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In `F33f`_`[computational fluid dynamics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Computational_fluid_dynamics]`_`f (CFD), the `!SIMPLE algorithm`! is a widely used `F33f`_`[numerical procedure`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Numerical_algorithm]`_`f to solve the `F33f`_`[Navier–Stokes equations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Navier–Stokes_equation]`_`f. `*SIMPLE`* is an acronym for Semi-Implicit Method for Pressure Linked Equations.

The SIMPLE algorithm was developed by Prof. `F33f`_`[Brian Spalding`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Brian_Spalding]`_`f and his student `F33f`_`[Suhas Patankar`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Suhas_Patankar]`_`f at `F33f`_`[Imperial College London`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Imperial_College_London]`_`f 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.`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f]

Many popular books on computational fluid dynamics discuss the SIMPLE algorithm in detail.`:cite-ref-2[`F5bf`_`[2`#cite-note-2]`_`f]`:cite-ref-3[`F5bf`_`[3`#cite-note-3]`_`f] A modified variant is the `*SIMPLER`* algorithm (SIMPLE Revised), that was introduced by Patankar in 1979.`:cite-ref-4[`F5bf`_`[4`#cite-note-4]`_`f]

>>Contents

• `F0af`_`[Algorithm`#algorithm]`_`f

• `F0af`_`[See also`#see-also]`_`f

• `F0af`_`[References`#references]`_`f

-─

>>Algorithm

The algorithm is `F33f`_`[iterative`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Iterative]`_`f. The basic steps in the solution update are as follows:

1. Set the boundary conditions.

2. Compute the gradients of velocity and pressure.

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

4. Compute the uncorrected mass fluxes at faces.

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

6. Update the pressure field: p k + 1 = p k + urf ⋅ ⋅ p ′ {\\displaystyle p^{k+1}=p^{k}+{\\text{urf}}\\cdot p^{'}} where urf is the under-relaxation factor for pressure.

7. Update the boundary pressure corrections p b ′ {\\displaystyle p_{b}^{'}} .

8. Correct the face mass fluxes: m ˙ ˙ f k + 1 = m ˙ ˙ f ∗ ∗ + m ˙ ˙ f ′ {\\displaystyle {\\dot {m}}_{f}^{k+1}={\\dot {m}}_{f}^{*}+{\\dot {m}}_{f}^{'}}

9. Correct the cell velocities: v → → k + 1 = v → → ∗ ∗ − − Vol ∇ ∇ p ′ a → → P v {\\displaystyle {\\vec {v}}^{k+1}={\\vec {v}}^{*}-{\\frac {{\\text{Vol}}\\ \\nabla p^{'}}{{\\vec {a}}_{P}^{v}}}} ; where ∇ ∇ p ′ {\\displaystyle {\\nabla p^{'}}} is the gradient of the pressure corrections, a → → P v {\\displaystyle {{\\vec {a}}_{P}^{v}}} is the vector of central coefficients for the discretized linear system representing the velocity equation and Vol is the cell volume.

10. Update density due to pressure changes.

>>See also

• `F33f`_`[PISO algorithm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=PISO_algorithm]`_`f

• `F33f`_`[SIMPLEC algorithm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=SIMPLEC_algorithm]`_`f

>>References

`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f `:citerefmanganibianchini2007`aMangani, L.; Bianchini, C. (2007). `*Heat transfer applications in turbomachinery`* (PDF). Proceedings of the OpenFOAM International Conference 2007. Retrieved 2016-03-16.

`:cite-note-2`!2.`! `F0af`_`[↑`#cite-ref-2]`_`f `:citerefpatankar1980`a`F33f`_`[Patankar, S. V.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Suhas_Patankar]`_`f (1980). `*Numerical Heat Transfer and Fluid Flow`*. `F33f`_`[Taylor & Francis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Taylor_&_Francis]`_`f. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-89116-522-4.

`:cite-note-3`!3.`! `F0af`_`[↑`#cite-ref-3]`_`f `:citerefferzigerperic-m-2001`a`F33f`_`[Ferziger, J. H.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=J._H._Ferziger]`_`f; Peric, M. (2001). `*Computational Methods for Fluid Dynamics`*. `F33f`_`[Springer-Verlag`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Springer-Verlag]`_`f. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-3-540-42074-3.

`:cite-note-4`!4.`! `F0af`_`[↑`#cite-ref-4]`_`f `:citereftannehillanderson-d-a-pletcher-r-h-1997`aTannehill, J. C.; `F33f`_`[Anderson, D. A.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dale_A._Anderson]`_`f; Pletcher, R. H. (1997). `*Computational Fluid Mechanics and Heat Transfer`*. `F33f`_`[Taylor & Francis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Taylor_&_Francis]`_`f. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 9781560320463.

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