49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition
4 - 7 January 2011, Orlando, Florida
AIAA 2011-471
Speed of Sound and Real Gas Solution Algorithm for
Supercritical Flows
Seon-Young Moon 1, Jae-Ryul Shin 2 and Jeong-Yeol Choi 3
Pusan National University, Busan 609-735, Korea
The definition of the speed of sound is reexamined since it is crucial in the numerical
analysis of compressible real gas flows. The thermodynamic speed of sound (TSS), ath, and
the characteristic speed of sound (CSS), ach, are derived using generalized equation of state
(EOS). It is found that the real gas EOS, for which pressure is not linearly dependent on
density and temperature, results in slightly different TSS and CSS. in this formalism, Roe's
approximate Riemann solver was derived again with corrections for real gases. The results
show a little difference when the speeds of sound are applied to the Roe's scheme and
Advection Upstream Splitting Method (AUSM) scheme, but a numerical instability is
observed for a special case using AUSM scheme. It is considered reasonable to use of CSS
for the mathematical consistency of the numerical schemes. The approach is applicable to
multi-dimensional problems consistently.
Nomenclature
a
a,b
A
B,C
c
cp
cv
d
e
speed of sound
gas specific constants
flux Jacobian matrix
Virial coefficients
speed of sound
specific heat per unit mass at constant pressure
specific heat per unit mass at constant volume
infinitesimal variation
total energy per unit volume, e = ρ(ε + 1 u 2 )
2
F
h
convectional flux vector
enthalpy per unit mass, h ≡ ε + pv
H
Total enthalpy per unit mass, H = h + 1 u 2 = (e + p ) / ρ
2
index subscript for species
subscripts for left and right interface
momentum per unit volume, m=ρu
subscript for mixture
Molecular weight (mass per unit mole)
pressure
Eigenvector matrix and its inverse matrix
vector of conservative variables
specific gas constant, R=R/Mw
universal gas constant, R≡8.314 J/mol⋅K
entropy per unit mass
time
k
L,R
m
mix
Mw
p
P,P-1
Q
R
R
s
t
1
Research Assistant, Department of Aerospace Engineering.
Research Assistant, Department of Aerospace Engineering, Student member AIAA.
3
Professor, Department of Aerospace Engineering, Senior member AIAA, Email: [email protected].
1
American Institute of Aeronautics and Astronautics
2
Copyright © 2011 by by the Authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
T
temperature
q
heat per unit mass
u
fluid velocity
v
specific volume, volume per unit mass, v=1/ρ
molar volume, volume per unit mole, v = v ⋅ Mw
v
W
vector of characteristic variables
X
spatial coordinate
mass fraction of k-th species
yk
α,β temporary variables
specific heat ratio, γ=cp/cv
γ
equivalent gamma
Γ
difference right and left interfaces
δ
internal energy per unit mass
ε
λ, λ1,2,3 eigenvalues
vector of eigenvalues
Λ
density, mass per unit volume
ρ
compressibility factor
σ
I. Introduction
I
deal gas equation of state (EOS), which states that the pressure is linearly proportional to the density and
temperature, holds at the limiting case where the volume of a particle of gas and the interaction between the
particles are assumed being negligible. The ideal gas EOS is widely used in many gas-dynamic applications without
a problem, but there are some problems where the ideal gas assumption cannot be applied. The non-ideal effects,
called as the real gas effects arise typically at high pressure and/or low temperature problems where the volume of
the gas particles and their interactions are not negligible. The real gas problems involve the multiple-phase problems
where the distinction between the phases is ambiguous, and the ‘condensation and evaporation’ becomes important.
Representative examples of the real gas problems are steam engines using super-heated steam, cryogenic systems,
high-pressure combustion and explosions of explosives.[1] Recent methods for the ‘cavitation’ problems also use
the real gas EOS to manipulate the multiple phases as a single phase.[2]
Here, it should be noted that a terminology of ‘real gas’ is different from the ‘real gas effects’ that is used in the
hypersonic aerodynamics. Anderson[3] addressed that the ‘real gas effects’ used in the hypersonic aerodynamics
refers the vibrational and rotational relaxations, the dissociation and the ionization of gases at high temperature, thus
the ‘high temperature effects’ should be used instead of ‘real gas effect.’ Because, the term ‘real gas’ has been used
in the thermodynamic sciences since 1800s to refer other scientific issues. Even though, it is pretty ambiguous until
now because the hypersonic problems also includes some case where the pressure is not linearly proportional to
density and temperature. In the present paper, the terminology, ‘real gas’ is restricted presently to the classical
meaning of the gas in which the pressure is not linearly proportional to the density and temperature.
The EOS is considered as a one of the governing equations of the compressible flows, in addition to the
conservation equations of mass, momentum and energy. The ideal gas EOS that defines the linear relation between
