Incompressible Vortex Research Articles

Reynolds and Mach Number Effects on Propulsive Performance for a Pitching Airfoil

In this work, numerical simulations are conducted to investigate the effects of Reynolds (Re) and Mach (Ma) numbers on thrust characteristics for a pitching airfoil. The results show that as Re increases, the thrust performance is improved. The inverse square root of Reynolds number is demonstrated to be a universal scaling law for both thrust coefficients and drag-to-thrust crossovers across Re=1×103∼1×106 and Ma=0∼0.3. In contrast, as Ma increases above 0.1, the thrust performance is significantly reduced, especially when Ma approaches 0.3. The thrust coefficients follow the square law of the Mach number, which also exhibits universality. Furthermore, the incompressible vortex projection method and the compressible vortex projection method are employed for thrust decomposition. Before this, the full expression of the compressible vortex projection method is derived. It is revealed that as Re increases, the improvement of thrust performance is primarily attributed to the reduced flow viscous effect and, meanwhile, benefits from the formation of more intense boundary layers and vortex structures in near wake. As Ma increases, the enhancement of local flow compressibility ultimately leads to the divergence of added mass drag and thereby reduces the thrust performance.

Circulation production model and unified formation number of compressible vortex rings generated by a shock tube

To investigate the formation number of compressible vortex rings (CVRs), a shock-tube apparatus with an open end is designed to generate CVRs and the flow structures are measured by using particle image velocity (PIV) and time-resolved schlieren techniques. A series of experiments were designed by varying the two governing factors: Mach numbers of the incident shock (Ms= 1.28, 1.48, and 1.59) and driven section length (DL = 100, 200, and 300 mm). By theoretically analyzing the shock diffraction problem, a slipstream model is proposed to predict the circulation generation of CVRs. Comparing with the PIV results, this model well predicts the circulation for Ms=1.28 but slightly underestimates the circulation for Ms= 1.48 and 1.59. Then, an alternative model based on the variation of Ms is proposed and well predicts the circulation generation. Based on the general definition of the vortex formation time and the circulation production model, we newly define the physical formation time of CVRs and then determine the formation number (denoted by F*) when CVRs pinch off. The formation number of CVRs (F*≈3.5) is found to coincide with the optimal vortex formation number originated from incompressible vortex rings (ICVRs). This consistency generalizes the principle of optimal vortex formation into compressible flows. However, both the PIV and schlieren results demonstrate that the CVRs for different Ms pinch off in different modes. With the aim of modulating F* of CVRs, a converging nozzle is designed, and we found that F*≈3.5 is remained for Ms = 1.28 but F* = 5.5 and 6.0 is obtained for Ms= 1.48 and 1.59. Furthermore, an extension of the Kelvin–Benjamin variational principle is explored to explain the unified formation number of CVRs and ICVRs.

Three-dimensional unsteady axisymmetric vortex solutions to the Bellamy-Knights equation and the distribution of boundary conditions

The Bellamy-Knights equation gives three-dimensional unsteady axisymmetric incompressible vortex solutions that are disconnected from the known exact steady solutions. In this paper, by introducing a new parameter that allows a change of the time scale, the Bellamy-Knights equation is transformed into an integro-differential equation that continuously connects with the equation for steady vortices. The new equation is viewed as describing the one-dimensional motion of a particle under conserved and nonconserved forces and leads to new regular solutions, in which the particle initially undergoes almost sinusoidal oscillations around the minimum of potential followed by a very rapid approach to the extrema of the potential. The regular solutions distribute discretely in the space of boundary conditions and are classified into types that are specified by the pattern of sequence of small and large oscillations of the axial component of velocity. Extending the solutions to singular ones, the boundary conditions for solutions of the same type are shown to form a continuous one-dimensional set, which we call a branch. The set of branches forms dense self-similar patterns with accumulation points, some of which coalesce to a single point. In an arbitrary neighborhood of the accumulation or coalescent point, any two distinct boundary conditions whose distance in the space of boundary conditions is arbitrarily small give mutually distinctive solutions with a finite difference at some region of the spatial distance. This feature of the equations signifies the extreme sensitivity of inflowing vortex solutions to boundary conditions.

