Mach Number & Reynolds Number

J. Gordon Leishman

doi:https://doi.org/10.15394/eaglepub.2022.1066.n13

17 Mach Number & Reynolds Number

Introduction

The dimensionless (or non-dimensional) similarity parameters, Mach number and Reynolds number, have already been introduced. These parameters typically arise from dimensional analysis in nearly all fluid-flow problems. But what do they mean, how are their values interpreted, and how are they used? Of all the ways to categorize aerodynamic flows, the distinction based on Mach number is among the most useful; it can help quantify the degree of compressibility effects in a flow. The Reynolds number is another useful aerodynamic parameter that quantifies the relative importance of inertia versus viscous effects in a flow. Hence, classifying flows and operating conditions by Reynolds number is also useful. Mach and Reynolds numbers are so frequently encountered and used in aerodynamics that a broader understanding of their meanings and applications is essential for engineers.

Learning Objectives

Better appreciate the significance of Mach number in understanding the effects of compressibility on aerodynamic flows.
Understand the significance of the Reynolds number in how it affects aerodynamic flows and other characteristics.
Know how to calculate the Mach number and Reynolds number.

Definition of Mach Number

The Mach number is a quantity defined as the ratio of the local flow velocity to the local speed of sound in the same fluid at the same state, and is given by

(1) $\begin{equation*} M = \frac{|V|}{a} \end{equation*}$

where $|V|$ is the magnitude of the flow velocity (i.e., the flow speed or the true speed of a flight vehicle) and $a$ is the speed of sound. Usually, the magnitude sign is dropped by recognizing that both quantities are in units of speed, and so

(2) $\begin{equation*} M = \frac{V}{a} \end{equation*}$

Remember that the Mach number is a dimensionless (and, therefore, unitless) parameter. It is just one of a series of dimensionless parameters encountered in engineering, known as similarity parameters. The Mach number is named after Ernst Mach, an Austrian physicist renowned for his contributions to various fields of physics, including the study of shock waves. In general, the flow velocity $V$ value can vary from point to point in a flow. The speed of sound may also vary from point to point, particularly in high-speed flows where compressibility effects and temperature variations are present. Therefore, the Mach number can also vary from point to point.

Using the principles of thermodynamics, at a constant temperature, the speed of sound in a gas is given by the square root of the ratio of the change in pressure to the change in density resulting from a disturbance, i.e.,

(3) $\begin{equation*} a = \sqrt{ \gamma \left( \frac{ \partial p}{\partial \varrho} \right)} \end{equation*}$

For an ideal gas, the equation of state is $p = \varrho \, R \, T$ , where ${\gamma}$ is the ratio of specific heats, $R$ is the gas constant (in appropriate engineering units), and $T$ is absolute temperature, so

(4) $\begin{equation*} a^2 = \gamma \left( \frac{ \partial p}{\partial \varrho} \right) = \gamma \left( \frac{\partial (\varrho \, R \, T)}{\partial \varrho} \right) = \gamma \, R \, T \end{equation*}$

Therefore, the speed of sound is proportional to the square root of the absolute temperature, i.e.,

(5) $\begin{equation*} a = \sqrt{\gamma R T } \end{equation*}$

Remember that using the correct values of ${\gamma}$ and $R$ for the specific gas used, in the proper units, is essential for accurate numerical calculations of $a$ .

Significance of the Speed of Sound

A necessary consequence of the compressibility of a gas is that disturbances produced at one point propagate to another point at a finite speed, i.e., at the speed of sound $a$ . As an object (in this case, a point pressure or acoustic source) accelerates from rest, it emits disturbances as spherical pressure waves, as shown in Figure 1. The effect can be visualized by considering the points (open dots, from right to left) as generating disturbances at earlier times, with the solid dot indicating the current time. The speed of sound is the same in all directions in a uniform, constant-temperature fluid, so these waves manifest in three-dimensional space as a series of offset concentric spheres.

A moving point source illustrates compressibility effects in flow. When it becomes supersonic, pressure disturbances merge into a wavefront called a Mach cone.

As the point source moves faster, the pressure waves appear to bunch closer together in the direction of motion and to spread apart in the opposite direction, a phenomenon known as the Doppler effect. Eventually, as the source’s speed approaches the speed of sound, it catches up with the previously produced pressure waves. These waves merge to form a wavefront perpendicular to the direction of travel, thereby creating the essence of a shock wave. The shock waves then bend back to form a Mach line for supersonic motion; although named after Ernst Mach, these oblique waves were first theorized by Christian Doppler.

