CFD - Revision history

NikLebedenko: corrected Moukalled name

2022-02-12T13:16:40Z

corrected Moukalled name

← Older revision		Revision as of 13:16, 12 February 2022
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	** Absolutely brilliant. Please read either this or Moukalled before anything else.		** Absolutely brilliant. Please read either this or Moukalled before anything else.
	**It is usually behind a paywall, but you can sometimes find an old freely available copy by simply [[google:versteeg+cfd+filetype:pdf\|googling]] <code>versteeg cfd filetype:pdf</code>		**It is usually behind a paywall, but you can sometimes find an old freely available copy by simply [[google:versteeg+cfd+filetype:pdf\|googling]] <code>versteeg cfd filetype:pdf</code>
	* '''RECOMMENDED: Fluid ~~Mechanics and Its Applications~~''' - commonly known as just "Moukalled" - https://link.springer.com/book/10.1007/978-3-319-16874-6		* '''RECOMMENDED: The Finite Volume Method in Computational Fluid Dynamics''' - commonly known as just "Moukalled" - https://link.springer.com/book/10.1007/978-3-319-16874-6
	** Slower paced than Versteeg - recommended for those who are a little shaky on their fluid dynamics knowledge. Also contains some more OpenFOAM-specific knowledge.		** Slower paced than Versteeg - recommended for those who are a little shaky on their fluid dynamics knowledge. Also contains some more OpenFOAM-specific knowledge.
	**It is usually behind a paywall, but you can sometimes find an old freely available copy by simply [[google:moukalled+cfd+filetype:pdf\|googling]] <code>moukalled cfd filetype:pdf</code>		**It is usually behind a paywall, but you can sometimes find an old freely available copy by simply [[google:moukalled+cfd+filetype:pdf\|googling]] <code>moukalled cfd filetype:pdf</code>

NikLebedenko: Added Moukalled to reading list

2022-02-11T21:16:35Z

Added Moukalled to reading list

← Older revision		Revision as of 21:16, 11 February 2022
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	<gallery widths="687" heights="336">		<gallery widths="687" heights="336">
	File:Cell-neighbours.png\|An arbitrary cell P neighboured by cells W and E on the west and east sides respectively		File:Cell-neighbours.png\|An arbitrary cell P neighboured by cells W and E on the west and east sides respectively
	File:Linear-interp.png\|A graph showing cell values (grey circles) and the linear interpolated face values (~~dotted~~ blue circles)		File:Linear-interp.png\|A graph showing cell values (grey circles) and the linear interpolated face values (blue circles)
	</gallery>The continuity equation tells us that mass is not created or destroyed. For an arbitrary cell P, knowing that the mass flow rate through a given face is <math>{\rho}UA</math>, we can write		</gallery>The continuity equation tells us that mass is not created or destroyed. For an arbitrary cell P, knowing that the mass flow rate through a given face is <math>{\rho}UA</math>, we can write

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	<math>0 = (UA)_w - (UA)_e</math> - '''(1)'''		<math>0 = (UA)_w - (UA)_e</math> - '''(1)'''

	At the moment, we are using values of velocity at the faces between the cells. But in a CFD solver, we only store the values at the cell centroid. The process of calculating face values using cell values on either side of the face is called ''interpolation''. The most obvious interpolation ''scheme'' (method) is linear interpolation - that is, assume that all variables vary linearly between two points, as in the image above. Note: interpolation isn't unique to the variable <math>U</math> - it can be applied to all flow variables.		At the moment, we are using values of velocity at the faces between the cells. But in a CFD solver, we only store the values at the cell centroid. The process of calculating face values using cell values on either side of the face is called ''interpolation''. The most obvious interpolation ''scheme'' (method) is linear interpolation - that is, assume that all variables vary linearly between two points, as in the image above. Note: interpolation isn't unique to the variable <math>U</math> - it can be applied to all flow variables.
	~~After this point, the continuity equation will only work for cells that aren't on the boundary. The continuity equation becomes~~

