KUNDU Fluid Mechanics 2 Episode 13 ppt

Compare, for example, the viscous dissipation and the shear production terns: where U is the scale for mean velocity, L is a length scale for example, thc width of the boundary layer, an

and any tensor a Ui / a x is equal to the product of uiui and symmetric part of a Vi / h j ,

namely, Eij ; this is proved in Chapter 2, Section 11 ) If the mean flow is given by U ( y ) ,

then W ( a U i / a x j ) = E ( d U / d y ) We saw in the preceding scction that is likely

to be negative if d U / d y is positive The fifth term uiu,(aUi/axj) js therefore likely

to be negative in shear flows By analogy with the fourth term, it must represcnt an energy loss to the agency that generates turbulent stress, namely the fluctuating field Indced, we shall see in the following section that this term appears on the right-hand side of an equation for the rate of change of turbulent kinetic energy, but with the

sign reversed Therefore, this term generally results in a loss of mean kinetic energy

and a gain of turbulent kinetic energy We shall call this term the shearproduction of

turbulence by the interaction of Reynolds stresses and the mcan shear

The sixth term represents the work done by gravity on the mean vertical motion For example, an upward mean motion results in a loss of mean kinetic energy, which

is accompanied by an increase in the potential energy of the mean field

Thc two viscous terms in Eq (13.32), namcly, the viscous transport

2ua(Ui E i j ) / a x j and the viscous dissipation -2uEijEij, are smallin afully turbulent

flow at high Reynolds numbers Compare, for example, the viscous dissipation and the shear production terns:

where U is the scale for mean velocity, L is a length scale (for example, thc width of the boundary layer), and u,, is the rms value of the turbulent fluctuation; we have also assumed that urms and U are of the same order, since experiments show that

urms is a substantial fraction of U The direct influence of viscous terms is therefore negligible on the meun kinetic energy budget We shall see in the following section that this is not true for the turbulent kinetic energy budget, in which the viscous terms

play a major role What happens is the following: The mean flow loses energy to the turbulent field by means of the shear production; the furbulent kinetic energy so

generated is then dissipated by viscosity

7 Kinelic K n e w Budget of Turbulcnl How

An equation for the turbulent kinetic energy is obtained by first finding an equation for aulat and taking the scalar product with u The algebra becomes compact if we

use the “comma notation,” introduced in Chapter 2, Section 15, namely, that a comma

denotes a spatial derivative:

3 A

axi ’ where A is any variable (This notation is very simple and handy, but it may take a little practice to get used to it It is used in this book only if the algebra would become cumbersome otherwise There is only one other place in the book where this notation

has been applied, namely Section 5.6 With a little initial patience, the reader will

quickly see the convenience of this notation.)

Equations of motion for the total and mean flows are, respectively,

= (I' + p ) , i - gll - u(T + T' - q 1 ) ] S i : 3 + u(U; + ~ i ) , j j ,

Subtracting, we obtain thc equation of motion for the turbulent vclocity u ; :

where we have used thc continuity equation uiqi = 0 and U i = 0

The first and second terms on the right-hand si& of Eq (13.33) givc

-ui - ~ , i = GT)j 7

Po Po

u j g a T ' S i 3 = gawT' - Thc last term on the right-hand side of Eq (13.33) gives

v u ; u i j j = v { u i u i j j + $ ( u i , j + u j , i ) ( u i , j - u j , ; ) } :

where we have added the doubly contracted product of a symmetric tensor ( U ; J + u j , i )

and an antisymmetric tensor ( u i , ~ - u j , i ) , such a product bcing zero In the first term

on thc right-hand side, we can write u i ~ j = ( U ~ J + u j ~ ) , j because ofihe continuity equation Then wc can writc

= v { u i ( u i , j + j : i ) , j + (ui,,j + U j : i ) ( i , , j - ;Ui.,j - $ u j : i ) I

= v { [ u i ( u i j + j , i ) ~ : j - i ( i j + ujyiI2I-

Defining the fluctuating strain rate by

The fmt three terms on the right-hand side are in Ihc flux divergence form and con- sequently represent the spatial transport of turbulent kinetic energy The first two

terms represent the transport by turbulence itself, whereas the third lerm is viscous

then represent the rate of generation of turbulent kinetic energy by thc interaction of

the Reynolds stress with the mean shear U i , j Therefore,

convective motions cause an increase of turbulent kinetic energy (Figure 13.9) We

shall call g a m thc buoyantproduction of turbulent kinetic energy, keeping in mind that it can also be a buoyant "destruction" if the turbulcnt heat flux is downward Therefore,

