Sonderforschungsbereich/Transregio 40 – Annual Report 2016
167
A discontinuous Galerkin method for
tabulated general equations of state
By F. Föll, S. Baab†, C.-D. Munz , B. Weigand † A N D G. Lamanna†
Institute of Aerodynamics and Gas Dynamics, University of Stuttgart,
Pfaffenwaldring 21, 70569 Stuttgart, Germany
†Institute of Aerospace Thermodynamics, University of Stuttgart,
Pfaffenwaldring 31, 70569 Stuttgart, Germany
While for many technical applications, multi-phase flows can be treated as incompressible, this assumption fails in cases with high pressure and temperature near the critical
point as they can be found e.g. in rocket combustion chambers. In this paper we present
a high order density-based numerical scheme for the direct numerical simulation of compressible multi-phase flows with tabulated equations of state. Initially we will describe the
numerical framework, which is based on a high order discontinuous Galerkin spectral
element method. To incorporate multi-phase effects like phase-transition, we use the so
called dense-gas approach, which assumes a phase equilibrium between the saturated
liquid and the saturated vapour. Real gas effects are handled by a tabulated equation of
state, which prevents expensive iterations and which is highly accurate by an adaptive
approximation procedure. Spurious oscillations may arise in conservative density based
formulations at material or phase interfaces. This issue appears to be more severe in
the context of high-order discretizations and associated lower dissipation properties.
To address this issue, we show for at those interfaces a non-conservative double-flux
method.
1. Introduction
The direct numerical simulation (DNS) of multi-phase flows is usually performed on
the basis of the incompressible Navier-Stokes equations. Typical technical applications
concern droplets in an gaseous environment at ambient pressure. In such a configuration, the liquid itself can be considered to be almost incompressible, resulting in the separation of thermodynamics and hydrodynamics. Due to the incompressibility assumption
thermodynamic effects are not directly coupled to the flow variables and phase changes
are modelled as source terms [1]. However, in the context of fuel injection processes,
more extreme ambient conditions have to be faced that are characterized by an augmented pressure and temperature and thus an incompressible modelling is no longer
admissible. Especially for large pressure, temperature or density gradients in the flow
field, the thermodynamic effects have to be fully coupled to the fluid flow requiring the
compressible flow equations.
The simulation of multi-phase flows is always characterized by large jumps in the material properties. For compressible flows special attention has to be paid on the different
equations of states (EOS) on either side of the material interface [2]. Apart from this,
the resolution and tracking of the interface itself is a task of great importance for incompressible and compressible flows. To handle these difficulties different strategies can be
168
F. Föll, S. Baab, C.-D. Munz, B. Weigand & G. Lamanna
followed [3]. Sharp and diffuse interface approaches are distinguished according to the
way the interface is resolved and the thermodynamic transition is modeled.
Sharp interface approaches treat the interface as a discontinuity and have to prevent
any density smearing. The ghost fluid approach of Fedkiw et al. [4] is such a sharp
interface scheme beside moving grid techniques or cut-cell methods. A difficulty in all
these approaches is to keep the numerical approximation oscillation-free for general
equations of state and density ratios as well as to handle strong shocks [5].
Diffuse interface approaches treat the interface as a smooth transition. For this purpose, the EOS are modified in the diffuse zone to preserve thermodynamic consistency.
However spurious oscillations at material interfaces must still be considered, especially
when using density based methods. For fluids obeying the perfect gas EOS, Abgrall [2]
introduced a numerical method that freezes the thermodynamic state to guarantee that
the pressure stays constant across the interface. This scheme is not fully conservative,
but it allows the simulation of strong shock waves.
Our objective is the validation of different numerical approaches for compressible
multi-phase/species flow to figure out the proper numerics for thermodynamic consistent simulations in the transcritical area. In this paper we first present our macro-scale
model, which is based on a density based high order discontinuous Galerkin spectral element method, see [6]. To incorporate multi-phase effects like phase-transition, we use
here a dense-gas approach, which uses a phase equilibrium assumption in the multiphase area by Maxwell-line construction. Real gas effects are handled by a accurate
and efficient tabulated EOS based on automatic mesh refinement (AMR). By simulating
nitrogen jets of Mayer et al. [7] with variable thermodynamic properties we recognized
spurious oscillations in our fully-conservative density based DG formulations. Especially
the transcritical cases with density ratios of more than ten generates oscillations and
instabilities. To stabilize the simulations at these regions, we will present a generalized
non-conservative double-flux method in the context of tabulated real EOS and DG methods [8, 9].
The outline of the paper is as follows. In the next section, the governing equations
are presented. This is followed by the description of the numerical methods and thermodynamic modelling. In section four we focus on the generalized double-flux method to
prevent spurious oscillations. Finally, results of a super-critical jet simulation from Mayer
et al. [7] and one-dimensional results with the double-flux improvement are presented.
2. Governing equations
The compressible Navier-Stokes equations for a real non-reactive gas are,
∂ρ
+ ∇ · (ρu) = 0 ,
∂t
(2.1)
∂ρu
+ ∇ · (ρuu + pσ − τ ) = 0 ,
(2.2)
∂t
∂e
+ ∇ · [(e + p) u] = ∇ · (τ · u − q) ,
(2.3)
∂t
1
e = ǫ + ρu · u, ǫ = f (ρ, p, Yk ), p = f (ρ, ǫ, Yk ) ,
(2.4)
2
where ρ is the density, u is the velocity vector, p is the static pressure, e is the total
energy, ǫ is the internal energy per volume, σ is the unit tensor, q is the specific heat
flux, τ is viscous stress tensor for a Newtonian fluid. The system (2.1)-(2.3) has to be
DG method with a tabulated general EOS
169
closed with an EOS (2.4) relating the pressure to the internal energy.
To incorporate more than one species, the system is extended by k transport equations
for k species,
∂ρYk
+ ∇ · (ρuYk ) = ∇ (ρDk ∇Yk ) ,
∂t
where Dk is the species diffusion coefficient and Yk =
species.
ρk
ρ
(2.5)
is the mass fraction for each
3. Numerical methods and thermodynamic modelling
3.1. Discontinuous Galerkin method
In this section we describe the the macro-scale solver, which is based on a high order
discontinuous Galerkin spectral element method. We apply here a density-based approach for the solution of the compressible Navier-Stokes equations together with an
appropriate EOS.
The description of the method is kept brief, for more details, we refer to [6, 10, 11]. The
approach is suitable for general systems of conservation equations, in this paper we
restrict ourself to the conservation equations of the following form
U t + F (U , ∇U ) = 0,
(3.1)
where U is the vector of the solution unknowns, F is the corresponding flux containing
the viscous and the heat conduction fluxes. The nabla operator in the physical space is
T
∂
∂
∂
, ∂y
, ∂z
.
∇ = ∂x
In a three-dimensional domain we subdivide the computational space into non-overlapping hexahedral elements. Each element is mapped onto a reference cube element
E := [−1, 1]3 by a mapping x(ξ) where ξ = (ξ, η, ζ)T is the master element coordinate
vector. The mapping onto the master element, E, transforms (3.1) to the system
JU t + ∇ξ · F (U , ∇ξ U ) = 0,
(3.2)
T
∂
∂
∂
, ∂η
, ∂ζ
.
with the Jacobian J and the nabla operator in the reference space ∇ξ = ∂ξ
In each element, the solution and the fluxes are then approximated as polynomials
according to
Uh =
N
X
i,j,k=0
Û ijk ψijk (ξ) and
Fm
h
=
N
X
m
F̂ ijk ψijk (ξ),
(3.3)
i,j,k=0
where the superscript m = {1, 2, 3} denotes the flux in the direction of the Cartesian
coordinates. The basis function ψijk (ξ) = li (ξ)lj (η)lk (ζ) is the tensor product of the
one-dimensional Lagrange polynomial l of degree N . As interpolation nodes we choose
Gauss-Legendre or Gauss-Legendre-Lobatto points. Due to the nodal character of the
m
Lagrange basis, the degrees of freedom Û ijk and F̂ ijk are evaluations of the solution
and flux vectors at the interpolation nodes. To obtain the discontinuous Galerkin formulation, the approximations (3.3) are employed in Eq. (3.2), which is then multiplied by a
test function φ identical to the basis function ψ and then integrated in space. Integration
by parts of the volume integral of the flux yields the weak formulation, with three contributing parts, the volume integral of the time derivative of the solution (a), a volume
170
F. Föll, S. Baab, C.-D. Munz, B. Weigand & G. Lamanna
EOS bak ends:
(a )
–
–
–
(b )
(c)
Analytial EOS
Ideal gas
Peng-Robinson
Redlih-Kwong
CoolProp [14℄
RefProp [15℄
F IGURE 1. EOS table for Nitrogen, here: CoolProp
integral (b) and a surface integral (c) of the fluxes.
