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## Documentation

With our lattice Boltzmann method (LBM), it is possible to input additional modeling of geometry in the form of surface roughness and porous media to better capture its effects on the wind comfort simulation for an accurate analysis. Additionally, it is possible to include a rotating wall as an advanced concept.

## Surface Roughness

Surface roughness has an influence on friction resistance. While the effect of roughness for laminar flows is negligible, turbulent flows are highly dependent on wall roughness. Because roughness changes the thickness of the viscous sublayer and modifies the law-of-wall for mean velocity. As a result, the turbulent friction factor increases with the roughness ratio.

In LBM simulations, wind flow is affected by the frictious surfaces as well as the obstacles, such as terrain, buildings, trees, etc. To apply the roughness effects on any surface in LBM analysis, one should use equivalent sand grain roughness height. Simply type the roughness coefficient and select the surface or a component. This will apply roughness on surface elements of selected entities.

In the following table, equivalent sand-grain roughness of some materials and terrain types can be seen:

With the help of the following article you can learn more about how surface roughness can affect your PWC results which is applicable for LBM simulations as well:

In order to maintain the proper amount of shear in the Inlet ABL profile, the corresponding roughness has to be defined. Currently, this must be input as “Sand-Grain Equivalent” Roughness defined by the parameter $$k_r$$. However, the  ABL definition is based on the Aerodynamic Roughness $$z_0$$. For example, the roughness of some objects is specified as :

Performing proper conversion from $$z_0$$ to $$k_r$$ is thus required to define the proper roughness parameter as Sand-Grain Equivalent. Based on the most general relation  between $$k_r$$ and $$z_0$$:

$$g(\frac{k_r * u_{tau}}{\ nu_0}) – B = g(\frac{z_0* u_{tau}} {\ nu_0})\tag{1}$$

where $$B$$ = 8.5 and $$g()$$ is the universal wall function describing the wall-normalized velocity from the wall-normalized height:

$$u+ = g(y+)\tag{2}$$

Because  of the above formulation there is no explicit conversion between $$k_r$$  and $$z_0$$. For $$g()$$ different modeling can be done. In the logarithmic  layer the relation is usually simplified as:

$$g(y+) = \frac{ln(y+)} {\kappa} + C \tag{3}$$

where $$\kappa$$ ~ 0,41 and $$C$$ ~ 5.1. Taking this simplification following approximate solution can be found:

$$k_r = z_0 * {e} {^{B * kappa}} \tag{4}$$

$$k_r = z_0 * 32.6224271 \tag{5}$$

Thus, the Aerodynamic roughness for the terrain category must be multiplied by 32.622 to be input as equivalent sand grain roughness.

## Porous Objects

Porous media is a medium filled with solid particles, which lets fluid pass through. The arrangement of the flow path inside the porous medium can be regular or irregular.

A porous medium can be classified as follows:

• Consolidated medium: Solid-body has internal pores. Fluid passes through the pores.
• Unconsolidated medium: A pile of solid particles is packed inside a bed. Fluid flows around the particles.

Porous media is used to model permeable obstructions such as trees, hedges, windscreens, and other wind mitigation measures. When air flows through a porous body, a pressure gradient along the direction of the flow is generated. Using porous media simplification reduces CAD and mesh complexity, and saves computational time and expenses.

Within the LBM analysis type, SimScale allows users to model porous objects using the following two models:

### Darcy-Forchheimer Model

The pressure loss due to porosity is modeled by the empirical Darcy-Forchheimer equation where, $$\Delta p/\Delta x$$ is pressure gradient, $$\mu$$ is dynamic viscosity, $$\rho$$ is density, $$u$$ is velocity vector, $$F_\varepsilon$$ is friction form coefficient and $$K$$ is permeability.

$$\overline{\frac{\Delta {p}}{\Delta x}}=- \frac{\mu}{K}.\overline{u}-\rho.\frac{F_\varepsilon}{\sqrt{K}}.|\overline{u}|.\overline{u}\tag{6}$$

The first and the second term on the right-hand side of the equation are the Darcy and Forchheimer terms respectively. The Darcy term accounts for the friction drag which has a linear relation to the local velocity vector. The Forchheimer term accounts for the inertial drag or the form drag which has a quadratic relation to the local velocity vector.

To be able to define a porous media in SimScale, one should define $$K$$ and $$F_\varepsilon$$. Users can extract these coefficients. Just use a minimum of 3 data points and predict the $$K$$ and $$F_\varepsilon$$ to fit the line. An example is shown below:

Curve equation:

$$\frac{\mathrm dP}{\mathrm d x} = \frac{0.0000181}{K}.u+1.\frac{F_\varepsilon}{K^{0.5}}.u^2 \tag{7}$$

Using the curve-fitting method, missing coefficients were calculated as follows:

• $$K$$ = 0.00007135065
• $$F_\varepsilon$$ = 0.01890935

Once the relevant coefficients are found, assign them in Darcy-Forchheimer porous object definition and select the porous media geometry. This selection can be in the form of faces, volumes or geometry primitives.

While the isotropic type adds specific resistance in every direction, directional adds the specific resistance only on specified direction/s and assigns the remaining direction/s an infinite resistance (such as a wall).

### Tree Model

Tree models are used to model the vegetation (single trees, bushes, hedges, forest canopies, etc.) as porous mediums. The user can either define the porosity as a custom tree model or choose one of the 5 most common trees in EU cities:

• Planetree,
• Oak,
• Sycamore,
• Silver birch,
• Chestnut.

Custom tree model is also possible, and requires user input for assigning:

• Leaf Area Index (LAI),
• Average tree height,
• Drag coefficient

Leaf area index (LAI) is a dimensionless number, which is used to compare plant canopies. It can be simply defined as the total leaf area per unit ground area.

Default tree models require only the assignment of tree height since the solver applies related LAI and drag coefficient automatically. The following table displays the default trees, and their respective LAI along with the drag coefficient:

The above information obtained$$^2$$ is a compiled data ranging over 70 years from 500 different locations.

In the tree model, we used the modified Darcy-Forchheimer equation. By assigning a high permeability value, we neglected the Darcy portion and simplified the equation as follows:

$$\frac{\Delta \overline{p}}{\Delta x}=-\rho.\frac{F_\varepsilon}{\sqrt{K}}.|\overline{u}|.\overline{u}\tag{8}$$

Next, modified the equation to define it with respect to Drag Coefficient $$C_d$$ and Leaf Area Density $$LAD$$:

$$\frac{\Delta \overline{p}}{\Delta x}=-\rho.LAD.C_d.|\overline{u}|.\overline{u}\tag{9}$$

Leaf area density is calculated with respect to leaf area index $$LAI$$ and height of the vegetation $$h$$:

$$LAD=\frac{LAI}{h}\tag{10}$$

## Rotating Walls

Rotating walls can be found under Advanced concepts in LBM simulation: Figure 6: In LBM simulations an additional option exists for modeling rotating parts of the model, like the wheels of a car, or the impeller of a turbine.

Here the origin, axis of rotation, and the rotational velocity of the body must be defined accordingly: