Compressible flow

Compressible fluid mechanics is a combination of the fields of traditional fluid mechanics and thermodynamics. It is related to the more general study of compressibility. In fluid dynamics, a flow is considered to be a compressible flow if the density of the fluid changes with respect to pressure. In general, this is the case where the Mach number (defined as the ratio of the flow speed to the local speed of sound) of the flow exceeds 0.3. Below Mach .3 fluid flows experience less than a 5% change in density.

Definition

Compressible flow theory is distinguished from incompressible flow theory in that the density can no longer be considered a constant. As such, where incompressible flow theory is governed mainly by the conservation of mass and conservation of momentum equations, compressible flows require that the conservation of energy and conservation of entropy equations be solved simultaneously. Maintaining assumption of a calorically perfect gas, these equations can be solved to obtain temperature, pressure and density profiles that vary with local Mach number.

These definitions, though they seem to be inconsistent, are all saying one and the same thing: the Mach number of the flow is high enough so that the effects of compressibility can no longer be neglected.

Subsonic Compressible Flows

Compressible Flow Correction Factors

Due to the complexities of compressible flow theory, many times it is easier to first calculate the incompressible flow characteristics, and then employ a correction factor to obtain the actual flow properties. Several correction factors exist towards this end with varying degrees of complexity and accuracy.

Prandtl–Glauert transformation

The Prandtl-Glauert transformation is found by linearizing the potential equations associated with compressible, inviscid flow. It was discovered that the linearized pressures in such a flow were equal to those found from incompressible flow theory multiplied by a correction factor. The Prandtl-Glauert correction factor will always underestimate the magnitude of the pressure within the fluid. This correction factor is given below. :

$c_{p}={\frac {c_{p0}}{\sqrt {|1-{M}^{2}|}}}.$

where

• c_p is the compressible pressure coefficient
• c_p0 is the incompressible pressure coefficient
• M is the Mach number.

This correction factor works well for all Mach numbers M<.7 and M>1.3.

Karmen-Tsien Correction Factor

The Karmen-Tsien transformation is a nonlinear correction factor to find the pressure coefficient of a compressible, inviscid flow. It is an empirically derived correction factor that tends to slightly overestimate the magnitude of the fluid's pressure. In order to employ this correction factor, the incompressible, inviscid fluid pressure must be known from previous investigation.

$C_{P}={\frac {C_{P0}}{{\sqrt {1-M^{2}}}+{\frac {C_{P0}}{2}}(1-{\sqrt {1-M^{2}}})}}$

where

• c_p is the compressible pressure coefficient
• c_p0 is the incompressible pressure coefficient
• M is the Mach number.

This correction factor is valid for M<.8.

Supersonic Flows

For many other flows, their nature is qualitatively different to subsonic flows. A flow where the local Mach number reaches or exceeds 1 will usually contain shock waves. A shock is an abrupt change in the velocity, pressure and temperature in a flow; the thickness of a shock scales with the molecular mean free path in the fluid (typically a few micrometers).

Shock Waves

Shocks form because information about conditions downstream of a point of sonic or supersonic flow cannot propagate back upstream past the sonic point.

Transonic Flows=

Whitcomb area rule The behaviour of a fluid changes radically as it starts to move above the speed of sound (in that fluid), ie. when the Mach number is greater than 1. For example, in subsonic flow, a stream tube in an accelerating flow contracts. But in a supersonic flow, a stream tube in an accelerating flow expands. To interpret this in another way, consider steady flow in a tube that has a sudden expansion: the tube's cross section suddenly widens, so the cross-sectional area increases.

Applications

Nozzles

In subsonic flow, the fluid speed drops after the expansion (as expected). In supersonic flow, the fluid speed increases. This sounds like a contradiction, but it isn't: the mass flux is conserved but because supersonic flow allows the density to change, the volume flux is not constant. This effect is utilized in De Laval nozzles.

See also

• Gas dynamics
• Transonic flow
• Supersonic flow
• Hypersonic flow
• Fanno flow
• Rayleigh flow
• Mach number

References

• Shapiro, Ascher H. The Dynamics and Thermodynamics of Compressible Fluid Flow, Volume 1. Ronald Press. ISBN 978-0-471-06691-0.
• Anderson, John D. Modern Compressible Flow. McGraw-Hill. ISBN 0071241361.
• Liepmann, H. W. Elements of Gasdynamics. Dover Publications. ISBN 0486419630. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• von Mises, Richard. Mathematical Theory of Compressible Fluid Flow. Dover Publications. ISBN 0486439410.
• Saad, Michael A. Compressible Fluid Flow. Prentice Hall. ISBN 0-13-163486-0.
• Hodge, B. K. Compressible Fluid Dyanmics with Personal Computer Applications. Prentice Hall. ISBN 013308552X. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Lakshminarayana, B. Fluid Dynamics and Heat Transfer of Turbomachinery. Wiley-Interscience. ISBN 978-0-471-85546-0.

External links

• NASA Beginner's Guide to Compressible Aerodynamics