation on how to set up a simulation (like the specifications, parameters, the types of materials, the density etc. etc.), but maybe I am missing something? And after that, how can I relate that to lift, thrust, and other properties involved in making sure, say, an airplane could fly? -- [[User:Kanzure|Kanzure]] 12:11, 3 March 2008 (MST)<br />
: Alright. I am starting to understand. If anybody needs some help, don't hessitate to ask. :) -- [[User:Kanzure|Kanzure]] 10:17, 9 March 2008 (MDT)<br />
<br />
== Femwater link broken ==<br />
<br />
I tried to fix the dead Femwater link, but the spam filter prevented me. So someone with higher privileges needs to<br />
change it from:<br />
* Femwater -- [http://www.cee.odu.edu/model/femwater.php Femwater code]<br />
to<br />
* Femwater -- [http://www.epa.gov/ceampubl/gwater/femwater/index.htm Femwater code]<br />
[[User:Slffea|Slffea]] 11:16, 12 June 2008 (MDT)<br />
<br />
:I fixed it. I also had the same problem with spam denial at first. It was due to an old link on the page that has been added to a spam database that CFD-Wiki uses. The old link was not directly related to CFD so I removed it and then it worked. I assume that it will now also work for you when the old spam related link is gone. --[[User:Peter|Peter]] 02:10, 13 June 2008 (MDT)<br />
::Thanks. I thought that removing the link would fix things, but decided to defer it to someone else. [[User:Slffea|Slffea]] 10:47, 13 June 2008 (MDT)</div>Slffeahttps://www.cfd-online.com/Wiki/Talk:CodesTalk:Codes2008-06-12T17:16:23Z<p>Slffea: Femwater link broken</p>
<hr />
<div>I added a short note at the top of this page about limiting the length of the code descriptions - this is the result of a discussion in the wiki forum late last year - and I added a link to the copyright page to hopefully head off any more submissions of copyrighted material. I haven't started to look to hard at the individual pages yet - I'll start that later this week. Finally, I have removed all of the links to nonexistent pages in the free solvers section. I'm not sure how that looks. Any opinions? --[[User:Jasond|Jasond]] 15:48, 6 June 2007 (MDT)<br />
<br />
:It looks great I think. It is very good that someone works on the code section. It is one of the most frequented sections. About the description of the codes. What I think is most important is that we never accept any advertisements in these descriptions. Everything should be verifiable and objective truths. Hence, just writing that a code is accurate and uses state-of-the-art models and numerics is not good. Describing which models and methods that are used should be okay though, as long as adjectives like "good", "accurate", "best" etc. are used very sparingly. About the copyright issue. We must of course ensure that we don't break any copyrights. However, I don't think that companies will have any problems with using their descriptions of their codes, on the opposite, that is probably what they want. Some descriptions have also been written by the code companies themselves. For example, the [[Gridgen]] description was written by John Chawner and Rick Matus, two of the top people in Pointwise. It is really not that good that companies themselves write these decriptions since that has a tendency to always produce advertising material. Pointwise have been very good at avoiding advertisements and unverifiable adjectives though. --[[User:Jola|Jola]] 03:22, 7 June 2007 (MDT)<br />
<br />
::I guess that I generally agree. However, the webpage copying looks to be more widespread than I initially thought. I would rather have no description pages at all than have pages that are cut-and-paste jobs. On the [[Gridgen]] page, I guess my objection is that it is rather long - and they have their own web presence for that sort of thing. It does fit your requirements, though, and maybe we should add the part about "verifiable and objective truths" to the text on the code page. --[[User:Jasond|Jasond]] 13:55, 7 June 2007 (MDT)<br />
<br />
:::Yes, just having cut-and-paste copies of web-sites that can just as well be linked to directly is not good. We will just have old data that takes time to maintain and gives nothing extra. And just as you talked about if the original authors are not aware of it we will also break their copyrights. I think that you have started a good job of cleaning this section up. I will try to add something about the CFD-Wiki policies and the "verifiable and objective truths" requirement unless you or someone else have done it tomorrow. Now I have to go to bed, it is running late over here in Sweden --[[User:Jola|Jola]] 15:46, 7 June 2007 (MDT)<br />
<br />
::::As of right now, there are only a few issues left (as far as I know): [[Delaundo]]- which is a public domain code (so I left the text as is even though it is cut-and-pasted), [[vtk]],[[vtk.Net]] - Tony posted these but does give specific attribution, and there are a few pages that are basically nothing. I leave these as is unless there is opinion otherwise. I would still like to shorten the [[Gridgen]] page, but I won't do that unless you specifically agree that it needs to be shortened (and this is the last mention of it from me). I think I'll leave the policy modification to you, but I might add a little more text to my "note to contributors" - I want to keep things reasonable, but I don't want to discourage contribution. For the free codes, it might be a good idea to figure out a way to include license information on the main [[Codes]] page - maybe break the list into a "GPL" list and an "Other free license" list. Wouldn't it also be nice to have some sort of information on the language used? --[[User:Jasond|Jasond]] 10:54, 8 June 2007 (MDT)<br />
<br />
== What does a typical cfd software toolchain look like? ==<br />
<br />
Can anybody help me figure out the file formats and programs involved in completing cfd calculations etc.? Starting with blender, the open source 3D modeling program, I can get a mesh file (gmsh can read this), but then the majority of the cfd simulators do not seem to have a lot of documentation on how to set up a simulation (like the specifications, parameters, the types of materials, the density etc. etc.), but maybe I am missing something? And after that, how can I relate that to lift, thrust, and other properties involved in making sure, say, an airplane could fly? -- [[User:Kanzure|Kanzure]] 12:11, 3 March 2008 (MST)<br />
: Alright. I am starting to understand. If anybody needs some help, don't hessitate to ask. :) -- [[User:Kanzure|Kanzure]] 10:17, 9 March 2008 (MDT)<br />
<br />
== Femwater link broken ==<br />
<br />
I tried to fix the dead Femwater link, but the spam filter prevented me. So someone with higher privileges needs to<br />
change it from:<br />
* Femwater -- [http://www.cee.odu.edu/model/femwater.php Femwater code]<br />
to<br />
* Femwater -- [http://www.epa.gov/ceampubl/gwater/femwater/index.htm Femwater code]<br />
[[User:Slffea|Slffea]] 11:16, 12 June 2008 (MDT)</div>Slffeahttps://www.cfd-online.com/Wiki/Cfd_simulation_of_vortex_sheddingCfd simulation of vortex shedding2008-04-19T00:01:02Z<p>Slffea: /* Example of 2 Dimensional Problem */ Fixed labeling</p>
<hr />
<div>To numerically simulate [[Vortex shedding|vortex shedding]], [[CFD]] is used to calculate the unsteady flow that arises from a fluid moving past an obstruction. [[Downwind]] of the obstruction are regions of lower pressure which causes the fluid which initially was deflected around the obstruction to get sucked into these regions, begin to circulate, and form [[vortices]].<br />
<br />
For a [[Newtonian fluid]], it is sufficient to use the [[Navier-Stokes equations]] as the governing equations, and choose either the [[Finite element|finite element method]], '''FEM''', or [[Finite difference|finite differencing]], '''FD''', to do the simulation. Vortex shedding is more likely in [[Incompressible flow|incompressible fluids]] than [[Compressible flow|compressible fluids]], so the former is used when capturing this phenomenon. Although both types of fluids are represented by the [[Navier-Stokes equations]], their methods of solution differ significantly. For instance, when using '''FD''' for incompressible fluids, it is common to have an explicit expression for the velocity as you integrate the equations in time. In [[Compressible flow|compressible fluids]], the governing equations are often written in terms of the [[conservative form]] of [[Euler's equation]], with velocity and pressure extracted from the conserved quantities(e.g. mass, momentum, energy). Also, compressible flows usually involve the calculation of additional quantities such as density and temperature.<br />
