Improved Entrainment Turbulent Boundary Method Layer for

Improved Entrainment Turbulent Boundary Method Layer for

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R. & M. No. 3643
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.i' '!( !:, J
MINISTRY OF TECHNOLOGY
AERONAUTICAL RESEARCH COUNCIL
REPORTS
AND
MEMORANDA
Improved Entrainment Method for Calculating
Turbulent Boundary Layer Development
By M. R. HEAD and V. C. PATEL
Cambridge University Engineering Department
L O N D O N : H E R MAJESTY'S S T A T I O N E R Y O F F I C E
1970
PRICE 14s Od [70p] NET
Improved Entrainment Method for Calculating
Turbulent Boundary Layer Development
By M. R. HEAD and V. C. PATEL
Cambridge University Engineering Department
Reports and Memoranda No. 3643*
March, 1968
Summary.
A simple integral method is presented for the calculation of two-dimensional incompressible turbulent
boundary layers.
Use is made of established relationships to determine values of the entrainment coefficient for equilibrium layers, and the entrainment coefficient for non-equilibrium conditions is obtained by multiplying
the corresponding equilibrium value by a suitable empirical function. This increases the entrainment when
the rate of growth of the layer is less than that of the corresponding equilibrium layer, and decreases it
when the rate of growth is greater. This variation of entrainment is in accordance with observation, and a
simple physical explanation !s proposed to account for it. This explanation further suggests how the
effects of flow convergence may be taken into account.
Comparisons with measured boundary-layer developments show the general accuracy of the method.
LIST OF CONTENTS
Section
1.
Introduction
2.
The Original Entrainment Method
3.
Shortcomings of the Original Method
4.
Development of the Present Method
5.
Physical Considerations
6.
Extension of the Calculation Method to deal with Three-Dimensional Effects
7.
Cases Treated
*Replaces A.R.C. 31 043.
8.
Discussion
9.
Conclusions
Acknowledgements
References
Appendix
Table 1
Illustrations--Figs. 1 to 16
Detachable Abstract Cards
1: Introduction.
At a recent symposium held at Stanford ~, some thirty different methods of calculating the development
of the turbulent boundary layer were compared and assessed. Most of the methods had been developed
comparatively recently and some showed very satisfactory agreement with at least a majority of the wide
range of experimental data against which the calculations were tested. Among the most successful methods,
those of Bradshaw, Nash and Felsch are probably worthy of mention. An early method due to one of the
present authors 3 showed up reasonably satisfactorily and was placed in the middle third, and it is a
development of this method that is presented here. As it now stands the method would appear to be at
least as accurate as the best presented at Stanford while being sufficiently simple to enable a hand calculation to be completed in an hour or two, and a computer calculation in a very few seconds.
Although the modifications to the original method were arrived at largely empirically, some justification
on physical grounds can be provided, as will be seen later. A brief account of the development of the
present method is given in the following sections. Comparisons with experiment are presented towards
the end of the report.
2. The Original Entrainment Method.
In this, the very simple assumption was made that the rate at which free-stream fluid was incorporated
into the boundary layer was determined by the velocity defect in the outer part of the layer, and two
measured boundary-layer developments were analysed and used to relate the non-dimensional rate of
entrainment to the boundary layer form parameter H*
-=
were used to relate H* to the more conventional form parameter H
. The same sets of measurements
~ ~-
. Thus, the non-dimensional
entrainment rate CE was given by
ldQ
1 d
C~ = ~ d--x= U d-~ [ U(6-3*)-] = F(H*)
and H* = G(H).
The two functions F and G were presented as curves which could be approximated by simple analytic
expressions for computer calculations.
It will be seen that the functions F and G serve as an auxiliary equation for the calculation of H develop-
ment. If the value of H" is known at any point, the non-dimensional entrainment is also known, and from
this the increment in U(8-8*) over a step can be obtained. The momentum integral equation gives a
corresponding increment in UO, and the values of U(6 - 8*) and UO at the end of the step give the value of
H* there, and hence the corresponding value of H, which enables the calculation to proceed.
3. Shortcomings of the Original Method.
The assumption that the entrainment is uniquely related to the boundary-layer form parameter is
obviously likely to prove an over-simplified one, and it'is perhaps surprising that it gives satisfactory
results in quite a wide range of cases, including, in particular, layers with injection. Where the method
does fall down is in the case of equilibrium flows and flows where a strong adverse pressure gradient is
followed by a region of zero pressure gradient. In both these cases the trends are correctly predicted but
actual magnitudes are appreciably in error. For example, the method would predict a value of H of
approximately 1.8 for Bradshaw's equilibrium flow a = -0.255, as compared to the measured value of
approximately 1.59, and when this adverse pressure gradient is followed by a region of constant pressure,
the value of H tends to the zero pressure gradient value, but much too slowly.
4. Development of the Present Method.
The foregoing observations, if closely examined, are sufficient to suggest that. if the entrainment in the
equilibrium layer is taken as datum, then, for layers proceeding to separation, the entrainment should be
reduced, and for layers where H is decreasing it should be increased.
We can secure this result by multiplying C~ for the equilibrium layer by a suitable function of some
parameter that measures the departure from equilibrium conditions. There are a large number of para-
1 d(UO)/[1 d(UO)]
dx
U dx
~
meters that could be used for this purpose; the one actually chosen is the ratio U
for which we use the symbol rl. If rl = 1 then we have equilibrium conditions; ifr~ > I then we have a
layer with Ro increasing more rapidly than in the equilibrium case, and H increasing; if rl < 1 we have a
layer with Ro increasing less rapidly than in the equilibrium case, and H decreasing.
We therefore write CE = (Ce)eqF(rO, where (CE)~q is the entrainment coefficient for the equilibrium
layer and F(rO is a function that must satisfy the following conditions.
(i) For rl = 1 (i.e. equilibrium conditions) F(rl) = 1.
(ii) For r 1 > 1 (i.e. layers with R o growing more rapidly than the corresponding equilibrium layer)
F(rl) < 1, but however large r 1 becomes F(r 0 must remain positive, since there is no reason to
expect that the entrainment should become negative in a boundary layer that is growing rapidly.
(iii) For r 1 < 1 (i.e. layers with R o growing less rapidly than the corresponding equilibrium layer)
F(rl) > l, but however small, or even negative, r l should become, F(rl) should remain finite and
probably not greater than (say) 2.
Now, a simple function that satisfies these requirements is shown in Fig. 1. This particular function
was chosen quite arbitrarily, and although a more suitable function may well exist, there is no indication
from the calculations so far performed that this should be the case.
With the function F(r 1) determined, CE can be found for given values of H and Ro if (Ce)~q is known as
function of these variables. In equilibrium layers, H and H* vary only slowly with x and we can therefore
write
(c~)eq = ~ ~
Further,
u(~-8
eq
.
.
U dx
.
.
eq
U
dx
~q
(1)
1 d(UO)1
U
-~x ]ea can readily be found from the ~ - G relation given by Nash s by making use of
the momentum integral equation and a skin-friction relation such as that given by Thompson 6 and shown
in Fig. 2.
I
1 d(UO)
U
dx ] eq is shown as a
function of H and Ro in Fig. 3.
To complete the calculation method all that is required is a relation between H* and H, and this is
provided by Thompson's profile family6, 6 being taken for the present purpose as the value ofy for which
u = 0.995. H* is shown as a function of H and Ro in Fig. 4.
U
The only further point to be remarked is that, instead of using equation (1), it may be somewhat more
convenient to use an equation giving the rate of change of H* explicitly. Thus
o eH *
l e(vo)]
(2)
The derivation of equation (2) and details of the procedure used to determine
in the Appendix.
The method may now be described as follows.
The momentum integral equation is used in the form
1 d(UO) _ c~-'H+l't
U
dx
I
1 d(UO)
U
dx l eqaregiven
) 0 dU
2
(3)
U dx
to find the increment in UO over a step.
The value of r I is obtained from equation (3) and the corresponding value of
U
dxx _1eq' F(h) is
obtained from the value ofrl and H* is known from the values of H and Ro. Equation (2) can thus be used
to find the increment in H*.
The values of H* and Ro at the end of the step give the value of H and the calculation can proceed.
For hand calculations we use Figs. 1, 2 and 3 and 4.
For computer calculations Fig. 1 is replaced by the relations
1
Fif0-2rl_l
and
for
r1>/1,
5 - 4 r l for
F(rl)- 3_2r 1
rl < 1,
Fig. 2 may be approximated by the analytic relation
c$ = exp (a H + b)
where
a = 0'019521-0'386768c+0"028345c2-0"000701c 3
b=O.191511-O.834891c+O.O62588cZ-O.O01953c3
and
c = logeR0
and Fig. 3 is replaced by the use of equations A. 1, A.2 and A.3 given in the Appendix. It is still necessary,
however, to store Fig. 4, and in Table 1 values of H(H*, Ro) are given at sufficiently close intervals for
linear interpolation to be used with acceptable accuracy.
