Block Format Keyword Describes the ideal gas equation of state for stiffed gas
P
(
μ
,
E
)
=
(
γ
−
1
)
(
1
+
μ
)
E
−
γ
P
*
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadcfadaqadaWdaeaapeGaeqiVd0MaaiilaiaadweaaiaawIca
caGLPaaacqGH9aqpdaqadaWdaeaapeGaeq4SdCMaeyOeI0IaaGymaa
GaayjkaiaawMcaamaabmaapaqaa8qacaaIXaGaey4kaSIaeqiVd0ga
caGLOaGaayzkaaGaamyraiabgkHiTiabeo7aNjaadcfadaahaaWcbe qaaiaacQcaaaaaaa@4C31@
.
This equation of state was originally designed to model water for underwater
explosions.
Format
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
/EOS/STIFF-GAS /mat_ID /unit_ID
eos_title
γ
P 0
P sh
P
*
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadcfadaahaaWcbeqaaiaacQcaaaaaaa@382F@
Definition
Field
Contents
SI Unit Example
mat_ID
Material identifier.(Integer, maximum
10 digits)
unit_ID
Unit identifier .(Integer, maximum 10 digits)
eos_title
EOS title.(Character, maximum 100
characters)
γ
Heat capacity ratio
γ
=
C
p
C
v
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape
Gaeq4SdCMaeyypa0ZaaSaaa8aabaWdbiaadoeapaWaaSbaaSqaa8qa
caWGWbaapaqabaaakeaapeGaam4qa8aadaWgaaWcbaWdbiaadAhaa8 aabeaaaaaaaa@3D36@
.(Real)
P 0
Initial pressure.(Real)
[
Pa ]
P sh
Pressure shift.(Real)
[
Pa ]
P
*
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadcfadaahaaWcbeqaaiaacQcaaaaaaa@382F@
EOS additional pressure
term.(Real)
[
Pa ]
Example (Water)
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
g mm ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW06/7/1
water
# RHO_I
0.001
# NU PMIN
0 0
/EOS/STIFF-GAS/7/1
STIFF_GAS_WATER
# GAMMA P0 PSH P_STAR
6.1 0.10 0 368.85
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata
Comments
Where,
E
=
E
i
n
t
V
0
µ
=
ρ
ρ
0
−
1
The stiffened gas EOS introduces an
additional pressure term (
P
*
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaCa
aaleqabaGaaiOkaaaaaaa@37A7@
) with respect to the ideal gas equation of state. The EOS is
stated in this form because the speed of sound in water is:This way, by considering
P
*
≫
P
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaCa
aaleqabaGaaiOkaaaakiablUMi=iaadcfaaaa@39E3@
, the speed of sound will not be affected by
P
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaaaa@36CC@
fluctuations and
c
s
≈
γ
P
*
ρ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa
aaleaacaWGZbaabeaakiabgIKi7oaakaaabaWaaSaaaeaacqaHZoWz
caWGqbWaaWbaaSqabeaacaGGQaaaaaGcbaGaeqyWdihaaaWcbeaaaa a@3F0A@
.
P
*
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaCa
aaleqabaGaaiOkaaaaaaa@37A7@
can be calculated with:As
P
*
≫
P
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaCa
aaleqabaGaaiOkaaaakiablUMi=iaadcfaaaa@39E3@
,
P
*
≈
ρ
c
s
2
γ
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaCa
aaleqabaGaaiOkaaaakiabgIKi7oaalaaabaGaeqyWdiNaam4yamaa
DaaaleaacaWGZbaabaGaaGOmaaaaaOqaaiabeo7aNbaaaaa@3FAB@
leads to a good estimate of
P
*
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaCa
aaleqabaGaaiOkaaaaaaa@37A7@
.
The stiffened gas EOS 1 can be derived from a polynomial EOS:Where,
C
0
=
−
γ
P
*
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa
aaleaacaaIWaaabeaakiabg2da9iabgkHiTiabeo7aNjaadcfadaah
aaWcbeqaaiaacQcaaaaaaa@3CF9@
C
1
=
C
2
=
C
3
=
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa
aaleaacaaIXaaabeaakiabg2da9iaadoeadaWgaaWcbaGaaGOmaaqa
baGccqGH9aqpcaWGdbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaaG imaaaa@3EF1@
C
4
=
C
5
=
γ
−
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa
aaleaacaaI0aaabeaakiabg2da9iaadoeadaWgaaWcbaGaaGynaaqa
baGccqGH9aqpcqaHZoWzcqGHsislcaaIXaaaaa@3ECB@
E
0
=
P
0
−
C
0
C
4
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa
aaleaacaaIWaaabeaakiabg2da9maalaaabaGaamiuamaaBaaaleaa
caaIWaaabeaakiabgkHiTiaadoeadaWgaaWcbaGaaGimaaqabaaake
aacaWGdbWaaSbaaSqaaiaaisdaaeqaaaaaaaa@3EE3@
For example, for water,
ρ
=
0.001
g
.
m
m
−
3
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdiNaey
ypa0JaaGimaiaac6cacaaIWaGaaGimaiaaigdacaWGNbGaaiOlaiaa
d2gacaWGTbWaaWbaaSqabeaacqGHsislcaaIZaaaaaaa@41B1@
,
c
s
=
1500
m
m
.
m
s
−
1
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa
aaleaacaWGZbaabeaakiabg2da9iaaigdacaaI1aGaaGimaiaaicda
caWGTbGaamyBaiaac6cacaWGTbGaam4CamaaCaaaleqabaGaeyOeI0 IaaGymaaaaaaa@4255@
and
γ
=
6.1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaey
ypa0JaaGOnaiaac6cacaaIXaaaaa@3AD1@
can be found in literature. 1 This leads to
P
*
=
368.852
M
P
a
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaCa
aaleqabaGaaiOkaaaakiabg2da9iaaiodacaaI2aGaaGioaiaac6ca
caaI4aGaaGynaiaaikdacaWGnbGaamiuaiaadggaaaa@4072@
. Thus, the stiffened gas EOS will give the following
behavior at atmospheric pressure:Figure 1 .
Equations of state are used by
Radioss to compute the hydrodynamic pressure and are
compatible with the material laws: