Incompressible flow, in general terms of fluid mechanics, refers to a fluid that maintains constant density during a flow. To an extent, all fluid flows have some change in density when subjected to an external force or internal viscous forces; however, density variation is more prevalent in some analysis results than others. The ability to keep density constant in equations of fluid dynamics can greatly reduce the computational burden, and by assuming a flow as incompressible, the engineer is stating that this technique will not significantly compromise the accuracy of the solution. The incompressible flow assumption has large implications within aerospace engineering, and the application in which the analysis is being performed will be essential in determining whether the assumption can be utilized. This article will address the subject and answer the following questions:

- What is incompressible flow?
- What are Euler's and Bernoulli's equations?
- What are examples of flows relevant to aerospace engineering that can be modeled as incompressible flows?

*Created By:* Jason Corman

## Overview Edit

Most incompressible flows within aerospace engineering are in the field of aerodynamics, where compressibility effects of air flow can be neglected if the Mach number is below 0.3.^{[1]} In this velocity range, the maximum change in density of air is less than 5%, so it is assumed that this variation is negligible. Beyond Mach 0.3, the variation in density can no longer be ignored, and analysis must involve the principles of compressible flow. Liquids, in most cases, can be regarded as incompressible fluids because of the immense force required to change their density. There are, however, certain applications of liquid flows in which compressibility effects cannot be ignored, such as water hammer, cavitation, and high-pressure hydraulics. The underlying principle of the incompressible flow assumption is that all materials (solids, liquids, gases) are compressible, even if the change in density is on the order of 10^{-9}. In some instances, it is very clear that density can remain constant in an analysis without creating inaccuracies in the solution. In other instances, it becomes a judgment call of the engineer, and that is why incompressible flow is considered an assumption rather than an actual physical occurrence.^{[4]}

## Basic Equations in Fluid Dynamics Edit

The analysis of a flow takes into consideration the following principles:

- Conservation of mass (or continuity equation),
- Newton's 2nd law (or momentum equation),
- Conservation of angular momentum, and
- First law of thermodynamics and Second law of thermodynamics.
^{[4]}

By solving all these equations for a system, a flow can be understood completely. Incompressible flow perhaps has the greatest implications in the momentum equation, which is very dependent on density.

### Momentum Equation Edit

The momentum equation can be solved using either integral or differential analysis. However, differential analysis is most widely used, in the form of the Navier-Stokes equations:^{[2]}

In their most general form, the Navier-Stokes equations take into consideration steady and unsteady, viscous and inviscid, compressible and incompressible flows. Without any assumptions, these equations are very complicated to solve and computationally tedious. Therefore, in the correct scenario, assumptions about the system become very important.

## Incompressible, Inviscid Flow Edit

Friction within a flow is introduced by the fluid's viscosity, or the extent to which its particles interact to hinder flow. Inviscid flow is an assumption in which these frictional effects internal to the flow are neglected. Another representation of this assumption is with the Reynolds number, which approaches infinity for inviscid flow (the inertial effects greatly outweigh the viscous effects).^{[5]} Incompressible, inviscid flow is one of the fundamental foundations in fluid mechanics. Simply put, it is the most simplified type of flow because it does not introduce compressible or friction effects. This becomes noticeably helpful in the above equations, where density can be taken outside the differential in the left-side terms and the friction forces disappear on the right side. Simplification of these terms also diminishes the need to calculate Reynolds, number, surface relative roughness, etc. When flow is considered steady, or the properties don't change with time, the Navier-Stokes equations can be reduced to:^{[4]}

This equation is known as Euler's equation, and relates changes in pressure to changes in velocity. This is the basis dynamic pressure, q, which is involved in many aerodynamics equations:^{[2]}

Euler's equation can also be transformed to Bernoulli's equation, which relates flow properties at two points on the same streamline.

### Flow Along a StreamlineEdit

Integrating Euler's equation along a streamline results in:^{[4]}

^{[4]}

This form can be used to analyze piping systems at two different points, which is the basis for one of the flow measurement techniques mentioned below (pitot tube). It is very important, however, to understand when these equations apply.

## Validity of Incompressible Flows Edit

There is a large difference in validity of incompressible flow that depends on whether the flow is internal or external. Internal flows are more likely to be exposed to large external forces, which may introduce compressibility effects in both liquids and gases (pneumatics, water hammer, hydraulics, gas pipelines, etc.). For external flows, especially air flow across a wing section, and other flows relevant to aerospace engineering, incompressible flow is valid for low speeds (M < 0.3). In respect to the previous equations, validity is achieved * only* under the following restrictions:

- Steady flow
- Incompressible flow
- Inviscid flow
- Flow along a streamline (+)

(+) This condition is only needed for Bernoulli's equation.^{[2]}

As an important reminder, incompressible and inviscid flows are assumptions and do not physically occur in nature. This principle is more important at a microscopic level rather than macroscopic, such as large fixed wing aircraft design. Slight density changes in microfluidics could have a greater impact than they would at a larger scale. Hence, it is **VERY** important to know the application before assuming a flow compressible or incompressible.^{[3]}

## Aerospace Engineering Applications Edit

Many types of flows within aerospace engineering can benefit from an incompressible flow, constant density analysis.### Low-Speed Aerodynamics Edit

The most prevalent application is within low-speed aerodynamics. The mechanics of flight are easily calculated when the incompressible flow is assumed. These flows include:

- Atmospheric flow over an airfoil (wing) during flight, especially NLF airfoils which operate best at low speeds
^{[6]} - Testing components/models in a low-speed wind tunnel
^{[1]} - Analyzing flow through low-speed wind tunnel convergences/divergences
- Velocity measurement through pitot tubes
^{[1]} - Flow over most wind turbine blades.
^{[4]}

### Aircraft Components Edit

Liquid flows through machinery play a large role in aircraft performance and its abilities. Many of these applications are modeled as incompressible flows, which include:

- Fuel flow from reservoir to aircraft engine
- Flow through hydraulic lines (with small loads)
^{[4]}

### Other Edit

Since the scope of an aerospace engineer does not lie only in the skies, other applications such as water flow about Naval ship hulls can be assumed to be incompressible.

## Resources Edit

[1] Anderson, J D. *Introduction to Flight*, 4th Ed. McGraw-Hill, 2007. Pgs 143-162

[2] Anderson, J D. *Fundamentals of Aerodynamics*, 4th Ed. McGraw-Hill, 2007. Pgs 96-291.

[3] Beskok, A. *Micro Flows: Fundamentals and Simulation*. Springer-Verlag, 2002. Pgs 52-56

[4] Fox, R W and McDonald, A T. *Introduction to Fluid Mechanics*, 6th Ed. John Wiley & Sons, Inc, 2004. Pgs 4, 26-40, 211-223, 232-248.

[5] Katz, J. and Plotkin, A. (1991). Low-speed Aerodynamics: From Wing Theory to Panel Methods. Series in Aeronautical and Aerospace Engineering. New York: McGraw-Hill, Inc.

[6] Lee, JM. "Development of Subsonic Transports with Natural Laminar Flow Wings." AIAA 98-0406.