Viscous Cutoff Research Articles

Conical focusing: mechanism for singular jetting from collapsing drop-impact craters

Fast microjets can emerge out of liquid pools from the rebounding of drop-impact craters, or when a bubble bursts at its surface. The fastest jets are the narrowest and are a source of aerosols both from the ocean and from a glass of champagne, of importance to climate and the olfactory senses. The most singular jets, which we observe experimentally at a maximum velocity of $137\pm 4\ {\rm m}\ {\rm s}^{-1}$ and a diameter of $12\ \mathrm {\mu }{\rm m}$ , under reduced ambient pressure, are produced when a small dimple forms at the crater bottom and rebounds without pinching off a small bubble. The radial collapse and rebounding of this dimple is purely inertial, but highly sensitive to initial conditions. High-resolution numerical simulations reveal a new focusing mechanism, which drives the fastest jet within a converging conical channel, where an entrained air sheet provides effective slip at the outer boundary of the conically converging flow into the jet. This configuration bypasses any viscous cutoff of the jetting speed and explains the extreme sensitivity to initial conditions observed in detailed experiments of the phenomenon.

A Kolmogorov‐Brutsaert structure function model for evaporation into a turbulent atmosphere

AbstractIn 1965, Brutsaert proposed a model that predicted mean evaporation rate from rough surfaces to scale with the 3/4 power law of the friction velocity ( ) and the square‐root of molecular diffusivity (Dm) for water vapor. In arriving at these results, a number of assumptions were made regarding the surface renewal rate describing the contact durations between eddies and the evaporating surface, the diffusional mass process from the surface into eddies, and the cascade of turbulent kinetic energy sustaining the eddy renewal process itself. The working hypothesis explored here is that is a direct outcome of the Kolmogorov scaling for inertial subrange eddies modified to include viscous cutoff thereby bypassing the need for a surface renewal assumption. It is demonstrated that Brutsaert's model for may be more general than its original derivation implied.

Coalescence-induced jumping of micro-droplets on heterogeneous superhydrophobic surfaces

The phenomenon of droplets coalescence-induced self-propelled jumping on homogeneous and heterogeneous superhydrophobic surfaces was numerically modeled using the volume of fluid method coupled with a dynamic contact angle model. The heterogeneity of the surface was directly modeled as a series of micro-patterned pillars. To resolve the influence of air around a droplet and between the pillars, extensive simulations were performed for different droplet sizes on a textured surface. Parallel computations with the OpenMP algorithm were used to accelerate computation speed to meet the convergence criteria. The composition of the air-solid surface underneath the droplet facilitated capturing the transition from a no-slip/no-penetration to a partial-slip with penetration as the contact line at triple point started moving to the air pockets. The wettability effect from the nanoscopic roughness and the coating was included in the model by using the intrinsic contact angle obtained from a previously published study. As the coalescence started, the radial velocity of the coalescing liquid bridge was partially reverted to the upward direction due to the counter-action of the surface. However, we found that the velocity varied with the size of the droplets. A part of the droplet kinetic energy was dissipated as the merged droplet started penetrating into the cavities. This was due to a different area in contact between the liquid and solid and, consequently, a higher viscous dissipation rate in the system. We showed that the effect of surface roughness is strongly significant when the size of the micro-droplet is comparable with the size of the roughness features. In addition, the relevance of droplet size to surface roughness (critical relative roughness) was numerically quantified. We also found that regardless of the viscous cutoff radius, as the relative roughness approached the value of 44, the direct inclusion of surface topography was crucial in the modeling of the droplet-surface interaction. Finally, we validated our model against existing experimental data in the literature, verifying the effect of relative roughness on the jumping velocity of a merged droplet.

Dynamics of Collapse of Free Surface Bubbles

We experimentally study the collapse of cavities formedby bubbles of different radius R at the free surface of five different fluids. The tangential and normal velocities of the disturbance front on the bubble surface were measured andshown that the tangential velocities scale as the capillary velocity (σ/ρR)1/2. The total time for the completion of cavity collapse was measured and shown to scale as capillary time scale R3/2/(σ/ρ)1/2. We hence infer thatthe cavity collapse is dominated by surface tension and inertia with viscosity not playing a role in the regime of our investigation. Such a behavior of cavity dynamics is different from the behavior of jet velocity discussed by Sangeeth et.al.[9] wherein the jet velocity showed a clear viscous cut off. The rate of change of cavity volume was found to be constant with time, the implications of which needs to be investigated further.

