SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>False diffusion</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">False diffusion</span></span>

</h1>

</header>

<p><b>False diffusion</b> is a type of error observed when the <a href="Upwind_scheme" title="Upwind scheme">upwind scheme</a> is used to approximate the <a href="Convection" title="Convection">convection</a> term in <a href="Convection%E2%80%93diffusion_equation" title="Convection–diffusion equation">convection–diffusion equations</a>. The more accurate <a href="Finite_difference#central_difference" title="Finite difference">central difference scheme</a> can be used for the <a href="Convection" title="Convection">convection</a> term, but for grids with cell <a href="Peclet_number" class="mw-redirect" title="Peclet number">Peclet number</a> more than 2, the central difference scheme is unstable and the simpler upwind scheme is often used. The resulting error from the upwind differencing scheme has a diffusion-like appearance in two- or three-dimensional co-ordinate systems and is referred as "false diffusion". False-diffusion errors in numerical solutions of convection-diffusion problems, in two- and three-dimensions, arise from the numerical approximations of the convection term in the conservation equations. Over the past 20 years many <a href="Numerical_analysis" title="Numerical analysis">numerical</a> techniques have been developed to solve convection-diffusion equations and none are problem-free, but false diffusion is one of the most serious problems and a major topic of controversy and confusion among <a href="Numerical_analysis" title="Numerical analysis">numerical analysts</a>.

</p>

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

<p>False diffusion is defined as an error having a diffusion-like appearance, obtained when the <i>upwind scheme</i> is used in multidimensional cases to solve for the distribution of transported properties flowing non-orthogonally to one or more of the system's major axes. The error is absent when the flow is orthogonal or parallel to each major axis.

</p>

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

<p>In figure 1, <i>u</i> = 2 and <i>v</i> = 2 m/s everywhere so the <a href="Velocity_field" class="mw-redirect" title="Velocity field">velocity field</a> is uniform and perpendicular to the <a href="Diagonal" title="Diagonal">diagonal</a> (XX). The boundary conditions for <a href="Temperature" title="Temperature">temperature</a> on north and west wall is 100 ̊C and for east and south wall is 0 ̊C. This region is meshed into 10×10 equal grids. Take two cases, (i) with <a href="Diffusion_coefficient" class="mw-redirect" title="Diffusion coefficient">diffusion coefficient</a> ≠ 0 and, case (ii) with diffusion coefficient = 0.

</p>

<p>In this case, heat from west and south walls is carried by <a href="Convection" title="Convection">convection</a> flow towards north and east walls. Heat is also diffused across the diagonal XX from upper to lower triangle. Figure 2 shows the approximate temperature distribution.

</p>

<p>In this case heat from west and south walls is convected by flow towards north and east. There will be no diffusion across the diagonal XX but, when the upwind scheme is applied the results are similar to case (i) where actual diffusion is occurring. This error is known as false diffusion.

</p>

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

<p>In early approaches, <a href="Derivative" title="Derivative">derivatives</a> in the <i><a href="Differential_form" title="Differential form">differential form</a></i> of the governing <i><a href="Transport_equation" class="mw-redirect" title="Transport equation">transport equation</a></i> were replaced by finite difference approximations, usually central differencing approximations with second order accuracy. However, for large Peclet numbers (generally > 2) this approximation gave inaccurate results. It was recognized independently by several investigators<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><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> that the less expensive but only first order accurate <i>upwind scheme</i> can be employed but that this scheme produces results with false diffusion for multidimensional cases. Many new schemes have been developed to counter false diffusion but a reliable, accurate and economical discretisation scheme is still unavailable.

</p>

<div class="mw-heading mw-heading2"><h2 id="Reducing_errors">Reducing errors</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Finer_mesh">Finer mesh</h3></div>

<p>False diffusion with the <i>upwind scheme</i> is reduced by increasing the mesh density. In the results of figure 3 and 4 the false diffusion error is lowest in figure 4(b) with finer mesh size.

