Statistics of quasi-geostrophic point vortices

The statistics of quasi-geostrophic point vortices is investigated theoretically and numerically, in order to understand fundamental aspects of quasi-geostrophic turbulence. The numerical computations are performed using the fast special-purpose computer for molecular dynamics simulations, MDGRAPE-2/3. The most probable vortex distributions are determined based on the maximum entropy theory. The theoretical predictions agree well with the numerical results.     

Flow around two tandem square cylinders near a plane wall

An experimental study has been conducted to investigate the flow around two identical square cylinders in tandem arrangement and placed near a plane wall at a Reynolds number of 6,300. The inter-cylinder spacing ratio was varied from S ^* = 0.5 to 6, and the cylinder-to-wall gap ratio from G ^* = 0.25 to 2. Totally, 42 cases were considered to systematically examine the effects of wall proximity and the mutual interference between the two cylinders in the normalized gap–spacing (G ^*–S ^*) plane. The flow fields were captured using digital particle image velocimetry, in conjunction with measurements of the fluid forces (drag and lift) acting on the downstream cylinder using a piezoelectric load cell. The results show that the flow is highly dependent on the combined values of G ^* and S ^*. Categories relating to G ^* could be broadly classified as small-gap regime (G ^* < 0.5) at which periodic vortex shedding from the cylinders is suppressed, intermediate-gap regime (0.5 < G ^* < 1) where vortex shedding occurs but is under the influence of the wall proximity, and large-gap regime (G ^* > 1) where the wall effects become negligible. Similarly, the flow interference between the two cylinders can be divided into three basic categories as a function of S ^*, namely, shielding regime at S ^* < 1, reattachment regime at 1 < S ^* < 3, and impinging regime at S ^* > 3. Variations of force coefficients, amplitude spectra, Strouhal numbers, and Reynolds shear stress with G ^* and S ^* are presented to characterize the different flow regimes.     

Unsteady near wake of a flat disk normal to a wall

The unsteady wake of a flat disk (diameter D) located at a distance of H from a flat plate has been experimentally investigated at a Reynolds number Re _D  = 1.3 × 10⁵. Tests have been performed for a range of gap ratio (H/D), spanning from 0.3 to 1.75. The leading edge of the flat plate is either streamlined (elliptical) or blunt (square). These configurations have been studied with PIV, high speed PIV and multi-arrayed off-set fluctuating pressure measurements. The results show a progressive increase of the complexity of the flow and of the interaction as the gap ratio decreases. For large values of H/D (1.75), the interaction is weak and the power spectral densities (PSD) exhibit a strong peak associated with the vortex shedding events (St = 0.131) – St = fD/U _∞ is the Strouhal number. For lower values of H/D (0.75), the magnitude of the wall fluctuating pressure increases significantly. A large band contribution is associated with the unsteady wake structure and turbulence. A slight increase of the shedding frequency (St = 0.145) is observed. A critical value of the gap ratio (about 0.35) has been determined. Below this critical value, a three-dimensional separated region is observed and the natural vortex shedding process is very strongly altered. These changes induce a great modification of the fluctuating pressure at the wall. Each interaction reacts in a different way to perturbed upstream conditions. In particular, the disk is an overwhelming perturbation for the lowest H/D value studied here and the relative influence of the upstream turbulence on the wall fluctuating pressure below the near wake region is moderate.     

