By Margaret Beck and C. Eugene Wayne
The time scales over which fluids evolve have a critical effect on the physical systems in which they occur. These time scales arise from an interplay of different effects, some of which—like Lagrangian coherent structures—tend to stabilize the flow, while others, such as shear (inviscid) damping or viscous damping, tend to break down structures, at least on small-length scales. A variety of recent investigations, many of which involve a dynamical systems point of view, have begun to shed light on the origin of these time scales in the case of two-dimensional fluid flows. Since the basic questions of existence and uniqueness for two-dimensional fluid flows are well understood, one can ask more detailed queries about their evolution. Moreover, these flows have a tendency to form large, vortical structures on both laboratory and geophysical scales, as seen in Figure 1, a satellite photo of the Gulf Stream. Dynamical systems theory is well suited to answer these types of questions since invariant families of solutions often appear to organize the dynamics, effectively creating the multiple time scales and observed asymptotic behavior [3, 5]. In simple settings, invariant manifolds [4, 9] can even characterize this organization.
A particularly important example of this is the two-dimensional incompressible Navier-Stokes equation with small viscosity, \(0 \ll \nu < 1 \):
\[ \partial_t \omega + {\bf u}\cdot \nabla \omega = \nu \Delta \omega, \quad \omega = \omega(x,t), \quad x \in \Omega \subseteq {\bf R}^2.\]
Figure 1. Satellite photo of the Gulf Stream. Image courtesy of NASA.
Here, \(\omega\) is the vorticity of the fluid and \(\bf{u}\) is the fluid velocity, recoverable from the vorticity via the Biot-Savart law. In other words, \(\omega = (\nabla \times \mathbf{u}) \cdot (0,0,1)\). When \(\nu=0\), the equation reduces to the Euler equation, which has infinitely many stationary solutions. Though these no longer remain stationary states for positive (but small) vorticity, it is reasonable to believe that they still play an important role in the longtime evolution of the Navier-Stokes equation. However, most stationary states of the Euler equation are surprisingly never observed in the Navier-Stokes evolution. Instead, a small number of the Euler states become quasi-stationary states of Navier-Stokes, and only a subset of these seem to have long-term influence. As a first “guess” at the time scales over which the viscosity makes itself felt, one can note that the two-dimensional Navier-Stokes equations on \(\mathbb{R}^2\) have a family of exact solutions known as the Oseen vortices, given by
\[\omega^O(x,t) = \frac{A}{1+t} e^{-\frac{|x|^2}{4 \nu (1+t)}}.\]
From this formula, it seems as if the viscosity should be perceptible on a time scale
\[t_{visc} \sim \frac{1}{\nu}.\]
However, numerical experiments indicate that vortices and other large-scale characteristic structures emerge in the flow on a much shorter time scale. For instance, in the numerical simulation of Figure 2 [10], the viscous time scale would be \(t_{visc} \sim 1500\), but large-scale vortical structures emerge on a much shorter time scale. Understanding the origin of these scales is currently a question of great interest.
Figure 2. A numerical simulation of a two-dimensional flow at six different times. Image credit: [10], by permission of John Wiley & Sons, Inc.
There is presently no mathematical theory predicting which of the Euler solutions will play the most important role in the viscous evolution. However, a finite subset of these quasi-stationary states correspond to an explicit family that decays on the viscous time scale \(\mathcal{O}(e^{-\nu t})\) and can be described by the lowest four Fourier modes, \(\{e^{\pm ix}, e^{\pm iy}\}\). Bar states (also known as Komogorov flow, a type of shear flow) are solutions that vary only in the \(x\) or only in the \(y\) direction, while dipoles vary in both directions. Researchers have observed, both experimentally and numerically, that most initial conditions lead to solutions which originally experience rapid evolution to either a bar state or a dipole, followed by slow decay to the background rest state (zero solution). A classical approach to analyzing such behavior begins by linearizing the Navier-Stokes equation about a bar state or dipole and attempting to determine the rate of convergence to the state, which should correspond to the observed initial period of rapid evolution. This type of linearization near a bar state [5] suggests that the rapid evolution occurs on the time scale \(\mathcal{O}(e^{-\sqrt{\nu}t})\), at least at the linear level. Interestingly, the linearization leads to a highly non-self-adjoint operator, making it unclear whether the multiple time scale phenomenon is spectral or pseudospectral. A dynamical systems perspective is useful in this analysis because it permits a separation between the decay rate to the invariant family, at \(\mathcal{O}(e^{-\sqrt{\nu}t})\), and the decay rate within the family, at \(\mathcal{O}(e^{-\nu t})\).
