AIAA 2008-4567
44th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit 21 - 23 July 2008, Hartford, CT
A Flamelet/Progress-variable Approach for the Simulation of Turbulent Combustion of Real Gas Mixtures L. Cutrone∗ Centro Italiano Ricerche Aerospaziali, CIRA via Maiorise – 81043 Capua, Italy
P. De Palma†, G. Pascazio‡, M. Napolitano§ Dipartimento di Ingegneria Meccanica e Gestionale, DIMeG Centro di Eccellenza in Meccanica Computazionale, CEMeC Politecnico di Bari, Via Re David 200 – 70125 Bari, Italy
Abstract The industrial and scientific communities are devoting major research efforts to identify and assess critical technologies for new advanced propulsive concepts: combustion at high pressure has been assumed as a key issue to achieve better propulsive performance and lower environmental impact, as long as the replacement of hydrogen with a hydrocarbon, to reduce the costs related to ground operations (propellant handling, infrastructure and procedures) and increase flexibility. For the class of engines of interest in this work, namely, liquid-propellant rocket engines, the pressure is always supercritical, whereas the temperature could be either sub- or super-critical; however, propellants are typically injected into an environment that exceeds the critical temperature and pressure for both fuel and oxidizer, therefore a fast transition to a supercritical state is observed. In such a condition, it is possible to neglect the liquid phase and treat the liquid as a “dense” gaseous jet. However, the ideal gas equation of state is not suitable for computing the correct p − v − T relationship for oxygen and fuel at the operating pressure and temperature typical of LOx/HC rocket combustion chambers. Therefore, a suitable equation of state together with adequate model equations for the transport properties are employed. Starting from this background, the current work provides a model for the numerical simulation of highpressure turbulent combustion employing detailed chemistry description, embedded in a Reynolds averaged Navier-Stokes equations solver with a Low Reynolds number k − ω turbulence model. Keywords: compressibility factor, high pressure combustion, flamelet, progress-variable
Nomenclature C Cp G h p T
progress variable constant pressure specific heat, J/(kg·K) gibbs energy, J/kg enthalpy, J/kg pressure, Pa temperature, K
u velocity magnitude, p m/s uτ friction velocity, τw /ρ, m/s ui , uj , uk velocity components, m/s Vm molar volume, m3 /mole + y1 (yn uτ )/ν Z mixture fraction
∗ Research
engineer, ATER, AIAA member,
[email protected], +390823623108 AIAA member,
[email protected], +390805963226 ‡ Professor,
[email protected], +390805963221 § Professor, AIAA senior member,
[email protected], +390805963464
† Professor,
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Copyright © 2008 by Luigi Cutrone, Italian Aerospace Research Center, CIRA. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
Le Pr Re Sc Tu
Lewis number Prandtl number Reynolds number Schimdt number p turbulence intensity, 2k/3u2
ER FPV RCM SLF
Equivalence ratio: (m ˙ f /m ˙ ox ) / (m ˙ f /m ˙ ox )st Flamelet Progress/Variable Rocket Combustion Modeling workshop Steady Laminar Flamelet
Symbols
Subscripts f fuel in inlet ox oxidizer t turbulent st stoichiometric
χ λT µ ν ρ
Conventions
I.
