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# A Guide to Understanding Turbulent Kinetic Energy

### Key Takeaways

• Turbulent kinetic energy measures the intensity of turbulence in a flow.

• Turbulent budget terms help in identifying the stability or irresoluteness of the flow.

• Dissipation occurs when the eddies interact and the viscous force converts kinetic energy into heat.

In fluid system design, turbulence plays a critical role in heat or mass transfer, dispersion, mixing, and similar applications. However, turbulence modeling and analysis can be a challenge. This is due to the eddies and kinetic energy induced during flow, which makes the prediction of flow attributes challenging. Turbulent kinetic energy ensures statistical analysis of the different flow factors to define their behavior in a mechanical system.

CFD has mostly been relied on to calculate turbulent kinetic energy in an external or internal flow system. Through the accurate calculation of energy and dissipation rates for different flow and boundary conditions, precise CFD modeling of complex turbulent flow systems can be ensured. In this article, we will discuss turbulent kinetic energy and its mathematical description to ensure a validated turbulent model.

## What Is Turbulent Kinetic Energy?

Turbulent kinetic energy is the quantitative measure of the intensity of turbulence for a given flow. It can be measured as the root mean square of the fluctuation in flow velocity. In fluid dynamics, it can simply be defined as the mean kinetic energy per unit mass for a turbulent flow.

Mathematically, kinetic energy can generally be expressed as the following equation. Note that m= mass and v= velocity.

So, in the case of turbulent flow with time-averaged velocity components u’, v’, and w’, the turbulent kinetic energy (k) is:

The velocity component is the difference between the instantaneous and mean velocity, i.e.,. These components represent the fluctuations associated with the turbulence in each direction. These fluctuations are time-dependent, and this needs to be accounted for in the calculation of the mean and variance of turbulent velocity component with the time scale T, which can be presented as:

respectively

The above turbulent kinetic energy equation also indicates that ‘k’ is simply half the sum of all Reynold stresses. Thus, employing Boussinesq approximation for the Reynolds averaged Navier-Stokes equation, the equation for k can be deduced as:

In the above equation, the related terms are:

• A - Local derivative
• B - Advection
• C - Pressure diffusion
• D - Turbulent transport
• E - Molecular viscous transport
• F - Production
• G - Dissipation ()
• H - Buoyancy flux

### Turbulent Kinetic Energy Budget

The above equation facilitates the assessment of mechanisms such as advection, production, pressure dissipation, diffusion, and buoyancy flux. Each of these budget terms plays an important role in identifying the stability of flow or its ability to produce or diffuse turbulence in the flow system. By analyzing the contribution of each of these budget terms in the turbulent kinetic energy equation, it is possible to get an insight into the dynamics of turbulent flow and its attributes for complex turbulence modeling such as for large eddies simulations.

The different turbulent kinetic energy budget terms from the above TKE equation can be grouped for simplified analysis as:

1. Transport term:

Advection + Turbulent transport + Pressure diffusion + Molecular transport =

1. Production term:

Buoyancy flux + Production due to shear =

1. Dissipation term:

### Dissipation Rate

Energy dissipation is a common outcome of turbulence. In a turbulent flow, the eddies interact with each other, during which the kinetic energy dissipates into heat with the help of viscous force. The dissipation rate for turbulent kinetic energy can be derived from the above turbulent kinetic energy equation:

In the initial phase of dissipation, large eddies are produced. But, with time, as the flow is allowed to come to rest, these eddies break down into smaller eddies, which dissolve into even smaller units until no flow pattern is visible. This happens at the Kolmogorov microscale, which is the smallest scale in turbulent flow, when the viscous force is much more prevalent. Thus, the eddies dissipate into thermal energy.

## CFD for Turbulent Kinetic Energy Simulation

Turbulence modeling and simulation should take into account the dissipation, diffusion, transport, production, and all other factors that can influence the turbulence in flow. CFD solvers make solving the governing equations in CFD simulations easier for methods including large eddy simulation, direct numerical simulation, and Reynold Averaged- Navier Stokes simulation. With simulation platforms such as Omnis, it is possible for engineers and designers to examine the turbulence model to accurately analyze the accuracy of the simulation.

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