# A Discussion on the Laminar Flow of Water Through a Pipe

### Key Takeaways

• In laminar flow, water flows through a pipe in a straight, parallel path without intermixing the layers.

• The pressure difference and viscous force balance each other in a fully developed laminar flow.

• CFD tools can help analyze the shear stress and pressure drop in the pipe flow with accurate simulation to ensure laminar behavior. In laminar flow, water flows through a pipe in a straight, parallel path without intermixing the layers

The flow regime has been a subject of intense study in the design of fluid systems. While applications involving heat transfer, mixing, and distribution may operate under a turbulent regime, laminar flow can be observed in the design of aircraft or to obtain steady flow through taps or pipes. As such, the study of these flow regimes and the importance of the Reynolds number for appropriate fluid modeling holds great significance.

In a flow system, achieving laminarity in a system can require serious consideration of the viscosity, velocity, pressure, and other factors associated with the fluid. The design of the laminar flow of water thus requires intricate calculation and simulation in computational fluid dynamics (CFD) solver applications. This article will further explore the laminar flow of water through a pipe system and explore the governing equations associated with it.

## Discussing Laminar Flow of Water

Laminar flow can be distinguished from turbulent flow when the fluid layer moves in a straight, parallel path without intermixing and creating any eddies or swirls. The fluid is usually viscous with lower velocity. The laminar flow can be commonly observed in canals or ducts where water flows without lateral mixing. With the increase in flow velocity, the laminar flow can slowly transition into a turbulent one. This can be expressed ideally in numerical terms with the help of Reynolds number. For a Reynolds number of 2300 or less, the characteristics of water flow can be considered laminar. ### Re=ρvD/μ

where,

ρ is the density of the fluid

𝒗 is the fluid velocity

D is the hydraulic diameter of the pipe

μ is the fluid viscosity

When analyzing the water flow through a pipe, molecular diffusion can be observed to be slow for a pipe with a larger diameter. However, the significance can be high for small diameter pipes.

## Fully Developed Laminar Flow of Water

Let us consider a flow through a cylindrical pipe. In a long, straight pipe section with a constant diameter, the laminar flow can be considered fully developed. If we neglect the gravitational effect in the pipe, the resulting cross-sectional velocity profile will remain the same at any point. The fully developed laminar flow in a horizontal pipe is a result of the pressure difference and viscous force balancing each other. This balance can be expressed as: On simplification, we can get: Where,

Δp - pressure difference

𝜏 - shear stress

L - length of the pipe

R - radius of the pipe

At the centerline of the pipe, the shear stress is zero (𝜏=0) while at the pipe wall, the shear stress (𝜏w) is maximum. Based on the above energy balance equation, it can be noted that Δp and l are not functions of r. Thus,

𝜏 = C.r , given C is a constant

At the pipe wall, Where C=2𝜏/D

Thus, the relation between the pressure difference and wall shear stress can be expressed as: ## Velocity Profile and Pressure Drop Analysis

The velocity profile for laminar water flow through a straight, cylindrical pipe is a function of the velocity of the flow and the radius of the pipe. It can be expressed as: Where v(max) is the velocity at the centerline of the flow or maximum velocity in the pipe of radius, R. The maximum velocity is twice the average velocity (v) of the pipe flow, i.e.: However, equating the shear stress and force balance equation, we have the formula for centerline velocity: On simplification of average and maximum velocity equations, we get the pressure drop for the laminar flow of water in a pipe as: ## Validating the Laminarity of Water Flow With CFD Solvers

The design of water flow systems including distribution networks require information on inlet/outlet velocity, fluid viscosity, pipe diameter, length, etc., which are the critical design parameters. With the help of CFD tools, the simulation of a fully developed laminar flow for a pipe system can be performed with ease. The numerical approach of a CFD solver also facilitates the calculation of the Reynolds number and the solving of the governing Navier-Stokes equation. Through the analysis of pressure and velocity distribution in the system, the potential turbulent effect can be identified. By generating an appropriate mesh for complex numerical simulations, the water flow system can conform to the laminar attributes.

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