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Compressible Flow
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DOI: 10.1615/AtoZ.c.compressible_flow

创建日期： 2 February 2011

最后修改日期： 16 March 2011
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All fluids are compressible and when subjected to a pressure field causing them to
flow, the fluid will expand or be compressed to some degree. The acceleration of
fluid elements in a given pressure gradient is a function of the fluid density, ρ,
whereas the degree of compression is determined by the isentropic bulk modulus
of compression, κ. The speed of sound in a medium is given by, a = (κ/ρ)1/2 and
compressibility effects are apparent when the flow velocity, u, becomes significant
compared to the local speed of sound. The local Mach number M = u/a is the primary
parameter which characterizes the effects of compressibility. Under normal
atmospheric conditions, the speed of sound in water is 1500 ms−1 and that in air is
345 ms−1. Thus, it can be expected that compressibility manifests itself in gas flows
more readily than in liquid flows and the discussion below deals predominantly with
gas flows. Transients in hydraulic systems are an example of compressible liquid

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flow which is of some importance. The case of liquid-gas mixtures is of interest and
is discussed below.

The role of Mach number in compressible gas flow may be derived from the
governing equations of motion and state. However, the physics of these processes
are clear when gas flow from one chamber to another is considered. Flow from a
constant pressure reservoir, a, is produced by reducing the pressure in chamber b
below that in a (Figure 1).

Figure 1. Flow of element C from chamber A to chamber B.

An element of gas, c, will accelerate from a to b, and while doing so increases its
volume and decreases temperature. The local velocity of sound reduces as a result
of this fall in temperature as a = (γRT)1/2, where γ is the ratio of specific heats, T is the
absolute temperature and R is the specific gas constant. Initially, at low-pressure
differentials, the flow is essentially incompressible and pressure falls as the gas
passes through the throat, d. But it recovers somewhat upon diffusion in the nozzle
into chamber b. When the pressure in b is reduced further, this fact is conveyed by
sound waves which travel back through the nozzle into a. The flowfield then
responds by passing more gas through the nozzle. This process will continue as the
pressure in b is reduced up to a point when the local speed of sound has fallen to a
level equal to the local flow velocity. This will first occur at the throat, d, and
henceforth, the flowfield upstream of the throat will remain frozen as sound waves
with the information about conditions in b cannot travel through the throat against
the flow. From then on, mass flow through the nozzle remains constant and the
nozzle is said to be choked. Reducing the pressure in b further does not increase the
mass flow. The flow at the throat has a unity Mach number, i.e., a = u; upstream the
flow is subsonic, M < 1, and downstream the flow becomes supersonic, M > 1. (See
also Nozzles.)

The same conclusion may be drawn from a one-dimensional isentropic analysis of

the steady nozzle flow in Figure 1. This leads to the expression below for du, the

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change in velocity, with u, resulting from a change, dA, in the nozzle cross-sectional
area, A.

(1)

At low Mach numbers, M << 1, u varies inversely with A as demanded by

incompressible continuity. For subsonic conditions, M < 1, u increases with
decreasing A and vice versa. However, the reverse is true for the supersonic case
when M > 1. Also, it is only possible for U to keep increasing through the throat, dA =
0, if M = 1; otherwise du must be zero. The conclusion is therefore the same as that
arrived at above; the nozzle is choked and the Mach number is unity at the
minimum area if the pressure ratio exceeds a particular value.

The isentropic nature of the flow is based on the assumption that no heat transfer
between elements of the gas occurs and that the expansion is reversible, i.e., the
normal conditions for an isentropic change. Thus, no heat conduction or viscous
effects occur. This is true in the free-stream remote from the walls and boundary
layer. However, when shock waves or low-density effects are present, this will not be
the case.

In any flowfield, a narrow stream tube may be taken such that conditions may be
considered one-dimensional (see Figure 1). The Steady Flow Energy Equation (i.e., a
control volume for energy) may be applied from the stagnation conditions at the
inlet (suffix 0) to any point along the stream tube to give the result:

(2)

where h is the specific enthalpy of the fluid. For a perfect gas, h = γRT/(γ − l). Hence,
the variation of local static temperature, T, throughout the flowfield—in terms of the
local stagnation temperature and local Mach number—immediately follows.

(3)

The assumption of isentropic flow then gives the local pressure, P, and density, ρ, in
terms of the stagnation values of these quantities. Thus, the ratio of stagnation
pressure to local pressures.

(4)

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Values calculated from these relationships are tabulated in many texts. Anderson
(1990), Shapiro (1954), Liepmann and Roshko (1960) have provided such tables. As
mentioned in the physical description of nozzle flow, information about boundary
conditions is transmitted by sound waves travelling through the flow to adapt the
flowfield. In supersonic flow, these waves will always be swept downstream. If a
body is introduced into supersonic flow, there has to be a mechanism whereby the
upstream flow becomes aware of the presence of the body. Shock waves, being
discontinuities in flow parameters, provide this mechanism. In the case of slender
and pointed bodies, the shock wave may attach to the leading edge whereas in
blunt bodies, the shock wave detaches and stands upstream as a normal shock
(Figure 2).