<br />
==Example of 2 Dimensional Problem==<br />
The images below were created for the case of 2-D flow past a square obstruction with a [[Reynolds number]] of 1000 and freestream boundary conditions on the top, bottom, and right boundaries. The quantities of the Navier-Stokes equations were non-dimensionalized and the [[Schemes_by_Leonard_-_structured_grids|QUICKEST]] finite differencing algorithm by B.P. Leonard was used. The grid was uniform with either a 204 x 90 node mesh or a 102 x 45 node mesh.<br />
<br />
<br><br />
[[Image:vortex_prs.jpg|thumb|200px|left|pressure at 95.4s and 3180 timesteps.]]<br />
<br><br />
[[Image:vortex_vx.jpg|thumb|200px|left|velocity x at 95.4s and 3180 timesteps.]]<br />
<br><br />
[[Image:vortex_vy.jpg|thumb|200px|left|velocity y at 95.4s and 3180 timesteps.]]<br />
[[Image:Vortex_streaklines-0ang.jpg|400px|thumb|center|[[streaklines]] for square obstruction at 96s and 2400 timesteps for 0 angle of attack flow.]]<br />
<br><br />
<br><br />
[[Image:Vortex_pathlines-0ang.jpg|400px|thumb|center|[[pathlines]] for triangle obstruction at 96s and 2400 timesteps for 0 angle of attack flow.]]<br />
<br><br />
<br><br />
[[Image:Vortex_streaklines-15ang.jpg|400px|thumb|center|[[streaklines]] for square obstruction at 96s and 2400 timesteps for 15 angle of attack square flow.]]<br />
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<br><br />
<br><br />
<br><br />
<br><br />
<br><br />
<br><br />
<br />
==Pressure Corrector==<br />
The incompressibility of the fluid is enforced through the [[Continuity equation|continuity equation]] using pressure as a corrective quantity whose gradient in either the x or y direction is added to the velocity in the respective direction. Calculating the pressure correction involves solving the [[Discrete Poisson equation|discrete Poisson equation]] for pressure.<br />
<br />
==Staggered grid==<br />
Use of a [[staggered grid]], where pressure and velocities in the x and y directions are calculated at alternating grid points, is commonly used with [[upwinding]] to avoid oscillatory behavior.<br />
<br />
== See also ==<br />
*[[Flow across a square cylinder]]<br />
<br />
==References==<br />
*Davis, R. W., and E. F. Moore, <I>A numerical study of vortex shedding from rectangles</I>, Journal of Fluid Mechanics, 116, 475 (1982).<br />
<br />
*Davis, R. W., E. F. Moore, and L. P. Purtell, <I>A numerical-experimental study of confined flow around rectangular cylinders</I>, Physics of Fluids, 27, 46 (1984).<br />
<br />
*Hoffman, Joe D., <I> Numerical Methods for Engineers and Scientist, fourth edition </I>, McGraw-Hill Inc., New York, 1992.<br />
<br />
*Leonard, B. P., <I>A stable and accurate convective modelling procedure</I>, Computer Methods in Applied Mechanics and Engineering, 19, pages 59-98 (1979).</div>Slffeahttps://www.cfd-online.com/Wiki/Cfd_simulation_of_vortex_sheddingCfd simulation of vortex shedding2008-04-18T23:58:02Z<p>Slffea: Sent in better images</p>
<hr />
<div>To numerically simulate [[Vortex shedding|vortex shedding]], [[CFD]] is used to calculate the unsteady flow that arises from a fluid moving past an obstruction. [[Downwind]] of the obstruction are regions of lower pressure which causes the fluid which initially was deflected around the obstruction to get sucked into these regions, begin to circulate, and form [[vortices]].<br />
<br />
For a [[Newtonian fluid]], it is sufficient to use the [[Navier-Stokes equations]] as the governing equations, and choose either the [[Finite element|finite element method]], '''FEM''', or [[Finite difference|finite differencing]], '''FD''', to do the simulation. Vortex shedding is more likely in [[Incompressible flow|incompressible fluids]] than [[Compressible flow|compressible fluids]], so the former is used when capturing this phenomenon. Although both types of fluids are represented by the [[Navier-Stokes equations]], their methods of solution differ significantly. For instance, when using '''FD''' for incompressible fluids, it is common to have an explicit expression for the velocity as you integrate the equations in time. In [[Compressible flow|compressible fluids]], the governing equations are often written in terms of the [[conservative form]] of [[Euler's equation]], with velocity and pressure extracted from the conserved quantities(e.g. mass, momentum, energy). Also, compressible flows usually involve the calculation of additional quantities such as density and temperature.<br />
<br />
==Example of 2 Dimensional Problem==<br />
The images below were created for the case of 2-D flow past a square obstruction with a [[Reynolds number]] of 1000 and freestream boundary conditions on the top, bottom, and right boundaries. The quantities of the Navier-Stokes equations were non-dimensionalized and the [[Schemes_by_Leonard_-_structured_grids|QUICKEST]] finite differencing algorithm by B.P. Leonard was used. The grid was uniform with either a 204 x 90 node mesh or a 102 x 45 node mesh.<br />
<br />
<br><br />
[[Image:vortex_prs.jpg|thumb|200px|left|pressure at 95.4s and 3180 timesteps.]]<br />
<br><br />
[[Image:vortex_vx.jpg|thumb|200px|left|velocity x at 95.4s and 3180 timesteps.]]<br />
<br><br />
[[Image:vortex_vy.jpg|thumb|200px|left|velocity y at 95.4s and 3180 timesteps.]]<br />
[[Image:Vortex_streaklines-0ang.jpg|400px|thumb|center|[[streaklines]] at 96s and 2400 timesteps for 0 angle of attack flow.]]<br />
<br><br />
<br><br />
[[Image:Vortex_pathlines-0ang.jpg|400px|thumb|center|[[pathlines]] at 96s and 2400 timesteps for 0 angle of attack flow.]]<br />
<br><br />
<br><br />
[[Image:Vortex_streaklines-15ang.jpg|400px|thumb|center|[[streaklines]] at 16s and 400 timesteps for 15 angle of attack flow.]]<br />
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<br><br />
<br><br />
<br><br />
<br><br />
<br><br />
<br><br />
<br><br />
<br><br />
<br />
==Pressure Corrector==<br />
The incompressibility of the fluid is enforced through the [[Continuity equation|continuity equation]] using pressure as a corrective quantity whose gradient in either the x or y direction is added to the velocity in the respective direction. Calculating the pressure correction involves solving the [[Discrete Poisson equation|discrete Poisson equation]] for pressure.<br />
<br />
==Staggered grid==<br />
Use of a [[staggered grid]], where pressure and velocities in the x and y directions are calculated at alternating grid points, is commonly used with [[upwinding]] to avoid oscillatory behavior.<br />
<br />
== See also ==<br />
*[[Flow across a square cylinder]]<br />
<br />
==References==<br />
*Davis, R. W., and E. F. Moore, <I>A numerical study of vortex shedding from rectangles</I>, Journal of Fluid Mechanics, 116, 475 (1982).<br />
<br />
*Davis, R. W., E. F. Moore, and L. P. Purtell, <I>A numerical-experimental study of confined flow around rectangular cylinders</I>, Physics of Fluids, 27, 46 (1984).<br />
<br />
*Hoffman, Joe D., <I> Numerical Methods for Engineers and Scientist, fourth edition </I>, McGraw-Hill Inc., New York, 1992.<br />
<br />
*Leonard, B. P., <I>A stable and accurate convective modelling procedure</I>, Computer Methods in Applied Mechanics and Engineering, 19, pages 59-98 (1979).</div>Slffeahttps://www.cfd-online.com/Wiki/File:Vortex_pathlines-0ang.jpgFile:Vortex pathlines-0ang.jpg2008-04-18T23:35:43Z<p>Slffea: </p>
<hr />
<div></div>Slffeahttps://www.cfd-online.com/Wiki/Cfd_simulation_of_vortex_sheddingCfd simulation of vortex shedding2007-10-10T00:52:32Z<p>Slffea: /* Example of 2 Dimensional Problem */</p>
<hr />
<div>To numerically simulate [[Vortex shedding|vortex shedding]], [[CFD]] is used to calculate the unsteady flow that arises from a fluid moving past an obstruction. [[Downwind]] of the obstruction are regions of lower pressure which causes the fluid which initially was deflected around the obstruction to get sucked into these regions, begin to circulate, and form [[vortices]].<br />
