# Exploring the Hydrodynamic Boundary Layer

April 28, 2022

### Key Takeaways

• The hydrodynamic boundary layer depends on the shear stress in a viscous fluid.

• The shear stress and viscous effect are negligible outside the hydrodynamic boundary layer.

• The Prandtl number can be used to explore the relationship between hydrodynamic and thermal boundary layers. Fluid in contact with a surface meets frictional and viscous resistance to form a velocity gradient, which defines the hydrodynamic boundary layer

A boundary layer is simply a thin layer of fluid that comes into contact with the surface, for instance, of pipe walls or flat plates during flow in a fluid system. However, the study of this simple boundary layer is an important part of fluid dynamics.

In CFD analysis, boundary layer analysis facilitates the replication of the behavior of liquid or gases along boundaries, allowing systems designers to analyze influences like shear stress or heat transfer. While the flow can be laminar or turbulent based on where the flow is examined (laminar at the upstream portion of the flow and turbulent at the downstream portion of the flow), in all conditions, a hydrodynamic boundary layer is formed. In this article, we will discuss this hydrodynamic boundary layer and its importance when working with thermal boundary layers and different flow structures.

## What Is a Hydrodynamic Boundary Layer?

Consider a flat plate that is isothermally heated to temperature T。and the incompressible fluid flows over it with freestream velocity ‘v’ at temperature ‘T’. The velocity of the flow in this case is dictated by the viscosity and the frictional force between the surface and fluid particles. During the flow, due to the frictional force, a layer of fluid gets attached to the surface wall. At a no-slip condition, the velocity of this layer is zero. With the increase in distance, the velocity of the flow increases. At a certain point, the velocity becomes one with the freestream velocity, v. This area where the velocity transitions from zero to the free stream velocity, influenced by the shear stress in the fluid, is called the hydrodynamic boundary layer. As the region indicates the flow velocity distribution, it is can also be called the momentum or velocity boundary layer.

The relationship between the velocity gradient, shear stress, and viscosity in the hydrodynamic boundary layer can be expressed as:

Shear Stress = Viscosity x Velocity Gradient

## Hydrodynamic and Thermal Boundary Layers

Hydrodynamic and thermal boundary layers greatly influence each other’s attributes. Given that the temperature of the fluid is different from that of the surface wall, the temperature of the fluid is influenced during flow. As the fluid flows through the surface of the flat plate, a temperature gradient is defined as the fluid temperature reaches to become similar to that of the freestream, thus forming a thermal boundary layer. The temperature gradient is indicative of the heat transfer taking place among the fluid layers over time. Heat transport is a decisive factor in identifying the flow rate and viscosity of the fluid. For instance, high temperatures can decrease the viscosity and thus increase the fluid flow rate. Fluid flowing over a flat plate forms a temperature gradient as the fluid temperature reaches that of the freestream, creating a thermal boundary layer

The Prandtl number is one way the thermal and hydrodynamic boundary layers can be characterized. In simple terms, the Prandtl number is a dimensionless parameter that is the ratio of momentum and thermal diffusivity. Mathematically, it can be expressed as: Note that Cp is the specific heat capacity, µ is the dynamic viscosity, and k is the thermal conductivity.

According to the above formula, when the value of the Prandtl number is small, i.e., Pr<<1, the thermal diffusivity is dominant. This means heat conduction is more compared to the convection effect. Similarly, when Pr>>1, momentum diffusivity is dominant, i.e., the convective effect is more compared to heat conduction.

## Hydrodynamic Boundary Layer Analysis Using a CFD Solver

When exploring the boundary conditions in complex fluid systems, designers need a detailed understanding of hydrodynamic boundary layers and the associated shear stress, diffusivity, and viscosity-momentum relationship. The analysis of these factors can be facilitated by a CFD solver. Through CFD analysis of the governing equations, accurate boundary layer simulation can be achieved for laminar or turbulent hydrodynamic conditions.

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