4
5. Physical Considerations.
It is well established that the outer part of the turbulent boundary layer consists of a series of billows
or clumps or eddies of turbulent fluid (see, for example, Fig. 1 in Reference 2). It is the existence of these
that is responsible for the phenomenon of intermittency. Smoke observations show that the billows (as
we shall call them) are essentially three-dimensional in character, with a cross-stream dimension that is
generally similar to that in the streamwise direction. Since the outer boundary of the turbulent flow is normally defined by the outer boundaries of these billows, it seems reasonable to suppose that the closeness
with which they are packed will define the distribution of intermittency; the closer the packing (in terms
of boundary-layer thickness) the narrower the spread of intermittency and the more it will be confined to
the region of small velocity defect close to the boundary-layer edge.
In an equilibrium layer, we should expect an equilibrium spacing of billows, and a corresponding
distribution of intermittency, which, in terms of boundary-layer thickness should be largely independent
of Reynolds number, though it may be expected to depend upon the value of the pressure gradient parameter re.
Now, any retardation of the flow beyond that occurring in equilibrium conditions will lead to a closer
packing of the billows and hence a reduction in the spread of intermittency, while any relaxation will lead
to a wider spacing and hance an increased spread of intermittency. It is to be expected that, for a given
distribution of mean velocity in the outer part of the layer (i.e. a given value of H), these changes in the
distribution of intermittency will be reflected in corresponding changes in the entrainment.
By whatever mechanism the entrainment of free-stream fluid is assumed to occur, it would seem that
it must be reduced if the intermittency distribution is confined to a narrow region of small velocity defect,
and increased by the presence of a more irregular boundary, which pentrates more deeply into the layer.
We may then reasonably assume that the packing of the billows and the resulting distribution of
intermittency will have an important effect on the entrainment, over and above that exerted by the defect
of mean velocity in the outer part of the layer. The ratio rl may be takenas a measure of the closeness of
this packing.
The foregoing account of the role played by the packing of billows is, of course, purely speculative but
it has some degree of plausibility and accords with observations of the smoke-filled boundary layer.
Certain aspects of the present hypothesis can readily be checked by making measurements of intermittency
in a variety of situations, and it is hoped to undertake such an investigation in the future.
From the arguments outlined above we may also deduce that convergence of the flow will increase the
closeness of packing of the billows and hence reduce the entrainment, while divergence of the flow will
have the opposite effect. This suggestion forms the basis for the extension of the present calculation
method described in the next section.
6. Extension of the Calculation Method to deal with Three-Dimensional Effects.
It is comparatively seldom that boundary-layer developments have been measured in accurately twodimensional conditions, as reference to the Stanford data shows. Some measure of flow convergence or
divergence is almost invariably present in the experiment, leading to rates of boundary-layer growth that
are respectively greater and less than would be obtained in two-dimensional conditions.
At the end of the previous section it was suggested that flow convergence should lead to a reduction of
entrainment and that divergence should lead to an increase. We now make the tentative assumption that
the influence of convergence on entrainment is precisely similar to that of an increased rate of growth in
two-dimensional conditions. In other words, the same parameter U1
1.
measure of the entrainment for given values of H and Ro, U
d(UO)]
-~x [ F
L U1 d(UO)]
dx _1eq is used
as a
d(UO)
d ~ now representing the rate of growth
in conditions of flow convergence or divergence. This very simple assumption leads, as we shall see, to
predictions of H development that are in very much better agreement with experiment than predictions
based on the assumption of two-dimensional conditions.
The following shows the modifications to the equations required to take account of observed departures
from two-dimensionality. If we assume that such departures are confined to pure convergence or divergence the momentum integral equation becomes
1 d(UO) _ cY_(H+ 1) 0 dU
0
U dx
z
U dx X-Xo'
(4)
where Xo represents the distance from the origin to the point of convergence of the external streamlines.
For divergent flow x - Xo is positive ; for convergent flow it is negative. In general, Xo may be expected to
vary with x.