<br />
For a [[Newtonian fluid]], it is sufficient to use the [[Navier-Stokes equations]] as the governing equations, and choose either the [[Finite element|finite element method]], '''FEM''', or [[Finite difference|finite differencing]], '''FD''', to do the simulation. Vortex shedding is more likely in [[Incompressible flow|incompressible fluids]] than [[Compressible flow|compressible fluids]], so the former is used when capturing this phenomenon. Although both types of fluids are represented by the [[Navier-Stokes equations]], their methods of solution differ significantly. For instance, when using '''FD''' for incompressible fluids, it is common to have an explicit expression for the velocity as you integrate the equations in time. In [[Compressible flow|compressible fluids]], the governing equations are often written in terms of the [[conservative form]] of [[Euler's equation]], with velocity and pressure extracted from the conserved quantities(e.g. mass, momentum, energy). Also, compressible flows usually involve the calculation of additional quantities such as density and temperature.<br />
<br />
==Example of 2 Dimensional Problem==<br />
The images below were created for the case of 2-D flow past a square obstruction with a [[Reynolds number]] of 1000 and freestream boundary conditions on the top, bottom, and right boundaries. The quantities of the Navier-Stokes equations were non-dimensionalized and the [[Schemes_by_Leonard_-_structured_grids|QUICKEST]] finite differencing algorithm by B.P. Leonard was used. The grid was uniform with 102 points in the x direction and 45 points in y.<br />
<br />
<br><br />
[[Image:vortex_prs.jpg|thumb|200px|left|pressure at 16s and 400 timesteps.]]<br />
<br><br />
[[Image:vortex_vx.jpg|thumb|200px|left|velocity x at 16s and 400 timesteps.]]<br />
<br><br />
[[Image:vortex_vy.jpg|thumb|200px|left|velocity y at 16s and 400 timesteps.]]<br />
[[Image:Vortex_streaklines-0ang.jpg|400px|thumb|center|[[streaklines]] at 8s and 200 timesteps for 0 angle of attack flow.]]<br />
<br><br />
<br><br />
[[Image:Vortex_streaklines-0ang2.jpg|400px|thumb|center|[[streaklines]] at 16s and 400 timesteps for 0 angle of attack flow.]]<br />
<br><br />
<br><br />
[[Image:Vortex_streaklines-15ang.jpg|400px|thumb|center|[[streaklines]] at 16s and 400 timesteps for 15 angle of attack flow.]]<br />
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<br><br />
<br><br />
<br><br />
<br />
==Pressure Corrector==<br />
The incompressibility of the fluid is enforced through the [[Continuity equation|continuity equation]] using pressure as a corrective quantity whose gradient in either the x or y direction is added to the velocity in the respective direction. Calculating the pressure correction involves solving the [[Discrete Poisson equation|discrete Poisson equation]] for pressure.<br />
<br />
==Staggered grid==<br />
Use of a [[staggered grid]], where pressure and velocities in the x and y directions are calculated at alternating grid points, is commonly used with [[upwinding]] to avoid oscillatory behavior.<br />
<br />
== See also ==<br />
*[[Flow across a square cylinder]]<br />
<br />
==References==<br />
*Davis, R. W., and E. F. Moore, <I>A numerical study of vortex shedding from rectangles</I>, Journal of Fluid Mechanics, 116, 475 (1982).<br />
<br />
*Davis, R. W., E. F. Moore, and L. P. Purtell, <I>A numerical-experimental study of confined flow around rectangular cylinders</I>, Physics of Fluids, 27, 46 (1984).<br />
<br />
*Hoffman, Joe D., <I> Numerical Methods for Engineers and Scientist, fourth edition </I>, McGraw-Hill Inc., New York, 1992.<br />
<br />
*Leonard, B. P., <I>A stable and accurate convective modelling procedure</I>, Computer Methods in Applied Mechanics and Engineering, 19, pages 59-98 (1979).</div>Slffeahttps://www.cfd-online.com/Wiki/Cfd_simulation_of_vortex_sheddingCfd simulation of vortex shedding2007-10-10T00:48:18Z<p>Slffea: /* Example of 2 Dimensional Problem */ Formatting</p>
<hr />
<div>To numerically simulate [[Vortex shedding|vortex shedding]], [[CFD]] is used to calculate the unsteady flow that arises from a fluid moving past an obstruction. [[Downwind]] of the obstruction are regions of lower pressure which causes the fluid which initially was deflected around the obstruction to get sucked into these regions, begin to circulate, and form [[vortices]].<br />
<br />
For a [[Newtonian fluid]], it is sufficient to use the [[Navier-Stokes equations]] as the governing equations, and choose either the [[Finite element|finite element method]], '''FEM''', or [[Finite difference|finite differencing]], '''FD''', to do the simulation. Vortex shedding is more likely in [[Incompressible flow|incompressible fluids]] than [[Compressible flow|compressible fluids]], so the former is used when capturing this phenomenon. Although both types of fluids are represented by the [[Navier-Stokes equations]], their methods of solution differ significantly. For instance, when using '''FD''' for incompressible fluids, it is common to have an explicit expression for the velocity as you integrate the equations in time. In [[Compressible flow|compressible fluids]], the governing equations are often written in terms of the [[conservative form]] of [[Euler's equation]], with velocity and pressure extracted from the conserved quantities(e.g. mass, momentum, energy). Also, compressible flows usually involve the calculation of additional quantities such as density and temperature.<br />
<br />
==Example of 2 Dimensional Problem==<br />
The images below were created for the case of 2-D flow past a square obstruction with a [[Reynolds number]] of 1000 and freestream boundary conditions on the top, bottom, and right boundaries. The quantities of the Navier-Stokes equations were non-dimensionalized and the [[Schemes_by_Leonard_-_structured_grids|QUICKEST]] finite differencing algorithm by B.P. Leonard was used. The grid was uniform with 102 points in the x direction and 45 points in y.<br />
<br />
<br><br />
[[Image:vortex_prs.jpg|thumb|200px|left|pressure at 16s and 400 timesteps.]]<br />
<br><br />
[[Image:vortex_vx.jpg|thumb|200px|left|velocity x at 16s and 400 timesteps.]]<br />
<br><br />
[[Image:vortex_vy.jpg|thumb|200px|left|velocity y at 16s and 400 timesteps.]]<br />
[[Image:Vortex_streaklines-0ang.jpg|400px|thumb|center|[[streaklines]] at 8s and 200 timesteps for 0 angle of attack flow.]]<br />
<br><br />
<br><br />
[[Image:Vortex_streaklines-0ang2.jpg|400px|thumb|center|[[streaklines]] at 16s and 400 timesteps for 0 angle of attack flow.]]<br />
<br><br />
<br><br />
[[Image:Vortex_streaklines-15ang.jpg|400px|thumb|center|[[streaklines]] at 16s and 400 timesteps for 15 angle of attack flow.]]<br />
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<br><br />
<br><br />
<br><br />
<br><br />
<br><br />
<br><br />
<br />
==Pressure Corrector==<br />
The incompressibility of the fluid is enforced through the [[Continuity equation|continuity equation]] using pressure as a corrective quantity whose gradient in either the x or y direction is added to the velocity in the respective direction. Calculating the pressure correction involves solving the [[Discrete Poisson equation|discrete Poisson equation]] for pressure.<br />
<br />
==Staggered grid==<br />
Use of a [[staggered grid]], where pressure and velocities in the x and y directions are calculated at alternating grid points, is commonly used with [[upwinding]] to avoid oscillatory behavior.<br />
<br />
== See also ==<br />
*[[Flow across a square cylinder]]<br />
<br />
==References==<br />
*Davis, R. W., and E. F. Moore, <I>A numerical study of vortex shedding from rectangles</I>, Journal of Fluid Mechanics, 116, 475 (1982).<br />
<br />
*Davis, R. W., E. F. Moore, and L. P. Purtell, <I>A numerical-experimental study of confined flow around rectangular cylinders</I>, Physics of Fluids, 27, 46 (1984).<br />
<br />
*Hoffman, Joe D., <I> Numerical Methods for Engineers and Scientist, fourth edition </I>, McGraw-Hill Inc., New York, 1992.<br />
<br />
*Leonard, B. P., <I>A stable and accurate convective modelling procedure</I>, Computer Methods in Applied Mechanics and Engineering, 19, pages 59-98 (1979).</div>Slffeahttps://www.cfd-online.com/Wiki/File:Vortex_streaklines-15ang.jpgFile:Vortex streaklines-15ang.jpg2007-10-10T00:46:06Z<p>Slffea: Streaklines for 15 angle of attack flow
