Appendix_A.pdf

Appendices

595

Appendix A

Boundary layer survey

NOTE: This section appears to have been part of the chapter on the

boundary layer (Chapter 4 at the end of Section 4.9.4) but was pulled

out as a separate file and labeled “Appendix A.” It has no title of its

own. The file name is “highre.tex,” presumably for “high Reynolds

number.” –B. Coles

In 1962, I undertook an extensive survey of experimental data in low-

speed turbulent boundary layers at constant pressure, in an attempt

to identify a fully developed (standard, normal, ideal, equilibrium,

asymptotic) state and determine its properties. My objective at that

time was to establish a point of departure for a study of compressibil-

ity. My survey appeared as Appendix A of a RAND report (COLES

1962), but was never published outside of the subliterature. Because

the work is not readily accessible, I will summarize here my methods

and conclusions. I set out to test a large number of mean-velocity pro-

files for their consistency with the momentum-integral equation and

the momentum-defect law, which is to say the departure of the outer

part of the profile from the logarithmic law of the wall. The next four

figures

are copied from Appendix A to show the test method. Given

a profile and a value for

, I first determined a value for

in equation

(xxx)

that would put one point belonging to the hypothetical log

Presented here as Figures A.2, and A.3

Given as equation (A-1) on p. 54 of COLES 1962 Rand Report, available at

597

598

APPENDIX A. BOUNDARY LAYER SURVEY

Figure A.1:  Fit of typical profile to logarithmic

formula (figure from Appendix A of COLES 1962).

region on the straight line defined by the particular constants of the

time,

= 0

41 and

= 5

0, as shown in FIGURE A.1. I then drew a

parallel straight line through the point of maximum departure of the

profile from the log line. Great precision was not needed for this oper-

ation, and there was no formal curve-fitting. (Both the non-linear re-

gression scheme of Levenberg and Marquardt (SECTION X) and the

corner modification proposed by Sandham (SECTION Y) were still

in the future.) The vertical distance between the two lines, labeled

∆

u/u

, is a measure of the strength of the wake component of the

profile. I found that this quantity, although is represents only about

ten percent of

∞

for flow at constant pressure, was distinguished

by an almost exquisite sensitivity to the history and environment of

each particular flow. This property in turn made possible not only a

close classification of boundary-layer flows at constant pressure, but

a refinement and rationalization of the similarity laws for the profile.

The upper part of FIGURE A.2 shows ∆

u/u

as a function of

the local Reynolds number

for flows that I classified as normal.

http://www.rand.org/content/dam/rand/pubs/reports/2006/R403.pdf

599

Figure A.2:  Upper plot: Strength of the wake component

in equilibrium turbulent flow. Lower plot: Momentum

balance for the data of the upper plot (figures from

Appendix A of COLES 1962).

600

APPENDIX A. BOUNDARY LAYER SURVEY

The lower part of the figure compares two estimates of the surface

friction

for the same data; first, the momentum-integral result

ρu

∞

dθ/dx

, where

is the momentum thickness, and second, the

result from the graphical procedure for

in the form

ρu

. The

agreement is generally within ten percent, and usually better. I take

this agreement as strong evidence in favor of identifying

with

(

/ρ

)

By way of contrast, FIGURE A.3 shows the corresponding

quantities for flows that I classified as abnormal. I noted as a sig-

nificant point of technique that the normal data were obtained for

the most part in closed wind tunnels, either on plates having blunt

leading edges or some equally effective tripping means, or on tunnel

walls having a long approach length. Some of the anomalous data

were obtained in open-jet tunnels using models not equipped with

adequate side plates. In view of the generally poor momentum bal-

ance in FIGURE A.3, I blamed the anomalies in these flows for the

most part on three-dimensionality of the mean flow.

