Effect of local forcing on a turbulent boundary layer

An experimental study is performed to analyze flow structures behind local suction and blowing in a flat-plate turbulent boundary layer. The local forcing is given to the boundary layer flow by means of a sinusoidally oscillating jet issuing from a thin spanwise slot at the wall. The Reynolds number based on the momentum thickness is about Re _θ =1700. The effects of local forcing are scrutinized by altering the forcing frequency (0.011 ≤ f ⁺≤ 0.044). The forcing amplitude is fixed at A ₀=0.4. It is found that a small local forcing reduces the skin friction and the skin friction reduction increases with the forcing frequency. A phase-averaging technique is employed to capture the large-scale vortex evolution. An organized spanwise vortical structure is generated by the local forcing. The cross-sectional area of vortex and the time fraction of vortex are examined by changing the forcing frequency. An investigation of the random fluctuation components reveals that turbulent energy is concentrated near the center of vortical structures. Received: 17 March 2000/Accepted: 3 April 2001     

An experimental study of a spanwise structure around a reattachment region of a two-dimensional backward-facing step

 The flow field downstream of a two-dimensional backward-facing step is usually assumed to be independent of the direction along the span of the step. This assumption is made even though it is well known that the flow exhibits a three-dimensional vortex structure. This state of affairs is no doubt due to the lack of detailed information concerning the characteristics of the vortex structure. In this paper, we report our investigations of the flow structure around a reattachment region using an ultrasound velocity profiler to measure the spanwise velocity component as a function of the spanwise coordinate and time. The flow field is found to be very complex both in space and time. The low-frequency component of the spanwise velocity fluctuation becomes dominant in the near-wall region, with peaks in the power spectrum at frequencies fh/Uc=0.05 and fh/Uc=0.012. Using multiple ultrasound transducers, we also find that a streamwise vortex exists in the flow. Received: 20 March 2000 / Accepted: 15 January 2001 Published online: 29 November 2001     

Experimental investigations of a swirling jet in both stationary and rotating surroundings

The ‘plug’ flow emerging from a long rotating tube into a large stationary reservoir was used in the experimental investigation of swirling jets with Reynolds numbers, Re = 600, 1,000 and 2,000, and swirl numbers, S = ΩR/U, in the range 0–1.1, to cover flow regimes from the non-rotating jet to vortex breakdown. Here Ω is the nozzle rotation rate, R is the radius of the nozzle exit, and U is the mean mass axial velocity. The jet was more turbulent and eddies shed faster at larger Re. However the flow criticality and shear layer morphology remained unchanged with Re. After the introduction of sufficient rotation, co-rotating and counter-winding helical waves replaced vortex rings to become the dominant vortex structure. The winding direction of the vortex lines suggests that Kelvin–Helmholtz and generalized centrifugal instability dominated the shear layer. A quantitative visualization study has been carried out for cases where the reservoir was rotating independently with S _a  = Ω _a R/U = ±0.35, ±0.51 and ±0.70 at Re = 1,000 and 2000, where Ω _a is the rotation rate of the reservoir. The criterion for breakdown was found to be mainly dependent on the absolute swirl number of the jet, S. This critical swirl number was slightly different in stationary and counter-swirl surroundings but obviously smaller when the reservoir co-rotated, i.e. S _c  = 0.88, 0.85 and 0.70, respectively. These results suggest that the flow criticality depends mainly on the velocity distributions of the vortex core, while instabilities resulting from the swirl difference between the jet and its ambient seem to have only a secondary effect.     

Near-ground tornado-like vortex structure resolved by particle image velocimetry (PIV)

