Non-equilibrium turbulence scalings and self-similarity in turbulent planar jets.

We study the self-similarity and dissipation scalings of a turbulent planar jet and the theoretically implied mean flow scalings. Unlike turbulent wakes where such studies have already been carried out (Dairay et al. 2015 J. Fluid Mech. 781, 166-198. (doi:10.1017/jfm.2015.493); Obligado et al. 2016 Phys. Rev. Fluids 1, 044409. (doi:10.1103/PhysRevFluids.1.044409)), this is a boundary-free turbulent shear flow where the local Reynolds number increases with distance from inlet. The Townsend-George theory revised by (Dairay et al. 2015 J. Fluid Mech. 781, 166-198. (doi:10.1017/jfm.2015.493)) is applied to turbulent planar jets. Only a few profiles need to be self-similar in this theory. The self-similarity of mean flow, turbulence dissipation, turbulent kinetic energy and Reynolds stress profiles is supported by our experimental results from 18 to at least 54 nozzle sizes, the furthermost location investigated in this work. Furthermore, the non-equilibrium dissipation scaling found in turbulent wakes, decaying grid-generated turbulence, various instances of periodic turbulence and turbulent boundary layers (Dairay et al. 2015 J. Fluid Mech. 781, 166-198. (doi:10.1017/jfm.2015.493); Vassilicos 2015 Annu. Rev. Fluid Mech. 95, 114. (doi:10.1146/annurev-fluid-010814-014637); Goto & Vassilicos 2015 Phys. Lett. A 3790, 1144-1148. (doi:10.1016/j.physleta.2015.02.025); Nedic et al. 2017 Phys. Rev. Fluids 2, 032601. (doi:10.1103/PhysRevFluids.2.032601)) is also observed in the present turbulent planar jet and in the turbulent planar jet of (Antonia et al. 1980 Phys. Fluids 23, 863055. (doi:10.1063/1.863055)). Given these observations, the theory implies new mean flow and jet width scalings which are found to be consistent with our data and the data of (Antonia et al. 1980 Phys. Fluids 23, 863055. (doi:10.1063/1.863055)). In particular, it implies a hitherto unknown entrainment behaviour: the ratio of characteristic cross-stream to centreline streamwise mean flow velocities decays as the -1/3 power of streamwise distance in the region, where the non-equilibrium dissipation scaling holds.

Attached flow structure and streamwise energy spectra in a turbulent boundary layer.

On the basis of (i) particle image velocimetry data of a turbulent boundary layer with large field of view and good spatial resolution and (ii) a mathematical relation between the energy spectrum and specifically modeled flow structures, we show that the scalings of the streamwise energy spectrum E_{11}(k_{x}) in a wave-number range directly affected by the wall are determined by wall-attached eddies but are not given by the Townsend-Perry attached eddy model's prediction of these spectra, at least at the Reynolds numbers Re_{τ} considered here which are between 10^{3} and 10^{4}. Instead, we find E_{11}(k_{x})∼k_{x}^{-1-p} where p varies smoothly with distance to the wall from negative values in the buffer layer to positive values in the inertial layer. The exponent p characterizes the turbulence levels inside wall-attached streaky structures conditional on the length of these structures. A particular consequence is that the skin friction velocity is not sufficient to scale E_{11}(k_{x}) for wave numbers directly affected by the wall.

Statistical independence of the initial conditions in chaotic mixing.

Experimental evidence of the scalar convergence towards a global strange eigenmode independent of the scalar initial condition in chaotic mixing is provided. This convergence, underpinning the independent nature of chaotic mixing in any passive scalar, is presented by scalar fields with different initial conditions casting statistically similar shapes when advected by periodic unsteady flows. As the scalar patterns converge towards a global strange eigenmode, the scalar filaments, locally aligned with the direction of maximum stretching, as described by the Lagrangian stretching theory, stack together in an inhomogeneous pattern at distances smaller than their asymptotic minimum widths. The scalar variance decay becomes then exponential and independent of the scalar diffusivity or initial condition. In this work, mixing is achieved by advecting the scalar using a set of laminar flows with unsteady periodic topology. These flows, that resemble the tendril-whorl map, are obtained by morphing the forcing geometry in an electromagnetic free surface 2D mixing experiment. This forcing generates a velocity field which periodically switches between two concentric hyperbolic and elliptic stagnation points. In agreement with previous literature, the velocity fields obtained produce a chaotic mixer with two regions: a central mixing and an external extensional area. These two regions are interconnected through two pairs of fluid conduits which transfer clean and dyed fluid from the extensional area towards the mixing region and a homogenized mixture from the mixing area towards the extensional region.

