On the gas dynamics of a rotating impeller


Material Information

On the gas dynamics of a rotating impeller
Series Title:
Physical Description:
16 p. : ill. ; 28 cm.
Busemann, Adolf, 1901-
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


Subjects / Keywords:
Propellers, Aerial   ( lcsh )
Aerodynamics -- Research   ( lcsh )
federal government publication   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


Abstract: It is shown that for a compressible flow with constant entropy the pressure rise maintains the direct relation to the circulation around the blades existing for incompressible flow. In contrast, however, the torque, and with it the power consumption, is increased because of sound waves traveling to infinity already at subsonic circumferential speeds.
Statement of Responsibility:
by A. Busemann.
General Note:
"Translation of "Zur gasdynamik des drehenden Schaufelsterns." from Zeitschrift für angewandte Mathematik und Mechanik, vol. 18, issue 1, Feb. 1938."
General Note:
"Report date March 1956."

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Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003807542
oclc - 127116384
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Full Text
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By A. Busemann


Centrifugal pumps and turbines may sometimes be treated by using
plane incompressible flow where the fluid springs from a source or runs
into a sink in the center of the impeller. However, application of
conformlal'mapping is not so simple fo;r thnis type Ijf flou ais fory recti-
linear flow, because a steady flow abou~lt the blade sections exists only
in a coordinate system rotating with the blade. In a coordinate
system at rest, the velocity field outside of the singularities is free
from sources and rotation, but the impeller rotates in it. By applying
appropriate distributions of sources, sinks, and vortices inside the
blade contour, a flow can be found which does not pass through the
rotating blade contour. The action of the flow on these singularities
inside the blades creates the torque.

For blowers with high circumferential velocities, the same consider-
ations would have to be extended by the requirements of a compressible
flow. Though there exist corresponding laws of forces on sources, sinks,
and vortices, the conditions became considerably more involved because
the freedom from vortices in the regular domain of the flow concerns the
velocity field whereas freedom from sources exists only in steady flow and
involves the stream-density field (the product of velocity and gas density).
It becomes impossible to superimpose two fields free irom sources and
vortices into a new field which is free from sources and vortices;
furthermore, in a compressible flow one can operate with distributed
singularities only, since a certain region around point sources and point
vortices is void of any velocity field. For these reasons, the laws of
forces on singularities have only a very limited range of application in
gas dynamics.

It is known that the velocity fields of incompressible flow and
magnetic fields are similar with respect to the distribution of the
vector and to the field energy. Accordingly, one should anticipate the
forces on corresponding singularities to be equal so that a displacement
of the singularities produces the amount of work required for the change
in field energy connected with it. Contrary to this expectation the

w"Zur Gasdynamik des drehenden Schaufelsterns." Zeitschrift f(Gr
angewandte Mathematik und Mechasnik, vol. 18, issue 1, Feb. 1958, pp. 31-38,
dedicated to the memory of the late editor Erich Trefftz.

NACA TM 1585

hydrodynamic and the magnetic forces are always opposite in sign. The
result is that, on the one hand, the forces on magnetic sources and sinks
(north and south poles) and those on hydrodynamic vortices are such that
their displacement work balances the field energy. On the other hand,
for the displacement of hydrodynamic sources and sinks and of magnetic
vortices (around conductors of electric currents), the opposite sign of
the displacement work would be required to balance the field energy.
The difference is restored for the magnetic vortex according to Faraday's
induction law by the well-known fact that the electric current flowing
in the conductor is opposed, during any displacement, by induced elec-
tromotive forces, the overcoming of which requires a supply of electrical
energy. Likewise, additional energy must be supplied during displace-
ments of the hydrodynam~ic sources and sinks; in this case the energy
supply is caused by higher pressures at the location of the sources and
lower pressures at the location of the sinks compared to the pressures
of the steady flow. These pressure differences are precisely what
constitutes the pressure heads or pressure differentials for rotating
machinery .