The angle of the Mach wave, $\mu$ , can easily be calculated because it will be apparent from the velocity components that

(6) $\begin{equation*} \sin \mu = \frac{a_{\infty}}{V_{\infty}} = \frac{1}{M_{\infty}} \end{equation*}$

where ${V_{\infty}}$ is the velocity of the source, $a_{\infty}$ is the speed of sound, and $M_{\infty}$ is the Mach number of the source. Therefore, the Mach angle is simply

(7) $\begin{equation*} \mu = \sin^{-1} \left(\frac{1}{M_{\infty}} \right) \end{equation*}$

Mach Number Regimes

It will now be apparent that in a compressible flow, there is an essential distinction between flows involving velocities less than the speed of sound (subsonic flow) and flows involving velocities greater than the speed of sound (supersonic flow). If $M$ is the local Mach number at a point P in the flow, then by definition, the flow is locally:

Subsonic if $M < 1$ .
Sonic if ${M = 1}$ .
Supersonic if $M > 1$ .

The freestream Mach number $M_{\infty}$ is defined as the ratio of the freestream velocity (that is, the velocity far upstream of the airfoil) to that of the freestream sonic velocity, $a_{\infty}$ , i.e.,

(8) $\begin{equation*} M_{\infty} = \frac{V_{\infty}}{a_{\infty}} \end{equation*}$

Figure 2 illustrates the categorization of flight vehicles by flight Mach number. The speed of sound at MSL ISA is about 1,117 ft/s (340 m/s, 768 mph, 1,236 km/h). Few air-breathing vehicles can sustain supersonic flight speeds of $M_{\infty} > 3$ .

Categorization of aerodynamic flows and flight vehicle speeds using the Mach number.

Check Your Understanding #1 – Calculating the Mach number

An airplane is cruising at a true airspeed of 500 knots at 30,000 ft. Calculate the flight Mach number for these conditions. Assume ISA standard conditions.

Show solution/hide solution.

The airspeed ${V_{\infty}}$ is given as 500 knots, equivalent to 843.9 ft/s. The flight Mach number will be the airspeed divided by the speed of sound at this altitude. Using the International Standard Atmosphere (ISA) model, for which a handy online ISA calculator can be found here, the standard temperature at this altitude is -47.83^oF or 411.84 R. According to the calculator, the corresponding speed of sound, a_{\infty, at this altitude and temperature, is 994.86 ft/s. This latter value can also be determined using

$a_{\infty} = \sqrt {\gamma \, R \, T} = \sqrt{1.4 \times 1716.49 \times 411.84} = 994.86~\mbox{ft/s}$

where the gas constant for air is 1716.49 ft lb slug $^{-1}$ R $^{-1}$ . Therefore, the flight Mach number is

$M_{\infty} = \frac{V_{\infty}}{a_{\infty}} = \frac{843.9}{994.85} = 0.84$

Physical Interpretation of Mach Number

In terms of a physical interpretation, the Mach number can also be expressed as

(9) $\begin{equation*} M = \frac{V}{a} = \sqrt{ \frac{\mbox{\small Inertia effects}}{\mbox{\small Elastic effects}} } \end{equation*}$

The elastic effects can be expressed in terms of the bulk modulus of elasticity, $E$ , also called the volume modulus. The bulk modulus is a material property that characterizes a fluid’s compressibility, i.e., the ability of an applied pressure to change the fluid’s volume and density. An increase in pressure decreases volume, increasing density.

In terms of the bulk modulus, the speed of sound is given by

(10) $\begin{equation*} a = \sqrt{ \frac{E}{\varrho}} \end{equation*}$

An increase in the bulk modulus indicates an increasingly incompressible fluid, i.e., a higher speed of sound.

Using the result in Eq. 10, gives

(11) $\begin{equation*} M^2 = \frac{V^2}{a^2}= \frac{ \varrho \, V^2}{E} \end{equation*}$

Therefore, the Mach number can also be expressed as

(12) $\begin{equation*} M = \sqrt{ \frac{ \varrho \, V^2 \, L^2}{E \, L^2} } \end{equation*}$

where $L$ is a characteristic length scale. On the numerator, the grouping $\varrho \, V^2 L^2$ represents an inertial force in the flow. The term $E \, L^2$ in the denominator represents an elastic force. The Mach number, therefore, represents a relative measure of inertial to elastic forces in a flow, i.e., a measure of the compressibility of the flow medium.

