Reynolds-averaged Navier–Stokes equations#Derivation of RANS equations

The basic tool required for the derivation of the RANS equations from the instantaneous Navier–Stokes equations is the Reynolds decomposition. Reynolds decomposition refers to separation of the flow variable (like velocity $u$) into the mean (time-averaged) component ($\backslash overline\{u\}$) and the fluctuating component ($u^\{\backslash prime\}$). Because the mean operator is a Reynolds operator, it has a set of properties. One of these properties is that the mean of the fluctuating quantity is equal to zero $(\backslash bar\{u\text{'}\})$ {{cite book|author= Henk Tennekes |title= A first course in turbulence|year=1992|publisher=MIT Press|location=Cambridge, Mass. [u.a.]|isbn=978-0-262-20019-6|edition=14. print.| author2=Lumley, J. L. }} Some authors prefer using $U$ instead of $\backslash bar\{u\}$ for the mean term (since an overbar is sometimes used to represent a vector). In this case, the fluctuating term $u^\backslash prime$ is represented instead by $u$. This is possible because the two terms do not appear simultaneously in the same equation. To avoid confusion, the notation $u$, $\backslash bar\{u\}$, and $u\text{'}$ will be used to represent the instantaneous, mean, and fluctuating terms, respectively.

The properties of Reynolds operators are useful in the derivation of the RANS equations. Using these properties, the Navier–Stokes equations of motion, expressed in tensor notation, are (for an incompressible Newtonian fluid):

$$\backslash frac\{\backslash partial\; u\_i\}\{\backslash partial\; x\_i\}\; =\; 0$$

$$\backslash frac\{\backslash partial\; u\_i\}\{\backslash partial\; t\}\; +\; u\_j\; \backslash frac\{\backslash partial\; u\_i\}\{\backslash partial\; x\_j\}$$

= f_i

- \frac{1}{\rho} \frac{\partial p}{\partial x_i}

+ \nu \frac{\partial^2 u_i}{\partial x_j \partial x_j}

where $f\_i$ is a vector representing external forces.

Next, each instantaneous quantity can be split into time-averaged and fluctuating components, and the resulting equation time-averaged,

{{efn|

Splitting each instantaneous quantity into its averaged and fluctuating components yields,

$$\backslash frac\{\backslash partial\; \backslash left(\; \backslash bar\{u\_i\}\; +\; u\_i^\backslash prime\; \backslash right)\}\{\backslash partial\; x\_i\}\; =\; 0$$

$$\backslash frac\{\backslash partial\; \backslash left(\; \backslash bar\{u\_i\}\; +\; u\_i^\backslash prime\backslash right)\}\{\backslash partial\; t\}$$

+ \left( \bar{u}_j + u_j^\prime\right) \frac{\partial \left( \bar{u}_i + u_i^\prime\right)}{\partial x_j}

= \left( \bar{f}_i + f_i^\prime\right)

- \frac{1}{\rho} \frac{\partial \left(\bar{p} + p^\prime\right)}{\partial x_i}

+ \nu \frac{\partial^2 \left( \bar{u}_i + u_i^\prime\right)}{\partial x_i \partial x_j}.

Time-averaging these equations yields,

$$\backslash overline\{\backslash frac\{\backslash partial\; \backslash left(\; \backslash bar\{u\}\_i\; +\; u\_i^\backslash prime\; \backslash right)\}\{\backslash partial\; x\_i\}\}\; =\; 0$$

$$\backslash overline\{\backslash frac\{\backslash partial\; \backslash left(\; \backslash bar\{u\}\_i\; +\; u\_i^\backslash prime\backslash right)\}\{\backslash partial\; t\}\}$$

+ \overline{\left( \bar{u}_j + u_j^\prime\right) \frac{\partial \left( \bar{u_i} + u_i^\prime\right)}{\partial x_j}}

= \overline{\left( \bar{f}_i + f_i^\prime\right)}

- \frac{1}{\rho} \overline{\frac{\partial \left(\bar{p} + p^\prime\right)}{\partial x_i}}

+ \nu \overline{\frac{\partial^2 \left( \bar{u}_i + u_i^\prime\right)}{\partial x_i \partial x_j}}.