Unsteady turbulence cascades.

We have run a total of 311 direct numerical simulations (DNSs) of decaying three-dimensional Navier-Stokes turbulence in a periodic box with values of the Taylor length-based Reynolds number up to about 300 and an energy spectrum with a wide wave-number range of close to -5/3 power-law dependence at the higher Reynolds numbers. On the basis of these runs, we have found a critical time when (i) the rate of change of the square of the integral length scale turns from increasing to decreasing, (ii) the ratio of interscale energy flux to high-pass filtered turbulence dissipation changes from decreasing to very slowly increasing in the inertial range, (iii) the signature of large-scale coherent structures disappears in the energy spectrum, and (iv) the scaling of the turbulence dissipation changes from the one recently discovered in DNSs of forced unsteady turbulence and in wind tunnel experiments of turbulent wakes and grid-generated turbulence to the classical scaling proposed by G. I. Taylor [Proc. R. Soc. London, Ser. A 151, 421 (1935)1364-502110.1098/rspa.1935.0158] and A. N. Kolmogorov [Dokl. Akad. Nauk SSSR 31, 538 (1941)]. Even though the customary theoretical basis for this Taylor-Kolmogorov scaling is a statistically stationary cascade where large-scale energy flux balances dissipation, this is not the case throughout the entire time range of integration in all our DNS runs. The recently discovered dissipation scaling can be reformulated physically as a situation in which the dissipation rates of the small and large scales evolve together. We advance two hypotheses that may form the basis of a theoretical approach to unsteady turbulence cascades in the presence of large-scale coherent structures.

Topologies of velocity-field stagnation points generated by a single pair of magnets in free-surface electromagnetic experiments.

The velocity fields generated by a static pair of magnets in free-surface electromagnetically forced flows are analyzed for different magnet attitudes, ionic currents, and brine depths. A wide range of laminar velocity fields is obtained despite the forcing simplicity. The velocity fields are classified according to their temporal mean flow topology, which strongly depends on the forcing geometry but barely on its strength, even through the bifurcation to unsteady regimes. The mean flow topology possesses a major influence on the critical Reynolds number Rec under which the steady velocity fields remain stable. The qualitative comparison of the dependence of Rec on the topology is in agreement with previous works. The unsteady configurations evidence the advection of smaller flow structures by the largest scales, commonly known as "sweeping."

Axisymmetric turbulent wakes with new nonequilibrium similarity scalings.

The recently discovered nonequilibrium turbulence dissipation law implies the existence of axisymmetric turbulent wake regions where the mean flow velocity deficit decays as the inverse of the distance from the wake-generating body and the wake width grows as the square root of that distance. This behavior is different from any documented boundary-free turbulent shear flow to date. Its existence is confirmed in wind tunnel experiments of wakes generated by plates with irregular edges placed normal to an incoming free stream. The wake characteristics of irregular bodies such as buildings, bridges, mountains, trees, coral reefs, and wind turbines are critical in many areas of environmental engineering and fluid mechanics.

Fractal space-scale unfolding mechanism for energy-efficient turbulent mixing.

Using top-end high fidelity computer simulations we demonstrate the existence of a mechanism present in turbulent flows generated by multiscale or fractal objects and which has its origin in the multiscale or fractal space-scale structure of such turbulent flow generators. As a result of this space-scale unfolding mechanism, fractal grids can enhance scalar transfer and turbulent diffusion by one order of magnitude while at the same time reduce pressure drop by half. This mechanism must be playing a decisive role in environmental, atmospheric, ocean, and river transport processes wherever turbulence originates from multiscale or fractal objects such as trees, forests, mountains, rocky riverbeds, and coral reefs. It also ushers in the concept of fractal design of turbulence which may hold the power of setting entirely new mixing and cooling industrial standards.

