Iform = 4

Block Format Keyword This boundary can simulate gas inlet conditions for multi-material ALE laws (formulation: I_form = 0, 1, 10, or 11).

The boundary sub-material states is calculated from a state at a stagnation point which is provided by the user. When using this feature, it is no longer necessary to use imposed velocity (/IMPVEL) where the velocity is computed by numerical scheme.

Description

The user provides stagnation state

α_{s t a g n a t i o n} = α_{0}

ρ_{s t a g n a t i o n} = ρ_{0}

and

E_{s t a g n a t i o n} = E_{0}

which corresponds to state for which v=0. From the Ideal Gas EOS: (1)

P_{0} = C_{0} + (1 + μ) \cdot (γ - 1) \cdot E_{0}

Where, $C_{4} = γ - 1$ . It can be deduced that $P_{s t a g n a t i o n} = C_{0} + C_{4} \cdot E_{0}$ .

At each cycle, Radioss computes gas inlet state

ρ_{i n}, E_{i n}, P_{i n}

such as Bernoulli theory is satisfied 1 using velocity at inlet face.

Format

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
/MAT/LAW51/`mat_ID`/`unit_ID`
`mat_title`
Blank Format
`I`_form

#Global Parameters

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`Scale`_time		`P`_EXT

#Material1 Parameters

(1)	(3)	(5)	(7)	(8)	(9)
$α_{0}^{mat_1}$	$ρ_{0}^{mat_1}$	$E_{0}^{m a t_ 1}$	`fct_ID`_α1	`fct_ID`_{$ρ$ 1}	`fct_ID`_E1
$C_{1}^{m a t_1}$			$C_{4}^{m a t_1}$
	$C_{0}^{m a t_1}$

#Material2 Parameters

(1)	(3)	(5)	(7)	(8)	(9)
$α_{0}^{mat_2}$	$ρ_{0}^{mat_2}$	$E_{0}^{m a t_ 2}$	`fct_ID`_α2	`fct_ID`_{$ρ$ 2}	`fct_ID`_E2
$C_{1}^{m a t_2}$			$C_{4}^{m a t_2}$
	$C_{0}^{m a t_2}$

#Material3 Parameters

(1)	(3)	(5)	(7)	(8)	(9)
$α_{0}^{mat_3}$	$ρ_{0}^{mat_3}$	$E_{0}^{m a t_ 3}$	`fct_ID`_α3	`fct_ID`_{$ρ$ 3}	`fct_ID`_E3
$C_{1}^{m a t_3}$			$C_{4}^{m a t_3}$
	$C_{0}^{m a t_3}$

Definitions

Field	Contents	SI Unit Example
`mat_ID`	Material identifier. (Integer, maximum 10 digits)
`unit_ID`	Unit Identifier. (Interger, maximum 10 digits)
`mat_title`	Material title. (Character, maximum 100 characters)
`I`_form	Formulation flag. = 4 Gas Inlet (computed from data at stagnation point). (Integer)
`Scale`_time	Abscissa scale factor for input functions. 2 Default = 1 (Real)
`P`_EXT	External (ambient) pressure. 3 (Real)	$[Pa]$
$α_{0}^{mat_i}$	Initial volumetric fraction. 4 (Real)
$ρ_{0}^{mat_i}$	Initial density at stagnation point. 1 (Real)	$[\frac{kg}{m^{3}}]$
$E_{0}^{m a t_ i}$	Initial energy at stagnation point. 5 (Real)	$[\frac{J}{m^{3}}]$
`fct_ID`_αi	(Optional) Volumetric fraction scaling function. $f_{α_{i}} (t)$ identifier. 6 = 0 $α^{m a t_{i}} (t) = α_{0}^{m a t_{i}}$ > 0 $α^{m a t_{i}} (t) = α_{0}^{m a t_{i}} f_{α_{i}} (t)$ (Integer)
`fct_ID`_{$ρ$ i}	(Optional) Density fraction scaling function. $f_{ρ_{i}} (t)$ identifier = 0 $ρ^{m a t_{i}} (t) = ρ_{0}^{m a t_{i}}$ > 0 $ρ^{m a t_{i}} (t) = ρ_{0}^{m a t_{i}} . f_{ρ_{i}} (t)$ (Integer)
`fct_ID`_Ei	(Optional) Energy fraction scaling function. $f_{E_{i}} (t)$ identifier. = 0 $E^{m a t_{i}} (t) = E_{0}^{m a t_{i}}$ > 0 $E^{m a t_{i}} (t) = E_{0}^{m a t_{i}} f_{E_{i}} (t)$ (Integer)
$C_{0}^{m a t_ 3}$	Coefficient for perfect gas EOS. 5 (Real)	$[Pa]$
$C_{4}^{m a t_ i}$	Perfect gas ( $γ - 1$ ) constant. 5 (Real)
$C_{0}^{m a t_ i}$	Coefficient for perfect gas EOS. 5 (Real)	$[Pa]$

