Vortex Sheet Research Articles

On the mother bodies of steady polygonal uniform vortices. Part I: numerical experiments

ABSTRACT The existence of an integral relation between self-induced velocity of a uniform, planar vortex and Schwarz function of its boundary opens the way to understand the kinematics of the vortex by analysing the internal singularities of that function. In general, they are branch cuts and form the so-called “mother body” of the vortex, because they generate the same external velocities of the vortex, by means of a relation identical to the Biot–Savart law for a vortex sheet. The jump of the Schwarz function across the cuts plays the role of the (complex) density of circulation. This paper investigates the singularities of polygonal vortices, which are highly nontrivial steady vortices widely present in Nature, and having fascinating properties, some of them still not well understood. By means of the equation of the dynamics of the Schwarz function specialised for steady vortices, a numerical tool based on elementary properties of the holomorphic functions is used for detecting the internal singularities and evaluating their strengths. In this way, it is shown that an nagonal vortex possesses n internal branch cuts. In a reference system having origin on the centre of vorticity of the vortex and real axis crossing one of its vertices, these cuts start from the origin and are directed along the n roots of the unity, so that they are aligned with the vertices. The positions of the branch points and the values assumed by the Schwarz function in these points are calculated by evaluating this function just outside the vortex boundary. Once the conditions on the branch points are defined, a power series representation of the Schwarz function is proposed, that is able to explain the behaviour of its real and imaginary parts in neighbourhoods of these points. Some conjectures about the external singularities are also discussed.

Roughness Effect on the Flow Past Axisymmetric Bodies at High Incidence

The flow at low Mach numbers and high angles of attack over axisymmetric configurations is not symmetric. The mechanism that triggers the asymmetry is a combination of a global (temporal) instability and a convective (spatial) instability. This latter instability is caused by roughness and other geometrical imperfections, which lead to roll angle dependent forces. The flow at these conditions has a complex vortex sheet structure, with two or three different flow regions. An accurate simulation by means of Computational Flow Dynamics (CFD) is thus very challenging, and many researchers have therefore employed Large Eddy Simulation (LES) codes. This study demonstrates that Unsteady Reynolds Averaged Navier-Stokes (URANS) methods are a suitable alternative, if Scale Adaptive Simulation (SAS) is used. This method is capable of capturing the main flow features, provided that fine meshes, which achieve geometrical similarity between the meshed geometry and the real object, and small-time steps are used. It is also demonstrated that, by using URANS methods in combination with SAS, strong differences in the global and local forces depending on the surface roughness of the model are obtained, a result which coincides with several wind tunnel tests.

Dipole- and vortex sheet-based models of fish swimming

Elucidating the hydrodynamics of fish swimming is critical to identifying the processes underlying fish orientation and schooling. Due to their mathematical tractability, models based on potential flow are preferred in the study of bidirectional interactions of fish with their surroundings. Dipole-based models that assimilate fish to pairs of vortices are particularly enticing, but yet to be thoroughly validated. Here, we embark on a computational fluid dynamics (CFD) campaign informed by experimental data to validate the accuracy of dipole-based models. The locomotory patterns of a fish undergoing carangiform swimming are reconstructed from existing experimental data, which are used as inputs to CFD simulations of a fish swimming in a channel flow. We demonstrate that dipole-based models are accurate in capturing key features of the fluid flow, but cannot predict the elongated flow streamlines around the fish that are evident in CFD. To address this issue, we propose an alternative model that replaces each vortex in the pair with a sheet along the fish length. Using a pair of vortex sheets that span approximately 80% of the fish body length with a separation distance of approximately 50% of the body width, the model is successful in predicting the fluid flow around the swimming fish for a range of background flow speeds and channel widths. The proposed model shows improved accuracy at the cost of a mildly increased computational effort, thereby constituting an ideal basis for research on fish hydrodynamics.