	<math>0 = (U_{\small{W}}i_{\small{W}} + U_{\small{P}}(1-i_{\small{W}})~~) \cdot A_w - (~~U_{\small{P}}i_{\small{E}} + U_{\small{E}}(1-i_{\small{E}})) \~~cdot A_e~~</math>		In our case, the cells and faces are all perfectly uniformly spaced, so if we use linear interpolation to work out the face values,

			<math>U_w = {1 \over 2} (U_{\small{W}} + U_{\small{P}}), \quad U_e = {1 \over 2} (U_{\small{P}} + U_{\small{E}})</math>

			This is the same as saying that the face values are the average of the neighbouring cell values. Let's use a more general equation to represent linear interpolation.

			<math>U_w = i_{\small{W}}U_{\small{W}} + (1-i_{\small{W}})U_{\small{P}}, \quad U_e = i_{\small{E}}U_{\small{P}} + (1-i_{\small{E}})U_{\small{E}}</math>

	where <math>i_{\small{W}} = \frac{x_w - x_{\small{W}}}{x_{\small{P}} - x_{\small{W}}}</math> and <math> i_{\small{E}} = \frac{x_e - x_{\small{P}}}{x_{\small{E}} - x_{\small{P}}} </math>		where <math>i_{\small{W}} = \frac{x_w - x_{\small{W}}}{x_{\small{P}} - x_{\small{W}}}</math> and <math> i_{\small{E}} = \frac{x_e - x_{\small{P}}}{x_{\small{E}} - x_{\small{P}}} </math>

			''Don't be intimidated by the complicated-looking fractions!'' <math>i_{\small{W}}</math> simply represents how far the face <math>w</math> is from cell <math>W</math>, divided by the distance between <math>W</math> and <math>P</math>. This way, if the face is exactly half-way between <math>W</math> and <math>P</math>, then <math>i_{\small{W}} = {1 \over 2}</math>, giving the same equation as before.
			If instead, the face was closer to <math>W</math> than to <math>P</math>, then <math>i_{\small{W}} < {1 \over 2}</math>, and vice versa.

			After this point, the continuity equation will only work for cells that aren't on the boundary. We also factor out the areas of each face, since they are just constants. The continuity equation becomes

			<math>0 = (U_{\small{W}}i_{\small{W}} + U_{\small{P}}(1-i_{\small{W}})) \cdot A_w - (U_{\small{P}}i_{\small{E}} + U_{\small{E}}(1-i_{\small{E}})) \cdot A_e</math>

	We now expand and simplify the equation, combining the constants into single variables.		We now expand and simplify the equation, combining the constants into single variables.
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	* '''RECOMMENDED: An Introduction to Computational Fluid Dynamics: The Finite Volume Method''' - commonly known as just "Versteeg" - https://books.google.co.uk/books/about/An_Introduction_to_Computational_Fluid_D.html?id=RvBZ-UMpGzIC		* '''RECOMMENDED: An Introduction to Computational Fluid Dynamics: The Finite Volume Method''' - commonly known as just "Versteeg" - https://books.google.co.uk/books/about/An_Introduction_to_Computational_Fluid_D.html?id=RvBZ-UMpGzIC
	** Absolutely brilliant. Please read it before anything else.		** Absolutely brilliant. Please read either this or Moukalled before anything else.
	**It is usually behind a paywall, but you can sometimes find an old freely available copy by simply [[google:versteeg+cfd+filetype:pdf\|googling]] <code>versteeg cfd filetype:pdf</code>		**It is usually behind a paywall, but you can sometimes find an old freely available copy by simply [[google:versteeg+cfd+filetype:pdf\|googling]] <code>versteeg cfd filetype:pdf</code>
			* '''RECOMMENDED: Fluid Mechanics and Its Applications''' - commonly known as just "Moukalled" - https://link.springer.com/book/10.1007/978-3-319-16874-6
			** Slower paced than Versteeg - recommended for those who are a little shaky on their fluid dynamics knowledge. Also contains some more OpenFOAM-specific knowledge.
			**It is usually behind a paywall, but you can sometimes find an old freely available copy by simply [[google:moukalled+cfd+filetype:pdf\|googling]] <code>moukalled cfd filetype:pdf</code>
	* Fluid Mechanics 101 - YouTube channel: https://www.youtube.com/channel/UCcqQi9LT0ETkRoUu8eYaEkg/		* Fluid Mechanics 101 - YouTube channel: https://www.youtube.com/channel/UCcqQi9LT0ETkRoUu8eYaEkg/
	** Has some very well explained and illustrated videos on a random selection of CFD topics.		** Has some very well explained and illustrated videos on a random selection of CFD topics.

