Draft:Scale Analysis of Air and Water Interaction in Urban Drainage Systems

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Scale Analysis of Air and Water Interaction in Urban Drainage Systems

Abstract

Urban drainage systems are crucial in managing stormwater and wastewater in cities, but their interaction with air, particularly during high-velocity flows, introduces complexities in fluid mechanics and environmental processes. This report focuses on the application of scale analysis to the interaction between air and water in urban drainage systems, examining key phenomena such as water flow, turbulence, air entrainment, and pressure dynamics. The scale analysis simplifies these interactions into dimensionless parameters, providing insights into the behavior of the drainage system during extreme events like storms.

Introduction

Urban drainage systems are designed to handle the collection, transport, and discharge of stormwater and wastewater in cities. During intense rain events, the interaction between water flows and air within the drainage system can cause pressure fluctuations, air entrainment, and turbulence, which can lead to system inefficiencies or failures. The complexity of this air-water interaction is challenging to model, and scale analysis provides a useful tool to simplify the equations governing these phenomena and help design more robust systems.

This project applies scale analysis to study air and water interaction in urban drainage systems, focusing on factors such as flow regimes, air entrainment, and pressure dynamics during storm events. The report also highlights how these factors influence the design and performance of urban drainage networks.

Governing Equations

1. Continuity Equation

For any incompressible fluid flow, such as water in a drainage system, the continuity equation governs the conservation of mass:

${\partial t \over \partial t}+\nabla .(\rho v)=0$

For incompressible flow, where the density ρ\rhoρ is constant:

$\nabla .v=0$

Where:

• $v$ is the velocity of water (m/s)

In urban drainage systems, this equation ensures that the mass of water entering and leaving a control volume remains constant.

2. Momentum Equation (Navier-Stokes Equation)

The momentum equation for the water flow in drainage systems is governed by the Navier-Stokes equation, which accounts for the forces acting on the fluid:

$\rho \left({\frac {\partial \mathbf {v} }{\partial t}}+\left(\mathbf {v} \cdot \nabla \right)\mathbf {v} \right)=-\nabla P+\mu \nabla ^{2}\mathbf {v} +\mathbf {F}$

Where:

• ρ is the density of water (kg/m³),
• $v$ is the velocity field (m/s),
• $P$ is the pressure (Pa),
• μ is the dynamic viscosity (Pa·s),
• $F$ represents body forces (e.g., gravity).

In the context of drainage systems, body forces such as gravitational acceleration significantly affect the flow, especially during high-velocity events like storms.

3. Air Entrainment

When water flows at high speeds through drainage systems, it can entrain air, leading to complex interactions. The amount of air entrained is related to the flow velocity and the surface roughness of the drainage channels. The rate of air entrainment $Q_{a}$ can be approximated as:

$Q_{a}=C_{d}A{\sqrt {2gH}}$

Where:

• $C_{d}$ is the drag coefficient,
• $A$ is the cross-sectional area of the flow (m²),
• $g$ is the acceleration due to gravity (9.81 m/s²),
• $H$ is the hydraulic head (m).

4. Energy Equation

The energy conservation equation for fluid flow is represented by the Bernoulli equation, applicable for inviscid flow:

${\frac {P}{\rho }}+{\frac {v^{2}}{2}}+gz={\text{constant}}$

Where:

• $P$ is the pressure (Pa),
• $v$ is the fluid velocity (m/s),
• $g$ is the acceleration due to gravity (m/s²),
• $z$ is the elevation head (m).

This equation is crucial for understanding the distribution of energy within the drainage system, particularly when dealing with changes in elevation, velocity, and pressure.

Scale Analysis

1. Dimensionless Parameters

To simplify the governing equations for air-water interaction, several dimensionless groups are introduced to scale the problem and analyze different flow regimes:

Froude Number (Fr)

The Froude number is a dimensionless number that represents the ratio of inertial forces to gravitational forces in open channel flows. It is given by:

${\text{Fr}}={\frac {v}{\sqrt {gL}}}$

Where:

• $v$ is the flow velocity (m/s),
• $g$ is the gravitational acceleration (m/s²),
• $L$ is the characteristic length (m), typically the hydraulic radius or depth of flow.

