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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

Fundamentals of pair diffusion in kinematic simulations of turbulence.

We demonstrate that kinematic simulation (KS) of three-dimensional homogeneous turbulence produces fluid element pair statistics in agreement with the predictions of L F. Richardson [Proc. R. Soc. London, Ser. A 110, 709 (1926)] even though KS lacks explicit modeling of turbulent sweeping of small eddies by large ones. This scaling is most clearly evident in the turbulent diffusivity's dependence on rms pair separation and, to a lesser extent, on the pair's travel time statistics. It is also shown that kinematic simulation generates a probability density function of pair separation which is in good agreement with recent theory [S. Goto and J. C. Vassilicos, New J. Phys. 6, 65 (2004)] and with the scaling of the rms pair separation predicted by L. F. Richardson [Proc. R. Soc. London, Ser. A 110, 709 (1926)]. Finally, the statistical persistence hypothesis (SPH) is formulated mathematically and its validity tested in KS. This formulation introduces the concept of stagnation point velocities and relates these to fluid accelerations. The scaling of accelerations found in kinematic simulation supports the SPH, even though KS does not generate a Kolmogorov scaling for the acceleration variance (except for a specific case and a limited range of outer to inner length-scale ratios). An argument is then presented that suggests that the stagnation points in homogeneous isotropic turbulence are on average long-lived.

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.

Acceleration statistics as measures of statistical persistence of streamlines in isotropic turbulence.

We introduce the velocity Vs of stagnation points as a means to characterize and measure statistical persistence of streamlines. Using theoretical arguments, direct numerical simulations (DNS), and kinematic simulations (KS) of three-dimensional isotropic turbulence for different ratios of inner to outer length scales L/eta of the self-similar range, we show that a frame exists where the average Vs = 0 , that the rms values of acceleration, turbulent fluid velocity, and Vs are related by La'/u'2 approximately (V's/u')(L/eta)(2/3+q) , and that V's/u' approximately (L/eta)q with q = -1/3 in Kolmogorov turbulence, q = -1/6 in current DNS, and q = 0 in our KS. The statistical persistence hypothesis is closely related to the Tennekes sweeping hypothesis.

Diffusivity dependence of ozone depletion over the midnorthern latitudes.

The mixing and reaction properties of advected chemicals (and passive scalars) are determined by the fractal dimension D of the interface between the chemicals. We show that the scaling of the amount m of reacted chemicals with diffusivity kappa is m(0)-m(kappa) proportional, proportional to kappa(1-D/2) in the two-dimensional case. This relation is valid in a range of times and diffusivities where the diffusive length scales of the chemicals are within the range of scales where the chemical interface has a well-defined fractal dimension. We apply the relation to the problems of chlorine deactivation and ozone depletion over the midnorthern latitudes. We determine numerically the fractal dimension of an interface advected by stratospheric winds. This allows us, first, to explain the diffusivity dependence of chlorine deactivation and ozone depletion that was previously observed in numerical simulations (Tan et al., J. Geophys. Res., [Atmos.] 103, 1585 (1998)) and, second, to extrapolate the results of such simulations down to realistically low diffusivities.

Mixing in fully chaotic flows.

Passive scalar mixing in fully chaotic flows is usually explained in terms of Lyapunov exponents, i.e., rates of particle pair separations. We present a unified review of this approach (which encapsulates also other nonchaotic flows) and investigate its limitations. During the final stage of mixing, when the scalar variance decays exponentially, Lyapunov exponents can fail to describe the mixing process. The failure occurs when another mixing mechanism, first introduced by Fereday et al. [Phys. Rev. E 65, 035301 (2002)], leads to a slower decay than the mechanism based on Lyapunov exponents. Here we show that this mechanism is governed by the large-scale nonuniformities of the flow which are different from the small scale stretching properties of the flow that are captured by the Lyapunov exponents. However, during the initial stage of mixing, i.e., the stage when most of the scalar variance decays, Lyapunov exponents describe well the mixing process. We develop our theory for the incompressible and diffusive baker map, a simple example of a chaotic flow. Nevertheless, our results should be applicable to all chaotic flows.

Scalings of scalar structure functions in a velocity field with coherent vortical structures.

In planar turbulence modeled as an isotropic and homogeneous collection of two-dimensional noninteracting compact vortices, the structure functions S(p)(r) of a statistically stationary passive scalar field have the following scaling behavior in the limit where the Péclet number Pe-->infinity: S(p)(r) approximately const+ln(r/L Pe(-1/3)) for L Pe(-1/3)<

Inertial particle segregation by turbulence.

We study collections of heavy and light small spherical particles initially well mixed with each other, subjected to linear (Stokes) drag force and gravity, and falling through a fluid turbulence. We introduce the segregation power spectrum, which we use to define the segregation length scale. Kinematic simulation predicts that the turbulence can segregate heavy and light falling particles and leads to a well-defined segregation length scale. The properties of this length scale and of the segregation power spectrum used to define it are discussed and, where possible, explained.

Kinematic simulation of turbulent dispersion of triangles.

As three particles are advected by a turbulent flow, they separate from each other and develop nontrivial geometries, which effectively reflect the structure of the turbulence. We investigate here the geometry, in a statistical sense, of three Lagrangian particles advected, in two dimensions, by kinematic simulation (KS). KS is a Lagrangian model of turbulent diffusion that makes no use of any delta correlation in time at any level. With this approach, situations with a very large range of inertial scales and varying persistence of spatial flow structure can be studied. We first demonstrate that the model flow reproduces recent experimental results at low Reynolds numbers. The statistical properties of the shape distribution at a much higher Reynolds number is then considered. The numerical results support the existence of nontrivial shape statistics, with a high probability of having elongated triangles. Even at the highest available inertial range of scales, corresponding to a ratio between large and small scale L/eta=17,000, a perfect self-similar regime is not found. The effects of the parameters of the synthetic flow, such as the exponent of the spectrum and the effect of the sweeping affect our results, are also discussed. Special attention is given to the effects of persistence of spatial flow structure.

Scalar variance decay in chaotic advection and Batchelor-regime turbulence.

The decay of the variance of a diffusive scalar in chaotic advection flow (or equivalently Batchelor-regime turbulence) is analyzed using a model in which the advection is represented by an inhomogeneous baker's map on the unit square. The variance decays exponentially at large times, with a rate that has a finite limit as the diffusivity kappa tends to zero and is determined by the action of the inhomogeneous map on the gravest Fourier modes in the scalar field. The decay rate predicted by recent theoretical work that follows scalar evolution in linear flow and then averages over all stretching histories is shown to be incorrect. The exponentially decaying scalar field is shown to have a spatial power spectrum of the form P(k) approximately k(-sigma) at wave numbers small enough for diffusion to be neglected, with sigma<1.

Turbulent wakes of fractal objects.

Turbulence of a windtunnel flow is stirred using objects that have a fractal structure. The strong turbulent wakes resulting from three such objects which have different fractal dimensions are probed using multiprobe hot-wire anemometry in various configurations. Statistical turbulent quantities are studied within inertial and dissipative range scales in an attempt to relate changes in their self-similar behavior to the scaling of the fractal objects.