KUNDU Fluid Mechanics 2 Episode 9 doc

At higher Reynolds numbers, when the flow has boundary layer characteristics, the flow downsheam of separation is unsteady and How strong an adverse p r e s m gradient the boundary laye

334 HOumhy I d p m und lielated Tw’a

over a flat plate where U ( d U / d x ) = 0 Using definition (10.17) for 8 , Eq (10.43)

(3/2)U v / 6 0.646

CJ = (1/2)PU2 - (1/2)U2 &’

which is also very closc to the exact solution of Eq (10.37)

pohlhausen found that a fourthdegme polynomial was necessary to exhibit sen- sitivity of the velocity profile to the pressure gradient; Adding mother term below

w (10.44) e(y/814 requires m additional boundary condition, azu/ay2 = o at

y = 8 Wilh the assumption of a form for thc velocity profile, Eq (10.43) may be

reduced to an equation with one unknown, 8 ( x ) with V ( x ) , or tfie pressure gradient

-4 This equation was solved approximately by Pohlhausen in 1921 This is

described in Yih (1977, pp 357-360) Subsequent improvements by Holstein and

Bohlen (1940) are recounted in Schlichting (1979, pp 357-360) and Rosenhead (1988, pp 293-297) Sherman (1990, pp 322-329) mlated the approximate solution

due to Thwaites

So far we have consided the boundary layer on a flat plate, for which the pssm

gradienl of the external stream is m Now suppose that the surface of the body is

curved (Figure 10.1 1 ) Upstream of the highest point the streamlines ofthe outer flow

converge, resulting in au increase of the free siream velocity V ( x ) and a consequent

fall of pressure with x Downstream of the highest point the streamlines diverge,

resultinginadecreawof U ( x ) andariseinpressure.Iuthissection weshallinvestigate theefFectofsuchapressuregradientcmtheshapeoftheboundarylayerprofileu(x, y) Thc boundary layer equation is

aU aU 1 ap a2u

ax ay p a x ay2’

where the pressure gradient is found from the external velocity field as d p / d x

= -pU(dU/dx), wilhx taken along the surface of the body At the wall, theboundary

layer equation becomes

In an accelerating stream d p / d x < 0, and therefore

< 0 (accelerating)

( 10.45)

As the velocity profile has to blend in smoothly with the external profilc, the slope

valuc to zero; therefore, a2u/i)y2 slightly below thc boundary layer edge is negative Equation (10.45) then shows that a2u/ay3 has the same sign at both the wall and the boundary layer edge, and presumably throughout the boundary layer In contrast, for

a decelerating external stream, the curvature of the velocity profile at thc wall is

the exbrnal stream ccascs to flow nearly parallcl to the boundary surfacc This is discussed in the next section

8 Separation

We have sccn in the last section that the boundary layer in a decelerating stream has a point of inflection and grows rapidly The existencc of the point of inflcction implics a slowing down of the region next to the wall, a conscquence of the uphill pressure gradient Under a strong enough adverse pressure gradient, the flow ncxt

Figme l O J 2 Stmmlincs and vclaciiy pfiles near a -on piru S P o i d inoection is indicated

by 1 The dashed linerepmenm u = 0

to the wall mses direction, resulting in a region of backward flow (Figure 1.0.22)

The z e v e z s e d flow meets the forward flow at some point S at which the fluid near the

d i c e is transported out into the mainstream We say hat the flow sepamtes h

the wall The separation point S is defined as the boundary between the forward flow and backward flow of the fluid near the wall, where the stcess vanishes:

It is apparent h m the figure that one streamline intersects the wall at a definite angle

at the point of separation

At lower Reynolds numbem the ~wersed flow downstream of the paint of sep

d o n forms part of a large steady vortex behind the surface (see Figure 10.15 in

Section 9 for the range 4 < Re < 40) At higher Reynolds numbers, when the flow has boundary layer characteristics, the flow downsheam of separation is unsteady and

How strong an adverse p r e s m gradient the boundary layer can withstand with- out undergoing sepamtion depends on the geometry of lhe flow, and whether the

boundary layer is laminar M turbulent A steep pressure gdient., such as that behind

a blunt body, invariably leads to a quick separation In contrast., the boundary layer on

the trailing surface of a thin body can overcome the weak pressure gradients involved

Therefore,toavoidseparationandlargedcag,thetrailingsectionofasubmergedbody

should be g m d u d l y reduced in size, giving it a so-called stnamlined shape

Evidence indicates Ihat the point of separation is insensitive to the Rcynolds number as long as the boundary layer is laminar However, a rmnsirion fo furbuknce