Inhomogeneous Couette–Poiseuille shear flow

The paper studies a Couette–Poiseuille exact solution for describing inhomogeneous steady-state flows of a viscous incompressible vertical vortex fluid. The fluid moves in an infinite horizontal layer, this motion being conditioned by the displacement of the lower boundary and the specification of the pressure gradient. An exact solution to the hydrodynamics equations describing the three-dimensional inhomogeneous Couette–Poiseuille shear flow is obtained. It is polynomial in all the coordinates, and it belongs to the Lin–Sidorov–Aristov family. Consideration of the inhomogeneity of the velocity field leads to the recording of counterflows. Pressure variation along the horizontal coordinates results in an additional stationary stagnation point as compared to the isobaric flow of a viscous incompressible fluid.

A targeted essentially non-oscillatory (TENO) SPH method and its applications in hydrodynamics

In this paper, we propose a targeted essentially non-oscillatory (TENO) SPH scheme. In this scheme, a 5-point stencil consisting of 3 candidate sub-stencils is adopted to implement the TENO reconstruction as originally proposed in the mesh-based methods. In the original TENO reconstruction, these five points are fixed and equidistant. However, it is difficult for SPH to determine these five points due to the random particle distribution. To remedy this issue, for two interacting particles, we first consider them as two existing stencil points. Then, for the other three points, we search for the closest particles to their exact positions. Subsequently, considering the gradient of these particles, we can finally obtain the values of these target points. Afterwards, the primitive values of points are reconstructed by TENO reconstruction where the polynomial interpolations are modified. Numerical results show that the proposed TENO-SPH scheme can be applied to reproduce some compressible flows involving shocks and small-scale structures, some incompressible vortex flows and free surface flows with superior accuracy.

Numerical Method for Determining Three-Dimensional Response of a Wing to Incident Turbulence

A method is developed to numerically determine the response of three-dimensional wings to turbulent upwash. The method uses a generalized spatial-domain gust response function in the place of the Sears function that differentiates the strip experiencing the gust and the strip producing force. This generalized gust response function is computed with an incompressible unsteady vortex lattice method. The vortex lattice operates in the time-domain to simulate the development and relaxation of the wake to steady state and then switches to a frequency-domain approach to evaluate the gust response. The total wing response to homogenous isotropic turbulent inflows is numerically integrated to produce stochastic turbulent lift spectra. The numerically generated transfer functions are compared to the Sears and Filotas gust response functions. The effects of finite span and sweep on both the transfer functions and lift spectra are investigated.

On the algebra and groups of incompressible vortex dynamics

An algebraic representation for 2D and 3D incompressible, inviscid fluid motion based on the continuous Nambu representation of Helmholtz vorticity equation is introduced. The Nambu brackets of conserved quantities generate a Lie algebra. Physically, we introduce matrix representations for the components of the linear momentum (2D and 3D), the circulation (2D) and the total flux of vorticity (3D). These quantities form the basis of the vortex-Heisenberg Lie algebra. Applying the matrix commutator to the basis matrices leads to the same physical relations as the Nambu bracket for this quantities expressed classically as functionals. Using the matrix representation of the Lie algebra we derive the matrix and vector representations for the nilpotent vortex-Heisenberg groups that we denote by VH(2) and VH(3). It turns out that VH(2) is a covering group of the classical Heisenberg group for mass point dynamics. VH(3) can be seen as central extension of the abelian group of translations. We further introduce the Helmholtz vortex group V(3), where additionally the angular momentum is included. Regarding application-oriented aspects, the novel matrix representation might be useful for numerical investigations of the group, whereas the vector representation of the group might provide a better process-related understanding of vortex flows.

Non-equilibrium behavior of large-scale axial vortex cores

A logical basis for incorporating pressure non-equilibrium and turbulent eddy viscosity in an incompressible vortex model is presented. The infrasonic acoustic source implied in our earlier work has been examined. Finally, this non-equilibrium turbulent vortex core is shown to dissipate mechanical energy more slowly than a Burgers vortex, helping us to explain the persistence of axial vortices in nature. Recent molecular dynamics simulations replicate aspects of this non-equilibrium pressure behavior.

