Primary and secondary bifurcation in baroclinic instability

In modelling atmospheric flows the baroclinic instability of the flow in a differentially heated rotating annulus plays a central role. This paper deals with an experimental study using LDV and flow visualization techniques. Usually the temperature difference, T, was kept fixed while the angular velocity, , was varied. On crossing the stability boundary, the primary bifurcation, the basic flow gives way to a baroclinic wave flow. For a given annulus geometry the wave number, m, of the first wave pattern was found to be uniquely defined by T. The measured critical values of , _crit, agree reasonably well with those obtained by other authors. On increasing above _crit the wave number changed, this process showing hysteresis. The situation might indicate secondary bifurcation phenomena. Flow visualization using aluminium particles shows surface flow details.This paper is dedicated to Prof. Dr. K. Gersten on the occasion of his 60th birthday     

Explosive instability of geostrophic vortices. Part 1: baroclinic instability

In a quasi-geostrophic model, we study the baroclinic instability of a two-layer vortex. The singular unstable modes for potential vorticity anomalies are compared with the classical normal modes. Short-time singular modes are explosively unstable and, at short times, depend only on the baroclinic component of the flow. As time progresses, they evolve towards the normal modes and their sensitivity to flow parameters is explored. Asymptotic solutions are provided.     

Convective instability in a medium with spiral turbulence

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 61–66, January–February, 1990.     

Particle behavior in homogeneous isotropic turbulence

Direct numerical simulations were conducted to investigate the behavior of heavy particles in homogeneous isotropic turbulence. The present study focused on the effect of particle inertia and drift on the autocorrelations of the particle velocity and the fluid seen by particles and the dispersion characteristics of particles. The Lagrangian integral time scale of particles monotonically increased as the magnitude of the particle response time increased, while that of the fluid seen by particles remained relatively constant; it reached a maximum when the particle response time was close to the Kolmolgorov time scale of the flow. Particle dispersion increased as the particle inertia increased for small particles, while for larger particles, it decreased as particle inertia increased; particle eddy diffusion coefficient was maximal, and greater than that of the fluid by about 30%, at the preferential concentration. The concentration field of the particles with τp/τk≈1.0 showed that particles tend to collect in regions of low vorticity (high strain) due to preferential concentration. As the drift velocity of a particle is increased it crosses the paths of fluid elements more rapidly and will tend to lose correlation with its previous velocity faster than a fluid element will. And the correlation of particle velocities along the drift direction is more persistent than that perpendicular to the direction of drift. Simulations also showed that the continuity effect and the crossing-trajectory effect are weakened for particles with infinite inertia.     

Behavior of homogeneous turbulence in channels with variable cross-sectional area

On the basis of the equation describing the behavior of the spectral tensor of the energy of the pulsation velocity for one-dimensional flow in a channel with variable cross-sectional area, we obtained a system of equations for the meansquare components of the pulsation velocity vector and integral scales of the turbulence length in various directions, enabling us to use these parameters in the initial section of the channel for determining their behavior along the channel. We made use of some ideas of A. N. Kolmogorov and J. Rotta concerning the possibility of describing viscous and nonlinear terms in the equations for the components of the tensor of Reynolds stresses in terms of the energy of pulsation motion and the integral scale of the turbulence length. The resulting system, in the special cases of very low intensity of turbulence, leads to the results of the linear theory; for constant cross-sectional area (or for a very high intensity of turbulence, when the damping affects the turbulence much more strongly than does the deformation effect) it describes the known empirical laws of the degeneration of turbulence beyond grids. We made a comparison with the data available in the literature on the behavior of the characteristics of turbulence in channels with variable cross-sectional area.     

Modeling of pressure-strain rate in homogeneous turbulence

Direct numerical simulations were used to study the slow pressure strain rate term in homogeneous, axisymmetric turbulence during its relaxation towards isotropy. A strong Reynolds number dependence was found on the relaxation rate. A model for the rapid pressure strain rate tensor was developed using the classical approach of expansion in the Reynolds stress tensor. The model is tensorially complete and satisfies strong realizability. Rapid distortion analysis has been used to calibrate the model coefficients. The range of applicability of the model has been tested using direct numerical simulations.     

