Wakes Of Two-dimensional Bluff Bodies Research Articles

A universal three-dimensional instability of the wakes of two-dimensional bluff bodies

Linear stability analysis of a wide range of two-dimensional and axisymmetric bluff-body wakes shows that the first three-dimensional mode to became unstable is always mode E. From the studies presented in this paper, it is speculated to be the universal primary 3D instability, irrespective of the flow configuration. However, since it is a transition from a steady two-dimensional flow, whether this mode can be observed in practice does depend on the nature of the flow set-up. For example, the mode E transition of a circular cylinder wake occurs at a Reynolds number of $\mathit{Re}\simeq 96$, which is considerably higher than the steady to unsteady Hopf bifurcation at $\mathit{Re}\simeq 46$ leading to Bénard–von-Kármán shedding. On the other hand, if the absolute instability responsible for the latter transition is suppressed, by rotating the cylinder or moving it towards a wall, then mode E may become the first transition of the steady flow. A well-known example is flow over a backward-facing step, where this instability is the first global instability to be manifested on the otherwise two-dimensional steady flow. Many other examples are considered in this paper. Exploring this further, a structural stability analysis (Pralits et al.J. Fluid Mech., vol. 730, 2013, pp. 5–18) was conducted for the subset of flows past a rotating cylinder as the rotation rate was varied. For the non-rotating or slowly rotating case, this indicated that the growth rate of the instability mode was sensitive to forcing between the recirculation lobes, while for the rapidly rotating case, it confirmed sensitivity near the cylinder and towards the hyperbolic point. For the non-rotating case, the perturbation, adjoint and structural stability fields, together with the wavelength selection, show some similarities with those of a Crow instability of a counter-rotating vortex pair, at least within the recirculation zones. On the other hand, at much higher rotation rates, Pralits et al. (J. Fluid Mech., vol. 730, 2013, pp. 5–18) have suggested that hyperbolic instability may play a role. However, both instabilities lie on the same continuous solution branch in Reynolds number/rotation-rate parameter space. The results suggest that this particular flow transition at least, and probably others, may have a number of different physical mechanisms supporting their development.

Scale-by-scale turbulent energy budget in the intermediate wake of two-dimensional generators

It is first established, on the basis of new X-wire measurements, that the equilibrium similarity of the terms in the scale-by-scale (s-b-s) budget of the turbulent energy $\overline{q^2}$q2¯ is reasonably well approximated on the axis of the intermediate wake of a circular cylinder. The similarity, which scales on the Taylor microscale λ and $\overline{q^2}$q2¯, is then used to determine s-b-s energy budgets from the data of Antonia, Zhou, and Romano [“Small-scale turbulence characteristics of two-dimensional bluff body wakes,” J. Fluid Mech. 459, 67–92 (2002)] for 5 different two-dimensional wake generators. In each case, the budget is reasonably well closed, using the locally isotropic value of the mean energy dissipation rate, except near separations comparable to the wavelength of the coherent motion (CM). The influence of the initial conditions is first felt at a separation Lc identified with the cross-over between the energy transfer and large scale terms of the s-b-s budget. When normalized by $\overline{q^2}$q2¯ and Lc, the mean energy dissipation rate is found to be independent of the Taylor microscale Reynolds number. The CM enhances the maximum value of the energy transfer, the latter exceeding that predicted from models of decaying homogeneous isotropic turbulence.