Numerical simulations of self-propelled jumping upon drop coalescence on non-wetting surfaces

AbstractCoalescing drops spontaneously jump out of plane on a variety of biological and synthetic superhydrophobic surfaces, with potential applications ranging from self-cleaning materials to self-sustained condensers. To investigate the mechanism of self-propelled jumping, we report three-dimensional phase-field simulations of two identical spherical drops coalescing on a flat surface with a contact angle of 180°. The numerical simulations capture the spontaneous jumping process, which follows the capillary–inertial scaling. The out-of-plane directionality is shown to result from the counter-action of the substrate to the impingement of the liquid bridge between the coalescing drops. A viscous cutoff to the capillary–inertial velocity scaling is identified when the Ohnesorge number of the initial drops is around 0.1, but the corresponding viscous cutoff radius is too small to be tested experimentally. Compared to experiments on both superhydrophobic and Leidenfrost surfaces, our simulations accurately predict the nearly constant jumping velocity of around 0.2 when scaled by the capillary–inertial velocity. By comparing the simulated drop coalescence processes with and without the substrate, we attribute this low non-dimensional velocity to the substrate intercepting only a small fraction of the expanding liquid bridge.

Turbulent magnetohydrodynamic dynamo action in a spherically bounded von Kármán flow at small magnetic Prandtl numbers

Turbulent magnetohydrodynamic (MHD) dynamo action in a spherically bounded electrically conducting flow is investigated numerically. A large-scale two-vortex flow driven by a constant body force is simulated. The numerical setup models the spherical Madison Dynamo Experiment, which uses an impeller-driven flow of liquid sodium. The study focuses on small magnetic Prandtl numbers (Pm), the regime relevant to liquid sodium experimental flows. The critical magnetic Reynolds number (Rmc) of the dynamo model is determined. It initially rises steeply quasi-linearly as a function of the Reynolds number (Re) by about a factor of 10. Finally, it starts to flatten for Pm ≲ 0.1. Further investigations yield that the initial rise of the stability curve is caused in concert with large- and small-scale fluctuations of the velocity field. As an inertial range of turbulence develops with increasing Re, small-scale dynamo modes become unstable, indicating a transition from large-scale (dipolar) to small-scale dynamo action. It is argued that the flattening of the stability curve is related to a saturation of detrimental large-scale velocity fluctuations, the activation of small-scale dynamo action, and the separation of resistive and viscous cutoff scales for Pm < 1. Moreover, it is shown that only the turbulent fluctuations obtained by subtracting the precomputed mean flow from the dynamically evolving flow can act as a small-scale dynamo.

Large-eddy simulations of turbulent mixing layers using the stretched-vortex model

The stretched-vortex subgrid model is used to run large-eddy simulations of temporal mixing layers at various Reynolds and Schmidt numbers, with different initial and boundary conditions. A self-similar flow is obtained, during which the growth rate, mean velocity and Reynolds stresses are in accord with experimental results. However, predictions of the amount of mixed fluid, and of the variation in its composition across the layer, are excessive, especially at high Schmidt number. More favourable comparisons between experiment and simulation are obtained when the large-scale flow is quasi-two-dimensional; however, such states are not self-similar and not sustainable. Present model assumptions lead to predictions of the continued subgrid spectrum with a viscous cutoff that is dependent on grid resolution.

Dynamo Effect in the Kraichnan Magnetohydrodynamic Turbulence

The existence of a dynamo effect in a simplified magnetohydrodynamic model of turbulence is considered when the magnetic Prandtl number approaches zero or infinity. The magnetic field is interacting with an incompressible Kraichnan-Kasantzev model velocity field augmented with a viscous scale cutoff. An approximate system of equations in the different scaling ranges can be formulated and solved, so that the solution tends to the exact one when the viscous and magnetic-diffusive cutoffs approach zero. In this approximation we are able to determine analytically the conditions for the existence of a dynamo effect and give an estimate of the dynamo growth rate. Among other things we show that in the large Prandtl number case the dynamo effect is always present. Our analytical estimates are in good agreement with previous numerical studies of the Kraichnan-Kasantzev dynamo by D. Vincenzi.

