Wuppertal. Turbulent flows and the Navier-Stokes equations

A report from the PhD student Marc Stromberg (University of Wuppertal)

Fluid dynamics is one of the important fields of applied and computational mathematics. It has a wide range of applications such as in aviation, car construction, space industry, modeling the flow of blood in medicine, weather forecasting or even football (For the last one search for the so called Magnus effect and find a jaw-dropping goal by Roberto Carlos as one of the prime examples). In addition scientists have proposed fluid based models for financial markets or cosmology. Moreover the phenomenon of turbulence is one that has been observed in many experiments but remains hardly understood from a theoretical point of view. Richard Feynman has called it ”the most important unsolved problem in classical physics”. It is even more astonishing, that we are aware of this gap in our understanding for well over hundred years and still a satisfying solution is not in sight.

Given the variety of different problems, which can be described in the realm of fluid dynamics it might be surprising that the behavior of fluids can be described by a one system of equations, the so called Navier-Stokes equations, named after Claude Louis Marie Henri Navier (1785-1835) and George Gabriel Stokes (1819-1903) even though they have been formulated in the 19th century by various different scientists. They are a (nonlinear) system of partial differential equations. The so called incompressible Navier-Stokes equations are given by

Here

• v(x, t) is the velocity of the fluid (represented by a continuum) at any point x ∈ R³ and time t,
• p(x, t) is the pressure at any point in time,
• ρ, ν ≥ 0 is are constants called the density and viscosity,
• f is an external force.

Usually we consider this set of equations combined with an initial condition u(·, 0) = u₀ with a function u₀ describing the flow at some time t = 0 (Cauchy problem). There is another compressible form of the equations, where the second equation becomes div ρv = 0. If we assume the density of the fluid to be constant, which is seen as a suitable model up to velocities in the range up to Mach 0.5 the compressible and incompressible equations of coincide. Of course there are applications, where this assumption is not feasible (just consider supersonic aviation), however of the theory of the Navier-Stokes equations is centered around this formulation and therefore this article will do so as well. The classical theory for the Navier-Stokes equations relies on the so called energy-inequality, that states, that the term ||u||_L² is decreasing in time. This of course requires the lack of external forces, i.e. f = 0, which we will assume going forward.

Mathematically, these equations are a combination of two different partial differential equations (PDEs), the Euler-equations for ideal fluids

which form a so called (non-linear) system of hyperbolic conservation laws (the conserved quantity here is the kinetic energy given by ||u||_L² ) and the heat-equation or diffusion-equation, which states

which is a linear parabolic PDE. Note that Euler’s equations are the difficult part of the problem, since it involves a nonlinear term. In fact the diffusion part presents an opener, as the Galerkin method that is used for constructing solutions originates from the theory of parabolic equations. This theory was originally developed by Jean Leray who published his work ”Sur le mouvement d’un liquide visqueux emplissant l’espace” in 1934. Conversely, the theory of Navier-Stokes can be employed for finding solutions to Euler’s equations by using solutions to the Navier-Stokes equations for decreasing values of ν. Then, it can be demonstrated, that under certain conditions the sequence of solutions to Navier-Stokes has a limit in a certain space. This is called the vanishing-viscosity limit and has been proven by Kato in his paper ”Remarks on zero viscosity limit for non-stationary Navier-Stokes flows with boundary” from 1984.

Weak solutions

We now want to construct space-periodic solutions to the Cauchy problem of the Navier-Stokes equations. Compared to the case of boundary conditions this has the advantage, that it does not become necessary to introduce the Stokes-operator explicitly. (The Stokes-operator is the linear operator we obtain by applying an orthogonal projector, that results from the Helmholtz-decomposition, towards the Laplacian. In case of periodic boundary conditions, the projector and the Laplacian can be commuted and therefore we can work with the Laplacian).

