A HEAT TRANSFER TEXTBOOK - THIRD EDITION Episode 2 Part 4 ppt

§6.7 Turbulent boundary layers 315Figure 6.17 Fluctuation of u and other quantities in a turbu-lent pipe flow.. §6.7 Turbulent boundary layers 317Figure 6.18 The shear stress, τyx, in a

one or two enormous vortices of continental proportions These hugevortices, in turn, feed smaller “weather-making” vortices on the order ofhundreds of miles in diameter These further dissipate into vortices ofcyclone and tornado proportions—sometimes with that level of violencebut more often not These dissipate into still smaller whirls as they inter-act with the ground and its various protrusions The next time the wind

blows, stand behind any tree and feel the vortices In the great plains,

where there are not many ground vortex generators (such as trees), youwill see small cyclonic eddies called “dust devils.” The process continuesright on down to millimeter or even micrometer scales There, momen-tum exchange is no longer identifiable as turbulence but appears simply

as viscous stretching of the fluid

The same kind of process exists within, say, a turbulent pipe flow athigh Reynolds number Such a flow is shown in Fig.6.17 Turbulence

in such a case consists of coexisting vortices which vary in size from asubstantial fraction of the pipe radius down to micrometer dimensions.The spectrum of sizes varies with location in the pipe The size andintensity of vortices at the wall must clearly approach zero, since thefluid velocity goes to zero at the wall

Figure6.17shows the fluctuation of a typical flow variable—namely,velocity—both with location in the pipe and with time This fluctuationarises because of the turbulent motions that are superposed on the aver-

age local flow Other flow variables, such as T or ρ, can vary in the same

manner For any variable we can write a local time-average value as

where T is a time that is much longer than the period of typical

fluctua-tions.9 Equation (6.81) is most useful for so-called stationary processes—ones for whichu is nearly time-independent.

If we substitute u = u + u in eqn (6.81), where u is the actual local

velocity and u  is the instantaneous magnitude of the fluctuation, weobtain

9Take care not to interpret this T as the thermal time constant that we introduced

in Chapter 1 ; we denote time constants are asT

§6.7 Turbulent boundary layers 315

Figure 6.17 Fluctuation of u and other quantities in a

turbu-lent pipe flow

This is consistent with the fact that

u or any other average fluctuation= 0 (6.83)since the fluctuations are defined as deviations from the average

We now want to create a measure of the size, or lengthscale, of

turbu-lent vortices This might be done experimentally by placing two

velocity-measuring devices very close to one another in a turbulent flow field

When the probes are close, their measurements will be very highly

corre-lated with one one another Then, suppose that the two velocity probes

are moved apart until the measurements first become unrelated to one

another That spacing gives an indication of the average size of the

tur-bulent motions

Prandtl invented a slightly different (although related) measure of the

lengthscale of turbulence, called the mixing length,  He saw  as an

average distance that a parcel of fluid moves between interactions It

has a physical significance similar to that of the molecular mean free

path It is harder to devise a clean experimental measure of  than of the

correlation lengthscale of turbulence But we can still use the concept of

 to examine the notion of a turbulent shear stress.

The shear stresses of turbulence arise from the same kind of tum exchange process that gives rise to the molecular viscosity Recallthat, in the latter case, a kinetic calculation gave eqn (6.45) for the lami-nar shear stress

where  was the molecular mean free path and u was the velocity

differ-ence for a molecule that had travelled a distance  in the mean velocity

gradient In the turbulent flow case, pictured in Fig.6.18, we can think ofPrandtl’s parcels of fluid (rather than individual molecules) as carrying

the x-momentum Let us rewrite eqn (6.45) in the following way:

• The shear stress τ yx becomes a fluctuation in shear stress, τ yx  ,resulting from the turbulent movement of a parcel of fluid

•  changes from the mean free path to the mixing length

• C is replaced by v = v + v , the instantaneous vertical speed of the

fluid parcel

• The velocity fluctuation, u , is for a fluid parcel that moves a

dis-tance  through the mean velocity gradient, ∂u/∂y It is given by

§6.7 Turbulent boundary layers 317

Figure 6.18 The shear stress, τyx, in a laminar or turbulent flow

Notice that, while u  = v  = 0, averages of cross products of fluctuations

(such asu  v  or u 2) do not generally vanish Thus, the time average of

the fluctuating component of shear stress is

In addition to the fluctuating shear stress, the flow will have a mean shear

stress associated with the mean velocity gradient, ∂u/∂y That stress is

µ(∂u/∂y), just as in Newton’s law of viscous shear.

It is not obvious how to calculatev  u (although it can be measured),

so we shall not make direct use of eqn (6.86) Instead, we can try to model

v  u  From the preceding discussion, we see that v  u should go to zero

when the velocity gradient, (∂u/∂y), is zero, and that it should increase

when the velocity gradient increases We might therefore assume it to be

proportional to (∂u/∂y) Then the total time-average shear stress, τ yx,

can be expressed as a sum of the mean flow and turbulent contributions

that are each proportional to the mean velocity gradient Specifically,

where ε m is called the eddy diffusivity for momentum We shall use this

characterization in examining the flow field and the heat transfer.The eddy diffusivity itself may be expressed in terms of the mixinglength Suppose that u increases in the y-direction (i.e., ∂u/∂y > 0).

