# Accounting Eddy Actions Using Turbulent Kinematic Viscosity in Fluid Flow Modeling

### Key Takeaways

• The rough motion of a fluid due to eddies and swirls is called turbulent flow.

• The turbulent kinematic viscosity has no physical existence and is considered a flow property, not a fluid properly.

• The effective kinematic viscosity of a fluid can be expressed as the sum of kinematic viscosity without turbulent effect or turbulent kinematic viscosity.

As the fluid flow velocity increases, laminar flow transitions to turbulent flow

In fluid systems, fluid flow can be laminar or turbulent. The turbulence in the flow regime is caused by differences in the velocity of the fluid layers. The flow resistance acting on the flow is greater in turbulent flow and is called Reynolds stress. The turbulent kinematic viscosity is a physical quantity of significance in turbulent flows. The turbulent kinematic viscosity, otherwise called eddy viscosity, is dependent on the state of the flow. In this article, we will explore turbulent flow and turbulent kinematic viscosity.

## Fluid Flows

Fluid flows are of two types: laminar or turbulent.

### Laminar Flow

A fluid flow that is uniform, even, and in order is considered a laminar flow. The laminar flow is deterministic in nature. The future behavior of the laminar flow can be predetermined from the knowledge of flow characteristics at an earlier time. The average laminar motion is in one direction even though there are irregularities and disturbances within the flow.

The uniform laminar flow of a viscous fluid can be modeled as a fluid flow containing distinct and stable layers. Each layer moves over the other in the same direction. The top layer moves at the highest velocity and the layer sticking to the boundary flows at the lowest speed. Internal friction is the reason for the velocity difference. Viscosity is used as a measure of the internal friction within a fluid. However, as the fluid flow velocity increases, the flow regime becomes turbulent.

## Turbulent Flow

As fluid flow velocity increases, the laminar flow transitions to turbulent flow. The increase in the fluid flow velocity causes the fluid layers to mix up. With increasing velocity, more fluid layers mix up and disrupt the smooth flow. The flow becomes non-uniform and disturbed with eddies and swirls. The rough motion of the fluid due to these eddies and swirls is called turbulent flow. The turbulent flow is characterized by significant velocities in different directions. The velocity directions are different from the overall direction of the flow.

### Viscosity

Viscosity is an important quantity to discuss in a turbulent flow. The fluids with high viscosity resist the turbulence in flow or transition from laminar to turbulent flow slowly. Reynolds number is significant in categorizing fluid systems where the viscosity of the fluid influences its flow velocity and flow pattern.

Let’s look at the viscosity in a turbulent flow.

## Momentum and Energy Transfer in Turbulent Flow

In turbulent flow, eddying motions are present in all sizes. Most mechanical energy in the fluid flow is utilized to form the eddies, which dissipate energy in the form of heat in the fluid. As a result of this heat dissipation, the drag force of turbulent flow is higher than that of laminar flow.

The unsteady eddying motions that are moving relative to other eddies are characteristics of turbulent flow. The eddies create fluctuations in fluid pressure as well as fluid velocity. Eddies that interact with each other exchange energy and momentum.

The eddies present in the center of a pipe with high velocity interact with eddies near the wall boundary with lower velocities. The mixing of the eddies evens out the momentum difference. The eddy action is analogous to the viscosity smoothing out the momentum differences through molecular interactions. To represent the eddy action, the terms turbulent kinematic viscosity or eddy viscosity are used.

## Turbulent Kinematic Viscosity

The turbulent kinematic viscosity is a model viscosity that accounts for the action of eddies in smoothing out the momentum gradients. The turbulent kinematic viscosity is the quantity that models the dissipation of energy and transport in the fluid flows of a turbulent nature.

The turbulent kinematic viscosity is proportional to:

1. The density of the fluid
2. The eddy velocity scale
3. The eddy length scale

The turbulent kinematic viscosity has no physical existence and is considered a flow property (not a fluid property) in turbulent flow.

## Effective Viscosity of a Fluid

The effective kinematic viscosity of a fluid can be expressed as the sum of kinematic viscosity without turbulent effect and turbulent kinematic viscosity. As the characteristics of the fluid flow significantly depend on fluid viscosity, it is important to understand these characteristics while modeling fluid flow. The influence of the turbulence effect on viscosity cannot be ignored, and the turbulent parameter also needs to be considered in the model.

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