Turbulence Properties Calculator

Author: Krishnendu G S

Introduction

In CFD simulations, specifying turbulence variables at the inlets is often a challenge due to the uncertainty in the exact nature of incoming turbulence. Instead of directly estimating these complex turbulence properties, it is often more practical to use intuitive parameters such as turbulent intensity, turbulent length scale, or eddy viscosity ratio, which can be more easily related to the physical characteristics of the problem. The turbulent length scale helps in estimating the properties, but it must be chosen carefully. If it’s too large, it suggests eddies bigger than the problem itself; if too small, turbulence energy dissipates too quickly. Similarly, the eddy viscosity ratio is particularly useful in low-turbulence cases where estimating a characteristic turbulent length scale is challenging. For internal flows, it is often better to estimate the turbulent length scale. Because each of these parameters serves a different purpose and is more suitable for specific flow conditions, there is no universal method to compute all turbulence variables at once, unlike simpler fluid parameters. Instead, estimating turbulence properties step by step ensures consistency with the CFD model while maintaining physical relevance. This calculator provides a structured approach, allowing you to compute the necessary parameters based on the most appropriate inputs for your specific case.

Calculation of Turbulence Properties

To determine the required turbulence properties, this calculator uses a set of well-established equations and estimations. The following sections explain the key equations used in the calculator:

Turbulent intensity ( $I$ )

Turbulent intensity is a dimensionless measure of the fluctuations in velocity within a turbulent flow, expressed as a percentage of the mean flow velocity. It provides insight into the level of turbulence in a given flow field.

$I = \frac{u'}{U}\times 100\%$

where u′ is the root-mean-square (RMS) of velocity fluctuations, and U is the mean velocity of the flow.

Typical estimations of turbulence intensity

Low-Turbulence Case:
- Observed in flows originating from stationary fluids, such as external flow around cars, submarines, and aircraft.
- High-quality wind tunnels can also achieve very low turbulence levels.
- Typical turbulence intensity: well below 1%.
Medium-Turbulence Case:
- Found in moderately complex devices like large pipes, ventilation systems, and low-speed flows (low Reynolds number).
- Typical turbulence intensity: 1% – 5%.
High-Turbulence Case:
- Occurs in high-speed flows within complex geometries, such as heat exchangers and rotating machinery (e.g., turbines and compressors).
- Typical turbulence intensity: 5% – 20%.
Fully Developed Pipe Flow:
- For fully developed duct flow, the turbulence intensity at the core can be estimated using the empirical relation

$I = 0.16Re^{-\frac{1}{8}}$

where Re is the Reynolds number based on the pipe’s hydraulic diameter.

Turbulent length scale ( $T_L$ )

The turbulent length scale represents the size of the largest eddies in a turbulent flow. It defines the scale at which energy-containing eddies transfer energy to smaller eddies in the turbulence cascade.

Some of the commonly used methods for defining the length scale are outlined below:

Classical Definition (Based on classical book on Turbulence Modeling by Wilcox):
$T_{L\text{classical}} = C_\mu \frac{k^{3/2}}{\epsilon}$
Mixing Length Model (used in Fluent and OpenFOAM):
$\begin{aligned}T_{L\text{mixing}} &= C_\mu^{3/4} \frac{k^{3/2}}{\epsilon}\\ &\approx 1.8\times T_{L\text{classical}}\end{aligned}$
Dimensional Definition (used in CFX):
$\begin{aligned}T_{L\text{dimensional}} &= \frac{k^{3/2}}{\epsilon}\\ &\approx 11\times T_{L\text{classical}}\end{aligned}$

where C $_{\mu}$ is an empirical constant (typically 0.09), k is the turbulent kinetic energy, and $\epsilon$ is the turbulent dissipation rate.

Estimating the Turbulent Length Scale

Turbine Inlet (Complex Flow Geometries):
- At the inlet to a turbine stage or other complex geometries, the turbulent length scale is commonly set as a percentage of the channel height.
- A typical value is around 5% of the channel height.
Grid-Generated Turbulence (Flow Inside Computational Grid):
- In cases of grid-generated turbulence, the turbulent length scale is often set to be close to the size of the grid bars.
Fully Developed Pipe Flow (Internal Flow):
- For pipe flow, the turbulent length scale can be estimated using the hydraulic diameter ( $D_h$ ).
- In fully developed pipe flow, the turbulence length scale is approximately 3.8% of the hydraulic diameter: $T_L = 0.038 D_h$
- For turbulence models based on the mixing-length approach (e.g., Fluent, Openfoam), this becomes 7% of the hydraulic diameter: $T_L = 0.07 D_h$

Turbulent Kinetic Energy ( $k$ )

Turbulent kinetic energy represents the average kinetic energy per unit mass associated with eddies in a turbulent flow. It is physically characterized by the root-mean-square (RMS) velocity fluctuations. It is defined as:

$k = \frac{1}{2} (\overline{u'^2} + \overline{v'^2} + \overline{w'^2})$

where $\overline{u'}$ , $\overline{v'}$ , and $\overline{w'}$ are the turbulent velocity fluctuations in each direction (m/s).

In CFD simulations, turbulence intensity ( $I$ ) is commonly used to estimate the initial condition for turbulent kinetic energy (k).

