The power law scheme was first used by
Suhas Patankar
Suhas V. Patankar (born 22 February 1941) is an Indian mechanical engineer. He is a pioneer in the field of computational fluid dynamics (CFD) and Finite volume method. He is currently a Professor Emeritus at the University of Minnesota. He is ...
(1980). It helps in achieving approximate solutions in
computational fluid dynamics
Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate th ...
(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
Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate th ...
(CFD). This scheme is based on the analytical solution of the
convection diffusion equation
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convec ...
. This scheme is also very effective in removing
False diffusion
False diffusion is a type of error observed when the upwind scheme is used to approximate the convection term in convection–diffusion equations. The more accurate central difference scheme can be used for the convection term, but for grids wit ...
error.
Working
The power-law scheme
interpolates the face value of a variable,
, using the exact solution to a one-dimensional convection-diffusion equation given below:
:
In the above equation Diffusion Coefficient,
and both the density
and velocity remains constant u across the interval of integration.
Integrating the equation, with Boundary Conditions,
:
:
Variation of face value with distance, x is given by the expression,
![Graph depicting relation between face value and distance](https://upload.wikimedia.org/wikipedia/commons/5/58/Graph_depicting_relation_between_face_value_and_distance.JPG)
where Pe is the Peclet number given by
Peclet number is defined to be the ratio of the rate of
convection
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convec ...
of a physical quantity by the flow to the rate of
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
of the same quantity driven by an appropriate gradient.
The variation between
and x is depicted in Figure for a range of values of the Peclet number. It shows that for large Pe, the value of
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,
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
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 :
where
and
are the properties on the left node and right node respectively.
The central coefficient is given by
.
Final coefficient form of the discrete equation:
References
{{reflist
Computational fluid dynamics