Axisymmetric Turbulence Research Articles

On the Effects of Tokamak Plasma Edge Symmetries on Turbulence Relaxation

The plasma edge of a tokamak configuration is characterized by turbulent dynamics leading to enhanced transport. We construct a simplified 3D Hasegawa–Wakatani model reducing to a single partial differential equation for the turbulent electric potential dynamics. Simulations demonstrate how the 3D turbulence relaxes on a 2D axisymmetric profile, corresponding to the so-called interchange turbulence. The spectral features of this regime are found to be strongly dependent on the initialization pattern. We outline that the emergence of axisymmetric turbulence is also achieved when the corresponding mode amplitude is not initialized. Then, we introduce the symmetries of the magnetic X-point of a tokamak configuration. We linearize the governing equation by treating the poloidal field as a small correction. We show that it is not always possible to solve the electric potential dynamics following a perturbative approach. This finding, which is due to resonance between the modes of the background and the poloidal perturbation, confirms that the X-point symmetries can alter the properties of turbulent transport in the edge region.

The Decay of Energy and Scalar Variance in Axisymmetric Turbulence

We review the recent progress in our understanding of the large scales in homogeneous (but anisotropic) turbulence. We focus on turbulence which emerges from Saffman-like initial conditions, in which the vortices possess a finite linear impulse. Such turbulence supports long-range velocity correlations of the form uiu′j=O(r−3), where u and u′ are separated by a distance r, and these long-range interactions dominate the dynamics of large eddies. We show that, for axisymmetric turbulence, the energy and integral scales evolve as u⊥2~u//2~t−6/5 and l⊥~l//~t2/5, where ⊥ and // indicate directions that are perpendicular and parallel to the symmetry axis, respectively. These predictions are consistent with the evidence of direct numerical simulations. Similar results are obtained for the passive scalar variance, where we find that θ2~t−6/5. The primary point of novelty in our discussion of passive scalar decay is that it is based in real (rather than spectral) space, making use of an integral invariant which is a generalization of the isotropic Corrsin integral.

The decay of stably stratified grid turbulence in a viscosity-affected stratified flow regime

The decay of stably stratified turbulence generated by a towed rake of vertical plates is investigated by direct numerical simulations (DNS) of temporally evolving grid turbulence in a linearly stratified fluid. The Reynolds number $Re_M=U_0M/\nu$ is 5000 or 10 000 while the Froude number $Fr_M=U_0/MN$ is between 0.1 and 6 ( $U_0$ : towing speed; $M$ : mesh size; $\nu$ : kinematic viscosity; $N$ : Brunt–Väisälä frequency). The DNS results are compared with the theory of stably stratified axisymmetric Saffman turbulence. Here, the theory is extended to a viscosity-affected stratified flow regime with low buoyancy Reynolds number $Re_b$ , and power laws are derived for the temporal variations of the horizontal velocity scale ( $U_H$ ) and the horizontal and vertical integral length scales ( $L_H$ and $L_V$ ). Temporal grid turbulence initialized with the mean velocity deficit of wakes exhibits a $k^{2}$ energy spectrum at a low-wavenumber range and invariance of $U_H^2L_H^2L_V$ , which are the signatures of axisymmetric Saffman turbulence. The decay of various quantities follows the power laws predicted for low- $Re_b$ Saffman turbulence when $Fr_M$ is sufficiently small. However, the decay of $U_H^2$ at $Fr_M=6$ is no longer expressed by a power law with a constant exponent. This behaviour is related to the scaling of kinetic energy dissipation rate $\varepsilon$ , for which $\alpha =\varepsilon /(U_H^3/L_H)$ is constant during the decay for $Fr_M\leq 1$ while it varies with time for $Fr_M=6$ . We also examine the experimental data of towed-grid experiments by Praud et al. (J. Fluid Mech., vol. 522, 2005, pp. 1–33), which is shown to agree with the theory of low- $Re_b$ Saffman turbulence.