Then, when a fluid parcel moves downward into slower moving fluid,

it has u  (∂u/∂y) If that parcel moves upward into faster fluid, the sign changes The vertical velocity fluctation, v , is positive for anupward moving parcel and negative for a downward motion On average,

u  and v for the eddies should be about the same size Hence, we expectthat

Turbulence near walls

The most important convective heat transfer issue is how flowing fluidscool solid surfaces Thus, we are principally interested in turbulence nearwalls In a turbulent boundary layer, the gradients are very steep nearthe wall and weaker farther from the wall where the eddies are largerand turbulent mixing is more efficient This is in contrast to the gradualvariation of velocity and temperature in a laminar boundary layer, whereheat and momentum are transferred by molecular diffusion rather thanthe vertical motion of vortices In fact,the most important processes inturbulent convection occur very close to walls, perhaps within only afraction of a millimeter The outer part of the b.l is less significant.Let us consider the turbulent flow close to a wall When the boundarylayer momentum equation is time-averaged for turbulent flow, the result

§6.7 Turbulent boundary layers 319

In the innermost region of a turbulent boundary layer — y/δ  0.2,

where δ is the b.l thickness — the mean velocities are small enough

that the convective terms in eqn (6.90a) can be neglected As a result,

∂τ yx /∂y 0 The total shear stress is thus essentially constant in y and

must equal the wall shear stress:

τ w τ yx = ρ (ν + ε m ) ∂u

Equation (6.91) shows that the near-wall velocity profile does not

de-pend directly upon x In functional form

u = fnτ w , ρ, ν, y (6.92)

(Note that ε mdoes not appear because it is defined by the velocity field.)

The effect of the streamwise position is carried in τ w, which varies slowly

with x As a result, the flow field near the wall is not very sensitive

to upstream conditions, except through their effect on τ w When the

velocity profile is scaled in terms of the local value τ w, essentially the

same velocity profile is obtained in every turbulent boundary layer.

Equation (6.92) involves five variables in three dimensions (kg, m, s),

so just two dimensionless groups are needed to describe the velocity

where the velocity scale u ∗ ≡3τ w /ρ is called the friction velocity The

friction velocity is a speed characteristic of the turbulent fluctuations in

the boundary layer

Equation (6.91) can be integrated to find the near wall velocity profile:

To complete the integration, an equation for ε m (y) is needed

Measure-ments show that the mixing length varies linearly with the distance from

the wall for small y

where κ = 0.41 is called the von Kármán constant Physically, this says that the turbulent eddies at a location y must be no bigger that the dis-

tance to wall That makes sense, since eddies cannot cross into the wall

The viscous sublayer Very near the wall, the eddies must become tiny;

 and thus ε m will tend to zero, so that ν ε m In other words, inthis region turbulent shear stress is negligible compared to viscous shearstress If we integrate eqn (6.94) in that range, we find

= (u ∗ )2y ν

a large increase in the wall shear stress)

The log layer Farther away from the wall,  is larger and turbulent

shear stress is dominant: ε m ν Then, from eqns (6.91) and (6.89)

§6.7 Turbulent boundary layers 321

Assuming the velocity gradient to be positive, we may take the square

root of eqn (6.97), rearrange, and integrate it:



du =

2

τ w ρ

for B 5.5 Equation (6.99) is sometimes called the log law

Experimen-tally, it is found to apply for (u ∗ y/ν)  30 and y/δ  0.2.

Other regions of the turbulent b.l For the range 7 < (u ∗ y/ν) < 30,

the so-called buffer layer, more complicated equations for , ε m , or u are

used to connect the viscous sublayer to the log layer [6.7, 6.8] Here, 

actually decreases a little faster than shown by eqn (6.95), as y3/2 [6.9]

In contrast, for the outer part of the turbulent boundary layer (y/δ 

0.2), the mixing length is approximately constant:  0.09δ Gradients

in this part of the boundary layer are weak and do not directly affect

transport at the wall This part of the b.l is nevertheless essential to

the streamwise momentum balance that determines how τ w and δ vary

along the wall Analysis of that momentum balance [6.2] leads to the

following expressions for the boundary thickness and the skin friction

To write these expressions, we assume that the turbulent b.l begins at

x = 0, neglecting the initial laminar region They are reasonably accurate

for Reynolds numbers ranging from about 106 to 109 A more accurate

formula for C f, valid for all turbulent Rex, was given by White [6.10]:

C f (x) = ! 0.455

ln(0.06 Re x )"2 (6.102)

