On variational features of vortex flows

Ideal incompressible fluid is a Hamiltonian system which possesses an infinite number of integrals, the circulations of velocity over closed fluid contours. This allows one to split all the degrees of freedom into the driving ones and the “slave” ones, the latter to be determined by the integrals of motions. The “slave” degrees of freedom correspond to “potential part” of motion, which is driven by vorticity. Elimination of the “slave” degrees of freedom from equations of ideal incompressible fluid yields a closed system of equations for dynamics of vortex lines. This system is also Hamiltonian. The variational principle for this system was found recently (Berdichevsky in Thermodynamics of chaos and order, Addison-Wesly-Longman, Reading, 1997; Kuznetsov and Ruban in JETP Lett 67, 1076–1081, 1998). It looks striking, however. In particular, the fluid motion is set to be compressible, while in the least action principle of fluid mechanics the incompressibility of motion is a built-in property. This striking feature is explained in the paper, and a link between the variational principle of vortex line dynamics and the least action principle is established. Other points made in this paper are concerned with steady motions. Two new variational principles are proposed for steady vortex flows. Their relation to Arnold’s variational principle of steady vortex motion is discussed.      

Using the perturbation method to solve the problem of separated incompressible flow past thin airfoils

A model for separated incompressible flow past thin airfoils in the neighborhood of the “shockless entrance” condition is constructed based on the averaging of the vortex shedding flow past the airfoil edges. By approximation of the vortex shedding by two vortex curves, determination of the average hydrodynamic parameters is reduced to a twofold solution of an integral singular equation equivalent to the equation describing steady-state nonseparated airfoil flow. In this case, the calculation time is two orders of magnitude smaller than the time required for the solution of the corresponding evolution problem. The results of a test calculation using the proposed method are in fair agreement with available results of calculations and experiments. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 49–63, May–June, 2006.     

Vortex diffraction on a swept wing

The forces acting on a swept wing in the presence of a vortex induced by a delta wing, as well as the velocity field in the vicinity of the swept wing, have been measured. By means of the “frozen,” vortex model and a specially-developed numerical panel method, the forces and moments acting on the wing are calculated from the known velocity field. Comparison of the calculated and measured force characteristics makes it possible to determine the extent to which the model fits the physical flow pattern. It is shown that for the intense vortex considered in this study the model gives results which disagree sharply with the experimental data. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 98–105, November–December, 1998. The study was supported by the International Scientific and Technical Center under grant No. 201.     

Numerical investigation on the flowfield of ＂swallowtail＂ cavity for supersonic mixing enhancement

A ＂swallowtail＂ cavity for the supersonic combustor was proposed to serve as an efficient flame holder for scramjets by enhancing the mass exchange between the cavity and the main flow. A numerical study on the ＂swallow- tail＂ cavity was conducted by solving the three-dimensional Reynolds-averaged Navier-Stokes equations implemented with a k-e turbulence model in a multi-block mesh. Turbu- lence model and numerical algorithms were validated first, and then test cases were calculated to investigate into the mechanism of cavity flows. Numerical results demonstrated that the certain mass in the supersonic main flow was sucked into the cavity and moved spirally toward the combustor walls. After that, the flow went out of the cavity at its lateral end, and finally was efficiently mixed with the main flow. The comparison between the ＂swallowtail＂ cavity and the conventional one showed that the mass exchanged between the cavity and the main flow was enhanced by the lateral flow that was induced due to the pressure gradient inside the cavity and was driven by the three-dimensional vortex ring generated from the ＂swallowtail＂ cavity structure.     

Primary and secondary vortical structures contribution in the entrainment of low Reynolds number jet flows

Particle image velocimetry measurements and time-resolved visualization are used for the reconstruction of the Kelvin–Helmholtz vortex passing in the near field of a round jet and of a lobed jet. For the round jet, the entrainment is produced in the braid region, where streamwise structures develop. In the Kelvin–Helmholtz ring, entrainment is dramatically affected by the attenuation of the streamwise structures. As for the lobed jet, the special geometry introduces a transverse shear leading to a breakdown of the Kelvin–Helmholtz structures into “ring segments.” Streamwise structures continuously develop at the resulting discontinuity regions and control the lobed jet self-induction. In this case, the entrainment rate is less affected by the primary structures dynamics.     

Asymptotic behavior of the far region of turbulent wake vortices

An approximate mathematical model, formulation of the problem, and its approximate solution are proposed for the far region of a turbulent vortex wake past a moving body, where the departure of the horizontal velocity component from the uniform flow is slight. It is assumed that the single important parameter that defines the main flow characteristics in this region is the vortex momentum per unit length produced in the fluid by the lift equal to the weigth of the moving body uncompensated by the buoyancy force. Thus, the flow is self-similar, and the self-similarity law determines the intensity, shape, and location of vortex lines as functions of the downstream distance with accuracy up to a constant factor, which cannot be determined theoretically and should be obtained by comparison of theory with experiment. A boundary-value problem is formulated to determine the flow structure of vortex lines (vorticity distribution). A solution of the problem is obtained numerically in the limit of “vanishing turbulent viscosity.” The variation in the maximum velocity of a vortex line with distance, determined by self-similarity, is in agreement with available experimental data. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 13–23, March–April, 1999.     

