Session G10: Instability: Wakes I - Cylindrical Objects

Cylinder wakes in quasi-two-dimensional flows with surface friction I: instability and scaling

We measured the frequency of vortex shedding produced by cylinders in a quasi-two-dimensional system with homogenous drag. The system is characterized by the Reynolds number Re $=$ U$_{\mathrm{0}}$D/$\nu $ (U$_{\mathrm{0}}=$ flow speed without the obstacle, D $=$ cylinder diameter, $\nu =$ kinematic viscosity), and a dimensionless drag parameter, $\alpha $* $=$ D$^{\mathrm{2}}$/(L$_{\mathrm{s}}^{\mathrm{2}}$Re) (L$_{\mathrm{s}}=$ length scale above which drag force exceeds viscous force). We investigated the scaling of the Strouhal number St $=$ fD/U$_{\mathrm{o\thinspace }}$(f $=$ vortex shedding frequency) and compared it with conventional measurements in flows without homogenous drag. The dynamics bifurcates above a critical diameter D$_{\mathrm{c}}\sim $L$_{\mathrm{s}}$,$_{\mathrm{\thinspace }}$indicating that the effect of surface friction becomes important. Increased fluctuations beyond the bifurcation indicate the onset of a previously unobserved instability associated with the drag. Also, near some critical parameters, shear instability without vortex shedding is observed, with vortex streets appearing at both higher and lower Reynolds number; the mode at lower Reynolds number has not previously been observed.   

Cylinder wakes in quasi-two-dimensional flows with surface friction II: effects of film thickness

Vortex shedding in a quasi-two-dimensional system with homogeneous drag (Ekman friction) is observed to have different phenomenology than in systems without friction. To understand why, we studied the wakes of circular cylinders in a vertical soap film channel and measured thickness profiles (pachymetry) of the film in the cylinder wake. The kinematic viscosity and drag coefficients in this system both depend on the thickness of the soap film, which varies over the wake. To measure thickness, broad-spectrum light is reflected off the film, and the resulting interference pattern of intensity vs. wave number is measured. The spacing in wave number of the interference minima is proportional to the film thickness, giving high-accuracy thickness measurements with a precision on the order of 0.2{\%}. Pachymetry profiles transverse to the mean flow were measured at five longitudinal positions for various values of Reynolds number and drag parameter. Possible causes for differences in the dynamics from conventional systems could be: ambiguity in the specifications of Reynolds number or non-Newtonian effects arising from viscosity gradients, elastic effects particular to soap films, or surface friction. The pachymetry results favor the latter explanation.   

An iterative methodology for the computation of perturbation fields induced by harmonic forcing of the linearised Navier-Stokes equations in complex geometries and application to forced cylinder wakes

An efficient, iterative methodology is developed for the computation of the perturbation fields induced by harmonic forcing of the linearised Navier-Stokes equations in complex geometries. The problem is formulated in the frequency domain and the resulting system of equations is solved iteratively until convergence. This approach offers distinct advantages: convergence is monitored easily, and the solution from one value of frequency can be used as a restart field for another, nearby, frequency. It is also straightforward to implement in any implicit code that solves the steady Navier-Stokes equations iteratively. The method can be extended to solve the optimal forcing problem, i.e. to find the forcing fields that will maximise the energy of the flow perturbations for a particular frequency. In the present study, the method is applied to investigate the wake behind a cylinder with pulsating approaching flow. The perturbation velocity and pressure fields induced by external forcing are computed and the mechanisms that drive the energy growth of the developed structures in the wake are examined. It is shown that perturbations grow by extracting energy from two sources: the underlying base flow field and the externally provided energy that maintains the imposed oscillation.   

A Zoology of unstable modes in a stratified cylinder wake

Although the dynamics of a cylinder wake is well known and extremely rich for a homogeneous fluid, very few studies have been focused on stratified wakes despite the obvious extensive number of applications for geophysical flows and submarine wakes. The presence of the stratification may largely modify the dynamics of the wake. The study is devoted to understand the effect of the tilt and also of a strong stratification. So extensive experimental and numerical results have been investigated to describe the full dynamics of a tilted cylinder wake. For weak stratification and small tilt angle, the classical mode A found for a homogeneous fluid is still present, but for a large tilt angle, an instability appearing far from the cylinder is created. The case of a cylinder towed a very stratified fluid has been finally investigated. The dynamics is strongly modified and for moderate tilt angles, a new unstable mode appears with a structure similar to the Kelvin-Helmholtz billows (observed in the critical layer of a tilted stratified vortex), whereas for large tilt angles, another unstable mode characterized by a strong shear appears generated without a 2D von Karman structure. This reveals the rich dynamics of the cylinder wake in the presence of a stable stratification.   