Like the turbulent momentum boundary layer, the turbulent thermalboundary layer is characterized by inner and outer regions In the in-ner part of the thermal boundary layer, turbulent mixing is increasinglyweak; there, heat transport is controlled by heat conduction in the sub-layer Farther from the wall, a logarithmic temperature profile is found,and in the outermost parts of the boundary layer, turbulent mixing is thedominant mode of transport

The boundary layer ends where turbulence dies out and uniform stream conditions prevail, with the result that the thermal and momen-tum boundary layer thicknesses are the same At first, this might seem

free-to suggest that an absence of any Prandtl number effect on turbulentheat transfer, but that is not the case The effect of Prandtl number isnow found in the sublayers near the wall, where molecular viscosity andthermal conductivity still control the transport of heat and momentum

The Reynolds-Colburn analogy for turbulent flow

The eddy diffusivity for momentum was introduced by Boussinesq [6.11]

in 1877 It was subsequently proposed that Fourier’s law might likewise

be modified for turbulent flow as follows:

q = −k ∂T

∂y −

another constant, whichreflects turbulent mixing

where ε h is called the eddy diffusivity of heat This immediately suggests

yet another definition:

turbulent Prandtl number, Prt ≡ ε m

§6.8 Heat transfer in turbulent boundary layers 323

Equation (6.103) can be written in terms of ν and εm by introducing Pr

and Prt into it Thus,

Before trying to build a form of the Reynolds analogy for turbulent

flow, we must note the behavior of Pr and Prt:

• Pr is a physical property of the fluid It is both theoretically and

actually near unity for ideal gases, but for liquids it may differ from

unity by orders of magnitude

• Pr t is a property of the flow field more than of the fluid The

nu-merical value of Prtis normally well within a factor of 2 of unity It

varies with location in the b.l., but, for nonmetallic fluids, it is often

near 0.85

The time-average boundary-layer energy equation is similar to the

time-average momentum equation [eqn (6.90a)]

and in the near wall region the convective terms are again negligible This

means that ∂q/∂y 0 near the wall, so that the heat flux is constant in

y and equal to the wall heat flux:

The constant A depends upon the Prandtl number It reflects the thermal

resistance of the sublayer near the wall As was done for the constant

B in the velocity profile, experimental data or numerical simulation may

be used to determine A(Pr) [6.12,6.13] For Pr≥ 0.5,

A(Pr) = 12.8 Pr 0.68 − 7.3 (6.109)

To obtain the Reynolds analogy, we can subtract the dimensionlesslog-law, eqn (6.99), from its thermal counterpart, eqn (6.108):

T w − T (y)

q w /(ρc p u ∗ ) − u(y) u ∗ = A(Pr) − B (6.110a)

In the outer part of the boundary layer,T (y) T ∞ and u(y) u ∞, so

C f = A(Pr) − B (6.110d)Rearrangment of the last equation gives

q w

(ρc p u ∞ )(T w − T ∞ ) = C f 2

1+ [A(Pr) − B]4C f 2 (6.110e)The lefthand side is simply the Stanton number, St= h (ρc p u ∞ ) Upon substituting B = 5.5 and eqn (6.109) for A(Pr), we obtain the Reynolds-Colburn analogy for turbulent flow:

1+ 12.8Pr0.68 − 1 4C f 2 Pr≥ 0.5 (6.111)This result can be used with eqn (6.102) for Cf , or with data for C f,

to calculate the local heat transfer coefficient in a turbulent boundary

layer The equation works for either uniform T w or uniform q w This isbecause the thin, near-wall part of the boundary layer controls most ofthe thermal resistance and that thin layer is not strongly dependent onupstream history of the flow

Equation (6.111) is valid for smooth walls with a mild or a zero

pres-sure gradient The factor 12.8 (Pr 0.68 − 1) in the denominator accounts

for the thermal resistance of the sublayer If the walls are rough, thesublayer will be disrupted and that term must be replaced by one thattakes account of the roughness (see Sect.7.3)

§6.8 Heat transfer in turbulent boundary layers 325

Other equations for heat transfer in the turbulent b.l.

Although eqn (6.111) gives an excellent prediction of the local value of h

in a turbulent boundary layer, a number of simplified approximations to

it have been suggested in the literature For example, for Prandtl numbers

not too far from unity and Reynolds numbers not too far above transition,

the laminar flow Reynolds-Colburn analogy can be used

The best exponent for the Prandtl number in such an equation actually

depends upon the Reynolds and Prandtl numbers For gases, an exponent

of−0.4 gives somewhat better results.

A more wide-ranging approximation can be obtained after

introduc-ing a simplifed expression for C f For example, Schlichting [6.3, Chap XXI]

shows that, for turbulent flow over a smooth flat plate in the low-Re range,

Somewhat better agreement with data, for 2× 105  Re x  5 × 106, is

obtained by adjusting the constant [6.15]:



k L



dx



§6.8 Heat transfer in turbulent boundary layers 325

Other equations for heat transfer. ..