Universal dissipation scaling for nonequilibrium turbulence.

It is experimentally shown that the nonclassical high Reynolds number energy dissipation behavior, C(ε)≡εL/u(3)=f(Re(M))/Re(L), observed during the decay of fractal square grid-generated turbulence (where Re(M) is a global inlet Reynolds number and Re(L) is a local turbulence Reynolds number) is also manifested in decaying turbulence originating from various regular grids. For sufficiently high values of the global Reynolds numbers Re(M), f(Re(M))~Re(M).

Rapid growth of cloud droplets by turbulence.

Assuming perfect collision efficiency, we demonstrate that turbulence can initiate and sustain the rapid growth of very small water droplets in air even when these droplets are too small to cluster, and even without having to take gravity and small-scale intermittency into account. This is because the range of local Stokes numbers of identical droplets in the turbulent flow field is broad enough even when small-scale intermittency is neglected. This demonstration is given for turbulence which is one order of magnitude less intense than is typical in warm clouds but with a volume fraction which, even though small, is nevertheless large enough for an estimated a priori frequency of collisions to be ten times larger than in warm clouds. However, the time of growth in these conditions turns out to be one order of magnitude smaller than in warm clouds.

Acceleration-based classification and evolution of fluid flow structures in two-dimensional turbulence.

Zero acceleration points (ZAPs) and flow structures around them are studied in a direct numerical simulation of two-dimensional energy-cascading stationary homogeneous isotropic turbulence with an extended k(-5/3) energy spectrum. A well-defined classification of ZAPs in terms of the acceleration gradient tensor's (∇a) invariants emerges naturally as a result of well-defined properties of and relations between these invariants at ZAPs. About half of all ZAPs are anti-ZAPs [with det(∇a)<0 ] and the number of vortical and straining ZAPs [with det(∇a)>0 ] is about the same. Vortical and straining ZAPs are swept by the local fluid velocity to a good statistical approximation whereas anti-ZAPs, which are present in every vortical and straining ZAP's creation and destruction events, are not. The average lifetime of ZAPs seems to scale with the time scale of the smallest eddies in the turbulence, though ZAPs (in particular vortical ones) are able to survive up to a few integral time scales. Our ZAP classification can also be applied to extended flow regions and it turns out that vortical and straining regions are mediated by regions containing anti-ZAPs. A discussion of the length scales and sizes characterizing these regions and the distances between ZAPs is also given.

Defining a new class of turbulent flows.

We apply a method based on the theory of Markov processes to fractal-generated turbulence and obtain joint probabilities of velocity increments at several scales. From experimental data we extract a Fokker-Planck equation which describes the interscale dynamics of the turbulence. In stark contrast to all documented boundary-free turbulent flows, the multiscale statistics of velocity increments, the coefficients of the Fokker-Planck equation, and dissipation-range intermittency are all independent of Rλ (the characteristic ratio of inertial to viscous forces in the fluid). These properties define a qualitatively new class of turbulence.

Strong polymer-turbulence interactions in viscoelastic turbulent channel flow.

This paper is focused on the fundamental mechanism(s) of viscoelastic turbulence that leads to polymer-induced turbulent drag reduction phenomenon. A great challenge in this problem is the computation of viscoelastic turbulent flows, since the understanding of polymer physics is restricted to mechanical models. An effective state-of-the-art numerical method to solve the governing equation for polymers modeled as nonlinear springs, without using any artificial assumptions as usual, was implemented here on a three-dimensional channel flow geometry. The capability of this algorithm to capture the strong polymer-turbulence dynamical interactions is depicted on the results, which are much closer qualitatively to experimental observations. This allowed a more detailed study of the polymer-turbulence interactions, which yields an enhanced picture on a mechanism resulting from the polymer-turbulence energy transfers.

Stagnation point von Kármán coefficient.

