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2.016 Hydrodynamics Reading #5 2.016 Hydrodynamics Prof. A.H. Techet Fluid Forces on Bodies 1. Steady Flow In order to design offshore structures, surface vessels and underwater vehicles, an understanding of the basic fluid forces acting on a body is needed. In the case of steady viscous flow, these forces are straightforward. Lift force, perpendicular to the velocity, and Drag force, inline with the flow, can be calculated based on the fluid velocity, U , force coefficients, C D and C , the object’s dimensions or area, A , and fluid density, ρ . For L viscous flows the drag and lift on a body are defined as follows 1 2 F Drag = U A C D (5.1) 2 1 2 F Lift = C L (5.2) 2 These equations can also be used in a quiescent (stationary) fluid for a steady translating body, where U is the body velocity instead of the fluid velocity, since U is still the relative velocity of the fluid with respect to the body. The drag force arises due to viscous rubbing of the fluid. The fluid may be thought of as comprised of several “layers” which move relative to one another. The layer at the surface of the body “sticks” to the surface due to the no-slip condition . The next layer of fluid away from the surface rubs against the layer below, and this rubbing requires a certain amount of force because of viscosity. One would expect that in the absence of viscosity, the force would go to zero. Jean Le Rond d'Alembert (1717-1783) performed a series of experiments to measure the drag on a sphere in a flowing fluid, and on the basis of the potential flow analysis he expected that the force would approach zero as the viscosity of the fluid approached zero. However, this was not the case. The net force seemed to converge on a non-zero value as the viscosity approached zero. Hence, the vanishing of the net force in the potential flow analysis is known as d'Alembert's Paradox. version 3.0 updated 8/30/2005 -1- © 2005 A. Techet

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2.016 Hydrodynamics Reading #5 D’Allembert’s Paradox for a fixed sphere in uniform inflow: Force on a sphere (radius a ) in an unbounded STEADY moving fluid with velocity U is explored in the following f discussion. U f z r x θ The corresponding 3D potential function for a sphere in uniform inflow is simply: a 3 φ ( r , ) = U r + 2 r 2 c o s (5.3) f The hydrodynamic force on the body due to the unsteady motion of the sphere is given as a surface integral of pressure around the body. Pressure formulation comes from the unsteady form of Bernoulli. Force in the x-direction is 1 2 F =− ρ + | ∇ | n d S (5.4) x x ∫∫ t 2 B Here, the time derivative of the potential is zero since the flow is steady and velocity is not a function of time.
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