Compressible Flows about an Airfoil

Below a Mach number of approximately 0.3, air and other gases can be considered incompressible. However, if the freestream Mach number becomes sufficiently high (usually $M_{\infty} \ge 0.75$ ), then the flow may locally exceed the speed of sound as it moves over a wing or other parts of a flight vehicle, i.e., $M > 1$ . This situation gives rise to a mixed subsonic/supersonic flow field, known as transonic flow, as shown in Figure 3. Most jet transport aircraft operate in transonic conditions, where the freestream Mach number is subsonic but local Mach numbers on some wing sections are supersonic. Today, these aircraft utilize supercritical wing sections, enabling them to fly efficiently at airspeeds approaching the speed of sound.

Transonic and supersonic flow over a wing exhibits significant changes as the freestream Mach number increases, with distinct flow characteristics in each regime.

Shock waves occur in fully supersonic flows, where the Mach number, $M$ , exceeds unity everywhere. They involve abrupt changes in pressure, density, and temperature in the flow, as well as irreversible processes driven by viscosity and thermal conduction within the shock wave. Therefore, the formation of shock waves leads to losses of pressure, momentum, and energy, resulting in wave drag on a flight vehicle.

At hypersonic speeds, which are usually defined for $M_{\infty} > 5$ , not only do strong shock waves form, but kinetic heating and chemical dissociation of the air can also occur. However, hypersonic flows remain poorly understood and are the subject of ongoing research.

Supersonic Flow Around an Aircraft

Figure 4 shows a fascinating schlieren image of the multitude of Mach (shock) waves generated by an aircraft during supersonic flight, with the Mach cones clearly visible. Notice the powerful shock waves (Mach cone) produced at the nose and tail of the aircraft, which are responsible for the sonic booms heard on the ground. The flight Mach number is easily determined from the Mach angle, which, in this case, is about 70 degrees because

(13) $\begin{equation*} \sin \mu = \frac{1}{M_{\infty}} \end{equation*}$

then

(14) $\begin{equation*} M_{\infty} = \frac{1}{\sin \mu} = \frac{1}{\sin 70^{\circ}} \approx 1.1 \end{equation*}$

so the aircraft is just flying supersonically. Notice also the turbulence and sound waves in the wake of the aircraft from the engine and its hot exhaust, the latter of which appears as a shimmering effect because of the turbulence.

A dramatic image of the Mach waves generated by an aircraft in supersonic flight was obtained using a background-oriented schlieren system.

The images in Figure 5 are schlieren flow visualizations of a fighter airplane model in a supersonic wind tunnel. Note that schlieren images are necessarily circular because the technique employs parabolic mirrors. Notice the build-up in the number and intensity of the almost normal shock waves as Mach 1 is approached. In supersonic flight, the Mach cone becomes increasingly swept back as the flight Mach number increases. The angle $\mu$ is used to denote the Mach angle, as given by

(15) $\begin{equation*} \mu = \sin^{-1} \left( \frac{1}{M_{\infty}} \right) \end{equation*}$

which is the half-apex angle of the Mach cone. Therefore, the Mach angle is 90 $^{\circ}$ at $M_{\infty} = 1$ , and its value decreases quickly beyond Mach 1. Notice that even by a Mach number of 2, the Mach cone is already swept back 60 $^{\circ}$ to a half apex angle of $\mu = 30^{\circ}$ . At hypersonic speeds, the Mach angle is so acute that it can approach the body surface, making a blunt or bluff body shape a better design for deflecting shock waves away from the surface.

Motiv: Slirbilder av vindtunnelmodell av flygplan J 35 i skala 1:30. Tagna i vindtunnel S4 på Flygtekniska försöksanstalten, FFA. Ur Flygvapenmuseums bildarkiv. — Schlieren flow-visualization images show the formation of shock waves around a fighter aircraft model in a wind tunnel as the flow transitions from transonic to supersonic.

What is a schlieren flow visualization system?