Large-eddy simulation of turbulent natural convection in a cylindrical cavity using an off-lattice Boltzmann method

In the present work, a characteristic-based off-lattice Boltzmann method with the large-eddy simulation (LES) as the turbulence model is developed for the simulation of turbulent natural convection. A double-distribution-function approach is used to resolve flow and thermal fields, and the proposed framework is developed, in three-dimensional curvilinear coordinates. The solver is verified using three benchmark cases, namely, the turbulent Taylor–Green vortex flow, natural convection in a periodic tall cavity, and Rayleigh–Bénard convection. Due to the absence of an inlet in this kind of closed cavity flow, initial perturbations are proposed and verified, which accelerate transition to a turbulent state. The turbulent natural convection in a cylindrical cavity is simulated for a Rayleigh number of Ra=7.5×105, and the flow and thermal characteristics are analyzed. A grid sensitivity study is conducted and an appropriate mesh resolution is selected, that is, further verified using the LES index of quality-of-resolution. The resulting turbulent flow and the associated thermal plume are analyzed using instantaneous and time-averaged mean and second-order statistics, vortical structures, turbulence anisotropy maps, energy budgets, frequency spectra, and the mean and root mean square of temperature and Nusselt numbers. The results indicate that the thermal plume region is highly anisotropic, whereas the rest of the annulus contains single-component axisymmetric turbulence. The production and convection of turbulence are dominant on top of the inner cylinder in the thermal plume region, whereas diffusion is dominant closer to the outer cylinder. The azimuthal profiles of mean Nusselt number for the inner and the outer cylinders are observed to be negatively correlated. Furthermore, natural convection in the cylindrical cavity is simulated for Ra=4×103 to 5×106 and the effect of the Rayleigh number on the mean Nusselt number and flow patterns is studied.

Transition from axisymmetric to three-dimensional turbulence

In purely axisymmetric turbulence sustained by a linear forcing mechanism, a stable, entirely poloidal flow is observed when the toroidal component of the forcing is below a threshold. We investigate using numerical simulations whether this state persists when toroidal variations of the flow are continuously reintroduced and the forcing is purely poloidal. It is shown how the pressure–strain correlation allows the redistribution of the energy towards the toroidal component. A simple statistical model allows to capture the main physical effects on the level of the global energy balance. This model is then used to investigate the stability of the poloidal state for various toroidal-to-poloidal forcing strengths and different degrees of axisymmetry.

Reynolds stress anisotropy in flow over two-dimensional rigid dunes

Characteristics of turbulence anisotropy in flow over two-dimensional rigid dunes are analysed. The Reynolds stress anisotropy is envisaged from the perspective of the stress ellipsoid shape. The spatial evolutions of the anisotropic invariant map (AIM), anisotropic invariant function, eigenvalues of the scaled Reynolds stress tensor and eccentricities of the stress ellipsoid are investigated at various streamwise distances along the vertical. The data plots reveal that the oblate spheroid axisymmetric turbulence appears near the top of the crest, whereas the prolate spheroid axisymmetric turbulence dominates near the free surface. At the dune trough, the axisymmetric contraction to the oblate spheroid diminishes, as the vertical distance below the crest increases. At the reattachment point and one-third of the stoss-side, the oblate spheroid axisymmetric turbulence formed below the crest appears to be more contracted, as the vertical distance increases. The AIMs suggest that the turbulence anisotropy up to edge of the boundary layer follows a looping pattern. As the streamwise distance increases, the turbulence anisotropy at the edge of the boundary layer approaches the plane-strain limit up to two-thirds of the stoss-side, intersecting the plane-strain limit at the top of the crest and thereafter moving towards the oblate spheroid axisymmetric turbulence.