An experimental study on the vortical structures and behaviour of jets issuing from inclined coaxial nozzles

An experimental study on inclined coaxial jets using laser-induced fluorescence and particle image velocimetry is presented here. The Reynolds numbers of the inner primary jet and outer secondary jet were Re = 2,500 and between Re = 500 and 2,000 (based on gap size), respectively, which corresponded to secondary-to-primary jet velocity ratios (VR) of VR = 0.5–2.0. The secondary-to-primary jet area ratio was 2.25, and 45° and 60° incline-angles were studied. Flow visualizations show that relatively independent inclined primary and secondary jet vortex roll-ups were formed at VR = 0.5. At VR = 1.0, regular pairings and mergings between primary and secondary jet vortex roll-ups led to large-scale entrainment of secondary jet and ambient fluids into the primary jet column and conferred a “serpentile”-shaped outline upon it. While the “serpentile”-shaped outline continued to exist at VR = 2.0, it was a result of stronger secondary jet inner vortex roll-ups which “pinched” the primary jet column regularly. These flow behaviours are observed to intensify with an increase in the incline-angle used. Velocity measurements demonstrate that inclined coaxial nozzles promoted vectoring of the primary jet momentum towards the longer nozzle lengths when velocity-ratio and/or incline-angle were increased. Lastly, peak velocity and higher turbulence intensity levels due to augmented vortical interactions are also detected along shorter nozzle lengths.     

Turbulent boundary layers on axially-inclined cylinders. II. Circumferentially averaged wall-pressure wavenumber–frequency spectra

The boundary layer on a long cylinder with its axis at small inclinations to the freestream (an idealisation of “streamers” used in underwater seismic surveys) has been studied experimentally. For the range of incidence 0–6°, there is no evidence of vortex shedding at typical Reynolds numbers. The circumferentially averaged fluctuating wall pressure decreases with increasing incidence, showing that increases in extraneous noise in seismic measurements when the streamer is at incidence are not caused by the changes in boundary layer structure. Wavenumber–frequency spectra of the circumferentially averaged wall pressure show a convective ridge that persists for the range of incidences studied. Received: 10 June 2000/Accepted: 4 July 2001     

The effects of acceleration in jets: kinematics of the near field vortices

Direct and large-eddy simulations (DNS/LES) of accelerating round jets are used to analyze the effects of acceleration on the kinematics of vortex rings in the near field of the jet (x/D < 12). The acceleration is obtained by increasing the nozzle jet velocity with time, in a previously established (steady) jet, and ends once the inlet jet velocity is equal to twice its initial value. Several acceleration rates (α = 0.02–0.6) and Reynolds numbers (Re _D = 500–20000) were simulated. Acceleration maps were used to make a detailed study of the kinematics of vortex rings in accelerating jets. One of the effects of the acceleration is to cause a number of new primary and secondary vortex merging events that are absent from steady jets. As the acceleration rate α increases, both the number of primary merging events between rings and the axial position where these take place decreases. The statistics for the speed of the starting ring that forms at the start of the acceleration phase for each simulation, agree well with the statistics for the “front” speed observed by Zhang and Johari (Phys Fluids 8:2185–2195, 1996). Acceleration maps and flow visualizations show that during the acceleration phase the near field coherent vortices become smaller and are formed at an higher frequency than in the steady jet, and their (mean) shedding frequency increases linearly with the acceleration rate. Finally, it was observed that the acceleration decreases the spreading rate of the jet, in agreement with previous experimental works.      

Effect of wing–wake interaction on aerodynamic force generation on a 2D flapping wing

This paper is motivated by the works of Dickinson et al. (Science 284:1954–1960, 1999) and Sun and Tang (J Exp Biol 205:55–70, 2002) which provided two different perspectives on the influence of wing–wake interaction (or wake capture) on lift generation during flapping motion. Dickinson et al. (Science 284:1954–1960, 1999) hypothesize that wake capture is responsible for the additional lift generated at the early phase of each stroke, while Sun and Tang (J Exp Biol 205:55–70, 2002) believe otherwise. Here, we take a more fundamental approach to study the effect of wing–wake interaction on the aerodynamic force generation by carrying out simultaneous force and flow field measurements on a two-dimensional wing subjected to two different types of motion. In one of the motions, the wing at a fixed angle of attack was made to follow a motion profile described by “acceleration-constant velocity-deceleration”. Here, the wing was first linearly accelerated from rest to a predetermined maximum velocity and remains at that speed for set duration before linearly decelerating to a stop. The acceleration and deceleration phase each accounted for only 10% of the stroke, and the stroke covered a total distance of three chord lengths. In another motion, the wing was subjected to the same above-mentioned movement, but in a back and forth manner over twenty strokes. Results show that there are two possible outcomes of wing–wake interaction. The first outcome occurs when the wing encounters a pair of counter-rotating wake vortices on the reverse stroke, and the induced velocity of these vortices impinges directly on the windward side of the wing, resulting in a higher oncoming flow to the wing, which translates into a higher lift. Another outcome is when the wing encounters one vortex on the reverse stroke, and the close proximity of this vortex to the windward surface of the wing, coupled with the vortex suction effect (caused by low pressure region at the center of the vortex), causes the net force on the wing to decrease momentarily. These results suggest that wing–wake interaction does not always lead to lift enhancement, and it can also cause lift reduction. As to which outcome prevails depend very much on the flapping motion and the timing of the reverse stroke.     