the thermodynamic variables has been used conveniently in the compressible fluid dynamics. Most of the modern
numerical methods are developed base on the linear relation, ideal gas EOS. However there are many gas-dynamic
applications where the ideal gas EOS may not be applicable, and the numerical methods should be corrected for
these problems based on the non-linear real gas EOS.
Among the numerical methods for the solution of compressible flows a shock-capturing scheme is the most
fundamental part. There are a number of shock-capturing schemes those can be classified into several categories,
such as FCT (Flux Corrected Transport) methods, flux-splitting methods and etc. Among these, flux-splitting
methods, generally called as upwind schemes, may be the most widely spread. The flux-splitting methods can be
classified again into several categories, such as FVS (Flux Vector Splitting), Riemann solvers (also called as FDS,
Flux Difference Splitting) and AUSM (Advection Upstream Splitting Methods)-variants.[4,5] Among the various
flux splitting schemes, Roe’s approximate Riemann solver may be the most widely one in reality.[6] Even though
many latest schemes have good characteristics and Roe’ scheme has some problems, Roe’ scheme or its variants
were already adopted in many commercial and non-commercial fluid dynamics packages. So, revision of Roe’
scheme is inevitable if a real gas correction necessary for shock capturing schemes.
The basic principle of the flux splitting schemes is the discretization of flux functions based on the direction of
convectional transport speed across the cell interface. In case of AUSM-variant schemes, the flux splitting is
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determined by cell interface Mach number. Thus, the consideration of real gas EOS becomes relatively simple,
because the real gas EOS is considered only for the definition of speed of sound, as was done by Edwards et al.[2]
However most of the flux splitting schemes requires a flux Jacobian matrix for the transport speed, the
implementation of real gas EOS becomes quite complex due to the necessity for the derivatives for pressure and
energy. Also, the non-linearity of real gas EOS changes the mathematical characteristics of the flux functions those
were the basis for the derivation of the schemes.
There were previous studies for the upwind schemes with real gas effects, but most of them were focused in the
high temperature effects, and real gas EOS was treated less importantly.[7-10] Even in case when the real gas EOS
was considered, the influence on the thermodynamic aspect of the flux functions and the change of mathematical
characteristics were treated insufficiently.[11-13] In this study, Roe’ approximate Riemann solver will be derived
again with real gas EOS, and more attention will be paid for the thermodynamic aspect and the change of
mathematical characteristics. Also the attention will be given to the correction of the existing codes for the practical
applications where real gas EOS is necessary.
II. Classification of Gases
The ideal gas is defined as a gas for which the ideal gas assumption, (known as Boyle-Charle’s law) in equation
(1) holds.
pv
= constant = R
(1)
T
The ideal gas is also called as perfect gas. The perfect gas is classified into calorically perfect gas and thermally
perfect gas. Calorically perfect gas is that gas for which specific heat is constant.
(2)
dε = cv dT , dh = c p dT , pv = RT
And, the thermally perfect gas is the gas for which specific heat is a function of temperature only.
dε = c v (T )dT , dh = c p (T )dT , pv = RT
(3)
Thus the ideal gas assumption also holds for a bit complex situation for the mixture of perfect gases, generally the
cases of chemically reacting flows.
dε = cv (T )dT , dh = c p (T )dT , pv = RT
cv (T ) = ∑k =1 y k cvk (T ) , R = R / Mwmix
N
(4)
In the meanwhile, real gas is a gas that does not satisfy the relation (1), even with a single composition and
constant specific heat. The basic form of real gas EOS may be the Van der Waals EOS that accounts for the
influences of the molecular volume and the inter-molecular forces.[1]
(p +
a
v2
)(v − b) = RT
(5)
where a and b are gas specific constants. However the Van der Waals EOS does not phase change correctly, and
many revised EOS’ are suggested. Among them the most general form of real gas EOS may be the Virial EOS,
which represents the relation between the pressure, density and temperature as a polynomial.[1]
pv
B (T ) C (T )
(6)
= 1+
+ 2 + ... = σ(T , v)
RT
v
v
Here, the right-hand-side polynomial is named as a compressibility factor used to show the deviation from the ideal
gas condition. The compressibility factor is generally a function of two thermodynamic variables and has a value of
unity for ideal gas condition. Thus the general form of real gas EOS can be written as equation (7), with different
definition of the compressibility factor.
pv = σ(T , v)RT
or
p = σ(T , ρ)ρRT
For example the SRK EOS can be written as equation (7) with a compressibility factor defined as.
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(7)
1
σ=
( ρ,T )
1−
bρ
Mw
−
a
(8)
 Mw