The scaling properties of two-dimensional compressible magnetohydrodynamic turbulence

Understanding the phenomenology captured in direct numerical simulation (DNS) of magnetohydrodynamic (MHD) turbulence rests upon models and assumptions concerning the scaling of field variables and dissipation. Here compressible MHD turbulence is simulated in two spatial dimensions by solving the isothermal equations of resistive MHD on a periodic square grid. In these simulations it is found that the energy spectrum decreases more slowly with k, and the viscous cutoff length is larger, than would be expected from the 1941 phenomenology of Kolmogorov (K41). Both these effects suggest that the cascade time is modified by the presence of Alfvén waves as in the phenomenology of Iroshnikov and Kraichnan (IK). Motivated by this, these scaling exponents are compared with those of the IK-based model of Politano and Pouquet [Phys. Rev. E 52, 636 (1995)], which is an extension of the model of She and Leveque [Phys. Rev. Lett. 72, 336 (1994)]. However, the scaling exponents from these simulations are not consistent with the model of Politano and Pouquet, so that neither IK nor K41 models would appear to describe the simulations. The spatial intermittency of turbulent activity in such simulations is central to the observed phenomenology and relates to the geometry of structures that dissipate most intensely via the scaling of the local rate of dissipation. The framework of She and Leveque implies a scaling relation that links the scaling of the local rate of dissipation to the scaling exponents of the pure Elsässer field variables (z±=v±B∕μoρ). This scaling relation is conditioned by the distinct phenomenology of K41 and IK. These distinct scaling relations are directly tested using these simulations and it is found that neither holds. This deviation suggests that additional measures of the character of the dissipation may be required to fully capture the turbulent scaling, for example, pointing towards a refinement of the phenomenological models. It may also explain why previous attempts to predict the scaling exponents of the pure Elsässer fields in two-dimensional magnetohydrodynamic turbulence by extending the theory of She and Leveque have proved unsuccessful.

The Onset of a Small-Scale Turbulent Dynamo at Low Magnetic Prandtl Numbers

We study numerically the dependence of the critical magnetic Reynolds number Rmc for the turbulent small-scale dynamo on the hydrodynamic Reynolds number Re. The turbulence is statistically homogeneous, isotropic, and mirror-symmetric. We are interested in the regime of low magnetic Prandtl number Pm = Rm/Re < 1, which is relevant for stellar convective zones, protostellar disks, and laboratory liquid-metal experiments. The two asymptotic possibilities are Rmc → const as Re → ∞ (a small-scale dynamo exists at low Pm) or Rmc/Re = Pmc → const as Re → ∞ (no small-scale dynamo exists at low Pm). Results obtained in two independent sets of simulations of MHD turbulence using grid and spectral codes are brought together and found to be in quantitative agreement. We find that at currently accessible resolutions, Rmc grows with Re with no sign of approaching a constant limit. We reach the maximum values of Rmc ~ 500 for Re ~ 3000. By comparing simulations with Laplacian viscosity, fourth-, sixth-, and eighth-order hyperviscosity, and Smagorinsky large-eddy viscosity, we find that Rmc is not sensitive to the particular form of the viscous cutoff. This work represents a significant extension of the studies previously published by Schekochihin et al. (2004a) and Haugen et al. (2004a) and the first detailed scan of the numerically accessible part of the stability curve Rmc(Re).

Magnetic Field Structure and Stochastic Reconnection in a Partially Ionized Gas

We consider stochastic reconnection in a magnetized, partially ionized medium. Stochastic reconnection is a generic effect, due to field line wandering, in which the speed of reconnection is determined by the ability of ejected plasma to diffuse away from the current sheet along magnetic field lines, rather than by the details of current sheet structure. We consider the limit of weak stochasticity, so that the mean magnetic field energy density is greater than either the turbulent kinetic energy density or the energy density associated with the fluctuating component of the field. We consider field line stochasticity generated through a turbulent cascade, which leads us to consider the effect of neutral drag on the turbulent cascade of energy. In a collisionless plasma, neutral particle viscosity and ion-neutral drag will damp mid-scale turbulent motions, but the power spectrum of the magnetic perturbations extends below the viscous cutoff scale. We give a simple physical picture of the magnetic field structure below this cutoff, consistent with numerical experiments. We provide arguments for the reemergence of the turbulent cascade well below the viscous cut-off scale and derive estimates for field line diffusion on all scales. We note that this explains the persistence of a single power law form for the turbulent power spectrum of the interstellar medium, from scales of tens of parsecs down to thousands of kilometers. We find that under typical conditions in the ISM stochastic reconnection speeds are reduced by the presence of neutrals, but by no more than an order of magnitude.

