Abstract
AbstractThe development of incompressible turbulent flow through a pipe of wavy cross-section was studied numerically by direct integration of the Navier–Stokes equations. Simulations were performed at Reynolds numbers of $4.5\times 10^{3}$ and $10^{4}$ based on the hydraulic diameter and the bulk velocity. Results for the pressure resistance coefficient ${\it\lambda}$ were found to be in excellent agreement with experimental data of Schiller (Z. Angew. Math. Mech., vol. 3, 1922, pp. 2–13). Of particular interest is the decrease in ${\it\lambda}$ below the level predicted from the Blasius correlation, which fits almost all experimental results for pipes and ducts of complex cross-sectional geometries. Simulation databases were used to evaluate turbulence anisotropy and provide insights into structural changes of turbulence leading to flow relaminarization. Anisotropy-invariant mapping of turbulence confirmed that suppression of turbulence is due to statistical axisymmetry in the turbulent stresses.
Highlights
IntroductionSchiller (1922) pointed out that, for special geometries, deviations from the correlation based on the concept of hydraulic diameter might occur, and demonstrated
Schiller (1922) outlined the concept for the determination of skin-friction losses in pipes of non-circular cross-sections based on the hydraulic diameter, Dh = 4A/P, and the Blasius (1913) correlation for dimensionless coefficient of resistance, λ of=circupl/a(r(1c/ro2s)sρ-UseB2c)tDiohn/.LH=ere0.A3,16P4,Rep−m,0.2U5,B, suggested ρ, ν and originally to hold for pipes Rem denote, respectively, pipe cross-sectional area, wetted perimeter, pressure drop over the pipe length L, bulk velocity, fluid density, kinematic viscosity of the fluid and Reynolds number Rem = DhUB/ν
Schiller succeeded in correlating the experimental data obtained in pipes of square, equilateral triangle and rectangular cross-sections in the Reynolds-number range up to Rem 6 × 104. These findings were further supported by Nikuradse (1930) for a wider class of cross-sectional geometries to form the basis for the determination of skin-friction losses in pipes of complex shapes
Summary
Schiller (1922) pointed out that, for special geometries, deviations from the correlation based on the concept of hydraulic diameter might occur, and demonstrated This with results obtained in pipes of threaded and wavy cross-sections, as shown in figure 1. In light of the data shown in figure 2 and considering (1.1), Schiller’s (1922) results imply that a wavy cross-section decreases the turbulent dissipation rate, , as Rem increases Such a tendency is favourable for achieving turbulent drag reduction. With significant tripping of the initial flow, by blocking the flow over 30 % of the pipe cross-sectional area, was it possible to realize a fully developed turbulent state with the pressure resistance coefficient in very close agreement with experiments (see table 2).
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