The entrainment equation is now given by
1 d ru(3-6*)] = C F _ H . _ _ . O
U dx
(5)
x - Xo
From the definition of H* it is readily shown that
dH*
0 d~-
1 d [U(6-6")]
U dx
H* d(UO)
U dx
Substituting from (4) and (5) we then have
0 dH* C ~ _ H * I 1 d(UO) 1
dx =
(J -~x 2D'
where
I I d(Ud)l
U
--
2D
=c_g_i_(H+l) O dU
2
U d--x-"
It will be noted that the convergence terms have disappeared.
From the suggestion made above, that the effect of convergence on entrainment should be adequately
accounted for by treating the additional rate of growth as though it arose from normal two-dimensional
causes, the entrainment coefficient is now given by
C~=H*[ 0l
where
r 2 is the ratio
Hence,
0.-dx
U
=H*[u
d(UO)J
-~X eqF(r2)'
d~
'
~,,pt
U
(6)
--
r, ]
-~x J
'7'
Thus, in conditions where the development of 0 is known, we need not find explicitly the flow convergence
1
or divergence at each point but only the experimental values of ~
d(UO)
d~x-' Two-dimensional values of the
same quantity, required to determine r 1, follow from the values of H and R, and the pressure gradient
at any point.
In any particular case. then, where the experimental developnwnl or 0 does not sallsly the two-dimensional momentum integral equation we may use equation (7) in place of equation (3), feeding into the
calculation the measured UO development in place of the assumption of two-dimensionality.
It may readily be shown that, when we use the measured 0 development as the basis for our calculation
~fH, there is no point in recalculating the 0 development (except, possibly, as a check on the computation
procedure); the use of the momentum integral equation with calculated values of H and cl, and convergence or divergence terms corresponding to them, should simply lead to the measured 0 development
being reproduced.
This would not be precisely the case if we fed into the H calculations values of convergence or divergence determined from the use of measured H and cI values, and it would then seem necessary to recalculate
the 0 development.
In the calculations to be described, the simpler procedure outlined above has been adopted wherever
it has been thought worthwhile to take three-dimensionality into account. Thus, the calculated and
experimental 0 development for these cases automatically coincide.
7. Cases Treated.
• A considerable range of cases has been considered and a representative selection of results is shown here.
Three basically different types of flow have been treated :
(i) Equilibrium flows.
Bradshaw (a = - 0.255)
Bradshaw (a = - 0.15)
Herring and Norbury (/~ = -0.35)
• (ii) Separating flows.
Schubauer and Spangenberg A
Schubauer and Spangenberg E
Schubauer and Spangenberg B
Schubauer and Klebanoff
Ludwieg and Tillman
(strong adverse pressure gradient)
Goldberg 6
(iii) Flows with H decreasin9
Tillmann (reattaching flow)
Bradshaw's relaxing layer
Goldberg 3
The results are shown in Figs. 5 to 16 with (in some cases) the results of Bradshaw, Ferriss and Atwell 1
included for comparison*. For most of the cases both two-dimensional and three-dimensional calculations
were performed.
In only one case, that of Goldberg 3, has any modification been made to the procedure described in
earlier sections. In this case, where H first rises rapidly, then falls, agreement with experiment is greatly
improved if we introduce an expression which limits the rate of change of CE, in other words if we relax
the stringent coupling we have imposed between CE and local conditions. The expression we have used is
ACE =
ACE n o m
0'
1 + 400 ACE n o m AX
where ACE is the change of entrainment coefficient over the step length Ax, and ACE norais the change in
Cz that would follow from the use of equation (1) or (6) at the beginning and end of the step. The use of
this expression, which was introduced at a fairly late stage, appears to have a negligible effect on the flows
calculated earlier.
*The authors are indebted to Mr. P. Bradshaw for providing detailed results of the calculations.
8. Discussion.
From an examination of Figs. 5 to 16 three points emerge very clearly.
First, it is apparent that, where the flow is closely two-dimensional, as evidenced by the agreement
between the measured 0 development and that obtained from the two-dimensional calculation, the
predicted H development is also in good agreement with the measurements (see, for example, Figs. 5, 6,
8, 9, 13 and 15).
Second, where departures from two-dimensionality are small, the use of the three-dimensional correction may convert agreement which is merely satisfactory to agreement which is very nearly perfect (see, for
example, Figs. 5, 6, 7 and 10). Where there are gross departures from two-dimensionality, the threedimensional correction appears always to operate in the correct sense, though the final agreement may
still not be completely satisfactory. (See Figs. 11 and 12). The possibility cannot of course be dismissed
that in these cases ihctors other than simple flow convergence or divergence may be operative. For
example, the stabilising influence of surface curvature in the flow measured by Schubauer and Klebanoff
may well result in a reduction of entrainment that is not considered here, though Thompson v has indicated
how such an effect might be taken into account.
