power law scheme

The power-law scheme{{cite book|last=Versteeg|first=H.K.|title=An introduction to computational fluid dynamics: the finite volume method|year=2007|publisher=Prentice Hall|location=Harlow|isbn=9780131274983|edition=2nd |author2=Malalasekera, W.}}{{cite book|last=Patankar|first=Suhas V.|title=Numerical heat transfer and fluid flow|year=1980|publisher=Taylor & Francis|location=Bristol, PA|isbn=9780891165224|edition=14. printing.}} interpolates the face value of a variable, $\backslash phi\backslash ,$, using the exact solution to a one-dimensional convection-diffusion equation given below:

:$\backslash frac\{\backslash partial\}\{\backslash partial\; x\}(\backslash rho\; u\; \backslash phi)\backslash ,=\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\}\backslash Gamma\backslash frac\{\backslash partial\; \backslash phi\}\{\backslash partial\; x\}$

In the above equation Diffusion Coefficient, $\backslash Gamma$ and both the density $\backslash rho$ and velocity remains constant u across the interval of integration.

Integrating the equation, with Boundary Conditions,

:$\backslash phi\_0\backslash ,=\; \backslash phi\; |\_\{(x=0)\}$

:$\backslash phi\_L\backslash ,=\; \backslash phi\; |\_\{(x=L)\}$

Variation of face value with distance, x is given by the expression,

$\backslash frac\{\backslash phi(x)-\backslash phi\_0\}\{\backslash phi\_L-\backslash phi\_0\}\backslash ,=\; \backslash frac\{\backslash exp(\backslash text\{Pe\}\; \backslash frac\{x\}\{L\})-1\}\{\backslash exp(\backslash text\{Pe\})-1\}$

File:Graph depicting relation between face value and distance.JPG

where Pe is the Peclet number given by

$\backslash text\{Pe\}\backslash ,=\; \backslash frac\{\backslash rho\; u\; L\}\{\backslash Gamma\}$

Peclet number is defined to be the ratio of the rate of convection of a physical quantity by the flow to the rate of diffusion of the same quantity driven by an appropriate gradient.

The variation between $\backslash phi\backslash ,$ and x is depicted in Figure for a range of values of the Peclet number. It shows that for large Pe, the value of $\backslash phi\backslash ,$ at x=L/2 is approximately equal to the value at upwind boundary which is assumption made by the upwind differencing scheme. In this scheme diffusion is set to zero when cell Pe exceeds 10.

This implies that when the flow is dominated by convection, interpolation can be completed by simply letting the face value of a variable be set equal to its upwind or upstream value.

When Pe=0 (no flow, or pure diffusion), Figure shows that solution, $\backslash phi\backslash ,$ may be interpolated using a simple linear average between the values at x=0 and x=L.

When the Peclet number has an intermediate value, the interpolated value for $\backslash phi\backslash ,$ at x=L/2 must be derived by applying the power law equivalent.

The simple average convection coefficient formulation can be replaced with a formula incorporating the power law relationship :

class="wikitable" |+Power Law Relationship
$a\_l$	$a\_r$
$D\_l\; \backslash max\; [0,(1-0.1|\backslash text\{Pe\}\_l|)^5]\; +\; \backslash max[F\_l,0]$	$D\_r\; \backslash max\; [0,(1-0.1|\backslash text\{Pe\}\_r|)^5]\; +\; \backslash max[-F\_r,0]$

where $F\; =\; \backslash rho\; u,\backslash quad\; D\; =\; \backslash Gamma\; /\; L,\backslash quad\; L\; =\; x\_r\; -\; x\_c\; =\; x\_c\; -\; x\_l,\backslash quad\; \backslash text\{and\}\; \backslash quad\backslash text\{Pe\}=F\; /\; D$

$F\_l,D\_l$ and $F\_r,D\_r$ are the properties on the left node and right node respectively.

The central coefficient is given by $a\_c=a\_l+a\_r+(F\_r-F\_l)$.

Final coefficient form of the discrete equation: $a\_c\; \backslash phi\_c\; =\; a\_l\; \backslash phi\_l\; +\; a\_r\; \backslash phi\_r$

{{reflist}}

Category:Computational fluid dynamics