Draft:Scale Analysis of Flow Through a Woven Mesh

Woven meshes are widely used in applications where fluid needs to pass through a porous structure, such as filters, separators, and in aerospace surfaces. A woven mesh consists of interlaced fibers, creating a pattern of pores that allow fluid to flow through. Under-standing how fluid interacts with this geometry is critical to optimizing mesh design for different industrial applications.

This article focuses on performing a scale analysis of fluid flow through a woven mesh, where the objective is to analyze the flow at different characteristic scales, derive govern-ing equations for fluid behavior, and model both viscous and inertial effects.

Mesh geometry and fluid flow

Woven mesh characteristics

A woven mesh is composed of interlaced fibers or wires arranged in a specific pattern that creates a series of pores through which fluid can pass. The geometry of the mesh significantly influences the flow behavior. Key geometric parameters of a woven mesh include:

• Fiber diameter, d: The diameter of the individual fibers or wires that form the mesh. The size of the fibers affects both the resistance to flow and the scale at which viscous forces dominate.

• Mesh spacing, S: The distance between adjacent fibers, also referred to as the pore size or aperture of the mesh. It is the primary determinant of the mesh's permeability and how much fluid can pass through per unit area.

• Porosity, ϕ: Defined as the fraction of the total area that is open to flow. Porosity can be calculated as the ratio of the pore area to the total area:

ϕ =A_pore⁄A_total

Porosity plays a crucial role in determining the permeability of the mesh and how effectively it allows fluid to pass.

• Weave pattern (plain, twill, etc.): The arrangement of fibers in the mesh. Different weave patterns can result in variations in flow characteristics. For instance, a plain weave has an alternating over-under pattern, whereas a twill weave allows fibers to cross over more than one other fiber, which results in different mechanical and flow properties.

Effect of mesh geometry on flow

The structure of the mesh introduces two characteristic scales of fluid flow:

• Microscale, which is associated with the flow around individual fibers.

• Macroscale, which is concerned with the flow through the mesh as a whole.

At the microscale, the fiber diameter d and the viscosity µ dominate flow behavior, particularly at low Reynolds numbers where viscous forces are significant. On the other hand, at the macroscale, parameters such as the mesh spacing S and the overall porosityϕ play a more prominent role in determining the flow characteristics.

When fluid flows through a woven mesh, it interacts with the fibers in a complex manner, resulting in pressure drop and velocity variations that depend on both the mesh geometry and the flow regime (laminar or turbulent).

Applications of woven meshes

Woven meshes are used in a variety of engineering applications, including:

• Filtration: In industries such as oil refining and water treatment, woven meshes act as filters that separate particles from fluids based on pore size.
• Aerospace: Meshes are used in heat exchangers, fluid distribution systems, and fuel filters.
• Textiles: In protective clothing, meshes with specific patterns are used to allow breathability while maintaining strength.
• Medical Devices: Woven meshes are employed in medical implants and stents, where fluid flow through the mesh is critical for the device's functionality.

Governing equations

The behavior of fluid flow through a woven mesh is governed by the Navier-Stokes equa-tions, which describe the motion of incompressible Newtonian fluids. The equations are expressed as follows:

ρ(∂u⁄∂t+ u · ∇u)= −∇P + µ∇²u

∇· u = 0 Where:

• u is the fluid velocity vector, which describes the speed and direction of the fluid particles at any point in space.
• ρ is the fluid density, representing the mass per unit volume of the fluid.
• µ is the dynamic viscosity of the fluid, which quantifies the internal resistance to flow due to friction between fluid layers.
• P is the pressure in the fluid, which drives the flow when a pressure gradient exists.

Reynolds number and flow regimes

The dimensionless Reynolds number is used to characterize the flow regime, either laminar or turbulent. It is defined as:

Re =ρUL⁄µ

Where:

• U is the characteristic velocity of the fluid.
• L is the characteristic length scale of the flow, which could be the fiber diameter d or the mesh spacing S depending on the scale of interest.

- **Low Reynolds Number (Laminar Flow)**: At low Re, the flow is dominated by viscous forces, and the fluid moves in smooth, predictable streamlines around the fibers.

This regime is common when dealing with very fine meshes or low-velocity fluids.

- **High Reynolds Number (Turbulent Flow)**: At high Re, inertial forces dominate, and the flow becomes turbulent. Vortices may form behind the fibers, leading to chaotic fluid motion, higher pressure drops, and increased drag. This regime is more common in coarser meshes or higher velocity flows.

