The development of the turbulent flow in a bent pipe

Abstract

The three-dimensional incompressible turbulent flow through a slender bent pipe of simple cross-section is analysed, the pipe gradually bending the rapid flow through a substantial angle. The ratio of the relative radius of curvature to the magnitude of the turbulent fluctuations is crucial: analysis of the entry region involving exact solutions of the governing equations shows three different downstream developments, depending on the magnitude of that ratio. The main velocity components are found in each case, and one downstream development studied in detail is when turbulence dominates the flow.The main novel points and results are as follows. (i) The present physical situation which arises commonly in industrial settings has been little studied previously by theory or experiments. (ii) The working applies for any two-tier mixing-length model. (iii) As a most surprising feature, the fully developed flow far downstream is not unique, being found to depend instead on the global flow behaviour (thus the centreline velocity is not determined simply by the pressure drop, in contrast to the laminar case). (iv) A quite accurate predictive tool based on approximation is suggested for the downstream flow. (v) Crossflow maxima are found to occur very close to the walls, as observed in experiments. (vi) Other comparisons are made with experimental data and prove generally favourable.