Two-component Turbulence Research Articles

PIV study of large-scale flow organisation in slot jets

The paper reports on particle image velocimetry (PIV) measurements in turbulent slot jets bounded by two solid walls with the separation distance smaller than the jet width (5–40%). In the far-field such jets are known to manifest features of quasi-two dimensional, two component turbulence. Stereoscopic and tomographic PIV systems were used to analyse local flows. Proper orthogonal decomposition (POD) was applied to extract coherent modes of the velocity fluctuations. The measurements were performed both in the initial region close to the nozzle exit and in the far fields of the developed turbulent slot jets for Re⩾10,000. A POD analysis in the initial region indicates a correlation between quasi-2D vortices rolled-up in the shear layer and local flows in cross-stream planes. While the near-field turbulence shows full 3D features, the wall-normal velocity fluctuations day out gradually due to strong wall-damping resulting in an almost two-component turbulence. On the other hand, the longitudinal vortex rolls take over to act as the main agents in wall-normal and spanwise mixing and momentum transfer. The quantitative analysis indicates that the jet meandering amplitude was aperiodically modulated when arrangement of the large-scale quasi-2D vortices changed between asymmetric and symmetric pattern relatively to the jet axis. The paper shows that the dynamics of turbulent slot jets are more complex than those of 2D, plane and rectangular 3D jets. In particular, the detected secondary longitudinal vortex filaments and meandering modulation is expected to be important for turbulent transport and mixing in slot jets. This issue requires further investigations.

A comparison of turbulence-reduced drift coefficients of importance for the modulation of galactic cosmic-ray protons in the supersonic solar wind

The study of the modulation of cosmic rays in the heliosphere relies heavily on a thorough understanding of the transport of these charged particles in the turbulent solar wind. Drift effects due to gradients and the curvature of the background magnetic field have long been known to be reduced in the presence of turbulence, and as such, several forms for the drift coefficient that include the effect of turbulence have been proposed. The present study aims to investigate the qualitative effects of various turbulence-reduced drift coefficients on cosmic ray intensities computed using an ab initio 3D steady-state cosmic-ray modulation code. Results from a two-component turbulence transport models are used as inputs for the basic turbulence quantities. Furthermore, an expression for the perpendicular mean free path is derived here from a modification of the non-linear guiding center theory of Matthaeus et al. (2003) assuming a 2D turbulence power spectrum with a k-1 energy range wavenumber dependence, and is used in conjunction with the various proposed turbulence-reduced drift coefficients. Cosmic-ray intensities computed using different drift coefficients but assuming the same turbulence conditions are found to differ widely. This study emphasises the need to gain a better understanding of the effect of turbulence on drifts in the heliosphere.

S.23.03 Anti-alcohol properties of the positive allosteric modulators of the GABAB receptor

The study of the modulation of cosmic rays in the heliosphere relies heavily on a thorough understanding of the transport of these charged particles in the turbulent solar wind. Drift effects due to gradients and the curvature of the background magnetic field have long been known to be reduced in the presence of turbulence, and as such, several forms for the drift coefficient that include the effect of turbulence have been proposed. The present study aims to investigate the qualitative effects of various turbulence-reduced drift coefficients on cosmic ray intensities computed using an ab initio 3D steady-state cosmic-ray modulation code. Results from a two-component turbulence transport models are used as inputs for the basic turbulence quantities. Furthermore, an expression for the perpendicular mean free path is derived here from a modification of the non-linear guiding center theory of Matthaeus et al. (2003) assuming a 2D turbulence power spectrum with a k-1 energy range wavenumber dependence, and is used in conjunction with the various proposed turbulence-reduced drift coefficients. Cosmic-ray intensities computed using different drift coefficients but assuming the same turbulence conditions are found to differ widely. This study emphasises the need to gain a better understanding of the effect of turbulence on drifts in the heliosphere.