Stabilization of absolute instability in spanwise wavy two-dimensional wakes

AbstractControlling vortex shedding using spanwise-varying passive or active actuation (namely three-dimensional control) has recently been reported as a very efficient method for regulating two-dimensional bluff-body wakes. However, the mechanism of how the designed controller regulates vortex shedding is not clearly understood. To understand this mechanism, we perform a linear stability analysis of two-dimensional wakes, the base flow of which is modified with a given spanwise waviness. Absolute and convective instabilities of the spanwise wavy base flows are investigated using Floquet theory. Two types of base-flow modification are considered: varicose and sinuous. Both of these modifications attenuate absolute instability of two-dimensional wakes. In particular, the varicose modification is found to be much more effective in the attenuation than the sinuous one, and its spanwise lengths resulting in maximum attenuation show good agreement with those in three-dimensional controls. The physical mechanism of the stabilization is found to be associated with the formation of streamwise vortices from tilting of two-dimensional Kármán vortices and the subsequent tilting of these streamwise vortices by the spanwise shear in the base flow. Finally, the sensitivity of absolute instability to spanwise wavy base-flow modification is investigated. It is shown that absolute instability of two-dimensional wakes is much less sensitive to spanwise wavy base-flow modification than to two-dimensional modification. This suggests that the high efficiency observed in several three-dimensional controls is not due to the sensitive response of the wake instability to the spanwise waviness in the base flow.

Active flow control for reduction of fluctuating aerodynamic forces of a blunt trailing edge profiled body

Vortex shedding in the wake of two-dimensional bluff bodies is usually accompanied by three dimensional instabilities. These instabilities result in streamwise and vertical vorticity components which occur at a certain spanwise wavelength. The spanwise wavelength of the instabilities ( λ Z ) depends on several parameters, including profile geometry and Reynolds number. The objective of the present work is to study the three dimensional wake instabilities for a blunt trailing edge profiled body, comprised of an elliptical leading edge and a rectangular trailing edge, and to manipulate these instabilities to control the aerodynamic forces. Results of numerical simulations of flow around the body at Re( d) = 400, 600, and 1000, as well as planar Laser Induced Fluorescence (LIF) flow visualizations at Re( d) = 600 and 1000 are analyzed to determine the wake vorticity structure and λ Z . Based on the findings of these analyses, an active flow control mechanism for attenuation of the fluctuating aerodynamic forces on the body is proposed. The flow control mechanism is comprised of a series of trailing edge injection ports distributed across the span, with a spacing equal to λ Z . Injection of a secondary flow leads to amplification of the three dimensional instabilities and disorganization of the von Kármán vortex street. Numerical simulations indicate that the flow control mechanism can attenuate the fluctuating aerodynamic forces at lower Reynolds numbers (Re( d) = 400 and 600) where λ Z is constant in time. However, the control mechanism loses its effectiveness at Re( d) = 1000, due to the temporal variations of λ Z .

