Chapter%207%203D%20Irrotational%20Flows

Chapter%207%203D%20Irrotational%20Flows - VII....

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VII. Three-Dimensional Irrotational Flows 7.1 General Problems A typical problem of a three-dimensional irrotational incompressible flow around a rigid body (in Fig. 7.1) is posed graphically as follows. Governing equation in the (R, θ , ϕ ) coordinates s given by 2 Φ = + + = 0 (7.1) The boundary condition on a moving body may be given as ∇ Φ n = V b n (7.2) V b = velocity of the body due to, say, its rotation and translation. Note: (7.2) is a scalar equation. It only involves the normal component. We cannot specify tangential velocity condition for inviscid flow. The condition at infinity may be given by the uniform flow condition, ∇Φ→ U z y ∇Φ⋅ n =V b n Fig. 7.1 A uniform irrotational flow over a rigid 3-D body. x z R y x ϕ θ r Fig. 7.2 Spherical coordinates (R, θ , ϕ ) and polar-cylindrical coordinates (z, r, ϕ ).
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∇ Φ U as |x | . (7.3) Method of separation of variables for solution. Let Φ (R, θ , ϕ ) = P( θ )Q( ϕ ), (7.4) Multiply the equation (7.1) by R 2 sin 2 θ /(FPQ), we obtain R 2 sin 2 θ { + } + = 0. (7.5) Setting = - m 2 (7.6) (7.5) becomes R 2 sin 2 θ { + } = m 2 or + - = 0. (7.7) Setting = l ( l +1) (7.8) Eqn. (7.7) becomes + [ l ( l +1) - ] P = 0. (7.9) The solutions to (7.6) and (7.8) are easily found as Q( ϕ ) = e ±im ϕ (7.10) F(R) = A R l +1 + B R - l . (7.11) The solution for P( θ ) is given by associated Legendre functions (or spherical functions) of the first kind, ( ) m x P l where -1 x =cos θ 1. The number l is called the degree and the number m is called the order of the function ( ) m x P l .
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l and m, ( ) m x P l can be expressed in terms of the Legendre functions of the first kind, P l (x), as ( ) m x P l = (-1) m (1-x 2 ) m/2 P l (x), (7.12) and ( ) m x P - l = (-1) m (1-x 2 ) m/2 ( ) m x P l (7.13) Since P l (x) is a polynomial of degree l , ( ) m x P l is zero for m> l or m<- l . Define spherical harmonicsc Y l m ( θ , ϕ ) = (cos ) m P θ l e ±im ϕ , (7.14) recalling that the solution for the potential can be expressed as Φ (R, θ , ϕ ) = 0 m = = l l -l (A l m R l + B l m R -( l +1) ) Y l m ( θ , ϕ ) (7.15) The coefficients A l m and B l m are determined from the boundary conditions (7.2) and (7.3). 7.2 Axisymmetric Flows i. Solution for velocity potential The solution to general 3-D problems is quite complicated. The determination of coefficients A l m and B l m is quite involved. To gain physical insights into three-dimensional flows, we next focus on axisymmetric flows. Axisymmetry: = 0 for f= velocity, pressure, density, and temperature, In general, the axisymmetry of a flow does not necessarily require u ϕ = 0. For simplicity, however, we consider axisymmetric flows with u
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Chapter%207%203D%20Irrotational%20Flows - VII....

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