Nonlinear Turbulence Research Articles

Reduced-order models for closed-loop wake control

We review a strategy for low- and least-order Galerkin models suitable for the design of closed-loop stabilization of wakes. These low-order models are based on a fixed set of dominant coherent structures and tend to be incurably fragile owing to two challenges. Firstly, they miss the important stabilizing effects of interactions with the base flow and stochastic fluctuations. Secondly, their range of validity is restricted by ignoring mode deformations during natural and actuated transients. We address the first challenge by including shift mode(s) and nonlinear turbulence models. The resulting robust least-order model lives on an inertial manifold, which links slow variations in the base flow and coherent and stochastic fluctuation amplitudes. The second challenge, the deformation of coherent structures, is addressed by parameter-dependent modes, allowing smooth transitions between operating conditions. Now, the Galerkin model lives on a refined manifold incorporating mode deformations. Control design is a simple corollary of the distilled model structure. We illustrate the modelling path for actuated wake flows.

Numerical Studies on the Effects of Stagnation Pressure and Temperature on Supersonic Flow Characteristics in Cold Spray Applications

Low-temperature particle coating requires supersonic flow. The characteristics of this supersonic flow are investigated using a nonlinear turbulence model. The low-temperature, supersonic particle deposition technique is valuable because its rapid and dense coating minimizes thermal damage to both particles and substrate. It has excellent potential for industrial production of low-cost thin films. Stagnation pressures and temperatures of the supersonic nozzle range from 4 < P o < 45 bar and 300 < T o < 1500 K, respectively. The exit Mach number, M e, and velocity, V e, range from 0.6 to 3.5 and 200 to 1400 m/s, respectively. The effects of stagnation pressure (P o) and stagnation temperature (T o) on supersonic flow impinging upon a substrate are described. In other words, the energy loss through shockwaves and shear interactions between the streaming jet and surrounding gas are quantified as functions of P o and T o. P o is decreased because of friction (loss ranges from 40 to 60%) while T o is nearly conserved. To realize the nozzle exit condition of P e = P amb, we demonstrate that P o should be adjusted rather than T o, as T o has little effect on exit pressures. On the other hand, T o is more influential than P o for varying the exit velocity. Various nozzle geometries yielding different flow characteristics, and hence, different operating conditions and coating performances are investigated. The corresponding supersonic flows for three types of nozzles (under-, correctly , and over-expanded) are simulated, and their correctly expanded (or shock-free) operating conditions are identified. Diamond shock structures induced by the pressure imbalance between the exiting gas and the surrounding atmosphere are captured.

Application of Nonlinear Turbulence Models for Marine Propulsors

The realistic simulation of cavitation on a marine propeller is important for the efficient design of the propeller. However, the flow characteristic that occurred on the marine propeller is complicated and difficult to predict due to the combined effects of turbulence, cavitation, and multiphase phenomena. There is still currently no turbulence model that can predict these combined effects satisfactory. The nonlinear turbulence model is therefore modified and applied to predict the cavitation on a marine propeller for the first time in this work. It is found that the nonlinear turbulence model can predict the cavitation and hence the thrust and torque coefficients much more accurately than the existing Reynolds-averaged Navier–Stokes turbulence models including the Reynolds-stress model.

Response of the geomagnetic activity to solar wind turbulence during solar cycle 23

The solar wind–magnetosphere coupled system is characterized by dynamical processes. Recent works have shown that nonlinear couplings and turbulence might play a key role in the study of solar wind–magnetosphere interaction processes. Within this framework, this study presents a statistical analysis aimed to investigate the relationship between solar wind MHD turbulence and geomagnetic activity at high and low latitudes as measured by the AE and SYM-H indices, respectively. This analysis has been performed for different phases of solar cycle 23. The state of turbulence was characterized by means of 2-D histograms of the normalized cross-helicity and the normalized residual energy. The geomagnetic response was then studied in relation to those histograms. The results found clearly show that, from a statistical point of view, solar cycle 23 is somewhat peculiar. Indeed, good Alfvénic correlations are found unexpectedly even during solar activity maximum. This fact has implications on the geomagnetic response as well since a statistical relationship is found between Alfvénic fluctuations and auroral activity. Conversely, solar wind turbulence does not seem to play a relevant role in the geomagnetic response at low latitudes.

