Stagnation Inlet

The stagnation inlet boundary is an inflow condition at which you prescribe the total values for pressure and temperature as well as the flow direction. The stagnation conditions refer to the conditions in an imaginary plenum, far upstream, in which the flow is completely at rest.

For incompressible flows, Bernoulli's equation is used to relate total pressure, static pressure, and velocity magnitude. For compressible ideal-gas flows, isentropic relations are used, and characteristic variables help determine the boundary properties of the flow.

Pressure Constraints

Simcenter STAR-CCM+ provides the following options for constraining the pressure on the boundary:

Non-Reflecting—prevents spurious numerical reflection of the solution into the solution domain
Pressure-Jump—imposes a pressure rise or pressure loss at the boundary obtained from a specified fan curve, a porous inertial/viscous resistance, or a pressure loss coefficient.

Boundary Inputs

For a stagnation inlet boundary, you specify the following variables:


Inputs	Incompressible Equation of State	Compressible Equation of State
supersonic static pressure $P_{s, s p e c}^{\sup}$		✓
total pressure $P_{t, s p e c}$	✓	✓
total temperature $T_{t, s p e c}$	✓	✓
inflow direction $θ_{spec}$	✓	✓
For Non-Reflecting: number of modes $n_{s p e c}$	✓	✓
For Pressure Jump—Fan: fan curve $F_{s p e c}$	✓	✓
For Pressure Jump—Porous: porous inertial resistance $α_{s p e c}$ porous viscous resistance $β_{s p e c}$	✓	✓
For Pressure Jump—Loss Coefficient: pressure loss coefficient $K_{s p e c}$	✓	✓

The total pressure $P_{t, s p e c}$ and the total temperature $T_{t, s p e c}$ are specified with respect to a reference frame. The reference frame is the laboratory, the region, or the local reference frame. The flow direction is either given as normal to the boundary, as individual angle components, or directly as flow direction angles.

Computed Values

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

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

Compressible and Non-Isothermal Equation of State

The computation of the velocity magnitude differs depending on whether an ideal gas or a non-ideal gas equation of state is used.

Ideal Gas

Simcenter STAR-CCM+ computes the negative Riemann invariant by using extrapolated values from the interior of the domain:

Figure 1. EQUATION_DISPLAY

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

(761)

where:

$v^{e x t}$ is the boundary velocity that is extrapolated from the interior of the domain.
$n = a / | a |$ with $a$ being the outward pointing face area vector.
$c^{e x t}$ is the velocity of sound that is extrapolated from the interior of the domain.
$γ = \frac{C_{p}}{C_{v}}$ is the specific heat ratio.

Equalizing:

Figure 2. EQUATION_DISPLAY

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

(762)

Substituting the following Eqn. (763) to Eqn. (765) into Eqn. (762) and assuming a perfect gas

(C_{p} = c o n s t .)

Figure 3. EQUATION_DISPLAY

\begin{matrix} v \cdot n & = & | v | \cdot (θ_{s p e c} \cdot n) \end{matrix}

(763)

Figure 4. EQUATION_DISPLAY

\begin{matrix} C_{p} T_{t, s p e c} = & C_{p} T_{s} + \frac{{| v |}^{2}}{2} \end{matrix}

(764)

Figure 5. EQUATION_DISPLAY

c = \sqrt{γ R T_{s}} = \sqrt{(γ - 1) C_{p} T_{s}} = \sqrt{(γ - 1) C_{p} (T_{t, s p e c} - \frac{{| v |}^{2}}{2 C_{p}})}

(765)

results in a quadratic equation in boundary velocity magnitude $| v |$ that is solved for $| v |$ .

Non-Ideal Gas

For a non-ideal gas equation of state, the boundary velocity magnitude is extrapolated from the cell layer adjacent to the boundary in the interior of the domain:

Figure 6. EQUATION_DISPLAY

| v | = {| v |}^{e x t}

(766)

For both types of compressible equations of state— the ideal gas law and the non-ideal gas law—the onward procedure for calculating the boundary values is the same.

By using the specified inflow direction, the boundary velocity vector is given by:

Figure 7. EQUATION_DISPLAY

v = | v | \cdot θ_{s p e c}

(767)

The relation for static enthalpy:

Figure 8. EQUATION_DISPLAY

H (T_{s}, P_{s}) = H (T_{t, s p e c}, P_{t, s p e c}) - \frac{{| v |}^{2}}{2}

(768)

Together with the isentropic relation:

Figure 9. EQUATION_DISPLAY

S (T_{s}, P_{s}) = S (T_{t, s p e c}, P_{t, s p e c})

(769)

are solved for static temperature $T_{s}$ and static pressure $P_{s}$ , where $S$ is the fluid entropy. For a compressible ideal gas, the isentropic relation is a simple expression that can readily be solved for $T_{s}$ and $P_{s}$ .

