Three-dimensional bifurcating internal flow is studied for a single mother tube branching into two equal but diverging daughter tubes. The mother tube is straight and of circular cross-section, containing a fully developed incident motion, while the diverging daughters are straight and of semi-circular cross-section. This basic configuration is treated first by direct numerical simulation and secondly by slenderflow modelling, for a variety of Reynolds numbers and angles of divergence. The direct simulations and modelling highlight different forms of three-dimensional separation or flow reversal as well as enhanced upstream and downstream influence and pressure loss induced by the bifurcations especially at increased divergence angles. Comparisons between the results from the simulations and those from the slender-flow modelling show relatively close agreement at medium values of Reynolds number. In particular, as the angle of divergence increases for a given Reynolds number, there is generally first an increase in flow attachment on to the inner divider wall(s) and then, at higher angles, an increasing trend to flow reversal at the corners formed by the junctions of the outer wall with the divider; longitudinal vortex motion is also enhanced then. The agreement persists over a surprisingly wide range of divergence angles.

A numerical study of the generation of Tollmien-Schlichting (T–S) waves due to the interaction between a small free-stream disturbance and a small localized variation of the surface geometry has been carried out using both finite–difference and spectral methods. The nonlinear steady flow is of the viscous–inviscid interactive type while the unsteady disturbed flow is assumed to be governed by the Navier–Stokes equations linearized about this flow. Numerical solutions illustrate the growth or decay of the T–S waves generated by the interaction between the free-stream disturbance and the surface distortion, depending on the value of the scaled Strouhal number. An important result of this receptivity problem is the numerical determination of the amplitude of the T–S waves.

Abstract. In this part-review part-new work, studies on branching tube flows are described. These are based on modelling for increased flow rates as well as on direct numerical simulations and are motivated by applications to the cardiovascular system, lung airways and cerebral arteriovenous malformations. Small pressure differentials acting across a multiple branching are considered first, followed by substantial pressure differentials in a side branching, multiple branching or basic three-dimensional branching. All cases include a comparison of results between the modelling and the direct simulations. Wall shear, pressure variation, influence lengths, and separation or its suppression are examined, showing in particular sudden spatial adjustment of the pressure between mother and daughter tubes, nonunique flow patterns and a linear increase of flow rate with increasing number of daughters, dependent on the specific conditions. The agreement between modelling and direct simulations is generally close at moderate flow rates, suggesting their combined use in the biomedical applications.

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Receptivity of a laminar boundary layer to the interaction of time-harmonic free-stream disturbances with a three-dimensional roughness element is studied. The three-dimensional nonlinear triple–deck equations are solved numerically to provide the basic steady-state motion. At high Reynolds numbers, the governing equations for the unsteady motion are the unsteady linearized three-dimensional triple-deck equations. These equations can only be solved numerically. In the absence of any roughness element, the free-stream disturbances, to the first order, produce the classical Stokes flow, in the thin Stokes layer near the wall (on the order of our lower deck). However, with the introduction of a small three-dimensional roughness element, the interaction between the hump and the Stokes flow introduces a spectrum of all spatial disturbances inside the boundary layer. For supercritical values of the scaled Strouhal number, S0 > 2, these Tollmien–Schlichting waves are amplified in a wedge-shaped region, 15° to 18° to the basic-flow direction, extending downstream of the hump. The amplification rate approaches a value slightly higher than that of two-dimensional Tollmien–Schlichting waves, as calculated by the linearized analysis, far downstream of the roughness element.