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.     

The mean-field method in the problem of acoustic wave propagation in a turbulent atmosphere

We analyze the problem of acoustic-wave propagation in a turbulent atmosphere using the mean-field method. The equation for the sound pressure is written with accuracy up to terms that are linear with respect to the Mach number of the turbulent air flow. An expression for the attenuation constant of the mean field is obtained. For the turbulence model described by the von Kármán correlation function of fluctuations, the attenuation coefficient of the mean field is numerically studied in detail. It is shown that under typical conditions of the near-ground atmospheric layer, the predominant contribution to scattering of acoustic waves is given by the turbulent motion of the air masses. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 51, No. 5, pp. 413–424, May 2008.     

Differential integral equation for the spectrum of isotropic homogeneous turbulence

The double and triple velocity correlations for isotropic homogeneous turbulence are constructed and used in v. Kármán-Howarth differential equation for isotropic correlations; it is reduced to a differential integral equation for the spectrum of turbulence; it contains only one dependent function (5.2). The equation of energy, which follows from the above equation, can be reduced toHeisenberg's type of equation for the spectrum of turbulence (5.6). Eddies of the same order of magnitudeinteract; they partly generate partly destroy votricity. Small eddies (large wave numbersk)act on large eddies by mainly destroying them.     

Controlling the Bénard-von Kármán instability in the wake of a cylinder by driving the pressure at the front stagnation point

We report an experimental method for inhibiting vortex shedding generated by the Bénard von Kármán instability (BvK) in the wake of a cylinder. We show that monitoring the pressure at the front stagnation point of a circular cylinder can completely suppress the Bénard-von Kármán instability for Reynolds numbers in the range 48.5<Re<150. We then study some properties of the BvK instability in the presence of suction at the front stagnation point and mention that this method can be used to generate well-controlled localized vortical structures in the form of vortex pairs. Received 2 August 1999     

Random particle methods applied to broadband fan interaction noise

Predicting broadband fan noise is key to reduce noise emissions from aircraft and wind turbines. Complete CFD simulations of broadband fan noise generation remain too expensive to be used routinely for engineering design. A more efficient approach consists in synthesizing a turbulent velocity field that captures the main features of the exact solution. This synthetic turbulence is then used in a noise source model. This paper concentrates on predicting broadband fan noise interaction (also called leading edge noise) and demonstrates that a random particle mesh method (RPM) is well suited for simulating this source mechanism. The linearized Euler equations are used to describe sound generation and propagation. In this work, the definition of the filter kernel is generalized to include non-Gaussian filters that can directly follow more realistic energy spectra such as the ones developed by Liepmann and von Kármán. The velocity correlation and energy spectrum of the turbulence are found to be well captured by the RPM. The acoustic predictions are successfully validated against Amiet’s analytical solution for a flat plate in a turbulent stream. A standard Langevin equation is used to model temporal decorrelation, but the presence of numerical issues leads to the introduction and validation of a second-order Langevin model.     

Transition scenario to turbulence in thin vibrating plates

A thin plate, excited by a harmonic external forcing of increasing amplitude, shows transitions from a periodic response to a chaotic state of wave turbulence. By analogy with the transition to turbulence observed in fluid mechanics as the Reynolds number is increased, a generic transition scenario for thin vibrating plates, first experimentally observed, is here numerically studied. The von Kármán equations for thin plates, which include geometric non-linear effects, are used to model large amplitude vibrations, and an energy-conserving finite difference scheme is employed for discretisation. The transition scenario involves two bifurcations separating three distinct regimes. The first regime is the periodic, weakly non-linear response. The second is a quasiperiodic state where energy is exchanged between internally resonant modes. It is observed only when specific internal resonance relationships are fulfilled between the eigenfrequencies of the structure and the forcing frequency; otherwise a direct transition to the last turbulent state is observed. This third, or turbulent, regime is characterized by a broadband Fourier spectrum and a cascade of energy from large to small wavelengths. For perfect plates including cubic non-linearity, only third-order internal resonances are likely to exist. For imperfect plates displaying quadratic nonlinearity, the energy exchanges and the quasiperiodic states are favored and thus are more easily obtained. Finally, the turbulent regime is characterized in the light of available theoretical results from wave turbulence theory.     

