Fluid Flows and Infinite Dimensional Manifolds (Part 3)

Or: Twisting on the Borderline

guest post by Tim van Beek

flow around a symmetric airfoil, lamiar to turbulent

In Part 2 of this series, I told you what ideal incompressible fluids are. Then I explained how the equation of motion for such fluids, Euler’s equation, can be seen as the equation for geodesics in \mathrm{SDiff}(M), the infinite dimensional manifold of volume preserving diffeomorphisms. Last time I promised to talk about the role of pressure in Euler’s equation. I also mentioned that Arnold used this geometric setup to put a bound on how good weather forecasts can be. I will try to fulfill both promises in the next blog post!

But last time I also mentioned that the ideal fluid has serious drawbacks as a model. This is an important topic, too, so I would like to explain this a little bit further first, in this blog post.

So, this time we will talk a little bit about how we can get viscosity, and therefore turbulence, back into the picture. This will lead us to the Navier–Stokes equation. Can ideal fluids, which solve Euler’s equation, model fluids with a very small viscosity? This depends on what happens to solutions when one lets the viscosity go to zero in the Navier–Stokes equation, so I will show you a result that answers this question in a specific context.

I’ll also throw in a few graphics that illustrate the transition from laminar flow to turbulent flow at boundaries, starting with the one above. These are all from:

• Milton Van Dyke, An Album of Fluid Motion, Parabolic Press, 12th edition, 1982.

flow around a circular cylinder, laminar to turbulent

Re-introducing viscosity: The Navier-Stokes equation

The motion of an incompressible, homogeneous, ideal fluid is described by Euler’s equation:

\displaystyle{ \partial_t u + u \cdot \nabla u = - \nabla p }

Ideal fluids are very nice mathematically. Nicest of all are potential flows, where the velocity vector field is the gradient of a potential. In two dimensions can be studied using complex analysis! One could say that a whole ‘industry’ evolved around the treatment of these kinds of fluid flows. It was even taught to some extend to engineers, before computers took over. Here’s a very nice, somewhat nostalgic book to read about that:

• L. M. Milne-Thomson, Theoretical Aerodynamics, 4th edition, Dover Publications, New York, 2011. (Reprint of the 1958 edition.)

The assumption of ‘incompressibility’ is not restrictive for most applications involving fluid flows of water and air, for example. Maybe you are a little bit surprised that I mention air, because the compressibility of air is a part of every day life, for example when you pump up a cycle tire. It is, however, not necessary to include this property when you model air flowing at velocities that are significantly lower than the speed of sound in air. The rule of thumb for engineers seems to be that one needs to include compressibility for speeds around Mach 0.3 or more:

Compressible aerodynamics, Wikipedia.

However, the concept of an ‘ideal’ fluid takes viscosity out of the picture—and therefore also turbulence, and the drag that a body immersed in fluid feels. As I mentioned last time, this is called the D’Alembert’s paradox.

The simplest way to introduce viscosity is by considering a Newtonian fluid. This is a fluid where the viscosity is a constant, and the relation of velocity differences and resulting shear forces is strictly linear. This leads to the the Navier–Stokes equation for incompressible fluids:

\displaystyle{ \partial_t u + u \cdot \nabla u - \nu \Delta u = - \nabla p }

If you think about molten plastic, or honey, you will notice that the viscosity actually depends on the temperature, and maybe also on the pressure and other parameters, of the fluid. The science that is concerned with the exploration of these effects is called rheology. This is an important research topic and the reason why producers of, say, plastic sheets, sometimes keep physicists around. But let’s stick to Newtownian fluids for now.

curved wall, lamiar to turbulent

Sending Viscosity to Zero: Boundary Layers

Since we get Euler’s equation if we set \nu = 0 in the above equation, the question is, if in some sense or another solutions of the Navier-Stokes equation converge to a solution of Euler’s equation in the limit of vanishing viscosity? If you had asked me, I would have guessed: No. The mathematical reason is that we have a transition from a second order partial differential equation to a first order one. This is usually called a singular perturbation. The physical reason is that a nonzero viscosity will give rise to phenomena like turbulence and energy dissipation that cannot occur in an ideal fluid. Well, the last argument shows that we cannot expect convergence for long times if eddies are present, so there certainly is a loophole.