&Zap hunahy rclyer sepamtbn; that is, a turbulent boundary layer is more capable

of withstanding an adverse p % s m gradient This is because the velocity profile

in a turbulent boundary layer is "fuller" (Figure 10.13) and has more energy Fa example, the laminar boundary layer over a circular cylinder separates at 82" from frequently chaotic

Figure 10.13 Coinparison of laminar and turbulcnt vclocity pmfiles in a boundary layer

Figure 10.14 Separation or flow in B highly divergent chsmncl

the forward stagnation point, whercas a turbulent layer ovcr the same body separates

at 125" (shown later in Figure 10.15) Experiments show that the pressure rcmains

fairly uniform downstrcarn of separation and has a lower value than thc pressures on the forward face of the body The resulting drag due to pressure forccs is calledfimn

drag, as it depends crucially on the shape of the body For a blunt body the form drag

is larger than the skin €riction drag because of the occurrence of separation (For a streamlined body, skin friction is generally larger than the form drag.) As long as the separation point is located at the same place on the body, h e drag coefficient

of a blunt body is nearly constant at high Reynolds numbers However, the drag coefficient drops suddcnly when the boundary layer undergoes transition to turbulence

(see Figure 10.20 in Section 9) This is because thc separation point thcn moves downstream, and thc wake becomes narrower

Separation takes place not only in external flows, but also in internal flows such as

thal in a highly divergent channel (Figure 10.14) Upstream of the throat the prcssure gradient is favorable and the flow adheres to the wall Downstream of the h a t a

large enough adverse pressure gradient can cause separation

The boundary layer equations are valid only as Iar downstream as the point of

separalion Bcyond it the boundary layer becoma so thick that the basic underly-

ing assumption bccome invalid Moreover, the parabolic character of the boundary layer equations q u j n x that a numerical integration is possible only in the dkc-

tion of advection (along which information is propagated), which is rcpstrecun within

the w e d flow region A farward (downstream) integration of the boundary layer equation therefore breaks down after the separation point Last, we can no longer apply potential thcory to find the pressure distribution in the separated region, as the effective boundary or thc irrotational flow is no longer the solid surface but some

unknown shape cncompassing part of the body plus the separated regia

In gcncral, analytical soluticms of viscous flows can be found (possibly in terms of

perturbation series) only in two limiting cases, namely Re << 1 and Re >> 1 Tn

the Re << 1 limit the inertia forax are negligible over most of the flow field; the

Stokes-Oseen solutions discusscd in the p d n g chapter are of this type In the

w i t c limit of Re >> 1 , the viscous forces are neagible everywhere except close

io thc surfacc, and a solution may be attempted by matching an irrotational outcr

flow with a boundary layer near the surface In the intexmediate range of Reynolds numbers, finding aualytical solutions becomes almost an impossible task, and one has

to depend on experimentation and numerical solutions Some of these experimental

flow patterns will be described in thi section, taking the flow over a circular cylinder

as an example Instead of discussing only the intermediate Reynolds number range,

we shall describe the experimental data for the entire range of small to very high

Reynolds numbers

Low Reynolds Numbers

ZRt us start with a consideration of the creeping flow around a circular cylinder, charactcrizcd by Rc < 1 (Hen: we shall define Re = U,d/u, based on h e upstream velocity and the cylinder diamctcr.) Vorlicity is gcnmed close to the surface because

of the neslip boundary conditioL In the Stokes approximation this vorticity is sim- ply diffuscd, not advccted, which results in a lore and d t symmetry The Oseen

approximation partially takes into account the advection of vorticity, and resulk in an

asymmetric velocity distributionfurihm the body (which was ShowninFigure 9.17)

Thc vorticity distribution is qualitatively analogous to the dye distribution c a u d by

a s o w of colored fluid at the position of the body The color diffuscs symmelrisally

in very slow flows, but at higher flow speeds h c dye source is mn6ned behind a

parabolic boundary with thc dyc source at the focus

A q Re is increased beyond l., the Oseen approximation breaks down, and the vor-

ticity iu inueasingly coujined behind the cylinder becawc of advection For Re > 4,

two small auacbed or “standing” eddies appcar behind the cylinder The wake is com-

pletely laminar and the vortices act like ‘Wuidynamic rollers” over which the main

stream flows (Figure 10.1 5) The eddies gct longer as Re is increased

4 e R e c 4 0 Re<4

80 <Re e 200

laminar boundary layer turbulcnt boundary layer

nt

Figure 10.15 Some regimes or flow over a circular cylindcr

von Karman Vortex Street

A very interesting sequcnce of events begins to develop when the Reynolds number is

incrcased beyond 40, at which point the wake behind the cylinder becomes unstable Pholographs show that the wake develops a slow oscillation in which the velocity

is periodic in time and downstrcam distance, with the amplitudc of the oscillation

increasing downstrcam The oscillating wake rolls up into two staggered rows of

vortices with opposite scnse of rotation (Figure 10.16) von Karman investigated the

phenomenon as a problem of supcrposition olirrotational vortices; he concluded that

a nonstaggered row of vortices is unstable, and a staggered row is stable only if the

ratio of lateral distance between the vorlices to their longitudinal distance is 0.28

Because of thc similarity of the wake with footprints in a street, the staggered row of vortices behind a blunt body is called a von Kurmara vorrex street The vortices move downstream at a speed smaller than the upstream velocity U, This mcans that the vortex pattern slowly follows thc cylinder iC it is pulled Lhrough a stationary fluid

In the range 40 < Re < 80, the vortex street does no1 interact wilh thc pair

of attached vortices As Re is increased beyond 80 the vortex street €oms closer to

h e cylinder, and the attached eddics (whose downstream length has now grown to be about twice thc diameter of thc cylinder) themselves begin to oscillate Finally the attached eddies periodically break off alternatcly from the two sides of the cylinder

Figure 10.16 von Karman vortex street downstream of a circular cylinder at Re = 55 Flow visualized by

condensedmilk.S.’IBneda, Jour:Phys.Soc., Jlrpanu): 1714-1721,1%5,andreprintedwiththepermission

of The Physical society of Ja~#m and Dr !Watosh ‘Taneda

an oscillating “lift” or lateral force If the frequency of vortex shedding is close

to the natural frequency of some mode of vibration of the cylinder body, then an appreciable lateral vibration has been observed to result Engineering structures such

as suspension bridges and oil drilling platforms are designed so as to break up a coherent shedding of vortices from cylindrical structures This is done by including spiral blades protruding out of the cylinder surface, which break up the spanwise coherence of vortex shedding, forcing the vortices to detach at different times along the length of these structures (Figure 10.17)

The passage of regular vortices causes velocity measurements in the wake to have

a dominant periodicity The frequency n is expressed as a nondimensional parameter

known as the Strouhal number, defined as

Experiments show that for a circular cylinder the value of S remains close to 0.2 1 for a large range of Reynolds numbcrs For small values of cylinder diameter and moderate values of U,, the rcsulting frequencies of the vortex shedding and oscillating lift lie

in the acoustic range For example, at U, = 10m/s and a wire diameter of 2mm, the frequency corresponding to a Strouhal number of 0.21 is n = 1050 cyclcs per second The “singing” of telephone and transmission lincs has been attributed to this

two-dimensional computations and the data are asymptotic to S = 0.2417

Below Re = 200, the vortjces in the wake i laminar and continue to be so for

very large distances downstnam Above 200, thc vortex street becomcs unstable and irregular, and the flow within the vortices themselves becomes chaotic However, the flow in the wake continues to have a strong frequency component corresponding to

a Strouhal number d S = 0.21 Above a very high Reynolds number, say 5000, thc periodicity in the wake becomcs imperceptible, and the wake m a y bc described as completely turbulent

Striking examples of vortex streets have also been obscrved in the atmosphere Figure 1.0.18 shows a satellite photograph of the wakc bchind several isolated moun- tain peaks, through which the wind is blowing toward thc southeast Thc mountains picrce through the cloud Icvel, and the flow pattern becomes visible by thc cloud pattern The wakes behind at least two mountain peaks display the characteristics ofa von Karman vortex street Thc strong density stratification in this flow has prcvented

a vertical motion, giving the flow the two-dimensional character necessary for the formation of vortex streets

High Reynolds Numbers

At high Rcynolds numbers thc frictional elTects upstream of scparation are confined near the surface of the cylinder, and the boundary layer approximation becomes

valid a far downstream as thc point of scpamtion For Re c 3 x 16, the boundary

layer remains laminar, although the wake may be completely turbulent Thc laminar boundary layer separates at % 82” from thc forward stagnation point (Figure 10.15) The pressure in the wake downstream or the point of separation is nearly constant and lower than Lhc upstream pressure (Figure 10.19) As Lhc drag in this range is primarily due to the asymmetry in thc pressure distribution caused by scparation, and as the

point or separation remains fairly stationary in this range, the drag coeflicient also

stays constant at C D 21 1.2 (Figure 10.20)

Importanl changcs take place bcyond the critical Reynolds number or

Re, - 3 x lo-’ (circular cylindcr)

