A method of quadrature for calculation of the laminar and turbulent boundary layer in case of plane and rotationally sym...

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Title:
A method of quadrature for calculation of the laminar and turbulent boundary layer in case of plane and rotationally symmetrical flow
Series Title:
NACA TM
Physical Description:
40 p. : ill. ; 28 cm.
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English
Creator:
Truckenbrodt, Erich, 1917-
United States -- National Advisory Committee for Aeronautics
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Laminar boundary layer -- Research   ( lcsh )
Aerodynamics -- Research   ( lcsh )
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federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

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Abstract:
Abstract: For calculation of the characteristic parameters of the boundary layer (momentum-loss thickness and form parameter for the velocity profile), two quadrature formulas are given which are valid for the laminar as well as for the turbulent state of flow. These formulas cover both the two-dimensional and the rotationally symmetrical case. The calculation of the momentum-loss thickness is carried out by a simple integration of the energy theorem. The equation for the form parameter is obtained by coupling of the momentum theorem with the energy theorem. Knowledge of the derivatives of the velocity distribution and of the radius of the body along the length x is not necessary.
Bibliography:
Includes bibliographic references (p. 30).
Statement of Responsibility:
by E. Truckenbrodt.
General Note:
"Translation of Ein quadraturverfahren zur Berechnung der laminaren und turbulenten Reibungsschicht bei ebener und rotationssymmetrischer Strömung." from Ingenieur-Archiv, Band XX, Viertes Heft, 1952."
General Note:
"Report date May 1955."

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University of Florida
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S' I .

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

TECHNICAL MEMORANDUM 1579


A METHOD OF QUADRATURE FOR CALCULATION OF THE LAMINAR

AND TURBULENT BOUNDARY LAYER IN CASE OF PLANE

AND ROTATIONALLY SYMMETRICAL FLOW*

By E. Truckenorodt


1. SUMMARY


For calculation of the characteristic parameters of the boundary
layer (momentum-loss thickness and form parameter for the velocity pro-
file), two quadrature formulas are given which are valid for the laminar
as well as for the turbulent state of flow. These formulas cover both
the two-dimensional and the rotationally symmetrical case.

The calculation of the momentum-loss thickness is carried out by a
simple integration of the energy theorem. The equation for the form
parameter is obtained by coupling of the momentum theorem with the energy
theorem. Knowledge of the derivatives of the velocity distribution and
of the radius of the body along the length x is not necessary.


2. INTRODUCTION


The calculation of the laminar and turbulent boundary layer in case
of pressure drop and pressure rise is the decisive problem in the deter-
mination of the flow loss in channels and pipes and of the flow drag of
bodies. We shall treat below the plane problem as well as the rotationally
symmetrical one. In these considerations, we shall limit ourselves to
incompressible flows.

Whereas theoretical treatment of the laminar boundary layer has
been fundamentally clarified, one is still dependent on semiempirical
connections for the treatment of the turbulent boundary layer. Since the
exact methods in the laminar case are rather troublesome, various approxi-
mation methods have been developed just like for the turbulent case. All
these methods are based on the momentum equation given for the first time
by Th. von KarmAn (ref. i).

*"Ein Quadraturverfahren zur Berechnung der laminaren und turbulenten
Reibungsschicht bei ebener und rotationssymmetrischer Stromung." Ingenieur-
Archiv, Band XX, Viertes Heft, 1952, pp. 16-228.






NACA TM 1579


The approximation method for the plane laminar case has been per-
fected by K. Pohlhausen (ref. 2). Later on, it was essentially improved
by H. Holstein and T. Bohlen (ref. 5): they introduced as the desired
parameter not the boundary-layer thickness as Pohlhausen had done but
the momentum-loss thickness. More recently, K. Wieghardt (ref. 4) der-
ived an energy equation in addition to the momentum equation which he
uses for developing a two-parameter calculation method. A. Walz (ref. 5)
simplified this latter method by reverting to the one-parameter condition
as in the methods of Pohlhausen and Holstein-Bohlen. The rotationally
symmetrical method which is analogous to the Pohlhausen method was indi-
cated by S. Tomotika (ref. 6). Its simplification in the sense of the
Holstein-Bohlen statement was carried out by F. W. Scholkemeyer (ref. 7).

A method for calculation of the plane turbulent boundary layer was
indicated for the first time by E. Gruschwitz (ref. 8). Aside from the
momentum theorem, Gruschwitz also uses a semiempirical equation obtained
from certain energy considerations for determination of a form parameter
marking the velocity profile in the boundary layer; this form parameter
characterizes the sensitivity to separation of the boundary layer. The
empirical parameters appearing iii Gruschwitz' were investigated once more
by A. Kehl (ref. 9). Another method which was similar to Gruschwitz'
method was developed by E. Buri (ref. 10). In the United States, a
method by A. E. von Doenhoff and N. Tetervin (ref. 11) proved to be use-
ful. In this method, too, an empirical equation, in addition to the
momentum theorem, is used for determination of a form parameter char-
acterizing the velocity profile. Starting from this report, H. C. Garner
(ref. 12), England, developed a method which is superior to that of
v. Doenhoff-Tetervin in its numerical evaluation. The transfer of the
momentum theorem to the rotationally symmetrical case was performed by
C. B. Millikan (ref. 15).

Owing to recent investigations by H. Ludwieg and W. Tillmann (ref. 14)
and J. Rotta (ref. 15) oh the theoretical properties of turbulent flows,
particularly of the wall shear stress and the energy loss in the boundary
layer, it is possible to find a better basis for and to improve the
existing semiempirical methods.