Darcy's Law and permeability

At low Reynolds numbers, when the flow through the mesh is predominantly laminar, Darcy's law can be applied to describe the relationship between the pressure gradient and the fluid velocity. Darcy's law is given by:

u = −K⁄µ∇P

Where:

• K is the permeability of the mesh, a property that depends on the geometry of the mesh (porosity ϕ and mesh spacing S).
• ∇P is the pressure gradient driving the flow.
• u is the superficial velocity (flow velocity averaged over the entire cross-sectional area, including the solid parts of the mesh).

Permeability K can be estimated for woven meshes using empirical relations such as the Kozeny-Carman equation, which relates permeability to porosity and the geometric properties of the mesh:

K ≈ϕ³S²⁄C(1-ϕ²)

Where C is a constant that depends on the weave pattern and fiber arrangement.

Forchheimer Equation and inertial effects

For higher Reynolds numbers, where inertial effects become significant, Darcy's law is no longer sufficient to describe the flow. In this case, the Forchheimer equation is used, which introduces a nonlinear term to account for the increased resistance due to inertial forces:

−∇P =µ⁄Ku + βρ|u|u

Where:

• β is the Forchheimer coefficient, which depends on the mesh geometry and the flow conditions.
• The second term βρ|u|u accounts for the inertial drag that becomes significant at higher velocities.

The Forchheimer equation is particularly useful for modeling flow through woven meshes in applications involving high-velocity fluids or coarse meshes with larger pore sizes.

Microscale analysis

At the microscale, the flow of fluid around individual fibers is influenced primarily by the local interaction between the fluid and the fibers. This analysis focuses on the forces acting on the fluid as it navigates around each fiber and the corresponding fluid velocity and pressure fields.

Reynolds number at the microscale

The dimensionless Reynolds number at the fiber scale, denoted as Re_d, is based on the fiber diameter d and the characteristic fluid velocity near the fiber, u_f. It is defined as:

Re_d =ρu_fd⁄µ

Where:

• ρ is the fluid density,
• u_f is the velocity of the fluid near the fiber,
• d is the fiber diameter,
• µ is the dynamic viscosity of the fluid.

This Reynolds number helps classify the flow regime at the microscale:

• **Low Re_d (Creeping Flow)**: For Re_d < 1, the flow is dominated by viscous forces, and inertial effects are negligible. This regime is characterized by smooth, steady, and laminar flow around the fibers. This situation often occurs when the fluid velocity is low or when the fibers are very fine.
• **High Re_d (Inertial Flow)**: For Re_d ≫1, inertial effects become significant, and the flow is no longer laminar. Vortex shedding and flow separation may occur, resulting in fluctuating forces on the fibers and turbulent wake regions.

Flow regime: low Reynolds number

At low Reynolds numbers, particularly in the creeping flow regime (Re_d ≪1), the motion of the fluid around the fibers is governed primarily by viscous forces. In this regime, the Stokes flow assumption holds, and the Navier-Stokes equations reduce to the linear Stokes equations. The drag force on an individual fiber can be described using the Stokes drag law:

F_d = 4πµu_fL

Where:

• F_d is the drag force on the fiber,
• µ is the dynamic viscosity of the fluid,
• u_f is the fluid velocity near the fiber,
• L is the length of the fiber.

This law assumes that the flow around the fiber is steady and that inertial effects are negligible, which is valid for very low Reynolds numbers. The drag force is directly proportional to the fluid velocity and fiber length, and it provides a measure of the resistance the fluid experiences as it moves around the fibers.

Flow patterns around fibers

At low Re_d, the flow pattern around the fibers is smooth and streamlined, with no sepa-ration or vortices. The streamlines wrap around the fibers, and the pressure distribution is symmetric. This behavior is typical in systems where the fibers are closely spaced and the flow rate is low, such as in fine filtration applications.

Flow regime: high Reynolds number

For higher Reynolds numbers, Re_d > 1, inertial forces start to dominate the flow behavior. In this regime, the flow around the fibers becomes more complex, with phenomena such as vortex shedding and flow separation occurring. The drag on the fibers increases due to the formation of turbulent wake regions behind the fibers, which leads to an increase in the overall resistance to flow.

Inertial drag models, such as the Oseen or more complex empirical models, are re-quired to accurately capture the effects of vortex shedding and flow separation. These models account for both viscous and inertial forces:

Fd = CD1⁄2ρu f2d2

Where:

• C_D is the drag coefficient, which depends on the Reynolds number and the shape of the fibers.
• ρ is the fluid density,
• u_f is the fluid velocity near the fiber,
• d is the fiber diameter.

