2.8 Flow in a volume

We presented conservation of momentum, Eq. (2.19 ), in terms of the material derivative eqn. To solve the equation in this form, would require a method that tracks particles of fluid as they move around.

Generally it is easier to solve the equation by fixing the volume of space and solving for the fluid motion through it. To enable this, we replace the material derivative for eqn using the following expression:

|----------------------| |Du @ u | |---= ---- + r ( uu) | --Dt-----@t------------- \relax \special {t4ht=
Eq. (2.26 ) is derived from the material derivative Eq. (2.14 ) by

PICT\relax \special {t4ht=

Conservation of momentum can then be written

@--u- @t + r ( uu) = r ☐☐☐+ b: \relax \special {t4ht=
A term of the form eqn or eqn represents the bulk motion of the fluid at velocity eqn, which transports the property eqn (a tensor of any rank) by advection6. For example, if the property eqn is temperature, then advection will transport heat from one region of the flow domain to another.

Advection contains a divergence derivative so represents a flux across a surface per unit volume. The example in Eq. (2.27 ) represents the flux of momentum, where the advected property is itself eqn (or eqn, depending whether eqn is associated with bulk flow or the advected property).

Outer product of two vectors

The advection term in Eq. (2.27) includes a product of two vectors eqn. This is the outer product of two vectors7 which produces a tensor by

 0 1 axbx axby axbz ab = B@ aybx ayby aybz CA : azbx azby azbz \relax \special {t4ht=

Inner product of two tensors

The inner product of two tensors, e.g. eqn produces a tensor eqn where the components are (replacing eqn with eqn)

Yij = TixSxj + TiySyj + TizSzj: \relax \special {t4ht=

Identity tensor

The identity tensor eqn is the tensor equivalent of unity (one) such that for any tensor eqn,

 0 1 1 0 0 I = B@ 0 1 0 CA : 0 0 1 \relax \special {t4ht=

6The term “convection” is sometimes used, but it has conflicting meanings, so we avoid it here.
7sometimes written “eqn”, but we write it as a product with no symbol, similar to a scalar multiplication.

Notes on CFD: General Principles - 2.8 Flow in a volume