Taken overall there can be little doubt that the present correction for convergence or divergence makes
a useful contribution to the calculation of nominally two-dimensional layers and may have important
implications for the calculation of three-dimensional layers.
Finally, from the comparisons with the c:~'c'~dations by Bradshaw et al it appears that the latter are
fairly consistently in error in the prediction ol ~km friction, and, in this respect at least, the present method
is clearly superior.
As a general comment it may be said that the comparisons with experiment confirm the validity og the
0 dU
assumption that in two-dimensional flow Ce is determined by local values of H, Ro and ~ d-~x'which is
the basic assumption of the present method. Only in the case of G91dberg 3 (Fig. 16) is the validity of this
dU
and H change very raptdly,
assumption seriously open to question. In this particular case, where U
the introduction of an expression which tends to limit the rate of change of CE brings the calculations into
satisfactory agreement with the measurements, but whether this particular axisymmetric experiment
is sufficiently accurate and representative of two-dimensional conditions to justify the precise form of
this correction is open to question. If this type of pressure distribution is of practical significance, then
accurate two-dimensional experiments are certainly called for.
9. Conclusions.
The method of calculation presented here has been shown to give very satisfactory agreement with
experiment in a wide variety of flow situations.
The proposed method of taking into account the effects of flow convergence and divergence has been
shown to produce a very considerable improvement in the general accuracy of prediction.
It is evident from the present results that methods which do not take the history of the layer explicitly
into account (except in so far as this is reflected in local values of H and Ro) may yet be capable of giving
accurate results in a very wide variety of conditions, and the speed and simplicity of the present method
should make it attractive for many applications. Computing times on the Cambridge Titan computer are
of the order of one to five seconds.
Acknowledgements.
Preliminary work on the present method was undertaken while the first author was working as vacation
ment of Dr. J. J. Cornish and the assistance of Mr. J. Halligan. The report was revised while the first
author was Visiting Professor at IIT Kanpur, India.
REFERENCES
No.
Author(s)
Title, etc.
1 P. Bradshaw, D. H. Ferriss
and N. P. Atwell
2
H. Fiedler and M. R. Head
3
M.R. Head
. . . . . .
S. J. Kline, H. K. Moffatt and
M. V. Morkovin
5 J.F. Nash
6
. . . . . .
Calculation of boundary-layer development using the turbulent
energy equation.
J. Fluid Mech. Vol. 28, p. 593. 1967.
..
intermittency measurements in the turbulent boundary layer.
J. Fluid Mech. Vol. 25, p. 719. 1966.
Entrainment in the turbulent boundary layer.
A.R.C.R. & M. 3152. 1958.
Report on the AFOSR-IFP-Stanford Conference on computation
of turbulent boundary layers.
J. Fluid Mech. Vol. 36, p. 481. 1969.
Turbulent boundary-layer behaviour and the auxiliary equation.
A.R.C.C.P. 835. 1965.
B.G.J. Thompson
....
A new two-parameter family of mean velocity profiles for incompressible boundary layers on smooth walls.
A.R.C.R. & M. 3463. 1965.
7 B.G.J. Thompson
....
The calculation of shape-factor development in incompressible
turbulent boundary layers with or without transpiration.
AGARDograph 97, p. 159. 1965.
O
APPENDIX
lit Determination o[
[,
U
dx _] eq
N a s h 5 gives the following expression relating n and G for equilibrium layers
G = 6.1 ( n + 1.81) ~ - 1.7,
6" dp
%w dx
where
n -
and
HG = - - H - -1 ( c ~ ) ½
(A.I)
This expression m a y be written
n =
- -
- 1.81.
(A.2)
F r o m the m o m e n t u m integral equation we have
UI d(UO) - c2s [ 1 - ( H + l) c.f uO dU
i~ ~
=
-~
'
.
Hence
[
F o r given values of H and
of n from (A.2).
(ii)
U
1 d(UO)
U
dx
I
=
1
l + - -Hf f+- - ~1
eq
.
Ro we can obtain cf, and hence G from (A.1), and the corresponding value
d/dxo l] eq then follows from (A.3).
