Unsteady entrance heat transfer in laminar pipe flows with convection from the ambient

The characteristics of unsteady entrance heat transfer in the combined entrance heat transfer region of laminar pipe flows resulting from time-varying inlet temperature are numerically investigated. Three non-dimensional parameters,Nu ₀, a^*, andf are identified in the study. Also, their effects on the non-dimensional duct wall temperature, fluid bulk temperature, and duct wall heat flux are discussed in great detail. Comparisons are made with the zero thermal capacity wall solution.     

Numerical analysis of unsteady cavitating turbulent flow and shedding horse-shoe vortex structure around a twisted hydrofoil

Cavitating turbulent flow around hydrofoils was simulated using the Partially-Averaged Navier–Stokes (PANS) method and a mass transfer cavitation model with the maximum density ratio (ρ_l/ρ_v,clip) effect between the liquid and the vapor. The predicted cavity length and thickness of stable cavities as well as the pressure distribution along the suction surface of a NACA66(MOD) hydrofoil compare well with experimental data when using the actual maximum density ratio (ρ_l/ρ_v,clip = 43391) at room temperature. The unsteady cavitation patterns and their evolution around a Delft twisted hydrofoil were then simulated. The numerical results indicate that the cavity volume fluctuates dramatically as the cavitating flow develops with cavity growth, destabilization, and collapse. The predicted three dimensional cavity structures due to the variation of attack angle in the span-wise direction and the shedding cycle as well as its frequency agree fairly well with experimental observations. The distinct side-lobes of the attached cavity and the shedding U-shaped horse-shoe vortex are well captured. Furthermore, it is shown that the shedding horse-shoe vortex includes a primary U-shaped vapor cloud and two secondary U-shaped vapor clouds originating from the primary shedding at the cavity center and the secondary shedding at both cavity sides. The primary shedding is related to the collision of a radially-diverging re-entrant jet and the attached cavity surface, while the secondary shedding is due to the collision of side-entrant jets and the radially-diverging re-entrant jet. The local flow fields show that the interaction between the circulating flow and the shedding vapor cloud may be the main mechanism producing the cavitating horse-shoe vortex. Two side views described by iso-surfaces of the vapor volume fraction for a 10% vapor volume, and a non-dimensional Q-criterion equal to 200 are used to illustrate the formation, roll-up and transport of the shedding horse-shoe vortex. The predicted height of the shedding horse-shoe vortex increases as the vortex moves downstream. It is shown that the shape of the horse-shoe vortex for the non-dimensional Q-criterion is more complicated than that of the 10% vapor fraction iso-surface and is more consistent with the experiments. Further, though the time-averaged lift coefficient predicted by the PANS calculation is about 12% lower than the experimental value, it is better than other predictions based on RANS solvers.     

Large eddy simulation of a vortex ring impinging on a three-dimensional circular cylinder

A vortex ring impacting a three-dimensional circular cylinder is studied using large eddy simulation (LES) for a Reynolds number Re = 4 × 10⁴ based on the initial translation speed and diameter of the vortex ring. We have investigated the evolution of vortical structures and identified three typical evolution phases. When the primary vortex closely approaches to the cylinder, a secondary vortex is generated and its segment parts move inward to the primary vortex ring. then two large-scale loop-like vortices are formed to evolve in opposite directions. Thirdly, the two loop-like vortices collide with each other to form complicated small-scale vortical structures. Moreover, a series of hair-pin vortices are generated due to the stretching and deformation of the tertiary vortex. The trajectories of vortical structures and the relevant evolution speeds are analyzed. The total kinetic energy and enstrophy are investigated to reveal their properties relevant to the three evolution phases.     