scalar dissipation rate, 1/s, molar fraction of the i-th species turbulence length scale, mm viscosity, Pa s µ/ρ density, kg/m3
Introduction
Numerical modeling of near-critical mixing and combustion processes of fluids is particularly challenging since in addition to standard closure problems several difficulties arise due to non-ideal thermodynamic effects and to singularities of the transport properties. Close to the critical point, propellant mixture properties exhibit liquid-like density, gas-like diffusivities, and pressure-dependent solubilities; surface tension and heat of vaporization approach zero, whereas the isothermal compressibility and constant-pressure specific heat increase significantly. Depending on the injector type, fluid properties, and flow characteristics, two opposite behaviors may be observed. At subcritical chamber pressures, the injected liquid jet undergoes the standard cascade of processes associated with atomization. Ligaments detach from the jet surface and form droplets which break up and vaporize. In this case, dynamic forces and surface tension lead to the formation of a heterogeneous spray evolving through a wide range of thermophysical conditions. Concerning the second behavior, when the chamber pressure approaches or exceeds the critical pressure of the propellant, the injected liquid undergoes a ”transcritical” change of state since interfacial fluid temperatures locally rise above the saturation or critical temperature of the mixture. Therefore, diminished intermolecular forces promote diffusion-dominated processes before atomization takes place. The jet vaporizes forming a continuous fluid with large gradients. Cold tests, employing a jet of liquid nitrogen in gaseous nitrogen, show that the structure of the injected fluid appears more like a turbulent gaseous jet than a liquid spray.1 The purpose of this work is to develop an Eulerian single-phase numerical method for the simulation of mixing and combustion of liquid propellants in operating conditions typical of rocket combustion chambers. For this class of engines, the pressure of the injected fluid is always supercritical whereas its temperatures may be either sub- or super-critical. However, the fluid is typically injected into a chamber in which both pressure and temperature exceed the critical ones for both fuel and oxidizer, so that a fast transition to the supercritical state is observed.2 In such conditions, it is possible to describe the liquid phase as a ”dense” gaseous jet. On the other hand, the ideal-gas equation of state cannot predict the correct p − v − T relation for the oxidizer and the fuel at the operating pressure and temperature typical of rocket combustion chambers: for example, the density of oxygen predicted using the ideal-gas equation of state at supercritical conditions may be one fourth of the real value. Therefore, a suitable equation of state, together with adequate model equations for the transport properties must be implemented. In the present work, the Peng–Robinson3 equation of state has been employed in conjunction with a progress variable flamelet approach4 for the combustion modeling as described in the following sections.
II.
Governing Equations
The Reynolds-Averaged Navier-Stokes (RANS) equations for an axisymmetric multi-component chemically reacting system of n species can be expressed as ∂F − Fv ∂Q ∂E − Ev + + = S, ∂t ∂x ∂y
(1)
e ρk, ui , ρe uj , ρH, where E, F , and Ev , Fv are the inviscid and viscous fluxes vectors,5 respectively, Q=(ρ, ρe en ) is the vector of the conservative variables, ρ, (e e indicate the Favre-average mean value of ρω, ρR ui , u ej ), H 2 of 15
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density, velocity and total enthalpy; k and ω are the turbulent kinetic energy and its specific dissipation en is a generic set of conserved variables related to the transport of the species (usually the species rate; R e and a reactive scalar, C). e mass fraction Yen , or, as described in the following, the mixture fraction, Z Two fundamental issues have been identified as causes of difficulties in achieving convergence using a timemarching scheme when dealing with low-speed flows: the round-off errors associated with the calculation of the pressure gradient in the momentum equation and the numerical stiffness arising from eigenvalue disparity. The former can be easily overcome by decomposing the pressure into a constant and a fluctuating part, p = p0 + pg , where the constant pressure, p0 , should be taken so as to comprise the majority of p. It is then possible to substitute in the flux vector the pressure p with its gauge value pg , noting that ∇ (p0 + pg ) = ∇pg . The second problem will be circumvented by preconditioning the RANS equations (1) en ) is the vector of the primitive adding a pseudo time derivative Γ∂Qv /∂τ , where Qv = (e p, u ei , u ej , Te, k, ω, R variables and ρ′ pet ρTe ρk 0 ρRen 0 0 ρ′ pet u ρ 0 ρTe u ei ρk u ei 0 ρRen u ei ei ′ ρ pet u ρ ρTe u ej ρk u ej 0 ρRen u ej 0 ej ∂Q e 5ρ e e e e e e ′ pe − 1 ρe e = ρhpet + Hρ (2) Γ= + Hρ Hρ u ρe u + ρ h h ρ + 0 ρ h + Hρ i j k en en , k t Te Te R R 3 ∂Qv ′ ρ + kρk 0 kρRen 0 0 kρTe kρ pet ′ ωρ pet ρk ω ρ ωρRen 0 0 ωρTe ′ e e e e ρk Rn 0 ρ + Rn ρRen 0 0 Rn ρTe Rn ρ pet is the preconditioning matrix. Of course, when the convergence is reached in pseudo time, ∂Qv /∂τ = 0, the new system is equivalent to Eqs. (1). In order to preserve consistency of the model, the preconditioning matrix and all the other Jacobian matrices will be evaluated using the same thermodynamic relationship. II.A.