+ b  RM wT

ρ


where,
2
=
a
0.42748 Ru Tc
pc
2
2
0.48 + 1.574ω − 0.176ω and b =
1 + f ( ω ) (1 − Tr0.5 )  , f ( ω ) =
2
0.08664 Ru Tc
.
pc
Fig. 1 is an example of compressibility of nitrogen computed by the SRK EOS.
Figure.1 Compressibility factor of nitrogen computed by SRK EOS.
III. Thermodynamic Relations for Real Gas EOS
For real gas, the internal energy and enthalpy are functions of two thermodynamic variables for real gas.
ε = ε(T , v) , h = h(T , p )
(6)
The infinitesimal variation of internal energy and enthalpy is defined as,
 ∂ε 
 ∂ε 
 ∂ε 
dε = 
 dT +   dv = c v (T )dT +   dv
 ∂T  v
 ∂v  T
 ∂v  T
(7a)
 ∂h 
 ∂h 
 ∂h 
dh = 
 dT +   dp = c p (T )dT +   dp
 ∂T  p
 ∂p  T
 ∂p  T
(7b)
Thus, the internal energy and enthalpy is defined by integration as a sum of ideal term and excess term.
ε(T , v) = ε id (T ) + ε ex (T,v ) , ε id (T ) = ∫ c v dT
(8a)
h(T , p) = h id (T ) + h ex (T,p ) , h id (T ) = ∫ c p dT
(8b)
Here the ideal parts are functions of temperature only and the excess parts reflect the real gas effects. The excess
terms can be evaluated calculated more precisely from the definitions of the internal energy and the enthalpy, and
their total derivatives.
ε ≡ q + w , h ≡ ε + pv
dε = dq + dw = Tds − pdv
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(9)
(10a)
dh = dε + pdv + vdp = Tds + vdp
(10b)
In the meanwhile the total derivatives are
 ∂ε 
 ∂ε 
dε =   ds +   dv
 ∂s  v
 ∂v  s
(11a)
 ∂h 
 ∂h 
dh =   ds +   dp ,
 ∂s  p
 ∂p  s
(11b)
and the total variation of entropy is defined as,
 ∂s 
 ∂s 
 ∂s 
 ∂s 
ds = 
 dT +   dv = 
 dT +   dp .
T
v
∂
T
∂
∂

v
 T

p
 ∂p  T
(12)
So, substituting the equation (12) into (11) and comparing the coefficients terms with equation (7) and (10) after
applying the Maxwell relation, the total derivatives of internal energy and enthalpy is refined defined again as
equation (13).
  ∂p 

dε = c v dT + T 
 − p  dv
  ∂T  v

(13a)

 ∂v  
dh = c p dT + v − T 
  dp
 ∂T  p 

(13b)
The derivatives terms can be calculated if a specific EOS is supplied.
On the other hand, the speed of sound is defined thermodynamically as,
ath2 = (∂p ∂ρ)s
(14)
By following the similar thermodynamic process, the speed of sound is determined.
a th2 = Γth
c p 1  ∂p 
p
, Γth =


ρ
c v ρ  ∂T  v
 ∂v 
p

 ∂T  p
(15)
By applying the general form of the real gas EOS, the equivalent gamma is determined using the derivatives of the
compressibility factor.[14]
Γth =
c p  σv 
 ∂σ 
 ∂σ 
1 −
 , σ v = v  , σ T = T 

