Compressible Flow Lecture 1

FLUID DYNAMICS II – COMPRESSIBLE FLOW

1. INTRODUCTION Compressible flow is defined as a fluid flow with variable density, as opposed to incompressible flows where the density is assumed to be constant. In reality, all fluids are compressible to some extent, but for almost all liquids and some gas flows under certain conditions, the density changes are so small that the assumption of constant density still holds. In these flows, Bernoulli’s equation may be used, which relates the pressure distribution and the local velocity changes along a single streamline.

constantvρ½p 2 =+

For compressible flows, the change in density corresponds to a change in pressure . The relationship between and is given by the compressibility of the fluid :

dρdp dp dρ

τ

dpdρ

ρ1τ =

The fractional change in density is then given by dpτρdρ = . High speed flows thus generally incur large pressure differences. For gas velocities less than ~0.3 of the speed of sound, the associated pressure changes are small, such that will also be small. Low speed gas flows are thus assumed to be incompressible.

dρ

ME2101 – Fluid Dynamics 1

A flow is considered to be compressible if the change in pressure, , results in a fractional change in density, dp ρdρ , that is too large

to ignore. Generally this occurs for gas speeds larger than ~0.3 of the speed of sound. The Mach number is the parameter that we will use to determine the effects of compressibility. The Mach number is also very useful for determining the property ratios ahead of and behind shock waves, to be discussed later. 2. GENERAL DEFINITIONS • Density, ρ : Mass per unit volume (kg/m3). • Specific volume, ν = 1 / ρ: Volume per unit mass (m3/kg). • Pressure p: (Pa).

A normal stress which is isotropic (pressure at a point in a fluid is independent of the orientation of the surface passing through the point). Pressure is therefore a scalar and it always acts normal to the surface.

• Temperature (oK).

H2O at 100oC and freezes at 0oC at ambient atmospheric conditions (p=101.5 kPa). If pressure changes, these temperatures will change. The boiling point is the temperature at which the partial pressure of the water vapour equals atmospheric pressure. If atmospheric pressure is reduced, temperature required for boiling is reduced.


• Perfect gas

The gas flows will be assumed to be thermally perfect, that is, the gas obeys the equation of state TRρp = (The perfect gas law). We will also assume that the flow is calorifically perfect, that is, the specific heats and are constant such that the relations and

pc vc

Tch p= Tcv=e hold. Note that ratio vp CC=γ A fluid that is both thermally and calorifically perfect is called a perfect gas. All other gases are known as imperfect or real gases.

• Dynamic or Absolute Viscosity, µ: (Ns/m2)

Calculated using Sutherland’s Law:

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

CTCT

TT ref

refref

23

µµ

Where: µref =1.716x10-5 Ns/m2 and Tref = 273.15 K and C = 110.4 K for air.

Kinematic viscosity (m2/s): ρµυ =

• Stagnation properties

The stagnation properties refer to the maximum values of each property that the fluid would achieve if brought to rest adiabatically. The subscript indicates a stagnation condition. Thus , , and are the maximum values of enthalpy, temperature, density and pressure that will occur anywhere in the flow field. Derivations for the values of the stagnation properties will be discussed later.

0

0h 0T 0ρ 0p


3. COMPRESSIBLE FLOWS All fluids to some extent are compressible, ie. variable density, hence the fact we can hear sound waves in both air and water. Liquids can be considered practically incompressible (constant density), however, while gases undergo significant changes in density if pressure or temperature are abruptly changed. Gases must be treated as compressible for the following cases, which involve appreciable changes in density:

• Velocity is greater than 0.3 times the speed of sound, a. • Large accelerations of gas • Large changes in elevation • Natural convection due to heat transfer.

Also curious, counter intuitive, phenomena can occur with compressible flows, such as:

• Fluid accelerates due to friction • Fluid decelerates in converging duct • Fluid temperature decreases with heating

3.1 Governing Flow Equations For incompressible, constant density flows, Bernoulli’s equation, which assumes constant density, can be valid. This equation is obviously not valid for compressible flows. The 1D Euler equation for inviscid compressible flows is:

0=++ gdzududpρ (1)


Integrating this expression, with constant density, yields Bernoulli’s equation. 3.1.1 Isothermal Flows (T=constant) Substituting the perfect gas law, p=ρRT, into equation 1 gives:

0=++ gdzudup

dpRT

Integrating, in the limits between two arbitrary points 1 and 2 on the same streamline, with constant T and assuming no change in elevation gives:

2ln

22

21

1

2 uuppRT −

=⎟⎟⎠

⎞⎜⎜⎝

⎛

3.1.2 Adiabatic Flows (No heat transfer) Most compressible flows cannot be assumed to be isothermal and it is more rigorous to assume that a compressible flow is an adiabatic, isentropic process (isentropic, constant entropy, flow is adiabatic and reversible). Thus:

c constant ρpγ == (2)

Then from equation (2):

γ1

γ1

pcρ1 −

= (3)


Substituting equation (3) into Euler’s equation and integrating gives:

0duuρdp

=+

0duudppc γ1

γ1

=+∴−

constant2

u

γ11

pc2

1γ1

γ1

=+−

∴+−

(4)

Substituting the constant, c, from equation (3) into this equation gives:

constant2

u

γ1-γ

pρp 2

1γ1

γ1

γ =+⎟⎟⎠

⎞⎜⎜⎝

⎛+−

constant2

uρp

1γγ 2

=+−

∴

Thus, between two points on a streamline:

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=−∴

2

2

1

121

22 ρ

pρp

1γγ2uu (5)

This is the compressible version of Bernoulli’s equation.


