Explanation Given information: The flow is a fully developed turbulent flow. Write the expression for velocity profile in a fully developed laminar flow. u = u max ( 1 − r 2 R 2 ) …… (I) Here, Velocity at radial distance r is u , maximum velocity at the axis is u max , radial distance from centre and radius of the pipe is R . Write the expression for shear stress in fully developed laminar flow in circular pipe. τ l a m = μ d u d r …… (II) Here, shear stress at radial distance r is τ , dynamic viscosity is μ , and velocity gradient along radial direction is d u d r . Write the expression for shear stress in fully developed turbulent flow. τ t u r = μ d u d r + ρ l 2 d u d r …… (III) Here, shear stress for fully developed turbulent flow in pipe is τ t u r , dynamic viscosity is μ , and velocity gradient along radial direction is d u d r , density of the fluid is ρ , and Prandtl’s mixing length for time averaging is l . Calculation: Substitute u max ( 1 − r 2 R 2 ) for u in Equation (II). τ l a m = μ d [ u max ( 1 − r 2 R 2 ) ] d r = μ u max [ − 1 R 2 d r 2 d r ] = − ( 2 μ u max R 2 ) r

8th Edition

ISBN: 9780073398273

Author: Frank M. White

Publisher: McGraw-Hill Education

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Fluid Dynamics

The study of the flow of gases and liquids, usually in and around solid surfaces, is known as fluid dynamics in physics and engineering. Fluid dynamics, for example, can be used to evaluate the movement of air through an airplane wing or the surface of a…

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Chapter 6, Problem 6.3WP

To determine

The shear stress varies linearly from zero at the axis to maximum at the wall in case of fully developed turbulent flow at least in the time mean.

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Chapter 6, Problem 6.2WP

Chapter 6, Problem 6.4WP

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