The precise formulation of the last statement depends on the boundary conditions one chooses. One way is this: Let u be a smooth solution of Euler’s equation in \mathbb{R}^3 with sufficiently fast decay at infinity (this is our boundary condition), then the kinetic energy E

\displaystyle{ E =  \frac{1}{2} \int \| u \|^2 \; \mathrm{d x} }

is conserved for all time. This is not the only conserved quantity for Euler’s equation, but it’s a very important one.

But now, suppose u is a smooth solution of the Navier–Stokes equation in \mathbb{R}^3 with sufficiently fast decay at infinity. In this case we have

\displaystyle{ \frac{d E}{d t} =  - \nu \int \| \nabla \times u \|^2 \mathrm{d x} }

So, the presence of viscosity turns a conserved quantity into a decaying quantity! Since the 20th century, engineers have taken these effects into account following the idea of ‘boundary layers’ introduced by Ludwig Prandtl, as I already mentioned last time. Actually the whole technique of singular perturbation theory has been developed following this ansatz. This has become a mathematical technique to get asymptotic expansions of solutions of complicated nonlinear partial differential equations.

The idea is that the concept of ‘ideal’ fluid is good except at boundaries, where effects due to viscosity cannot be ignored. This is true for a lot of fluids like air and water, which have a very low viscosity. Therefore one tries to match a solution describing an ideal fluid flow far away from the boundaries with a specific solution for a viscous fluid with prescribed boundary conditions valid in a thin layer on the boundaries. This works quite well in applications. One of the major textbooks about this topic has been around for over 60 years now and has reached its 10th edition in German. It is:

• H. Schlichting and K. Gersten: Boundary-Layer Theory, 8th edition, Springer, Berlin, 2000.

Since I am also interested in numerical models and applications in engineering, I should probably read it. (I don’t know when the 10th edition will be published in English.)

flow on a sphere, lamiar to turbulent

Sending Viscosity to Zero: Convergence Results

However, this approach does not tell us under what circumstances we can expect convergence of solutions u_{\nu} to the viscous Navier–Stokes equations with viscosity \nu > 0 to a solution u_{0} of Euler’s equation with zero viscosity. That is, are there such solutions, boundary and initial conditions and a topology on an appropriate topological vector space, such that

\lim_{\nu \to 0} u_{\nu} = u_{0} \; \text{?}

I asked this question over on Mathoverflow and got some interesting answers. Obviously, a lot of brain power has gone into this question, and there are both interesting positive and negative results. As an example, let me describe a very interesting positive result. I learned about it from this book:

• Andrew J. Majda and Andrea L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2001.

It’s Proposition 3.2 in this book. There are three assumptions that we need to make in order for things to work out:

• First, we need to fix an interval [0, T] for the time. As mentioned above, we should not expect that we can get convergence for an unbounded time interval like [0, \infty].

• Secondly, we need to assume that solutions u_{\nu} of the Navier–Stokes equation and a solution u_0 of Euler’s equation exist and are smooth.

• Thirdly we will dodge the issue of boundary layers by assuming that the solutions exist on the whole of \mathbb{R}^3 with sufficiently fast decay. As I already mentioned above, a viscous fluid will of course show very different behavior at a boundary than an ideal (that is, nonviscous) one. Our third assumption means that there is no such boundary layer present.

We will denote the L^{\infty} norm on vector fields by \| \cdot \| and use the big O notation.

