Structural similarity for fully developed turbulence in smooth tubes
Authors: Morrison W. R. B. , Kronauer R. E. 
Published: March 2006

The structure of fully developed turbulence in smooth circular tubes has been studied in detail in the Reynolds number range between 10,700 and 96,500 (R based on centre velocity and radius). The data was taken as longitudinal and transverse correlations of the longitudinal component of turbulence in narrow frequency bands. By taking Fourier transforms of the correlations, crosspower spectral densities are formed with frequency, ω, and longitudinal or transverse wave-number, kx or kz, as the independent variables. In this form the data shows the distribution of turbulence intensity among waves of different size and inclination, and permits an estimate of the phase velocity of the individual waves.

Data taken at radii where the mean velocity profile is logarithmic show that the waves of smaller size (higher (k2x + k2z)½) decrease in intensity more rapidly with distance from the wall than the larger waves, and also possess lower phase velocity. This suggests that the waves might constitute a geometrically similar family such that the variation of intensity with wall distance is a unique function with a scale established by (k2x + k2z)−½). The hypothesis fits the data very well for waves of small inclination, α = tan−1(kx/kz), and permits a collapse of the intensity data at the several radii into a single ‘wave-strength’ distribution. The function of intensity with wall distance which effects this collapse has a peak at a wall distance roughly equal to 0·6(k2x + k2z)−½). For waves whose inclination is not small, it would not be expected that the intensity data could collapse in this way since the measured longitudinal component of turbulence represents a combination of two turbulence components when resolved in the wave co-ordinate system.

Although the similarity hypothesis is strictly true only for data taken where the mean velocity profile is logarithmic, a simple correction procedure has been discovered which permits the extension of the similarity concept to the sublayer region as well. This procedure requires only that the observed total turbulence intensity at any station in the sublayer be reduced by a factor which depends solely on the y+ distance from the wall (i.e. on the distance from the wall, scaled by the viscid parameters of the sublayer). The correction factor is independent of Reynolds number and applies equally to waves of all sizes. In this way, all of the turbulence waves down to the very smallest of any significance, are found to satisfy slightly modified similarity conditions.

From the data taken a t Reynolds numbers between 96,500 and 46,000 wave ‘strength’ is seen to be distributed more or less uniformly over a range bounded at one extreme by the largest waves which the tube can contain (k2x + k2z ≅ (2/a)2, where a is the tube radius) and at the other extreme by the smallest waves which can be sustained against the dissipative action of viscosity (k2x + k2z ≅ (0·04v/Uτ)2, where Uτ is the shear velocity). As the Reynolds number of the flow is lowered, the spread between the bounds becomes smaller. If the data is projected to a Reynolds number of order lo3 the bounds coalesce and turbulence should no longer be sustainable.