M&AE 3050 October 3, 2008Incompressible Potential FlowD. A. CaugheySibley School of Mechanical & Aerospace EngineeringCornell UniversityIthaca, New York 14853-7501These notes provide, as a supplement to our textbook [2], a description of two-dimensional, in-compressible, potential flows, and the associated development of several fundamental aerodynamicresults.1 IntroductionThe analysis of high Reynolds number flows past streamlined bo dies is greatly simplified by the factthat viscous effects are generally important only within thin regions immediately adjacent to solidboundaries. Outside these viscous boundary layers the flow is well approximated as inviscid. Thefollowing sections will provide some background for analyzing these inviscid flows, at least in theincompressible limit.2 Incompressible Potential FlowThe vorticity ζ is the curl of the velocity field Vζ = ∇ × V . (1)It is not too difficult to show that the vorticity at any point in the flow corresponds to twice therotation rate of an infinitesimal fluid particle centered at that point. To demonstrate this for a two-dimensional flow, we let u and v represent the x- and y- components of the velocity field, relativeto the fluid velocity at a given point. Then the instantaneous rates of rotation of two mutually-orthogonal axes (taken, for convenience, to be parallel to the x- and y-axes, respectively), can becomputed as (see Fig. 1)˙θx=∂v∂x(2)and˙θy= −∂u∂y, (3)whence the average rotation rate˙θ is˙θ =12˙θx+˙θy=12∂v∂x−∂u∂y. (4)The quantity above within the parentheses is exactly the vorticity of this two-dimensional flow,which can be written1ζz=∂v∂x−∂u∂y. (5)1For a two-dimensional flow only one component of the vorticity vector is non-zero: the component normal to theplane of the motion.2 INCOMPRESSIBLE POTENTIAL FLOW 2xy∆∆xyFigure 1: Relative velocity field in the vicinity of a point, illustrating the relationship b etweenderivatives of relative velocity and instantaneous rotation rates of a pair of mutually perpendicularaxes.A flow in which the vorticity is everywhere zero is said to be irrotational , and these irrotationalflows play an important role in aerodynamics.Now, if viscous stresses are negligible there is no mechanism to introduce rotation of fluid elements.Thus, an initially irrotational flow must remain irrotational for all time.2This statement, calledKelvin’s Theorem, is a fundamental basis of much of aerodynamic theory.Any irrotational flow field can be represented as the gradient of a potential function, so we canrepresent the velocity asV = ∇φ , (6)where the scalar function φ is called the velocity potential. It is clear from the vector identity∇ × (∇φ) = 0 (7)that any potential field is irrotational. Since an irrotational flow can be represented by a velocitypotential, the terms irrotational and potential are synonymous.In Cartesian coordinates (x, y), Eq. (6) gives the components (u, v)Tof the velocity vector asu =∂φ∂xand v =∂φ∂y(8)for flows in two dimensions.For constant density flows, recall that the continuity equation requires∇ · V = 0 , (9)which, upon introducing the velocity potential, becomes∇ · (∇φ) = ∇2φ = 0 . (10)2In addition to having negligibly small viscous stresses, we must also ensure that the pressure forces do not inducerotation. This simply requires that the pressure forces acting on a fluid element must act through the center of mass,which is guaranteed if the density is constant. In the more general case of compressible flows, all that is required is thatsurfaces of constant pressure also be surfaces of constant density; so compressible flows also satisfy this requirementif they are well approximated as isentropic.2 INCOMPRESSIBLE POTENTIAL FLOW 3Thus, the velocity potential for an incompressible flow is seen to satisfy Laplace’s Equation. In twodimensions, the Cartesian-coordinate version of this can be written∂2φ∂x2+∂2φ∂y2= 0 . (11)Two-dimensional, constant-density flows can also be described in terms of the stream function ψ,which is defined to be constant along stream lines. That is, along a streamline we must havedψ =∂ψ∂xdx +∂ψ∂ydy = 0 , (12)which requiresdydxψ=const.=−∂ψ∂x∂ψ∂y=vu. (13)Thus, if we define the stream function such thatu =∂ψ∂yand v = −∂ψ∂x, (14)the function ψ will be constant along stream lines.Since the stream function, as defined by Eqs. (14), can be used to define rotational, as well as irrota-tional flows, the condition of irrotationality supplies an additional constraint. Thus, for irrotationalflows, we must have∂u∂y−∂v∂x=∂∂y∂ψ∂y−∂∂x−∂ψ∂x=∂2ψ∂y2+∂2ψ∂x2= 0 .(15)Thus, for irrotational flows the stream function is seen also to satisfy Laplace’s equation.Note the duality between the velocity potential and the stream function. The velocity potentialautomatically satisfies irrotationality and continuity requires that it satisfy the Laplace equation.On the other hand, the stream function automatically satisfies continuity, and it is irrotationalitythat requires it to satisfy the Laplace equation.For some of the elementary solutions to be discussed in the following sections, it is convenient todescribe the flow in the cylindrical coordinates defined byr =px2+ y2and θ = tan−1yx, (16)orx = r cos θ and y = r sin θ . (17)In terms of r and θ the components of the gradient of the velocity potential, corresponding to theradial and circumferential components of velocity, arevr=∂φ∂rand vθ=1r∂φ∂θ, (18)and Laplace’s equation for the velocity potential φ becomes∂2φ∂r2+1r∂φ∂r+1r2∂2φ∂θ2= 0 . (19)2 INCOMPRESSIBLE POTENTIAL FLOW 4VRnFigure 2: Flow field represented in cylindrical coordinates, matching the curvature of the streamlineat a point.2.1 The Bernoulli EquationRecall that, for inviscid, incompressible flow the Bernoulli equation requires thatp +12ρV2= pT, (20)where pTis constant along a streamline. We show here that for irrotational flows the total pressurepTis the same for all streamlines; i.e., the total pressure is the same constant throughout the flow .If we differentiate the Bernoulli equation in the direction n normal to a streamline we have∂p∂n+ ρV∂V∂n=∂pT∂n. (21)At the same time, the force balance normal to the streamline requires∂p∂n=ρV2R, (22)where R is the radius of curvature of the streamline.