1 Buoyant production = gawT' - I (1 3.36)

T h e buoyant generation of turbulent kinetic energy lowcrs the potential cnergy

of thc mean field This can be understood from Figure 13.9, where it is seen that h e heavier fluid has moved downward in the final state as a rcsult of the heat flux This

can also be demonslrated by dcriving an equalion for thc mean potential cnergy, in

which thc term gcrwT’appears with a negutive sign on the right-hand side Thcrefore, the huoyunt generution of turbulent kinetic energy by the upward heat flux occurs at

thc expense of the mean potenrial energy This is in contrast to the shear pmduction

of turbulent kinetic energy, which occurs at lhe expensc or the mean kineric energy

The sixth knn 2 v m is the viscous dissipation of turbulent kinetic energy, and

Evidence suggests that the largc eddies in a turbulent flow arc anisotropic, i n the scnse that thcy are “aware” of thc direction of mean shear or of background density gradient In a complctcly isotropic field the off-diagonal components of the Reynolds stress cliuj are zero (see Section 5 here), as is the upward heat flux wT‘ because there

i s no prcltrence between thc upward and downward dircctions In such an isolropic

F i p 13.10 Large eddics oriented dong the principal dirations or a parallel shear flow Note thal h e vortcx aligned wih the a-axis has a posilive u when M is negalive and a ncgative u when u is positivc, resulting in W e 0

case no turbulent energy can be extracted from the mean field Therefore, turbulence must dcvelop anisotropy if it has to sustain itself against viscous dissipation

A possible mechanism of generating anisotropy in a turbulent shear flow is dis-

cussed by Tennekes and Ludey (1 972, p 41) Consider a parallcl shear flow U ( y )

shown in Fiprc 13.10, in which the fluid elements translate, rotate, and undergo shearing deformation The nature of deformation of an elemcnt depcnds on the ori-

entation of the clement An element oriented parallel to the xy-axes undergoes only

a shear strain rate Ex,, = f dU/dy, but no linear strain rate (Exx = Eyp = 0) The strain rate tensor in the xy-coordinate system is therefore

I-

As shown in Chapter 3, Section 10, such a symmetric tensor can be diagonalized by

rotating the coordinate system by 45" Along thesc principal axes (denoted by a and

/ in Figure 13 IO), the strain rate tensor is

so that there is a linear extension rate of Emu = f dU/dy, a linear compression rate

of Epp = -: dU/dy, and no shear (Eap = 0) The kinematics of strctching and compression along the principal directions in a parallel shear flow is discussed further

in Chapter 3, Section 10

The large eddies with vorticity oriented along the a-axis intensify in stren,@h due

to the vortex stretching, and the ones with vorticity oriented along the &axis decay in strength The net effect of the mean shear on the turbulent field is therefore to cause

a predominance of eddics with vorticity oriented along the a-axis As is evident in Figure 13.10, thesc cddies are associated with apositive u when u is negative, and with

a negative u whcn u is positivc, resulting in a positive value for the shear production The largest cddies are of order of the width of the shear flow, for examplc the diameter of a pipe or the width of a boundary layer along a wall or along the uppcr surface of thc ocean Thew eddies extract kinetic energy from the mean field The

eddies that are somewhal smaller than thcse are straincd by the velocity field of the largest cddies, and exhact energy from h e larger eddics by the same mechanism of vortcx stretching The much smaller eddies arc cssenliaUy advectcd in the velocity field of the large eddics, as the scales of the strain rate field of the large cddies are much larger than the size of a small eddy Thcrdore, the small eddies do not interact with either thc large eddics or the mean ficld The kinetic energy is therefore cascuded down J.om large to snzall eddies in n series of snwll steps This process of energy

criscude is essentially inviscid, as the vortex stretching mechanism arises jmna the nonlinear terms of the equations af motion

h B fully turbulent shcar flow (i.e., for large Reynolds numbers), therefoE, the viscosity of the fluid does not alTect the s h c i production, if all other variablcs are

held constant The viscosity does, howevcr, determine thc scales at which turbulent

cnergy is dissipated into hcat From the expression E = 2ueijeij, it is clcar that the

encrgy dissipation is effective only at very small scales, which have high fluctuating strain rates The continuous strclching and cascade generate long and thin filaments, somewhat like “spaghetti.” When these fi lamcnls become thin enough, molecular diffusive effects arc able to smear out their vclocity gradients These arc the small-

est scales in a turbulent flow and are responsible for the dissipation of the lurbulent kinclic energy Figure 13.1 1 illustrates the deformation of a fluid particle in a tur- bulent motion, suggesting that molecular effccls can act on thin filaments generatcd

by continuous stretching The large mixing rates in a turbulent flow, therefore, are essentially a result of the turbulent fluctuations generating thc large suijiuces on which

the molecular diffusion finally acts

It is clear that E docs not depend on u, but is dctermined by the inviscid properties

of the large cddies, which supply thc cnergy to thc dissipating scales Suppose 1 is a

typical length scale of the large eddies (which may bc taken equal to the integral length

