Notes on Computational Fluid Dynamics: General Principles

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2.8 Flow in a volume

We presented conservation of momentum, Eq. (2.19 ), in terms of the material derivative $\text{[math]}$ . 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 $\text{[math]}$ using the following expression:

|----------------------| |Du @ u | |---= ---- + r ( uu) | --Dt-----@t------------- \relax \special {t4ht=

(2.26)

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=

(2.27)

A term of the form $\text{[math]}$ or $\text{[math]}$ represents the bulk motion of the ﬂuid at velocity $\text{[math]}$ , which transports the property $\text{[math]}$ (a tensor of any rank) by advection ⁶. For example, if the property $\text{[math]}$ 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 $\text{[math]}$ (or $\text{[math]}$ , depending whether $\text{[math]}$ 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 $\text{[math]}$ . 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=

(2.28)

Inner product of two tensors

The inner product of two tensors, e.g. $\text{[math]}$ produces a tensor $\text{[math]}$ where the components are (replacing $\text{[math]}$ with $\text{[math]}$ )