Vortex dynamics impact on the wake flow of a marine rudder with leading-edge tubercles

The impact of two tubercle leading-edge (TLE) modifications on the turbulent wake of a reference marine rudder at Reynolds number 2.26 × 106 was analysed numerically using Detached Eddy Simulations (DES). This paper studies the counter-rotating vortex pair formation around the TLE and their impact on the wake structures behind the rudder to find out if the vortex interaction can accelerate the tip vortex dissipation. According to the results, the tubercles enhanced lift for angles of attack (AOA) 10º and above, but at the cost of a drag penalty which reduced the rudders’ lift-to-drag ratio. The formation of the distinctive stream-wise counter-rotating vortex pairs occurred behind the tubercles, which then interacted with the dominant tip vortex. Due to the inherent spanwise flow component of finite-span lifting surfaces the counter-rotating vortex pairs were generated at unequal strength and soon merged into singular vortices co-rotating with the tip vortex. The vortices facilitated flow compartmentalisation over the rudder suction side which broke up the trailing-edge vortex sheet and confined the spanwise flow separation over the rudder surface as AOA increased. The tubercles confined flow separation closer to the rudder tip which reduced the lift generation in the tip area and minimised the initial tip vortex strength. Large elements of stream-wise counter-rotating vorticity formed around the localised stall cells of the TLE rudders that interacted with the tip vortex downstream, introducing elliptical instabilities further weakening the tip vortex and changing its trajectory.

Numerical Investigation of Secondary Vortex Structures in a Rotor Wake

In hovering rotor computational fluid dynamics simulations, a common phenomenon is the breakdown of the primary vortex system due to secondary vortices, which are characterized by S-shaped vortical structures appearing between the primary blade-tip vortices. Since the presence of the secondary vortices is strongly influenced by the numerical settings, the degree to which secondary vortices physically occur warrants investigation. Therefore, this study investigates how the numerical settings affect the development of secondary vortices in hover simulations using the rotating blade geometry from previous experiments, where the focus is on temporal convergence. The computed near-wake vortical flow field is qualitatively compared to the experimental results. The calculations are then used to elucidate the secondary vortex development process, where it is shown that the secondary vortices develop by the interaction of the vortex sheet with the primary vortex structure. A number of parametric studies are done on the rotating blade geometry to investigate how the numerical settings affect the secondary vortex development process. Although the focus is on understanding the impact of temporal convergence, the effect of the off-body grid structure, grid rotation, mesh resolution, numerical dissipation, and time-step scaling is also studied. In contrast to previous work, the resulting wake breakdown from using the different numerical settings is compared both qualitatively and quantitatively. For relatively high flow solver subiteration convergence, the off-body grid structure and grid rotation have little effect on the development of the secondary vortex structures. However, at low solver subiteration convergence, larger differences occur between the torus and no torus grid structures. A parametric sweep on subiterations shows that as the subiteration residual drop increases the vortex sheet is broken down less and, correspondingly, both the rotor performance data and peak number of secondary vortices converge. The analysis shows that when the subiteration convergence or grid resolution is not high enough, the vortex sheet and primary vortex are broken down, which disrupt the development process and leads to fewer secondary vortices. To better understand the rotating blade case, a separate vortex ring study is done to observe how vortex breakdown is influenced by off-body grid structure and different levels of subiteration convergence. In agreement with the rotating blade case, when all flow solvers are sufficiently converged, vortex breakdown is minimized and the solution to the rotating and stationary torus is identical.

A time domain boundary element method for acoustic scattering by lined surfaces in a subsonic uniform mean flow

A time domain boundary integral equation with Burton-Miller reformulation is presented for acoustic scattering by surfaces with liners in a uniform mean flow. The Ingard-Myers impedance boundary condition is implemented using a broadband multipole impedance model which is in turn converted into time domain differential equations to augment the boundary integral equation. The coupled integral-differential equations are solved numerically by a March-On-in-Time (MOT) scheme. While the Ingard-Myers condition is known to support Kelvin-Helmholtz instability due to its use of a vortex sheet interface between the flow and the lined surface, it is found that by neglecting a second-order derivative term in the current time domain impedance boundary condition formulation, the instability can be effectively avoided in computation. Proposed formulation and implementation are validated with numerical examples. Moreover, a minimization procedure for finding the poles and coefficients of the broadband multiple impedance expansion is formulated by which, unlike the commonly used vector-fitting method, passivity of the model is ensured. Numerical tests show the proposed minimization approach is effective for modeling liners that are commonly used in aeroacoustic applications.

Influence factors and prediction model of enstrophy dissipation from the tip leakage vortex in a multiphase pump.