Figure 2. Shock waves around sharp and blunt bodies.

The flow passing through a normal shock is subject to large gradients in

temperature and the assumption of isentropic flow is not tenable. The energy
conservation equation, together with the mass continuity and momentum
equations, and the equation of state for an ideal gas lead to relationships between
like properties on either side of the shock; u2/u1, T2/T1, P2/P1 (suffices 1 and 2 refer to
static conditions upstream and downstream). These may be given in terms of Ms =
u1/a1, the flow Mach number relative to the stationary shock wave, as the
independent variable. Tables of these quantities are given in all standard texts. For a
normal shock, the downstream flow is always subsonic in shock relative coordinates.

When supersonic flow expands around a surface convex to the flow (Figure 3), the
information is again transmitted along the sound waves, called Mach waves, which
travel into the flow.

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Figure 3. Expansion around a corner.

Conditions along a streamline obey the isentropic equation mentioned previously.

However, there is now a direct relationship between the angle the flow has turned
through and the local flow Mach number. This is the Prandtl-Meyer relationship,
which gives the Prandtl-Meyer angle, ν(M), as a function of the Mach number. The
angle ν is the angle the flow has turned through from sonic conditions (M = 1) to the
local Mach number. Thus, if flow at M1 turns through a further angle, θ, the change
in Prandtl-Meyer angle equals q and this enables the new Mach number M2 to be
determined

(5)

The Mach waves travel into the flow causing the turning as shown in Figure 3 and
have an inclination to the local flow, μ, called the Mach angle, given by μ = sin−1(l/M).

When two such convex surfaces are arranged to constitute a nozzle, the Mach waves
can be seen to interact. This is shown in Figure 4 where it is clear that the streamline
on the center line has been subjected to Mach waves from both surfaces, which give
no net turning. Nevertheless, the local Mach number will increase according to the
sum of the modulus of the turning angles from both walls for there is no physical
difference in the flow being turned and expanded by the upper or lower wall.

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Figure 4. Interaction of Mach waves in a nozzle.

The process described can be used to design nozzle shapes by calculating the
trajectory of the Mach waves and the resulting turning of the flow and is a simple
example of the Method of Characteristics for solving supersonic flowfields.

In two dimensions, the shock waves may be inclined and determination of the
conditions behind the shock wave is achieved by considering those components of
velocity along and perpendicular to the shock separately. The momentum along the
shock is unaltered whereas the normal component may be considered as in a
normal shock. The results are often given graphically as β, the angle of the shock to
the flow as a function of the flow deflection angle, θ. The upstream Mach number.
M1, is the independent variable. Such a plot is given in Figure 5.

Figure 5. Inclined shock properties from Anderson (1990).

It can be seen that there are two solutions of β for each value of θ at a given free-
stream Mach number, M1. For attached shocks, it is usually the lower value of β—the
weak shock—which is relevant. Flow is predominantly supersonic behind the weak
shock. When the flow deflection angle is increased above a certain angle, there is no
solution. Thus, for sharp bodies of large angle, the shock wave cannot be attached
and a normal shock is formed (Figure 6).

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Figure 6. Examples of regular and Mach reflections.

Inclined shocks may reflect off solid boundaries in a "regular" manner, which
satisfies the condition that the flow remains in contact with the wall after reflection.
As there is a restriction on the maximum value of θ, as shown in Figure 5, some
regular reflections are not possible and a normal shock forms which is normal to the
wall. This automatically satisfies the requirement that the flow remains in contact
with the wall. The normal shock then leaves the wall and curves to join the incident
shock as shown in Figure 6. These shock formations have their counterparts in
supersonic jet plumes. Here, there may be regular reflections from the centerline or
a normal shock, called a Mach disk in the axisymmetric situation, could be present.
These are also shown in Figure 6.

Isentropic flow which has constant stagnation enthalpy can be shown to be

irrotational. This results from Crocco's theorem, and can apply to a wide range of
compressible flows, such as nozzle flow. The irrotational nature of the flow means
that the velocity may be derived by taking the gradient of a potential function (i.e., a
scalar function of position). Thus, the governing equations of motion—the Euler
equations—may be represented by a differential equation for the potential function,
commonly called the velocity potential equation. In subsonic flow, the equation is
elliptical whereas in supersonic flow, the equation is hyperbolic. In the latter case, it
can be shown that along Mach lines certain quantities are invariant. In Figure 4, two
Mach lines or characteristics arise from any point in the flowfield. These will be at
angles θ + μ and θ − μ, where θ is the flow angle and μ, the Mach angle. The quantities

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(6)

(7)

are constant along the −ve and +ve characteristics, respectively. Given an initial
upstream boundary condition for the flow, it is possible to march downstream,
taking into account constraining walls, to solve the flowfield. In the past, this
method of characteristics has been solved graphically but is solved numerically at
present. The method of characteristics may also be employed in axisymmetric and
three-dimensional flowfields. The characteristics are the mathematical counterparts
of the sound waves discussed earlier.

The velocity potential equation may be linearized to solve problems where

deviations in a uniform stream are small. This applies to thin airfoils in a uniform
stream. In this case, it can be shown that with a simple transformation of geometry
the linearized equation is the same as that for incompressible flow, i.e., the Laplace
equation.

As a result, the pressure coefficient, Cp, for Mach number, M, is related to the
incompressible value, CpO, by

(8)

This is known as the Prandtl-Glauert Rule. In supersonic flow, the pressure

coefficient from linearized theory gives the airfoil surface pressure coefficient as a
function of deflection angle, α, of the airfoil surface

(9)

Transonic flow occurs beyond the point when the airfoil, for example, becomes
critical—i.e., when sonic conditions appear on the airfoil. Shock waves can then exist
in the supersonic patch which occurs. This usually results in separation of the airfoil
boundary layer, with a consequent increase in drag (Figure 7). In the transonic
region, the Mach number is extremely sensitive to small changes in flow area.

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Figure 7. Shock formation on a transonic airfoil from Shapiro (1954)

In hypersonic flow, the free-stream velocity is much greater than the local velocity of
sound. This may roughly occur at Mach numbers greater than 5. When expanding
from fixed stagnation conditions, the flow velocity tends to the constant value
where aO is the stagnation velocity of sound. In the hypersonic limit, the
density ratio across a normal shock approaches a constant and the angle of an
oblique shock is linearly related to the deflection angle. Compressible flows are
described in standard texts. Anderson (1990), Anderson (1989), Liepmann and
Roshko (1957) and Shapiro (1954).

Compressible effects are also very important in long ducts subjected to large
pressure ratios. Choking of the ducts may occur if this is of sufficient length, as is
often the case. Here, viscous effects are important and in the analysis it is usual to
assume that fully-developed viscous flow is present. If the flow is adiabatic and the
viscous effects are characterized by a constant friction factor, f, this leads to the
Fanno flow solution. The salient features of this flow are.

1. For a given subsonic inlet Mach number to the duct, there is a maximum length
of the duct, Lmax, in nondimensional terms as given by f L/D max, where D is the
duct diameter at which the flow becomes sonic, i.e., it is choked.

2. Similarly, if supersonic flow enters the duct, the flow decelerates and there is also
a nondimensional length at which the flow chokes.

Conditions in the duct relative to the choked conditions for different inlet Mach
numbers for Fanno flow are tabulated in most texts. These are widely-used for the
design of flow in pipework. Even if choked exit conditions do not prevail, the
compressible effects may be deduced from these tables. Another extreme is

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Rayleigh flow, which is frictionless but with heat addition. Shapiro (1954) has
covered a wide range of such flows.

Two-phase bubbly flow can have interesting properties since it is highly

compressible, due to the gaseous component, and a high density, due to the liquid
present. Thus, the velocity of sound may be much lower than the gas or the liquid in
isolation. The subject is covered by Van Wijngaarden (1972) and Drew (1983). The
flow does act as a compressible fluid of low speed of sound and exhibits shock
structure. This velocity may be as low as 20 ms−1 in bubbly water; however, the
bubbles may slip with respect to the liquid.

A wide range of unsteady compressible flow phenonema exist and these are
covered in the works of Glass and Sislian (1994) and Kentfield (1993). (See also Shock
tubes.)

参考文献列表

1. Anderson, J. D. Jr. (1989) Hypersonic and High Temperature Gas Dynamics,

McGraw Hill New York.
2. Anderson, J. D. Jr (1990) Modem Compressible Flow, McGraw Hill New York.
3. Drew, D. A. (1983) Mathematical Modeling of Two-Phase Flow, Ann. Rev. Fluid
Mech. 15, 261-91.
4. Glass, I. I. and Sistian, J. (1994) Non Stationary Flows and Shock Waves, Oxford
Engineering Series 39, Oxford University Press.
5. Kentfield, J. A. C. (1993) Nonsteady, One-Dimensional Internal, Compressible
Flows, Oxford Engineering Science Series, 31, Oxford University Press. New York.
6. Liepmann, H. W. and Roshko. A. (1956) Elements of Gasdynamics, Galcit
Aeronautical Series, Wiley, New York.
7. Shapiro, A. H. (1954) Compressible Fluid Flow, Volumes I and II, Ronald Press, New
York.
8. Wijngaarden, L. V. (1972) One-Dimensinal Flow of Liquids Containing Small Gas
Bubbles, Ann. Rev Fluid Mech 4, 369. DOI: annurev.fl.04.010172.002101

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