Finally: Resolution of d'Alembert's Paradox Johan Hoffman and Claes Johnson January 20, 2006 Abstract We propose a new resolution to d'Alembert's Paradox from 1752 com- paring the mathematical prediction of zero drag (resistance to motion) through an ideal (zero viscosity) incompressible fluid, with massive ob- servations of non-zero drag in fluids with very small viscosity, such as air and water. Our resolution is fundamentally different from the accepted resolution suggested by Prandtl in 1904 based on boundary layer effects of vanishing viscosity. We base our resolution on computational solution of the Euler equations describing ideal incompressible flow by noting that the zero drag potential solution considered by dAlembert, is unstable and instead a turbulent (approximate) solution develops with non-zero drag, even without boundary layer effects. We claim that our resolution is bet- ter than Prandtl's in the case of very small viscosity. 1 Introduction How wonderful that we have met with a paradox. Now we have some hope of making progress. (Nils Bohr) We present a new resolution of d'Alembert's Paradox from 1752 ([1]), which states that a body can move through an ideal (zero viscosity) incompressible fluid without any resistance (drag) to the motion. This statement is paradox- ical because all experience shows that motion through fluids with very small viscosity, such as air and water, meets a drag which is far from zero. Our res- olution is fundamentally different from the accepted resolution from 1904 by Ludwig Prandt ([2]), called the father of modern fluid mechanics, which builds on boundary layer effects from very small viscosity. We base our resolution on computational solution of the Euler equations de- scribing ideal incompressible fluid flow, which shows that the potential solution with zero drag considered by d'Alembert is unstable and instead a turbulent solution develops with non-zero drag. This solves the paradox in the origi- nal setting of d'Alembert and Euler, assuming the fluid to be ideal with zero viscosity, by showing that the zero drag potential solution cannot be realized physically, because it is unstable, and thus cannot be observed. We solve the Euler equations with a slip boundary condition at solid boundaries prescribing 1 the normal velocity to be zero but letting the tangential velocity be free, which means that no boundary layers are created. Nevertheless, a turbulent solution to the Euler equations with non-zero drag develops, starting from the poten- tial solution. Our resolution is thus completely different from Prandtl's and we claim that our resolution is more to the point for the very small viscosities met in a wide range of turbulent flows in aero- and hydro-dynamics. We also claim that our resolution is more satisfactory from a scientific point of view than Prandtl's, because we do no suggest that a very small cause (very small viscosity) can have a large effect (change the drag), as Prandtl does, which is close to saying that anything can happen from virtually nothing, and which can be very hard to either prove or disprove. We show instead that the potential solution is unstable and that a turbulent solution develops, even without influence from boundary layer effects of very small viscosity. We do not claim that boundary layer effects never influence the global flow, e.g. by separation, but we do claim that these effects may be small for very small viscosities, which fits with the observation that the so-called skin-friction tends to zero as the viscosity tends to zero. To solve the Euler equations numerically we use an adaptive finite element method with automatic control of the error in the drag ([8, 4, 5, 7, 9]), and we find that the computed drag is stable under mesh refinement. In [6] we also use a skin-friction boundary condition to model the effect of turbulent boundary layers, with zero skin-friction corresponding to a slip boundary condition. Let- ting the skin friction tend to zero, we obtain good agreement with experimental drag coefficients for varying viscosity (Reynolds number) including the so-called drag crisis occuring for very small viscosities ([10]). An outline of this note is as follows: We first recall the Euler equations for ideal incompressible fluid flow, and the stationary irrotational potentional solu- tion with zero drag considered by dAlembert. We then present computational results and point to basic features. Finally, we compare our new resolution of d'Alembert's Paradox with that of Prandtl, and leave to the reader to judge which resolution may be closer to the truth. 