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Upwind differencing scheme for convection
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<h2 id="description">Description</h2> <p>By taking into account the direction of the <a href="/facts/Fluid_dynamics/y7uzSCfg">flow</a>, the upwind differencing scheme overcomes that inability of the <a href="/facts/Central_differencing_scheme/m0m6MWjX">central differencing scheme</a>. This scheme is developed for strong convective flows with suppressed diffusion effects. Also known as ‘Donor Cell’ Differencing Scheme, the convected value of property ϕ {\displaystyle \phi } at the cell face is adopted from the upstream node. </p><p>It can be described by Steady convection-diffusion partial Differential Equation:<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>: 103 <a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>[<i>circular reference</i>] ∂ ∂ t ( ρ ϕ ) + ∇ ⋅ ( ρ u ϕ ) = ∇ ⋅ ( Γ ∇ ϕ ) + S ϕ {\displaystyle {\frac {\partial }{\partial t}}(\rho \phi )+\nabla \cdot (\rho \mathbf {u} \phi )\,=\nabla \cdot (\Gamma \nabla \phi )+S_{\phi }} </p><p><a href="/facts/Continuity_equation/GVav2Vet">Continuity equation</a>: ( ρ u A ) e − ( ρ u A ) w = 0 {\displaystyle \left(\rho uA\right)_{e}-\left(\rho uA\right)_{w}=0\,} <a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>: 104 <a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>[<i>circular reference</i>] </p><p>where ρ {\displaystyle \rho } is density, Γ {\displaystyle \Gamma } is the diffusion coefficient, u {\displaystyle \mathbf {u} } is the velocity vector, ϕ {\displaystyle \phi } is the property to be computed, S ϕ {\displaystyle S_{\phi }} is the source term, and the subscripts e {\displaystyle e} and w {\displaystyle w} refer to the "east" and "west" faces of the cell (see Fig. 1 below). </p><p>After <a href="/facts/Discretization/JvjSzwyk">discretization</a>, applying continuity equation, and taking source term equals to zero we get<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>[<i>circular reference</i>] </p><p>Central difference discretized equation<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>: 105 </p> <table><tbody><tr><td> F e ϕ e − F w ϕ w = D e ( ϕ E − ϕ P ) − D w ( ϕ P − ϕ W ) {\displaystyle F_{e}\phi _{e}-F_{w}\phi _{w}\,=D_{e}(\phi _{E}-\phi _{P})-D_{w}(\phi _{P}-\phi _{W})} </td> <td></td> <td>1</td></tr></tbody></table> <table><tbody><tr><td> F e − F w = 0 {\displaystyle F_{e}-F_{w}\,=0} </td> <td></td> <td>2</td></tr></tbody></table> <p>Lower case denotes the face and upper case denotes node; E {\displaystyle E} , W {\displaystyle W} , and P {\displaystyle P} refer to the "East," "West," and "Central" cell. (again, see Fig. 1 below). </p><p>Defining variable F as convection mass <a href="/facts/Flux/Aw3P5Bvb">flux</a> and variable D as <a href="/facts/Diffusion/KbAxroBa">diffusion</a> <a href="/facts/Fluid_conductance/lwVNuo0K">conductance</a> F = ρ u A {\displaystyle F\,=\rho uA} and D = Γ A δ x {\displaystyle D\,={\frac {\Gamma A}{\delta x}}} </p><p><a href="/facts/Peclet_number/lIGehNcg">Peclet number</a> (Pe) is a <a href="/facts/Dimensionless_number/TINrpQIq">non-dimensional</a> <a href="/facts/Parameter/zFvVFMv9">parameter</a> determining the comparative strengths of convection and diffusion </p><p>Peclet number: P e = F D = ρ u Γ / δ x {\displaystyle Pe\,={\frac {F}{D}}\,={\frac {\rho u}{\Gamma /\delta x}}} </p><p>For a Peclet number of lower value (|Pe| < 2), diffusion is dominant and for this the central difference scheme is used. For other values of the Peclet number, the upwind scheme is used for convection-dominated flows with Peclet number (|Pe| > 2). </p><p>For positive flow direction </p><p> u w > 0 u e > 0 {\displaystyle {\begin{aligned}u_{w}>0\\u_{e}>0\end{aligned}}} Corresponding upwind scheme equation:<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>: 115 </p> <table><tbody><tr><td> F e ϕ P − F w ϕ W = D e ( ϕ E − ϕ P ) − D w ( ϕ P − ϕ W ) {\displaystyle F_{e}\phi _{P}-F_{w}\phi _{W}\,=D_{e}(\phi _{E}-\phi _{P})-D_{w}(\phi _{P}-\phi _{W})} </td> <td></td> <td>3</td></tr></tbody></table> <p>Due to strong convection and suppressed diffusion<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>: 