Cosmic-Ray Modulation: an Ab Initio Approach

A better understanding of cosmic-ray modulation in the heliosphere can only be gained through a proper understanding of the effects of turbulence on the diffusion and drift of cosmic rays. We present an ab initio model for cosmic-ray modulation, incorporating for the first time the results yielded by a two-component turbulence transport model. This model is solved for periods of minimum solar activity, utilizing boundary values chosen so that model results are in fair to good agreement with spacecraft observations of turbulence quantities, not only in the solar ecliptic plane but also along the out-of-ecliptic trajectory of the Ulysses spacecraft. These results are employed as inputs for modelled slab and 2D turbulence energy spectra. The latter spectrum is chosen based on physical considerations, with a drop-off at the very lowest wavenumbers commencing at the 2D outerscale. There currently exist no models or observations for this quantity, and it is the only free parameter in this study. The modelled turbulence spectra are used as inputs for parallel mean free path expressions based on those derived from quasi-linear theory and perpendicular mean free paths from extended nonlinear guiding center theory. Furthermore, the effects of turbulence on cosmic-ray drifts are modelled in a self-consistent way, employing a recently developed model for drift along the wavy current sheet. The resulting diffusion coefficients and drift expressions are applied to the study of galactic cosmic-ray protons and antiprotons using a three-dimensional, steady-state cosmic-ray modulation code, and sample solutions in fair agreement with multiple spacecraft observations are presented.

THE MODIFICATION OF THE NONLINEAR GUIDING CENTER THEORY

We modify the NonLinear Guiding Center (NLGC) theory (Matthaeus et al. 2003) for perpendicular diffusion by replacing the spectral amplitude of the two-component model magnetic turbulence with the 2D component one (following Shalchi 2006), and replacing the constant $a^2$, indicating the degree particles following magnetic field line, with a variable $a^{\prime 2}$ as a function of the magnetic turbulence. We combine the modified model with the NonLinear PArallel (NLPA) diffusion theory (Qin 2007) to solve perpendicular and parallel diffusion coefficients simultaneously. It is shown that the new model agrees better with simulations. Furthermore, we fit the numerical results of the new model with polynomials, so that parallel and perpendicular diffusion coefficients can be calculated directly without iteration of integrations, and many numerical calculations can be reduced.

SMALL-SCALE GRADIENTS OF CHARGED PARTICLES IN THE HELIOSPHERIC MAGNETIC FIELD

Using numerical simulations of charged-particles propagating in the heliospheric magnetic field, we study small-scale gradients, or "dropouts," in the intensity of solar energetic particles seen at 1 AU. We use two turbulence models, the foot-point random motion model and the two-component model, to generate fluctuating magnetic fields similar to spacecraft observations at 1 AU. The turbulence models include a Kolmogorov-like magnetic field power spectrum containing a broad range of spatial scales from those that lead to large-scale field-line random walk to small scales leading to resonant pitch-angle scattering of energetic particles. We release energetic protons (20 keV–10 MeV) from a spatially compact and instantaneous source. The trajectories of energetic charged particles in turbulent magnetic fields are numerically integrated. Spacecraft observations are mimicked by collecting particles in small windows when they pass the windows at a distance of 1 AU. We show that small-scale gradients in the intensity of energetic particles and velocity dispersions observed by spacecraft can be reproduced using the foot-point random motion model. However, no dropouts are seen in simulations using the two-component magnetic turbulence model. We also show that particle scattering in the solar wind magnetic field needs to be infrequent for intensity dropouts to form.

AN AB INITIO MODEL FOR THE MODULATION OF GALACTIC COSMIC-RAY ELECTRONS

The modulation of galactic cosmic-ray electrons is studied using an ab initio three-dimensional steady state cosmic-ray modulation code in which the effects of turbulence on both the diffusion and drift of these cosmic-rays are treated as self-consistently as possible. A significant refinement is that a recent two-component turbulence transport model is used. This model yields results in reasonable agreement with observations of turbulence quantities throughout the heliosphere. The sensitivity of computed galactic electron intensities to choices of various turbulence parameters pertaining to the dissipation range of the slab turbulence spectrum, and to the choice of model of dynamical turbulence, is demonstrated using diffusion coefficients derived from the quasi-linear and extended nonlinear guiding center theories. Computed electron intensities and latitude gradients are also compared with spacecraft observations.

The role of the Kubo number in two-component turbulence

We explore the random walk of magnetic field lines in two-component turbulence by using computer simulations. It is often assumed that the two-component model provides a good approximation for solar wind turbulence. We explore the dependence of the field line diffusion coefficient on the Kubo number which is a fundamental and characteristic quantity in the theory of turbulence. We show that there are two transport regimes. One is the well-known quasilinear regime in which the diffusion coefficient is proportional to the Kubo number squared, and the second one is a nonlinear regime in which the diffusion coefficient is directly proportional to the Kubo number. The so-called percolative transport regime which is often discussed in the literature cannot be found. The numerical results obtained in the present paper confirm analytical theories for random walking field lines developed in the past.

