Koopman analysis of the long-term evolution in a turbulent convection cell

We analyse the long-time evolution of the three-dimensional flow in a closed cubic turbulent Rayleigh-B\'{e}nard convection cell via a Koopman eigenfunction analysis. A data-driven basis derived from diffusion kernels known in machine learning is employed here to represent a regularized generator of the unitary Koopman group in the sense of a Galerkin approximation. The resulting Koopman eigenfunctions can be grouped into subsets in accordance with the discrete symmetries in a cubic box. In particular, a projection of the velocity field onto the first group of eigenfunctions reveals the four stable large-scale circulation (LSC) states in the convection cell. We recapture the preferential circulation rolls in diagonal corners and the short-term switching through roll states parallel to the side faces which have also been seen in other simulations and experiments. The diagonal macroscopic flow states can last as long as a thousand convective free-fall time units. In addition, we find that specific pairs of Koopman eigenfunctions in the secondary subset obey enhanced oscillatory fluctuations for particular stable diagonal states of the LSC. The corresponding velocity field structures, such as corner vortices and swirls in the midplane, are also discussed via spatiotemporal reconstructions.Comment: 32 pages, 9 figures, article in press at Journal of Fluid Mechanic

Statistics of the Energy Dissipation Rate and Local Enstrophy in Turbulent Channel Flow

Using high-resolution direct numerical simulations, the height and Reynolds number dependence of higher-order statistics of the energy dissipation rate and local enstrophy are examined in incompressible, fully-developed turbulent channel flow. The statistics are studied over a range of wall distances, spanning the viscous sublayer to the channel flow centerline, for friction Reynolds numbers $Re_\tau = 180$ and $Re_\tau = 381$. The high resolution of the simulations allows dissipation and enstrophy moments up to fourth order to be calculated. These moments show a dependence on wall distance, and Reynolds number effects are observed at the edge of the logarithmic layer. Conditional analyses based on locations of intense rotation are also carried out in order to determine the contribution of vortical structures to the dissipation and enstrophy moments. Our analysis shows that, for the simulation at the larger Reynolds number, small-scale fluctuations of both dissipation and enstrophy become relatively constant for $z^+ \gtrsim 100$.Comment: Accepted by Physica

Patterned turbulence and relaminarization in MHD pipe and duct flows

We present results of a numerical analysis of relaminarization processes in MHD duct and pipe flows. It is motivated by Julius Hartmann's classical experiments on flows of mercury in pipes and ducts under the influence of magnetic fields. The simulations, conducted both in periodic and non‐periodic settings, provide a first detailed view of flow structures that have not been experimentally accessible. The main novelty of the analysis is very long (tens to hundreds of hydraulic diameters) computational domains that allows to discover new flow regimes with localized turbulent spots near the side walls parallel to the magnetic field. The computed critical parameters for transition as well as the friction coefficients are in good agreement with Hartmann's data. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/109939/1/603_ftp.pd

Subcritical instability of liquid metal channel flow in the presence of a spanwise magnetic field

The linear and nonlinear evolution of perturbations is investigated in a magnetohydrodynamic channel flow with electrically insulating walls. The applied magnetic field is parallel to the walls and orthogonal to the stream. Linear optimal perturbations and their maximum amplifications over finite time intervals are computed using a scheme based on the direct and adjoint governing equations. It is shown that dominant optimal perturbations are no more the classical streamwise modes and how the flow is two-dimenzionalized for high enough Hartmann numbers. For fixed Reynolds and Hartmann numbers, direct numerical simulations are applied to investigate how the transition to turbulence is affected by the magnetic field. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/60899/1/4140005_ftp.pd

Direct numerical simulation of transition in MHD duct flow

Transition in the flow of electrically conducting fluid in a square duct with insulating walls is studied by direct numerical simulations. A uniform magnetic field is applied in the transverse direction. Moderate values of the Reynolds ( Re = 5000 ) and Hartmann ( Ha = 0 … 30 ) numbers are considered that correspond to the classical Hartmann & Lazarus [1] experiments. It is shown that the laminarization begins in the Hartmann layers, whereas the sidewall layers remain turbulent. Complete re‐laminarization occurs in the range of R = Re / Ha ≈︁ 220 , which is in agreement with the H. & L. experiments. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/89488/1/659_ftp.pd

Effect of wall conductivity on turbulent channel flow under spanwise magnetic field

The effect of wall conductivity on turbulence in electrically conducting fluid in the presence of a constant magnetic field is considered. A channel flow with a spanwise magnetic field is analyzed using high-resolution direct numerical simulations performed for the case of low magnetic Reynolds number. It is found that the effect of suppression of wall-normal momentum transfer and reduction of wall friction identified earlier for the flow with perfectly insulating walls is enhanced if the walls are electrically conducting. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/78302/1/515_ftp.pd

Wall-attached convection under strong inclined magnetic fields

We employ a linear stability analysis and direct numerical simulations to study the characteristics of wall-modes in thermal convection in a rectangular box under strong and inclined magnetic fields. The walls of the convection cell are electrically insulated. The stability analysis assumes periodicity in the spanwise direction perpendicular to the plane of the homogeneous magnetic field. Our study shows that for a fixed vertical magnetic field, the imposition of horizontal magnetic fields results in an increase of the critical Rayleigh number along with a decrease in the wavelength of the wall modes. The wall modes become tilted along the direction of the resulting magnetic fields and therefore extend further into the bulk as the horizontal magnetic field is increased. Once the modes localized on the opposite walls interact, the critical Rayleigh number decreases again and eventually drops below the value for onset with a purely vertical field. We find that for sufficiently strong horizontal magnetic fields, the steady wall modes occupy the entire bulk and therefore convection is no longer restricted to the sidewalls. The above results are confirmed by direct numerical simulations of the nonlinear evolution of magnetoconvection.Comment: 25 pages, 18 figure

Effects of strong fringing magnetic fields on turbulent thermal convection

We study the influence of fringing magnetic fields on turbulent thermal convection in a horizontally extended rectangular domain. The magnetic field is created in the gap between two semi-infinite planar magnetic poles, with the convection layer located near the edge of the gap. We employ direct numerical simulations in this setup for fixed Rayleigh and small Prandtl numbers, but vary the fringe-width by controlling the gap between the magnetic poles and the convection cell. The magnetic field generated by the magnets is strong enough to cease the flow in high magnetic flux region of the convection cell. We observe that as the local vertical magnetic field strength increases, the large scale structures become thinner and align themselves perpendicular to the longitudinal sidewalls. We determine the local Nusselt and Reynolds numbers as functions of the local Hartmann number (based on the vertical component of the magnetic field) and estimate the global heat and momentum transport. We show that the global heat transport decreases with increasing fringe-width for strong magnetic fields but decreases with increasing fringe-width for weak magnetic fields. In the regions of large vertical magnetic fields, the convective motion becomes confined to the vicinity of the sidewalls. The amplitudes of these wall modes show a non-monotonic dependence on the fringe-width.Comment: 19 pages, 11 figure