A liquid flows in a 30 cm diameter pipe at a Reynolds number of 10⁶ . If the friction factor is 0.025, the thickness of the laminar sublayer, in mm is

Question

A liquid flows in a 30 cm diameter pipe at a Reynolds number of 106 . If the friction factor is 0.025, the thickness of the laminar sublayer, in mm is

Prepp Team · Accepted Answer

Understanding Laminar Sublayer in Pipe Flow

In turbulent flow within a pipe, the fluid velocity fluctuates significantly. However, very close to the pipe wall, viscous effects dominate, and the flow becomes smooth or laminar. This thin region is known as the laminar sublayer or viscous sublayer.

The thickness of the laminar sublayer is often characterized in terms of dimensionless wall units, denoted by $y^+$. The value of $y^+$ is defined as:

$ y^+ = \frac{y u^*}{\nu} $

where:

$ y $ is the distance from the wall.
$ u^* $ is the friction velocity.
$ \nu $ is the kinematic viscosity of the fluid.

The laminar sublayer is typically considered to be the region where $ y^+ \lesssim 5 $. However, sometimes the viscous sublayer boundary is taken at $ y^+ \approx 11.6 $ or other values depending on the context or correlation being used. The calculation matching the provided option uses a value close to 11.6.

Key Formulas for Laminar Sublayer Thickness

The thickness of the laminar sublayer, denoted by $ \delta_l $, can be found from the definition of $ y^+ $. If we consider $ y = \delta_l $ at the edge of the sublayer with a specific $ y^+ $ value, then:

$ \delta_l^+ = \frac{\delta_l u^*}{\nu} $

So, the laminar sublayer thickness is:

$ \delta_l = \frac{y^+ \nu}{u^*} $

The friction velocity $ u^* $ is related to the mean flow velocity $ U $ and the Darcy-Weisbach friction factor $ f $ by the formula:

$ u^* = U \sqrt{\frac{f}{8}} $

The Reynolds number $ Re $ for flow in a pipe of diameter $ D $ is defined as:

$ Re = \frac{U D}{\nu} $

From the Reynolds number definition, the mean velocity $ U $ can be expressed as:

$ U = \frac{Re \cdot \nu}{D} $

Substitute the expression for $ U $ into the formula for $ u^* $:

$ u^* = \left(\frac{Re \cdot \nu}{D}\right) \sqrt{\frac{f}{8}} $

Now substitute this expression for $ u^* $ into the formula for $ \delta_l $:

$ \delta_l = \frac{y^+ \nu}{\left(\frac{Re \cdot \nu}{D}\right) \sqrt{\frac{f}{8}}} = \frac{y^+ \nu \cdot D}{Re \cdot \nu \sqrt{\frac{f}{8}}} $

The kinematic viscosity $ \nu $ cancels out:

$ \delta_l = \frac{y^+ D}{Re \sqrt{\frac{f}{8}}} $

Step-by-Step Calculation of Laminar Sublayer Thickness

Given values:

Pipe diameter, $ D = 30 \text{ cm} = 0.3 \text{ m} $
Reynolds number, $ Re = 10^6 $
Friction factor, $ f = 0.025 $

We use the derived formula for laminar sublayer thickness $ \delta_l = \frac{y^+ D}{Re \sqrt{f/8}} $. As noted, the provided option aligns with using $ y^+ \approx 11.6 $.

Calculate the term $ \sqrt{f/8} $:

$ \sqrt{\frac{f}{8}} = \sqrt{\frac{0.025}{8}} = \sqrt{0.003125} \approx 0.0559017 $

Substitute the values into the formula using $ y^+ = 11.6 $:

$ \delta_l = \frac{11.6 \times 0.3}{10^6 \times 0.0559017} $

$ \delta_l = \frac{3.48}{55901.7} $

$ \delta_l \approx 0.00006225 \text{ m} $

Convert the thickness from meters to millimeters:

$ \delta_l \approx 0.00006225 \text{ m} \times 1000 \text{ mm/m} $

$ \delta_l \approx 0.06225 \text{ mm} $

This calculated value of approximately 0.062 mm matches one of the given options.

Explanation Based on Provided Answer

While the laminar sublayer is strictly defined for $ y^+ \le 5 $, the viscous sublayer boundary is often extended to $ y^+ \approx 11.6 $ or even higher in various engineering correlations and practices. The calculation yielding 0.062 mm corresponds to using $ y^+ = 11.6 $ in the formula for boundary layer thickness.

Revision Table: Key Concepts in Turbulent Pipe Flow

Concept	Description	Formula/Significance
Turbulent Flow	Flow characterized by random velocity fluctuations.	High Reynolds number ($ Re $).
Laminar Sublayer	Thin layer near the wall where flow is dominated by viscosity and is essentially laminar.	Region where $ y^+ \le 5 $ (typically).
Viscous Sublayer	Includes the laminar sublayer, where viscous stresses are dominant. Boundary often considered up to $ y^+ \approx 11.6 $.	Region where $ y^+ \lesssim 11.6 $.
Friction Velocity ($ u^* $)	A characteristic velocity scale related to wall shear stress.	$ u^* = \sqrt{\tau_w / \rho} $ or $ u^* = U \sqrt{f/8} $.
Reynolds Number ($ Re $)	Dimensionless number indicating the ratio of inertial forces to viscous forces.	$ Re = \rho U D / \mu = U D / \nu $.
Friction Factor ($ f $)	Dimensionless number relating pressure drop to kinetic energy of the flow.	Darcy-Weisbach $ f $.

Additional Information on Boundary Layers

In turbulent flow, the boundary layer near a solid surface is conceptually divided into multiple regions:

Laminar Sublayer (Viscous Sublayer): The region closest to the wall ($y^+ \lesssim 5$) where viscous shear stress dominates and the velocity profile is nearly linear, $u \approx (u^{*2}/\nu)y$.
Buffer Layer: A transition region ($5 \lesssim y^+ \lesssim 30$) where both viscous and turbulent shear stresses are significant.
Turbulent Core (Outer Layer): The region farthest from the wall ($y^+ \gtrsim 30$) where turbulent shear stress dominates and the velocity profile follows a logarithmic or power law.

The thickness of these layers is dependent on the friction velocity and kinematic viscosity, which in turn depend on the overall flow conditions (like Reynolds number, pipe diameter, and surface roughness, reflected in the friction factor).

The calculation of laminar sublayer thickness is important for understanding heat and mass transfer near the wall, as transport mechanisms are different in the laminar and turbulent regions.

Prepp Team · Answer

0.025

Prepp Team · Answer

0.0031

Prepp Team · Answer

1.00