Validation of symmetry-induced high moment velocity and temperature scaling laws in a turbulent channel flow.

The symmetry-based turbulence theory has been used to derive new scaling laws for the streamwise velocity and temperature moments of arbitrary order. For this, it has been applied to an incompressible turbulent channel flow driven by a pressure gradient with a passive scalar equation coupled in. To derive the scaling laws, symmetries of the classical Navier-Stokes and the thermal energy equations have been used together with statistical symmetries, i.e., the statistical scaling and translation symmetries of the multipoint moment equations. Specifically, the multipoint moments are built on the instantaneous velocity and temperature fields other than in the classical approach, where moments are based on the fluctuations of these fields. With this instantaneous approach, a linear system of multipoint correlation equations has been obtained, which greatly simplifies the symmetry analysis. The scaling laws have been derived in the limit of zero viscosity and heat conduction, i.e., Re_{τ}→∞ and Pr>1, and they apply in the center of the channel, i.e., they represent a generalization of the deficit law, thus extending the work of Oberlack et al. [Phys. Rev. Lett. 128, 024502 (2022)0031-900710.1103/PhysRevLett.128.024502]. The scaling laws are all power laws, with the exponent of the high moments all depending exclusively on those of the first and second moments. To validate the new scaling laws, the data from a large number of direct numerical simulations (DNS) for different Reynolds and Prandtl numbers have been used. The results show a very high accuracy of the scaling laws to represent the DNS data. The statistical scaling symmetry of the multipoint moment equations, which characterizes intermittency, has been the key to the new results since it generates a constant in the exponent of the final scaling law. Most important, since this constant is independent of the order of the moments, it clearly indicates anomalous scaling.

Turbulence Statistics of Arbitrary Moments of Wall-Bounded Shear Flows: A Symmetry Approach.

The calculation of turbulence statistics is considered the key unsolved problem of fluid mechanics, i.e., precisely the computation of arbitrary statistical velocity moments from first principles alone. Using symmetry theory, we derive turbulent scaling laws for moments of arbitrary order in two regions of a turbulent channel flow. Besides the classical scaling symmetries of space and time, the key symmetries for the present work reflect the two well-known characteristics of turbulent flows: non-Gaussianity and intermittency. To validate the new scaling laws we made a new simulation at an unprecedented friction Reynolds number of 10 000, large enough to test the new scaling laws. Two key results appear as an application of symmetry theory, which allowed us to generate symmetry invariant solutions for arbitrary orders of moments for the underlying infinite set of moment equations. First, we show that in the sense of the generalization of the deficit law all moments of the streamwise velocity in the channel center follow a power-law scaling, with exponents depending on the first and second moments alone. Second, we show that the logarithmic law of the mean streamwise velocity in wall-bounded flows is indeed a valid solution of the moment equations, and further, all higher moments in this region follow a power law, where the scaling exponent of the second moment determines all higher moments. With this we give a first complete mathematical framework for all moments in the log region, which was first discovered about 100 years ago.

Shear-Thinning in Oligomer Melts-Molecular Origins and Applications.

We investigate the molecular origin of shear-thinning in melts of flexible, semiflexible and rigid oligomers with coarse-grained simulations of a sheared melt. Entanglements, alignment, stretching and tumbling modes or suppression of the latter all contribute to understanding how macroscopic flow properties emerge from the molecular level. In particular, we identify the rise and decline of entanglements with increasing chain stiffness as the major cause for the non-monotonic behaviour of the viscosity in equilibrium and at low shear rates, even for rather small oligomeric systems. At higher shear rates, chains align and disentangle, contributing to shear-thinning. By performing simulations of single chains in shear flow, we identify which of these phenomena are of collective nature and arise through interchain interactions and which are already present in dilute systems. Building upon these microscopic simulations, we identify by means of the Irving-Kirkwood formula the corresponding macroscopic stress tensor for a non-Newtonian polymer fluid. Shear-thinning effects in oligomer melts are also demonstrated by macroscopic simulations of channel flows. The latter have been obtained by the discontinuous Galerkin method approximating macroscopic polymer flows. Our study confirms the influence of microscopic details in the molecular structure of short polymers such as chain flexibility on macroscopic polymer flows.

