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6 2. NON-PULSATILE AND PULSATILE INTERNAL FLOW 2π · f α=r· υ (2.1) where r is the pipe radius, f the frequency and υ is the kinematic viscosity. A small value of the Womersley number implies that the effects of unsteadiness are less important as compared to viscous effects. For low Womersley number (α <1), the velocity is nearly a quasi-steady Poiseuille flow which is in phase with the pressure gradient. As the Womersley number increases, the velocity starts to show a phase lag with respect to the pressure gradient. For all pulsat- ing flows, a phase shift between the axial flow and the pressure gradient occurs, see for example Dec et al. (1991) and Ramaprian & Tu (1983). For pure os- cillating very low Reynolds number flow the phase lag is 90◦ and for turbulent flow the phase lag is less than 90◦ according to Ramaprian & Tu (1983). For pure oscillating very low Reynolds number flow, the ∂ -term in the momentum ∂t equation (4.2) is balanced by the ∂p -term. This can be showed by assuming ∂xi one-dimensional oscillating flow and that the pressure gradient is described by ∂p = pˆ · eiωt where pˆ is a constant and ω is the frequency of the oscillation. ∂x When this is inserted into the momentum equation (4.2) and solved for U, the following result is obtained U = −ipˆ · eiωt. As can be seen, the phase shift ω between the velocity and the pressure is 90◦. For pulsatile turbulent flow, the time derivate term will be balanced both by the pressure gradient term and remaining terms in the momentum equations, including the Reynolds-stresses. Flow pulsations also affect the axial velocity profile in the pipe. The phase shift between the pressure and the bulk velocity gives overshoots in boundary layer which depends on that the flow in the region near the wall has lower momen- tum than the bulk flow, and therefore, it is more sensitive to changes in the axial pressure gradient. This effect increases with increasing frequency due to the inertia effect. Ramaprian & Tu (1983) carried out an experimental study of turbulent oscillatory flow in a long circular pipe. The oscillating frequencies correspond to a Womersley number of approximately 43 and 116, respectively. They compared the results with theoretical results for laminar oscillating pipe flow at the same frequency. The laminar flow shows sharper peaks of the over- shoots in the boundary layer than the turbulent flow. Also, the peaks occur closer to wall for the theoretical laminar case. The experiments also show that the phase lag between pressure and velocity is smaller for turbulent flow than for the laminar flow. Dec et al. (1991) investigated a cyclic oscillating veloc- ity field in a tail pipe of a pulse combustor using Laser Doppler Velocimetry (LDV). The oscillating frequencies varied from 67 to 101 Hz and the ampli- tude was up to 5 times the mean velocity. The mean Reynolds number was 3750. The turbulent flow shows a similar behaviour to the theoretical laminar oscillating pipe flow, with a phase shift between the pressure and the velocity and an overshoot near the wall. The flow near the wall reverses earlier thanPDF Image | Numerical computations of the unsteady flow in a radial turbine
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