σ 
cv 
 ∂T  v
 ∂v  T
(16)
IV. One-dimensional Euler Equations for Real Gas
One-dimensional Euler equations for real gas can be summarized in vector form.
ρ
 ρu 
∂Q ∂F




+
= 0 , Q = ρu  , F = ρu 2 + p 
∂t ∂x
 e 
(e + p )u 
(17)
ε(T ) = ∫ cv dT
(18)
If the ideal gas assumption applies,
R
m 2 
ρ 2  R 
e
−
e − u  =
cv 
2  c v 
2ρ 
.
R
= ρ ε = ρf (ε(T ))
cv
p=
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(19)
A. Eigenvalues and Characteristic Speed of Sound
And, the pressure becomes a simple function of conservative variables. At least, the pressure is simply reduced to a
product of density and a function of internal energy. Then, the flux vector F is homogenous function of degree 1 of
Q. That is,
F = AQ , where A =
∂F
and F (λQ) = λF (Q) .
∂Q
(20)
This result becomes a fundamental basis for FVS schemes. One the contrary for real gas, the pressure is not a
linear function of density and temperature as shown in equation (7), and cannot be simply reduced to a explicit
function of the conservative variables. Thus, the flux vector F is not a homogenous function of degree 1 of Q, and
the basis of FVS schemes vanishes. This result means that the FVS schemes should not be applied for real gas flows,
even though the numerical procedure can be still applicable technically.
However, unlike the FVS schemes, the Roe’ scheme is not restricted by this condition and can be considered.
The conditions for the Roe’s scheme are;[6]
1 ) FR − FL = A (Q R ,Q L )(QR − Q L )
2 ) A (Q,Q) = A(Q) ≡ ∂F ∂Q , Q R = Q L = Q
3 ) real eigenvalues and linearly independent eigen vectors
So, the first step is to find the Jacobian matrix, eigen-values and eigenvectors. The Jacobian matrix can be derived as
simply as ideal gas flow.

0
∂F  2
A=
= − u + pρ
∂Q 
( pρ − H )u
Here, the derivatives of pressures are defined as,
pρ =


2u + pρu

H + pρu u ( pe + 1)u 
1
∂p
∂p
pρu =
∂ρ ρu ,e
∂ρu
0
pe
∂p
∂ρ ρ,ρu
pe =
ρ ,e
(21)
(22)
If we let the three independent eigenvalues as,
Λ = {λ1 , λ 2 , λ 3 } = {u , u + a, u − a}
(23)
a characteristic speed of sound is defined for the determination of the eigenvalues.
2
a ch
= pρ + p e h + upρu = pρ + p e (h − u 2 )
(24)
It is clear that this definition is same as equation (14) if the ideal gas assumption applies, but it is not confirmed yet
for real gas. In the meanwhile the eigenvector matrix and its inverse matrix is derived same procedure for ideal gas.

 1

P= u
u 2 − pρ

pe
P −1
pρ

 1− 2
a

p
au
−
ρ
= 
2
2a
 p + au
ρ

 2a 2
1



u−a 
H − ua 

1
u+a
H + ua
pe u
2
a
− pe u + a
2a 2
− pe u − a
2a 2
pe 

a2 
pe 
2a 2 

Pe 
2a 2 
(25)
−
(26)
B. Characteristic Form of Roe’s Scheme
By assuming the flux function, the Jacobian matrix and the variable vector satisfy the condition (1) for Roe’s
scheme, the numerical flux function for Roe’s approximate Riemann solver is written as,[6]
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[
]
~
1
F1 2 = F (QR ) + F (QL ) − δF .
2
Here, the artificial damping is determined as.
(27)
δF = P Λ P −1δQ = P Λ δW = P Λ ( P −1MδV )
(28)


 1 
 1 
 1 




= λ1 ∂w1  u  + λ 2 ∂w2  u + a  + λ 3 ∂w3  u − a 
u 2 − pρ 
 H + ua 
 H − ua 

pe 
Here, the M is a Jacobian matrix of conservative variable vector Q with respect to primitive variable vector V.
1 0 0
ρ 
δQ 