Quadratic and cubic eddy-viscosity models in turbulent supercavitating flow computation

Quadratic and cubic non-linear eddy-viscosity turbulence models (NLEVM) with low Reynolds number (Re) correction were presented to provide better description of anisotropic turbulence stresses in the numerical prediction of supercavitating flows, which are accompanied with large density ratio and large-scaled swirling flow structures. The applications of the NLEVM were carried out through a self-developed cavitation codes, coupled with a cavitation model based on the transport equation of liquid phase. These NLEVM were verified capable of capturing more accurate macroscopic shape and hydrodynamic property of supercavity by the benchmark problems of supercavities over simple objects. Finally, the cubic NLEVM was further applied to the numerical prediction of supercavitating flow around a complex submerged vehicle. The corresponding cavitation behaviors were explored in detail to provide beneficial experience for further research.

Nonlinear turbulence theory and simulation of Buneman instability

In the present paper, the weak turbulence theory for reactive instabilities, formulated in a companion paper [P. H. Yoon, Phys. Plasmas 17, 112316 (2010)], is applied to the strong electron-ion two-stream (or Buneman) instability. The self-consistent theory involves quasilinear velocity space diffusion equation for the particles and nonlinear wave kinetic equation that includes quasilinear (or induced emission) term as well as nonlinear wave-particle interaction term (or a term that represents an induced scattering off ions). We have also performed one-dimensional electrostatic Vlasov simulation in order to benchmark the theoretical analysis. Under the assumption of self-similar drifting Gaussian distribution function for the electrons it is shown that the current reduction and the accompanying electron heating as well as electric field turbulence generation can be discussed in a self-consistent manner. Upon comparison with the Vlasov simulation result it is found that quasilinear wave kinetic equation alone is insufficient to account for the final saturation amplitude. Upon including the nonlinear scattering term in the wave kinetic equation, however, we find that a qualitative agreement with the simulation is recovered. From this, we conclude that the combined quasilinear particle diffusion plus induced emission and scattering (off ions) processes adequately account for the nonlinear development of the Buneman instability.

Implementation of a Nonlinear Subfilter Turbulence Stress Model for Large-Eddy Simulation in the Advanced Research WRF Model

Abstract Two formulations of a nonlinear turbulence subfilter-scale (SFS) stress model were implemented into the Advanced Research Weather Research and Forecasting model (ARW-WRF) version 3.0 for improved large-eddy simulation performance. The new models were evaluated against the WRF model’s standard Smagorinsky and 1.5-order turbulence kinetic energy (TKE) linear eddy-viscosity SFS stress models in simulations of geostrophically forced, neutral boundary layer flow over both flat terrain and a shallow, symmetric transverse ridge. Comparisons of simulation results with similarity profiles indicate that the nonlinear models significantly improve agreement with the expected profiles near the surface, reducing the overprediction of near-surface stress characteristic of linear eddy-viscosity models with no near-wall damping. Comparisons of simulations conducted using different mesh sizes indicate that the nonlinear model simulations at coarser resolutions agree more closely with the higher-resolution results than corresponding lower-resolution simulations using the standard WRF SFS stress models. The nonlinear models produced flows featuring a broader range of eddy sizes, with less spectral power at lower frequencies and more spectral power at higher frequencies. In simulated flow over the transverse ridge, distributions of flow separation and reversal near the surface simulated at higher resolution were likewise better depicted in coarser-resolution simulations using the nonlinear models.

Numerical study of the cavitating flows over underwater vehicle with large angle of attack

A Homogenous-Equilibrium-Model based cavitation code was developed to numerically analyze the three-dimensional cavitating flows over underwater vehicle navigating with large angle of attack. Rayleigh-Plesset equation based cavitation model was used together with non-linear eddy-viscosity turbulence model. The computed cavity shapes and pressure distributions along the vehicle body were found to generally accord with experimental results at different cavitation numbers and angles of attack. The forces acting on the vehicle body was studied qualitatively to explain why the body may get damaged during navigation. It was also discovered interestingly that, the variation trend of drag-force along with cavitation number at the conditions of large angle was completely opposite to that at zero angle.

Interactive Workshop Discusses Nonlinear Waves and Chaos: Eighth International Nonlinear Wave Workshop; La Jolla, California, 1–5 March 2010

Nonlinear waves and chaos were the focus of a weeklong series of informal and interactive discussions at the Eighth International Nonlinear Wave Workshop (NWW8), held in California. The workshop gathered nonlinear plasma and water wave experts from the United States, France, Czech Republic, Germany, Greece, Holland, India, and Japan. Attendees were from the fields of space, laboratory, and fusion plasma physics, astrophysics, and applied mathematics.Special focus was placed on nonlinear waves and turbulence in the terrestrial environment as well as in the interstellar medium from observational, laboratory, and theoretical perspectives. Discussions covered temperature anisotropies and related instabilities, the properties and origin of the so‐called dissipation range, and various coherent structures of electromagnetic as well as electrostatic nature. Reconnection and shocks were also topics of discussion, as were properties of magnetospheric whistler and chorus waves. Examples and analysis techniques for superdiffusion and subdiffusion were identified. On this last topic, a good exchange of ideas and results occurred between a water wave expert and a plasma expert, with the rest of the audience listening intently.

Theoretical and Experimental Investigation of Water-in-Oil Transverse Dispersion in Porous Media

Summary Water production is controlled by the size and distribution of water saturation around wells. A recent discovery shows that not employing hydrodynamic mixing in numerical simulators may underestimate the water transition zone (Duan and Wojtanowicz 2006). This paper reports continuing research into mechanisms causing expansion of the water-saturation transition zone (transverse dispersion) in a segregated flow of oil and water approaching a vertical well's completion. The mechanisms—including nonlinear flow, turbulence, shear rate, and flow baffling at grains—all contribute to the instability of the oil/water interface, resulting in hydrodynamic mixing. Interface instability because of shearing rate has been demonstrated in our recent study on the Hele-Shaw model (Duan and Wojtanowicz 2007). In this paper, we mathematically model the effect of flow baffling and demonstrate transverse dispersion experimentally using a linear physical sandpack. A simple model of tortuous flow was developed to demonstrate the effect of two-phase-flow baffling in granular porous media. The model shows that the change in flow momentum of the two fluids at the point of collision with rock grains becomes the major factor causing water dispersion. A series of segregated-flow runs (top, oil; bottom, water) was carried out using a physical model packed with different porous media at a constant pressure drop. The runs were videotaped and analyzed for saturation distribution using a color-intensity-recognition software. The results clearly demonstrate onset of transverse dispersion of water into the flowing oil. Further dispersion, however, was overshadowed by the dimensional and end-point effects of the model. With a numerical estimation procedure, the initial dispersion rate—computed from the 1D flow model—is the essential data needed to estimate total dispersion in radial inflow to wells.

A robust implicit multigrid method for RANS equations with two-equation turbulence models

The design of a new, truly robust multigrid framework for the solution of steady-state Reynolds-Averaged Navier-Stokes (RANS) equations with two-equation turbulence models is presented. While the mean-flow equations and the turbulence model equations are advanced in time in a loosely-coupled manner, their multigrid cycling is strongly coupled (FC-MG). Thanks to the loosely-coupled approach, the unconditionally positive-convergent implicit time-integration scheme for two-equation turbulence models (UPC) is used. An improvement to the basic UPC scheme convergence characteristics is developed and its extension within the multigrid method is proposed. The resulting novel FC-MG-UPC algorithm is nearly free of artificial stabilizing techniques, leading to increased multigrid efficiency. To demonstrate the robustness of the proposed algorithm, it is applied to linear and non-linear two-equation turbulence models. Numerical experiments are conducted, simulating separated flow about the NACA4412 airfoil, transonic flow about the RAE2822 airfoil and internal flow through a plane asymmetric diffuser. Results obtained from numerical simulations demonstrate the strong consistency and case-independence of the method.

