Turbulent Mach Number

While the Mach number is widely understood as a measure of an object's speed relative to sound, such as a supersonic jet, the chaotic, swirling nature of turbulence within that flow presents a more complex challenge. How do we quantify the intensity of these internal motions and, more importantly, determine when the turbulence itself begins to compress the fluid? The answer lies in a powerful and distinct parameter: the turbulent Mach number ($M_t$). This concept addresses a critical gap in fluid dynamics, as the overall speed of a flow does not, by itself, reveal the behavior of the turbulence within it.

This article explores the fundamental role of the turbulent Mach number in understanding and modeling high-speed flows. The first chapter, "Principles and Mechanisms," will define $M_t$, distinguish it from the conventional freestream Mach number, and introduce Morkovin's Hypothesis—a key principle that simplifies turbulence modeling. Following this, the chapter "Applications and Interdisciplinary Connections" will demonstrate the far-reaching impact of this parameter, from the engineering of quieter aircraft to the astrophysical processes that govern star formation and feed supermassive black holes.

To understand the world of high-speed flow, we must first appreciate the concept of the Mach number. We are familiar with it in the context of a jet airplane: a Mach 2 fighter is traveling at twice the speed of sound. This number, the ratio of the object's speed to the speed of sound in the surrounding air, tells us when the air can no longer be treated as an incompressible fluid. At high Mach numbers, the air itself is compressed and rarefied, creating shock waves and fundamentally changing the nature of flight.

But what about the turbulence, the chaotic, swirling maelstrom of eddies that makes up the wake of the jet or the flow inside its engine? This turbulence is not a single object with a single velocity. It is a complex field of fluctuating motions. Can we assign a Mach number to this "weather" within the flow? The answer is a resounding yes, and this concept, the ​​turbulent Mach number​​, is the key to unlocking a deeper understanding of compressible flows.

Let's imagine a turbulent flow. At any point, the velocity is not steady but is composed of a mean, average velocity, $\overline{u}$, and a fluctuating part, $u'$. These fluctuations are the essence of turbulence. To characterize their intensity, we look at the average kinetic energy they carry, a quantity physicists call the ​​turbulent kinetic energy​​, or $k$. It's defined as half the mean-square of the fluctuation velocity: $k = \frac{1}{2}\overline{u_i' u_i'}$.

If $k$ represents the average energy of the turbulent motions, then a characteristic speed of these fluctuations must be related to $\sqrt{k}$. From the definition, the root-mean-square (rms) speed of the eddies is $u'_{\mathrm{rms}} = \sqrt{\overline{u_i' u_i'}} = \sqrt{2k}$. This is the typical speed with which the turbulent eddies are swirling and tumbling relative to the mean flow. The ​​turbulent Mach number​​, $M_t$, is simply the Mach number of these eddies: the ratio of their characteristic speed to the local speed of sound, $a$.

This definition seems simple enough, but it harbors a profound distinction. The turbulent Mach number $M_t$ is fundamentally different from the familiar freestream Mach number $M_\infty$.

• The ​​freestream Mach number​​, $M_\infty$, tells us about the compressibility of the mean flow. Is the river as a whole moving so fast that its average properties, like density, are changing significantly?
• The ​​turbulent Mach number​​, $M_t$, tells us about the compressibility of the turbulence itself. Are the individual eddies within the river churning so violently that they are creating their own local compressions and expansions?

The crucial insight is that these two numbers can be completely independent. Consider two fascinating, contrasting scenarios drawn from fluid dynamics:

1. ​​Supersonic Flight:​​ Imagine air flowing at $M_\infty = 2.0$ over a flat plate, like the wing of a supersonic jet. The mean flow is highly compressible. However, in the turbulent boundary layer near the surface, the velocity fluctuations might be relatively weak, giving a turbulent kinetic energy of, say, $k \approx 15 \, \mathrm{m}^2/\mathrm{s}^2$. In the cold air at high altitude ($T \approx 240 \, \mathrm{K}$), the speed of sound is about $a \approx 310 \, \mathrm{m/s}$. This gives a turbulent Mach number of $M_t = \sqrt{2 \times 15} / 310 \approx 0.018$. Here, the overall river is supersonic, but the eddies within it are placid and behave as if they were in an incompressible fluid.

2. ​​A Cool Jet Exhaust:​​ Now, picture a jet of gas exiting into still air at a low freestream Mach number, $M_\infty = 0.25$. The mean flow is nearly incompressible. But suppose this is a high-intensity, cooled jet where the turbulent mixing is extremely violent ($k \approx 4000 \, \mathrm{m}^2/\mathrm{s}^2$) and the local temperature is very low ($T \approx 180 \, \mathrm{K}$), making the speed of sound low ($a \approx 269 \, \mathrm{m/s}$). The turbulent Mach number is now $M_t = \sqrt{2 \times 4000} / 269 \approx 0.33$. In this case, the river is flowing slowly, but the eddies are so ferocious that they are beginning to compress the fluid locally. The turbulence itself is becoming compressible, even though the mean flow is not.

