KUNDU Fluid Mechanics 2 Episode 7 pdf

Iritwnal Mimes in a Continmu&- L9rati@d IYuid In this chapter we have considered gravity waves at the surface or at a density dis- continuity; these waves propagate only in the horizont

We shall see in the next section that it has the significance of being the frequency of

oscillation if a fluid particle is vertically displaced

Afier substitution of Eq (7.136), the equations of motion (7.130)-(7.134) become

In deriving thc equations for a stratified fluid, we have assumcd that p is a

function of temperature T and concentration S of a constituent, but not of pressurc

At first this docs not seem to be a good assumption The compressibility effects in the atmosphere are ccrtaiuly not negligible; even in the ocean the density changes due to the huge changes in the background pressure are as much as 4% which is 4 0 times the density changes due to the variations of the salinity and temperature The effects

of compressibility, however, can be handled within the Boussinesq approximation if

we regard p in the defiilition of N as the background potential density, that is the

density distribution from which the adiabatic changes of density due to the changes

of prcssure have been subtracted out The concept oi potential dcnsity is explained

in Cliaptcr 1 Oceanographers account for compressibility effects by converting all their density measurements to thc standard atmospheric pressure; thus, when they

report variations hi density (what thcy call “sigma lee”) they arc generally reporting

variations due only to changes in temperature and salinity

A useful equation for stratified flows is the one involving only U L The u and li

can be eliminated by taking the time derivative of the continuity equation (7.144) and using the horizontal momentum equations (7.140) and (7.141) This givcs

(7.145)

where Vi

p’ from Eqs (7.142) and (7.143) gives

a’/ax2 + a 2 / a y 2 is the horizuntal Laplacian operalor Elimination o€

(7.14)

Finally, p’ can be eliminated by laking Vi of Eq (7.146), and using Eq (7.145) This gives

which can bc written as

(7.147)

where V’ a 2 / ~ x ’ + a 2 / a y 2 + iI2/3z2 = Vi + a2/az2 is the three-dimensional Laplacian operator The w-equation will be used in the following section to dcrive the dispersion relation for internal gravity wavcs

19 Iritwnal Mimes in a Continmu&- L9rati@d IYuid

In this chapter we have considered gravity waves at the surface or at a density dis- continuity; these waves propagate only in the horizontal direction Because every horizontal direction is alike, such waves are isotmpic, in which only the magnitude

of thc wavenumber vector matters By taking tbe x-axis along the direction of wave

propagation, we obtained a dispersion relation w(k) that depends only on the m g - nitude of the wavcnumber We found that phases and groups propagate in the same direction, although at Merent spccds I€, on the other hand, the fluid is continuously

suatified, then the internal wavcs can propagate in any direction, at any angle to the vertical Tn such a case the direction of the wavenumber vector becomes important Consequently, we can no longer treat the wavenumber, phase velocity, and group velocity as scalars

a2 -v2w + N’VAW = 0, at1

Any flow variable q can now be written as

i(k.r+/p+nc-cut) - i ( K x-cut)

SI = 9 0 e - 90e

where 40 is the amplilude and K = ( k , I , m) is the wavenumber vector with com- ponents k , 1 , and m in the three Cartesian directions Wc cxpect that in this case the

direction of wavc propagation should matter becausc horizontal directions are basi-

cally differcnt from the vertical direction, along which the all-important gravity acts Internal waves in a continuously stratified fluid therefore nnisotmpic, for which

the fresuency is a function of all three components of K This can be written in the following two ways:

w = ~ ( k 1 m) = o(K) (7.148)

However, the waves are still horizontally isotropic because thc dependence of the wave field on k and I is similar, although the dependence on k and ni is dissimilar

The propagation of internal waves is a bamlinic process, in which the surfaces of

constant pressurc do not coincidc with the surfaces of constant density It was shown

in Section 5.4, in connection with the demonstration of Kelvin’s circulation theorem,

that baroclinic proccsses generate vorticity InteinaX waves in a continuously stratiJied jluid are therefam not iirotutional Waves at a density interface constitute a limiting cme in which all the vorticity is concentrated in the form of a velocity discontinuity

rrt the integace The Lrrplace equation can ther-efoiv be used bo describe thejiowfield

within each Zayx However; internal waves in a continuous1.y szrutijied jhid cnnnot

be described by the Lpluce equution

The first taqk is to derive the dispersion relation We shall simplify the analysis

by assuming that N is depth indcpendent, an assumption that may seem ~mmdistic at fist hi the ocean, for example, N is large a1 a depth of %200 in and small elscwhere

(see Figure 14.2) Figmz 14.2 shows that N .e 0.01 evcrywhere but N is largest between ~ 2 0 0 m and 2km However, Ihc results obtained by treating N as constant

;ire locdly valid if N varies slowly over the vcrtical wavelength 25r/ni of the motion The so-called WKB approximation of internal waves, in which such a slow variation

of N ( z ) is not neglected, is discussed in Chaptcr 14

Consider a wave propagating in three dimensions, for which the vertical vcloc- ity is

= u,,, ei(k.r+ly+m~-wr) (7.149) where tug is the amplitude of fluctuations Substituting into the governing equation

a2

-V2w + N'VAW = 0 , at'

gives the dispersion relation

(7.147)

(7.150)

For simplicity of discussion we shall orient the xz-plane so as to contain the wave- number vector K No generality is lost by doing this because the medium is hoii- zontally isotropic For this choice of referencc axes wc have 1 = 0; t h a t is, the wave motion is two dimensional and invariant in the y-direction, aid k rcpresents the eiilirc horizontal wavcnumber We can then write Eq (7.150) as

Iequency : N is the mmimirm possible fi-equeiicy if iiiteinul waves in a strutified.fluid

Before discussing the dispersion relation further, let LIS explore particle motion

in an incompressible internal wave Thc fluid motion can be written as

(7.153)

= ug ei(kx+l.v-mz nrt)

showing that pcirticle motion isperpeizdicular to the wuvenzmber vector (Figure 7.32)

Note that- only two conditions have been used to derive this result, namely the incom-

pressible continuity equation and a trigonometric behavior in ull spatial directions As

such, thc rcsult is valid for many other wavc systems that meet these two conditions These waves are called shear wuves (or transverse waves) because the fluid moves

parallcl to the constant phase lines Surface or interfacial gravity wa17es do not have

this property because the field varies exponenriafly in the vertical

We can now intcrpret 8 in thc dispersion relation (7.152) as the angle bctween the

p h c l c motion and the vertical direction (Figure 7.32) The maximum frequency w =

N occurs when 8 = 0, that is, when the particles move up and down vertically This

case cornsponds lo m = 0 (sce Eq (7.15 1)) showing that the motion is independent

of the z-coordinate Thc resulting motion consists of a series of vertical colurmis, all oscillating at the buoyancy frcquency N , the flow field varying in thc horizontal direction only

Figure 7.33 Blocking in shrmgly sb-&licd flow Thc circular region represents a two-dimensional body

with its axis along Ihe y direction

The w = 0 Limit

At the opposite cxtreme we have w = 0 when 8 = x / 2 , that is, when the particle motion is completely horizontal In this limit our inkrnal wave solution (7.151) would

seem to rcquire k = 0, that is, horizontal independcnce of the motion However, such

a conclusion is not valid; purc horizontal motion is not a limiting ca$e of internal waves, and it is necessary to examine the basic equations to draw any conclusion lor

this case An examination of the governing set (7.1.40)-(7.144) shows that a possible

steady solution is w = p' = p' = 0, with u aid v any functions of r and y satisfying

(7.155)

The z-dependence of u and v is arbitrary The motion is thercfore two-dimensioixd

in the horizontal plane, with the motion in the various horizontal planes decoupled from each othcr This is why clouds in the upper atmosphere seem to move in flat horizontal sheets, as often observed in airplane flights (Gill, 1982) For a similar

I-eason a cloud pattern pierced by a mountain peak soinetimes shows Kurman vurrex streets, a two-dimensional feature; see ihe striking photograph in Figure 10.18 A i-eslriction of strong stratification is necessary for such almost horizontal flows, for

Eq (7.143) suggests that the vertical motion is small if N is large

The forcgoing discussion leads to the interesting phenomcnon of blocking in

a strongly stratified fluid Coiisidcr a two-dimensional body placed in such a fluid, with its axis horizontal (Figure 7.33) The two dimensionality of the body requires

a v / 8 y = 0, so that Ihc continuity Eq (7.155) rcduces to au/ax = 0 A horizontal layer of fluid ahcad af thc body, bounded by tangcnts abovc and below it, is therefore blocked (For photographic evidencc see Figure 3.18 in the book by ?inner (1973).)

This happens bccause lhc strong stratification suppresses the M: ficld and pi-events the fluid horn going around and over thc body

In the casc of isotropic gravity wavcs at a free surface and at a density discontinuity,

we found hat c and c, are in the same dircction, although their rnagnitudcs can bc diflerent This couclusioii is no longer valid for thc anisoiropic intcrnal wavcs in a continuously stratified fluid In fact, as we shall see shortly, lhcy are peipendiculur to

each olhcr, violating all our intuitions acqilired by obscrving surface gravity waves!

In three dimensions, the dcfinition cg = dw/dk has to be generalized to

whmc K/ K represents the unit vcctor in the direction of K (Note that c # i, (m/k) +

iL(w/ntj, as explained in Section 3.) It lbllows fromEqs (7.157) and (7.158) that

(7.159)

showing thal phase and group velocity veclors are peipeidiculai:

Equations (7.157) and (7.1 58) show that the horizontal components of c and cg

are in the same direction, while thcir vcrtical components are equal and opposite In fact, c and cg form two sides of a right-angled triangle whose hypotenuse is horizontal (Figuie 7.34) Consequently thc phase velocity has an upward component when thc

p u p velocity has a downward component, and vice versa Equations (7.154) and

(7.159) are consistent because c and K are parallel and cg and u are parallel The fact that c and cg arc pcrpendicular, and havc opposite vertical components, is illustrated in Figure 7.35 It shows that the phasc lines are propagating toward the left and upward, whereas the wave groups are propagating to the left and downward Wave cmsts are constantly appearing at one cdge 01 the group, propagating through the g~vup, and vanishing at the other cdge

The group velocity here has the usual significance of being the velocity ofprop-

agation of energy of a certain sinusoidal Component Supposc a source is oscillating

at 1requcncy w Thcn its energy will only be found radially outward along four beams

C

Figure 7W Oricnhtion o f p h c m d gmup velocity in inkriirl wavcs

Figurr! 7.35 Illustration of phase and group propagation in internal waves Positions of a wave group at

two timcs are shown Thc phase line PF’ at time tl propagates to PP at tz

oriented at an angle 0 with the vertical, where cos 8 = o / N This has been verified

in a laboratory experiment (Figure 7.36) The source in this casc was a vertically oscillatiiig cylinder with its axis perpendicular to the planc of paper The €Teguency was w < N The light and dark lines in the photograph are lines of constant density,

inade visible by an optical technique The experiment showcd that the cnergy radiated

along four beams that became morc vertical as the frequency was increased, which agrees with cos0 = o / N

23 Knergy C‘omsideradiorix of lirlcinal Maues in a

In this section we shall derive the various cormnonly used expressions for potential energy of a continuously stratified fluid, and show that they are equivalent We then show that the encrgy flux p‘u is cg times the wave energy

A mechanical energy equation for internal waves can be derived from

Eqs (7.140)-(7.142) by multiplying the first equation by pou, the sccond by pov, the third by f i w , and summing the results This gives

$& + v 2 + w2) 1 + gp‘w + v (p’u) = 0 (7.160)

Hem the continuit yequation has beenused to write u ap’/ax+v ap’/iIy+w =

V (p’u), which reprcsents thc net work done by pressure €orces Another interpreta-

tion is that V - (p’u) is the divergence of the enerKyJIux p’u, which inust change the

21 Energy coneidcrniionv of Internal W- in a Stra@fkd Fkid 25 1

r

Figure 73 Waves generated in a stratified fluid of uniform buoyancy frequency N = 1 rad/s The

forcing agency is a horizontal cylinder, with its axis perpendicular to the plane of the paper, oscillathg

vertically at frequency w = 0.71 rads With w / N = 0.71 = cos8, this agrees with the observed angle of

8 = 45" made by the beams with the horizontal The vertical dark line in the upper half of the photograph is

the cylinder support and should be ignored The light and dark radial lines represent contours of constant p'

and are therefore constant phase lines The schematic diagram below the photograph shows the directions

of E and e, for the four beams Reprinted with the permission of Dr T Neil Stevenson, University of

Manchester

wave energy at a point As the first term in Eq (160) is the rate of change of kinetic

energy, we can anticipate that the second term gp’w must be the rate of changc of

potential energy This is consistent with the energy principle derived in Chapter 4 (see Eq (4.62)), except that pJ and p’ replace p and p because we have subtracted

the mean state of rest here Using the density equation (7.143), the rate of change of potential energy can be written as

(7.161)

which shows that the potential energy per unit volume must be the positive quan-

tity E,, = gZpR/2poN2 The potential energy can also be expressed in terms of the displacement f of a fluid particle, given by w = a ( / a t Using the density equation

infinitely deep fluids, for whichthe average potential energy of the entire wakrcolumn

per unit horizontal area was shown to be

gives Eq (7.164) Nok that for surface or interfacial waves Ek and E, represent

kinetic and potential energies o€ the entire water column, per unit horizontal area In

a continuously stratified fluid, lhcy represent energies p r unit volume

We shall now demonstrate that the average kinetic and potential energies are equal for internal wave motion Substitute periodic solutions

[u, U I , pJ, p’] = [i, 6, j j , 81 ei(kx+mz-cur)

Thcn all vari:tblcs can be expmsed in terms of M’:

avcrage potential ciiergy per unit volume is

(7.1 71 j

Using Eqs (7.157) and (7.170) group velocity times wave energy is

Nni K3

c,E = -[[i,m - i;k]

which reduces to Eq (7.171) on using the dispcrsion relation (7.1Sl) Tt follows that

(7.172)

I

I F=c,E

I

This result also holds for surface or intcrfacial gravity waves However, in that case

F reprcsents the flux per unit width pcrpendicular to the propagation direction (inte-

grated over thc cnlire depth), and E represents the energy per unit horizontal area In

Eq (7.1 72), on die othcr hand, F is die flux per unit m a , and E is the encrgy per unit volume

Lximises

1 Consider statioimy surface gravity waves in a rectangular container of length

L and breadth by containing water of undisturbed depth H Show that the velocity potenlial

#I = A cos(mrrx/L) cos(nrry/h) cosh k(z + H) e-irur,

satisfies V2#I = 0 and the wall boundary conditions, if

( r n ~ / L ) ~ + ( n ~ / b ) ~ = k’

Here in and n are integers To satisfy the free surface boundary condition, show that the allowable frequencies must be

J = g k t a n h k H

[Hint: combine the two boundary conditions (7.27) and (7.32) inlo a single equalion

2 This is a continuation of Exercise 1 A lake has the following dimensions

4 Plot the group velocity of surface gravity waves, including surface tension 0 ,

as a function of A Assuming deep water, show that the group velocity is

For water at 20 ’C ( p = 1000kg/m3 and 0 = 0.074N/m), veiiry thal

5 A r/wmcliize is a thin layer in the upper ocean ilcross which tcmperature and,

consequently, density change rapidly Suppose Lhc thermocline in n very deep ocean

is at a depth of lOOm €om the ocean surface, and that the temperalurc drops across

it froin 30 to 20’C Show thal the reduced gravity is g’ = 0.025 m/s2 Neglecting

Coriolis effects, show that the specd or pi-opagation of long gravity wavcs on such a

hennocline is 1.58 m / s

~ g n l i l l = 17.8 C ~ S

6 Consider internal waves in a continuously stratified fluid or buoyancy fre-

quency N = 0.02 s - ~ and average density 800kg/m3 What is the direction of ray

paths if the frequency of oscillation is OJ = 0.01 s-'? Find the energy flux per unit

area if the amplitude of vertical velocity is 6 = 1 c d s and the horizontal wavelength

is K meters

7 Consider jntcrnal waves at a dcnsiry interface bctween two infinitely deep

fluids Using the expressions given in Section 15, sliow that thc average kinetic energy

per unit horizontal m a is Ek = (p2 - p l ) g a 2 / 4 This result was quotedbut not proved

in Section 15

8 Considcr waves in a finite layer overlying an infinitely decp fluid discussed

in Section 16 Using the constants given in Eqs (7.116)-(7.119)1 prove the dispcrsion relation (7.120)

9 Solve the equation governing spherical waves i12p/ar2 = ( c 2 / r 2 ) ( a / 8 r ) ( f 2 J p / & ) subject to the initial conditions: p ( r 0) = e-r, ( 8 p / a r ) ( r , 0 ) = 0

l,itimzhw Cihd

Gill, A (: 982) Amospherr4wnn Dynamics, New York Acdcmic I'ress

Kinsman B l196.5) Wtnd Wairs, tinglewood CUTS, New Jersey: F'rciitic~Hall

LcBlond I? H and L A Mysak (1978) Wirves in the Ocean Amsterd;lm: Elscvicr Scientific Publishing Liepmmi, H W and A Roshko ( 1957) Elenienfs qf Guscfynamics New Ymk Wilcy

Lighthill M J (1978) Wuves in Fluidv London: Cmhridge Univcrsity Prcss

Phillips 0 M (1977) The Dynmiiics of ihe Upper Ocem, Imdim: Cambridge Uiiivelxity Prcss

Tunier J S (1973) &uiy.uncy Eflecfs in Fluids, Loiidon: Canibridgc Univcrsity h s s

Whitham, G U (1974) Linear undNonlinear WUIW New York Wiley

Chapter 6

Dynamic Similarity

1 Iri~rodrrction 256 7

2 Ahridinici~ioriiil Runrrielim

&6ermiri~~~nrri &$hrrUiat

E(pri/ions 251

3 I)irireinsii~iml !lki1rix 261

4 Biirkirgfim k Pi ‘1 ‘heorrm 262

5 Abrulirnaisioriol R t r r ~ m t m rind Ijyicunic Sirndari~ 264

hetlic7joii of h v i - Hchtivior 6nm Wmcnsiord Considrmitiom 265

6 (’iimmeii/.T im %fidel Zsting 266

l<xtunple 8.1 267

.Sigru/icrm-i: of C‘oiriinm ,~onr/irrirrL~ii~r~rrml hirnimc?~i?rs 268

I~cpioldr Yimhrx 268

Rniicle Yuiiibcr 268

Intcrd Fmiidt: Number 268

RiclmnLqm Aimher 269

Mwh N i u i h s 270

I’ra~idtl R’i~nhrx 270

ficrc&es 270

Litemlure Cited 270

Siipplementul Reading 270

1 Indimduelion

Two flows having different values of length scales, flow speeds, or fluid properties can apparently be different but still “dynamically similar” Exactly what is meant

by dynamic siinilarity will be explained later in this chapter At this point it is only necessary to know that in a class of dynamically similar flows we can predict flow

properties if we have experimental data on one of them In this chapiptcr, we shall

determine circuinstances under which two flows can be dynamically similar to one anohcr We shall see that equality of certain relevant nondimensional parameters is

a requirement for dynamic similarity What these nondimensional parameiers should

be depends on the nature of the problem For example, one nondimensional parameter must involve the fluid viscosity if the viscous effects are important in the problem The principle of dynamic similarity is a1 the hcart of cxperimental fluid mcchan- ics, in which the data should be unified and presented in tenns of nondimensional pmametcrs Thc concept of similarity is also indispensable for designing modcls in which tests can bc conducted for predicting flow propcrties of full-scale objecls such

as aircraft, submarines, and dams An understanding of dynamic similarity is also

important in theoretical fluid mechanics, cspecially when simplifications are to be

256

made Undcr various limiting situations certain variables can be eliminated from our consideration, rcsulting in very useful relationships in which only the constants need

to be determined from cxperiments Such a procedurc is used extensively in turbu- lence theory and leads, for example, to the well-known K - 5 / 3 spcctral law discussed

in Chapter 13 Analogous arguments (applied IO a different problem) are pmsented

in Section 5 of the present chapter

Nondhneiisional paranetem for a problem can be determincd in two ways They can be deduced directly from the governing di.lTerential equations if these equations arc known: this method is illustrated in thc next section If, on the other hand, the governing differential equations are unknown, then the nondimensional parametcrs

ca11 be detcrmined by pcr1ormhig a simple dimcnsional aiialysis on the variables involved This method is illustrated in Section 4

The fonnulattion of all problems in fluid mechanics is in tcrms of the conscrvation laws inlass, momentum, and energy), constitutivc equations and cquations of state

to define thc fluid, aid boundary c.onditions to spccify the problcm Most oftcn, the

conservation laws are written as partial diffcrcntial eqwitions and the conservation

of momentum and cnergy may include the constitutivc cquations for s t ~ e s s and heat flux, respectively Each tenn in the various equations has certain dimcnsions in terns

of units of rncasurements Of course, all or the tenns in any givcn equation must have

thc same dimcnsions Now, dimensions or units of measuiwncnt are human con- structs for our conveniencc No system d units has any inherent superiority over any other, despite lhc fact that in this text wc exhibit a preferencc for the units ordained

by Napoleon BoiiaparZe (of France) over those ordained by King Henry V U (of Englanclj The point here is that any physical problem must bc expressible in com- pletely dimcnsionless form Momover the parameters uscd to render the dependent and indepcndent variables dinlensionless must appear in the equations or boundary conditions One cannot define “refcrence” quantities that do not appcar in the prob- Icm: spurious dimensionless pararncters will be the result If the procedure is done properly, there will be a reduction in the parametric dcpendence of the formulation, gcnerally by the numbcr of bidepcndent units This is described i n Sections 3 and 4

in this chhaptcr The parametric reductioii is called a similitude Similitudcs greatly facilitate conelatioi~ of experimcntal data In Chapter 9 we will encounter a situation

in which thcre are no naturally occurring scales for length or time that can be used

to render the formulation of a particular problem dimnensionlcss As thc axiom that

a dimcnsionless formulation is a physical necessity still holds, we must look for a

dinicnsionless conibiiiation of the independent variablcs This rcsults in a contraction

of the dhnensionality of h e s p x c requircd for thc solution, that is, a rcduction by onc in the number of kdependent variblcs Such a reduction is called a similarity and

resu11.s in what is callcd a similarity solution

2 RloritCirrierixii~tcrrzcll I + ~ r - ~ i r t d i J r s 1Mwniiried.fiwri

To illuswitc the method of dcterininiiig nondimensional paramcters from h e gov-

erning diffcrcntial equations, consider a flow in which bolh viscosity and gravity are

important An exaniplc of such a flow is h e motion of a ship, whcre the drag experi- enced is cnuscd both by the gencration of surfacc waves and by friction on the surface