Probing plasma turbulence by modulating the electron temperature gradient

The local value of a/LTe, a turbulence drive term, was modulated with electron cyclotron heating in L-mode discharges on DIII-D [J. L. Luxon, Nucl. Fusion 42, 614 (2002)] and the density and electron temperature fluctuations in low, intermediate, and high-k regimes were measured and compared with nonlinear gyrokinetic turbulence simulations using the GYRO code [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)]. The local drive term at ρ∼0.6 was reduced by up to 50%, which produced comparable reductions in electron temperature fluctuations at low-k. At intermediate k, kθ∼4 cm−1 and kθρs∼0.8, a very interesting and unexpected result was observed where density fluctuations increased by up to 10% when the local drive term was decreased by 50%. Initial comparisons of simulations from GYRO with the thermal diffusivity from power balance analysis and measured turbulence response are reported. Simulations for the case with the lowest drive term are challenging as they are near the marginal value of a/LTe for trapped electron mode activity.

Measurements of the cross-phase angle between density and electron temperature fluctuations and comparison with gyrokinetic simulations

This paper presents new measurements of the cross-phase angle, αneTe, between long-wavelength (kθρs&amp;lt;0.5) density, ñe, and electron temperature, T̃e, fluctuations in the core of DIII-D [J. L. Luxon, Nucl. Fusion 42, 614 (2002)] tokamak plasmas. The coherency and cross-phase angle between ñe and T̃e are measured using coupled reflectometer and correlation electron cyclotron emission diagnostics that view the same plasma volume. In addition to the experimental results, two sets of local, nonlinear gyrokinetic turbulence simulations that are performed with the GYRO code [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] are described. One set, called the pre-experiment simulations, was performed prior to the experiment in order to predict a change in αneTe given experimentally realizable increases in the electron temperature, Te. In the experiment the cross-phase angle was measured at three radial locations (ρ=0.55, 0.65, and 0.75) in both a “Base” case and a “High Te” case. The measured cross-phase angle is in good qualitative agreement with the pre-experiment simulations, which predicted that ñe and T̃e would be out of phase. The pre-experiment simulations also predicted a decrease in cross-phase angle as Te is increased. Experimentally, this trend is observed at the inner two radial locations only. The second set of simulations, the postexperiment simulations, is carried out using local parameters taken from measured experimental profiles as input to GYRO. These postexperiment simulation results are in good quantitative agreement with the measured cross-phase angle, despite disagreements with transport fluxes. Directions for future modeling and experimental work are discussed.

Magnetic pumping of whistler waves by tether current modulation

Magnetic excitation of whistlers by a square array of electrodynamic tethers is discussed. The array is made of perpendicular rows of tethers that carry equal, uniform, and time‐modulated currents at equal frequency with a 90° phase shift. The array would fly vertical in the orbital equatorial plane, which is perpendicular to the geomagnetic field B0 when its tilt is ignored. The array radiates a whistler wave along B0. A parametric instability due to pumping by the background magnetic field through the radiated wave gives rise to two unstable coupled whistler perturbations. The growth rate is maximum for perturbations with wave vector at angles 38.36° and 75.93° from B0. For an experiment involving a wavefront that moves with the orbiting array, which might serve to study nonlinear wave interactions and turbulence in space plasmas, characteristic values of growth rate and parameters, such as the number of tethers and their dimensions and distances in the array, are discussed for low Earth orbit ambient conditions.

HOW DOES RADIO AGN FEEDBACK FEED BACK?

Radio AGN feedback is often assumed to work, but detailed physical models of this process are not well developed. This paper examines a possible path for radio AGN feedback to heat the gas in and around galaxies and perhaps suppress star formation. Nearby radio AGN are almost all FR-I radio galaxies, and it is argued that such outflows become decelerated and fully turbulent in their early stages. A 3D non-linear MHD turbulence calculation is made to determine the time required for this outflowing turbulent energy to be converted into heat. Using radio and x-ray observations of nearby FR-I AGN for normalization results in a time for the onset of heating of about 100 million years. This is comparable to the gas cooling times in and around galaxies, and the resulting location of heat deposition is consistent with FR-I outflows.