M =
= u ρ 0  , V =  u 
δV 

 p 
eρ ρu e p 
(29)
And, the variation of the characteristic variable set is


α δ ρ − β 2 δp


a
 ∂w1  

1
∂W = ∂w2  =  2 (1 − α)a 2 δ ρ + ρ a δ u + β δp  .
 2a

∂w3   1

2
 2 (1 − α)a δ ρ − ρ a δ u + β δ p 
 2a

(30)
δρ = ρ R − ρ L , δu = u R − u L , δp = p R − p L
(31)
{
{
}
}
where,
Here, temporary variables are defined as
pe (h − eρ )
β = pe ⋅ e p .
(32)
a2
These variables are unity for ideal gas, but should be determined after the determination of pressure and energy
derivatives for real gas.
In the above equations, the pressure derivatives with respect to conservative variables are derived by chain-rule,
since the pressure is not an explicit function of the conservative variables and the temperature should be used as an
intermediate variable. Thus by assuming pressure from the EOS as,
(33)
p = p (ρ, T ) .
α=
The pressure derivatives are derived in the following manner.
pρ =
∂p
∂p
∂p
=
+
∂ρ m,e ∂ρ T ∂T
∂T
ρ ∂ρ
,
(34)
m ,e
Here, the temperature derivatives with respect to the conservative variables are derived from the definition of the
total energy by assuming the internal energy as a function of temperature and density. The resulting pressure
derivatives are determined as equation (35) by applying the general form of real gas EOS in equation (7).


e
pρ = (σ − σ v ) RT + pe  u 2 − + σT RT 
ρ


pe =
R
(σ + σT ) , p m = −upe
cv
By substituting the pressure derivatives into equation (24), the characteristic speed of sound is derived as,
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(35a)
(35b,c)
2

σ
R (σ + σT )
p
, Γch =  (1 − v ) +

ρ
σ
σ
cv



(36)


This derivation of the speed of sound does not seem to be identical to the thermodynamic derivation in equation (15)
and (16), and needs further examination.
In the meanwhile, the energy derivatives should be derived with respect to the primitive variables in equation
(29). Thus the EOS is considered as a definition function for temperature in this case. All the energy derivatives in
equation (29) are derived by following the similar way of the pressure derivatives. Applying the energy derivatives
along with the pressure derivatives, the temporary variables in equation are proved to be unity.
2
ach
= Γch
α ≡ 1 and β ≡ 1
(37)
Therefore, the formulation of the numerical flux for the real gas becomes identical to that of ideal gas except the
pressure derivatives and the speed of sound. It should be noted that all the variables in the formulation of numerical
flux should be determined by an average values satisfying the condition (1) for Roe’s scheme except the differences
defined in equation (31).
C. Roe’s Average for Real Gas
Presently, the average state satisfying the condition (1) is not decided yet. The averages values, called as Roe’s
average were derived based on the Roe’s finding that the conservative variables in Q and the flux functions in F are
quadratic functions of an intermediate variables defined as a vector Z as follows.[6]
ρ
 ρu 
1




