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Two-Dimensional Flow: Lift and Drag Questions How can we determine the pressures at the surface of a body? How does lift occur? How to explain D’Alemberts paradox ?- That for a perfect inviscid fluid the drag force is zero. Hydrodynamic Theory: The elements Establish two functions (1)The Potential function - defined so that /x=velocity in the x-direction /x=velocity in the y-direction (2) The stream function defined so that no fluid parcel crosses a streamline Apply the continuity and momentum concepts Apply the principle of irrotationality Find distributions of and that satisfy these principles throughout the fluid Apply the boundary conditions on the surface of the body The Theory of Lift: Circulation Consider a steady, two-dimensional flow in concentric circles in a perfect, incompressible fluid. When combined with a uniform linear flow such a combination will produce lift. p+(dp/dr)dr (r+dr)d dr e p+dp/dr(r/2) rd p r V Volume Element for a Concentric Circular Flow d There is a small fluid element a distance r from the center, a length dr, unit thickness perpendicular to the flow, a width rd at the inner face, and (r +dr) d at the outer face. The radially outward acting centrifugal forces on this element must be balanced by the pressure forces on the sides. The centrifugal force is: (rd +(r+dr)d)dr/2(V2/r)= rddrV2/r Where, the higher order term has been dropped The strength of the circulatory flow is called the circulation defined as the line integral of Vcos around a closed path in the fluid. V cos dl is the angle between the velocity V and the path at a point on the contour dl is an element of length along the contour Now consider a circular path where V=A/r Since the velocity is tangent to the path, is 0 and cos is 1. V (1)rd A / r (r )d Ad Integrating through 2 gives = 2A or A= /2 V=/2r dl r d Circulation with concentric circle flow An important point is that outside of a vortex core the circulation is constant. Since V = /(2r) and for a potential vortex V = A/r, there is not radial dependence of circulation ( = 2A). The superposition of a vortex flow and a uniform linear flow can produce lift PP+ Case of uniform flow with no circulation Case of rotation with no uniform flow Case of Uniform Flow with Rotation Producing Lift The pressure forces positive in the radial inward direction are: Inner Face -prd Outer Face (p+(dp/dr)dr)(r+dr)d Side Faces -(p +dp/dr(dr/2))(r+dr)d-rd Neglecting higher order terms the total of the pressure forces is: rddp Equating the pressure and centrifugal forces: rddp= rddrV2/r dp = V2dr/r Substituting for dp in the Euler equation dp = -VdV gives: -VdV = V2dr/r or dV/V = -dr/r Integrating we obtain: Log V = -Log r + constant Log(Vr)=constant, and Vr=constant V = Constant/r = A/r Such vortices are called potential vortices Summary of Pressure Force Calculations Inner Face: -prd Outer Face: (p+dp)(r+dr)d (pr+pdr+dpr+dpdr)d prd+pdrd+dprd Side Faces: -(p+dp/2)(r+dr)d-rd -p+dp/2)(rd+drd-rd) -(p+dp/2)drd -drpd-dp/2(drd) = neglect product of differentials Inner Face -prd Outer Face Side Faces prd+pdrd+dprd The total of the forces is rddp -drpd In summary, for a cylinder in steady 2-d flow in a perfect fluid: The drag is zero If there is no circulation, the lift is zero If circulation exists around a body in uniform flow, lift is produced perpendicular to the freestream The magnitude of the circulation can be calculated from the Kutta-Joukowski Law Assume a constant increase of velocity V on the upper surface of a lifting body with cord c and an equal reduction V on the lower surface pl = p0 + /2 V20 -(V20 + 2V0 V +V2) pl= p0 + /22V0V +V2 pu = p0 - /2(V20 + 2V0 V +V2) -V20 pu= p0 - /22V0V +V2 The lift per unit span is: pl - pu = 2V0Vc The circulation is: dVdx 2Vc Lift per unit span is: V0 Kutta-Joukowski Law L = Lift per unit span = V = L/(V) = Total Lift/ (SV) Where S is the span This indicates that heavy aircraft with smaller wing spans, moving slowly, will create the greatest circulation This is why heavy aircraft during take off or landing present the greatest hazard to following aircraft