New scaling for compressible wall turbulence

Classical Mach-number(M) scaling in compressible wall turbulence was suggested by van Driest(Van Driest E R.Turbulent boundary layers in compressible fluids.J Aerodynamics Science,1951,18(3):145-160) and Huang et al.(Huang P G,Coleman G N,Bradshaw P.Compressible turbulent channel flows:DNS results and modeling.J Fluid Mech,1995,305:185-218).Using a concept of velocity-vorticity correlation structure(VVCS),defined by high correlation regions in a field of two-point cross-correlation coefficient between a velocity and a vorticity component,we have discovered a limiting VVCS as the closest streamwise vortex structure to the wall,which provides a concrete Morkovin scaling summarizing all compressibility effects.Specifically,when the height and mean velocity of the limiting VVCS are used as the units for the length scale and the velocity,all geometrical measures in the spanwise and normal directions,as well as the mean velocity and fluctuation(r.m.s) profiles become M-independent.The results are validated by direct numerical simulations(DNS) of compressible channel flows with M up to 3.Furthermore,a quantitative model is found for the M-scaling in terms of the wall density,which is also validated by the DNS data.These findings yield a geometrical interpretation of the semi-local transformation(Huang et al.,1995),and a conclusion that the location and the thermodynamic properties associated with the limiting VVCS determine the M-effects on supersonic wall-bounded flows.     

A turbulence characteristic length scale for compressible flows

The current RANS models are generally established and calibrated under incompressible condition and these kinds of models could succeed in predicting many features of incompressible flows. However, these models extended to the high-speed, compressible flows are always less accurate. In the paper, a compressible von Kármán length scale is proposed for compressible flows considering the variable densities. It contains no empirical coefficients and is based on phenomenological theory. In the turbulent kinetic equation, the extra unclosed terms induced by non-constant densities are treated as dissipation terms and the equation is closed algebraically via the introduction of the von Kármán length scale. The original and the proposed von Kármán length scale lead to two different kinds of SAS (scale adaption simulation) models, KDO (turbulence kinetic energy dependent only) and CKDO (compressible KDO), respectively. Compressible mixing layer with significant compressibility is studied within standard k–?, k–ω, KDO turbulence models and their compressible versions. The compressibility effects such as the reduced mixing layer thickness, growth rate and turbulence intensity can be reproduced by CKDO. The new length scale can improve the performances of the model in predicting the mixing layer thickness, stream-wise velocity and Reynolds shear stresses when the convective Mach number is 0.8. Besides, the new length scale also leads to accurate computed growth rate when the convective Mach number ranges from 0.1 to 1.0.     

A unified wall function for compressible turbulence modelling

Turbulence modelling near the wall often requires a high mesh density clustered around the wall and the first cells adjacent to the wall to be placed in the viscous sublayer. As a result, the numerical stability is constrained by the smallest cell size and hence requires high computational overhead. In the present study, a unified wall function is developed which is valid for viscous sublayer, buffer sublayer and inertial sublayer, as well as including effects of compressibility, heat transfer and pressure gradient. The resulting wall function applies to compressible turbulence modelling for both isothermal and adiabatic wall boundary conditions with the non-zero pressure gradient. Two simple wall function algorithms are implemented for practical computation of isothermal and adiabatic wall boundary conditions. The numerical results show that the wall function evaluates the wall shear stress and turbulent quantities of wall adjacent cells at wide range of non-dimensional wall distance and alleviate the number and size of cells required.     

Magnetohydrodynamic turbulence: Generalized formulation and extension to compressible cases

A general framework that incorporates the Iroshnikov-Kraichnan (IK) and Goldreich-Sridhar (GS) phenomenalogies of magnetohydrodynamic (MHD) turbulence is developed. This affords a clarification of the regimes of validity of the IK and GS models and hence help resolve some controversies on this aspect. This general formulation appears to have a certain robustness as revealed here by its form invariance with respect to inclusion of compressible effects. Generalizations of the IK and GS spectra to compressible MHD turbulence are given. These two branches are shown to merge with the MHD shockwave spectrum, as to be expected, in the infinite compressibility limit.     

