…beauty is not symmetry of parts- that’s so impotent -as Mishima says, beauty is something that attacks, overpowers, robs, & finally destroys…

― John Geddes

In much of physics a great deal is made of the power of symmetry. The belief is that symmetry is a powerful tool, but also a compelling source of beauty and depth. In fluid mechanics the really cool stuff happens when the symmetry is broken. The power and depth of consequence comes from the asymmetric part of the solution. When things are symmetric they tend to be boring and uninteresting, and nothing beautiful or complex arises. I’ll be so bold as to say that the power of this essential asymmetry hasn’t been fully exploited, but could be even more magnificent. Fluid mechanics at its simplest is something called Stokes flow, basically motion so slow that it is solely governed by viscous forces. This is the asymptotic state where the Reynolds number (the ratio of inertial to viscous forces) is identically zero. It’s a bit oxymoronic as it is never reached, it’s the equations of motion without any motion or where the motion can be ignored. In this limit flows preserve their basic symmetries to a very high degree.

Basically nothing interesting happens and the whole thing is basically a giant boring glop of nothing. It is nice because lots of pretty math can be done in this limit. The equations are very well behaved and solutions have tremendous regularity and simplicity. Let the fluid move and take the Reynolds number away from zero and cool things almost immediately happen. The big thing is the symmetry is broken and the flow begins to contort, and wind into amazing shapes. Continue to raise the Reynolds number and the asymmetries pile up and we have turbulence, chaos and our understanding goes out the window. At the same time the whole thing produces patterns and structures of immense and inspiring beauty. With symmetry fluid mechanics is dull as dirt; without symmetry it is amazing and something to marveled at.

So, let’s dig a bit deeper into the nature of these asymmetries and the opportunity to take them even further. The fundamental asymmetry in physics is the arrow of time, and its close association with entropy. The connection with asymmetry and entropy is quite clear and strong for shock waves where the mathematical theory is well-developed and accepted. The simplest case to examine is Burgers’ equation, $u_t + u u_x = 0$, or its conservation form $u_t + 1/2 \left[u^2 \right]_x = 0$. This equation supports shocks and rarefactions, and their formation is determined by the sign of $u_x$. If one takes the gradient of the governing equation in space, you can see the solution forms a Ricatti equation along characteristics, $\left( u_x \right)_t + u u_{xx} + \left( u_x \right)^2 = 0$. The solution on characteristics tells one the fate of the solution, $u_x\left(t\right) = \frac{u_x\left(0\right)}{1 + t u_x}$. The thing to recognize that the denominator will go to zero if $u_x<0$ and the value of the derivative will become unbounded, i.e., form a shock.

The process in fluid dynamics is similar. If the viscosity is sufficiently small and the gradients of velocity are negative, a shock will form. It is inevitable as death and taxes. Moving back to Burgers’ equation briefly we can also see another aspect of the dynamics that isn’t so commonly known, the presence of dissipation in the absence of viscosity. Without viscosity for the rarefied flow where gradients diminish there is no dissipation. For a shock there is dissipation, and the form of it will be quite familiar by the end of the post. If one forms an equation for the evolution of the energy in a Burgers’ flow and looks at the solution for a shock via the jump conditions a discrepancy is uncovered, the rate of kinetic energy dissipation is $\ell \left(1/2 u^2\right)_t = \frac{1}{12}\left(\Delta u\right)^3$. The same basic character is shared by shock waves and incompressible turbulent flows. It implies the presence of a discontinuity in the model of the flow. On the one hand the form seems to be unavoidable dimensionally, on the other it is a profound result that provides the basis of the Clay prize for turbulence. It gets to the core of my belief that to a very large degree the understanding of turbulence will elude us as long as we use the intrinsically unphysical incompressible approximation. This may seem controversial, but incompressibility is an approximation to reality, not a fundamental relation. As such its utility is dependent upon the application. It is undeniably useful, but has limits, which are shamelessly exposed by turbulence. Without viscosity the equations governing incompressible flows are pathological in the extreme. Deep mathematical analysis has been unable to find singular solutions of the nature needed to explain turbulence in incompressible flows.

