SIMPLE algorithm: Difference between revisions

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{{Short description|Computational fluid dynamics algorithm}}

In [[computational fluid dynamics]] (CFD), the '''SIMPLE algorithm''' is a widely used [[numerical algorithm|numerical procedure]] to solve the [[Navier-StokesNavier–Stokes equation]]s. ''SIMPLE'' is an acronym for Semi-Implicit Method for Pressure Linked Equations.

The SIMPLE algorithm was developed by Prof. [[Brian Spalding]] and his student [[Suhas Patankar]] at [[Imperial College London|Imperial College]], London in the early 1970s. Since then it has been extensively used by many researchers to solve different kinds of fluid flow and heat transfer problems.<ref>{{cite web|url=https://engineering.purdue.edu/ME608/webpage/project-reports/SIMPLE-DrivenCavity.pdf |title=SIMPLE solver for driven cavity flow problem |format=PDF |accessdate=2011-08-21}}</ref><ref>{{cite conference |last1=Mangani |first1=L. |last2=Bianchini |first2=C. |conference=[[Proceedings of the OpenFOAM International Conference 2007]] |year=2007 |url=https://flore.unifi.it/retrieve/handle/2158/418277/15222/OFIC-07.pdf |title=Heat transfer applications in turbomachinery |accessdateaccess-date=2016-03-16}}</ref>

Many popular books on computational fluid dynamics discuss the SIMPLE algorithm in detail.<ref>{{cite book |last=Patankar |first=S. V. | authorlinkauthor-link = Suhas Patankar |title=Numerical Heat Transfer and Fluid Flow |publisher=[[Taylor & Francis]] |year=1980 |isbn=978-0-89116-522-4}}</ref><ref>{{cite book |last=Ferziger |first=J. H. | authorlinkauthor-link = J. H. Ferziger |author2=Peric, M. |title=Computational Methods for Fluid Dynamics|publisher=[[ Springer-Verlag]] |year=2001 |isbn= 978-3-540-42074-3}}</ref>

A modified variant is the ''SIMPLER'' algorithm (SIMPLE Revised), that was introduced by Patankar in 1979.<ref>{{cite book |last=Tannehill|first=J. C.| authorlinkauthor2 = Anderson, D. A. |author2-link = [[Dale A. Anderson|Anderson, D. A.]] |author3=Pletcher, R. H. |title=Computational Fluid Mechanics and Heat Transfer |url=https://archive.org/details/computationalflu0000tann|url-access=registration|publisher=[[Taylor & Francis]] |year=1997 |isbn=9781560320463 }}</ref>

== Algorithm ==

The algorithm is [[iterative]]. The basic steps in the solution update are as follows:

# Set the boundary conditions.

# Compute the gradients of velocity and pressure.

# Solve the discretized momentum equation to compute the intermediate velocity field.

# Compute the uncorrected mass fluxes at faces.

# Solve the pressure correction equation to produce cell values of the pressure correction.

# Update the pressure field: <math> p^{k + 1} = p^k + \text{urf} \cdot p^{'} </math> where urf is the under-relaxation factor for pressure.

# Update the boundary pressure corrections <math> p_b^{'} </math>.

# Correct the face mass fluxes: <math>\dot m_f^{k + 1} = \dot m_f^{*} + \dot m_f^{'} </math>

# Correct the cell velocities: <math> \vec v^{k + 1} = \vec v^{*} - \frac{{\text{Vol} \ \nabla p^{'} }}{{\vec a_P^v }} </math> ; where <math> {\nabla p^{'} } </math> is the gradient of the pressure corrections, <math> {\vec a_P^v } </math> is the vector of central coefficients for the discretized linear system representing the velocity equation and Vol is the cell volume.

# Update density due to pressure changes.

== See also ==

* [[PISO algorithm]]

* [[SIMPLEC algorithm]]

==References==

{{reflistReflist}}

[[Category:Computational fluid dynamics]]