D~fim?nliul Kqualioris

258 Q?rtUrnk Sbriituri!y

ofthe hull All other effects such as surface tension and compressibility are neglected The governing differential equation is the NavierStokes equation

and two other equations for u and v The equation can be nondimensionalized by

defining a characteristic length scale 1 and a characteristic velocity scale U In the

present problem we can take 1 to be the length of the ship at the waterline and U

to be the free-stream velocity at a large distance from the ship (Figure 8.1) The choice of these scales is dictated by their appeamnce in the boundary conditions; U

is the boundary condition on the variable u and I occurs in the shape function of

the ship hull Dynamic similarity requires that the flows have geometric similarity

of the boundaries, so that all characteristic lengths are proportional; for example,

in Figure 8.1 we must have d / l = d l / l ~ Dynamic similarity also requires that the flows should be kinemutically similar, that is, they should have geometrically similar streamlines The velocities at the same relative location are therefore proportional,

if the velocity at point P in Figure 8 I a is U / 2 , then the velocity at the correspond- ing point PI in Figure 8.lb must be U1/2 All length and velacity scales are then pmporbional in a class of dynamically similarjows (Alternatively, we could take

the characteristic length to be the depth d of the hull under water Such a choice is,

however, unconventional.) Moreover, a choice of 1 as the length of the ship makes

the nondimensional distances of interest (that is, the magnitude o € x / l in the region around the ship) of order one Similarly, a choice of U as the frec-stream velocity

rnakes the maximum value of the nondimensional velocity zr/U of ordcr one For

reasons that will become more apparent in the later chapters, it is of value to have all

dimensionless variables of finite order Approximations m y then be based on any extreme size of the dimensionless parameters that will preface some of the terms Accordingly, we intmduce the following nondimensional variables, denoted by primes:

It is clear that the boundary conditions in terms of the nondimensional variables in

Bq (8.2) are independent of 1 and U For example, consider the viscous flow over a

circular cylinder of radius R We choose the vclocity scale U to be tbe free-stream velocity and the lcngth scalc to be the radius R In terms of nondimensional velocity

11' = u / U and the nondimensional coordinate T' = r / R the boundary condition at

infinity is II' + 1 as r' 3 00, and thc condition a1 the surface of the cylinder is

u I = O a t r ' = 1

There are instances where the shape fiinclion of a body may requirc two length scales, such as a lcnglh I and a thickness d An additional dimensionlcss parameter,

d / l would result to describc h e slenderncss of the body

Normalization, that is dimensionless represcnlation of thc pressure, depcnds on lhe doininant effect in the flow unless the flow is pressure-gradient drivcn In the latter case [or flow jn ducts or tubcs, the pressure should bc made dhncnsionless

by a characteristic prcssure difference in the duct so that thc dirnensionlcss teim

is finite Tn other cases when the flow is not prcssure-gradicnl driven, the pressure

is a passive variablc and should be normalized to balance the dominant effect in the flow Because pressurc enters only a gradicnl, the prcssive itself is not of consequencc; only prcssure differences are important The conventional practice is

to render y - pw dimensionless Dcpcnding on the nature or the flow, this could bc

in terms of' viscous stress , u U / l a hydrostatic pressure pgl, or ils in the preceding a dynamic pressure pU2

Substitution of Eq (8.2) into Eq (8.1) gives

The nondiniensional p,arameters U / / v and U / J $ have been givcn special

Both Rc and R have to bc equal for dynamic similarity of two flows in which both

viscous and gravitational effects are impoilant Notc that thc mere presence of gravity

does not make thc gravitational efkcts dyiainicdly important For flow around an object in a homogeneous fluid, gi-avily is important only if surface waves are gencrated

Othemisc, [he effccl of gravity is simply to add a hydrostatic pressure to lhe entire

system, which can be eliminated by absorbing gravity into the pressure lenn Under dynamic similarity h e nondimensional solutions are identical Thcrefore, the local pressure a1 point x = (x y , E) must be of the rorm

where (p - p x , ) / p U 2 is called the pressure coeficient Similar relations also hold for any other nondimensional flow variable such as velocity u/ U and acceleration

a l / U 2 11 follows that in dynamically similar flows the nondimensional local flow

variables arc identical at corresponding points (that is, for idcntical values of x / l )

In the foregoing analysis we have assumed that the imposed boundary conditions are steady However, we have retained the time derivative in Eq (8.3) becausc the

rcsulting flow can still be unsteady; for example, unstablc waves can arise sponta- ncously under steady boundary conditions Such unsteadiness must have a time scale

proportional to l / U , as msumed in Eq (8.2) Consider now a situation in which the

imposed boundsuy conditions are unsteady To be specific, consider an object having

a characteristic length scale 1 oscillating with a frequency w in a fluid at rest at infinity

This is a problem having an imposed length scale and an imposed time scale 1 / w In such a c a x a velocity scale can be derived from B and 1 to be U = 20 The preceding

analysis then goes through, leading to thc conclusion that Re = U l / u = wl'/u and