- E ( d U / d y )

(Kolmogorov microscale)

A y r e 13.11

thc scale bccomcs of odcr of thc Kolmogorov microscalc

Successivc &limnations ol‘a marked h i d cleinenl Di flusivc cll’cc~s causc smearing whcn

520 ' l i u b ~ l e c

scale defined h m a spatial correlation function, analogous to the integral time scale defined by Eq (13.10)), and u' is a typical scale of the fluctuating velocity (which

may be taken equal to the rms fluctuating speed) Then the time scale of large eddies

is of order l / d Observations show that the large eddies lose much of their energy during the time they turn over one or two times, so that the rate of energy transferred from large eddies is proportional to un times their frequency u'/Z The dissipation rate must then be of order

(13.38)

signifying that the viscous dissipation is determined by the inviscid large-scale dynamics of the turbulent ficld

Kolmogorov suggested in 1941 that the size of the dissipating eddies depends

on hose parameters that are relevant to the smallest eddies These parameters are the rate E at which energy has to be dissipated by the eddies and the diffusivity u that does the smearing out of the vclocity gradients As the unit of E is cm2/s3, dimensional reasoning shows that the lcngth scale formed from E and u is

(1 3.39)

which is called the Kolmogomv micmscale A decrease afv merely decreases the scale

at which viscous dissipation takes place, and not the rate of dissipatian E Estimates show that is of the order of millimeters in the ocean and the atmosphere Tnlaboratory flows the Kolmogorov microscale is much smaller because of the largerrate of viscous dissipation Landahl and Mollo-Christensen (1986) give a nice illustration of this

Suppose we are using a 100-W household mixer in 1 kg of water As all the power is used to generate the turbulence, the rate of dissipation is E = 100 W/kg = 100 m2/s3

Using u = m2/s for water, we obtain q = mm

In Section 4 we &fined the wavenumber spectrum S(K), representing turbulent kinetic energy a, a function of the wavenumber vector K Tf the turbulence is isotropic, then the spectrum becomes independent of the orientation a€ the wavenumber vector

and depends on its magnitude K only In that case we can wrile

oc

-

u3 =$ S ( K ) d K

In this section we shall derive the Form d S ( K ) in a certain rangc of wavenumbers

in which h e turbulence is nearly isotropic

Somewhat vaguely, wc shall associate a wavenumber K with an eddy of size K-'

Small cddies are therefore represented by large wavenumbers Suppose I is the scale

of thc large eddics, which may bc h e width of the boundary laycr A1 the relativcly small scales represented by wavenumbers K >> 1 - I , there is no direct interaction between thc turbulence and the motion of the large, encrgy-containing eddies This is because the small scalcs have been gencrakd by a long series of small steps, losing

information at each stcp The spectrum in this range oJfarge wtsvenumhers is nerrrly

isotropic, as only thc large eddies are aware of the directions of mcan gradients Thc spcctruin here docs not depend on how much encrgy is present at large scales (wherc

most 01 the energy is contained), or the scales at which most of thc cnergy is present

The spectrum in this range depends only on the parametcrs that determine thc nature

o€ h e small-scale flow, so that we can write

Thc range of wavenumbers K >> 1-' is usually called the equifihri-ium rcmge The

dissipating wavenumbers with K - r j - I , beyond which the spectrum falls off very rapidly, form the high end of thc equilibrium range (Figure 13.12) The lower cnd

of this range, for which 1-' << K << q-' is called the inertial subrange, as only

the n-dnsfer of encrgy by inertial forces (vortex stmtching) takes place in this rangc Both production and dissipation are small in the inertial subrangc The production of energy by large eddics causes a peak of S at a ccrlah K 2 1 ' - I , and the dissipation

of energy causes a sharp drop of S for K =- I)'-' The question is, how does S vary with K between the two limits in the inertial subrange?

