Mach Number & Reynolds Number

J. Gordon Leishman

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

14 Mach Number & Reynolds Number

Introduction

The non-dimensional similarity parameters Mach number and Reynolds number have already been introduced. They will usually appear from the process of dimensional analysis applied to almost all fluid flow problems. These are two of the most important parameters to understand aerodynamic flows, so they require special consideration and discussion. For example, of all the various ways of categorizing different aerodynamic flows, the distinction based on the Mach number is one of the most useful; it is often used to help quantify the degree of compressibility effects in a flow. The Reynolds number is another useful aerodynamic parameter that can help quantify the relative effects of viscosity to inertia effects in a flow. Hence, the classifications of flows based on the Reynolds number are also helpful. The terms Mach number and Reynolds number are so frequently encountered and used in aerodynamics that understanding what they mean and how they are used is essential for engineers.

Learning Objectives

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

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 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) 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 non-dimensional (i.e., a unitless) parameter and is just one of a series of non-dimensional parameters that are encountered in engineering called similarity parameters. Mach number is named after Ernst Mach, an Austrian physicist noted for his contributions to various fields of physics and 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 differ from one point to point in a flow, especially in a high-speed flow where the effects of compressibility are present and temperature changes occur. As previously discussed, the speed of sound is proportional to the square root of the absolute temperature, i.e.

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

where $\gamma$ is the ratio of specific heats, and $R$ is the gas constant (in appropriate engineering units). Remember that using the correct values of $\gamma$ and $R$ for whatever gas is being used is essential.

In a compressible flow, there is a great distinction between flows involving velocities less than that of sound (subsonic flow) and flow velocities greater than that of sound (supersonic). 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 free-stream Mach number $M_{\infty}$ is defined as the ratio of the free-stream velocity (that is, the velocity far upstream of the airfoil) to that of the free-stream sonic velocity, $a_{\infty}$ , i.e.,

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

The free-stream sonic velocity $a_{\infty}$ would be evaluated using the free-stream temperature $T_{\infty}$ , i.e.,

(5) $\begin{equation*} a_{\infty} = \sqrt{\gamma R T_{\infty} } \end{equation*}$

The speed of sound at MSL ISA is about 1,117 ft/s (340 m/s, 768 mph, 1,236 km/h).

Of all the ways to categorize aerodynamic flows about a wing or flight vehicle, the free-stream Mach number is one of the most useful.

Worked Example #1 – Calculating Mach Number

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

The airspeed $V_{\infty}$ is given as 500 knots, which is 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 at here, the standard temperature at this altitude is -47.83 F 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

$a_{\infty} = \sqrt {\gamma \, R \, T} = \sqrt{1.4 \times 1716.59 \times 411.84} = 994.86~\mbox{ft/s}$

where the gas constant for air is 1716.59 ft lb slug $^{-1}$ R $^{-1}$ . Therefore, the flight Mach number is

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

If the free-stream 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 a mixed subsonic/supersonic flow field called transonic flow, as shown in the figure below. Most jet transport aircraft fly in transonic conditions where the free-stream Mach number is subsonic, but the local Mach numbers at some points on the wings are supersonic. Today, these aircraft use supercritical wing sections, allowing them to fly efficiently at airspeeds near the speed of sound.

The development of compressibility effects, transonic flow and then supersonic flow over a wing section with increasing free-stream Mach number. Notice the fundamental differences in the flow characteristics as the Mach number increases.

For fully supersonic flows in that $M > 1$ everywhere, shock waves occur. A shock wave involves an abrupt flow change in pressure, density, and temperature. In addition, there is an irreversible process caused by viscosity and thermal conduction effects inside the shock wave. Therefore, the formation of shock waves leads to losses of pressure, momentum, and energy and manifests as a source of drag on a flight vehicle called wave drag.

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

Mach Number as a Measure of Compressibility

It should be clear why the Mach number can be used as a measure of compressibility effects in a flow. First, however, it is essential to note that the free-stream Mach number $M_{\infty}$ by itself is not a reliable indicator of whether the local flow can be considered incompressible or not. This is because as the flow approaches and passes around a body, such as an airfoil, it will be locally accelerated, and its local velocity (particularly near its leading edge or nose) will increase. For example, suppose the free-stream Mach number is 0.3. As the flow accelerates over the upper surface of an airfoil, then the local Mach number may be as high as 0.5 or 0.6, or even higher, depending on the airfoil’s angle of attack.