Matching theory to characterize sound emission during vortex reconnection in quantum fluids

In a concurrent work, Villois et al. 2020 reported the evidence that vortex reconnections in quantum fluids follow an irreversible dynamics, namely vortices separate faster than they approach; such time-asymmetry is explained by using simple conservation arguments. In this work we develop further these theoretical considerations and provide a detailed study of the vortex reconnection process for all the possible geometrical configurations of the order parameter (superfluid) wave function. By matching the theoretical description of incompressible vortex filaments and the linear theory describing locally vortex reconnections, we determine quantitatively the linear momentum and energy exchanges between the incompressible (vortices) and the compressible (density waves) degrees of freedom of the superfluid. We show theoretically and corroborate numerically, why a unidirectional density pulse must be generated after the reconnection process and why only certain reconnecting angles, related to the rates of approach and separations, are allowed. Finally, some aspects concerning the conservation of centre-line helicity during the reconnection process are discussed.

Evaluation of a nonlinear variational multiscale method for fluid transport problems

Diverse transport problems, especially those based on fluid flow models, are intrinsically multiscale and nonlinear, characteristics that often lead to intricate dynamics such as the development of instabilities and turbulence. Computational simulations that resolve all scales in these problems are often unfeasible, prompting to coarse-grained simulation strategies in which small-scale features are modeled instead of resolved. Variational Multiscale (VMS) methods, and particularly residual-based Large-Eddy Simulation (LES) approaches, have proven effective and robust for the coarse-grained simulation of complex transport problems. VMS methods avoid the assumption of separable nonlinearity and the reliance on empirical small-scale models by using a variational decomposition of scales together with a residual-based approximation of the small-scales. Evaluation of a nonlinear VMS approach, denoted as VMSn, is presented for the coarse-grained simulation of transient-advective-diffusive-reactive (TADR) transport problems arising from fluid flow models. In contrast to classical VMS approaches that neglect the effect of the small scales on the transport operator, VMSn treats the inter-dependence between large- and small-scales upfront. The treatment of inter-scale coupling involves the solution of a local algebraic nonlinear system describing the evolution of the small-scales. The VMSn approach is complemented with two algebraic approximations of the small-scales: one based on the main diagonal of the transport matrices and another that preserves transport fluxes and is suitable for generic TADR systems. The suitability of the VMSn approach for handling general TADR problems and regimes is evaluated with benchmark incompressible, compressible, and magnetohydrodynamic laminar flow problems, the incompressible Taylor-Green vortex flow, the turbulent free jet, and the two-temperature arc in crossflow. Simulation results show that VMSn leads to minor improvements in accuracy with respect to the classical VMS for the laminar flow problems, but to significantly greater accuracy for the turbulent flows and the unsteady plasma flow problems, while using the same cohesive numerical formulation.

Aeroelastic Reduced Order Model: Kriging-Corrected Potential Flow

A novel aeroelastic reduced order model is developed for two-dimensional gust interaction modeling in transonic flows. Such conditions exhibit a key nonlinearity: sonic shock motion. The proposed approach comprises an aerodynamics reduced order model coupled to a structural model. The aerodynamics model consists of corrected integrated force coefficients computed by an unsteady potential flow method (incompressible vortex panels). The corrections are obtained from kriging surrogates trained using inviscid compressible computational fluid dynamics data. Two gust lengths (namely, long and short) are simulated for this purpose and serve as the upper and lower bounds of the model. The developed aeroelastic reduced order model is evaluated via modeling untried gust cases, the lengths of which fall between the training cases. Applied to the FFAST Crank and NACA0012 airfoils, the approach accurately reproduces aeroelastic response—with deviations never exceeding 8.0 and 10.5%, respectively. In terms of computational cost, the trained models were more than an order of magnitude faster than the computational fluid dynamics simulations.

Speed of a von Kármán point vortex street in a weakly compressible fluid

The modification due to weak fluid compressibility of the speed of travel of the incompressible point vortex streets of von Karman is determined in analytical form as a function of the street aspect ratio.