These examples reveal that $M_t$, not $M_\infty$, is the true arbiter of whether the physics of the turbulence must include compressibility.

This discovery leads to a wonderful simplification, a gift from nature known as ​​Morkovin's Hypothesis​​. This hypothesis states that for a wide range of flows, as long as the turbulent Mach number $M_t$ is small (typically $M_t \lesssim 0.3$), the direct effects of compressibility on the structure and dynamics of the turbulence are negligible. The turbulence behaves, for all intents and purposes, as if it were incompressible.

To appreciate what this "pact" allows us to ignore, we must understand what new physics compressibility introduces into turbulence. We can think of any velocity field as a combination of two types of motion: ​​solenoidal​​ (swirly, rotational motion like a vortex) and ​​dilatational​​ (squeezy, compressive motion like a sponge). Incompressible turbulence is purely solenoidal. Compressibility introduces the dilatational part, which brings two new players onto the stage of the turbulent energy budget:

• ​​Pressure-Dilatation ($\Pi = \overline{p' \theta'}$):​​ This term represents the work done by fluctuating pressure ($p'$) on the fluctuating rate of volume change, or dilatation ($\theta' = \nabla \cdot \mathbf{u}'$). In incompressible flow, pressure fluctuations can only shuffle energy between different directions—a graceful ballet that pushes turbulence toward isotropy. The mathematical object representing this, the pressure-strain tensor, is trace-free. But in compressible flow, pressure can also do work on the fluid's volume, creating or destroying turbulent kinetic energy. This term represents a reversible exchange between the kinetic energy of the turbulence and the internal energy (heat) of the gas.

• ​​Dilatational Dissipation ($\varepsilon_d$):​​ All turbulence dissipates, its kinetic energy ultimately turning into heat due to viscosity. In incompressible flow, this happens as viscosity acts on the shearing, vortical motions (solenoidal dissipation). Compressibility opens a new channel for this decay: viscosity acting on the squeezing and expanding motions. This is an irreversible conversion of kinetic energy into heat, a true dissipative process.

Morkovin's hypothesis works because the strength of both these new effects—the reversible energy exchange and the new dissipative channel—is not large. Rigorous theory and extensive computer simulations show that their magnitude, relative to the dominant incompressible processes, scales with the square of the turbulent Mach number: $O(M_t^2)$.

This quadratic scaling is the heart of the matter. If $M_t = 0.2$, then $M_t^2 = 0.04$. This means these new compressible effects are only about 4% as strong as their incompressible counterparts. For many engineering and scientific purposes, this is a negligible contribution. Thus, within "Morkovin's Quiet Kingdom" ($M_t \lesssim 0.3$), we can use our well-established models for incompressible turbulence, even if the flow is embedded in a supersonic mean flow, as long as we account for the variation in the mean density.

What happens when we leave the quiet kingdom? When turbulence is sufficiently intense or the gas is cold enough, $M_t$ can climb above 0.3, and Morkovin's pact is broken. The physics of turbulence changes dramatically.

The eddies become so violent that they can generate their own miniature shock waves, or ​​shocklets​​. The flow is no longer a collection of interacting vortices but is now populated by a sea of sharp, dissipative discontinuities. This has profound consequences:

• ​​A Steeper Energy Cascade:​​ The classic Kolmogorov theory of incompressible turbulence predicts that the energy spectrum follows a $k^{-5/3}$ power law. In shock-dominated supersonic turbulence, the spectrum steepens to something closer to $k^{-2}$. This signifies a different mechanism of energy transfer: instead of a gradual cascade through ever-smaller eddies, energy is abruptly dissipated in the shocks themselves.

• ​​Extreme Intermittency:​​ Energy dissipation in turbulence is always "intermittent," meaning it's concentrated in small, localized regions. In incompressible flow, these are vortex filaments. In supersonic turbulence, dissipation is concentrated in the geometrically thin sheets of shocks, leading to a far more extreme and spotty distribution of energy loss.

• ​​Shock-Turbulence Interaction:​​ When a turbulent eddy from a low-$M_t$ region encounters a strong shock wave from the mean flow, it is violently transformed. The shock compresses the eddy in the direction of the flow. According to the principles of Rapid Distortion Theory, the velocity fluctuations in the flow direction are damped, while the transverse fluctuations are largely unaffected. An initially isotropic, spherical-looking eddy is squashed into a pancake-like, highly anisotropic one. The ratio of the streamwise to transverse turbulent stresses immediately downstream of the shock is approximately $(\rho_1/\rho_2)^2$, where $\rho_1/\rho_2$ is the density ratio across the shock. Since density always increases across a shock, this ratio is always less than one, signifying a powerful suppression of motion along the flow direction.