The near-ground flow structure of tornadoes is of utmost interest because it determines how and to what extent civil structures could get damaged in tornado events. We simulated tornado-like vortex flow at the swirl ratios of S = 0.03–0.3 (vane angle θ_v = 15°–60°), using a laboratory tornado simulator and investigated the near-ground-vortex structure by particle imaging velocimetry. Complicated near-ground flow was measured in two orthogonal views: horizontal planes at various elevations (z = 11, 26 and 53 mm above the ground) and the meridian plane. We observed two distinct vortex structures: a single-celled vortex at the lowest swirl ratio (S = 0.03, θ_v = 15°) and multiple suction vortices rotating around the primary vortex (two-celled vortex) at higher swirl ratios (S = 0.1–0.3, θ_v = 30°–60°). We quantified the effects of vortex wandering on the mean flow and found that vortex wandering was important and should be taken into account in the low swirl ratio case. The tangential velocity, as the dominant velocity component, has the peak value about three times that of the maximum radial velocity regardless of the swirl ratio. The maximum velocity variance is about twice at the high swirl ratio (θ_v = 45°) that at the low swirl ratio (θ_v = 15°), which is contributed significantly by the multiple small-scale secondary vortices. Here, the results show that not only the intensified mean flow but greatly enhanced turbulence occurs near the surface in the tornado-like vortex flow. The intensified mean flow and enhanced turbulence at the ground level, correlated with the ground-vortex interaction, may cause dramatic damage of the civil structures in tornadoes. This work provides detailed characterization of the tornado-like vortex structure, which has not been fully revealed in previous field studies and laboratory simulations. It would be helpful in improving the understanding of the interaction between the tornado-like vortex structure and the ground surface, ultimately leading to better predictions of tornado-induced wind loads on civil structures.     

Near-field tip vortex behind a swept wing model

The near-field flow structure of a tip vortex behind a sweptback and tapered NACA 0015 wing was investigated and compared with a rectangular wing at the same lift force and Re=1.81×10⁵. The tangential velocity decreased with the downstream distance while increased with the airfoil incidence. The core radius was about 3% of the root chord c _r, regardless of the downstream distance and α for α<8°. The core axial velocity was always wake-like. The core Γ_c and total Γ_o circulation of the tip vortex remained nearly constant up to x/c _r=3.5 and had a Γ_c/Γ_o ratio of 0.63. The total circulation of the tip vortex accounted for only about 40% of the bound root circulation Γ_b. For a rectangular wing, the axial flow exhibited islands of wake- and jet-like velocity distributions with Γ_c/Γ_o=0.75 and Γ_o/Γ_b=0.70. For the sweptback and tapered wing tested, the inner region of the tip vortex flow exhibited a self-similar behavior for x/c _r≥1.0. The lift force computed from the spanwise circulation distributions agreed well with the force-balance data. A large difference in the lift-induced drag was, however, observed between the wake integral method and the inviscid lifting-line theory.     

Flow structures and force characteristics for flat plate in oscillatory flows withKc number from 2 to 40 and in combined flows

The evolution of wake structures and variation of the forces on a flat plate in harmonic oscillatory and in-line combined flows are obtained numerically by improved discrete vortex method. For the oscillatory oncoming flow cases, wyenKc number varies from 2 to 40, the vortex pattern changes from a “harmonic wave” shaped (in a range of smallKc number) to a slight inclined “harmonic wave” shaped (in a range of moderateKc numbers), then to inclined vortex clusters with an angle of 50° to the oncoming flow direction (atKc=20), at last, asKc number becomes large, the vortex pattern is like a normal Karman vortex street. The well predicted drag and inertia force coefficients are obtained, which are more close to the results of Keulegan & Carpenter's experiment as compared with previous vortex simulation by other authors. The existence of minimum point of inertia force coefficientC _m nearKc=20 is also well predicted and this phenomenon can be interpreted according to the vortex structure. For steady-oscillatory in-line combined flow cases, the vortex modes behave like a vortex street, exhibit a “longitudinal wave” structure, and a vortex cluster shape corresponding to the ratios ofU _m toU ₀ which are ofO (10⁻¹)O(1) andO(10), respectively. The effect on the prediction of forces on the flat plate from the disturbance component in a combined flow has been demonstrated qualitatively. In addition to this, the lock in phenomenon of vortex shedding has been checked. The project supported by National Natural Science Foundation of China & LNM, Institute of Mechanics, CAS     

Formation of a 2D vortex pair and its 3D breakup: an experimental study

 Vortex pairs are studied using a dye tracing visualisation technique and a particle tracking velocimetry system. The vortex pairs are produced by gravity induced inlets of water issued through a uniform gap. The inlet Reynolds number is Re=Ud/ν≈875 in all tests (d being the gap width and U the cross sectional mean velocity), i.e. the flow is in the laminar regime. Initially, the dipolar vortex structure is two-dimensional, but after travelling a distance of a few times its own width, the flow structure becomes unstable, breaks up and changes into a three-dimensional flow structure. The breakup appears to be caused by an axial flow in the core centres of each vortex of the dipolar structure. These axial flows are induced by boundary effects related to the von Karman viscous pump. After the breakup, it is believed that a vortex ring is formed through reconnection of rudiments from the dipolar structure mediated by the wall induced vorticity. Received: 20 November 1995/Accepted: 14 November 1996     

