R. Fernandez-Feria

Structure of trailing vortices: Comparison between particle image velocimetry measurements and theoretical models

The velocity field of the trailing vortex behind a wing at different angles of attack has been measured through the stereo particle image velocimetry technique in a water tunnel for Reynolds numbers between 20 000 and 40 000, and for several distances to the wing tip. After filtering out the vortex meandering, the radial profiles of the axial and the azimuthal velocity components and of the radial profiles of the vorticity were compared to the theoretical models for trailing vortices by [G. K. Batchelor, J. Fluid Mech. 20, 645 (1964)] and by [D. W. Moore and P. G. Saffman, Proc. R. Soc. London, Ser. A 333, 491 (1973)], whose main features are conveniently summarized. We take into account the downstream evolution of these profiles from just a fraction of the wing chord to more than ten chords. The radial profiles of the vorticity and the azimuthal velocity are shown to fit quite well to Moore and Saffman’s trailing vortex model, while Batchelor’s model does not fit so well, especially in the tails of the p...

The onset of absolute instability of rotating Hagen–Poiseuille flow: A spatial stability analysis

A spatial, viscous stability analysis of Poiseuille pipe flow with superimposed solid body rotation is considered. For each value of the swirl parameter (inverse Rossby number) L>0, there exists a critical Reynolds number Rec(L) above which the flow first becomes convectively unstable to nonaxisymmetric disturbances with azimuthal wave number n=−1. This neutral stability curve confirms previous temporal stability analyses. From this spatial stability analysis, we propose here a relatively simple procedure to look for the onset of absolute instability that satisfies the so-called Briggs–Bers criterion. We find that, for perturbations with n=−1, the flow first becomes absolutely unstable above another critical Reynolds number Ret(L)>Rec(L), provided that L>0.38, with Ret→Rec as L→∞. Other values of the azimuthal wave number n are also considered. For Re>Ret(L), the disturbances grow both upstream and downstream of the source, and the spatial stability analysis becomes inappropriate. However, for Re

Solution breakdown in a family of self-similar nearly inviscid axisymmetric vortices

Many axisymmetric vortex cores are found to have an external azimuthal velocity v, which diverges with a negative power of the distance r to their axis of symmetry. This singularity can be regularized through a near-axis boundary layer approximation to the Navier-Stokes equations, as first done by Long for the case of a vortex with potential swirl, v ∼r -1 . The present work considers the more general situation of a family of self-similar inviscid vortices for which v ∼r m-2 , where m is in the range 0 < m < 2. This includes Longs vortex for the case m = 1. The corresponding solutions also exhibit self-similar structure, and have the interesting property of losing existence when the ratio of the inviscid near-axis swirl to axial velocity (the swirl parameter) is either larger (when 1 < m < 2) or smaller (when 0 < m < 1) than an m-dependent critical value. This behaviour shows that viscosity plays a key role in the existence or lack of existence of these particular nearly inviscid vortices, and supports the theory proposed by Hall and others on vortex breakdown.

Dynamics of the wing-tip vortex in the near field of a NACA 0012 aerofoil

Vortex meandering (or wandering) is a typical feature of wing-tip vortices that consists in a random fluctuation of its vortex centreline. This meandering of the vortex is quite significant a few chords downstream the wing, and was originally thought to be due to free stream turbulence [1], then to instabilities of the vortex core [2]. But, independently of the controversy about its origin [3], the quantitative characterization of the vortex wandering phenomenon is a subject of current research [4]-[6]. In this work we have undertaken a systematic visualization of the trailing vortex behind a NACA0012 airfoil at several distances near the wing tip for different angles of attack and different Reynolds numbers to characterize the structure of the vortex meandering phenomenon as well as its frequency, wavelength, and amplitude. The technique is similar to that used by Roy and Leweke [5], but we characterize the downstream evolution of these vortex meandering characteristics and, therefore, the dynamics of the wing-tip vortex in the near field.

Spatial stability and the onset of absolute instability of Batchelor's vortex for high swirl numbers

Batchelor’s vortex has been commonly used in the past as a model for aircraft trailing vortices. Using a temporal stability analysis, new viscous unstable modes have been found for the high swirl numbers of interest in actual large-aircraft vortices. We look here for these unstable viscous modes occurring at large swirl numbers ( q> 1.5), and large Reynolds numbers (Re > 10 3 ), using a spatial stability analysis, thus characterizing the frequencies at which these modes become convectively unstable for different values of q, Re, and for different intensities of the uniform axial flow. We consider both jet-like and wake-like Batchelor’s vortices, and are able to analyse the stability for Re as high as 10 8 . We also characterize the frequencies and the swirl numbers for the onset of absolute instabilities of these unstable viscous modes for large q.