Given our three assumptions, Proposition 3.2 says that:

\displaystyle{ \mathrm{sup}_{0 \le t \le T} \; \| u_{\nu} - u_0 \| = O(\nu) }

It also gives the convergence of the derivatives:

\displaystyle{ \int_0^T \| \nabla (u_{\nu} - u_0) \| \; d t =  O(\nu^{1/2}) }

This result illustrates that the boundary layer ansatz may work, because the ideal fluid approximation could be a good one away from boundaries and for fluids with low viscosity like water and air.

flow around a symmetric airfoil, laminar to turbulent

So, when John von Neumann joked that “ideal water” is like “dry water”, I would have said: “Well, that is half right”.

15 Responses to Fluid Flows and Infinite Dimensional Manifolds (Part 3)

  1. John Baez says:

    Nice post! You wrote:

    The idea is that the concept of ‘ideal’ fluid is good except at boundaries, where effects due to viscosity cannot be ignored. This is true for a lot of fluids like air and water, which have a very low viscosity. Therefore one tries to match a solution describing an ideal fluid flow far away from the boundaries with a specific solution for a viscous fluid with prescribed boundary conditions valid in a thin layer on the boundaries.

    What are these boundary conditions like? My guess is that for a viscous fluid, the velocity of its flow at a boundary should be zero. Obviously the normal component of the velocity should vanish, but due to the ‘stickiness’, I’d guess the tangential component should vanish too.

    A practical website on nuclear power plant design shows this picture:

    Obviously for turbulent flow this velocity profile must show some sort of time-averaged velocity.

    • Ali Moharrer says:

      John,

      I am reading Alexandre Chorin’s ‘Vorticity and Turbulence’ (1994) and noting his remarkable suggestion that in the very high Reynolds number (Re ) range, the inertial velocity field dissociates from the ‘time parameter’ and turns into an equilibrium statistical mechanics problem.

      • nick says:

        The use of tools from statistical mechanics for problems in turbulence is a fascinating area of research, and is similar to certain ideas that have been discussed on this blog. Hopefully Tim Van Beek will discuss this subject in future posts in this series.

    • John Baez says:

      That sounds interesting! Of course in real life turbulent flow gradually slows down and comes to a halt, so it’s not really in equilibrium. Is he saying there’s some sort of ‘quasi-equilibrium’?

      • Ali Moharrer says:

        Hi John, It is assumed that there is an external energy source to sustain the (hard) turbulent flow which is similar to maintaining an energy cascading process like the Kolmogorov spectrum. As I understand Chorin’s work, the assertion for the equilibrium mode is on a par with a kind of stable energy cascade, i.e., the external energy source ultimately driving the creation of smaller scale eddies until a universal dissipative structure is established. Interestingly he also provides a correlation between equilibrium and enstrophy (see his concluding thoughts at the end of section 5.1 of his Vorticity and Turbulence, where he makes an attempt to provide an effective and analogical representation of ‘entropy’ disguised under ‘enstrophy’. Do I make sense?

      • John Baez says:

        Ali wrote:

        Do I make sense?

        Yes. Saying we’re looking a fluid flowing around a spherical obstacle at high Reynolds number, with boundary conditions that say the flow is laminar and has fixed velocity far from the obstacle. Then we could try to model turbulence using a probability measure on the space of velocity vector fields obeying these boundary conditions. We would want to show there’s one that’s invariant under time evolution. And we could even hope it’s formally analogous to a Gibbs state, though it’s not a Gibbs state.

        • Ali Moharrer says:

          Hoping is good! As I understand the conditions under which this Gibbs State could be achieved, is something a like a very distinct limiting case of renormalization of the (energy) scales below and above the Gibbs measure. That limiting case could be an asymptotic limit of two different flow statistics. Chorin in one of his papers, (On Turbulence Modeling, A. Chorin, 1994) discusses conditions under which this Gibbs State could hold. He discusses a limiting vortex temperature (after Onsager’s hydrodynamic treatment of turbulence, 1949) that could accept values from minus infinity to positive infinity. The temperature here is an equivalent analogue of temperature in the statistical mechanical sense. In 2 dimensions, the statistical theory finds strong numerical and observation supports, with Jupiter Great Red Spot being a prominent example of stable, time invariant, 2D turbulent flows for vortex equilibria, see P Marcus 1988.