16 seconds, 400 steps
102 x 45 grid
dx = .266
dy = .266
dt = .04</p>
<hr />
<div>Streaklines for 15 angle of attack flow<br />
16 seconds, 400 steps<br />
102 x 45 grid<br />
dx = .266<br />
dy = .266<br />
dt = .04</div>Slffeahttps://www.cfd-online.com/Wiki/Cfd_simulation_of_vortex_sheddingCfd simulation of vortex shedding2007-10-10T00:14:48Z<p>Slffea: /* Example of 2 Dimensional Problem */ Added better results.</p>
<hr />
<div>To numerically simulate [[Vortex shedding|vortex shedding]], [[CFD]] is used to calculate the unsteady flow that arises from a fluid moving past an obstruction. [[Downwind]] of the obstruction are regions of lower pressure which causes the fluid which initially was deflected around the obstruction to get sucked into these regions, begin to circulate, and form [[vortices]].<br />
<br />
For a [[Newtonian fluid]], it is sufficient to use the [[Navier-Stokes equations]] as the governing equations, and choose either the [[Finite element|finite element method]], '''FEM''', or [[Finite difference|finite differencing]], '''FD''', to do the simulation. Vortex shedding is more likely in [[Incompressible flow|incompressible fluids]] than [[Compressible flow|compressible fluids]], so the former is used when capturing this phenomenon. Although both types of fluids are represented by the [[Navier-Stokes equations]], their methods of solution differ significantly. For instance, when using '''FD''' for incompressible fluids, it is common to have an explicit expression for the velocity as you integrate the equations in time. In [[Compressible flow|compressible fluids]], the governing equations are often written in terms of the [[conservative form]] of [[Euler's equation]], with velocity and pressure extracted from the conserved quantities(e.g. mass, momentum, energy). Also, compressible flows usually involve the calculation of additional quantities such as density and temperature.<br />
<br />
==Example of 2 Dimensional Problem==<br />
The images below were created for the case of 2-D flow past a square obstruction with a [[Reynolds number]] of 1000 and freestream boundary conditions on the top, bottom, and right boundaries. The quantities of the Navier-Stokes equations were non-dimensionalized and the [[Schemes_by_Leonard_-_structured_grids|QUICKEST]] finite differencing algorithm by B.P. Leonard was used. The grid was uniform with 102 points in the x direction and 45 points in y.<br />
<br />
<br><br />
[[Image:vortex_prs.jpg|thumb|200px|left|pressure at 16s and 400 timesteps.]]<br />
<br><br />
[[Image:vortex_vx.jpg|thumb|200px|left|velocity x at 16s and 400 timesteps.]]<br />
<br><br />
[[Image:vortex_vy.jpg|thumb|200px|left|velocity y at 16s and 400 timesteps.]]<br />
<br><br />
<br><br />
[[Image:Vortex_streaklines-0ang.jpg|400px|thumb|center|[[streaklines]] at 8s and 200 timesteps for 0 angle of attack flow.]]<br />
<br><br />
<br><br />
[[Image:Vortex_streaklines-0ang2.jpg|400px|thumb|center|[[streaklines]] at 16s and 400 timesteps for 0 angle of attack flow.]]<br />
<br><br />
<br><br />
[[Image:Vortex_streaklines-15ang.jpg|400px|thumb|center|[[streaklines]] at 16s and 400 timesteps for 15 angle of attack flow.]]<br />
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<br><br />
<br><br />
<br><br />
<br><br />
<br><br />
<br><br />
<br />
==Pressure Corrector==<br />
The incompressibility of the fluid is enforced through the [[Continuity equation|continuity equation]] using pressure as a corrective quantity whose gradient in either the x or y direction is added to the velocity in the respective direction. Calculating the pressure correction involves solving the [[Discrete Poisson equation|discrete Poisson equation]] for pressure.<br />
<br />
==Staggered grid==<br />
Use of a [[staggered grid]], where pressure and velocities in the x and y directions are calculated at alternating grid points, is commonly used with [[upwinding]] to avoid oscillatory behavior.<br />
<br />
== See also ==<br />
*[[Flow across a square cylinder]]<br />
<br />
==References==<br />
*Davis, R. W., and E. F. Moore, <I>A numerical study of vortex shedding from rectangles</I>, Journal of Fluid Mechanics, 116, 475 (1982).<br />
<br />
*Davis, R. W., E. F. Moore, and L. P. Purtell, <I>A numerical-experimental study of confined flow around rectangular cylinders</I>, Physics of Fluids, 27, 46 (1984).<br />
<br />
*Hoffman, Joe D., <I> Numerical Methods for Engineers and Scientist, fourth edition </I>, McGraw-Hill Inc., New York, 1992.<br />
<br />
*Leonard, B. P., <I>A stable and accurate convective modelling procedure</I>, Computer Methods in Applied Mechanics and Engineering, 19, pages 59-98 (1979).</div>Slffeahttps://www.cfd-online.com/Wiki/File:Vortex_streaklines-0ang2.jpgFile:Vortex streaklines-0ang2.jpg2007-10-09T23:55:30Z<p>Slffea: Streaklines for 0 angle of attack flow
16 seconds, 400 steps
102 x 45 grid
dx = .266
dy = .266
dt = .04</p>
<hr />
<div>Streaklines for 0 angle of attack flow<br />
16 seconds, 400 steps<br />
102 x 45 grid<br />
dx = .266<br />
dy = .266<br />
dt = .04</div>Slffeahttps://www.cfd-online.com/Wiki/File:Vortex_streaklines-0ang.jpgFile:Vortex streaklines-0ang.jpg2007-10-09T20:36:34Z<p>Slffea: Streaklines for 0 angle of attack flow
8 seconds, 800 steps
102 x 45 grid
dx = .266
dy = .266
dt = .04</p>
<hr />
<div>Streaklines for 0 angle of attack flow <br />
8 seconds, 800 steps<br />
102 x 45 grid<br />
dx = .266<br />
dy = .266<br />
dt = .04</div>Slffeahttps://www.cfd-online.com/Wiki/Cfd_simulation_of_vortex_sheddingCfd simulation of vortex shedding2007-05-30T23:18:56Z<p>Slffea: /* Example of 2 Dimensional Problem */ Improved linking</p>
<hr />
<div>To numerically simulate [[Vortex shedding|vortex shedding]], [[CFD]] is used to calculate the unsteady flow that arises from a fluid moving past an obstruction. [[Downwind]] of the obstruction are regions of lower pressure which causes the fluid which initially was deflected around the obstruction to get sucked into these regions, begin to circulate, and form [[vortices]].<br />
<br />
For a [[Newtonian fluid]], it is sufficient to use the [[Navier-Stokes equations]] as the governing equations, and choose either the [[Finite element|finite element method]], '''FEM''', or [[Finite difference|finite differencing]], '''FD''', to do the simulation. Vortex shedding is more likely in [[Incompressible flow|incompressible fluids]] than [[Compressible flow|compressible fluids]], so the former is used when capturing this phenomenon. Although both types of fluids are represented by the [[Navier-Stokes equations]], their methods of solution differ significantly. For instance, when using '''FD''' for incompressible fluids, it is common to have an explicit expression for the velocity as you integrate the equations in time. In [[Compressible flow|compressible fluids]], the governing equations are often written in terms of the [[conservative form]] of [[Euler's equation]], with velocity and pressure extracted from the conserved quantities(e.g. mass, momentum, energy). Also, compressible flows usually involve the calculation of additional quantities such as density and temperature.<br />
<br />
==Example of 2 Dimensional Problem==<br />
The images below were created for the case of 2-D flow past a square obstruction with a [[Reynolds number]] of 250 and freestream boundary conditions on the top, bottom, and right boundaries. The quantities of the Navier-Stokes equations were non-dimensionalized and the [[Schemes_by_Leonard_-_structured_grids|QUICKEST]] finite differencing algorithm by B.P. Leonard was used. The grid was uniform with 204 points in the x direction and 90 points in y.<br />
<br />
<br><br />
[[Image:vortex_prs.jpg|thumb|200px|left|pressure at 20s and 1000 timesteps.]]<br />
<br><br />
[[Image:vortex_vx.jpg|thumb|200px|left|velocity x at 20s and 1000 timesteps.]]<br />
<br><br />
[[Image:vortex_vy.jpg|thumb|200px|left|velocity y at 20s and 1000 timesteps.]]<br />
<br><br />
<br><br />
[[Image:vortex_streaklines.jpg|400px|thumb|center|[[streaklines]] at 20s and 1000 timesteps.]]<br />
<br />
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<br><br />
<br><br />
