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Numerical solution of the convection–diffusion equation
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<h2 id="equation">Equation</h2> <h3>General</h3> <p>The following convection diffusion equation is considered here<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p><p> c ρ [ ∂ T ( x , t ) ∂ t + ϵ u ∂ T ( x , t ) ∂ x ] = λ ∂ 2 T ( x , t ) ∂ x 2 + Q ( x , t ) {\displaystyle c\rho \left[{\frac {\partial T(x,t)}{\partial t}}+\epsilon u{\frac {\partial T(x,t)}{\partial x}}\right]=\lambda {\frac {\partial ^{2}T(x,t)}{\partial x^{2}}}+Q(x,t)} </p><p>In the above equation, four terms represents <a href="/facts/Transient_state_(chemical_engineering)/YdH9qtMr">transience</a>, <a href="/facts/Convection/E8vsCPhu">convection</a>, <a href="/facts/Diffusion/KbAxroBa">diffusion</a> and a source term respectively, where </p> <ul><li>T is the temperature in particular case of <a href="/facts/Heat_transfer/1IV3WbFL">heat transfer</a> otherwise it is the variable of interest</li> <li>t is time</li> <li>c is the specific heat</li> <li>u is velocity</li> <li>ε is porosity that is the ratio of liquid volume to the total volume</li> <li>ρ is mass density</li> <li>λ is thermal conductivity</li> <li><i>Q</i>(<i>x</i>,<i>t</i>) is source term representing the capacity of internal sources</li></ul> <p>The equation above can be written in the form </p><p> ∂ T ∂ t = a ∂ 2 T ∂ x 2 − ϵ u ∂ T ∂ x + Q c ρ {\displaystyle {\frac {\partial T}{\partial t}}=a{\frac {\partial ^{2}T}{\partial x^{2}}}-\epsilon u{\frac {\partial T}{\partial x}}+{\frac {Q}{c\rho }}} </p><p>where <i>a</i> = <i>λ</i>/<i>cρ</i> is the diffusion coefficient. </p> <h3>Solving the convection–diffusion equation using the finite difference method</h3> <p>A solution of the transient convection–diffusion equation can be approximated through a <a href="/facts/Finite_difference/aY1txFhj">finite difference</a> approach, known as the <a href="/facts/Finite_difference_method/EoG81Qn9">finite difference method</a> (FDM). </p> <h4>Explicit scheme</h4> <p>An explicit scheme of FDM has been considered and stability criteria are formulated. In this scheme, temperature is totally dependent on the old temperature (the initial conditions) and θ, a weighting parameter between 0 and 1. Substitution of <i>θ</i> = 0 gives the explicit <a href="/facts/Discretization/JvjSzwyk">discretization</a> of the unsteady conductive heat transfer equation. </p><p> T i f − T i f − 1 Δ t = a T i − 1 f − 1 − 2 T i f − 1 + T i + 1 f − 1 h 2 − ϵ u T i + 1 f − 1 − T i − 1 f − 1 2 h + Q i f − 1 c ρ {\displaystyle {\frac {T_{i}^{f}-T_{i}^{f-1}}{\Delta t}}=a{\frac {T_{i-1}^{f-1}-2T_{i}^{f-1}+T_{i+1}^{f-1}}{h^{2}}}-\epsilon u{\frac {T_{i+1}^{f-1}-T_{i-1}^{f-1}}{2h}}+{\frac {Q_{i}^{f-1}}{c\rho }}} </p><p>where </p> <ul><li>Δ<i>t</i> = <i>t</i><i>f</i> − <i>t</i><i>f</i> − 1</li> <li>h is the uniform grid spacing (mesh step)</li></ul> <p> T i f = ( 1 − 2 a Δ t h 2 ) T i f − 1 + ( a Δ t h 2 + ϵ u Δ t 2 h ) T i − 1 f − 1 + ( a Δ t h 2 − ϵ u Δ t 2 h ) T i + 1 f − 1 + Q i f − 1 c ρ Δ t {\displaystyle T_{i}^{f}=\left(1-{\frac {2a\Delta t}{h^{2}}}\right)T_{i}^{f-1}+\left({\frac {a\Delta t}{h^{2}}}+{\frac {\epsilon u\Delta t}{2h}}\right)T_{i-1}^{f-1}+\left({\frac {a\Delta t}{h^{2}}}-{\frac {\epsilon u\Delta t}{2h}}\right)T_{i+1}^{f-1}+{\frac {Q_{i}^{f-1}}{c\rho }}\Delta t} </p> <h5>Stability criteria</h5> <p>The inequalities h < 2 a | ϵ u | , Δ t < a ϵ 2 u 2 / 4 + a 2 / h 2 < h 2 2 a < h | ϵ u | {\displaystyle {\begin{aligned}h&<{\frac {2a}{|\epsilon u|}},&\Delta t&<{\frac {a}{\epsilon ^{2}u^{2}/4+a^{2}/h^{2}}}<{\frac {h^{2}}{2a}}<{\frac {h}{|\epsilon u|}}\end{aligned}}} follow from setting Q i f → 0 {\displaystyle Q_{i}^{f}\to 0} and requiring the <a href="/facts/Ansatz/3epoB3Rg">ansatz</a> T i f → T 0 f exp ( − 1 i θ ) {\displaystyle T_{i}^{f}\to T_{0}^{f}\exp({\sqrt {-1}}i\theta )} not to gain amplitude as f increases for any θ {\displaystyle \theta } . They set a stringent maximum limit to the time step that represents a serious limitation for the explicit scheme. This method is not recommended for general transient problems because the maximum possible time step has to be reduced as the square of h. </p> <h4>Implicit scheme</h4> <p>In implicit scheme, the temperature is dependent at the new time level <i>t</i> + Δ<i>t</i>. After using implicit scheme, it was found that all coefficients are positive. It makes the implicit scheme unconditionally stable for any size of time step. This scheme is preferred for general purpose transient calculations because of its robustness and unconditional stability.