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

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

This preview shows pages 1–4. Sign up to view the full content.

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 ).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
∇ 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 .
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online