Session M38: Turbulence: Theory/Modeling

Reynolds Stress Closure for Inertial Frames and Rotating Frames

In a rotating frame-of-reference, the Coriolis acceleration and the mean vorticity field have a profound impact on the redistribution of kinetic energy among the three components of the fluctuating velocity. Consequently, the normalized Reynolds (NR) stress is not objective. Furthermore, because the Reynolds stress is defined as an ensemble average of a product of fluctuating velocity vector fields, its eigenvalues must be non-negative for all turbulent flows. These fundamental properties (realizability and non-objectivity) of the NR-stress cannot be compromised in computational fluid dynamic (CFD) simulations of turbulent flows in either inertial frames or in rotating frames. The recently developed universal realizable anisotropic prestress (URAPS) closure for the NR-stress depends explicitly on the local mean velocity gradient and the Coriolis operator [1]. The URAPS-closure is a significant paradigm shift from turbulent closure models that assume that dyadic-valued operators associated with turbulent fluctuations are objective. [1] Koppula, K.S., A. Benard, and C. A. Petty, 2011, Turbulent Energy Redistribution in Spanwise Rotating Channel Flows, Ind. Eng. Chem. Res., 50 (15), 8905-8916.   

Toward a generalized equation for the Reynolds stress: Turbulence momentum balance in non-canonical flows.

Recently, we developed a theoretical basis for determination of the Reynolds stress in canonical flows. Writing momentum balance for a control volume moving at the local mean velocity, along with a differential transform $\frac{\partial }{\partial x}=C_{1} U\frac{\partial }{\partial y}$, a turbulence momentum balance is discovered which includes the Reynolds stress as a function of root turbulence parameters: $\frac{\partial (u'v')}{\partial y}=-C_{1} U\frac{\partial u'^{2}}{\partial y}+\nu_{m} \frac{\partial^{2}u_{rms} '}{\partial y^{2}}$. Then, the Reynolds stress can simply be computed by integrating in the y-direction using the right-hand side (RHS). This is obviously a far simplification of complex modeling of the Reynolds stress, but contains the correct physics, as borne out by comparisons with experimental and DNS data in canonical flows in our earlier works (e.g. in APS 2016). The RHS contains only two parameters, U and u'. In this work, we seek extensions of this solution to non-canonical flows such as wakes, flow over a step, and mixing layers. Comparisons with experimental and DNS data will be presented.   

Multiple stages of decay in two-dimensional turbulence

We report measurements of the free decay of turbulence in a quasi-two-dimensional laboratory flow. We observe three clearly distinguished stages of decay, each characterized by an exponential decrease of the kinetic energy with time, but with different decay constants. Using filtering techniques, we identify the physics that controls each stage of decay. The first, most rapid stage is not due to the merger of like-sign vortices as has often been suggested but rather to the rapid relaxation of downscale spectral energy leakage. The second stage is a manifestation of dynamical inverse energy cascade processes, and lasts until the separation of scales becomes small. The final stage of decay appears to be dominated by the vertical stratification in our experiment. Our results clarify the dynamical processes at work in decaying two-dimensional turbulence.   

How inertia and topology influence single-particle irreversibility in turbulence

It has recently been suggested that the irreversibility of a single fluid particle in turbulence may be quantified by considering the evolution of its power, which has been shown to posses a negatively skewed Probability Density Function (PDF). This has been interpreted in terms of flight-crash events, wherein fluid particles tend to gain energy slowly and lose it fast. Furthermore, it has been argued that the dynamical origin of the negative skewness is the predominance of vortex stretching over compression in turbulence. Here, we consider the irreversibility of a single inertial particle in turbulence. We find that weak inertia can enhance the irreversibility through the way that inertia causes the particles to preferentially interact with the topology of the turbulent flow. When inertia is moderate to strong, the irreversibility is significantly reduced since the inertial particles have a damped response to the flight-crash events in the flow. By following the fluid power along inertial particle trajectories, we also probe whether vortex stretching really is the cause of the negative skewness of the fluid power, or whether the dynamical origin is in fact the self-amplification of the strain field.   

Time irreversibility of fully developed turbulence: multifractal statistics of Lagrangian power

The irreversible energy cascade of fully developed turbulence is a prototype for systems far from equilibrium. Recently, time irreversibility in turbulence has been discovered at the level of single Lagrangian trajectory, whose rate of kinetic energy change - the Lagrangian power - displays an asymmetric distribution with a power-law dependence on the Reynolds number. In this contribution the statistics of Lagrangian power, obtained from extensive direct numerical simulations at different Reynolds numbers, is shown to be well described by the Multifractal model of turbulence. The predictions of the multifractal model are also compared with the results from a shell model of turbulence, which allows to reach very high Reynolds numbers. Surprisingly in this case the even moments of power, insensitive to time asymmetry, are well described by the model, while the odd moments display different scaling exponents. The relevance of our finding for modeling Lagrangian turbulence are discussed.   