The principal construction does not change compared to the approach for homogeneous Dirichlet or Neumann boundary conditions, as the only difference it that we have to construct an orthonormal basis of the solenoidal L²-vectorfields, that consists of eigenfunctions to the Stokes operator. For the periodic case we can write down such eigenfunctions immediately by using sin and cos. Afterwards, the procedure becomes identical.

For a weak formulation one assumes the existence of a classical solution as a starting point and derives a weaker condition, that such a solution would also have to fulfill. Then we proceed to find a functions, that obeys this weaker condition. To do this we take the scalar product of the Navier-Stokes equation with a test function ϕ, that is only non-zero on a finite time interval [0, T], fulfills the required boundary conditions and has vanishing divergence. By using integration by parts (this is where the boundary condition is relevant) we can eliminate the pressure field, include the initial condition as well as reduce the Laplacian towards a term only requiring first-order derivatives. This yields a weak formulation is given by

for all test functions ϕ. Here, u₀ is the initial condition, : denotes the matrix scalar-product and ϕ is a test function
as required earlier. The natural space for weak solutions is given by L∞([0, T],L²) ∩L²([0, T],H¹). By a density argument we can demonstrate that it suffices to regard test functions given by the form . Here, d_k is a smooth function in time and w_k is an (L²-)orthonormal basis of the Laplacian-operator. (For boundary conditions this becomes the Stokes-operator). Now we insert an ansatz for our solution u as well as for the test-function ϕ. After some computations, one obtains a system of ordinary differential equations (ODEs) depending on the time variable given by

the so called the Galerkin equations. Here every term except for d_k is constant in time. To this system we can find a solution at least locally in time and with some additional computations we can demonstrate, that the solutions to the system, called Galerkin approximations, do not blow up and is even smooth on the entire real axis. The idea here is to establish some estimate for the L∞-norm of the u_N.

The next step is that we now want to extract a sub-sequence of the Galerkin approximations that converges in some weak sense. After we have established uniform boundaries for the Galerkin approximations in these normed spaces, we find a sub-sequence that converges with respect to the weak-* topology by the theorem of Banach-Alaoglu. The weak convergence now coincides with the convergence of the sequence of the integral expressions applied to u_N for any ϕ towards the integral expression applied to u for the terms that are linear in u, i.e. we have

The nonlinear term requires some additional attention. The key here is to observe, that (u · ∇)v is linear in both u and v. We want to find a strongly convergent sub-sequence, since in this allows to demonstrate that the integral expression applied to u_N converges to the expression applied to u. This is done via the so called Aubin-Lions-lemma, which is a compactness criterion in Bochner-spaces, i.e. spaces of the form L^p(I,E), where I is a finite interval and E a Banach-space. Afterwards we have established the convergence of all integral-expressions, which allows to conclude, that we have indeed constructed a weak solution to the Navier-Stokes equations.

The uniqueness of solutions to this weak formulation is unfortunately still unknown to this day. The argument for the uniqueness for the heat-equation uses the linearity of all integral expressions and afterwards applies the fundamental lemma of variational calculus. However, due to the nonlinear term and the smaller class of test functions used we cannot use these arguments here. This is not only of theoretical interest,but has some quite significant ramifications for numerical methods, namely finite element methods (FEMs), which use the weak formulation as a starting point for finding weak solutions numerically. This problem naturally leads us to the concept of strong solutions, which shall be discussed in the following section.

Strong solutions and the millennium problem

While the existence of weak solutions is very helpful, these are far from sufficient. The most pressing issue is the lack of uniqueness, but we also ideally want solutions that are smooth in time and space at least for smooth initial conditions. Therefore we introduce strong solutions, which are solutions in the (more regular) space L∞([0, T],H¹) ∩ L²([0, T],H²). These solutions have a lot of useful properties most importantly the uniqueness and stability (i.e, that the solution depends continuously on the initial data u₀).