Linear and non-linear features of the Taylor–Green dynamo

The Taylor–Green flow is a model flow sharing many properties with the von Kármán flow, in which experimental turbulent dynamo action has recently been achieved. We present here recent numerical results on the Taylor–Green dynamo instability, both in the linear and non-linear regime. Various properties are considered, such as the influence of turbulence, the energy transfer between different scales, the spatial structure of the neutral mode, the nature of the bifurcation and the saturation mechanisms. We also discuss the role of the velocity fluctuations on the dynamo onset. To cite this article: Y. Ponty et al., C. R. Physique 9 (2008).     

Simulations of turbulent asymptotic suction boundary layers

A series of large-eddy simulations of a turbulent asymptotic suction boundary layer (TASBL) was performed in a periodic domain, on which uniform suction was applied over a flat plate. Three Reynolds numbers (defined as ratio of free-stream and suction velocity) of Re = 333, 400 and 500 and a variety of domain sizes were considered in temporal simulations in order to investigate the turbulence statistics, the importance of the computational domain size, the arising flow structures as well as temporal development length required to achieve the asymptotic state. The effect of these two important parameters was assessed in terms of their influence on integral quantities, mean velocity, Reynolds stresses, higher order statistics, amplitude modulation and spectral maps. While the near-wall region up to the buffer region appears to scale irrespective of Re and domain size, the parameters of the logarithmic law (i.e. von Kármán and additive coefficient) decrease with increasing Re, while the wake strength decreases with increasing spanwise domain size and vanishes entirely once the spanwise domain size exceeds approximately two boundary-layer thicknesses irrespective of Re. The wake strength also reduces with increasing simulation time. The asymptotic state of the TASBL is characterised by surprisingly large friction Reynolds numbers and inherits features of wall turbulence at numerically high Re. Compared to a turbulent boundary layer (TBL) or a channel flow without suction, the components of the Reynolds-stress tensor are overall reduced, but exhibit a logarithmic increase with decreasing suction rates, i.e. increasing Re. At the same time, the anisotropy is increased compared to canonical wall-bounded flows without suction. The reduced amplitudes in turbulence quantities are discussed in light of the amplitude modulation due to the weakened larger outer structures. The inner peak in the spectral maps is shifted to higher wavelength and the strength of the outer peak is much less than for TBLs. An additional spatial simulation was performed, in order to relate the simulation results to wind tunnel experiments, which – in accordance with the results from the temporal simulation – indicate that a truly TASBL is practically impossible to realise in a wind tunnel. Our unique data set agrees qualitatively with existing literature results for both numerical and experimental studies, and at the same time sheds light on the fact why the asymptotic state could not be established in a wind tunnel experiment, viz. because experimental studies resemble our simulation results from too small simulation boxes or insufficient development times.     

Numerical simulation and experiments of a control method to suppress the Bénard von Kármán instability

We report both two-dimensional numerical simulations and experimental results that confirm the robustness of a new method for inhibiting vortex shedding associated to the Bénard-von Kármán (BvK) instability in the wake of a cylinder. Using the SIMPLER algorithm on a 2D channel, we solve the Navier-Stokes equations and we show that pressure suction at the front stagnation point of a circular cylinder, modelled here through a point sink located at the front stagnation point, can completely suppress the Bénard-von Kármán instability for super-critical Reynolds numbers. Comparison with recent experimental results are in close agreement. Received 7 March 2002 / Received in final form 12 September 2002 Published online 29 November 2002     

偶极玻色-爱因斯坦凝聚体在类方势阱中的Bénard-von Kármán涡街

对偶极玻色-爱因斯坦凝聚体（Bose-Einstein condensate,BEC）在类方势阱中的Bénard-von Kármán涡街现象进行了数值研究.结果表明,当障碍势在BEC中的运动速度与尺寸在适当范围内时,系统中会出现稳定的两列涡旋对阵列,即Bénard-von Kármán涡街.研究了偶极相互作用强弱、障碍势尺寸以及运动速度对尾流中产生的涡旋结构的影响,得到了相图结构.对障碍势所受拖拽力进行计算,分析了涡旋对产生的力学机理.     

On the third moments in helical turbulence

The evolution of the correlation characteristics in homogeneous helical turbulence is studied. Additional Kármán-Howarth-type equations describing the evolution of the mixed correlation tensor of the velocity and vorticity are obtained. In the helical scaling region, the solution of the obtained equation gives the exact relation between the antisymmetric component of a rank-three correlation tensor and the average dissipation of helicity; this relation is a kind of analog of Kolmogorov’s well-known 4/5 law [A. N. Kolmogrov, Dokl. Akad. Nauk SSSR 32(1), 19 (1941)]. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 10, 768–772 (25 May 1996)     

Chaotic dynamics of flexible beams with piezoelectric and temperature phenomena

The Euler–Bernoulli kinematic model as well as the von Kármán geometric non-linearity are used to derive the PDEs governing flexible beam vibrations. The beam is embedded into a 2D temperature field, and its surface is subjected to action of the electric potential. We report how an increase of the exciting load amplitude yields the beam turbulent behavior, and how the temperature changes a scenario from a regular/laminar to spatio-temporal/turbulent dynamics. Both classical Fourier analysis and Morlet wavelets are used to monitor a strong influence of temperature on regular and chaotic beam dynamics.     