Transition from non-swirling to swirling axisymmetric turbulence

Strictly axisymmetric turbulence, i.e., turbulence governed by the Navier-Stokes equations modified such that the flow is invariant in the azimuthal direction, is a system intermediate between two-and three-dimensional turbulence. We investigate statistically steady states of this system by direct numerical simulation using a forcing protocol, which allows the injected energy in the toroidal and in the poloidal directions to be tuned independently. A sharp transition between a two-dimensional two-component (2D2C, non-swirling) flow and a two-dimensional three-component (2D3C, swirling) flow is observed. We derive a statistical model which reproduces this transition.

Statistical theory of helical turbulence

A statistical theory for homogeneous helical turbulence is developed under the condition of strong symmetry. The latter describes reflectional symmetry in planes through and normal to the helical unit vector eξ, which can be achieved by demanding that the mean velocity is zero. The two-point velocity correlation, the pressure–velocity correlation, and the two-point triple correlation are expressed by scalar functions in the helical unit vector system. By introducing the continuity equation for the correlations, the number of unknown functions can be reduced to such an extent that ultimately, a single scalar transport equation remains. Furthermore, a two-point Poisson equation is derived to express the pressure–velocity correlation in terms of the triple correlation. From the two-point version of the transport equation, the single-point limit is derived. Using the single-point equation, it can be shown that all velocity components are generally non-zero. Therefore, it is concluded that the phenomenon of vortex stretching is present in the helical coordinate system. Finally, the theory of axisymmetric turbulence is derived as a limiting case of helical turbulence to show consistency with former work.

Effects of aspect ratio on higher-order moments, conditional statistics, TKE budget and anisotropy in narrow open channel flow

The purpose of the present study is to investigate the influence of aspect ratio on the higher- order statistics of velocity fluctuations, conditional statistics, turbulent kinetic energy (TKE) fluxes, TKE budget, and Reynolds stress anisotropy in hydraulically rough narrow open channel flow (OCF). In the experiments, the aspect ratios were maintained low and varied between 2.5 and 4. The velocities were measured with an acoustic Doppler velocimeter (ADV). The higher-order moments, quadrant analysis, and TKE budget from the present experiments were compared with available literature data in wide and narrow OCFs to elucidate the effect of the aspect ratio. In this analysis, the third-order moments of velocity fluctuations were found to be sensitive to the aspect ratio in the outer region. The fractional contributions of all quadrant events are approximately equal in magnitude in lower aspect ratio flows, whereas ejections and sweeps are the dominant events as the aspect ratio increases. The upward transfer of TKE flux increases in the outer layer with increase in aspect ratio. In the inner layer, the TKE production and rate of dissipation are found to be increasing with decreasing aspect ratio. Therefore, the production and dissipation of TKE are dependent on the aspect ratio of the flow. The analysis of Reynolds stress AIM reveals that for low aspect ratio flows turbulence tends to attain rod-like axisymmetric turbulence only in the intermediate layer whereas for higher aspect ratio, turbulence attains rod-like axisymmetric turbulence throughout the depth.

On the highly swirling flow through a confined bluff-body

We address highly swirling, confined-bluff-body-flow evolving through a burner; particularly, contributions of the swirling motion to a central-recirculation-zone (CRZ) downstream the injector. Previous studies suggest that flame stability reduces in combustors lacking this zone; careful consideration of the CRZ is thus desirable. We use Reynolds-Averaged-Navier–Stokes (RANS) and Large-Eddy-Simulation (LES) to simulate the flow, and the influence of the swirl device was included in defining the inflow conditions for LES simulations. We use mean velocity profiles and turbulence statistics to test results. There is qualitative agreement between computed and reported experimental data, and we document quantitative differences obtained with the RANS models. LES velocity field results are mostly within 3% of the experimental data, better than the latest reported LES data, reinforcing the suitability of our approach. We took advantage of the quality of the LES mesh, which solves 95.6% of the resolved-turbulence-energy, to present the vorticity structures showing the precessing vortex motion on the CRZ boundaries. Anisotropic states of the Reynolds-stress were characterized with the aid of an anisotropy invariant map, a novelty for this type of burner; the turbulence states considerably vary inside the burner, behaving isotropically in the center of the CRZ, whereas axisymmetric turbulence is predominant in the other areas of the CRZ. The results reinforce the importance of applying appropriate turbulence models and inflow conditions for simulations involving confined-bluff-body-flows in order to capture the main flow fields and structures in the CRZ.

