Understanding Compressibility of a flow

Understanding Compressibility of a flow
Comprendiendo la comprensibilidad de un flujo
Recibido: marzo de 2016 | Revisado: abril de 2016 | Aceptado: mayo de 2016
Luis A. Arriola1
A b s t r ac t
The coverage of the physical phenomena
experienced in compressible flow is somehow
difficult to understand. However, here we
provide a brief explanation of the assumptions
used in the analysis of compressible flows.
A strong foundation for more advanced and
focused study and understanding of what
causes compressible flows to differ from
incompressible flows and how they can be
analyzed is vital to resolve realistic problems
in aerodynamics. Compressibility issues arise
when fluids are moving at velocities higher than
the speed of sound which is the equivalent of
Mach number 0.3. It is vital to understand
the behavior of the fluid beyond this number
because compressibility factors affect the fluid
and it would be necessary to account for such
factors in order to get results closer to the real
solution.
Keywords: compressible, flow, incompressible,
aerodynamics, Mach number, speed, sound
Resumen
1 Facultad de Ingeniería y Arquitectura
Escuela Profesional de Ciencias Aeronáuticas
Universidad de San Martín de Porres,
Lima - Perú
[email protected]
La cobertura de los fenómenos físicos experimentados en el flujo compresible es algo difícil
de entender. Sin embargo, aquí damos una explicación breve acerca de las hipótesis empleadas
en el análisis de flujo compresible. Una base sólida para un estudio más avanzado y enfocado, y
la comprensión de las causas de flujo compresible para diferenciar de los flujos incompresibles
y la forma en que se pueden analizar es vital para
resolver problemas reales en aerodinámica. Los
asuntos de comprensibilidad aparecen cuando
los fluidos se están moviendo con una velocidad mayor que la velocidad del sonido la cual
es equivalente al valor del número Mach 0.3.
Es importante entender el comportamiento
del fluido por encima de este valor de número
Mach porque los factores de comprensibilidad
afectan al fluido y se necesitaría tomar en cuenta
tales factores para conseguir resultados cercanos
a la solución real.
Palabras clave: compresible, flujo, incompresible, aerodinámica, número Mach, velocidad,
sonido
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enero-junio | 2016 | issn 1812-6049
57
Luis A. Arriola
Introduction
However, the simple definition of compressible flow as one in which the density is
variable requires more elaboration. Consider
a small element of fluid of volume “v” as
shown in Figure 1. The pressure exerted on
the sides of the element by the neighboring
fluid is p.
High speed flows are frequently encountered in engineering applications. Flow
features such as shock waves in nozzles or
supersonic wind tunnels, expansion waves,
and oblique or bow shock waves in front of
rapidly moving objects are all examples of intriguing phenomena that occur due to high
speeds and compressibility of fluids. Compressible fluid flow theory is intended to cover compressible behavior of fluids. Here, we
Figure 1. Small element of fluid of volume v
provide a brief introduction to compressible
flows, including fundamentals of isentropic
compressibility and isothermal compressibility. This phenomenon takes place in certain
regions of speed of the Assume
flow such
high is now increased by an infinitesimal amount dp. The volume of
theaspressure
subsonic, transonic, sonic, supersonic and
Figure 1. Small element of fluid of volume v
hypersonic velocities. will be correspondingly compressed by the amount dv. Since the volume is redu
Assume the pressure is now increased by
However, normal shocks,
oblique
and
exnegative quantity. The compressibility
ofamount
the fluid,
Ʈ,The
isofdefined
by Anderson, Jr. (20
an infinitesimal
volume
Figure
1. Small element
ofdp.
fluid
volume vof
pansion shocks may appear in these phenothe element will be correspondingly commena contributing to the complexity of the
pressed by the amount dv. Since the volusolution. Properties of the flow may change
me is reduced, dv is a negative quantity. The
dramatically right after the shocks affecting
compressibility of the fluid, Ʈ, is defined by
the whole solution of the fluid flow in this
as
Assume the pressure is nowAnderson,
increased Jr.
by(2003)
an infinitesimal
amount dp. The volume of the
area and the surrounding
ompressible flow is routinely definedDiscussion
aswill
variable
density flow; compressed
this is in by
contrast
be correspondingly
the amount
1to𝑑𝑑𝑑𝑑𝑣𝑣𝑣𝑣 dv. Since the(2)volume is reduced
τ= - 𝑣𝑣𝑣𝑣 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
compressible flow, where the
density is assumed
to be
constant
throughout.
