Temporal evolution of an initially Gaussian velocity and vorticity fields under the Navier-Stokes dynamics. Zoom Image
Temporal evolution of an initially Gaussian velocity and vorticity fields under the Navier-Stokes dynamics.

Turbulence is (almost) everywhere!

Many flows in nature and technology are turbulent, i.e. they display a complex and irregular behavior in space and time. The Earth's atmosphere and oceans may serve as two examples in which turbulence plays a key role. On an astrophysical scale, the Sun's convective core as well as the ejected solar wind both are turbulent. Turbulence is also present in our everyday life; when stirring milk in coffee, we benefit from its properties, whereas we may suffer its consequences during a turbulent flight.

One of the most fascinating aspects of turbulence is that it is a tangible phenomenon, yet its mathematical description turns out to be incredibly difficult, withstanding most concepts that modern physics has developed so far. The combination of its broad relevance and the scientific challenge it poses makes it one of "the most important unsolved problem of classical physics".

In our group, we explore turbulent flows with the help of computer simulations and develop new concepts to tackle this long standing problem. We furthermore use these methods to address open questions of neighboring fields. The following selection gives an overview over recent and ongoing work.

Fundamental aspects of turbulent flows

Turbulent flows show the peculiar property of statistically broken scale invariance. For example, probing the velocity fields at large scales, the statistics is almost Gaussian. Focusing on small scales, however, extreme velocity fluctuations are orders of magnitude more probable than in a Gaussian field. This phenomenon, known as intermittency, is the statistical reverberation of the nonlinear and nonlocal nature of the Navier-Stokes equation. The combination of nonlinear and nonlocal interactions lead to the emergence of small-scale coherent structures such as vortex tubes or strain sheets (see visualization at the top of this page). In our group we study these phenomena by means of direct numerical simulation of the Navier-Stokes equations and use the results as input for the formulation of analytical statistical theories of turbulence.

  • M. Wilczek, C. Meneveau, Pressure Hessian and viscous contributions to the velocity gradient statistics based on Gaussian random fields, J. Fluid Mech. 756, 191 (2014) [J. Fluid Mech.]
  • M. Wilczek, Y. Narita, Wavenumber-frequency spectrum for turbulence from a random sweeping hypothesis with mean flow, Phys. Rev. E  86, 066308 (2012) [Phys. Rev. E]
  • R. Friedrich, A. Daitche, O. Kamps, J. Lülff, M. Voßkuhle , M. Wilczek, The Lundgren-Monin-Novikov Hierarchy: Kinetic Equations for Turbulence, Com. Ren. Phy. 13, 929 (2012) [Com. Ren. Phy.]
Volume rendering of the temperature field in Rayleigh-Benard convection. Hot fluid (red) rises up and cold fluid (blue) falls down. Image adopted from J. Fluid. Mech. 781, 276-297 (2015). Zoom Image
Volume rendering of the temperature field in Rayleigh-Benard convection. Hot fluid (red) rises up and cold fluid (blue) falls down. Image adopted from J. Fluid. Mech. 781, 276-297 (2015). [less]

Convection

In many natural settings, such as the Earth's core or atmosphere, fluid flow is driven by a heat source. In the prototypical situation in which the flow is heated from below and cooled from above, convection can arise. Sufficiently far from its onset, the convective flow becomes turbulent. From the viewpoint of fundamental turbulence research turbulent convection is one of the most interesting systems to study due to the complex interaction of large-scale flow patterns and turbulent fluctuations. In our group, we aim at disentangling this dynamics as well as at establishing rigorous statistical formulations of turbulent convection.