Spectral Broadening of Tonal Sound Propagating Through an Axisymmetric Turbulent Shear Layer

In aeroacoustics, spectral broadening refers to the scattering of tonal sound fields by turbulent shear layers, whereby the interaction of the sound with turbulent flow results in power lost from the tone and distributed into a broadband field around the tone frequency. Fan and turbine tone spectral broadening is known colloquially as “haystacking”. Recently a new analytical model has been derived to predict weak spectral broadening of a tone radiated through a circular jet. A key part of the modeling is the choice of the two-point turbulent velocity cross-correlation function, which is used to provide a statistical description of the turbulence in the shear layer. A new cross-correlation function for an axisymmetric turbulent shear layer formed by a circular jet, based on the theory for homogeneous axisymmetric turbulence, has been developed. Validation results of weak-scattering calculated using this correlation function show better agreement with measurements when compared with the results calculated using a correlation function based on the theory for homogeneous isotropic turbulence.

Dissipation rate estimation in the turbulent boundary layer using high-speed planar particle image velocimetry

The challenge of turbulent kinetic energy dissipation rate estimation, in general, is associated with errors embedded in measured velocity vectors and spatial resolution of appropriate optical method. Different optical methods together with different filtering procedures may yield completely different estimations. To evaluate the sources for such discrepancies, key parameters and factors which directly affect the accuracy of the dissipation rate estimation are briefly discussed. A joint effect of random error, velocity vector spacing, IW size and IW overlap on the dissipation rate estimation is theoretically and experimentally shown. In the present study, we demonstrate that high-speed planar particle image velocimetry (PIV) enables accurate estimation of the dissipation rate if high spatiotemporal resolution and appropriate local temporal filtering procedure are utilized. We consider a synthetic linear shear flow and a turbulent boundary layer flow using an assumption of locally axisymmetric turbulence when estimating the dissipation rate. The calculated dissipation rate profiles are compared with measurements by the state-of-the-art optical methods such as planar smoke image velocimetry (SIV), Stereo PIV, Tomo PIV, Tomo PTV–VIC+, and DNS results. The advantages of the implemented technique compared to others and the temporal filtering procedure are discussed.

Cascades of energy and helicity in axisymmetric turbulence

A spectral analysis of strictly axisymmetric turbulence is performed. Both freely decaying and statistically steady flows are considered. In helical flows we identify a dual cascade, where energy is transferred towards the large scales and helicity to the smallest ones. It is shown that even in the absence of net helicity, a dual cascade persists, transferring energy backward and positively and negatively polarized helicity fluctuations forward.

Direct numerical simulation of axisymmetric turbulence

How does strictly axisymmetric turbulence behave? Numerical simulations indicate that large-scale structures are formed, in agreement with theoretical predictions based on statistical mechanics.

Rapid distortion theory analysis on the interaction between homogeneous turbulence and a planar shock wave

