Hydraulics Prof. B.S. Thandaveswara

Indian Institute of Technology Madras

(e) Compressible and Incompressible Flow

Compressiblity is a measure of the change of volume of a fluid or gas under external

forces. Modulus of elasticity K can be defined as

PKV∆

= −∆ 0

V 0

PP

∆=

⎛ ⎞ ⎛ ⎞∆⎜ ⎟ ⎜ ⎟ρ⎝ ⎠ ⎝ ⎠

(1.4 e - 1)

In other words

VP K ∆∆ = − 0

V( )0 (1.4 e - 2)

p K = ρ ρ∆∆

(1.4 e - 3)

in which, V∆ 0

V 0

⎛ ⎞⎜ ⎟⎝ ⎠

, denotes the change of volume brought about by a pressure increase

p∆ . Generally, the compressibility of the liquid is very small. k is the bulk modulus of

elasticity.

Example

The bulk modulus of elasticity for Fresh water is 2.19 G Pa, Sea water 2.27G Pa (G Pa

=Giga Pascals =109) at temperature 20°C ( 293.15 K ) and 15°C ( 288.15 K )

respectively. This is equivalent of 0.005% . i.e., the pressure increase of one

atmosphere causes a relative change of 1/20000 and this is normally neglected in free

surface flows.

When the hydraulic transients occur then the water is taken as compressible. Velocity of

propagation of a pressure wave for rigid pipes is K /ρ .

Hydraulic Transient: When flow in a pipe line changes from one steady condition to

another steady condition. Such a phenomenon is common in Penstock (high pressure

pipelines) of high head hydro electric power system and in water supply pipeline.

The dimensionless number describing the compressibility of fluid is Known as Mach

number ( M ).

Hydraulics Prof. B.S. Thandaveswara

Indian Institute of Technology Madras

vM =

c is ratio of Velocity of the flow v

Velocity of the sound in that media c (1.4 e - 4)

in which, v is the velocity of flow, c is the velocity of sound in the fluid media. Based on this Flow is classified as Subsonic when M < 1, Transonic when 0.8 < M < 1.2, Supersonic when M > 1 and

Hypersonic when M > 5.

The fluid may be taken as incompressible when

1 2M 12

(1.4 e - 5)

The flow of water under atmospheric pressure can be treated as incompressible flow.

In other words if the Mach number is small compared to unity , the flow velocity is small

compared to the speed of sound If c = 335.3 m/s ( the velocity of sound in air ) then the

change of density

∆ 1 2= M2

ρρ

(1.4 e - 6)

is equal to 0.05 for a velocity of v = 100.58 m/s. This value may be taken as the upper

limit for treating the gas flow as incompressible.

Mach angle

1Shock cone angle sin =M

For example if M = 1.5, = 41.81

M = 3 , 19.47

M = 6, = 9.594

β

β

β

β

β

°

= °

°

Supersonic jet (X43A supersonic combustion ramjet (scramjet)) with Mack number > 7

has been successfully tested in the 2nd half of March 2004. It flew at a speed of 7700

kmph.