20.35 Incompressible and compressible fluid flow

In post 17.15, we saw how fluid flow could be explained by two equations – Bernoulli’s equation (conservation of mechanical energy) and a continuity equation (conservation of mass). But, conservation of mechanical energy assumes that the viscosity of the fluid is negligible because viscosity dissipates mechanical energy (post 17.17) and so is similar to friction which dissipates mechanical energy when two solid surfaces are in contact (post 16.19). We will return to dealing with the problem of viscosity in post 20.36.

This post is concerned with another assumption in both Bernoulli’s equation and the continuity equation – both assume that the density of the fluid is constant. In the continuity equation of post 17.15, volume is conserved because mass is conserved and density (mass/volume, post 16.44) is constant. The assumption of constant density is equivalent to the assumption that the fluid is incompressible. If a fixed mass of fluid is compressed, its volume decreases with the result that its density must increase (post 16.44). At the molecular level, compression moves the molecules in a fluid closer together, so that there is less empty space (post 16.37). In post 17.5, we saw that, in many circumstances, liquids can be considered as incompressible; this is the same as saying that their bulk modulus is so high that the effects of compression are negligible (post 20.20). Brake fluid, in a car, transmits a force because its fluid is almost incompressible (post 17.5). But compression of gases is not negligible, as explained in posts 20.23, 20.24 and 20.25.

Despite the non-negligible compression of gases, we used Bernoulli’s equation to describe the flow of air over an aerofoil, to explain how planes fly, in post 17.16. If air is a gas and, therefore, compressible, is the analysis of post 17.16 valid?

The answer is that incompressible flow can occur for a compressible fluid. This happens when pressure changes in the fluid, predicted by Bernoulli’s equation, change the speed of flow without causing appreciable compression. When a bird flaps it wings, in flight (post 17.32), it is much more likely to move air than it is to compress it because air moves easily and there is nothing to contain it which might lead it to be compressed. This is in complete contrast to air is a closed cylinder that has nowhere to flow when it is compressed by a piston (posts 20.23, 20.24 and 20.25).

Does compressible fluid flow ever occur? The answer is yes – when the flow is sufficiently fast. Then the flowing fluid meets a fluid which cannot move away fast enough to ensure that all the pressure of the flowing fluid leads to further flow. As a result, compression occurs. It is often assumed that air flow is incompressible for air speeds of less than 0.3 times the speed of sound. At this speed, the density of the air increases by less than about 5%. This speed is sometimes said to have a Mach number of 0.3; the Mach number is the speed of flow divided by the speed of sound. When a plane flies at the speed of sound, this speed is sometimes called Mach 1, because the air is moving at the speed of sound relative to the plane (post 16.4). The Mach number is named after the Austrian philosopher and physicist Ernst Mach (1838-1916) whose ideas on relative motion were a prequel to Einstein’s special theory of relativity.

Why is the speed of sound involved in whether pressure changes cause flow or density changes? The reason is that sound travels, in air, as a pressure wave (post 18.13). And, like a wave in water (post 18.10), a wave in air does not transport air – it transports energy. In the case of a sound wave, the energy is store in compressed air, in the same way as energy is stored in a spring (post 16.49). So, when air speed reaches the speed of sound, no flow occurs but air is compressed. The closer the speed of moving air approaches the speed of sound, the more important compression and, therefore, changes in density become.

So, compressible fluid mechanics is needed in high speed aerodynamics to explain the flight of bullets and planes that fly faster than Mach 0.3. But be careful, treating air as incompressible below Mach 0.3 is simply a convention based on the assumption that a 5% increase in density is negligible – it is not based on any theory of fluid flow. Compression can also occur during flow of liquids; for example, in water jets used to cut materials in some manufacturing processes.

In conclusion, incompressible flow can occur in compressible fluids. And compressible flow can occur in fluids (like water) that are often considered to be incompressible.

Related posts

17.15 Fluid flow

Follow-up posts

20.36 The Navier-Stokes equation

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