Problem 13 - A Heat Transfer Textbook

Question

It is known from flow measurements that the transition to turbulence occurs when the Reynolds number based on mean velocity and diameter exceeds 4000 in a certain pipe. Use the fact that the laminar boundary layer on a flat plate grows according to the relation $$ \frac{\delta}{x}=4.92 \sqrt{\frac{v}{u_{\max } x}} $$ to find an equivalent value for the Reynolds number of transition based on distance from the leading edge of the plate and \(u_{\max }\). (Note that \(u_{\max }=2 \bar{u}_{\mathrm{av}}\) during laminar flow in a pipe.)

Step-by-Step Solution

Answer

The equivalent Reynolds number of transition based on \(x\) is 4000.

Step 1: Understand the Problem

We need to find an equivalent Reynolds number for the transition from laminar to turbulent flow on a flat plate using given relations for a flat plate and a pipe. We know the critical Reynolds number for a pipe is 4000, and the maximum velocity is related to the average velocity.

Step 2: Relationship Between Parameters

First, recognize the given relation: \( \frac{\delta}{x} = 4.92 \sqrt{\frac{v}{u_{\max} x}} \). Here, \(\delta\) is the boundary layer thickness, \(x\) is the distance from the leading edge, \(v\) is the kinematic viscosity, and \(u_{\max}\) is the maximum velocity. For a pipe, \(u_{\max} = 2 \bar{u}_{\mathrm{av}}\).

Step 3: Calculate Reynolds Number

The Reynolds number for flow over a flat plate is defined as \( \text{Re}_x = \frac{u_{\max} x}{v} \). Our task is to find the equivalent value of this Reynolds number for transition to turbulence. Rearrange the given equation: \(4.92 \sqrt{\frac{v}{u_{\max} x}} = \frac{\delta}{x}\), leading to \( \frac{v}{u_{\max} x} = \left(\frac{\delta}{4.92 x}\right)^2 \).

Step 4: Equate to Critical Reynolds Number for Pipe

Use \( u_{\max} = 2 \bar{u}_{\mathrm{av}} \) to find \( \text{Re}_D = \frac{2 \bar{u}_{\mathrm{av}} D}{v} = 4000 \). Equate this relation to the flat plate form: \( \text{Re}_x = \frac{u_{\max} x}{v} = \frac{2 \bar{u}_{\mathrm{av}} x}{v} \).

Step 5: Solve for Reynolds Number for Flat Plate

At transition, equate the Reynolds number: \( \text{Re}_x = 4000 \) (as an equivalent) based on the relation: \( \text{Re}_x = \frac{2 \bar{u}_{\mathrm{av}} x}{v} = 4000 \). Hence, the equivalent Reynolds number of transition based on distance \( x \) is 4000.

Key Concepts

Laminar Flow

Laminar flow is a flow regime characterized by smooth, parallel layers of fluid that do not mix. This type of flow occurs at lower velocities and is highly predictable. In laminar flow, each layer of fluid moves smoothly past the adjacent layers with little to no disruption between them. This orderly flow results in low energy loss and occurs when the Reynolds number is below a critical threshold, typically around 2000 for flow in pipes. Understanding laminar flow is crucial in applications where minimizing energy loss is essential, such as in the transportation of viscous fluids.

• The motion is orderly with a high degree of uniformity.
• Navigates quietly without turbulence or eddies.
• Applicable in precise flow situations like chemical reactors.

Turbulent Flow

Turbulent flow occurs when fluid flows in a chaotic manner, with eddies and vortices creating unpredictable movements. This type of flow generally happens at high velocities when the Reynolds number exceeds the critical value, commonly seen as 4000 in pipe flow scenarios. Unlike laminar flow, turbulent flow is associated with significant mixing and greater energy dissipation due to its chaotic nature. This high-energy and disorderly motion enhances heat and mass transfer, making it applicable in systems like engines and reactors where such characteristics are advantageous.

• Characterized by chaotic and irregular fluid motion.
• Involves significant mixing and increased frictional effects.
• Common in natural settings, like rivers and ocean currents.

Reynolds Number

The Reynolds number ( ext{Re}) is a dimensionless quantity that predicts the flow regime of a fluid, whether it will be laminar or turbulent. It is calculated as the ratio of inertial forces to viscous forces in a fluid and is determined by the formula \[ ext{Re} = \frac{u L}{v} \]where \(u\) is the flow velocity, \(L\) is a characteristic length (like pipe diameter), and \(v\) is the kinematic viscosity of the fluid. A low Reynolds number indicates laminar flow, while a high Reynolds number signifies turbulent flow. The critical Reynolds number varies based on the system, such as 2000 in pipes and around 500,000 for flow over flat plates.

• Critical in determining and predicting flow patterns.
• High Re suggests turbulence, low Re suggests laminar behavior.
• Helps in designing pipes, ducts, and open channels.

Boundary Layer

The boundary layer is a thin sheet of fluid that forms along a solid surface, where viscous effects are significant. When fluid flows past a solid surface, the velocity of the layer of fluid in immediate contact with the surface is zero due to the no-slip condition. Gradually, the velocity increases to match the free stream velocity, resulting in a velocity gradient. This region where the velocity changes from zero to approximately 99% of the free stream velocity is called the boundary layer. The thickness of the boundary layer affects drag and heat transfer rates across surfaces.

• Influences the overall aerodynamic and hydrodynamic performance.
• Boundary layer separation can lead to increased drag.
• Key concept in improving efficiency in vehicles and aircraft.

Kinematic Viscosity

Kinematic viscosity, a fundamental property in fluid dynamics, measures a fluid's internal resistance to flow under gravitational forces. It is defined as the ratio of dynamic viscosity to density, expressed mathematically as \[ u = \frac{\mu}{\rho} \] where \(u\) is the kinematic viscosity, \(\mu\) is the dynamic viscosity, and \(\rho\) is the fluid density. Kinematic viscosity is essential for calculating the Reynolds number, influencing whether a flow is laminar or turbulent. This property helps engineers and scientists design systems that leverage specific flow characteristics to optimize processes and instrument functionality.

• Determines flow behavior under gravitational effects.
• Crucial for simulation and analysis in fluid mechanics.
• Affects time-dependent processes like sedimentation.