The similarity theory of turbulence, and flow between parallel planes and through pipes

Abstract

Before any distribution of mean velocity can be calculated from mixture length theories of turbulent motion, some assumption must be made concerning the mixture length. The only theory so far proposed which yields a formula for the length involved is Kármán’s similarity theory (Kármán 1930 a , b , 1932, 1934 a , pp.7-9 ,1934 b ; see also Noether 1931, 1933; Betz 1931; Dedebant, Schereschewsky and Wehrle 1934; Prandtl 1935). The assumptions of the theory have not, however, been sufficiently tested yet, and further research is needed. Moreover, even in the cases of the fairly simple mean motions to which the theory has so far been applied, there are regions in the field of flow where the assumptions break down, so that care is needed in the applications. It is assumed (1) that the turbulence mechanism is independent of the viscosity (except in the viscous layers near the walls); (2) that in comparing the turbulence mechanisms at two different points, consideration of the fields of turbulent flow may be restricted to the immediate neighbourhoods of these points; and (3) that the turbulence flow patterns at different points are similar (relative to frames of reference moving with the mean velocities at the points), and differ only in the scales of length and time (or velocity). The last assumption implies constant correlation between any two turbulent velocity components: if u, v, w denote the turbulent velocity components, the ratios u 2 ‾ : v 2 ‾ : w 2 ‾ : uv ‾ : vw ‾ : wu ‾ should, in fact, all be constant. Kármán (1934 a , fig. 7; 1934 b , fig. 6) has published curves of uv ‾ / u 2 ‾ for two-dimensional (mean) flow between parallel walls, obtained from measurements by Reichardt and by Wattendorf and Kuethe. The experimental points are rather scattered, but the curves show a fairly constant value for the ratio except near the walls and over the middle 0·6 h of the channel, where 2 h denotes the distance between the walls. Near the walls there is a marked departure from a constant value only when the viscous layer is approached. (The width of the viscous layer is of the order of 30 v / U r , where v is the kinematic viscosity and U r is the so-called “friction velocity”, defined by ρ U 2 r = r 2 0 , r 0 being the intensity of the wall friction.) In the middle of the channel the ratio falls to zero, showing that there is no correlation between u and v , a result to be expected from considerations of symmetry.