Exact relation for correlation functions in compressible isothermal turbulence

Compressible isothermal turbulence is analyzed under the assumption of homogeneity and in the asymptotic limit of a high Reynolds number. An exact relation is derived for some two-point correlation functions which reveals a fundamental difference with the incompressible case. The main difference resides in the presence of a new type of term which acts on the inertial range similarly as a source or a sink for the mean energy transfer rate. When isotropy is assumed, compressible turbulence may be described by the relation -2/3ε(eff)r = F(r)(r), where F(r) is the radial component of the two-point correlation functions and ε(eff) is an effective mean total energy injection rate. By dimensional arguments, we predict that a spectrum in k(-5/3) may still be preserved at small scales if the density-weighted fluid velocity ρ(1/3)u is used.     

Density-PDFs and Lagrangian statistics of highly compressible turbulence

In isothermal, highly compressible turbulent flows, density fluctuations follow a log-normal distribution. We establish a connection between these density fluctuations and the probability-density-functions (PDF) of Lagrangian tracer particles advected with the flow. Our predicted particle statistics is tested against large scale numerical simulations, which were performed with 512³ collocation points and 2 million tracer particles integrated over several dynamical times.     

A priori assessment of the subgrid scale viscous/scalar dissipation closures in compressible turbulence

An appraisal is made of several subgrid scale (SGS) viscous/scalar dissipation closures via a priori analysis of direct numerical simulation data in a temporally evolving compressible mixing layer. The effects of the filter width, the compressibility level and the Schmidt number are studied for several models. Based on the scaling of SGS kinetic energy, a new formulation for SGS viscous dissipation is proposed. This yields the best overall prediction of the SGS viscous dissipation within the inertial subrange. An SGS scalar dissipation model based on the proportionality of the turbulent time scale with the scalar mixing time scale also performs the best for the filter widths in the inertial subrange. Two dynamic methods are implemented for the determination of the model coefficients. The one based on the global equilibrium of dissipation and production is shown to be more satisfactory than the conventional dynamic model.     

Multi-fractal formulation of compressible fully developed turbulence: Parabolic-profile approximation for the singularity spectrum

A parabolic-profile approximation (PPA) for the singularity spectrum D (h) in the multi-fractal model for compressible fully developed turbulence (FDT) is considered and is then extrapolated to the Kolmogorov microscale regime. The generalization of Kolmogorov’s “4/5th law” relating the third-order velocity structure function to the mean energy dissipation rate ε to compressible FDT is considered. The PPA is also shown to afford, unlike the generic multi-fractal model, an analytical calculation of probability distribution function (PDF) of velocity gradients and to describe intermittency corrections for this PDF that complement those provided by homogeneous-fractal model.     

Statistics and structures of pressure and density in compressible isotropic turbulence

We study statistics and structures of pressure and density in the presence of large-scale shock waves in a forced compressible isotropic turbulence using high-resolution numerical simulation. The spectra for pressure and density exhibit a ?2 scaling over an operational definition of the inertial range. Both the numerical simulation and a heuristic PDF model reveal that the PDFs of pressure increment exhibit a ?2 power law region for the separation in the operational definition of inertial range, quantitatively similar to the PDF of pressure gradient, which also displays a ?2 power law region. Moreover, the statistical relation between density increment and pressure increment has been investigated through a shock-relation model. There is a positive correlation between the vorticity magnitude and pressure, which is different from the case of incompressible turbulence. We argue that this difference is due to large-scale shock waves, another type of intermittent structures in addition to vortex structures in incompressible turbulence.     

Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves

Flows in which shock waves and turbulence are present and interact dynamically occur in a wide range of applications, including inertial confinement fusion, supernovae explosion, and scramjet propulsion. Accurate simulations of such problems are challenging because of the contradictory requirements of numerical methods used to simulate turbulence, which must minimize any numerical dissipation that would otherwise overwhelm the small scales, and shock-capturing schemes, which introduce numerical dissipation to stabilize the solution. The objective of the present work is to evaluate the performance of several numerical methods capable of simultaneously handling turbulence and shock waves. A comprehensive range of high-resolution methods (WENO, hybrid WENO/central difference, artificial diffusivity, adaptive characteristic-based filter, and shock fitting) and suite of test cases (Taylor–Green vortex, Shu–Osher problem, shock-vorticity/entropy wave interaction, Noh problem, compressible isotropic turbulence) relevant to problems with shocks and turbulence are considered. The results indicate that the WENO methods provide sharp shock profiles, but overwhelm the physical dissipation. The hybrid method is minimally dissipative and leads to sharp shocks and well-resolved broadband turbulence, but relies on an appropriate shock sensor. Artificial diffusivity methods in which the artificial bulk viscosity is based on the magnitude of the strain-rate tensor resolve vortical structures well but damp dilatational modes in compressible turbulence; dilatation-based artificial bulk viscosity methods significantly improve this behavior. For well-defined shocks, the shock fitting approach yields good results.     