</p>

<div class="mw-heading mw-heading3"><h3 id="Other_schemes">Other schemes</h3></div>

<p>False diffusion error also can be reduced by using schemes such as the <i><a href="Power_law_scheme" title="Power law scheme">power law scheme</a></i>, <i><a href="QUICK_scheme" title="QUICK scheme">QUICK scheme</a></i>, <i>exponential scheme</i>, and <i>SUCCA</i>, and others.<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><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>

</p>

<div class="mw-heading mw-heading3"><h3 id="Improving_the_upwind_scheme">Improving the upwind scheme</h3></div>

<p>False diffusion with the simple <i>upwind scheme</i> occurs because the scheme does not take into account grid/flow direction inclination. An approximate expression for the false-diffusion term in two dimensions has been given by de Vahl Davis and Mallinson(1972)<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p>

/* start https://en.wikipedia.org/ */

.mw-parser-output table.numblk{border-collapse:collapse;border:none;margin-top:0;margin-right:0;margin-bottom:0}.mw-parser-output table.numblk>tbody>tr>td{vertical-align:middle;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2){width:99%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table{border-collapse:collapse;margin:0;border:none;width:100%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:first-child,.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:last-child{padding:0 0.4ex}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:nth-child(2){width:100%;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{padding:0}.mw-parser-output table.numblk>tbody>tr>td:last-child{font-weight:bold}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child{font-weight:unset}.mw-parser-output table.numblk>tbody>tr>td:last-child::before{content:"("}.mw-parser-output table.numblk>tbody>tr>td:last-child::after{content:")"}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::before,.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::after{content:none}.mw-parser-output table.numblk>tbody>tr>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:none;border-right:none;border-bottom:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:thin solid;border-right:thin solid;border-bottom:thin solid}.mw-parser-output table.numblk:target{color:var(--color-base,#202122);background-color:#cfe8fd}@media screen{html.skin-theme-clientpref-night .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}

/* end https://en.wikipedia.org/ */

</style><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 {\tau _{fc}^{\star }={\frac {\rho \ U\,\Delta x\,\Delta y\sin 2\theta }{4(\Delta y\sin ^{3}\theta +\Delta x\cos ^{3}\theta )}}}}">

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tau _{fc}^{\star }={\frac {\rho \ U\,\Delta x\,\Delta y\sin 2\theta }{4(\Delta y\sin ^{3}\theta +\Delta x\cos ^{3}\theta )}}}}</annotation>

</semantics>

</math></span><img src="./0d6dbf773745bee81582d1d465db911baa69b21d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:30.773ex; height:6.509ex;" alt="{\displaystyle {\tau _{fc}^{\star }={\frac {\rho \ U\,\Delta x\,\Delta y\sin 2\theta }{4(\Delta y\sin ^{3}\theta +\Delta x\cos ^{3}\theta )}}}}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span></td></tr></tbody></table>

<p>where <i>U</i> is the resultant velocity and <i>θ</i> is the angle made by the velocity vector with the <i>x</i> direction. False diffusion is absent when the resultant flow is aligned with either of the sets of grid lines and is greatest when the flow direction is 45˚ to the grid lines.

</p>

<div class="mw-heading mw-heading3"><h3 id="Determining_the_accuracy_of_approximation_for_the_convection_term">Determining the accuracy of approximation for the convection term</h3></div>

<p>Using <i><a href="Taylor_series" title="Taylor series">Taylor series</a></i> for <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 \phi _{W}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi _{W}}</annotation>

</semantics>

</math></span><img src="./3686421995fcf2f755f38ab319a363cf92924cb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.34ex; height:2.509ex;" alt="{\displaystyle \phi _{W}}" loading="lazy"></span> and <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 \phi _{P}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi _{P}}</annotation>

</semantics>

</math></span><img src="./5eeda598a995a2028169a20e3da0abd63712ac4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.852ex; height:2.509ex;" alt="{\displaystyle \phi _{P}}" loading="lazy"></span> at the time <i>t</i> + <i>kt</i> are

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<msub>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi _{Wk}=\phi _{wk}-\left({\frac {\delta x_{i}}{2}}\right)\left({\frac {\partial \phi }{\partial x}}\right)_{wk}+{\frac {1}{2!}}\left({\frac {\delta x_{i}}{2}}\right)^{2}\left({\frac {\partial ^{2}\phi }{\partial x^{2}}}\right)_{wk}+\cdots .}</annotation>