Statistical symmetries and its impact on new decay modes and integral invariants of decaying turbulence

Classical decay laws of isotropic turbulence usually derived from the von Kármán–Howarth equation are essentially based on two paradigms. First, scaling symmetries of space and time, both tracing back to the Navier–Stokes equations in the limit of large Reynolds numbers (or r?η), give rise to a temporal power-law decay for the turbulent kinetic energy and at the same time an algebraic growth of the integral length scale at an exponent that is uniquely coupled to the latter energy decay. Second, global invariants such as Birkhoff or Loitsianskii integrals determine the exponent of both power laws. We presently show that this class of decay laws may be considerably extended considering the entire set of multi-point correlation equations that admit a much wider class of symmetries. It was recently shown that these new symmetries are of paramount importance, e.g. in deriving the logarithmic law of the wall being an analytic solution of the multi-point equations. For the present case, it is particularly an additional scaling group, which we call statistical scaling group, that gives rise to two additional families of ‘canonical’ decay laws including those with an exponential characteristic for both the kinetic energy and the integral length scale. Finally, a second rather generic group admitted by all linear differential equations corresponding to the superposition principle induces an infinite set of scaling laws of rather complex form that may match rather generic initial conditions. All scaling laws are analyzed in the light of the above-mentioned integral invariants that have been further extended in the present contribution to an exponential-type invariant.     

Nonlinear size-dependent longitudinal vibration of carbon nanotubes embedded in an elastic medium

In this paper, we study the longitudinal linear and nonlinear free vibration response of a single walled carbon nanotube (CNT) embedded in an elastic medium subjected to different boundary conditions. This formulation is based on a large deformation analysis in which the linear and nonlinear von Kármán strains and their gradient are included in the expression of the strain energy and the velocity and its gradient are taken into account in the expression of the kinetic energy. Therefore, static and kinetic length scales associated with both energies are introduced to model size effects. The governing motion equation along with the boundary conditions are derived using Hamilton's principle. Closed-form solutions for the linear free vibration problem of the embedded CNT rod are first obtained. Then, the nonlinear free vibration response is investigated for various values of length scales using the method of multiple scales.     

On applying the extended intrinsic mean spin tensor to modelling the turbulence in non-inertial frames of reference

Modelling the turbulent flows in non-inertial frames of reference has long been a challenging task. Recently we introduced the notion of the “extended intrinsic mean spin tensor” for turbulence modelling and pointed out that, when applying the Reynolds stress models developed in the inertial frame of reference to modelling the turbulence in a non-inertial frame of reference, the mean spin tensor should be replaced by the extended intrinsic mean spin tensor to correctly account for the rotation effects induced by the non-inertial frame of reference, to conform in physics with the Reynolds stress transport equation. To exemplify the approach, we conducted numerical simulations of the fully developed turbulent channel flow in a rotating frame of reference by employing four non-linear K-ε models. Our numerical results based on this approach at a wide range of Reynolds and Rossby numbers evince that, among the models tested, the non-linear K-ε model of Huang and Ma and the non-linear K-ε model of Craft, Launder and Suga can better capture the rotation effects and the resulting influence on the structures of turbulence, and therefore are satisfactorily applied to dealing with the turbulent flows of practical interest in engineering. The general approach worked out in this paper is also applied to the second-moment closure and the large-eddy simulation of turbulence.     

