Sec. 8.6 / Boundary-Layer Theory

(b) First, calculate the Reynolds number: Re : 10 x 40110 6 : 4 x 103. We selectn : 9.Equation (8.6.25) becomes

6tt1 dA: 0.281 (vlU-)'ta dx

Assume6 : 0 at; : 0 anci integrate. Tiris provides

5 : 0.433xRe,1/5

/ I0 x 40 \-1l5: 0.433(40 )l '- ' ' , '" | : 0.33 m. .\. 10"

This value is I5% too low.The drag force is lound to be

r,,:0.071Rer'/'x *pUZLw/10 x 40\ I/s

I

\10-" ) 2

This value is250k too low. Obviously, the power-law equations are in significant error.

S.S.6 E-acmFrl*r fficarc"rdery-Layer Eq*atE*msThe solution presented in Section 8.6.3 for the laminar boundary layer was an approximatesolution Llsing a cubic polynomial to approximate the velocity profile. In this section wesimplify the Navier-Stokes equations and present a more accurate solution for the iaminarboundary layer on a flat plate with a zero pressure gradient.

The x-component Navier-Stokes equation for a steady, incompressible, plane flow is(see Eq. 5.3.14 and ignore the gravity term)

411',1

d fo ^,,,1, .),,'[, _ lt),,r'lr,,o -,t*J,p"tlaJ L' la,/ )"'

9 -,, d6: rro?u;-Equating this to the ro of Eq.8.6.24,we find that

(8.6.43)

In boundary-layer theory the boundary layer is assumed to be very thin (see Figure 8.24),so there is no pressure variation in the y-direction in the boundary layer; that is, p : p(x).In addition (this is a very important point), the pressure p(x) is given by the inviscid flowsolution as the wail pressure; hence the pressure is not an unknown. This leaves only twounknowns, u ando. Equation 8.6.43 provides us with one equation and the continuity equation

6u i)u ldn (a'u A2rz\U-+L1-:----L I lrl -

+ ..l6x 0y p Ax [d*' dy' )

6u 6u

-*-:00x 0y

KEY COhJCEPTThere is nopressure variationin the y-directionin the boundarylayer.

(8.6.44)

4OZ Chapter 8 / External Flows

KEY COI\ICEPTThe pressuregradient dp/dxis known fromthe inviscid flowsalution.

0u OLr ldo 0:uIt- i ---=1 iV-----d.r 0y pdx 0y'

provides us with the other. The y-component Navier-stokes equation is not of use inboundary layer theory since all the terms are negligibly sma11 (a 11 u as inferred fromFigure 8.34).

In addition to the simplification that a known pressure gradient provides, Azuli)x2 is

much, much less than the large gradients that exist in the y-direction (refer to the sketch ofFigure 8.24); consequently, neglecting 02ulOf , the boundaryJayer equation that must be

solved is

(8.6.45)

where the pressure gra<iient dplclx is assrimed to be known from the inviscid flowsolution. This is olten relerred to as the Prontltl hountlary-layer equatiott, named afterLudwig Prandtl (1875-1953). Neither term on the left can be neglected, they-componenta may be small, but the velocity gradient 6ul0y rs quite large; hence the product mnstbe retained.

I-et us now focus on flow over a flzrt plate with zero pressure gradient. In addition, 1et us

irriroduce the stream lunclion:

at,U-

dx(8.6.45)

The boundary-layer equation in terms of the stream function, becomes

A,! A',1., \rlt a'1rtt _ (8.6.41)^)dy (tl(ty dx dy-

In this form the -v zind y dependence cannot be separated. If we transform this equation (suciltranslormations are selected by trial and error and experience) by letting

F:r rl:y (8 6.48)

thele results

a{0y

rl- llJV--=

0y'

W:,_! v.t

(8.6.4e)

Tlris eqr,ration lnay appear to be a more difficult equation to solve than Eq. 8.6.47. bLrt by

observing the position ol 5t in this equation, we separate variables by letting

ittb 6'l.r ;hJt i)'A A',lr lU -I --

-:: -!'- J

:r- I'L a- ttt )-1 .'r-i \i .rius,,'l .'tt t'5

, .]1(a,1,\;t ;lLS \\t tl ,/

g(€. lt) : Ji,.G rfnl ( 8.6. iL t)