It is the aim of the present report to develop a calculation method
which is equally valid for the four cases of laminar and turbulent as
well as plane and rotationally symmetrical flow. The most recent results
will be taken into consideration.


5. THE MOMENTUM THEOREM AND ENERGY THEOREM


As is well known, Prandtl's oPundary-layer equations and Bernoulli's
equation represent the fundamental equations for calculation of boundary





NACA TM 1379


layers. As stated in the Introduction, we are going to consider incom-
pressible flows. Since, however, the following derivations can be given
for variable density p without particular difficulties, we shall make
the transfer to incompressible flow only in the final result.

We assume that, in the rotationally symmetrical case, the radius
of the body or pipe R is large compared to the thickness of the boundary
layer 5. The equations mentioned then read



ax 6y x 6y



+(puR) + 6(pvR) 0 (2)
6x oy



6p dU
=- Pau (3)
ox dx


Therein, u and v signify the velocity components within the boundary
layer


0o y &


in x and y direction, U the velocity outside of the boundary layer
(y 2 ), figure 1. p is the pressure, assumed constant across the
boundary-layer thickness, T the shear stress, and p the density of
the flowing medium. For the plane (two-dimensional) case, the radius R
is to be omitted in equation (2) as well as in all following formulas.

The pertaining boundary conditions are


y = O : u = 0, v = v0, T = TO, P =
(4)
y = 6 : u = U, T = O, p = Pa

If V0 J 0, one is dealing with the case of suction (v0 < 0) or of
blowing (v0 > 0).





NACA TM 1579


We now combine the three equations (1) to (5) by adding to equa-


tion (1) which has been multiplied by ur the equation (2)
been multiplied by ur+l/(r + 1)R; one may choose arbitrarily
r = 0, 1, 2. One then obtains


1 6A(pur+2R) + i-(pur+lv) = (r + 1)urgU dU
R 8x y + dx


+ C)
by


The continuity equation (2) is integrated over the distance y
wall, and one obtains


pvp
pv = puR dy + p
Sdx J0


which has




(5)



from the



(6)


If we now assume the velocity distribution
distribution T(y) in y direction to be
tion (5) over y from y = 0 to y = 6.
over, substitute equation (6)


u(y) and the shear-stress
known, we may integrate equa-
We then obtain, if we, more-


1 ._(p Ur+2Rfr) + gr dU er
PaUr+2R dx U dx Pa U

Therein, the newly introduced abbreviations signify


fr =
0 Pa U



gr = (r + 1)



er = (r + 1)


r+1
1 (- dy
W


0


U(


u u
U p U


T 'dy
y PaUj2





NACA TM 1579


Equation (7) is valid quite generally for laminar and turbulent, incom-
pressiole and compressiole, plane and rotationally symmetrical flow with
and without suction or blowing. As said before, the radius R is to be
omitted in the plane case.

If r is put equal to zero, there results the well-known momentum
equation of von KarmAn. K. Wieghardt (ref. 4) and J. Rotta (ref. 15) have
shown that, for r = 1, equation (7) may ue given a physical interpreta-
tion, namely that of the energy equation. We shall now assume that the
flowing medium is incompressible, p = Constant, and also that neither
suction nor blowing occur v0 = 0. Then r becomes

r = 0: momentum theorem (v. KarmAn, Pohlhausen)


1 d(U2R
u2R d-x
lRM


S5* dU T
U dx U2


= fo= J/


5* = go =


u U


j 1 dy


as momentum-loss thickness




as displacement thickness


TO
eO as wall shear stress
pU2

r = 1: energy theorem (Wieghardt, Rotta)


1 d (U3R5 ) = 2 +t
UR dx pU3


with


(10)




(11)


(12)





NACA TM 1579


with


8 = fl


d+ t
pU3


JO


el
2


as energy-loss thickness


pu2 by \


as shear-stress work1


In the laminar case, the shear-stress work equals the energy converted
into heat (dissipation d). In case of turbulent flows, not all energy
is converted into heat; one part still remains as-turbulence energy (t)
which is, however, usually negligibly small. The fact that, in the case
r = 1, the second term on the left side of equation (7) vanishes is
important since, with the density assumed to be constant, gl = 0.

If we now introduce the boundary-layer thickness ratios


H = *
.


and


8
H= &
fl


(15)


we may write for equations (9) and (12)


I d (u2Rc ) + H B dU = -,
U2R dx U dx pU2


1 d(u55R) = 2 d + t
UR dx pu3


(16)



(17)


We subtract equations (16) from (17) and find2
iThis equation is obtained by partial integration from (8).

2For the laminar case, a corresponding formula has already been
given by A. Walz (ref. 5). In the turbulent case, too, one can show that
the empirical equations indicated by E. Gruschwitz, A. E. von Doenhoff
and N. Tetervin, and also H. C. Garner for calculation of the form param-
eter of the velocity profile (in Gruschwitz' paper 1 = 1 (uO/U)2, in
those of the others H) have a structure very similar to that of
equation (18).


(13)


(14)


ul u.' dy
U \u






NACA TM 1379 7


d (- dU d + t +0
S= (H 1)B + 2 (18)
dx Ud pU pu2


As will be shown in the following section, the shear stress TO/pU2 and
the shear-stress work (d + t)/pU3 may be expressed as functions of the
dimensionless quantities U-/v (Reynolds number formed with the momen-
tum thickness) and H. Furthermore, a one-parameter condition is valid
in good approximation for the velocity distributions in the boundary
layer; that is, there exists a fixed relation between the boundary-layer
thickness ratios R and H. With consideration of these facts, the
equations (17) and (18) represent two equations for the two unknowns 3
and H.