The interactions between homogeneous turbulence and a planar shock wave are analytically investigated using rapid distortion theory (RDT). Analytical solutions in the solenoidal modes are obtained. Qualitative answers to unsolved questions in a report by Andreopoulos et al. (Annu. Rev. Fluid Mech., vol. 524, 2000, pp. 309–345) are provided within the linear theoretical framework. The results show that the turbulence kinetic energy (TKE) is increased after interaction with a shock wave and that the contributions to the amplification can be interpreted primarily as the combined effect of shock-induced compression, which is a direct consequence of the Rankine–Hugoniot relation, and the nonlinear effect, which is an indirect consequence of the Rankine–Hugoniot relation via the perturbation manner. For initial homogeneous axisymmetric turbulence, the amplification of the TKE depends on the initial degree of anisotropy. Furthermore, the increase in energy at high wavenumbers is confirmed by the one-dimensional spectra. The enstrophy is also increased; its increase is more significant than that of the TKE because of the significant increase in enstrophy at high wavenumbers. The vorticity components perpendicular to the shock-induced compressed direction are amplified more than the parallel vorticity component. These results strongly suggest that a high resolution is needed to obtain accurate results for the turbulence–shock-wave interaction. The integral length scales ($L$) and the Taylor microscales ($\unicode[STIX]{x1D706}$) are decreased for most cases after the interaction. However, $L_{22,3}(=\,L_{33,2})$ and $\unicode[STIX]{x1D706}_{22,3}(=\,\unicode[STIX]{x1D706}_{33,2})$ are amplified. Here, the subscripts 2 and 3 indicate the perpendicular components relative to the shock-induced compressed direction. The dissipation length and TKE dissipation rate are amplified.

Conserved scalar mixing in a confined-opposed-jets flow

The focus of this paper is on the mixing of a conserved passive scalar for Sc = 1 (Sc is the Schmidt number) in axisymmetric turbulence for which the initial injections of turbulent kinetic energy and scalar variance are similar. Two confined-opposed-jets (COJ) are experimentally studied through simultaneous PIV (particle image velocimetry) and PLIF (planar laser induced fluorescence) measurements, for different flow regimes. One-point transport equation for the scalar variance is assessed through experimental data, along the common axis of the two opposed jets, and different physical phenomena are revealed (production, diffusion, dissipation). The production of scalar variance is equilibrated by the diffusion term (∼75%) and the mean dissipation of the scalar variance (∼25%). To further assess the scalar behaviour at each scale in this anisotropic, but axisymmetric, flow, a scale-by-scale scalar variance budget equation is derived for axisymmetric turbulence. This equation reduces to Yaglom's 4/3 law, under additional restrictions. The equation is assessed through experimental data, in the impingement region between the two COJ. In particular, the anisotropic energy transfer along different directions is quantified. It is shown that for scales smaller than the size of the central region, Δ, the cascade of the scalar variance is completely inhibited, independently of the particular direction. For scales larger than Δ, the apparent aspect of the energy transfer is that of an inverse cascade, with positive values of the scalar variance transfer. Nonetheless, inhomogeneity of the flow and mixing at those scales is directly responsible for these positive values.

FORWARD MODELING OF REDUCED POWER SPECTRA FROM THREE-DIMENSIONAL K-SPACE

We present results from a numerical forward model to evaluate one-dimensional reduced power spectral densities (PSD) from arbitrary energy distributions in $\mathbf{k}$-space. In this model, we can separately calculate the diagonal elements of the spectral tensor for incompressible axisymmetric turbulence with vanishing helicity. Given a critically balanced turbulent cascade with $k_\|\sim k_\perp^\alpha$ and $\alpha<1$, we explore the implications on the reduced PSD as a function of frequency. The spectra are obtained under the assumption of Taylor's hypothesis. We further investigate the functional dependence of the spectral index $\kappa$ on the field-to-flow angle $\theta$ between plasma flow and background magnetic field from MHD to electron kinetic scales. We show that critically balanced turbulence asymptotically develops toward $\theta$-independent spectra with a slope corresponding to the perpendicular cascade. This occurs at a transition frequency $f_{2D}(L,\alpha,\theta)$, which is analytically estimated and depends on outer scale $L$, critical balance exponent $\alpha$ and field-to-flow angle $\theta$. We discuss anisotropic damping terms acting on the $\mathbf{k}$-space distribution of energy and their effects on the PSD. Further, we show that the spectral anisotropies $\kappa(\theta)$ as found by Horbury et al. (2008) and Chen et al. (2010) in the solar wind are in accordance with a damped critically balanced cascade of kinetic Alfv\'en waves. We also model power spectra obtained by von Papen et al. (2014) in Saturn's plasma sheet and find that the change of spectral indices inside $9\,R_\mathrm{s}$ can be explained by damping on electron scales.