<br><br />
<br><br />
<br><br />
<br><br />
<br />
==Pressure Corrector==<br />
The incompressibility of the fluid is enforced through the [[Continuity equation|continuity equation]] using pressure as a corrective quantity whose gradient in either the x or y direction is added to the velocity in the respective direction. Calculating the pressure correction involves solving the [[Discrete Poisson equation|discrete Poisson equation]] for pressure.<br />
<br />
==Staggered grid==<br />
Use of a [[staggered grid]], where pressure and velocities in the x and y directions are calculated at alternating grid points, is commonly used with [[upwinding]] to avoid oscillatory behavior.<br />
<br />
== See also ==<br />
*[[Flow across a square cylinder]]<br />
<br />
==References==<br />
*Davis, R. W., and E. F. Moore, <I>A numerical study of vortex shedding from rectangles</I>, Journal of Fluid Mechanics, 116, 475 (1982).<br />
<br />
*Davis, R. W., E. F. Moore, and L. P. Purtell, <I>A numerical-experimental study of confined flow around rectangular cylinders</I>, Physics of Fluids, 27, 46 (1984).<br />
<br />
*Hoffman, Joe D., <I> Numerical Methods for Engineers and Scientist, fourth edition </I>, McGraw-Hill Inc., New York, 1992.<br />
<br />
*Leonard, B. P., <I>A stable and accurate convective modelling procedure</I>, Computer Methods in Applied Mechanics and Engineering, 19, pages 59-98 (1979).</div>Slffeahttps://www.cfd-online.com/Wiki/Talk:Discrete_Poisson_equationTalk:Discrete Poisson equation2007-05-24T16:09:30Z<p>Slffea: </p>
<hr />
<div>I noticed that you have used footnote references in this page. This requires a special add-on to be installed. I will try to install it later today. --[[User:Jola|Jola]] 02:06, 24 May 2007 (MDT)<br />
:Now the cite add-on is installed and the footnotes in this article seem to work --[[User:Jola|Jola]] 09:45, 24 May 2007 (MDT)<br />
::Thanks. That looks a lot better. And this will be a big benefit for future articles.[[User:Slffea|Slffea]] 10:09, 24 May 2007 (MDT)</div>Slffeahttps://www.cfd-online.com/Wiki/Cfd_simulation_of_vortex_sheddingCfd simulation of vortex shedding2007-05-24T15:56:47Z<p>Slffea: improved linking</p>
<hr />
<div>To numerically simulate [[vortex shedding]], [[CFD]] is used to calculate the unsteady flow that arises from a fluid moving past an obstruction. [[Downwind]] of the obstruction are regions of lower pressure which causes the fluid which initially was deflected around the obstruction to get sucked into these regions, begin to circulate, and form [[vortices]].<br />
<br />
For a [[Newtonian fluid]], it is sufficient to use the [[Navier-Stokes equations]] as the governing equations, and choose either the [[Finite element|finite element method]], '''FEM''', or [[Finite difference|finite differencing]], '''FD''', to do the simulation. Vortex shedding is more likely in [[Incompressible flow|incompressible fluids]] than [[Compressible flow|compressible fluids]], so the former is used when capturing this phenomenon. Although both types of fluids are represented by the [[Navier-Stokes equations]], their methods of solution differ significantly. For instance, when using '''FD''' for incompressible fluids, it is common to have an explicit expression for the velocity as you integrate the equations in time. In [[Compressible flow|compressible fluids]], the governing equations are often written in terms of the [[conservative form]] of [[Euler's equation]], with velocity and pressure extracted from the conserved quantities(e.g. mass, momentum, energy). Also, compressible flows usually involve the calculation of additional quantities such as density and temperature.<br />
<br />
==Example of 2 Dimensional Problem==<br />
The images below were created for the case of 2-D flow past a square obstruction with a [[Reynolds number]] of 250 and freestream boundary conditions on the top, bottom, and right boundaries. The quantities of the Navier-Stokes equations were non-dimensionalized and the [[QUICKEST]] finite differencing algorithm by B.P. Leonard was used. The grid was uniform with 204 points in the x direction and 90 points in y.<br />
<br />
<br><br />
[[Image:vortex_prs.jpg|thumb|200px|left|pressure at 20s and 1000 timesteps.]]<br />
<br><br />
[[Image:vortex_vx.jpg|thumb|200px|left|velocity x at 20s and 1000 timesteps.]]<br />
<br><br />
[[Image:vortex_vy.jpg|thumb|200px|left|velocity y at 20s and 1000 timesteps.]]<br />
<br><br />
<br><br />
[[Image:vortex_streaklines.jpg|400px|thumb|center|[[streaklines]] at 20s and 1000 timesteps.]]<br />
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<br><br />
<br><br />
<br />
<br />
==Pressure Corrector==<br />
The incompressibility of the fluid is enforced through the [[Continuity_Equation|continuity equation]] using pressure as a corrective quantity whose gradient in either the x or y direction is added to the velocity in the respective direction. Calculating the pressure correction involves solving the [[Discrete Poisson equation|discrete Poisson equation]] for pressure.<br />
<br />
==Staggered grid==<br />
Use of a [[staggered grid]], where pressure and velocities in the x and y directions are calculated at alternating grid points, is commonly used with [[upwinding]] to avoid oscillatory behavior.<br />
<br />
== See also ==<br />
*[[Flow across a square cylinder]]<br />
<br />
==References==<br />
*Davis, R. W., and E. F. Moore, <I>A numerical study of vortex shedding from rectangles</I>, Journal of Fluid Mechanics, 116, 475 (1982).<br />
<br />
*Davis, R. W., E. F. Moore, and L. P. Purtell, <I>A numerical-experimental study of confined flow around rectangular cylinders</I>, Physics of Fluids, 27, 46 (1984).<br />
<br />
*Hoffman, Joe D., <I> Numerical Methods for Engineers and Scientist, fourth edition </I>, McGraw-Hill Inc., New York, 1992.<br />
<br />
*Leonard, B. P., <I>A stable and accurate convective modelling procedure</I>, Computer Methods in Applied Mechanics and Engineering, 19, pages 59-98 (1979).</div>Slffeahttps://www.cfd-online.com/Wiki/User:SlffeaUser:Slffea2007-05-23T23:09:40Z<p>Slffea: </p>
<hr />
<div>[[Image:slffeascreenshot.jpg|thumb|right|SLFFEA Screenshot]]<br />
[[Image:Slfcfd3.jpg|thumb|right|SLFCFD Pressure field for [[incompressible flow]]<br />
around square obstruction.]]<br />
<br />
I am San Le, writer of [http://www.slffea.com/ SLFFEA], which deals with<br />
'''Finite Element Analysis''' in '''solid mechanics''' and [http://www.slfcfd.com/ SLFCFD],<br />
which is for '''Computational Fluid Dynamics'''. On May 23rd, I created the page<br />
[[Cfd simulation of vortex shedding]].</div>Slffeahttps://www.cfd-online.com/Wiki/Discrete_Poisson_equationDiscrete Poisson equation2007-05-23T23:05:26Z<p>Slffea: Took this page from Wikipedia which I originally wrote</p>
<hr />
<div>In [[mathematics]], the '''discrete Poisson equation''' is the [[finite difference]] analog of the [[Poisson equation]]. In it, the [[discrete Laplace operator]] takes the place of the [[Laplace operator]]. The discrete Poisson equation is frequently used in [[numerical analysis]] as a stand-in for the continuous Poisson equation, although it is also studied in its own right as a topic in [[discrete mathematics]].<br />
<br />
==On a two-dimensional rectangular grid==<br />
Using the [[finite difference]] numerical method to discretize<br />
the 2 dimensional Poisson equation (assuming a uniform spatial discretization) on an ''m x n'' grid gives the following formula:<br />
<br />
:<math><br />
( {\nabla}^2 u )_{ij} = \frac{1}{dx^2} ( u_{i+1,j} + u_{i-1,j} + u_{i,j+1} + u_{i,j-1} - 4 u_{ij}) = g_{ij}<br />
</math><br />
<br />
where <math> 2 \le i \le m-1 </math> and <math> 2 \le j \le n-1 </math>. The preferred arrangement of the<br />
solution vector is to use [[natural ordering]] which, prior to removing boundary elements, would look like:<br />
<br />
:<math><br />
\begin{bmatrix} U \end{bmatrix} =<br />
\begin{bmatrix} u_{11} , u_{21} , \ldots , u_{m1} , u_{12} , u_{22} , \ldots , u_{m3} , \ldots , u_{mn}<br />
\end{bmatrix}^{T}<br />
</math><br />
<br />
This will result in an '' mn x mn '' linear system:<br />
<br />
:<math><br />
\begin{bmatrix} A \end{bmatrix} \begin{bmatrix} U \end{bmatrix} = \begin{bmatrix} b \end{bmatrix}<br />
</math><br />
<br />
where<br />
<br />