Characteristics of counter-rotating vortex rings formed ahead of a compressible vortex ring

Characteristics of high Mach number compressible vortex ring generated at the open end of a short driver section shock tube is studied experimentally using high-speed laser sheet-based flow visualization. The formation mechanism and the evolution of counter rotating vortex ring (CRVR) formed ahead of the primary vortex ring are studied in details for shock Mach number (M) 1.7, with different driver section lengths. It has been observed that the strength of the embedded shock, which appears at high M, increases with time due to the flow expansion in the generating jet. Strength of the embedded shock also varies with radius; it is strong at smaller radii and weak at larger radii; hence, it creates a velocity gradient ahead of the embedded shock. At critical Mach number (M _c ≥ 1.6), this shear layer rolls up and forms a counter rotating vortex ring due to Biot-Savart induction of the vortex sheet. For larger driver section lengths, the embedded shock and the resultant shear layer persists for a longer time, resulting in the formation of multiple CRVRs due to Kelvin–Helmholtz type instability of the vortex sheet. CRVRs roll over the periphery of the primary vortex ring; they move upstream due to their self-induced velocity and induced velocity imparted by primary ring, and interact with the trailing jet. Formation of these vortices depends strongly upon the embedded shock strength and the length of the generating jet. Primary ring diameter increases rapidly during the formation and the evolution of CRVR due to induced velocity imparted on the primary ring by CRVR. Induced velocity of CRVR also affects the translational velocity of the primary ring considerably.     

Three-dimensional numerical simulation of a vortex ring impacting a bump

A vortex ring impinging on a three-dimensional bump is studied using large eddy simulation for a Reynolds number Re = 4 × 10⁴ based on the initial translation speed and diameter of the vortex ring. The effects of bump height on the vortical flow phenomena and the underlying physical mechanisms are investigated. Based on the analysis of the evolution of vortical structures, two typical kinds of vortical structures, i.e., the wrapping vortices and the hair-pin vortices, are identified and play an important role in the flow state evolution. The circulation of the primary vortex ring reasonably elucidates some typical phases of flow evolution. Furthermore, the mechanism of flow transition from laminar to turbulent state has been revealed based on analysis of turbulent kinetic energy.     

Experimental flow study over a blunt-nosed axisymmetric body at incidence

The flowfield over a blunt-nosed cylinder was examined experimentally at a low subsonic speed for Re=1.88×10⁵ and angles of attack up to 40°. Velocity measurements were carried out (employing a seven-hole Pitot tube) as well as wall static pressure and wall shear-stress measurements. Surface flow visualization was applied using liquid crystals and a mixture of oil–TiO₂. For all the examined cases no flow asymmetries were found. For high angles of attack (20° and above) a separation “bubble” appears at the leeside of the nose area (streamwise flow separation). The basic feature of the circumferential pressure distribution at the after body area for these angles of attack is a plateau close to the suction peak and a fast recovery next to it. One streamwise vortex on each side of the symmetry plane is formed as well as a separation bubble about 90° far from this plane, where the cross-flow primary separation line is located. Each cross-flow primary separation line starts at the leeside nose area and moves towards the windward side along the cylindrical after body. The space between the two primary separation lines close to the wall is characterized by high flow fluctuations on the leeside, compared to the low fluctuations of the windward side.     

Vortex rings in radially confined domains

The dynamics of vortex rings generated within confined domains are relevant to important hydrodynamic processes such as flow past heart valves or severe arterial constrictions. However, despite their importance, these flows have not received much attention to date. This study examines the development and evolution of radially confined vortex rings. Time-resolved digital particle image velocimetry was used to investigate two levels of radial confinement and a range of vortex ring strengths. We found that for severely confined vortex rings, the formation time and peak circulation values were unaffected for L/D ₀?<?4 cases and slightly affected for larger L/D ₀ cases. After pinch-off, circulation decay was observed with an approximately constant normalized circulation decay rate. We found that with increasing circulation strength, the nondimensional time delay between the pinch-off and the onset of circulation decay reduced due to an increased vortex ring diameter within the confinement domain and a reduction in the necessary time for the surface induced and core vorticity regions to interact. This study uncovers the dynamics of radially confined vortex rings and show that the nondimensional rate of circulation decay is dependent on the vortex ring confinement ratio (ratio of the vortex ring orifice diameter to the diameter of the outer cylinder), and the time delay between the vortex pinch-off and the onset of circulation is dependent on the vortex ring circulation strength.     

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.     