Equation of state and thermodynamic properties
The Peng Robinson (PR) equation of state3, 6 (EOS) is one of the most frequently used cubic equations of state for two reasons: its straightforward implementation and its accuracy.6 The PR equation of state is: p=
a RT − , 2 Vm − b (Vm + 2Vm b − b2 )
(3)
where R is the universal gas constant and Vm is the molar volume. The parameters a and b account for the effects of attractive and repulsive forces between the molecules and, as indicated by Twu et al.,7 a proper temperature dependence of a is essential for the reproduction of vapor pressures. Recently, Harstad, Miller, and Bellan6 have presented a computationally efficient form of the PR-EOS. In particular, the parameters a and b can be obtained as: Ns Ns Ns X X X χi b i , (4) χi χj aij (T ), b= a= i
i
j
where Ns is the number of the species, χi is the molar fraction of the species i, aij (T ) and bi are functions of the temperature and of the pure component parameters, as described in Refs. 6, 8. To ensure self-consistency in the model, all of the thermodynamic properties of the flow have to be calculated from the same equation of state. The properties of interest for the present fluid dynamic simulations are the specific enthalpy, h, and the constant pressure specific heat, Cp . Each of these properties can be obtained through various derivatives and functions of the Gibbs energy (G), which is defined as: G(T, p) =
Z
V m,u Vm
′
′
p(Vm , T, χi )dVm + pVm − RT +
X i
χi G0α + RT ln (χi ) ,
(5)
where the superscript 0 represents the “low pressure” reference condition for the integration as generally used in the departure function formalism described by Prausnitz et al.,9 and Vm,u is given below. Note that the integral is ill-defined for a zero-pressure reference condition; hereinafter the reference condition has been
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chosen to be p0 = 1 bar such that Vm,u = RT /(p0 ). From this relation, the following expressions for molar enthalpy and molar heat capacity can been obtained: ∂a ∂G 0 a − T , (6) = h + pV − RT + K h=G−T m 1 ∂T p,χ ∂T and
Cp =
2 (∂p/∂T )Vm ,χ ∂2a ∂h 0 = C − T K1 , − R − T p ∂T p,χ (∂p/∂v)T ∂T 2 " √ # Vm + 1 − 2 b 1 √ , K1 = √ ln 2 2b Vm + 1 + 2 b
(7)
(8)
where the partial derivatives of a with respect to temperature are evaluated analytically. II.B.
Evaluation of transport properties
The viscosity and the thermal conductivity of a gas are strongly dependent on the pressure near the critical point and at high pressure. The classical methods, based on the ideal-gas equation of state which are suitable for low pressure conditions, generally fail in the evaluation of the transport parameters of a real gas at supercritical conditions. Therefore, two advanced models have been implemented in this work: the method of Cho and Chung10 and the method of Ely and Hanley11 for the viscosity and the thermal conductivity evaluation, respectively. The method proposed by Cho and Chung relies on the development of a base formulation valid at low pressures, which is used as the starting point to develop an expression for viscosity that is valid at high (supercritical) pressure and low (transcritical) temperature. The method of Ely and Hanley, based on an extended corresponding state procedure, was developed to estimate the thermal conductivity of non-polar fluids, pure ones or mixtures, over a wide range of densities and temperatures. The estimation technique is based on Eucken’s proposal to separate the thermal conductivity into contributions from the interchanges of both translation and internal energy. For the latter, the modified Eucken representation was used,12 while for the translational component, a corresponding state method using methane as the reference component was selected.
III.