The real key to understanding the issues goes to a fundamental misunderstanding about shock waves and compressibility. First, it would be very good to elaborate how the same dynamic manifests itself for the compressible Euler equations. For intents and purposes the way to look at shock formation in the Euler equations acts just like Burgers’ equation for the nonlinear characteristics. In its simplest form the Euler equations have three fundamental characteristic modes, two being nonlinear associated with acoustic (sound) waves, one being linear and associated with material motion. The nonlinear acoustic modes act just like Burgers’ equation, and propagate at a velocity of $u\pm c$ where $u$ is the fluid velocity, and $c$ is the speed of sound.

Once the Euler equations are decomposed into the characteristics and the flow is smooth everything follows as Burgers’ equation. Along the appropriate characteristic, the flow will be modulated according to the nonlinearity of the equations, which is slightly different than Burgers’ in an important manner. The nonlinearity now depends on the equation of state in a key was, the curvature of an isentrope, $G=\left.\partial_{\rho\rho} p\right|_S$. This quantity is dominantly and asymptotically positive (i.e., convex), but may be negative. For ideal gases $G=\left(\gamma + 1\right)/2$. For convex equations of state shocks then always form given enough time if the velocity gradient is negative just like Burgers’ equation. One key thing to recognize is that the formation of the shock does not depend on the underlying Mach number of the flow. A shock always forms if the velocity is negative even as the Mach number goes to zero (the incompressible limit). Almost everything else follows as with Burgers’ equation including the dissipation relation associated with a shock wave, $T d S=\frac{G}{12c}\left(\Delta u\right)^3$. Once the shock forms, the dissipation rate is proportional to the cube of the jump across the shock. In addition this limit is actually most appropriate in the zero Mach number limit (i.e., the same limit as incompressible flow!).

Shocks aren’t just supersonic phenomena; they are a result of solving the equations in a limit where this viscous terms are small enough to neglect (i.e., the high Reynolds’ number limit!). So just to sum up, the shock formation along with intrinsic dissipation is most valid in the limits where we think of turbulence. We see that this key effect is a direct result of the asymmetric effect of a velocity gradient on the flow. For most flows where the equation of state is convex, the negative velocity gradient sharpens flow features into shocks that dissipate energy regardless of the value of viscosity. Positive velocity gradients smooth the flow and modify the flow via rarefying the flow. Note that physically admissible non-convex equations of state (really isolated regions in state space) have the opposite character. If one could run a classical turbulence experiment where the fluid is non-convex, the conceptual leap I am suggesting could be tested directly because the asymmetry in turbulence would be associated with positive rather than negative velocity gradients.

Now we can examine the basic known theory of turbulence that is so vexing to everyone. Kolmogorov came up with three key relations for turbulent flows. The spectral nature of turbulence is the best known where one looks at the frequency decomposition of the turbulence flow, and finds a distinct region where the decay of energy shows -5/3 slope. There is a lesser know relation of velocity correlations known as the 2/3 law. I believe the most important relation is known as the 4/5 law for the asymptotic decay of kinetic energy in a high Reynolds number turbulent flow. This equation implies that dissipation occurs in the absence of viscosity (sound familiar?).

The law is stated as $4/5 \left< K_t \right>\ell = \left<\left(\Delta_L u\right)^3 \right>$. The subscript $L$ means longitudinal where the differences are taken in the direction the velocity is moving over a distance $\ell$. This relation implies a distinct asymmetry in the equations that means negative gradients are intrinsically sharper than positive gradients. This is exactly what happens in compressible flows. Kolmogorov derived this relation from the incompressible flow equations and it has been strongly confirmed by observations. The whole issue associated with the (in)famous Clay prize is the explanation of this law in the mathematical admissible solutions of the incompressible equations. This law suggests that the incompressible flow equations must support singularities that are in essence like a shock. My point is that the compressible equations support exactly the phenomena we seek in the right limits for turbulence. The compressible equations have none of the pathologies of the incompressible equations and have a far greater physical basis and remove the unphysical aspects of the physical-mathematical description.

The result is a conclusion that the incompressible equations are inappropriate for understanding what is happening fundamentally in turbulence. The right way to think about it is that turbulent relations are supported by the basic physics of compressible flows in the right asymptotic limits of zero Mach number, high Reynolds number limits.

Symmetry is what we see at a glance; based on the fact that there is no reason for any difference…

― Blaise Pascal