Sec. 8.6 / Boundary-Layer Theory d:,{i:3

The velocity components can then be shown to be, r-rsing Eqs. 8.6.48 and 8.6.50,

dtltLt -- -- -- U F'lq)

61:

d4' tFC, (8'6's1)p:--- - ,fh(rt:'- F)

Substitute Eq.8.6.50 into Eq.8.6.49 and an or:dinary nonlinear, differential equation,-^^-,1+^. i+ i^lf,JulrJ. lr lJ

trd,F +)d,F :n- dn' - dn'(8.6.s2)

This equation repiaces the partial differential equation (8.6.47). Now let us state the bound-ary conditions.

Theboundaryconditions[zr(x,0):0, a(x,0):0 andu(x,], > 6) : U..ltaketheform

F : F' : 0 atrl :0 and F, :latlargeq (8.6.s3)

The boundary-va1ue probiem, consisting of the ordinary differential equation (8.6.52) andboundary conditions (8.6.53), can now be solved numerically. The results are tabulated inTable 8.5. The last two columns are used to provide rs and ro, respectively.

Defining the bounciary-iayer thickness to be the point where u:0.99U*, we see fromTable8.5thatthisoccurswhere4:5.Hence,ietting4:5andl:6inEq.8.6.48,wehave

5:5 (8.6.s4)

Using

6u 0u 0r1 : t)-- t1dy 0'q 6y(8.6. s s)

Table 8.5 Solution for the Laminar Boundary Layer with dpldr : 0

lul7t-]'^l-1 F F'-ulU, :tnf'-fl F'\l v.\' 2

I 0.1656 0.3298 0.0821 0.32302 0.6500 0.6298 0.300s 0.26683 t.397 0.8461 0.5708 0.16144 2.306 0.9555 0.7581 0.06425 3.283 0.9916 0.8379 0.01596 4.280 0.9990 0.8572 0.00247 s.279 0.9999 0.8604 0.00028 6.219 1.0000 0.8605 0.0000

U.vx

404 Chapter 8 / External Flows

the wall shear in a laminar boundary layer with dpldx : 0is

aul,, : *;1, o- 0.3320U)

The local skin friction coefficient is

0.664' JR..

(8.6.56)

(8.6.s7)

and the skin friction coefficient is

1 ta^ I.JJ( :-

JRt.(8.6.58)

Numerically integrating Eqs. 8.6.6 anci 8.6.7, the displacement and moilrentLrm thicknessesare found to be

0 :0.644 (8.6.se)

vxU-

frxan:p8e #.37

Atmospheric air at 30'C flows over a B-m-long, 2-m-wide flat plate at 2 mls. Assume thatiaminar flow exists in the boundary iayer over the entire iength. At x : 8 m, calcrilate(a)the maximirm value ol'u, (b) the wail shear. and (c) the flow rate throtrgh the layer.

1<i.1 niso. calcuiaie the cirag iorce on the piaic.

Soluticn(a) The y-component of velocity has been assumed to be smail in boundary-1ayer theory.Its maximrrm value at .r : 8 nr is lound, using 8.6.51. to be

,:ffx){nr'-n)ip

1l

fr

:;:

itr.l

p

x 0.86 : 0 00112 ryrb1.6x105x2

where 0.86 comes from Table 8.5 Compare o with U, :2mls.

(b) The wall shear at -r : 8 m is lound using Eq. g.6.56 to be

r,, - 0.332pLr)

:0.332 x 1.16kg/m3 x22m2ls2 i.6 x 10-5m'?/s2m2lsx8m

:2mx2n/s2mls x 3.28 = 0.105m3.s

(d) The drag force is derer.mined ro be

: 0.00154 Pa

rc) The flow rate til'ough rlre boundary layer al r : g m is given by

f' [i r'Q: I u Xtud.v:rt,.|"" I tr..F',lnJo Iu* Jo*n- *'t

where we have substituied for u and y from Eqs. g.6.51 and g.6.4g.I o' n, - ,F. rhe flow rale is

-' -"" / v'u'J r urru u'v'-

Q:zuLr-ff"

Sec. 8.6 / Boundary-Layer Theory 4*ij

Recognizing that

.,Dxvl.oxlo*

KEY C6ruCEPTA strong negativepressure gradientntay relaminarizea turbulentboundary layer.