Derivation of equation (2) in main text.
o dH*
ax
=
(A.3)
0 d [U(6-6")]= 1 d
~
~
] E
[uO-6")]
6-6* 1 d(UO)
o u ax
= CE_H, U d(UO)dx
1 d(UO)
= (C~)eqF(rO- H* 7~ dx
~I -~x
[F(rl)-rx] .
eq
10
TABLE 1
H(H*, Ro)
2"5 ~< loglo Ro <~3"2
Loglo Ro 2"500
H
2"600
2"700
2"800
2-900
3"000
3"100
3'200
*
3.600
3.650
3.700
3.750
3.800
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3.850
3.900
3.950
4.000
4.100
0
0
0
0
2"500
0
0
0
2"655
2.400
0
0
2"800
2"540
2.330
0
0
2"620
2"450
2"280
0
2"800
2.515
2"375
2"230
0
2"640
2-450
2"320
2"180
2"795
2"545
2"380
2"270
2"115
2"635
2"460
2"330
2'235
2"085
4-200
4.300
4.400
4.600
4.800
2"340
2"260
2"185
2"085
2"010
2.275
2'195
2.130
2"030
1"960
2"220
2.135
2"085
1-985
1.910
2.170
2"090
2"040
1-950
1-880
2-125
2"050
2"005
1"915
1"845
2"090
2"020
1"965
1.880
1"810
2"055
1"985
1"940
1"855
1.780
2"025
1"955
1.915
1-830
1-750
5-000
5.500
6-000
6.500
7.000
1.960
1-885
1-840
1"750
1"725
1.905
1.795
1-735
1"700
1"670
1-865
1"750
1"695
1.655
1.625
1.825
1"715
1.655
1"620
1"590
1.795
1"675
1.625
1"585
1"555
1"765
1"650
1-590
1-555
1-525
1-725
1-625
1-565
1.525
1"490
1.700
1"600
1-540
1.495
1"470
8-000
9-000
1-000"1
1-000"1
1-200"1
1.675
1"635
1.590
1"545
1"505
1.625
1"580
1.535
1"485
1"455
1.580
1.535
1.485
1"445
1"415
1"540
1"495
1"450
1-415
1"385
1-505
1-460
1-420
1.380
1"355
1-480
1"435
1"395
1'355
1"330
1.455
1"405
1"365
1"335
1.305
1"435
1"385
1"345
1"320
1"285
1"300'1
1'400'1
1"470
1"430
1-425
1"390
1"390
1"350
1"355
1.320
1"325
1"290
1'295
1"270
1.270
1"245
1.250
1"230
11
TABLE
1--continued
H(H*, Ro)
3'3 ~ loglo Ro <~4-0
Loglo Ro 3.300
3.400
3.500
3.600
3.700
3.800
3.900
4.000
3.600
3.650
3.700
3.750
3"800
0
0
0
0
2'795
0
0
0
0
2"665
0
0
0
2-835
2"565
0
0
0
2-680
2"480
0
0
0
2"590
2"425
0
0
0
2"510
2"370
0
0
2-850
2"450
2"330
0
0
2"605
2'405
2'290
3"850
3.900
3.950
4.000
4"100
2'555
2"395
2"290
2-195
2"060
2"480
2"340
2"250
2"165
2"035
2"415
2"290
2'210
2"135
2'015
2"360
2"250
2"180
2"105
1'995
2"315
2"215
2-145
2"085
1"975
2"265
2"185
2'115
2"060
1"955
2"235
2"150
2'090
2"045
1'940
2"200
2"120
2"070
2"025
1'910
4.200
4.300
4.400
4.600
4.800
2'000
1"935
1'890
1'805
1'725
1"975
1-915
1-865
1'785
1'700
1"955
1-890
1-850
1"765
1"670
1-935
1"875
1"830
1"745
1'660
1"915
1"860
1'815
1'735
1"650
1"900
1'845
1"800
1'720
1"635
1-875
1"835
1"790
1-705
1'630
1'870
1'820
1"775
1"700
1-620
5-000
5.500
6-000
6-500
7-000
1"675
1"575
1-520
1-475
1'445
1'655
1'560
1"500
1"455
1"430
1"635
1'540
1"480
1"440
1"415
1'620
1"525
1"465
1"425
1"400
1"610
1"510
1"455
1"415
1"385
1"600
1"500
1"440
1"405
1"375
1"585
1'485
1"435
1"395
1-370
1"580
1"480
1"420
1'385
1'360
8-000
9-000
1-000"1
1 '000" 1
1.200" 1
1"415
1'365
1'330
1'305
1.270
1"395
1'350
1'320
1'290
1.255
1"380
1"340
1.310
1'280
1'245
1-365
1"325
1"295
1'270
1'235
1"350
1.315
1.285
1"260
1-230
1"340
