M&AE 3050 November 25, 2008Effects of ViscosityD. A. CaugheySibley School of Mechanical & Aerospace EngineeringCornell UniversityIthaca, New York 14853-7501These notes provide, as a supplement to our textbook [2], a more complete description of viscouseffects, especially those associated with skin friction.1 The Boundary LayerThe principal insight that allowed quantitative analysis of aerodynamic problems, even in the pre-computer era, was the realization that viscous effects generally are limited to very thin regions of aflow in the immediate vicinity of bodies immersed in flows at large Reynolds numbers. This allowsone to analyze separately the flow external to the boundary layers using inviscid, potential theory,then analyze the flow within the boundary layers subject to the pressure distribution imposed on theboundary layer by the outer, inviscid, flow. This is a consistent approach so long as the boundarylayers remain thin – i. e., that they do not separate from the body surface.Viscous effects contribute to aerodynamic drag through two very different mechanisms. Skin frictiondrag results from the integrated effect of the fluid shear stress acting over the wetted area of thebody, while pressure drag results from changes in the pressure distribution acting on the body dueto the fact that the boundary layer is not infinitesimally thin. Pressure drag becomes especiallyimportant when the boundary layers separate from the surface of the body, creating large wakes ofrelatively low pressure fluid. A study of the flow in the boundary layer will help us to understandboth these contributions to drag.2 The Boundary Layer EquationsThe Navier-Stokes equations describe the flow of a constant-density fluid, including the effects ofviscosity. For simplicity we write them for steady flow in two dimensions, x and y, as follows∂u∂x+∂v∂y= 0 , (1)u∂u∂x+ v∂u∂y=−1ρ∂p∂x+µρ∂2u∂x2+∂2u∂y2, (2)u∂v∂x+ v∂v∂y=−1ρ∂p∂y+µρ∂2v∂x2+∂2v∂y2, (3)where u and v are the x- and y-components of the velocity, ρ and p are the fluid density and pressure,respectively, and µ is the coefficient of viscosity.2 THE BOUNDARY LAYER EQUATIONS 2xyu(y)δ (x)ULFigure 1: Sketch illustrating development of the viscous boundary layer in the flow past a flat plate,including typical boundary-layer profile, u(y) as a function of distance y normal to the plate.Now, we consider a scaling argument that will suggest which of the terms in these equations will beimportant within the boundary layer. We consider the flow near a solid surface, taken for simplicityto be y = 0, and consider the magnitudes of the various terms in Eqs. (1) – (3). We assume thatvariation in quantities across the boundary layer scales with the boundary layer thicknes s δ, whilevariation in the x-direction, i.e., in the direction of flow, scales with the length L; see Fig. 1.Now, the derivative∂u∂xwill scale as U/L, while the derivative∂v∂ywill scale as v/δ. These twoderivatives must be of the same order (for them to balance in Eq. (1), the continuity equation) sowe must havev ∼δLU . (4)Remember that our basic hypothesis is that the boundary layer will be thin (i.e., that δ/L will besmall), so the continuity equation tells us that the normal velocity v will be small compared to thevelocity U in the free stream.Now, the press ure changes must scale with ρU2(since they are determined by the inviscid flowoutside the viscous layer), so the left-hand side and the term involving the derivative of pressure inEq. (2) will all scale as U2/L. The viscous terms on the right-hand side will scale as µU/L2andµU/δ2, respectively, so for thin layers the second derivative with respect to x will be small comparedto the second derivative with respect to y. Thus, in the boundary layer the x-momentum equationreduces tou∂u∂x+ v∂u∂y=−1ρ∂p∂x+µρ∂2u∂y2. (5)This equation, in fact, gives us our first estimate for the thickness of the boundary layer. In orderfor any viscous effects to be significant, the final term in Eq. (5) must b e the same order as the otherterms; i.e., we must haveµρUδ2∼U2L,which requiresδL∼rµρUL=1√ReL. (6)Thus, our scalings are, in fact, consistent with the boundary layer being thin for large values of theReynolds number.A study of this scaling also helps us to understand the resolution of D’Alembert’s paradox – i.e.,to see how viscous effects can remain important even in the limit as the Reynolds number goes toinfinity. We consider the flow of a fluid of fixed density past an object of fixed s ize with a fixedfreestream velocity, in the limit as the viscosity of the fluid goes to zero. The simplest argumentwould say that the viscous stresses would have to go to zero as the viscosity goes to zero, so a fluidof vanishingly small viscosity should behave like a perfect (inviscid) fluid. This is consistent withD’Alembert’s paradox, since we know that even fluids with very small viscosities (read: flows atvery large Reynolds numbers) do not b ehave in this way. But, this simplistic view assumes that the3 SKIN FRICTION DRAG 3velocity gradients don’t change as the viscosity is reduced – and our scaling arguments show that asthe Reynolds number increases the boundary layer gets thinner, so the velocity gradients get larger.And, if the boundary layer equation is to balance, the layer gets thinner at just the right rate toallow the viscous stress es always to balance the convective acceleration terms.Turning to the y-momentum equation, we see that the terms on the left-hand side both scale asU2δ/L2, while the term involving the derivative of pressure scales as U2/δ. Thus, the terms onthe left-hand side are of order (δ/L)2relative to the pressure term, and are therefore negligible.The dominant viscous term is again the one involving the second derivative with respect to thenormal coordinate y, and it is of order µU/(ρδL). Thus, the dominant viscous term viscous is oforder µ/(ρUL) = 1/ReLrelative to the pressure term, and also is negligible for large values of theReynolds number ReL. Thus, the y-momentum equation reduces to∂p∂y= 0 . (7)This result shows that the pressure must b e constant across the boundary layer, and its value canbe determined from the outer, inviscid flow, at the edge of the layer. The Bernoulli equation gives−1ρdpdx= UedUedx(8)where the subs cript ()eis used to denote the velocity specified at the edge of