NikLebedenko: /* Getting Started with CFD */

2022-02-02T00:40:31Z

Getting Started with CFD

← Older revision		Revision as of 00:40, 2 February 2022
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	= Getting Started with CFD =		= Getting Started with CFD =
			[[OpenFOAM installation instructions\|OpenFOAM installation instructions here]]

	Skip this section if you dislike broad-strokes philosophical discussion.		Skip this section if you dislike broad-strokes philosophical discussion.

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	The remainder of this wiki is dedicated to free and open source software, including OpenFOAM.		The remainder of this wiki is dedicated to free and open source software, including OpenFOAM.

	~~[[OpenFOAM installation instructions\|OpenFOAM installation instructions here]]~~

	= Crash course on the Finite Volume Method (FVM) =		= Crash course on the Finite Volume Method (FVM) =

NikLebedenko: /* Optional: Further reading */ clarified links

2022-02-02T00:33:11Z

Optional: Further reading: clarified links

← Older revision		Revision as of 00:33, 2 February 2022
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	Hopefully this has been an understandable / intuitive explanation of roughly what most CFD solvers do. If you want to explore the topics mentioned here in more detail, here are some resources that might help:		Hopefully this has been an understandable / intuitive explanation of roughly what most CFD solvers do. If you want to explore the topics mentioned here in more detail, here are some resources that might help:

	* An Introduction to Computational Fluid Dynamics: The Finite Volume Method - https://books.google.co.uk/books/about/An_Introduction_to_Computational_Fluid_D.html?id=RvBZ-UMpGzIC		* '''RECOMMENDED: An Introduction to Computational Fluid Dynamics: The Finite Volume Method''' - commonly known as just "Versteeg" - https://books.google.co.uk/books/about/An_Introduction_to_Computational_Fluid_D.html?id=RvBZ-UMpGzIC
	** ~~I highly recommend this book for further reading -~~ it is usually behind a paywall, but you can sometimes find an old freely available copy by simply [[google:versteeg+cfd+filetype:pdf\|googling]] <code>versteeg cfd filetype:pdf</code>		** Absolutely brilliant. Please read it before anything else.
			**It is usually behind a paywall, but you can sometimes find an old freely available copy by simply [[google:versteeg+cfd+filetype:pdf\|googling]] <code>versteeg cfd filetype:pdf</code>
	* Fluid Mechanics 101 - YouTube channel: https://www.youtube.com/channel/UCcqQi9LT0ETkRoUu8eYaEkg/		* Fluid Mechanics 101 - YouTube channel: https://www.youtube.com/channel/UCcqQi9LT0ETkRoUu8eYaEkg/
	** Has some very well explained and illustrated videos on a random selection of CFD topics.		** Has some very well explained and illustrated videos on a random selection of CFD topics.
	* ANSYS Fluent Theory Guide - find it with google.		* ANSYS Fluent Theory Guide - find it with [[google:ansys+fluent+theory+guide+filetype:pdf\|google]].
	** Usually a bit easier to find a copy of, but isn't ideal for someone who isn't already familiar with CFD and the finite volume method.		** Usually a bit easier to find a copy of, but isn't ideal for someone who isn't already familiar with CFD and the finite volume method. See Versteeg for that.
	* OpenFOAM: A little user manual - https://~~www~~.~~researchgate.net~~/~~publication~~/~~340174689_OpenFoam_~~-~~_a_little_user_manual~~		* OpenFOAM: A little user manual - https://github.com/ParticulateFlow/OSCCAR-doc/blob/master/openFoamUserManual_PFM.pdf
	** Click "Download full-text PDF", since the preview isn't available.
	** More specific to OpenFOAM and its code syntax, but contains some discussion of the fundamental algorithms which are used everywhere, e.g. SIMPLE, PISO, PIMPLE, as well as some turbulence models.		** More specific to OpenFOAM and its code syntax, but contains some discussion of the fundamental algorithms which are used everywhere, e.g. SIMPLE, PISO, PIMPLE, as well as some turbulence models.
	* OpenFOAM wiki - http://openfoamwiki.net/index.php/Main_Page		* OpenFOAM wiki - http://openfoamwiki.net/index.php/Main_Page

