Compressible flow

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Compressible flow (gas dynamics) ) is the branch of fluid mechanics that deals with flows having significant changes in fluid density. Gases, but not liquids, display such behavior. To distinguish between compressible and incompressible flow in gases, the Mach number (the ratio of the speed of the flow to the speed of sound) must be greater than about 0.3 before significant compressibility occurs. The study of compressible flow is relevant to high-speed aircraft, jet engines, gas pipelines, commercial applications such as abrasive blasting, and many other fields.

History[edit]

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 a simpler machine. At the beginning of the 19th century, investigation into the behavior of fired bullets led to improvement in the accuracy and capabilities of guns and artillery.[1] As the century progressed, inventors such as Gustaf de Laval advanced the field, while researchers such as Ernst Mach sought to understand the physical phenomenon 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.[1] Theodore von Kármán, a student of Prandtl, continued to improve the understanding of supersonic flow. Other notable figures (Meyer, Crocco, and Shapiro) also contributed significantly to the principles considered fundamental to the study of modern gas dynamics.

Accompanying the improved conceptual understanding of gas dynamics 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 only barrier that existed for supersonic flight was a technological barrier. Amongst other factors, conventional airfoils 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-1A. Piloted by Chuck Yeager, the X-1A achieved supersonic speed in October 1947.[2] This achievement paved the way to the future of modern aircraft, missiles, and spacecraft.

Historically two paths of research have been used, in order to further gas dynamics knowledge. Experimental gas dynamics comes in the form of wind tunnel model experiments and shock tubes with the use of optical techniques to document the findings. Computational fluid dynamics applies supercomputing power to analyze a variety of geometries and flow characteristics. Both internal and external flows can be evaluated. Although not a complete substitute for experimental confirmation, computational gas dynamics is an inexpensive alternative that continues to increase in capability.

Mach Number and Sonic Flows[edit]

Mach number (M) is defined as the speed of an object to the speed of sound. M can range from 0 to ∞, but this broad range is broken up into several flow regimes. These regimes are subsonic, transonic, supersonic, hypersonic, and hypervelocity flow. For instance, in air at room temperature, the speed of sound is about 340 m/s (760 mph). The figure below illustrates the spectrum of Mach number flow regimes.

Mach Number Flow Regimes

As an object accelerates from subsonic toward supersonic speed, certain regimes of wave phenomena occur. To illustrate these changes, the figure below shows a stationary point (M=0) that emits symmetric sound waves. One can think of this point as a “boom box” floating in the air and projecting sound waves in all directions. From this stationary point, the boom box begins to accelerate to a subsonic speed. As the boom box accelerates, the sound waves it creates pile up in the direction of motion and stretch out in the opposite direction. When the boom box reaches sonic speed (M=1), it is travelling at the same speed as the sound waves it creates. Therefore, an infinite number of these waves stack up in the direction of motion to form a shock wave. Upon achieving supersonic flow, the boom box leaves its pressure waves behind. When this occurs, the pressure waves create an angle known as the Mach wave angle (or Doppler angle), µ:

 \mu = \arcsin(a/V) = \arcsin(1/M)

where a represents the speed of sound in air and V represents the velocity of the object. Although named for Austrian physicist Ernst Mach, these oblique waves were actually first discovered by Christian Doppler.

Explanation of Sonic Motion

One-Dimensional Flow[edit]

One-dimensional (1-D) flow refers to the 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 analyzing the 1-D channel flow, a number of assumptions are made:

Converging-Diverging Laval Nozzles[edit]

As the speed of a flow accelerates from the subsonic to the supersonic regime, the physics of nozzle and diffuser flows is altered. Using the conservation laws of fluid dynamics and thermodynamics, the following relationship for channel flow is developed (combined mass and momentum conservation):

 dP(1-{M}^2) = \rho{V}^2\left(\frac {dA} {A}\right) ,

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 (dA<0) increases the velocity of the flow and a diverging duct (dA>0) decreases velocity of the flow. For supersonic flow, the opposite occurs due to the change of sign of (1-M2). A converging duct (dA<0) now decreases the velocity of the flow and a diverging duct (dA>0) 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.

Table showing the reversal in the physics of nozzles and diffusers with changing Mach Numbers

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.

Nozzle de Laval diagram

Maximum Achievable Velocity of a Gas[edit]

Ultimately, because of the energy conservation law, a gas is limited to a certain maximum velocity based on its energy content. The maximum velocity, Vmax, that a gas can attain is:

V_{max} = \sqrt {2c_{p} T_{t}}

where cp is the specific heat of the gas and Tt is the stagnation temperature of the flow.

Isentropic Flow Mach Number Relationships[edit]

Using conservations laws and thermodynamics, a number of relationships of the form

 \frac{{property_1}}{{property_2}}=f(M,\gamma)

can be obtained, where M is the Mach number and γ is the ratio of specific heats (1.4 for air). See table of Isentropic Flow Mach Number Relationships.

Isentropic Flow Relationship Table. Equations to relate the field properties in isentropic flow.