In urban drainage systems, the Froude number helps characterize whether the flow is subcritical (Fr<1) or supercritical (Fr>1). Supercritical flows are more likely to entrain air and generate turbulence.

Reynolds Number (Re)

The Reynolds number characterizes the flow regime in terms of inertial and viscous forces:

${\text{Re}}={\frac {\rho vL}{\mu }}$

Where:

• ρ is the density of water (kg/m3),
• v is the flow velocity (m/s),
• L is the characteristic length (m),
• μ is the dynamic viscosity (Pa . s).

High Reynolds numbers indicate turbulent flow, which dominates during storm events when large volumes of water pass through the drainage system.

Weber Number (We)

The Weber number compares inertial forces to surface tension forces, playing a key role in the entrainment of air bubbles:

${\text{We}}={\frac {\rho v^{2}L}{\sigma }}$

Where:

• $\sigma$ is the surface tension of water (N/m).

When the Weber number is high, inertial forces dominate, leading to the formation of air bubbles and air pockets in the drainage system.

Euler Number (Eu)

The Euler number is a dimensionless pressure number that compares the pressure forces to inertial forces:

${\text{Eu}}={\frac {P}{\rho v^{2}}}$

The Euler number is important for understanding pressure changes in the system, particularly in regions where air and water interact, such as drainage inlets or manholes.

Assumptions

1. Incompressible Flow: Both water and air are treated as incompressible fluids.
2. Steady-State Analysis: The flow is assumed to be steady during the analysis, even though real-world storm events may cause transient conditions.
3. Negligible Heat Transfer: Heat transfer between the air and water is considered negligible, focusing solely on the fluid mechanics of the interaction.
4. Turbulence in High Reynolds Number Flows: For high Re, the flow is considered turbulent, which is typical in stormwater events.

Design Considerations

1. Channel Design

Urban drainage systems must be designed to minimize the formation of air pockets and manage the interaction between air and water effectively. Trapezoidal or rectangular channels are often used to ensure that water flows smoothly, reducing turbulence and air entrainment.

2. Air Ventilation

Manholes and drainage inlets must include air vents to allow trapped air to escape during storm events. Without proper ventilation, air pockets can form and cause pressure surges that may damage the system.

3. Slope and Flow Control

The slope of drainage channels must be carefully designed to balance the flow velocity and prevent air entrainment. Steeper slopes result in higher velocities, which can lead to supercritical flow and increased air-water interaction.

Results and Discussion

The scale analysis using dimensionless parameters reveals that:

• Froude number analysis suggests that supercritical flows (high Fr\text{Fr}Fr) are more prone to air entrainment, leading to the formation of air bubbles and pockets in the drainage system.
• Reynolds number scaling indicates that most urban drainage systems operate in the turbulent regime during storm events, causing complex air-water interactions.
• Weber number analysis shows that for high-speed flows, the surface tension of the water is overcome by inertial forces, further increasing the likelihood of air entrainment.
• Euler number analysis helps in understanding how pressure fluctuations occur within drainage systems, particularly in areas with air entrainment or rapid changes in flow velocity.

Conclusion

This report demonstrates the use of scale analysis to understand the air and water interactions in urban drainage systems. By introducing dimensionless parameters such as the Froude, Reynolds, Weber, and Euler numbers, the analysis simplifies the complex behavior of air-water flows during storm events. These insights are critical for designing more efficient urban drainage systems that can handle extreme weather events without failure due to pressure surges or air pocket formation.

References

1. White, F.M. (2011). Fluid Mechanics. McGraw-Hill.
2. Munson, B.R., Young, D.F., & Okiishi, T.H. (2013). Fundamentals of Fluid Mechanics. Wiley.
3. Sturm, T.W. (2001). Open Channel Hydraulics. McGraw-Hill.
4. Bejan, A. (2013). Convection Heat Transfer. Wiley.