Dean instability and secondary flow structure in curved rectangular ducts

The pressure driven, fully developed turbulent flow of incompressible viscous fluid (water) in 120° curved ducts of rectangular cross-section is investigated experimentally and numerically. Three different types of curved duct (A-CL, B-SL and C-IL) with continuously varying curvature conform to blade profile as the inner and outer curvature walls to simplify and guide the impeller design of pumps. After validating the numerical method against Particle Image Velocimetry (PIV) measurements, the flow development in the ducts is analyzed in detail by Computational Fluid Dynamics (CFD) for a wide range of Reynolds numbers (Re = 2.4 × 10⁴–1.4 × 10⁵) and aspect ratios (Ar > 1.0, =1.0 and <1.0). The results clearly depict the existence of multiple Dean vortices along the duct: while the axial velocity profile is more related to an inner Dean vortex (called split base vortex), the wall pressure is more influenced by the Dean vortex attached to the inner curvature wall (called ICW Dean vortex). The induced multiple Dean vortices and the secondary flow patterns in the duct cannot be faithfully predicted by using traditional techniques. Therefore, a new criterion based on the vortex core velocities is proposed. With this approach, the effects of Re, Cr and Ar on the Dean instabilities in curved ducts are carefully studied. Decreasing Re promotes the generation of Dean vortices closer to the duct inlet, a trend that is as opposed to laminar flow. In addition, a new pair of vortices called entrainment Dean vortex occurs near the outlet of the curved duct with Ar = 1.0, which has not been previously reported in the literature.     

Effect on the flow and heat transfer characteristics for sinusoidal pulsating laminar flow in a heated square cylinder

Two-dimensional numerical simulation is performed to understand the effect of flow pulsation on the flow and heat transfer from a heated square cylinder at Re = 100. Numerical calculations are carried out by using a finite volume method based on the pressure-implicit with splitting of operators algorithm in a collocated grid. The effects of flow pulsation amplitude (0.2 ≤ A ≤ 0.8) and frequency (0 ≤ f _p  ≤ 20 Hz) on the detailed kinematics of flow (streamlines, vorticity patterns), the macroscopic parameters (drag coefficient, vortex shedding frequency) and heat transfer enhancement are presented in detail. The Strouhal number of vortices shedding, drag coefficient for non-pulsating flow are compared with the previously published data, and good agreement is found. The lock-on phenomenon is observed for a square cylinder in the present flow pulsation. When the pulsating frequency is within the lock-on regime, time averaged drag coefficient and heat transfer from the square cylinder is substantially augmented, and when the pulsating frequency in about the natural vortex shedding frequency, the heat transfer is also substantially enhanced. In addition, the influence of the pulsating amplitude on the time averaged drag coefficient, heat transfer enhancement and lock-on occurrence is discussed in detail.     

Self-similar collapse of 2D and 3D vortex filament models

In this article, a very simple toy model for a candidate blow-up solution of the Euler equation by Boratav and Pelz (vortex dodecapole) is investigated. In this model, vortex tubes are replaced with straight vortex filaments of infinitesimal thickness, and the entire motion is monitored by tracing the motion of a representative point on one vortex filament. It is demonstrated that this model permits a self-similar collapse solution which provides the time dependence of the length scale as (t _c ? t)^1/2, (t < t _c), where the collapse time t _c depends on the initial configuration. From the conservation of circulation, this time dependence implies that vorticity ω scales as (t _c ? t) ^?1, which agrees with the one observed in the direct numerical (pseudo spectral) simulations of the vortex dodecapole. Finally, possible modification of the model is considered.     

Scalar and vector multipole and vortex solitons in the spatially modulated cubic–quintic nonlinear media

We derive scalar and vector multipole and vortex soliton solutions in the spatially modulated cubic–quintic nonlinear media, which is governed by a (3+1)-dimensional N-coupled cubic–quintic nonlinear Schrödinger equation with spatially modulated nonlinearity and transverse modulation. If the modulation depth \(q=1\), the vortex soliton is constructed, and if \(q=0\), the multipole soliton, including dipole, quadrupole, hexapole, octopole and dodecagon solitons, is constructed, respectively, when the topological charge \(k=1\)–5. If the topological charge \(k=0\), scalar solitons can be obtained. Moreover, the number of layers for the scalar and vector multipole and vortex solitons is decided by the value of the soliton order number n.     

Collision detection for complicated polyhedra using the fast multipole method or ray crossing

The purpose of this paper is a comparison of two different methods that can be used for collision detection. One method is called the ray-crossing method, a commonly used geometrical approach. The other method is the fast multipole method, usually used for boundary element methods, which is also applied for collision detection purposes here. Both methods are especially of interest when the collision for arbitrarily shaped polyhedra has to be detected. Here, both methods are described and compared for different examples of complex shaped polyhedra with up to 5 × 10⁵ faces and more than 5 × 10⁵ test points regarding efficiency and required calculation time.     

Numerical analysis of interaction between non-gray radiation and forced convection flow over a recess using the full-spectrum k-distribution method

In the present work, the interaction between non-gray radiation and forced convection in a laminar radiating gas flow over a recess including two backward and forward facing steps in a duct is investigated numerically. Distributions of absorption coefficients across the spectrum (50 cm^?1 < η < 20,000 cm^?1) are obtained from the HITRAN2008 database. The full-spectrum k-distribution method is used to account for non-gray radiation properties, while the gray radiation calculations are carried out using the Planck mean absorption coefficient. To find the divergence of radiative heat flux distribution, the radiative transfer equation is solved by the discrete ordinates method. The effects of radiation–conduction parameter, wall emissivity, scattering coefficient and recess length on heat transfer behaviors of the convection–radiation system are investigated for both gray and non-gray mediums. In addition, the results of gray medium are compared with non-gray results in order to judge if the differences between these two approaches are significant enough to justify the usage of non-gray models. Results show that for air mixture with 10 % CO₂ and 20 % H₂O, use of gray model for the radiative properties may cause significant errors and should be avoided.     

Lattice Boltzmann simulation of natural convection in nanofluid-filled 2D long enclosures at presence of magnetic field

In this paper, the effects of a magnetic field on natural convection flow in filled long enclosures with Cu/water nanofluid have been analyzed by lattice Boltzmann method. This study has been carried out for the pertinent parameters in the following ranges: the Rayleigh number of base fluid, Ra = 10³–10⁵, the volumetric fraction of nanoparticles between 0 and 6 %, the aspect ratio of the enclosure between A = 0.5 and 2. The Hartmann number has been varied from Ha = 0 to 90 with interval 30 while the magnetic field is considered at inclination angles of θ = 0°, 30°, 60° and 90°. Results show that the heat transfer decreases by the increment of Hartmann number for various Rayleigh numbers and the aspect ratios. Heat transfer decreases with the growth of the aspect ratio but this growth causes the effect of the nanoparticles to increase. The magnetic field augments the effect of the nanoparticles at high Rayleigh numbers (Ra = 10⁵). The effect of the nanoparticles rises for high Hartmann numbers when the aspect ratio increases. The rise in the magnetic field inclination improves heat transfer at aspect ratio of A = 0.5.     

Numerical simulation of secondary vortex chamber effect on the cooling capacity enhancement of vortex tube

A vortex tube with additional chamber is investigated by computational fluid mechanics techniques to realize the effects of additional chamber in Ranque–Hilsch vortex tube and to understand optimal length for placing the second chamber in order to have maximum cooling effect. Results show that by increasing the distance between two chambers, both minimum cold and maximum hot temperatures increase and maximum cooling effect occurs at Z/L = 0.047 (dimensionless distance).     

POD analysis of a finite-length cylinder near wake

The near wake of a wall-mounted finite-length square cylinder with an aspect ratio of 7 is investigated based on the proper orthogonal decomposition (POD) of the PIV data measured in three spanwise planes, i.e., z/d = 6, 3.5 and 1.0, near the cylinder free end, mid-span and fixed end (wall), respectively. The Reynolds number based on free-stream velocity (U _∞) and cylinder width (d) is 9,300. A two-dimensional (2D) square cylinder wake is also measured and analyzed at the same Reynolds number for the purpose of comparison. The structures of various POD modes show marked differences between the two flows. While the coefficients, a ₁ and a ₂, of the POD modes 1 and 2 occur within an annular area centered at a ₁ = a ₂ = 0 in the 2D wake, their counterparts are scattered all over the entire circular plane at z/d = 1.0 and 3.5 of the finite-length cylinder wake. Flow at z/d = 6 is dominated by POD mode 1, which corresponds to symmetrical vortex shedding and accounts for 54.0 % of the total turbulent kinetic energy (TKE). On the other hand, the POD modes 1 and 2, corresponding to anti-symmetrical vortex shedding, are predominant, accounting for about 45.0 % of the total TKE, at z/d = 3.5 and 1. It has been found that the flow structure may be qualitatively and quantitatively characterized by the POD coefficients. For example, at z/d = 6, a larger a ₁ corresponds to a smaller length of flow reversal zone and a stronger downwash flow. At z/d = 3.5 and 1, two typical flow modes can be identified from a ₁ and a ₂. While large a ₁ and/or a ₂ correspond to anti-symmetrical vortex shedding, as in a 2D cylinder wake, small a ₁ and a ₂ lead to symmetrical vortex shedding. Any values between the large and small a ₁ and/or a ₂ correspond to the flow structure between these two typical flow modes. As such, the probability of occurrence of a flow structure may be determined from the distribution of the POD coefficients.     

Caractérisation et contrôle de l'asymétrie tourbillonnaire autour d'un côneAerodynamic caracteristics and vortex asymmetry control around a circular cone

The structure of the vortex asymetry around a circular cone at high incidence and Reynolds numbers of order 10⁴, is related to the existence of at least four vortices shed from the apex and exhibiting vortex breakdown points correlated with the sign of the lateral force. Control of the lateral force by injection points out three regimes, and not two as usually admitted, corresponding to inversions of the lateral force direction. To cite this article: O. Rodriguez, J. Pruvost, C. R. Mecanique 332 (2004).     

Osgood’s Lemma and Some Results for the Slightly Supercritical 2D Euler Equations for Incompressible Flow

We investigate the (slightly) super-critical two-dimensional Euler equations. The paper consists of two parts. In the first part we prove well-posedness in C ^s spaces for all s > 0. We also give growth estimates for the C ^s norms of the vorticity for ${0 < s \leqq 1}$ . In the second part we prove global regularity for the vortex patch problem in the super-critical regime. This paper extends the results of Chae et al. where they prove well-posedness for the so-called LogLog-Euler equation. We also extend the classical results of Chemin and Bertozzi–Constantin on the vortex patch problem to the slightly supercritical case. Both problems we study in the setting of the whole space.     

An advanced design method for propellers behind bodies of revolutionCalcul de l'hélice fonctionnant en arrière d'un corps à symétrie axiale

The flow around the unshrouded marine propellers operating in the wake of an axisymmetric body is rotational and tridimensional. An inverse method based on the model of inviscid and rotational fluid and coupling two complementary steps (axisymmetric computation + 3D panel method) is proposed for the design of the marine propellers. The meridional flow computation leads to the determination of axisymmetrical stream sheets as well as the approximate camber surface of the blades and gives a good estimation of the surface of the free vortex wake. The new aspects developed in this method are the involvement of the contraction and the stretching of the free vortex wake, the rotational character of the incoming flow in the axisymmetric computation with tridimensional effects due to 3D panel method. To cite this article: N. Settou, B. Viney, C. R. Mecanique 331 (2003).     

Wingtip vortex control via the use of a reverse half-delta wing

The effect of a 65° sweep reverse half-delta wing (RHDW), mounted at the squared tip of a rectangular NACA 0012 wing, on the tip vortex was investigated experimentally at Re?=?2.45?×?10⁵. The RHDW was found to produce a weaker tip vortex with a lower vorticity level and, more importantly, a reduced lift-induced drag compared to the baseline wing. In addition to the lift increment, the RHDW also produced a large separated wake flow and subsequently an increased profile drag. The reduction in lift-induced drag, however, outperformed the increase in profile drag and resulted in a virtually unchanged total drag in comparison with the baseline wing. Physical mechanisms responsible for the RHDW-induced appealing aerodynamics and vortex flow modifications were discussed.     

Control of vortex shedding in an axisymmetric bluff body wake

In the present experimental study the effect of a control disc mounted at the rear of an axisymmetric blunt-based body of revolution, first studied by Mair, is investigated in the Reynolds number range 3×10³≤Re_D≤5×10⁴ . As the distance of the control disc from the blunt base is increased, four vortex shedding regimes are identified: at small distances there is no effect, then a sharp increase of vortex shedding activity and total drag is observed, followed by an interval with reduced activity and drag and finally at large distances a regime where the flow around the main body and disc become essentially independent, i.e. where the drag forces of the two elements become additive. The near and far wake velocity fields corresponding to the different regimes are documented with time- and phase-averaged hot-wire and LDA measurements, with spectral analysis of the data and with flow visualizations of the near wake. The results are used to develop an improved understanding of the instability mechanism leading to high vortex shedding activity.     