Achieving Supersonic Flow[edit]

As previously mentioned, in order for a flow to become supersonic, it must pass through a duct with a minimum area, or sonic throat. Additionally, an overall pressure ratio, Pb/Pt, of approximately 2 is needed to attain Mach 1. Once it has reached Mach 1, the flow at the throat is said to be “choked.” Because changes downstream can only move upstream at sonic speed, the mass flow through the nozzle cannot be affected by changes in downstream conditions after the flow is choked.

Non-Isentropic 1D Channel Flow of a Gas - Normal Shock Waves[edit]

Normal shock waves are shock waves that are perpendicular to the local flow direction. These shock waves occur when pressure waves build up and coalesce into an extremely thin shockwave that converts useful energy into heat. Because a loss of energy occurs over the thin shock wave, the shock is considered non-isentropic and enthalpy increases across the shock. When analyzing a normal shock wave, one-dimensional, steady, and adiabatic (stagnation temperature does not change across the shock wave) flow of a perfect gas is assumed.

The Rankine-Hugoniot Equations relate conditions before and after a normal shock wave.

Normal shock waves can occur in two reference frames: the standing normal shock and the moving shock. The flow before a normal shock wave must be supersonic, and the flow after a normal shock must be subsonic. The Rankine-Hugoniot equations are used to solve for the flow conditions.

Two-Dimensional Flow[edit]

Although one-dimensional flow can be directly analyzed, it is merely a specialized case of two-dimensional flow. It follows that one of the defining phenomena of one-dimensional flow, a normal shock, is likewise only a special case of a larger class of oblique shocks. Further, the name “normal” is with respect to geometry rather than frequency of occurrence. Oblique shocks are much more common in applications such as: aircraft inlet design, objects in supersonic flight, and (at a more fundamental level) supersonic nozzles and diffusers. Depending on the flow conditions, an oblique shock can either be attached to the flow or detached from the flow in the form of a bow shock.

Attached shock wave shown on a X-15 Model in a supersonic wind tunnel
Bowshock example for a blunt body

Oblique Shock Waves[edit]

Diagram of obstruction

Oblique shock waves are similar to normal shock waves, but they occur at angles less than 90° with the direction of flow. When a disturbance is introduced to the flow at a nonzero angle (δ), the flow must respond to the changing boundary conditions. Thus an oblique shock is formed, resulting in a change in the direction of the flow.

Shock polar diagram

Shock Polar Diagram[edit]

Based on the level of flow deflection (δ), oblique shocks are characterized as either strong or weak. Strong shocks are characterized by larger deflection and more entropy loss across the shock, with weak shocks as the opposite. In order to gain cursory insight into the differences in these shocks, a shock polar diagram can be used. With the static temperature after the shock, T*, known the speed of sound after the shock is defined as,

 {A}^*= \sqrt{\gamma R{T}^*}

with R as the gas constant and γ as the specific heat ratio. The Mach number can be broken into Cartesian coordinates

 {M}^*_{2x} = \frac{{V_x}}{{{a}^*}}
 {M}^*_{2y} = \frac{{V_y}}{{{a}^*}}

with Vx and Vy as the x and y-components of the fluid velocity V. With the Mach number before the shock given, a locus of conditions can be specified. At some δ max the flow transitions from a strong to weak oblique shock. With δ = 0°, a normal shock is produced at the limit of the strong oblique shock and the Mach wave is produced at the limit of the weak shock wave.

Oblique Shock Reflection[edit]

Due to the inclination of the shock, after an oblique shock is created, it can interact with a boundary in three different manners, two which are explained below.

Solid Boundary[edit]

Incoming flow is first turned by angle δ with respect to the flow. This shockwave is reflected off the solid boundary, and the flow is turned by – δ to again be parallel with the boundary. It is important to note that each progressive shock wave is weaker and the wave angle is increased.

Irregular Reflection[edit]

An irregular reflection is much like the case described above, with the caveat that δ is larger than the maximum allowable turning angle. Thus a detached shock is formed and a more complicated reflection occurs.

Prandtl-Meyer Expansion Fans[edit]

Prandtl-Meyer Expansion Fan Diagram

To this point, the only flow phenomena that have been discussed are shock waves, which slow the flow and increase its entropy. It is possible to accelerate supersonic flow in what has been termed a Prandtl-Meyer expansion fan, after Ludwig Prandtl and Theodore Meyer. The mechanism for the expansion is shown in the figure below.

As opposed to the flow encountering an inclined obstruction and forming an oblique shock, the flow expands around a convex corner and forms an expansion fan through a series of isentropic Mach waves. The expansion “fan” is composed of Mach waves that span from the initial Mach angle to the final Mach angle. Flow can expand around either a sharp or rounded corner equally, as the increase in Mach number is proportional to only the convex angle of the passage (δ).

The Prandtl-Meyer expansion can be seen as the physical explanation of the operation of the Laval nozzle. The contour of the nozzle creates a smooth and continuous series of Prandtl-Meyer expansion waves.

