COMPRESSIBLE FLOW

Stagnation State Relations

• When the fluid is approximated as an ideal gas with constant specific heats;

Stagnation State Relations

• For ideal gases with constant specific heats,

P0 is related to the static pressure of the fluid by;

Stagnation density to static density is expressed as;

SPEED OF SOUND AND MACH NUMBER

• For the most part, we have limited our consideration so far to flows for which density variations and thus compressibility effects are negligible.

• We lift this limitation and consider flows that involve significant changes in density.

• Such flows are called compressible flows, and they are often encountered in devices that involve the flow of gases at very high velocities.

SPEED OF SOUND AND MACH NUMBER

• An important parameter in the study ofcompressible flow is the speed of sound (orthe sonic speed), defined as the speed atwhich an infinitesimally small pressurewave travels through a medium.

• The pressure wave may be caused by a smalldisturbance, which creates a slight rise inlocal pressure.

SPEED OF SOUND AND MACH NUMBER

• The amplitude of theordinary sonic wave is verysmall and does not cause anyappreciable change in thepressure and temperature ofthe fluid.

• Therefore, the propagation ofa sonic wave is not onlyadiabatic but also very nearlyisentropic.

SPEED OF SOUND AND MACH NUMBER

• Then the thermodynamicrelation reduces to;

• Finally combining the followingequations;

SPEED OF SOUND AND MACH NUMBER

• When the fluid is an ideal gas and withthe help of Tds equations;

• This quantity is called the speed of sound because sound waves are weak pressure waves.

SPEED OF SOUND AND MACH NUMBER

• Noting that the gas constant R has afixed value for a specified ideal gas

• the specific heat ratio k of an idealgas is, at most, a function oftemperature,

• we see that the speed of sound in aspecified ideal gas is a function oftemperature alone.

SPEED OF SOUND AND MACH NUMBER

• A second important parameter in theanalysis of compressible fluid flowisthe Mach number Ma, named afterthe Austrian physicist Ernst Mach(1838–1916).

• It is the ratio of the actual speed ofthe fluid (or an object in still fluid) tothe speed of sound in the same fluidat the same state:

SPEED OF SOUND AND MACH NUMBER

• Note that the Mach numberdepends on the speed of sound,which depends on the state of thefluid.

• Therefore, the Mach number of anaircraft cruising at constant velocityin still air may be different atdifferent locations.

• Fluid flow regimes are often described in terms of the flow Mach number.

SPEED OF SOUND AND MACH NUMBER

• The flow is called

• sonic when Ma = 1,

• subsonic when Ma < 1,

• supersonic, when Ma > 1,

• hypersonic when Ma >> 1,

• and transonic when Ma ≅ 1.

Variation of Fluid Velocity with Flow Area

• In the remainder of this section we investigatethese couplings more thoroughly, and we developrelations for the variation of static-to-stagnationproperty ratios with the Mach number forpressure, temperature, and density.

• This is an important relation for isentropic flow inducts since it describes the variation of pressurewith flow area.

Variation of Fluid Velocity with Flow Area

• We note that A, ρ, and V are positive quantities.For subsonic flow (Ma < 1), the term 1 – Ma2 ispositive; and thus dA and dP must have the samesign.

• That is, the pressure of the fluid must increase asthe flow area of the duct increases and mustdecrease as the flow area of the duct decreases.

• Thus, at subsonic velocities, the pressuredecreases in converging ducts (subsonic nozzles)and increases in diverging ducts (subsonic

• diffusers).

Variation of Fluid Velocity with Flow Area

• In supersonic flow (Ma > 1), the term 1 – Ma2 isnegative, and thus dA and dP must have oppositesigns.

• That is, the pressure of the fluid must increase asthe flow area of the duct decreases and mustdecrease as the flow area of the duct increases.

• Thus, at supersonic velocities, the pressuredecreases in diverging ducts (supersonic nozzles)and increases in converging ducts (supersonicdiffusers).

Variation of Fluid Velocity with Flow Area

• Another important relation for the isentropic flowof a fluid is obtained by substituting ρV = –dP/dVfrom.

• This equation governs the shape of a nozzle or adiffuser in subsonic or supersonic isentropic flow.Noting that A and V are positive quantities, weconclude the following:

Variation of Fluid Velocity with Flow Area

• Thus the proper shape of a nozzle dependson the highest velocity desired relative tothe sonic velocity. To accelerate a fluid, wemust use a converging nozzle at subsonicvelocities and a diverging nozzle atsupersonic velocities.

• The velocities encountered in most familiarapplications are well below the sonicvelocity, and thus it is natural that wevisualize a nozzle as a converging duct.However, the highest velocity we canachieve with a converging nozzle is thesonic velocity, which occurs at the exit ofthe nozzle.

Variation of Fluid Velocity with Flow Area

• If we extend the converging nozzleby further decreasing the flow area,in hopes of accelerating the fluid tosupersonic velocities, as shown inFig. we are up for disappointment.

• Now the sonic velocity will occur atthe exit of the converging extension,instead of the exit of the originalnozzle, and the mass flow ratethrough the nozzle will decreasebecause of the reduced exit area.

Variation of Fluid Velocity with Flow Area

• Based on Eq. which is anexpression of the conservation ofmass and energy principles, wemust add a diverging section to aconverging nozzle to accelerate afluid to supersonic velocities. Theresult is a converging– divergingnozzle.

Mach number relations; Property Relations forIsentropic Flow of Ideal Gases• Relation between T0 and T

• Relation between P0 and P

• Relation between ρ0 and ρ

Property Relations for IsentropicFlow of Ideal Gases• The properties of a fluid at a location where the Mach number is

unity (the throat) are called critical properties, Setting Ma = 1

• The critical properties of compressible flow should not be confusedwith the thermodynamic properties of substances at the critical point.

The critical property ratios

(k=ϒ)

k=ϒ