arXiv subjects

Rainer Grauer

Publications and source records attributed to Rainer Grauer.

47 records · Page 3Linked to original sources

arXiv2012

Longitudinal and Transverse structure functions in high Reynolds-number turbulence

Using exact relations between velocity structure functions (Hill, Hill and Boratav, and Yakhot) and neglecting pressure contributions in a first approximation, we obtain a closed system and derive simple order-dependent rescaling relationships between longitudinal and transverse structure functions. By means of numerical data with turbulent Reynolds numbers ranging from $\Re_λ=320$ to $\Re_λ=730$, we establish a clear correspondence between their respective scaling range, while confirming that their scaling exponents do differ. This difference does not seem to depend on Reynolds number. Making use of the Mellin transform, we further map longitudinal to (rescaled) transverse probability density functions.

physics.flu-dyn↗

arXiv2012

Turbulence properties and global regularity of a modified Navier-Stokes equation

We introduce a modification of the Navier-Stokes equation that has the remarkable property of possessing an infinite number of conserved quantities in the inviscid limit. This new equation is studied numerically and turbulence properties are analyzed concerning energy spectra and scaling of structure functions. The dissipative structures arising in this new equation are curled vortex sheets contrary to vortex tubes arising in Navier-Stokes turbulence. The numerically calculated scaling of structure functions is compared with a phenomenological model based on the She-Lévêque approach. Finally, for this equation we demonstrate global well-posedness for sufficiently smooth initial conditions in the periodic case and in $\mathbb R^3$. The key feature is the availability of an additional estimate which shows that the $L^4$-norm of the velocity field remains finite.

physics.flu-dyn↗

arXiv2012

A novel code for numerical 3-D MHD studies of CME expansion

A recent third-order, essentially non-oscillatory central scheme to advance the equations of single-fluid magnetohydrodynamics (MHD) in time has been implemented into a new numerical code. This code operates on a 3-D Cartesian, non-staggered grid, and is able to handle shock-like gradients without producing spurious oscillations. To demonstrate the suitability of our code for the simulation of coronal mass ejections (CMEs) and similar heliospheric transients, we present selected results from test cases and perform studies of the solar wind expansion during phases of minimum solar activity. We can demonstrate convergence of the system into a stable Parker-like steady state for both hydrodynamic and MHD winds. The model is subsequently applied to expansion studies of CME-like plasma bubbles, and their evolution is monitored until a stationary state similar to the initial one is achieved. In spite of the model's (current) simplicity, we can confirm the CME's nearly self-similar evolution close to the Sun, thus highlighting the importance of detailed modelling especially at small heliospheric radii. Additionally, alternative methods to implement boundary conditions at the coronal base, as well as strategies to ensure a solenoidal magnetic field, are discussed and evaluated.

astro-ph.SR↗

arXiv2011

Numerical Simulation of Current Sheet Formation in a Quasi-Separatrix Layer using Adaptive Mesh Refinement

The formation of a thin current sheet in a magnetic quasi-separatrix layer (QSL) is investigated by means of numerical simulation using a simplified ideal, low-$β$, MHD model. The initial configuration and driving boundary conditions are relevant to phenomena observed in the solar corona and were studied earlier by Aulanier et al., A&A 444, 961 (2005). In extension to that work, we use the technique of adaptive mesh refinement (AMR) to significantly enhance the local spatial resolution of the current sheet during its formation, which enables us to follow the evolution into a later stage. Our simulations are in good agreement with the results of Aulanier et al. up to the calculated time in that work. In a later phase, we observe a basically unarrested collapse of the sheet to length scales that are more than one order of magnitude smaller than those reported earlier. The current density attains correspondingly larger maximum values within the sheet. During this thinning process, which is finally limited by lack of resolution even in the AMR studies, the current sheet moves upward, following a global expansion of the magnetic structure during the quasi-static evolution. The sheet is locally one-dimensional and the plasma flow in its vicinity, when transformed into a co-moving frame, qualitatively resembles a stagnation point flow. In conclusion, our simulations support the idea that extremely high current densities are generated in the vicinities of QSLs as a response to external perturbations, with no sign of saturation.

physics.plasm-ph↗

arXiv2010

Conditional Eulerian and Lagrangian velocity increment statistics of fully developed turbulent flow

Conditional statistics of homogeneous isotropic turbulent flow is investigated by means of high-Reynolds number direct numerical simulations performed with $2048^3$ collocation points. Eulerian as well as Lagrangian velocity increment statistics under several conditions are analyzed and compared. In agreement with experimental data longitudinal probability density functions $P(δ^{||}_l u|ε_l)$ conditioned on a scale-averaged energy dissipation rate are close to Gaussian distributions over all scales within the inertial range of scales. Also transverse increments conditioned on either the dissipation rate or the square of the vorticity have quasi-Gaussian probability distribution functions (PDFs). Concerning Lagrangian statistics we found that conditioning on a trajectory averaged energy-dissipation rate $ε_τ$ significantly reduces the scale dependence of the increment PDFs $P(δ_τu_i|ε_τ)$. By means of dimensional arguments we propose a novel condition for Lagrangian increments which is shown to reduce even more the flatness of the corresponding PDFs and thus intermittency in the inertial range of scales. The conditioned Lagrangian PDF corresponding to the smallest increment considered is reasonably well described by the K41-prediction of the PDF of acceleration. Conditioned structure functions show approximately K41-scaling with a larger scaling range than the unconditioned ones.

physics.flu-dyn↗

arXiv2007

A semi-implicit Hall-MHD solver using whistler wave preconditioning

The dispersive character of the Hall-MHD solutions, in particular the whistler waves, is a strong restriction to numerical treatments of this system. Numerical stability demands a time step dependence of the form $Δt\propto (Δx)^2$ for explicit calculations. A new semi--implicit scheme for integrating the induction equation is proposed and applied to a reconnection problem. It it based on a fix point iteration with a physically motivated preconditioning. Due to its convergence properties, short wavelengths converge faster than long ones, thus it can be used as a smoother in a nonlinear multigrid method.

physics.comp-ph↗

arXiv2007

Numerical simulations of possible finite time singularities in the incompressible Euler equations: comparison of numerical methods

The numerical simulation of the 3D incompressible Euler equation is analyzed with respect to different integration methods. The numerical schemes we considered include spectral methods with different strategies for dealiasing and two variants of finite difference methods. Based on this comparison, a Kida-Pelz like initial condition is integrated using adaptive mesh refinement and estimates on the necessary numerical resolution are given. This estimate is based on analyzing the scaling behavior similar to the procedure in critical phenomena and present simulations are put into perspective.

physics.flu-dyn↗

arXiv2007

Density-PDFs and Lagrangian Statistics of highly compressible Turbulence

We report on probability-density-functions (PDF) of the mass density in numerical simulations of highly compressible hydrodynamic flows and the corresponding structure formation of Lagrangian particles advected by the flows. Numerical simulations were performed with $512^3$ collocation points and 2 million tracer particles integrated over several dynamical times. We propose a connection between the PDF of the Lagrangian tracer particles and the predicted log-normal distribution of the density fluctuations in isothermal systems.

physics.flu-dyn↗

arXiv2007

A Corrsin type approximation for Lagrangian fluid Turbulence

In Lagrangian turbulence one is faced with the puzzle that 2D Navier-Stokes flows are nearly as intermittent as in three dimensions although no intermittency is present in the inverse cascade in 2D Eulerian turbulence. In addition, an inertial range is very difficult to detect and it is questionable whether it exists at all. Here, we investigate the transition of Eulerian to Lagrangian probability density functions (PDFs) which leads to a new type of Lagrangian structure function. This possesses an extended inertial range similar to the case of tracer particles in a frozen turbulent velocity field. This allows a connection to the scaling of Eulerian transversal structure functions.

physics.flu-dyn↗

arXiv2007

Impact of the floating-point precision and interpolation scheme on the results of DNS of turbulence by pseudo-spectral codes

In this paper we investigate the impact of the floating-point precision and interpolation scheme on the results of direct numerical simulations (DNS) of turbulence by pseudo-spectral codes. Three different types of floating-point precision configurations show no differences in the statistical results. This implies that single precision computations allow for increased Reynolds numbers due to the reduced amount of memory needed. The interpolation scheme for obtaining velocity values at particle positions has a noticeable impact on the Lagrangian acceleration statistics. A tri-cubic scheme results in a slightly broader acceleration probability density function than a tri-linear scheme. Furthermore the scaling behavior obtained by the cubic interpolation scheme exhibits a tendency towards a slightly increased degree of intermittency compared to the linear one.

physics.comp-ph↗

arXiv1996

Adaptive Mesh Refinement for Singular Current Sheets in Incompressible Magnetohydrodynamic Flows

The formation of current sheets in ideal incompressible magnetohydrodynamic flows in two dimensions is studied numerically using the technique of adaptive mesh refinement. The growth of current density is in agreement with simple scaling assumptions. As expected, adaptive mesh refinement shows to be very efficient for studying singular structures compared to non-adaptive treatments.

comp-gas↗