Moderately Strong Vorticity in a Bathtub-Type Flow

Theoretical and numerical analysis is performed for an inviscid axisymmetric vortical bathtub-type flow. The level of vorticity is kept high so that the image of the flow on the radial–axial plane (r–z plane)is not potential. The most significant findings are: (1) the region of validity of the strong vortex approximation is separated from the drain by a buffer region, (2) the power-law asymptote of the stream function, specified by Δψ∼r ^4/3Δz, appears near the axis when vorticity in the flow is sufficiently strong and (3) the local Rossby number in the region of the power-law is not very sensitive to the changes of the initial vorticity level in the flow and the global Rossby number. Received 3 April 2000 and accepted 29 September 2000     

Effects of the corner radius on the near wake of a square prism

The near wake of square cylinders with different corner radii was experimentally studied based on particle imaging velocimetry (PIV), laser doppler anemometry (LDA) and hotwire measurements. Four bluff bodies, i.e., r/d=0 (square cylinder), 0.157, 0.236, 0.5 (circular cylinder), where r is corner radius and d is the characteristic dimension of the bluff bodies, were examined. A conditional sampling technique was developed to obtain the phase-averaged PIV data in order to characterize quantitatively the effect of corner radii on the near-wake flow structure. The results show that, as r/d increases from 0 to 0.5, the maximum strength of shed vortices attenuates, the circulation associated with the vortices decreases progressively by 50%, the Strouhal number, St, increases by about 60%, the convection velocity of the vortices increases along with the widening of the wake width by about 25%, the vortex formation length and the wake closure length almost double in size. Meanwhile, both the vortex wavelength, λ _x , and the lateral spacing, λ _y , decrease as r/d increases, but the ratio of λ _y to λ _x is approximately 0.29, irrespective of r/d, which is close to the theoretical value of 0.281 for a stable Karman vortex street. The decrease in wavelength is probably responsible for the change in the flow structure from the approximately circular-shaped vortex at r/d=0 to the laterally stretched vortex at r/d=0.5. The leading edge corner radius is more important than the trailing one in influencing the near wake structure since it determines to a great extent the behavior of the streamlines, the separation angle and the base pressure. It is further found that the ratio of the mean drag coefficient to the total shed circulation, C _d/Γ₀, approaches a constant, about 0.25 for different bluff bodies in the subcritical flow regime. The streamwise evolution of vortices and the streamwise fluctuating velocity along the centerline for rounded cylinders are also discussed.     

Accuracy of out-of-plane vorticity measurements derived from in-plane velocity field data

 A study of the errors in out-of-plane vorticity (ω _z ) calculated using a local χ² fitting of the measured velocity field and analytic differentiation has been carried out. The primary factors of spatial velocity sampling separation and random velocity measurement error have been investigated. In principle the ω _z error can be decomposed into a bias error contribution and a random error contribution. Theoretical expressions for the transmission of the random velocity error into the random vorticity error have been derived. The velocity and vorticity field of the Oseen vortex has been used as a typical vortex structure in this study. Data of different quality, ranging from exact velocity vectors of analytically defined flow fields (Oseen vortex flow) sampled at discrete locations to computer generated digital image frames analysed using cross-correlation DPIV, have been investigated in this study. This data has been used to provide support for the theoretical random error results, to isolate the different sources of error and to determine their effect on ω _z measurements. A method for estimating in-situ the velocity random error is presented. This estimate coupled with the theoretically derived random error transmission results for the χ² vorticity calculation method can be used a priori to estimate the magnitude of the random error in ω _z . This random error is independent of a particular flow field. The velocity sampling separation is found to have a profound effect on the precise determination of ω _z by introducing a bias error. This bias error results in an underestimation of the peak vorticity. Simple equations, which are based on a local model of the Oseen vortex around the peak vorticity region, allowing the prediction of the ω _z bias error for the χ² vorticity calculation method, are presented. An important conclusion of this study is that the random error transmission factor and the bias error cannot be minimised simultaneously. Both depend on the velocity sampling separation, but with opposing effects. The application of the random and bias vorticity error predictions are illustrated by application to experimental velocity data determined using cross-correlation DPIV (CCDPIV) analysis of digital images of a laminar vortex ring. Received: 31 October 1997/Accepted: 6 February 1998     

