Free-Stream

The free-stream boundary allows you to model free-stream compressible flow conditions at a far-field boundary.

The free-stream boundary condition is based on extrapolation of Riemann invariants under the assumption of irrotational, quasi-1D flow in the boundary-normal direction. This condition generally applies to external flows when the boundary is placed sufficiently far from the body. Internal flows often have walls, possibly of irregular shape, immediately next to the flow boundary. These walls can produce boundary layers, vortices, or other multi-dimensional flow structures such that the irrotational, quasi-1D flow assumption breaks down.

To set the free-stream conditions of external flows, the following options are available:

Mach Number + Pressure + Temperature—explicitly defines the free-stream values for Mach number, working pressure, and static temperature.
Altitude + Length Scale + Reynolds Number—extracts the free-stream values for working pressure and static temperature at the specified altitude from an atmosphere table. The Mach number is calculated from these values using the specified length scale and Reynolds number.
Altitude + Mach Number—explicitly defines the free-stream Mach number and extracts the free-stream values for working pressure and static temperature at the specified altitude from an atmosphere table.

Boundary Inputs

The free stream boundary is valid only for compressible flows. At the boundary, you specify the following variables:


Inputs	Compressible Equation of State
Mach number $M_{s p e c}$	✓
working pressure $P_{s p e c}$	✓
static temperature $T_{s, s p e c}$	✓
altitude $h_{s p e c}$	✓
length scale $L_{s p e c}$	✓
Reynolds number ${Re}_{s p e c}$	✓
flow direction $θ_{s p e c}$	✓

You specify the inflow direction $θ_{s p e c}$ as normal to the boundary, as individual angle components, or directly as flow direction angles. Flow conditions are specified relative to the flow conditions's reference frame, which can be either laboratory, the region, or the local reference frame.

The working pressure

P_{s p e c}

is always expressed relative to the reference pressure. It represents the difference between the absolute pressure and the reference pressure.

In flows without the influence of gravity, the working pressure is equal to the static gauge pressure. If the gravity model is active, the working pressure is equal to the piezometric pressure, as shown in Eqn. (862) for variable and constant density flows. When a turbulence model is used, the working pressure implicitly includes the contribution of $\frac{2}{3} ρ k$ . However, this contribution can be considered negligible in most flows of engineering interest.

Computed Values

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

velocity $v$
static pressure $P_{s}$
static temperature $T_{s}$

Compressible Equation of State

To compute $v$ , $P_{s}$ , and $T_{s}$ at the boundary faces, Simcenter STAR-CCM+ requires the following free-stream variables, that are calculated depending on the specified free-stream condition as tabulated below:


Free-stream variable	Mach Number + Pressure + Temperature	Altitude + Length Scale + Reynolds Number	Altitude + Mach Number
Mach number $M_{\infty}$	$M_{s p e c}$	$\frac{{Re}_{s p e c} μ_{\infty}}{ρ_{\infty} L_{s p e c} c_{\infty}}$	$M_{s p e c}$
working pressure $P_{\infty}$	$P_{s p e c}$	$P (h_{s p e c})$ from atmosphere table	$P (h_{s p e c})$ from atmosphere table
static temperature $T_{s, \infty}$	$T_{s, s p e c}$	$T_{s} (h_{s p e c})$ from atmosphere table	$T_{s} (h_{s p e c})$ from atmosphere table
flow direction $θ_{\infty}$	$θ_{s p e c}$	$θ_{s p e c}$	$θ_{s p e c}$
speed of sound $c_{\infty}$	$\sqrt{γ R_{gas} T_{s, \infty}}$	$\sqrt{γ R_{gas} T_{s, \infty}}$	$\sqrt{γ R_{gas} T_{s, \infty}}$
velocity $v_{\infty}$	$c_{\infty} M_{\infty} θ_{\infty}$	$c_{\infty} M_{\infty} θ_{\infty}$	$c_{\infty} M_{\infty} θ_{\infty}$
entropy $S_{\infty}$	$\frac{T_{s, \infty}^{\frac{γ}{γ - 1}}}{P_{\infty}}$	$\frac{T_{s, \infty}^{\frac{γ}{γ - 1}}}{P_{\infty}}$	$\frac{T_{s, \infty}^{\frac{γ}{γ - 1}}}{P_{\infty}}$

where:

$γ = \frac{C_{p}}{C_{v}}$ is the specific heat ratio.
$R_{gas}$ is the specific gas constant.

Additionally, the magnitude of the boundary normal velocity $v_{n}$ is obtained from characteristics as:

Figure 1. EQUATION_DISPLAY

v_{n} = \frac{(R^{+} - R^{-})}{2}

(850)

$R^{+}$ and $R^{+}$ are the positive and negative Riemann invariant, respectively, defined as:

Figure 2. EQUATION_DISPLAY

R^{+} = v^{e x t} \cdot n + 2 \frac{c^{ext}}{γ - 1}

(851)

Figure 3. EQUATION_DISPLAY

R^{-} = v_{\infty} \cdot n - 2 \frac{c_{\infty}}{γ - 1}

(852)

where:

ext indicates that the value is extrapolated from the adjacent cell.
$n = a / | a |$ with $a$ being the outward pointing face area vector.

The speed of sound at the boundary is calculated as:

Figure 4. EQUATION_DISPLAY

c = (R^{+} - R^{-}) \frac{γ - 1}{4}

(853)

Inflow: Under subsonic inflow conditions, the face value of velocity, static pressure, and static temperature are computed as:; Figure 5. EQUATION_DISPLAY

$v = v_{\infty} + (v_{n} - v_{\infty} \cdot n) \cdot n$
(854); Figure 6. EQUATION_DISPLAY

$P_{s} = \begin{matrix} \frac{{T_{s}}^{\frac{γ}{γ - 1}}}{S_{\infty}} \end{matrix}$
(855); Figure 7. EQUATION_DISPLAY

$T_{s} = \begin{matrix} \frac{c^{2}}{γ R_{gas}} \end{matrix}$
(856); Under supersonic inflow conditions, the velocity, static pressure, and static temperature are taken to be the respective free-stream value.
Outflow: Under subsonic outflow conditions, the face value of velocity, static pressure, and static temperature are computed as:; Figure 8. EQUATION_DISPLAY

$v = v^{e x t} + (v_{n} - v^{e x t} \cdot n) \cdot n$
(857); Figure 9. EQUATION_DISPLAY

$P_{s} = \begin{matrix} \frac{{T_{s}}^{\frac{γ}{γ - 1}}}{S^{ext}} \end{matrix}$
(858); Figure 10. EQUATION_DISPLAY

$T_{s} = \begin{matrix} \frac{c^{2}}{γ R_{gas}} \end{matrix}$
(859); Under supersonic outflow conditions, the velocity, static pressure, and static temperature are extrapolated from the adjacent cell.