Coherence and Reynolds stresses in the turbulent wake behind a curved circular cylinder

The turbulent wake behind a curved circular cylinder is investigated based on data obtained from a direct numerical simulation. Here, emphasis is placed in the assessment of two approaches for simplified modelling: reduced-order modelling (ROM) and Reynolds-averaged Navier–Stokes equations. To this end, the instantaneous vortical structures, the proper orthogonal decomposition (POD) of the flow, and relevant Reynolds stress components have been analysed. The results show that despite the complexity of the instantaneous vortical structures, the wake dynamics are governed by the quasi-periodic shedding of primary vortices. Between 24% and 50% of the kinetic energy in the POD is captured by the two most energetic modes, and about 200 modes are needed to capture 90% of the kinetic energy. These findings suggest that, as long as the large-scale structures of the von Kármán vortex shedding are concerned, the present case can be approached by ROM; but a detailed representation of the flow dynamics without an eddy viscosity model that accounts for the unresolved turbulent fluctuations would require a large amount of degrees of freedom. Concerning the Reynolds stresses, their magnitude varies considerably depending on the depth at which they have been sampled. This dependence is related to the strength of the vortex shedding, and the intensity of the secondary flows induced by the curvature of the cylinder. As a consequence of the combination of these two effects, the correlation between streamwise and vertical velocity fluctuations is highest in the wake behind the midspan of the curved cylinder, and the correlation between cross-flow and vertical velocity fluctuations reaches large values in the lower wake.     

Impact of nanoparticles and radiative heat flux in von Kármán swirling flow of Maxwell fluid

In this paper, the classical von Kármán swirling flow problem due to a rotating disk is modeled and studied for the rate type Maxwell nanofluid together with heat and mass transfer mechanisms. The model under consideration predicts the relaxation time characteristics. The novel aspects of thermophoresis and Brownian motion features due to nanoparticles are investigated by employing an innovative Buongiorno’s model. The analysis further explores the impact of linear Rosseland radiation on heat transfer characteristics. The concept of boundary layer approximations is utilized to formulate the basic governing equations of Maxwell fluid. The dimensionless form of a system of ordinary differential equations is obtained through similarity approach adopted by von Kármán. The system of equations is integrated numerically in domain [0,∞) by using bvp midrich scheme in Maple software. The obtained results intimate that higher rotation raises the radial and angular velocity components. The nano-particles concentration enhances with Brownian motion parameter. Further, the heat transfer rate at the disk surface diminishes with thermophoresis parameter. The achieved numerical computations of velocity profiles, friction coefficient and Nusselt number are matched in limiting cases with previously published literature and an outstanding agreement is observed.     

Historical note: On the simulation of dispersive wave propagation by elasticity models

Attention is called to E. Schrödinger's elegant analytical solution [Annalen der Physik 44 (1914) p.916] of the initial-value problem for the Born–von Kármán model of an infinite one-dimensional chain of uniformly spaced particles of mass M with nearest-neighbour coupling by harmonic springs. This model has recently served as the starting point for a computer study of the transition to partial differential equations describing dispersive wave propagation in inhomogeneous media [Askes et al., Phil. Mag. 88 (2008) p.3415]. Schrödinger's solution allows the main features of the limit process involved in this transition to be studied in a straightforward way.     

A Reynolds stress model for turbulent flows of viscoelastic fluids

A second-order closure is developed for predicting turbulent flows of viscoelastic fluids described by a modified generalised Newtonian fluid model incorporating a nonlinear viscosity that depends on a strain-hardening Trouton ratio as a means to handle some of the effects of viscoelasticity upon turbulent flows. Its performance is assessed by comparing its predictions for fully developed turbulent pipe flow with experimental data for four different dilute polymeric solutions and also with two sets of direct numerical simulation data for fluids theoretically described by the finitely extensible nonlinear elastic – Peterlin model. The model is based on a Newtonian Reynolds stress closure to predict Newtonian fluid flows, which incorporates low Reynolds number damping functions to properly deal with wall effects and to provide the capability to handle fluid viscoelasticity more effectively. This new turbulence model was able to capture well the drag reduction of various viscoelastic fluids over a wide range of Reynolds numbers and performed better than previously developed models for the same type of constitutive equation, even if the streamwise and wall-normal turbulence intensities were underpredicted.     

Kinematic dynamo simulations of von Kármán flows: application to the VKS experiment

The VKS experiment has evidenced dynamo action in a highly turbulent liquid sodium von Kármán flow [R. Monchaux et al., Phys. Rev. Lett. 98, 044502 (2007)]. However, the existence and the onset of a dynamo happen to depend on the experimental configuration. Performing kinematic dynamo simulations on real flows, we study the influence of the configuration on dynamo action, namely the sense of rotation and the presence of an annulus in the shear layer plane. The 3 components of the mean velocity fields are measured in a water prototype for different VKS configurations through Stereoscopic Particle Imaging Velocimetry. Experimental data are then processed in order to use them in a periodic cylindrical kinematic code. Even if the kinematic predicted mode appears to be different from the experimental saturated one, the results concerning the existence of a dynamo and the thresholds are in qualitative agreement, showing the importance of the flow characteristics.