In the range 3 x l.05 -= Re < 3 x lo6, the laminar boundary layer hecomcs unstable and undergoes transition to turbulcnce We have seen in thc preceding scction that

ofHowpaataChdarQ+k%r 343

9 LksqMwn

Figore 10.18 A von Kannan vortex street downstream of mountain peaks in a strongly stratified atmo-

sphexe There are several mountain peaks along the linear, light-colored feature Nnning diagonally in the

upper lefi-hand corner of the photograph North is upward, and the wind is blowing toward the southeast

R E Thomson and J E R m e r , Monfhly Wenther Review 105: 873-884,1977, and reprinted with the

permission of the American Meteorlogical Society

because of its greater energy, a turbulent boundary layer, is able to overcome a larger

adverse pressure gradient In the case of a circular cylinder the turbulent boundary

layer separates at 125" from the forward stagnation point, resulting in a thinner wake

and a pressure distribution more similar to that of potential flow Figure 10.19 com-

pares the pressure distributions around the cylinder for two values of Re, one with a

laminar and the other with a turbulent boundary layer It is apparent that the pressures

w i t h the wake are higher when the boundary layer is turbulent, resulting in a sudden

drop in the drag coefficient from 1.2 to 0.33 at the point of transition For values of

Re > 3 x lo6, the separation point slowly moves upstream as the Reynolds number

is increased, resulting in an increase of the drag coefficient (Figure 10.20)

It should be noted that the critical Reynolds number at which the boundary

layer undergoes transition is strongly affected by two factors, namely the intensity

of fluctuations existing in the approaching stream and the roughness or the surface,

an increase in eilher of which decreases Re,, The value of 3 x lo5 is found to be

valid for a smooth circular cylinder at low levels of fluctuation of the oncoming stream

9 lhacnption oJF1ow p m l u (.~imdar C:i.linder 345

Before concluding this section we shall note an inlercsting anecdotc about the

von Karman vortex strect The pattern was investigated expcrimentally by the French

physicist Henri BCnard, well-hown for his observations of the instability of a layer

of fluid healed from below In 1954 von Karman wrotc that BCnard became "jealous

because thc vortex street was connected with my name, and several times claimed

priority [or carlier observation of the phenomenon In reply 1 oncc said '1 agrec that

what in Berlin and London is called Karman Street in Paris shall be called Avenue

de Henri Rinard.' After this wisecrack wc made peace and became good friends."

von Karman also says that the phenomenon has been known for a long timc and is

evcn found in old paintings

We close this scction by noting h a t this flow illustratcs three instanccs where the

solution is countcrintuitive First, small causes can havc large effects If wc solve for

the flow of a fluid with zero viscosity around a circular cylinder, we obtain the results

of Chapter 6, Section 9 The inviscid Flow has fore-aft symmctry and the cylindcr

experiences zero drag The bottom two pancls of Figure 10.15 illustrate the flow for

small viscosity For viscosity m small as you choosc, in the limit viscosity tends

to zero, the flow musl look like the last panel in which there is substantial fore-aft

asymmetry, a significant wake, and significanl drag This is because of the necessity

of a boundary laycr and the satisfaction of the no-slip boundary condition on thc

sur€ace so long as viscosity is not cxactly zero When viscosity is exactly zero, there

is no boundary layer and there is slip at the surface Thc rcsolution of d'Alembcrt's

paradox is through the boundary layer, a singular perturbation of the NavierSlokcs

equations in the direction normal to thc boundary

The sccond instance of counterintuitivity is that symmetric problems can have

nonsymmelric solutions This is evident in the intermediate Rcynolds number middle

pancl of Figure 10.15 Beyond a Reynolds number or 2 4 0 the symmetric wakc

becomes unstable and a pattcrn of alternating vorticcs called a von Karman vortcx

street is establishcd Yct the cquations and boundary conditions are symmetric about a

central planc in the flow If one were to solve only a half-problem, assuming symmctry,

a solution would hc obtained, but it would be unstable to infinitesimal disturbanccs

and unlikely to bc scen in the laboratory

Thc third instance of counterintuitivity is that there is a range or Reynolds num-

bers where roughening the surracc of the body can reduce its drag This is true for

all blunt bodies, such as a spherc (to be discussed in the next scction) In this range

of Rcynolds numbers, the boundary laycr on thc surface of a blunt body is laminar,

but sensitive to disturbanccs such as surface roughness, which would cause earlier

transition of the boundary layer to turbulence than would occur on a smooth body

Although, as we shall see, the skin friction of a turbulent boundary layer is much

largcr than that of a laminar boundary layer, most of the drag is causcd by incomplete