$k = \frac{3}{2} (I U)^2$

where U is the initial velocity, and $I$ is the initial turbulence intensity (%).

Using length scale and eddy viscosity ratio:

Classical Definition (Based on classical book on Turbulence Modeling by Wilcox):
$k = \left[\frac{\mu_t}{\mu} \times \frac{\mu}{\rho T_L}\right]^2$
Mixing Length Model (used in Fluent and OpenFOAM):
$k = \left[{\frac{\mu_t}{\mu}\times \frac{\mu}{\rho C_{\mu}^{1/4}T_L}}\right]^2$
Dimensional Definition (used in CFX):
$k = \left[{\frac{\mu_t}{\mu}\times \frac{\mu}{\rho C_{\mu}T_L}}\right]^2$

where C $_{\mu}$ is an empirical constant (typically 0.09), $\rho$ is the density, $\mu$ is the molecular dynamic viscosity, $\frac{\mu_t}{\mu}$ is the viscosity ratio, and $T_L$ is the turbulent length scale.

Turbulent Dissipation Rate ( $\epsilon$ )

The turbulence dissipation rate represents the rate at which turbulent kinetic energy is converted into thermal energy due to viscous effects. It is given by:

Using length scale:

Classical Definition (Based on classical book on Turbulence Modeling by Wilcox):
$\epsilon = C_{\mu} \frac{k^{3/2}}{T_L}$
Mixing Length Model (used in Fluent and OpenFOAM):
$\epsilon = C_{\mu}^{3/4} \frac{k^{3/2}}{T_L}$
Dimensional Definition (used in CFX):
$\epsilon = \frac{k^{3/2}}{T_L}$

Using eddy viscosity ratio:

$\epsilon = C_\mu \frac{\rho k^2}{\mu}\left[{\frac{\mu_t}{\mu}\right]^{-1}$

where C $_{\mu}$ is an empirical constant (typically 0.09), k is the turbulent kinetic energy, $T_L$ is the turbulent length scale, $\rho$ is the density, $\mu$ is the molecular dynamic viscosity, and $\frac{\mu_t}{\mu}$ is the viscosity ratio.

Specific Turbulence Dissipation Rate or Turbulence frequency ( $\omega$ )

The specific turbulence dissipation rate represents the rate at which turbulent kinetic energy is dissipated into thermal internal energy per unit volume and time.

It is defined as:

$\omega = \frac{\epsilon}{C_{\mu} k}$

Using length scale:

Classical Definition (Based on classical book on Turbulence Modeling by Wilcox):
$\omega = \frac{\sqrt{k}}{T_L}$
Mixing Length Model (used in Fluent and OpenFOAM):
$\omega =C_{\mu}^{-1/4} \frac{\sqrt{k}}{T_L}$
Dimensional Definition (used in CFX):
$\omega =C_{\mu}^{-1} \frac{\sqrt{k}}{T_L}$

Using eddy viscosity ratio:

$\omega = \frac{\rho k}{\mu}\left[{\frac{\mu_t}{\mu}\right]^{-1}$

where C $_{\mu}$ is an empirical constant (typically 0.09), $\epsilon$ is the turbulent dissipation rate, k is the turbulent kinetic energy, $T_L$ is the turbulent length scale, $\rho$ is the density, $\mu$ is the molecular dynamic viscosity, and $\frac{\mu_t}{\mu}$ is the viscosity ratio.

Eddy Viscosity Ratio or viscosity ratio ( $\frac{\mu_t}{\mu}$ )

The eddy viscosity ratio represents the ratio of turbulent viscosity ( $\mu_t$ ) to molecular dynamic viscosity ( $\mu$ ), indicating the dominance of turbulent effects over molecular viscosity in a given flow.

The turbulent viscosity is given by:

$\mu_t = \rho C_{\mu} \frac{k^2}{\epsilon} = \rho\frac{k}{\omega}=\frac{\mu_t}{\mu}\times \frac{\mu}{\rho}$

The turbulent viscosity ratio is:

$\frac{\mu_t}{\mu} = \rho C_{\mu} \frac{k^2}{\mu \epsilon} = \rho\frac{k}{\mu\omega}$

where $C_\mu$ is a turbulence model constant which usually has a value of 0.09, k is the turbulent kinetic energy, $\rho$ is the density, $\mu$ is the molecular dynamic viscosity, $\epsilon$ is the turbulent dissipation rate, and $\ omega$ is the specific turbulence dissipation rate.

References

[1] Versteeg, H. K. (2007). An introduction to computational fluid dynamics the finite volume method, 2/E. Pearson Education India.

[2] CFD-Online Turbulence length scale

[3] (2025) ANSYS Fluent 2025 R1, “User Guide”.

[4] ANSYS, Inc. “ANSYS CFX-Solver Modelling guide.” (2025)

[5] OpenCFD Ltd. (2021). OpenFOAM User Guide: Version 2112. OpenCFD Ltd. https://www.openfoam.com/documentation/user-guide/

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Introduction

Calculation of Turbulence Properties

Turbulent intensity ()

Typical estimations of turbulence intensity

Turbulent length scale ()

Estimating the Turbulent Length Scale

Turbulent Kinetic Energy ()

Turbulent Dissipation Rate ()

Specific Turbulence Dissipation Rate or Turbulence frequency ()

Eddy Viscosity Ratio or viscosity ratio ()

References

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