## Compressibility Effects in Pipe Flow

**Introduction**: In the design of mall scale models of aircraft or missiles for testing in supersonic tunnels, It 1s sometimes desired to represent jet engine nacelles by means of simple hollow pipes allowing a free flow through the made. Such a flow may be either subsonic or supersonic, depending on the Kachsnci Reynolds numbers and the maternal taper of the pipe. It is desirable to be able to product where type of flav will be obtaued, primarily in order to determine whether or not the front external shock wave will be attached to the lip of the pipe. The nature and position of this shock will, in general, have some effect on the pressure distribution of the wing or other surfaces in proximity to the nacelle.

Compressible flow is the area of fluid mechanics that deals with fluids in which the fluid density varies significantly in response to a change in pressure. Compressibility effects are typically considered significant if the Mach number (the ratio of the flow velocity to the local speed of sound) of the flow exceeds 0.3, or if the fluid undergoes very large pressure changes. The most distinct differences between the compressible and incompressible flow models are that the compressible flow model allows for the existence of shock waves and choked flow.

**Definition:** Compressible flow describes the behavior of fluids that experience significant variations in density. For flows in which the density does not vary significantly, the analysis of the behavior of such flows may be simplified greatly by assuming a constant density. This is an idealization, which leads to the theory of incompressible flow. However, in the many cases dealing with gases (especially at higher velocities) and those cases dealing with liquids with large pressure changes, the significant variations in density can occur, and the flow should be analyzed as a compressible flow if accurate results are to be obtained.

**Compressible Flow Phenomena:** Two of the most distinctive phenomena which occur in compressible are the possibility of choked flow(see Internal Flows) and the presence of acoustic waves, which may also be referred to as either compression or expansion waves, depending on whether they lead to an increase or decrease in pressure.

**Shock Waves: **Shock waves are one of the most common examples of compressible flow phenomena. A shock is characterized by a discontinuous change in the thermodynamic properties. In one dimensional flow, shock waves can form when a series of compression waves coalesce, or when a membrane separating two regions of differing pressure is suddenly removed. This is the technique often used to produce shock waves in shock tubes (see Shock Tubes).

In two and three dimensional supersonic flows, oblique shock waves occur as a result of a change in direction of the flow. A classic example of these shock waves are those shock waves that form off the nose of a supersonic aircraft.

**Aerodynamics:** Aerodynamics is a subfield of fluid dynamics and gas dynamics, and is primarily concerned with obtaining the forces that air exerts on an object. For Mach numbers greater than about 0.3, density changes are significant, and the flow should be considered compressible for an accurate representation of reality.

**Subsonic Aerodynamics:** Due to the complexities of compressible flow theory, it is often easier to calculate the incompressible flow characteristics first, and then employ a correction factor to obtain the actual flow properties. Several correction factors exist with varying degrees of complexity and accuracy.

**Prandtl–Glauert transformation**: The Prandtl-Glauert transformation is found by linearizing the potential equations associated with compressible, in viscid flow. The Prandtl–Glauert transformation or Prandtl–Glauert rule (also Prandtl–Glauert–Ackeret rule) is an approximation function which allows comparison of aero dynamical processes occurring at different Mach numbers. 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.

Where:

- cp is the compressible pressure coefficient
- cp is the incompressible pressure coefficient
- M is the Mach number.