Inlet

Inlet#

The inlet of a flow is usually defined by a pressure gradient or a imposed velocity, the BCs most frequently used to represent this kind of boundary are described below:

Uniform Velocity#

This boundary condition forces an uniform velocity and pressure profile. Its implementation consists on setting the populations to its equilibrium fixing velocity and pressure values.

(1)#\[f_{\bar{i}}\left(t+\Delta t\right)=f_{i}^{\mathrm{eq}}\left(\rho,\mathbf{u}\right)\]

Use Case

Uniform velocity is used to assure a constant velocity and density inlet, which increases the simulation stability and allows for the calculation of the pressure coefficient.

Synthetic Eddy Method#

The synthetic eddy method (SEM) builds a turbulent inlet with transient aspect, it was first introduced by Jarrin et al.[1] and presented in an LBM framework by Buffa et al.[2]. It requires the user to provide the flow average velocity profile \(\bar{u}_{\alpha}\) and Reynolds stress tensor \(R_{\alpha\beta}\) associated to the desired turbulent intensity \(\left\{I_{u},I_{v},I_{w}\right\}\). That way, it is possible to reproduce atmospheric boundary layer profiles at inlet.

Note

The input data for the SEM can be obtained from experimental wind tunnel data or by performing a precursor simulation that reproduces a wind tunnel configuration necessary to generate the wind profile of the desired terrain category.

In the SEM, \(N\) synthetic eddies are distributed in random positions \(\left\{x_{i},y_{i}z_{i}\right\}\) in a virtual domain as illustrated below:

../../_images/schematic_sem.svg

The variable \(L\) is a representative length scale provided as input data that represents the radius of influence of each eddy. That way, the interval to which eddies will be distributed in \(x\) is \(\left[-L,L\right]\), in \(y\) is \(\left[0-L,n_{y}+L\right]\), and in \(z\) is \(\left[0-L,n_{z}+L\right]\). The inlet of the simulation domain is then given by the velocity field in the plane \(x=0\). There, the velocity is given by:

(2)#\[u_{\alpha}\left(0,y,z\right) = \bar{u}_{\alpha}\left(z\right) + {u}'_{\alpha}\left(0,y,z,t\right)\]

where \(u'\) is a fluctuation velocity responsible to provide turbulent intensity at inlet, this vector field is calculated as:

(3)#\[u'_{\alpha}\left(x,y,z,t\right)=A_{\alpha\beta}\left(z\right)\tilde{u}_{\beta}\left(x,y,z,t\right)\]

in which, \(A_{\alpha\beta}\) is the Cholesky decomposition of the prescribed Reynolds stress tensor with a user defined scaling factor \(K\), introduced by Buffa et al.[2] as a tunning parameter for turbulent intensity:

(4)#\[\begin{split}A_{\alpha\beta} = K\left( \begin{matrix} \sqrt{R_{11}} & 0 & 0 \\ R_{21}/A_{11} & \sqrt{R_{22}-A_{21}^{2}} & 0 \\ R_{31}/A_{11} & \left(R_{32}-A_{21}A_{31}/A_{22}\right) & \sqrt{R_{33}-A_{31}^{2}-A_{32}^{2}} \end{matrix} \right)\end{split}\]

and \(\tilde{u}_{\alpha\beta}\) is given by:

(5)#\[\tilde u_{\beta}\left(x,y,z,t\right) = \frac{1}{\sqrt{N_{\mathrm{eff}}}}\sum_{i=1}^{N}\epsilon_{i\beta}f\left(\frac{x-x_{i}}{L}\right)f\left(\frac{y-y_{i}}{L}\right)f\left(\frac{z-z_{i}}{L}\right)\]

where \(N_{\mathrm{eff}}\) is the effective number of vortices acting at a domain position, considering \(L\) as a radius of acting. It can be calculated with:

(6)#\[N_{\mathrm{eff}} = \left(\frac{N}{2L\left(n_{y}+2L\right)\left(n_{z}+2L\right)}\right)\frac{4}{3}\pi L^{3}\]

Where \(n_{y}\) and \(n_{z}\) are the domain size in \(y\) and \(z\).

Note

The use of \(N_{\mathrm{eff}}\) is an adjustment developed in the current solver, in Buffa et al.[2] the value \(N\) is used instead.

The function \(f\left(x\right)\) is a gaussian function:

(7)#\[f\left(x\right)=2\mathrm{exp}\left(\frac{-1}{2\left(0.225\right)^{2}}x^{2}\right)\]

The tensor \(\epsilon_{\alpha\beta}\left[N:3\right]\) is composed of random signs -1 or 1 for the directions of each eddy. The evolution of inlet boundary occurs through the displacement of the eddies positions. They move alongside flow direction with the average inlet velocity \(U_{\infty}\), hence:

(8)#\[x_{i}\left(t+\Delta t\right) = x_{i}\left(t\right) + U_{\infty}\Delta t\]

When \(x_{i}\left(t+\Delta t\right)>L\) the positions \(y_{i}\) and \(z_{i}\) are randomized along with \(\epsilon_{i\beta}\). At direction \(x\), periodicity is adopted, so when \(x_{i}\left(t+\Delta t\right)>L\), the new \(x_{i}\) position will be \(x_{i}\left(t+\Delta t\right)-2L\). The average flow velocity \(U_{\infty}\) is given by:

(9)#\[U_{\infty}=\frac{1}{n_{z}}\int_{0}^{n_{z}}\bar{u}\left(z\right)dz\]

Following the above approach, it is possible to produce a transient velocity field whose statistics return the input data of average velocity and Reynolds stress tensor. Knowing the velocity field at \(x\), the rate-of-strain \(S_{\alpha\beta}\) is calculated through a finite difference scheme and the density is assumed constant \(\rho=\rho_{0}\). such that \(\partial \rho/ \partial \mathbf{n}=0\). For the obtained \(S_{\alpha\beta}\), the SGS viscosity can be calculated and from all known macroscopic variables, the populations at inlet can be built within the LBM framework.

Note

In finite volume methods is also found formulations to which \(\partial \rho/ \partial \mathbf{n}=0\), however as reported by Buffa et al.[2], the LBM is related to a weakly compressible flow, and a purely solenoidal inlet condition is not appropriate. Hence, the density perturbations at the inlet must be freeze to minimize the radiated noise.