The drag coefficient C_D typically decreases as the Reynolds number increases, reflect-ing the transition from laminar to turbulent flow. The presence of vortices and eddies behind the fibers contributes to the higher drag observed at higher Re_d.

Implications for mesh performance

At the microscale, the performance of the mesh in allowing fluid flow is determined by the balance between viscous and inertial forces around individual fibers. In low Red regimes, the flow is predictable and manageable, while at higher Red regimes, the in-creased drag due to vortex shedding may require more energy to maintain the desired flow rate. This has significant implications for the design of woven meshes in applications such as filtration, where the flow regime must be carefully controlled to ensure optimal performance.

Macroscale analysis

At the macroscale, the woven mesh can be treated as a porous medium, and the overall flow behavior is influenced by the combined effect of many fibers. The parameters of interest at this scale are the mesh spacing S, porosity ϕ, and the pressure gradient across the mesh. The analysis at this scale considers the bulk flow through the porous structure and the resulting pressure drop.

Reynolds Number at the macroscale

The Reynolds number at the macroscale is based on the mesh spacing S and the average fluid velocity through the mesh, um. It is defined as:

Re_S =ρu_mS⁄µ

Where:

• ρ is the fluid density,
• u_m is the macroscopic fluid velocity through the mesh,
• S is the mesh spacing (the characteristic length scale at the macroscale),
• µ is the dynamic viscosity of the fluid.

This Reynolds number determines the overall flow regime through the mesh:

• **Low Re_S (Darcy Flow)**: For Re_S < 1, the flow is dominated by viscous forces, and the mesh can be treated as a porous medium with Darcy's law describing the relationship between pressure gradient and flow velocity.
• **High Re_S (Non-Darcy Flow)**: For higher Reynolds numbers, inertial effects become significant, and Darcy's law must be extended to account for these effects using the Forchheimer equation.

Darcy's Law for low Reynolds number flow

At low Reynolds numbers (Re_S < 1), the flow through the mesh can be described using Darcy's law. This empirical law relates the macroscopic velocity through the mesh to the pressure gradient driving the flow:

um = −K⁄µ∇P

Where:

• u_m is the macroscopic fluid velocity through the mesh,
• K is the permeability of the mesh, which depends on the mesh geometry (porosity ϕ and mesh spacing S),
• ∇P is the pressure gradient across the mesh,
• µ is the dynamic viscosity of the fluid.

Permeability of the mesh

The permeability K quantifies how easily fluid can flow through the mesh. For woven meshes, permeability is typically estimated using empirical relations such as the Kozeny-Carman equation:

K ≈ϕ³S²⁄C(1-ϕ²)

Where:

• ϕ is the porosity of the mesh,

• S is the mesh spacing,
• C is a shape factor that depends on the geometry and weave pattern of the mesh.

The Kozeny-Carman equation provides a good approximation for permeability in cases where the flow is predominantly viscous, and the mesh has a regular, well-defined structure.

Forchheimer Equation for higher Reynolds numbers

As the Reynolds number increases and inertial effects become significant, Darcy's law no longer holds, and the Forchheimer equation is used to describe the flow:

−∇P =µ⁄Kum + βρ|um|um

Where:

• β is the Forchheimer coefficient, which accounts for the additional resistance to flow due to inertial effects,
• The second term βρ|u_m|u_m represents the nonlinear increase in pressure drop due to inertial drag at higher velocities.

The Forchheimer equation extends Darcy's law to account for non-linearities intro duced by inertia, and it becomes essential in applications where the flow velocity through the mesh is high or the mesh spacing is large.

Conclusion

The analysis of fluid flow through a woven mesh involves understanding both microscale effects around individual fibers and macroscale effects that treat the mesh as a porous medium. At low Reynolds numbers, the flow can be modeled using Darcy's law, while higher Reynolds numbers require accounting for inertial losses using the Forchheimer equation. This thesis provides a framework for analyzing and optimizing woven mesh structures in fluid systems.

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Article prepared by

1. Gurjujhar Singh Sidhu (20135043), IIT (BHU) Varanasi
2. Jyotiska Saha (21135067), IIT (BHU) Varanasi
3. Niraj Kumar (21135087), IIT (BHU) Varanasi
4. Piyush Gupta (21135094), IIT (BHU) Varanasi
5. Piyush Gupta (21135095), IIT (BHU) Varanasi