### Turbulence Parameter Calculator at Inlet Boundary

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09 July

When modelling turbulent flows in CFD, turbulence models require the specification of turbulence variable values at the inlet boundaries. There are several ways to provide turbulence parameters at boundaries. This calculator gives all the turbulence values based on inlet conditions like velocity or mass flow rate, inlet area and fluid properties. It calculates all turbulence values like Turbulent Kinetic Energy ($k$), Turbulent Dissipation ($\epsilon$), Specific Rate of Dissipation ($\omega$), Turbulent Viscosity Ratio ($\mu_t/\mu$), Turbulence Intensity ($\textit{I}$) and Turbulence Length Scale ($\textit{l}$).

#### Turbulence Parameters Calculator at Inlet Boundary

 Input Parameters Inlet Flow Specification Method Velocity Mass Flow Rate Velocity [m/s] Inlet Area [m2] Inlet Perimeter [m] Fluid Density [kg/m3] Fluid Viscosity (Dynamic) [kg/m-s] Estimated Reynolds Numbers Turbulent Kinetic Energy ($k$) [m2/s2] Turbulent Dissipation ($\epsilon$) [m2/s3] Specific Rate of Dissipation ($\omega$) [1/s] Turbulent Viscosity Ratio ($\mu_t/\mu$) Turbulence Intensity ($\textit{I}$) [%] Turbulence Length Scale ($\textit{l}$) [m] Hydraulic Diameter ($\textit{d}$) [m]

##### Background Information

In case of CFD modelling of turbulent flow all CFD solvers needs turbulence quantities to be specified at inflow boundaries. The turbulent quantities at inlet depends on flow velocity, inlet geometry (area, perimeter and hydraulic diameter) and fluid properties.

There are many ways one can provide turbulence conditions at inlet. Most of the CFD solvers has one or all of below methods:

• Turbulent kinetic energy ($k$) and turbulent dissipation ($\epsilon$)
• Turbulent kinetic energy ($k$) and specific rate of dissipation ($\omega$)
• Intensity ($I$) and length scale ($l$)
• Intensity ($I$) and viscosity ratio ($\frac{\mu_{t}}{\mu}$)
• Intensity ($I$) and hydraulic diameter ($d$)

When providing the turbulence values at inlet, one has to make sure to provide good approximation to avoid solution convergence issues and unphysical turbulence values. Following section gives details about equations used in this calculator to calculate the turbulence properties.

##### Reynolds Number ($Re$)

Reynolds number is the ratio of inertia forces to viscous forces. This number helps to determine if the flow is laminar or turbulent.

$Re = \frac{Inertia Forces}{Viscous Forces}= \frac{\rho V L}{\mu}$

Where, $V$ is mean fluid velocity, $\rho$ is the density of fluid, $L$ is characteristic length (hydraulic diameter or traveled velocity of fluid) and $\mu$ is dynamic viscosity of fluid.

##### Turbulence Intensity ($I$)

Turbulence intensity is defined as ratio of root mean square of the velocity fluctuations $u'$, to the mean flow velocity $u_{avg}$. A turbulence intensity of 1% or less is generally considered low and turbulence intensities and greater than 10% are considered high. The turbulence intensity at the core of a fully developed duct flow can be estimated as:

$I \equiv \frac{u'}{ u_{avg}}=0.16 (Re)^{-1/8}$

##### Turbulence Length Scale ($l$)

Turbulent length scale represents the size of the large eddies in turbulent flows. In fully-developed duct flows, l is restricted by the size of the duct, since the turbulent eddies cannot be larger than the duct. An approximate relationship between l and the physical size of the duct is:

$l = 0.07 L$

Where, $L$ is the diameter of pipe in case of circular ducts and hydraulic diameter ($D_{h}$) in case of non-circular ducts.

##### Turbulent Kinetic Energy ($k$)

Turbulence kinetic energy is the mean kinetic energy per unit mass associated with eddies in turbulent flow. Physically, the turbulence kinetic energy is characterised by measured root-mean-square (RMS) velocity fluctuations. Turbulence kinetic energy can be calculated (for smooth duct) using following equation:

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

Where, $u_{avg}$ is the mean flow velocity and $I$ is turbulence intensity.

##### Turbulent Dissipation Rate ($\epsilon$)

Turbulence dissipation, is the rate at which turbulence kinetic energy is converted into thermal internal energy. It is given by:

$\epsilon= (C_{\mu})^{3/4}\frac{k^{3/2}}{l}$

Where, $C_{\mu}$ is imperial constant specified in turbulence models (approximately 0.09), $k$ is turbulent kinetic energy and $l$ is turbulent length scale.

##### Specific Dissipation Rate ($\omega$)

Specific dissipation rate is the rate at which turbulence kinetic energy is converted into thermal internal energy per unit volume and time. It is given by:

$\omega = (C_{\mu})^{-1/4}\frac{k^{3/2}}{l}$

Where $C_{\mu}$ is imperial constant specified in turbulence models (approximately 0.09), $k$ is turbulent kinetic energy and $l$ is turbulent length scale.

##### Turbulence Viscosity Ratio ($\frac{\mu_{t}}{\mu}$)

The turbulent viscosity ratio is simply the ratio of turbulent to laminar (molecular) viscosity. Turbulent viscosity ratio is given by:

$\frac{\mu_{t}}{\mu} = (C_{\mu})\rho\frac{k^{2}}{\mu\epsilon}$

Where $C_{\mu}$ is imperial constant specified in turbulence models (approximately 0.09), $\rho$ is density, $k$ is turbulent kinetic energy, $\mu$ is dynamic viscosity of fluid and $\epsilon$ is turbulence dissipation.

 Turbulence Modeling for CFD By David C. Wilcox A First Course in Turbulence By Henk Tennekes and John L. Lumley Turbulent Flows By Stephen B. Pope Computational Fluid Mechanics and Heat Transfer By Richard H. Pletcher, John C. Tannehill and Dale Anderson

##### Sanket Dange

Sanket is working as a CFD engineer in CCTech. He has worked on numerous CFD projects in the filed of automobile and HVAC. He has worked as a tutor at LearnCAx and taught to students through LeanrCAx online as well as offline course modules.

Currently he is working in the field of CFD software development which is mainly based on ANSYS FLUENT. Sanket has written blogs on various topics in CFD. He also has interest in open source CFD technologies.

Sanket holds a Bachelors degree in Mechanical engineering from University of Pune.

##### Subhransu Majhi

Subhransu is man of all treads. He has worked on almost all aspects of CFD. Started with CCTech's education brand LearnCAx, he is now moved into cloud based CFD simulation product. Subhranshu has unique combination of skill sets including strong knowledge of physcis, numerical methods and an excellent programming skill sets.

Subhranshu has extensively worked on CFD software customization and automation. He has worked on data center cooling software development. He has great passion for automation and customization and has written many blogs covering complex topic like UDF and CFD software automation.

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