</semantics>

</math></span><img src="./03b7cd4a9ef33a9e16aecdaeb51e5eb29cc7f464.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:61.713ex; height:6.509ex;" alt="{\displaystyle \phi _{Wk}=\phi _{wk}-\left({\frac {\delta x_{i}}{2}}\right)\left({\frac {\partial \phi }{\partial x}}\right)_{wk}+{\frac {1}{2!}}\left({\frac {\delta x_{i}}{2}}\right)^{2}\left({\frac {\partial ^{2}\phi }{\partial x^{2}}}\right)_{wk}+\cdots .}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_2a" class="reference nourlexpansion" style="font-weight:bold;">2a</span></td></tr></tbody></table>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<msub>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi _{Pk}=\phi _{wk}-\left({\frac {\delta x_{i}}{2}}\right)\left({\frac {\partial \phi }{\partial x}}\right)_{wk}+{\frac {1}{2!}}\left({\frac {\delta x_{i}}{2}}\right)^{2}\left({\frac {\partial ^{2}\phi }{\partial x^{2}}}\right)_{wk}+\cdots .}</annotation>

</semantics>

</math></span><img src="./f00f90e7187e777d0a8db0fb03608a8d4eb69a39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:61.225ex; height:6.509ex;" alt="{\displaystyle \phi _{Pk}=\phi _{wk}-\left({\frac {\delta x_{i}}{2}}\right)\left({\frac {\partial \phi }{\partial x}}\right)_{wk}+{\frac {1}{2!}}\left({\frac {\delta x_{i}}{2}}\right)^{2}\left({\frac {\partial ^{2}\phi }{\partial x^{2}}}\right)_{wk}+\cdots .}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_2b" class="reference nourlexpansion" style="font-weight:bold;">2b</span></td></tr></tbody></table>

<p>according to the upwind approximation for convection (UAC),<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 {\phi _{wk}=\phi _{Wk}}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\phi _{wk}=\phi _{Wk}}}</annotation>

</semantics>

</math></span><img src="./bc2be809dda98b23769f05fba68db4140778a8f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.945ex; height:2.509ex;" alt="{\displaystyle {\phi _{wk}=\phi _{Wk}}}" loading="lazy"></span>. Neglecting the higher order in equation (2a), the error of convected flux due to this approximation is <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 {-\rho _{w}u_{w}\delta y_{i}\left({\frac {\delta x_{i}}{2}}\right)\left({\frac {\partial \phi }{\partial x}}\right)_{wk}}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {-\rho _{w}u_{w}\delta y_{i}\left({\frac {\delta x_{i}}{2}}\right)\left({\frac {\partial \phi }{\partial x}}\right)_{wk}}}</annotation>

</semantics>

</math></span><img src="./b4a23afc3113f2365a8e125ae2f6cc752be35831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.581ex; height:6.176ex;" alt="{\displaystyle {-\rho _{w}u_{w}\delta y_{i}\left({\frac {\delta x_{i}}{2}}\right)\left({\frac {\partial \phi }{\partial x}}\right)_{wk}}}" loading="lazy"></span>. It has the form of flux of <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 {\phi }}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\phi }}</annotation>

</semantics>

</math></span><img src="./cf2f7c773dfea3549179d15219e6e06b74054be1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle {\phi }}" loading="lazy"></span> by false diffusion with a diffusion co-efficient<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{fc,UAC}^{\star }={\rho _{w}u_{w}\left({\frac {\Delta x_{i}}{2}}\right)}}</annotation>

</semantics>

</math></span><img src="./1f783abdce22f58fc43cc1f1cf20d2ece2a27a7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.222ex; height:6.176ex;" alt="{\displaystyle \tau _{fc,UAC}^{\star }={\rho _{w}u_{w}\left({\frac {\Delta x_{i}}{2}}\right)}}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span></td></tr></tbody></table>

<p>The subscript <i>fc</i> is a reminder that this is a false diffusion arising from the estimate of the convected flux at the instant <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 t+k\,\Delta t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t+k\,\Delta t}</annotation>

</semantics>

</math></span><img src="./3787534d29a38f99d0ce5cb5ccb31ce1921c507f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.054ex; height:2.343ex;" alt="{\displaystyle t+k\,\Delta t}" loading="lazy"></span> using UAC.

