Unsteady entrance flow in a 90° curved tube

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Abstract
A numerical model enabling the prediction of the axial and secondary velocity fields in three-dimensional configurations at moderate Reynolds numbers and Womersley parameters is presented. Steady and unsteady entrance flows in a 90° curve tube (δ = ⅙) under various flow conditions are analysed. The good quality agreement between axial and secondary velocities for a sinusoidally varying flow rate at a Womersley parameter of α = 7.8, obtained from a finite-element calculation, and those obtained from laser-Doppler measurements justify the use of the numerical model.Halfway into the deceleration phase for a sinusoidally varying flow rate (200 < Re < 800, α = 7.8) a strong resemblance is found to the steady flow case (Re = 700). In contrast with steady flow, near the inner wall reversed axial flow regions are found halfway into and at the end of the deceleration phase. Throughout the flow cycle the Dean-type secondary flow field highly influences axial flow resulting in a shift of the maximal axial velocity towards the outer wall, C-shaped axial isovelocity lines and an axial velocity plateau near the inner wall. Further downstream in the curved tube the Dean-type secondary vortex near the plane of symmetry is deflected towards the sidewall (’tail’–formation), as is also found for steady flow. An increase of the Womersley parameter (α = 24.7) results in a constant secondary flow field which is probably mainly determined by the steady component of the flow rate. A study on the flow phenomena occurring for a physiologically varying flow rate suggests that the diastolic phase is only of minor importance for the flow phenomena occurring in the systolic phase. Elimination of the steady flow component (− 300 < Re < 300) results in a pure Dean-type secondary flow field (no ‘tail’-formation) for α = 7.8 and in a Lyne-type secondary flow field for α = 24.7. The magnitude of the secondary velocities for α = 24.7 are of O(10−2) as compared to the secondary velocities for α = 7.8.

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