In a multiphase pump, tip clearance is the required distance between the blade tip and the pump body wall of the impeller, forming tip leakage vortex (TLV), causing unstable flow and energy dissipation. In the present work, the enstrophy dissipation theory is innovatively applied to quantitatively study the energy dissipation of the TLV. The flow rate, tip clearance, and inlet gas void fraction (IGVF) play a crucial role in affecting the enstrophy dissipation of the TLV. The results show that increasing flow rate, tip clearance, and IGVF significantly exacerbate the TLV pattern and raise the TLV scale, which gradually raises volume enstrophy dissipation and decreases wall enstrophy dissipation. As the flow rate increases, the separation angle between the primary TLV trajectory and the blade gradually decreases, and widely dispersing the enstrophy dissipation near the shroud. However, as the tip clearance increases, the tip separated vortex scale increases and extends to the suction surface, raising the velocity gradient. Besides, as the IGVF increases, the secondary TLV develops from a continuous sheet vortex to a scattered strip vortex, increasing the significantly increasing the enstrophy dissipation. Considering the flow rate, tip clearance, and IGVF as independent variables, simple and multiple nonlinear regression models have the ability to predict the enstrophy dissipation of the TLV accurately.

Acoustic energy balances for sound radiated from duct exit with mean flow

An old model problem for the exchange of energy between sound field and mean flow by vortex shedding has been worked out in numerical detail. The analytically exact solution of the problem of reflection, diffraction and radiation of acoustic modes in a semi-infinite annular duct with uniform subsonic mean flow, including shedding of unsteady vorticity from the duct exit, allows a precise formulation of Myers’ energy for perturbations of an inviscid mean flow. The transmitted power [Formula: see text] in the duct and the radiated power [Formula: see text] in the far field differ by the amounts of hydrodynamic far field powers [Formula: see text] inside and [Formula: see text] outside the wake (vortex sheet) emanating from the duct edge, plus the power [Formula: see text] that disappears into the vortex sheet. This last component represents the source term in Myers’ energy equation. This is evidence of the non-conserved character of acoustic energy in mean flow, owing to the coupling of the acoustic field with the mean flow. [Formula: see text], [Formula: see text] and [Formula: see text] are always positive. This is normally the case too for [Formula: see text] and [Formula: see text]. But for not too high frequencies or other circumstances where shed vorticity produces more sound than was necessary for its creation, [Formula: see text] and even [Formula: see text] may also be negative.

Vortex Sheet Sensitivity to Low-Level Vertical Shear and Airmass Temperature Perturbation

Abstract A theoretical, numerical-modeling-based examination of the sensitivity of vortex sheets along airmass boundaries to the following three characteristics is presented: 1) boundary-normal component of the vertical wind shear, 2) boundary-parallel component of the vertical wind shear, and 3) temperature perturbation within the parent air mass of the boundary. The overall aim of this work is to advance understanding of the sensitivity of micro-α- to meso-γ-scale vortex generation along airmass boundaries to the ambient environment. Density currents are simulated in a 2D domain that does not allow baroclinic generation of near-surface vertical vorticity (ζns) with parameterized latent heating for convection initiated at the associated airmass boundary and Coriolis turned on. Despite the absence of baroclinically generated ζns, with Coriolis turned on and without any boundary-parallel shear, ζns more than two orders of magnitude larger than planetary vorticity is generated along the boundary and located within the cold air. The magnitude of ζns is found to increase with increasing boundary-normal shear with statistically significant intra-experiment separations. Near-surface vertical vorticity ζns is found to scale inversely with boundary-parallel shear with a transition to negative leading-edge ζns in several of the larger boundary-normal shear simulations. An inverse and statistically significant relationship is found between ζns and the temperature perturbation within the parent air mass of the boundary (Δθ), and is a direct consequence of the dependence of boundary propagation speed on Δθ. Significance Statement Research presented in this article aims to contribute to an improved understanding of the environmental controls on the generation of small-scale vortices along airmass boundaries. Vertical shear, both along and across an airmass boundary, as well as temperature of the air mass on the cool side of an airmass boundary are found to regulate the magnitude of near-surface vertical vorticity available to small-scale vortices.

On coherent vortical structures in wave breaking

The flow generated by the breaking of free-surface waves in a periodic domain is simulated numerically with a gas–liquid Navier–Stokes solver. The solver relies on the volume-of-fluid method to account for different phases, and the interface tracking is carried out by using novel schemes based on a tailored total-variation-diminishing limiter. The numerical solver is proved to be characterized by a low numerical dissipation, thanks to the use of a scheme that guarantees energy conservation in the discrete form. Both two- and three-dimensional simulations have been performed, and the analysis is presented in terms of energy dissipation, air entrainment, bubble fragmentation, statistics and distribution. Particular attention is paid to the analysis of the mechanisms of viscous dissipation. To this purpose, coherent vortical structures, such as vortex tubes and vortex sheets, are identified, and the different behaviours of the vortex sheets and tubes at various Reynolds numbers are highlighted. The correlation between vortical structures and energy dissipation demonstrates clearly their close link both in the mixing zone and in the pure water domain, where the coherent structures propagate as a consequence of the downward transport. Notably, it is found that the dissipation is identified primarily by the vortex sheets, whereas the vortex tubes govern mainly the intermittency.

