Notes on Computational Fluid Dynamics: General Principles

2.14 Diffusion

The momentum equation for a homogeneous, incompressible, Newtonian fluid is presented in Eq. (2.47 ). In the case of zero body force, , and  constant, Eq. (2.47 ) becomes

(2.49)

This is a typical transport equation characterised by:

• the local rate of change , described in Sec. 2.5 ;
• advection of by , described in Sec. 2.8 ;
• diffusion of by ;
• a “source” due to , described in Sec. 2.12 .

Note that, since  constant, , where denotes the Laplace operator.

The term is a special form of divergence in which the flux includes the surface normal gradient denoted by the operator where .

The term models diffusion across the surface of the fluid element. Diffusion generally represents the transport of a fluid property — here, momentum — due to fluctuating motions that are not captured by the bulk motion that is represented by the continuum velocity .

Fluctuations include any random motion of particle constituents of matter, e.g. molecules, and turbulent structures. Through these motions, particles can pass across a surface boundary, transporting property through a gradient of (above, left).

Particles carrying higher move into regions of particles with lower and vice versa. Through particle collisions, high values of tend to reduce and low values increase (right).

Laplacian

“Laplacian”¹¹ describes a term of the form where is a diffusivity coefficient.¹² A Laplacian term is conservative since all variables are to the right of a divergence, as described in Sec. 2.9 . It is also bounded since it tends to decrease high values and increase low values as shown above.

The Laplacian represents a flux due to across the surface, per unit volume, as

(2.50)

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