A related work also analyzes the rapid convergence to bars and dipoles using a dynamical systems perspective [3]. Researchers take the two-dimensional Navier-Stokes equation, written in Fourier space, and formally project that system onto the lowest eight modes: the lower four contain the bars and dipoles and the next four model the effects of all higher modes. They then use classical dynamical systems techniques, including invariant manifolds and estimates involving Duhamel’s formula, to study the resulting eight-dimensional ordinary differential equation (ODE). This method focuses on understanding the effects of perturbing the domain from a square torus, represented by a parameter \(\delta = 1\), to a rectangular torus, represented by \(\delta \neq 1\). The parameter \(\delta\) controls whether a particular invariant manifold is a center \((\delta = 1)\), stable \((\delta <1)\), or unstable \((\delta>1)\) manifold, which then determines if the dominant quasi-stationary state was a dipole, \(y\)-bar state, or \(x\)-bar state, respectively. In this ODE model, the initial period of rapid decay notably occurred on the time scale \(\mathcal{O}(e^{-t/\nu})\) instead of time scale \(\mathcal{O}(e^{-\sqrt{\nu}t})\), which researchers observed in the previously-mentioned work.
The dynamical systems perspective sheds light not only on the question of multiple time scales in fluids, but on other aspects of their motion as well. For example, one can analyze the stability and interaction of vortices in the planar Navier-Stokes equation with limit \(\nu \to 0\) using a point vortex model [11]. A key aspect of that work is its ability to capture the higher-order effects of vortex interaction, showing that for motions in which the centers of vorticity were initially well-separated, the essentially inviscid motion of the vortex cores accurately described the overall nature of the flow until the distance between vortices became comparable to the size of the vortex core. Interestingly, such configurations of near point vortices appear in a host of experimental circumstances (see Figure 3) [1].
Figure 3. Experimental illustrations of point vortices. Image credit: [1], by permission of the Royal Society.
In work more closely related to the above discussion about bar state metastability [6], researchers studied solutions of the two-dimensional Navier-Stokes equations in a neighborhood of Couette flow, a particular type of shear flow in a channel. Using careful partial differential equation (PDE) estimates in Gevrey spaces, they were able to treat the full nonlinear problem. This work is particularly interesting because it identifies precisely different time scales associated with an initial period of so-called inviscid damping—in which the Euler equations essentially govern flow—followed by a rapid evolution due to enhanced diffusion and then a final, slow period of convergence to the Couette flow, during which viscosity dominates. The intermediate period of enhanced diffusion relates to the hypocoercivity in [5, 8], and is further connected to the phenomenon of Taylor dispersion, which also occurs in the channel setting but for different boundary conditions. Originally studied in the 1950s, Taylor dispersion is another example of a situation in which shearing in the ambient flow field enhances dispersive or dissipative effects.
Researchers have recently attacked this problem from two different perspectives using dynamical systems ideas. In [7], hypocoercivity methods are used to analyze the decay enhancement in a variety of shearing flows. In [2], more classical dynamical systems methods like invariant manifolds play a key role, but not in an entirely straightforward way. Although the PDE does not seem to possess an invariant manifold—in fact, evidence suggests that it does not—its solutions are shown to be well-approximated by solutions to an ODE that does possess a center manifold, on which the enhanced diffusion can be computed explicitly. This matches Taylor’s original formal calculations from the 1950s.
References
[1] Aref, H. (2008). Something old, something new. Phil. Trans. R. Soc. A, 366, 2649-2670.
[2] Beck, M., Chaudhary, O., & Wayne, C.E. (2015). Analysis of enhanced diffusion in Taylor dispersion via a model problem. In Hamiltonian partial differential equations and applications (pp. 31-71). Vol. 75 of Fields Institute Communications. New York, NY: Springer, Inc.
[3] Beck, M., Cooper, E., & Spiliopoulos, K. (2017). Selection of quasi-stationary states in the Navier-Stokes equation on the torus. Preprint, arXiv:1701.04850.
[4] Beck, M., & Wayne, C.E. (2011). Using global invariant manifolds to understand metastability in the Burgers equation with small viscosity. SIAM Review, 53(1), 129-153.
[5] Beck, M., & Wayne, C.E. (2013). Metastability and rapid convergence to quasi-stationary bar states for the two-dimensional Navier-Stokes equations. Proc. Roy. Soc. Edinburgh Sect. A, 143(5), 905-927.
[6] Bedrossian, J., Masmoudi, N., & Vicol, V. (2016). Enhanced dissipation and inviscid damping in the inviscid limit of the Navier-Stokes equations near the two dimensional Couette flow. Arch. Ration. Mech. Anal., 219(3), 1087-1159.
[7] Bedrossian J., & Zelati, M.C. (2015). Enhanced dissipation, hypoellipticity, and anomalous small noise inviscid limits in shear flows. Preprint, arXiv:1510.08098.
[8] Gallagher, I., Gallay, T., & Nier, F (2009). Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator. Int. Math. Res. Not. IMRN, 12, 2147-2199.
[9] Haller, G. (2015). Lagrangian coherent structures. Annual Review of Fluid Mechanics, 47, 137-162.
[10] Robinson, M., & Monaghan, J.J. (2012). Direct numerical simulation of decaying two-dimensional turbulence in a no-slip square box using smoothed particle hydrodynamics. Int. J. Numer. Meth. Fluids, 70, 37-55.
[11] Thierry, G. (2012). Interacting vortex pairs in inviscid and viscous planar flows. In Mathematical aspects of fluid mechanics (pp. 173-200). Vol. 402 of London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press.
Margaret Beck is an associate professor of mathematics at Boston University. C. Eugene Wayne is a professor of mathematics at Boston University.