On the Equation Satisfied by a Steady Prandtl-Munk Vortex Sheet

We show that the vorticity distribution obtained by minimizing the induced drag on a wing, the so called Prandtl-Munk vortex sheet, is not a travelling-wave weak solution of the Euler equations, contrary to what has been claimed by a number of authors. Instead, it is a weak solution of a non-homogeneous Euler equation, where the forcing term represents a “tension” force applied to the tips. This is consistent with a heuristic argument due to Saffman. Thus, the notion of weak solution captures the correct physical behavior in this case. The Prandtl-Munk vortex sheet is notorious in fluid dynamics as the one that generates a circulation distribution that minimizes the induced drag on a plane wing [1, 8]. The vorticity associated to the Prandtl-Munk vortex sheet is given by ω0(x1, x2) = x1 √ 1− x1 χ(−1,1)(x1)⊗ δ0(x2), (1) where χ(−1,1) represents the characteristic function of the interval (−1, 1) and δ0 is the Dirac delta at x2 = 0. The associated velocity, given by the Biot-Savart law, can be evaluated at points on the sheet by means of a principal value integral and is found to be constant and equal to (0,−1/2) [8]. Hence a natural candidate for a weak solution is determined by the steadily translating vorticity profile: ω(x1, x2, t) = ω0(x1, x2 + t 2 ). (2) A number of authors [3, 6] have claimed, albeit with some doubts, that (2) is indeed a weak solution of the steady Euler equations. However, the numerical experiments performed by Krasny [3] suggest that the (physically correct) solution of the Euler equation with (1) as initial data is not steady, actually rolling-up at the tips for positive time. Later, Saffman [8] gave a heuristic argument, based on vortex principles, as to why the steady solution (2) is not dynamically consistent. This scenario suggests that the Prandtl-Munk vortex profile (1) is an example of nonuniqueness for the weak formulation of vortex sheet evolution. We shall show that (2) is actually a weak solution of a nonhomogeneous Euler equation, with forcing term consistent with that obtained by Saffman, thus ruling out the Prandtl-Munk vortex sheet as an example of nonuniqueness. It is not known to which degree unphysical flows may satisfy the standard notion of weak solution to the Euler equations. Examples of (unphysical) weak solutions with kinetic energy growing in time have been obtained [9, 11] in a situation considerably more irregular than vortex sheet dynamics. The main point of the present ∗Received: April 15, 2002; Accepted (in revised version): April 25, 2002. †Departamento de Matematica, IMECC-UNICAMP. Caixa Postal 6065, Campinas, SP 13081-970, Brasil (mlopes@ime.unicamp.br). ‡Departamento de Matematica, IMECC-UNICAMP. Caixa Postal 6065, Campinas, SP 13081-970, Brasil (hlopes@ime.unicamp.br). §Departamento de Matematica, PUC-Rio, Rua Marquês de São Vicente, 225 Rio de Janeiro RJ, 22453-900, Brasil (msouza@mat.puc-rio.br). Present address: Depto. de Matematica Aplicada, UFF, Rua Mario Santos Braga s/n, Niteroi, RJ 24020-140, Brasil. 68 M. C. LOPES FILHO, H. J. NUSSENZVEIG LOPES, AND M. O. SOUZA 69 paper is the fact that the notion of weak solution is discriminating enough to distinguish the physically correct evolution for the Prandtl-Munk vortex sheet, something which cannot be done through the Birkhoff-Rott model alone. A different example of nonphysical weak solution in the context of vortex sheets has been proposed and numerically studied, see Lopes Filho et al. [4]. The standard definition of a weak solution of the Euler equations uses velocity as the main dynamic variable [5]. We observe that it is not easy to check, using this notion of weak solution, whether (2) is or is not a weak solution of the Euler equations. Indeed, despite the fact that the vorticity associated to the Prandtl-Munk vortex sheet has the particularly simple form (1), the associated velocity is fairly complicated [8]. For this reason, we choose to work with the vorticity equation, and we will use the weak vorticity formulation originally introduced by Delort [2] and reformulated by Schochet [10] stated as follows: a measure ω ∈ L∞([(0,∞);BM(R2) ∩ H−1 loc (R)) is a weak solution of the inhomogeneous vorticity formulation of the incompressible 2D Euler equations with initial data ω0 ∈ BMc(R) ∩ H−1(R2), and forcing F ∈ L∞([(0,∞); W−1,1 loc (R)) if, for any test function φ ∈ C∞ c ([(0,∞)× R), we have: ∫ ∞ 0 (∫ R2 φtdω(x, t) + ∫ R2 ∫ R2 Hφdω(x, t)⊗ dω(y, t) ) dt+ ∫ R2 φ(x, 0)dω0(x) = ∫ ∞ 0 ∫ R2 ∇⊥φ · dF (x, t)dt, (3) where Hφ(x, y, t) ≡ ∇φ(x, t)−∇φ(y, t) 4π|x− y| · (x− y)⊥ |x− y| . In the homogeneous case, this formulation was shown to be equivalent to the standard velocity one, as long as the velocity is (Lloc(R)), by Delort [2]. However, the proof carries over in a straightforward manner to the inhomogeneous case. Using the formulation above, we shall show that the velocity associated to (2) solves, in a weak sense, the system of equations: { ut + u · ∇u = −∇p + F div u = 0, with the forcing term F (x1, x2) = −π/8   δ0(x1 − 1, x2 + t/2)− δ0(x1 + 1, x2 + t/2) 0   . (4) The factor in (4) should be compared with equation (7) in § 6.3 of Saffman [8], keeping in mind that, in our case, we have U = 1/2 and a = 1. Thus, in accordance with Saffman’s analysis, the forcing given by (4) can be thought of as a tension applied near the tips of the vortex sheet. We begin by choosing a test function of the form φ(x1, x2, t) = α(t)ζ ( x1, x2 + t 2 ) , where α and ζ are C∞ c in [0,∞) and R, respectively. We notice that finite sums of test functions having this form are dense in the set of all C∞ c ([0,∞) × R) test functions. We now compute the right-hand side of (3) with ω given by (2): 70 ON THE EQUATION SATISFIED BY A STEADY PRANDTL-MUNK VORTEX SHEET (i) ∫ ∞