1"305
1"280
1'250
1.225
1.330
1"295
1'270
1'245
1.220
1"325
1.290
1-265
1.240
1"215
1"300"1
1"400" 1
1-235
1-215
1"220
1-205
1"210
1"190
1'205
1"185
1-195
1'180
1"190
1-175
1'190
1"170
1'185
1"170
12
TABLE
1---continued
H(H*, Ro)
4.2 ~< loglo Ro
Logl9 Ro] 4"200
<~5.4
4"400
4"600
4"800
5"000
5"200
5'400
*
H
3.600
3.650
3.700
3.750
3.800
0
0
2.490
2.330
2.230
0
2.665
2.415
2.270
2.180
0
2.535
2.350
2.225
2.145
0
2.455
2.305
2.19
2.120
0
2.405
2.270
2.170
2-095
2.764
2.365
2"245
2.150
2-080
2.640
2.330
2-230
2.140
2.070
3-850
3.900
3.950
4.000
4.100
2.145
2-075
2.025
1.985
1.885
2.110
2.040
1.995
1.960
1.865
2.075
2.015
1.975
1.935
1.850
2.050
1.995
1.955
1.920
1.840
2-035
1.980
1.845
1.905
1.835
2.025
1.975
1.935
1.895
1.835
2"015
1.970
1-930
1-890
1-830
4.200
4-300
4.400
4.600
4.800
1.850
1.800
1-760
1-680
1.610
1-830
1.785
1-745
1-670
1.605
1.815
1.775
1.730
1-660
1.600
1.805
1.765
1-720
1-655
1-595
1.795
1-760
1.715
1.650
1.595
1-790
1.750
1.710
1.650
1.590
1.785
1.745
1.705
1.645
1-590
5.000
5-500
6.000
6.500
7.000
1.565
1.465
1.410
1.370
1-345
1.555
1.460
1.400
1.365
1.335
1.550
1.450
1-400
1.360
1.330
1.545
1.450
1.400
1-355
1.325
1.540
1-450
1.395
1.355
1.325
1.540
1.445
1.395
1.355
1.320
1.540
1.445
1.395
1.355
1-320
8"000
9-000
1"000"1
1"100"1
1"200"1
1.310
1.280
1.250
1.230
1.210
1.305
1.270
1.245
1.225
1.205
1.300
1.260
1.240
1.220
1.205
1.295
1.260
1.235
1-215
1-200
1.290
1.260
1-230
1.215
1-195
1-285
1.255
1-230
1.215
1.195
1.285
1.255
1.230
1.215
1"195
1.300"1
1.400"1
1.185
1.170
1.185
1.170
1.185
1.170
1-185
1.170
1.185
1.170
1.180
1.170
1-180
1-170
13
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2.
cf(Ro, H ).
15
, + • am• • : i i • • a i I r l -mm~; I I I I I I I I I I I I [ ] I I I ~ l i P . I I I I I I l l h ' l
l l l H l l l l l l a l l l l l l l l l l l l l l l l l l l n l l l l l l l t l l l l l l l i l i i l l l l r
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lh'll
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2.5
3
3.5
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i"OGBo P'o
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16
4.5
5
5"5
3"0
3'95 3'9 3"85 3"8 3"75
3"7
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2"4
2"2
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L OGIo Re
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H*(Ro, H ).
17
4"5
5
5"5
2.0
H
1.5
I
1.0
I
?
0
2-0
BRADSHAW
2-D CALCULATION
3-D CALCULATION
•
~
H
3-D
2-D
__. .......
-(.......,,~.~
CALCULATION
CALCULATION
BRADSHAW
!.0
6
4
T~~----- - - - - - ~ ~ . . _
1'5
0
2
4
8
6
0.0021
0-003
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-
4
6
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2
4
6
/J
40
25
35
aox 10-3
20
30
Re xlO "3
15
25
I0
20
5
0
2
4
6 FEET
FIG. 5. Equilibrium flow measured by Bradshaw
(a = -0.15).