1o(s)- F$)1

1.6 x 10-sm2/sx 8m

IFr: =pL/.)LwC,"2"'tI: - xl.16kg/m3 x22m2ls2 xgmx2mx

: 0.049 N

i.JJ

&.6.7 Fressur*-GradEerlt Effe*tsIn the foregoing sections we have concentrated our study of boundary layers on the flatplate with a zero presslrre -eradient. This is the simplest boundary-layer flow and allows us tomodel many flows of engineering interest. The inclusio., of u p."ssrie gradient, even thoughrelatively low, markedly alters the boundary-layer flou,. In fict, a strong negative pressuregradient (such as flow in a contraction) may relaminarize a turbulent bounJary layer; thatis, the turbulence production in the viscous wall layer which sustains the turbulence ceasesand a laminar boundary layer is reestablished. A positive pressure gradient quickly causesthe boundary layer to thicken and eventual)y to separate. These two e{fects are shown in thephotograpirs of Figure 8.27.

vU-x

405 Chapter B / External Flows

XqY CSFJfEFTFor a negativepressure gradient,there is a reducedtendency for theflow to separate.

Flgur* E"27 Influence of a strong pressure gradient on a turbulent flow: (a) a strongnegative pressure gradient may reiamin arize a flow; (b) a strong positive pressure gridientcauses a boundary layer to thicken. (photograph by R. E. Falcoi

The flow about any plane body with curvature, such as an airfoil, can be modeled asflow over a flat plate with a nonzero pressure gradient. The boundary-layer thickness is somuch smaller than the radius of curyature that the additional curvature terms drop out ofthe differential equations. The inviscid flow solution at the wall provides the pressuie gradi-ent dpldx and the velocity U(x) at the edge of the boundary-layer. For axisymmetric flows,such as flolv over the nose of an aircraft, boundary layer equations in cylindrical coordinatesmust be utilized.

The pressure gradient determines the value of the second derivative a2ulay2 at the wali.Fronr the boundary-layer equation (3.6.45) at the wail, u : a :0, so that

.l^ r2 .. ILty u ul. -p-tdx d!'lr:o (8.6.60)

for either a laminar or a turbulent boundary-layer flow. For a zero pressure gradient, thesecond derivative is zero at the wall; then since the first derivative is greatest at the wall anddecreases as y increases. the second derivative mnst be negative for positi.re i,.Theprofilesare sketched in Figure 8.28a.

For a negative (favorable) pressure gradient, the slope of the velocity prolile near thewall is relatively large with a negative second derivative at the wall and throughout the layer.The momentum near the wall is larger than that of the zero pressure gradient flow, as showr.rin Figure 8'28b, and thus there is a reduced tendency for the flow to separate. Turbulenceproduction is discouraged, and relaminarization may occur for a sufficiently large negativepressure gradient over a sufficient distance.

Sec. 8.6 / Boundary-Layer Theory 4,.*V

M:::.

0u/0y(a) dp/dx = 0

(b) dp/dx < 0 (a favorable gradient)

tc I dp/dx > 0 ( en unfavorable gradient)

1, ,r l

d'u/dt,'LI

Figur* 8.28 Influence of

0u/0y(d) dp/dx > 0 (separated ffow)

the pressure gradient.

