Wall

A wall boundary represents an impermeable surface that confines fluid or solid regions.

You can use the wall boundary conditions to model different types of walls. For viscous flows, the no-slip condition is applied by default. The no-slip conditions means that the fluid sticks to the wall and moves with the same velocity as the wall. Thus, for a stationary wall, the fluid has zero velocity at the wall. You can also model the motion of a wall by specifying a tangential velocity component. Depending on the geometry, you can model a translational or rotational motion of the wall in the plane of the boundary. For the no-slip condition, this means that the fluid moves at the same velocity as the specified tangential velocity component. Alternatively, the wall can be modelled as a slip wall representing an impenetrable but traction-free surface.

For situations where the slip or the no-slip wall boundary condition do not apply—such as rarefied flows where the Knudsen number is between 0.01 and 0.1—Simcenter STAR-CCM+ provides partial-slip condition models. The following partial-slip models are available:

Maxwell Slip—allows you to model a partial slip at the wall. It is usually used for microfluidics applications.
Von Smoluchowski Slip—allows you to model a temperature slip at the wall. It is applied in conjunction with the Maxwell Slip model.

In the case of turbulent flow, different wall treatments to compute the wall shear stress are employed. See Wall Treatment.

When you take heat transfer to or from a wall into account, that is, you solve for temperature in a fluid and/or a solid, different thermal specifications are available that determine the heat flux-temperature relationship at the wall.

Boundary Inputs

For a wall boundary, depending on your thermal and tangential velocity specification, you specify the following variables:


Inputs
(relative) velocity $v_{spec}$
(relative) wall rotation $ω_{s p e c}$
static temperature $T_{spec}$
Lennard-Jones characteristic length $σ_{s p e c}$
mean free path $L_{s p e c}$
momentum accommodation factor $σ_{v, s p e c}$
thermal accommodation factor $σ_{t, s p e c}$

In the case of heat transfer, you specify additional boundary input variables.

The shear stress specification determines whether you model a no-slip wall or a slip wall. The tangential velocity specification sets the velocity of the wall, which in turn influences the computation of the fluid velocity.

The tangential velocity of a wall boundary is specified relative to the reference frame, which can be either the laboratory, the region, or the local reference frame.

Computed Values

For a wall boundary, Simcenter STAR-CCM+ computes the following values at the boundary faces:

Wall fluid velocity $v$
Wall pressure $P_{s}$
Static temperature at the wall $T_{s}$

No-Slip Wall

On a no-slip wall, the fluid moves at the same velocity as the wall. In other words, the fluid velocity relative to the wall velocity is zero. Depending on the tangential velocity specification, the fluid velocity on the wall is computed differently:

Fixed
Figure 1. EQUATION_DISPLAY

$v = 0$
(836)
Vector
Figure 2. EQUATION_DISPLAY

$v = v_{spec}$
(837)
Rotation Rate
Figure 3. EQUATION_DISPLAY

$v = Ω \times r$
(838)

where $Ω$ is the rotational speed vector that is given by:
Figure 4. EQUATION_DISPLAY

$Ω = ω_{s p e c} \times d$
(839)

where $ω_{s p e c}$ is the specified angular velocity in $rad / s$ and $d$ is the axis vector.

$r$ is the position vector that is defined as:
Figure 5. EQUATION_DISPLAY

$r = r_{f} - r_{o}$
(840)

where $r_{f}$ is the location of the boundary face and $r_{o}$ is the location of the rotation axis origin.

The boundary fluid velocity is then corrected to match the normal grid flux at the boundary:

Figure 6. EQUATION_DISPLAY

v = v - (v \cdot a - G) \frac{a}{{| a |}^{2}}

(841)

where $a$ is the outward pointing face area vector.

This correction ensures consistency between the normal component of wall fluid velocity and the grid-flux.

Slip Wall

On a slip wall, the fluid slides along the wall without any shear forces. The fluid velocity does not correspond to the wall velocity and is computed as:

Figure 7. EQUATION_DISPLAY

v = v^{ext} - (v^{ext} \cdot a - G) \frac{a}{{| a |}^{2}}

(842)

where $v^{ext}$ is the fluid velocity extrapolated from the interior of the domain.

Partial-Slip Wall

The partial slip formulation consists of velocity- and temperature-dependent boundary conditions.

The boundary condition for the tangential slip velocity, by Maxwell, is:

Figure 8. EQUATION_DISPLAY

v = \frac{2 - σ_{v, s p e c}}{σ_{v, s p e c}} L_{s p e c} \frac{\partial v}{\partial n} |_{w a l l}

(843)

where:

$σ_{v, s p e c}$ is the specified tangential momentum accommodation factor, an empirical parameter ranging from 0 through 1, denoting the gas-surface momentum exchange.
$n$ is normal distance from the wall.

$L_{s p e c}$ is the specified mean free path of the gas. If an energy model is selected, the mean free path can be calculated using kinetic theory as:

Figure 9. EQUATION_DISPLAY

L = \frac{κ T}{{\sqrt{2} π p σ}_{s p e c}^{2}}

(844)

where:

$κ$ is the Boltzmann constant, 1.38066 × 10^-23J/K.
$T$ is temperature.
$p$ is pressure.
$σ_{s p e c}$ is the specified Lennard Jones characteristic length or collision diameter.

The temperature-dependent slip boundary condition, by von Smoluchowski, is:

Figure 10. EQUATION_DISPLAY

T_{s} - T_{w a l l} = \frac{2 - σ_{t, s p e c}}{σ_{t, s p e c}} \cdot \frac{2 γ}{γ + 1} \cdot \frac{L}{\Pr} \cdot \frac{\partial T}{\partial n} |_{w a l l}

(845)

where:

$T_{s}$ is the fluid temperature at the wall.
$T_{w a l l}$ is the wall temperature.
$σ_{t, s p e c}$ is the specified thermal accommodation factor, an empirical parameter ranging from 0 through 1, denoting the gas-surface energy exchange.
$γ$ is the specific heat ratio.
$\Pr$ is the Prandtl number.

The static pressure at the wall is linearly extrapolated from the interior of the domain:

Figure 11. EQUATION_DISPLAY

P_{s} = P_{s}^{ext}

(846)

The computation of the wall fluid temperature depends on the thermal specification at the wall:

Adiabatic

An adiabatic wall does not permit heat transfer across the boundary. The fluid temperature at the wall:

Figure 12. EQUATION_DISPLAY

T_{s} = T_{s}^{ext}

(847)

as well as density and total enthalpy:

Figure 13. EQUATION_TITLE

\begin{matrix} ρ & = & ρ^{ext} \\ H_{t} & = & H_{t}^{ext} \end{matrix}

(848)

are extrapolated from the interior of the domain.

Temperature

The temperature at the wall can be set to a user-specified temperature:

Figure 14. EQUATION_DISPLAY

T_{s} = T_{spec}

(849)

Convection

The convection wall boundary takes into account a convective heat flux from the environment to the external side of the boundary. The static fluid temperature at the wall

T_{s}

is computed as part of the solution of the governing fluid flow and energy equations.

Heat Flux

For this thermal specification, you specify the heat flux density at a wall. The fluid temperature at the wall $T_{s}$ is computed as part of the solution of the governing fluid flow and energy equations.