fluid_mechanics
486 Chapter 9 ■ Flow over Immersed Bodies SOLUTION Whether the flow is laminar or turbulent, it is true that the drag force is accounted for by a reduction in the momentum of the fluid flowing past the plate. The shear is obtained from Eq. 9.26 in terms of the rate at which the momentum boundary layer thickness, , increases with distance along the plate as For the assumed velocity profile, the boundary layer momentum thickness is obtained from Eq. 9.4 as or by integration where d is an unknown function of x. By combining the assumed shear force dependence 1Eq. 12 with Eq. 2, we obtain the following differential equation for d: or q This can be integrated from d 0 at x 0 to obtain or in dimensionless form 0 d 1 (2) (3) (Ans) Strictly speaking, the boundary layer near the leading edge of the plate is laminar, not turbulent, and the precise boundary condition should be the matching of the initial turbulent boundary layer thickness 1at the transition location2 with the thickness of the laminar boundary layer at that point. In practice, however, the laminar boundary layer often exists over a relatively short portion of the plate, and the error associated with starting the turbulent boundary layer with d 0 at x 0 can be negligible. The displacement thickness, d*, and the momentum thickness, , can be obtained from Eqs. 9.3 and 9.4 by integrating as follows: q d* a1 u U b dy d 1 a1 u U b dY 0 1 d 11 Y 17 2 dY d 8 0 0 t w rU 2 d dx u U a1 u U b dy d 1 u U a1 u U b dY Y 1 7 11 Y 1 7 2 dY 7 72 d 0.0225rU 2 a n 14 Ud b 7 2 dd rU 72 dx d 1 4 dd 0.231 a n U b 14 dx d 0.370 a n U b 15 x 4 5 d x 0.370 15 Re x Thus, by combining this with Eq. 3 we obtain d* 0.0463 a n U b 15 x 4 5 0 0 (Ans) __ y Similarly, from Eq. 2, (4) (Ans) The functional dependence for d, d*, and is the same; only the constants of proportionality are different. Typically, 6d* 6 d. By combining Eqs. 1 and 3, we obtain the following result for the wall shear stress (Ans) This can be integrated over the length of the plate to obtain the friction drag on one side of the plate, d f , as or δ Y = 1.0 0.8 0.6 0.4 0.2 0 0 Laminar F I G U R E E9.6 7 72 d 0.0360 a n U b 15 x 4 5 14 t w 0.0225rU 2 n c U10.37021nU2 15 x d 4 5 0.0288rU 2 Re x 15 / / d f bt w dx b10.0288rU 2 2 a n 15 Ux b dx 0 ( ) 1 __ u__ = __ y 7 U δ Turbulent 0.5 u__ U d f 0.0360rU 2 where A b/ is the area of the plate. 1This result can also be obtained by combining Eq. 9.23 and the expression for the momentum thickness given in Eq. 4.2 The corresponding friction drag coefficient, C Df , is C Df d f 1 2rU 2 A 0.0720 15 Re / (Ans) COMMENT Note that for the turbulent boundary layer flow the boundary layer thickness increases with x as d x 4 5 and the shear stress decreases as t w x 15 . For laminar flow these dependencies are x 1 2 and x 12 , respectively. The random character of the turbulent flow causes a different structure of the flow. Obviously the results presented in this example are valid only in the range of validity of the original data—the assumed velocity profile and shear stress. This range covers smooth flat plates with 5 10 5 6 Re / 6 10 7 . 1.0 A 0 Re / 15
9.2 Boundary Layer Characteristics 487 0.014 0.012 0.010 Completely turbulent 5 × 10 –3 ε = 3 × 10 –3 2 × 10 –3 C Df 0.008 0.006 0.004 Turbulent Transitional 1 × 10 –3 5 × 10 –4 2 × 10 –4 1 × 10 –4 5 × 10 –5 2 × 10 –5 5 × 10 –6 0.002 1 × 10 –6 Laminar Turbulent smooth plate 0 10 5 10 6 10 7 10 8 10 9 Re F I G U R E 9.15 Friction drag coefficient for a flat plate parallel to the upstream flow (Ref. 18, with permission). The flat plate drag coefficient is a function of relative roughness and Reynolds number. In general, the drag coefficient for a flat plate of length / is a function of the Reynolds number, Re / , and the relative roughness, e/. The results of numerous experiments covering a wide range of the parameters of interest are shown in Fig. 9.15. For laminar boundary layer flow the drag coefficient is a function of only the Reynolds number—surface roughness is not important. This is similar to laminar flow in a pipe. However, for turbulent flow, the surface roughness does affect the shear stress and, hence, the drag coefficient. This is similar to turbulent pipe flow in which the surface roughness may protrude into or through the viscous sublayer next to the wall and alter the flow in this thin, but very important, layer 1see Section 8.4.12. Values of the roughness, e, for different materials can be obtained from Table 8.1. The drag coefficient diagram of Fig. 9.15 1boundary layer flow2 shares many characteristics in common with the familiar Moody diagram 1pipe flow2 of Fig. 8.23, even though the mechanisms governing the flow are quite different. Fully developed horizontal pipe flow is governed by a balance between pressure forces and viscous forces. The fluid inertia remains constant throughout the flow. Boundary layer flow on a horizontal