prcssurc rccovcry on the downstream side of a blunt body as shown in Figurc 10 19,

wthcr than by skin friction In fact, it is because the skin friction of a turbulcnt bound-

ary layer is much largcr, as a result of a larger velocity gI'adiml 511 the surface, that

a turbulcnt boundary layer can remain attached [arther on thc downstrcam sidc of a

blunt body, leading to a narrower wakc and morc complete pressure recovery and thus

reduced drag The drag reduction atwibutcd to thc turbulcnt boundary layer is shown

in Figmr: 10.20 for a circular cylinder and Figure 10.21 for a spherc

Several features ofthe description of flow over a circular cylinder qualitatively apply

to flows over other two-dimensional blunt bodies For cxamplc, a vortex street is observed in a flow perpendicular to a Rat plate The flow over a three-dimensional body, however, has one fundamental difference in that a regular vortex street is absent For flow around a sphere at low Reynolds numbers, there is an attached eddy in the

form of a doughnut-shaped ring; in fact, an axial section of the flow looks similar to

that shown in Figure 10.15 for the range 4 e Re c 40 For Re > 130 the ring-eddy oscillates, and some of it breaks off periodically in the form of distorted vortex

The behavior of the boundary layer around a sphere is similar to that around

a circular cylinder In particular it undergoes transition to turbulence at a critical Reynolds number of

loops

Recr - 5 x lo5 (sphere),

which corresponds to a sudden dip of the drag coefficient (Figure 10.21) As in the

case of a circular cylinder, the separation point slowly moves upstream forpostcritical Reynolds numbers, accompanied by a risc in the drag coefficient The behavior of the

separation point lor flow around a sphere at subcritical and supercritical Reynolds

numbers is responsible for the bending in the flight paths of sports balls, as explained

in the following section

o l s p t s balls bend in the air

11 Ihncimica aj.Sprh lkxh 347

11 i&tmrriic:n of Spowh Hullx

The discussion of the preceding section could be used to explain why the trajectories

of sports balls (such as those involved in tennis, cricket, and bascball games) bend in

the air The bending is commonly known as swing, swerve, or curve The problem has

been investigated by wind tunnel tests and by stroboscopic photographs of flight paths

in ficld tests, a summary of which was given by Mchta (1985) Evidence indicates

that the mechanics of bending is different for spinning and nonspinning balls The

following discussion givcs a qualitative explanation of the mechanics of flight path

bending (Readers not intemted in sports may omit this section!)

Cricket Bdl Dynamics

The cricket ball has a promincnt (1-mm high) seam, and tcsts show that the oricntation

ofthe seam is responsible for bending of thc ball’s flight path It is known to bend when

thrown at high spceds of around 30 m/s, which is equivalent to a Reynolds number of

Re = 1 05 Hcre we shall define the Reynolds number a, Re = U,d/u, based on the

translational speed U, of the ball and its diameter d The operating Reynolds number

is somewhar less than the critical value of Re, = 5 x l(9 nccessary for transition of

the boundary layer on a smooth sphere into turbulencc However, the presence of

the seam is ablc to trip the laminar boundary Iaycr into turbulence on one side of

the ball (the lower sidc in Figure 10.22), while the boundary layer on the other side

remains laminar Wc have seen in the preceding sections that because of greater energy

a turbul cnt boundary layer separates lam Typically, the boundary layer on thc laminar

side scparates at 2 85’, whereas that on thc turbulent side separates at 120‘ Compared

to region B, thc surface pressure near rcgion A is therefore closer to that given by

the potcntial flow theory (which predicts a suction pressure of (Pmin - p x ) / ( i p U & )

= - 1.25; see Eq (6.79)) In other words, thc prcssurcs are lower on side A, resulting

in a downward force on the ball (Notc that Figurc 10.22 is a view of the flow pattcrn

looking downward on the ball, so that it corrcsponds to a ball that bends to the left in its

flight The flight of a cricket ball oricnted as in Figure 10.22 is called an “outswinger”

Re - lo5

m = O M 6 kg

d = 7 2 m

I!Xgurc 10.22 The swing of a cricket ball The seam is oriented in such a r a y that the lateral force on the

hall is downward in UIC l i p

348 Boundary h p r s and Related 7bpieR

Figve 10.23 Smoke photograph of flow over a cricket ball Flow is from left to right Seam angle is 40” flow speed is 17 m/s, Re = 0.85 x 1 6 R Mehta, Ann Rev Fluid Mech 17 151-189.1985 Photograph reproduced with permission from theAnnua1 Review of Fluid Mechanics, Vol 17 @ 1985 Annual Reviews

w w AnnualReviews org

in cricket literature, in contrast to an “inswinger” for which the seam is oriented in

the opposite direction so as to generate an upward force in Figure 10.22.)