Investigation of the velocity fluctuation spectrum of a vortex flow of vibrationally excited molecular gas in a glow discharge

The flow velocity fluctuations in a gas laser are measured under non-self-maintained glow discharge conditions. Spectra of the pulsating velocity signal on the frequency interval up to 2 kHz and the time dependence of the signal under glow and neutral flow conditions for pure nitrogen, a mixture of nitrogen and helium, and a trial CO₂-N₂-He mixture are obtained. Deformation (due to the action of the discharge) of the spectra of the transverse velocity component in the wake of a right circular cylinder is observed. In order to analyze the data obtained a model of the dynamic response of the pulsating velocity is constructed and a model of the Kármán vortex street behind a body is generalized with allowance for variations of the oscillation phase in the street. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 81–91, March–April, 2000. The work was carried out with partial financial support from the Russian Foundation for Basic Research (projects Nos. 96-01-00372 and 99-01-01199).     

Investigation of flow mechanism of a robotic fish swimming by using flow visualization synchronized with hydrodynamic force measurement

When swimming in water by flapping its tail, a fish can overcome the drag from uniform flow and propel its body. The involved flow mechanism concerns 3-D and unsteady effects. This paper presents the investigation of the flow mechanism on the basis of a 3-D robotic fish model which has the typical geometry of body and tail with periodic flapping 2-freedom kinematical motion testing in the case of St = 0.78, Re = 6,600 and phase delay mode (φ = −75°), in which may have a greater or maximum propulsion (without consideration of the optimal efficiency). Using a special technique of dye visualization which can clearly show vortex sheet and vortices in detail and using the inner 3-component force balance and cable supporting system with the phase-lock technique, the 3-D flow structure visualized in the wake of fish and the hydrodynamic force measurement were synchronized and obtained. Under the mentioned flapping parameters, we found the key flow structure and its evolution, a pair of complex 3-D chain-shape vortex (S–H vortex-rings, S₁–H₁ and S₂–H₂, and their legs L₁ and L₂) flow structures, which attach the leading edge and the trailing edge, then shed, move downstream and outwards and distribute two anti-symmetric staggering arrays along with the wake of the fish model in different phase stages during the flapping period. It is different with in the case of St = 0.25–0.35. Its typical flow structure and evolution are described and the results prove that they are different from the viewpoints based on the investigation of 2-D cases. For precision of the dynamic force measurement, in this paper it was provided with the method and techniques by subtracting the inertial forces and the forces induced by buoyancy and gravity effect in water, etc. from original data measured. The evolution of the synchronized measuring forces directly matching with the flow structure was also described in this paper.     

Scale-adaptive simulation of flow past wavy cylinders at a subcritical Reynolds number

The Xu & Yan scale-adaptive simulation (XYSAS) model is employed to simulate the flows past wavy cylinders at Reynolds number 8 × 10 3.This approach yields results in good agreement with experimental measurements.The mean flow field and near wake vortex structure are replicated and compared with that of a corresponding circular cylinder.The effects of wavelength ratios λ/D m from 3 to 7,together with the amplitude ratios a /D m of 0.091 and 0.25,are fully investigated.Owing to the wavy configuration,a maximum reduction of Strouhal number and root-meansquare (r.m.s) fluctuating lift coefficients are up to 50% and 92%,respectively,which means the vortex induced vibration (VIV) could be effectively alleviated at certain larger values of λ/D m and a /D m.Also,the drag coefficients can be reduced by 30%.It is found that the flow field presents contrary patterns with the increase of λ/D m.The free shear layer becomes much more stable and rolls up into mature vortex only further downstream when λ/D m falls in the range of 5-7.The amplitude ratio a /D m greatly changes the separation line,and subsequently influences the wake structures.     