To demonstrate the existence of strong solutions must establish the limit of the sequence of Galerkin approximations
to be in higher order Sobolev spaces. This is done by again using the theorem of Banach-Alaoglu but now we want the sequence u_N to be bounded in norms ||u||_H^m for m ≥ 2. The critical estimate is the following inequality

This is an ordinary differential inequality (ODI) in the variable ||∇u||² _L² for which the comparison principle is applicable. This allows us to use the solution of the corresponding ODE as an upper bound for ||∇u||² _L² as a function in time. The solution to this ODE is an increasing function, which allows to estimate the function over an interval by the value at its final time. Unfortunately, the solution to the ODE contains a singularity, which means, that we are only able to estimate the Galerkin equations on a certain finite time interval [0, T] by this method. Of course, this does not imply, that ||∇u_N||² _L² also has to blow up as it is possible (and due to the fact, that we just estimate the second order term as non-negative), that this estimate is quite bad and with a better estimate blow-up scenarios could be avoided. In the case that either ||∇u₀|| or ||u₀|| · ||∇u₀|| are smaller than a certain constant, it can be established that there are strong solutions globally in time.

This graphic illustrates potential blowup-scenarios for the Navier-Stokes equations. We start with two different initial conditions represented by the green and blue lines. Here the blue line represents the solution that blows up. After the blowup we can no longer guarantee the uniqueness of the solution and we get a another solution (violet line), that is also a weak solution with the same initial data. Similarly we get a third weak solution after the second blowup given by the black line. The dotted red line represents the constant for which we know, that if the initial data are smaller, we have strong solutions global in time. It represents a ”point of no return” for blowup-scenarios, since if the H¹-norm ever dips below that constant the solution will remain strong and can never blow up again.

For the transition from strong solutions to actually classically differentiable functions we have to improve the regularity of the solutions even further. It can be demonstrated, that for u₀ ∈ H^m it follows u ∈ L∞([0, T],H^m) ∩ L²([0, T],H^m+1) by similar arguments as the existence proof for strong solutions. By the Sobolev-Lemma we conclude that smooth initial conditions imply smooth in space solutions. For the time regularity, a similar result holds true, however we do not get differentiability in 0, but only differentiability on an interval [ϵ, T] for any ϵ > 0.

An outlook to my own research

The previous sections have been a short summary of the classical theory to Navier-Stokes equations, that i have elaborated in my masters thesis in detail. Now I am eager to get an alternative point of view towards fluid dynamic via the Boltzmann equation and its numerical solvers given by Lattice-Boltzmann-methods (LBMs) during my PhD-studies.

In the (macroscopic) Navier-Stokes model we treat a fluid as a continuum, where we can assign a value for velocity and pressure at any point in time and use macroscopic quantities such as density and viscosity for describing properties of the fluid. This is of course not an accurate description of the real world, since a fluid consists of a finite amount particles with nothing in between. However due to how small these particles are compared to the macroscopic fluid this approximation is reasonable and results in a fairly accurate description.

The theory for the Boltzmann-equation describes the fluid by a probability density function in the state space, a six-dimensional space that consists of three dimensional vectors of coordinates and momentum. Using the Liouville equation we can describe the time-evolution of probability density function. However, this does not take into account collisions between particles. This is done via the so called collision-integral. This term is nonlinear in the probability density function and not easy to grasp from a theoretical point of view. Many applications use a simplified expression for the collision term, that given the difference between the distribution f and a thermodynamic equilibrium, which is represented by a Maxwellian distribution, over a relaxation time τ , which in theory is a function of f but assumed constant for many applications. This approach is known as the Bhatnagar-Gross-Krook (BGK) collision operator and is a useful approximation for many application, but not necessarily for turbulent flows. One advantage of LBM schemes that is worth mentioning is that these schemes are highly adaptable to more complex geometric boundaries. I hope to contribute to the development and improvement of LBM methods and schemes for the Navier-Stokes over the next years as part of my PhD studies and hope to give updates on my
progresses.