On the formation of vortex rings and pairs

The axisymmetric vortex sheet model developed by Nitsche & Krasny (1994) has been extended to study the formation of vortex rings (pairs) at the edge of circular (2D) tube and opening. Computations based on this model are in good agreement with the experiments (Didden (1979) for circular tube and Auerbach (1987) for 2D tube and opening). Using this new model, evidences are provided to show that the main failure of the similarity theory (the false prediction of axial trajectory of vortex ring) is due to its ignorance of the self-induced ring velocity (mutual induction for vortex pair). We further reason why the similarity theory succeeds in its prediction of radial movement of vortex ring. The effects of various parameters such as turning angle α and piston speedU _p (t) on the formation of vortex ring are investigated. Numerical result shows that turning angle α has no effect on circulation shed τ. We also discuss Glezer (1988)'s summary on the influence ofU _p upon the shedding circulation, and finally give the variation of core distribution of vortex ring with α andU _p (t). The project is supported by National Natural Science Foundation of China and Doctoral Program of Institution of Higher Education     

An experimental study of the unsteady vortex structures in the wake of a root-fixed flapping wing

An experimental study was conducted to characterize the evolution of the unsteady vortex structures in the wake of a root-fixed flapping wing with the wing size, stroke amplitude, and flapping frequency within the range of insect characteristics for the development of novel insect-sized nano-air-vehicles (NAVs). The experiments were conducted in a low-speed wing tunnel with a miniaturized piezoelectric wing (i.e., chord length, C = 12.7 mm) flapping at a frequency of 60 Hz (i.e., f = 60 Hz). The non-dimensional parameters of the flapping wing are chord Reynolds number of Re = 1,200, reduced frequency of k = 3.5, and non-dimensional flapping amplitude at wingtip h = A/C = 1.35. The corresponding Strouhal number (Str) is 0.33, which is well within the optimal range of 0.2 < Str < 0.4 used by flying insects and birds and swimming fishes for locomotion. A digital particle image velocimetry (PIV) system was used to achieve phased-locked and time-averaged flow field measurements to quantify the transient behavior of the wake vortices in relation to the positions of the flapping wing during the upstroke and down stroke flapping cycles. The characteristics of the wake vortex structures in the chordwise cross planes at different wingspan locations were compared quantitatively to elucidate underlying physics for a better understanding of the unsteady aerodynamics of flapping flight and to explore/optimize design paradigms for the development of novel insect-sized, flapping-wing-based NAVs.     

UNSTEADY LIFT AND SOUND PRODUCED BY AN AIRFOIL IN A TURBULENT BOUNDARY LAYER

An analysis is made of the unsteady lift exerted on a stationary rigid body immersed in an incompressible, plane-wall turbulent boundary layer. The lift is expressed as a surface integral over the body involving theupwash velocity induced by the “free” vorticity Ω (found by taking explicit account of the interaction of the body with the flow and excluding the bound vorticity) and a harmonic function X₂that depends only on the shape of the body. The upwash velocity is the free-field velocity given in terms of Ω by the Biot–Savart formula, augmented by the velocity field of a conventional distribution of image vortices in the wall. The function X₂can be interpreted as the velocity potential of flow past the body, produced by motion of the wall at unit speed towards the body. Detailed predictions are made of the lift on a slender airfoil placed in the outer region of the boundary-layer. When the airfoil chord is large compared to the boundary-layer thickness, vortex shedding into the wake causes the magnitude of the net upwash velocity near the trailing edge to be small. The main contributions to the surface integral are then from the nose region, where the upwash velocity may be estimated independently of the fluctuations near the trailing edge. Analytical results for a thin plate airfoil of chord 2a at distance h from the wall show that the lift increases as a/h increases; it is ultimately independent of a and scales with the ratio of h to the hydrodynamic wavelength. Application is made to determine the sound generated by the airfoil in a weakly compressible boundary layer flow over a finite elastic plate.     

