Chapter8_SM - Chapter 8 Potential Flow and Computational...

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Chapter 8 Potential Flow and Computational Fluid Dynamics 8.1 Prove that the streamlines ψ (r, θ ) in polar coordinates, from Eq. (8.10), are orthogonal to the potential lines φ ). Solution: The streamline slope is represented by r streamline potential line dr v / r rd v (1/r)( / ) dr ∂φ ∂ θ∂φ == = ⎛⎞ ⎜⎟ ⎝⎠ | 1 Since the slope = 1 /( slope), the two sets of lines are orthogonal . Ans. 8.2 The steady plane flow in the figure has the polar velocity components v = Ω r and vr = 0. Determine the circulation Γ around the path shown. Solution: Start at the inside right corner, point A, and go around the complete path: Fig. P8.2 ππ Γ= = + v 21 22 12 1 1 0( ) ( ) 0( ) ( ) R RR R R R Vd s ( ) or: . Ans Γ= Ω π 8.3 Using cartesian coordinates, show that each velocity component (u, v, w) of a potential flow satisfies Laplace’s equation separately if 2 = 0. Solution: This is true because the order of integration may be changed in each case: ∂φ ∂∂ ∇= = ∇ = = 2 Example: ( ) (0) 0 . xx x uA n s 8.4 Is the function 1/r a legitimate velocity potential in plane polar coordinates? If so, what is the associated stream function )?
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Chapter 8 Potential Flow and Computational Fluid Dynamics 563 Solution: Evaluation of the laplacian of ( 1/r ) shows that it is not legitimate: ∂∂ ⎡⎤⎡⎤ ⎛⎞ ∇= = −= ⎜⎟ ⎢⎥⎢⎥ ⎝⎠ ⎣⎦ 2 2 11 1 1 1 rr r r r r r r Ans 3 1 0 Illegitimate r . 8.5 Consider the velocity distribution u = Ax , v = By , w = 0 . Find the conditions on A and B , if any, for which the flow has ( a ) a velocity potential; and ( b ) a stream function. Solution: ( a ) For the flow to possess a velocity potential, curl( V ) must be zero: 22 1 ; , Solve ( ) Partial answer ( ) 2 u Ax v By Ax By a xy φφ φ == = + () 0 0 0 ; Y e s , e x i s t s . z vu ωφ = =−= However, unless A and B are related, this velocity potential is not realistic. ( b ) For the flow to possess a stream function, ∇• V must be zero: ψψ ψ = = = ;, S o l v e uA x v A y A x y A n sb yx += = + = 0 This is only true if . ( ) uv AB B A So, to satisfy continuity, Ans .( a ) should be modified to = ( A /2)(x 2 – y 2 ) Ans .( a ) 8.6 Given the plane polar coordinate velocity potential = Br 2 cos(2 θ ), where B is a constant. ( a ) Show that a stream function also exists. ( b ) Find the algebraic form of ( r, ). ( c ) Find any stagnation points in this flow field. Solution: ( a ) First find the velocities from and then check continuity: 1 2 cos(2 ); 2 sin(2 ); r v Br v Br Then check 1 1 ( ) ( ) (4 cos2 ) ( 4 cos2 ) 4 cos2 4 cos2 0 r rv v Br B B r r r θθ = = + Yes, continuity is satisfied, so exists. Ans.( a ) (b) Find from its definition:
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564 Solutions Manual Fluid Mechanics, Fifth Edition 2 1 2 cos(2 ); 2 sin(2 ) Solve sin(2 ) .( ) r vB r v B r rr B rA θ n s b ψ θθ ψθ ∂∂ == = = = ( c ) By checking to see where v r and v = 0 from part ( a ), we find that the only stagnation point is at the origin of coordinates , r = 0. These functions define plane stagnation flow , Fig. 8.19 b .
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Chapter8_SM - Chapter 8 Potential Flow and Computational...

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