Time resolved PIV analysis of flow over a NACA 0015 airfoil with Gurney flap

A NACA 0015 airfoil with and without a Gurney flap was studied in a wind tunnel with Re _c = 2.0 × 10⁵ in order to examine the evolving flow structure of the wake through time-resolved PIV and to correlate this structure with time-averaged measurements of the lift coefficient. The Gurney flap, a tab of small length (1–4% of the airfoil chord) that protrudes perpendicular to the chord at the trailing edge, yields a significant and relatively constant lift increment through the linear range of the C _L versus α curve. Two distinct vortex shedding modes were found to exist and interact in the wake downstream of flapped airfoils. The dominant mode resembles a Kàrmàn vortex street shedding behind an asymmetric bluff body. The second mode, which was caused by the intermittent shedding of fluid recirculating in the cavity upstream of the flap, becomes more coherent with increasing angle of attack. For a 4% Gurney flap at α = 8°, the first and second modes corresponded with Strouhal numbers based on flap height of 0.18 and 0.13. Comparison of flow around ‘filled’ and ‘open’ flap configurations suggested that the second shedding mode was responsible for a significant portion of the overall lift increment.     

A rotating elliptic airfoil in fluid at rest and in a parallel freestream

This paper reports results of DPIV measurements on a two-dimensional elliptic airfoil rotating about its own axis of symmetry in a fluid at rest and in a parallel freestream. In the former case, we examined three rotating speeds (Re _c,Ω = 400, 1,000 and 2,000), and in the later case, four rotating speeds (Ro _c,Ω = 2.4, 1.2, 0.6 and 0.4), together with two freestream velocities (Re _c,u  = 200 and 1,000) and two starting configurations of the airfoil (i.e., chord parallel to (α ₀ = 0°) or normal (α ₀ = 90°) to the freestream). Results show that a rotating airfoil in a stationary fluid produces two distinct types of vortex structures depending on the Reynolds number. The first type occurs at the lowest Reynolds number (Re _c,Ω = 400), where vortices shed from the two edges or tips of the airfoil dissipated quickly, resulting in the airfoil rotating in a layer of diffused vorticity. The second type occurs at higher Reynolds numbers (i.e., Re _c,Ω = 1,000 and 2,000), where the corresponding vortices rotated together with the airfoil. Due to the vortex suction effect, the torque characteristics are likely to be heavily damped for the first type because of the rapidly subsiding vortex shedding, and more oscillatory for the second type due to persistent presence of tip vortices. In a parallel freestream, increasing the tip-speed ratio (V/U) of the airfoil (i.e., decreasing the Rossby number, Ro _c,Ω) transformed the flow topology from periodic vortex shedding at Ro _c,Ω = 2.4 to the generation of a “hovering vortex” at Ro _c,Ω = 0.6 and 0.4. The presence of the hovering vortex, which has not been reported in literature before, is likely to enhance the lift characteristics of the airfoil. Freestream Reynolds number is found to have minimal effect on the vortex formation and shedding process, although it enhances shear layer instability and produces more small-scale flow structures that affect the dynamics of the hovering vortex. Likewise, initial starting configuration of the airfoil, while affecting the flow transient during the initial phase of rotation, has insignificant effect on the overall flow topology. Unfortunately, technical constraint of our apparatus prevented us from carrying out complimentary force measurements; nevertheless, the results presented herein, which are more extensive than those computed by Lugt and Ohring (1977), will provide useful benchmark data, from which more advanced numerical calculations can be carried out to ascertain the corresponding force characteristics, particularly for those conditions with the presence of hovering vortex.     

Experimental investigation into vortex structure and pressure drop across microcavities in 3D integrated electronics

Hydrodynamics in microcavities with cylindrical micropin fin arrays simulating a single layer of a water-cooled electronic chip stack is investigated experimentally. Both inline and staggered pin arrangements are investigated using pressure drop and microparticle image velocimetry (μPIV) measurements. The pressure drop across the cavity shows a flow transition at pin diameter–based Reynolds numbers (Re _d ) ~200. Instantaneous μPIV, performed using a pH-controlled high seeding density of tracer microspheres, helps visualize vortex structure unreported till date in microscale geometries. The post-transition flow field shows vortex shedding and flow impingement onto the pins explaining the pressure drop increase. The flow fluctuations start at the chip outlet and shift upstream with increasing Re _d . No fluctuations are observed for a cavity with pin height-to-diameter ratio h/d = 1 up to Re _d ~330; however, its pressure drop was higher than for a cavity with h/d = 2 due to pronounced influence of cavity walls.     

