Upwind differencing scheme for convection

By taking into account the direction of the flow, the upwind differencing scheme overcomes that inability of the central differencing scheme. This scheme is developed for strong convective flows with suppressed diffusion effects. Also known as ‘Donor Cell’ Differencing Scheme, the convected value of property $\backslash phi$ at the cell face is adopted from the upstream node.

It can be described by Steady convection-diffusion partial Differential Equation:{{Cite book |last1=Versteeg | first1 = H. K. |url=https://www.worldcat.org/oclc/76821177 |title=An Introduction to Computational Fluid Dynamics: the Finite Volume Method |date=2007 |publisher=Pearson Education Ltd |first2=W. | last2 = Malalasekera |isbn=978-0-13-127498-3 |edition=2nd |location=Harlow, England |oclc=76821177}}{{rp|p=103}}Central differencing scheme#Steady-state convection diffusion equation{{Circular reference|date=April 2023}}

$$\backslash frac\{\backslash partial\}\{\backslash partial\; t\}(\backslash rho\backslash phi)+\backslash nabla\; \backslash cdot\; (\backslash rho\; \backslash mathbf\{u\}\; \backslash phi)\backslash ,=\; \backslash nabla\; \backslash cdot\; (\backslash Gamma\; \backslash nabla\; \backslash phi)\; +\; S\_\{\backslash phi\}$$

Continuity equation: $\backslash left(\backslash rho\; u\; A\; \backslash right)\_\{e\}\; -\; \backslash left(\backslash rho\; u\; A\; \backslash right)\_w\; =\; 0\; \backslash ,${{rp|p=104}}Central differencing scheme#Formulation of Steady state convection diffusion equation{{Circular reference|date=April 2023}}

where $\backslash rho$ is density, $\backslash Gamma$ is the diffusion coefficient, $\backslash mathbf\{u\}$ is the velocity vector, $\backslash phi$ is the property to be computed, $S\_\backslash phi$ is the source term, and the subscripts $e$ and $w$ refer to the "east" and "west" faces of the cell (see Fig. 1 below).

After discretization, applying continuity equation, and taking source term equals to zero we getCentral differencing scheme#Formulation of Steady state convection diffusion equation{{Circular reference|date=April 2023}}

Central difference discretized equation{{rp|p=105}}

{{NumBlk||$$F\_\{e\}\; \backslash phi\_\{e\}-F\_\{w\}\; \backslash phi\_\{w\}\backslash ,=\; D\_\{e\}(\backslash phi\_\{E\}-\backslash phi\_\{P\})-D\_\{w\}(\backslash phi\_\{P\}-\backslash phi\_\{W\})$$ |{{EquationRef|1}}}}

{{NumBlk||$$F\_\{e\}\; -\; F\_\{w\}\; \backslash ,\; =\; 0$$|{{EquationRef|2}}}}

Lower case denotes the face and upper case denotes node;

$E$, $W$, and $P$ refer to the "East," "West," and "Central" cell.

(again, see Fig. 1 below).

Defining variable F as convection mass flux and variable D as diffusion conductance

$$F\backslash ,=\; \backslash rho\; u\; A$$ and $$D\backslash ,=\; \backslash frac\{\backslash Gamma\; A\}\{\backslash delta\; x\}$$

Peclet number (Pe) is a non-dimensional parameter determining the comparative strengths of convection and diffusion

Peclet number:

$$Pe\; \backslash ,=\; \backslash frac\{F\}\{D\}\; \backslash ,=\; \backslash frac\{\backslash rho\; u\}\{\backslash Gamma\; /\; \backslash delta\; x\}$$

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).