Vortex states in a non-Abelian magnetic field

A type-II superconductor survives in an external magnetic field by admitting an Abrikosov lattice of quantized vortices. This is an imprint of the Aharonov-Bohm effect created by the Abelian U(1) gauge field. The simplest non-Abelian analogue of such a gauge field, which belongs to the SU(2) symmetry group, can be found in topological insulators. Here we discover a superconducting ground state with a lattice of SU(2) vortices in a simple two-dimensional model that presents an SU(2) "magnetic" field (invariant under time-reversal) to attractively interacting fermions. The model directly captures the correlated topological insulator quantum well, and approximates one channel for instabilities on the Kondo topological insulator surface. Due to its simplicity, the model might become amenable to cold atom simulations in the foreseeable future. The vitality of low-energy vortex states born out of SU(2) "magnetic" fields is promising for the creation of incompressible vortex liquids with non-Abelian fractional excitations.

Coupled adjoint aerostructural wing optimization using quasi-three-dimensional aerodynamic analysis

This paper presents a method for wing aerostructural analysis and optimization, which needs much lower computational costs, while computes the wing drag and structural deformation with a level of accuracy comparable to the higher fidelity CFD and FEM tools. A quasi-three-dimensional aerodynamic solver is developed and connected to a finite beam element model for wing aerostructural optimization. In a quasi-three-dimensional approach an inviscid incompressible vortex lattice method is coupled with a viscous compressible airfoil analysis code for drag prediction of a three dimensional wing. The accuracy of the proposed method for wing drag prediction is validated by comparing its results with the results of a higher fidelity CFD analysis. The wing structural deformation as well as the stress distribution in the wingbox structure is computed using a finite beam element model. The Newton method is used to solve the coupled system. The sensitivities of the outputs, for example the wing drag, with respect to the inputs, for example the wing geometry, is computed by a combined use of the coupled adjoint method, automatic differentiation and the chain rule of differentiation. A gradient based optimization is performed using the proposed tool for minimizing the fuel weight of an A320 class aircraft. The optimization resulted in more than 10 % reduction in the aircraft fuel weight by optimizing the wing planform and airfoils shape as well as the wing internal structure.

The Effect of Streamtube Contraction on the Characteristics of a Streamwise Vortex

The ingestion of a vortex by an aero-engine is potentially an area of concern for current and future aircraft-engine configurations. However, there are very little experimental data on the characteristics of a streamwise vortex undergoing ingestion through a contracting streamtube. To address this dearth of knowledge, the ingestion of a streamwise vortex has been studied experimentally using stereoscopic particle image velocimetry (stereo PIV). A subscale model of an aircraft intake has been used to generate a contracting capture streamtube, and an isolated streamwise vortex has been generated upstream of the intake using semispan NACA 0012 and delta wing vortex generators (VGs). A range of contraction ratios, vortex Reynolds numbers, and vortex initial conditions have been examined. Measurements on planes perpendicular to the freestream flow show that the vortex undergoes notable levels of intensification through the contraction streamtube. The characteristics of the vortex are dependent on the streamtube contraction level, the initial aerodynamic characteristics of the vortex, and the trajectory that the vortex follows inside the capture streamtube. Results from inviscid, incompressible vortex filament theory have been compared with the experimental data. At relatively low streamtube contraction ratios this theory provides a good estimate of the vortex characteristics. However, at higher contraction levels, there are notable levels of diffusion, which render the vortex less intense than that anticipated from vortex filament theory.

The topology of Liouville foliation for the Borisov-Mamaev-Sokolov integrable case on the Lie algebra so(4)

In 2001, A.V. Borisov, I. S.Mamaev, and V.V. Sokolov discovered a new integrable case on the Lie algebra so(4). This system coincides with the Poincare equations on the Lie algebra so(4), which describe the motion of a body with cavities filled with an incompressible vortex fluid. Moreover, the Poincare equations describe the motion of a four-dimensional gyroscope. In this paper topological properties of this system are studied. In particular, for the system under consideration the bifurcation diagrams of the momentum mapping are constructed and all Fomenko invariants are calculated. Thereby, a classification of isoenergy surfaces for this system up to the rough Liouville equivalence is obtained.