:<math><br />
A =<br />
\begin{bmatrix}<br />
~D & -I & ~0 & ~0 & ~0 & \ldots & ~0 \\<br />
-I & ~D & -I & ~0 & ~0 & \ldots & ~0 \\<br />
~0 & -I & ~D & -I & ~0 & \ldots & ~0 \\<br />
\vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \vdots \\<br />
~0 & \ldots & ~0 & -I & ~D & -I & ~0 \\<br />
~0 & \ldots & \ldots & ~0 & -I & ~D & -I \\<br />
~0 & \ldots & \ldots & \ldots & ~0 & -I & ~D<br />
\end{bmatrix}<br />
</math><br />
<br />
<math> I </math> is the '' m x m '' [[identity matrix]], and <math> D </math>, also '' m x m '', is given by:<br />
<br />
:<math><br />
D =<br />
\begin{bmatrix}<br />
~4 & -1 & ~0 & ~0 & ~0 & \ldots & ~0 \\<br />
-1 & ~4 & -1 & ~0 & ~0 & \ldots & ~0 \\<br />
~0 & -1 & ~4 & -1 & ~0 & \ldots & ~0 \\<br />
\vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \vdots \\<br />
~0 & \ldots & ~0 & -1 & ~4 & -1 & ~0 \\<br />
~0 & \ldots & \ldots & ~0 & -1 & ~4 & -1 \\<br />
~0 & \ldots & \ldots & \ldots & ~0 & -1 & ~4<br />
\end{bmatrix}<br />
</math><br />
<br />
<ref>Golub, Gene H. and C.F. Van Loan, '' Matrix Computations, 3rd Ed.'',<br />
The Johns Hopkins University Press, Baltimore, 1996, pages 177-180.</ref><br />
For each <math> u_{ij} </math> equation, the columns of <math> D </math> correspond to the <math> u </math> components:<br />
<br />
:<math><br />
\begin{bmatrix}<br />
u_{1j} , & u_{2j} , & \ldots, & u_{i-1,j} , & u_{ij} , & u_{i+1,j} , & \ldots , & u_{mj}<br />
\end{bmatrix}^{T}<br />
<br />
</math><br />
<br />
while the columns of <math> I </math> to the left and right of <math> D </math> correspond to the <math> u </math> components:<br />
<br />
:<math><br />
\begin{bmatrix}<br />
u_{1,j-1} , & u_{2,j-1} , & \ldots, & u_{i-1,j-1} , & u_{i,j-1} , & u_{i+1,j-1} , & \ldots , & u_{m,j-1}<br />
\end{bmatrix}^{T}<br />
</math><br />
<br />
and<br />
<br />
:<math><br />
\begin{bmatrix}<br />
u_{1,j+1} , & u_{2,j+1} , & \ldots, & u_{i-1,j+1} , & u_{i,j+1} , & u_{i+1,j+1} , & \ldots , & u_{m,j+1}<br />
\end{bmatrix}^{T}<br />
</math><br />
<br />
respectively.<br />
<br />
From the above, it can be inferred that there are <math>n</math> block columns of <math> m </math> in <math> A </math>. It is important to note that prescribed values of <math> u </math> (usually lying on the boundary) would have their corresponding elements removed from <math> I </math> and <math> D </math>. For the common case that all the nodes on the boundary are set, we have <math> 2 \le i \le m - 1 </math> and <math> 2 \le j \le n - 1 </math>, and the system would have the dimensions '' (m - 2) (n - 2) x (m - 2) (n - 2) '', where <math> D </math> and <math> I </math><br />
would have dimensions '' (m-2) x (m-2) ''.<br />
<br />
== Example ==<br />
<br />
For a '' 5 x 5 '' ( <math> m = 5 </math> and <math> n = 5 </math> ) grid with all the boundary nodes prescribed,<br />
the system would look like:<br />
<br />
:<math><br />
\begin{bmatrix} U \end{bmatrix} =<br />
\begin{bmatrix} u_{22}, u_{32}, u_{42}, u_{23}, u_{33}, u_{43}, u_{24}, u_{34}, u_{44}<br />
\end{bmatrix}^{T}<br />
</math><br />
<br />
with<br />
<br />
:<math><br />
A =<br />
\begin{bmatrix}<br />
~4 & -1 & ~0 & -1 & ~0 & ~0 & ~0 & ~0 & ~0 \\<br />
-1 & ~4 & -1 & ~0 & -1 & ~0 & ~0 & ~0 & ~0 \\<br />
~0 & -1 & ~4 & ~0 & ~0 & -1 & ~0 & ~0 & ~0 \\<br />
-1 & ~0 & ~0 & ~4 & -1 & ~0 & -1 & ~0 & ~0 \\<br />
~0 & -1 & ~0 & -1 & ~4 & -1 & ~0 & -1 & ~0 \\<br />
~0 & ~0 & -1 & ~0 & -1 & ~4 & ~0 & ~0 & -1 \\<br />
~0 & ~0 & ~0 & -1 & ~0 & ~0 & ~4 & -1 & ~0 \\<br />
~0 & ~0 & ~0 & ~0 & -1 & ~0 & -1 & ~4 & -1 \\<br />
~0 & ~0 & ~0 & ~0 & ~0 & -1 & ~0 & -1 & ~4<br />
\end{bmatrix}<br />
</math><br />
<br />
and<br />
<br />
:<math><br />
b =<br />
\begin{bmatrix}<br />
dx^2 g_{22} + u_{12} + u_{21} \\<br />
dx^2 g_{32} + u_{31} ~~~~~~~~ \\<br />
dx^2 g_{42} + u_{52} + u_{41} \\<br />
dx^2 g_{23} + u_{13} ~~~~~~~~ \\<br />
dx^2 g_{33} ~~~~~~~~~~~~~~~~ \\<br />
dx^2 g_{43} + u_{53} ~~~~~~~~ \\<br />
dx^2 g_{24} + u_{14} + u_{25} \\<br />
dx^2 g_{34} + u_{35} ~~~~~~~~ \\<br />
dx^2 g_{44} + u_{54} + u_{45} \\<br />
\end{bmatrix}<br />
</math><br />
<br />
As can be seen, the boundary <math> u </math>'s are brought to the right-hand-side<br />
of the equation. <ref>Cheny, Ward and David Kincaid, '' Numerical Mathematics and Computing 2nd Ed.'',<br />
Brooks/Cole Publishing Company, Pacific Grove, 1985, pages 443-448 </ref> The entire system is '' 9 x 9 ''<br />
while <math> D </math> and <math> I </math> are '' 3 x 3 '' and given by:<br />
<br />
:<math><br />
D =<br />
\begin{bmatrix}<br />
~4 & -1 & ~0 \\<br />
-1 & ~4 & -1 \\<br />
~0 & -1 & ~4 \\<br />
\end{bmatrix}<br />
</math><br />
<br />
and<br />
<br />
:<math><br />
-I =<br />
\begin{bmatrix}<br />
-1 & ~0 & ~0 \\<br />
~0 & -1 & ~0 \\<br />
~0 & ~0 & -1<br />
\end{bmatrix}<br />
</math><br />
<br />
== Methods of Solution ==<br />
<br />
Because <math> \begin{bmatrix} A \end{bmatrix} </math> is block tridiagonal and sparse, many methods of solution<br />
have been developed to optimally solve this linear system for <math> \begin{bmatrix} U \end{bmatrix} </math>.<br />
Among the methods are a generalized [[Thomas algorithm]],<br />
[[cyclic reduction]], [[successive overrelaxation]], and [[Fourier transform]]s. A theoretically optimal <math> O(n) </math> solution can be computed using [[multigrid methods]].<br />
<br />
== Applications ==<br />
<br />
In [[Computational fluid dynamics]], for the solution of an incompressible flow problem, the incompressiblity condition acts as a constraint for the pressure. There is no explicit form available for pressure in this case due to a strong coupling of the velocity and pressure fields. In this condition, by taking the divergence of all terms in the momentum equation, one obtains the pressure poisson equation. For an incompressible "Flow" this constraint is given by:<br />
:<math><br />
\frac{ \partial v_x }{ \partial x} + \frac{ \partial v_y }{ \partial y} + \frac{\partial v_z}{\partial z} = 0<br />
</math><br />
<br />
where <math> v_x </math> is the velocity in the <math> x </math> direction, <math> v_y </math> is<br />
velocity in <math> y </math> and <math> v_z </math> is the velocity in the <math> z </math> direction. Taking divergence of the momentum equation and using the incompressibility constraint, the pressure poisson equation is formed given by:<br />
:<math><br />
\nabla^2 p = f(\nu,V)<br />
</math><br />
<br />
where <math> \nu </math> is the kinematic viscosity of the fluid and <math> V </math> is the velocity vector. <br />
<ref><br />
Fletcher, Clive A. J., ''Computational Techniques for Fluid Dynamics: Vol I'', 2nd Ed., Springer-Verlag, Berlin, 1991, page 334-339.<br />
</ref>.<br />
<br />
==Footnotes==<br />
<div class="references-small"><br />
<references/><br />
</div><br />
<br />
==References==<br />
<br />
*Cheny, Ward and David Kincaid, '' Numerical Mathematics and Computing 2nd Ed.'', Brooks/Cole Publishing Company, Pacific Grove, 1985.<br />
<br />
*Golub, Gene H. and C.F. Van Loan, '' Matrix Computations, 3rd Ed.'', The Johns Hopkins University Press, Baltimore, 1996.<br />
<br />
*Hoffman, Joe D., '' Numerical Methods for Engineers and Scientists, 4th Ed.'', McGraw-Hill Inc., New York, 1992.<br />
<br />
*Sweet, Roland A., '' SIAM Journal on Numerical Analysis, Vol. 11, No. 3 '', June 1974, 506-520.<br />
<br />
[[Category:Finite differences]]<br />
[[Category:Numerical differential equations]]</div>Slffeahttps://www.cfd-online.com/Wiki/Cfd_simulation_of_vortex_sheddingCfd simulation of vortex shedding2007-05-23T23:02:03Z<p>Slffea: improved writing</p>
<hr />
<div>To numerically simulate [[Vortex Shedding]], [[CFD]] is used to calculate the unsteady flow that arises from a fluid moving past an obstruction. [[Downwind]] of the obstruction are regions of lower pressure which causes the fluid which initially was deflected around the obstruction to get sucked into these regions, begin to circulate, and form [[vortices]].<br />
<br />
