Wall function

The wall function models the velocity profile between two points of computational domain. The coupling with LES is then performed through velocity adjustments near wall boundaries.

Log Law

Why the profile is logarithmic

Before writing the law down, it helps to see why a logarithmic profile is the right shape. Consider the inertial (overlap) region of an attached turbulent boundary layer: far enough from the wall that molecular viscosity no longer sets the dynamics, yet close enough that the outer-flow geometry has not yet taken over. In this region there is no externally imposed length scale; the only quantities that can govern the mean shear are the friction velocity \(u^{*}\) (the velocity scale built from the wall stress) and the distance to the wall \(z\) itself (Pope[1]).

Dimensional reasoning then forces the form of the velocity gradient. The only combination of \(u^{*}\) and \(z\) with the units of a shear rate is \(u^{*}/z\), so

\[ \frac{\partial u}{\partial z} = \frac{u^{*}}{\kappa\, z}, \]

where the constant of proportionality is written as \(1/\kappa\) with \(\kappa\) the von Karman constant. Integrating in \(z\) gives a logarithm, which is exactly the law of the wall below. The physical reading is that each decade of distance from the wall contributes the same velocity increment, a direct consequence of the eddies growing in proportion to their distance from the surface.

This derivation also makes the model’s limits explicit. The argument assumed a single velocity scale set by the wall stress and no other length scale, so the log law holds only where the boundary layer is attached, the pressure gradient is weak and the flow is in equilibrium with the local wall stress. Those are the same assumptions introduced in the wall-stress modeling idea, and they are why the more general TBL models below are needed near separation and reattachment.

Limits of validity

The log law inherits its validity from the dimensional argument: a single velocity scale and no competing length scale. It fails wherever a second scale enters: under a strong adverse pressure gradient, at separation and reattachment, and at the sharp corners of bluff bodies. In those regions reach for the TBL models below, which carry the pressure-gradient term the log law discards.

Following the log law of the wall, the mean velocity profile for a given terrain category and roughness can be written as (Davidson[2]):

(1)\[ u\left(z\right)=\frac{u^{*}}{\kappa} \mathrm{ln}\left(\frac{z}{z_{0}}\right) \]

where \(\kappa=0.41\) is the Von Kármán constant and \(z_{0}\) the roughness length. In a numerical simulation, the friction velocity \(u^{*}\) can be calculated from the tangential velocity at the established reference point \(\mathrm{P_{N}}\):

(2)\[ u^{*} = \kappa u_{t,\mathrm{P_{N}}} \left[\mathrm{ln}\left(\frac{z_{\mathrm{P_{N}}}}{z_{0}}\right)\right]^{-1} \]

Roughness length and terrain categories (CWE)

The roughness length \(z_{0}\) is the single most important parameter linking this wall function to computational wind engineering, so it is worth understanding what it represents. A smooth wall has the velocity vanish at the surface; a rough wall does not, because the bumps, vegetation and buildings that make up the terrain shed their own small-scale wakes and shift the effective origin of the logarithmic profile upward. The roughness length \(z_{0}\) is the height at which the extrapolated log profile of (1) reaches zero mean velocity. A larger \(z_{0}\) means a rougher surface, a thicker retarded layer near the ground and, for a given wind speed aloft, a higher friction velocity (Davidson[2]).

This is exactly how the atmospheric boundary layer (ABL) over different terrains is parametrised. Wind-engineering standards group terrains into categories, each with a representative \(z_{0}\): open sea and flat coast at the smooth end (\(z_{0}\) of the order of \(10^{-3}\,\mathrm{m}\)), open farmland in the middle, and suburban then dense urban terrain at the rough end (\(z_{0}\) reaching the order of \(1\,\mathrm{m}\)) (1991-1-4[3], Kozmar[4]). Because a single value of \(z_{0}\) selects the whole inflow profile shape, you can reproduce a target terrain category in a Nassu simulation simply by setting \(z_{0}\) on the wall model, without meshing the individual roughness elements.

