Structure and Branching of Unstable Modes in a Swirling Flow

Swirling has a significant effect on the main characteristics of flow and can lead to its fundamental restructuring. On the flow axis, a stagnation point with zero velocity is possible, behind which a return flow zone is formed. The apparent instability leads to the formation of secondary vortex motions and can also be the cause of vortex breakdown. In the paper, a swirling flow with a velocity profile of the Batchelor vortex type has been studied on the basis of the linear hydrodynamic stability theory. An effective numerical method for solving the spectral problem has been developed. This method includes the asymptotic solutions at artificial and irregular singular points. The stability of flows was considered for the values of the Reynolds number in the range 10≤Re≤5×106. The calculations were carried out for the value of the azimuthal wavenumber parameter n=−1. As a result of the analysis of the solutions, the existence of up to eight simultaneously occurring unstable modes has been shown. The paper presents a classification of the detected modes. The critical parameters are calculated for each mode. For fixed values of the Reynolds numbers 60≤Re≤5000, the curves of neutral stability are plotted. Branching points of unstable modes are found. The maximum growth rates for each mode are determined. A new viscous instability mode is found. The performed calculations reveal the instability of the Batchelor vortex at large values of the swirl parameter for long-wave disturbances.

Physiologic blood flow is turbulent

Contemporary paradigm of peripheral and intracranial vascular hemodynamics considers physiologic blood flow to be laminar. Transition to turbulence is considered as a driving factor for numerous diseases such as atherosclerosis, stenosis and aneurysm. Recently, turbulent flow patterns were detected in intracranial aneurysm at Reynolds number below 400 both in vitro and in silico. Blood flow is multiharmonic with considerable frequency spectra and its transition to turbulence cannot be characterized by the current transition theory of monoharmonic pulsatile flow. Thus, we decided to explore the origins of such long-standing assumption of physiologic blood flow laminarity. Here, we hypothesize that the inherited dynamics of blood flow in main arteries dictate the existence of turbulence in physiologic conditions. To illustrate our hypothesis, we have used methods and tools from chaos theory, hydrodynamic stability theory and fluid dynamics to explore the existence of turbulence in physiologic blood flow. Our investigation shows that blood flow, both as described by the Navier–Stokes equation and in vivo, exhibits three major characteristics of turbulence. Womersley’s exact solution of the Navier–Stokes equation has been used with the flow waveforms from HaeMod database, to offer reproducible evidence for our findings, as well as evidence from Doppler ultrasound measurements from healthy volunteers who are some of the authors. We evidently show that physiologic blood flow is: (1) sensitive to initial conditions, (2) in global hydrodynamic instability and (3) undergoes kinetic energy cascade of non-Kolmogorov type. We propose a novel modification of the theory of vascular hemodynamics that calls for rethinking the hemodynamic–biologic links that govern physiologic and pathologic processes.

A space–time variational approach to hydrodynamic stability theory

We present a hydrodynamic stability theory for incompressible viscous fluid flows based on a space–time variational formulation and associated generalized singular value decomposition of the (linearized) Navier–Stokes equations. We first introduce a linear framework applicable to a wide variety of stationary- or time-dependent base flows: we consider arbitrary disturbances in both the initial condition and the dynamics measured in a ‘data’ space–time norm; the theory provides a rigorous, sharp (realizable) and efficiently computed bound for the velocity perturbation measured in a ‘solution’ space–time norm. We next present a generalization of the linear framework in which the disturbances and perturbation are now measured in respective selected space–time semi-norms ; the semi-norm theory permits rigorous and sharp quantification of, for example, the growth of initial disturbances or functional outputs. We then develop a (Brezzi–Rappaz–Raviart) nonlinear theory which provides, for disturbances which satisfy a certain (rather stringent) amplitude condition, rigorous finite-amplitude bounds for the velocity and output perturbations. Finally, we demonstrate the application of our linear and nonlinear hydrodynamic stability theory to unsteady moderate Reynolds number flow in an eddy-promoter channel.

THE PHYSICAL ORIGIN OF SEVERE LOW-FREQUENCY PRESSURE FLUCTUATIONS IN GIANT FRANCIS TURBINES

The physical origin of severe low-frequency pressure fluctuation frequently observed in Francis hydraulic turbines under off-design conditions, which greatly damages the structural stability of turbines and even power stations, is analyzed based on the hydrodynamic stability theory and our Reynolds-averaged Navier-Stokes equation simulation (RANS) of the flow in the entire passage of a Francis turbine. We find that spontaneous unsteady vortex ropes, which induce severe pressure fluctuations, are formed due to the absolute instability of the swirling flow at the conical inlet of the turbine's draft tube.

On the frequency selection of finite-amplitude vortex shedding in the cylinder wake

In this paper it is shown that the two-dimensional time-periodic vortex shedding régime observed in the cylinder wake at moderate Reynolds numbers may be interpreted as a nonlinear global structure and its naturally selected frequency obtained in the framework of hydrodynamic stability theory. The frequency selection criterion is based on the local absolute frequency curve derived from the unperturbed basic flow fields under the assumption of slow streamwise variations. Although the latter assumption is only approximately fulfilled in the vicinity of the obstacle, the theoretically predicted frequency is in good agreement with direct numerical simulations for Reynolds numbers Re > 100.