The Small‐Scale Structure of Magnetohydrodynamic Turbulence with Large Magnetic Prandtl Numbers

We study the intermittency and field-line structure of the MHD turbulence in plasmas with very large magnetic Prandtl numbers. In this regime, which is realized in the interstellar medium, some accretion disks, protogalaxies, galaxy-cluster gas, the early universe, etc., magnetic fluctuations can be excited at scales below the viscous cutoff. The salient feature of the resulting small-scale magnetic turbulence is the folded structure of the fields. It is characterized by very rapid transverse spatial oscillation of the field direction, while the field lines remain largely unbent up to the scale of the flow. Quantitatively, the fluctuation level and the field-line geometry can be studied in terms of the statistics of the field strength and of the field-line curvature. In the kinematic limit, the distribution of the field strength is an expanding lognormal, while that of the field-line curvature K is stationary and has a power tail ~K-13/7. The field strength and curvature are anticorrelated, i.e., the growing fields are mostly flat, while the sharply curved fields remain relatively weak. The field, therefore, settles into a reduced-tension state. Numerical simulations demonstrate three essential features of the nonlinear regime. First, the total magnetic energy is equal to the total kinetic energy. Second, the intermittency is partially suppressed compared to the kinematic case, as the fields become more volume-filling and their distribution develops an exponential tail. Third, the folding structure of the field is unchanged from the kinematic case: the anticorrelation between the field strength and the curvature persists, and the distribution of the latter retains the same power tail. We propose a model of back-reaction based on the folding picture that reproduces all of the above numerical results.

New Regime of Magnetohydrodynamic Turbulence: Cascade below the Viscous Cutoff

ABSTRACTIn astrophysical situations, e.g. in the interstellar medium (ISM), neutralscan provide viscous damping on scales much larger than the magnetic diffusionscale. Through numerical simulations, we have found that the magnetic fieldcan have a rich structure below the dissipation cutoff scale. This implies thatmagnetic fields in the ISM can have structures on scales much smaller than parsecscales. Our results show that the magnetic energy contained in a wavenumberband is independent of the wavenumber and magnetic structures are intermittentand extremely anisotropic. We discuss the relation between our results and theformation of the tiny-scale atomic structure (TSAS). Subject headings: ISM:general—ISM:structure—MHD—turbulence1. IntroductionIn the interstellar medium (ISM), flows are complicated and dynamic. Observationssuggest thatthe ISMisin aturbulent state(Larson1981; Myers 1983; Scalo1984; Armstrong,Rickett & Spangler 1995; Stanimirovic & Lazarian 2001).Hydrodynamic turbulence can be described by so-called energy cascade model, in whichenergy injected at a scale L cascades down to progressively smaller and smaller scales. Ul-timately, the energy will reach the molecular

Scaleless algebraic energy spectra for the incompressible Navier–Stokes equation; relation to other nonlinear problems

