Unit 1: Concepts of compressible flow - Subjective Questions

Compressible flow is a fluid flow in which variations in density are significant and must be included in the governing equations.

Scope in aerodynamics:

• High-speed flight of aircraft, missiles, and spacecraft
• Flow through nozzles, diffusers, compressors, and turbines
• Propagation of acoustic and pressure waves
• Shock-wave and expansion-wave phenomena
• Supersonic wind tunnels and jet propulsion systems

The importance of compressibility is primarily determined by the Mach number:

where is the flow velocity and is the local velocity of sound.

• For , density changes are generally below about , so the flow may be treated as incompressible.
• For , changes in density become important.
• At transonic and supersonic speeds, compressibility effects such as choking, shock waves, and expansion waves dominate the flow.

An isentropic flow is a flow that is both adiabatic and reversible. Therefore, the entropy of each fluid particle remains constant:

Principal assumptions:

• The fluid behaves as a perfect gas.
• The flow is adiabatic, so there is no heat transfer.
• The flow is frictionless and internally reversible.
• There are no shock waves or other dissipative effects.
• Specific heats are usually assumed constant.

For a perfect gas, the isentropic relations are:

where . These relations connect pressure, temperature, and density without explicitly considering heat transfer or entropy generation.

For steady one-dimensional flow through a stream tube, conservation of mass gives:

Between two sections:

The differential form is:

This equation shows that area, velocity, and density can all vary in compressible flow.

For steady, inviscid, one-dimensional flow without body forces, the momentum equation is:

or

Physical interpretation:

• A favorable pressure gradient, , accelerates the flow.
• An adverse pressure gradient, , decelerates the flow.
• Unlike incompressible flow, changes in velocity are accompanied by potentially significant changes in density.

For a steady-flow control volume, the energy equation per unit mass is:

For adiabatic flow with no shaft work and negligible change in potential energy:

Thus, at any point:

Here, is the stagnation enthalpy. For a calorically perfect gas, , giving:

Therefore:

Using , , and :

Hence:

The stagnation temperature is the temperature attained when the flow is brought to rest adiabatically.

Reference velocities simplify the comparison of different compressible-flow states.

Important reference velocities include:

• Local velocity of sound:
  
  It defines the local Mach number .

• Stagnation velocity of sound:
  
  It is based on the stagnation temperature of the flow.

• Critical velocity of sound:
  
  At the critical state, and therefore .

• Maximum theoretical velocity: If a gas expands adiabatically until its static temperature approaches zero, its stagnation enthalpy is converted into kinetic energy:
  

The corresponding nondimensional velocity parameters include and . These parameters are useful in nozzle-flow and high-speed-flow calculations.

A stagnation state is the state a moving fluid would attain if it were brought to rest adiabatically and reversibly. Its properties are denoted by , , and .

From the energy equation:

For an isentropic perfect gas:

Rearranging gives the pressure relation:

Also, using the isentropic temperature-density relation:

Therefore:

These relations apply only when the deceleration to rest is isentropic. Across a shock, stagnation pressure decreases because entropy is generated, although stagnation temperature remains constant for adiabatic flow without work.

The velocity of sound is the speed at which an infinitesimal pressure disturbance travels through a medium. For a small disturbance:

The derivative is evaluated at constant entropy because acoustic disturbances are approximately adiabatic and reversible.

For a perfect gas undergoing an isentropic process:

Differentiating:

Therefore:

Using :

Important conclusions:

• Sound speed increases with absolute temperature.
• It depends on the gas properties through and .
• For a perfect gas, it does not directly depend on pressure when temperature is specified.
• It provides the reference speed used to define the Mach number.

The Mach number is the ratio of flow velocity to the local velocity of sound:

where is the flow speed and is the local sound speed.

Physical significance:

• It compares the speed of fluid motion with the speed at which pressure information propagates.
• When , pressure disturbances can travel both upstream and downstream.
• When , the flow moves at the local acoustic speed.
• When , disturbances cannot propagate upstream against the flow.
• Mach number determines the response of a flow to area changes and pressure gradients.
• It is the principal parameter used to classify compressible-flow regimes.

Mach number is not simply a measure of velocity because sound speed varies with temperature. The same velocity may correspond to different Mach numbers at different temperatures.

The critical state, denoted by a superscript , is the state at which the local Mach number is unity:

The stagnation-temperature relation is:

Setting gives:

Therefore:

Applying the isentropic relations:

The critical speed is:

Critical conditions are especially important in choked nozzle flows, where the mass flow rate reaches its maximum value for fixed stagnation conditions and throat area.

Mach number:

It is the ratio of local flow velocity to local sound speed.

Critical-state velocity parameter:

Here, is the critical speed of sound associated with the same stagnation state. Unlike the ordinary Mach number, does not generally indicate the ratio of velocity to the local sound speed.

Critical Mach number of an aerofoil:

• It is the freestream Mach number at which the local flow first reaches somewhere on the aerofoil.
• The local sonic point usually appears where the flow has accelerated over the surface.
• The aircraft's critical Mach number is normally less than unity.
• Above this value, local supersonic regions and shock waves may form even though the freestream remains subsonic.

Thus, a critical state describes a local sonic thermodynamic state, whereas critical Mach number is an aerodynamic characteristic of a body.

