Forward and Inverse Energy Cascade in Fluid Turbulence Adhere to Kolmogorov's Refined Similarity Hypothesis.

We study fluctuations of the local energy cascade rate Φ_{ℓ} in turbulent flows at scales (ℓ) in the inertial range. According to the Kolmogorov refined similarity hypothesis (KRSH), relevant statistical properties of Φ_{ℓ} should depend on ε_{ℓ}, the viscous dissipation rate locally averaged over a sphere of size ℓ, rather than on the global average dissipation. However, the validity of KRSH applied to Φ_{ℓ} has not yet been tested from data. Conditional averages such as ⟨Φ_{ℓ}|ε_{ℓ}⟩ as well as of higher-order moments are measured from direct numerical simulations data, and results clearly adhere to the predictions from KRSH. Remarkably, the same is true when considering forward (Φ_{ℓ}>0) and inverse (Φ_{ℓ}<0) cascade events separately. Measured ratios of forward and inverse cascade probability densities conditioned on ε_{ℓ} also confirm the applicability of the KRSH to analysis of the fluctuation relation from nonequilibrium thermodynamics.

A localized momentum constraint for non-equilibrium molecular dynamics simulations.

A method which controls momentum evolution in a sub-region within a molecular dynamics simulation is derived from Gauss's principle of least constraint. The technique for localization is founded on the equations by Irving and Kirkwood [J. Chem. Phys. 18, 817 (1950)] expressed in a weak form according to the control volume (CV) procedure derived by Smith et al. [Phys. Rev. E. 85, 056705 (2012)]. A term for the advection of molecules appears in the derived constraint and is shown to be essential in order to exactly control the time evolution of momentum in the subvolume. The numerical procedure converges the total momentum in the CV to the target value to within machine precision in an iterative manner. The localized momentum constraint can prescribe essentially arbitrary flow fields in non-equilibrium molecular dynamics simulations. The methodology also forms a rigorous mathematical framework for introducing coupling constraints at the boundary between continuum and discrete systems. This functionality is demonstrated with a boundary-driven flow test case.

The method of planes pressure tensor for a spherical subvolume.

Various formulas for the local pressure tensor based on a spherical subvolume of radius, R, are considered. An extension of the Method of Planes (MOP) formula of Todd et al. [Phys. Rev. E 52, 1627 (1995)] for a spherical geometry is derived using the recently proposed Control Volume formulation [E. R. Smith, D. M. Heyes, D. Dini, and T. A. Zaki, Phys. Rev. E 85, 056705 (2012)]. The MOP formula for the purely radial component of the pressure tensor is shown to be mathematically identical to the Radial Irving-Kirkwood formula. Novel offdiagonal elements which are important for momentum conservation emerge naturally from this treatment. The local pressure tensor formulas for a plane are shown to be the large radius limits of those for spherical surfaces. The radial-dependence of the pressure tensor computed by Molecular Dynamics simulation is reported for virtual spheres in a model bulk liquid where the sphere is positioned randomly or whose center is also that of a molecule in the liquid. The probability distributions of angles relating to pairs of atoms which cross the surface of the sphere, and the center of the sphere, are presented as a function of R. The variance in the shear stress calculated from the spherical Volume Averaging method is shown to converge slowly to the limiting values with increasing radius, and to be a strong function of the number of molecules in the simulation cell.

Control-volume representation of molecular dynamics.

A molecular dynamics (MD) parallel to the control volume (CV) formulation of fluid mechanics is developed by integrating the formulas of Irving and Kirkwood [J. Chem. Phys. 18, 817 (1950)] over a finite cubic volume of molecular dimensions. The Lagrangian molecular system is expressed in terms of an Eulerian CV, which yields an equivalent to Reynolds' transport theorem for the discrete system. This approach casts the dynamics of the molecular system into a form that can be readily compared to the continuum equations. The MD equations of motion are reinterpreted in terms of a Lagrangian-to-control-volume (LCV) conversion function ϑ(i) for each molecule i. The LCV function and its spatial derivatives are used to express fluxes and relevant forces across the control surfaces. The relationship between the local pressures computed using the volume average [Lutsko, J. Appl. Phys. 64, 1152 (1988)] techniques and the method of planes [Todd et al., Phys. Rev. E 52, 1627 (1995)] emerges naturally from the treatment. Numerical experiments using the MD CV method are reported for equilibrium and nonequilibrium (start-up Couette flow) model liquids, which demonstrate the advantages of the formulation. The CV formulation of the MD is shown to be exactly conservative and is, therefore, ideally suited to obtain macroscopic properties from a discrete system.

Pressure dependence of confined liquid behavior subjected to boundary-driven shear.

Non-equilibrium molecular dynamics simulations of boundary-driven sheared Lennard-Jones liquids at variable pressure up to 5 GPa (for argon) reveal a rich out-of-equilibrium phase behavior with a strong degree of shear localization. At the lowest apparent shear rate considered (wall speed ~1 m s(-1)) the confined region is an homogeneously sheared solid (S) with no slip at the walls. This transforms at higher shear rates to a non-flowing plug with slip at the walls, referred to as the plug slip (PS) state. At higher shear rate a central localized (CL) state formed in which the shear gradient was localized in the center of the film, with the rest of the confined sample in a crystalline state commensurate with the wall lattice. The central zone liquidlike region increased in width with shear rate. A continuous rounded temperature profile across the whole system reflects strong dynamical coupling between the wall and confined region. The temperature rise in the confined film is consistent with the Brinkman number. The transition from the PS to CL states typically occurred at a wall speed near where the shear stress approached a critical value of ~3% of the shear modulus, and also near the peak in the traction coefficient, µ. The peak traction coefficient values computed, ~0.12-0.14 at 1000 MPa agree with those found for traction fluids and occur when the confined liquid is in the PS and CL states. At low wall speeds slip can occur at one wall and stick at the other. Poorly wetting liquids manifest long-lived asymmetries in the confined liquid properties across the system, and a shift in solid-liquid phase co-existence to higher shear rates. A non-equilibrium phase diagram based on these results is proposed. The good agreement of the tribological response of the Lennard-Jones fluid with that of more complicated molecular systems suggests that a corresponding states scaling of the tribological behavior could apply.

The equivalence between volume averaging and method of planes definitions of the pressure tensor at a plane.

It is shown analytically that the method of planes (MOP) [Todd, Evans, and Daivis, Phys. Rev. E 52, 1627 (1995)] and volume averaging (VA) [Cormier, Rickman, and Delph, J. Appl. Phys. 89, 99 (2001)] formulas for the local pressure tensor, P(α, y)(y), where α ≡ x, y, or z, are mathematically identical. In the case of VA, the sampling volume is taken to be an infinitely thin parallelepiped, with an infinite lateral extent. This limit is shown to yield the MOP expression. The treatment is extended to include the condition of mechanical equilibrium resulting from an imposed force field. This analytical development is followed by numerical simulations. The equivalence of these two methods is demonstrated in the context of non-equilibrium molecular dynamics (NEMD) simulations of boundary-driven shear flow. A wall of tethered atoms is constrained to impose a normal load and a velocity profile on the entrained central layer. The VA formula can be used to compute all components of P(αß)(y), which offers an advantage in calculating, for example, P(xx)(y) for nano-scale pressure-driven flows in the x-direction, where deviations from the classical Poiseuille flow solution can occur.