Negative probability of random multiplier in turbulence

The random multiplicative process (RMP), which has been proposed for over 50 years, is a convenient phenomenological ansatz of turbulence cascade. In the RMP, the fluctuation in a large scale is statistically mapped to the one in a small scale by the linear action of an independent random multiplier (RM). Simple as it is, the RMP is powerful enough since all of the known scaling laws can be included in this model. So far as we know, however, a direct extraction for the probability density function (PDF) of RM has been absent yet. The reason is the deconvolution during the process is ill-posed. Nevertheless, with the progress in the studies of inverse problems, the situation can be changed. By using some new regularization techniques, for the first time we recover the PDFs of the RMs in some turbulent flows. All the consistent results from various methods point to an amazing observation--the PDFs can attain negative values in some intervals; and this can also be justified by some properties of infinitely divisible distributions. Despite the conceptual unconventionality, the present study illustrates the implications of negative probability in turbulence in several aspects, with emphasis on its role in describing the interaction between fluctuations at different scales.   

Multi-scale analysis of polymer-diluted turbulent flow using a new elastic dumbbell model with incorporation of variable non-affinity.

We consider turbulent flows diluted with the polymers. The polymer chains are modeled as elastic dumbbells and represented by Brownian dynamics. The motion of solvent fluid is pursued by DNS. Affinity in the motion of the bead-spring configuration with the fluid surrounding the dumbbells is commonly assumed, but it results in emergence of Elasto-inertial turbulence (EIT) regime. When the polymers are highly stretched, molecular motions may not precisely correspond to the macroscopic deformation (de Gennes 1986). We develop a new dumbbell model in which the affine constraint is removed and non-affinity is introduced by allowing slippage of the dumbbells against the solvent. This is done by adopting the lower-convective derivative in addition to the upper-convective derivative in the governing equation for the motion of the dumbbells. We conduct its assessment in the forced homogeneous isotropic turbulence. It is shown that the dumbbells obtained from the case with complete affinity are rotated and converted to the alignment of the dumbbells in the complete non-affine case, and vice versa. This alteration of configurations is repeated quasi-periodically with the intervals comparable to the relaxation time. The largest stretching of the dumbbells and elastic energy production are achieved in the complete non-affine dumbbells. Occurrence of EIT is eliminated and de Gennes hypothesis is justified.   

Effects of polymers on the spatial structure of turbulent flows

It is well known that the addition of minor amounts of polymers to a turbulent water flow can significantly change its properties. One of the most prominent effects is the observed drastic reduction of drag in wall-bounded flows that is utilized in many engineering applications. Much of the research on polymers in turbulence has focused on their influence on the turbulent energy cascade and their interaction with the energy transfer processes. Much less investigated are their effects on the spatial structure of turbulent flows. In a classical von-Kárman swirling flow setup, we used Lagrangian particle tracking to obtain three-dimensional particle trajectories, velocities, and accelerations and find that polymers have a significant effect on Lagrangian measures of the turbulence structure such as radial distribution functions and the curvature of particle trajectories. We find that not only do the statistical distributions change, but also that polymers appear to affect the spatial statistics well beyond the size of the polymers themselves.   

Reynolds number scaling of straining motions in turbulence.

Strain is an important fluid motion in turbulence as it is associated with the kinetic energy dissipation rate, vorticity stretching, and the dispersion of passive scalars. The present study investigates the scaling of the turbulent straining motions by evaluating the flow in the eigenframe of the local strain-rate tensor. The analysis is based on DNS of homogeneous isotropic turbulence covering a Reynolds number range \textit{Re}$_{\lambda }$~$=$~34.6 - 1131. The resulting flow pattern reveals a shear layer containing tube-like vortices and a dissipation sheet, which both scale on the Kolmogorov length scale, $\eta $. The vorticity stretching motions scale on the Taylor length scale, while the flow outside the shear layer scales on the integral length scale. These scaling results are consistent with those in wall-bounded flow, which suggests a quantitative universality between the different flows. The overall coherence length of the vorticity is 120$\eta $ in all directions, which is considerably larger than the typical size of individual vortices, and reflects the importance of spatial organization at the small scales. Transitions in flow structure are identified at \textit{Re}$_{\lambda }$~$\approx $~45 and 250. Below these respective Reynolds numbers, the small-scale motions and the vorticity stretching motions appear underdeveloped.   

Orientation patterns of non-spherical particles in turbulence

Turbulent strains tend to align non-spherical particles advected by turbulence. When two such particles spend extended time near each other, they might reasonably be expected to converge toward the same orientation. We show here that this intuition fails in general. Orientations of nearby particles can be very different in turbulence because the distribution of relative orientations of nearby particles has power-law tails. We measure the moments of this distribution in experiments and numerical simulations, and explain their anomalous scaling as a function of centre-of-mass distance by analysing a statistical model. Our analysis builds on a description of the relative motion in a phase space that includes not only the usual spatial coordinates, but also the angular degrees of freedom. In this phase space the dynamics evolves to a fractal attractor. We explain how its geometry determines the anomalous scaling exponents. Our results provide a foundation for understanding collisions of non-spherical particles in turbulence, since relative orientations are critical in modeling collision rates and outcomes.. This talk is based on L. Zhao, K. Gustavsson, R. Ni, S. Kramel, G. A. Voth, H. I. Andersson, and B. Mehlig, arxiv:1707.06037