Combustion Model
For the prediction of the heat release and the effect of the density change on the flow field, a flamelet/progress variable (FPV) model is used.4 In this flamelet-based model, a non-premixed flame is considered as an ensemble of laminar flamelets and their chemical state is obtained from the solution of the steady laminar flamelet (SLF) equations13 assuming a unity Lewis number for all the species:14 χ(Z) ∂ 2 Yn 2 ∂Z 2 χ(Z) ∂ 2 T ∂Yn χ(Z) P ∂Cp ∂T − C − + n p,n 2 ∂Z 2 2Cp ∂Z ∂Z ∂Z −
= = −
ω˙ n , 1 P hn ω˙ n , Cp n
(9)
where Z is the mixture fraction, ω˙ n is the chemical production rate per unit mass of species n and the scalar dissipation rate is χ(Z) = 2αZ ∇Z · ∇Z. The thermodynamic closure of the system of equations (9) is achieved using the same relationships introduced in the previous sections, II.A and II.B, preserving the consistency of the overall approach when dealing with mixtures of reacting real gases. Assuming for χ(Z) the functional form corresponding to the distribution of the scalar dissipation rate in a counterflow diffusion flame, given by an inverse error function,15 the system of equations (9) can be solved as a function of a single parameter, namely, the stoichiometric scalar dissipation rate χst , obtaining the following state relationships: φ = Gφ (Z, χst ), (10) with φ = (T, Yn )T denoting the vector of the independent thermodynamical and chemical variables of Eqs. (9). A typical solution of the flamelet equations is reported in Figure 1: along the so-called “S-shaped” 4 of 15 American Institute of Aeronautics and Astronautics
2800
upper brach (stable)
2400
2400
Temperature, T [K]
Maximum temperature, Tmax [K]
2800
χq
2000 1600
middle brach 1200
2000 1600 1200 800
800
400
lower brach (only mixing)
400 1
10
100
Stoichiometric dissipation rate, χ st [s-1]
1000
0
0
(a) S-shaped curve Figure 1. maximum the upper the lower,
0.2
0.4
0.6
0.8
1
Mixture fraction, Z (b) Flamelet solutions
Solution of the steady flamelet equations for methane/air combustion.16 (a): S-shaped curve shows the temperature as function of the stoichiometric scalar dissipation rate, χst = χ(Zst ). The turning point between and the middle branch is denoted by χq ; (b): three temperature profiles for χst =10 s−1 , corresponding to middle and upper branches, respectively.
curve, all the thermochemical state of the system, i.e., all the solution of Eqs (9), are identified by the maximum temperature value: the upper and the lower branches of the S-shaped curve correspond to the stable burning and non burning state, respectively, whereas the middle branch describes unstable reacting states. The turning point between the upper and middle branches is indicated by χq and corresponds to the scalar dissipation rate at which quenching occurs. A unique parameterization of the entire S-shaped curve is not possible using the scalar dissipation rate as parameter, since multiple solutions of Eqs. (9) exists for χst < χq . In order to overcome this ambiguity of the SLF model, in the FPV model a reaction progress parameter Λ has been introduced. The reaction progress parameter is based on a reactive scalar, the progress variable C, and is defined to be independent of the mixture fraction. This parameter is chosen so that it uniquely identifies each single state along the Sshaped curve (also the unstable states on the middle-branch). The definition of C is something arbitrary since different choices are possible in a multi-species chemical system. In this work, according to what described in Ref. 17, the progress variable C is a linear combination of the principal products of the combustion, and the progress parameter Λ is the stoichiometric value of the progress variable. Flamelets experiencing a transition from the burning to the extinguished flame state, or those which are likely to re-ignite, are then projected horizontally onto the S-shaped curve (and not vertically, as in the SLF model), obtaining a new interpretation of such a curve. Under this assumption, the state relations (10), obtained from the SLF model, are re-written as: φ = Fφ (Z, Λ) . (11) e are obtained using a In the simulation of a turbulent reacting flow, the Reynolds-Averaged quantities φ presumed joint PDF of Z and Λ, which are modeled using a beta-distribution and a Dirac-distribution: ZZ g g ′′2 , Λ) ′′2 )δ(Λ − Λ)dZdΛ e Z, e Z e = e Z e φ( Fφ (Z, Λ)β(Z; Z, , (12)