Kinetic theory of binary gas mixtures with large mass disparity

The Boltzmann equations for a binary mixture of gases are considered in the asymptotic limit when their molecular weight ratio and the light gas Knudsen number are small quantities. A first mass‐ratio expansion reduces the cross‐collision operator of the light gas Boltzmann equation to a Lorentz form, uncoupling its kinetic behavior from that of the heavy gas. The light gas distribution function is then determined to first order in the Knudsen number, independently of the degree of nonequilibrium characterizing the heavy gas, whose influence is felt only through its hydrodynamic quantities. All transport coefficients arising are determined variationally for arbitrary interaction potentials using Sonine polynomial expansions as trial functions. A remarkable feature of this analysis is that it yields binary transport information (i.e., diffusion and thermal diffusion coefficients) from considering only the Boltzmann equation for the light gas. A second mass expansion reduces the cross‐collision operator of the heavy gas equation to a Fokker–Planck form. The corresponding coefficients involve integrals over the light gas distribution function determined previously and are evaluated explicitly in terms of the hydrodynamic quantities and transport coefficients of the light gas. The heavy gas distribution function can be determined by solving a Fokker–Planck equation at dilutions large enough to make heavy–heavy collisions negligible, or by a new Knudsen number expansion when the molar fraction of the heavy gas is of order 1. In this latter case, the heavy gas kinetic behavior is independent of the light gas, being characterized by the same transport coefficients of the pure heavy gas. The problem is then reduced to a set of two‐fluid hydrodynamic equations.

On the development of three-dimensional vortex breakdown in cylindrical regions

Three-dimensional and axisymmetric numerical simulations of the incompressible Navier-Stokes equations have been conducted to study the appearance and development of vortex breakdown in a family of columnar vortex flows in a straight pipe without wall friction. The numerical simulations show that the basic form of breakdown is axisymmetric, and a transition to helical breakdown modes is shown to be caused by a sufficiently large pocket of absolute instability inside the original axisymmetric “bubble” of recirculating flow. Depending on the values of the Reynolds and swirl parameters, two distinct unstable modes corresponding to azimuthal wave numbers n= +1 and n = +2 have been found to yield a helical or a double-helical breakdown mode, respectively. By means of a simple linear spatial stability analysis carried out in the sections of the pipe where the basic axisymmetric flow presents reverse flow, we have identified the frequencies and the dominant azimuthal wave numbers observed in the three-dimensional simulations.

An explicit projection method for solving incompressible flows driven by a pressure difference

Abstract A finite-difference method for solving the incompressible time-dependent three-dimensional Navier–Stokes equations in open flows where Dirichlet boundary condition (BC) for the pressure are given on part of the boundary is presented. The equations in primitive variables ( v ,p) are solved using a projection method on a non-staggered grid with second-order accuracy in space and time. On the inflow and outflow boundaries the pressure is obtained from its given value at the contour of these surfaces using a two-dimensional form of the pressure Poisson equation, which enforces the incompressibility constraint ∇· v =0 . The obtained pressure in these surfaces is used as Dirichlet BCs for the three-dimensional Poisson equation inside the domain. The solenoidal requirement imposes some restrictions on the choice of the open surfaces. However, these restrictions are usually met in most flows of interest driven by a pressure (or a body force) difference, to which the present numerical method is mainly intended. To check the accuracy of the method, it is applied to several examples including the flow over a backward-facing step, and the three-dimensional pressure driven flow in a circular pipe.

Nonlinear waves in the pressure driven flow in a finite rotating pipe

To investigate the nature of nonlinear waves appearing in an axially rotating pipe, we have performed a series of time-depending, three-dimensional numerical simulations of the incompressible Navier–Stokes equations in a rotating long pipe. As a difference with some previous works on the subject, which look for several given types of traveling wave solutions in pipes of infinite length, we leave the flow to evolve freely after a pressure difference is set between two points, one on each end of the finite rotating pipe. We use a recently developed numerical method that allows us to simulate numerically the three-dimensional flow produced in a pipe when Dirichlet boundary condition for the pressure is given on part of the inlet and outlet sections of the pipe. This technique is further improved here so that the pressure is only fixed at just one point on each one of the open boundaries of the pipe. Thus, no restrictions on the flow properties are given in these sections, allowing the free entrance and exit ...

Axisymmetric instabilities of Bödewadt flow

A spatial linear stability analysis of Bodewadt’s self-similar solution for the rotating flow over a flat plate is performed. In particular, considered is the stability of axisymmetric perturbations propagating towards the axis of rotation, which are the most important ones observed experimentally. Viscous and nonparallel effects on the stability of the perturbations are retained up to the order of the inverse of the local Reynolds number R. The resulting parabolic stability equations are solved numerically using a spectral collocation method varying the nondimensional frequency q and R. The instability region on the (q,R)-plane is discussed and compared with existing experimental data and direct numerical simulation results. The circular waves observed experimentally and in numerical simulations are shown to correspond to an inertial instability mode which becomes stabilized as R decreases below a critical value.