AN AB INITIO MODEL FOR COSMIC-RAY MODULATION

A proper understanding of the effects of turbulence on the diffusion and drift of cosmic rays (CRs) is of vital importance for a better understanding of CR modulation in the heliosphere. This study presents an ab initio model for CR modulation, incorporating for the first time the results yielded by a two-component turbulence transport model. This model is solved for solar minimum heliospheric conditions, utilizing boundary values chosen so that model results are in reasonable agreement with spacecraft observations of turbulence quantities in the solar ecliptic plane and along the out-of-ecliptic trajectory of the Ulysses spacecraft. These results are employed as inputs for modeled slab and two-dimensional (2D) turbulence energy spectra. The modeled 2D spectrum is chosen based on physical considerations, with a drop-off at the very lowest wavenumbers. There currently exist no models or observations for the wavenumber where this drop-off occurs, and it is considered to be the only free parameter in this study. The modeled spectra are used as inputs for parallel mean free path expressions based on those derived from quasi-linear theory and perpendicular mean free paths from extended nonlinear guiding center theory. Furthermore, the effects of turbulence on CR drifts are modeled in a self-consistent way, also employing a recently developed model for wavy current sheet drift. The resulting diffusion and drift coefficients are applied to the study of galactic CR protons and antiprotons using a 3D, steady-state CR modulation code, and sample solutions in fair to good agreement with multiple spacecraft observations are presented.

Effects of an axisymmetric contraction on a turbulent pipe flow

AbstractThe flow in an axisymmetric contraction fitted to a fully developed pipe flow is experimentally and numerically studied. The reduction in turbulence intensity in the core region of the flow is discussed on the basis of the budgets for the various turbulent stresses as they develop downstream. The contraction generates a corresponding increase in energy in the near-wall region, where the sources for energy production are quite different and of opposite sign compared to the core region, where these effects are caused primarily by vortex stretching. The vortices in the pipe become aligned with the flow as the stretching develops through the contraction. Vortices which originally have a spanwise component in the pipe are stretched into pairs of counter-rotating vortices which become disconnected and aligned with the mean flow. The structures originating in the pipe which are inclined at an angle with respect to the wall are rotated towards the local mean streamlines. In the very near-wall region and the central part of the contraction the flow tends towards two-component turbulence, but these structures are different. The streamwise and azimuthal stresses are dominant in the near-wall region, while the lateral components dominate in the central part of the flow. The two regions are separated by a rather thin region where the flow is almost isotropic.

A study on turbulence modulation via an analysis of turbulence anisotropy-invariants

We investigate the turbulence modulation by particles in a turbulent two-phase channel flow via an analysis of turbulence anisotropy-invariants. The fluid turbulence is calculated by a large eddy simulation with a point-force two-way coupling model and particles are tracked by the Lagrangian trajectory method. The channel turbulence follows the two-component turbulence state within the viscous sub-layer region and outside the region the turbulence tends to follow the right curve of the anisotropy-invariant. The channel turbulence, interacting with heavy particles, is modulated to the two-component turbulence limit state near the wall and is separate from the axisymmetric turbulence state in the turbulence anisotropy-invariants map. The fluctuations of streamwise component are transferred to the other two components and hence the anisotropy decreases due to particle modulation. The study has deepened the understanding of the turbulence modulation mechanism in two-phase turbulent flows.

Some Swirling-flow Challenges for Turbulent CFD

The paper examines some of the continuing challenges, within a RANS framework, of computing turbulent swirling flows such as are encountered in industry and the environment. The principal focus is on modelling turbulent transport processes but serious problems also arise in handling numerical issues, too. Recent researches of two of these types of flow by the authors and their colleagues in the Turbulence Mechanics Group at Manchester are examined; namely, the confined flow within a rotor–stator disc cavity and the trailing wing-tip vortex. The former flow, while geometrically axisymmetric, has been found to create multiple rotating vortices necessitating a three-dimensional time-dependent analysis. The wing-tip vortex is extremely sensitive to the choice of turbulence model and only a second-moment closure that complies with the constraints of two-component turbulence has been found capable of handling both the flow over the wing and the wake vortex. Moreover, because of the large distances downstream of the aircraft to which, for practical cases, computations need to be carried, the numerical strategy is brought into question. Finally, arising from these two test cases, outline remarks are made about a swirling flow that poses one of the major computational challenges of the twenty-first century.

