Stationary Velocity Field Research Articles

Mixing to a molecular level in turbulent media

An equation is obtained for the mean product of concentrations of two species in an isotropic stationary random velocity field. The velocity field is assumed to be a normal random process δ-correlated with respect to time. The process of mixing up to a molecularlevel is shown to be associated with diffusion in the Lagrangian coordinates. The mean species concentration in the Lagrangian variables has been found. The estimates have been made for the rate of growth of the admixture cloud size in the Lagrangian and Eulerian coordinates. The characteristic times of mixing of preseparated species have been estimated. An expression is obtained for the total product of a slow irreversible second-order chemical reaction at arbitrary initial distributions of reagents. The equilibrium structural function of the passive admixture is given in terms of the structural function of the velocity field.

Equation for the structure function of a turbulent stationary isotropic velocity field and its solution in the inertial scale interval

A closed equation is obtained for the structure function of a turbulent stationary isotropic velocity field and the equation is solved in the inertial scale interval.

Eulerian direct interaction applied to the statistical motion of particles in a turbulent fluid

Eulerian direct interaction is used to close Liouville's equation central to the transport of particles in a turbulent fluid where the dominant drag force is derived from the particle and local fluid velocities. The reliability of the equation is then tested by comparison of solutions with those of a computer simulation of particle motion with Stokes drag in a random velocity field. Using an empirical drag law accurate for particle Reynolds numbers up to 500, formulae are derived for the statistical moments central to particle dispersion for a weak drag force operating in a Gaussian isotropic stationary velocity field. These show for instance that the long time particle diffusion coefficient is in general greater than the equivalent value based on Stokes drag.

Diffusion of weak magnetic fields by isotropic turbulence

The diffusion of slowly varying, weak magnetic fields by a statistically isotropic and stationary velocity field in a perfectly conducting fluid is studied by Eulerian analysis. The characteristic wavenumber and variance of the velocity field are k0 and 3v20, thus defining the eddy-circulation time τ0 = 1/v0k0. The velocity field is assumed constant on intervals of duration 2τ1 and statistically independent for distinct intervals. Thus the correlation time is τ1. The α-effect dynamo mechanism in the quasi-linear approximation is corroborated. Both the quasilinear and the direct-interaction approximations give identical diffusion of magnetic and passive scalar fields in reflexionally invariant turbulence. This result is found to be exact for τ1/τ0 → 0 but is demonstrated to be incorrect in general for finite τ1/τ0 because of effects of helicity fluctuations. The nature of the failure of the direct-interaction approximation is exhibited by an exactly soluble model system. Analysis based on a double-averaging device shows that longrange, persistent helicity fluctuations in reflexionally invariant turbulence give an anomalous negative contribution to the magnetic diffusivity which depends on the helicity covariance function. We term this the α2 effect. The magnitude of the effect depends sensitively on the turbulence statistics. If the characteristic scales of the helicity fluctuations are sufficiently larger than τ0 and 1/k0, the magnetic diffusivity is negative, implying unstable growth, while a passive scalar field diffuses normally. On the other hand, a crude estimate suggests that the α2 effect is small in normally distributed turbulence.

Turbulent self-diffusion

A method relating Lagrangian and Eulerian statistics for the diffusion of a particle in a homogeneous stationary velocity field, which was developed for plasma diffusion by Taylor and McNamara and further explained by Montgomery, is shown to be equivalent to a method proposed by Corrsin and used by Saffman. The results are compared with numerical simulations with good agreement for velocity fields which have properties similar to real turbulence.

On characteristic functionals of a random velocity field of helical motions of a viscous fluid: PMM vol. 39, n≗2, 1975, pp. 379–381

A class of exact solutions is derived for the characteristic functional of a random velocity field of helical motions of a viscous incompressible fluid. The solution [1, 2] of the equation for the characteristic functional of the velocity field derived in variational derivatives relates to a statistically stationary velocity field in a perfect incompressible fluid, and of the form of a Gaussian functional with constant spectral density of energy.

An approach to the autoignition of a turbulent mixture

This paper considers the turbulent homogeneous mixing of two reactants undergoing a one step, second order, irreversible, exothermic chemical reaction with a rate constant of the Arrhenius type. A statistically stationary turbulent velocity field is assumed given and unaffected by mass or heat production due to the chemical reaction. Relative density fluctuations are neglected. A Hopf-like functional formalism is presented, with application to both statistically inhomogeneous and statistically homogeneous flows. Single and double point probability density function differential equations are derived from those functional equations. The limit of very large activation energies is considered; a low degree of statistical correlation between temperature and concentration fields during the ignition period is hypothesized. After making use of the homogeneity assumption a closure problem is still present due to the nonlocalness of the molecular diffusion term. The problem is rendered closed by assuming a Gaussian conditional expected value for the temperature at a point given the temperature at a neighboring point. The closure is seen to preserve very important mathematical and physical properties. A linear first order hyperbolic differential equation with variable coefficients for the probability density function of the temperature field is obtained. A second Damköhler number based on Taylor's microscale turns out to be an important controlling parameter. A numerical integration for different values of the second Damköhler number and the initial stochastic parameters is carried out. The mixture is seen to evolve towards an eventual thermal runaway, the detailed behavior however being different for different systems. Some peculiarities during the ignition period evolution are uncovered.

Isotropic Turbulent Mixing under a Second-Order Chemical Reaction

The decay of a dynamically passive reactant in a stationary isotropic turbulent velocity field is investigated using the direct-interaction and (modified) quasi-normal closures. The reactant undergoes an isothermal second-order chemical reaction of single species. The dynamical behavior can be characterized approximately by the ratio of the Damkohler to the Peclet numbers. For slow reactions, only the convective mixing action is important and the closures can be applied without restrictions just as in the turbulent mixing case. For the closures to yield consistent results with fast reactions, the initial mean concentration must be at least twice the initial rms fluctuations. Then the effect of reactive nonlinear interaction is so meager that the two modes of closure make little difference. Otherwise, application of the closure schemes which evoke initial Gaussian conditions is unwarranted and thus presents a convergence difficulty. Simple decay expressions of the mean concentration and rms fluctuations are obtained for the asymptotic range of slow and fast reactions.

Comparison of Closure Approximation Theories in Turbulent Mixing

Using various closure approximation theories, the evolution of scalar quantity spectrum in a statistically stationary isotropic turbulent velocity field is investigated to provide a unified comparison at Peclet number of 50. The phenomenological Heisenberg (eddy diffusivity) transfer function and the analytical approaches including the truncation, the quasi-normal, and the recent direct-interaction closure approximations are examined. As already known, the truncation and quasi-normal closures result in a negative-spectrum within short decay times. A modified quasi-normal closure formula is suggested as a means of approximating the simultaneous covariances, and the results thereof are comparable with the direct-interaction counterparts.

Decay of Scalar Quantity Fluctuations in a Stationary Isotropic Turbulent Velocity Field

The direct interaction approximation suggested by Kraichnan is used to study the scalar mixing problem in a stationary isotropic turbulent velocity field. The main part of this paper deals with the decay of scalar quantity fluctuations in a simplified stationary velocity field. The numerical integration of the direct interaction equations was carried out with several different initial spectra for Peclet numbers of 50 and 500. The results seem to demonstrate naturalness of the direct interaction approximation; however, no comparison with experiments can be made at present.