Z
Z
Z
∂
([F h · n] φ) dS = 0.
(JU h φ) dξ − (F h · ∇ξ φ) dξ +
∂t
} |Ω
{z
} | ∂Ω
{z
}
| Ω {z
a
(3.4)
c
b
The integrals are evaluated using Gauss-Legendre or Gauss-Legendre-Lobatto quadratures. To obtain an approximation of the flux at the element surface, a numerical flux
function G = G(U L , U R ) is introduced to approximate the term [F h · n]. It depends on
the states of the solution left and right of the interface, U L and U R , respectively. In case
of the viscous and heat conduction fluxes the gradients are needed in addition. For the
evaluation of the numerical fluxes at the element faces, a Riemann problem is defined.
The discrete formulation is discretized in time using an explicit fourth-order Runge-Kutta
scheme. For the viscous fluxes the approach of e. g. Bassi and Rebay [12, 13] is used.
The DG method with high order accuracy is favourable in smooth parts of the flow. At
discontinuities or strong gradients we introduce in the coarse DG grid cells a number
of sub-cells, on which a second-order finite volume scheme with the same number of
degrees of freedom is applied. A modal Persson indicator [16] is used to switch between
DG and FV cells.
3.2. Equation of state
The evaluation and the generation of the EOS table is done within the EOS library that
is developed at IAG. It is based on the tabulation method of Dumbser et al. in [17]. The
improved library supports several EOS evaluation programs as back ends and an easy
implementation of analytic EOS. The general use of tabulated EOS can be divided into
two steps.
Building step: In the building step, that is done prior to the simulation, we start the
adaptive tabulation using a quad-tree approach. The polynomial coefficients for the tabulated quantity x are computed using a L2 projection within each cell of the quad-tree.
Z 1Z 1
Z 1Z 1
ϕk (ξ, η)ϕm (ξ, η)dξdη =
ϕk (ξ, η)x(ρ, ǫ)dξdη
(3.5)
0
0
0
0
The integrals on both sides are evaluated using Gauss-Legendre quadrature rules. The
EOS table is refined till the L∞ -Error of the polynominal approximation does not satisfy
DG method with a tabulated general EOS
171
F IGURE 2. Dense-gas approach: Maxwell-line construction
a specific criterion, e.g.
max
ph (ρ, ǫ) − p(ρ, ǫ)
p(ρ, ǫ)
> Ξ,
(3.6)
∞
with Ξ being the maximal permittable approximation error of the EOS table.
Simulation step: During the simulation an iteration of a specific quantity (e. g. temperature) is replaced by an interpolation of the polynomial basis. The valid cell for the EOS
interpolation is found by a grid traversal through the quad-tree until the lowest level is
reached. An example visualization of a EOS table can be seen in Fig. 1.
3.3. Dense-gas approach
To maintain the hyperbolic character of the multi-phase solver an important part is the
EOS approximation within the multi-phase region that is enclosed by the liquid and
vapour saturation lines. A popular approximation in this region is to assume a fluid at
thermodynamic equilibrium with a Maxwell-line construction, see Fig. 2. In this case,
the pressure p and temperature T are set constant within the multi-phase region. The
Maxwell-line is constructed in a way, such that the area of the isotherme above and
below the line are the same. A linear interpolation of the specific inner energy ǫ is used
between the saturated liquid and vapour states based on the vapour mass quality Xv .
This method removes the elliptic part of the EOS formulation. This step is essential for
the direct usage of a hyperbolic flow solver framework. The vapour mass quality Xv is
defined as
ρ − ρsat,vap
Xv =
(3.7)
ρsat,liq − ρsat,vap
using the densities on the liquid saturation line ρsat,liq and on the vapour saturation line
ρsat,vap . This implies that for each state in the multi-phase region the corresponding saturation states have to be iterated. Thereby, the inner energy ǫ can be defined by linear
172
F. Föll, S. Baab, C.-D. Munz, B. Weigand & G. Lamanna
F IGURE 3. left: classical Riemann-problem; right: multi-phase Riemann-problem
interpolation of the saturation points as
ǫ = (1 − Xv )ǫsat,vap + Xv ǫsat,liq .
