Overview

Teaching: 0 min
Exercises: 25 min

Questions

• Can you calculate the thickness of a boundary layer

Objectives

• Calculate the thickness of boundary layers

• Compare boundary layer growth for various fludis

The Problem

Compare (non dimensionally) the rates of growth of the laminar boundary layer over a smooth flat plate in the following cases:

• (a) Flat plate placed in a water stream flowing at 2m/s
• (b) Flat plate placed in an air stream flowing at 2m/s
• (c) Flat plate placed in an air stream flowing at 8m/s

Density of water and air are \(1000\) and \(1.2 kg/m^3\) and viscosities are \(1\times10^{-3}\) and \(19\times10^{-6} Ns/m^2\) respectively

Challenge

What is the growth rate of a laminar boundary layer?

Solution

\(\frac{\delta}{x} = \frac{5}{\sqrt{Re}} \therefore\) \(\delta=\frac{5}{\sqrt{U}}\sqrt{\frac{\mu}{\rho}}\sqrt{x}\)

Growth rate generalized

From the challenge above we can now explore the growth rate for various fluids at various velocities

Challenge

What is the growth rate of a laminar boundary layer for water?

Answer

\(\sqrt{\frac{\mu}{\rho}}=0.001\) units

Challenge

What is the growth rate of a laminar boundary layer for air?

Answer

\(\sqrt{\frac{\mu}{\rho}}=0.004\) units

Putting this together

So for a) and b):

Since \(U\) is the same we can show that the ratio of air to water: \(\nu = \frac{u}{\rho} = 4\)

Comparing b) and c)

Since \(\rho\) and \(\mu\) are the same, we only need to consider the ratio between the square roots of \(U\) i.e. \[\frac{b}{c} = \frac{\sqrt{2}}{\sqrt{8}} = \frac{1}{2}\]

Key Points

• The boundary layer grows the fastest in case (b)

• The boundary layer grows half as fast in case (c)

• The boundary layer grows the slowest in (a)