Fr = U / n = w f i have to be duplicated for dynamic similarity of two flows in which viscous and gravitational effects are important

All nondimensional qumtitics are identical for dyiiamically similar flows For Row around an immersed body, we can dcfiiie a nondimensional drag coemcient

where D is the drag expcrienced by the body; use of the factor of 1/2 in Eq (8.6) is conventional but not necessary Tnstead of writing CD in terms of a length scale I , it

is customarry to dcfiiie h c drag coefficient more generally as

D

p U 2 A/2 '

where A is a characteristic area For blunt bodies such as spheres and cylinders, A

is taken to be a cross section pcrpendicular to the flow Therefore, A = nd'/4 for a

sphcre of diameter d, and A = bd for a cylinder of di'meter d and length h, with the axis of the cylinder perpendicular to the flow For flow over a flat plate, on the other hand, A is taken to be the "wettcd area", that is, A = hf; here, 1 is the length of the

place in the direction of flow ,and b is thc width perpendicular to the flow

The values of the drag cocfficient CD are identical for dynamically similar Rows

In thc present example in which the drag is caused both by gravitational and viscous effects, we must have a functional rclation of the form

In many complicated flow problems the precise form of the differentid equations may not bc known I n this case the conditions for dynamic similiirity can be detcrmined

by means of a dimensional analysis of the variablcs involved A formal method of dimensional analysis is presented in the following section Here we introduce certain

ideas that are necded Ior performing a Iormal dimensional analysis

The underlying principle in dimensional analysis is that of dimensional homo- geneify, which statcs that all ternis in an equation must have the same dimcnsion This

is a basic check that we constantly apply when we derive an equation; if the t e r m do not have the samc dimension, then the equation is not correct

Fluid flow probleim without clcc~romagnetic forces ‘and chemical reactions involve only mcchanical variables (such as velocity and density) and thermal vari- ables (such as temperature and specific heat) The dimensions of all these vari- ables can be expresscd in terms of four basic dimensions-inass M, length L, time T, and tempcralure 8 We shall denote the dimension of a variable 4

by [ q ] For example, the dimension of velocity is [u] = L/T, that of pres-

sure is [p1 = [force]![area’l = MLT-*/L’ = M/LT’, and that of specific heal

is [C] = [energy]/[mass][tempcrature] = MLT-2L/MB = L2/8T2 When thermal effects are not considered, all variables can bc expressed in tcrms of tbree funcla- niental dimensions, namely, M, L, and T Tf tcmperature is considered only in coni- bination with Boltzmann’s constant ( k e ) or a gas constant ( R e ) , then the units of the combination are simply L2/T’ Then only the thrce dimensions M, L and T arc required

The method of dimensional analysis presented hcre uses the idca of a “dimen- sional matrix” and its rank Consider thc pressure drop Ap in a pipeline, which is

cxpected to & p e d on the inside diametcr d of the pipe, its length I , the average size

e of the wall roughness elemenls the average flow velocity U , the fluid density p,

and the fluid viscosity p We can write the Iunctional dependence as

Where wc have written the variables Ap d , on thc lop and their dimensions in a

vertical coliunn undcmeath For example, [Apl = ML-’T-2 An array of dinlensions such as Eq (8.10) is called a dimensional ntutrix The r-unk r of any matrix is defined

to be the size of the largest square submatrix that has a nonzero determinant Testing the determinant of tbe first three rows and columns, we obtain

l o ’

; 1 -3 -1 1-1 0 -1

commonly a nondimensioizalproduct (The symbol n is used because the nondimen- sional perainekr can be written as aproducrof the variables q1, q,,, raised to some power, as we shall sm.) Thus, Eq (8.1 1) can he written 5 ~ s a functional relationship

# ( I l l ? np, ? = 0 (8.12)

It will bc seen shortly that thc nondimensional parameters are not uniquc However,

(n - r ) of them are independent and form a coirplete set

The method of forming nondimensional parainetcrs proposed by Buckinghdm is

best illustrated by an example Consider again the pipe flow problcm expressed by

f W d , 1, e U P, 1.4 = 0, (8.13) whose dimensional matrix (8.10) has a rank of r = 3 Since there arc n = 7 variables

in the problem, the number of iiondimensional paramcters must bc - r = 4 We

= - I

first select any 3 (= I - ) of the variables as 'repeating variables", which we want to be repeated in all of our nondhneiisional parameters These repeating variables must have different dimensions, and among them must contain all the fundamental dimensions

M, L, and T In many fluid flow problems we choose a characteristic velocity, a characteristic length, and a fluid property as the repeating variables For thc pipe flow problem, let us choose U , d , and p as the repeating variables Although other choices would result in a different set of nondimcnsional products, we can always obtain other complete sets by combining the ones we have Therefore, any choice of the repeating variables is satisfactory