One major finding of the study, which perhaps should have

been anticipated, was that ∆(

u/u

) decreases, and hence that the

traditional defect law fails, as the Reynolds number

decreases be-

low a value of about 5000. In fact, the wake component in FIGURE

A.2 disappears entirely and rather abruptly by the time

has de-

creased to a value near 500. This behavior was present in all of the

data, I therefore do not view it normal or not. The behavior might be

viewed as a residual effect of transition, because I would not expect

such a high degree of commonality in such a diverse population of

data. A better hypothesis is that the flows are fully turbulent and

in equilibrium, in the special sense that two characteristic scales

and

ν/u

are emerging and separating from each other as discrete

parameters for turbulent flow near a wall. It is possible, although dif-

ficult to prove experimentally, that the constants in the log law are

also evolving. If so, I doubt that the graphical classification scheme

of FIGURE A.1 is seriously compromised.

In the RAND report I also looked at the strength of the wake

component in the presence of high stream turbulence and in the

flow downstream from very strong tripping devices. These effects

601

Figure A.3:  Upper plot: Strength of the wake component

in anomalous turbulent flow. Lower plot: Momentum

balance for the data of the upper plot (figures from

Appendix A of COLES 1962).

602

APPENDIX A. BOUNDARY LAYER SURVEY

will not be discussed here. (

Elsewhere?

) I turn instead to my second

major finding, which was that the quantity ∆

u/u

seemed to decrease

substantially at Reynolds numbers larger than the upper limit in

FIGURES A.2–A.3. This second finding, if correct, signals a serious

and perhaps fatal defect in the defect law.

In 1962, almost the only reliable data in low-speed boundary-

layer flow at high Reynolds numbers were the careful and extensive

measurements by SMITH and WALKER (1958), which offered them-

selves by default as definitive for Reynolds numbers

from 15000

to 50000. I found no evidence that these measurements might be af-

fected by pressure gradient, stream turbulence, or three-dimension-

ality. My conclusion at the time was that these data could only be

questioned on some other ground. Failing this, the defect law is not

valid at the high level of precision attempted in my survey.

Fortunately, there is other ground. Because the freestream ve-

locities in the experiments by Smith and Walker reached 110 meters

per second, I propose here to make one more test of these and certain

other data, a test based on the premise that the apparent problem

with the defect law may be solved by considering the effect of com-

pressibility. I cannot recall why I did not test this hypothesis in my

1962 report, except that suitable descriptions of mean-velocity pro-

files, including my own failed description of 1962, were not part of the

machinery of the time. In particular, in 1962 the proposal by VAN

DRIEST (1951) was still a decade or more away from being gener-

ally accepted as the best available means for organizing the effects

of compressibility. This proposal will be examined to what follows.

Standard methods exist for data processing in studies of tur-

bulent boundary layers in compressible fluids. The fluid is invariably

assumed to be a perfect gas, with equation of state

ρRT .

(A.1)

The two specific heats

and

are taken as constants, as are their

combinations

−

and

. The instrument of choice is

the impact or total-pressure tube. Almost without exception, each

measurement of velocity begins with the local Mach number

which is inferred from the ratio of impact pressure to static pressure.

603

If the flow is supersonic, the operational equation (LIEPMANN and

ROSHKO 1957) is

′

(

+ 1

)

−

(

+ 1

−

+ 1

)

−

(A.2)

where the prime denotes probe impact pressure behind a normal

shock wave, and

is the static pressure at the probe entrance in

the absence of the probe. The latter pressure is usually measured at

an adjacent wall or is computed by assuming isentropic expansion

to a Mach number

∞

in the free stream. In either case the static

pressure is taken as constant through the boundary layer. If the flow

is subsonic and thus free of shock waves, equation (A.2) is replaced

(

1 +

−

)

−

(A.3)

Finally, if the Mach number is much less than unity, the last equation

reduces to Bernoulli’s integral,

ρu

(A.4)

The last two equations are the ones plotted in FIGURE 1.1

of the introduction. I will assume in what follows that accuracy in

measurement of

and

′

and hence of the local Mach number

have been accurately measured. I will also ignore for the moment

any corrections for effects of turbulence or mean-flow gradients on

the probe readings.