115 ϕ e = ϕ P ϕ w = ϕ W {\displaystyle {\begin{aligned}\phi _{e}\,=\phi _{P}\\\phi _{w}\,=\phi _{W}\end{aligned}}} </p><p>Rearranging equation (3) gives [ ( D w + F w ) + D e + ( F e − F w ) ] ϕ P = ( D w + F w ) ϕ W + D e ϕ E ) {\displaystyle [(D_{w}+F_{w})+D_{e}+(F_{e}-F_{w})]\phi _{P}\,=(D_{w}+F_{w})\phi _{W}+D_{e}\phi _{E})} </p><p>Identifying coefficients, a P = [ ( D w + F w ) + D e + ( F e − F w ) ] a W = ( D w + F w ) a E = D e {\displaystyle {\begin{aligned}a_{P}&=[(D_{w}+F_{w})+D_{e}+(F_{e}-F_{w})]\\a_{W}&=(D_{w}+F_{w})\\a_{E}&=D_{e}\end{aligned}}} </p><p>For negative flow direction u w < 0 u e < 0 {\displaystyle {\begin{aligned}u_{w}<0\\u_{e}<0\end{aligned}}} </p><p>Corresponding upwind scheme equation:<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>: 115 </p> <table><tbody><tr><td> F e ϕ E − F w ϕ P = D e ( ϕ E − ϕ P ) − D w ( ϕ P − ϕ W ) {\displaystyle F_{e}\phi _{E}-F_{w}\phi _{P}\,=D_{e}(\phi _{E}-\phi _{P})-D_{w}(\phi _{P}-\phi _{W})} </td> <td></td> <td>4</td></tr></tbody></table> <p> ϕ w = ϕ P ϕ e = ϕ E {\displaystyle {\begin{aligned}\phi _{w}=\phi _{P}\\\phi _{e}=\phi _{E}\end{aligned}}} </p><p>Rearranging equation (4) gives [ ( D e − F e ) + D w + ( F e − F w ) ] ϕ P = D w ϕ W + ( D e − F e ) ϕ E {\displaystyle [(D_{e}-F_{e})+D_{w}+(F_{e}-F_{w})]\phi _{P}=D_{w}\phi _{W}+(D_{e}-F_{e})\phi _{E}} </p><p>Identifying coefficients, a W = D w a E = D e − F e {\displaystyle {\begin{aligned}a_{W}&=D_{w}\\a_{E}&=D_{e}-F_{e}\end{aligned}}} </p><p>We can generalize coefficients as<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>: 116 a W = D w + max ( F w , 0 ) a E = D e + max ( 0 , − F e ) {\displaystyle {\begin{aligned}a_{W}&=D_{w}+\max(F_{w},0)\\a_{E}&=D_{e}+\max(0,-F_{e})\end{aligned}}} </p> <h2 id="use">Use</h2> <p>Solution in the central difference scheme fails to <a href="/facts/Convergence_(mathematics)/DTok170z">converge</a> for Peclet number greater than 2 which can be overcome by using an upwind scheme to give a reasonable result.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a>: Fig. 5.5, 5.13 Therefore the upwind differencing scheme is applicable for Pe > 2 for positive flow and Pe < −2 for negative flow. For other values of Pe, this scheme doesn’t give effective solution. </p> <h2 id="assessment">Assessment</h2> <p>Conservativeness<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a>: 118(5.6.1.1) </p><p>The upwind differencing scheme formulation is conservative. </p><p>Boundedness<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a>: 118 (5.6.1.2) </p><p>As the coefficients of the discretised equation are always positive hence satisfying the requirements for boundedness and also the coefficient matrix is diagonally dominant therefore no irregularities occur in the solution. </p> <p>Transportiveness<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a>: 118. (5.6.1.3) </p><p>Transportiveness is built into the formulation as the scheme already accounts for the flow direction. </p><p>Accuracy </p><p>Based on the backward differencing formula, the accuracy is only first order on the basis of the <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> truncation error. It gives error when flow is not aligned with grid lines. Distribution of transported properties become marked giving diffusion-like appearance, called as the <a href="/facts/False_diffusion/HsYds4Qb">false diffusion</a>. Refinement of grid serves in overcoming the issue of false diffusion. With decrease in the grid size, false diffusion decrease thus increasing the accuracy. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Central_differencing_scheme/m0m6MWjX">Central differencing scheme</a></li> <li><a href="/facts/Finite_difference/aY1txFhj">Finite difference</a></li> <li><a href="/facts/Upwind_scheme/cFtNTsub">Upwind scheme</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Versteeg, H. K.; Malalasekera, W. (2007). An Introduction to Computational Fluid Dynamics: the Finite Volume Method (2nd ed.). Harlow, England: Pearson Education Ltd. ISBN 978-0-13-127498-3. OCLC 76821177. <a href="978-0-13-127498-3" target="_blank">978-0-13-127498-3</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Central