(3.8)
Analogously, the viscosity µ and the heat transfer coefficient λ can be obtained by linear
interpolation of the values on the saturation lines.
3.4. Riemann solvers and sound speed approximation
In contrast to a sharp interface method [18], where states in the multi-phase region are
prevented due to the use of multi-phase Riemann solvers between liquid and gas states,
the dense-gas method can be considered as a diffuse interface approach, where states
in the multi-phase region are produced due to the use of standard Riemann solvers everywhere, see Fig. 3. However, there exists a constraint regarding this. Riemann solvers
with explicit approximation of a contact discontinuity, like HLLC or Roe-Pike, can only be
used in the bulk- and not in the multi-phase region. This is due to the wrong estimation of
the contact discontinuity by disregarding evaporation or condensation effects. Therefore
for our dense-gas simulations we use
– a HLLC solver [19] in the bulk-phase and
– a Rusanov solver [19] in the multi-phase region.
The thermodynamic definition of the sound speed
c2 =
∂ρ
∂p
(3.9)
and Fig. 2 (left) indicate an imaginary sound speed in the spinodal region. By considering the dense-gas approach, the sound speed is zero in the binodal region. Since our
hyperbolic solver needs a well defined sound speed, one has to define a thermodynamic
reasonable sound speed in this region. A popular and widely used model is the Wood
sound speed [20] which can be defined in an isentropic ∂ρ/∂p|s or isothermal ∂ρ/∂p|T
sense [21].
For the isentropic case and with further assumptions [22, 23] one can get a sound
speed definition in the multi-phase region
−1
∂ρ αv
αl
2
cW,s :=
+
.
(3.10)
= ρ·
∂p s
ρl · c2l
ρv · c2v
4. Spurious oscillations at material interfaces
The DNS of compressible multi-phase flows is a rather challenging task as the two fluids often differ significantly in their material properties as well as in their thermodynamic
DG method with a tabulated general EOS
173
behaviour expressed by the EOS. The material interface represents a discontinuity in
the EOS that may lead to spurious pressure and velocity oscillations at this location, especially when density-based flow solvers in conservative formulation are used without
any special interface treatment [2,17]. To avoid this undesired effect, different strategies
exist. One popular method is the double-flux method of Abgrall [24], which is not strictly
conservative. In the literature the double-flux method has been applied to DG methods
with analytical EOS [8, 24], e.g. ideal gas EOS and to FV methods (WENO) with more
advanced Peng-Robinson or Redlich-Kwong EOS [9].
Here, we want to show how to apply the double-flux method to DG methods with tabulated real EOS and multiple species. The idea behind the double-flux method is the
solution of two Riemann fluxes at one interface
L
R
Gi+1/2
6= Gi+1/2
(4.1)
L
:= Gi+1/2 (UL , UR , Z(cp , cv , Yk )L )
Gi+1/2
(4.2)
R
Gi+1/2
(4.3)
:= Gi+1/2 (UL , UR , Z(cp , cv , Yk )R ) .
and
Uin+1 = Uin −
∆t L
R
.
Gi+1/2 − Gi−1/2
∆x
(4.4)
Here G denotes the numerical flux in normal direction on each face, Z(cp , cv , Yk ) the
thermodynamic state, which is a function of two or more thermodynamic variables depending on how many species are regarded, and n the current time step. The main
features of the double-flux method in combination with tabulated EOS and DG method
are:
– definition of a pseudo ideal gas EOS which will be used during the RK-steps
– assumption that the thermodynamic state in one DG cell is constant
– assumption that the thermodynamic state during the RK-steps is constant
– at the end of each time-step an energy relaxation must be done via DG-projection