Prandtl-Meyer Compression Fans[edit]

Basic PM Compression diagram

A Prandtl-Meyer compression is the opposite phenomenon to a Prandtl-Meyer expansion. If the flow is gradually turned through an angle of δ, a compression fan can be formed. This fan is a series of Mach waves that eventually coalesce into an oblique shock. Because the flow is defined by an isentropic region (flow that travels through the fan) and an anisentropic region (flow that travels through the oblique shock), a slip line results between the two flow regions.

Applications[edit]

Supersonic Wind Tunnels[edit]

Supersonic wind tunnels are used for testing and research in supersonic flows, approximately over the Mach number range of 1.2 to 5. The operating principle behind the wind tunnel is that a large pressure difference is maintained upstream to downstream, driving the flow.

Supersonic Wind Tunnel Classification List

Wind tunnels can be divided into two categories: continuous-operating and intermittent-operating wind tunnels. Continuous operating supersonic wind tunnels require an independent electrical power source that drastically increases with the size of the test section. Intermittent supersonic wind tunnels are less expensive in that they store electrical energy over an extended period of time, then discharge the energy over a series of brief tests. The difference between these two is analogous to the comparison between a battery and a capacitor.

Blowdown supersonic wind tunnel schematic
Langley indraft supersonic wind tunnel vacuum sphere

Blowdown type supersonic wind tunnels offer high Reynolds number, a small storage tank, and readily available dry air. However, they cause a high pressure hazard, result in difficulty holding a constant stagnation pressure, and are noisy during operation.

Indraft supersonic wind tunnels are not associated with a pressure hazard, allow a constant stagnation pressure, and are relatively quite. Unfortunately, they have a limited range for the Reynolds number of the flow and require a large vacuum tank. There is no dispute that knowledge is gained through research and testing in supersonic wind tunnels; however, the facilities often require vast amounts of power to maintain the large pressure ratios needed for testing conditions. For example, Arnold Engineering Development Complex has the largest supersonic wind tunnel in the world and requires the power required to light a small city for operation. For this reason, large wind tunnels are becoming less common at universities.

Supersonic Aircraft Inlets[edit]

Perhaps the most common application for oblique shocks is in high-speed aircraft inlets. The purpose of the inlet is to slow incoming supersonic flow to the subsonic regime before it enters the turbojet engine, with the caveat of minimizing losses across the shock. Knowledge of normal and oblique shocks suggests that this be accomplished with a series of weakening oblique shocks followed by a very weak normal shock, usually less than M = 1.4. This may sound relatively straight forward, but there is one rather large issue to be dealt with when designing a supersonic aircraft inlet: acceleration. Between taking off, maneuvering, and cruising, an aircraft travels at a range of Mach numbers. In order to ensure efficient flight, the aircraft intake must be capable of variable geometry. If it is not, the shock waves will not reflect properly through the inlet and negatively affect performance.

McDonnell Douglas F-15C
Concorde on Bristol
Lockheed SR-71 Blackbird

Although variable geometry is a universally recognized approach to improve aircraft efficiency and performance over a range of Mach numbers, there is no one method to achieve variable geometry. The F-15 Eagle employs wedge inlets with adjustable flaps to control the flow. For subsonic flow, the flaps are completely closed and for supersonic flow, the flaps are open. The Concorde employed an external-compression inlet, using a series of oblique shocks followed by a normal shock to slow the flow sufficiently for the turbojet engine. Perhaps the most recognizable supersonic aircraft, the SR-71, used a hydraulically actuated cone to reduce the speed of the supersonic flow through the aircraft inlet.

Natural Gas Pipeline[edit]

Constructing natural gas line in winter, Finland

Natural gas pipelines are used to transport natural gas from extraction sites to refinement or chemical processing facilities. In the United States there are more than 210 natural gas pipeline systems with more than 305,000 miles of intrastate transmission pipelines.[3] Two compressible flow phenomenon characterize the flow through these pipelines: friction (Fanno flow) and (Rayleigh flow) and heat transfer. Natural gas pipelines are buried in the ground at a constant temperature of 15 °C. However, the friction generated by the flow offsets the heat loss to the Earth, thus resulting in an isothermal flow.

Friction factor as a function of Mach number in Fanno Flow

The relationship between fLmax/D and Mach number for Fanno flow suggests that only subsonic flow can be used in the long pipes used to transport natural gas (even these pipes must be broken into shorter segments with compressor stations at the discontinuities in the pipeline). Additionally using conservation, an equation can be derived to describe the flow.

 \frac{{dM}}{{M}}=\frac{{fdx}}{{D}} \frac{{\gamma M^2}}{{2 - 2 \gamma M^2}}

This equation describes flow that chokes at M* = 0.87 for natural gas γ = 1.32; however choking requires an infinite heat flux. Therefore, a combination of intuition and mathematics explains why it is most economically feasible that subsonic natural gas is pumped through long sections of pipe to reach its intended destination.

See also[edit]

References[edit]

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