Numerical algorithms for water waves with background flow over obstacles and topography

We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiple-valued velocity potentials but single-valued stream functions. We prove that the resulting second-kind Fredholm integral equations are invertible, possibly after a physically motivated finite-rank correction. In an angle-arclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravity-capillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually self-intersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid, including near the free surface and solid boundaries. To assess the accuracy of the time evolution, we monitor energy conservation and the decay of Fourier modes and compare the numerical results of the two methods to each other. We implement several solvers for the discretized linear systems and compare their performance. The fastest approach employs a graphics processing unit (GPU) to construct the matrices and carry out iterations of the generalized minimal residual method (GMRES).

Vortex ring connection to a free surface

We perform numerical simulations of the interaction between a vortex ring and a free surface, and provide a new interpretation of the mechanism by which the vortex ring connects to the free surface. Large vorticity gradients at the free surface result in the diffusion of surface-tangential vorticity out of the fluid. This is accompanied by the diffusion of opposite-signed surface-normal vorticity away from the connection line, along the free surface, which results in the attachment of the vortex ring to the free surface. Compared to existing descriptions, this interpretation explains better how the solenoidal property that vortex lines do not end in the fluid is maintained. By including an interface vortex sheet at the free surface, the important property of vorticity conservation is maintained throughout the interaction. The upper part of the vortex ring simply diffuses out of the fluid, and into the interface vortex sheet, with the total circulation remaining constant.

Computation of the Added Masses of Bodies by Using T-Schemes for Numerical Reconstruction of the Vortex Sheet Intensity

Computation of the Added Masses of Bodies by Using T-Schemes for Numerical Reconstruction of the Vortex Sheet Intensity

Rotor-stator aerodynamic interaction of centrifugal compressor at pulsating backpressure conditions

The performance of a centrifugal compressor is strongly influenced by pulsating backpressure condition, but little knowledge of the mechanism has been obtained. This study investigates aerodynamic interaction between impeller and volute at pulsating condition via experimentally validated numerical method. Results show that the features of volute-induce distortion at impeller exit are closely coupled with pulsating backpressure at different operational conditions. Specifically, the volute-induce pressure distortion varies out-of-phase with the mass flow rate near surge condition (OP2), but in phase near choke condition (OP1). The circumferential flow distortion at the impeller outlet manifests a notably unsteady effect at pulsating conditions for both OP1 and OP2. Comparing with the quasi-steady cases, the magnitude of the distortion is alleviated maximally by 2.3% and 2.7% for OP1 and OP2, respectively. The evolution of secondary flow patterns in the impeller show that an array of vortex tubes appears in the shroud region of the inducer downstream the main blade leading edge. It is resulted from the rolling-up of vortex sheet due to the K-H instability. The pulsating backpressure can profoundly depress the development of the vortex street. Therefore, the blockage effect by the vortex street in the inducer is alleviated.

Numeric evaluation from cantilever beam and plate

As soon as fluid passes over a solid surface vortices are formed at the surface of structure to satisfy the no slip boundary conditions at the fluid-structure interface. Vortices that are formed at fluid-structure interface diffuse in fluid domain. In the present study, the strength of vortices is evaluated on the surface of an oscillating structural element (cantilever beam and plate) that is interacting with oscillating fluid flow. In the analysis, the analysis of structural element is carried out by finite element method and analysis of fluid domain is carried out by panel method. It is assumed that the deformed geometry of the structure at any moment is same shape as the vortex sheet. The structure is replaced by a virtual vortex sheet of uniformly distributed point vortices on the surface. A few numerical examples are presented to show the variation of vortex strength, lift and pressure coefficients for different type flow passes over the oscillating rigid and flexible structural element in fluid medium. The strength of vortex depends on fluid flow characteristics around the structural element and motion of structure in fluid medium.