NikLebedenko: /* Optional: Further reading */ clarified things

2022-02-02T00:26:45Z

Optional: Further reading: clarified things

← Older revision		Revision as of 00:26, 2 February 2022
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	* An Introduction to Computational Fluid Dynamics: The Finite Volume Method - https://books.google.co.uk/books/about/An_Introduction_to_Computational_Fluid_D.html?id=RvBZ-UMpGzIC		* An Introduction to Computational Fluid Dynamics: The Finite Volume Method - https://books.google.co.uk/books/about/An_Introduction_to_Computational_Fluid_D.html?id=RvBZ-UMpGzIC
	** I highly recommend this book for further reading - it is usually behind a paywall, but you can sometimes find an old freely available copy by simply googling: versteeg cfd filetype:pdf		** I highly recommend this book for further reading - it is usually behind a paywall, but you can sometimes find an old freely available copy by simply [[google:versteeg+cfd+filetype:pdf\|googling]] <code>versteeg cfd filetype:pdf</code>
	* Fluid Mechanics 101 - YouTube channel: https://www.youtube.com/channel/UCcqQi9LT0ETkRoUu8eYaEkg/		* Fluid Mechanics 101 - YouTube channel: https://www.youtube.com/channel/UCcqQi9LT0ETkRoUu8eYaEkg/
	** Has some very well explained and illustrated videos on a random selection of CFD topics.		** Has some very well explained and illustrated videos on a random selection of CFD topics.

NikLebedenko: Made links a bit clearer

2022-02-01T22:38:27Z

Made links a bit clearer

← Older revision		Revision as of 22:38, 1 February 2022
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	This page is maintained by Nik Lebedenko - if you want to ask questions / see a mistake / request more details to be added to this page, feel free to email me or message me on slack.		This page is maintained by Nik Lebedenko - if you want to ask questions / see a mistake / request more details to be added to this page, feel free to email me or message me on slack.

	Computational Fluid Dynamics (CFD) is the process of solving fluid mechanics equations with the use of computers. Essentially F=ma, but with many simultaneous equations.		Computational Fluid Dynamics (CFD) is the process of solving fluid mechanics equations with the use of computers, usually with the Finite Volume Method.

			Essentially F=ma, but with many simultaneous equations.

			[[OpenFOAM installation instructions\|OpenFOAM installation instructions here]]

			= Getting Started with CFD =
			Skip this section if you dislike broad-strokes philosophical discussion.

			The path you should take for CFD differs depending on what your end goal is, and what budget you have. The main choice is between OpenFOAM and a commercial package (ANSYS Fluent / CFX, STAR-CCM, etc).