The state equation (A.1) and the condition

∞

re-

duce the number of independent thermodynamic variables from three

to one. One thermodynamic quantity must therefore be measured or

assumed. The usual choice is the local stagnation temperature

the local static temperature

. The definition of

for a perfect gas

with

u/a

(

γRT

)

[

1 +

(

−

)

]

(A.5)

604

APPENDIX A. BOUNDARY LAYER SURVEY

If the flow is laminar, there exist under certain conditions an

energy integral, which is to say a relation between temperature and

velocity, or more accurately between enthalpy and kinetic energy,

that satisfies the equations of motion and is valid independent of

position in the flow. Suppose that the wall temperature

is constant

and the Prandtl number

(A.6)

is equal to unity. With the further restriction that there is no heat

transfer, a primitive energy integral, first found by BUSEMANN

(1931), is

= constant =

∞

(A.7)

There is no restriction on pressure gradient. A different integral ob-

tains if there is heat transfer, still with

= constant and

= 1,

but now with the restriction of constant pressure. For these con-

ditions a generalization of Busemann’s energy integral was found

by CROCCO (1932), and independently by BUSEMANN (1935)

(

check

);

= constant =

(A.8)

where the parameter

will be shown shortly to be a measure of heat

transfer at the wall.

The relations just given are often summarily adopted as a

model for turbulent flow. FERNHOLZ and FINLEY (1980) recom-

mend a relationship between temperature and velocity originally pro-

posed by WALZ (1966) (

check

∞

(A.9)

This expression is formally identical with the Crocco-Busemann in-

tegral (A.8) but has no physical basis in turbulent flow except that

it is capable of satisfying the two boundary conditions

= 0

, T

∞

(A.10)

605

It follows that

∞

, B

= 1

−

∞

(A.11)

The derivative of equation (A.9) at the wall is

∞

(

∂T

∂y

)

∞

(

∂u

∂y

)

(A.12)

Hence

= 0 corresponds to adiabatic flow at constant wall temper-

ature. The discussion hereafter will be limited to this special case of

zero heat transfer. (

A reference

The energy integral (A.9) with

= 0 becomes

∞

(

−

∞

)

∞

(A.13)

In most experiments in compressible fluids, as already pointed out,

the measured quantity is the Mach number, and it is necessary to

prepare equation (A.13) for this situation. For a perfect gas, the

temperature, velocity, and Mach number are related by the definition

;

ρu

γpM

(A.14)

Whether or not the pressure

depends only on

, the ratio

p/ρ

is always equal to

. Hence a suitable normalized form of equa-

tion (A.14) is

∞

(A.15)

When this equation is used to eliminate the velocity in equation (A.13),

the result is

= 1

−

(

−

∞

)

∞

(A.16)

For the sake of symmetry, the energy integral (A.13) can be rewritten

= 1 +

(

∞

−

)

∞

(A.17)

606

APPENDIX A. BOUNDARY LAYER SURVEY

Within this formulation, the Mach number determines the tempera-

ture, and the temperature determines the velocity (and the density).

In dimensional form, the argument assumes a knowledge of

and

one of the three parameters

∞

, together with use of the

rigorous definitions (A.5) and (A.14).

To recapitulate, the preceding discussion refers to impact-probe

measurements of the Mach-number profile in adiabatic boundary lay-

ers. Nothing in the discussion requires that the flow be specified as

laminar or turbulent. If the flow is laminar and adiabatic and the

Prandtl number is unity, the relationships are rigorous within the

boundary-layer approximation (

∂p/∂y

= 0) and the usual consider-

ations of experimental accuracy. Moreover, the wall temperature is

equal to the free-stream stagnation temperature, according to equa-

tion (A.7). It is a difficulty, readily overcome by numerical means

if the flow is laminar, that the Prandtl number for common gases is

nearly constant but usually at a value near 0.7 rather than unity. This

difficulty is usually expressed by introduction of a recovery factor

defined as

−

∞

−

∞

(A.18)

where

, the recovery temperature, is the wall temperature

when

there is no heat transfer. In practice, the recovery factor is close to

unity, although it is different for laminar and turbulent flows.

(Paragraph on high thermal price, case

→

∞

Pr >

etc.)

Experimenters may sometimes have direct access to the veloc-

ity in high-speed flow through laser Doppler velocimetry or particle-

image velocimetry or the like; all these techniques face formidable

difficulties in supersonic flow. In such cases, the temperature can be

estimated from equation (A.17) in order to determine the density,

which is required for any test of momentum balance.

(Section on experimental

profile.)