Schlieren is an optical technique to visualize variations in a fluid’s refractive index. It can help study the flow patterns in gases or liquids. The word “schlieren” is derived from the German word for “streaks” or “striae.” In a schlieren system, a point light source illuminates the fluid-flow region. The flow must exhibit density variations.

These so-called “density gradients” cause slight changes in the fluid’s refractive index, which in turn bend or “refract” the light passing through it. A knife-edge, such as a razor blade, is placed in the light path after interacting with the fluid flow, which “cuts off” part of the refracted light. An imaging system, such as a camera, then captures the remaining light. Higher- or lower-density regions appear as light and dark areas or streaks, providing insights into phenomena such as shockwaves, boundary layers, turbulence, and other flow characteristics.

Why Wing Sweep is Important

The Mach angle is essential in supersonic wing design because pressure disturbances in a supersonic flow are confined to the cone region as determined by the Mach angle. Unlike a subsonic flow, there is no upstream influence beyond the Mach cone in a supersonic flow; pressure disturbances are only transmitted along the Mach cone and downstream. The consequence is profound: if the wing’s leading edge is swept back behind the Mach cone, the wing experiences relatively lower drag (i.e., lower wave drag), as shown in the schlieren images in Figure 6. Otherwise, if the shock wave reaches the wing, it will disrupt the boundary-layer flow, increasing drag and potentially leading to flow separation.

Schlieren images show that sweeping wings behind the Mach wave prevent adverse shock interactions. (NACA image.)

Supersonic Airfoils

Unlike subsonic airfoils, supersonic airfoils generate shock waves that affect the flow field and aerodynamic performance. Oblique-compression shock waves occur at the airfoil’s leading and trailing edges, as shown in Figure 7. At the points of maximum thickness, expansion waves form, increasing the Mach number and decreasing the pressure after the expansion is complete. Rarefaction shock waves form at the trailing edge, increasing the Mach number and returning the pressure to the freestream value. The higher pressure on the lower front half of the airfoil and the upper rear half generates the lift.

Representative flow pattern about a supersonic double-wedge (diamond) airfoil.

Definition of Reynolds Number

The Reynolds number is another important parameter used in categorizing aerodynamic flows. Osborne Reynolds conducted fundamental experiments to improve the understanding of fluid dynamics and developed a statistical-mathematical framework for studying turbulence. The Reynolds number is given the symbol $Re$ and is defined as

(16) $\begin{equation*} Re = \frac{\varrho V L}{\mu} \end{equation*}$

where $\varrho$ and $\mu$ are the fluid’s density and viscosity, respectively, $V$ is a reference velocity, and $L$ is a characteristic length. Notice that when quoting the value of the Reynolds number, the characteristic length must be defined, e.g., the Reynolds number based on chord length, diameter, etc.

The Reynolds number is a dimensionless parameter and another similarity parameter used in aerodynamics. The Reynolds number is essential for understanding aerodynamics because it governs the relative magnitudes of inertial and viscous effects in the flow. Hence, the Reynolds number affects the drag and aerodynamic efficiency of a flight vehicle.

The reference velocity is often the freestream velocity ${V_{\infty}}$ , and the density is the freestream density $\varrho_{\infty}$ . For a wing, the reference length is the wing chord ${c}$ ( $L = c$ ), so the Reynolds number based on the chord is

(17) $\begin{equation*} Re_c = \frac{\varrho_{\infty} V_{\infty} c}{\mu_{\infty}} \end{equation*}$

This latter form of the Reynolds number, which people first encounter in aerodynamics, is one of many encountered in practice. This form is often used to scale flow situations, such as comparing an aircraft model in a wind tunnel to one in flight, to achieve full-size dynamic flow similarity.

When the values of air density and viscosity are substituted into the equation for the Reynolds number for representative flight vehicles and flight speeds, the resulting Reynolds numbers span many orders of magnitude and are often expressed in the units of “millions.”

Check Your Understanding #2 – Calculating the Reynolds number

The wing of an airplane has a mean aerodynamic chord $\overline{c}$ = 2.75 m, and it is in cruise flight at a freestream Mach number of 0.82 at an altitude of 30,000 ft. Calculate the Reynolds number for these conditions. Assume ISA standard conditions.

Show solution/hide solution.

The Reynolds number will be

$Re_c = \frac{\varrho_{\infty} V_{\infty} \overline{c}}{\mu_{\infty}} = \frac{0.45904 \times 248.65 \times 2.75}{1.4876\times 10^{-5} } = 2.11 \times 10^7$

So, the Reynolds number is about 27 million. Again, the atmospheric properties have been evaluated using the International Standard Atmosphere (ISA), for which an online ISA calculator is available here.

Figure 8 summarizes the ranges of Mach and Reynolds numbers (based on chord length) for various flight vehicles and other systems. Notice the logarithmic scales. Tiny flying creatures, such as birds and insects, encounter Reynolds numbers so low during flight that most of their energy is spent overcoming viscous forces. The Reynolds numbers for jet airplanes range from 10 $^7$ to 10 $^8$ , i.e., from 10 to 100 million. Model aircraft (and many UAVs) will have Reynolds numbers much smaller in the 10 ${^3}$ to 10 $^5$ range.

Summary of the range of Reynolds and Mach numbers encountered by various flight articles.

It will also be apparent that the Reynolds and Mach numbers are interrelated by the velocity, $V$ . For example, the Reynolds number based on the chord can be written in terms of the freestream Mach number as

(18) $\begin{equation*} Re_c = \frac{\varrho_{\infty} V_{\infty} c }{\mu_{\infty} } = \frac{\varrho_{\infty} c }{\mu_{\infty} } a_{\infty} M_{\infty} = \left( \frac{\varrho_{\infty} a_{\infty} }{\mu_{\infty} } \right) c \, M_{\infty} \end{equation*}$

Using the values for $\varrho_{\infty}$ , $a_{\infty}$ and $\mu_{\infty}$ for air at standard sea-level conditions then

(19) $\begin{equation*} Re_c = 23.26\times 10^6 M_{\infty}~\mbox{~per foot of chord} \end{equation*}$

or in SI units

(20) $\begin{equation*} Re_c = 7.096\times 10^6 M_{\infty}~\mbox{~per meter of chord} \end{equation*}$

This interdependence of $Re_c$ and $M_{\infty}$ poses an interesting dilemma in the investigation of Reynolds number and Mach number on items being tested in a conventional wind tunnel because $Re_c$ and $M_{\infty}$ cannot easily be varied separately. Nevertheless, in practice, the effects of the Reynolds number are more critical at low Mach numbers, whereas at higher Mach numbers, it is less important.

Physical Interpretation of Reynolds Number

It has been mentioned that the Reynolds number is important in aerodynamics because it governs the relative magnitudes of inertial to viscous effects in the flow. This fact can be seen by writing the Reynolds number as

(21) $\begin{equation*} Re = \frac{\varrho_{\infty} V_{\infty} L}{\mu_{\infty}} = \frac{\varrho_{\infty} V_{\infty} L (V_{\infty} L)}{\mu_{\infty} (V_{\infty} L)} = \frac{\varrho_{\infty} V_{\infty}^2 L^2}{\mu_{\infty} (V_{\infty}/L) L^2} \equiv \frac{\mbox{\small Inertial effects}}{\mbox{\small Viscous effects}} \end{equation*}$

On the numerator, the grouping $\varrho_{\infty} V_{\infty}^2 L^2$ has units of force (pressure times an area), so in this case, it represents an inertial force, i.e., after the flow is moving, it has the propensity to keep moving. The coefficient of viscosity, $\mu$ , is the shear force per unit area per unit velocity gradient, so the ratio $V_{\infty}/L$ has dimensions of a velocity gradient, which is expected based on Newton’s law of viscosity. Therefore, the grouping in the denominator represents a viscous force that acts to retard its motion. The Reynolds number, therefore, represents a relative measure of inertial to viscous effects in a flow.

This outcome, which explains the effects of the Reynolds number, is significant in aerodynamics. For example, the results in Figure 9 show the lift-to-drag ratio of several airfoil sections as a function of the Reynolds number based on chord length, i.e.,

(22) $\begin{equation*} Re_c = \frac{\varrho_{\infty} V_{\infty} \, c}{\mu_{\infty}} \end{equation*}$

where ${c}$ is the chord or distance from the leading edge to the wing’s trailing edge. The lift-to-drag ratio, i.e., the ratio of lift to drag, is a measure of aerodynamic efficiency; therefore, these results reveal a significant effect of Reynolds number variation.

Lift-to-drag ratio increases with Reynolds number and drops significantly at lower values. Note the logarithmic scales.

The operation of the airfoil (or wing) at higher chord Reynolds numbers, e.g., higher flow speeds, will improve aerodynamic efficiency because the inertial effects dominate over the viscous effects. Notice that the Reynolds number can also be increased by increasing the scale (size) of the wing. However, at lower Reynolds numbers, the relative effects of viscosity are greater, resulting in higher profile drag and a lower lift-to-drag ratio.

The underlying physics behind these preceding effects on airfoil efficiency is tied to the developing boundary layers on the airfoil’s surface, as shown in Figure 10. Boundary layers are thin regions immediately adjacent to surfaces where viscous effects are significant. The boundary layers are relatively thinner at higher Reynolds numbers, resulting in reduced profile drag. However, they are relatively thicker at lower Reynolds numbers, increasing profile drag.

A boundary layer as it develops over the surface of an airfoil.

It is now apparent that the Reynolds number is a fundamental parameter in the study of boundary layers and other viscous phenomena. A local Reynolds number can characterize the developing nature of the boundary layer. Instead of the Reynolds number being based on chord length, the length can be measured in terms of the distance ${x}$ from the point where the flow initially develops, i.e.,

(23) $\begin{equation*} { Re_ x = \frac{\varrho_{\infty} V_{\infty} x}{\mu_{\infty} } = \frac{V_{\infty} x}{\nu_{\infty} } } \end{equation*}$

For example, in the case of an airfoil, ${x}$ is measured from the stagnation point at the nose to the downstream distance ${x}$ .

Experiments have shown that at Reynolds numbers based on $Re_x$ above about $5 \times 10^{5}$ , the boundary layer becomes turbulent, which increases skin friction drag. This behavior arises because natural flow disturbances develop even over perfectly smooth surfaces and cause a transition from a laminar to a turbulent boundary layer. Numerous experiments have investigated the transition from laminar to turbulent flow. In all cases, the transition corresponds to a specific value (or a narrow range) of the Reynolds number, depending on the downstream distance. As turbulent mixing progresses, the transition from a fully laminar to a fully turbulent boundary layer occurs over a finite distance; thus, it is a gradual process rather than a sudden event.

Summary & Closure

The Mach number is defined as the ratio of the fluid velocity to the speed of sound in the fluid, while the Reynolds number is defined as the ratio of inertial forces to viscous forces. The Mach number is a measure of compressibility effects. The Mach number also determines whether a flow is subsonic, transonic, or supersonic. The Reynolds number helps determine whether the flow is laminar or turbulent and quantifies the relative importance of inertial and viscous effects. Understanding the importance of these two parameters is crucial for comprehending aerodynamic flows and accurately predicting flow behavior under various conditions. Both parameters are routinely used in aerodynamic problem-solving for flow around airfoils, wings, and complete aircraft, so their fundamental significance must be recognized.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

Calculate the Mach number and Reynolds number of the wing of a Cessna 172 in cruise flight at 5,000 ft (or use 1,800 m). Hint: Use the standard properties of the atmosphere.
Do some research to determine the likely range of Mach and Reynolds numbers a rocket will encounter as it leaves the Earth’s atmosphere.
The speed of sound in helium is nearly three times that of air. Explain why and what some interesting consequences might be.
Study the image below, which is a schlieren flow visualization of the flow field produced when a bullet is fired from a gun. What is the bullet’s approximate flight Mach number? Can you explain the origin of the other (circular) waves?

Additional Online Resources

To improve your understanding of the use of similarity parameters in engineering, navigate to some of these online resources:

A film by Shell Oil on transonic flight.
Airplanes approaching the sound barrier – another film by Shell Oil.
High-speed flight: Part 1, and Parts 2 & 3.
A film of wind-tunnel testing of Concorde.
Supersonic flight and sonic booms – what they sound like!
Top 5 sonic booms caught on camera!
Video on Reynolds number.
Video: Reynolds’ experiment.
Video: Physics of Life – The Reynolds Number and Flow Around Objects
Smarter Every Day: Shockwaves generated by bullets.

License

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Introduction to Aerospace Flight Vehicles Copyright © 2022–2026 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Digital Object Identifier (DOI)

https://doi.org/https://doi.org/10.15394/eaglepub.2022.1066.n13