For plane flow, one may regard instead of the field of the stream-
lines the field of the potential lines which are orthogonal to the
streamlines. In the case of incompressible flow, this field is in the
regular domain free from sources and vortices; however, sources and
vortices are interchanged, since a streamline source represents a
potential-line vortex and vice versa. Comparing plane potential lines
with the plane field of magnetic lines of force, one finds now perfect
agreement including the sign of the forces and the induction law. If
one wants to determine the additional pressure difference between two
points of the field which alters Bernoulli's pressure difference of the
steady flow, one must draw a line connecting these points and observe
on it the variation with time in the number of the potential lines which
intersect this connecting line. The connecting line must not pass over
source points of the potential lines. This is the same rule which
applies for the determination of induced electromotive forces. In order
to ascertain whether the connecting line has moved over a source, these
sources cannot be allowed to appear and disappear. For magnetic sources,
this prerequisite is obviously satisfied by the fact that a north pole
can be created only by separating a north and a south pole of equal
strength. The sources of the potential lines represent vortices of the
streamlines and it follows from Helmholtz' vortex theorems that, in two-
dimensional flow, a vortex rotating clockwise can be produced only by
separating a vortex rotating clockwise from another one rotating counter-
clockwise. For these plane fields, the similarity is therefore very far
reaching. For magnetic and hydrodynamic fields in space, the similarity
is subject to limitations because lines of forces remain lines in space
whereas the potential lines change to potential surfaces insofar as
unique surfaces orthogonal to the streamlines exist. Generally, one is
therefore limited to correlate only the Tydrodynamic velocity field and

NACA TM 1585

the magnetic force field. If one identifies velocity and magnetic-field
intensity, then the source matches the magnetic pole and the hydrodynamic
vortex matches the wire through which current flows; with respect to the
induction law, however, it is just the wire with electric current and the
hydrodynamic source which can be compared.

Whereas the laws about forces on sources, sinks, and vortices almost
maintain their form in transfer from hydrodynamics to gas dynamics but
lose their major field of application, the hydrodynamical induction Law
must correspond to a similarly formulated gas-dynamical induction law
and at the same time maintain to some extent its applicability, since
it deals only with the pressure difference of unsteady flow as compared
to steady flow. It is just this pressure difference which is related to
the pressure rise in blowers. Thus, the author will give below the
general derivation of the gas-dynamical induction law as he presented it
for the first time in the summer of 1936 in a colloquium on gas dynamics
at the DVL under the chairmanship of E. Trefftz. In a second part, he
will show that the torque at the impeller in nonviscous compressible
flow, even for circumferential velocities below sonic velocity, does not
result from the effective pressure rise alone as in nonviscous incom-
pressib~le flow, but that, on the contrary, even without effective pressure
rise, energy quantities may be radiated from the rotating impeller.


1. Derivation of the Gas-Dynamical Induction Law

In the revision of the section "Hydrodynamics" for the 8th edition
of A. F~ppl, "Vorlesungen iiber technische Mechanik" (Lectures on technical
mechanics), volume IV, page 417, I derived the "hydrodynamical induction
law." If one limits the application to a loss-free gas flow of constant
entropy, the derivation may be transferred directly to gases.

For a nonviscous gas, free from gravity, of pressure p, density p,
and velocity W vith the components u, v, v in the directions of
the spatial coordinates x, y, z, one obtains, in dependence on the
time t, the following equations of motion:

ap du ap dv Sp dw
ax at ay at dz at

For constant entropy s there applies for the enthalpy 1 of the gas:

di -p (2)

NACA TM 1385

By substitution of this relationship into equation (1) there result the
following equations:

a1i du ai dv_ ai du *5
ax at ay dt dz dt

For a certain time t = to one can combine these three partial deriva-
tives into the following total differential of the enthalpy in space:

-idu dx+dv dw dz(4)
at at at

If one introduces the convention that the components dx, dy, dz of a
line element in space (which are used in equation (4) only for the
time t = to) shall be for all times, the components of a line element
attached to the gas which connects the adjacent points 01 and G2
moved with the gas, the following transformation of equation (4) holds:

-di = (u dx + v dy + w dz) u du v av v dw (5)

since in this case the interchange of the differentiation

S(dx) = d )= du, etc., is valid. If one places the points GI and G2

attached to the gas farther apart, one may integrate the differentials
indicated in equation (5) along a line from GI to G2 moved with the
gas, so that the following difference in enthalpy is then obtained:

il 12 G.0 (u dx + v dy + w dz) l 2 Y 12)