In contrast to the existing methods, we shall calculate the momentum-
loss thickness not from the momentum theorem (16) but from the energy
theorem (17). This offers decisive advantages for the performance of the
integration indicated in section 4. We resort to the momentum theorem
in connection with the energy theorem for calculating the parameter H,
equation (18), which is characteristic for the velocity profile.


4. SHEAR STRESS, SHEAR-STRESS WORK, AND BOUNDARY-LAYER
THICKNESS RATIO


In order to be able to operate with the equations (17) and (18), one
must know the dependence of the shear stress and the shear-stress work on
the quantities Uo/v and H, and likewise the connection H(H).


(a) Laminar Flow

Assuming single-parameter velocity profiles we may write


S= fY.H\


whence then follows


To- = (H) with a1 = U (19)
pU2 U8
b O










d 0(H)
pU3 u6


with


H = f 1]d
0 U J


S= p/i'


H= 0 a
0


\ 2
U
y
6-


d -
6


NACA TM 1379


(20)


ul dJ
uI (d


For evaluation of these formulas, we select the so-called Hartree pro-
files; these are the exact profiles for the velocity variation U ~ xm.
In figure 2, the quantities a, 0, H are plotted against H 3. It is
of interest that 0 is almost independent of H. For the case without
pressure gradient, one has a = 0.220, 3 = 0.173, and H = 2.60.


(b) Turbulent Flow

Recently, H. Ludwieg and W. Tillmann (ref. 14), as well as J. Rotta
(ref. 15), dealt in detail with the determination of the wall-shear stress
in case of turbulent boundary layers with pressure gradient.

Ludwieg-Tillmann indicate for the range of the Reynolds numbers
1 x 10l < U9/v < 4 x 104 the following interpolation formula


T0
pU2


S0.123 10-0.678H
U o0.268
VI1


(22)


Rotta finds for the wall-shear stress the relation


U 1 idH U" B
-. n 1 + B
u* K \ v '


(23)


The fol-


(21)


%We took over the numerical values of A. Walz (ref. 5).
lowing identities are valid: a2 E*, B1 = D* and R= H32.





NACA TM 1579


Therein u* = To/p signifies the shear velocity. Thus




u 2 ~(24)
OU2 \ )

is valid. In equation (25) 1/< = 2.5 and B = B(II) is a function

of the quantity Ii H The function B was evaluated and
H u*
graphically represented by Rotta.

We calculated the shear-stress values for various values of H and
plotted them against the Reynolds number U3/v in figure 5. The agree-
ment between the values according to Ludwieg-Tillmann and to Rotta is
quite satisfactory.

J. Rotta also dealt with the calculation of the shear-stress work
(dissipation and turbulence energy). For the dissipation, there applies


d In H U) + (25)
pU U/Lh

The function appearing therein, G = G(11), has been evaluated by Rotta.
With consideration of equation (25), one may write for equation (25) also



5 = ( 2 1 + u(G B1 (25a)


Since u f= fiiUH) and I = H 1 I, the dissipation may also be
U \v H u*
represented as a function of the Reynolds number U9/v and of the form
parameter H; this has been done in figure 4. It is found that the
differences for different values H are only slight; whereas, for the
wall-shear stress, they are very considerable. We see from figure 4
that the dissipation may be approximated by the statement

d (26)
pU5 U3) n
v





NACA TM 1379


A suitable value for n is n = 1/6. The dependence of the value Ot
on H is only slight so that we may assume for it the constant value
S= 0.56 x 10-2.

For calculation of the turbulence energy, J. Rotta indicates an
approximation formula. With the aid of this formula, one can show that
the turbulence energy is negligibly small compared to the dissipation;
this fact has already been pointed out by Rotta. We put, therefore,
for our further calculations

t =0 (27)
pU3

The boundary-layer thickness ratio H (energy-loss
thickness/momentum-loss thickness) may also be determined from J. Rotta's
results. The equation


12 (H 1)2
H= H + (28)
Il2 H

is valid. Therein, 12 is a form parameter which is in a fixed relation
(indicated oy Rotta) with 1I. We calculated according to equation (28)
the values R in dependence on the quantity H and on the Reynolds num-
ber Us/v. The influence of the Reynolds number was shown to be vanish-
ingly small. The result of our calculation is represented in figure 5.
K. Wieghardt (ref. 16) also has dealt with the connection R(H). He finds


H =AH with A = 1.269 and B = 0.579 (29)
H -B

The resulting curve also has been plotted in figure 5. Except for
the larger values of H, the agreement with Rotta's curves is satisfactory.


5. THE APPROXIMATION METHOD


We shall summarize the result of the previous section: The repre-
sentations

T-= f (L, H)- d + t U H) and i = h(H)
p p 3 pu V






NACA TM 1379


given in figures 2 to 5, are valid. In the laminar case, the
values TO/pU2 and d/pU3 are inversely proportional to the Reynolds
number.


(a) The Calculation of the Momentum-Loss Thickness

As mentioned above, we shall determine the momentum-loss thickness
from the energy theorem (17).

We write for the shear-stress work according to equations (20), (26),
and (27)


d + t ((H)
S(30o)
pU5 U- )n


with n = 1 being valid for the laminar, n = 1/6 for the turbulent
case.

