Geoscience Reference
In-Depth Information
4
Turbulent fluxes
4.1 Introduction
We saw in Chapter 3 that averaging produces new types of fluxes in the
momentum and scalar conservation equations. In this chapter we'll discuss those
produced by ensemble averaging; we'll follow the usual convention in refer-
ring to these as turbulent fluxes . We'll use turbulent heat flux
to illustrate the
concepts.
To obtain some insight into the nature of heat flux in turbulent flow, we multiply
the fluid temperature Eq. (1.32) by ρc p
=
κ/α , here a constant, and write it in
“flux form”:
ρc p T
∂ρc p T
∂t
k ∂ T
∂x i
∂ H i
∂x i .
∂x i
=−
u i
˜
=−
(4.1)
The heat flux H i is a turbulent vector variable: it fluctuates in space and time and is
different in every realization. Equation (4.1) says it has a contribution from the fluid
motion, ρc p T
u i , with ρc p T the enthalpy of the fluid (joules m 3 ); and a contri-
bution from thermal conduction,
˜
k∂ T/∂x i , with k the fluid thermal conductivity
(watts m 1 K 1 ) .
In our pipe-flow example of Chapter 1 the radial component of heat flux is
the most important. At the wall it is due solely to conduction, since the fluid
velocity vanishes there; above the wall it is dominated by the turbulent radial
velocity.
Hereafter in this chapter we shall deal with the temperature equation (1.32) rather
than with (4.1) , its product with ρc p . Although by the usual convention a flux refers
to an extensive property, in using Eq. (1.32) we shall make an exception by referring
to the flux of temperature.
Thermodynamicists (e.g., Bohren and Albrecht , 1998 ) have long railed against this use of “heat,” but here it is
the conventional term.
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