KUNDU Fluid Mechanics 2E Episode 16 ppt

In general w , r, U,, E, and cyc are all Cunctions of y , so that for the entire wing The drag force Di induced by the trailing vortices is called the induced drug, which is zero for a

Figurn 15.24 Lift and induced drag on a wing element dy

Because the aspect ratio is assumed large, E is small Each element d y of the finite wing may then be assumed to act as though it is an isolated two-dimensional section set in a stream of uniform velocity Ue, at an angle of attack a, According to the

Kutta-Zhukhovsky lift theorem, a circulation r supcrimposed on the actual resulrant

velocity U, generates an elementary mrodynamic force d L , = pUJ d y , which acts

normal to U, This force may be resolved into two components, the conventional lift force d L normal to the direction of flight and a component dDi parallel to thc

direction of flight (Figure 15.24) Therefore

d L = d L , c o s ~ = p U , r d y c 0 ~ ~ 2 1 p U r d y , dDi = d L , s i n & = p U C r d y s i n & 2 1 p w r d 4 ’

In general w , r, U,, E, and cyc are all Cunctions of y , so that for the entire wing

The drag force Di induced by the trailing vortices is called the induced drug, which is

zero for an airfoil of infinite span It arises because a wing of finite span continuously crcatcs trailing vortices and the rate dgeneration of the kinetic energy of thc vortices must equal the ratc of work done against the induced drag, namcly Di U For this reason the induced drag is also known as the vortex drug It is analogous to the wuve drug

experienced by a ship, which continuously radiates gravity waves during its motion

As we shall see, the induccd drag is the largest part of the total drag experienced by

an airfoil

A basic reason why there must be a downward velocity behind the wing is the following: The fluid exerts an upward lift force on the wing, and therefore b e wing exerts a downward force on the fluid The fluid must therefore constantly gain downward momentum as it goes past tbc wing (See thc photograph of the spinning baqeball (Figure 10.25), which exerts an upward force on thc fluid.)

For a given r(y), it is apparent that w(y) can be determined from Eq (15.13) and Di can then be determined f o m Eq (15.15) However, r(y) itself depends on the distribution of w(y), essentially because the cffective angle of attack is changcd due to w(y) To see how r ( y ) m a y be cstirnated, Tist note that the lilt coefficient lor

a two-dimensional Zhukhovsky airfoil is nearly C L = 217(a + b) For a finite wing

we may assume

(15.16)

where (a - w / V ) is the effectivc angle of attack, - s ( y ) is the angle of attack for: zero

lift (found from experimental dah such as Figure 15.18), and K is a constant whose value is nearly 6 for most airfoils (K = 2;r for a Zhukhovsky airfoil.) An expression for the circulation can be obtained by noting that the lift coefficient is related to the

circulation as C L L / ( $ p V * c ) = r / ( ; V c ) , so that I’ = i V c C L The assumption

Eq (151.6) is then equivalent to thc assumption that the circulation for a wing of finitc span is

( 15.1 7)

For a given U , a, c ( y ) , and #? ( y ) , Eqs (15.13) and (15.17) define an integral equation

for dekrmjning r(y) (An integral equation is one in which the unknown function

appears under an integral sign.) The problem can be solved numerically by iterativc

techniques Instead of pursuing this approach, in the next scction we shall assumc

that r(y) is givcn

Lancheater versus Prandtl

Thcre is some controversy in the literature about who should get more credit for developing modem wing theory Since Prandtl in 1918 first published the thcory in

a mathematical form, textbooks for a long time have called it the “Randtl Lifting Line Theory.” Lanchester was bitter about this, because he felt that his contributions werc not adequatcly recognizcd The controversy has been discussed by von Karman

(1 954, p 50), who witncssed the dwclopment of the thcory He givcs a lot ofcrcdit to Lanchester, but falls short of accusing his teacher Prandtl of bcing delibcrately unfair Here we shall note a few facts that von M a n brings up

Lanchester was thc first person to study a wing of finite span He was also the h-st person to conceive that a wing can be rcplaced by a bound vortex, which bends backward to foim the tip vortices Last, Lanchestcr was the first to recognize that thc

minimum power necessary to fly is that requircd to generate the kinctic energy field

of the downwash field It secms, then, that Lanchester had conceived all of the basic

ideas of the wing theory, which he published in 1907 in the form of a book called

”Aerodynamics.” In Tact, a figurc from his book looks very similar to our Figure 15.21

Many ol these ideas werc cxplaincd by Lanchester in his talk at Gijttingen, long

before Prandtl published his theory Prandtl, his graduate student von Karman, and

Carl Runge were all present Runge, well-known €or his numerical integration scheme

of ordinary differential equations, served as an interpreter, because ncithcr Lanchcstcr

qor Prandtl could speak the other’s language As von Karman said: “both Prandtl and Runge learned very much from these discussions.”

However, Prdndtl did not want to recognize Lanchester for priority of ideas,

saying that he conceivcd of thcm before he saw Lanchester’s book Such controversies cannot bc scttlcd And grcat mcn havc been involvcd in controversies before For cxamplc, astTophyskist Stcphcn Hawking (1 988), who occupicd Newton’s chair at Cambridge (after Lighthill), described Newton to be a rather mean man who spent much of his later years in unfair attempts at discrediting Leibniz, in trying to force the Royal astronomer to release some unpublished data that he needed to verify his predictions, and in heated disputes with his lifelong nemesis Robert Hook

ln view of the fact that Lanchester’s book was already in print when Prandtl pub- lished his thcory, and the fact that Lanchcstcr had all the ideas but not a formal mathc- matical thcory, wc havc called it the “Li.liing Line Theory or Prandtl and Lanchester.”

I 1 Resulk for Ellipdic C’imulalion Ilistribution

Thc induced drag and other properties of a finite wing depend on thc distribution oT

T(y) Tfie circulation distribution: however, dcpcnds in a complicated way on the wing planform, angle of attack, and so on Tt can be shown that, for a given total lift and wing area, the induced drag is a minimum whcn thc circulation distribution is :lliptic (See, for e.g., Ashley and Landahl, 1965, Tor a proof.) Here we shall simply

assume an elliptic distribution of the form (see Figure 15.22b)

and deteminc thc rcsulting expressions for downwash and induced drag

The total lift Torce on a wing is then

Writing y = ( y - y i ) + y1 in the numerator, wc obtain

The first integral has the valuc n/2 The second integral can be reduced to a standard

form (listcd in any mathematical handbook) by substituting x = y - y1 On setting

limits the second integral turns out to be zero, although the integrand is not an odd

function The downwash at y~ is thereforc

(15.20) r0

W ( Y l ) = - 9

2s

which shows that, €or an elliptic circulation distribution, the induced velocity at the

wing is constant along the span

Using Eqs (15.18) and (15.20), the induced drag is found as

Tn terms of the lift Eq (15.19), this bccornes

which can be written as

Equation (15.21) shows that Cui + 0 in the two-dimensional limit A + m More

important, it shows that thc induced dmg coeflicient increases us rhe square of the

liJr coe#cienl Wc shall see in the following section that the induced drag generally

makes the largest contribution to the total drag of an airfoil

Since an elliptic circulation distribution minimizes the induced drag, it is of inter-

est to detcrmine the circumstances under which such a circulation can be established

Considcr an element d y of thc wing (Figurc 15.25) The lift on thc element is

d L = pUI'dy = C ~ f p U ~ ~ d y , (1 5.22)

where c d y is an elemcntary wing area Now if the circulation distribution is clliptic,

then the downwash is independent of y In addition, if the wing profile is gcomel-

rically similar at every point along the span and has thc same geometrical angle of

Figure 15.25 Wing of elliptic planibrm

attack a, ihen the egective angle or atlack and hence thc lift coerficient CL will be

indcpendent of y Equation (1 5.22) shows that the chord length c is then simply pro- portional to r, and so c(y) is also elliptically distributed Thus, an untwisted wing with clliptic planform, or composed of two semiellipscs (Figure 15.25), will generate

an elliptic circulation distribution However, the same effcct can also be achieved with nonelliptic planrorms if the anglc of attack varies along the span, that is, if the wing

is givcn a "twist."

1.2 I@ m d !)rug Charackri.stics oJAi~foi1.s

Before an aircrart is built its wings are tested in a wind tunnel, and the results are generally given as plots of C, and CD vs the angle of attack A typical plot is shown in

Figure 15.26 It is seen that, in a range of incidcnce angle from a = -4' to a = 12",

the variation of CL with a is approximatcly linear, a typical value of dCL/da being

xO.1 per degree Thc lift reaches a maximum value at an incidence of %IS" If the anglc of attack is increased further, thc steep adversc prcssure gradient on the upper surface of the airfoil causa the flow to separate nearly at thc lcading edge, and a very large wakc is rormed (Figurc 15.27) The lift coefficient drops suddenly, and thc wing

is said to s/ull Beyond thc stalling incidcnce the lift cocfficient levels off again and remains at aO.74.8 for rairly large anglcs of incidencc

The maximum liR coefficient dcpcnds largely on the Reynolds number Re At lower values ofRe - 105-1 Oh, the flow separatcs before the boundary layerundergocs

transition, and a very large wake is formcd This givcs maximum lift cocfficients t0.9

At largcr Reynolds numbers, say Re > lo7, the boundary layer undergoes transition

to turbulent flow before it separatcs This produces a somewhat smaller wakc, and

maximum lift coefficients of =z 1.4 are obtained

The angle of attack at zero lifi, denoted by -b here, is a function of the scclion camber (For a Zhukhovsky airfoil, b = 2(camber)/chord.) The effect of increasing the airfoil camber is to raisc the entire graph of CL vs a, thus increasing thc maximum values of CL without stalling A cambcrcd profile dclays stalling csscntially becausc

654 Aed-mumics

Figure 15.26 Ut and drag codkients vs angle of attack

F@re 15.27 Stalling of an airfoil

its leading edge points into the airstream while the rest of the airfoil is inclined to the

stream Rounding the airfoil nose is very helpful, for an airfoil of zero thickness would undergo separation at the leading cdge Trailing edge flaps act to increase the camber when thcy are deployed Then the maximum lift coefficient is increased, allowing for lower landing speeds

Various terms are in common usage to describe the different components of the drag The total drag of a body can be divided into africrion drug due to the tangential stresses on the surface and pressure drag due to the normal stresses The pressure drag can be furthcr subdivided into an induced drag and afiwm drag The induced drag is Lhc “drag due to lift” and results from the work done by the body to supply the kinetic energy of the downwash field as the trailing vortices incrcase in lcngth The form drag

is defined as thc parc of the total pressure drag that remains a h the induced drag is subtracted out (Sometimes the skin friction and form drags are grouped together and

called the projfe drug, which rcpresents the drag due to the “profile” alone and not

due to the fmitcness of the wing.) The form drag depends strongly on h c shape and

orientation of the airfoil and can be minimized by good design In contrast, relatively little can be done about the induced drag if the aspect ratio is fixed

Normally thc induced drag constitutes the major part of the total drag of a wing

4s Coi is ncarly proportional to Ci, and CL is nearly proportional to a, it rollows

that Coi oc a2 This is why the drag cwfficient in Figure 15.26 seems to increase quadratically with incidence

For high-spced aircraft, the appearance of shock waves can adversely affect the behavior of thc lift and drag characteristics In such caqes the maximumJlow speeds can be cbsc to or higher than the speed of sound even when the aircraft is flying at subsonic speeds Shock waves can form when thc local flow speed exceeds the local specd dsound To reduce their effect, the wings are given asweepbackangle, as shown

in Figure 15.2 The maximum flow speeds depcnd primarily on the component of the

oncoming stream perpendicular to the leading edge; this component is rcduced as a

result of the sweepback As a result, increased flight speeds are achievable with highly swept wings This is particularly true when the aircraft fits at supersonic speeds, in which there is invariably a shock wave in rmnt of the nose of the fuselage, extending downstream in the €om of a cone Highly swept wings are h e n used in ordcr that the wing does not pcnetrate this shock wave For flight spceds exceeding Mach numbers

of order 2, thc wings have such large sweepback angles h a t they resernblc the Greek

letter A; thcse wings are somctimes called delta wings

13 Pmpulxive Mechnniumw of’l+ish and B i d s

The propulsive mechanisms or many animals utilize the aerodynamic principle of lift

generation on winglike surfaces We shall now describe some of the basic ideas of

this interesting subject, which is discussed in more detail by Lighthill (1986)

Locomotion of Fish

First consider the caqe of a fish It develops a forward thrust by horizontally oscillating its tail fmm side tu side The tail has a cross section resembling that of a symmctric airfoil (Figure 15.28a) One-half of the oscillation is represented in Figure 15.28bb, which shows the top view of tbe tail The sequence 1 to 5 represents the positions of the tail during the tail’s motion to the left A quick change of orientutiun occurs at one extreme position of thc oscillation during 1 to 2; the tail then moves to the Icft during 2 to 4, and another quick change of orientation occurs at the othcr extreme during 4 to 5

Suppose the tail is moving to the left at speed V, and the fish is moving forward

at speed U The fish controls thesc magnitudes so that the resultant fluid velocity U,

(relative to the tail) is inclined to the tail surface at a positive “angle of attack.” Thc

resulting lift L is perpendicular to U, and has a [orward component L sin 8 (It is casy

to verify that there is a similar forward propulsive force when h e tail moves from IcIt

to right.) This thrust, working at the rate U L sin 8 , propels the fish To achieve this propulsion, the tail of thc Esh pushcs sideways on the water against a force of L cos 8 ,

which rcquires work at the ratc VLcosO As V / U = tan0, idcally the conversion

or energy is perfect-all of thc oscillatory work done by the fish tail goes into the

(b) Top view of tail motion Figure 15.28 Propulsion of fish (a) Cross section of the Pail along AA is a symmetric airfoil Fivc

positions of Ihc tail during its motion 10 the left lirc shown in (b) The lin force I, is normal to the resulkml speed U, of water with respect 10 the tail

translational mode In practice, however, this is not the case because of the presence

of induced drag and other effccts that generate a wake

Most fish stay afloat by controlling the buoyancy of a swim hladdcr inside their stomach In contrast, some large marinc mammals such as whales and dolphins develop buth a forward thrust and a vertical lift by moving their tails vem'cally

They arc able to do this bccause thcir tail surface is horizonrul, in contrast to thc

vertical tail shown in Figure 15.28

night of Birds

Now consider the flight of birds, who flap their wings to gencrate horh the lift to support their body weight and the forward thrust to overcome h c drag Figurc 15.29

shows a vertical section of the wing positions during the upstroke and downstroke

of the wing (Birds have cambered wings, but this is not shown in the figure.) The angle of inclination of the wing with the airstream changes suddenly at the end of each stroke, as shown Thc important point is that the upstroke is inclincd at a greater angle

to the airstream than the downstroke As the figure shows, thc downstroke dcvelops a lift force L perpendicular to the ~ s u l t a n t velocity of thc air relative LO the wing Both

a forward thrust and an upward force result from the downstroke In contrast, very little aerodynamic force is developed during the upstroke, as the resultant vclocity

is then nearly parallel to the wing Birds thcreforc do most of the work during the

downstroke, and the upstroke is "easy."

14 LYuiling against h e M n d

People have sailed without the aid of an engine €or thousands of years and havc known how to arrive at a destination against the wind Actually, it is not possiblc

to sail cxactly against the wind, but: it is possiblc lo sail at ~ 4 0 4 5 ” to the wind Figurc 15.30 shows how this is made possible by the aerodynamic lift on the sail, which is a piece of large stretched cloth The wind speed is U , and the sailing speed

is V, SO that the apparent wind speed relative to the boat is U, II the sail is properly orientcd, this givcs rise to a lift force perpendicular to U, and a drag force parallel to

UT The rcsultant forcc F can be rcsolved into a driving component (thrust) along the motion of the boat and a lateral component The driving component performs work

in moving the boat; most of this work goes into overcoming the frictional drag and

in generating the gravity waves that radiate outward The latcral componcnt does not cause much sideways drift because of the shape of the hull It is clcar that the thrust decrcases as thc angle 0 dccrea9es and normally vanishes whcn 0 is ~40-45’ The energy for sailing comes from the wind field, which loses kinetic energy aftcr passing througb thc sail

In the foregoing discussion we havc not considered the hydrodynamic forces cxerted by the water on the bull At constant sailing spccd the net hydrodynamic ibrce must bc equal and opposite to thenei aerodynamic force onthe sail The hydrodynamic force can be dccornposed inlo a drag (parallel to the dirccrion of motion) and a lift Thc lift is provided by the “keel,” which is a thin vcrlical surface extending

downward from the bottom 01 the hull For thc keel to act as a lifting surfacc, the longitudinal axis or the boat points at a small angle to thc direction o€ motion or the

boat, as indicatcd near thc bottom right part of Figure 15.30 This “angle of attack”

658 Aennly7uamicmr

Fiprc 15.30 Principlc ora sailboat

is generally <3" and is not noticeable The hydrodynamic lift developed by the keel opposes the aerodynamic lateral force on the sail It is clear that without the keel the latcral aerodynamic force on the sail would topple the boat around its longitudinal axis

To arrive at a destination directly against the wind, one has to sail in a zig-zag path, always maintaining an angle of %45" to the wind For example, if the wind is corning from the east, we can fist proceed northeastward as shown, then change the orientation of the sail to proceed southeastward, and so on In practice, a combination

of a number of sails is used for effective maneuvering The mechanics of sailing yachts is discussed in Herreshoff and Newman (1966)

l!kCfViSt?#

1 Consider an airfoil section in the xy-plane, the x-axis being aligned with the

chordline Examine the pressure forces on an element ds = (dx, dy) on the surface, and show that the net force (per unit span) in the y-direction is

Fy = - lc pu dx + l f ' p l d x ?

where pu and pl are the pressures on the upper and the lower surfaces and c is the chord lenglh Show that this relation can be rearranged in the form

where C , = (p - p m ) / ( $ p V 2 ) , and the integral represents the m a enclosed in a

C , vs x / c diagram, such as Figure 15.8 Neglect shear stresses [Note that Cy is not

exactly thc lift coefficient, since the airstrcam is inclined at a small angle a with the

x-axis.]

2 The measured pressure distribution over a scction of a two-dimensional airfoil

a1 4" incidcncc ha$ the following form:

Upper Surface: C , is constant at -0.8 from the leading edge to a distance equal to 60% or chord and then increases linearly to 0.1 at the trailing edge

Lower Sudace: C,, is constant at -0.4 from the leading edge to a distance

equal to 60% of chord and then increases lincarly to 0.1 at the trailing edge Using the iesul ts of Exercise 1, show that the lift coefficient is nearly 0.32

3 The Zhukhovsky transformation z = [ + h 2 / [ transforms a circle of radius

h, centcrcd at the origin o€ the (-plane, into a flat plate of length 4h in the z-plane The circulation around ihe cylinder is such that the Kutta condition is satisfied at the trailing edge ofthe flat plate If the platc is inclined at an angle a to a uniform stmam

U , show that

(i) The complex velocity in the [-plane is

where r = 4ic U h sin a Notc that this represcnts flow 0 \ 7 e r a circular cylinder with circulation, in which thc oncoming velocity is oriented at an angle a

(iij The velocity components at point P (-2b, 0) in the (-plane arc [iU cosa,

(iii) The coordinates of the transformed point P' in the xy-plane arc [ - 5 h / 2 , 0 ]

(iv) Thc velocity components at [-5h/2? 01 in the xy-plane are [U cos a, 3U sin a]

4 In Figure 15.13, the angle at A' has been markcd 2p Prove this [Hinr : Locatc

5 Consider a cambered Zhukhovsky airfoil determined by h e following thc center of thc circular arc in the z-planc.]

6 A thin Zhukhovsky airfoil has a lift coefficicnt or 0.3 at zcro incidence What

is thc lili coefficient at 5'' incidence?

7 An untwisted elliptic wing of 20-m span supports a weight of 80,000N in a levcl flight at 300 km/hr Assuming sea level conditions, find (i) the induced drag and (ii) the circulation around sections halfway along each wing

8 The circulation across the span of a wing follows the parabolic law

Andmon, John D., Jr (1991) Fundamenfals of Aerodynamics, New York McGraw-Hill

Anderson, John D., Jr (1998) A History oJAemdynamics, London: Cmbridgc University Press

Batchclor G K (1 967) An Znrmducrion f a Fluid Dynmics, London: Cambridge Univcrsity Prcss

Kmrncheti, K (1980) Principles rJfIdeal-Fl~~Ul Aerodynamics, Melhournc, FL: Kriegcr Publishing Co

Kuclhc, A M and C Y Chow (1998) Foundafians of Aendynamics: Basis of Aerodynamic Design,

Ncw York: Wilcy

Pmndtl, I, (1952) Essentials rJfFluid Dynamics, London: Blackie & Sons Ltd (This is the English edition

of the original German cdilion It is vcry easy to understand, and much or it is still relevant today.) Printcd in New York by H a h r Publishing Co If this is unavailable sec the following reprints in paperhack hat contain much if no1 all ofthis material:

Prandtl, L and 0 G actjens (1934) [original puhlication date) Fundamentals of Hydro and Aemme- chanics, New York Dover Puhl Co.; and

Prandil, L and 0 G Tietjens (1934) loriginal publication daic] Applied Hydni and Aemmechnic.s

New York: Dover Puhl Co This contains many original flow pholopphs from Prandtl's labmaiory

Chapter 16

Compressible Flow

1

2

3

4

5

6

1

Critt:nori lor R'cglctx of (~oqmsihility

(~lr~hdication of Cor~iprcssiblc

l h v 3 663

I :S!ful Thcrmodyruurii: Hclntions 664

S p d OJSOIJJUL 665

0ne-l)iinc.nriorrrrl llow 667

Comiriiuty I.ipariori 668

Eric:~p I :(pition 668

T h i i d l i and Euler Equations 669

ErrHm:ts 662

h i c I:+itu)ri.sji)r 8 * MmeIitiim I?iriciple for a C O I I ~ ) ~ 9 M i m c 670 10 S k y w i h r i wid So n.ir I'm p d c s 67 I kt)lc 16.1: 1xiim)pii: I:lo~ of Brft:ct Cas ( y = I 4) 673

.4ni ci-lhloci[y lkl(hw.9 in 11 12 (~ri(?-~~irri~n~ion(il Iwrihnpir b'lou 676

Exrirriplc 16.1 679

rVorn7d Shock Mire 680

Normal Shock I'rnpriptiug iii R Stdl \ic.diiirri 683

Slm:k Stmctiirc 684

I h k h w m 9 685

( : o r i \ c ~ ~ n t - l ~ i v : ~ ~ i ~ Soxxlc 685

Exumplc 16.2 687

lirblc 16.2: Ow-Dirnensiond N o r d Shock I(chrior~ ( y = 1.4) 689

~ & L S r f f i z k h n and Ihwtuig in CorLsturit-Am I~UCLS 690

I<ffcxi of I h i 'liansft s 693

Choking by 1;riction or Iknt Sdditiori 693 Much O h ! 694

OMque S titick U.irvc! 696

(kncratim of Ohliqiic Shtw:k U"(:s 698

'rh(: Vi& Shock Iiruit 699

Expunsion and (!onymxsion in Siq)mwiic llou: 700

Tliiri 4i$il ' I heo or^ b7 S iq~i?m~)riic Fhu 702

l<G?~i.S~?S 704

Lihruhm t X d 705

Siqq~lmcri~al Ihufirig 706

coll\-elgent Yomlc 685

E I I ~ ~ or ~ i ~ l i ~ ) ~ ~ 691

To this point we havc neglected the elTects of density variations due to pressurc

changes In this chapter we shall examine some elementary aspwts or flows in which

the compressibility effects are important Thc subject of comprcssible flows is also

called gas dynamics which has wide applications in high-specd flows around objects

of cngineering interest These include extemalfluws such as those around airplanes

and internal Jhws in ducts and passages such as n o r ~ l e s and diffusers used in jet

66 1

engines and rocket motors Compressibility effccts are also important in astrophysics Two popular books dealing with compressibility effects in enginering applications are those by Liepmann and Roshko (1957) and Shapiru (1953), which discuss in fuurtbcr dctail most of the material presented here

Our study in this chapter will be rather superficial and elemcntq bwause this book is essentially about incompressible flows However, this small chapter on com- pressible flows is added because a complete ignorancc about compressibility effects

is rather unsatisfying Several startling and fascinating phenomena arise in compress- ible flows (especially in the supersonic range) that go against our intuition developcd

from a knowledge of incomprcssible flows Discontinuitics (shock waves) appear

within the flow, and a ralhcr strange circumslance arises in which an increase or flow

area accelerates a (supersonic) stream Friction can also make the flow go faster and adding heat can lower thc temperature in subsonic duct flows We will sec this latcr in

h i s chapter Some understanding of these phenomena, which have no counterpart in low-speed flows, is desirable even if the reader may not make much immediate usc of this knowledge Except for our treatment of friction in constant area ducts, we shall

limit our study to that of frictionless flows outside boundary layers Our study will, however, havc a great dcal of practical valuc becausc the boundary layers arc espe- cially thin in high-speed flows Gravitational effects, which are minor in high-speed flows, will be neglected

Criterion for Neglect of Compressibility Effects

Compressibility effects are determined by the magnitude of the Mach number

defined as

where u is the spced of flow, and c is the spccd of sound given by

wherc the subscript ‘Y’ signifies that the partial derivative is taken at constant cntropy

To see how Iargc the Mach number has to be For the comprcssibility effects to bc appreciable in a steady flow, consider the one-dimensional version of the continuity equation V (pu) = 0, that is,

h s s u r e changes can be estimated from the definition of c, giving

“Mach’s principle,” which states that propertics of space had no indepcndent exis- tencc but are dctennined by the mass distribution within it Strangely, Mach never

acceptcd either thc theory of relativity or the atomic structure of matter.)

Classification of Compressible Flows

Compressible flows can be classificd in various ways, one of which is based on the Mach numbcr M A common way of classifying flows is as follows:

(i) IncompressibleJEow: M < 0.3 cverywhcre in the flow Density variations duc

LO pressurc changes can be ncglected The ga medium is compressible but the density may be regarded as constant

(ii) SubwnicJow: M exceeds 0.3 somcwhere in the flow, but does not cxceed I

anywherc Shock waves do not appear in the flow

(iii) T‘unsonicfluw: Thc Mach number in thc flow lics in the rangc 0.8-1.2 Shock

wavcs appear and lead to a rapid increasc of the drag Analysis or transonic flows is difficult because the governing equations are inhcrcntly nonlinear, and also because a separation of the inviscid and viscous aspccts of thc flow

is orten impossiblc (The word “transonic” was invented by von Karman and

Hugh Dryden, although thc latter argued in favor of having two s’s in the word

664 ( ? ~ ~ n p ~ v ~ i b l e Mtiw

von Karman (1954, p 116) stated hat “T first introduced the term in a report to the U.S Air Force I am not sure whether the general who read the ~ 7 0 r d knew what it meant, but his answer contained the word, so it seemed to be oficially accepted.”)

(iv) SupersonicJlow: M lies in the range 1-3 Shock waves are generally prcsent

In many ways analysis of a flow that is supersonic everywhere is easier than an analysis of a subsonic or incompressible flow as we shall see This is because information propagates along certain directions, called characteristics, and a determination of these directions greatly facilitates the compuktion of the flow

IiCld

(v) HypersonicJlow: A4 > 3 The very high flow speeds cause severe heating in

boundary layers, resulting in dissociation of molecules and other chemical effects

Useful Thermodynamic Relations

As density changes are accompanied by temperature changes, thcrmodynamic prin- ciples will be constantly used here Most of the necessary concepts and relations have been summarized in Sections 8 and 9 of Chapter 1, which m a y be reviewed before proceeding further Some of the most frequently used relations, valid for a perfect gas with constant specific heats, are listed here for quick reference:

Equation of stute p = p R T , Internul energy e = C,T, Enthalpy h = C,T,

Y R

y - I ’

C , - C , = R , Speed o f sound

Some important propertics or air at ordinary temperatures and prcssures are

We know that a pressure pulse in an incompressible flow behaves in the same way

as that in a rigid body, where a displaced particlc simultaneously displaces all the

particles in the medium The effects of pressure or other changes are therefore instantly

felt throughout the mcdium A comprcssible fluid, in contrast, bchaves similarly to

an elastic solid, in which a displaced particle compresses and increases the density of

adjacent particles that move and increasc the density of the neighboring particles, and

so on In this way a disturbance in the form of an elastic wave, or a pressure wave,

travels through Lhe medium The speed of propagation is faster when the medium is

more rigid If thc amplitude ofthe elastic wave is infinitesimal, it is callcd an acoustic

wave: or a sound wave

We shall now find an cxpression for the speed o i propagation of sound

Figure 16 l a shows an infinitcsimal pressurc pulse propagating to the l d t with speed c

into a still fluid The fluid properties ahead ofthe wave are p, T, and p , while the flow

speed is u = 0 The properties behind the wave arc p + d p , T + d T , and p + dp,

whereas the flow speed is du directed to the left Wc shall see that a “compression

wavc” (for which the fluid pressure rises after the passage of the wave) must movc

the fluid in the dircction of propagation, as shown in Figure 16.la In contrast, an

“expansion wave” moves the fluid ”backwards.”

To make the analysis steady, we superimpose a velocity c, dirccted to h c right,

on the entire system (Figure 16.lb) The wave is now stationary, and the fluid enters the wave with velocity c and leaves with a velocity c - du Consider an area A on

the wavefront A mass balance gives

where viscous stresses have been neglected Herc, Apc is the mass €low rate The first

term on the right-hand side represents the rate of outflow of x-momentum, and the second term represents the rate af i d o w of x-momentum Simplifying the momentum equation, we obtain

the irreversible entropy production is proportional to the squures of the velocity and

temperature gradients (see Chapter 4, Section 15) and is therefore negligible for weak waves The particles do undergo small temperature changes, but the changes

are due to adiabatic expansion or compression and are not duc to heat transfer from the neighboring particles The entropy of a fluid particle then remains constant as a weak wave passes by This will also be demonstrated in Section 6, whcre it will be shown that the entropy change across the wave is dS a ( d ~ ) ~ , implying that dS goes

to zero much faster than the rate at which the amplitude d p tends to zero

It follows that the derivative dp/dp in Eq (1 6.8) should be replaccd by the partial dcrivative at constant cntropy, giving

(16.9)

For a perfect gas, the use of p/pY = const and p = p R T reduces the speed of sound

( 16.9) to

(1 6.10)

For ah at 15 “ C , this gives c = 340m/s We note that the nonlinear terms that we

have ncglccted do change thc shape of a propagating wavc depending on whether it

is a compression or expansion, as follows Because y > 1, the isentropic relations show that if dp > 0 (compression), thcn d T > 0: and from Eq (16.10) the sound

speed c is increascd Therefore, the sound speed behind thc h n t is gmater than that

at the front and the back of the wave catches up with the front of the wave Thus the wave stcepens as it travels The opposite is true [or an cxpansion wave, for which

d p < 0 and dT < 0 so c decreases The back of the wave falls farther behind the front so an cxpansion wave flattcns as it travels

Finite amplitude waves, across which there is a discontinuous change of pressure, will bc considcrcd in Section 6 These are called shock wuves Tt will be shown that the finitc waves are not iscntropic and that thcy propagate through a still fluidfuster than thc sonic spccd