Pitch-angle scattering in pure two-dimensional and two-component turbulence

Aims. It has been demonstrated that quasilinear theory is inappropriate for describing pitch-angle scattering and parallel spatial diffusion, if a turbulence model with a strong two-dimensional component is assumed. We aim to revisit this problem using a more systematic and reliable approach. Methods. We consider particle transport in pure two-dimensional turbulence, by using a vector-potential description of pitch-angle scattering. Results. It is demonstrated that pitch-angle scattering in pure two-dimensional turbulence behaves subdiffusively and, thus, parallel scattering is expected to behave superdiffusively. In the slab/2D composite model, the total pitch-angle Fokker-Planck coefficient reaches the slab limit when very long timescales are considered.

Analytical description of stochastic field-line wandering in magnetic turbulence

A nonperturbative nonlinear statistical approach is presented to describe turbulent magnetic systems embedded in a uniform mean magnetic field. A general formula in the form of an ordinary differential equation for magnetic field-line wandering (random walk) is derived. By considering the solution of this equation for different limits several new results are obtained. As an example, it is demonstrated that the stochastic wandering of magnetic field-lines in a two-component turbulence model leads to superdiffusive transport, contrary to an existing diffusive picture. The validity of quasilinear theory for field-line wandering is discussed, with respect to different turbulence geometry models, and previous diffusive results are shown to be deduced in appropriate limits.

Separation of Magnetic Field Lines in Two‐Component Turbulence

The problem of the separation of random magnetic field lines in collisionless astrophysical plasmas is closely related to the problem of the magnetic field line random walk and is highly relevant to the transport of charged particles in turbulent plasmas. In order to generalize treatments based on quasi-linear theory, here we examine the separation of nearby magnetic field lines by employing a nonperturbative technique based on the Corrsin independence hypothesis. Specifically, we consider the case of two-component turbulence in which the magnetic field fluctuations are a mixture of one-dimensional (slab) and two-dimensional ingredients, as a concrete example of anisotropic turbulence that provides a useful description of turbulence in the solar wind. We find that random field trajectories can separate in general through three regimes of the behavior of the running diffusion coefficient: slow diffusive separation, an intermediate regime of superdiffusion, and fast diffusive separation at large distances. These features are associated with the gradual, exponential divergence of field lines within islands of two-dimensional turbulence, followed by diffusive separation at long distances. The types of behavior are determined not by the Kubo number but rather a related ratio that takes the turbulence anisotropy into account. These results are confirmed by computer simulations. We discuss implications for space observations of energetic charged particles, including ‘‘dropouts’’ of solar energetic particles.

Reynolds number effects in the near-wall region of turbulent channel flows

The continuity and momentum equations do not imply a Reynolds number dependence of turbulence data when wall variables are used for normalization. However, experimental and numerical results show a Reynolds number dependence of turbulence intensity very close to the wall. The cause of this is explained. It results from the behavior of a sink term in the dissipation rate equation which shows a Reynolds number dependence in the limit of two-component two-dimensional turbulence as it exists close to walls. Away from the near-wall region the Reynolds number dependence originates from the streamwise pressure gradient which enters into the equations for the turbulent kinetic energy and turbulent dissipation rate through the gradient production processes. The low-Reynolds number effects in turbulent channel flow were investigated experimentally using the laser Doppler anemometry (LDA) measuring technique. A new method was used to eliminate the influence of the limited spatial resolution of the LDA measuring control volume. Results are presented for the limiting behavior of the turbulent intensity near the wall and its variation with the Reynolds number. The present LDA measurements confirm the trend in the data of direct numerical simulations.

Nonlinear eddy viscosity modelling for turbulence and heat transfer near wall and shear-free boundaries

Abstract New turbulence and turbulent heat flux models are proposed for capturing flow and thermal fields bounded by walls or free surfaces. The models are constructed using locally definable quantities only, without any recourse to topographical parameters. For the flow field, the proposed model is a cubic nonlinear k–e–A three equation eddy viscosity model. It employs dependence on Lumley's stress flatness parameter A, by solving its modelled transport equation as the third variable. Since A vanishes at two-component turbulence boundaries, introducing its dependency enables a turbulence model to capture the structure of turbulence near shear-free surfaces as well as wall boundaries. To close the modelled A equation, an up-to-date second-moment closure is applied. For the thermal field, an explicit algebraic second-moment closure for turbulent heat flux is proposed. The new aspect of this heat flux model is the use of nonlinear Reynolds stress terms in the eddy diffusivity tensor. This model complies with the linearity and independence principles for passive scalar. The proposed models are tested in fully developed plane channel, open channel and plane Couette–Poiseuille flows at several fluid Prandtl numbers. The results show the very encouraging performance of the present proposals in capturing anisotropic turbulence and thermal fields near both wall and shear-free boundaries in the range of 0.025⩽Pr⩽95.

