Turbulent Pipe Flow with Rough and Porous Walls

Abstract

The characteristics of turbulence in shear flow near rough and porous walls are investigated theoretically and experimentally. Flows of this nature are present, for instance, in the case of a erodible particulate stream-bed. The fluctuating flow generated in the porous wall by the external turbulent flow is analyzed using a macroscopic, linearized equation of motion, with the boundary condition at the surface defined by a pressure distribution of the same form as that observed at smooth walls. Two different specifications for the wall pressure are applied, one as- a random function of space and time and the other as a sinusoidal wave. The intensity of the longitudinal velocity fluctuations at the wall surface, normalized with respect to shear velocity, is shown to approach an asymptotic value of 0.38 as the permeability increases, whereas the fraction of the total energy dissipation that occurs within the porous medium has a maximum for a characteristic dimensionless combination of permeability, viscosity and external velocity. The Reynolds stress is identically zero throughout the porous medium. Experiments were conducted in two pipes, 10" in diameter, one with 1/8" spherical roughness elements and the other with a 1.20" thick porous lining, 5 5 for Reynolds numbers between 10 and 5 x 105. The rough pipe behavior with respect to friction factor and mean velocity distribution is in good agreement with classical experiments for sand roughness. The porous pipe has very high friction factors, between 0.06 and 0.08, which increase with Reynolds number. Both friction factor and the displacement of mean velocity profile with respect to the smooth law indicate an equivalent relative roughness of the order of 0.10. A value of 0.40 for Karman constant K is consistent with the observations, but deviations from the logarithmic velocity law occur at a distance from the wall less than 10% of the radius. Both the eddy viscosity and the velocity defect distributions show systematic variations in the core region depending on the nature of the wall. Measurements of turbulence intensity in the longitudinal and radial directions made with hot wire anemometers show a universal distribution, in velocity. However, the intensities relative to the local velocity increase with the effective roughness of the wall. The ratio between the radial and longitudinal intensities agrees with the smooth wall distribution throughout most of the pipe, except very close to the wall where it remains at a constant level of 0.6 for the rough and porous cases. Energy spectra for both components of turbulence indicate a definite change with distance to the wall. In normalized form, the spectral measurements for both rough and porous walls are in substantial agreement with previous smooth wall measurements.