Anti-diffusion interface sharpening technique for two-phase compressible flow simulations

In this paper we propose an interface sharpening technique for two-phase compressible-flow simulations based on volume-of-fluid methods. The idea of sharpening the two-fluid interface is to provide a correction algorithm which can be applied as post-processing to the volume-fraction field after each time step. For this purpose an anti-diffusion equation, i.e. a diffusion equation with a positive diffusion coefficient, is solved to counter-act the numerical diffusion resulting from the underlying VOF discretization. The numerical stability and volume-fraction boundedness in solving the anti-diffusion equation are ensured by a specified discretization scheme. No interface reconstruction and interface normal calculation are required in this method. All flow variables are updated with the sharpened volume-fraction field for ensuring the consistency of the variables, and the update of the phase mass, momentum and energy is conservative. Numerical results for shock-tube and shock-bubble interactions based on the ideal-gas EOS and shock contact problems based on the Mie–Grüneisen EOS show an improved interface resolution. The large-scale interface structures are in good agreement with reference results, and finer small-scale interface structures are recovered in a consistent manner as the grid resolution increases. As compared with reference high grid-resolution numerical results based on AMR algorithms, the interface roll-up phenomena due to the Richtmyer–Meshkov instability and the Kelvin–Helmholtz instability are recovered reliably for shock-bubble interactions involving different ideal gases.     

Kolmogorov turbulence, Anderson localization and KAM integrability

A modified Shastry-Sutherland model with fourth-nearest-neighbor dimerization is studied with strong coupling perturbation theory. One-triplet excitations are calculated up to the second order. Bose-Einstein condensation of one-triplet excitations may induce different ordered states, i.e. collinear antiferromagnetic state, Néel state or incommensurate state when the gap closes by changing the physical parameters or by an external magnetic field. Two-triplet bound states in the S = 1 and S = 2 channels are shown to exist around the symmetric lines of K _x = ±K _y and K _x ± K _y = ±??. For ferromagnetic dimer-dimer interaction, the energies of the bound states in the S = 1 (S = 2) channel lie above (below) the two-triplet continuum. By calculating the existence region, the degeneracy, the symmetry and the dispersion relation, a whole image and main features of the bound states are obtained. Experimental results of (CuX)LaNb₂O₇ with X = Cl, Br are interpreted.     

A class of finite difference schemes with low dispersion and controllable dissipation for DNS of compressible turbulence

In this paper, a class of finite difference schemes which achieves low dispersion and controllable dissipation in smooth region and robust shock-capturing capabilities in the vicinity of discontinuities is presented. Firstly, a sufficient condition for semi-discrete finite difference schemes to have independent dispersion and dissipation is derived. This condition enables a novel approach to separately optimize the dissipation and dispersion properties of finite difference schemes and a class of schemes with minimized dispersion and controllable dissipation is thus obtained. Secondly, for the purpose of shock-capturing, one of these schemes is used as the linear part of the WENO scheme with symmetrical stencils to constructed an improved WENO scheme. At last, the improved WENO scheme is blended with its linear counterpart to form a new hybrid scheme for practical applications. The proposed scheme is accurate, flexible and robust. The accuracy and resolution of the proposed scheme are tested by the solutions of several benchmark test cases. The performance of this scheme is further demonstrated by its application in the direct numerical simulation of compressible turbulent channel flow between isothermal walls.     

Contribution to the nonlinear theory of sound and hydrodynamic turbulence of a compressible liquid

The interaction of sound with hydrodynamic turbulence has been studied in detail. The sound absorption decrement, the correlation time and length and the frequency diffusion coefficient for the acoustic wave packet are calculated. The spectral composition of the sound radiated by a unit, turbulent volume and the spectral energy density of sound in equilibrium with the turbulence are studied. The region of applicability of the kinetic equation for sound with a linear dispersion low is found.     