4. MACH WAVES AND THE SPEED OF SOUND Consider the wave patterns below for two dimensional flow, which were first presented by Ernst Mach. The figures show the pattern of infinitesimally weak pressure disturbances (ie. sound waves) propagated by a small particle (point source) moving at speed u through a still medium whose sound velocity is a. As the particle moves, it continually interacts (collides) with other fluid particles, and sends out spherical pressure waves that emanate from every point along its path. The behaviour of the wave fronts is different depending on whether the particle speed is subsonic or supersonic. Consider the figure 1 where the particle moves subsonically, (u < a) :

Figure 1: Pressure disturbances for a subsonic particle


Since u < a then M = u/a < 1. The spherical disturbances move out radially in all directions and do not catch up with each other. The sound waves also move well out in front of the particle, since they travel a distance in the time interval , while the particle has only moved . A subsonic body thus makes its presence felt everywhere in the flow field.

δta δtδtu

For the case of a particle moving at the sonic speed, (u=a), as shown in figure 2, the pressure disturbances move at exactly the speed of the particle, and thus coalesce at the position of the particle into a “wave front locus”, called a Mach wave. No pressure disturbances propagate ahead of the particle. Thus an observer to the left of the particle would not feel the presence of, or hear the sound waves from, the particle until it collided with them.

Figure 2: Pressure disturbances for a sonic particle


The case of a particle moving at supersonic speed, (u > a), is depicted in the figure 3.

Figure 3: Pressure disturbances for a supersonic particle

supersonic motion, the spherical pressure disturbances cannot

Incatch up with the fast-moving particle that created them. The sound waves trail behind the particle, and are tangent to a conical locus called the Mach cone. The half-angle of the Mach cone is:

M1µsin =

For the limiting case of sonic flow, 1M = , , the Mach cone

e particle, as

o90µ =becomes a plane front moving with th shown in Figure 2.


The disturbance caused by the supersonic particle cannot be felt unless an observer is in the zone of action inside the Mach cone. The region outside the Mach cone is called the zone of silence. There is thus a fundamental difference between subsonic (u < a) and supersonic (u > a) flows. In subsonic flows, the pressure disturbance propagates upstream of its location, and the upstream flow receives information of the presence of the particle. However, in supersonic flows no information is received, since pressure disturbances cannot propagate upstream into a supersonic flow field. In order to define the mathematical definition of the speed of sound, a, in a gas, in terms of the static temperature and pressure of the gas, consider figure 4, which shows the infinitesimally small changes in the flow properties across a sound wave (isentropic).

Weak pressure wave with frontal area A.

p p + dp ρ ρ + dρ a a - du T T + dT

control volume

Figure 4: Change in flow properties across a sound wave propagating in at sonic velocity, a, in a gas flow of velocity du.


The pressure wave moves at the speed of sound, a, in a fluid flowing at a low subsonic speed, du. If we analyse the problem from the reference of a fixed pressure wave such that the problem becomes a steady one, the conservation of mass (continuity) gives:

( ) ( uaAAa )δδρρρ −+=

( )δρδδρρδρρ uauaAAa −+−=∴

0=−−∴ δρδρδδρ uua Ignoring second order terms ( δρδu ) which for a weak disturbance are vanishingly small, gives:

0=− ua ρδδρ ie: ua ρδδρ = (6) Now, the linear momentum equation can be derived:

( ) ( ) ( ) AaAuaApppA ρδρρδδ 22 −+−=+−

ρδρδδρρδρδ 222 22 auaauaap −−+−=−∴ Substituting equation (6) and neglecting second order terms:

δρδρδρδ 2222 aaap −=+−=−

Thus: for vanishingly weak pressure disturbances. ρddpa =2

For an isentropic process we have, from equation (2):


c constant ρpγ ==

Such that:

( ) γρ

γρρ

γρρρ γ

γγ

γ ppcdcda ==== −− 112

And, given the perfect gas equation, p=ρRT, we have:

RTa γ= (7) The ratio of the velocity, u, of a gas to the speed of sound, a, in that gas is called the Mach number, M, ie:

auM =

Historically, the compressible flow regime has been subdivided into a number of flow categories, depending on the Mach number range, ie:

• Compressible subsonic flow: 0.3 < M < ~0.6* • Compressible transonic flow: 0.6* < M < ~1.1* • Compressible supersonic flow: 1.1* < M < 5.0 • Compressible hypersonic flow: M > 5.0


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Compressible Flow Lecture 1