S = AE2f3K-j/3 1-1 << K << q - ' ,

~ ~ ~ _ _ _ _ _ _ _

where A 21 1.5 has been €ound to be a universal constant, valid for all turbulent flows Quation (13.40) is usually called Kalmoguraw's K - 5 / 3 law If the Reynolds number of the flow is large, then the dissipating eddics are much smaller than the energy-containing eddies, and the inertial subrange is quite broad

Because very large Reynolds numbcrs are difficult to generate in the laboratory, the Kolmogorov spectral law was not verified for many years In fact, doubts were being raised about its theoretical validity The first confirmation of the Kolmogorov law camc from the oceanic observations of Grant etaf (1 962), who obtained a vdocity spectrum in a tidal flow through a narrow passage between two islands near the west coast of Canada The velocity fluctuations were measured by hanging a hot film

anemometer from the bottom of a ship Based on the depth of water and the average flow velocity, the Reynolds number was of order lo8 Such large Reynolds numbers

are typical of gcophysical flows, since the length scales are very large The K-s/3

law has since been verified in the ocean over a wide range of wavenumbers, a typical behavior being sketched in Figure 13.12 Notc that the spectrum drops sharply at

K q - 1, where viscosity begins to affect the spectral shape The figure also shows

that the spectrum departs f o m the K-'I3 law for small values of the wavenumber,

where thc turbulence production by large eddies bcgins to affect the spectral shape

Laboratory cxperimcnts are also in agreemcnt with the Kolmogorov spectral law,

although in a namwcr range of wavenumbers because the Reynolds number is not as

large as in geophysical flows The K 'l3 law has become one of the most important

results of turbulence theory

(1 3.40)

Nearly parallel shear flows are divided into two classes-wall-free shear flows and wall-bounded shear flows In this section we shall examine some aspects of turbulent flows hat are free of solid boundaries Common examples of such flows are jets, wakes, and shear layers (Figure 13.1 3) For simplicity we shall consider only plane two-dimensional Bows hisymmetric flows are discussed in Townsend (1 976) and

Tennekes and Lumlcy (1 972)

Intermittency

Consider aturbulent flow confined to a limited region To be specific we shall consider the example of a wake (Figm 13.13b), but our discussion also applies to a jet, a shear

layer, or the outer part of a boundary layer on a wall The fluid outside the turbulent

region is either in irrotational motion (as in the case of a wake or H boundary layer), OT

nearly static (as jn the case of a jet) Observations show that the instantaneous interface

between thc turbulent and nonturbulent fluid is very sharp In fact, the thickness of the interface must equal the size of h e smallest scalcs in the flow, namely the Kolmogorov

microscale The interface is highly contorted due to the presence of eddies of various sizes However, a photograph exposed for a long time does not show such an irregular and sharp interface but rather a gradual and smooth transition region

Measurements at a fixed point in the outer part or the turbulcnt region (say at

point P in Figure 13.1 3b) show pcriods of high-hequency fluctuations as the point P

moves into the turbulcnt flow and quiet pcriods as the point moves out ofthe turbulcnt

region Intermittency y is defincd as the faction of time the flow at apoint is turbulent The variation of y across a wake is sketched in Figure 13.13b, showing lhdt y = 1

near the center where thc flow is always turbulent, and y = 0 at the outer edge of the flow

Entrainment

A flow can slowly pull the surrounding irrotational fluid inward by “€rictional” effects;

the process is called enminment The source of this “friction” is viscous in laminar

flow and inertial in turbulent flow The entminment of a laminar jet was discussed in Chapter 10, Section 12 The entrainment in a turbulcnt flow is similar, but the rate is much larger After the irrotational fluid is drawn inside a turbulent region, the new fluid must be made turbulent This is initiated by small eddies (which are dominated

by viscosity) acting at the sharp intcrface between the turbulent and the nonturbulent fluid (Figure 13.14)

Thc foregoing discussion 01 intennittcncy and entrainment applies not only to wall-he shear flows but also to the outer edge of boundary layers

Self-Preservation

Far downstream, experiments show that the mean field in a wall-he shear flow becomcs approximately self-similar at various downswam distances As the mean field is affected by the Reynolds stress through thc equations of motion, this means that the various turbulent quantities (such as Reynolds stress) also must reach self-similar states This is indeed found to be approximately true (Townsend, 1976) The flow is then in a state of “moving equilibrium,” in which both the mean and the turbulent

fields are determined solely by the local scales of lcngth and velocity This is called

self-preservation Tn the self-similar statc, the mean velocity at various downstream

viscous eddies

irrotational fluid

irrotational fluid

turbulent fluid turbulent fluid

4

Figure 13.14 htrainmcnt of a nonturbulent fluid and its assimilation into turhuleni fluid by viscous aciion at lhc interke

