Fluid Mechanics: Fundamentals and Applications
Fluid Mechanics: Fundamentals and Applications
4th Edition
ISBN: 9781259696534
Author: Yunus A. Cengel Dr., John M. Cimbala
Publisher: McGraw-Hill Education
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Chapter 10, Problem 97P

The stramwise velocity component of steady, incompressible, laminar, flat plate boundary layer thickness δ is approximated by the simple linear expression, u = U y / δ for y < δ , and u=U for y > δ (Fig. 10-97). Generate expressions for displacement thickness momentum thickness as functions of δ , based on this linear approximation, Compare the approximate values of δ * / δ and θ / δ to the δ * / δ and θ / δ obtained from the Blasius solution.

Answer: 0.500, 0.167

Chapter 10, Problem 97P, The stramwise velocity component of steady, incompressible, laminar, flat plate boundary layer
FIGURE P10-97

Expert Solution & Answer
Check Mark
To determine

The comparison between different ratio when approximation method and blasius methods are used simultaneously.

Answer to Problem 97P

The approximation value for δ*δ is 0.5.

The approximation value for θδ is 0.167.

The blasius solution for δ*δ is 0.35.

The blasius solution for θδ is 0.135.

Explanation of Solution

Given information:

Write the expression for velocity profile if y<δ.

  u=Uyδ  ...... (I)

Here, boundary layer thickness is δ, velocity function is u, maximum velocity of the laminar flow over the flat plate is U.

Write the expression for velocity profile if y>δ.

  u=U  ...... (II)

Here, maximum velocity of the laminar flow over the flat plate is U.

Write the expression for displacement thickness.

  δ*=0δ(1 u y<δ U)dy+δ(1 u y>δU)dy  ...... (III)

Write the expression for momentum thickness.

  θ=0δu y<δU(1 u y<δ U)dy+δuy>δU(1 u y>δU)dy  ...... (IV)

Write the expression for Blasius solution for boundary layer thickness.

  δ=4.91x Rex  ...... (V)

Here, any location on flat plate is x, Reynolds number is Rex.

Write the expression for Blasius solution for displacement thickness.

  δ*=1.72x Rex  ...... (VI)

Write the expression for Blasius solution for momentum thickness.

  θ=0.664x Rex  ...... (VII)

Write the expression for the ratio of displacement thickness to boundary layer thickness.

  R=δ*δ  ...... (VIII)

Write the expression for momentum thickness to boundary layer thickness.

  S=θδ  ...... (IX)

Calculation:

Substitute Uyδ for uy<δ, and U for uy>δ in Equation (III).

  δ*=0δ( 1 ( Uy δ ) U )dy+δ(1 U U)dy=0δ(1yδ)dy+δ(0)dy=[yy22δ]0δ=δδ22δ

  δ*=δ2

The displacement thickness δ* is δ2.

Substitute Uyδ for uy<δ, and U for uy>δ in Equation (IV).

  θ=0δ Uy δ U( 1 Uy δ U )dy+δUU(1 U U)dy=0δyδ(1yδ)dy+δUU(0)dy=0δyδ(1yδ)dy+0

  θ=[ y 2 2δ y 3 3 δ 2 ]0δ=( δ 2 2δ δ 3 3 δ 2 )0=δ2δ3=δ6

The momentum thickness θ is δ6.

Substitute δ2 for δ* in Equation (VIII).

  R=( δ 2 )δ=δ2δ=120.5

Ratio of displacement thickness to boundary layer thickness is 0.5.

Substitute δ6 for θ in Equation (IX).

  S=( δ 6 )δ=δ6δ=16=0.167

Ratio of displacement thickness to boundary layer thickness is 0.167.

Divide Equation (VI) with Equation (V) when blasius solution is considered.

  δ*δ=( 1.72x Re x )( 4.91x Re x )=1.724.91=0.35

Divide Equation (VII) with Equation (V) when blasius solution is considered.

  θδ=( 0.664x Re x )( 4.91x Re x )=0.6644.91=0.135

Conclusion:

The approximation value for δ*δ is 0.5.

The approximation value for θδ is 0.167.

The blasius solution for δ*δ is 0.35.

The blasius solution for θδ is 0.135.

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