2 The Euler equations We consider the motion of an ideal incompressible fluid occupying a fixed volume Ω in R3 with boundary Γ. We want to find the fluid velocity u(x; t) and pressure p(x; t) for all points x = (x1; x2; x3) 2 Ω and time t > 0, assuming that the fluid flow through the boundary Γ and the initial velocity u(x; 0) are given. We assume that Γ is divided into a part Γ0 corresponding to a solid (inpenetrable) boundary, and a remaining part corresponding to inflow and outflow. The mathematical model for the motion of the fluid takes the form of the Euler equations formulated by Leonard Euler in 1755 ([11]) expressing conservation of momentum (Newton's second law) and conservation of mass, combined with 2 a boundary condition (g) and an initial condition (u0) for the velocity: u_ + (u · r)u + rp = 0; in Ω × I; r · u = 0; in Ω × I; (1) u · n = g; on Γ × I; u(·; 0) = u0; in Ω: Here n is the outward unit normal to Γ and the given boundary flow g satisfies RΓ g ds = 0 and g = 0 on Γ0. Requiring u · n = 0 on Γ0 corresponds to a slip boundary condition (bc) with the normal velocity vanishing, while the tangential velocity is free. This is to be compared to the no-slip bc u = 0 in Navier{Stokes equations (with non-zero viscosity ν > 0 including the term −ν∆u in the momentum equation), where also the tangential velocity is required to vanish reflecting that the fluid sticks to a solid boundary. Prandtl's resolution of the Paradox is connected to the no-slip bc, whereas we advocate that the slip bc is more relevant, in the case of very small viscosity. 3 Potential Flow around a Circular Cylinder Following d'Alembert we consider stationary (time-independent) potential flow around an (infinitely) long cylinder of diameter 1 oriented along the x3-axis and immersed in an ideal incompressible fluid filling R3 with velocity (1; 0; 0) at infinity. This models, for example, the flow of air around a tall cylindrical high- rise subject to a strong wind, or the flow of water around a pillar of a bridge in a strong current. The potential velocity is given as u = rφ, where φ satisfies Laplace's equation ∆φ = 0 outside the cylinder and appropriate conditions at infinity, that is, 1 φ(x1; x2; x3) = (r + ) cos(θ); (2) r where (x1; x2) = (r cos(θ); r sin(θ)) is expressed in polar coordinates (r; θ). Us- ing standard Calculus one verifies that u is irrotational, that is r × u = 0, since r × rφ = 0, and that (u; p) solves the Euler equations, where the pressure p is 1 2 determined by Bernoulli's Law stating that 2 juj + p is constant for stationary irrotational flow. In Fig. 1 we plot the streamlines of u in a section of the cylin- der, which are the curves followed by fluid particles, and the pressure. We notice that the potential flow (in each section) has one separation point at the back of the cylinder, where the flow separates from the cylinder boundary. We also notice that the both velocity and pressure are symmetric in the flow direction (x1-direction), which means that the drag of the cylinder is zero; the build up of pressure in front of the cylinder is balanced by the same strong pressure behind, and thus the drag is zero. The cylinder thus seems to be "pushed through the fluid” by the strong pressure behind, which of course is counter-intuitive and in fact is never observed in practice, where the pressure behind always is much lower than up front, with resulting non-zero drag. According to d'Alembert's potential solution there would be no wind load on a high-rise and no force on a 3 bridge pillar from a strong current, which is in contradiction with all practical experience. One can extend this result to flow around a body of arbitrary shape, since there is always a corresponding potential solution. We have thus met a scientific Paradox, which we have to resolve to save fluid mechanics as a mathematical science from collapse. But what do to? Evidently, something must be wrong with the potential solution, since it gives zero drag. But what? It cannot be Newton's second law or mass conservation. Figure 1: Potential solution of the Euler equations for flow past a circular cylin- der; colormap of the pressure (left) and streamlines together with a colormap of the magnitude of the velocity (right) . Prandtl in 1904 claimed that the Paradox is due to the assumption of zero viscosity. Prandtl stated that even if the viscosity is very small, it is not equal to zero, which means that one has to consider the Navier-Stokes equations with no-slip bc (instead of Eulers equations with slip bc), for which in a thin boundary layer close a solid boundary the fluid velocity will change quickly from zero at the boundary to the free stream value outside the layer.