Here S(x) is thc width offlow, U c ( x ) is thc centerline velocity for the jet and the wake,

and U I and U2 arc the velocities of the two strcams in a shear layer (Figure 13.13)

We shall now dcrive how the centerline vclocity and width in a planc jct must vary

if we assume that thc mean velocity profiles at various downstream distanccs arc self

similar This can be done by cxamining the equations of motion in differential form

An alternatc way is to examine an integral form of the equation of motion, derived in

Chapter 10, Section 12 It was shown there that the momentum flux M = p I U 2 dy

across the jet is independent of x , while the maTs flux pJU dy increaqes downstream due to entrainment Exactly the same constraint applies to a turbulcnt jet For the sake of readers who find cross references annoying, the integral constraint For a two-dimensional jet is rederived hcre

Consider a control volume shown by thc dotted line in Figurc 13.13a in which thc horizontal surfaces of the control volume arc assumed to be at a large distance from the jct axis At these large distances, there is a mean V field toward the jet axis due to entrainmcnt, but no U field Therefore, the flow oFx-momentum through the horizon- tal surfaccs ollhe control volume is zero Thc pressure is uniform throughout the flow, and the viscous forces are negligible The nct force on the surfxc of the control vol- ume is therefore zero The momentum principle for a control volumc (see Chapter 4,

Section 8) states that thc net x-directed force on the boundary equals the ne# rate of outflow of x-momentum through the control surfaces As thc net force here is zcro,

the influx of x-momentum must equal the outflow of x-momentum That is

00

U 2 d y = independcnt oix: (1 3.42) where M is the momentum flux of thc jct (=integral of inass flux p U d y times velocity U) The momentum flux is the basic externally controlled parameter for a jet and is known from an evaluation of Eq (13.42) at the orifice opening The mass

fluid

The assumption of selr similarity can now bc used to predict how S and U, in a jet should vary with x Substitution or the self-similarity assumplion (1 3.41) into the integral constraint (1 3.42) gives

M = d _

526 Turhulerrcc!

The preceding intcgral is a constant because it is completely expressed in terms of

thc nondimensional function f(y/6) As A4 is also a constant, we must have

At this point we make anothcr important assumption We assume that the Reynolds number is large, so that the gross characteristics of the flow are independent

of thc Reynolds numbcr This is callcd Reynolds number similurirq The assumption

is expected to be valid in a wall-free shear flow, as viscosity does not directly affect the motion; a d m a s e of v, for cxample, merely decreases the scale of the dissipat-

ing eddies, as discussed in Section 8 (The principle is not valid near a smooth wall, and as a consequcnce the drag coefficient for a smooth flat plate does not become independent of the Reynolds number as Re + x; see Figure 10.10.) For large Re, then, U, is independcnt of viscosity and can only depend on x, p , and M :

A dimensional analysis shows that

(13.44)

so that Eq ( I 3.43) requires

This should be compared with the 6 o( x2I3 behavior of a Zumimr jet, derived in

Chapter 10, Scction 12 Experiments show that the width of a turbulent jet does grow

linearly, with a spreading angle of 4':

For two-dimensional wakes and shear layers, it can be shown (Townsend, 1976; Tennekes and Lumley, 1972) that the assumption of self similarity requires

u, - u, a x-"2, 6 o( f i (wake),

U, - iJ2 = const., B a x (shear layer)

The turbulent kinetic cnergy equation derived in Section 7 will now be applied to

a two-dimcnsional jct The encrgy budgct calculation uses the experimentally mea-

surcd distributions of turbulence intensity and Reynolds strcss across the jet There- fore, we present the distributions of these variables first Measurements show that the turbulent intensities and Reynolds stress arc distributed as in Figurc 13.15 Here

u2 is the intensity of fluctuation in thc downstream dircction x , 3 is the inten- sity along the cross-stream direction y , and 3 is the intensity in the z-direction;

q2 = (u2 + Y~ + w 2 ) / 2 is the turbulent kinctic encrgy per unit mass The Reynolds

stress is zero at h e center of tbe jet by symmetry, since there is no reason for u at the center to bc mostly of one sign if u is either positive or negativc The Reynolds stress

