Compressible swirling flow through a significantly bent duct

The three-dimensional vortical flow of an inviscid compressible fluid through a bend in a slender duct is considered. The duct of simple cross-section gradually bends the flow through a substantial angle. The flow is motivated by industry and involves high speeds and short time intervals. The relative radius of curvature and the magnitude of the inertial forces are important factors here. Weakly and fully non-linear coupling between the streamwise and cross-sectional velocities is considered, and numerical results are derived for a duct of rectangular cross-section in each case, based on a fourth-order compact-differencing approach to solving the three-dimensional Euler equations. The governing equations are parabolic in the streamwise direction, enabling a forward-marching approach. The bend leads to significant growth in vorticity which mixes the streamwise velocity. The weakly non-linear study reveals a role for compressibility which is absent in the fully non-linear study when the cross-sectional area is constant. In both cases there is relatively little pressure loss, in contrast to viscous flows. Linear growth of the total streamwise vorticity in the bend is shown analytically to occur in keeping with the numerical solutions. Far downstream behaviours in finite bends and bends which continue indefinitely are investigated. The latter case leads to a more strongly non-linear flow structure far downstream due to continued evolution of the vorticity.