CALCULATION OF STEADY AND OSCILLATING FLOWS IN TUBES USING A VORTICITY TRANSPORT ALGORITHM

Steady and oscillating axisymmetric tube flows are modelled using a vorticity transport algorithm. The axisymmetric convective –diffusive Navier–Stokes equations are solved using a splitting technique. Axisymmetric ring vortex filaments are introduced on the walls and subsequently convected and diffused throughout the flow field. An axisymmetric equation similar to the Oseen diffusion equation is used to diffuse the ring vortex filaments. Vorticity is reflected from the tube walls using two techniques. Results are presented for the developing Poiseuille flow and for the developed flow in the form of the entrance length and the axial velocity and vorticity profiles. Good agreement is achieved with a finite difference method in the developing region of Poiseuille flow. The developed flow results are compared with the analytical solutions. The developed profiles of velocity and vorticity have errors of less than 0ċ3 per cent for both methods of dealing with reflection of diffusion at the bounding surfaces and similar accuracy is obtained for the velocity profiles in oscillating flow except at the wall. Oscillating flow is produced with a discretized sinusoidal piston motion. Velocity profiles, boundary layer thickness and entrance length are presented for oscillating flow. Good agreement is achieved for low-Womersley-number non-dimensional frequency. At higher values of this parameter, flows are inaccurately simulated, because the number of piston positions used to discretize the piston motion is inversely proportional to the non-dimensional frequency.     

Entrance flow of a bingham fluid in a tube

The numerical solution of the entrance flow in a tube has been obtained for a Bingham fluid. The numerical procedure used is that of Patankar and Spalding [1]. The accuracy of the numerical results is demonstrated by comparing the fully-developed velocity profiles with analytical exact solutions. The results of the entrance flow in a tube for the case of a zero yield stress are compared with the entrance flow solution for a Newtonian fluid. Detailed results are presented for a wide range of yield numbers (=τ _y D/ūμ).     

An analysis of unsteady highly turbulent swirling flow in a model vortex combustor

This paper reports an experimental investigation of a non-reacting turbulent swirling flow in a practical vortex combustor. The flow was examined for the conditions characteristic of the presence of a breakdown zone and a strong flow instability appearing at swirl numbers S>0.5. Flow visualization techniques, LDA measurements and acoustic probes were employed to study the unsteady flow characteristics. Based on the experimental results a positive first helical mode of instability was identified with a wavelength and frequency depending on swirl. The wavelength was confirmed to grow monotonically with S, while the dominant frequency of the flow pulsations was found to have an unusual parabolic evolution with swirl, with a minimum at S _min=0.88. This finding was interpreted using a proposed kinematic model based on the contribution of two mechanisms: rotation and axial motion of the helical vortex. It was concluded that for S<S _min the instability frequency is essentially dominated by the axial translation of the spiral vortex being inversely proportional to S and therefore giving a decreasing trend. For S>S _min the frequency of the flow precession is more dependent on the angular transportation of the vortex core, which resulted in the expected growing dependence on S.     

An experimental study of the effects of pitch-pivot-point location on the propulsion performance of a pitching airfoil

An experimental investigation was conducted to characterize the evolution of the unsteady vortex structures in the wake of a pitching airfoil with the pitch-pivot-point moving from 0.16C to 0.52C (C is the chord length of the airfoil). The experimental study was conducted in a low-speed wind tunnel with a symmetric NACA0012 airfoil model in pitching motion under different pitching kinematics (i.e., reduced frequency k=3.8–13.2). A high-resolution particle image velocimetry (PIV) system was used to conduct detailed flow field measurements to quantify the characteristics of the wake flow and the resultant propulsion performance of the pitching airfoil. Besides conducting “free-run” PIV measurements to determine the ensemble-averaged velocity distributions in the wake flow, “phase-locked” PIV measurements were also performed to elucidate further details about the behavior of the unsteady vortex structures. Both the vorticity–moment theorem and the integral momentum theorem were used to evaluate the effects of the pitch-pivot-point location on the propulsion performance of the pitching airfoil. It was found that the pitch-pivot-point would affect the evolution of the unsteady wake vortices and resultant propulsion performance of the pitching airfoil greatly. Moving the pitch-pivot-point of the pitching airfoil can be considered as adding a plunging motion to the original pitching motion. With the pitch-pivot-point moving forward (or backward), the added plunging motion would make the airfoil trailing edge moving in the same (or opposite) direction as of the original pitching motion, which resulted in the generated wake vortices and resultant thrust enhanced (or weakened) by the added plunging motion.     

