Page 179 - Bird R.B. Transport phenomena

§5.4 Empirical Expressions for the Turbulent Momentum Flux 163

U)
in which /JL is the turbulent viscosity (often called the eddy viscosity, and given the
symbol E). AS one can see from Table 5.1-1, for at least one of the flows given there,
the circular jet, one might expect Eq. 5.4-1 to be useful. Usually, however, /x (0 is a
strong function of position and the intensity of turbulence. In fact, for some systems 2
/x (0 may even be negative in some regions. It must be emphasized that the viscosity JJL
is a property of the fluid, whereas the eddy viscosity /л и) is primarily a property of
the flow.
For two kinds of turbulent flows (i.e., flows along surfaces and flows in jets and
U)
wakes), special expressions for /JL are available:

(i) Wall turbulence: /x (0 = J у^Л 0 < Щ- < 5 (5.4-2)

This expression, derivable from Eq. 5.3-13, is valid only very near the wall. It is of con-
siderable importance in the theory of turbulent heat and mass transfer at fluid-solid
interfaces. 3
(ii) Free turbulence: /x = pK b(v - v ) (5.4-3)
(0
0 zmax z>min
in which K 0 is a dimensionless coefficient to be determined experimentally, b is the
width of the mixing zone at a downstream distance z, and the quantity in parentheses
represents the maximum difference in the z-component of the time-smoothed veloci-
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ties at that distance z. Prandtl found Eq. 5.4-3 to be a useful empiricism for jets and
wakes.

The Mixing Length of Prandtl

By assuming that eddies move around in a fluid very much as molecules move around
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in a low-density gas (not a very good analogy) Prandtl developed an expression for mo-
mentum transfer in a turbulent fluid. The "mixing length" / plays roughly the same role
as the mean free path in kinetic theory (see §1.4). This kind of reasoning led Prandtl to
the following relation:

dv r
* ~
IT (5 4 4)
If the mixing length were a universal constant, Eq. 5.4-4 would be very attractive, but in
fact / has been found to be a function of position. Prandtl proposed the following expres-
sions for /:
(i) Wall turbulence: / = к^у (у = distance from wall) (5.4-5)
(ii) Free turbulence: / = к Ь (b = width of mixing zone) (5.4-6)
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in which K] and к are constants. A result similar to Eq. 5.4-4 was obtained by Taylor 6 by
2
his "vorticity transport theory" some years prior to Prandtl's proposal.

2
J. O. Hinze, Appl. Sci. Res., 22,163-175 (1970); V. Kruka and S. Eskinazi, /. Fluid. Mech., 20, 555-579
(1964).
3
С S. Lin, R. W. Moulton, and G. L. Putnam, Ind. Eng. Chem., 45, 636-640 (1953).
L. Prandtl, Zeits. f. crngew. Math. u. Mech., 22, 241-243 (1942).
4
L. Prandtl, Zeits. f. angew. Math. u. Mech., 5,136-139 (1925).
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G. I. Taylor, Phil. Trans. A215,1-26 (1915), and Proc. Roy. Soc. (London), A135, 685-701 (1932).
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