Note that the nonlinear terms (like $\backslash overline\{u\_i\; u\_j\}$) can be simplified to

$$\backslash overline\{u\_i\; u\_j\}$$

= \overline{\left( \bar{u}_i + u_i^\prime \right)\left( \bar{u_j} + u_j^\prime \right) }

= \overline{\bar{u_i}\bar{u}_j + \bar{u}_i u_j^\prime + u_i^\prime\bar{u}_j + u_i^\prime u_j^\prime}

= \bar{u}_i\bar{u}_j + \overline{u_i^\prime u_j^\prime}

}}

to yield:

$$\backslash frac\{\backslash partial\; \backslash bar\{u\}\_i\}\{\backslash partial\; x\_i\}\; =\; 0$$

$$\backslash frac\{\backslash partial\; \backslash bar\{u\}\_i\}\{\backslash partial\; t\}$$

+ \bar{u}_j\frac{\partial \bar{u}_i }{\partial x_j}

+ \overline{u_j^\prime \frac{\partial u_i^\prime }{\partial x_j}}

= \bar{f}_i

- \frac{1}{\rho}\frac{\partial \bar{p}}{\partial x_i}

+ \nu \frac{\partial^2 \bar{u}_i}{\partial x_j \partial x_j}.

The momentum equation can also be written as,

{{efn|

This follows from the mass conservation equation which gives,

$$\backslash frac\{\backslash partial\; u\_i\}\{\backslash partial\; x\_i\}\; =\; \backslash frac\{\backslash partial\; \backslash bar\{u\}\_i\}\{\backslash partial\; x\_i\}\; +\; \backslash frac\{\backslash partial\; u\_i^\backslash prime\}\{\backslash partial\; x\_i\}\; =\; 0$$

}}

$$\backslash frac\{\backslash partial\; \backslash bar\{u\}\_i\}\{\backslash partial\; t\}$$

+ \bar{u}_j\frac{\partial \bar{u}_i }{\partial x_j}

= \bar{f}_i

- \frac{1}{\rho}\frac{\partial \bar{p}}{\partial x_i}

+ \nu \frac{\partial^2 \bar{u}_i}{\partial x_j \partial x_j}

- \frac{\partial \overline{u_i^\prime u_j^\prime }}{\partial x_j}.

On further manipulations this yields,

$$\backslash rho\; \backslash frac\{\backslash partial\; \backslash bar\{u\}\_i\}\{\backslash partial\; t\}$$

+ \rho \bar{u}_j \frac{\partial \bar{u}_i }{\partial x_j}

= \rho \bar{f}_i

+ \frac{\partial}{\partial x_j}

\left[ - \bar{p}\delta_{ij}

+ 2\mu \bar{S}_{ij}

- \rho \overline{u_i^\prime u_j^\prime} \right]

where, $$

\bar{S}_{ij} = \frac{1}{2}\left( \frac{\partial \bar{u}_i}{\partial x_j} + \frac{\partial \bar{u}_j}{\partial x_i} \right)

is the mean rate of strain tensor.

Finally, since integration in time removes the time dependence of the resultant terms, the time derivative must be eliminated, leaving:

$$\backslash rho\; \backslash bar\{u\}\_j\backslash frac\{\backslash partial\; \backslash bar\{u\}\_i\; \}\{\backslash partial\; x\_j\}$$

= \rho \bar{f_i}

+ \frac{\partial}{\partial x_j}

\left[ - \bar{p}\delta_{ij}

+ 2\mu \bar{S}_{ij}

- \rho \overline{u_i^\prime u_j^\prime} \right].