</p>

<div class="mw-heading mw-heading3"><h3 id="Skew_upwind_corner_convection_algorithm_(SUCCA)"><i>Skew upwind corner convection algorithm</i> (<i>SUCCA</i>)</h3></div>

<p><i>SUCCA</i> takes the local flow direction into account by introducing the influence of upwind corner cells into the discretized conservation equation in the general governing transport equation. In Fig 5, <i>SUCCA</i> is applied within nine cell grid cluster. Considering the SW corner inflow for cell P, the <i>SUCCA</i> equations for the convective transport of the conserved species <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 {\phi }}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\phi }}</annotation>

</semantics>

</math></span><img src="./cf2f7c773dfea3549179d15219e6e06b74054be1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle {\phi }}" loading="lazy"></span> are

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{P}\phi _{P}=\left({\dot {m}}_{w}-{\frac {({\dot {m}}_{s})^{2}}{{\dot {m}}_{w}}}\right)\phi _{W}+\left({\dot {m}}_{s}+{\frac {({\dot {m}}_{s})^{2}}{{\dot {m}}_{w}}}\right)\phi _{SW}+0.\phi _{S}{\text{ for }}0<\theta \leq 45}</annotation>

</semantics>

</math></span><img src="./b38d996bb4b4d7d5fbc5e5fdb1755d3f96a43453.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:75.505ex; height:7.509ex;" alt="{\displaystyle C_{P}\phi _{P}=\left({\dot {m}}_{w}-{\frac {({\dot {m}}_{s})^{2}}{{\dot {m}}_{w}}}\right)\phi _{W}+\left({\dot {m}}_{s}+{\frac {({\dot {m}}_{s})^{2}}{{\dot {m}}_{w}}}\right)\phi _{SW}+0.\phi _{S}{\text{ for }}0<\theta \leq 45}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_4" class="reference nourlexpansion" style="font-weight:bold;">4</span></td></tr></tbody></table>

<p>i.e.,

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{P}\phi _{P}=C_{w}\phi _{W}+C_{s}w\phi _{SW}}</annotation>

</semantics>

</math></span><img src="./522982d4c255f4ed3d66ad9aa3b1b5cffb4304c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.059ex; height:2.509ex;" alt="{\displaystyle C_{P}\phi _{P}=C_{w}\phi _{W}+C_{s}w\phi _{SW}}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_5" class="reference nourlexpansion" style="font-weight:bold;">5</span></td></tr></tbody></table>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{P}\phi _{P}=\left({\dot {m}}_{s}-{\frac {({\dot {m}}_{w})^{2}}{{\dot {m}}_{s}}}\right)\phi _{W}+{\left({\dot {m}}_{w}+{\frac {({\dot {m}}_{w})^{2}}{{\dot {m}}_{s}}}\right)\phi _{SW}}+0.\phi _{W}{\text{ for }}45<\theta <90}</annotation>

</semantics>

</math></span><img src="./77d633f53b53b4ec943d83193c1680139bc01311.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:78.141ex; height:7.509ex;" alt="{\displaystyle C_{P}\phi _{P}=\left({\dot {m}}_{s}-{\frac {({\dot {m}}_{w})^{2}}{{\dot {m}}_{s}}}\right)\phi _{W}+{\left({\dot {m}}_{w}+{\frac {({\dot {m}}_{w})^{2}}{{\dot {m}}_{s}}}\right)\phi _{SW}}+0.\phi _{W}{\text{ for }}45<\theta <90}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_6" class="reference nourlexpansion" style="font-weight:bold;">6</span></td></tr></tbody></table>

<p>i.e.,

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{P}\phi _{P}=C_{s}\phi _{S}+C_{s}w\phi _{SW}}</annotation>

</semantics>

</math></span><img src="./b6af9a81c1d6fc72fc474cb55d11e05c6a61fa86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.991ex; height:2.509ex;" alt="{\displaystyle C_{P}\phi _{P}=C_{s}\phi _{S}+C_{s}w\phi _{SW}}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_7" class="reference nourlexpansion" style="font-weight:bold;">7</span></td></tr></tbody></table>

<p>This formulation satisfies all the criteria of <a href="Convergence_(mathematics)" class="mw-redirect" title="Convergence (mathematics)">convergence</a> and stability.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p>

<p>In Fig. 6, as mesh is refined, the <i>upwind scheme</i> gives more accurate results but <i>SUCCA</i> offers a nearly exact solution and is more useful in avoiding multidimensional false diffusion errors.