There is not much evidence that the energy ansatz (A.9) is a

real improvement over other possible forms, such as the form

∞

, commonly used, or the form

, both of which have no

607

less claim to validity than the form (A.9). To see the effect of errors in

, suppose first that

, and

are known exactly. Hence so is

from

= 1 + (

−

2. So is

, from

(

γRT

)

. So is

, from

ρu

γpM

. If

is known but

is estimated rather than

measured, with a local relative error of

, then the local relative

errors in

, and

ρu

are

−

, and

−

2, respectively.

Van Driest.

The most widely accepted scheme for comparing

data for

= 0 with data for

= 0 is the scheme now called

Van Driest II. It has been endorsed by Fernholz and Finley in their

massive survey of data for

= 0, and by numerous other authors

(

Name some

.). These authors sometimes refer to the scheme as a

transformation, but I prefer to reserve this term for a relationship

based on the equations of motion, and to use the term “mapping” for

a relationship based on variables only. The compressibility mapping

proposed by VAN DRIEST (1951) begins with the mixing-length

expression

ρ`

(

)

(A.19)

together with Prandtl’s hypothesis

κy ,

(A.20)

and arrives at the ansatz

κy

(A.21)

The appearance of the combination (

) suggests that the physi-

cal velocity

can be replaced by an effective velocity

∗

, say, defined

by d

∗

∼

, or by its definite integral,

∗

∫

(

∗

)

u ,

(A.22)

where

∗

is a constant reference density included for dimensional

reasons. It is apparent that the Van Driest mapping (A.22) has the

effect of rotating the profile in the counter-clockwise direction in the

608

APPENDIX A. BOUNDARY LAYER SURVEY

usual semi-logarithmic coordinates, since

is small where

is small,

near the wall, and

is large where

is large, near the free stream.

This is the property that makes the scheme an attractive device in

any attempt to restore the defect law to respectability, as noted at

the beginning of this section.

An alternative expression for the mixing length is Karman’s

similarity hypothesis,

(A.23)

This form was proposed simultaneously and independently by Wilson

(1950), and both Wilson and Van Driest developed their hypothesis

into formulas for skin friction as a function of Mach number and

Reynolds number (

check

). These two authors were not alone; sev-

eral other authors, working independently, used the same or similar

methods and approximations to complete their analyses. The situ-

ation in 1953 was surveyed by COLES (1953) and by CHAPMAN

and KESTER (1953), and the various proposals to that time were

collected by Chapman and Kester in a celebrated figure that is re-

produced here as FIGURE 4.xx.

There were few competent mea-

surements in 1951, and it should not be surprising that the various

predictions filled a plot of

against

almost uniformly densely.

Chapman and Kester used the designations Van Driest I and

II and for the Prandtl and Karman forms, although they misplaced

the second one in their figure. A later survey by SPALDING and

CHI (1964) used the same designations, but reversed taking Karman

as I and Van Driest as II. In fact, Van Driest and Wilson were fully

informed very early about their respective contributions, according

to the proceedings of a Navy conference in 1951. Van Driest in 1956

described both models, Prandtl first and Karman second, without

attributing either. The confusion was made permanent by Spalding

and Chi, and made permanent by FERNHOLZ and FINLEY (1978?)

(

Check all this.

) It is only necessary to know that Van Driest’s

first and only analysis was based on the Prandtl model and is now

universally referred to as Van Driest II. Details follow.

Unclear reference.

609

The definition (A.22) is readily integrated in closed form for

the energy integral (

)

for adiabatic flow, putting

∞

for

ρ/ρ

∞

The result is the Van Driest mapping for velocity;

(

∗

)

∗

∞

= sin

−

(

∞

)

(A.24)

where

, defined by (

check

)

−

∞

(

−

)

∞

1 +

(

−

)

∞

(A.25)

lies between 0 and 1 and in any case requires

> T

∞

. Given

the specified energy integral, the dimensionless mean-velocity pro-

file

u/u

∞

can be replaced at each value of

∗

∞

sin

−

(

∞

)

sin

−

(

)

(A.26)

without regard to the definition of

∗

. Incidentally, the sin

−

operator

is the unique signature of the Van Driest mapping for any energy law

that has

quadratic in

The mapping defined by equation (A.26) is not based on any

observable process or mechanism. Whether or not the accepted sim-

ilarity laws remain valid for the mean velocity profile after the Van

Driest mapping is a question to be settled experimentally. Before the

evidence can be tested, the friction velocity

and some other con-

stants have to be redefined in a plausible way for the case of variable

density.