			OpenFOAM Pros vs commercial packages:

			* Free
			* Open Source
			* You can edit any part of the software, any setting, any way you want to.
			* No limits, no "paid version only" features. (ANSYS limits you to 2 CPU threads and 100k cells for simulation)

			OpenFOAM Cons vs commercial packages:

			* Difficult to install
			* Difficult to use
			* Difficult to learn - less of a learning curve, more like a cliff.
			* No graphical user interface
			* Mesh generation software is terrible. IMO, the best way to generate a mesh in OpenFOAM, is not to generate a mesh in OpenFOAM
			** Pro tip: use ANSYS meshing software (either the mechanical mesher, or the Fluent mesher) to generate CFD meshes. It will be 10x quicker (including setup time), take 5x less RAM, and be a higher quality than what you could achieve with OpenFOAM. Then import it into OpenFOAM with <code>fluentMeshToFoam <options> <.msh file></code>. ANSYS does not put a cap on the mesh size you can generate, only the mesh size you can simulate is limited.
			** There are other free & open source meshing programs, e.g. Gmsh, but I have not tried them.

			OpenFOAM throws you into the deep end, and forces you to learn nearly everything at once. If you stick through to the end of the learning cliff, you will understand much more about CFD than you would have otherwise learned with a commercial package.

			If you just need to do one or two small sims, a commercial package e.g. ANSYS is your best choice. (Just not SOLIDWORKS CFD simulations).

			The remainder of this wiki is dedicated to free and open source software, including OpenFOAM.

			[[OpenFOAM installation instructions\|OpenFOAM installation instructions here]]

	= Crash course on the Finite Volume Method (FVM) =		= Crash course on the Finite Volume Method (FVM) =
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	=== Overview ===		=== Overview ===

	~~Every~~ CFD ~~solver works~~ by splitting up the region of interest (a process called ''meshing'' the ''domain'') into a bunch of small boxes (called the ''cells'' in the ''mesh''), and solving a set of equations in each of those cells, taking into account how the flow variables (velocity, temperature, pressure) are related between neighbouring cells. Most commonly:		The most common method of doing CFD is with the Finite Volume Method (FVM). An informal derivation of some flow equations along with application of the FVM to them is in this section.

			''It is also possible to do CFD with the Finite Element Method (FEM), which is more typically associated with Structural Analysis / Elastodynamics problems, but we won't discuss that here.''

			CFD solvers work by splitting up the region of interest (a process called ''meshing'' the ''domain'') into a bunch of small boxes (called the ''cells'' in the ''mesh''), and solving a set of equations in each of those cells, taking into account how the flow variables (velocity, temperature, pressure) are related between neighbouring cells. Most commonly:

	* '''The continuity equation''': mass is not created or destroyed (conservation of mass).		* '''The continuity equation''': mass is not created or destroyed (conservation of mass).
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	= Fluid-structure interaction (FSI)=		= Fluid-structure interaction (FSI)=
			[[FSI installation instructions\|FSI installation instructions here]]

	This technique allows complex interactions between flexible structures and fluid flows. This has been used to predict fin flutter at hypersonic speeds for [[Griffin 1]].		This technique allows complex interactions between flexible structures and fluid flows. This has been used to predict fin flutter at hypersonic speeds for [[Griffin 1]].

	Simulations of this kind are achievable with the combination of the following free open-source software packages:		Simulations of this kind are achievable with the combination of the following free open-source software packages:

	*OpenFOAM ([https://www.openfoam.com/ website], [~~https://develop.openfoam.com/Development/openfoam/-/wikis/precompiled/debian install on Ubuntu~~]) - a C/C++ library for the solution of fluid mechanics problems formulated with the Finite Volume Method (FVM),		*OpenFOAM ([https://www.openfoam.com/ website], [[OpenFOAM installation instructions\|OpenFOAM installation instructions here]]) - a C/C++ library for the solution of fluid mechanics problems formulated with the Finite Volume Method (FVM),
	** Important: there are two main versions of OpenFOAM, provided at OpenFOAM.com and OpenFOAM.org. I have only used the .com version, but everything here should also be possible with the .org version.		** Important: there are two main versions of OpenFOAM, provided at OpenFOAM.com and OpenFOAM.org. I have only used the .com version, but everything here should also be possible with the .org version.
	*FEniCS ([https://fenicsproject.org/ website~~], [https://fenicsproject.org/download/ install on Ubuntu~~]) - a Python interface to the DOLFIN C++ library for the solution of solid mechanics problems formulated with the Finite Element Method (FEM), and		*FEniCS ([https://fenicsproject.org/ website]) - a Python interface to the DOLFIN C++ library for the solution of solid mechanics problems formulated with the Finite Element Method (FEM), and
	*preCICE ([https://precice.org/index.html website~~], [https://precice.org/quickstart.html install on Ubuntu~~]) - a very powerful library which allows the coupling of arbitrary solvers using human-readable syntax.		*preCICE ([https://precice.org/index.html website]) - a very powerful library which allows the coupling of arbitrary solvers using human-readable syntax.
	**Note: in order to use OpenFOAM and FEniCS with preCICE, it is also necessary to download their respective adapters. See [https://precice.org/adapter-openfoam-overview.html this page for OpenFOAM] and [https://precice.org/adapter-fenics.html this page for FEniCS] from the preCICE website for further information.		**Note: in order to use OpenFOAM and FEniCS with preCICE, it is also necessary to download their respective adapters. See [https://precice.org/adapter-openfoam-overview.html this page for OpenFOAM] and [https://precice.org/adapter-fenics.html this page for FEniCS] from the preCICE website for further information.

NikLebedenko: /* Fluid-structure interaction (FSI) */ reworded slightly

2022-01-26T00:54:40Z

Fluid-structure interaction (FSI): reworded slightly

← Older revision		Revision as of 00:54, 26 January 2022
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	= Fluid-structure interaction (FSI)=		= Fluid-structure interaction (FSI)=
	'''''IMPORTANT:''''' Although I hope that these instructions will be easily understood by anyone, <u>this is not a beginner's project!</u> If you want a proper introduction to CFD, please read the section above.

	This technique allows complex interactions between flexible structures and fluid flows. This has been used to predict fin flutter at hypersonic speeds for [[Griffin 1]].		This technique allows complex interactions between flexible structures and fluid flows. This has been used to predict fin flutter at hypersonic speeds for [[Griffin 1]].

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	== Step-by-step instructions==		== Step-by-step instructions==
			'''''IMPORTANT:''''' Although I hope that these instructions will be easily understood by anyone, <u>this is not a beginner's project!</u> If you want a better introduction to CFD, please read the crash-course section above, or any of the linked resources.

	This section has moved to [[FSI installation instructions\|a separate page]].		This section has moved to [[FSI installation instructions\|a separate page]].

NikLebedenko: /* Step-by-step instructions */ removed redundant note

2022-01-26T00:52:45Z

Step-by-step instructions: removed redundant note

← Older revision		Revision as of 00:52, 26 January 2022
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	== Step-by-step instructions==		== Step-by-step instructions==
	'''''IMPORTANT:''''' Although I include the links to the official websites for Ubuntu installation instructions, I recommend you <u>follow my instructions for installation</u> (unless my instructions no longer work, or you are a masochist), since the official instructions contain some mistakes and omissions.

	This section has moved to [[FSI installation instructions\|a separate page]].		This section has moved to [[FSI installation instructions\|a separate page]].

