Investigation on Energy Distribution in Steady and Unsteady Flow Instabilities through a Bend Square Pipe

Fluid flow analysis through a bend pipe is extensively conducted in practical and cell separation operations. It is observed that flow behaviors in the bend pipe are influenced by some parameters such as curvature, aspect ratio, etc. As a result, various phenomena, steady solution branches, unsteady solutions, energy transfer are changed. In this paper, the acts of flows are performed together for fixed curvature, δ = 0.2, and Prandtl number, Pr = 7.0 (water). Here, for a wide variety of Dean numbers (100 ≤ Dn ≤ 1000) and three fixed Grashof numbers, Gr = 100, 500, and 1000; time-independent solutions with linear stabilities are investigated first where only the first steady branch exhibits linear stability out of two steady solution branches obtained. Then, different flow transitions between the required range of Dean numbers (Dn) and several Grashof numbers (Gr) are investigated using time-dependent solutions. Power spectrum density (PSD) is further revealed in order to gain a deeper understanding of periodic and multi-periodic flows. Flow velocity contours including axial flow (AF) and secondary flow (SF) and their temperature profiles (TP) are also exposed. The SFs reveal that two- to four-vortex flows are produced due to the turning of steady branch and the flow instabilities. Furthermore, the energy transfer between the cooled and heated sidewalls of the pipe is calculated. Finally, a link between centrifugal and body force with the energy transfer has been shown in this research which reveals that the fluid has merged that certainly rises the overall energy transfer.

Recirculating flow in a basin with closed f/h contours

A general circulation model is used to study the time evolution of a rotating, weakly baroclinic fluid in a basin with sloping sidewalls. Contours of f/h, where f is the Coriolis parameter and h is the depth of the fluid, are closed in this model. The fluid is forced by a localized source of positive vorticity. The initial response is a narrow, recirculating cell that resembles a β-plume modified by bathymetry. Such cells have been found in previous studies and have been linked to the recirculation cells observed in the subpolar North Atlantic. However, this is not a steady solution in this basin with closed f/h contours, and the circulation evolves into a gyre that encircles the basin. The time at which this transition occurs depends on the Rossby number, with higher Rossby numbers transitioning earlier. Based on the budget of potential vorticity, an argument is made that the western boundary is not long enough to drain significant vorticity from the flow and therefore a bathymetric β-plume is not a steady solution. A similar argument suggests that the Labrador Sea cannot sustain steady, linear, barotropic recirculations either. We speculate that the observed recirculations depend on inertial separation at sharp bathymetric gradients to break the assumption of linearity, which leads to significant viscous dissipation.

On the critical free-surface flow over localised topography

Flow over bottom topography at critical Froude number is examined with a focus on steady, forced solitary wave solutions with algebraic decay in the far field, and their stability. Using the forced Korteweg–de Vries (fKdV) equation the weakly nonlinear steady solution space is examined in detail for the particular case of a Gaussian dip using a combination of asymptotic analysis and numerical computations. Non-uniqueness is established and a seemingly infinite set of steady solutions is uncovered. Non-uniqueness is also demonstrated for the fully nonlinear problem via boundary-integral calculations. It is shown analytically that critical flow solutions have algebraic decay in the far field both for the fKdV equation and for the fully nonlinear problem and, moreover, that the leading-order form of the decay is the same in both cases. The linear stability of the steady fKdV solutions is examined via eigenvalue computations and by a numerical study of the initial value fKdV problem. It is shown that there exists a linearly stable steady solution in which the deflection from the otherwise uniform surface level is everywhere negative.