For this purpose, integrate the mixing-length equation (A.21)

formally, with d

= (

∗

/ρ

)

∗

, to obtain the modified law of the

wall,

∗

(

∗

)

(

∗

)

+ constant

(A.27)

Unclear equation reference.

610

APPENDIX A. BOUNDARY LAYER SURVEY

where

∗

is a constant reference length also included for dimensional

reasons. Equation (A.27) is typical of mixing-length formulas in that

it is at best an unclear description of a fragment of the mean-velocity

profile. The choices for

∗

and

∗

and the value of the constant in

equation (A.27) are customarily resolved by an extension of the map-

ping to the wall, taking care to be consistent with the earlier treat-

ment of an incompressible fluid in SECTION X. First, write the

functional dependence in the profile formula (A.27) in dimensionless

form as

(

∗

)

∗

(

∗

)

(A.28)

Near the wall, this becomes approximately

(

∗

)

∗

(A.29)

and

approach zero, the definition (A.22) and its integral (A.24)

both lead to

∗

(

∗

)

u .

(A.30)

Finally, the requirement of Newtonian friction at the wall implies, to

the same approximation,

(A.31)

When

∗

is eliminated between equations (A.29) and (A.30), and

u/y

is eliminated between this result and equation (A.31), the constant

of integration

∗

emerges in terms of well-defined quantities,

∗

(

)

(A.32)

At the same time, a generalized friction velocity

emerges as

(

)

(A.33)

611

and equation (A.27) becomes

(

∗

)

∗

(

)

+ constant

(A.34)

This form and equation (A.24) strongly suggest, although they do

not require, defining

∗

(A.35)

whereupon

∗

(A.36)

very near the wall.

As usual, this reasoning for

→

0 is not part of the mixing-

length argument, which applies only for fully turbulent flow outside

the sublayer. Given the choices (A.33) and (A.35), then in a usual

notation equation (A.27) becomes

c ,

(A.37)

where now

∗

(A.38)

and

∗

∞

= sin

−

(

∞

)

(A.39)

The fragile derivation just given, with Prandtl’s equation (A.20)

for

, is commonly referred to as Van Driest II. The choice for

∗

, and

∗

is important because it controls the dependence of the

generalized

and

∞

and

. What is wanted is the particu-

lar choice that minimizes this dependence. There is substantial evi-

dence, for example, in papers by FENTER and STALMACH (1957),

ROTTA (1960), MOORE and HARKNESS (1964), MAISE and MC-

DONALD (1968), MICHEL, QUEMART, and ELENA (1969), DAN-

BERG (1971), SQUIRE (1971), and FERNHOLZ (1976) that use of

wall quantities as in equations (A.37)–(A.39) is very nearly optimum

612

APPENDIX A. BOUNDARY LAYER SURVEY

from this point of view, at least for adiabatic flow at constant pres-

sure at Mach numbers up to perhaps 5.

Most of these authors have also gone beyond the mixing-length

argument to consider a more general fit to a defect law or to a

combined wall-wake formulation of the mean profile, in the manner

adopted by COLES (1968) for low-speed flow; i.e., a fit to

+ 2

sin

η ,

(A.40)

where

(A.41)

The present method for determining the strength of the wake

component is the third in an evolving series. In 1962 the fit of the

mean-velocity profile used only one point. In 1968 I tried to involve

a fit of the entire profile to equation (A.40) but had to finesse the

problem of a misfit near

by omitting data for

y/δ

greater than

some threshold value noted in the tabulation in the “Young person’s

guide.” In this monograph, I have made room for the omitted data by

using the Sandham scheme (SECTION 4.9.4) for rounding the profile

near

. In addition, the constants

and

now have new val-

ues based on Zagarola’s pipe measurements (SECTION 2.5.7). The

constants

and

are here given their new incompressible values,

= 0

435 and

= 6

10. The parameters

, and

(need equa-

tion) are then determined by a three-parameter least-squares fit of

the experimental data to equation (A.40), after eliminating Π tem-

porarily with the aid of the constraint imposed by the local friction

law,

∞

+ 2

(A.42)

The quality of Van Driest scaling, when universal constant val-

ues are assumed for

and

, can be tested in different ways. One test

is to compare values inferred for the local friction coefficient (

does

this make sense

= 2

∞

(

∞

)

(A.43)

613

with values obtained by other means. A second test is to compare

values obtained for the profile parameter Π with corresponding val-

ues for low-speed flow. This second comparison will be made first and

will lead to the conclusion is that there is very little effect of com-

pressibility on the shape of the mean-velocity profile in Van Driest

II coordinates, at least for Mach numbers up to about 3.

Such tests are not new. The first paper to compare various

mappings of

was the extensive survey by SPALDING and CHI

(1964). Tests were also carried out by JACKSON et al (1965), PE-

TERSEN (19xx), MILES and KIM (19xx), DANBERG (1971), HOP-

KINS and INOUE (1971), and WINTER and GAUDET (1973).

These efforts are not necessarily redundant, since they differed in

their choice of data, viscosity law, and handling of temperature. The

consensus is that the Van Driest scheme is at least as good as any

other when taken as a high-level technical application.

Full profile fits and reports of wake strength have been car-

ried out by (Winter, Gaudet, others). Among the most satisfactory

studies to my mind is one by D. Collins at JPL, for which I was con-

sultant and eventually co-author (COLLINS, COLES, and HICKS

1978). The invariance of the defect law under Van Driest mapping

was strongly supported by these data for Mach numbers up to 2.2.

FERNHOLZ and FINLEY (1977) in their massive catalog of

boundary-layer measurements involving compressibility, did not in-

clude the operation of curve fitting for the mean velocity profile. In

a second volume (1980), they provided numerous plots in Van Driest

coordinates, but still without a fitting operation. The survey has a

large clerical component limited mainly to major issues such as ef-

fects of flow history and the validity of empirical energy integrals.

I have relied very heavily on this survey in the new analysis that

follows. (

Say how to get data

I have used this scheme before in connection with work by

D. Collins at JPL, in which I participated as consultant and co-

author. The objective was to document a set of flows for LDV mea-

surements of

′

in supersonic flow, a quantity that was then under

a cloud, and perhaps still is. The next few paragraphs are borrowed

614

APPENDIX A. BOUNDARY LAYER SURVEY

from that report.

Values for viscosity are obtained from the Sutherland viscosity

law,

(

)(

)

(A.44)

where

= 291

75 K,

= 110 K, and

= 1

827

−

gm/cm-sec.

Integral thicknesses for the boundary layer are computed from

∗

∫

(

−

ρu

∞

)

dy ,

(A.45)

and

∫

ρu

∞

(

−

∞

)

dy .

(A.46)

The boundary-layer form parameter

is defined as

∗

(A.47)

For two-dimensional mean flow, the surface friction can be ob-

tained from von K ́arm ́an’s momentum-integral equation, (

What is

cap

= 2

dθ

−

(

2 +

−

∞

)

γM

∞

(A.48)

The accuracy of equation (A.48) is expected to be low, primarily be-

cause of difficulty in differentiating experimental data for

(

) and

∞

(

) (Table A3).

For the JPL measurements, the second term in

equation (A.48) is at most 3 percent of the first term, and is uncer-

tain by a comparable amount. Hence this term has been discarded.

Values for

= 2

dθ/dx

are listed in Table 3, which compares values

obtained for

by this and several other methods.

Table A3, and Table 3 mentioned later, are both in COLLINS, COLES, and

HICKS, 1978, available at https://apps.dtic.mil/dtic/tr/fulltext/u2/a053790.pdf

615

Note that these measurements at JPL were the last hurrah of

the 20-inch tunnel before it was dismantled and moved to Langley

in 1980. Hence there was ample time to do the work well. Note also

that the boundary layer experiments reported in my thesis at Caltech

were almost the first tests conducted in this tunnel in 1951, 29 years

earlier.