Figure 10.23, photograph of a cricket ball in a wind tunnel experiment, clearly

shows the delayed separation on the seam side Note that the wake has been deflected upward by the presence of the ball, implying that an upward force has been exerted

by the ball on the fluid It follows that a downward force has been exerted by the fluid

on the ball

In practice some spin is invariably imparted to the ball The ball is held along the seam and, because of the round arm action of the bowler, some backspin is always imparted along the seam This has the important effect of stabilizing the orientation

of the ball and preventing it from wobbling A typical cricket ball can generate side forces amounting to almost 40% of its weight A constant lateral force oriented in the same direction causes a deflection proportional to the square of time The ball

therefore travels in a parabolic path that can bend as much as 0.8 m by the time it reaches the batsman

It is known that the trajectory of the cricket ball does not bend if the ball is thrown too slow or too fast In the former case even the presence of the seam is not enough

to trip the boundary layer into turbulence, and in the latter case the boundary layer

on both sides could be turbulent; in both cases an asymmetric flow is prevented It is

also clear why only a ncw: shiny ball is able to swing, because the rough surface of an old ball causes the boundary layer to become turbulcnt on both sides Fast bowlers in cricket maintain one hemisphere of the ball in a smooth state by constant polishing

It therdorc sccms that most of the known facts about the swing of a micket ball have bccn adcquately explained by scicntific rcsearch The feature that has not been explained is the universally obscrved fact that a cricket ball swings more in humid conditions Thc changcs in density and viscosity due to changes in humidity can change the Rcynolds number by only 2%, which cannot explain this phenomcnon

Tennir Ball Dynamics

Unlike the crickcr ball, the path of the tennis ball bcnds because of spin A ball hit

with topspin curves downward, whcreas a ball hit with underspin travcls in a much flatter trajectory Thc dircction of the lateral force is therefore in the same sense as that of thc Magnus effect experienced by a circular cylinder in potential flow with circulation (see Chapter 6, Section 10) The mechanics, however, is different The potential flow argument (involving the Bernoulli cquation) offered to account for the

lateral force around a circular cylindcr cannot explain why a n.egurive Magnus cffcct

is univcrsally obscrved at lower Reynolds numbers (By a negativc Magnus effect we mcan a lateral force opposite to that experienced by a cylindcr with a circulation of the same sense as the rotation of the sphcrc.) The correct argument seems to be the asymmelric boundary layer scparation caused by the spin In fact, the phenomenon was not properly explained until the boundary layer concepts wcrc undcrstood in thc twcnticth ccntury Some pioneering experimental work on the bending paths

of spinning spheres was conducted by Robins about two hundred ycars ago; the

deflection of rotating spheres is sometimes called the Robins eflect

Experimental data onnonrotating spheres (Figure 10.21) shows that thc boundary layer on a sphere undergoes transition at a Reynolds number of % Rc = 5 x lo5,

indicated by a sudden drop in the drag cocflicient As discussed in the preceding scction, chis drop i s duc lo thc triinsition of thc laminar boundary layer to turbulence

An important point for our discussion here is that for supercritical Reynolds numbers the separation point slowly moves upstream, as evidenced by the increase of the drag coefficient after the sudden drop shown in Figure 10.2 1

With this background, wc arc now in a position to understand how a spinning hall generates a negative Magnus effect at Re e Recr and a positive Magnus effect

at Re > Re,, For a clockwise rotation of the ball, the fluid velocity relutive ra the

case (Figure 10.24a), this causes a transition of thc boundary laycr on thc lowcr sidc, whilc thc boundary layer on the upper side remains laminar The result is a delayed sqaration and lower pressure on the bottom surface, and a conscqucnt downward force on the ball The force here j s in a sense opposite to that of thc Magnus cffect The rough surface of a tennis ball lowcrs thc critical Reynolds number, so that lor a well-hit tennis ball the boundary laycrs on both sidcs of the ball have already undergone transition Due to the higher relative velocity, thc flow ncar the bottom has

a higher Reynolds number, and is therefore farther along the Rc-axis of Figure 10.21,

in the rmge AB in which the separation point mows upstrcam with an increase of

turbulent

turbulent

(a) Re c Re, (b) Re > Re,

Figure 10.24 Bending of rotating sphcrcs, in which F indicates the forcc cxcrtcd by the fluid: (a) ncgative Magnus effect; and (b) positive Magnus efiect A wcll-hi1 lcnnis ball is likely to display h e positive Magnus

c ~ c c t

the Reynolds number The scparation therefore occurs earlier on the bottom side, resulting in a higher pressure there than on the top This causes an upward lift force and a positive Magnus efiect Figure 10.24b shows that a tcnnis ball hit with undcr- spin generates an upward forcer this overcomes a large fraction of the weight of the ball, resulting in a much Battcr trajectory than that of a tennis ball hit with topspin

A “slice serve,” in which the ball is hit tangentially on the right-hand side, curves io

the left duc to the same effect (Presumably soccer balls curve in the air due to similar dynamics.)