Solid/free-surface juncture boundary layer and wake

The Reynolds-averaged flow for a solid/free-surface juncture boundary layer and wake is documented. The three mean-velocity components and five of the Reynolds stresses are measured for a surface-piercing flat plate in a towing tank using a laser-Doppler velocimeter system for both boundary-layer and wake planes in regions close to the free surface. The experimental method is described, including the foil-plate model, laser-Doppler velocimeter system, conditions, and uncertainty analysis. The underlying flow data is in excellent agreement with benchmark data. Inner (near the plate and wake centerplane and below the free surface) and outer (near the free surface) regions of high streamwise vorticity of opposite sign are observed, which transport, respectively, high mean velocity and low turbulence from the outer to the inner and low mean velocity and high turbulence from the inner to the outer portions of the boundary layer and wake. For the wake, the inner region of vorticity is relatively weak. The physical mechanism for the streamwise vorticity is analyzed with regard to the Reynolds-averaged streamwise vorticity equation. The anisotropy of the crossplane normal Reynolds stresses closely correlates with the vorticity and, additionally, indicates similarity, i.e., its nature is such that it only depends on the proximity to the plate and free surface boundaries or wake centerplane symmetry plane. Free-surface effects on the Reynolds stresses are analyzed with regard to the behavior close to the free surface of the turbulent kinetic energy and the normal components of the anisotropy tensor and the anisotropy invariants. Close to the free surface, the turbulent kinetic energy is nearly constant and increases for the inner and outer portions, respectively, of the boundary layer and wake and the normal components of the anisotropy tensor and the anisotropy invariants roughly correspond to the limiting values for two-component turbulence. The similarities and differences between the present results and analysis with those from related studies are discussed. The data and analysis should have practical application with regard to the development of turbulence models for computational fluid dynamics methods for the Reynolds-averaged Navier–Stokes equations.

Two-Component Phase-Averaged Turbulence Statistics Downstream of a Rotating Spoked-Wheel Wake Generator

Measurements of the axial and tangential components of the unsteady turbulent flow downstream of a rotating spoked-wheel wake generator have been obtained. The results of this study have implications for the use of this type of wake generator to produce simulated turbine guide vane wakes. Instantaneous velocity information was phase averaged based on a signal synchronized with the bar-passing frequency. Mean velocity profiles and phase-averaged Reynolds stress results were found to be consistent with measurements obtained behind a stationary cylinder. Reynolds stresses were significantly higher than corresponding measurements obtained in large-scale research turbomachines, however. Phase-averaged triple velocity correlations, also calculated from the digital velocity records, reveal the sign and magnitude of skewness in the velocity probability density distributions for the two components. Large crossflow gradients observed in the triple correlations in the wake indicate the importance of the tangential-component fluctuations in the net turbulent transport of turbulent energy across the wake. Streamwise-component wake velocity spectra for low values of reduced bar-passing frequency include a peak associated with vortex shedding from the cylindrical wake-generating bars at a shedding Strouhal number of 0.2. For higher bar-passing frequencies, the energy associated with vortex shedding is shifted to lower frequencies and becomes broadband from the stationary reference frame viewpoint.

Global existence for a particular convection diffusion system

This system can be thought of a parabolic regularization of a nonstrictly hyperbolic conservation law or it can be considered as a Burgers type system related to the two component turbulence model which was introduced in [l]. In [4], Dlotko utilizes Galerkin arguments to guarantee global existence to the two component Burgers system. Other recent work related to Burgers systems appears in [3, 5, 12, 6, 71. One way to approach (1.la-c) would be to apply the methods of Hoff and Smoller [lo] and to utilize an entrophy-enstrophy flux pair to obtain a global solution. However, we choose a different approach; we set (1.la-c) within the framework of analytic semigroups and the associated theory of abstract semilinear evolution equations. This approach has been previously applied to the two-component turbulence system of Burgers, [6, 71.