2~u22+v '22 22) ()

If one introduces, instead of the points C1 and G2 attached to the
gas, the points fixed in space Pi and P2, which at the time t = to
coincide with the former, one can prolong the line between 01 and G2

NACA TM 1385 5

with those gas points sweeping over the points Pl and P2 after the
time t = to fg ) The integration along this line, which is like-
wise attached to the gas (and is therefore determined in the development
with time) but connects the fixed points Pi and P2, includes beyond
the integral required in equation (6) contributions which result from the
shift of the integration LFnits and have the value tj;2 _j2 dt. If
one subtracts the latter, one obtains the following relation:

ilr rP2 2 d (u dx + v dy + w dz) (ul2 + yl2 + 12) +

(u2 '22 d 22

ig + (u2 4 y2 + Y22) .1+ (l l l

Bt(u dx + v dy + v dz) (7)

The right side of the equation represents the induced pressure rise which
is generated between the points Pl and P2. Note that one deals here
not with a partial differentiation with respect to time but with a total
differentiation along a line moved with the gas. The selection of the
line at the time to is arbitrary. If this line, once selected, is not
moved exactly with the gas, there originates an error which is proportional
to the vo~rtices located between the line moved exactly with the gas and the
wrong connecting line.

2. Applications of the Induction Law

In a steady gas flow free from vortices, the right side qf equation (7)
disappears because in the first place the integral, due to the freedomn
frcan vorticity, is independent of the path and equals the potential differ-
ence 62 1 and in the second place, this potential difference, due to
the steady state, is independent of the time.

NACA T1M 1858

For a steady gas flow with vortices, the right side of equation (7)
disappears only for points Pl and P2 which lie on the same stream-
line. If one draws the connecting line from Pl to P2 along this
streamline, all its points remain on this streamline later on, too. The
entire integration path varies with time only in that it extends beyond
the downstream point P2 in a flat loop which makes, however, no contri-
bution to the integral. The value of the integral on the remaining piece
from Pl to P2 is again independent of the time so that the right side
of equation (7) disappears. The vanishing right side of equation (7)
establishes on the left side the validity of the Bernoulli equation for
the gas flow.

For a flow which is free from vortices but variable with time, the
value of the integral is at every instant, independently of the path,
equal to the potential difference #2 91. However, since the velocity.
distribution varies with time, the potential difference also becomes
dependent on the time, and one obtains the well-known relationship:

12 + u2 '2 22-i 2 '2 1 2"1


i + u 2 2 -Constant )
2 aU t

This is the generalized Bernoulli equation for the unsteady gas flow
free from vortices.

A further application of the induction law is possible for nonsteady
gas flows which vary periodically with time. In this case, it is easier
to determine, instead of the pressure rise at every instant, the mean
value of the pressure rise for the period of the duration T. If one
integrates the right side of equation (7) with respect to t over the
period and then divides by the duration of this period, one obtains the
time average:

h = 1 "2 ( dx + v dy + w dz) + 1J 1 (u dx + v dy + v dz)
21 t L22t+

NACA TM~ 1585

Since the velocity field is the same at the times to and t, + T, the
mean pressure rise indicated above represents the circulation in one
instantaneous field on a closed line from Pl via P2 back to Pl
divided by the time T.

Figure 2 shows how to apply this rule to the periodic flow in a
rotor. One recognizes that the circulation required by equation (9)
matches exactly the circulation r around the blade. In the case of
m blades and an angular velocity of the impeller m, the period is given
by T = 2g-; accordingly, the pressure rise of the rotor amounts to:

h Ea) (9a)
T 21(

Thus, the significance of the circulation around the blades of the
impeller with respect to the pressure rise is established for all gas
flows without increase in entropy. However, it remains to be investi-
gated whether also the torque and therewith the power consumption
depends in the same manner on the blade circulation.