-

- - -

Y

A g r e 13.15 Skckh of ohsewed variation of turbulent intensily and Rcynolds stress across a jcl

reaches a maximum magnitude roughly where a U / a y is maximum This is also close

to the region whcrc Ihe turbulent kinetic energy reaches a maximum

Consider now the kinetic energy budget For a two-dimensional jet under the boundary layer assumption a/ax << a/i1y, Eq (13.34) becomes

where - - the left-hand side represcnts i)q'/at = 0 Here the viscous transport and a term

(!' - u2 ) ( a U / a x ) arising out of thc shear produclion have been ncglected on the

right-hand sidc because thcy are small Thc balance of terms is analyzed in Townsend (1 976), and thc results are shown in Figurc 13.16, where T denotes turbulent transport rcpresented by the fourth term on the right-hand side of (1 3.46) The shear production

is zero at the center whcre bolh a U / a y and iiij are zero, and reacbes a maximum close

to the position of thc maximum Reynolds strcss Near the ccntcr, the dissipation is primarily balanced by the downstream advection - U ( i f q 2 / a x ) , which is positivc

Secausc the turbulcnt inlensity y' decays downstream Away from the center, but not

too close to the outer edge of thc jet, the production and dissipation terms balance In the outcr parts of thc jet, the transport term balances the cross-stream advection Tn

this region V is ncgative (i.e., toward the ccntcr) due to entrainment orthc s m u n d i n g

fluid, and alsoq2 decreases withy Thercfore the cross-stream advcction - V ( a q * / a y )

is negativc, signifying that the entrainment velocity V tends to decrease thc turbulent kinetic cncrgy at the outer edge of the jet The stationary statc is thereforc maintained

by the transport term 1 carrying turbulent kinctic energy away from thc center (whcre

T -= 0) into the outer parts of the jet (whcre T > 0)

3 1 Wall-Bounded Shear Flow

The gross characteristics of free shear flows, discussed in the preccding section, are independent of viscosity This is not truc of a turbulent flow bounded by a solid wall,

in which the presence of viscosity dfccts the motion near the wall The cffect of viscosity is reflected in the fact that the drag coeficient of a smooth flat plate depends

on the Rcynolds number even for Re + 30, as seen in Figure 10.10 Therefore, the concept of Reynolds number similarity, which says that the gross characteristics

are independent of Re when Re + 00, no longer applies In this section we shall examine how the properties of a turbulent flow near a wall arc affectcd by viscosity Bcfore doing his, we shall examine how the Reynolds stress should vary with distance

from thc wall

Consider first a fully developed turbulent flow in a channel By ‘’fully developed”

we mcan that the flow is no longer changing in x (see Figure 9.2) Then the mean cquation of motion is

whcre i = p ( d U / d y ) - piE is the total smss Because a P / a x is a function of x

alone and a t / a y is a function of y alone, both of them must bc constants The stress

distribution is then linear (Figwc 13.17a) Away from thc wall 5 is due mostly io the

Rcynolds stress, but close to the wall the viscous contribution dominates In fact, at the wall thc velocity fluctuations and consequently the Rcynolds stresses vanish, so

that the stress is cntirely viscous

Consider the flow ncar the wall of a channel, pipe, or boundary layer Le1 U, be the

ke-stream vclocity in a boundary layer or the centerline velocity in a channel and pipe Let S bc the width of flow, which may bc the width of thc boundary laycr, the channel half width, or the radius of the pipe Assume that the wall is smooth, so that the height of the surface roughness elements is too small to affect the flow Physical considerations suggest that thc velocity profile near the wall dcpends only on the parametcrs that are rclevant near the wall and does not depend on the free-stream velocity U, or the thickness of the flow S Very ncar a smooth surface, then, wc expect that

where to is the shear stress at the wall To express Eq (13.47) in terms of dimen- sionless variables, notc that only p and q) involve the dimension of mass, so that thcsc two variablcs must always occur togethcr in any nondimensional p u p The

important ratio

(13.48)

530 l i ~ ~ u i t ! I l m ~

has thc dimension of velocity and is called thefiction velocity Equation (13.47) can

then be written as

u = U ( U * , V1 y) ( 1.3.49) This relates four variables involving only the two dimcnsions of length and time According to thc pi theorem (Chapter 8, Section 4) there must bc only 4 - 2 = 2

nondimensional groups U / u , and yu*/u, which should be related by some universal functional ronn

- = f (f-) = f(y+)

where y+ yuJu is the distance nondimensionalized by the viscous scurle u/u*

Equation (13.50) is called thc law ofthe wall, and states that U/u* must bc a universal function of yu,/u near a smooth wall