A scanning PIV method for fine-scale turbulence measurements

A hybrid technique is presented that combines scanning PIV with tomographic reconstruction to make spatially and temporally resolved measurements of the fine-scale motions in turbulent flows. The technique uses one or two high-speed cameras to record particle images as a laser sheet is rapidly traversed across a measurement volume. This is combined with a fast method for tomographic reconstruction of the particle field for use in conjunction with PIV cross-correlation. The method was tested numerically using DNS data and with experiments in a large mixing tank that produces axisymmetric homogeneous turbulence at \(R_\lambda \simeq 219\). A parametric investigation identifies the important parameters for a scanning PIV set-up and provides guidance to the interested experimentalist in achieving the best accuracy. Optimal sheet spacings and thicknesses are reported, and it was found that accurate results could be obtained at quite low scanning speeds. The two-camera method is the most robust to noise, permitting accurate measurements of the velocity gradients and direct determination of the dissipation rate.

Numerical investigation on the partial return to isotropy of freely decaying homogeneous axisymmetric turbulence

The return to isotropy of freely decaying homogeneous axisymmetric turbulence is numerically studied using the Eddy-Damped Quasi-Normal Markovian closure. The model, whose classical formulation has been extended to moderately anisotropic flows by Cambon, Jeandel, and Mathieu [“Spectral modelling of homogeneous non-isotropic turbulence,” J. Fluid Mech. 104, 247–262 (1981)], allows for an accurate description of the turbulence decay for very long times. More specifically, the observed results encompass both the high and low Reynolds number asymptotic regimes. Such an analysis escapes both wind tunnel experiments and direct numerical simulations possibilities at the present time. Anisotropy generation mechanisms considered in the present paper do not affect the nature of nonlinear interactions and the related energy cascade mechanisms. Their influence on the decay regime is quantified by the investigation of the features of the initial three dimensional kinetic energy spectrum. The initial anisotropy level is always chosen in agreement with experimental grid turbulence observations. The present results show that the relaxation towards an isotropic state is observed in the inertial range of the energy spectrum E(k, t) during the initial high Reynolds regime, while the large scales conserve the imposed anisotropy level if the permanence of large eddies hypothesis is verified. A direct consequence is that the ratio between axial and transverse kinetic energies γ increases when the low Reynolds asymptotic regime is reached. Saturation effects, i.e., effects related to the boundedness of the physical domain under consideration, are also investigated. Isotropic saturation (same cut off scale in every direction) leads to the relaxation towards a fully isotropic state in all cases, whereas anisotropic saturation (unbounded domain in the axial direction) leads to an amplification of the initial anisotropy.

Numerical investigation of flow through a triangular duct: The coexistence of laminar and turbulent flow

Experimental studies of turbulent flow through a triangular duct with small apex angle of 11.5° performed by Eckert and Irvine (1956) show flow laminarisation in the corner region of the duct. This effect is re-investigated using direct numerical simulation (DNS). In order to analyze the impact of duct corners on the flow behavior, results for the friction factor and the mean velocity profiles arising from different non-circular duct geometries, namely a square duct, an equilateral triangular duct and isosceles triangular ducts with an apex angle of 11.5° and 4° are compared with the results for circular pipe flow. Within the cross-sections of the ducts with the small apex angles, regions of essentially laminar and turbulent properties are found to exist simultaneously. From analysis of the turbulent quantities performed exemplary for the 11.5°-triangle we gain further insights into the mechanisms responsible for flow laminarization which are supported by analytical considerations for statistically axisymmetric turbulence.