g ′′2 is its variance and Λ e the mean progress parameter. The application where Ze is the mean mixture fraction, Z e which is rather difficult. Here, the of this model then requires the solution of the transport equation for Λ, e is replaced by the mean progress table is re-interpolated to a different set of independent parameters and Λ g ′′2 , and C. e e Z e variable, C. The flamelet library then provides the mean scalars as function of Z, In addition to the solution of the Reynolds-Averaged Navier-Stokes equations, the FPV model requires
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g ′′2 , and C: e and Z e the solution of the following transport equations for Z " # ∂Z e e ∂ρZ ∂ ∂ t e = uj Z) D + DZ , (ρe + e ρ ∂t ∂xj ∂xj ∂xj " # ∂Z 2 g g ′′2 ′′2 ∂ ∂ρZ ∂ t t g ′′2 e , D + Dg − (ρe + uj Z ) = ρ ρe χ + 2ρD ∇ Z t e Z Z ′′2 ∂t ∂xj ∂xj ∂xj " # ∂C e e ∂ ∂ ∂ρC t e = uj C) D + DCe ρ + ρω˙ C , (ρe + ∂t ∂xj ∂xj ∂xj
(13)
where D is the diffusion coefficient for all the species, which is equal to D = ν/Pr under the hypothesis of t t t unity Lewis numbers, DZ = DC ˙ C is e = DZ e = ν/Sct are the turbulent mass diffusion coefficients and ω g ′′2 the source term for the progress variable, obtained by the flamelet library using Eq. (12). The gradient transport assumption for turbulent fluxes is used and the mean scalar dissipation rate, χ et , appearing as a g ′′2 with γ = 2. sink term in the equation for the variance of the mixture fraction, is calculated as χ et = γǫ/k Z ′′2 g e C e and Z The solution of Eq. (12), for several values of Z, within their validity ranges, provides the f mean composition Yn that can be stored in a chemical library and used at run-time in order to calculate density, enthalpy, and the transport properties.
IV. IV.A.
Test Cases
RCM-1B 2001 test case
The RCM-1B 2001 test case2 is a very good benchmark case because it allows one to separate the effect of real-gas modeling and turbulent mixing from combustion. It consists in the simulation of the injection of a dense cryogenic jet of liquid nitrogen in a warm environment of gaseous nitrogen. Table 1 provides the main thermodynamic data of the critical point for nitrogen. The geometry of the test facility is shown in Figure 2. The test chamber is a cylindrical vessel with an inner diameter equal to 122 mm, the length of the vessel being 1000 mm. The injector is located at the center of the face plate and its inner diameter is d =2.2 mm. The tube length is 90 mm. Since the tube length to tube diameter is more than 40, a fully developed turbulent velocity profile is expected at the end of the tube (point T2). The operating conditions for the RCM-1B 2001 test are reported at point T1, in Table 2. Table 1. Critical thermodynamic data for nitrogen.
Critical temperature, Tc (K) Critical pressure, pc (bar) Critical density, ρc (kg/m3 ) Acentric factor, ω Normal boiling point (K) Dipole moment, µ (debye)
126.192 33.958 313.300 0.0372 77.355 0.0
The 2-D axisymmetric steady computation has been performed using a computational grid with about 30,000 points with y + of the first cell at the wall lower than one. Two additional simulations were performed using a coarser grid with about 7,500 points and a finer one with about 120,000 points. The results (not shown here) indicate that the intermediate grid level achieves grid independence. Table 2. RCM-1B 2001 test case.
Chamber pressure (MPa) Temperature (K) Density (kg/m3 )
5.98 128.7 514.0
Velocity (m/s) Reynolds number Inlet Turbulence Intensity, T uin (-) Inlet Turbulence Length, ℓT,in (mm)
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0.736 23,250 0.5% 15
Figure 2. Test chamber and injector.
200
400
Density, kg/m3
Density, kg/m3
500
300
200
150
100
100 -4
-3
-2
-1
0
1
2
3
4
50 -4
-3
-2
-1
R/d
0
1
2
3
4
R/d
(a) x/d = 5.
(b) x/d = 25 .