We now substitute this expression into equation (17) and obtain with

n
x U3+2nRl+ngl+nUn)3 (31)



d = 2(1 + n)B(H)fnU5+2nRl+n (52)
dx

If we now assume that H is known from x and that H, too, is known
by the unique relation H(H), equation (52) may be integrated with respect
to x, and there results, if we introduce in addition the quantity

e = (U8 (n)
) -U (33)


and take equation (51) into consideration

Kl + E(H)US+2nRl+ndx'
e(x) = xl (54)
H( +n 35+2n l+n
H(x) U(x) R(x)






NACA TM 1379


As a new abbreviation we introduced


E(H) = 2(1 + n)B(H)Hn (35)

We plotted this function in figure 6 for the two cases of laminar and
turbulent flow.

By way of approximation, we shall now assume constant mean values
for E(H) and H. This assumption proves correct in a particularly
satisfactory manner for the turbulent case. We then obtain from equa-
tion (34)


Cl + A x US+2nRl+ndx'

9(x) = U 2= 1nl+ (36)



Herein


A E 2(1 + n) (37)
fl+n H

signifies a mean value suitable for the range xl < x' < x.

The integration constant is determined to be


Cl = U3+2nR1+n Ul 1n (38)

with n = 1/6 if, starting from the point xl, a turbulent boundary
layer is present. As the laminar or, respectively, turbulent mean value
for A, we shall choose the value which results when we assume U = UD
to be constant and consider the flat plate for fully laminar or fully
turbulent flow. Equation (36) then becomes, with x1 = 0 and C1 = 0


Op = Unp = Ax (39)

Between the momentum-loss thickness Sp(l) and the drag coefficient cf
of a plate of the length I wetted on one side and approached by a flow
of the velocity Um there exists the connection






NACA TM 1579


_p()_ Cf (40)
1 2
If we put in equation (59) x = 1, there follows by comparison of equa-
tions (59) and (40)

A = (U n 1+n (37a)

According to the existing flow state, cf is to be taken for laminar
or turbulent flow (fig. 7). The notation (57a) offers the additional
advantage that the surface roughness also can easily be taken into con-
sideration, merely by substitution of the corresponding cf values of
a rough plate. For smooth surfaces, there result the values of table 1
for the constant A.

TABLE 1.- THE QUANTITIES n AND A
FOR LAMINAR AND TURBULENT FLOW

Laminar Turbulent
(Blasius) (Falkner)4

n 1 1
6

A 0.441 0.760 x 10-2


We now solve equation (56) for the momentum thickness 9 and
obtain if we introduce, in addition, dimensionless quantities and take
equation (57a) into consideration the following expression


S + ()+nf U 5+2n l+nd 1'+n
L _1 2t (41)(5) d
=C* x+0 (.0L

R

V. M. Falkner, Aircraft Engineering, 19, 1945, p. 65. cf = 0.o6


is valid.





NACA TM 1579


The integration constant is determined as


C i Ul R1). l+n Ft
1 W 2


(42)


Summarizing, we repeat once more: If the flow, starting from the initial
point xl = O, is fully laminar or fully turbulent, one has Cl* = 0.
One has'to insert accordingly the laminar or turbulent drag coefficients.
If a laminar starting length precedes the turbulent boundary layer, the
value for in equation (42) is to be taken from the laminar calcula-
tion (second formula in equation (42)). For the laminar case n = 1,
for the turbulent case n = 1/6 is valid. The two-dimensional case
results if one omits in equations (41) and (42) the radius R. As can
be seen from equation (41), the quantity n does not play a significant role
in the turbulent case so that one may assume for rough calculations also
n = 0.


(b) The Calculation of the Form Parameter

We shall determine the form parameter H which is characteristic
for the velocity profile in the boundary layer from equation (18); we
write this equation as follows:


e(x) d F(H)r(x) + G(H)
dx


(43)


Here e is given according to equation (55). For the other abbreviations


r(x) = dU
U dx


(44)


F(H) = (H 1)H


(45)


G(H) = (2


d+ t
pU3


pU2


' U 2 R 2d l+n






NACA TM 1579


are valid. Furthermore, we transform equation (45) by introducing the
substitution



L(B) = / L(H) (46)
F(l)


and the abbreviation



K(H) G( ) K(L) (47)
F(H)


We then obtain a differential equation for the new form parameter L


e(x)d = r(x) K(L) (48)
dx

We determined the quantity L by graphical integration of the func-
tion 1/F(H) over H. The value L = 0 we have placed in the domain of
vanishing pressure gradients (constant pressure, flat plate). In fig-
ures 8 and 9, we represent the relation H(L), also for the laminar
case a(L), and, in figures 10 and 11, the relation K(L) for the cases
of laminar and turbulent flow. Whereas no influence of the Reynolds num-
ber exists for the function K(L) in the laminar case, that influence
is rather considerable for the turbulent case. One can show that it is
possible to give to the equations for calculation of the form parameter
appearing in the reports of E. Gruschwitz, A. Kehl, and H. C. Garner the
form of our equation (48)5. The resulting connections for H(L) and K(L)
have also been plotted in the figures named above. The differences
between the individual methods are therefore based on the deviations of
the curves H(L) and K(L).

For solution of equation (48), we set up a linear expression for K(L),
(compare the figures 10 and 11):


K(L) = a(L b) (49)

The quantities a and b are obtained, for instance, from table 2. In
the turbulent case, b is, in addition, dependent on the Reynolds num-
ber and therewith on the length x.

5A detailed comparison of the individual methods has been carried out
in an unpublished report of the author.