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> The disadvantage of this method is that more procedures are involved and due to larger Δ<i>t</i>, truncation error is also larger. </p> <h4>Crank–Nicolson scheme</h4> <p>In the <a href="/facts/Crank%25E2%2580%2593Nicolson_method/6RnaoWIP">Crank–Nicolson method</a>, the temperature is equally dependent on t and <i>t</i> + Δ<i>t</i>. It is a second-<a href="/facts/Orders_of_approximation/FxajXyHk">order</a> method in time and this method is generally used in <a href="/facts/Diffusion/KbAxroBa">diffusion</a> problems. </p> <h5>Stability criteria</h5> <p> Δ t < h 2 a {\displaystyle \Delta t<{\frac {h^{2}}{a}}} </p><p>This time step limitation is less restricted than the <a href="/facts/Explicit_method/jnJhWfuh">explicit method</a>. The <a href="/facts/Crank%25E2%2580%2593Nicolson_method/6RnaoWIP">Crank–Nicolson method</a> is based on the central differencing and hence it is second-order accurate in time.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h2 id="finite-element-solution-to-convectiondiffusion-problem">Finite element solution to convection–diffusion problem</h2> <p>Unlike the conduction equation (a finite element solution is used), a numerical solution for the <a href="/facts/Convection%25E2%2580%2593diffusion_equation/jHoHG5Xj">convection–diffusion equation</a> has to deal with the convection part of the governing equation in addition to diffusion. When the <a href="/facts/P%25C3%25A9clet_number/lIGehNcg">Péclet number</a> (Pe) exceeds a critical value, the spurious oscillations result in space and this problem is not unique to finite elements as all other <a href="/facts/Discretization/JvjSzwyk">discretization</a> techniques have the same difficulties. In a finite difference formulation, the spatial oscillations are reduced by a family of discretization schemes like <a href="/facts/Upwind_scheme/cFtNTsub">upwind scheme</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> In this method, the basic shape function is modified to obtain the upwinding effect. This method is an extension of <a href="/facts/Runge%25E2%2580%2593Kutta/YRjbyHpY">Runge–Kutta</a> discontinuous for a convection-diffusion equation. For time-dependent equations, a different kind of approach is followed. The <a href="/facts/Finite_difference_scheme/EoG81Qn9">finite difference scheme</a> has an equivalent in the <a href="/facts/Finite_element_method/eUcfP1vn">finite element method</a> (<a href="/facts/Galerkin_method/oYm8Iylo">Galerkin method</a>). Another similar method is the characteristic Galerkin method (which uses an implicit algorithm). For scalar variables, the above two methods are identical. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Advanced_Simulation_Library/89AzuD0o">Advanced Simulation Library</a></li> <li><a href="/facts/Convection%25E2%2580%2593diffusion_equation/jHoHG5Xj">Convection–diffusion equation</a></li> <li><a href="/facts/Double_diffusive_convection/MIHRLG3w">Double diffusive convection</a></li> <li><a href="/facts/An_Album_of_Fluid_Motion/SV9b4Fa9">An Album of Fluid Motion</a></li> <li><a href="/facts/Lagrangian_and_Eulerian_specification_of_the_flow_field/sYLvgboC">Lagrangian and Eulerian specification of the flow field</a></li> <li><a href="/facts/Fluid_animation/xKwmVRdu">Fluid animation</a></li> <li><a href="/facts/Finite_volume_method_for_unsteady_flow/mn5XPUpE">Finite volume method for unsteady flow</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>“Discontinuous Finite in Fluid Dynamics and Heat transfer” by Ben Q. Li, 2006. <a href="/wiki/Heat_transfer" target="_blank">/wiki/Heat_transfer</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"The Finite Difference Method For Transient Convection Diffusion", Ewa Majchrzak & Łukasz Turchan, 2012. <a href="/wiki/Finite_Difference_Method" target="_blank">/wiki/Finite_Difference_Method</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>H.Versteeg & W. Malalasekra, "an Introduction to Computational Fluid Dynamics" 2009, pages 262–263. <a href="/wiki/Computational_Fluid_Dynamics" target="_blank">/wiki/Computational_Fluid_Dynamics</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>H.Versteeg & W. Malalasekra, "an Introduction to Computational Fluid Dynamics" 2009, page no. 262. <a href="/wiki/Computational_Fluid_Dynamics" target="_blank">/wiki/Computational_Fluid_Dynamics</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Ronald W. Lewis, Perumal Nithiarasu & Kankanhally N. Seetharamu, "Fundamentals for the finite element method for heat and fluid flow". <a href="/wiki/Finite_element_method" target="_blank">/wiki/Finite_element_method</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>