We use the Wiener expansion, a.k.a. the Wiener–Hermite (W–H) expansion, which is based on weighted integrals of the white noise process (also called Feynman path integrals) to represent turbulence, near to Gaussian. Fully developed turbulence-fluctuation Reynolds (Re) numbers greater than 10 4 are treated. The energy transfer function T( k, q) (transfer from wavenumber q to k) is shown, and for the equilibrium 5/3 law energy spectrum, it is seen that the dependence on k 0, the energy range wavenumber, drops out so that dimensionally the energy spectrum must be algebraic, 5/3 in the inertial subrange (ISR) at equilibrium, for driven turbulence. An interesting consequence of the k 0 independence is the loss of the last evidence of the large scale drive of the turbulence. The form of the driving force has no influence. The same conclusion follows for decaying turbulence, with the slight modification that rate of energy loss (viscous decay) is a (slow) function of time. The integral over the transfer (the net energy gain/loss) converges independently of k v, the viscous cutoff so that the turbulence is scaleless (see Barenblatt and Zel'dovich [Barenblatt, G.I., Zel'dovich, Ya.B., 1972. Self-similar solutions as intermediate asymptotics. Ann. Rev. Fluid Mech. 4, 285–312.] for a full discussion of the influence of this property). The scaleless character is a necessary ingredient of fractals. Fractals of course have the property that they preserve their qualitative character at any magnification, and this characteristic would be upset by the presence of a scale external to the process. The transfer from wavenumbers of order k dominates that from wave numbers of order k 0 because of the following. First, the volume of phase space available for transfer, of order the large wavenumber k, is enormous compared with the small region in the vicinity of the energy spectrum peak, k 0. Second, the pressure term in conjunction with incompressibility yields geometric coefficients (in the transfer) which are proportional to the square of the (small) wavenumber, q, in the region k 0. The effect of moderate compressibility (which will be discussed) is to destroy this second small behavior of the transfer in the energy range at a threshold Mach number; then a different ISR spectrum can develop, as is known to be the case. This opens the way for processes with other phenomena, e.g., buoyancy effects, to yield other spectra (than the 5/3). We introduce a compressible flow model by relieving incompressibility and calculate the resulting transfer. There is a critical (fluctuation) Reynolds number, Re c′=2.5(Ma′) −8. If Re′<Re c′, there is only the compressible spectrum, −6/3. If Re′ is larger, then there is a critical wavenumber k c=2 k 0(Ma′) −6, forming a knee. For k< k c, the spectrum is −5/3, and for k> k c, it will be −6/3. There are geophysical (and of course astrophysical) processes where this condition is fulfilled. The discussion here gives, to the author's knowledge, the first explanation of the 5/3 spectrum which goes beyond dimensional analysis (with additional ad-hoc assumptions concerning appropriate parameters, specifically an assumed independence of k 0).

Time-dependent geometry and energy distribution in a spiral vortex layer.

The purpose of this paper is to study how the geometry and the spatial distribution of energy fluctuations of different length scales in a spiral vortex layer are related to each other in a time-dependent way. The numerical solution of Krasny [J. Comput. Phys. 65, 292 (1986)], corresponding to the development of the Kelvin-Helmholtz instability, is analyzed in order to determine some geometrical features necessary for the analysis of Lundgren's unstrained spiral vortex. The energy distribution of the asymptotic solution of Lundgren characterized by a similar geometry is investigated analytically (1) in the wavelet radius-scale space, with a wavelet selective in the radial direction, and (2) in the wavelet azimuth-scale space, with a wavelet selective in the azimuthal direction. Energy in the wavelet radius-scale space is organized in "blobs" distributed in a way determined by the Kolmogorov capacity of the spiral D(K) in [1,2] (which determines the rate of accumulation of spiral turns). As time evolves these blobs move towards the small scale region of the wavelet radius-scale space, until their scale is of the order of the diffusive length scale square root[nut], where t is the time and nu is the kinematic viscosity. In contrast, energy in the wavelet azimuth-scale space is not localized, and is characterized by a shear-augmented viscous cutoff proportional to sqaure root[nut(3)]. An accelerated viscous dissipation of the enstrophy and energy of Lundgren's spiral vortex is found for D(K) > 1.75, but not for D(K) < or =1.75.

On the universal form of energy spectra in fully developed turbulence

Several experimentally measured energy spectra of fully developed incompressible turbulent flows with Taylor microscale Reynolds numbers ranging from 130–13 000 have been reexamined. It is shown that all spectra collapse to a universal curve after a simple rescaling of space and time. This rescaling does not involve any multifractal-type transformation, and the rescaled wave numbers scatter around the Kolmogorov dissipation wave numbers derived from experimentally measured quantities. An empirical formula for the resulting universal form is proposed, indicating a nontrivial form of the energy spectrum near the viscous dissipation cutoff.

