Question

How does the friction factor \(f\) vary along the flow direction in the fully developed region in (a) laminar flow and (b) turbulent flow?

Short answer

Answer: In both laminar and turbulent flows, the friction factor (f) remains constant along the flow direction in the fully developed region.

Step-by-step solution

Understanding Laminar and Turbulent Flow

Laminar flow is characterized by smooth and steady flow, where fluid particles move in parallel layers or "streamlines". On the other hand, turbulent flow is characterized by random and chaotic motion of fluid particles creating swirls and eddies.

Understanding Friction Factor

The friction factor (f) is a dimensionless number that describes the resistance offered by the boundary walls of a pipe or a channel to the fluid flow taking place within it. In fully developed flow, the flow properties such as velocity and pressure remain constant in the direction of flow.

Variation of Friction Factor in Laminar Flow

In laminar flow (Reynolds number Re < 2000), the friction factor is only dependent on the Reynolds number and is given by the Hagen-Poiseuille equation: \[f = \frac{16}{Re}\] where \(Re = \frac{VD}{\nu}\) is the Reynolds number, with V being the average velocity, D the diameter of the pipe, and \(\nu\) the kinematic viscosity of the fluid. Since the Hagen-Poiseuille equation shows that f is only dependent on the Reynolds number, and the flow is fully developed, the friction factor f remains constant along the flow direction in the case of laminar flow.

Variation of Friction Factor in Turbulent Flow

In turbulent flow (Reynolds number Re > 4000), the friction factor is dependent on both the Reynolds number and the relative roughness of the pipe (\(\frac{\epsilon}{D}\)), where \(\epsilon\) is the average roughness height of the internal surface of the pipe and D is the diameter of the pipe. Various empirical correlations, such as the Colebrook-White equation or the Moody diagram, can be used to determine the friction factor in turbulent flow. However, these correlations still show that f is dependent on both Reynolds number and relative roughness. In fully developed turbulent flow, fluid properties such as velocity and pressure remain constant in the direction of flow. Therefore, both Reynolds number and relative roughness remain constant along the flow direction. Consequently, the friction factor f also remains constant along the flow direction in the case of turbulent flow. In conclusion, in both laminar and turbulent flows, the friction factor (f) remains constant along the flow direction in the fully developed region.

Key concepts

Laminar Flow

In fluid dynamics, laminar flow refers to a smooth and orderly movement of fluid. It is often visualized as layers of fluid gliding past each other. Each layer moves at a particular speed and in a straight parallel path. This orderly flow reduces energy loss due to friction. In laminar flow, there is a low velocity and less mixing between the layers. Since the movement is smooth, fewer energies are spent overcoming friction. This makes laminar flow predictable and often easier to handle in calculations, especially in pipes and channels.

Laminar flow generally occurs at low velocities or with fluids that are less viscous. It's common in systems where precision and stability are important, such as in microfluidic devices or in the flow of oil in narrow pipelines.

• Occurs when Reynolds number (Re) is less than 2000.
• Streamlines are stable and parallel.
• Friction factor is determined using the Hagen-Poiseuille equation: \( f = \frac{16}{Re} \).

Turbulent Flow

Turbulent flow is characterized by chaotic property changes and introduces a lot of complexities in fluid dynamics. Unlike laminar flow, turbulent flow involves irregular fluctuations and mixing. Instead of moving in parallel layers, the fluid particles move in swirling eddies of various sizes, directions, and speeds, leading to greater energy dissipation due to internal friction.

This erratic movement makes predicting flow characteristics more difficult. Turbulent flow is more common at higher velocities and with fluids of higher viscosity or in larger pipes.

• Occurs when Reynolds number (Re) is greater than 4000.
• Flow is random and characterized by eddy currents.
• The friction factor is dependent on the Reynolds number and the relative roughness of the pipe surface.

Engineers often utilize empirical correlations like the Colebrook-White equation or look up the Moody diagram to determine the friction factor in turbulent flow. This is important for designing systems where turbulent flow is predominant, such as in large water distribution networks.

Reynolds Number

The Reynolds number is a critical dimensionless parameter in determining the nature of flow within a fluid system. It helps decide whether the flow will be laminar or turbulent. It combines the effects of fluid velocity, density, viscosity, and a characteristic length (like the diameter of a pipe) to assess flow behavior.

Mathematically, it is expressed as:
\[ Re = \frac{VD}{u} \] where:

• \( V \) is the velocity of the fluid.
• \( D \) is the characteristic length, often the diameter of a pipe.
• \( u \) is the kinematic viscosity of the fluid.

The Reynolds number categorizes flow with an approximate criterion set at 2000 for laminar flow and 4000 for turbulent flow. Flows between these numbers are considered transitional. Knowing the Reynolds number assists engineers and scientists in designing systems that optimize flow efficiency and minimize energy losses. It provides a benchmark for controlling and predicting the behavior of the fluid in various environments.

Fully Developed Flow

Fully developed flow describes a situation where the flow characteristics are steady and unchanging along the length of a channel or pipe. This term indicates that the velocity profile, along with pressure and friction factor, remain constant as the fluid moves through the pipe.

In a fully developed flow, both axial velocity and pressure are stabilized. There are no fluctuations or disturbances along the direction of the flow. This makes it easier to predict and control, thereby simplifying calculations significantly. This stability ensures that all changes in flow parameters occur only across the radius of the pipe rather than along its length.

• In laminar flows, the velocity profile is parabolic.
• In turbulent flows, the profile is flatter due to mixing, but consistent.

Maintaining a fully developed flow is critical in many engineering applications as it impacts the efficiency and design of fluid systems. Whether analyzing laminar or turbulent flow, assuming a fully developed condition typically allows for more accurate system designs and fluid management.