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Unformatted text preview: Integral Thicknesses 1 Deﬁnitions The details of the velocity proﬁle u(y) at any x location are rarely signiﬁcant in engineering
applications. The most signiﬁcant quantities are integral thicknesses which describe the mass
ﬂux, momentum ﬂux, and kinetic energy ﬂux in the shear layer: 6* = / <1 — pu >dy displacement thickness
peue 0 = / (1 — 3) pu dy momentum thickness
as peue
u2 pu _ . . 6* = / 1 — —2 dy kinetic energy thickness
“5 peue 2 Integral Thickness Interpretation These thicknesses appear when comparing the mass, momentum, and kinetic energy ﬂows
in a shear layer and a corresponding potential ﬂow. 2.1 Mass ﬂow comparison Figure 1 shows the mass ﬂux passing between the vertical extent y = 0 . . .ye for inviscid and
viscous ﬂows with the same edge velocity. Figure 1: Comparison of inviscid and viscous mass ﬂows  . ye 
m1 : /dm 2 0 pudy : m1 = peueye m1 _ peue6* . . ye ye
mv = /dm = 0 pudy = peueye — /0 (pane—pawl; The viscous mass ﬂow is decreased by an amount equal to the mass defect peu66*. 2.2 Momentum ﬂow comparison Figure 2 shows the momentum ﬂux carried by the mass ﬂow passing between 3; = 0 . . .ye of
the inviscid case. The viscous case capture height is increased by 6* so that the comparison
is done at the same mass ﬂows. In each case, the momentum ﬂow can be considered to be
the force acting on a barrier which arrests the ﬂow velocity to zero. momentum extractor
(barrier) Figure 2: Comparison of inviscid and viscous momentum ﬂows, at the same mass ﬂow ye
F1 = fudr'n = pu2dy = F1 = peuiye
0 I ye+6‘ 2 2 yea—Hr 2
FV = fudm = /0 pa dy = peueye — f0 (ue—wudy = F: — peuee The viscous momentum ﬂow is decreased by an amount equal to the momentum defect peugﬁ. 2.3 Kinetic energy ﬂow comparison Figure 3 shows the kinetic energy ﬂux carried by the mass ﬂow passing between 3/ = 0 . . .ye
of the inviscid case. The viscous case capture height is again increased by 6* so that the
comparison is done at the same mass ﬂows. In each case, the kinetic energy ﬂow can be considered to be the power generated on an array of perfect windmills which reversibly bring
the ﬂow velocity to zero. 1 , 961 1
P1 = /§u2dm = /O §W3dy = P1 = Epeugye
_ 1 2 . “+6.1 3 1 3 glad—6‘1 2 2 1 3 *
PV — dm 2 /0 Ep'u dy — ipeueye —/O 5 (ue—u )pudy = P1 — ipeueﬂ The viscous kinetic energy flow is decreased by an amount equal to the kinetic energy defect
1 3 >1<
Epeueﬁ . K.E. extractor
(windmill array) mom Figure 3: Comparison of inviscid and viscous kinetic energy ﬂows, at the same mass ﬂow ...
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 Fall '03
 MarkDrela
 Dynamics, Aerodynamics

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