Influence of polymer additives on turbulent energy cascading in forced homogeneous isotropic turbulence studied by direct numerical simulations

Direct numerical simulations （DNS） were performed for the forced homogeneous isotropic turbulence （FHIT） with/without polymer additives in order to elaborate the characteristics of the turbulent energy cascading influenced by drag-reducing effects. The finite elastic non-linear extensibility-Peterlin model （FENE-P） was used as the conformation tensor equation for the viscoelastic polymer solution. Detailed analyses of DNS data were carried out in this paper for the turbulence scaling law and the topological dynamics of FHIT as well as the important turbulent parameters, including turbulent kinetic energy spectra, enstrophy and strain, velocity structure function, small-scale intermittency, etc. A natural and straightforward definition for the drag reduction rate was also proposed for the drag-reducing FHIT based on the decrease degree of the turbulent kinetic energy. It was found that the turbulent energy cascading in the FHIT was greatly modified by the drag-reducing polymer additives. The enstrophy and the strain fields in the FH1T of the polymer solution were remarkably weakened as compared with their Newtonian counterparts. The small-scale vortices and the small-scale intermittency were all inhibited by the viscoelastic effects in the FHIT of the polymer solution. However, the scaling law in a fashion of extended self-similarity for the FHIT of the polymer solution, within the presently simulated range of Weissenberg numbers, had no distinct differences compared with that of the Newtonian fluid case.

The influence of sub-grid scale motions on particle collision in homogeneous isotropic turbulence

The absence of sub-grid scale(SGS) motions leads to severe errors in particle pair dynamics, which represents a great challenge to the large eddy simulation of particle-laden turbulent flow. In order to address this issue,data from direct numerical simulation(DNS) of homogenous isotropic turbulence coupled with Lagrangian particle tracking are used as a benchmark to evaluate the corresponding results of filtered DNS(FDNS). It is found that the filtering process in FDNS will lead to a non-monotonic variation of the particle collision statistics, including radial distribution function, radial relative velocity, and the collision kernel. The peak of radial distribution function shifts to the large-inertia region due to the lack of SGS motions, and the analysis of the local flowstructure characteristic variable at particle position indicates that the most effective interaction scale between particles and fluid eddies is increased in FDNS. Moreover,this scale shifting has an obvious effect on the odd-order moments of the probability density function of radial relative velocity, i.e. the skewness, which exhibits a strong correlation to the variance of radial distribution function in FDNS.As a whole, the radial distribution function, together with radial relative velocity, can compensate the SGS effects for the collision kernel in FDNS when the Stokes number based on the Kolmogorov time scale is greater than 3.0. However,it still leaves considerable errors for St< 3.0.

Direct numerical simulation study of the interaction between the polymer effect and velocity gradient tensor in decaying homogeneous isotropic turbulence

Direct numerical simulation of decaying homogeneous isotropic turbulence （DHIT） of a polymer solution is performed. In order to understand the polymer effect on turbulence or additive-turbulence interaction, we directly investigate the influence of polymers on velocity gradient tensor including vorticity and strain. By visualizing vortex tubes and sheets, we observe a remarkable inhibition of vortex structures in an intermediate-scale field and a small-scale field but not for a large scale field in DHIT with polymers. The geometric study indicates a strong relevance among the vorticity vector, rate-of-strain tensor, and polymer conformation tensor. Joint probability density functions show that the polymer effect can increase ＂strain generation resistance＂ and ＂vorticity generation resistance＂, i.e., inhibit the generation of vortex sheets and tubes, ultimately leading to turbulence inhibition effects.

Large-eddy simulations of a forced homogeneous isotropic turbulence with polymer additives

Large-eddy simulations （LES） based on the temporal approximate deconvolution model were performed for a forced homogeneous isotropic turbulence （FHIT） with polymer additives at moderate Taylor Reynolds number. Finitely extensible nonlinear elastic in the Peterlin approximation model was adopted as the constitutive equation for the filtered conformation tensor of the polymer molecules. The LES results were verified through comparisons with the direct numerical simulation results. Using the LES database of the FHIT in the Newtonian fluid and the polymer solution flows, the polymer effects on some important parameters such as strain, vorticity, drag reduction, and so forth were studied. By extracting the vortex structures and exploring the flatness factor through a high-order correlation function of velocity derivative and wavelet analysis, it can be found that the small-scale vortex structures and small-scale intermittency in the FHIT are all inhibited due to the existence of the polymers. The extended self-similarity scaling law in the polymer solution flow shows no apparent difference from that in the Newtonian fluid flow at the currently simulated ranges of Reynolds and Weissenberg numbers.