For a [[Newtonian Fluid]], it is sufficient to use the [[Navier-Stokes equations]] as the governing equations, and choose either the [[finite element method]], '''FEM''', or [[finite difference method|finite differencing]], '''FD''', to do the simulation. Vortex shedding is more likely in [[incompressible fluids]] than [[compressible fluids]], so the former is used when capturing this phenomenon. Although both types of fluids are represented by the [[Navier-Stokes equations]], their methods of solution differ significantly. For instance, when using '''FD''' for incompressible fluids, it is common to have an explicit expression for the velocity as you integrate the equations in time. In [[compressible fluids]], the governing equations are often written in terms of the [[conservative form]] of [[Euler's equation]], with velocity and pressure extracted from the conserved quantities(e.g. mass, momentum, energy). Also, compressible flows usually involve the calculation of additional quantities such as density and temperature.<br />
<br />
==Example of 2 Dimensional Problem==<br />
The images below were created for the case of 2-D flow past a square obstruction with a [[Reynolds number]] of 250 and freestream boundary conditions on the top, bottom, and right boundaries. The quantities of the Navier-Stokes equations were non-dimensionalized and the [[QUICKEST]] finite differencing algorithm by B.P. Leonard was used. The grid was uniform with 204 points in the x direction and 90 points in y.<br />
<br />
<br><br />
[[Image:vortex_prs.jpg|thumb|200px|left|pressure at 20s and 1000 timesteps.]]<br />
<br><br />
[[Image:vortex_vx.jpg|thumb|200px|left|velocity x at 20s and 1000 timesteps.]]<br />
<br><br />
[[Image:vortex_vy.jpg|thumb|200px|left|velocity y at 20s and 1000 timesteps.]]<br />
<br><br />
<br><br />
[[Image:vortex_streaklines.jpg|400px|thumb|center|[[streaklines]] at 20s and 1000 timesteps.]]<br />
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<br />
==Pressure Corrector==<br />
The incompressibility of the fluid is enforced through the [[continuity equation]] using pressure as a corrective quantity whose gradient in either the x or y direction is added to the velocity in the respective direction. Calculating the pressure correction involves solving the [[Discrete Poisson equation|discrete Poisson equation]] for pressure.<br />
<br />
==Staggered grid==<br />
Use of a [[staggered grid]], where pressure and velocities in the x and y directions are calculated at alternating grid points, is commonly used with [[upwinding]] to avoid oscillatory behavior.<br />
<br />
== See also ==<br />
*[[Flow across a square cylinder]]<br />
<br />
==References==<br />
*Davis, R. W., and E. F. Moore, <I>A numerical study of vortex shedding from rectangles</I>, Journal of Fluid Mechanics, 116, 475 (1982).<br />
<br />
*Davis, R. W., E. F. Moore, and L. P. Purtell, <I>A numerical-experimental study of confined flow around rectangular cylinders</I>, Physics of Fluids, 27, 46 (1984).<br />
<br />
*Hoffman, Joe D., <I> Numerical Methods for Engineers and Scientist, fourth edition </I>, McGraw-Hill Inc., New York, 1992.<br />
<br />
*Leonard, B. P., <I>A stable and accurate convective modelling procedure</I>, Computer Methods in Applied Mechanics and Engineering, 19, pages 59-98 (1979).</div>Slffeahttps://www.cfd-online.com/Wiki/Cfd_simulation_of_vortex_sheddingCfd simulation of vortex shedding2007-05-23T22:53:34Z<p>Slffea: Created page.</p>
<hr />
<div>To numerically simulate [[Vortex Shedding]], [[CFD]] is used to calculate the unsteady flow that arises from a fluid moving past an obstruction. [[Downwind]] of the obstruction are regions of lower pressure which causes the fluid which initially was deflected around the obstruction to get sucked into these regions, begin to circulate, and form [[vortices]].<br />
<br />
For a [[Newtonian Fluid]], it is sufficient to use the [[Navier-Stokes equations]] as the governing equations, and choose either the [[finite element method]], '''FEM''', or [[finite difference method|finite differencing]], '''FD''', to do the simulation. Vortex shedding is more likely in [[incompressible fluids]] than [[compressible fluids]], so the former is more likely to be simulated to capture this phenomenon. And although both types of fluids are represented by the [[Navier-Stokes equations]], their methods of solution differ significantly. For instance, when using '''FD''' for incompressible fluids, it is common to have an explicit expression for the velocity as you integrate the equations in time. In [[compressible fluids]], the governing equations are often written in terms of the [[conservative form]] of [[Euler's equation]], with velocity and pressure extracted from the conserved quantities(e.g. mass, momentum, energy). Also, compressible flows usually involve the calculation of additional quantities such as density and temperature.<br />
<br />
==Example of 2 Dimensional Problem==<br />
The images below were created for the case of 2-D flow past a square obstruction with a [[Reynolds number]] of 250 and freestream boundary conditions on the top, bottom, and right boundaries. The quantities of the Navier-Stokes equations were non-dimensionalized and the [[QUICKEST]] finite differencing algorithm by B.P. Leonard was used. The grid was uniform with 204 points in the x direction and 90 points in y.<br />
<br />
<br><br />
[[Image:vortex_prs.jpg|thumb|200px|left|pressure at 20s and 1000 timesteps.]]<br />
<br><br />
[[Image:vortex_vx.jpg|thumb|200px|left|velocity x at 20s and 1000 timesteps.]]<br />
<br><br />
[[Image:vortex_vy.jpg|thumb|200px|left|velocity y at 20s and 1000 timesteps.]]<br />
<br><br />
<br><br />
[[Image:vortex_streaklines.jpg|400px|thumb|center|[[streaklines]] at 20s and 1000 timesteps.]]<br />
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<br />
==Pressure Corrector==<br />
The incompressibility of the fluid is enforced through the [[continuity equation]] using pressure as a corrective quantity whose gradient in either the x or y direction is added to the velocity in the respective direction. Calculating the pressure correction involves solving the [[Discrete Poisson equation|discrete Poisson equation]] for pressure.<br />
<br />
==Staggered grid==<br />
Use of a [[staggered grid]], where pressure and velocities in the x and y directions are calculated at alternating grid points, is commonly used with [[upwinding]] to avoid oscillatory behavior.<br />
<br />
== See also ==<br />
*[[Flow across a square cylinder]]<br />
<br />
==References==<br />
*Davis, R. W., and E. F. Moore, <I>A numerical study of vortex shedding from rectangles</I>, Journal of Fluid Mechanics, 116, 475 (1982).<br />
<br />
*Davis, R. W., E. F. Moore, and L. P. Purtell, <I>A numerical-experimental study of confined flow around rectangular cylinders</I>, Physics of Fluids, 27, 46 (1984).<br />
<br />
*Hoffman, Joe D., <I> Numerical Methods for Engineers and Scientist, fourth edition </I>, McGraw-Hill Inc., New York, 1992.<br />
<br />
*Leonard, B. P., <I>A stable and accurate convective modelling procedure</I>, Computer Methods in Applied Mechanics and Engineering, 19, pages 59-98 (1979).</div>Slffeahttps://www.cfd-online.com/Wiki/File:Vortex_vy.jpgFile:Vortex vy.jpg2007-05-23T22:34:03Z<p>Slffea: </p>
<hr />
<div></div>Slffeahttps://www.cfd-online.com/Wiki/File:Vortex_vx.jpgFile:Vortex vx.jpg2007-05-23T22:33:09Z<p>Slffea: </p>
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<div></div>Slffeahttps://www.cfd-online.com/Wiki/File:Vortex_prs.jpgFile:Vortex prs.jpg2007-05-23T22:32:42Z<p>Slffea: </p>
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<div></div>Slffeahttps://www.cfd-online.com/Wiki/File:Vortex_streaklines.jpgFile:Vortex streaklines.jpg2007-05-23T22:22:33Z<p>Slffea: </p>
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<div></div>Slffeahttps://www.cfd-online.com/Wiki/User:SlffeaUser:Slffea2007-04-21T03:24:46Z<p>Slffea: </p>
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<div>[[Image:slffeascreenshot.jpg|thumb|right|SLFFEA Screenshot]]<br />
[[Image:Slfcfd3.jpg|thumb|right|SLFCFD Pressure field for [[incompressible flow]]<br />
around square obstruction.]]<br />
<br />
I am San Le, writer of [http://www.slffea.com/ SLFFEA], which deals with<br />
'''Finite Element Analysis''' in '''solid mechanics''' and [http://www.slfcfd.com/ SLFCFD],<br />
which is for '''Computational Fluid Dynamics'''.</div>Slffeahttps://www.cfd-online.com/Wiki/User:SlffeaUser:Slffea2007-04-12T21:22:27Z<p>Slffea: </p>
<hr />
<div>[[Image:slffeascreenshot.jpg|thumb|right|SLFFEA Screenshot]]<br />
