Compression shocks in two-dimensional gas flows

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Title:
Compression shocks in two-dimensional gas flows
Series Title:
NACA TM
Physical Description:
17 p. : ill ; 27 cm.
Language:
English
Creator:
Busemann, Adolf, 1901-
Vanier, J
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Aerodynamics   ( lcsh )
Shock waves   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
Contains early work presenting treatment of shock waves by introducing the shock polars.
Bibliography:
Includes bibliographic references (p. 4).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by A. Busemann.
General Note:
"Report date February 1949."
General Note:
"Translation of "Verdichtungsstösse in ebenen Gasströmungen." From Vorträge aus dem Gebiete der Aerodynamik und verwandter Gebiete, 1929, pp. 162-169."
General Note:
"Translated by J. Vanier, National Advisory Committee for Aeronautics."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003804321
oclc - 123551737
System ID:
AA00009239:00001


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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM NO. 1199


COMPRESSION SHOCKS IN TWO-DIMENSIONAL GAS FLOWS*

By A. Busemann


The following arguments on the compression shocks in gas flow start
with a simplified representation of the results of the study made by
Th. Meyer as published in the Forschungsheft 62 of the VDI, supplemented
by several amplifications for the application.

In the treatment of compression shocks, the equation of energy,
The equation of continuity, the momentum equation, the equation of state
of the particular gas, as well as the condition of the second law of
thermodynamics that no decrease of entropy is possible in an isolated
system, must be taken into consideration. The result is that, in those
cases where the sudden change of state according to the second law of
therncdynamics is possible, there always occurs a compression of the gas
-'hich is uniquely determined by the other conditions.

First, it will be shown that the resulting relations can be easily
expressed if the thermodynamic and the pure dynamic relations have been
previously transformed so that pure thermodynamics, as well as pure
dynamics, can be expressed simultaneously in one diagram. Since the
static pressure p itself represents a state quantity as well as a dynamic
quantity, one axis of the diagram may represent a p-axis. From the equation
of energy for steady flow from a tank follows the heat conduction being
disregarded the conventional relation that the kinetic energy of the

unit mass w2 (v = velocity) is equal to the difference of the heat
2
content of the tank io and the momentary heat content 1. Hence, when
a w-axis is chosen as the other axis, the lines w = Constant correspond
to definite heat contents 1 and the diagram can be used as a distorted
p,i diagram exactly like any other state diagram utilizing two state-
quantities as axes. The following general relations in this diagram can
be easily proved for adiabatic flow (fig. 1). For constant entropy, the
negative differential quotient -dp/dw represents the rate of flow pw
(p = gas density), as obtained from Bernoulli's equation: -- = i-d.
p 2
The slope of the line of constant entropy accordingly represents for each
point the rate of flow, that is, the reciprocal value of the surface
necessary for the discharge of the unit mass. It immediately follows
that the tanrent to the entropy line on the p-axis cuts the momentum p + p -

"Verdichtungsstosse in ebenen Gasstrbmungen." Vortrage aus dem
Gebiete der Aerodynamik und verwandter Geblete, Aachen 1929, -p. 162-169.







NACA TM No. 1199


From these simple relations in the p,v diagram, the statement is
readily proved that normal compression shocks or compression shocks 4n
one-dimensional flow are possible only between those points which have a
common tangent on their entropy lines. Such states fulfill the equation
of state of the particular gas, because they are located on its p,i dia-
gram; they comply with the e-uation of energy, because the equation is
used to identify the w-exis; they satisfy continuity, because their entropy
lines have the same direction; and they have identical momentum, because
the tangents have equal Intercepts on the p-axis (fig. 2).

The second law of thermodynamics contributes the fact that the later
one be the state of greater entropy. Since the cross section necessary
for the unit mass increases with the speed at supersonic velocity, and
hence the rate of flow decreases, the upward concave part of the entropy
lines signifies supersonic velocity, the upwardly convex part subsonic
velocity. Normal compression shocks have, therefore, supersonic speed
as initial state and subsonic speed as terminal state.

Extension of the arguments to include two-dimensional flow simply
involves the substitution of the w-axis for a u,v or velocity plane,
against which the pressures p are plotted, the surface of equal entropy
being obtained as surface of rotation of the entropy lines of the p,w dia-
gram (fig. 3). In isentropic flow, all states are situated on one single
surface of constant entropy. As stated elsewhere (reference 1), the gas
flow is unusually sensitive in cross sections in which a relatively
maximum rate of flow exists. In one-dimensional flow, the absolute
maximum rate of flow is through cross sections in which the flow velocity
equals the sonic velocity. In the p,w diagram, they are represented by
the point of inflexion of the entropy lines as the point of maximum slope
of the entropy line. In two-dimensional flow, all such cross sections are
normal to the directions of the curves of the main tangents on the. saddle-
like curved region of the entropy surface. Then sensitive cross sections
with the relatively maximum rate of flow appear as steady sound waves in
two-dimensional supersonic flow, when minor disturbances (such as
roughening with a file) are applied at the boundary walls of the flow
(fig. 1).

The curves of the main tangents projected on the plane of the velocity
then give a network of lines by means of which the supersonic flow in the
prescribed channels can be pursued (fig. 5).

If the streamlines in a supersonic flow are deflected at a finite
angle toward the flow, say, by the boundary wall, for example, no stagnation
point need occur at this point like in the subsonic flow. The supersonic
flow can rather achieve the deflection by an oblique compression shock
(fig. 6), if the angle of deflection does not exceed a certain amount.
But this is accompanied by an entropy rise without which momentum, energy,
and continuity theorem cannot be fulfilled. The terminal states after a






NACA TM No. 1199


compression shock are therefore no longer located on the surface of constant
entropy, but within the pressure dome of constant initial entropy by reason
of the entropy rise. On assuming the direction of the compression shock,
or normal to it, the direction of the velocity variation, as given, it
results in a p,w diagram above the particular straight line, in which
the terminal state can be identified as the normal compression shock
exactly like in figure 2 (fig. 7).

For a given velocity w1 all terminal states after compression shocks
lie on the tangential plane at the pressure dome in point pl, X1 (fig. 8).
In the tangential plane, the terminal states appear again as points of
relatively maximum entropy on all rays through pl 1. In the projection
on the velocity plane, the line connecting all terminal states w2 possible

from X1 is termed shock polar. The shock polars give the possible
deflections as well as the position of the shock surface perpendicular to
the velocity difference Xl y2 for each deflection. Figure 9 represents
a shock polar diagram for air with k 1.405, showing the shock polars from
different starting points on the u-axis, along with the curves of constant
entropy of the terminal state and indicated by the pressure ratio p'o/Po.
By multiplication with p'o/Po it affords, for perfect gases, the height of
the other surfaces of constant entropy from the heights of the initial
adiabatic surface.

With these diagrams, it is possible to follow two-dimensional flows
even in cases where compression shocks occur. For illustration, figure 10
shows a flat plate with a given angle of attack and figure 11 shows
a symmetrical flow past a biconvex airfoil, and In figure 12, a schlieren
record of real flow past such an airfoil. This example demonstrates that
supersonic flows in which shocks occur, can also be treated graphically
in close agreement with reality. Minor deflections may be treated by
the methods of adiabatic flow.

Strong compression shocks present a certain difficulty if neighboring
stream filaments pass through compression shocks of dissimilar intensity. Such
flows are no longer irrotational and can then be treated approximately by
the methods of potential flow only if the vortex strength is concentrated
in certain stream lines. Each strip between two such stream lines can then
be treated separately as potential flow and the bordering stream lines
plotted in such a way that equal pressure and equal velocity direction
appear in both adjacent strips. In the examples (figs. 10 and 11), the
departure from potential flow was regarded as disappearingly small.







NACA TM No. 1199


Discussion

Mr. Burgers, Delft, asked whether it was possible to draw a con-
clusion from the compression shocks about the wave resistance of bodies
at supersonic speeds.

Busemann: To compute the magnitude of wave resistance it is per-
missible for slender profiles to work with adiabatic compression shocks,
as given by Riemann (reference 2). The result is then invariably a
positive pressure on the surfaces which push the flow aside, and negative
pressure on the surfaces which contribute room to the flow. From this
the wave resistance (reference 3) follows immediately. The question of
where the work of resistance in the gas remains can be answered from the
compression shocks, even for slender profiles. As figure 11 indicates, com-
pression shocks are obtained, the strength of which abates simultaneously
with the disappearance of the wave field with increasing distance from
the profile. By integration with respect to all stream filaments, it
can be proved that the heating of the gas on traversing the compression
shocks precisely represents the work of resistance. The resistance momentum
follows likewise as momentum of the forward movement in the wake produced by
the shocks.


Translated by J. Vanier
National Advisory Committee
for Aeronautics




REFERENCES


1. Prandti, L., and Busemann, A.: Niiherungsverfahren zur zeichnerischen
Ermittlung von ebenen Stromungen mit Uberschallgeschwindigkeit.
Honnegger, Stodola-Festschrift, Zurich 1929, page 499.

2. Riemann Weber: Partielle Differentialgleichung, 5th Edition, 1912,
page 481.

3. Ackeret: Z. f. Flugtechnik u. Motorluftschiffahrt, 1925, page 72.







NACA TM No. 1199 5


Figure 1.- Relations in the p,w diagram.




















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Figure 2.- Iormal compression shock in the p,w diagram.


NACA TM No. 1199


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NACA TM No. 1199


Figure 3.- p,u,v diagram for plane flow with constant entropy.






































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NACA TM No. 1199


Figure 4.- Schlieren photograph of steady sound waves.










NACA TM No. 1199


Figure 5.- Graphical representation of flow of figure 4.















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Figure 6.- Compression shock at deflection by a finite angle.







12 NACA TM No. 1199


Figure 7.- p,u,v diagram with entropy rise.







NACA TM No. 1199


Figure 8.- Shock polar in the tangential plane at the p dome.







NACA TM No. 1199


Figure 9.- Shock polar diagram of air (k = 1.405). Shock polars, solid lines;
Po'/Po curves, broken lines.








NACA TM No. 1199


Figure 10.- Flow past a flat plate with angle of attack.


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Figure 11.- Flow past a biconvex profile.








NACA TM No. 1199


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