The power law scheme was first used by Suhas Patankar (1980). It helps in achieving approximate solutions in computational fluid dynamics (CFD) and it is used for giving a more accurate approximation to the one-dimensional exact solution when compared to other schemes in computational fluid dynamics (CFD). This scheme is based on the analytical solution of the convection diffusion equation. This scheme is also very effective in removing False diffusion error.
The power-law scheme[1] [2] interpolates the face value of a variable,
\phi
\partial | |
\partialx |
(\rhou\phi)=
\partial | \Gamma | |
\partialx |
\partial\phi | |
\partialx |
In the above equation Diffusion Coefficient,
\Gamma
\rho
Integrating the equation, with Boundary Conditions,
\phi0=\phi|(x=0)
\phiL=\phi|(x=L)
Variation of face value with distance, x is given by the expression,
\phi(x)-\phi0 | |
\phiL-\phi0 |
=
| ||||||
\exp(Pe)-1 |
where Pe is the Peclet number given by
Pe=
\rhouL | |
\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
\phi
\phi
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,
\phi
When the Peclet number has an intermediate value, the interpolated value for
\phi
The simple average convection coefficient formulation can be replaced with a formula incorporating the power law relationship :
where
F=\rhou, D=\Gamma/L, L=xr-xc=xc-xl, and Pe=F/D
Fl,Dl
Fr,Dr
The central coefficient is given by
ac=al+ar+(Fr-Fl)
Final coefficient form of the discrete equation:
ac\phic=al\phil+ar\phir