The first approximate cxpression for c was found by Newton, who assumed that

dp was proportional to dp, as would be truc if the process undergone by a fluid particle was isothermal In this nianner Ncwton arrived at thc expression c = m

He attributed the disc~pancy of this formula with expcrimental measuremcnts as duc to “unclcan ak.” The science of thcrmodynamics was virtually nonexistcnl at the timc, so that the idea of an iscntropic process was unknown to Newton The correct cxpression for the sound s p e d was first givcn by Laplace

3 llusic I?quatir,nsfiw Oni?-l)irni?mional Flow

In this section we begin our study of certain compressible flows that can bc analyzcd

by a one-dimcnsional approximation Such a simplification is valid in flow lhrough a duct whose ccnterlinc does not have a largc curvature and whose cross section does not vary abruptly The overall behavior in such flows can hc studied by ignoring the variation of velocity and other properties across the duct and replacing thc properly distributions by their avcrage values ovcr the cross section (Figurc 16.2) The arca or

the duct is taken as A ( x ) , and the flow propertics are taken a p ( x ) , p ( x ) , u ( x ) , and

so on Unsteadiness can be introduced by including 2 as an additional independent variable Thc forms of the basic equations in a one-dimensional compressible flow are discussed in what follows

Consider a control volume within the duct, shown by the dashed line in F i p 16.2

The first law of thermodynamics for a control volume fixed in space is

where u2/2 is the kinetic energy per unit mass The first term on thc left-hand side rep- resents the rate of change of “stored energy” (the sum of internal and kinetic energies) within the control volumc, and the second term represents the flux of encrgy out of the control surface The first term on thc right-hand side represents the rate of work done

on the control surface, and the second term on the right-hand side repwents the hcat

input through the control surface Body forccs havc been neglected in Eq (16.12) (Here, q is the heat flux per unit area per unit time, and dA is directed along the outward normal, so that 1 q - d A is the rate of ourJow of heat.) Equation (16.12) can

easily be derived by intcgrating the differential form given by Eq (4.65) ovcr the control volume

Assume steady state, so thal the first term on thc left-hand side of Eq (16.12) is zero Writing ri = plul A , = p p ~ A 2 (where the subscripts denote sections 1 and 2)

thc second term on the left-hand sidc in Eq (16.12) gives

The work donc on thc control surfaces is

ujrijdAj = u l p l A l - U Z P ~ A ~

J

Herc, we havc assumcd no-slip on the sidewalls and €rictional stresses on thc endfaces

1 and 2 arc: negligible The rate of heat addition to h e control volumc is

Thisis thcenergy cquation, which is validevenifthcre are frictional or nonequilibrium

conditions (e.g., shock waves) between scctions 1 and 2 It is apparent that thc sum u j enthalpy and kinetic eneQxy rem.ains constant in an udiahaticjuw Therefore, enthalpy plays the same rolc in a flowing system that internal energy plays in a nonflowing

system Thc differcnce between thc lwo types of systems is IheJlOw work p u izquircd

to push matter across a section

Bernoulli and Euler Equations

For inviscid flows, the steady form of the momcnlum cyuation is the Euler equation

(16.16)

Tntegrating along a streamline, we obtain the Bernoulli equation for a compress- ible flow:

.I-uz + J = const.,

which agrees with Eq (4.78)

equation To see this, note that this is an isentropic flow, so that the T dS equation

For adiabaticfrictionless flows the Bemuulli equation is identical to the energy

which is identical to the adiabatic [om of h e energy equation (16.15) The collapse

of the momentum and energy equations is expected because the constancy of entropy has eliminated one of the flow variablcs

Momentum Principle for a Control Volume

If the centerline of the duct is straight, then the sleady form o€ the momentum principle

for a finite control volume, which cuts across the duct at sections 1 and 2, gives

piAi - mA2 + F E fiuZA2 - piu;Ai, (16.18)

wherc F is the x-component of the rcsultant force exerted on the fluid by thc walls The momentum principle (16.18) is applicable even when there are frictional and

dissipative processes (such as shock waves) within the control volume:

If frictional processes are absent, then Eq (16.18) reduces to the Eu1e.r equa- tion (16.16) To see this, consider an infinitesimal area change between sections l

and 2 (Figure 16.3) Thcn the averagc pressure exerted by the walls on the control surface is ( p + i d p ) , so that F = d A ( p + s d p ) Then Eq (16.1 8) bccomes

pA - ( p + d p ) ( A + d A ) + ( p + i d p ) d A = puA(u + d u ) - &A,

where by canceling terms and neglecting second-order terms, this Educes to the Euler cquation (16.16)