Explanation Given information: The duct is two dimensional, converging and compressible. The bottom wall of the duct is horizontal and flat. The streamline along x-axis ( ψ p ) is 0 . Write the expression for the velocity component along x-direction. u = u 1 + C u x Here, the velocity component along x-axis at point 1 is u 1 , the velocity coefficient is C u , and the distance along x-axis is x . Write the expression for the velocity component along y-axis. v = − ( ρ 1 C u + u 1 C p ) y − 2 C u C p x y ρ 1 + C p x Here, the density coefficient is C p , the density at point 1 is ρ 1 , and the distance along y-axis is y . Write the expression for the density of the duct. ρ = ρ 1 + C p x Write the expression for the velocity component. C u = u 2 − u 1 Δ x Write the expression for the density coefficient. C p = ρ 2 − ρ 1 Δ x Write the expression to calculate the stream function for a two dimensional flow which is compressible with respect to y-axis. ∂ ψ p ∂ y = ρ u ........... (I) Write the expression for the stream function for a compressible flow which is in two dimensions with respect to x-axis. − ρ v = ∂ ψ p ∂ x ........... (II) Calculation: Substitute u 1 + C u x for u and ρ 1 + C p x for ρ in Equation (I). ∂ ψ p ∂ y = ( ρ 1 + C p x ) ( u 1 + C u x ) = ρ 1 u 1 + C p x u 1 + C p C u x 2 ∂ ψ p ∂ y = ρ 1 u 1 + ( C p u 1 + ρ 1 C u ) x + C p C u x 2 ........... (III) Integrate Equation (III) with respect to y . ∫ ∂ ψ p ∂ y = ∫ ( ρ 1 u 1 + ( C p u 1 + ρ 1 C u ) x + C p C u x 2 ) d y ψ p = ∫ ( ρ 1 u 1 + ( C p u 1 + ρ 1 C u ) x + C p C u x 2 ) d y ψ p = ρ 1 u 1 y + ( C p u 1 + ρ 1 C u ) x y + C p C u x 2 y + f ( x ) ........... (IV) Here, the constant function is f ( x ) . Substitute ρ 1 u 1 y + ( C p u 1 + ρ 1 C u ) x y + C p C u x 2 y + f ( x ) for ψ p in Equation (II). − ρ v = ∂ ( ρ 1 u 1 y + ( C p u 1 + ρ 1 C u ) x y + C p C u x 2 y + f ( x ) ) ∂ x = 0 + ( C p u 1 + ρ 1 C u ) y + 2 C p C u x y + f ′ ( x ) − ρ v = ( C p u 1 + ρ 1 C u ) y + 2 C p C u x y + f ′ ( x )

Fluid Mechanics: Fundamentals and Applications

4th Edition

ISBN: 9781259877827

Author: CENGEL

Publisher: MCG

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Chapter 9, Problem 69P

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