However, the flow may be treated as essentially incompressible for flows around wings and bodies where the local value of $M$ is less than about 0.3. To verify this, consider a fluid element initially at rest, which has density $\varrho_0$ . If this flow is accelerated to some velocity $V$ and some Mach number $M$ , the density of the fluid can be assumed to change according to the isentropic relationship

(6) $\begin{equation*} \frac{\varrho}{\varrho_0} = \left( 1 + \frac{\gamma-1}{2} M^2 \right)^{-1/(\gamma-1)} \end{equation*}$

where $\gamma=1.4$ for air, the result in Eq. 6 being plotted in the figure below.

Isentropic variation of the density ratio in a subsonic compressible flow showing that for Mach numbers less than 0.3 then the flow is essentially incompressible.

Notice that the variation of $\varrho/\varrho_0$ from unity is small at low Mach numbers. Therefore, for local Mach numbers below a threshold of about 0.3, i.e., the density varies less than 5% for $M \le 0.3$ , the flow can usually be treated as incompressible, and it is convenient to make this simplifying assumption in flow analysis. However, this incompressible flow assumption cannot be easily made for higher Mach numbers without introducing significant errors in predicting the flow properties. Therefore, in the case of compressible flows, the thermodynamic equation of state must be used to relate the flow properties.

It is also helpful to have on hand the corresponding isentropic flow equations for pressure and temperature. For the pressure ratio, then

(7) $\begin{equation*} \frac{p}{p_0} = \left( 1 + \frac{\gamma-1}{2} M^2 \right)^{-gamma/(\gamma-1)} \end{equation*}$

and for the temperature

(8) $\begin{equation*} \frac{T}{T_0} = \left( 1 + \frac{\gamma-1}{2} M^2 \right)^{-1} \end{equation*}$

Speed of Sound

Another important 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, called a point pressure or acoustic source) accelerates from rest, it will emit disturbances in the form of pressure waves, as shown in the figure below. 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.x

The movement of a point pressure or acoustic source helps explain the development of compressibility effects in a flow. As the source becomes supersonic it exceeds the speed at which disturbances propagate (speed of sound) and so pressure disturbances will coalesce along a wavefront called a Mach cone.

As the point source moves faster, the pressure waves bunch closer together in the direction of motion and spread apart in the other direction, i.e., a Doppler effect. Eventually, as the source’s speed approaches the speed of sound, the source catches up with the previously produced pressure waves, which coalesce and become perpendicular to the direction of travel, which is the essence of how a shock wave forms. For supersonic motion, the shock waves bend back to form a Mach wave line. 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

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

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

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

The photo below shows a fascinating schlieren image of the plethora of Mach (shock) waves an aircraft generates during supersonic flight, with the Mach cones identifiable. Notice the powerful shock waves (Mach cone) produced at the nose and tail of the aircraft, which are those 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

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

then in this case

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

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, that latter which appears as a shimmering effect because of the turbulence.

A rather dramatic image of the Mach wave generated by an aircraft in supersonic flight as obtained using a background oriented schlieren system.

Supersonic Airfoil

Unlike subsonic airfoils, supersonic airfoils produce shock waves, which affect the airflow and aerodynamic performance of the airfoil. Oblique compression shock waves occur at the leading and trailing edges of the airfoil. At the points of maximum thickness, expansion waves appear, which causes the Mach number to increase and the pressure to decrease after the expansion is complete. At the trailing edge, rarefaction shock waves form, which increase the Mach number and returns the pressure to the free-stream value. The lift is generated by the higher pressure on the lower front half of the airfoil and the upper rear half.

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 did fundamental experiments to understand fluid dynamics better and also developed the statistical, mathematical framework used in the study of turbulence. The Reynolds number is given the symbol $Re$ and is defined as

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

where $\varrho$ and $\mu$ are the fluid’s density and viscosity, respectively, $V$ is a reference velocity, and $L$ is a characteristic length. The Reynolds number is also a non-dimensional parameter and another similarity parameter used in aerodynamics. The Reynolds number is essential in understanding aerodynamics because it governs the relative magnitude of inertia effects to viscous effects in the flow. Hence, the Reynolds number tends to affect the drag and aerodynamic efficiency of flight vehicles.

More often than not, the reference velocity is the free-stream velocity $V_{\infty}$ , and the density is the free-stream 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

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

This latter form of the Reynolds number that students first encounter in their study of aerodynamics, but it is not the only form encountered in practice. This form is often used in the scaling of flow situations, such as between an aircraft model in a wind tunnel and the full-size version, to obtain what is called dynamic flow similarity.

When the values of the density and viscosity of air are substituted into the equation for the Reynolds number for representative flight vehicles and flight speeds, then it will be seen that the values of the Reynolds number will vary over many orders of magnitude and is often expressed in terms of “millions.”

Worked Example #2 – Calculating Reynolds Number

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

The Reynolds number will be

$Re_c = \frac{\varrho_{\infty} V_{\infty} \overline{c}}{\mu_{\infty}} = \frac{0.45904 \times 248.65 \times 2.75}{1.4876\times 10^{-5} } = 2.74 \times 10^7$

so a Reynolds number of about 27 million. Again, notice that the atmospheric properties have been evaluated using the International Standard Atmosphere (ISA) model, for which a handy online ISA calculator can be found here.

The figure below summarizes the range of Mach number and Reynolds numbers (based on the chord) for various flight vehicles and other things. Notice the logarithmic scales. Tiny flying things, such as birds and insects, encounter such low-flight Reynolds numbers that most of their energy goes into overcoming the effects of viscosity. 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 range 10 $^3$ to 10 $^5$ .

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

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

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

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

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

or in SI units

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

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 e $Re_c$ and $M_{\infty}$ cannot easily be varied separately. Nevertheless, in a practical sense, the effects of the Reynolds number are found to be more important at low Mach numbers, and at higher Mach numbers, the Reynolds number has somewhat less importance.

Physical Importance of Reynolds Number

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

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

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 then it has a propensity to keep moving. The coefficient of viscosity, $\mu$ , is the shear force per unit area per unit velocity gradient, so that the ratio $V_{\infty}/L$ has dimensions of a velocity gradient, which is expected based on Newton’s law of viscosity. Therefore, the grouping on the denominator represents a viscous force, which act in a way as to retard or slow its motion. The Reynolds number, therefore, represents a relative measure of inertial to viscous effects in a flow.

This outcome as it explains the effects of Reynolds number is very important in aerodynamics. For example, the results in the figure below shows the lift-to-drag ratio of several airfoil sections as a function of the Reynolds number based on chord length, i.e.,

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

where $c$ is the chord or the distance from the leading edge to the trailing edge of the wing. Lift to drag ratio, i.e. the ratio of the amount of lift to the amount of drag, is a measure of aerodynamic efficiency, so these results expose a significant effect of varying the Reynolds number.

The lift-to-drag ratio for an airfoil is generally higher at higher Reynolds numbers and substantially lower at lower Reynolds number. Notice 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 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 higher, which manifests as 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. Boundary layers are small regions immediately next to the surface where viscous flow effects are powerful. At higher Reynolds numbers, the boundary layers are relatively thin, which reduces profile drag. However, the boundary layers are thicker at lower Reynolds numbers, which correspondingly increases profile drag.

Boundary layers are thin regions near a surface where viscous effects in the flow are very strong. Boundary layers, in general, are very thin and their scales have been exaggerated in this figure.

It is now apparent that the Reynolds number is a fundamentally important parameter in the study of boundary layers and other viscous effects. In fact, the developing nature of the boundary layer can be characterized in terms of a local Reynolds number. 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.,

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

For example, in the case of an airfoil, then $x$ is measured from the stagnation point at the nose of the airfoil 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. The reason for this behavior is that natural flow disturbances develop (even over perfectly smooth surfaces) and causes a transition from a laminar to a turbulent boundary layer.

Many experiments have examined the transition from laminar to turbulent flows. In all cases, the transition correlates to a certain value (or small range) of the Reynolds number based on downstream distance. Because the turbulent mixing occurs progressively, the transition from a fully laminar boundary layer flow to one fully turbulent is a process that will occur over some distance, i.e., it is a process, not 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 used to determine whether a flow is subsonic, transonic, or supersonic, and the Reynolds number is used to determine whether the flow is laminar or turbulent. Understanding the significance of these parameters helps in understanding aerodynamic flows and making predictions about flow behavior in various conditions. Both parameters are used routinely in aerodynamic problem solving, and their fundamental significance must be appreciated and understood.

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 encountered by a rocket as it leaves the Earth’s atmosphere.
The speed of sound in helium is nearly three times that of air. Explain why and what might be some interesting consequences.
Study the image below, which is a schlieren flow visualization image of what is produced when a bullet is fired from a gun. What is the approximate flight Mach number of the bullet? Can 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.
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

License

Icon for the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License

Introduction to Aerospace Flight Vehicles 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