NikLebedenko: /* Crash course on the Finite Volume Method (FVM) */ added matrix inverse

2022-01-24T08:44:35Z

Crash course on the Finite Volume Method (FVM): added matrix inverse

← Older revision		Revision as of 08:44, 24 January 2022
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	<math> \boldsymbol{M}= \begin{bmatrix}		<math> \boldsymbol{M}= \begin{bmatrix}
	k_{1,1} & k_{1,2} & 0 & 0 \\		k_{1,1} & k_{1,2} & 0 & 0 \\
	k_{2,1} & k_{2,2} & k_{2,3} & 0 \\		k_{2,1} & k_{2,2} & k_{2,3} & 0 \\
	0 & k_{3,2} & k_{3,3} & k_{3,4} \\		0 & k_{3,2} & k_{3,3} & k_{3,4} \\
	0 & 0 & k_{4,3} & k_{4,4} \end{bmatrix} </math>,		0 & 0 & k_{4,3} & k_{4,4} \end{bmatrix} </math>,
	<math>\boldsymbol{U} = \begin{bmatrix} U_1 \\ U_2 \\ U_3 \\ U_4 \end{bmatrix}</math>, and		<math>\boldsymbol{U} = \begin{bmatrix} U_1 \\ U_2 \\ U_3 \\ U_4 \end{bmatrix}</math>, and
	<math>\boldsymbol{B} = \begin{bmatrix} -5A_w \\ 0 \\ 0 \\ 0 \end{bmatrix}</math>		<math>\boldsymbol{B} = \begin{bmatrix} -5A_w \\ 0 \\ 0 \\ 0 \end{bmatrix}</math>
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	<math> \boldsymbol{M}= \begin{bmatrix}		<math> \boldsymbol{M}= \begin{bmatrix}
	-0.5 & -0.5 & 0 & 0 \\		-0.5 & -0.5 & 0 & 0 \\
	0.5 & 0 & -0.5 & 0 \\		0.5 & 0 & -0.5 & 0 \\
	0 & 0.5 & 0 & -0.5 \\		0 & 0.5 & 0 & -0.5 \\
	0 & 0 & 0.5 & -0.5 \end{bmatrix} </math>		0 & 0 & 0.5 & -0.5 \end{bmatrix} </math>

	<math>\boldsymbol{B} = \begin{bmatrix} -5 \\ 0 \\ 0 \\ 0 \end{bmatrix}</math>		<math>\boldsymbol{B} = \begin{bmatrix} -5 \\ 0 \\ 0 \\ 0 \end{bmatrix}</math>

	We could now invert the matrix <math>\boldsymbol{M}</math>, and calculate the solution vector <math>\boldsymbol{U}</math> using <math>\boldsymbol{U=M^{-1}B}</math>. This yields a solution of U = 5 m/s everywhere. Hooray!		We could now invert the matrix <math>\boldsymbol{M}</math>, and calculate the solution vector <math>\boldsymbol{U}</math> using <math>\boldsymbol{U=M^{-1}B}</math>.

			<math> \boldsymbol{M^{-1}}= \begin{bmatrix}
			-1 & 1 & -1 & 1 \\
			-1 & -1 & 1 & -1 \\
			-1 & -1 & -1 & 1 \\
			-1 & -1 & -1 & -1 \end{bmatrix} </math>
			This yields a solution of U = 5 m/s everywhere. Hooray!

	''Think: does this result make sense, given our assumptions?''		''Think: does this result make sense, given our assumptions?''

NikLebedenko: removed all "WIP" markers

2022-01-24T08:39:23Z

removed all "WIP" markers

← Older revision		Revision as of 08:39, 24 January 2022
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	This page is ~~currently a work in progress~~ by Nik Lebedenko - if you want to ask questions / see a mistake / request more details to be added to this page, feel free to email me or message me on slack.		This page is maintained by Nik Lebedenko - if you want to ask questions / see a mistake / request more details to be added to this page, feel free to email me or message me on slack.

	Computational Fluid Dynamics (CFD) is the process of solving fluid mechanics equations with the use of computers. Essentially F=ma, but with many simultaneous equations.		Computational Fluid Dynamics (CFD) is the process of solving fluid mechanics equations with the use of computers. Essentially F=ma, but with many simultaneous equations.