Each nondimensional product is formed by combining the three repeating vari- ables with one of the remaining variables For example, let the first dimensional product be taken as

The cxponents a , b, and c are obtained from the requirement that I l l is dimensionless This requires

I l l = Uadbp"Ap

Equating indices, we obtain a = -2, b = 0, c = - 1, so that

A similar procedure gives

U , d , and p as the repeating variables Whatever nondimensional groiips we obtain, only four of these arc independent for the pipe flow problem described by Eq (8.13) Howevcr, the set in 3 (8.14) contains the most commonly used nondimensional parameters, which have familiar physical interpretation and have been given spe- cial names Several of the common dimensionless paramcters will be discussed in Section 7

The pi theorem is a formal method of forming dimensionless groups With some cxperience it becomes quite easy to form the dimensionless numbers by simple

264 I)muinie Siittilmf!?-

inspection For example, since there are thi-ee length scalcs d, e, and I in Eq (8.1 3),

we can Form two groups such as e / d and l l d We can also form A p / p U 2 as our

dependent nondimensional variable; the Bernoulli cquation tells us that p U 2 has

h e same units as p The nondimensional number that describes viscous effects is

well known to be p U d / p Thcrefore, with some experience, we can fiud all the

nondimensional variables by inspection alone, thus no formal analysis is needcd

5 ;Vimdimenn.iottaI hrutrmtders tmd Dpatnic lSirniIariv

Arranging the variables in terms of dimensionless products is especially useful in pixxenting experimcntal data Consider the case of drag on a sphere of diameter d

moving a1 a speed U through a fluid of density p and viscosity p The drag force can

be written as

D = f ( d U , P , /A) (8.15)

If we do not fonn dimensionless groups, we would havc to conduct an experiment

to determine D vs d , keeping U , p , and p fixed We would then have to conduct an experiment to detennine D as a function of U , keeping d, p and p fixed, and so on However, such a duplication of effort is unnecessary if we write Eq (8.15) in tern

of dimensionless groups A dimensional analysis of Eq (8.1.5) gives

Figurc 8.2 Note that the Reynolds number in Eq (8.16) is wriilcn as the independent

variable because it can be extcmally controlled in an experiment Tn contrast, the drag coefficient is written as a dependent variable

The idea of dimensionless products is intimately associated with the concept

of similarity Tn fact, a collapse of all thc data on a single graph such aqthe one in Figure 8.2 is possible only because in this problcm all flows having the same value

of Re = p U d / p are dynamically similar

For flow around a sphere, the pressure at any point x = ( x ~ y, z ) can be written as

P ( X ) - y, = f(d u PI p ; XI

A dimensional analysis gives the local pressurc coefficient:

(8.17)

requiring that nondimensional local Row variables be idcntical at corrcsponding points

in dynamically similar flows The difference between relations (8.16) and (8.17) should be uoted Equation (8.16) is a relation between averdl quantitics (scales of

motion), whereas (8.17) holds foculfy at a point

10-1 100 IO' 102 103 104 105 1V

Figun! 8.2 Dmg coefficient for a sphere The clwicteristic arca is taken as A = n d 2 / 4 Thc mason for

thc sudden drop or 4) at Rc - 5 x I d is thc transition of the Inminx bouiibry layer Lo :I turbulent oiic,

a expiaincd in Chapter 10

Prediction of Flow Behavior from Dimensional Considerations

An interesting observation in Figure 8.2 is that CD cx 1 /Re at small Reynolds numbers This can bc justified solely on dimensional grounds as follows AI sinall values of Reynolds numbers we expect that the incaia forces in the equations of motion must become negligible Then p drops out olEq (8.15) requiring

D = f ( d , U ? p ) The only diinensionless product that can be formed from the preceding is D / p U d Because there is no other- nondimensional parameter on which D / p U d can depend,

it can only be a constant:

D cx p U d (Re << 1) (3.18) which is equivalent to C,, cx 1/Re It is seen that the drug jbrce in u low Reynolds nuniber.flow is ZineurlypmportiunnI to the speed V; this is frequently called the Stokes law of resisrunce

At the opposite cxlrerne, Figure 8.2 shows that CD becomes independent of Re for values of Re > 1 03 This is because the drag is now due mostly to the formation

of a turbulent wake, in which h e viscosity only has an indirect influence on thc flow (This will be clear in Chapter 13, where we shall see that the only eflect of viscosity

as Rc + cm is lo dissipate the turbulent kinetic cnergy at increasingly smaller scales The overall flow is controllcd by inertia forces alonc.) In this limit p drops oul of

Eq (8.15), giving

D = .fW, U , P I