Wall shear stress and flow stability in a model fusiform aneurysm

Levels of wall shear stress in arterial blood flow are known to contribute to the integrity of wall tissue, and are believed to have a bearing on the pathogenesis of aneurysm disease. The flow of fluid through a model aneurysm was computed using a spectral element algorithm to study the distribution of wall shear stress and the stability of the flow to non-axisymmetric perturbations. Substantially higher mean and fluctuating wall shear stress components were found in the distal (downstream) region of the aneurysm bulge when compared to recent experimental measurements. A global stability analysis demonstrates that flow conditions consistent with a human aorta will produce non-axisymmetric flow in an aneurysm consistent with that modeled here. These results demonstrate the necessity for detailed numerical modeling of these flows to accurately resolve the near wall fluid stresses. References D. L. Fry. Acute vascular endothelial changes associated with increased blood velocity gradients. Circ. Res. , 22:165--197, 1968. G. E. Karniadakis, M. Israeli, and S. A. Orszag. High-order splitting methods for the incompressible Navier--Stokes equations. J. Comp. Phys. , 97:414--443, 1991. S. Kondo, N. Hashimoto, H. Kikuchi, F. Hazama, I. Nagata, and H. Kataoka. Cerebral aneurysms arising at nonbranching sites. an experimental study. Stroke , 28(2):398--403, 1997. D. N. Ku, D. P. Giddens, C. K. Zarins, and S. Glagov. Pulsatile flow and atherosclerosis in the human carotid bifurcation. Positive correlation between plaque location and low oscillating shear stress. Arterioscler. Thromb. Vasc. Biol. , 5:293--302, 1985. J. C. Lasheras. The biomechanics of arterial aneurysms. Ann. Rev. Fluid Mech. , 39:293--319, 2007. doi:10.1146/annurev.fluid.39.050905.110128 . R. B. Lehoucq, D. C. Sorenson, and C. Yang. ARPACK Users' Guide . SIAM: Philadelphia, PA, 1998. http://www.caam.rice.edu/software/ARPACK/UG/ug.html . S. Oyre, E. M. Pedersen, S. Ringgaard, P. Boesiger, and W. P. Paaske. In vivo wall shear stress measured by magnetic resonance velocity mapping in the normal human abdominal aorta. Eur. J. Vasc. Endovasc. Surg. , 13(3):263--271, 1997. R. B. Rutherford. Randomized EVAR trials and advent of level I evidence: A paradigm shift in management of large abdominal aortic aneurysms? Semin. Vasc. Surg. , 19(2):69--74, 2006. doi:10.1053/j.semvascsurg.2006.03.001 . A.-V. Salsac, S. R. Sparks, J.-M. Chomaz, and J. C. Lasheras. Evolution of the wall shear stresses during the progressive enlargement of symmetric abdominal aortic aneurysm. J. Fluid Mech. , 560:19--51, 2006. doi:10.1017/S00221120060036X . G. J. Sheard and K. Ryan. Pressure-driven flow past spheres moving in a circular tube. J. Fluid Mech. , 592:233--262, 2007. doi:10.1017/S0022112007008543 . A. Astradsson and J. Astrup. An intracranial aneurysm in one identical twin, but no anerusym in the other. Br. J. Neurosurg. , 15(2):168--171, 2001. T. B. M. Wilmink, C. R. G. Quick, and N. E. Day. The association between cigarette smoking and abdominal aortic aneurysms. J. Vasc. Surg. , 30:1099--1105, 1999. J. R. Womersley. Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J. Physiol. , 127:553--563, 1955. D. Barkley and R. D. Henderson. Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. , 322:215--241, 1996. H. M. Blackburn and J. M. Lopez. On three-dimensional quasi-periodic Floquet instabilities of two-dimensional bluff body wakes. Phys. Fluids , 15(8):L57--L60, 2003. doi:10.1063/1.1591771 . H. M. Blackburn and S. J. Sherwin. Formulation of a Galerkin spectral element-Fourier method for three-dimensional incompressible flow in cylindrical geometries. J. Comput. Phys. , 197:759--778, 2004. doi:10.1016/j.jcp.2004.02.013 .

Absolute and convective instabilities of two-dimensional bluff body wakes in ground effect

Local and global instabilities are investigated of wakes of general two-dimensional bluff bodies placed near and parallel to a plane boundary or ground. A spatio-temporal linear stability analysis is first applied to a four-parameter family of local wake profiles to investigate the fundamental local stability characteristics of the wake in ground effect. The analysis shows significant dependencies of the stability characteristics of the wake on the distance from the wake centreline to the ground (normalised by the wake width), and also on the velocity ratio of the near- and far-ground sides of the wake. The analysis is then compared with earlier experiments on a circular cylinder to examine, according to the transition scenario of the steep global modes, the streamwise variation of the local stability characteristics of the wake in ground effect. The comparison indicates that the near wake region of the cylinder changes from being absolutely unstable to being convectively unstable when the cylinder comes down into the near-ground range in which the von Kármán-type vortex shedding from the cylinder is suppressed, being qualitatively consistent with the transition scenario for general wake-type flows. A possible explanation is also given for the counter-intuitive relation between the thickness of the boundary layer on the ground and the critical gap distance for the cessation of the von Kármán-type vortex shedding in ground effect.

Three-dimensional transition in the wake of bluff elongated cylinders

Despite little supporting evidence, there appears to be an implicit assumption that the wakes of two-dimensional bluff bodies undergo transition to three-dimensional flow and eventually turbulence, through the same sequence of transitions as observed for a circular cylinder wake. Previous studies of a square cylinder wake support this assumption. In this paper, the transition to three-dimensional wake flow is examined for an elongated cylinder with an aerodynamic leading edge and square trailing edge. The three-dimensional instability modes are determined as a function of aspect ratio (, is more unstable than Mode A. These results suggest that the transition scenario for elongated bluff bodies may be distinctly different to short bodies such as circular or square cylinders. At the very least, the dominant spanwise wavelength in the turbulent wake is likely to be much longer than that for a circular cylinder wake. In addition, the reversal of the ordering of occurrence of the two modes with the different spatial symmetries is likely to affect the development of spatio-temporal chaos as a precursor to fully turbulent flow.In conjunction with prior work, the current results indicate that nearly all three-dimensional instabilities of the vortex street can be identified as one of only a handful of transition modes.