A theory of homogeneous isotropic turbulence

Summary Homogeneous and isotropic turbulence has been discussed in the present paper. An attempt has been made to find the simplifying hypothesis for connecting the higher order correlation tensor with the lower ones. Starting from the Navier-Stokes equations of motion for an incompressible fluid and following the usual method of taking the averages, a differential equation in Q and X, the defining scalar of the second order correlation tensor Q _x and the defining scalar of a third order isotropic tensor X _ijk , has been derived. The tensor X _ijk stands for a tensorial expression containing the derivatives of the third and the fourth order tensors. Then the hypothesis is used that X=F(Q), where F is an unknown function. To find the forms of F, Kolmogoroff's similarity principles have been used, and thus two forms for F(Q) corresponding to two regions of the validity of these principles have been deduced.     

Factorized cumulant expansion for homogeneous turbulence

A factorization approximation is introduced to the cumulant expansion of the characteristic functional for homogeneous turbulence. Under this approximation, an arbitrary nth order cumulant C^(itn) is expressed in terms of the cumulants C⁽²⁾,C⁽³⁾ and C⁽ⁿ⁻¹⁾, and thus we obtain a closed but untruncated system of equations for the cumulants. Using the factorized fourth-cumulant approximation, a closed set of equations for the energy spectrum function E(k,t) and the energy transfer function T(k,t) is derived. These equations are solved numerically and the similarity laws of the solutions are derived analytically. The statistical quantities such as the energy, the enstrophy, Taylor's microscale and the skewness of the velocity derivative are calculated numerically and the statistical laws of these quantities are discussed.     

Measurements of layer depth during baroclinic instability in a two-layer flow

This study investigates the baroclinic instability of a two-layer rotating fluid system. The instability is generated by releasing a cylinder of buoyant fluid at the surface of ambient fluid. The buoyant fluid is dyed so that its depth may be determined from its optical thickness. The system first adjusts until the horizontal density gradient is balanced by a flow along the front, and the adjusted state is then unstable to azimuthal waves. Contours of constant upper layer depth are examined, and the perturbation at each azimuthal wavenumber is determined. The initial wavenumber is well modelled by simple quasi-geostrophic theory. There is a clear high wavenumber cutoff, and a transfer of energy to larger scales with time.     

On the generation of large-scale homogeneous turbulence

An active turbulence generating grid, based on the rotating-vane design of Makita (1991), was developed for a large wind tunnel. At 2.14 m square, the grid is the largest of this type ever developed. To improve the isotropy of the turbulence generated, the grid was placed in the wind tunnel contraction. Measurements show that the grid produces a closely uniform mean flow and homogeneous isotropic turbulence to within two integral scales from the wall. By systematically varying the flow speed and parameters controlling the random motion of the vanes, grid turbulence with a wide variety of characteristics was produced and the dependence of those characteristics on the operating parameters of the grid revealed. Taylor Reynolds numbers of the grid turbulence varied from 100 to 1,360 and integral scales from 5 to almost 70 cm. The extreme cases represent some of the highest Reynolds number and largest scale homogeneous turbulent flows ever generated in a wind tunnel.     

The self-preservation of homogeneous shear flow turbulence

An analysis of the equations governing homogeneous shear flow shows the possibility of solutions which are self-preserving at all scales of motion, and that these solutions are dependent on the initial conditions. The appropriate velocity scale is the one obtained from the turbulence kinetic energy, q ²/2, while the length scale is the Taylor microscale, . Two cases of self-preserving flow are identified: one corresponding to constant mean shear, the other to a mean shear which is inversely proportional to time. For the first case (the only one considered in detail) the principal results of the postulated similarity are that is constant, while q ² varies exponentially with time. The ratio of the turbulence energy production rate to its dissipation rate remains constant. It is also shown that the energy spectra scale over all wavenumbers with q ² and , and that they have shapes determined by the initial conditions. The experimental evidence is generally consistent with the theory.     

Space-time description of stationary and homogeneous turbulence

A system of two functional equations is derived, describing turbulence that is stationary in time and homogeneous in space. A theory of perturbations is formulated and the character of hypotheses concerning termination of the series involved is sharpened.     

