Effect of stabilizing forces on turbulence Page: 3 of 12
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2 TN.A.C.A. Technical Memorandum No. 625
In addition I was hopeful that this aspect would be productive
of enhanced insight into the mechanism of turbulence.
The momentary status of this theory is as follows:
Assume a horizontal fluid flow of vertical laminas so that each
superposed stratum is lighter than the one below it. Then,
when the motion is turbulent it produces energy, e.g., the
heavier layer is raised and the lighter layer is lowered against
the lift of the heavier. Let the path in the vertical, traversed
by a fluid portion prior to intermingling anew with its new sur-
rounding be designated by 1. This is the so-called "mixing path
or distance." If -dp/dy expresses the decrease in density in
the vertical (y-direction), then the ultimate lift difference
of a par ticle shifted upward becomes = -gI_ Since this lift
difference increases from zero to this value while covering dis-
tance Z, we have an energy output of 2 d (positive hen,
as assumed here, dp/dy is negative!). Downward motions yield
the same expression. In order to embrace the volumes partici-
pating on the exchange, we draw a horizontal surface and at-
tribute to one fraction $ thereof an upward motion with speed
v1, and to another fraction 2 a downward motion with mean
speed v3. Accordingly the total amount on surface F is
F(p1v, + ,av2) (volume per second),-thus making available as per-
formance the product of the "apparent shearing stresses" called
forth by the exchange and the rate of slippage. When we bring
this performance into relation with a body of base F and
height l (Fig. 1),
and when T, is the apparent shear stress and u the mean rate
of the flow, this performance becomes = FT'- du. The selection
of a layer of height t is justified, for the previously con-
ceded fluid quantity is precisely transmitted in such a layer.
Conformal to 0. Reynolds the apparent shear stress is = pu'v',
the line denoting a formed average. Accordingly the mean of
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Prandtl, L. Effect of stabilizing forces on turbulence, report, June 1931; (digital.library.unt.edu/ark:/67531/metadc277441/m1/3/: accessed April 19, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.