Turbulent kinetic energy defines the energy of motion carried by fluid flow.
Physically, turbulent kinetic energy dissipation converts mechanical energy into heat through viscous forces and eddy production.
This behavior is described by including a dissipation term in the material derivative of the turbulent kinetic energy.
There are few areas of fluid dynamics more complex than turbulent flow. Turbulence produces complex, apparently random behavior that requires a mix of stochastic and deterministic methods to describe properly. Production of turbulence may be unwanted in some systems, so it is important to know the extent to which turbulence arises and how fast it dissipates during flow. Even if turbulence is produced in a certain system, it will eventually dissipate and become unnoticeable.
Turbulent kinetic energy dissipation is determined by viscous forces affecting the body of the fluid, leading to a breakdown of eddies until, eventually, eddies diminish in size and are unnoticeable. Multiple models have been developed to describe turbulent flow and turbulent kinetic energy dissipation, both for compressible and incompressible flows. In this article, we will look at the essential concepts involved in describing turbulent kinetic energy and how the dissipation term in a fluid’s kinetic energy equation is modeled.
Defining Turbulent Kinetic Energy
The kinetic energy of a flow that is accounted for in turbulent flow is defined in terms of fluctuations in the fluid’s flow rate velocity about its steady-state (or mean) flow rate. In terms of a continuous distribution of possible flow rates and the mean flow value, the variance in the flow rate can be defined in terms of a time-averaged mean-squared value:
Variance in the fluid flow rate
The overbar quantity is the mean flow rate in the system. The variance and the other quantities in the above equation can vary in space, which will account for the distribution of turbulent kinetic energy in different regions of a system. In other words, it accounts for the fact that turbulent flow may only exist in a specific region in a system, thus the turbulent kinetic energy would be confined in that region.
The above equation only applies to one component of the fluid’s velocity field. The total turbulent kinetic energy in three dimensions is just the variance for each dimension added in quadrature.
Turbulent kinetic energy per unit mass in terms of velocity variations
The behavior of turbulent flow and the turbulent kinetic energy dissipation rate depends on whether the fluid is compressible or incompressible. In both cases, a model is needed for ε as well as a numerical simulation scheme for calculating turbulent flow behavior.
For incompressible flows, turbulent kinetic energy obeys a simple flux conservation equation:
Material derivative of the turbulent kinetic energy. This equation accounts for the production of kinetic energy by an outside force and dissipation due to viscous forces
This equation is relatively simple to work with and can account for multiple aspects of turbulence in real systems. In particular, the T term accounts for heat transport thanks to its relation to the fluid’s enthalpy. In fact, dissipation can be linked back to enthalpy.
Once the material derivative in the above equation is expanded, the time-derivative of the kinetic energy term obeys the following equation that accounts for dissipation via viscous forces:
Turbulent kinetic energy transport, diffusion, production, and dissipation terms for a viscous fluid
Together, the cross-derivative turbulent transport terms and the dissipation terms account for eddy production during flow, producing vortical behavior that shears off from the main flow and eventually dissipates.
With the above definitions, we can also describe how the turbulent kinetic energy varies for compressible flows using the material derivative. In this case, the resulting equations are more complex. Expanding the material derivative gives a modified form of the previous equation, where the dissipation terms now depend on the density of the fluid, which may change in space and time.
Solving compressible flow problems and calculating turbulent kinetic energy dissipation in a simulation requires a model that defines the relationship between stress and strain in the compressible fluid. This will then be included in the above local derivative equation as part of the dissipation term. Some of the popular turbulent transport models that can account for dissipation include the k-ε, k-⍵, and Spalart-Allmaras models. One recent review of models that can be used to estimate turbulent kinetic energy dissipation is:
- Wang, G., et al. "Estimation of the dissipation rate of turbulent kinetic energy: A review." Chemical Engineering Science 229 (2021): 116133.
Calculating and Measuring Turbulent Kinetic Energy Dissipation
The usage of these equations in describing turbulent kinetic energy dissipation is complex, and these problems are intractable to solve by hand. However, some standard solution algorithms can be implemented in a CFD simulation application. Direct numerical simulation (DNS), Reynolds-averaged numerical simulation, and large eddy simulation techniques can be used to simulate turbulent flow and dissipation of turbulent flow via diffusion and eddy production processes. This behavior can be visualized in simulations both for compressible and incompressible flows, and a dissipation rate could be calculated over distance by examining how fluctuations decay in space.
Measurements can be performed in a variety of ways. Simpler measurements involve tracking floating markers along a flow, followed by some statistical analysis of the turbulence that arises during flow based on tracking the marker’s motion. Contemporary methods that produce highly accurate results involve hot-wire or hot-film anemometers, acoustic Doppler measurements, or laser Doppler measurements.
Recently, much more sophisticated measurement techniques have been developed, which can expedite data acquisition and the processing of turbulent kinetic energy dissipation measurements. One excellent recent example involves concurrent UAV and radar measurements of turbulent dissipation, followed by an estimation of the dissipation rate using regression analysis. Details can be found in this publication:
Systems designers that need to determine turbulent kinetic energy dissipation can analyze their systems with the set of fluid dynamics analysis and simulation tools in Omnis 3D Solver. The analysis tools from Cadence are ideal for defining and running CFD simulations with modern numerical approaches, including compressible and incompressible flows in complex systems.