On the use of the discontinuous Galerkin method for numerical simulation of two-dimensional compressible turbulence with shocks

In this paper, the discontinuous Galerkin (DG) method combined with localized artificial diffusivity is investigated in the context of numerical simulation of broadband compressible turbulent flows with shocks for under-resolved cases. Firstly, the spectral property of the DG method is analyzed using the approximate dispersion relation (ADR) method and compared with typical finite difference methods, which reveals quantitatively that significantly less grid points can be used with DG for comparable numerical error. Then several typical test cases relevant to problems of compressible turbulence are simulated, including one-dimensional shock/entropy wave interaction, two-dimensional decaying isotropic turbulence, and two-dimensional temporal mixing layers. Numerical results indicate that higher numerical accuracy can be achieved on the same number of degrees of freedom with DG than high order finite difference schemes. Furthermore, shocks are also well captured using the localized artificial diffusivity method. The results in this work can provide useful guidance for further applications of DG to direct and large eddy simulation of compressible turbulent flows.     

Quality assessment of localization technique performance in small volume in vivo H MR spectroscopy

A new phantom and evaluation method for experimental evaluation of ¹H-magnetic resonance spectroscopy single volume localization techniques regarding signal contamination (C), defined as the part of the signal originating outside the volume of interest, is presented. The quality assessment method is based on a spherical phantom with an oil/water interface in order to reduce susceptibility effects, and applied for stimulated-echo acquisition method (STEAM) and spin-echo (SE) sequences, echo times of 270, 135, and 10 ms, and cubic volumes of interest (VOI) of 1³, 1.5³, 2³, 2.5³, and 3³ cm³. To be able to mimic measurements of the contamination in three dimensions the physical gradients representing the three orthogonal directions for slice selection were shifted in the pulse sequences. Contamination values in one dimension differed between 6.5% and 8.4% in SE sequences, and between 0.7% and 13.8% in STEAM sequences. In STEAM sequences a decrease of C with increasing VOI size was observed while SE sequences showed comparable C values for the different VOI sizes tested. The total contamination in three dimensions were 19% and 18% in SE and STEAM sequences with a TE of 270 ms, and 7% in a STEAM sequence with a TE of 10 ms, respectively. The presented evaluation method is easily applied to the new phantom and showed high reproducibility.     

大气湍流对激光照射精度影响分析和计算

为了评估大气湍流效应对激光照射器照射精度的影响,理论分析和数值计算了由于大气湍流引入的激光照射误差随着照射距离、大气湍流强度、激光光束半径的变化关系。结果表明:随着照射距离和湍流强度的增加,以及激光光束半径的减小,大气湍流引起的激光照射误差增大,大气湍流对现有机载光电稳瞄系统照射精度影响为10 rad~30 rad。     

Adaptive detection technique for optical wireless communication over strong turbulence channels

Optical wireless communication (OWC) systems use the atmosphere as a propagation medium, so the atmospheric turbulence effects lead to fading related with signal intensity. The received signal of OWC over strong turbulence channels is assumed to be a mixture of K-distributed fading and Gaussian distributed thermal noise. Second-order spectral analysis is unable to separately estimate the mixed signal. In order to mitigate the fading induced by turbulence, the decision threshold-updating algorithm based on second and higher order cumulants is proposed and is able to operate in an unknown turbulence environment. The performance of the adaptive processing scheme has been evaluated by means of Monte Carlo simulations. Simulation results show the improvement of the bit error rate (BER) performance.     

Auditory localization in the vertical plane: accuracy and constraint on bodily movement

In two preliminary experiments, listeners were instructed to limit increasingly the movement of their heads and/or bodies while attempting to localize narrow bands of noise centered on 2.3 or 8.3 kHz. With increasing constraint on movement, the high-frequency band was incorrectly perceived as elevated above the horizon. The low-frequency band, when actually elevated above the horizon, was not so regularly perceived incorrectly as being below the horizon, a finding inconsistent with a previous report. A third experiment, which more closely replicated the task conditions and strategies of the previous study, did tend to reveal the anomalous low-frequency error. The error is explicable as a default response to which listeners whose sensitivity to the vertical dimension, in general, appears imperfect are prone. From various reports, it emerges that about 25% of presumed normally hearing people exhibit this insensitivity.