NACA TM 1579


If we substitute equation (49) into equation (48), we obtain a
linear differential equation of the first order for L which we can solve
in closed form. For this purpose, we make the substitution


p dx
S (x) C + A X US3+2nRl+ndx A

X1#


(50)


The last relation follows from the fact that in formula (56) for E the
denominator is exactly equal to the derivative of the numerator multiplied
by 1/A. The numerical values a/A for smooth surfaces are also plotted
in table 2.


Without dealing
which we perform, in
that the derivative


L = L
E


in detail with the intermediate calculation for
addition, a partial integration, in such a manner
dU/dk does no longer appear we find finally


+ n U(_)+ 1
Ul t .


(51)


At the initial point xl = 0, there is also 1i = 0. The first term in
equation (51) then disappears. Especially for the laminar case there
applies with b = 0 the simple expression


L = n U(' d-
-o u()


(52)


We shall report later on regarding the initial values for L at
the stagnation point xl = 0 and at the transition point.

As can be seen from equation (51), one may provide the new vari-
able t with an arbitrary factor without causing thereby a change in L.
We may therefore write


a
xA
I


= C +


(50a)


1+n x/1 +2n l+n

2 X11 I m jP ) (I


b(t') In U ) dE'
b' ) U1 --






NACA TM 1579


TABLE 2.- THE QUANTITIES a, b, AND a/A

FOR LAMINAR AND TURBULENT FLOW


IPressure drop.
2Pressure rise.


The calculation of t thus will be very simple, since the expression in
brackets already occurs in the calculation of the momentum-loss thickness
(eq. (41)).

A complete calculation is carried out as follows.

Prescribed:

General quantities: U(x), R(x), U--.


State of flow

l1, L.l

Desired:


n, Cf, a laminarr or turbulent); initial values
A


Momentum-loss thickness a(x); form parameters L, H(x).
First, one calculates the momentum-loss thickness according to equa-
tion (41) by performing a simple quadrature. Hence one forms, if one has
to calculate turbulent flows, the Reynolds number U-/v with which one
then calculates the quantity b according to table 2. Next, one deter-
mines according to equation (50a) the new variable t over which one
integrates the function b In U/U1. According to equation (51), one
then obtains the form parameter L and, by means of figure 8 or figure 9,
the boundary-layer thickness ratio H. Due to the large exponent a/A
in equation (50a), the quantity t increases very rapidly with growing


U1, R1,






NACA TM 1379


length x. This signifies, however, that the term (ft/ )LI in equa-
tion (51) loses its significance more and more with increasing distance;
therewith it is shown that the initial value L1 has, in case of larger
distances x (for instance in the neighborhood of the separation point),
only slight influence. The advantage of our method lies in the fact that
only simple quadratures have to be performed in the individual case.
Beyond that, no derivatives of the initial values U and R with respect
to x occur.


(c) The Initial Values

The initial values for the momentum-loss thickness as well as
for the form parameter L may be different according to the case to be
dealt with.

1. Flow toward a body with stagnation point.- The boundary-layer
calculation is started at the stagnation point xl = 0. There U = 0
and R = 0. From equation (355) then follows immediately


o9 = 0 (53)

In the two-dimensional as well as in the rotationally symmetrical case,
the potential velocity varies linearly with the length x, that is



U= ex with c = (54)


The constant c is different for the two-dimensional and for the rota-
tionally symmetrical case.

For the variation of the radius, there results also a linearity
in x, namely


R =x (55)

The integration constant in equations (54), (36), and (41) disappears


X1 = C1 = Cl* = O


(56)






NACA TM 1379


If we substitute equations (54) and (55) into equation (41), there
follows, if we put all quantities which additionally enter the formulas


in the rotationally symmetrical case into braces -

intermediate calculation


x->0: =


-, after a brief


1-n
1 cf (,,/x+n
1 n I,
2F2(2 + n) + 1 + +n *l+n
L +. Jj


Therein c* = signifies the dimensionless expression for
U, \Wx/0


(57)




c.


In the laminar case (n = 1), there results a value different from zero
for the momentum-loss thickness in the neighborhood of the stagnation
point


tO0


1 Cf
2 2(5 + (3) TC*


(58)


If one takes into consideration that c* -= c
U6


write for equation (58) also



= 1.528= -
Cf v c, (O) 3


and cf = 1.328, one may

S


.470
+r 14


(59)


In the turbulent case (n = 1/6), the momentum-loss thickness at the
stagnation point itself (x = O) has the value zero.

For the momentum-loss thickness ratio at the stagnation point for
equal velocity increase (dU/dx)o in the rotationally symmetrical and
in the two-dimensional case (c = Constant), one then has





NACA TM 1579


1
'/ orot\ 2(2 + n +n
60eb +
c=const + n


(60)


If the numerical values for n are substituted, there results in the
laminar case the value 0.867 and in the turbulent case 0.816. The exact
value for the laminar flow which one can calculate from the Hartree pro-
file is 0.84).

The initial value of the form parameter L is determined according
to equation (48) by putting therein according to equation (55) S- = 0,


r0 = K(LO)


(61)


P0 is obtained from the following boundary-layer determina-


9 dU Q A
FO = lim = lim =
x--O- U dx x-->0 x 2(2 + n) +


(62)


S+ n}


In the laminar case, there results with equation (59)


0 c .2 0.220
3+ (1}


(62a)


and from equation (49)


K(LO) =0_ 0.077
a a 3+ 1{


(63)


(See table 3.) Although, according to equation (62), in the turbulent
case a finite value different from zero does result for pO, it is of no
6It can be shown that this value follows also from equation (52).