The turbulent Mach number, born from a simple desire to characterize the speed of turbulent fluctuations, thus reveals itself to be a master parameter. It delineates the boundary between the familiar world of incompressible turbulence and the wild frontier of supersonic flow, a world of shocklets, steep energy spectra, and violent transformations. It guides our modeling strategies, telling us when simple approximations are valid and when we must confront the full, complex, and beautiful physics of compressibility.

We have journeyed through the principles of the turbulent Mach number, $M_t$, understanding it as the true measure of compressibility within the heart of turbulence itself. It is the ratio of the turbulent velocity fluctuations to the local speed of sound, a parameter that tells us not how fast the entire fluid is moving, but how violently it is churning relative to the speed at which information can travel through it. Now, the real fun begins. We are like children who have just learned the rules of chess; we are ready to see the game played out in all its intricate and beautiful variations. Where does this idea of a turbulent Mach number actually show up? The answer, you will see, is astonishingly broad. It appears in the roar of a jet engine, in the computational models that design our aircraft, and, most remarkably, in the grand cosmic theater where stars are born and supermassive black holes feast. It is a unifying concept, a secret key that unlocks doors in seemingly disconnected realms of science. Let us open a few of these doors.

Have you ever stood near a rushing river or listened to the wind howl? That sound is, in large part, the sound of turbulence. But why does turbulence make noise? The answer is compressibility. A perfectly incompressible fluid, a theoretical idealization, would be silent. Sound waves are, after all, propagating waves of compression and rarefaction. For turbulence to generate sound, its chaotic swirls and eddies must somehow create these compressions.

The great physicist James Lighthill taught us to think about this in a beautiful way. He imagined a region of turbulent flow as a collection of "acoustic sources" embedded in an otherwise still fluid. These sources are the turbulent fluctuations themselves. At low speeds, where the overall flow Mach number $M = U/c_0$ is much less than one, the fluid behaves as if it's nearly incompressible. The dominant pressure fluctuations are not sound waves but something else, a kind of "ghost" sound often called 'pseudosound'. These are just the local pressure variations needed to make the fluid swerve and eddy, as described by the incompressible fluid equations. They are strong in the immediate vicinity of the turbulence but die off very quickly with distance, unwilling to travel far from home.

Propagating sound, the kind that reaches our ears far from the source, is a different beast. It is generated by mechanisms that are incredibly inefficient at low speeds. For a compact region of turbulence, the primary mechanism is of a "quadrupole" nature, which you can imagine as two back-to-back pairs of speakers pushing and pulling out of phase. The upshot of a careful analysis is Lighthill's celebrated $U^8$ law: the acoustic power radiated by the turbulence scales with the eighth power of the characteristic velocity $U$. This steep scaling is a direct signature of the flow's reluctance to compress. Doubling the speed of a turbulent jet doesn't double the noise; it increases it by a factor of $2^8 = 256$! This is why a gently flowing stream whispers, while the exhaust of a jet engine, with its tremendously high velocities, produces a deafening roar. The turbulent motions in the jet are so fast that they can no longer be considered incompressible; they are effective at compressing the air and launching powerful sound waves.

Let us now turn to the world of engineering, specifically the design of high-speed aircraft. One might naively think that if an airplane is flying at Mach 2 ($M_{\infty} = 2$), then compressibility effects are everywhere and all-important. This is true for the large-scale flow—shock waves form, and the drag is immense. But what about the turbulence itself, for instance in the thin boundary layer of air clinging to the wing's surface? This is where the distinction between the freestream Mach number, $M_{\infty}$, and the turbulent Mach number, $M_t$, becomes absolutely crucial.

Inside that turbulent boundary layer, the velocity fluctuations, $u'$, are typically only a small fraction of the mean velocity. Furthermore, the temperature, and thus the local sound speed $a$, is much higher than in the freestream due to frictional heating. The result is that even when $M_{\infty}$ is high, the turbulent Mach number $M_t = u'/a$ often remains quite small—say, less than $0.3$. This observation, first articulated by Morkovin and now known as Morkovin's Hypothesis, is one of the most important guiding principles in high-speed fluid dynamics. It tells us that for many practical purposes, the internal physics of the turbulence is not directly affected by compressibility. The eddies interact with each other much as they would in a low-speed flow. This means that the turbulent transport of momentum and heat behave very similarly, and the turbulent Prandtl number, $Pr_t$, which measures their relative efficiency, remains roughly constant and close to one. This simplifies the task of modeling such flows immensely.

