Fully Developed Laminar Flow

Learning Objectives

• Describe Characteristics of Fully Developed Laminar Flow: You will be able to describe the characteristics of fully developed laminar flow, including the smooth, parallel layers of fluid, the parabolic velocity profile, and the conditions under which it occurs (e.g., low Reynolds number).
• Explain the Hydrodynamic Entrance Length: You will be able to explain the concept of the hydrodynamic entrance length and its significance in fluid flow through pipes.
• Describe Variation of Hydrodynamic Entry Length: You will be able to describe how the hydrodynamic entry length varies with the Reynolds number and the type of flow (laminar or turbulent).
• Estimate Hydrodynamic Entry Length: You will be able to estimate the hydrodynamic entry length for both laminar and turbulent flows using the provided formulas.

Fully developed laminar flow can be understood by imagining you're pouring honey into a narrow tube. As the honey flows, it moves smoothly and steadily, without any chaotic swirls or turbulence. This is similar to what happens in fully developed laminar flow. In this type of flow, the fluid moves in smooth, parallel layers, with each layer sliding gently past the others without mixing. The flow has a stable shape, meaning the speed of the fluid is highest in the center of the tube and gradually decreases to zero at the walls, creating a parabolic (curved) shape. This type of flow usually occurs at low speeds, which scientists describe using a number called the Reynolds number. If the Reynolds number is less than 2000, the flow is likely to be laminar. Because the flow is so smooth and steady, it's easy to predict how the fluid will behave. This predictability is important in situations where precise control is needed, such as in medical devices or chemical experiments.

When the flow in a pipe or duct is not fully developed, it means the speed of the fluid is still changing as it moves along. At the start, the fluid's speed isn't stable. Near the entrance of the pipe, the fluid particles are adjusting to the pipe walls, so the speed distribution isn't yet in its final shape. This happens near the entrance of the pipe, where the fluid is transitioning from a uniform speed to a fully developed, stable speed profile. How long this takes depends on factors like how fast the fluid is moving and the size of the pipe. The flow can be smooth (laminar), mixed (transitional), or chaotic (turbulent), but the key idea is that the speed profile is still changing. The boundary layer, which is the area near the pipe walls where the fluid speed changes from zero to its maximum, grows and merges until the flow becomes fully developed and stable.

When fluid flows through a pipe, it can behave differently depending on whether the flow is fully developed or still developing. In developing flow, the fluid's speed is not yet stable. Near the entrance of the pipe, the fluid particles are adjusting to the pipe walls, so the speed distribution is not in its final shape. This transition occurs in the entrance region, where the fluid moves from a uniform speed to a fully developed, stable speed profile. The length of this region depends on factors like the fluid's velocity and the pipe's diameter. The flow can be smooth (laminar), mixed (transitional), or chaotic (turbulent), but the key point is that the speed profile is still evolving. The boundary layer, which is the area near the pipe walls where the fluid speed changes from zero to its maximum, grows and merges until the flow becomes fully developed and stable.

In contrast, fully developed laminar flow can be understood by imagining you're pouring water into a narrow tube. As the water flows, it moves smoothly and steadily, without any chaotic swirls or turbulence. This is similar to what happens in fully developed laminar flow, where the fluid moves in smooth, parallel layers, with each layer sliding gently past the others without mixing. The flow has a stable shape, meaning the speed of the fluid is highest in the center of the tube and gradually decreases to zero at the walls, creating a parabolic (curved) shape. This type of flow usually occurs at low speeds, which scientists describe using a number called the Reynolds number. If the Reynolds number is less than 2300, the flow is likely to be laminar. Because the flow is so smooth and steady, it's easy to predict how the fluid will behave, which is important in situations where precise control is needed, such as in medical devices or chemical experiments.

Hydrodynamic Entrance Length

When fluid flows into a pipe, it doesn't immediately settle into a smooth, steady pattern. Instead, it takes some distance for the flow to become fully developed. This distance is called the hydrodynamic entrance length.

Image Source: Entrance length (fluid dynamics) - Wikipedia

What Happens in the Entrance Length?