On the basis of various direct numerical simulations (DNS) of turbulent channel flows the following picture is proposed. (i) At a distance y from either wall, the Taylor microscale lambda is proportional to the average distance l(s) between stagnation points of the fluctuating velocity field, i.e., lambda(y)=B(1)l(s)(y) with B(1) constant, for delta(nu) << y < or approximately equal to delta, where the wall unit delta(nu) is defined as the ratio of kinematic viscosity nu to skin friction velocity u(tau) and delta is the channel's half-width. (ii) The number density n(s) of stagnation points varies with height according to n(s)=(C(s)/delta(nu)(3))y(+)(-1) where y(+) identical with y/delta(nu) and C(s) is constant in the range delta(nu) << y < or approximately equal to delta. (iii) In that same range, the kinetic energy dissipation rate per unit mass, equals 2/3(E(+)((u(tau)(3)/kappa(s)y) where E(+) is the total kinetic energy per unit mass normalized by u(tau)(2) and kappa(s)=B(1)(2)/C(s) is the stagnation point von Kármán coefficient. (iv) In the limit of exceedingly large Reynolds numbers Re(tau) identical with delta/delta(nu), large enough for the Reynolds stress -(uv) to equal u(tau)(2) in the range delta(nu) << y << delta, and assuming that production of turbulent kinetic energy balances dissipation locally in that range and limit, the mean velocity U(+), normalized by u(tau), obeys (d/dy)U(+) approximately equal to 2/3(E(+)/kappa(s)y) in that same range. (v) It follows that the von Kármán coefficient kappa is a meaningful and well-defined coefficient and the log law holds in turbulent channel/pipe flows only if E(+) is independent of y(+) and Re(tau) in that range, in which case kappa approximately kappa(s). (vi) In support of (d/dy)U(+) approximately equal to 2/3(E(+)/kappa(s)y), DNS data of turbulent channel flows which include the highest currently available values of Re(tau) are best fitted by E(+) approximately equal to 2/3(B(4)y(+)(-2/15)) and (d/dy(+))U(+) approximately equal to (B(4)/kappa(s))y(+)(-1-2/15) with B4 independent of y in delta(nu) << y << delta if the significant departure from -(uv) approximately equal to u(tau)(2) at these Re(tau) values is taken into account.

Evolution of the acceleration field and a reformulation of the sweeping decorrelation hypothesis in two-dimensional turbulence.

From simulations of two-dimensional inverse energy cascading turbulence, we show that points with low acceleration values are predominantly advected by the local fluid velocity. The fluid velocity u in the global frame and the fluid velocity u in the frame moving with a low-acceleration point are approximately statistically independent. This property remains valid in high-acceleration regions but only in the direction of the local acceleration vector. In the perpendicular direction, the acceleration velocity V_a=u-xi is approximately independent of xi everywhere. These statistical independences constitute our formulation of the sweeping decorrelation hypothesis for two-dimensional inverse energy cascading turbulence.

Depletion of horizontal pair diffusion in strongly stratified turbulence: comparison with plane two-dimensional flows.

In this paper different arguments are put forward to explain why two-particle diffusion is depleted in the direction of stratification of a stably stratified turbulence. Kinematic simulations (KSs) which reproduce that depletion are used to shed light on the responsible mechanisms. The local horizontal divergence is studied and comparisons are made with two-dimensional kinematic simulation. The probability density function of the horizontal divergence of the velocity field is not a Dirac distribution in the presence of stratification but a Gaussian and this Gaussian does not depend on the Froude number. The number of stagnation points in the KS of three-dimensional strongly stratified turbulence is found virtually identical to what it is in KS of three-dimensional isotropic turbulence. However, the root mean square horizontal and vertical stagnation point velocities of the stratified turbulence are both larger than their counterparts in isotropic turbulence that latter getting progressively smaller as the Reynolds number increases. Therefore, the strong stratification destroys the persistence of the stagnation points. The main reason for the depletion, however, seems to have to be sought in the effect of stratification on the strain rate tensor. The stratification does lead to a depletion of the average square strain rate tensor, as well as of all average square strain rate eigenvalues. We conclude that it is these effects of stratification on the strain rate tensor that explain the depletion of the horizontal turbulent pair diffusion.