The inner part of the wall layer, right next to the wall, is dominated by viscous

effects (Figurn 13.18) and is called the viscous subluyer It used to be called the “lam-

inar sublayer:” until experiments revealed the prcsence of considerable flucluations within the layer In spite of the fluctuations, the Reynolds stresses arc still small here

because of thc dominancc of viscous cffects Bccause of the thinness of thc viscous sublayer, thc stress can bc taken as uniform within the layer and cqual to the wall shear stress 70 Thedorc the velocity gradient in the viscous sublayer is given by

which shows that the velocity distribution is linear Tntcgrdting, and using thc no-slip boundary condition, we obtain

Experim.ents show that the linear distribution holds up to yu,/v - 5 , which may be

taken to be thc limit of the viscous sublayer

We now explore the form of thc velocity distribution in the outer part of a turbulent layer The gross charactcrislics of the turbulcnce in the outcr rcgion are inviscid and resemble those of a wall-free turbulent flow The existence of Reynolds stresses in the outer region results in a drag on the flow and generates a velociry deJecr (U, - V),

which is cxpected to be proportional to the wall friction chardctcnzcd by ự Jt follows that the vclocity distribution in the outer region must have the form

U - [ J S

UX

where $ = y / S This is called thc vebcify defect law

Overlap Layer: Logarithmic Law

The velocity profiles in the inner and outcr parts of the boundary layer arc governed

by differ-cnl laws (13.50) and (13.52), in which the independent variable y is scaled differentlỵ Distances in the outer part are scaled by S, whereas thosc in the inner part are measured by the much smaller viscous sc:& v/u, In other words, the small

distances in the inner laycr are magnified by expressing them as y u , / v This is the typ- ical behavior in singular pcrturbation problems (see Chapter 10, Sections 14 and 16)

In these problems the inncr and outer solutions are matched togethcr in a region of

overlup by taking the limits y + 00 and e + 0 simulfmeouslỵ Instead of matching

vclocily, in this case it is more convenient to match thcir gradients (The derivation given here closely follows Tennekes and Lumley (1972).) From Eqs (13.50) and

(1 3.55)

valid for large y+ and small e As the Icft-hand side can only be a €unction of 6 and the right-hand side can only be a function of y+, both sides must be equal to the same

universal constanl, say l/k, where k is called the von Kamun consrunt Experiments show that k 21 0.41 Integration of Eq (1 3.55) givcs

(1 3.56)

Experiments show that A = 5.0 and B = -1 O for a smooth fiat plate, for which

These are the velocity distributions in the overlap Iuyer, also called the inertial sub-

Zuyer or simply thc lugariflzmic layer As the derivation shows, these laws are only

valid for large y7 and small y/S

The forcgoing method of justifying the logarithmic velocity distribution near a wall was first given by Clark B Millikan in 1938, before thc formal theory of singular perturbation problems was fully developed The logarithmic law, however, was h o w n from experiments conducted by the German researchers, and several derivations based

on semiempirical theories were proposed by Fhndtl and von Karman One such derivation by the so-callcd mixing length theory is presented in thc following section The logarithmic velocity distribution near a surface can be derived solcly on dimensional grounds In this layer the velocity gradient d U / d y can only depend on thc local distance y and on the only relevant velocity scale near the surface, namcly u*

(The layer is far enough from the wall so lhdt the direct effcct of u is not rclcvant and far enough from the outer part of the turbulent layer so that the effect of S is not relevant.) A dimensional analysis gives

the edge of the viscous sublayer at y = 10.7v/u, reduces Eq (13.59) to Eq (13.57)

(Excrcise 8) The logarithmic velocity distribution also applies to rough walls, as

discussed l a m in the section

The experimental data on the velocity distribution ncar a wall is sketchcd in Figure 13.18 It is a semilogarithmic plot in tcrms of the inner variables Tt shows that the lincar velocity distribution (13.51) is valid for J+ < 5, so that we can take the

viscoiis sublayer thickness to be

5 v

S, 2 -

u* (viscous sublayer thickness)

The logaritlmic velocity distribution (13.57) is seen to be valid for 30 < y; < 300 Thc upper limit on y , howevcr, depends on thc Reynolds numbcr and becomes larger as Re increases Therc is therefore a largc logarithmic ovcrlap region in flows

zt large Reynolds numbers Thc close analogy bctween the overlap rcgion in physical space and incrlial subrange in spectral space is cvident In both regions, therc is litlle production or dissipation; thcre is simply an "inertial" transfer across the rcgion by inviscid nonlinear proccsscs It is for this rcason that thc logarithmic laycr is called ihe inertial subluyer

