The different equations of motion of the central line of a slender vortex filament and their use to study perturbed vortices

ESAIM : Proceedings, Vol. 7, 1999, 270-279 Third International Workshop on Vortex Flows and Related Numerical Methods http://www.emath.fr/proc/Vol.7/ 270 The different equations of motion of the central line of a slender vortex filament and their use to study perturbed vortices D. Margerit and J-P. Brancher LEMTA (CNRS UMR 7563) 2 avenue de la forêt de Haye BP 160 54504 Vandoeuvre les Nancy, France. A comparison between the equation of motion of the central line of a slender vortex filament deduced from a matched asymptotic expansion[1] and the expansion of the equation of motion of the ad-hoc cut-off methods[2] with the cut-off length as the small asymptotic parameter is performed. It justifies the cut-off methods and gives the link between the cut-off lengths and the thickness of a viscous or inviscid vortex with an axial velocity component. The asymptotic equation of motion for an open filament is then simplified in case of a perturbed straight filament and different regimes are displayed. They depend of relatives values of the amplitude of the perturbation and the small thickness of the filament. 1 Introduction It is well known that the structure of flows often exhibits concentration of vorticity in the form of vortex sheets or of vortex filaments. They can be seen in aircraft wakes, jets, boundary layer, Direct Numerical Simulations of turbulent flows, tornados,... There are a considerable number of publications about the motion of a vortex filament and its stability. All field of methods have been used : formal asymptotic and linear stability analyses, direct numerical simulations, experimental investigations,...Theoretical studies have been performed either with Navier Stokes equations or simplified ones. In the later case, the field that covers the result of the study or its validity depends of the field of validity of the simplified equation that is used. This domain of validity needs to be given. Here, a general dimensionless description of a slender vortex filament is given as an introduction to all discussion. A number of ad-hoc equation of motion due to ad-hoc regularization of the singular line BiotSavart integral are given and compared with the results of a systematic matched asymptotic expansions. It is performed both for a closed vortex filament and an open one. Our first main result is to give a relation which makes a link between ad-hoc and ”exact” asymptotic approaches. The asymptotic equation of motion for an open filament is then simplified in case of a perturbed straight filament and different regimes are displayed. They depend of respective values of the amplitude of the perturbation and the small thickness of the filament. This is our second main result, that is sum up on a diagram showing the domain of validity of the various simplified equations. 2 Description of a vortex filament A slender vortex filament of thickness δ is a solenoidal field of vorticity ω(x, t) which is non-zero only in the neighbourhood of a three-dimensional curve C, called the central line. This curve is described parametrically (Fig.1) with the use of a function X = X(s, t), which denotes a point on the curve as a function of the parameter s and time t. Either s ∈ [−π, π[ for a vortex ring that has a closed central curve, or s ∈ [−∞, ∞[ Article published by EDP Sciences and available at http://www.edpsciences.org/proc or http://dx.doi.org/10.1051/proc:1999025 271 D. Margerit and J-P. Brancher Figure 1: The central curve and the local co-ordinates of the vortex filament. for an open central line. The local torsion of C is called T and K is the local curvature. The function σ is defined by σ(s, t) = |∂X/∂s| (1) where | | is the usual norm of R3 . For each point on the curve C, there is the Frenet frame (t, n, b) with respectively the tangent, normal and binormal vectors. The strength Γ of the filament is the flux of vorticity, that is the same in each section of the filament. For a vortex ring, S is the length of C. The vortex filament may have an axial flux of strength m. The thickness δ of the filament is of order l and the other length scales, for example : the radius of curvature, the ring length,... are of the same order L. Here l/L  1 as the vortex is slender. The parameter  is defined by  = l/L. Here, in fact, the exact value of l and L are not needed. The Reynolds number Re is defined as Re = Γ0 /ν where ν is the kinematic viscosity of the fluid and Γ0 −1/2 the initial circulation of the filament. The number α is defined such that Re = α. Thus, the inviscid case is obtained when α = 0 . Both inviscid α = 0 and viscous α = O(1) vortex filaments are studied. −1 The Swirl number Sw is defined as Sw = m∗0 = m0 /(Γl), where m0 is the initial axial flux of the filament. Dimensionless variables : X∗ K∗ δ∗ v∗ = X/L = LK = δ/L = v/(Γ/L) S∗ T∗ t∗ ω∗ = S/L = LT = t/(L2 /Γ) = ω/(Γ/L2 ) are introduced. From here on, all quantities are dimensionless and the asterisks are omitted. All dimensionless variable X, ω, v,... have expansion in . The velocity is of order −1 , which is called leading order, then there is first order and so on. 3 Equations of motion Two methods can be used to avoid the logarithmic singularity that appears in the velocity of a vortex filament without thickness : either a cut-off method [2], but it introduces an unknown length of cut off, or ESAIM: Proc., Vol. 7, 1999, 271 -279 D. Margerit and J-P. Brancher 272 a matched asymptotic expansion of a slender vortex [1]. These both methods are described in the following and then compared. 3.1 Cut-off methods The equation of motion of the central line C of a slender non-circular vortex ring was first expressed by different ad hoc methods. First there is Burger’s method[2] Ẋ(s, t) = Z 1 4π σ0 t(s0 , t) × (X(s, t) − X(s0 , t)) I |X(s, t) − X(s0 , t)| 3 ds0 (2) where Ẋ = ∂X/∂t, σ0 = σ(s0 , t) and I = [0, 2π[\[s − sc , s + sc [. The unknown small variable sc is a small parameter introduced to avoid the singularity. This ad hoc method was called the ’cut-off method’ by Crow[2], name that can be generalised to all methods that introduce a small ad hoc parameter sc to avoid the singularity in the integral. In this way, the slenderness of the vortex is taken into account. The parameter sc is called the cut-off length. In a similar way, 1 Ẋ(s, t) = 4π Z2π t(s0 , t) × (X(s, t) − X(s0 , t)) 0 σ0 h i3/2 ds 0 , t))2 + s2 (X(s, t) − X(s 0 c (3) can be used[8, 17, 16, 15, 18]. Finally, one often write[7] : Ẋ(s, t) = 1 4π Z2π σ0 0 t(s0 , t) × M0 M M0 M 3 f( M0 M )ds0 sc M0 M = X(s, t) − X(s0 , t) (4) (5) with f (χ) → 1 when χ → ∞ . For example : f (χ2 ) = −2χ2 + √ 4 πerf (χ2 )eχ √ χ4 πe (6) in Leonard vortex element method (VEM) [7], where erf is the error function. The Burger’s method of a non closed vortex filament is : 1 ∂X/∂t = 4π Z " t(s0 , t) × (X(s, t) − X(s0 , t)) 3 I |X(s, t) − X(s0 , t)| # ds0 (7) with I = [−∞, +∞[\[s − sc , s + sc [ These (...truncated)