Science Briefs

Turbulence

Fig. 1: Photo of sketch in da Vinci notebook
"Observe the motion of the surface of the water, which resembles that of hair, which has two motions, of which one is caused by the weight of the hair, the other by the direction of the curls; thus the water has eddying motions, one part of which is due to the principal current, the other to random and reverse motion". [Trans. Piomelli in Lumley, J.L., 1997. Some comments on turbulence. Phys. Fluids A 4, 203-211.]

For centuries, scientists have been fascinated with flows in nature. The legendary artist/scientist Leonardo da Vinci's view on turbulence is reproduced below. Most flows in nature have the tendency to become quite disorderly or turbulent. Reliable descriptions of the ocean and of the atmosphere thus depend in large measure on how well one is able to describe turbulence. Turbulence cannot exist by itself, however; it requires a continuous supply of energy. In Earth's atmosphere, there are at least two sources of stirring: temperature gradients that produce turbulent flows transporting heat from hot to cold regions, and wind shear that generates vortices.

Over the years, a major effort has been devoted at GISS toward the development of physically realistic and yet manageable models of turbulence. In a series of papers ("A dynamical model for turbulence", parts I to VI) that appeared in the journal Physics of Fluids, we have proposed, worked out, and tested a new model to treat turbulence. It was tested on a large variety of flows.

Three new papers deal with shear driven flows. In the first of these (paper VII), we derived the general mathematical properties of the velocity correlations that one can form using the properties of a mean flow (namely shear and vorticity). We have shown how to avoid the use of the rather cumbersome mathematical expressions that have been employed for about thirty years, by deriving much simpler expressions. This will considerably facilitate the use of the new model. We must remark that the possibility of formulation of an expression simpler than the traditional one was discussed many times in the literature, but to the best of our knowledge nobody has previously succeeded in working one out.

In paper VIII we derive the spectral forms of our new turbulent variables and discuss the fact that the match to laboratory data is quite satisfactory. Spectra of turbulent variables, like energy, render detailed information about how much energy is carried by individual eddies. This provides more information on the dynamics of the vortices (eddies) than the integrated quantities, which only yield the total amount of energy carried by all the vortices. Finally, in paper IX we work out the general expression for the Reynolds stresses, namely the correlations between the fluctuating velocities of the turbulent eddies. The latter act to drain energy from the large scales of motion in a fashion similar to ordinary viscosity, thus acting like an enhanced viscosity. Similarly, turbulence enhances the transport of heat. The proper quantification of the eddy-enhanced transports is the crucial challenge for any turbulence model.

The turbulence model is presently used at GISS to improve the description of turbulent mixing in the ocean. Preliminary results indicate that the new model produces a better match to the data than do previous, phenomenologically-based models of turbulence, even though this model has no adjustable parameters. Soon these results will be applied to the atmosphere.

Reference

Canuto, V.M., and M.S. Dubovikov 1999. A dynamical model for turbulence. VII. Complete system of five orthogonal tensors for shear driven flows. Phys. Fluids 11, 659-664.

Canuto, V.M., M.S. Dubovikov, and G. Yu 1999. A dynamical model for turbulence. VIII. IR and UV Reynolds stress spectra for shear-driven flows. Phys. Fluids 11, 665-677.

Canuto, V.M., M.S. Dubovikov, and G. Yu 1999. A dynamical model for turbulence. IX. Reynolds stresses for shear-driven flows. Phys. Fluids 11, 678-691.

Contact

Please address all inquiries about this research to Dr. Vittorio Canuto.