The value
tion






NACA TM 1379


help in calculating the value LO according to equation (61) since the
values K(L) for the Reynolds numbers U_ ->0 are not known. Since

the flow at the stagnation point always will be initially laminar, we
refer to section 5, c5, where we report on the initial values at the
transition point.

For the variation of the form parameter L with the length x at
the stagnation point, there applies also (as can be derived from equa-
tion (48))


() = 0 (64)
dx 0

2. Flow into a channel.- The boundary layer begins at a point
x1 = Xk = 0 where the velocity, and likewise the radius in the rotationally
symmetrical case, are of finite magnitude. Since again X1 = C1 = CI* = 0,
there follows immediately from equations (54), (36), and (41)


Gk = O and "k = 0 (65)

and from equation (44), when dU/dx #/


rk = 0 (66)

The form parameter then follows from equation (48) as


K(Lk) = 0 (67)

In the laminar case, this form parameter becomes


Lk = O (68)

that is, the value of the flat plate in longitudinal approach flow.
There is no difference between the two-dimensional and the rotationally
symmetrical case. (See table 3.) For the turbulent flow, there applies
what was said above in the discussion of the stagnation-point flow.

The variation of the form parameter L with x is obtained from
equation (48)





NACA TM 1579


(p)
Wk
dL k


1 dU


A dL k


(69)


1 1 dU
1+a- k
A


The numerical value a/A is to be taken from table 2.


TABLE 5.- LAMINAR INITIAL VALUES. THE EXACT VALUES INSOFAR

THEY DO NOT AGREE WITH THE APPROXIMATION

VALUES ARE PUT IN PARENTHESES



Stagnation-point flow Channel flow

Plane,
Plane Rotationally rotationally
symmetrical symmetrical

S 0 .2 0.271 0.23 k 0
S(.292) (.247)

LO .0260 .0195 Lk 0
(.0292) (.0208)

H0 2.25 2.52 Hk 2.60
(2.22) (2.50)

ao .45 .520 .220
(.560) (.524) k


5. Transition laminar-turbulent.- Behind a certain transition
region, the laminar boundary layer is transformed into the turbulent layer
at the point x, = x. From the theory of the origin of turbulence, com-
pare H. Schlichting (ref. 17), one can find, in agreement with measurements,
that the transition occurs at places which lie somewhat downstream with
respect to the velocity maximum. Since the phenomena in the transition
region have not yet been explored, determination of the initial values
for 9 and H or L which are required for the turbulent calculation
is only approximately possible. WAat is fundamental in our considerations






NACA TM 1379


has been shown in figure 12 on the example of the flat plate. Up to the
point xu the boundary layer is laminar and obeys the regularities
according to Blasius, that is, ~ ~xT. The corresponding H-value is
constant and amounts to H = 2.60. Starting from the point xu the
boundary layer is fully turbulent. Here applies approximately
4 ~ -Su + c(x xu). The corresponding H-value depends, in addition, to
some extent on the Reynolds number Um~o/ prevailing at the transition
point. According to measurements, for instance, by F. Schultz-Grunow
(ref. 18), also from the similar solutions in case of constant pressure
by J. Rotta (ref. 15), one finds about 1.2 < H < 1.4, with the H-value
being the smaller, the larger the Reynolds number. Thus the H-value
decreases compared to the laminar value. In the transition region, it
must therefore vary continuously from the laminar to the turbulent value.
In figure 13, we plotted the difference AC against U-/v. For the sake
of simplicity, we let xu coincide with xu; then the initial value for
the momentum-loss thickness of the turbulent calculation is equal to the
momentum-loss thickness which would result at the point xu if the flow
were fully laminar up to this point. We shall therefore put


St(xu) = xu = O(Xu) (70)

For the H-value, we write


Ht(xu) = Hu = Hi(xu) CH (71)

We shall now assume that AH may be taken from figure 15 also for the
cases with pressure gradient. This assumption we deem justified since
the transition point lies in the proximity of the point of vanishing
pressure gradient and, thus, the values of the flat plate may be used in
suitable approximation. Having thus determined Hu, we ascertain according
to figure 9 the value Lu = L(xu) which then enters equation (51) as L1.
We want here to point out once more that in places lying further downstream
from transition point the initial value for L is only of slight signif-
icance; therefore, a somewhat rougher estimate seems justified when it-is
a matter of determining the separation point.


(d) The Separation Point

Knowledge of the position of the separation point also is important.
Separation results when the wall-shear stress assumes the value zero.
Whereas this point in the laminar case corresponding to the prescribed
velocity profile is fixed by ca = 0 according to equation (19), it is






24 NACA TM 1379


not yet possible to make a perfectly clear statement regarding the separa-
tion point in the turbulent case. According to the wall-shear-stress
statements of Ludwieg-Tillmann and Rotta (compare fig. 3), the shear
stress decreases with increasing value H, however, without attaining
the value zero. As approximation rule one may, for instance, assume
according to v. Doenhoff-Tetervin (ref. 11) that the separation starts
at the earliest when H 1.8 7 and has certainly taken place when H
has attained the value 2.4. Table 4 presents a compilation of the occur-
ring values.

TABLE 4.- VALUES OF THE FORM PARAMETER

AT THE SEPARATION POINT


(e) Examples

The usefulness of the method described above is shown on one example
each of laminar and of turbulent flow.

1. Howarth flow (laminar).- As example for the laminar flow, we
choose the well-known Howarth flow (ref. 19). In this case, the velocity
distribution is


U x
u =1-
U0,


(72)


If equation (72) is substituted into equation (41) where the radius distri-
bution R is omitted, the quadrature may be carried out in closed form.
There results for the momentum-loss thickness


c f 1A -_( x

2\6 3I


(73)


Laminar Turbulent

H 4.058 1.8 to 2.4

L -0.018 -0.15 to -0.18


'This statement corresponds .approximately to Gruschwitz' assumption
n = 1 (u./U)2 = 0.8.