  • K. Petschel, S. Stellmach, M. Wilczek, J. Lülff, U. Hansen, Dissipation Layers in Rayleigh-Bénard convection: A unifying view, Phys. Rev. Lett 110, 114502 (2013) [Phys. Rev. Lett]
  • K. Petschel, S. Stellmach, M. Wilczek, J. Lülff, U. Hansen, Kinetic energy transport in Rayleigh-Bénard convection, J. Fluid Mech. 773: 395-417 (2015) [J. Fluid Mech.]
  • J. Lülff, M. Wilczek, R. Stevens, R. Friedrich, D. Lohse, Turbulent Rayleigh-Bénard convection described by projected dynamics in phase space, J. Fluid. Mech. 781, 276-297 (2015) [J. Fluid Mech.]
<span>Simulated wind-farm turbulence: volume rendering of low-velocity wake regions. Visualization courtesy of David Bock (NCSA, XSEDE, Extended Collaborative Support Services).<br /></span> Zoom Image
Simulated wind-farm turbulence: volume rendering of low-velocity wake regions. Visualization courtesy of David Bock (NCSA, XSEDE, Extended Collaborative Support Services).
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Atmospheric boundary layers

The understanding of atmospheric boundary layers plays a crucial role for climate research but also for technological applications such as wind energy conversion. It is, for example, necessary to quantify turbulent fluctuations to be able to predict fluctuations in power output of wind farms.
A variety of physical effects are important to understand atmospherical flows. In the simplest of all cases, a neutral atmospheric boundary layer can essentially be considered as a wall-bounded flow over a rough surface. In a collaboration with researchers at University of Twente and the Johns Hopkins University, we recently made some progress in quantifying the space-time structure of such flows by combining ideas from turbulence theory with computer simulations. As part of the WINDINSPIRE collaboration, we are extending these works to atmospheric boundary layers interacting with wind farms.

  • M. Wilczek, R. Stevens, C . Meneveau, Spatio-temporal spectra in the logarithmic layer of wall-turbulence: large-eddy simulations and simple models, J. Fluid Mech. 769, R1 (2015) [J. Fluid Mech.]
  • M. Wilczek, R. Stevens, C . Meneveau, Height-dependence of spatio-temporal spectra of wall-bounded turbulence - LES results and model predictions, J. Turb 16(10), 937-949 (2015) [J. Turb]
  • M. Wilczek, R. Stevens, Y. Narita, Charles Meneveau, A wavenumber-frequency spectral model for atmospheric boundary layers, J. Phys.: Conf. Ser. 524, 012104 (2014) [IOP]
Vorticity field from a simulation of active matter turbulence. In the initial stages, vortex rotors form from a spontaneous symmetry breaking (upper panel). They grow in number over the course of time. Finally, a regular vortex pattern forms (lower panel) and defects die out. Zoom Image
Vorticity field from a simulation of active matter turbulence. In the initial stages, vortex rotors form from a spontaneous symmetry breaking (upper panel). They grow in number over the course of time. Finally, a regular vortex pattern forms (lower panel) and defects die out. [less]

Active matter turbulence

Biophysical flows are perhaps among the most feature-rich in nature. Among the most fascinating systems are dense bacterial suspensions of self-propelling bacteria. The nematic and hydrodynamic interactions of its constituents give rise to the emergence of meso-scale vortex patterns reminiscent of two-dimensional turbulence.
From a theoretical point of view, continuum descriptions of these systems combine aspects from pattern formation with nonlinear and nonlocal advection of Navier-Stokes typ. The active stresses thereby to first order induce a linear instability on a mesoscopic scale. 
Far above linear onset, the dynamics becomes increasingly complex and eventually turbulent. The picture demonstrates one example for the wide range of dynamical states these systems can obtain: after an extensive period of quasi-turbulent behavior a quasi-stationary vortex lattice emerges. Linear and weakly nonlinear analysis is clearly insufficient to explain these fully nonlinear patterns. Active matter turbulence is the ideal testbed for novel methods that combine tools from nonlinear dynamics and pattern formation with the statistical mechanics of turbulence, which we develop in our group.

High-performance computing & scientific visualization

Besides experiments, numerical simulations are an indispensable tool to investigate turbulence. Such simulations are computationally very demanding as typically billions of degrees of freedom need to be tracked in time to represent a turbulent flow accurately.
In our group we develop an open-source parallel solver for the Navier-Stokes equations and variants thereof. The code furthermore allows to track millions of tracer particles to study transport properties of turbulence. In an ongoing collaboration with the Bodenschatz group we are benchmarking simulation results against the ones obtained from optical tracking experiments allowing for direct comparisons at an unprecedented level.
Please visit github for more information on our simulation code.

 
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