2
Q = ρu  , F = ρu + p  , Z = ρ  u 
 e 
(e + P)u 
 H 
(38)
This finding is based on the fact that pressure is a simple explicit function of conservative variables as shown in
equation (19) for ideal gas. However the pressure is no more a simple explicit function of conservative variables for
real gas and a general solution of average values for a real gas EOS does not exist any more. Therefore, the only
way is to use an approximation. For this purpose, the compressibility factor and its derivatives are assumed being
locally constant between the left and right interfaces. Then, it can be shown that the Roe’s Average formula in
equation (39) still hold for the real gas.
ρ1 / 2 = ρ L ρ R
u1 / 2 =
H1/ 2 =
ρR uR + ρL uL
ρR + ρL
(39)
ρR H R + ρL H L
ρR + ρL
For the complete calculation of the numerical flux, the average value of temperature should be calculated from the
definition of total enthalpy using the average values in equation (39). An iterative method may be used for the
evaluation of temperature since the temperature is an implicit function of density, velocity and total enthalpy.
V. Comparison of the Speeds of Sound
A. Comparison of the Thermodynamic and Characteristic Speed of Sound
The above two definitions of the speeds sound in (15) and (36) converge to the identical value for ideal gas,
constant compressibility factor, or if the pressure is linearly depends on density and temperature. However, the those
appear in different form, but is hard to prove analytically whether those are identical or not. Thus, the speeds of
sound are plotted in Fig. 2 with the fomular derive with van der Waals EOS. The material is nitrogen with molecular
weight of 28.01 g/mole, critical pressure 3.39 MPa, critical temperature 126.2 K. the subscript “ch” stands for the
characteristic speed of sound (CSS), “th” for the thermodynamic speed of sound (TSS), and “id” for the ideal gas. It
is shown in this figure that the two definitions lead to the different values. The CSS is always greater than TSS while
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the deviation is greater around the point of phase change or the critical point. The CSS and TSS converge to the
identical values for the ideal gas and liquid phase conditions.
Figure. 2 Various Speed of Sound computed by Van der Waals Equation of state.
B. Isentropic nozzle flows
The one dimensional solver was applied for the isentropic quasi-one dimensional supersonic nozzle flow for the
comparison of the two definition of the speed of sound. The nitrogen is considered with inflow conditions of 100
atm, 200 K and 450 m/s. The compressibility factor of this condition 0.7898, a bit far from the ideal gas condition.
The numerical solutions are compared with the analytic solution obtained by the Rankine-Hugoniot relation since all
the eigenvalues are positive in this case. The cross sectional are of the nozzle is defines by
(40)
=
s ( x)
1.398 + 0.347 tanh(0.8 x - 4) .
The inlet is located at x=0 and the exit is at 10.0, within where 100 evenly distributed grid point is allocated.
Pressure variation is plotted in Fig. 3 where all the result agrees quite well. Therefore, it is considered that effect of
the different definition of speed fo sound is not so significant, though the use of CSS results in about 25% better
convergence rate as shown in Fig. 4.
Figure 3. Comparison of Pressure profiles for real gas EOS and analytical
solution at far from ideal gas condition
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Figure 4. Comparison of Convergence histories for Quasi-one dimensional
nozzle flow computed by Characteristic speed of sound, Thermally speed of
sound
C. Isentropic nozzle flows
Comparison is also done for the Sod’s shock tube problem of pressure ration 10 and density ratio 1. The
reference temperature and pressure are 300K and 1 atm. AUSM scheme and Roe scheme are compared for this case.
Though this condition can be considered as an ideal gas condition resulting nearly same speed of sound, Fig. 5
shows that the AUSM with TSS results in solution instability around the contact surface. Therefore one should be
careful about using TSS in case of AUSM scheme that is sensitive to the definition of the speed of sound.
Figure 5. Comparison of density profiles with characteristic speed of sound and
thermally speed of sound at ideal gas condition
In overall it is considered that the use of CSS would be better choice regarding solution convergence an stability.
VI. Multi-Dimensional Application of Supercritical Injection Flow
Based on the above formulations, applications were conducted with SRK EOS. Constants of SRK EOS are
calculated from thermodynamic data of the critical condition of the fluid. All the variables were normalized using the
given reference state variables. The specific heat was assumed constant value and the temperature can be decided
with an iterative procedure.
As first application, two-dimensional cryogenic injection problem was tested. The theoretical and numerical
scheme were implemented to study the injection and mixing of cryogenic fluid under supercritical conditions.
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Schematics if shown in Fig. 6. The computational domain downstream of the injector measures a length of 30 Dinj
and a radius of 2 Dinj. The dimensions are sufficient to minimize the effect of the far-field boundary conditions on
the near-injector flow evolution. The entire grid system consists 302x102 points along the axial and radial directions,
respectively. The grids are clustered in the shear-layer and near the injector to resolve rapid property variations in
those regions.