As Eq (1 3.58) suggcsts, a logarithmic velocity distribution in the overlap region can also bc plotted in terms of the outer variables of (U - Uno)/u, vs y / 8 Such plots shuw that the logarithmic distribution is valid for y/S < 0.2 Thc logarithmic law, herefore, holds accuratcly in a rather small percentagc (-202) of thc total bound- ary layer thickness The gcneral defect law (13.52), where F ( e ) is not necessarily

!ogarithmic, holds almost everywhere cxcept in the inner part of thc wall layer Thc region 5 < yi < 30, whcre the velocity distribution is neither linear nor logarithmic, is callcd the bufler layer Neither the viscous slress nor thc Reynolds stress is negligiblc here This layer is dynamically very important, as the turbulence

production - i Z ( d U / d y ) reachcs a maximum here due to the large velocity gradients

Wosnik et ul (2000) very carefully reexamined turbulcnt pipe and channel flows

and compared their results with superpipe data and scalings developed by Zagarola and Smits j1998), and others Vcry briefly, Figure 13.18 is split into more regions

in that a "mesolaycr" is requircd between thc buffer laycr and the incrlial sublaycr Proper dcscription of the velocity in this mesolayer requires an offsct parameter in the logarithm of Eqs (13.56) This is obtaincd by generalizing Eq (13.55) to

wherc si = u / d , a+ = uu,/v

Equations (13.56) bccome

The valuc for u+ suggested by Wosnik et al that best fits h e supcrpipe data is

u+ = -8

534 ltubulem

The outer region of turbulent boundary layers (y+ > 100) is the subject of a

similarity analysis by Castillo and Georgc (2001) They found that 90% of a turbu- lent flow under all pressure gadients is charactcrized by a single pressurc gradient parameter,

A requircment for “equilibrium” turbulcnt boundary layer flows, to which their anal- ysis is mstricted, is that A = const., and this leads to similarity Examination of data from many sources led them to conclude that “ then: appear to be almost no

flows that are not in equilibrium .” Their most remarkable result is that only three valucs of A correlate thc data for all pressurc gradients: A = 0.22 (adverse pressure gradicnts); A = -1.92 (favorable pressure gradients); and A = 0 (zero pmssure gradient) A direct consequence of A = const is that S ( X ) - U&’’* Data was well correlated by this result for both favorable and adverse pressure gradicnts

Rough Surface

In dcriving the logarithmic law (1 3.57) we assumed that the flow in the inner laycr

is determined by viscosity This is true only in hydmdynumicaffy smooth surfaces,

for which Lhc averagc height of the surface roughness clements is smaller than the thickncss of the viscous sublayer For a hydrodynamically rough surface, on the othcr

hand, the roughness elements prolrude out of the viscous sublayer An example is thc flow near the surface of the earth, where the trecs and buildings act as rough-

ness elements This causes a wake behind each roughncss elemcnt, and the strcss is

transmitted to the wall by the “pressure drag” on the roughness elements Viscosity becomcs irrelevant for determining either the velocity distribution or thc overall drag

on thc surface This is why thc drag cocfficients for a rough pipe and a rough flat

surface becomc indepcndent of the Reynolds number as Re + 00

The velocity distribution near a rough surface is agah logarithmic, although it cannot be represented by Eq (13.57) To find its €om, we start with the general logarithmic law (13.59) The constant of integration can be dctermincd by noting that the mcan velocity U is cxpected to be ncgligiblc somewhere within the roughness elements (Figure 13.1.9b) We can thcrefore assume thal(13.59) applies for y > yo,

where yo is a measure of the roughness heights and is defined as the value of y at

which the logarithmic distribution givcs U = 0 Equation (1 3.59) then gives

11 Wiill-kwidd Shew Plam 535

Figurc 13.20 Sketch of observcd variation or lurhulent intensity and Rcynolds rlrcss acniss a channel

of half-width 6 Thc Icli pancls are plots as functions ol' lhc inncr variable y I , while the right pancls arc

?lois as hnctions of the outer variahlc y/S

turbulent velocity fluctuations are or order u* The longitudinal fluctuations are thc

largest because the shear production initially feeds thc cnergy into the u-component;

the energy is subscqucntly distributed into the latcral components zi and w (Inciden-

tally, in a convectivcly generated turbulencc thc turbulent energy is initially fed LO the

vertical compon.cnt.) The turbulent intensity initially rises as the wall is approached,