Tortuosity of unsaturated porous fractal materials.

The tortuosity of a capillary-condensed film of inviscid fluid adsorbed onto fractal substrates as a function of the filling fraction of the fluid has been calculated numerically. This acts as a way of probing the multiscale structure of the objects. It is found that the variation of tortuosity alpha with filling fraction varphi is found to follow a power law of the form alpha approximately varphi- for both deterministic and stochastic fractals. These numerically calculated exponents are compared to exponents obtained from a phenomenological scaling and good agreement is found, particularly for the stochastic fractals.

Sweep-stick mechanism of heavy particle clustering in fluid turbulence.

It is proposed that the inertial range clustering of small heavy particles in fluid turbulence occurs as a result of the sweep-stick mechanism which causes inertial particles to cluster so as to mimic the clusters of points where the fluid acceleration is perpendicular to the direction of highest contraction between neighboring particles. Direct numerical simulations of inertial particles subjected to linear Stokes drag and suspended in homogeneous isotropic turbulence support the validity of the sweep and stick properties on which the sweep-stick mechanism is based, and also support the clustering consequences of this mechanism. It also explains the observed Stokes-number dependence of inertial particle clustering.

Transport properties of saturated and unsaturated porous fractal materials.

Inviscid, irrotational flow through fractal porous materials is studied. The key parameter is the variation of tortuosity with the filling fraction phi of fluid in the porous material. Altering the filling fraction provides a way of probing the effect of the fractal structure over all its length scales. The variation of tortuosity with phi is found to follow a power law of the form alpha approximately phi (-E) for deterministic and stochastic fractals in two and three dimensions. A phenomenological argument for the scaling of tortuosity alpha with filling fraction phi is presented and is given by alpha approximately phi(D_{w}-2/D_{f}-d_{E}), where D_{f} is the fractal dimension, D_{w} is the random walk dimension, and d_{E} is the Euclidean dimension. Numerically calculated values of the exponents show good agreement with those predicted from the phenomenological argument for both the saturated and the unsaturated model.

Fast chemical reaction and multiple-scale concentration fields in singular vortices.

Two species involved in a simple, fast reaction tend to become segregated in patches composed of a single of these reactants. These patches are separated by a boundary where the stoichiometric condition is satisfied and the reaction occurs, fed by diffusion. Stirred by advection, this boundary and the concentration fields within the patches may tend to present multiple-scale characteristics. Based on this segregated state, this paper aims at evaluating the temporal evolutions of the length of the boundary and diffusive flux of reactants across it, when concentrations presenting initial self-similar fluctuations are advected by a singular vortex. First the two sources of singularity, i.e., the self-similar initial conditions and the singular vortex, are considered separately. On the one hand, self-similar initial conditions are imposed to a diffusion-reaction system, for one- and two-dimensional cases. On the other hand, an imposed singular vortex advects initially on/off concentration fields, in combination with diffusion and reaction. This problem is addressed analytically, by characterizing the boundary by a box-counting dimension and the concentration fields by a Hölder exponent, and numerically, by direct numerical simulations of the advection-diffusion-reaction equations. Second, the way the two sources hang together shows that, depending on the self-similar properties of the initial concentration fields, the vortex promotes the chemical activity close to its inner smoothed-out core or close to the outer region where the boundary starts to spiral. For all the considered situations, the length of the boundary and the global reaction speed are found to evolve algebraically with time after a short transient and a good agreement is found between the analytical and numerical scaling laws.

Multiscale laminar flows with turbulentlike properties.

By applying fractal electromagnetic force fields on a thin layer of brine, we generate steady quasi-two-dimensional laminar flows with multiscale stagnation point topology. This topology is shown to control the evolution of pair separation (Delta) statistics by imposing a turbulentlike locality based on the sizes and strain rates of hyperbolic stagnation points when the flows are fast enough, in which case Delta(2) approximately t(gamma) is a good approximation with gamma close to 3. Spatially multiscale laminar flows with turbulentlike spectral and stirring properties are a new concept with potential applications in efficient and microfluidic mixing.