Choosing a wall model

Which wall model should a wall boundary use, and what does the choice ask of the mesh?

Use no slip when the mesh resolves the boundary layer, with the first cell at $y^{+}$ between 1 and 5. Use wall function when the first cell lies clear of the viscous sublayer, at $y^{+}$ between 30 and 250, where the law of the wall is applied. Use slip where the wall should exert no shear. Use the compressible wall function on a heated wall or at speed. Giving the wall a temperature or a heat flux selects it.

Good practice is to target $y^{+} = 1$ on a wall-resolved mesh and $y^{+} = 100$ on a wall-modelled mesh.

Slip, no slip or wall function

The kind of a wall boundary condition is slip, no slip or wall function. Use no slip only where the mesh resolves the viscous sublayer, and a wall function elsewhere. A wall may also carry a velocity, a roughness, a temperature and an FSI modal model, fsi.

The best check of a mesh is a histogram of $y^{+}$ over the wall from a converged solution, plotted from the yplus surface output variable, which is written by default. $y^{+}$ scales approximately linearly with the first cell height. If the histogram sits 2.5 times above the target, reduce the first cell height by a factor of 2.5.

Wall function

The law of the wall depends on the turbulence model. Menter SST uses Spalding's law, a single expression that spans the viscous sublayer, the buffer layer and the log layer. SA-neg uses the Allmaras fit, which spans the same range; with the incompressible law of the wall it is applied where the first cell lies below $y^+ = 200$, and above that the log law alone is used. Because both laws hold through the sublayer, a wall-function wall tolerates a first cell anywhere from $y^+ \sim 1$ into the log layer, so a mesh whose first-cell height varies along the wall needs no switch between wall treatments. Even so, size the mesh for a first cell between $y^+ = 30$ and $250$.

By default a 'wall function' wall uses an incompressible law of the wall. The velocity profile is solved with the density and viscosity of the first cell off the wall, and no heat is transferred.

Giving the wall a temperature or a heat flux switches it to the compressible wall function. This is the Van Driest II / White–Christoph velocity transformation coupled to the Crocco–Busemann thermal law of the wall. It evaluates $y^+$ and the wall shear stress with the density and viscosity of the wall state, and produces the wall heat flux. With a heat flux the wall temperature is recovered; with a temperature, the heat flux. Both are available as the heatflux and twall surface output variables.

The velocity transformation is worth having at speed whether or not the wall exchanges heat, since the boundary layer of an adiabatic wall is still stratified by viscous heating. compressible wall function runs it on an adiabatic wall. The wall state is then evaluated at the recovery temperature $T_r$ rather than at a specified one, so the wall heat flux is zero by construction:

{
    "boundary_conditions": {
        "BC_2": {
            "zones": [2],
            "type": "wall",
            "kind": "wall function",
            "compressible wall function": True,
        },
    },
}

twall then reports $T_r$, and heatflux reports zero. The key is redundant alongside a temperature or a heat flux, which select the same transformation and add the thermal closure to it.

A constant-heat-flux wall is singular where the boundary layer is thin, at a plate leading edge for example. The modelled layer cannot carry an arbitrarily large flux there, and the wall temperature rises until it is limited internally. Check the twall output if a prescribed flux is large.

The compressible wall function, however it is selected, is not supported in the incompressible solver, and is rejected at input validation.

Rough compressible walls

Roughness may be combined with the compressible wall function. This is the combination an icing calculation needs. Ice accretion roughens the surface, which raises both the skin friction and the convective heat transfer that in turn drives the accretion, while an ice protection system imposes a wall heat flux or temperature. Both the roughness and the thermal state accept a field, so an ice roughness map and an electrothermal pad layout can be prescribed face by face from VTP files. Each field is a path relative to the directory zCFD is run from, and a deck whose file is missing or cannot be read is rejected at validation. A roughness map must carry a point array named Roughness, and the solver stops at setup if any of its values is negative or not a number.

The equivalent sand-grain height is given in mesh units, like every other length in the input deck. Nothing rescales the mesh, so a 0.5 mm ice roughness on a mesh in metres is 0.0005.

On the velocity side, the Van Driest transformed velocity obeys the incompressible rough-wall law, with the roughness Reynolds number $k_s^+$ formed from the density and viscosity of the wall, not of the first cell. That is the scaling the compressible wall function already uses for $y^+$ and the wall shear stress; on a heated, anti-iced wall the two differ substantially.

On the thermal side, the Crocco–Busemann closure is an exact Reynolds analogy, so on its own it would raise the Stanton number in the same ratio as the skin friction. Roughness elements shed momentum by pressure drag but carry heat only by conduction, so in reality the heat transfer rises much less. The default thermal model, 'owen thomson', therefore adds a thermal sublayer resistance

between the surface and the roughness crest. It is blended in over the same $k_s^+$ range as the velocity intercept, so a hydraulically smooth wall is unaffected. Setting thermal model to 'reynolds analogy' removes it.

On a rough wall the twall output is the temperature of the surface, not of the roughness crest on which the law of the wall is closed; the two differ by the drop across that resistance. heatflux is continuous through it, and so is unambiguous.

An anti-iced leading edge with a measured ice roughness map:

{
    "boundary_conditions": {
        "BC_3": {
            "type": "wall",
            "kind": "wall function",
            "heat flux": {"field": "ips_pads.vtp"},
            "roughness": {"type": "height", "field": "ice_roughness.vtp"},
        },
    },
}

Roughness

Roughness is applied by the wall-function treatment of both the compressible and the incompressible solver, for the Menter SST and SA-neg turbulence models. On a no slip or slip wall it has no effect.

Internally the models work in equivalent sand-grain height, so a roughness given as a length is converted on input, as height = length / 0.03. The roughness surface output variable reports that height, not the value written in the deck.

Moving walls

velocity applies a surface velocity, for surfaces in relative motion: a car moving over a road surface, for example, or a rotating tyre. A rotating wall velocity takes an axis, an origin and an omega, as a rotating frame does:

"BC_3": {"type": "wall", "kind": "wall function", "zones": [14, 20],
         "velocity": {"type": "rotating", "axis": [0.0, -0.9988, 0.0498],
                      "origin": [0.0, -0.6593, 0.2533], "omega": 168.6}},

Reference entries

• wall: kind, temperature, heat flux, compressible wall function, roughness, velocity and fsi