The continuity equation in differential form is:

For steady inviscid flow, the momentum equation is:

For an isentropic disturbance:

Substituting into the momentum equation:

Therefore:

Substituting this result into continuity gives:

Hence:

Implications:

• For subsonic flow, : acceleration requires decreasing area, so a converging passage acts as a nozzle.
• For supersonic flow, : acceleration requires increasing area, so a diverging passage acts as a nozzle.
• At : a smooth transition between subsonic and supersonic flow occurs at an area extremum, normally the throat of a converging-diverging nozzle.
• The opposite area changes produce diffusion or deceleration.

The steady adiabatic energy equation gives:

Using , , and gives:

For isentropic flow:

Therefore:

Similarly:

Thus:

These equations show that static pressure, temperature, and density decrease relative to their stagnation values as Mach number increases.

The principal waves in compressible flow are:

• Acoustic or Mach waves: Infinitesimally weak pressure disturbances that travel at the local speed of sound relative to the fluid. They produce negligible entropy change.
• Compression waves: Disturbances across which pressure, density, and temperature increase while velocity decreases. Compression waves tend to converge and may merge into a shock wave.
• Shock waves: Finite, abrupt compression waves involving irreversible changes in properties. Entropy increases and stagnation pressure decreases across a shock.
• Expansion waves: Continuous waves formed when supersonic flow turns around a convex corner. Pressure, density, and temperature decrease while velocity and Mach number increase.
• Contact or entropy waves: Surfaces across which pressure and normal velocity remain continuous, but density, temperature, or entropy may change.

Depending on geometry, shocks may be normal, oblique, or curved. Expansion disturbances generally spread out into a fan rather than combining into an expansion shock.

Consider a point source moving through a stationary fluid at a supersonic speed . The source continuously emits weak pressure disturbances that expand as spherical waves at speed .

After a time :

• The source travels a distance .
• A disturbance emitted initially travels a radial distance .

Because the source moves faster than its disturbances, the spherical wave fronts have a common tangent forming a cone called the Mach cone. If is the semi-vertical angle of the cone, geometry gives:

Since :

Consequences:

• A Mach cone exists only for .
• At , .
• As increases, the Mach angle decreases.
• Disturbances are confined within the downstream Mach cone, so the upstream flow is unaware of the approaching body.

Compressible-flow regimes are commonly classified as follows:

• Incompressible or low-subsonic flow, : Density variations are small and are often neglected.
• Subsonic compressible flow, approximately : Compressibility is important, but the flow remains subsonic everywhere in typical applications.
• Transonic flow, approximately : Subsonic and supersonic regions coexist. Local shock waves, rapid drag rise, and flow separation may occur.
• Supersonic flow, approximately : The flow speed exceeds sound speed. Mach waves, shock waves, expansion fans, and Mach cones are prominent.
• Hypersonic flow, : Strong shocks and very high temperatures occur. Chemical reactions, dissociation, ionization, and viscous interaction may become important.

The numerical boundaries are approximate. Actual behavior also depends on body geometry, gas properties, altitude, and temperature.

For isentropic flow at fixed stagnation conditions, increasing Mach number changes the static properties according to:

Thus, as increases under these conditions, static temperature, pressure, and density decrease.

Aerodynamic effects of increasing Mach number:

• Density variations become significant above approximately .
• Pressure disturbances become less able to propagate upstream.
• Near transonic conditions, local sonic regions and shocks appear.
• Wave drag rises rapidly in the transonic regime.
• Supersonic flow produces shock waves and expansion fans.
• At hypersonic speeds, aerodynamic heating and real-gas effects become substantial.

Therefore, Mach number changes both the thermodynamic state and the nature of aerodynamic force generation.

Compression waves:

• Form when supersonic flow turns toward itself, such as around a concave corner.
• Increase pressure, temperature, and density.
• Decrease velocity and Mach number.
• Individual weak compression waves tend to converge.

Shock waves:

• Result when compression waves merge into a finite discontinuity.
• Cause abrupt increases in pressure, temperature, density, and entropy.
• Reduce velocity, Mach number, and stagnation pressure.
• Are irreversible, although stagnation temperature remains constant in adiabatic flow without work.

Expansion waves:

• Form when supersonic flow turns away from itself around a convex corner.
• Spread as a Prandtl-Meyer expansion fan.
• Decrease pressure, temperature, and density.
• Increase velocity and Mach number.
• Are approximately isentropic because the expansion occurs through many infinitesimally weak waves.

An expansion shock does not occur in ordinary gas dynamics because it would violate the second law of thermodynamics.

The critical velocity parameter is defined as:

For the critical state:

Therefore:

At a general state:

Also:

Hence:

Substituting gives:

Thus:

At the critical state, and . The two parameters differ at all other states because uses local sound speed, whereas uses the critical sound speed associated with the stagnation state.

At the critical state, . The critical temperature ratio is:

The critical pressure ratio is:

For :

The critical density ratio is:

Therefore:

Thus:

These ratios identify sonic conditions for air. In a choked nozzle, the throat reaches these values relative to the upstream stagnation state.

Given:

The temperature ratio is:

The pressure ratio is:

The density ratio is:

The Mach angle is:

Therefore:

The result shows that stagnation properties differ substantially from static properties in supersonic flow, while disturbances are confined to a relatively narrow Mach cone.