A possible explanation of countergradient fluxes in homogeneous turbulence

This study addresses the phenomenon of persistent countergradient (PCG) fluxes of momentum and heat (density) as observed in homogeneous turbulence forced by shear and stratification. Countergradient fluxes may occur at large scales when stratification is strong. However, they always occur at small scales, independently of stratification. A conceptional model is introduced to explain PCG fluxes at small scales as the result of the collision of large-scale fluid parcels. The large parcels collide under the driving force of inclined vortex structures (in a shear-dominated flow) or of buoyancy (in a strongly stratified shear flow). This collision model also explains the PCG heat flux in an unsheared stratified flow with zero average momentum flux. It is found that the energy of the small-scale PCG motions is provided (i) by quick transport of kinetic energy from the scales of production to relatively slowly dissipating scales if the flow is shear-driven and (ii) by conversion of available potential energy to kinetic energy at small scales when the flow is stratified. The collision mechanism is an inherent property of the turbulence dynamics. Therefore, the PCG fluxes at small scales reflect a universal character of homogeneous turbulence, and are found over a large range of Reynolds numbers. The Prandtl (or Schmidt) number influences the rate of dissipation of temperature (or density) variance but not the dissipation rate of the velocity variance. In stratified flows, therefore, the number directly affects the strength of the PCG heat flux at small scales. It is found, however, that the PCG momentum flux is also altered slightly when the Prandtl number is large enough to sustain small buoyantly moving parcels after collision.     

Modulation of homogeneous turbulence seeded with finite size bubbles or particles

The dynamics of homogeneous, isotropic turbulence seeded with finite sized particles or bubbles is investigated in a series of numerical simulations, using the force-coupling method for the particle phase and low wavenumber forcing of the flow to sustain the turbulence. Results are given on the modulation of the turbulence due to massless bubbles, neutrally buoyant particles and inertial particles of specific density 1.4 at volumetric concentrations of 6%. Buoyancy forces due to gravity are excluded to emphasize finite size and inertial effects for the bubbles or particles and their interactions with the turbulence. Besides observing the classical entrapment of bubbles and the expulsion of inertial particles by vortex structures, we analyze the Lagrangian statistics for the velocity and acceleration of the dispersed phase. The turbulent fluctuations are damped at mid-range wavenumbers by the bubbles or particles while the small-scale kinetic energy is significantly enhanced. Unexpectedly, the modulation of turbulence depends only slightly on the dispersion characteristics (bubble entrapment in vortices or inertial sweeping of the solid particles) but is closely related to the stresslet component (finite size effect) of the flow disturbances. The pivoting wavenumber characterizing the transition from damped to enhanced energy content is shown to vary with the size of the bubbles or particles. The spectrum for the energy transfer by the particle phase is examined and the possibility of representing this, at large scales, through an additional effective viscosity is discussed.     

Lyapunov analysis for fully developed homogeneous isotropic turbulence

The present work studies the isotropic and homogeneous turbulence for incompressible fluids through a specific Lyapunov analysis. The analysis consists in the calculation of the velocity fluctuation through the Lyapunov theory applied to the local deformation using the Navier-Stokes equations, and in the study of the mechanism of energy cascade through the finite scale Lyapunov analysis of the relative motion between two particles. The analysis provides an explanation for the mechanism of energy cascade, leads to the closure of the von Kármán-Howarth equation, and describes the statistics of the velocity difference. Several tests and numerical results are presented.     

Direct numerical simulation of homogeneous stratified rotating turbulence

The effects of the Prandtl number on stratified rotating turbulence have been studied in homogeneous turbulence by using direct numerical simulations and a rapid distortion theory. Fluctuations under strong stable-density stratification can be theoretically divided into the WAVE and the potential vorticity (PV) modes. In low-Prandtl-number fluids, the WAVE mode deteriorates, while the PV mode remains. Imposing rotation on a low-Prandtl-number fluid makes turbulence two-dimensional as well as geostrophic; it is found from the instantaneous turbulent structure that the vortices merge to form a few vertically-elongated vortex columns. During the period toward two-dimensionalization, the vertical vortices become asymmetric in the sense of rotation. Communicated by S. Obi PACS 47.55.Hd