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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