Fluid Dynamics Sol ch8

Fluid Dynamics Sol ch8 - Chapter 8 Potential Flow and...

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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∂ == = ±² ³´ µ¶ | 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 = r and v r = 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= = +Ω + ± 21 2 2 12 1 1 0( ) ( ) 0( ) ( ) R RR R R R Vd s () or: . Ans Γ= Ω 22 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 ³´ ³ ´ ∇= = −= µ¶ µ ·¸·¸ ¹º ¹ º »¼ 2 2 11 1 1 1 rr . r r r r r r Ans 3 1 0 Illegitimate r 8.5 Consider the two-dimensional velocity distribution u = –By, v = + Bx, where B is a constant. If this flow possesses a stream function, find its form. If it has a velocity potential, find that also. Compute the local angular velocity of the flow, if any, and describe what the flow might represent. Solution: It does has a stream function, because it satisfies continuity: uv 0 0 0 (OK); Thus u By and v Bx xy y x ψ ∂ψ += + = = = == = Solve for ( ) . Ans −+ + 22 B c o n s t 2 It does not have a velocity potential, because it has a non-zero curl: ωφ ±² ==− = = vu 2 curl [B ( B)] 2B 0 thus . Ans V k k k does not exist The flow represents solid-body rotation at uniform clockwise angular velocity B . 8.6 If the velocity potential of a realistic two-dimensional flow is φ = Cln( x 2 + y 2 ) 1/2 , where C is a constant, find the form of the stream function ( x , y ). Hint : Try polar coordinates. Solution: Using polar coordinates is certainly an excellent hint! Then the velocity potential translates simply to = C ln( r ), which is a line source. Equation (8.12 b ) also shows that, ψθ Eq. (8.12 ): . bC A n s 1 y Ctan x 8.7 Consider a flow with constant density and viscosity. If the flow possesses a velocity potential as defined by Eq. (8.1), show that it exactly satisfies the full Navier-Stokes equation (4.38). If this is so, why do we back away from the full Navier-Stokes equation in solving potential flows?
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564 Solutions Manual Fluid Mechanics, Fifth Edition Solution: If V = φ , the full Navier-Stokes equation is satisfied identically: 2 d p becomes dt ρρ µ =−∇ + + ∇ V gV 2 2 V p ( gz) ( ), where the last term is .
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Fluid Dynamics Sol ch8 - Chapter 8 Potential Flow and...

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