On three-dimensional quasiperiodic Floquet instabilities of two-dimensional bluff body wakes

Previous studies dealing with Floquet secondary stability analysis of the wakes of circular and square cross-section cylinders have shown that there are two synchronous instability modes, with long (mode A) and short (mode B) spanwise wavelengths. At intermediate wavelengths another mode arises, which reaches criticality at Reynolds numbers higher than modes A or B. Here we concentrate on these intermediate-wave number modes for the wakes of circular and square cylinders. It is found that in both cases these modes possess complex-conjugate pair Floquet multipliers, and can be combined to produce either standing or traveling waves. Both these states are quasiperiodic.

Small-scale turbulence characteristics of two-dimensional bluff body wakes

Measurements have been made in nominally two-dimensional turbulent wakes generated by five different bluff bodies. Each wake has a different level of large-scale organization which is reflected in different amounts of large-scale anisotropy. Structure functions of streamwise (u) and lateral (v) velocity fluctuations at approximately the same value of Rλ, the Taylor microscale Reynolds number, indicate that inertial-range scales are significantly affected by the large-scale anisotropy. The effect is greater on v than u and more pronounced for the porous-body wakes than the solid-body wakes. In particular, ‘relative’ values of the scaling (or power-law) exponents indicate that the magnitude of the transverse exponents can exceed that of the longitudinal ones in the porous-body wakes. This is supported by the inertial-range behaviour of the spectra of u and v. The difference between the transverse and longitudinal exponents appears to depend on the large-scale anisotropy of the flow, as measured by the ratio of the variances of v and u and ratio of the integral length scales of v and u. The spanwise vorticity spectra are much less affected by the anisotropy than the spectra of u and v.

Properties of the mean recirculation region in the wakes of two-dimensional bluff bodies

The properties of the time- and span-averaged mean wake recirculation region are investigated in separated flows over several different two-dimensional bluff bodies. Ten different cases are considered and they divide into two groups: cylindrical geometries of circular, elliptic and square cross-sections and the normal plate. A wide Reynolds number range from 250 to 140000 is considered, but in all the cases the attached portion of the boundary layer remains laminar until separation. The lower Reynolds number data are from direct numerical simulations, while the data at the higher Reynolds number are obtained from large-eddy simulation and the experimental work of Cantwell & Coles (1983), Krothapalli (1996, personal communication), Leder (1991) and Lyn et al. (1995). Unlike supersonic and subsonic separations with a splitter plate in the wake, in all the cases considered here there is strong interaction between the shear layers resulting in Kármán vortex shedding. The impact of this fundamental difference on the distribution of Reynolds stress components and pressure in relation to the mean wake recirculation region (wake bubble) is considered. It is observed that in all cases the contribution from Reynolds normal stress to the force balance of the wake bubble is significant. In fact, in the cylinder geometries this contribution can outweigh the net force from the shear stress, so that the net pressure force tends to push the bubble away from the body. In contrast, in the case of normal plate, owing to the longer wake, the net contribution from shear stress outweighs that from the normal stress. At higher Reynolds numbers, separation of the Reynolds stress components into incoherent contributions provides more insight. The behaviour of the coherent contribution, arising from the dominant vortex shedding, is similar to that at lower Reynolds numbers. The incoherent contribution to Reynolds stress, arising from small-scale activity, is compared with that of a canonical free shear layer. Based on these observations a simple extension of the wake model (Sychev 1982; Roshko 1993a, b) is proposed.