The EOS for an ideal gas is given by
p = γρǫ
(4.5)
where γ := (κ − 1) is a constant and represents the ratios of the specific heat capacities
minus one of the fluid. Following this, for a general EOS p = f (ρ, ǫ, Yk ) one can define a
new dependent variable γ̃ := f (ρ, ǫ, Yk ) by
γ̃(ρ, ǫ, Yk ) :=
p
ρǫ
(4.6)
so that the pseudo ideal EOS can be rewritten as, see [25]
p(ρ, ǫ, Yk ) = γ̃(ρ, ǫ, Yk ) · ρǫ,
p
ǫ(ρ, p, Yk ) =
ρ · γ̃(ρ, p, Yk )
(4.7)
(4.8)
174
F. Föll, S. Baab, C.-D. Munz, B. Weigand & G. Lamanna
nitrogen
injetion
(superritial uid)
hamber
166
39.8e5
5.4
ρ[kg/m3 ]
p[N/m2 ]
v[m/s]
(superritial uid)
45
39.8e5
0
TABLE 1. Numerical setup of Mayer et al. [7] (Case 3)
For multi-species real EOS one has to take the reference state ǫ0 from the caloric EOS
K X
∂ǫ
∂ǫ
∂ǫ
dnk ,
(4.9)
dρ +
dT +
dǫ(ρ, T, nk ) =
∂ρ T,ni
∂T ρ,ni
∂nk ρ,T,
k=1
∆ǫ12 =
Z
ρ1
ρ0
∂ǫ
∂ρ
dρ +
Z
T1
T0
T,ni
∂ǫ
∂T
nj 6=nk
dT +
ρ,ni
K Z n1k
X
k=1
n0k
∂ǫ
∂nk
ρ,T,
nj 6=nk
dnk ,
(4.10)
into account. The arithmetic mean of the thermodynamic properties are calculated before each time step via Gauss quadrature
Z
1
Z=
Z(Ω)dξ
(4.11)
|Ω| Ω
=
N
X
wi wj wk Z(Ωi,j,k )
(4.12)
i,j,k
The main steps are summarized as followed:
(a) before the first RK-step: calculate the mean thermodynamic properties
n
for the current time step Z by using (4.12) with Uin or Win
R
L
(b) first RK-step: calculate the fluxes Gi+1/2
and Gi+1/2
with (4.2) and (4.3)
∗
(c) first RK-step: calculate the conservative values Ui with (4.4)
(d) first RK-step: calculate the primitive values Vi∗ with (4.7)
(e) repeat step (b)-(d) for each RK-step by using the thermodynamic properties
from (a) until the last RK-step is reached
(f) after the last RK-step: calculate the new thermodynamic properties
n+1
for the next time step Z
by using (4.12) with Ui∗ or Wi∗
n+1
(g) apply an energy relaxation step by using (4.8) to get Ui
n+1
(h) calculate Wi
with EOS table or pseudo ideal EOS (same result)
5. Results
5.1. Nitrogen jet simulation
Initially we present a calculation of the jet experiment super-critical simulation of Mayer
et al. [7]. The initial states for the injection and the chamber are listed in Tab. 1. The
simulation was performed with the EOS tabulation approach, the database is taken from
CoolProp. A number of n = 3 · 106 degree of freedoms were chosen with a polynomial
degree of p = 2. At strong density gradients we switched to a FV method. As subgridmodel and to stabilize the simulation we have performed a filtering approach [26]. The
DG method with a tabulated general EOS
175
Experiment
3
ρ[kg/m ]
160
FLEXI (DG)
OPENFOAM (FV)
120
INCA ("WENO")
80
40
0
10
x/D
20
30
F IGURE 4. Mayer (Case 3): left density field, right mean axial density distribution
nitrogen
ρ[kg/m3 ]
p[N/m2 ]
v[m/s]
left
(transritial uid)
432
39.8e5
5
right
(superritial uid)
45
39.8e5
5
TABLE 2. Material interface test case with one species, see [7] (Case 4)
results are shown in Fig. 4. The left plot of Fig. 4 shows the density field, the right plot
of Fig. 4 shows the mean axial density distribution. The values show a good agreement
with the experiments and the reference solution of the project partners. The trans-critical
test case of Mayer [7] was not stable for the fully conservative DG formulation. To cure
this problem a generalized non-conservative double-flux method is introduced in the
context of the tabulated real EOS. To show the validation we present one-dimensional
simulations with the FV and DG method. The simulations were performed with the EOS
tabulation approach, the database is taken from RefProp. For all simulations we have
chosen a number of n = 200 degree of freedoms.