0 0.2711 vE)

U.11






NACA TM 1579


In figure 14, we show the comparison of this approximation with the
exact values of Howarth. Within drawing accuracy, the agreement is
perfect.

For determination of the form parameter, we first calculate the new
variable ( according to equation (50a)


a 6 6
=ZA with Z = -(1 l- 1 ) (74)


The form parameter itself also may be calculated analytically according
to equation (52). Since we find ourselves, corresponding to the pre-
scribed velocity distributions, in the region of pressure increase, we
choose according to table 2 the value a/A = 8. We obtain after a brief
intermediate calculation



L= 1 i d' 1 dU 'r 1 Z Z8 dZ- 1 Zm
S0 ud t' 6z8 0 1 Z' m= m + 8

(75)

Taking equation (74) into consideration, we represented L(x) also in
figure 14. From these values we determined, with the aid of figure 8,
the values a for the wall-shear stress. They, too, are represented in
figure 14 and are there compared with the exact values of Howarth. The
agreement is quite satisfactory. We also showed the curve which results
according to the Pohlhausen method.8 One achieves, as already stated by
A. Walz, a considerably better determination of the separation point,
a = O, if one uses in addition to the momentum theorem also the energy
theorem, as in our method.

2. Profile NACA 65(216)-222 (turbulent).- As an example for the
turbulent calculation, we choose the profile NACA 65(216)-222 (approxi-
mately) for which measurements (ref. 11) as well as theoretical calcula-
tions have been carried out. The conditions refer to the upper side of
the profile placed at an angle of attack a = 10.10. The Reynolds num-
ber is Umt/v = 2.64 x 106. The graphical representation of the velocity
distribution, the momentum-loss thickness, and the form parameter according
to measurements, likewise the result of our calculation, are given in

8This curve we took from A. Walz (ref. 5).





NACA TM 1379


figure 15. We started our calculation at the first point measured, that
is, x/l = 0.075. The agreement between our approximation and the meas-
urement may be called satisfactory.' For comparison, we also plotted the
results according to the calculation of v. Doenhoff-Tetervin and according
to the methods of Garner, Gruschwitz, and Gruschwitz-Kehl. Regarding the
form parameter H, the last two methods deviate greatly from the other
methods and from the measurement.







































9As A. E. v. Doenhoff and N. Tetervin pointed out, the larger differ-
ences between measurement and calculation for the momentum-loss thickness
in the neighborhood of the separation point x/2 0.55 might be caused
by systematic measuring inaccuracies.






NACA TM 1579


APPENDIX


DISCUSSION OF THE FORMULA FOR CALCULATION OF

THE MOMENTUM-LOSS THICKNESS


As mentioned before, the momentum-loss thickness was calculated, so
far, from the momentum theorem (16). With the use of certain simplifica-
tions, it is possible to find for the calculation of the momentum-loss
thickness from the momentum theorem a formula similar to our equation (36).
Two possibilities exist:

1. The wall-shear stress is determined according to the laws of the
flat plate in longitudinal approach flow where the length x is expressed
by the momentum-loss thickness 8, and the approach-flow velocity U,
by the local velocity. For the laminar as well as for the turbulent case,
one may write



TO a




If one substitutes, moreover, in the momentum theorem a constant value
for the boundary-layer thickness ratio H, one can integrate the momentum
theorem in closed form, as was shown by E. Truckenbrodt in an unpublished
report (compare H. Schlichting, Boundary-layer theory, page 450; one
obtains the result

.x
C1 + (1 + n)a UaRl+n dx'
n x

V UaRl+n

where a = (1 + n)(H + 2) n.

The corresponding numerical values may be taken from table 5.





NACA TM 1379


TABLE 5


Laminar Turbulent


n 1 !(Blasius) I(Falkner)
4 6

A 0.441 ------ 0.0076
Factor ahead of the + n .441 ---O.- .007
Ineral (1 + n)a .441 0.0160 .0076
integral
inc .470 .0160 ----

3 + 2n 5.0 --- .53
Exponent of the velocity + 2 0 --- 3.6
distriutiona 8.2 4.0 5.67
b 5.0 4.0 ----


2. According to K. Pohlhausen for the laminar case and to E. Buri
for the turbulent case, the wall-shear stress and the boundary-layer ratio
are functions of the quantity


r dU
U dx


that is,


T0
pU2


fl(r)
()n
(n


H = f2(r)


If one puts


fx
X* = JX
0


A = 8Rl+n


Rl+n dx






NACA TM 1579


one can find after some intermediate calculations the following equation:


d = (
dx *


with


SA dU
U dx*


Therein


E(r) = (i + n)fl(r) [ + n


+ (1 + n)f2F(rr


represents a universal function for the laminar or, respectively, turbu-
lent state which, in the first case, may be calculated analytically (for
instance, for the Pohlhausen polynomials) and, in tie second case, may
be determined from measurements. In both cases, one may find by way of
approximation a linear connection between and r (compare A. Walz,
Lilienthal-Bericht 141 (1941) and E. Buri).


(r) = -br + c

If one substitutes this expression in the above equation, the integration
may be carried out in closed form and yields after a simple intermediate
calculation


( )n


x
Cl + cJ UbRl1+n dx'


U R1+n


The numerical values c and b are also indicated in table 5. (Compare
H. Schlichting, Boundary-layer theory, pp. 199 and 424.) The constants
according to equation (56) also nave been indicated for comparison.