On Bounded Two-Dimensional Globally Dissipative Euler Flows

We examine the two-dimensional Euler equations including the local energy (in)equality as a differential inclusion and show that the associated relaxation essentially reduces to the known relaxation for the Euler equations considered without local energy (im)balance. Concerning bounded solutions we provide a sufficient criterion for a globally dissipative subsolution to induce infinitely many globally dissipative solutions having the same initial data, pressure, and dissipation measure as the subsolution. The criterion can easily be verified in the case of a flat vortex sheet giving rise to the Kelvin--Helmholtz instability. As another application we show that there exists initial data for which associated globally dissipative solutions realize every dissipation measure from an open set in ${\mathcal C}^0(\mathbb{T}^2\times[0,T])$. In fact the set of such initial data is dense in the space of solenoidal $L^2(\mathbb{T}^2;\mathbb{R}^2)$ vector fields.

Remarks on stationary and uniformly rotating vortex sheets: flexibility results.

In this paper, we construct new, uniformly rotating solutions of the vortex sheet equation bifurcating from circles with constant vorticity amplitude. The proof is accomplished via a Lyapunov-Schmidt reduction and a second-order expansion of the reduced system. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.

Existence of stationary vortex sheets for the 2D incompressible Euler equation

AbstractWe construct a new type of planar Euler flows with localized vorticity. Let $\kappa _i\not =0$ , $i=1,\ldots , m$ , be m arbitrarily fixed constants. For any given nondegenerate critical point $\mathbf {x}_0=(x_{0,1},\ldots ,x_{0,m})$ of the Kirchhoff–Routh function defined on $\Omega ^m$ corresponding to $(\kappa _1,\ldots , \kappa _m)$ , we construct a family of stationary planar flows with vortex sheets that have large vorticity amplitude and concentrate on curves perturbed from small circles centered near $x_{0,i}$ , $i=1,\ldots ,m$ . The proof is accomplished via the implicit function theorem with suitable choice of function spaces.

Modal analysis of non-ducted and ducted propeller wake under axis flow

Modal decomposition techniques, flow field, and spectral analysis are employed to investigate the wake dynamics and destabilization mechanisms of a four-bladed marine propeller with or without a nozzle. Numerical simulations are conducted using the delayed detached eddy simulation model for the wake and the arbitrary mesh interface method for the blade rotation. The presence of the nozzle significantly reduces the wake's streamwise velocity, delays the wake destabilization, increases the wake length, and changes the morphologies of wake vortices. In particular, the hub vortex in the ducted propeller wake is broken down into chaotic turbulence by the perturbation of the backflow. Two modal decomposition methods, namely, proper orthogonal decomposition and dynamic mode decomposition, are used to decompose the vorticity magnitude in the rotor wake field. From modal analysis, the spatial scale of flow phenomena decreases with the increase in modal frequency. Underlying destabilization mechanisms in the wake correspond to some characteristic frequencies. The interaction of each sheet vortex with the previously shed tip (leakage) vortices occurs at blade passing frequency (BPF). The pairing of adjacent tip (leakage) vortices occurs at half-BPF. The long-wave instability of the hub vortex and the wake meandering are stochastic processes, each of which occurs at a frequency lower or equal to shaft frequency. These four destabilization mechanisms can approximately reconstruct the large-scale flow phenomena in the wake. Moreover, each sheet vortex's alternating connection and disconnection with the previously shed tip (leakage) vortices cause the short-wave instability of the tip (leakage) vortices and generate the secondary vortices. The radial expansion motion of large-scale helical vortices in the outer slipstream dominates the wake meandering phenomenon.

Planar potential flow on Cartesian grids

Potential flow has many applications, including the modelling of unsteady flows in aerodynamics. For these models to work efficiently, it is best to avoid Biot–Savart interactions. This work presents a grid-based treatment of potential flows in two dimensions and its use in a vortex model for simulating unsteady aerodynamic flows. For flows consisting of vortex elements, the treatment follows the vortex-in-cell approach and solves the streamfunction–vorticity Poisson equation on a Cartesian grid after transferring the circulation from the vortices onto the grid. For sources and sinks, an analogous approach can be followed using the scalar potential. The combined velocity field due to vortices, sinks and sources can then be obtained using the Helmholtz decomposition. In this work, we use several key tools that ensure the approach works on arbitrary geometries, with and without sharp edges. First, the immersed boundary projection method is used to account for bodies in the flow and the resulting body-forcing Lagrange multiplier is identified as the bound vortex sheet strength. Second, sharp edges are treated by decomposing the vortex sheet strength into a singular and non-singular part. To enforce the Kutta condition, the non-singular part can then be constrained to remove the singularity introduced by the sharp edge. These constraints and the Poisson equation are formulated as a saddle-point system and solved using the Schur complement method. The lattice Green's function is used to efficiently solve the discrete Poisson equation with unbounded boundary conditions. The method and its accuracy are demonstrated for several problems.