The torque of the impeller in plane flow may be determined from the
difference of the moments of momentum over a control circle outside the
Lapeller and one inside the impeller. If one forms in polar coordinates
the velocity components in the direction of the radius wr and of the
ci rcumfe rence wu, with wU counted positive in the direction of the
increasing angle 9, there results on the are element r dJI of a circle
of the radius r the mass flow pwrr d$. The momentum of this quantity
in circumferential direction is obtained by multiplication by wu, the
moment of momentum by multiplication by rwu. By integration of the
moment of momentum over the entire circumference one obtains a torque

D = r2% ,ru d (10)

If the entering fluid in the interior of the impeller is supplied
without rotation, a sufficiently small radius may exist on which circum-
ferential components of the velocity do not yet appear. In this case,
the integral over the external circle according to equation (10) already
yields the torque of the impeller. For incompressible flow with
p = const., the velocity components wr and vu do not correlate in


the absence of obstacles outside of the impeller, so that for the integra-
tion over wu the mean value of wr may be taken out from under the
integral sign. The mean value of wr multiplied by the density p and
the circumference of the circle 2rn, however, is simply the mass G dis-
charged per second through the impeller. Therefore, in the case of inean-
pressible flow, the torque may be split into the following factors:

D vu C=On--(1
2x 2xr m" Q-

H~ere m is the number of blades, and the circulation around each
of the m blades; the latter transformation results from equation (9a).
Thus, for an incompressible flow a discharge as well as a circulation
around the blades is necessary when a torque is to occur.

Since, according to the results of the previous section, the pres-
sure rise also depends directly on the blade circulation for compressible
flow, there would also be a certain justification for assuming that in
the case of compressible flow no correlation between discharge and blade
circulation enters equation (10). However, one single example where a
torque occurs without discharge or without blade circulation will be
sufficient for deciding; this question in the negative sense. In the
case of the example selected, there appears neither a discharge nor a
blade circulation, and yet one obtains at higher velocities a torque
different from zero.

In order to represent the flow for the case of vanishing discharge
and vanishing blade circulation at a large distance from the impeller,
one can replace the impeller by a rotating wavy cylinder. The simplest
wavy cylinder has a cross section in which on the circumference of a
circle of the radius R, m sinusoidal waves with the wave amplitude A
are superimposed (fig. J for the case m = 3). In case one wants to
represent the effect of an impeller with V1 blades more accurately, one
could still add further waves with the amplitudes A2, A etc., and
the numbers 2m, jm, etc. If the amplitude A, for the sakre of further
simplification of the calculation, is limited to small values compared
to the radius R or, more specifically, compared to the wave
length L = ---, one can neglect in equation (8) the square of the gas
velocity compared to the other summands. Considering constant entropy
according to equation (2) one then obtains

i + --- + --=Constant (12)
at p at


From the potential 4(~)the velocity components are obtained
in the following manner:

wr -

1 ae
vu -
r abJ


Expressed by these components the continuity equation of the unsteady
gas flow reads:

+ "

-- + Pr )
dt r or

P24 1 aG 1 a24 o(4
+p + --- + ---- =0 1)
r2r ar r2 4Ji

results the well-known equation of

Prom equations (12) and (14) there
sound propagation:


1 a0 1 a2
+ +
r ar 2 a92

-1 a*
.2 at2


in which a is the sonic velocity

of the gas according to the following

a2 9


Since, for small velocities, the pressure and the density deviate only
Little from the values of the gas at rest po and po, a may be
regarded as constant and equal to the value of the sonic velocity for
this state. By integration of the pressure at constant density, there
results from equation (12) the pressure

P = Po Po t


The impenetrability of the surface of the wavy cylinder furnishes the
boundary condition for the differential equation (15). At the time t =
the cylinder has the following radii r depending on the central angle


r = R A sin mt

NACA TM 1585

Due to the rotation at the angular velocity n (fig. 3), one obtains
the dependency of the cylinder radii on and t

r( i,t) = R A sin m(9 at) (19)

Hence, the spinning wavy cylinder produces a, radial pumping motion with
the velocity

wr= aP = Annu cos m(C wt)

Because of the smallness of the amplitude A, it is sufficient to pre-.
scribe this value for the radial component of the gas velocity. According
to equation (15), there results the boundary condition for QI:

w Amm cos m(9 cut) (20)
w; r=R

Since the integration proper of the differential equation for the
propagation of sound is known and we are here concerned only with the
application to the impeller, the detailed calculation and the determina-
tion of the integration constants may be omitted after stating that the
elimination of the constants was performed in such a manner that in the
asymptotic development of the solution for large radii only outgoing
waves (no incoming waves) were retained. The solution which thereby
became unique may be written in terms of the Bessel functions of the
first and second kind J, and Y, and reads

Y"mI) cos (rm m it + ) +J, '- sin (my mart + 6)
O = aA (21)

I m U)I\2\a, 2~

The phase angle 6 in equation (21), though in itself unessential,
has the value

tan 8 (22)

If one substitutes this solution into the equation (10) for the torque,
its integration can be achieved with the aid of the formula.

Je,(x)Ym'(x) Y,(x)J,'(x) -

One obtains thus the torque

D =(23)
mmR ,mLB

The power consumption E = Dau corresponds to the radiated sound output.

In order to represent the results in dimensionless form, we shall
use as the Mach number M of the gas flow the ratio of the circumferen-
tial velocity u and the sonic velocity a:

M u Rm (24)
a a

A coefficient for the resistance to the motion c, equal to the
torque D divided by the dynamic pressure q = p utegnrtn
surface of the cylinder F = 2Rx, and the radius R is introduced:

a = D (25)

ICamnpare Frank Mises: Differentialgleichungen der Physik (Differ-
ential equations in phy~sics), 1930, vol. 1, p. 414.

NACA TM~ 1385

and a second coefficient is formed for the radiated energy ce equal
to the radiated energy E = De divided by q, F, and the sonic
velocity a:

ce 1 E =CVM (26)

In this form the results of the calculation are plotted in figure 4
and ) with the Mach number used as the abscissa and the coefficients or
or ce used as the ordinate. The values m indicated at the individual
curves signify the number of waves on the circumference of the circle.
The ordinate has as its unit a value which is formed from the amplitude A
and the wave length L, or the amplitude A, the number of waves m, and
the radius R.

One recognizes from the figures that the torque and likewise the
radiated energy disappears only for incompressible flow and for gas flow
with very small velocities. In the subsonic domain there results for
growing m an ever increasing region in which no noteworthy torques
occur. The maximum of c, always lies at the Mach number 1.

It is interesting to compare the rotating wavy cylinder in a resting
gas and the resting wavy cylinder with a circulatory flow treated by
G. J. Taylor2. While it was found there that even at velocities higher
than sonic velocity damped perturbation waves occur, the above calcula-
tion, inversely, yields the result that even below sonic velocity the
disturbance extends to infinity. Both cases agree at m = m (that is,
for a cylinder radius large compared to the wave length) with the solu-
tion for a flat plate with sinusoidal waves treated before by J. Ackeret5.


It is shown by the example of the plane impeller that in a gas
flow with constant entropy as in an incompressible flow, the pressure
head of a rotating impeller depends only on the circulation around the
blades. In contrast to incompressible flow, however, one obtains for
thie impeller rotating freely in an infinite gas mass a larger torque
than would be necessary for production of the pressure rise because,
due to the periodic disturbance caused by the rotating impeller, sound

20. J. Taylor: ZAMM 10, 1930, p. 334.
5J. Ackeret: Helvetia Phys. Acta 1, Sol, 1928.

NACA TM 1585

NACA TM 1385

waves of finite energy travel away to infinity. This energy produces a
resistance which grows with a high initial power of the circumferential
velocities and, when sonic velocity is exceeded, gradually becomes the
wave resistance of bodies moved rectilinearly/ at supersonic velocity.

Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics

See footnote j on p. 12.

NACA 'IM 138)



w dt

t i= to +dt

t=t t
1 WI
Sw, dt

Figure 1.- IMoving path of integration.


Fiur 2. oaig melr

NACA 'lM 1385

Figure 3.- Cross section of the cylinder corrugated by m = 3 waves.

NACA '.IM 1585




\ L 1

I 2 3 4 5

Figure 4.- Resistance for rotation of the wavy cylinder.

c ==c
e 2UPouF :C

m= 4
,m= 3

( L



I2 3 4 5

Figure 5.- Energy radiation for rotation of the wavy cylinder.
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