When local supersonic conditions occur at the boundary, all boundary values are computed from the specified input values, including the supersonic static pressure $P_{s, s p e c}^{\sup}$ .

The fluid density

ρ

and the total enthalpy

H_{t}

at the boundary face are updated from the equation of state by using the now known boundary values for static pressure

P_{s}

, velocity

v

, and the static temperature

T_{s}

Figure 10. EQUATION_DISPLAY

\begin{matrix} ρ & = & ρ (P_{s}, T_{s}) \\ H_{t} & = & H_{s} (P_{s}, T_{s}) + \frac{{| v |}^{2}}{2} \end{matrix}

(770)

Incompressible and/or Isothermal Equation of State

The boundary pressure is calculated from the Bernoulli equation:

Figure 11. EQUATION_DISPLAY

P_{s} = P_{t, s p e c} - \frac{ρ}{2} {| v |}^{2}

(771)

where the boundary velocity magnitude is extrapolated from the interior of the domain:

Figure 12. EQUATION_DISPLAY

| v | = {| v |}^{e x t}

(772)

The boundary velocity vector is then obtained by multiplying the velocity magnitude with the specified inflow direction as given by Eqn. (767).

Incompressible and Non-Isothermal Equation of State

In addition to computing the static pressure and the velocity vector at the boundary according to Eqn. (771) and Eqn. (772), for a non-isothermal simulation, the static temperature is calculated as:

Figure 13. EQUATION_DISPLAY

T_{s} = T_{t, s p e c} - \frac{{| v |}^{2}}{2 C_{p}}

(773)

The density at the boundary face is then updated:

Figure 14. EQUATION_DISPLAY

ρ = ρ (T_{s})

(774)

The static and total enthalpies are given by:

Figure 15. EQUATION_DISPLAY

\begin{matrix} H_{s} & = & H_{s} (P_{s}, T_{s}) \\ H_{t} & = & H_{s} + \frac{{| v |}^{2}}{2} \end{matrix}

(775)

Pressure Constraints

Non-reflecting

The non-reflecting boundary condition approach assumes that the solution at the boundary is periodic and can be decomposed, circumferentially in three-dimensional space, into a specified number of Fourier modes $n_{s p e c}$ . The zeroth mode corresponds to the average solution and receives the standard boundary condition treatment. The working pressure that you specify at the outlet boundary corresponds to the zeroth mode.

In order to prevent reflections at the boundaries, the sum of the harmonics forming the remaining part of the solution are treated according to exact two-dimensional theory. Such treatment ensures that changes in dependent variables at the boundaries obey particular linear relationships derived by considering characteristic propagation of solutions to the linearized Euler equations [169], [170], [171], [208]. Enforcing these relationships imposes steady-state, non-reflecting boundary conditions such that all incoming modes are eliminated at the boundary and all outgoing modes are allowed to leave the domain unaffected.

The non-reflecting boundary condition is appropriate in situations where the flow normal to the boundary remains subsonic.

Pressure Jump

Depending on the various pressure jump options, the pressure rise or loss at the boundary is calculated as:

Fan
The pressure rise at the boundary is obtained from a fan curve $F_{s p e c}$ , which defines the pressure rise as a function of flow rate or flow velocity. The pressure rise is calculated depending on the specified pressure rise option as:
Standard
Figure 16. EQUATION_DISPLAY

$Δ P = P_{s, d o w n} - P_{t, u p}$
(776)

where:
- $P_{s, d o w n}$ is the static pressure downstream of the fan.
- $P_{t, u p}$ is the total pressure upstream of the fan.
Static to Static

Figure 17. EQUATION_DISPLAY

$Δ P = P_{s, d o w n} - P_{s, u p}$
(777)

where $P_{s, u p}$ is the static pressure upstream of the fan.
The pressure jump is usually imposed locally, which means that a separate pressure jump is applied at each face on the pressure outlet. Alternatively, a single pressure jump can be applied to all faces.
Porous
The pressure loss at the boundary is calculated as:
Figure 18. EQUATION_DISPLAY

$Δ P = α_{s p e c} ρ {| v |}^{2} + β_{s p e c} ρ | v |$
(778)

The pressure loss is applied in the direction of the flow.
Loss Coefficient
The pressure loss is calculated as:

Figure 19. EQUATION_DISPLAY

$Δ P = 0.5 K_{s p e c} ρ {| v |}^{2}$
(779)

The pressure loss is applied in the direction of the flow.