But what happens when Morkovin's Hypothesis breaks down? In regions with very intense turbulence or near shock waves, $M_t$ can become significant. Here, the direct effects of compressibility on turbulence can no longer be ignored. For example, a new mechanism for dissipation appears, called "dilatational dissipation," which comes from the viscous damping of the compressive motions. This effect is absent in incompressible turbulence. In the sophisticated turbulence models used in Computational Fluid Dynamics (CFD), engineers must add extra terms to account for these effects. And how do these correction terms scale? You guessed it: they are typically proportional to $M_t^2$. This scaling tells us that for small $M_t$, these are minor corrections, but they become progressively more important as $M_t$ grows. The turbulent Mach number thus serves as the natural parameter for building these more advanced models. It even finds its way into the very practical task of setting up a simulation, where it is used as a key parameter to define the turbulent state of a fluid entering the computational domain.

Now we take a leap, from the realm of human engineering to the cosmos itself. The vast spaces between stars are not empty; they are filled with the interstellar medium (ISM), a tenuous gas of hydrogen and helium. This gas is in a state of constant, violent, supersonic turmoil, stirred by supernova explosions and galactic winds. In these giant molecular clouds, the turbulent Mach number $\mathcal{M}$ (astronomers often use $\mathcal{M}$ instead of $M_t$, but the concept is identical) can be 10, 50, or even higher.

What is the consequence of such enormous turbulent Mach numbers? Imagine a parcel of gas being kicked around. It gets slammed by a shock wave (a strong compression), then expands into a rarefaction, then hit by another shock from a different direction. This series of multiplicative compressions and rarefactions has a profound effect: it produces an enormous range of densities. While most of the volume is filled with very low-density gas, a small fraction of the mass gets compressed into filaments and cores of exceedingly high density. The probability distribution of the gas density is no longer a simple bell curve; it becomes a "lognormal" distribution. The width of this distribution—how extreme the density fluctuations are—is governed directly by the turbulent Mach number. A well-established relation shows the variance of the logarithmic density, $\sigma_s^2$, scales as $\sigma_s^2 = \ln(1 + b^2\mathcal{M}^2)$, where $b$ is a parameter related to the type of turbulent stirring.

This is not just a curiosity; it is the very key to understanding how stars are born. Stars form when clouds of gas become so dense that their own gravity overwhelms all other support, leading to a runaway collapse. In a turbulent cloud, it is only the highest-density peaks, created by the supersonic turbulence, that can reach this critical threshold. A higher turbulent Mach number creates a wider density distribution, which means more of the gas mass resides in these extremely dense, star-forming cores. Therefore, the turbulent Mach number directly controls the rate at which a galaxy can form new stars. It is a breathtaking thought: the same parameter that fine-tunes engineering models for a jet wing also orchestrates the birth of stars across the universe.

The influence of $M_t$ doesn't stop there. Let's look at a single star, like our sun or a larger red giant. The outer layers of such stars are often boiling cauldrons of convection—a form of turbulence. This turbulent motion provides an additional source of pressure, beyond the ordinary gas pressure. This "turbulent pressure" pushes outward, effectively inflating the star's atmosphere like a balloon. The amount of this inflation depends directly on the turbulent pressure, which scales with the square of the turbulent Mach number, $M_t^2$. Astronomers trying to precisely measure the radii of stars must account for this effect, which can alter the apparent size of a star by several percent.

Finally, let us journey to the most extreme objects in the universe: supermassive black holes lurking at the centers of galaxies. These monsters grow by accreting surrounding gas. Simple theories of this accretion, like the classic Bondi model, assume the gas is smooth and uniform. But we know the gas is turbulent. The density is not uniform; it is clumpy. Since the accretion rate is highly sensitive to density, the process is dominated by the occasional infall of a dense clump, not the gentle inflow of average-density gas. How can we account for this "clumping"? Once again, the lognormal density distribution, whose width is set by $M_t$, comes to our rescue. A beautiful calculation shows that the clumping enhances the overall accretion rate by a factor of precisely $1 + b^2 \mathcal{M}^2$. This factor is nothing but $\exp(\sigma_s^2)$! The very same term that quantifies the breadth of density fluctuations also tells us how much more efficiently a black hole can feed. The turbulent Mach number, it turns out, sets the menu for the universe's most voracious eaters.

From the whisper of wind and the roar of jets to the birth of suns and the feeding of cosmic leviathans, the turbulent Mach number emerges again and again as a character of central importance. It reminds us of the profound unity of physics, where a single, simple idea can illuminate the workings of the universe on scales both human and astronomical.