5.2. Single species test case
The first test case is a single species simulation, where a material interface is generated
due to high density gradient. The test case is inspired by the experiments of Mayers et
al. [7], where cold Nitrogen is injected into warm nitrogen. A tube is filled with nitrogen
at two different thermodynamic states. The simulation domain is defined as
p x ∈ [−1, 1],
the discontinuity is placed at x = 0 and is smeared out with sgn(x) := x/ (x2 + ǫ2 ) to
be able to compare pure FV and DG solutions. The states are defined in Tab. 2. In Fig. 5
we show FV simulations with first and second order, with and without the double-flux
method. The figure shows mass density, velocity, pressure, temperature, heat capacity
ratio and mass fraction. The jump in the thermodynamic quantities is shown best in the
heat capacity ratios of cp /cv ≈ 15. The results show spurious oscillations in conservative
formulation, whereby the second order FV simulation performances better than the first
order one. The double-flux formulation is oscillation free. In Fig. 6 we have performed
fourth order DG simulations, with and without the double-flux method. As before we can
observe spurious oscillations in the conservative formulation in pressure and velocity.
We also recognize smaller oscillations by increasing the spatial order, due to the smaller
jumps in the Riemann solution. The double-flux formulation is again oscillation free.
176
F. Föll, S. Baab, C.-D. Munz, B. Weigand & G. Lamanna
Mass density [kg/m3]
500
Velocity [m/s]
x 10 6
4.4
Pressure [Pa]
20
400
4.2
300
10
200
0
4
3.8
100
-10
0
-1
3.6
0
1
-1
0
-1
1
Heat capacity ratio [-]
Temperature [K]
0
1
Mass fraction [-]
350
20
1.1
300
15
1.05
10
1
5
0.95
250
200
150
100
-1
0
x [m]
1
0
-1
0
x [m]
0.9
1
-1
0
x [m]
1
FV-Rusanov O(1) conservative
FV-Rusanov O(2) conservative
FV-Rusanov O(2) double flux
F IGURE 5. Material interface test case with one species, FV-solution: t = 0.2s
Mass density [kg/m3]
500
Velocity [m/s]
5.1
400
x 10 6
4.001
Pressure [Pa]
5.05
300
4
5
200
4.95
100
0
-1
3.999
4.9
0
1
Temperature [K]
-1
0
-1
1
Heat capacity ratio [-]
0
1
Mass fraction [-]
1.1
350
20
300
15
1.05
10
1
5
0.95
250
200
150
100
-1
0
x [m]
1
0
-1
0
x [m]
1
0.9
-1
0
x [m]
1
DG-Rusanov O(4) conservative
DG-Rusanov O(4) double flux
F IGURE 6. Material interface test case with one species, DG-solution: t = 0.2s
5.3. Multi species test case
The second test case is a multi species simulation, where a material interface is generated due to a mass fraction jump with same densities left and right. Nitrogen and argon
are considered as fluids, the diffusion coefficient in the species transport equation (2.5)
was set to D1 = 10−4 m2 /s. We want to indicate that this testcase is rather a generic
one, nevertheless we want to show that our approach also works for multi-species sim-
DG method with a tabulated general EOS
#
left
Nitrogen
right
1
1e5
5
ρ[kg/m3 ]
p[N/m2 ]
v[m/s]
177
Argon
1
1e5
5
TABLE 3. Material interface test case with two species
Mass density [kg/m3]
1.04
Velocity [m/s]
x 10 5 Pressure [Pa]
1.02
20
1.02
1.01
10
1
0
1
0.98
-10
0.99
0.96
-1
0
1
Temperature [K]
500
-1
2
0
Heat capacity ratio [-]
0
1
Mass fraction [-]
1.5
1.8
450
-1
1
1
1.6
400
0.5
1.4
350
0
1.2
300
-1
0
x [m]
1
1
-1
0
x [m]
1
-0.5
-1
0
x [m]
1
DG-Rusanov O(4) conservative
DG-Rusanov O(4) double flux
F IGURE 7. Material interface test case with two species, DG-solution: t = 0.1s
ulations in combination with DG method and tabulated EOS. The states are defined in
Tab. 3. In Fig. 7 we show fourth order DG simulations, with and without the double-flux
method. Unlike the single species test case, we now have same densities left and right.
The differences in the temperature and heat capacity ratio is forced by the mass fraction jump between nitrogen and argon. As before we can observe spurious oscillations
in the conservative formulation in pressure and velocity. The double-flux formulation is
oscillation free.
5.4. Multi species test case with shock
The last test case is taken from Hu et al. [3] and describes a two-species shock tube
problem. For this problem we have tabulated the ideal gas law and used the same
interface as for the real gas simulations. The states are defined in Tab. 4. Fig. 8 shows
two FV simulations, with and without the double-flux method. Here we also plotted a
quasi-exact reference solution. Again spurious oscillations occur in the conservative
case, whereas the double-flux formulation produces reasonable results.
6. Conclusions
The simulation of material of phase interfaces is difficult due to the inherent numerical smearing of the physical variables. For large density gradients spurious pressure
178
F. Föll, S. Baab, C.-D. Munz, B. Weigand & G. Lamanna
nitrogen
left right
ρ[kg/m3 ] 1 0.125
p[N/m2 ] 1
1
v[m/s]
0
0
κ
1.4
2
TABLE 4. Two-species shock tube problem
Mass density [-]
Velocity [-]
1
Pressure [-]
1
1
0.5
0.5
0.5
0
0
0
0
1
0
0
1
Heat capacity ratio [-]
x 10-3 Temperature [-]
1
Mass fraction [-]
1
2
6
0.8
1.8
4
0.6
0.4
1.6
0.2
1.4
0
2
0
~
x [-]
1
0
~
x [-]
1
0
~
x [-]
1
FV-HLLC O(2) double flux
FV-HLLC O(2) conservative
Exact
F IGURE 8. Multi species test case with shock: t = 0.2s
oscillations may occur. Based on a tabulated equation of state we showed that this
problem can be circumvented by introducing locally a double flux treatment that abandons locally the strict conservation. Validation examples clearly indicate a more robust
treatment and a better approximation. The use of this approach in now incorporated in
the three-dimensional solver to recalculate the jet flow and to compare the results.
Acknowledgments
Financial support has been provided by the German Research Foundation (Deutsche
Forschungsgemeinschaft – DFG) in the framework of the Sonderforschungsbereich
Transregio 40. Computational resources have been provided by the High Performance
Computing Center Stuttgart (HLRS).
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Sonderforschungsbereich/Transregio 40 – Annual Report 2016
181
CFD Simulations of Rocket Combustors with
Supercritical Injection
By M. Seidl†, R. Keller A N D P. Gerlinger
Institut für Verbrennungstechnik der Luft- und Raumfahrt, Universität Stuttgart
Pfaffenwaldring 38-40, 70569 Stuttgart
A consistent thermodynamic model is implemented into the finite volume CFD-code
TASCOM3D to simulate rocket combustors with supercritical injection conditions. The
efficient Soave-Redlich-Kwong cubic equation of state is used in this work to predict
the temperature-pressure-density-composition dependency. Empirical models based on
the corresponding-states principle are applied to calculate real gas fluid properties. Two
non-reacting experiments at supercritical pressure and a model rocket combustor with a
single coaxial injector and cryogenic, supercritical injection conditions are simulated to
validate the implemented approach. Simulations for all test cases are based on RANS
equations keeping computational costs low and allowing investigations of extensive parameter variations. The obtained results generally agree very well with experimental
measurements for all test cases. The numerical studies reveal a high sensitivity with
respect to the applied turbulence models and turbulent Prandtl and Schmidt numbers.
Though quantitatively good results could be obtained with test case dependent, individual numerical setups, a single, unique numerical setup valid for RANS simulations of
all test cases cannot be found.
1. Introduction
Rocket engines operate at extreme pressure-temperature conditions so that thermodynamics of the fluid can, at least locally, not be accurately described by formulae
valid for ideal gases only. Modifications at various locations in a CFD code are necessary to incorporate deviations from the ideal gas behavior. First, the well-known ideal
gas law has to be replaced by a real fluid equation of state (EOS) to describe the
temperature-pressure-density-composition dependency in a proper manner. For multicomponent problems, a simple cubic thermal EOS like the Soave-Redlich-Kwong (SRK)
or the Peng-Robinson (PR) EOS is usually applied in CFD simulations [1–3] to limit the
complexity and additional computational cost. Second, all thermodynamic parameters
like speed of sound, specific heat, species mass enthalpies, and thermodynamic derivatives like (∂ρ/∂p)T , which appear in the Jacobian matrices, should be consistently evaluated from the EOS via fundamental thermodynamic relations. Third, transport properties
(viscosity, thermal conductivity, binary mass diffusion coefficients) may require modifications depending on the problem of investigation. Fourth, flow phenomena like the Soret
and Dufour effect may have to be considered for some problems. Oefelein [4], however,
found that their contribution may be neglected for rocket combustors, where propellants
are commonly injected through shear-coaxial injectors.
† Institut für Verbrennungstechnik, Deutsches Zentrum für Luft- und Raumfahrt (DLR), Stuttgart