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  • Advanced Modeling LBM

    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.

    surface roughness in simscale
    Figure 1: Surface roughness applied in LBM analysis

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

    Materialε, equivalent sand-grain roughness \([m]\)
    Concrete, smooth wall0.0045
    Concrete, rough wall0.013
    Concrete, floor0.04
    Farmland with crops0.525
    Grass with shrubs0.265
    Grass and stone grid0.0225
    Case iron0.000254
    Commercial or welded steel0.00004572
    Cast iron0.00026
    Table 1: Materials and their respective sand-grain roughness in meters\(^1\)

    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 :

    Terrain description\(z_0\) \([m]\)
    Open sea, fetch at least 5 \(km\)0.0002
    Mud flats, snow: no vegetation, no obstacles0.005
    Open flat terrain: grass, few isolated obstacles0.03
    Low crops: occasional large obstacles, \(x \over\ H\) > 200.10
    High crops: scattered obstacles, 15 < \(x \over\ H\) < 200.25
    Parkland, bushes: numerous obstacles, \(x \over\ H\) ≈ 100.5
    Regular large obstacle coverage (suburb, forest)1.0
    Table 2: Aerodynamic roughness length in wind engineering is usually defined as “the height where the wind velocity is equal to zero”. It is an important aerodynamic parameter and reveals the exchange between the atmosphere and land surfaces.

    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:

    u \([m/s]\)dP/dx \([Pa/m]\)
    Table 3: Velocity and pressure gradient values for the curve fitting method

    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
    velocity graphic displaying pressure drop in simscale through advanced modelling
    Figure 2: Pressure drop per unit length over velocity graph

    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.

    porous media window in simscale advanced modelling in PWC
    Figure 3: Setting up porous media in SimScale using the Darcy-Forchheimer model

    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.
    tree selection tool in workbench advanced modelling in PWC
    Figure 4: Selecting the type of tree from the selection menu in SimScale 

    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:

    Tree TypeDrag CoefficientLeaf Area Index (LAI)
    Plane tree0.25.28
    Oak tree0.25.1657
    Silver birch0.23.2379
    Table 4: Default tree types with their respective drag coefficients and LAI

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

    visual description of leaf area index (lai) used in simscale advanced modelling in PWC
    Figure 5: Visual Description of height used to calculate Leaf Area Index (LAI)

    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 \(h\):


    Read more about modeling trees in SimScale in our blog:

    Rotating Walls

    Rotating walls can be found under Advanced concepts in LBM simulation:

    additional option under advanced concepts for rotating wall in LBM
    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:

    rotating zone covering everything that is supposed to rotate
    Figure 7: The rotating wall concept can only be applied to cylindrical bodies.

    You normally need a rotating wall for simulating pumps, turbines, wheels of a vehicle, or fans.

    Last updated: July 28th, 2021

    What's Next

    part of: Pedestrian Wind Comfort Analysis