Initially, when fluid first enters the pipe, it has a uniform speed across the entire pipe. As the fluid moves along the pipe, the speed near the walls slows down due to friction, while the speed in the center increases. This creates a velocity gradient, or a change in speed from the walls to the center. The boundary layer, which is the area near the pipe walls where the fluid speed changes from zero (at the wall) to the maximum (at the center), grows as the fluid moves along the pipe. Eventually, the boundary layer fills the entire pipe, and the flow becomes fully developed. In this state, the velocity profile is stable and doesn't change anymore.

Imagine water flowing smoothly through a pipe, like syrup pouring out of a bottle. This is called laminar flow. In this type of flow, the water moves in neat, orderly layers. The water in the middle of the pipe moves the fastest, while the water near the edges moves the slowest. This creates a shape like a parabola (a U-shape) if you were to look at the speed of the water from the center to the edge.

Now, think about water rushing through a pipe, like a fast-moving river. This is turbulent flow. Here, the water is all mixed up, with lots of swirls and eddies (little whirlpools). Because of this mixing, the speed of the water is more even across the pipe, except right near the edges where it slows down quickly. The speed profile is flatter compared to laminar flow.

Image Source: CFD Analysis of RAM Air Flow in an Aircraft Air Conditioning System

The illustration shows how fluid flows differently in laminar and turbulent conditions.

• Laminar Flow: The fluid moves in smooth, orderly layers. The speed is highest in the middle and lowest near the edges, creating a U-shaped profile.

• Turbulent Flow: The fluid is mixed up with swirls and eddies. The speed is more even across the pipe, except near the edges where it drops quickly, making the profile flatter.

When fluid first enters a pipe, it experiences the hydrodynamic entrance region, where the wall shear stress (τw) is highest at the pipe inlet due to the smallest boundary layer thickness (as shown in the image below). As the fluid moves along the pipe, the boundary layer thickens, causing the shear stress to decrease. This results in the highest pressure drop occurring in the entrance region, which increases the average friction factor for the entire pipe. However, this increase is negligible for long pipes. In the fully developed region of the pipe, the pressure gradient and the shear stress are balanced, leading to a stable and predictable flow.

Calculating Hydrodynamic Entrance Length

The length of the hydrodynamic entry region along the pipe $$L_h$$ is known as the hydrodynamic entry length, is the distance over which the fluid flow develops from an initial, non-uniform profile to a fully developed, stable profile. This length depends on the Reynolds number, which is a dimensionless quantity representing the ratio of inertial forces to viscous forces within the fluid. The Reynolds number characterizes the type of flow, whether laminar or turbulent, and influences how quickly the flow stabilizes. In laminar flow, the hydrodynamic entry length is typically longer because the fluid layers adjust more gradually. In turbulent flow, the entry length is shorter due to the rapid mixing and chaotic motion of the fluid.

For smooth, orderly flow (laminar flow), the entrance length can be estimated using the formula
$$L_h \approx 0.06 \times Re \times D$$where Re is the Reynolds number (a measure of flow type) and D is the pipe diameter.
For chaotic, mixed flow (turbulent flow), the entrance length is shorter and can be estimated as
$$L_h \approx 10 \times D$$

SOLVE THE FOLLOWING PROBLEMS

Problem 1.

A fluid flows through a pipe with a velocity of 0.5 m/s. The pipe has a diameter of 0.1 m. The fluid has a density of 1000 kg/m³ and a viscosity of 0.001 Pa⋅s.

Tasks:

• Compute the Reynolds number.
• Identify the type of flow (laminar, transitional, or turbulent).
• Estimate the hydrodynamic entry length.

Problem 2.

A fluid flows through a pipe with a velocity of 3 m/s, a diameter of 0.15 m, and a kinematic viscosity of 1×10⁻⁶ m²/s.

Tasks:

• Compute the Reynolds number.
• Identify the type of flow (laminar, transitional, or turbulent).
• Estimate the hydrodynamic entry length.

Problem 3.

A fluid flows through a pipe with a velocity of 1 m/s. The pipe has a diameter of 0.1 m. The fluid has a dynamic viscosity of 0.001 Pa⋅s. The fluid is known to have a kinematic viscosity of 1×10⁻⁶ m²/s.

Tasks:

• Determine the fluid density using the relationship between dynamic viscosity and kinematic viscosity.
• Compute the Reynolds number.
• Identify the type of flow (laminar, transitional, or turbulent).
• Estimate the hydrodynamic entry length.