Self-excited oscillations in the wake of two-dimensional bluff bodies and their control

The onset of Kármán-vortex shedding is studied experimentally in the wake of different two-dimensional bluff bodies, namely an oblong cylinder, circular cylinders and plates of rectangular cross-section. Different control measures, such as wake heating, transverse body oscillations and base bleed are investigated. As the steady-periodic Kármán shedding has previously been identified as a limit-cycle, i.e. as self-excited oscillations, the experiments are interpreted in the framework of the Stuart–Landau model. The coefficients of the Stuart–Landau equation for the characteristic vortex shedding amplitude, i.e. the linear temporal growth rate, linear frequency and the Landau constant, are fully determined for the two cylinders and in part for the plate. For this purpose transients are generated by suddenly switching transverse body oscillations or base bleed on or off. The analysis of these transients by a refined method based on complex demodulation provides reliable estimates of the model coefficients and yields an experimental validation of the concept that a global instability mode grows or decays as a whole. Also, it is demonstrated that the coefficients of the Stuart–Landau equation are independent of the experimental technique used to produce the transients.

Large structure in the far wakes of two-dimensional bluff bodies

Smoke-wire flow visualization and hot-wire anemometry have been used to study near and far wakes of two-dimensional bluff bodies. For the case of a circular cylinder at 70 < Re < 2000, a very rapid (exponential) decay of velocity fluctuations at the Kármán-vortex-street frequency is observed. Beyond this region of decay, larger-scale (lower wavenumber) structure can be seen. In the far wake (beyond one hundred diameters) a broad band of frequencies is selectively amplified and then damped, the centre of the band shifting to lower frequencies as downstream distance is increased.The far-wake structure does not depend directly on the scale or frequency of Kármán vortices shed from the cylinder; i.e. it does not result from amalgamation of shed vortices. The growth of this structure is due to hydrodynamic instability of the developing mean wake profile. Under certain conditions amalgamation can take place, but is purely incidental, and is not the driving mechanism responsible for the growth of larger-scale structure. Similar large structure is observed downstream of porous flat plates (Re ≈ 6000), which do not initially shed Kármán-type vortices into the wake.Measured prominent frequencies in the far cylinder wake are in good agreement with those estimated by two-dimensional locally parallel inviscid linear stability theory, when streamwise growth of wake width is taken into account. Finally, three-dimensionality in the far wake of a circular cylinder is briefly discussed and a mechanism for its development is suggested based on a secondary parametric instability of the subharmonic type.

Absolute instability in the near-wake of two-dimensional bluff bodies

The linear inviscid instability of a family of mean velocity profiles which is representative of the near wake of two-dimensional bluff bodies is studied on a locally parallel basis. Restricting attention to the incompressible case, both the influence of the centerline velocity and the ratio of wake width to individual mixing layer thickness are explored. The parameters are identified for which the flow is absolutely unstable locally, i.e. for which instability waves amplifying in the upstream direction exist. This feature plays an important role when the attempt is made to explain the selection of a particular vortex shedding mode, i.e. to predict the shedding frequency, for instance. Different possibilities of feedback loops, such as the one proposed by Koch, as well as other mode selection criteria, notably the one used by Pierrehumbert, and their relevance to particular types of wakes are discussed. For top-hat wakes bounded by thin mixing layers a new criterion is proposed which properly accounts for the influence of the initial mixing layer thickness. In addition, the effects of a recirculating region near the body and of initial mixing layer curvature on the near-wake stability are considered qualitatively. Finally, the possible bearing of these concepts on the observed mode competition between the Karman and the varicose mode is examined.