Comments

The provided stagnation point $ρ_{stagnation}, P_{stagnation}$ is used to compute gas inlet state. Bernoulli theorem is applied:
(2)
$P_{stagnation} = P_{in} + \frac{ρ_{in} v_{in}^{2}}{2}$

This leads to gas inlet state:(3)
$ρ_{in} = ρ_{stagnation} {[1 - \frac{γ - 1}{2 γ} \cdot \frac{ρ_{stagnation}}{P_{stagnation}} \cdot (1 + C_{d}) \cdot v_{in}^{2}]}^{\frac{1}{γ - 1}}$
(4)
$P_{i n} = P_{s t a g n a t i o n} {(\frac{ρ_{i n}}{ρ_{s t a g n a t i o n}})}^{γ}$
(5)
${(ρ e)}_{i n} = \frac{P_{a}}{γ - 1} {(\frac{ρ_{i n}}{ρ_{stagnation}})}^{γ - 1}$

Then the global material state is determined by computing a mean value:

Pressure

$Δ P_{i n} = \sum_{i} α^{m a t_{i}} (t) Δ P_{i n}^{m a t_i}$

Density

$ρ_{i n} = \sum_{i} α^{m a t_{i}} (t) ρ_{i n}^{m a t_i}$

Energy

${(ρ e)}_{i n} = \sum_{i} α^{m a t_{i}} (t) E_{i n}^{m a t_i}$
The optional scaling functions can be used such to scale the volumetric, density or energy fractions.
Parameter $P_{E X T}$ enables you to take ambient pressure into account in case you want to work with relative pressure $Δ P_{min}^{m a t_i}$ . This parameter is required by Radioss for correct energy integration at each cycle. Otherwise, numerical EOS solving is generally incorrect. It represents pressure which must be added to EOS calculation to obtain total (physical) pressure. It has no influence on pressure contour in animation files.
Example using linear EOS:

Total Pressure: $P = P_{amb} + C_{1} μ$ and also $P_{E X T} = 0$

Relative Pressure: $Δ P = C_{1} μ$ , and also $P_{E X T} = P_{a m b}$
Volumetric fractions enable the sharing of elementary volume within the three different materials.
For each material, $α_{0}^{m a t_ i}$ must be defined between 0 and 1.

Sum of initial volumetric fractions $\sum_{i = 1}^{3} α_{0}^{m a t _ i}$ must be equal to 1.

For automatic initial fraction of the volume, refer to /INIVOL.
Perfect gas EOS is $P (μ, E) = (γ - 1) (1 + μ)$ . Generally it can be written using this general form $P = C_{0} + C_{1} μ + C_{4} (1 + μ) E$ , where $C_{4} = (γ - 1)$ . This provides more flexibility, depending on whether pressure and energy are total or relative:(6)
$P (μ, E) = C_{4} (1 + μ) E$

Where, $C_{4} = (γ - 1)$ and $P_{E X T} = 0$ .

This leads to usual form from $Δ P (μ, E) = C_{0} + C_{4} (1 + μ) E$ .(7)
$Δ P (μ, E) = C_{0} + C_{4} (1 + μ) E$

Where, $C_{4} = (γ - 1)$ , $C_{0} = - P_{0}$ and $P_{E X T} = P_{a m b}$ .(8)
$Δ P (μ, Δ E) = C_{0} + C_{1} μ + C_{4} (1 + μ) Δ E$

Where, $C_{4} = (γ - 1)$ , $C_{1} = E_{0} (γ - 1)$ and $P_{E X T} = P_{a m b}$ .
$Δ P_{\min}^{m a t_ i}$ flag is the minimum value for the computed pressure.
Since $P = Δ P + P_{E X T}$ , defining $P_{E X T} = 0$ implies $Δ P \equiv P$ and $Δ P_{m i n} \equiv P_{m i n}$ .

The materials pressure must remain positive to avoid any tensile strength, then, $P_{m i n} = 0$ leads $Δ P_{m i n} = - P_{E X T}$ .

For solid materials, the default value for $Δ P_{\min}^{m a t_ i} = 10^{30}$ is suitable.
EOS parameters must be consistent with gas EOS from adjacent MM-ALE domain.