8
15
0
/
2
4
6
FEET
FIG. 6. Equilibrium flow measured by Bradshaw
(a = -0'255).
2.0
2"0
H
H
1"5
1'5
L
It
!'0
0
i
"
/
2
II
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i.o_
4
3
4
2
3
4
5
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.
-
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0.003
2- D CALCULATION
3-D CALCULATION
Cf
0"002 -
0"003
Cf
0"002
0-001
0-001
0
0
2
6
0
2
4
0
16
2-D CALCULATION ONLY
5
Ro xlO -s
4
12
Rex I0-~
0
2
4
FEET
FIG. 7. Equilibrium flow measured by Herring
and Norbury (fl = -0"35).
0w
0
I
2
3
FIG. 8. Measurements by Schubauer and
Spangenberg (flow A).
4 FEET 5
2.5
2.0
2-5
H
2-D CALCULATIONONLY I F
1 1
--~--
3 D CALCULATIONS
...... 2 D CALCULATIONS/ ?
2.0
i.s
1"5
1.0
I'0 0
O.OOz
5
I0
15
cf
0
4
8
12
0
4
8.
12
0.003
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0"003
Cf
0.00;>
0.001
0
b~
0
0-001
00
40
0
5
I0
15
25
./
20
RexlO-3
15
30
xl~3
20
I0
I0
0
/
3O
Y
0
/J
5
5
10 FEET
FIG. 9. Measurements of Schubauer and
Spangenberg (flow E).
15
0
FIG. 10.
,/
0
4
8 FEET
Measurements by Schubauer and
Spangenberg (flow B).
12
•
2"0
2.5
H
BRADSNAW
2-D CALCULATION
- -
-D CALCULATION, . / _
2.0
2- D CALCULATION
3"D CALCULATION
1.5
1"5
I'0
. . , , . ] . . . . , , ~ o . ~ , ' ~ 4/~
0
I
2
3
1.0
0
0.004
S
%
I0
15
20
25
30
cf
0 "002
i
0-003
0-001
Cf
0.002
%
0
ee~
50
o'.
00
bo
0'001
",
/
4
3 METRES
4
1141
0
4
0-003
0
5
I0
15
20
25
30
80
4O
Flox 10-3
30
6O
RoxlO"s
4O
f S
20
2O
- ......
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I0
0
5
I0
15
20 FEET25
FIG. 11. Measurements of Schubauer and
Klebanoff.
30
0
0
2
FIG. 12. Measurements by Ludwieg and Tillmann.
(Strong adverse pressure gradient).
2"0
H
/
2-D CALCULATION ONLY
1"5
2"5
......... 2- D CALCULATION
3-D CALCULATION
- - - - BRADSHAW
H
2"0
J
1.5
@
•
o
I'0 0
I0
20
0.004
I'0 0
O.003
cf
EXPTL Cf NOT GIVEN
•
40
30
I
2
O. 003
\
3
Q
@
cf
t-J
h,)
•
•
@
4
~'
0
•
5
6
5
6
@
J
0 "001
0.001
@
@
0.002
0.002
•
o
0
0
I0
0
20
40
30
6
/
R0xl63
4
3
0
4
15
I~ox 10-3
I0
V
@
S
@
@
er ~
0
0
0
!0
20
30 INCHES
FIG. 13. Measurements b) Goldberg.
(Pressure distribution 6).
40
0
2
3
4
5 METRES6
FIG. 14. Reattaching flow measured by Tillmann.
e~
l::z.
"SI
2"0
BRADSHAW
2-D CALCULATION
H
O'
1.5
~....~:,-
~
~
'
j,2'
/....
zr
I
o
4
5
7
6
.~--°.}w,~, ,~TE o, ~ , ~ . ~ TE,.
B
I'0
0.003
I0
0.004
Cf
I
20
30
40
20
30
40
20
30
INCHES 40
©
P~
0,002
v
..._.._....~...IF- - - - - ' - ' - o
0"003
Cf
0'002
t
>
0.001
0
g
r-"
e~
30
/
j
A
0
aexlO-3
./f
0
I0
6
Roxl()3
4
26
24
4
•
0-001
6
2B
,b
5
6
FEET
7
,j
8
00
FIG. 15. Relaxing flow measured by Bradshaw.
S
I0
FIG. 16. Measurements by Goldberg.
(Pressure distribution 3).
i
R. & M . No. 3643
© Crown copyright 1970
Published by
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