Description of inverse energy cascade in homogeneous isotropic turbulence using an eigenvalue method

A description of inverse energy cascade(from small scale to large scale)in homogeneous isotropic turbulence is introduced by using an eigenvalue method.We show a special isotropic turbulence,in which the initial condition is constructed by reversing the velocity field in space,i.e.,the time-reversed turbulence.It is shown that the product of eigenvalues of the rate-of-strain tensor can quantitatively describe the backward energy transfer process.This description is consistent to the velocity derivative skewness Sk.However,compared with Sk,it is easier to be obtained,and it is expected to be extended to anisotropic turbulence.Furthermore,this description also works for the resolved velocity field,which means that it can be used in engineering turbulent flows.The description presented here is desired to inspire future investigation for the modeling of the backward energy transfer process and lay the foundation for the accurate prediction of complex flows.

A closure model on velocity structure functions in homogeneous isotropic turbulence

Closure models started from Chou＇s work have been developed for more than 70 years, aiming at providing analytical tools to describe turbulent flows in the spectral space. In this study, a preliminary attempt is presented to introduce a closure model in the physical space, using the velocity structure functions as key parameters. The present closure model appears to qualitatively reproduce the asymptotic scaling behav- iors at small and large scales, despite some inappropriate behaviors such as oscillations. Therefore, further improvements of the present model are expected to provide appropriate descriptions of turbulent flows in the physical space.

THE SIZE EFFECT ON THE DISPERSION OF A PARTICLE IN A HOMOGENEOUS ISOTROPIC TURBULENCE

The mechanism of the response motion of a suspended particle to turbulent motion of its surrounding fluid is different according to si:e of turbulent eddies. The particle is dragged by the viscous force of large eddies, and meanwhile driven randomly by small eddies. Based on this understanding, the dispersion of a particle with finite size in a homogeneous isotropic turbulence is calculated in this study. Results show that there are two competing effects: when enhanced by the inertia of a particle, the long-term particle diffusivity is reduced by the finite size of the particle.

ON THE SOLUTIONS OF NAVIER-STOKES EQUATIONS AND THE THEORY OF HOMOGENEOUS ISOTROPIC TURBULENCE

The present paper is a further development of our previous work in solving the wholeproblem of the homogeneous isotropic turbulence from the nitial period to the final period ofdecay. An expansion method is developed to obtain the axinlly symmetrical solution of theNavier-Stokes equations of motion in the form of an infinite set of nonlinear partial differen-tial equations of the second order. For the present we solve the zeroth order approximation.By using the method of Fourier transform, we get a nonlinear nitegro-differential equationfor the amplitude function in the wave number space.It is also the dynamical equation forthe energy spectrum. By choosing a suitable initial condition, we solve this equation numerically. The energyspectrum function and the energy transfer spectrum function thus calculated satisfy the spec-trum form of the karman-Howarth equation exactly. We Lave computed the energy spectrumfunction, the energy transfer function the decay of turbulent energy, the integral scale, Taylormicroscale, the double and triple velocity correlations on the whole range from the initialperiod to the final period of decay. As a whole all these calculated statistical physicalquantities agree with experiments very wall except a few cases with small discrepancies at largeseparations.

Correlation between large-and small-scale velocity components under Fourier filters

By introducing the Fourier filters, we analyse the correlation between large- and small-scale velocity components in homogeneous isotropic turbulence theoretically. We show that different Fourier filters act similarly on this multiscale correlation with a ＂natural＂ mechanism of removing the physical correlations between large- and small-scale velocity components. This conclusion calls for the further investigation on the Hilbert-Huang decomposition to investigate the mechanism of Marusic et al （2008）.

A Memory-Saving Algorithm for Spectral Method of Three-Dimensional Homogeneous Isotropic Turbulence

Homogeneous isotropic turbulence has been playing a key role in the research of turbulence theory.And the pseudo-spectral method is the most popular numerical method to simulate this type of flow fields in a periodic box,where fast Fourier transform(FFT)is mostly effective.However,the bottle-neck in this method is the memory of computer,which motivates us to construct a memory-saving algorithm for spectral method in present paper.Inevitably,more times of FFT are needed as compensation.In the most memory-saving situation,only 6 three-dimension arrays are employed in the code.The cost of computation is increased by a factor of 4,and that 38 FFTs are needed per time step instead of the previous 9 FFTs.A simulation of isotropic turbulence on 20483 grid can be implemented on a 256G distributed memory clusters through this method.

A Liutex-based subgrid stress model for large-eddy simulation

The concept of vortex is crucial in both understanding and modeling of turbulence.For large eddy simulation(LES),the effect of small-scale eddies onto the large scales or the resolved flow field is modeled by subgrid stress models.Even though the rotating motions of fluids,i.e.,vortices or eddies are central in developing turbulent models,vortex identification methods are seldom used in these models.In this study,we develop a new subgrid model based on the Liutex vector,a new quantity introduced to decompose fluid motions into rigid rotation,pure shear and stretching,and thus identify vortices.The new model is then applied in a decaying homogeneous isotropic turbulence(DHIT)and a turbulent channel flow at Reynolds number Reτ=180.It is shown that the new model can predict accurate energy spectra compared with experiments in DHIT and give a well-matched velocity profile in turbulent channel flow without changing the form of the model.Future directions include improvement of the Liutex based model,for example developing anisotropic subgrid models,and its applications in various turbulent flows.