Baseball Dynamics

A baseball pitchcr uses different kinds of dcliveries, a typical Reynolds numbcr being

1.5 x lo5 One type of delivery is called a “curveball,” caused by sidcspin imparted

by the pitcher to bend away from the side of thc throwing arm A “screwball” has the opposite spin and curvedtrajectory The dynamics of this is similar to that or aspinning tennis ball (Figurc 10.24b) Figure 10.25 is a photograph of the flow over a spinning baseball, showing an asymmetric separation, a crowding together of strcamlines at the bottom, and an upward deflection of the wake that corresponds to a downward forcc on the ball

The knuckleball, on the other hand, is released without any spin In this case the path of the ball bends due to an asymmctric separation caused by the oricntation

of the seam, much like the cricket ball However, the cricket ball is Elcased with spin along thc seam, which stabilizes the orientation and results in a predictable bending The hucklcbdll, on the othcr hand, tumbles in its flight because a1 a lack

of stabilizing spin, rcsulting in an imgular orientation of the seam and a consequcnt irregular trajcctory

So far we havc considered boundary layers over a solid surface The concept 01

a boundary laycr, however, is more general, and the approximations involved are

applicable if thc vorticity is confined in thin layers wifhout the presence of a solid

surface Such a laycr can be in the form 01 a jet of fluid ejected from an orifice, a wakc

35 1

12 7bo-Dimensional Jets

Figure 10.25 Smoke photograph of flow around a spinning baseball Flow is from left to right, flow

speed is 21 m/s, and the ball is spinning counterclockwise at 15rev/s [Photograph by E N M Brown,

University of Notre Dame.] Photograph reproduced with permission, from the Annual Review of Ftuid

Mechanics, Vol 17 @ 1985 by Annual Reviews www.AnnualReviews.org

(where the velocity is lower than the upstream velocity) behind a solid object, or a

mixing layer (vortex sheet) between two streams of different speeds As an illustration

of the method of analysis of these “free shear flows:’ we shall consider the case of

a laminar two-dimensional jet, which is an efflux of fluid from a long and narrow

orifice The surrounding is assumed to be made up of the same fluid as the jet itself,

and some of this ambient fluid is carried along with the jet by the viscous drag at the

outer edge of the jet (Figure 10.26) The process of drawing in the surrounding fluid

from the sides of the jet by frictional forces is called entrainment

The velocity distribution near the opening of the jet depends on the details of

conditions upstream of the orifice exit However, because of the absence of an exter-

nally imposed length scale in the downstream direction, the velocity profile in the

jet approaches a seIf-similar shape not far from the exit, regardless of the velocity

distribution at the orifice

For large Reynolds numbers, the jet is narrow and the boundary layer approx-

imation can be applied Consider a control volume with sides cutting across the jet

axis at two sections (Figure 10.26); the other two sides of the control volume are

taken at large distances from the jet axis No external pressure gradient is maintained

in the surrounding fluid, in which d p / d x is zero According to the boundary layer

approximation, the same zero pressure gradient is also impressed upon the jet There

is, therefore, no net force acting on the surfaces of the control volume, which requires

that the rate of flow of x-momentum at the two sections across the jet are the same

Therefore

u2 dy = independent of x , (10.47)

wherc M is the momentum flux (= mass flux times velocity) of the jet Alterna-

t i v e ] ~ ~ Eq (10.47) may be established by adding u (au/ax + a u / a y ) = 0 to the x-momentum equation in the jet to obtain

2 p u - + P ( u g + u ~ ) = P q , 3U a2u

ax and integrating over all y Only the ikst term survives, yielding Eq (10.47) Momen- tum flux is the basic externally controlled parameter in a jet and is hown from an

evaluation of Eq (10.47) at the orifice opening The mass flux p s u dy across the jet must increase downstream, as is explained later

The boundary layer equations are

where the conditions at y = 0 specify symmetry Note that thc condition at infinity is

u = 0 but u # 0 because of the entrainment of the surrounding fluid (Figure 10.26)