30 Dinj
2 Dinj
Dinj
CL
Figure 6. Schematics of test problem.
As specific example, liquid nitrogen (N2) at a temperature of 120 K is injected through a circular tube with a
diameter of 254 µm into a supercritical nitrogen environment. A turbulent pipe flow with a bulk velocity of 15 m/s is
assumes at the injector exit. The ambient temperature remains at 300 K, but pressure is 93 atm, comparable to the
chamber pressure of many operational rocket engines. For reference, the critical pressure and temperature of
nitrogen are 34 atm and 126 K, respectively. Test condition summarized in Table 1 is considered, simulating the
experiments conducted by Chehroudi and Talley[10], where the subscript ∞ and inj denote the injection and ambient
conditions, respectively.
Table 1 Simulation Condition.
Condition
p∞ (MPa)
T∞ (K)
ρ∞ (kg/m3)
ρinj /ρ∞
Value
9.3
300
103
6.07
Condition
Tinj (K)
ρinj (kg/m3)
uinj (m/s)
Re
Value
120
626
15
42300
At the injector exit, a fully developed turbulent pupe flow is assumed. At the downstream boundary,
extrapolation of primitive variables from the interior may cause undesired reflection of waves propagating into the
computational domain. And the non-slip adiabatic conditions are enforced along solid walls. For a constant-density
jet, the shear-layer between the jet and the ambient fluid is susceptible to the Kelvin-Helmholz instability and
experiences vortex rolling, paring, and breakup A Cryogenic supercritical jet undergo qualitatively the same process,
but with additional mechanism arising from volume dilation and baroclinic torque.
Fig. 7 show snapshot of the density, temperature, compressibility factor, and mach number contour. The results
shows the break-up and phase changes from liquid jet into gas dispersion while showing the contributions of
turbulent eddy motions to the phase change processes. Temperature and density fields clearly demonstrate the
entrainment of lighter and warmer ambient gaseous nitrogen into the jet flow through vertical motions along with a
series of thread-like entities emerging from the jet surface.
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Fig. 7 Density, temperature, compressibility factor and Mach number contour
profile of two-dimensional cryogenic injection under supercritical condition.
As a second application, three-dimensional flow was considered. The real gas correction of the threedimensional numerical flux was carried out in similar way to two-dimensional case. The entire grid system consists
1.4 million points. The grids are also clustered in the shear-layer and near the injector to resolve rapid property
variations in those regions. Test condition is same as two-dimensional problem. For accurate simulation, the
numerical simulation was carried out with 5th order space acc-urate and second order time accurate numerical
methods developed for hybrid RANS/LES simulation was used with SRK equation of state. Fig. 8 shows
Instantaneous density contour profile of cryogenic Nitrogen injection under supercritical condition. Similarly, threedimensional simulation shows the vortex break-up and phase changes from liquid jet into gas dispersion, too.
Fig. 8
Three-dimensional instantaneous density contour profile of cryogenic nitrogen injection under
supercritical condition.
VII. Conclusion
Roe’s approximate Riemann solver was derived again with real gas EOS. In this derivation procedure, two
problematic points were observed for real gas EOS; 1) flux vector is not a linear function of the conservative
variable vector, 2) a general solution of average state satisfying the Rankine-Hugoniot does not exist for real gas
EOS. However, the systematic derivation was possible by assuming that the compressibility factor is locally
constant between the interfaces.
As a result, the final formulation of numerical damping becomes identical to the original Roe’s scheme for real
gas, except 1) corrected of pressure derivatives and 2) corrected of the speed of sound. The Roe’s average was still
applicable with the locally constant assumption of compressibility factor. Present paper is described for the original
form of Roe’s scheme, but the same way is also applicable to other forms of approximate Riemann solvers, such as
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the conservative formulation by Harten-Yee and the preconditioned equations.[16,17] Also, the same way of real
gas correction is applicable to multi-species reacting flow, even though the real gas EOS becomes getting more
complex.
The new formulation has been applied for some test problems including a one-dimensional shock tube, the quasi
one-dimensional nozzle flow. The results show a little difference when the speeds of sound are applied to the Roe's
scheme and Advection Upstream Splitting Method (AUSM) scheme, but a numerical instability is observed for a
special case using AUSM scheme. It is considered reasonable to use of CSS for the mathematical consistency of the
numerical schemes. The approach is applicable to multi-dimensional problems consistently. The new formulation
has been applied for some test problem including two-dimensional and three-dimensional injection problem. The
results of these problems show the break-up and phase changes from liquid jet into gas dispersion while showing the
contributions of turbulent eddy motions to the phase change processes.
Acknowledgments
This research was supported by NSL(National Space Lab) program through the National Research Foundation of
Korea funded by the Ministry of Education, Science and Technology. (Grant number 2008-2006283)
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