Stability of Flow around a Cylinder in Plane Poiseuille Flow

Simulation of Navier-Stokes equations is carried out to study the stability of flow around a cylinder in plane Poiseuille flow. The energy gradient method is employed to analyze the mechanism of instability of cylinder wake. The ratio of the channel width to the cylinder diameter is 30, and the Reynolds number based on the cylinder diameter and incoming centerline velocity is 26 and 100, respectively. The incoming flow is given as being laminar. It is found that the instability of the cylinder wake, starting near the front stagnation point upstream. The recirculation zone behind the cylinder has no effect on the stability of the wake. In the wake behind the recirculation zone, the flow stability is controlled by the energy gradient in the shear layer along the two sides of the wake. At high Re, the energy gradient of averaged flow in the channel interacts with the wake vortex, strengthening the wake vortex structure. Due to the large ratio of the channel width to the cylinder diameter, the disturbance caused by the cylinder mainly occurs in the vicinity of the centerline and has little effect on the flow near the wall. The velocity profile on the two sides of the cylinder wake in the downstream channel remains laminar (parabolic profile).   

Too big to grow: the saturation mechanism of the von Karman vortex street captured by a self-consistent model

The supercritical instability leading to the Karman vortex street in a cylinder wake is a well studied problem: the steady solution becomes linearly unstable and saturates into a limit cycle. However a simple physical picture for understanding the saturation amplitude is still missing. We present a simple self-consistent model that captures the saturation mechanism. The model shows that the main nonlinear effects of the saturation process are retained by the coupling of the mean flow and perturbation equations through the Reynolds stress, which is built only with the first harmonic calculated as the most unstable eigenmode. A simple physical picture is revealed, wherein the perturbation amplitude is such that the modified mean flow is neutrally stable. The mean flow velocity field and the Reynolds stress spatial structure are thus well approximated in a self-consistent manner without any DNS data. Moreover, the results show an accurate vortex shedding frequency prediction when compared to experiments.   

Investigation of the effect of the spanwise forcing on vortex shedding suppression in the flow past a cylinder

Controlling the wake vortex dynamics of bluff bodies efficiently is a fundamental problem in many applications. Earlier direct numerical simulations (Darekar, Sherwin, 2001) of three-dimensional bluff bodies have demonstrated that the introduction of a spanwise waviness at both the leading and trailing surfaces suppresses the vortex shedding and reduces the amplitude of the fluctuating aerodynamic forces. Under this motivation, direct numerical simulations and stability analysis of the flow past a three-dimensional cylinder in the supercritical regime were performed. Starting from a fully developed shedding, a sufficiently high spanwise forcing is introduced on the surface of the cylinder, in the regions where separation effects occur, resulting in the stabilisation of the near wake in a time-independent state. Numerical experiments were conducted to detect the critical values of the amplitude of the forcing capable of suppressing the vortex street, and three different physical structures of the wake were detected. Stability analysis of the linearised Navier-Stokes equations was then performed on the three-dimensional flows to investigate the role of the spanwise modulation on the absolute instability associated with the von K\'arman street.   

The Wake Analysis Behind a Foamed Cylinder

Particle Image Velocimetry (PIV) has been carried out to investigate the wake region behind a foamed and a finned cylinder. The purpose of this analysis is to develop one- and two- point correlations and to investigate the flow characteristics for these two cases. The experiments are conducted for two Reynolds numbers (based on the mean air velocity and the cylinder diameter) 2000 and 8000. The application of Proper Orthogonal Decomposition (POD) to the PIV velocity fields of the two cylinder types is also discussed. The POD computed for the measured velocity fields for both cases shows that the first two spatial modes contain most of the kinetic energy of the flow irrespective to the cylinder type. These two modes are also responsible for the large-scale coherence of the fluctuations. For foamed cylinder types, the first four eigenmodes of the velocity field were measured and their organizations were investigated. These eigenmodes disclose the overall mean flow structure, and the large- scale structure being essentially connected to the most robust flow motion.