Coherent structures in unsteady swirling jet flow

An LDA technique and phase-averaging analysis were used to study unsteady precessing flow in a model vortex burner. Detailed measurements were made for Re=15,000 and S=1.01. On the basis of the analysis of phase-averaged data and vortex detection by the λ₂-technique of Joeng and Hussain (1995), three precessing spiral vortex structures were identified: primary vortex (PV), inner secondary vortex (ISV), and outer secondary vortex (OSV). The PV is the primary and most powerful structure as it includes primary vorticity generated by the swirler; the ISV and OSV are considered here as secondary vortical structures. The jet breakdown zone is the conjunction of a pair of co-rotating co-winding spiral vortices, PV and ISV. The interesting new feature described is that the secondary vortices form a three-dimensional vortex dipole with a helical geometry. The effect of coupling of secondary vortices was suggested as a mechanism of enhanced stability reflected in their increased axial extent.     

Experimental investigation of a twice-shocked spherical gas inhomogeneity with particle image velocimetry

Results are presented from an experimental investigation into the interaction of a planar shock wave with a vortex ring. A free-falling spherical soap bubble is traversed by the incident shock wave and develops into a vortex ring as a result of baroclinically deposited vorticity (?r×?p 1 0{\nabla\rho\times\nabla p \neq 0}). The vortex ring translates with a velocity relative to the particle velocity behind the shock wave due to circulation. After the shock wave reflects from the tube end wall, it traverses the vortex ring (this process is called “reshock”) and deposits additional vorticity. Planar Mie scattering is used to visualize the atomized soap film at high frame rates (up to 10,000 fps). Particle image velocimetry (PIV) was performed for an argon bubble in nitrogen accelerated by a M = 1.35 shock wave. Circulation was determined from the PIV velocity field and found to agree well with Kelvin’s vortex ring model.     

Chaotic streamlines in the flow of knotted and unknotted vortices

This paper describes the motion and the flow induced by a thin tubular vortex coiled on a torus. The vortex is defined by the number of turns, p, that it makes round the torus symmetry axis and the number of turns, q, that it makes round the torus centerline. All toroidal filamentary vortices are found to progress along and to rotate round the torus symmetry axis in an almost steady manner while approximately preserving their shape. The flow, observed in a frame moving with the vortex, possesses two stagnation points. The stream tube emanating from the forward stagnation point and the stream tube ending at the backward stagnation point transversely intersect along a finite number of streamlines. This produces a three-dimensional chaotic tangle whose geometry depends primarily on the value of p. Inside this chaotic shell there are two major stability tubes: the first one envelopes the vortex whereas the second one runs parallel to it and possesses the same topology. When p > 2 there is an additional stability tube enveloping the torus centerline.     

Flow-induced force fluctuations on a sphere at high Strouhal number

An empirical model is developed to estimate the broadband unsteady force spectrum induced on a rigid sphere in a nominally steady, uniform flow. The Reynolds number is sub-critical, and the frequency range considered is above the low-mode Strouhal shedding frequency of the sphere (0.5⩽fd/U₀⩽100, where f is the frequency, d is the diameter, and U₀ is the mean flow speed). The model uses the separation of variables assumption for the cross-power spectral densities of the surface pressure fluctuations. The assumption is shown to be a proper engineering approximation except in the lower part of the considered frequency range. In addition, the flow-induced unsteady lift and drag forces are measured independently of each other using towed spheres in a basin of water. Both estimations, from the empirical model and the data measured in the tow tank, show that the dimensionless power spectral densities of broadband unsteady lift and drag forces are constant for fd/U₀<1, and (fd/U₀)⁻³ dependent for 1⩽fd/U₀⩽100. The model predicts that the broadband spectral density of the unsteady lift force is about 5 dB higher than that of the unsteady drag force, while the measured data show the level difference between 3 and 7 dB. The empirical model presented here has application in predicting the flow-induced noise of underwater hydrophones that sense acoustic particle velocity or acceleration.