Reply to "Comment on 'Statistical symmetries of the Lundgren-Monin-Novikov hierarchy' ".

In this Reply we respond to the criticism of Frewer et al. presented in the Comment on "Statistical symmetries of the Lundgren-Monin-Novikov hierarchy" by Waclawczyk et al. [Phys. Rev. E 90, 013022 (2014)]. We discuss physical interpretation of the statistical symmetries, and respond to criticism on the violation of the causality principle. We derive the Lundgren-Monin-Novikov equations for a flow with boundaries. Last, we stress that our work addressed the phenomenon of "external intermittency" (separation between laminar and turbulent flow), and not "internal intermittency" (strong fluctuations at small scales).

Colored-noise Fokker-Planck equation for the shear-induced self-diffusion process of non-Brownian particles.

In the literature, it is pointed out that non-Brownian particles tend to show shear-induced diffusive behavior due to hydrodynamic interactions. Several authors indicate a long correlation time of the particle velocities in comparison to Brownian particle velocities modeled by a white noise. This work deals with the derivation of a Fokker-Planck equation both in position and velocity space which describes the process of shear-induced self-diffusion, whereas, so far, this problem has been described by Fokker-Planck equations restricted to position space. The long velocity correlation times actually would necessitate large time-step sizes in the mathematical description of the problem in order to capture the diffusive regime. In fact, time steps of specific lengths pose problems to the derivation of the corresponding Fokker-Planck equation because the whole particle configuration changes during long time-step sizes. On the other hand, small time-step sizes, i.e., in the range of the velocity correlation time, violate the Markov property of the position variable. In this work we regard the problem of shear-induced self-diffusion with respect to the Markov property and reformulate the problem with respect to small time-step sizes. In this derivation, we regard the nondimensionalized Langevin equation and develop a new compact form which allows us to analyze the Langevin equation for all time scales of interest for both Brownian and non-Brownian particles starting from a single equation. This shows that the Fokker-Planck equation in position space should be extended to a colored-noise Fokker-Planck equation in both position and colored-noise velocity space, which we will derive.

Statistical symmetries of the Lundgren-Monin-Novikov hierarchy.

It was shown by Oberlack and Rosteck [Discr. Cont. Dyn. Sys. S, 3, 451 2010] that the infinite set of multipoint correlation (MPC) equations of turbulence admits a considerable extended set of Lie point symmetries compared to the Galilean group, which is implied by the original set of equations of fluid mechanics. Specifically, a new scaling group and an infinite set of translational groups of all multipoint correlation tensors have been discovered. These new statistical groups have important consequences for our understanding of turbulent scaling laws as they are essential ingredients of, e.g., the logarithmic law of the wall and other scaling laws, which in turn are exact solutions of the MPC equations. In this paper we first show that the infinite set of translational groups of all multipoint correlation tensors corresponds to an infinite dimensional set of translations under which the Lundgren-Monin-Novikov (LMN) hierarchy of equations for the probability density functions (PDF) are left invariant. Second, we derive a symmetry for the LMN hierarchy which is analogous to the scaling group of the MPC equations. Most importantly, we show that this symmetry is a measure of the intermittency of the velocity signal and the transformed functions represent PDFs of an intermittent (i.e., turbulent or nonturbulent) flow. Interesting enough, the positivity of the PDF puts a constraint on the group parameters of both shape and intermittency symmetry, leading to two conclusions. First, the latter symmetries may no longer be Lie group as under certain conditions group properties are violated, but still they are symmetries of the LMN equations. Second, as the latter two symmetries in its MPC versions are ingredients of many scaling laws such as the log law, the above constraints implicitly put weak conditions on the scaling parameter such as von Karman constant κ as they are functions of the group parameters. Finally, let us note that these kind of statistical symmetries are of much more general type, i.e., not limited to MPC or PDF equations emerging from Navier-Stokes, but instead they are admitted by other nonlinear partial differential equations like, for example, the Burgers equation when in conservative form and if the nonlinearity is quadratic.