[[Image:Slfcfd3.jpg|thumb|right|SLFCFD Pressure field for [[incompressible flow]]<br />
around square obstruction. Data displayed with [[gnuplot]]]]<br />
<br />
I am San Le, writer of [http://www.slffea.com/ SLFFEA], which deals with<br />
'''Finite Element Analysis''' in '''solid mechanics''' and [http://www.slfcfd.com/ SLFCFD],<br />
which is for '''Computational Fluid Dynamics'''.</div>Slffeahttps://www.cfd-online.com/Wiki/File:Slfcfd3.jpgFile:Slfcfd3.jpg2007-04-12T21:18:01Z<p>Slffea: </p>
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<div></div>Slffeahttps://www.cfd-online.com/Wiki/File:Slffeascreenshot.jpgFile:Slffeascreenshot.jpg2007-04-12T21:07:19Z<p>Slffea: </p>
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<div></div>Slffeahttps://www.cfd-online.com/Wiki/User:SlffeaUser:Slffea2007-04-12T20:57:26Z<p>Slffea: Created page</p>
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<div>[[Image:slffeascreenshot.jpg|thumb|right|SLFFEA [[screenshot]]]]<br />
[[Image:slfcfd.png|thumb|right|SLFCFD [[streakline | streaklines]] for [[incompressible flow]]<br />
around square obstruction. Data displayed with [[gnuplot]]]]<br />
<br />
I am San Le, writer of [http://www.slffea.com/ SLFFEA], which deals with<br />
'''Finite Element Analysis''' in '''solid mechanics''' and [http://www.slfcfd.com/ SLFCFD],<br />
which is for '''Computational Fluid Dynamics'''.</div>Slffeahttps://www.cfd-online.com/Wiki/CodesCodes2007-04-12T20:53:40Z<p>Slffea: /* Solvers */ Added Featflow and Femwater</p>
<hr />
<div>An overview of both free and commercial CFD software.<br />
<br />
== Free codes ==<br />
<br />
This section lists codes that are in the public domain, and codes that are available under GPL, BSD or similar licenses.<br />
<br />
=== Solvers ===<br />
<br />
*[[ADFC]] -- [http://adfc.sourceforge.net/index.html ADFC homepage]<br />
*[[Diagonalized Upwind Navier Stokes]] -- [http://duns.sourceforge.net DUNS homepage]<br />
*[[Dolfyn]] -- [http://www.dolfyn.net/dolfyn/index_en.html dolfyn homepage]<br />
*[[Edge]] -- [http://www.edge.foi.se/ Edge homepage]<br />
*[[ELMER]] -- [http://www.csc.fi/elmer/ ELMER homepage]<br />
*[[Featflow]] -- [http://www.featflow.de Featflow homepage]<br />
*[[Femwater]] -- [http://www.cee.odu.edu/model/femwater.php Femwater code]<br />
*[[FreeFEM]] -- [http://www.freefem.org FreeFEM homepage]<br />
*[[Gerris Flow Solver]] -- [http://gfs.sourceforge.net/ Gerris Flow Solver homepage]<br />
*[[IMTEK Mathematica Supplement (IMS)]] -- [http://www.imtek.uni-freiburg.de/simulation/mathematica/IMSweb/ IMTEK Mathematica Supplement (IMS) homepage]<br />
*[[iNavier]] -- [http://inavier.sourceforge.net/ iNavier Solver Home Page]<br />
*[[MFIX]] -- [http://www.mfix.org Computational multiphase flow homepage]<br />
*[[NaSt2D-2.0]] -- [http://home.arcor.de/drklaus.bauerfeind/nast/eNaSt2D.html NaSt2D-2.0 homepage]<br />
*[[NSC2KE]] -- [http://www-rocq1.inria.fr/gamma/cdrom/www/nsc2ke/eng.htm NSC2KE homepage]<br />
*[[OpenFlower]] -- [http://sourceforge.net/projects/openflower/ OpenFlower homepage]<br />
*[[OpenFOAM]] -- [http://www.openfoam.org/ OpenFOAM homepage]<br />
*[[OpenFVM]] -- [http://openfvm.sourceforge.net/ OpenFVM homepage]<br />
*[[PETSc-FEM]] -- [http://www.cimec.org.ar/petscfem PETSc-FEM homepage]<br />
*[[PP3D]] -- [http://www.featflow.de/ parpp3d++ homepage]<br />
*[[SLFCFD]] -- [http://slfcfd.sourceforge.net SLFCFD homepage]<br />
*[[Tochnog]] -- [http://tochnog.sourceforge.net Tochnog homepage]<br />
*[[Typhon solver]] -- [http://typhon.sf.net Typhon solver homepage]<br />
<br />
=== Grid generation ===<br />
<br />
*[[Delaundo]] -- [http://www.cerfacs.fr/~muller/delaundo.html Dalaundo homepage]<br />
*[[GMSH]] -- [http://www.geuz.org/gmsh/ GMSH hompage]<br />
*[[NETGEN]] -- [http://www.hpfem.jku.at/netgen/ NETGEN homepage]<br />
*[[SALOME]] -- [http://www.salome-platform.org SALOME homepage]<br />
*[[TETGEN]] -- [http://tetgen.berlios.de/ TETGEN hompage]<br />
*[[Triangle]] -- [http://www.cs.cmu.edu/~quake/triangle.html Trangle homepage]<br />
<br />
=== Visualization ===<br />
<br />
*[[GMV]] -- [http://www-xdiv.lanl.gov/XCM/gmv/ GMV homepage]<br />
*[[Gnuplot]] -- [http://www.gnuplot.info/ gnuplot homepage]<br />
*[[Mayavi]] -- [http://mayavi.sourceforge.net/ MayaVi homepage]<br />
*[[OpenDX]] <br />
*[[ParaView]] -- [http://www.paraview.org/HTML/Index.html ParaView homepage]<br />
*[[Vigie]] -- [http://www-sop.inria.fr/sinus/Softs/vigie.html Vigie homepage]<br />
*[[Visit]]<br />
*[[vtk]] -- [http://www.vtk.org vtk homepage]<br />
*[[vtk.Net]] -- [http://vtkdotnet.sourceforge.net/ vtk.Net homepage]<br />
<br />
=== Miscellaneous ===<br />
<br />
*[[Engauge Digitizer]]<br />
*[[Ftnchek]]<br />
*[[g3data]] -- [http://www.frantz.fi/index.php?page=software g3data homepage]<br />
*[[GIFMerge]] -- [http://www.the-labs.com/GIFMerge/ GIFMerge homepage]<br />
*[[Gifsicle]]<br />
<br />
== Commercial codes ==<br />
<br />
=== Solvers ===<br />
<br />
*[[ADINA-F]] -- [http://www.adina.com/index.html ADINA's homepage]<br />
*[[ADINA-FSI]] -- [http://www.adina.com/index.html ADINA's homepage]<br />
*[[ANSWER]] -- [http://www.acricfd.com/ ACRi's homepage]<br />
*[http://www.cfd-online.com/W/index.php?title=CFD%2B%2B CFD++] -- [http://www.metacomptech.com Metacomp Techonlogies' homepage]<br />
*[[CFD2000]] -- [http://www.adaptive-research.com/ Adaptive Research's homepage]<br />
*[[CFD-FASTRAN]] -- [http://www.esi-group.com/SimulationSoftware/advanced.html ESI Group's homepage]<br />
*[[CFD-ACE]] -- [http://www.esi-group.com/SimulationSoftware/advanced.html ESI Group's homepage]<br />
*[[CFX]] -- [http://www.ansys.com/ Ansys' homepage]<br />
*[[EFD]] -- [http://www.nika.biz/ NIKA's homepage]<br />
*[[FENSAP-ICE]] -- [http://www.newmerical.com/ NTI' homepage]<br />
*[[FINE]] -- [http://www.numeca.be/ Numeca's homepage]<br />
*[[FIRE]] -- [http://www.avl.com/ AVL's homepage]<br />
*[[FLACS]] -- [http://www.gexcon.com/index.php?src=flacs/overview.html GexCon's homepage]<br />
*[[FLOW-3D]] -- [http://www.flow3d.com/ Flow Science's homepage]<br />
*[[FLOWVISION]] -- [http://www.fv-tech.com FlowVision's homepage]<br />
*[[FLUENT]] -- [http://www.fluent.com Fluent's homepage]<br />
*[[FluSol]] -- [http://www.cfd-rocket.com FluSol's hompage]*[[J-FLO]] -- [http://www.newmerical.com NTI's homepage]<br />
*[[Flowz]]--[http://www.zeusnumerix.com Zeus Numerix's homepage ]<br />
*[[KINetics Reactive Flows]] -- [http://www.ReactionDesign.com Reaction Design's homepage]<br />
*[[KIVA]]--[http://www.lanl.gov/orgs/t/t3/codes/kiva.shtml Los Alamos homepage]<br />
*[[NOGRID FPM]] -- [http://www.no-grid.com NOGRIDS's homepage]<br />
*[[PHOENICS]] -- [http://www.cham.co.uk CHAM's homepage]<br />
*[[STAR-CD]] -- [http://www.cd-adapco.com CD-adapco's homepage]<br />
*[[STAR-CCMplus]] -- [http://www.cd-adapco.com CD-adapco's homepage]<br />
*[[Turb'Flow]] -- [http://www.fluorem.com Fluorem's hompage]<br />
<br />
=== Grid generation ===<br />
<br />
*[[ADINA-AUI]] -- [http://www.adina.com/index.html ADINA's homepage]<br />
*[[CFD-GEOM]] -- [http://www.esi-group.com/ ESI's homepage]<br />
*[[CFD-VISCART]] -- [http://www.esi-group.com/ ESI's homepage]<br />
*[[CFDExpert-GridZ ]] --[http://www.zeusnumerix.com/ Zeus Numerix's homepage]<br />
*[[Gridgen]] -- [http://www.pointwise.com/ Pointwise's homepage]<br />
*[[GridPro]] -- [http://www.gridpro.com/ PDC's homepage]<br />
*[[Harpoon]] -- [http://www.ensight.com/ CEI's homepage]<br />
*[[ICEM CFD ]] -- [http://www.ansys.com/ ANSYS' homepage]<br />
<br />
=== Visualization ===<br />
<br />
*[[ADINA-AUI]] -- [http://www.adina.com/index.html ADINA's homepage]<br />
*[[CFD-VIEW]] -- [http://www.esi-group.com/ ESI's homepage]<br />
*[[CFX-Post]] -- [http://www.ansys.com/ ANSYS' homepage]<br />
*[[EnSight]] -- [http://www.ensight.com/ CEI's homepage]<br />
*[[Fieldview]] -- [http://www.ilight.com/ Intelligent Light's homepage]<br />
*[[Tecplot]] -- [http://www.tecplot.com/ Tecplot's homepage]<br />
*[[ViewZ]] -- [http://www.zeusnumerix.com/ Zeus Numerix's homepage]<br />
<br />
=== Systems ===<br />
<br />
*[[ADINA]] -- [http://www.adina.com/index.html ADINA's homepage]<br />
*[[Flownex]] -- [http://www.flownex.com/ Flownex's homepage]<br />
<br />
== Online tools and services ==<br />
<br />
*[[CFDNet]] -- [http://www.cfdnet.com/ CFDNet homepage]</div>Slffea