A dynamic counterpart of Lamb vector in viscous compressible aerodynamics

The Lamb vector is known to play a key role in incompressible fluid dynamics and vortex dynamics. In particular, in low-speed steady aerodynamics it is solely responsible for the total force acting on a moving body, known as the vortex force, with the classic two-dimensional (exact) Kutta–Joukowski theorem and three-dimensional (linearized) lifting-line theory as the most famous special applications. In this paper we identify an innovative dynamic counterpart of the Lamb vector in viscous compressible aerodynamics, which we call the compressible Lamb vector. Mathematically, we present a theorem on the dynamic far-field decay law of the vorticity and dilatation fields, and thereby prove that the generalized Lamb vector enjoys exactly the same integral properties as the Lamb vector does in incompressible flow, and hence the vortex-force theory can be generalized to compressible flow with exactly the same general formulation. Moreover, for steady flow of polytropic gas, we show that physically the force exerted on a moving body by the gas consists of a transverse force produced by the original Lamb vector and a new longitudinal force that reflects the effects of compression and irreversible thermodynamics.

Pairing instabilities in topological insulator quantum wells

Topological insulator quantum wells with induced attractive interactions between electrons are candidate systems for the realization of novel vortex lattice states with time-reversal symmetry, and incompressible quantum vortex liquids with fractional excitations. We analyze the competition between different pairing channels stimulated by the superconducting proximity effect in these quantum wells, and calculate the helical triplet pairing instability that can produce the mentioned phases using perturbation theory. We discuss the phase diagram tunable by gate voltage.

Spatiotemporal instability of a variable-density Batchelor vortex

AbstractLinear and nonlinear impulse responses are computed, using three-dimensional numerical simulations, for an incompressible and variable density (inhomogeneous) Batchelor vortex at a moderately high Reynolds number, $\mathit{Re}= 667$. In the linear framework, the computed wavepacket is decomposed into azimuthal modes whose growth rates are determined along each spatiotemporal ray, in the laboratory frame. It is found that the Batchelor vortex undergoes a convective/absolute transition when the density ratio (inner/ambient), $s$, is varied solely (there is no need for an external counter flow to trigger this transition like that needed in the constant density case). More precisely, it is shown that the transition occurs for heavy vortices when the density ratio reaches a critical value, ${s}_{c} \simeq 1. 08$. For light vortices ($s\lt 1$) no transition was found. It is also shown that the first azimuthal mode that transits have an azimuthal wavenumber $m= \ensuremath{-} 2$ and the transition occurs for a swirl number (a measure of the azimuthal to axial velocity ratio), $q= 0. 57$. It is followed by $m= \ensuremath{-} 1$, then by $m= \ensuremath{-} 3$. When nonlinearities are allowed, it is found that they saturate the amplitude within the linear-response wavepacket, leaving the wavepacket fronts unaffected. The conclusions should thus be the same as those obtained in the linear case: the linear convective/absolute transition should coincide with the nonlinear one for the variable-density Batchelor vortex.

Aeroacoustics of the swinging corrugated tube: Voice of the Dragon

When one swings a short corrugated pipe segment around one's head, it produces a musically interesting whistling sound. As a musical toy it is called a "Hummer" and as a musical instrument, the "Voice of the Dragon." The fluid dynamics aspects of the instrument are addressed, corresponding to the sound generation mechanism. Velocity profile measurements reveal that the turbulent velocity profile developed in a corrugated pipe differs notably from the one of a smooth pipe. This velocity profile appears to have a crucial effect both on the non-dimensional whistling frequency (Strouhal number) and on the amplitude of the pressure fluctuations. Using a numerical model based on incompressible flow simulations and vortex sound theory, excellent predictions of the whistling Strouhal numbers are achieved. The model does not provide an accurate prediction of the amplitude. In the second part of the paper the sound radiation from a Hummer is discussed. The acoustic measurements obtained in a semi-anechoic chamber are compared with a theoretical radiation model. Globally the instrument behaves as a rotating (Leslie) horn. The effects of Doppler shift, wall reflections, bending of the tube, non-constant rotational speed on the observed frequency, and amplitude are discussed.