If a positive (unfavorable) pressure gradient is imposed on the flow, the second derivativeat the wall will be positive and the flow will be as sketched in part (c) or (d). If the unfa-vorable pressure gradient acts over a sufflcient distance, part (d) will probably represent theflow situation with the flow separated from the surface. Near the wall the higher pressuredownstream will drive the low momentum flow near the wail in the upstream direction,resulting in flow reversal, as shown. The point at which dulay : 0 at the wall locates thepoint of separation.

d- u/ d \''

408 Chapter 8 / External Flows

The problem of a laminar boundary layer with a pressure gradient can be solvedusing conventional numerical techniques. The procedure is relatively simple using thesimplified bor,rndary-layer equation (8.6.45) with a known pressure gradient. For a tur-bulent flow the Reynolds stress term must be included; research continues to deveiopmodels of turbulent quantities thiit will result in acceptable numerical soiutions. Exper-irnental results are often necessary for turbulent-flow problems, as was the situation forinternal flows.

8.7 SUMMARYThe drag and lift coefficients are dellned as

(8.7.1)

where the area is the projected area for blunt objects, and the chord times the length for anairloil.

Vortex shedding occurs lrom a cylinder whenever the Reynolcis number is in the range300 < Re < l0 000. The frequency of shedding is found from the Strouhai number

St:

r'vhere/'is the freqriency, in hertz (cycles/s).

(8.1.2)

Piane potential flor,vs al'e constructed by sr-rperimposing the following simple flows:

Drag Lil'tL,\ " lpVrA ' lpV'A

Uniform floi,v: ,1, :Linesource: *:

lrrotational vortex : tlt :

I\^"Lf^r. .t-vvllurlr . v -

rf : u..xat6: ' lnr' '\-f6: -0

, P - -^q2 - -- uosTr

fpV

Lr_yq

2r0f- lnr'a-

u-sltla/r

(8.7.3)

(8.7 4)

The streern'r function for the rotating cylinder is given by

f.yrin.re, : Lr.Y - 4sin0 + !- h,

rvhere the cylinder radius is

(8.7.5)

Fundamentals of Engineering Exam Review Problems 4Sg

The velocity components are

AA dtLrU-Ay d-r

For a laminar boundary layer on a flat plate with zero pressure gradient the exact soh-r-li ^,. nr-nrri rl ecrrvrr Hfvvauvu

For a turbuient flow from the leading edge, the pou,er-law profiie rvith 4 : 7 gives

I A,l aOL), : -- Ur,: -:t' [t0 " Ar

u : t,,8 ,, :0.664ff cl :1.33

,o : !r,pU) r, : )c, oui t

(8.7.6)

(8.7.1)

(8.7.e)

a :0.:s.-[ "l'"\ru- ) .. : o.ossl,'4 )"'' \v )c,:0073[ #)" (878)

where the wall shear and drag force per unit width are, respectively,

vLU-

FE SAIR& EEF&EFE E @ AFE- EJ t\& uf+EHi k E'rE E d!.L:* (} F E f\i {i E fd ts Effi [ E\i G HXA M ffi EVE EUS pRGffi E-Efffi S

8.1 The drag force on a streamlined shape is dueprimarily to:(A) The wake(B) The component of the pressure force

acting in the fiow direction(C) The shear stress(D) The separated region near the trailing

edge

A golf ball has dimples to increase its flighrdistance. Select the best reason that accountsfor the longer flight distance of a Cimpled ballcolnpared to that cl a smcoth ball.(A) The shearing stress is smaller on the

dimpled bali.(B) The dimpled ball has an effective smaller

diameter.(C) The wake on the dimpled ball is

sma11er.(D) The pressure on the front of the smooth

ball is larger.

Flood waters at 10'C flow over an B-mm-diameter fence wire with a velocity of 0.8 m/s.Which of the following is true?(A) It is a Stokes flow with no separation.(B) The separated region covers most of the

rear of the wire.(C) The drag is due to the r.elatively 1ow pres-

sure in the separated region.(D) The separated region covers only a small

arca at the rear of the wire.The drag on a 1O-m-diameter spherical waterstorage tank in an 80 km/h wind is approrimately:

8.2

(A) 6300 N(B) 4700 N

(c) 3200 N(D) 2300 N

8.5 A 4-m-long smooth cyiinder experiences a dragof 60 N when subjected to an atmospheric airspeed of 40 m/s. Estimate the diameter of thecylinder.(A) l2l mm(B) 63 mm

(C) 26 mm(D) 4.1 mm