An experimental investigation of flow past a dual step cylinder

Flow development in the wake of a dual step cylinder has been investigated experimentally using Laser Doppler Velocimetry and flow visualization. The dual step cylinder model is comprised of a large diameter cylinder (D) mounted at the mid-span of a small diameter cylinder (d). The experiments have been performed for a Reynolds number (Re _D ) of 1,050, a diameter ratio (D/d) of 2, and a range of large cylinder aspect ratios (L/D). The results show that the flow development is highly dependent on L/D. The following four distinct flow regimes can be identified based on vortex dynamics in the wake of the large cylinder: (1) for L/D ≥ 15, three vortex shedding cells form in the wake of the large cylinder, one central cell bounded by two cells of lower frequency, (2) for 8 < L/D ≤ 14, a single vortex shedding cell forms in the wake of the large cylinder, (3) for 2 < L/D ≤ 6, vortex shedding from the large cylinder is highly three-dimensional. When spanwise vortices are shed, they deform substantially and attain a hairpin shape in the near wake, (4) for 0.2 ≤ L/D ≤ 1, the large cylinder induces vortex dislocations between small cylinder vortices. The results show that for Regimes I to III, on the average, the frequency of vortex shedding in the large cylinder wake decreases with L/D, which is accompanied by a decrease in coherence of the shed vortices. In Regime IV, small cylinder vortices connect across the large cylinder wake, but these connections are interrupted by vortex dislocations. With decreasing L/D, the frequency of dislocations decreases and the dominant frequency in the large cylinder wake increases toward the small cylinder shedding frequency.     

Dynamics of hairpin vortices generated by a mixing tab in a channel flow

To better understand mixing by hairpin vortices, time-series particle image velocimetry (PIV) was applied to the wake of a trapezoidal-shaped passive mixing tab mounted at the bottom of a square turbulent channel (Re _h =2,080 based on the tab height). Instantaneous velocity/vorticity fields were obtained in sequences of 10 Hz in the tab wake in the center plane (x–y) and in a plane (x–z) parallel to the wall. Periodically-shed hairpin vortices were clearly identified and seen to rise as they advected downstream. Experimental evidence shows that the vortex-induced ejection of the near-wall viscous fluid to the immediate upstream is important to the dynamics of hairpin vortices. It can increase the strength of the hairpin vortices in the near tab region and cause generation of secondary hairpin vortices further downstream when the hairpin heads are farther away from the wall. Measurements also reveal the existence of a type of new secondary vortice with the opposite-sign spanwise vorticity. The distribution of vortex loci in the x–y plane shows that the hairpin vortices and the reverse vortices are spatially segregated in distinct layers. Turbulence statistics, including mean velocity profiles, Reynolds stresses, and turbulent kinetic energy dissipation rate distributions, were obtained from the PIV data. These statistical quantities clearly reveal imprints of the identified vortex structures and provide insight into mixing effectiveness. Received: 24 February 2000/Accepted: 24 October 2000     

Non-linear dynamics of semi-dilute polydisperse polymer solutions in microfluidics: effects of flow geometry

The non-linear dynamics of a semi-dilute (c/c* = 15) polydisperse polyethylene oxide (PEO) solution in microfluidics are studied experimentally using benchmark contraction–expansion flow geometries with three contraction–expansion ratios (4:1:4, 8:1:8 and 16:1:16) and two narrow channel lengths (L _c/D _h = 53 and 5.3, where L _c is the length of the narrow channel and D _h is its hydraulic diameter). Complex flows over a range of elasticity numbers (El), Weissenberg numbers (Wi) and Reynolds numbers (Re) are characterized using micro-particle image velocimetry ( \upmu\upmu-PIV) and pressure drop measurements. The evolution of vortex formation and dynamics has been visualized through a step-flow-rate experiment. Various flow dynamics regimes have been quantified and are presented in a Wi–Re diagram. The experimental results reveal that the contraction ratio can result in qualitatively different vortex dynamics of semi-dilute polymer solutions in microfluidics, whereas the length of the narrow channel merely affects the dynamics at a quantitative level. A single elasticity number, if defined by the size of the narrow channel, is not sufficient to account for the effects of contraction ratio on the non-linear vortex dynamics.