Varieties of Fully Resolved Spectra of Vertical Shear

The Multi-Scale profiler (MSP) resolves shear between vertical wavenumbers of 0.01 cpm and the viscous cutoff of small-scale turbulence. Observations from five sites reveal varied spectral shapes and amplitudes. Spectral amplitudes measured at low latitude do not increase toward the equator, contrary to Munk, and their shapes differ from the Garrett and Munk model by having weak maxima between 0.02 and 0.05 cpm. Moreover, at wavenumbers larger than 0.1 cpm these spectra roll off more steeply, k3−1, than do spectra at midlatitude. Of two average spectra from midlatitude, one is close to the Garrett and Munk model at low wavenumbers, and at 0.1 cpm it begins to roll off as k3−1. The second midlatitude spectrum has amplitudes well above Garrett and Munk at low wavenumbers, begins a k3−1 rolloff near 0.04 cpm, and has a well-developed turbulent range near 1 cpm. The decrease in the start of the rolloff is not linearly proportional to the increase in spectral amplitude at low wavenumber, unlike the spectra observed by Duda and Cox and models proposed by Munk and also Garrett. In spite of the diversity of shapes and amplitudes at low wavenumbers, all shear spectra have nearly the same amplitude at 0.14 cpm, which is in the rolloff range. The rolloff range cannot be a buoyancy subrange of three-dimensional turbulence because the largest overturns occur only at the high-wavenumber end of the range. Rather, the rolloff must be the signature of the high-wavenumber decay of the internal wave field. Near 0.5kE = (N3/ε)1/2 spectra change from the internal wave rolloff to the turbulent dissipation range, which is adequately represented by Nasmyth's “universal” spectrum. Midlatitude spectra with amplitudes close to the Garrett and Munk model have very weak turbulent spectra, but those with substantially larger low-wavenumber amplitudes have well-developed turbulent spectra with distinct inertial subranges. Owing to their steeper rolloffs, the low-latitude records also have weak dissipation spectra even though their spectra rise above Garrett and Munk at wavenumbers slightly less than the rolloff.

On the multifractal properties of the energy dissipation derived from turbulence data

Various difficulties can be expected in trying to extract from experimental data the distribution of singularities, the f(α) function, of the energy dissipation. One reason is that the multifractal model of turbulence implies a dependence of the viscous cutoff on the singularity exponent. It is an open question if current hot-wire probes can resolve the scales implied by exponents a significantly less than 1.Two exactly soluble models are used to show how spurious scaling can occur, due to finite Reynolds number effects. In the Gaussian model the true velocity signal is replaced by independent Gaussian random variables. The dissipation, defined as the square of the difference of successive variables, has trivial scaling in so far as all the moments of spatial averages of the dissipation behave asymptotically as a uniform dissipation. Still, contamination by subdominant terms requires that scaling exponents for high-order moments be identified over an increasingly large range of scales. If the available range is limited by the Reynolds number, scaling exponents for high orders will be systematically underestimated and spurious intermittency will be inferred. Burgers’ model is used to highlight further problems. At finite Reynolds numbers, regions with no small-scale activity (away from shocks) have a residual dissipation which contributes a spurious point (α = 1,f(α) = 1). In addition, when the f(α) function is obtained by Legendre transform techniques, convex hull effects generate an entire spurious segment.Finally, Burgers’ model also indicates that the relation between exponents of structure functions and exponents of local dissipation moments, which goes back to Kolmogorov's (1962) work, leads to an inconsistency for structure functions of low positive order.

Some observations of bispectral behavior of large ensembles of exact solutions to the Burgers equation for random initial conditions

Accurate bispectra for the Burgers equation have been computed into the high wave‐number domain. Although only the imaginary part ib(k,l) contributes to spectral transfer, new insights having implications for three‐dimensional turbulence can be gained by incorporating the real part a(k,l). For fixed l, for instance, the modulus of the bispectrum of the Burgers equation manifests a power‐law subrange corresponding to the inertial subrange of the energy spectrum and an exponential subrange corresponding to the viscous cutoff of the energy spectrum.

Effect of dissipation fluctuations on anomalous velocity scaling in turbulence.

We introduce a one-dimensional model of a turbulent velocity field. It consists of a superposition of successively smaller eddies, forming a Cayley-tree-like structure. The randomness of the eddy decay is modeled by a multiplier distribution p(s), whose parameters can be determined from experiment. In contrast to other cascade models, which are not stated in terms of the velocity field, every other quantity of the turbulent flow can now be calculated consistently. Scaling relations between the statistics of the velocity and the dissipation fields are shown to depend greatly on physical assumptions about the cascade. Specifically, we study the influence of the spatial coherence of eddies, and of the action of the viscous cutoff.