For x >> XO, the initial condition is rorgottcn, so wc scck a similarity solution of

a+ a2+ a$ a2$ a3$

as axay ax a4.2 - v- ay3

thc form

(10.49)

where m and n are unknown exponents, while u and h are constants chosen to make

f and r,~ dirncnsionless Substitution into Eq (1 0.48) gives

-

LJbx,n+ll-l [(m - n ) f ' 2 - mf'] = f"'

The left-hand sidc cannot dcpcnd explicitly on x , as the right-hand side does not do

so This requires that in + n - 1 = 0 A second condition relating m and n is found

by substituting Eq (10.49) into the momentum constraint (1 0.47), giving

oc:

M = pa2b'-Ix21n n L ff2dr,J = indcpcndcnt oFn,

which can be true only if 2m - n = 0 The exponents are therefore

m = f , n = 2 3 The valuc of n shows that the jet width increases as x 2 / 3

The factors u and b in Eq (10.49) can now be chosen so that r] and f are dimensionless These constants can depend only on the external parameter M and

fluid properties p and v Equation (1 0.49) requires that bx" should have thc dirncnsion

or length, so that h should havc the dimension of lengthlx" = (length)'l3 The

combination v 2 p / M has the unit of length and, accordingly, we choose

b = ( % ) I !3 ,

where the factor 48 is written for later algebraic convcnicnce Similarly Eq (10.49)

also requircs that ux"' = u x ' I 3 sbould have Lhc samc dimensions as the slreamrunc-

tion Dcnoting dimensions by 1 1, wc q u i r e [a] = [~+b/]/[x]'/~ = L513T-' The combination ( v M / ~ ) ' / ~ has this dimcnsion and accordingly, we take

(Integrate twice and substitute f = g’/g.) Thc velocity distribution is found as

which can be written as

where

is the velocity at the center of the jct It is appmnt that u,, + 00 as x + 0, showing that the origin is a singularity of the solution This is not important because the similarity solution is expected to be applicable to a real jet a,ymptotically as

x + 00 Note that if we &fine Re, = umx/u, then q = (y/x)&, modulo a finite factor (&) Further, $ = ( u ~ l r ~ ~ ) ~ / ~ f ( q ) modulo the samc &

The volume flux is

which increases downstream as thc jet entrains the surrounding fluid Far downstream, the volumc flux is much largcr than the original flux out of the orilice The externally imposed constraint in this problem is thc jet momentum flux M and not the mass flux

or centerline velocity, both of which vary with x

By drawing sketches of the profiles of u, uz, and u 3 , the reader can verify that, under similarity, thc constraint

must lead to

and

The last integral is proportional to the kinetic energy flux, which decreases down- stream bccausc of viscous dissipation Thus, the constancy of momentum flux, increase cf mass flux, and decay of enerH flux are all related

Entrainrncnt of mass is sccn by examination of

As r ] + f o o , tanh r ] + f l and scch’q + 0 Thus flow rmm thc top is downwards and flow from the bottom is upwards, both fccding thc jct additional mass

The laminar jet solution given here is not readily obscrvahlc bccausc the flow

easily breaks up into turbulencc Thc low critical Reynolds number for instability or

a jet or wake is associated with the existence of a point of inflcction i n thc vclocity profile, as discussed in Chapter 12 Nevertheless, the laminar solution has rcvcalcd several significant ideas (namely constancy of momentum flux and incrcasc of mass flux) that also apply to a turbulent jet However, the rdtc of spreading of a lurbulcnt jct is fastcr, being more like S o( x rather than S o( x2I3 (see Chapter 13)

The Wall Jet

An examplc or a two-dimensionaljet that also shares somc boundary layer character- istics is thc “wall jct.” The solution here is due to M B Glauert (1 956) We consider a fluid cxiting a narrow slot with its lower boundary bcing a planar wall taken along the x-axis (SCC Figurc 10.27) Near the wall J = 0 and the flow bchavcs like a boundary layer but far from the wall it bchaves like a free jet The boundary laycr analysis shows that for large Re, the jct is thin (S/x << 1) so ap/ay % 0 across it The

prcssurc is constant in the nearly stagnant outer fluid so p % const throughout thc flow The boundary layer cquations are

subjcct to the boundary conditions = 0: u = I: = 0; y + cc: u + 0 With

an initial velocity distribution forgotten sufficiently far downstream that Rc, + cc,

a similarity solution is availablc However, unlike the frcc jcr, the momentum flux

is not constant; instead, it diminishes downstream bccausc of h e wall shear strcss Onc relation connecting thc similarity exponcnts is obtained from the x-momcntum