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The methods used for solving two dimensional Diffusion problems are similar to those used for one dimensional problems. The general equation for steady diffusion can be easily derived from the general transport equation for property Φ by deleting transient and convective terms [1]
In order to find the cell face value a quadratic function passing through two bracketing or surrounding nodes and one node on the upstream side must be used. In central differencing scheme and second order upwind scheme the first order derivative is included and the second order derivative is ignored.
The spatial derivatives can then be approximated by two first order and a second order central finite differences. The resulting diffusion algorithm can be written as an image convolution with a varying kernel (stencil) of size 3 × 3 in 2D and 3 × 3 × 3 in 3D.
The convection–diffusion equation can be derived in a straightforward way [4] from the continuity equation, which states that the rate of change for a scalar quantity in a differential control volume is given by flow and diffusion into and out of that part of the system along with any generation or consumption inside the control volume: + =, where j is the total flux and R is a net ...
Lower case denotes the face and upper case denotes node; , , and 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 = and =
Fig 1:Flow domain illustrating false diffusion. In figure 1, u = 2 and v = 2 m/s everywhere so the velocity field is uniform and perpendicular to the diagonal (XX). The boundary conditions for temperature on north and west wall is 100 ̊C and for east and south wall is 0 ̊C.
For large Peclet numbers (|Pe| > 2) it uses the Upwind difference scheme, which first order accurate but takes into account the convection of the fluid. As it can be seen in figure 4 that for Pe = 0, it is a linear distribution and for high Pe it takes the upstream value depending on the flow direction.
The convection–diffusion equation describes the flow of heat, particles, or other physical quantities in situations where there is both diffusion and convection or advection. For information about the equation, its derivation, and its conceptual importance and consequences, see the main article convection–diffusion equation. This article ...