Combine and Spread

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Combine and Spread#

The combine/spread operations from IBM are performed through a special operator that can be seen as a discrete equivalent of the Dirac Delta function \(D_{h}(r_{\alpha})=\phi(r_{1})\phi(r_{2})\phi(r_{3})\) Peskin[1]. The IBM is then applied as follows. The flow velocity on the solid node has to be interpolated from the fluid nodes:

../../_images/wall0spread.svg

where the values from black grid nodes are interpolated in the blue mesh points.

(1)#\[\begin{split}&u_{\alpha}^{\mathrm{interp}}=\sum_{\mathrm{fluid\;nodes}}u_{\alpha}\phi\left(r_{x}\right)\phi\left(r_{y}\right)\phi\left(r_{z}\right)\\ &\rho_{\alpha}^{\mathrm{interp}}=\sum_{\mathrm{fluid\;nodes}}\rho_{\alpha}\phi\left(r_{x}\right)\phi\left(r_{y}\right)\phi\left(r_{z}\right)\end{split}\]

Note

This interpolated density is employed only in the force calculations and isn’t used for the estimation of pressure at the solid body surface, since it also accounts nodes internal to Lagrangian mesh.

where \(r_{\alpha}=|x_{\alpha}-X_{\alpha}|/\Delta x\) and \(\phi\) is the smoothed delta function. We adopt the smoothed 2-point function from Yang et al.[2]:

(2)#\[\begin{split}\phi\left(r\right)= \begin{cases} 3/4 - r^{2} & \text{, $|r| \leq 0.5$}\\ 9/8 - 3|r|/2 +r^{2}/2 & \text{, $0.5\leq |r| \leq 1.5$}\\ 0 & \text{, $1.5\leq |r|$}\\ \end{cases}\end{split}\]

The force necessary for imposition of a no-slip boundary condition is then given by an iterative process:

(3)#\[f_{\alpha,k+1}=f_{\alpha,k}+2.0\frac{\rho_{k}^{\mathrm{interp}}}{\Delta t}\left(u_{\alpha,k}^{\mathrm{target}}-u_{\alpha,k}^{\mathrm{interp}}\right)\]

where \(k\) is the sub time-step. The force \(f_{\alpha,0}\) and the target velocity on a solid node are null. From past experiences, only one iteration is sufficient to give a good accuracy when using the froce from the last time step as initial value. However, for high Reynolds LES simulations, it might be necessary to enforce more than one iteration to assure that there will be no fluid penetration.

Finally, the IB forces are spread to the fluid nodes. These forces are inputted into the same neighbooring fluid nodes used for velocity and density interpolation:

(4)#\[F_{\alpha}=\sum_{\mathrm{fluid\;nodes}}f_{\alpha}\phi\left(r_{x}\right)\phi\left(r_{y}\right)\phi\left(r_{z}\right)\Delta S\]

where \(\Delta S\) is the surface area of the solid node. In general, it can be assumed that \(\Delta S = A/N\), with \(A\) being the total surface area of the object and \(N\) the total number of IB nodes. However, the current solver adopts a more robust definition for the element area, which is better described in the Meshing section.

Todo

Reference meshing section.

The force field generated in Eulerian grid is computed in LBM evolution equation and gives a satisfactory representation of a solid body immersed in fluid flow.

Note

When the IBM is used along with multiblock, the area applied to spreading equation is relative to the grid in which the Lagrangian node is located. The area of an element located in lvl 1 will be four times its area if it was being measured relative to lvl 0.

Tree Effect#

The IBM might be also used to represent the effect of trees in the computational domain. In this case the spread is performed for a single iteration and the force to be spread is calculated as:

(5)#\[f_{\alpha}=-C_{\mathrm{D}}\mathrm{LAD}\frac{\rho_{k}^{\mathrm{interp}}}{2\Delta t}\left(\sqrt{u_{\beta,k}^{\mathrm{interp}}u_{\beta,k}^{\mathrm{interp}}}\right)u_{\alpha,k}^{\mathrm{interp}}\]

where \(C_{\mathrm{D}}\) is the drag coefficient and \(\mathrm{LAD}\) is the leaf area density in \(\mathrm{m}^{2}/\mathrm{m}^{3}\) Kang et al.[3].

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