</p>

<ul><li><a href="Computational_fluid_dynamics" title="Computational fluid dynamics">Computational fluid dynamics</a></li>

<li><a href="Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a></li>

<li><a href="Numerical_diffusion" title="Numerical diffusion">Numerical diffusion</a></li>

<li><a href="Finite_volume_method" title="Finite volume method">Finite volume method</a></li>

<li><a href="Taylor_series" title="Taylor series">Taylor series</a></li></ul>

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

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFCourantIsaacsonRees1952" class="citation journal cs1"><a href="Richard_Courant" title="Richard Courant">Courant, Richard</a>; <a href="Eugene_Isaacson" title="Eugene Isaacson">Isaacson, Eugene</a>; <a href="Mina_Rees" title="Mina Rees">Rees, Mina</a> (August 1952). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/10.1002/cpa.3160050303">"On the solution of nonlinear hyperbolic differential equations by finite differences"</a></span>. <i>Communications on Pure and Applied Mathematics</i>. <b>5</b> (3): <span class="nowrap">243–</span>255. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fcpa.3160050303">10.1002/cpa.3160050303</a>.</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="CITEREFTorrance1968" class="citation journal cs1">Torrance, Kenneth E. (1968). <a rel="nofollow" class="external text" href="https://archive.org/details/jresv72Bn4p281">"Comparison of finite difference computations of natural convection"</a>. <i>Journal of Research of the National Bureau of Standards: Mathematics and Mathematical Physics</i>. <b>72B</b>: <span class="nowrap">281–</span>301.</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="CITEREFVersteegMalalasekera,_W.2007" class="citation book cs1">Versteeg, H.K.; Malalasekera, W. (2007). <i>An introduction to computational fluid dynamics : the finite volume method</i> (2nd ed.). Harlow: Prentice Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780131274983</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="CITEREFPatankar1980" class="citation book cs1">Patankar, Suhas V. (1980). <i>Numerical heat transfer and fluid flow</i> (14. printing. ed.). Bristol, PA: Taylor & Francis. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780891165224</bdi>.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFPatankar1980" class="citation book cs1">Patankar, Suhas V. (1980). <i>Numerical heat transfer and fluid flow page no:108</i> (14. printing. ed.). Bristol, PA: Taylor & Francis. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780891165224</bdi>.</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFRaithby1976" class="citation journal cs1">Raithby, G. D. (1976-09-01). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016/0045-7825%2876%2990078-5">"A critical evaluation of upstream differencing applied to problems involving fluid flow"</a></span>. <i>Computer Methods in Applied Mechanics and Engineering</i>. <b>9</b> (1): <span class="nowrap">75–</span>103. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1976CMAME...9...75R">1976CMAME...9...75R</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0045-7825%2876%2990078-5">10.1016/0045-7825(76)90078-5</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0045-7825">0045-7825</a>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFCareyScanlonFraser1993" class="citation journal cs1">Carey, C.; Scanlon, T. J.; Fraser, S. M. (1993-05-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0307-904X%2893%2990048-L">"SUCCA—an alternative scheme to reduce the effects of multidimensional false diffusion"</a>. <i>Applied Mathematical Modelling</i>. <b>17</b> (5): <span class="nowrap">263–</span>270. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0307-904X%2893%2990048-L">10.1016/0307-904X(93)90048-L</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0307-904X">0307-904X</a>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFPatankar1980" class="citation cs2">Patankar, Suhas V. (1980), <i>Numerical Heat Transfer and Fluid Flow</i>, Taylor & Francis Group, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1980wdch.book.....P">1980wdch.book.....P</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780891165224</bdi></cite></li>

<li><cite id="CITEREFWesseling2001" class="citation cs2">Wesseling, Pieter (2001), <i>Principles of Computational Fluid Dynamics</i>, Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-67853-3</bdi></cite></li>

<li><cite id="CITEREFDate2005" class="citation cs2">Date, Anil W. (2005), <i>Introduction to Computational Fluid Dynamics</i>, Cambridge University Press, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005icfd.book.....D">2005icfd.book.....D</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780521853262</bdi></cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-05-26" href="https://en.wikipedia.org/wiki/?title=False_diffusion&oldid=1292397265">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>