Translation by Mary L. Mahler
National Advisory Committee
for Aeronautics






NACA TM4 1579


REFERENCES


1. Von Karman, Ti. : Z. angew. Math. Mech. 1, 1921, p. 255.

2. Pohlhausen, K.: Z. angew. Matii. Mech. 1, 1921, p. 252.

5. Holstein, H., and Bohien, T.: Lilienthal-Bericht S 10, 1940, p. 5.

4. Wieghardt, K.: Ing.-Arch. 16, 1948, p. 251.

5. Walz, A.: Ing.-Arch. 16, 1948, p. 245.

6. Tomotika, S.: Laminar Boundary Layer on the Surface of a Sphere in
a Uniform Stream. ARC-Rep. 1678, 1955.

7. Scholkemeyer, F. W.: Die Laminare Reibungsschicht an Rotationssym-
netrischen K6rpern. Diss. Braunschweig, 1945.

8. Gruschwitz, E.: Ing.-Arch. 2, 1951, p. 521.

9. Kehl, A.: Ing.-Arch. 12, 1945, p. 295.

10. Bari, E.: Eine Berechnungsgrundlage fir die Turbulente Grenzschicht
bei Beschleunigter und Verzogerter Stromung. Diss. Zurich, 1951.

11. Von Doenhoff, A. E., and Tetervin, N.: Determination of General
Relations for the Behavior of Turbulent Boundary Layers. NACA
Rep. 772, 1945.

12. Garner, H. C.: The Development of Turbulent Boundary Layers. ARC-
Rcp. 2155, 1944.

15. Millikan, C. B.: Trans. Am. Soc. Mech. Eng., Appl. Mech. 54, 1952,
[Io. 2, p. 29.

14. Ludwieg, H., and Tillmann, W.: Ing.-Arch. 17, 1949, p. 288.

15. Rotta, J.: Ing.-Arch. 19, 1951, p. 31 and 20, 1952, p. 195.

16. Wieghardt, K.: Zur Turbulenten Reibungsschicht bel Druckanstieg.
Untersuchungen und Mitteilungen der Deutschen Luftfahrtforschung,
UM 6617, 1944.

17. Schlichting, H.: Grenzschicht-Theorie, Karlsruhe, 1951.

18. Schultz-Grunow, F.: Luftfahrtforschung, 17, 1940, p. 259.

19. Howarth, L.: Proc. Roy. Soc. London, Ser. A 919, Bd. 164, 1938, p. 547.






NACA TM 1379


Y U(x)
S^ u(x, y)
8' x Plane


U


SU(x)


Sx RotationallyU
UmO


Figure 1.- Survey sketch.


a.3

0.5 1.7

04


0.3-

Q2 -


2D 3D 4.0


Figure 2.- Laminar friction-layer parameters (Hartree profiles).


H




S Constant pressure

N" %\\ a Separation-t
SI





NACA TM 1579


20



0H=1.2



1.8
06 -----^

04o- 2-- .0



0.2 :*1 _*
Rotta "
---Ludwieg,Tillimann
. I I I t I


103 2


5 104 2


U'
1/


105


Figure 3.- Turbulent wall-shear stress (according to H. Ludwieg,
W. Tillmann, and J. Rotta).





NACA TM 1579


d x
Pu


o.103 2 5 104 2 5 1o5
Figure 4.- Turbulent dissipation (according to J. Rotta).






NACA TM 1579


1.0 1.2 1.4 1.6 1.8 2.0 2.2 24

Figure 5.- Turbulent boundary-layer thickness ratio (according to
J. Rotta and K. Wieghardt).





NACA TM 1379


E,Exio2


Figure 6.-


The function E(H) for laminar and turbulent flow.


0.1 1-
10 5 2


S~l-lKner, Cf Ul
U5 2

5 106 2 5 107 2 5 108 2 5 109


Figure 7.- Drag law of the smooth flat plate in longitudinal approach flow.





NACA TM 1579


Figure 8.- Relation between the boundary-layer thickness ratio H and
the wall shear-stress coefficient a, on one hand, and the form
parameter L on the other for laminar flow.


-0.2


-0.1 0 0.1


Relation between the boundary-layer thickness
the form parameter L for turbulent flow.


ratio H and


Figure 9.-





NACA TM 1357


Figure 10.- The function K(L) for laminar flow.


- Truckenbrodt
- Approximation (49)
--- Garner
---- Gruschwitz
--- Gruschwitz-Kehl


Figure 11.- The function KiL) for turbulent flow.





NACA TM 1379


Lominar Turbulent
Ou x


XU-.xu


Figure 1,.- Survey sketch from the transition point of the fiat plate.


AH


Schultz-Grunow (flat plate)

05----Rotta (similar solution in the
case of constant pressure)


10 2 5 104 2 5 105 v


Figure 1.- Variation of the form parameter H in the range of
transition from laminar to turbulent flow.






NACA TM 1579


-0.02


Figure 14.- The laminar boundary-layer parameters of the Howarth flow.





NACA TM 1579


-.- Measurement.
{v. Doenhoff-Tetervin,
Garner
fGruschwitz,
LGruschwitz Kehl
Truckenbrodt


0 0,I 0.2 0.3' 0.4 0.5 0.6 j

Figure 15.- Turbulent boundary layer on the profile NACA 65(216)-222
a. = 10.10, T = 2.64 x 106. Measurement according to NACA Rep. 772.


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