FLOW PERIODICITY AND ACOUSTIC RESONANCE IN PARALLEL TRIANGLE TUBE BUNDLES

Parallel triangle tube arrays with pitch ratios in the range of 1·2–4·2 have been tested at Reynolds numbers up to 9×10⁴to investigate the vorticity shedding and acoustic resonance mechanisms. Three different components of flow periodicities have been observed. The flow periodicity with the highest Strouhal number (S₃) is the weakest component. It is caused by a shear-layer instability in small pitch ratio arrays and at low Reynolds numbers. The S₂component is associated with small-scale vortex shedding at the first row. The third component has the lowest Strouhal number (S₁) and is the strongest. It is generated by large-scale, alternating vortex shedding at deeper rows, and it becomes dominant at all rows at high Reynolds numbers. For tube arrays with pitch ratios less than 3·4, the onset of acoustic resonances could not be related to the natural flow periodicities mentioned above. This behaviour is in contrast with that of normal triangle arrays, but similar to the acoustic behaviour of in-line arrays. Strouhal number charts for the natural flow periodicities and for the onset of acoustic resonances have been developed.     

Three‐dimensional vortex simulation of unsteady flow in hydraulic turbines

On the basis of the Helmholtz decomposition, a grid‐free numerical scheme is provided for the solution of unsteady flow in hydraulic turbines. The Lagrangian vortex method is utilized to evaluate the convection and stretch of the vorticity, and the BEM is used to solve the Neumann problem to define the potential flow. The no‐slip boundary condition is satisfied by generating vortex sticks at the solid surface. A semi‐analytical regularization technique is applied to evaluate the singular boundary surface integrals of the potential velocity and its gradients accurately. The fast multipole method was extended to evaluate the velocity and velocity gradients induced by the discretized vortex blobs in the Lagrangian vortex method. The successful simulation for the unsteady flow through a hydraulic turbine's runner has manifested the effectiveness of the proposed method. Copyright © 2011 John Wiley & Sons, Ltd.     

The effect of a splitter plate on the flow around a finite prism

The effect of a wake-mounted splitter plate on the flow around a surface-mounted finite-height square prism was investigated experimentally in a low-speed wind tunnel. Measurements of the mean drag force and vortex shedding frequency were made at Re=7.4×10⁴ for square prisms of aspect ratios AR=9, 7, 5 and 3. Measurements of the mean wake velocity field were made with a seven-hole pressure probe at Re=3.7×10⁴ for square prisms of AR=9 and 5. The relative thickness of the boundary layer on the ground plane was δ/D=1.5–1.6 (where D is the side length of the prism). The splitter plates were mounted vertically from the ground plane on the wake centreline, with a negligible gap between the leading edge of the plate and rear of the prism. The splitter plate heights were always the same as the heights of prisms, while the splitter plate lengths ranged from L/D=1 to 7. Compared to previously published results for an “infinite” square prism, a splitter plate is less effective at drag reduction, but more effective at vortex shedding suppression, when used with a finite-height square prism. Significant reduction in drag was realized only for short prisms (of AR≤5) when long splitter plates (of L/D≥5) were used. In contrast, a splitter plate of length L/D=3 was sufficient to suppress vortex shedding for all aspect ratios tested. Compared to previous results for finite-height circular cylinders, finite-height square prisms typically need longer splitter plates for vortex shedding suppression. The effect of the splitter plate on the mean wake was to narrow the wake width close to the ground plane, stretch and weaken the streamwise vortex structures, and increase the lateral entrainment of ambient fluid towards the wake centreline. The splitter plate has little effect on the mean downwash. Long splitter plates resulted in the formation of additional streamwise vortex structures in the upper part of the wake.