Use case: terrain category from a single roughness length

A whole wind-engineering terrain category reduces to one number here. Setting \(z_{0}\) on the wall model selects the inflow profile shape for open sea, farmland, suburban, or dense urban terrain without meshing any individual roughness element. See Atmospheric Boundary Layer for how \(z_{0}\) maps to the terrain profile.

Thin boundary layer (TBL)

A different approach used for a smooth wall representation consists on modeling the region between \(\mathrm{P_{0}}\) and \(\mathrm{P_{N}}\) through the thin boundary layer equation (TBL) for normal \(n\) and tangential \(t\) directions (Suga et al.[5]):

(3)\[ \frac{\partial}{\partial x_{n}}\left(\left(\nu+\nu_{\mathrm{e}}\right)\frac{\partial u_{t}}{\partial x_{n}}\right)=\frac{\partial u_{t}}{\partial t}+\frac{\partial}{\partial x_{n}}\left(u_{n} u_{t}\right)+\frac{1}{\rho}\frac{\partial p}{\partial x_{t}} \]

where \(\nu_{\mathrm{e}}\) is the eddy-viscosity, which we assume to follow the Johnson-King mixing-length model with Van Driest damping function (Johnson and King[6]):

(4)\[ \nu_{\mathrm{e}}=\nu\kappa x_{n}^{+}\left(1 - e^{-\frac{x_{n}^{+}}{A^{+}}}\right)^{2} \]

where \(\kappa=0.41\) is the von Karman constant, and \(A^{+}=17\). For the implemented models, the TBL is simplified to:

(5)\[ \frac{\partial}{\partial x_{n}}\left(\left(\nu+\nu_{\mathrm{e}}\right)\frac{\partial u_{t}}{\partial x_{n}}\right)=+\frac{1}{\rho}\frac{\partial p}{\partial x_{t}} \]

then, writing the total viscosity as \(\nu_{\mathrm{T}}=\nu+\nu_{\mathrm{e}}\):

(6)\[ \frac{\partial \nu_{\mathrm{T}}}{\partial x_{n}}\frac{\partial u_{t}}{\partial x_{n}}+\nu_{\mathrm{T}}\frac{\partial^{2}u_{t}}{\partial x_{n}^{2}} = +\frac{1}{\rho}\frac{\partial p}{\partial x_{t}} \]

This ordinary differential equation has no closed-form solution for a general eddy-viscosity profile, so it is discretized on the wall-normal stencil and solved numerically. The finite-difference discretization and its tridiagonal (TDMA) solution are worked out on a separate page, since they are a self-contained numerical recipe rather than part of the modelling argument.

Note

Within IBM, the wall model acts by correcting the near-wall velocities rather than writing a wall-node tensor, as described in IBM Wall Model.

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[1]

Stephen B. Pope. Turbulent Flows. Cambridge University Press, 2000. doi:10.1017/CBO9780511840531.

[2] (1,2)

Peter Alan Davidson. Turbulence: an introduction for scientists and engineers. Oxford university press, 2015.

[3]

EN 1991-1-4. Eurocode 1: actions on structures-part 1–4: general actions-wind actions. Brussels (Belgium): Comité Européen de Normalization (CEN), 2005.

[4]

Hrvoje Kozmar. Physical modeling of complex airflows developing above rural terrains. Environmental Fluid Mechanics, 12:209–225, 2012.

[5]

Kazuhiko Suga, Tomoki Sakamoto, and Yusuke Kuwata. Algebraic non-equilibrium wall-stress modeling for large eddy simulation based on analytical integration of the thin boundary-layer equation. Physics of Fluids, 2019.

[6]

Dennis A Johnson and LS King. A mathematically simple turbulence closure model for attached and separated turbulent boundary layers. AIAA journal, 23(11):1684–1692, 1985.