Interaction of Mean and Oscillating Plasma Flows Across Confinement Mode Transitions

The interaction of mean plasma flows (driven by equilibrium E×B forces) and oscillating plasma flows - such as zonal flows (ZFs) and geodesic acoustic modes (GAMs) - (radially localized E × B plasma flows generated by non-linear turbulence interactions) is investigated in the ASDEX Upgrade tokamak using Doppler reflectometry. ZFs and GAMs play an important role in magnetic confinement devices by enhancing the velocity shearing of turbulent eddies which reduces turbulent cross-field transport. Although the GAM is universally observed across the plasma edge gradient region of ohmic and additionally heated low confinement L-mode regimes, it is not seen in the high confinement H-mode. Using slow power ramping of the additional ECRH and NBI heating systems the behaviour of the edge GAM, and its interaction with the mean flow shear, i.e. the radial electric field Er profile, across the L to H-mode transition are studied. As the edge Er well deepens with improving confinement, the associated negative Er shear region narrows and the radial extent of the GAM shrinks. The main feature, however, is the movement of spectral power out of the coherent GAM into large amplitude, low frequency random flow perturbations. It is speculated that these flow perturbations may play a role in the triggering of the confinement transition.

Kinetic Description of Nonlinear Plasma Turbulence

The kinetic description of nonlinear plasma turbulence is discussed for Vlasov plasmas with multiple species. Kadomtsev's formalism is reviewed, and the foundation from the Mori method of statistical physics is discussed. The coherent nonlinear term and nonlinear noise term are discussed. In addition to the diagonal term in memory function (eddy damping term), the off-diagonal term in the memory function and nonlinear noise are introduced simultaneously. The off-diagonal term in memory function affects the dynamics as the new nonlinear force, so that a nonlocal interaction in phase space occurs in wave-particle interactions. A set of special solutions for the eddy damping term and off-diagonal coherent term is obtained. The access to a nonlinear stationary state by using an incoherent emission term is explained. Quasi-modes, which are driven by the incoherent emission, are shown to accumulate at a frequency of ω( k )=2ω'( k /2), where ω'( k ) indicates the dispersion relation of the mode. The formulation ...

Turbulent Flow in Wavy Channels Simulated with Nonlinear Models and a New Implicit Formulation

This work examines the performance of linear and nonlinear eddy-viscosity models when used to predict the turbulent flow in periodically sinusoidal-wave channels. Two geometries are investigated, namely a converging-diverging channel and a channel with concave-convex walls. The numerical method employed for the discretization of the equations is the control-volume method in a boundary-fitted nonorthogonal coordinate system. The SIMPLE algorithm is used for correcting the pressure field. The classical wall function and a low Reynolds model are used to describe the flow near the wall. Comparisons between those two approaches using linear and nonlinear turbulence models are done. Here, a new implicit numerical treatment is proposed for the nonlinear diffusion terms of the momentum equations in order to increase the robustness. Results show that by decomposing and treating terms as presented, solutions using nonlinear models and the high Reynolds wall treatment, which combine accuracy and economy, are more stable and easier to be obtained.

Interplay between Gyrokinetic Turbulence, Flows, and Collisions: Perspectives on Transport and Poloidal Rotation

The impact of ion-ion collisions on confinement is investigated with the full-f and global gyrokinetic Gysela code through a series of nonlinear turbulence simulations for tokamak parameters. A twofold scan in the turbulence drive and in collisionality is performed, highlighting (i) a heat transport expressed in terms of critical quantities-threshold and exponent, (ii) a first evidence of turbulent generation of poloidal momentum, and (iii) the dominance of mean flow shear, mediated through the turbulent corrugation of the mean profiles, with regard to the oft-invoked zonal flow shear.

Validating a quasi-linear transport model versus nonlinear simulations

In order to gain reliable predictions on turbulent fluxes in tokamak plasmas, physics based transport models are required. Nonlinear gyrokinetic electromagnetic simulations for all species are still too costly in terms of computing time. On the other hand, interestingly, the quasi-linear approximation seems to retain the relevant physics for fairly reproducing both experimental results and nonlinear gyrokinetic simulations. Quasi-linear fluxes are made of two parts: (1) the quasi-linear response of the transported quantities and (2) the saturated fluctuating electrostatic potential. The first one is shown to follow well nonlinear numerical predictions; the second one is based on both nonlinear simulations and turbulence measurements. The resulting quasi-linear fluxes computed by QuaLiKiz (Bourdelle et al 2007 Phys. Plasmas 14 112501) are shown to agree with the nonlinear predictions when varying various dimensionless parameters, such as the temperature gradients, the ion to electron temperature ratio, the dimensionless collisionality, the effective charge and ranging from ion temperature gradient to trapped electron modes turbulence.