Passive Scalar Gradient Research Articles

Numerical study of comparison of vorticity and passive vectors in turbulence and inviscid flows.

The nonlinear vortex stretching in incompressible Navier-Stokes turbulence is compared with a linear stretching process of passive vectors (PVs). In particular, we pay special attention to the difference of these processes under long and short time evolutions. For finite time evolution, we confirm our previous finding that the stretching effect of vorticity is weaker than that of general passive vectors for a majority of the initial conditions with the same energy spectra. The above difference can be explained qualitatively by examining the Biot-Savart formula. In order to see to what extent infinitesimal time development explains the above difference, we examine the probability density functions (PDFs) of the stretching rates of the passive vectors in the vicinity of a solution of Navier-Stokes equations. It is found that the PDFs are found to have a Gaussian distribution, suggesting that there are equally many PVs that stretched less and more than the vorticity. This suggests the importance of the vorticity-strain correlation built up over finite time in turbulence. We also discuss the case of Euler equations, where the dynamics of the Jacobian matrix relating the physical and material coordinates is examined numerically. A kind of alignment problem associated with the Cauchy-Green tensor is proposed and studied using the results of numerical simulations. It is found that vorticity tends to align itself with the most compressing eigenvector of the Cauchy-Green tensor. A two-dimensional counterpart of active-passive comparison is briefly studied. There is no essential difference between stretching of vorticity gradients and that of passive scalar gradients and a physical interpretation is given to it.

Dynamics of the orientation of active and passive scalars in two-dimensional turbulence

The active nature of vorticity is investigated in order to understand its difference with a passive scalar. The direct cascade down to small scales is examined through both classical and new diagnostics (based on tracer gradient properties) in numerical simulations of freely decaying two-dimensional (2D) turbulence. During the transient evolution of turbulence, the passive scalar possesses a stronger cascade due to different alignment properties with the equilibrium orientations obtained in the adiabatic approximation by Lapeyre et al. [Phys. Fluids 11, 3729 (1999)] and Klein et al. [Physica D 146, 246 (2000)]. In strain-dominated regions, the passive scalar gradient aligns better with the equilibrium orientation than the vorticity gradient does, while the opposite is true in effective-rotation-dominated regions. A study of the kinematic alignment properties shows that this is due to structures with closed streamlines in the latter regions. However, in the final evolutionary stage of turbulence, both active and passive tracer gradients have identical orientations (i.e., there is a perfect alignment between the two gradients, all the more so when they are stronger). The effect of diffusion on the cascade is also studied.

Evolution of a scalar gradient’s probability density function in a random flow

The evolution of the probability density function (PDF) of the passive scalar gradient is studied in the limit of large Peclet and Prandtl numbers in d dimensions. Without diffusion, the closed Fokker-Planck equation can be derived and solved analytically leading to a number of conclusions. In particular, it allows the description of the restoration of the rotational symmetry and enables one to distinguish different regimes of evolution and different intervals of PDF behavior.

Fractal Patterns of Tracers Advected by Smooth Temporally Irregular Fluid Flows and their Analysis by Use of Random Maps

Temporally irregular, large spatial scale, fluid advection of passive tracers occurs in a wide variety of situations. Our main point is that these situations can be conveniently conceptualized as resulting from successive application of a sequence of random maps. This viewpoint is numerically convenient and also provides a useful theoretical framework for dynamical-systems-based analyses of the resulting fractal patterns. Examples of three different situations are discussed: (i) the fractal distribution of passive scalar gradients, (ii) the fractal pattern formed by a scum floating on the surface of a moving fluid, and (iii) the pattern of particles entrained as they flow past an obstacle in an open flow.

Similarity states of passive scalar transport in buoyancy-generated turbulence

The mixing of a passive scalar field by turbulence that is generated by buoyancy forces acting on an initial random density field is considered. Various asymptotic similarity states of the passive scalar field with and without a uniform mean passive scalar gradient are determined by dimensional arguments based on exact or near invariants of the density and passive scalar fields. The results of large-eddy numerical simulations are shown to support the derived scaling laws. The large-eddy simulations also demonstrate the different mixing properties of an active and passive scalar field.

The structure of a passive scalar field with a uniform mean gradient in rapidly sheared homogeneous turbulent flow

The effect of an arbitrarily oriented mean passive scalar gradient on one-point passive scalar statistics is studied in homogeneous turbulent shear flow in the limit of rapid shearing. By neglecting the nonlinear inertial transfer to small scales an analytical solution for individual Fourier modes is obtained for the case of unity Prandtl number. This solution is used to compute the development of one-point statistics for the velocity and scalar fields in the inviscid limit. Comparisons to direct numerical simulations of the full nonlinear equations for the same flow show that in addition to describing the early time response to the imposed shear, the linear solution gives reasonable estimates of several correlation coefficients for the developed shear flow.

Comment on "Chaotic fluid convection and the fractal nature of passive scalar gradients"

Comment on "Chaotic fluid convection and the fractal nature of passive scalar gradients"

Chaotic fluid convection and the fractal nature of passive scalar gradients.

The chaotic convection of passive scalars by an incompressible fluid is considered. (The convection is said to be chaotic if nearby fluid elements typically diverge from each other exponentially in time.) It is shown that during the time evolution the gradient squared of chaotically convected passive scalars typically concentrates on a fractal set. Considerations of the local stretching properties of the flow lead to a partition function which yields the dimension spectra of the resulting fractal measure. Fractal structure is a result of the nonuniform stretching of typical chaotic flows.