Vortex Shedding from Oscillating Bluff Bodies

When placed ih a fluid stream, some bodies generate separated flow over a substantial proportion of their surface and hence can be classified as bluff. On sharp-edged bluff bodies, separation is fixed at the salient edges, whereas on bluff bodies with continuous surface curvature the location of separation depends both on the shape of the body and the state of the boundary layer. At low Reynolds numbers, when separation first occurs, the flow around a bluff body remains stable, but as the Reynolds number is increased a critical value is reached beyond which instabilities develop. These instabilities can lead to organized unsteady wake motion, dis­ organized motion, or a combination of both. Regular vortex shedding, the subject of this article, is a dominant feature of two-dimensional bluff-body wakes and is present irrespective of whether the separating boundary layers are laminar or turbulent. It has been the subject of research for more than a century, and many hundreds of papers have been written. In recent years vortex shedding has been the topic of Euromech meetings reported on by Mair & Maull (1971) and Bearman & Graham (1980), and a comprehensive review has been undertaken by Berger & Wille (1972). Vortex shedding and general wake turbulence induce fluctuating pres­ sures on the surface of the generating bluff body, and if the body is flexible this can cause oscillations. Oscillations excited by vortex shedding are usually in a direction normal to that of the free stream, and amplitudes as large as 1.5 to 2 body diameters may be recorded. In addition to the generating body, any other bodies in its wake may be forced into oscillation. Broad-band force fluctuations, induced by turbulence produced in the flow around a bluff body, rarely lead to oscillations as severe as those caused by vortex shedding. Some form of aerodynamic instability, such that move-

Vortex Shedding from Oscillating Bluff Bodies

When placed ih a fluid stream, some bodies generate separated flow over a substantial proportion of their surface and hence can be classified as bluff. On sharp-edged bluff bodies, separation is fixed at the salient edges, whereas on bluff bodies with continuous surface curvature the location of separation depends both on the shape of the body and the state of the boundary layer. At low Reynolds numbers, when separation first occurs, the flow around a bluff body remains stable, but as the Reynolds number is increased a critical value is reached beyond which instabilities develop. These instabilities can lead to organized unsteady wake motion, dis­ organized motion, or a combination of both. Regular vortex shedding, the subject of this article, is a dominant feature of two-dimensional bluff-body wakes and is present irrespective of whether the separating boundary layers are laminar or turbulent. It has been the subject of research for more than a century, and many hundreds of papers have been written. In recent years vortex shedding has been the topic of Euromech meetings reported on by Mair & Maull (1971) and Bearman & Graham (1980), and a comprehensive review has been undertaken by Berger & Wille (1972). Vortex shedding and general wake turbulence induce fluctuating pres­ sures on the surface of the generating bluff body, and if the body is flexible this can cause oscillations. Oscillations excited by vortex shedding are usually in a direction normal to that of the free stream, and amplitudes as large as 1.5 to 2 body diameters may be recorded. In addition to the generating body, any other bodies in its wake may be forced into oscillation. Broad-band force fluctuations, induced by turbulence produced in the flow around a bluff body, rarely lead to oscillations as severe as those caused by vortex shedding. Some form of aerodynamic instability, such that move-

Aerodynamic behaviour of bodies in the wakes of other bodies

Wakes of two-dimensional bluff bodies are described, with emphasis on the properties of the wake which influence the loads on other bodies placed in the wake. The unsteady irrotational flow outside the true wake is included in the discussion. Some limited information on the wakes of three-dimensional bluff bodies is also considered. The interaction between two bodies is subdivided into two categories: (i) when the bodies are close together and the upstream body is influenced by the downstream one and (ii) when the bodies are so far apart that only the downstream body is affected. Experiments are described in which the load on an aerofoil in the wake of a two-dimensional bluff body was measured. The results are presented in the form of an aerodynamic admittance and these experiments are used to illustrate the type of problem associated with the determination of the loads on a bluff body in a wake. Experiments are also described which show the large variation of time-averaged load which can be developed on a body which is part of a closely packed complex of bodies, as the orientation of the complex to the wind is varied. Finally, some ideas for future research are outlined.

The formation of vortex streets

The formation of vortex streets in the wake of two-dimensional bluff bodies can be explained by considering the non-linear interaction of two infinite vortex sheets, initially a fixed distance, h, apart, in an inviscid incompressible fluid. The interaction of such sheets (represented in the calculation by rows of point-vortices) is examined in detail for various ratios of h to the wavelength, a, of the initial disturbance. The number and strength of the concentrated regions of vorticity formed in the interaction depend very strongly on h/a. The non-linear interaction of the two vortex sheets explains both the cancellation of vorticity and vortex-street broadening observed in the wakes of bluff bodies.