Rough Wall Function (k) - Boundary Condition

Rough Wall Function (k) - Boundary Condition   Description

Rough Wall Function (k) is a boundary condition, that provides a wall function for the turbulent viscosity (\(\nu_t\)) when modeling rough walls using wall functions, based on the turbulent kinetic energy (\(k\)) to account for the effects of surface roughness.

Rough Wall Function (k) is particularly useful for simulating flows over surfaces where roughness plays a significant role, such as in pipes, ducts, or external flows over rough terrain. Surface roughness affects the velocity profile and increases the wall shear stress, leading to higher turbulence levels in the near-wall region. Unlike nutkWallFunction (smooth walls) or nutURoughWallFunction (velocity-based), this BC incorporates roughness corrections for accurate modeling of turbulent viscosity on rough surfaces.

Rough Wall Function (k) - Boundary Condition   Understanding Rough Wall Function (k)

The boundary condition modifies the wall function for the kinematic viscosity based on the roughness height of the wall. It essentially adjusts the turbulent viscosity by considering the impact of the roughness on the flow close to the wall.

The turbulent viscosity \(\nu_t\) is calculated according to the following formulas:

\[\nu_{t_w} = \max (\min (\nu_w \frac{y^+ \kappa}{\ln (\max (E^\prime y^+, 1 + 10^{-4})) - 1}, 2\nu_{t_{lim}}), 0.5 \nu_{t_{lim}})\]
\[\nu_{t_{lim}} = \max ( \nu_{t_w}, \nu_w)\]
\[E^\prime = E \quad \text{ if } \quad K_s^+ \leq 2.25\]
\[E^\prime = \frac{E}{f_n} \quad \text{ if } \quad K_s^+ > 2.25\]
\[K_s^+ = \frac{u^* K_s}{\nu_w}\]
\[y^+ = \frac{u^* y}{\nu_w}\]
\[u^* = C_\mu^{1/4} \sqrt{k}\]
\[f_n = 1 + C_s K_s^+ \quad \text{if } \quad K_s^+ \geq 90\]
\[f_n = \left(\frac{K_s^+ - 2.25}{87.75} + C_s K_s^+ \right)^{\sin(0.4258 (\ln (K_s^+)) - 0.811)} \quad \text{if } \quad K_s^+ < 90\]
where:
  • \(k\) - Turbulent Kinetic Energy \([m^2/s^2]\)
  • \(y\) - Wall-Normal Height \([m]\)
  • \(y^+\) - Estimated Wall-Normal Height of the Cell Center in Wall Units
  • \(C_\mu\) - Empirical Model Constant \([-]\)
  • \(\nu_w\) - Kinematic Viscosity of Fluid Near Wall \([m^2/s]\)
  • \(\nu_{t_{w}}\) - Turbulent Viscosity Near Wall \([m^2/s]\)
  • \(\nu_{t_{lim}}\) - Limited Kinematic Viscosity Near Wall \([m^2/s]\)
  • \(\kappa\) - von Kármán constant \([-]\)
  • \(E\) - Wall Roughness Parameter \([-]\)
  • \(E`\) - Modified Wall Roughness Parameter \([-]\)
  • \(K_s\) - Sand-Grain Roughness Height \([m]\)
  • \(K_s^+\) - Sand-Grain Roughness Height in Wall Units
  • \(f_n\) - Roughness Function Parameter \([-]\)
  • \(C_s\) - Roughness Constant \([-]\)
  • \(u^*\) - Shear Velocity \([m/s]\)

Rough Wall Function (k) - Boundary Condition   Application & Physical Interpretation

Rough Wall Function (k) is crucial for accurately simulating flows over rough surfaces in turbulent conditions. By adjusting the kinematic eddy viscosity based on surface roughness, it allows for a more realistic representation of wall effects in CFD models.

This is particularly important in engineering applications such as pipelines, ducts, and channels where wall roughness plays a significant role in flow behavior.

Rough Wall Function (k) in Pipe Flow applications

Example applications: pipe or duct flow

These types of simulations can be solved using the simpleFoam (solver). In this case, we assume that the inner surface of the pipe is not perfectly smooth, but rather is characterized with some roughness.

Example Boundary Conditions set for Pipe Flow applications

Physics

Pressure

Velocity

\(k\)

\(\omega\)

\(\nu_t\)

Pipe (Wall)

Zero Gradient

No-Slip

Standard Wall Function

Standard Wall Function

Rough Wall Function (k)

Rough Wall Function (k) in Free Surface Flow applications

Example applications: hull flow analysis

These types of simulations can be solved using the interFoam (solver). In this case, the roughness of the hull is considered.

Example Boundary Conditions set for Free Surface Flow applications

Physics

Modified Pressure

Velocity

\(k\)

\(\omega\)

\(\nu_t\)

Hull (Wall)

Fixed Flux Pressure

Moving Wall Velocity

Standard Wall Function

Standard Wall Function

Rough Wall Function (k)

Rough Wall Function (k) - Boundary Condition   Rough Wall Function (k) in SimFlow

To define Rough Wall Function (k) on the wall boundary, the proper option must be selected from the drop-down menu in the Turbulence tab - Figure 1. Please note that the Turbulence tab will be visible only if the turbulence equations are activated (Turbulence panel).

Rough Wall Function (k) Boundary Condition SimFlow
Figure 1. Rough Wall Function (k) boundary condition in SimFlow
  • Roughness Height — represents the equivalent sand grain roughness height in [m]
  • Roughness Constant - Dimensionless constant that characterizes the type of roughness
    Roughness constant is a parameter that measures how uniform the roughness is. Inputs may range from 0.5 to 1:
    • 0.5 indicates a uniform wall roughness
    • 1 means that the roughness is strongly non-uniform, both in terms of spacing and height

Rough Wall Function (k) - Boundary Condition   Rough Wall Function (k) - Alternatives

In this section, we propose boundary conditions that are alternative to Rough Wall Function (k). While they may fulfill similar purposes, they might be better suited for a specific application and provide a better approximation of physical world conditions.

Boundary Condition

Description

Fixed Value

fixed value on the patch

Rough Wall Function

similar to Rough Wall Function (k) but calculates \(\nu_t\) based on velocity