Compressible flow
Compressible flow is the branch of fluid mechanics that deals with flows having significant changes in fluid density. While all flows are compressible, flows are usually treated as being incompressible when the Mach number is smaller than 0.3. The study of compressible flow is relevant to high-speed aircraft, jet engines, rocket motors, high-speed entry into a planetary atmosphere, gas pipelines, commercial applications such as abrasive blasting, and many other fields.
History
The study of gas dynamics is often associated with the flight of modern high-speed aircraft and atmospheric reentry of space-exploration vehicles; however, its origins lie with simpler machines. At the beginning of the 19th century, investigation into the behaviour of fired bullets led to improvement in the accuracy and capabilities of guns and artillery. As the century progressed, inventors such as Gustaf de Laval advanced the field, while researchers such as Ernst Mach sought to understand the physical phenomena involved through experimentation.At the beginning of the 20th century, the focus of gas dynamics research shifted to what would eventually become the aerospace industry. Ludwig Prandtl and his students proposed important concepts ranging from the boundary layer to supersonic shock waves, supersonic wind tunnels, and supersonic nozzle design. Theodore von Kármán, a student of Prandtl, continued to improve the understanding of supersonic flow. Other notable figures also contributed significantly to the principles considered fundamental to the study of modern gas dynamics. Many others also contributed to this field.
Accompanying the improved conceptual understanding of gas dynamics in the early 20th century was a public misconception that there existed a barrier to the attainable speed of aircraft, commonly referred to as the "sound barrier." In truth, the barrier to supersonic flight was merely a technological one, although it was a stubborn barrier to overcome. Amongst other factors, conventional aerofoils saw a dramatic increase in drag coefficient when the flow approached the speed of sound. Overcoming the larger drag proved difficult with contemporary designs, thus the perception of a sound barrier. However, aircraft design progressed sufficiently to produce the Bell X-1. Piloted by Chuck Yeager, the X-1 officially achieved supersonic speed in October 1947.
Historically, two parallel paths of research have been followed in order to further gas dynamics knowledge. Experimental gas dynamics undertakes wind tunnel model experiments and experiments in shock tubes and ballistic ranges with the use of optical techniques to document the findings. Theoretical gas dynamics considers the equations of motion applied to a variable-density gas, and their solutions. Much of basic gas dynamics is analytical, but in the modern era Computational fluid dynamics applies computing power to solve the otherwise-intractable nonlinear partial differential equations of compressible flow for specific geometries and flow characteristics.
Introductory concepts
There are several important assumptions involved in the underlying theory of compressible flow. All fluids are composed of molecules, but tracking a huge number of individual molecules in a flow is unnecessary. Instead, the continuum assumption allows us to consider a flowing gas as a continuous substance except at low densities. This assumption provides a huge simplification which is accurate for most gas-dynamic problems. Only in the low-density realm of rarefied gas dynamics does the motion of individual molecules become important.A related assumption is the no-slip condition where the flow velocity at a solid surface is presumed equal to the velocity of the surface itself, which is a direct consequence of assuming continuum flow. The no-slip condition implies that the flow is viscous, and as a result a boundary layer forms on bodies traveling through the air at high speeds, much as it does in low-speed flow.
Most problems in incompressible flow involve only two unknowns: pressure and velocity, which are typically found by solving the two equations that describe conservation of mass and of linear momentum, with the fluid density presumed constant. In compressible flow, however, the gas density and temperature also become variables. This requires two more equations in order to solve compressible-flow problems: an equation of state for the gas and a conservation of energy equation. For the majority of gas-dynamic problems, the simple ideal gas law is the appropriate state equation. Otherwise, more complex equations of state must be considered and the so-called non ideal compressible fluids dynamics establishes.
Fluid dynamics problems have two overall types of references frames, called Lagrangian and Eulerian. The Lagrangian approach follows a fluid mass of fixed identity as it moves through a flowfield. The Eulerian reference frame, in contrast, does not move with the fluid. Rather it is a fixed frame or control volume that fluid flows through. The Eulerian frame is most useful in a majority of compressible flow problems, but requires that the equations of motion be written in a compatible format.
Finally, although space is known to have 3 dimensions, an important simplification can be had in describing gas dynamics mathematically if only one spatial dimension is of primary importance, hence 1-dimensional flow is assumed. This works well in duct, nozzle, and diffuser flows where the flow properties change mainly in the flow direction rather than perpendicular to the flow. However, an important class of compressible flows, including the external flow over bodies traveling at high speed, requires at least a 2-dimensional treatment. When all 3 spatial dimensions and perhaps the time dimension as well are important, we often resort to computerized solutions of the governing equations.
Mach number, wave motion, and sonic speed
The Mach number is defined as the ratio of the speed of an object to the speed of sound. For instance, in air at room temperature, the speed of sound is about. M can range from 0 to ∞, but this broad range falls naturally into several flow regimes. These regimes are subsonic, transonic, supersonic, hypersonic, and hypervelocity flow. The figure below illustrates the Mach number "spectrum" of these flow regimes.These flow regimes are not chosen arbitrarily, but rather arise naturally from the strong mathematical background that underlies compressible flow. At very slow flow speeds the speed of sound is so much faster that it is mathematically ignored, and the Mach number is irrelevant. Once the speed of the flow approaches the speed of sound, however, the Mach number becomes all-important, and shock waves begin to appear. Thus the transonic regime is described by a different mathematical treatment. In the supersonic regime the flow is dominated by wave motion at oblique angles similar to the Mach angle. Above about Mach 5, these wave angles grow so small that a different mathematical approach is required, defining the hypersonic speed regime. Finally, at speeds comparable to that of planetary atmospheric entry from orbit, in the range of several km/s, the speed of sound is now comparatively so slow that it is once again mathematically ignored in the hypervelocity regime.
As an object accelerates from subsonic toward supersonic speed in a gas, different types of wave phenomena occur. To illustrate these changes, the next figure shows a stationary point that emits symmetric sound waves. The speed of sound is the same in all directions in a uniform fluid, so these waves are simply concentric spheres. As the sound-generating point begins to accelerate, the sound waves "bunch up" in the direction of motion and "stretch out" in the opposite direction. When the point reaches sonic speed, it travels at the same speed as the sound waves it creates. Therefore, an infinite number of these sound waves "pile up" ahead of the point, forming a Shock wave. Upon achieving supersonic flow, the particle is moving so fast that it continuously leaves its sound waves behind. When this occurs, the locus of these waves trailing behind the point creates an angle known as the Mach wave angle or Mach angle, μ:
where represents the speed of sound in the gas and represents the velocity of the object. Although named for Austrian physicist Ernst Mach, these oblique waves were first discovered by Christian Doppler.
One-dimensional flow
One-dimensional flow refers to flow of gas through a duct or channel in which the flow parameters are assumed to change significantly along only one spatial dimension, namely, the duct length. In analysing the 1-D channel flow, a number of assumptions are made:- Ratio of duct length to width is ≤ about 5,
- Steady vs. Unsteady Flow,
- Flow is isentropic,
- Ideal gas law
Converging-diverging Laval nozzles
where dP is the differential change in pressure, M is the Mach number, ρ is the density of the gas, V is the velocity of the flow, A is the area of the duct, and dA is the change in area of the duct. This equation states that, for subsonic flow, a converging duct increases the velocity of the flow and a diverging duct decreases velocity of the flow. For supersonic flow, the opposite occurs due to the change of sign of. A converging duct now decreases the velocity of the flow and a diverging duct increases the velocity of the flow. At Mach = 1, a special case occurs in which the duct area must be either a maximum or minimum. For practical purposes, only a minimum area can accelerate flows to Mach 1 and beyond. See table of sub-supersonic diffusers and nozzles.
Therefore, to accelerate a flow to Mach 1, a nozzle must be designed to converge to a minimum cross-sectional area and then expand. This type of nozzle – the converging-diverging nozzle – is called a de Laval nozzle after Gustaf de Laval, who invented it. As subsonic flow enters the converging duct and the area decreases, the flow accelerates. Upon reaching the minimum area of the duct, also known as the throat of the nozzle, the flow can reach Mach 1. If the speed of the flow is to continue to increase, its density must decrease in order to obey conservation of mass. To achieve this decrease in density, the flow must expand, and to do so, the flow must pass through a diverging duct. See image of de Laval Nozzle.