ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

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<title>SIMPLE algorithm</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">SIMPLE algorithm</span></span>

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<p>In <a href="Computational_fluid_dynamics" title="Computational fluid dynamics">computational fluid dynamics</a> (CFD), the <b>SIMPLE algorithm</b> is a widely used <a href="Numerical_algorithm" class="mw-redirect" title="Numerical algorithm">numerical procedure</a> to solve the <a href="Navier%E2%80%93Stokes_equation" class="mw-redirect" title="Navier–Stokes equation">Navier–Stokes equations</a>. <i>SIMPLE</i> is an acronym for Semi-Implicit Method for Pressure Linked Equations.

</p><p>The SIMPLE algorithm was developed by Prof. <a href="Brian_Spalding" title="Brian Spalding">Brian Spalding</a> and his student <a href="Suhas_Patankar" title="Suhas Patankar">Suhas Patankar</a> at <a href="Imperial_College_London" title="Imperial College London">Imperial College London</a> 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.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>Many popular books on computational fluid dynamics discuss the SIMPLE algorithm in detail.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

A modified variant is the <i>SIMPLER</i> algorithm (SIMPLE Revised), that was introduced by Patankar in 1979.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Algorithm">Algorithm</h2></div>

<p>The algorithm is <a href="Iterative" class="mw-redirect" title="Iterative">iterative</a>. The basic steps in the solution update are as follows:

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<ol><li>Set the boundary conditions.</li>

<li>Compute the gradients of velocity and pressure.</li>

<li>Solve the discretized momentum equation to compute the intermediate velocity field.</li>

<li>Compute the uncorrected mass fluxes at faces.</li>

<li>Solve the pressure correction equation to produce cell values of the pressure correction.</li>

<li>Update the pressure field: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{k+1}=p^{k}+{\text{urf}}\cdot p^{'}}">

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<annotation encoding="application/x-tex">{\displaystyle p^{k+1}=p^{k}+{\text{urf}}\cdot p^{'}}</annotation>

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</math></span><img src="./bed1f59faf6e0eb8017ceb942d4c218931947515.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:19.173ex; height:3.176ex;" alt="{\displaystyle p^{k+1}=p^{k}+{\text{urf}}\cdot p^{'}}" loading="lazy"></span> where urf is the under-relaxation factor for pressure.</li>

<li>Update the boundary pressure corrections <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{b}^{'}}">

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<annotation encoding="application/x-tex">{\displaystyle p_{b}^{'}}</annotation>

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</math></span><img src="./9f9ee1c96778e2d63114e625bb7746de47abc965.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:2.196ex; height:3.343ex;" alt="{\displaystyle p_{b}^{'}}" loading="lazy"></span>.</li>

<li>Correct the face mass fluxes: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {m}}_{f}^{k+1}={\dot {m}}_{f}^{*}+{\dot {m}}_{f}^{'}}">

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<annotation encoding="application/x-tex">{\displaystyle {\dot {m}}_{f}^{k+1}={\dot {m}}_{f}^{*}+{\dot {m}}_{f}^{'}}</annotation>

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</math></span><img src="./6e0ac5448409a36197edaa00d91e2b76139e7882.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.522ex; height:3.843ex;" alt="{\displaystyle {\dot {m}}_{f}^{k+1}={\dot {m}}_{f}^{*}+{\dot {m}}_{f}^{'}}" loading="lazy"></span></li>

<li>Correct the cell velocities: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}^{k+1}={\vec {v}}^{*}-{\frac {{\text{Vol}}\ \nabla p^{'}}{{\vec {a}}_{P}^{v}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}^{k+1}={\vec {v}}^{*}-{\frac {{\text{Vol}}\ \nabla p^{'}}{{\vec {a}}_{P}^{v}}}}</annotation>

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</math></span><img src="./8cb61bd8e73fed2b0c44f3ee756bb069f9db0a1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.371ex; height:6.843ex;" alt="{\displaystyle {\vec {v}}^{k+1}={\vec {v}}^{*}-{\frac {{\text{Vol}}\ \nabla p^{'}}{{\vec {a}}_{P}^{v}}}}" loading="lazy"></span> ; where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\nabla p^{'}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\nabla p^{'}}}</annotation>

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</math></span><img src="./ebe9c383e678dea1fa791c6e23a44ba7d32e34ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.869ex; height:3.176ex;" alt="{\displaystyle {\nabla p^{'}}}" loading="lazy"></span> is the gradient of the pressure corrections, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {{\vec {a}}_{P}^{v}}}">

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<annotation encoding="application/x-tex">{\displaystyle {{\vec {a}}_{P}^{v}}}</annotation>

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</math></span><img src="./2cecb75e7e967b20bc919e317048c43805ab548c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.696ex; height:3.009ex;" alt="{\displaystyle {{\vec {a}}_{P}^{v}}}" loading="lazy"></span> is the vector of central coefficients for the discretized linear system representing the velocity equation and Vol is the cell volume.</li>

<li>Update density due to pressure changes.</li></ol>

<ul><li><a href="PISO_algorithm" title="PISO algorithm">PISO algorithm</a></li>

<li><a href="SIMPLEC_algorithm" title="SIMPLEC algorithm">SIMPLEC algorithm</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFManganiBianchini2007" class="citation conference cs1">Mangani, L.; Bianchini, C. (2007). <a rel="nofollow" class="external text" href="https://flore.unifi.it/retrieve/handle/2158/418277/15222/OFIC-07.pdf"><i>Heat transfer applications in turbomachinery</i></a> <span class="cs1-format">(PDF)</span>. Proceedings of the OpenFOAM International Conference 2007<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-03-16</span></span>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFPatankar1980" class="citation book cs1"><a href="Suhas_Patankar" title="Suhas Patankar">Patankar, S. V.</a> (1980). <i>Numerical Heat Transfer and Fluid Flow</i>. <a href="Taylor_%26_Francis" title="Taylor & Francis">Taylor & Francis</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-89116-522-4</bdi>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFFerzigerPeric,_M.2001" class="citation book cs1"><a href="J._H._Ferziger" class="mw-redirect" title="J. H. Ferziger">Ferziger, J. H.</a>; Peric, M. (2001). <i>Computational Methods for Fluid Dynamics</i>. <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-42074-3</bdi>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFTannehillAnderson,_D._A.Pletcher,_R._H.1997" class="citation book cs1">Tannehill, J. C.; <a href="Dale_A._Anderson" title="Dale A. Anderson">Anderson, D. A.</a>; Pletcher, R. H. (1997). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/computationalflu0000tann"><i>Computational Fluid Mechanics and Heat Transfer</i></a></span>. <a href="Taylor_%26_Francis" title="Taylor & Francis">Taylor & Francis</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781560320463</bdi>.</cite></span>

</li>

</ol></div></div></div><div><div class="zim-footer">

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