Obviously,
real Ʈ, is defined by Anderson, Jr. (2003)
Compressible
flownegative
is routinely
defined
quantity.
The
compressibility
of theinfluid,
as variable density flow; this is in contrast
Physically,
thea compressibility
is the frace every flow of every fluid
is compressible
some the
greater
or lesser extent;
hence,
truly
to incompressible
flow,to where
density
tional change in volume of the fluid element
is assumed to be constant throughout. Obper
unit change
pressure. However, equanstant density (incompressible)
flow
is
a
myth.
However,
as
previously
mentioned,
for in
almost
viously, in real life every flow of every fluid is
transferred
into
or out of the
gas2through
the boundaries
of the
system.
tion
is not sufficiently
precise.
We
know Therefore, if the te
compressible to some greater
or lesser
extent;
liquid flows as well as for
the
flows
of
some
gases
under
certain
conditions,
the
density
changes
from experience that when a gas is compreshence, a truly constant density (incompresof the fluid element is held
constant
(due to some
then the
sed,
its temperature
tendsheat
to transfer
increase,mechanism),
desible) flow is a myth. However, as previously
compressibility
is the fractional
change in volume of the fluid elem
1
𝑑𝑑𝑑𝑑𝑣𝑣𝑣𝑣
e so small that the assumption of constant densityPhysically,
can be madethe
with
reasonable
accuracy.
In
such
pending on the amount of heat transferred
mentioned, for almost all liquid flows as well
τ= -Jr.𝑣𝑣𝑣𝑣(2003)
compressibility is defined into
by Anderson,
as the boundaor out of the
gas
𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑through
as for the flows of some gases
under
certain
change
in
pressure.
However,
equation
2
is
not
sufficiently
precise. We know from ex
ses, Bernoulli’s equation 1, defined by Anderson, Jr. (1989) can be applied
confidence.
rieswith
of the
system. Therefore, if the tempeconditions, the density changes are so small
rature
the fluid element
is increase,
held constant
that the assumption of constant
can
when a density
gas is compressed,
itsoftemperature
tends to
depending on the am
(due
to
some
heat
transfer
mechanism),
then
be made with reasonable accuracy. In such
the isothermal compressibility is defined by
cases, Bernoulli’s equation 1, defined by AnAnderson, Jr. (2003) as
derson, Jr. (1989) can be applied with confidence.
1 𝜕𝜕𝜕𝜕𝑣𝑣𝑣𝑣
Physically,
the
compressibility
is
the
fractional
change
of the fluid element
𝜏𝜏𝜏𝜏
=
−
( ) in volume (3)
𝑇𝑇𝑇𝑇
𝑣𝑣𝑣𝑣 𝜕𝜕𝜕𝜕𝑑𝑑𝑑𝑑 𝑇𝑇𝑇𝑇
(1)
𝑝𝑝𝑝𝑝 + 2 ρ 𝑉𝑉𝑉𝑉 2 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
(1)
change in pressure. However, equation 2 is not sufficiently precise. We know from experi
1
58
when a gas is compressed,
its |temperature
to increase, |depending
on the amoun
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V. XXI | No.tends
21 | enero-junio
2016 |
On the other hand, if no heat is added to or taken away from the fluid element (if the co
Understanding Compressibility of a flow
other hand, if no heat is added to or taken away from the fluid element (if the compression
the othertransport
hand, if mechanisms
no heat is added
ges in the
particular, we shall see
atic), and if no other On
dissipative
suchtoas viscosity
andpressure.
diffusionInare
or taken away from the fluid element (if the
that high-speed flows generally involve larcompression
is adiabatic),
if no other
ge pressure
For a given change in
nt (if the compression
is reversible),
then the and
compression
of the fluid
elementgradients.
takes place
dissipative transport mechanisms such as
pressure, dp, due to the flow, equation 7 deviscosity
and diffusion
(if thecompressibility
monstratesisthat
the resulting
change in denically as described
by Shapiro
(1953), are
andimportant
the
isentropic
defined
by
relatively
large pressure
gradients can
create
high velocities
without much change in density.
compression is reversible), then the compressity will be small for liquids (which have low
values of Ʈ), and large for gases (which have
on, Jr. (2003) as sion of the fluid element takes place isentroHence,
such
flows
are
usually assumed to be incompressible, where ρ is constant. On the other
pically as described by Shapiro (1953), and
high values of Ʈ). Therefore, for the flow of
the isentropic compressibility is defined by
liquids, relatively large pressure gradients can
hand,
for
the
flow
of
gases
attendant
large values
Ʈ, moderate to strong pressure
Anderson, Jr. (2003) as
createwith
hightheir
velocities
without
muchofchange
in density. Hence, such flows are usually
gradients
lead
to
substantial
changes
density via equation
7. Atρ the
assumed
to in
betheincompressible,
where
is same time, such pressure
1 𝜕𝜕𝜕𝜕𝑣𝑣𝑣𝑣
m). If the fluid element is assumed
to
have
unit
mass,
v
is
the
specific
volume
(volume
per
(4)
𝜏𝜏𝜏𝜏𝑠𝑠𝑠𝑠 = − 𝑣𝑣𝑣𝑣 (𝜕𝜕𝜕𝜕𝑑𝑑𝑑𝑑)𝑠𝑠𝑠𝑠
(4) hand, for the flow
constant. On the other
gradients create large velocity
in theattendant
gas. Such large
flowsvalues
are defined
of gaseschanges
with their
of as compressible flows,
ass), and the density is
Ʈ, moderate to strong pressure gradients
where the subscript s denotes that the partial
ρ is a variable. lead to substantial changes in the density via
derivative is taken at constantwhere
entropy.
equation 7. At the same time, such pressure
Compressibility
a property
of thev fluid.
m). If the fluid element
is assumed toishave
unit mass,
is the specific
volume
(volume
per
gradients
create
large velocity
changes in the
he subscript s denotes
thathave
the partial
derivative
taken
at constant
If isthe
velocity
of gasesgas.
isentropy.
less
than
0.3
than
the
speed as
of compressible
sound (M<0.3), the associated pressure
Liquids
very low
values of
compressibiSuch
flows
are
defined
1-10
m).
If
the
fluid
element
is
assumed
to
have
unit
2 mass, v is the specific volume (volume per
lityis(ƮT for water is𝜌𝜌𝜌𝜌5x10
ass), and the density
= 𝑣𝑣𝑣𝑣 m / N at 1 atm)
(5)
flows, where ρ is a variable.
whereas
gases
have
high
compressibility
(ƮT and
changes
are small,
even
though Ʈ is large
for
gases, dp in equation 7 may still be small enough
for
ssibility
is
a
property
of
the
fluid.
Liquids
have
very
low
values
of
compressibility
(Ʈ
T
ass), and the density
If the velocity of gases is less than 0.3 than
forisair is 10 -5 m2/ N at 1 atm). If the fluid
-10 m2/ N at 1
-5 m2(M<0.3),
the
speed
of
sound
the
associated
element
is assumed
havehigh
unit
mass,a vsmall
is dρ.(Ʈ
atm)
whereas
gasestohave
compressibility
airreason,
is 10
N
to dictate
For
this
the /low-speed
flow
of gases can be assumed to be
T for
ms5x10
of density,
equation
2 becomes
pressure changes are small, and even though
the specific volume (volume per unit mass),
1
Ʈ is
in equationthan
7 may
and the density is
inlarge
figurefor
2. gases,
This isdp
velocities
250still
mi/h (112 m/s). On the other
𝜌𝜌𝜌𝜌1=𝑑𝑑𝑑𝑑𝜌𝜌𝜌𝜌𝑣𝑣𝑣𝑣 incompressible as shown
(5) aless
be small enough to dictate
small dρ. For
Ʈ = 𝜌𝜌𝜌𝜌 (𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑1 )
(6)
reason,
flow of gases
can the associated pressure
𝜌𝜌𝜌𝜌 = 𝑣𝑣𝑣𝑣 hand, for flow(5)velocitiesthishigher
thanthe
0.3low-speed
of the(5)
(M>0.3),
6speed of sound
be assumed to be incompressible
as shown in
ms of density, equationIn2terms
becomes
Figure 2. This is velocities less than 250 mi/h
of density, equation
2 becomes
changes,
dp are relatively large, and having a large value of Ʈ for gases, large changes in density,
(112 m/s). On the other hand, for flow veloms of density, equation 2 becomes
cities higher than 0.3 of the speed of sound
1 𝑑𝑑𝑑𝑑𝜌𝜌𝜌𝜌
dρ are produced
7 the associated(6)pressure changes, dp
(6) via equation
Ʈ
=
(
)
(M>0.3),
ore, whenever the fluid experiences a𝜌𝜌𝜌𝜌change
in pressure, dp, the corresponding
change in
𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
1 𝑑𝑑𝑑𝑑𝜌𝜌𝜌𝜌
are
relatively
large,
and
Ʈ = 𝜌𝜌𝜌𝜌 (𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑)
(6)having a large value
of Ʈ for gases, large changes in density, dρ
y will be dρ, where from equation 6
Therefore, whenever the fluid experiences
are produced via equation 7
a change in pressure, dp, the corresponding
change in density will be dρ, where from
ore, whenever the equation
fluid experiences
a change in pressure, dp, the corresponding change in
6
the fluid
a change
dp, the corresponding change(7)
in
𝑑𝑑𝑑𝑑𝜌𝜌𝜌𝜌6 =
𝜌𝜌𝜌𝜌Ʈ𝑑𝑑𝑑𝑑𝑝𝑝𝑝𝑝in pressure,(7)
yore,
willwhenever
be dρ, where
fromexperiences
equation
y will be dρ, where from equation 6
To this point, we have considered just the
fluid itself, with compressibility being a property of the fluid. Now assume that the fluid
is in motion.
to Zucrow
and Hopoint, we have considered
justAccording
the fluid
itself,
with compressibility
being a property of (7)
the
𝑑𝑑𝑑𝑑𝜌𝜌𝜌𝜌
= 𝜌𝜌𝜌𝜌Ʈ𝑑𝑑𝑑𝑑𝑝𝑝𝑝𝑝
ffman (1976), such flows are initiated and
= 𝜌𝜌𝜌𝜌Ʈ𝑑𝑑𝑑𝑑𝑝𝑝𝑝𝑝
(7)Range of 0.3 M
maintained
by𝑑𝑑𝑑𝑑𝜌𝜌𝜌𝜌
forces
on
the fluid,tousually
Now assume that the
fluid is in
motion.
According
Zucrow andFigure
Hoffman
(1976), such
2. Compressibility
created by, or at least accompanied by, chanFigure 2. Compressibility Range of 0.3 M
are initiated and maintained by forces on the fluid, usually created by, or at least
point, we
considered
the fluidInitself,
with compressibility
a property of
the
panied
by,have
changes
in thejust
particular,
we shall
see| being
that high-speed
flows
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| pressure.
V. XXI | No.
21
| enero-junio
| 2016
point,
we have
just
fluid itself,
with compressibility
being
a property
of the
Now
assume
thatconsidered
the fluidgradients.
is
inthe
motion.
to Zucrow
and dp,
Hoffman
(1976),
lly involve
large
pressure
For aAccording
given change
in pressure,
due to the
flow,such
Eq.
59
Luis A. Arriola
Conclusion
In summary, a compressible flow is considered as one where the change in pressure,
dp, over a characteristic length of the flow,
multiplied by the compressibility via equation 7, results in a fractional change in den-
sity, dρ/ρ, which is too large to be ignored.
For most practical problems, if the density
changes by 5% or more, the flow is considered to be compressible.
References
Anderson, Jr., J. D. (1989). Introduction to
Flight (3rd ed.). New York: McGraw
Hill.
Anderson, Jr., J. D. (2003). Modern Compressible Flow with Historical Perspective. New York: McGraw Hill.
60
Shapiro, A. (1953). The dynamics and thermodynamics of compressible fluid flow.
New York: The Ronald Press Company.
Zucrow, M. & Hoffman, J. (1976). Gas
Dynamics. New York: Wiley
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