Benjamin Connell

18.325

3/15/00

G. I. Taylor and A. E. Green, 1937

In 1935 G. I. Taylor published his Statistical Theory of Turbulence, detailing a statistical method of describing turbulent flow. Experimental data were combined with the plan of statistical analysis in the generation formulae describing the decay of turbulence. A remaining challenge for Taylor was to corroborate the experimental work described with the statistical theory by tracking an explicit solution through time solved with the governing equations. The hope was that this solution would demonstrate the properties of turbulent flow defined experimentally. The development of the solution to the flow and comparison to the experimental results is the meat of Mechanism of the Production of Small Eddies from Large Ones.

Taylor describes the "fundamental process in turbulent flow" as "the grinding down of eddies produced by solid obstructions...into smaller and smaller eddies until...they die away owing to viscosity." He continues, "To explain this problem is, perhaps, the fundamental problem in turbulent motion."

A description is given of how the well observed diffusive property of turbulence is linked to this generation of small eddies from large ones. Starting from the assumption of diffusion in turbulent flow we know that two particles in the flow will tend to get further away from each other as time progresses. Imagine that these two particles are on the two ends of a vortex filament, a cylinder around which the fluid is circulating. Kelvin’s Theorem requires that the circulation around such a cylinder must remain constant for an ideal fluid (i.e. while ignoring viscosity). Also, the volume for such a cylinder must remain constant, as it is material and we’re assuming incompressibility. As the two particles diffuse, the distance between them (d) increases, and the following results:

G=2pvR=Constant or G=wA=Constant

The increasing of the distance between the two particles is thus causing an increasing flow velocity in a smaller eddy to maintain constant circulation around the material volume of the vortex filament. As we would suspect, the flow velocity cannot become infinite as the end particles get infinitely far apart. With the increase in vorticity, viscosity begins to play an active role in the equilibrium of the system, and the eddies are eventually burned up. This is not in violation of the above picture as Kelvin’s Theorem only holds for ideal (inviscid) fluids.

Taylor notes that mathematical solutions to turbulent flow done prior were in two dimensions, on a plane perpendicular to the axes of the cylinders pictured above. As that domain does not allow for diffusion of points along a vortex filament axis, the phenomenon of large eddies becoming smaller is not possible and not calculated. At this point Taylor concedes that in the pursuit of a mathematical description of turbulence decay "nothing but a complete solution to the equations of motion...will suffice."

The approach taken was to mathematically define an initial turbulent flow by three dimensional velocity components.

u(x,y,z)=A cos(ax) sin(by) sin(cz)

v(x,y,z)=B sin(ax) cos(by) sin(cz)

w(x,y,z)=C sin(ax) sin(by) cos(cz)

The following picture displays the nature of this flow in an x-y plane at z=p/2c:

The initial eddies are, therefore, ellipses with axes given by p/a, p/b, and p/c. The initial definitions of the flow velocities are used with the equations of equilibrium and continuity to develop expressions for u(x,y,z,t), v(x,y,z,t), and w(x,y,z,t), dependent on time as well as position. Starting with the Navier-Stokes equations, expressing equilibrium

  (1)

an expression for the pressure must be found. Using the above initial definitions of the flow, the time derivative drops out. Taking the divergence of both sides yields a Poisson equation for the pressure. With the expression of continuity,

  (2)

the last term of (1), the viscous term, becomes zero when the divergence is taken. The Poisson equation becomes

  (3)

The combination of terms in the development of the Poisson equation brought into the mix cosine terms periodic in p/2a, p/2b, and p/2c, suggesting the appearance of eddies half the initial length scale.

With the initial flow velocities and the pressure expression (3) used in the equilibrium equation (1), an expression for the time derivative of the (time dependent) components of the flow velocities is given. Integrating with respect to time, and using the initial flow for evaluating the constant, u(x,y,z,t), v(x,y,z,t), and w(x,y,z,t) are found

(4)

where q, A₃, and A₂ are constants dependent on the initial flow. Here, the appearance of the smaller eddies is evident, their contribution to the velocity increasing in time. Also with increasing time, a reduction of the contribution of the initial flow (the first term) is noted. Note that the degradation of the initial flow is scaled by the viscosity. Using streamlines, Fig.3 shows the development of the flow from initiation to the point where the largest eddies have disappeared.

The above expression for the flow, however, is not exact. It was derived by using the initial flow velocities in the general equations. Now that a more exact general form for the flow velocity has been obtained, it can be used in (1) to again obtain a better form for the description of the flow, and so on. A power series in t is being developed to describe the flow velocity, (4) contains the t⁰ and t¹ terms. The form obtained from the second iteration includes expressions of cos(3ax) (etc.) multiplied by t², suggesting even smaller eddies (p/3a) whose contributions become more significant than the linearly time-dependent (p/2a) eddies at large times. The third such iteration yields a result which Taylor describes as "too long to set out here," and he concludes that "it seems impossible to obtain significant results in the general case" for the initially chosen velocity field.

The subsequent approach was to choose the constants in the original flow so as to simplify the problem sufficiently to allow an adequate number of iterations of the above described procedure. The simplified case of the problem is carried through five iterations, giving expressions for the flow velocities good to t⁵.

The expression for the spatially-averaged rate of dissipation of energy (per unit volume) is given as

  (5)

where the vorticity, w, is defined by

The derived flow velocities are used in the above to obtain an expression for the mean rate of energy dissipation for the specific case being discussed. Balancing this energy dissipation rate with the time rate of change of the kinetic energy

a mean value of the square of the velocity components is obtained (by integration) accurate to t⁶.

The statistical theory developed an expression for mean rate of the dissipation of energy in terms of a value, l, which Taylor believed to be the size of the smallest eddies in a turbulent flow.

  (6)

Using (5) and (6), the value of l is found for the calculated flow.

Taylor tracked the rate of energy dissipation as a function of time for several Reynolds numbers, where

The findings were of an early increase in the rate of dissipation owing to the decreasing value of l. This is expected as the size of the smallest eddies is initially that of the largest eddies. As the smallest eddies decrease in size, they eventually reach a minimum value, owing to the effect of viscosity. The component of the energy dissipation rate dependent on the mean square of the velocity components decreases with time. When the minimum value of l is reached, its value stabilizes and the decreasing mean square of the velocity drives the energy dissipation rate down. The minimum value of l will, however, not be accurately calculated for higher Reynolds numbers, unless the appropriate number of iterations are enacted in the development of the flow velocity expressions. Taylor admits these limitations, but does include the questionable predictions in the graphical presentation for those of us who are interested (adding quite a bit of activity to the graphs).

The next step for Taylor was to compare the calculated time-dependent three-dimensional turbulent flow field to those fields measured experimentally and explained with the statistical theory. The experimental flow of comparison was turbulence induced by a mesh of spacing M in the path of the flow. It was found from the experimental results that correlation between the flow at two points went to zero at roughly d=M/2, where d is a taken as the distance between two points. For the calculated flow field d is considered to be p/2a, the radius of the largest eddies. Taylor initiates the comparison by setting the two d values equal to each other. Flow velocities are scaled by matching the total energy of the experimental system to the calculated system (in the form of mean square velocities). Scaling of Reynolds numbers are consequently developed. The temporal reference point chosen for the comparison was the state of fully developed turbulence, or when the smallest values of l had been reached. For the calculated fields, this point is represented by the maxima in the rate of energy dissipation as described above.

The key finding in the statistically managed experimental work was a consistency in the constant relating l to the length scale and Reynolds number in the form

(7)

Several different experiments yielded a value of A very close to 2.0. Taylor considers the possibility of the universality of the constant A in his Statistical Theory of Turbulence. At this point Taylor goes about calculating the relation (7) for his theoretical flow field. The results are presented in graphical form, but the value of the constant can be found. The three values presented corresponded to A=2.4, 2.7, 2.1. It must be noted that these values are calculated using minimum value of l that would be considered sketchy at best, judging from the presented convergence of the terms comprising the energy dissipation rate. There could well have been some room for biased interpretation of these l values, and Taylor exercises significant caution in praising the correlation.

Whether or not the final determination of the multiplicative constant was precisely enacted, Taylor did successfully solve the governing equations for a time-dependent three-dimensional turbulent flow. In the solution, smaller eddies appear from the initial condition of larger eddies. The contribution of these smaller eddies to the velocity components increases with time as the contribution of the larger eddies is reduced (in a manner scaled by viscosity). The reduction in the size of the eddies can be related to the picture of the extending vortex filament. In this sense Taylor has, with a solution to the governing equations, offered a mathematical explanation for the diffusion observed in turbulence.

There has been much interest and investigation into the spectrum of turbulence, and the transference of energy down the spectrum (from larger eddies to smaller). A very significant theory was set forth by A. N. Kolmogorov in 1941, concentrating on the local structure of turbulence. Kolmogorov’s idea was that in the intermediate regime of the spectrum, the only relevant length and time (velocity) scales were those of the active eddies (not the large eddies originally produced or the small eddies killed by viscosity). In this sense, he purported that the rate of energy transference for a given eddy size must be given solely by these values. By dimensional analysis, it was determined that the energy transference rate for an intermediate state of turbulence is given by the eddy size and velocity as

Following the theory to the smallest state of turbulence, where the energy is burned up by viscosity, the eventual energy dissipation rate is determined by the energy transference rate in the intermediate turbulence regime. Experimental validation of Kolmogorov’s theory marked another step in the understanding of the decay of turbulence.