The time evolution equation of Reynolds stress is given by:{{cite journal|author= P. Y. Chou |title=On velocity correlations and the solutions of the equations of turbulent fluctuation|journal=Quart. Appl. Math.|volume=3|year=1945 | pages=38–54 | doi=10.1090/qam/11999 |doi-access=free}}

\frac{\partial \overline{u_i^\prime u_j^\prime}}{\partial t}

+ \bar{u}_k \frac{\partial \overline{u_i^\prime u_j^\prime}}{\partial x_k} =

-\overline{u_i^\prime u_k^\prime}\frac{\partial \bar{u}_j}{\partial x_k}

-\overline{u_j^\prime u_k^\prime}\frac{\partial \bar{u}_i}{\partial x_k}

+\overline{ \frac{p^\prime}{\rho}\left( \frac{\partial u_i^\prime}{\partial x_j}

+\frac{\partial u_j^\prime}{\partial x_i} \right) }

- \frac{\partial }{\partial x_k} \left( \overline{u_i^\prime u_j^\prime u_k^\prime}

+ \frac{\overline{p^\prime u_i^\prime } }{\rho} \delta_{jk}

+ \frac{\overline{p^\prime u_j^\prime } }{\rho} \delta_{ik}

- \nu \frac{\partial \overline{u_i^\prime u_j^\prime}}{\partial x_k} \right)

-2 \nu \overline{\frac{\partial u_i^\prime}{\partial x_k} \frac{\partial u_j^\prime}{\partial x_k}}

This equation is very complicated. If $\backslash overline\{u\_i^\backslash prime\; u\_j^\backslash prime\}$ is traced, turbulence kinetic energy is obtained.

The last term $\backslash nu\; \backslash overline\{\backslash frac\{\backslash partial\; u\_i^\backslash prime\}\{\backslash partial\; x\_k\}\; \backslash frac\{\backslash partial\; u\_j^\backslash prime\}\{\backslash partial\; x\_k\}\}$ is turbulent dissipation rate. All RANS models are based on the above equation.

• A model for testing performance was determined that, when combined with the vortex lattice (VLM) or boundary element method (BEM), RANS was found useful for modelling the flow of water between two contrary rotation propellers, where VLM or BEM are applied to the propellers and RANS is used for the dynamically fluxing inter-propeller state.{{cite web |url=https://www.marinepropulsors.com/proceedings/2017/WA3-1.pdf |title=Application of a BEM/RANS Interactive Method to Contra-Rotating Propellers |last1=Su |first1=Yiran |last2=Kinnas |first2=Spyros A. |last3=Jukola |first3=Hannu |date=June 2017 | year=2017 |department= |website=www.marinepropulsors.com |series= |publisher=International Symposium on Marine Propulsion | agency=(Su: Ocean Engineering Group, Department of Civil, Architectural and Environmental Engineering The University of Texas at Austin; Jukola: Steerprop Ltd. P.O. Box 217, FI-26101 Rauma, Finland) |location=Espoo, Finland |page=1 |access-date=2 July 2021 |archive-url= |archive-date= |via=Google Scholar |quote= }}
• The RANS equations have been widely utilized as a model for determining flow characteristics and assessing wind comfort in urban environments. This computational approach can be executed through direct calculations involving the solution of the RANS equations, or through an indirect method involving the training of machine learning algorithms using the RANS equations as a basis. The direct approach is more accurate than the indirect approach but it requires expertise in numerical methods and computational fluid dynamics (CFD), as well as substantial computational resources to handle the complexity of the equations.{{cite journal |last1=BenMoshe |first1=Nir |last2=Fattal |first2=Eyal |last3=Leitl |first3=Bernd |last4=Arav |first4=Yehuda |title=Using Machine Learning to Predict Wind Flow in Urban Areas |journal=Atmosphere |date=June 2023 |volume=14 |issue=6 |pages=990 |doi=10.3390/atmos14060990 |doi-access=free }}

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• Favre averaging

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