For positive flow direction

$$\backslash begin\{align\}\; u\_\{w\}\; >\; 0\; \backslash \backslash \; u\_\{e\}\; >\; 0\; \backslash end\{align\}$$

Corresponding upwind scheme equation:{{rp|p=115}}

{{NumBlk||$$F\_\{e\}\; \backslash phi\_\{P\}-F\_\{w\}\; \backslash phi\_\{W\}\backslash ,=\; D\_\{e\}(\backslash phi\_\{E\}-\backslash phi\_\{P\})-D\_\{w\}(\backslash phi\_\{P\}-\backslash phi\_\{W\})$$|{{EquationRef|3}}}}

File:Upwind scheme for positive flow direction.jpg

Due to strong convection and suppressed diffusion{{rp|p=115}}

$$\backslash begin\{align\}\; \backslash phi\_\{e\}\; \backslash ,=\; \backslash phi\_\{P\}\; \backslash \backslash \; \backslash phi\_\{w\}\; \backslash ,=\; \backslash phi\_\{W\}\; \backslash end\{align\}$$

Rearranging equation (3) gives

$$[(D\_\{w\}+F\_\{w\})+\; D\_\{e\}+\; (F\_\{e\}-F\_\{w\})]\backslash phi\_\{P\}\backslash ,\; =(D\_\{w\}+F\_\{w\})\backslash phi\_\{W\}+D\_\{e\}\backslash phi\_\{E\})$$

Identifying coefficients,

$$\backslash begin\{align\}$$

a_{P} &= [(D_{w} + F_{w}) + D_{e} + (F_{e} - F_{w})] \\

a_{W} &= (D_{w} + F_{w}) \\

a_{E} &= D_{e}

\end{align}

For negative flow direction

Corresponding upwind scheme equation:{{rp|p=115}}

{{NumBlk||$$F\_\{e\}\; \backslash phi\_\{E\}-F\_\{w\}\; \backslash phi\_\{P\}\backslash ,=\; D\_\{e\}(\backslash phi\_\{E\}-\backslash phi\_\{P\})\; -\; D\_\{w\}(\backslash phi\_\{P\}-\backslash phi\_\{W\})$$|{{EquationRef|4}}}}

File:Upwind scheme for negative flow direction.jpg

$$\backslash begin\{align\}\; \backslash phi\_\{w\}\; =\; \backslash phi\_\{P\}\; \backslash \backslash \; \backslash phi\_\{e\}\; =\; \backslash phi\_\{E\}\; \backslash end\{align\}$$

Rearranging equation (4) gives

$$[(\; D\_e\; -\; F\_e\; )\; +\; D\_w\; +\; (\; F\_e\; -\; F\_w\; )]\; \backslash phi\_\{P\}\; =\; D\_w\; \backslash phi\_\{W\}\; +\; (\; D\_e\; -\; F\_e\; )\; \backslash phi\_\{E\}$$

Identifying coefficients,

$$\backslash begin\{align\}$$

a_{W} &= D_{w} \\

a_{E} &= D_{e} - F_{e}

\end{align}

We can generalize coefficients as{{rp|p=116}}

$$\backslash begin\{align\}$$

a_{W} &= D_{w} + \max(F_{w},0) \\

a_{E} &= D_{e} + \max(0,-F_{e})

\end{align}

File:Comparison of central differencing scheme with upwind differencing scheme for peclet no. greater than 2 "OR" less than -2.jpg

Solution in the central difference scheme fails to converge for Peclet number greater than 2 which can be overcome by using an upwind scheme to give a reasonable result.{{rp|at=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.

Conservativeness{{rp|p=118(5.6.1.1)}}

The upwind differencing scheme formulation is conservative.

Boundedness{{rp|p=118 (5.6.1.2)}}

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.

File:Accuracy and False Deviation variation with the grid size.jpg

Transportiveness{{rp|p=118. (5.6.1.3)}}

Transportiveness is built into the formulation as the scheme already accounts for the flow direction.

Accuracy

Based on the backward differencing formula, the accuracy is only first order on the basis of the Taylor series 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 false diffusion. Refinement of grid serves in overcoming the issue of false diffusion. With decrease in the grid size, false diffusion decrease thus increasing the accuracy.

{{reflist|33em}}

• Central differencing scheme
• Finite difference
• Upwind scheme

Category:Computational fluid dynamics

Category:Numerical differential equations