flat plate is governed by a balance between inertia effects and viscous forces. The pressure remains constant throughout the flow. 1As is discussed in Section 9.2.6, for boundary layer flow on curved surfaces, the pressure is not constant.2 It is often convenient to have an equation for the drag coefficient as a function of the Reynolds number and relative roughness rather than the graphical representation given in Fig. 9.15. Although there is not one equation valid for the entire Re / e/ range, the equations presented in Table 9.3 do work well for the conditions indicated. TABLE 9.3 Empirical Equations for the Flat Plate Drag Coefficient (Ref. 1) Equation Flow Conditions C Df 1.3281Re / 2 0.5 Laminar flow C Transitional with Re xcr 5 10 5 Df 0.4551log Re / 2 2.58 1700Re / C Df 0.4551log Re / 2 2.58 Turbulent, smooth plate C Df 31.89 1.62 log1e/24 2.5 Completely turbulent
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486 Chapter 9 ■ Flow over Immersed Bodies<br />
SOLUTION<br />
Whether the flow is laminar or turbulent, it is true that the drag<br />
force is accounted for by a reduction in the momentum of the<br />
<strong>fluid</strong> flowing past the plate. The shear is obtained from Eq. 9.26<br />
in terms of the rate at which the momentum boundary layer thickness,<br />
, increases with distance along the plate as<br />
For the assumed velocity profile, the boundary layer momentum<br />
thickness is obtained from Eq. 9.4 as<br />
or by integration<br />
where d is an unknown function of x. By combining the assumed<br />
shear force dependence 1Eq. 12 with Eq. 2, we obtain the following<br />
differential equation for d:<br />
or<br />
<br />
q<br />
This can be integrated from d 0 at x 0 to obtain<br />
or in dimensionless form<br />
0<br />
d <br />
1<br />
(2)<br />
(3) (Ans)<br />
Strictly speaking, the boundary layer near the leading edge of<br />
the plate is laminar, not turbulent, and the precise boundary<br />
condition should be the matching of the initial turbulent boundary<br />
layer thickness 1at the transition location2 with the thickness<br />
of the laminar boundary layer at that point. In practice, however,<br />
the laminar boundary layer often exists over a relatively<br />
short portion of the plate, and the error associated with starting<br />
the turbulent boundary layer with d 0 at x 0 can be negligible.<br />
The displacement thickness, d*, and the momentum thickness,<br />
, can be obtained from Eqs. 9.3 and 9.4 by integrating as follows:<br />
q<br />
d* a1 u U b dy d 1<br />
a1 u U b dY<br />
0<br />
1<br />
d 11 Y 17 2 dY d 8<br />
0<br />
0<br />
t w rU 2 d<br />
dx<br />
u<br />
U a1 u U b dy d 1 u<br />
U a1 u U b dY<br />
Y 1 7 11 Y 1 7 2 dY 7<br />
72 d<br />
0.0225rU 2 a n 14<br />
Ud b 7 2<br />
dd<br />
rU<br />
72 dx<br />
d 1 4<br />
dd 0.231 a n U b 14<br />
dx<br />
d 0.370 a n U b 15<br />
x 4 5<br />
d<br />
x 0.370<br />
15<br />
Re x<br />
Thus, by combining this with Eq. 3 we obtain<br />
d* 0.0463 a n U b 15<br />
x 4 5<br />
0<br />
0<br />
(Ans)<br />
__ y<br />
Similarly, from Eq. 2,<br />
(4) (Ans)<br />
The functional dependence for d, d*, and is the same; only the<br />
constants of proportionality are different. Typically, 6d* 6 d.<br />
By combining Eqs. 1 and 3, we obtain the following result for<br />
the wall shear stress<br />
(Ans)<br />
This can be integrated over the length of the plate to obtain the<br />
friction drag on one side of the plate, d f , as<br />
or<br />
δ<br />
Y =<br />
1.0<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
0<br />
0<br />
Laminar<br />
F I G U R E E9.6<br />
7 72 d 0.0360 a n U b 15<br />
x 4 5<br />
14<br />
t w 0.0225rU 2 n<br />
c<br />
U10.37021nU2 15 x d 4 5<br />
0.0288rU 2<br />
Re x<br />
15<br />
/<br />
/<br />
d f bt w dx b10.0288rU 2 2 a n 15<br />
Ux b dx<br />
0<br />
( ) 1 __<br />
u__ = __ y 7<br />
U δ<br />
Turbulent<br />
0.5<br />
u__<br />
U<br />
d f 0.0360rU 2<br />
where A b/ is the area of the plate. 1This result can also be obtained<br />
by combining Eq. 9.23 and the expression for the momentum<br />
thickness given in Eq. 4.2 The corresponding friction drag<br />
coefficient, C Df , is<br />
C Df <br />
d f<br />
1<br />
2rU 2 A 0.0720<br />
15<br />
Re /<br />
(Ans)<br />
COMMENT Note that for the turbulent boundary layer flow<br />
the boundary layer thickness increases with x as d x 4 5<br />
and the<br />
shear stress decreases as t w x 15 . For laminar flow these dependencies<br />
are x 1 2<br />
and x 12 , respectively. The random character<br />
of the turbulent flow causes a different structure of the flow.<br />
Obviously the results presented in this example are valid only<br />
in the range of validity of the original data—the assumed velocity<br />
profile and shear stress. This range covers smooth flat plates<br />
with 5 10 5 6 Re / 6 10 7 .<br />
1.0<br />
A<br />
0<br />
Re /<br />
15