  2. Tim van Beek says:

    The boundary condition is that the velocity at the wall is zero (relative to the wall). The velocity profile that we see in the graphic is one of the examples where the Naviers-Stokes equations have a solution in closed form, it is called “Hagen-Poiseuille” flow.

    This solution describes laminar flow only. It is an experimental fact that this laminar flow can turn into a turbulent flow, though. This is called in German “Rohrströmungsparadoxon” (I can try to look up the English term later :-)

    I believe that the profile that is labeled “Turbulent Flow” is actually the laminar Hagen-Poiseuille flow at the Reynolds number that marks the experimentally determined transition to turbulent flow.

  3. Tim van Beek says:

    BTW: The boundary layer is described by the “Prandtl boundary layer equations”.

  4. Dmitri Manin says:

    There are a number of other boundary layers in other interesting setups. The one I’m most familiar with is the Ekman boundary layer, which occurs in rotating fluids. Imagine a saucepan on a turntable that is rotating at 33 rpm as a solid, and then the turntable is switched to 45 rpm. While the water is adjusting to the new rotation rate, it continues in an almost solid rotation, except for a thin boundary layer at the bottom, where there is adjustment in the rotational velocity, and radial outflow (because of centrifugal force) modified by Coriolis effect.

    • Tim van Beek says:

      Dumb question: Is the Ekman boundary layer a special kind of boundary layer solution but with the usual boundary condition that the fluid velocity at the boundary is zero? Or has it a different boundary condition?

      • nick says:

        It depends on which Ekman layer you are referring to! For instance, the Ekman layer most commonly discussed in Oceanography 1 happens at the surface of the ocean, where the boundary condition is a prescribed surface (tangential) stress, parametrizing forcing (usually taken to be a constant wind). This phenomena has important implications for transport, since integration of the velocity profiles for all depths implies a non zero mass transport at \pm \pi/2 to the direction of the wind, (where the sign depends on what hemisphere you are in).

        An example of this is quite common during this time of year in California. Wind blows from the north, this leads to a transport of water to the right of the wind (f is greater than zero in the northern hemisphere) which causes surface waters to move offshore. Mass conservation then implies that water from depth moves up to replace the transported water, which tends to be much colder. Besides having biological implications, this makes surf sessions a bit chillier.

        See for instance, here http://en.wikipedia.org/wiki/Ekman_layer

        It is interesting to note that a phenomena so easy to mathematically model has been so difficult to observe. Indeed, Ekman presented his model in 1905, yet it was only very recently (somewhere in the ballpark of the 70s/80s) for the field techniques to reach a state where they could accurately measure the velocity profile in the Ekman layer. What people tend to observe is an “Ekman Spiral” that’s been flattened out vs what the model predicts.

        Some possible explanations include the additional vortex force due to the interaction of the Stokes drift and the Coriolis force and the fact that we are assuming the eddy viscosity is constant with depth.

        • Tim van Beek says:

          Thanks. I should have remembered the name Ekman from the book I browsed some time ago,

          * Vallis: “Atmospheric and Oceanic Fluid Dynamics”

  5. Robert says:

    You also need sufficiently fast decay to avoid viscosity becoming significant at large distances, which is pretty much the opposite of normal boundary layers.

    Basically, near an object in an infinite fluid, the appropriate length scale is the objects diameter, but at large distances, much greater than the diameter, the length scale is set by the distance from the object. if the velocity scale at large distances doesn’t decay fast enough, the Reynolds number will increase towards infinity, meaning that viscosity becomes dominant.

    As I recall, this happens with flow past an infinite cylinder, though I don’t have references to hand.

    • Tim van Beek says:

      That looks like a mathematical artefact that is of little importance for practical purposes: Just the right stuff to get mathematicians interested :-)

      I’ll see if I can find a reference…

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