differencing scheme#Steady-state convection diffusion equation <a href="/wiki/Central_differencing_scheme#Steady-state_convection_diffusion_equation" target="_blank">/wiki/Central_differencing_scheme#Steady-state_convection_diffusion_equation</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Versteeg, H. K.; Malalasekera, W. (2007). An Introduction to Computational Fluid Dynamics: the Finite Volume Method (2nd ed.). Harlow, England: Pearson Education Ltd. ISBN 978-0-13-127498-3. OCLC 76821177. <a href="978-0-13-127498-3" target="_blank">978-0-13-127498-3</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Central differencing scheme#Formulation of Steady state convection diffusion equation <a href="/wiki/Central_differencing_scheme#Formulation_of_Steady_state_convection_diffusion_equation" target="_blank">/wiki/Central_differencing_scheme#Formulation_of_Steady_state_convection_diffusion_equation</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Central differencing scheme#Formulation of Steady state convection diffusion equation <a href="/wiki/Central_differencing_scheme#Formulation_of_Steady_state_convection_diffusion_equation" target="_blank">/wiki/Central_differencing_scheme#Formulation_of_Steady_state_convection_diffusion_equation</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Versteeg, H. K.; Malalasekera, W. (2007). An Introduction to Computational Fluid Dynamics: the Finite Volume Method (2nd ed.). Harlow, England: Pearson Education Ltd. ISBN 978-0-13-127498-3. OCLC 76821177. <a href="978-0-13-127498-3" target="_blank">978-0-13-127498-3</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Versteeg, H. K.; Malalasekera, W. (2007). An Introduction to Computational Fluid Dynamics: the Finite Volume Method (2nd ed.). Harlow, England: Pearson Education Ltd. ISBN 978-0-13-127498-3. OCLC 76821177. <a href="978-0-13-127498-3" target="_blank">978-0-13-127498-3</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Versteeg, H. K.; Malalasekera, W. (2007). An Introduction to Computational Fluid Dynamics: the Finite Volume Method (2nd ed.). Harlow, England: Pearson Education Ltd. ISBN 978-0-13-127498-3. OCLC 76821177. <a href="978-0-13-127498-3" target="_blank">978-0-13-127498-3</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Versteeg, H. K.; Malalasekera, W. (2007). An Introduction to Computational Fluid Dynamics: the Finite Volume Method (2nd ed.). Harlow, England: Pearson Education Ltd. ISBN 978-0-13-127498-3. OCLC 76821177. <a href="978-0-13-127498-3" target="_blank">978-0-13-127498-3</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Versteeg, H. K.; Malalasekera, W. (2007). An Introduction to Computational Fluid Dynamics: the Finite Volume Method (2nd ed.). Harlow, England: Pearson Education Ltd. ISBN 978-0-13-127498-3. OCLC 76821177. <a href="978-0-13-127498-3" target="_blank">978-0-13-127498-3</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Versteeg, H. K.; Malalasekera, W. (2007). An Introduction to Computational Fluid Dynamics: the Finite Volume Method (2nd ed.). Harlow, England: Pearson Education Ltd. ISBN 978-0-13-127498-3. OCLC 76821177. <a href="978-0-13-127498-3" target="_blank">978-0-13-127498-3</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Versteeg, H. K.; Malalasekera, W. (2007). An Introduction to Computational Fluid Dynamics: the Finite Volume Method (2nd ed.). Harlow, England: Pearson Education Ltd. ISBN 978-0-13-127498-3. OCLC 76821177. <a href="978-0-13-127498-3" target="_blank">978-0-13-127498-3</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Versteeg, H. K.; Malalasekera, W. (2007). An Introduction to Computational Fluid Dynamics: the Finite Volume Method (2nd ed.). Harlow, England: Pearson Education Ltd. ISBN 978-0-13-127498-3. OCLC 76821177. <a href="978-0-13-127498-3" target="_blank">978-0-13-127498-3</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Versteeg, H. K.; Malalasekera, W. (2007). An Introduction to Computational Fluid Dynamics: the Finite Volume Method (2nd ed.). Harlow, England: Pearson Education Ltd. ISBN 978-0-13-127498-3. OCLC 76821177. <a href="978-0-13-127498-3" target="_blank">978-0-13-127498-3</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> </ol>