Impulse generated during unsteady maneuvering of swimming fish

The relationship between the maneuvering kinematics of a Giant Danio (Danio aequipinnatus) and the resulting vortical wake is investigated for a rapid, ‘C’-start maneuver using fully time-resolved (500 Hz) particle image velocimetry (PIV). PIV illuminates the two distinct vortices formed during the turn. The fish body rotation is facilitated by the initial, or “maneuvering” vortex formation, and the final fish velocity is augmented by the strength of the second, “propulsive” vortex. Results confirm that the axisymmetric vortex ring model is reasonable to use in calculating the hydrodynamic impulse acting on the fish. The total linear momentum change of the fish from its initial swimming trajectory to its final swimming trajectory is balanced by the vector sum of the impulses of both vortex rings. The timing of vortex formation is uniquely synchronized with the fish motion, and the choreography of the maneuver is addressed in the context of the resulting hydrodynamic forces.     

Numerical study of vortex reconnection for two anti-parallel vortex tubes

By direct numerical simulation of the Navier-Stokes equations we investigate the reconnection of two antiparallel vortex tubes. A new type of perturbation of the initial vorticity field is given which is different from that presented in Refs. [8] and [9]. The formation and the evolution of the “curved vortex belts”, their mutual action with the “bridges” are found. These are important phenomena not studied by others. The project supported by the LNM of Institute of Mechanics, Academia Sinica and The National Natural Science Foundation of China     

Dual leading-edge vortex structure for flow over a simplified butterfly model

The dye visualization experiments show that a dual leading-edge vortex (LEV) structure exists on the suction side of a simplified butterfly model of Papilio ulysses at α = 8°−12°. Furthermore, the results of particle image velocimetry (PIV) measurement indicate that the axial velocity of the primary (outer) vortex core reaches the lower extreme value while a transition from a “wake-like” to a “jet-like” axial velocity profile occurs. The work reveals for the first time the existence of dual LEV structure on the butterfly-like forward-sweep wing configuration.     

Experimental estimation of a D-shaped cylinder wake using body-mounted sensors

The effectiveness of a small array of body-mounted sensors, for estimation and eventually feedback flow control of a D-shaped cylinder wake is investigated experimentally. The research is aimed at suppressing unsteady loads resulting from the von-Kármán vortex shedding in the wake of bluff-bodies at a Reynolds number range of 100–1,000. A low-dimensional proper orthogonal decomposition (POD) procedure was applied to the stream-wise and cross-stream velocities in the near wake flow field, with steady-state vortex shedding, obtained using particle image velocimetry (PIV). Data were collected in the unforced condition, which served as a baseline, as well as during influence of forcing within the “lock-in” region. The design of sensor number and placement was based on data from a laminar direct numerical simulation of the Navier-Stokes equations. A linear stochastic estimator (LSE) was employed to map the surface-mounted hot-film sensor signals to the temporal coefficients of the reduced order model of the wake flow field in order to provide accurate yet compact estimates of the low-dimensional states. For a three-sensor configuration, results show that the estimation error of the first two cross-stream modes is within 20–40% of the PIV-generated POD time coefficients. Based on previous investigations, this level of error is acceptable for a moderately robust controller required for feedback flow control.     

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     

The flow in a planar-radial vortex chamber. 2. Vortex structure of the flow

A vortex structure of an air flow with a characteristic size of vortices comparable with the primary vortex size was observed in a vortex chamber of planar-radial geometry for the first time. The vortex component of the flow velocity along the chamber radius and its axis was calculated. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 41–49, January–February, 2000.     

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.     

Topological structures of vortex and helicity analysis

In this paper, we discuss the topological structures of the vortex filaments and vortex tubes with an exact solution of a straight spiral vortex tube. We find that there are some confusions about the calculation of the helicities of a knotted vortex filament and some linked vortex filaments by using different methods. We explain how to unify these methods and give the right results. This work is supported by the National Basic Research Important Project “Nonlinear Science”     

Model of vortex ring formation

A semi-empirical model of vortex ring formation during exhaustion of a pulsed submerged jet from a circular orifice is presented. Formulas for determining the parameters of the vortex ring, depending on the conditions of formation of the latter, are derived. The calculated characteristics of the vortex ring as functions of criteria determining the vortex formation process are demonstrated to be in good agreement with experimental data. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 25–36, November–December, 2008.