Page 119 - Aerodynamics for Engineering Students

102 Aerodynamics for Engineering Students

3 Transport equation for contaminant in two-dimensional flow field
In many engineering applications one is interested in the transport of a contaminant
by the fluid flow. The contaminant could be anything from a polluting chemical to
particulate matter. To derive the governing equation one needs to recognize that,
provided that the contaminant is not being created within the flow field, then the
mass of contaminant is conserved. The contaminant matter can be transported by
two distinct physical mechanisms, namely convection and molecular diffusion. Let C
be the concentration of contaminant (i.e. mass per unit volume of fluid), then the rate
of transport of contamination per unit area is given by

where i and j are the unit vectors in the x and y directions respectively, and V is the
diffusion coefficient (units m2/s, the same as kinematic viscosity).
Note that diffusion transports the contaminant down the concentration gradient
(i.e. the transport is from a higher to a lower concentration) hence the minus sign. It
is analogous to thermal conduction.
(a) Consider an infinitesimal rectangular control volume. Assume that no contam-
inant is produced within the control volume and that the contaminant is sufficiently
dilute to leave the fluid flow unchanged. By considering a mass balance for the
control volume, show that the transport equation for a contaminant in a two-
dimensional flow field is given by

dC dC dC
-+u-+v--v
dt ax ay
(b) Why is it necessary to assume a dilute suspension of contaminant? What form
would the transport equation take if this assumption were not made? Finally, how
could the equation be modified to take account of the contaminant being produced
by a chemical reaction at the rate of riz, per unit volume.

4 Euler equations for axisymmetric jlow
(a) for the flow field and coordinate system of Ex. 1 show that the Euler equations
(inviscid momentum equations) take the form:

5 The Navier-Stokes equations for two-dimensional axisymmetric jlow
(a) Show that the strain rates and vorticity for an axisymmetric viscous flow like that
described in Ex. 1 are given by:
.
.
dw
.du Ezz = z; u
r
Err = -- dr Y E$$ = -
dw au
[Hint: Note that the azimuthal strain rate is not zero. The easiest way to determine it
is to recognize that + id$ + iZ2 0 must be equivalent to the continuity equation.]
=