Figure 3. Density profiles: comparison between different levels of real gas corrections: —— Full correction (EOS, thermodynamic and transport data); – - – Partial correction (EOS and thermodynamic data); - - - - Only EOS correction; – - - – Ideal gas; • Experimental data.
Figures 3(a) and 3(b) show the comparison between the experimental data (symbols) and the computed density profiles at two sections orthogonal to the injection direction. Three lines are plotted: the first one (dashed-dotted line) refers to the results obtained using only the PR equation of state; the second one (dashed line) takes into account also the correction of the thermodynamic data (enthalpy and Cp ); whereas, the last one (solid line) refers to the complete model including also the transport properties at high-pressure conditions. Figure 3(a), referring to a section very close to the injection point (x = 11 mm), shows a good agreement between real-gas-model predictions and the experimental data. Moreover, large discrepancies between the ideal-gas and the real-gas predictions are clearly seen: the real-gas value at the jet centerline is about 500 kg/m3 whereas the ideal-gas value is only 89 kg/m3 . Moving downstream, the cold nitrogen jet mixes with the nitrogen of the chamber and warms up: real-gas effects decrease but a non-ideal behavior still exists, as demonstrated by the difference in the real and ideal gas predicted density profiles presented in Figure 3(b) at a radial section 55 mm downstream of the injection point. At this location, remarkable differences among the three levels of approximation can be seen: The first two levels of approximation (dashed-dotted and dashed lines) fail in the correct evaluation of the density profile, especially for the peak value, whereas, if the complete real-gas model is used (solid line), the difference between the computed and the experimental density profile reduces. Finally, Figure 4 shows that the constant pressure specific heat has a maximum along the centerline of 7 of 15 American Institute of Aeronautics and Astronautics
Cp r /Cp id
5
B
4 A
3
C
2 1 50
100 150 200 250 300 350 400 450
Temperature K A T=128.96
1.2
0.095
C T=149.82
B T=139.14
0.105
1.8
2.4
0.115
3
3.6
4.2
4.8
0.125
Figure 4. Computed specific heat factor (Cpr /Cpid ) contours.
the jet: this situation can be handled by the proposed model because it predicts a finite value for Cp at the critical point, whereas according to the theory it should go to infinity. In such a way it is possible to simulate the whole process in which a transition occurs from subcritical to supercritical conditions, as in the case of the injection of liquid propellants into high-pressure rocket combustion chambers. IV.B.
SANDIA Flame D
In the Sandia flame experiment, piloted partially premixed methane/air diffusion flames at different Reynolds numbers have been studied by Barlow & Frank.16 In the current work, the Sandia Flame D has been chosen for the validation of the combustion model because of the small degree of local extinction.18 Flame E (Re=33,600) and F (Re=44,800) have significant and increasing probability of local extinction above the pilot region, flame F being close to global extinction of the downstream part of the flame. The Reynolds number is based on the nozzle diameter, jet bulk velocity and kinematic viscosity of the fuel jet. The central jet nozzle has a diameter of dref = 7.2 mm and the annular pilot nozzle internal and external diameters are, respectively, 7.7 and 18.2 mm. The oxidizer air is supplied as a co-flow at 291 K. The main fuel jet consists of a mixture of methane and air in a volumetric ratio 1:3 with a stoichiometric mixture fraction of Zst = 0.351. The pilot stream is a lean premixed gas mixture of C2 H2 , H2 , CO2 , N2 and O2 with an equivalence ratio of 0.77, which corresponds to a similar equilibrium composition of the library ′′2 g e =0.27, Z e = 1. Point measurements are available at several stream-wise mixture at Z = 0.0075 and C locations and along the centerline. The statistical errors of the measurements for the mean major species and temperature are below 5%, and for CO below 10%. The considered computational domain is axisymmetric and includes a part of the burner: it extends 150 dref and 27 dref along the axial and radial directions, respectively. Two different levels of grid refinement have been considered, the coarser grid containing about 12,000 computational cells and the finer one about 44,000 cells. The turbulent inlet velocity has been assigned in accordance with the experimentally determined axial mean velocity profiles and turbulent quantities have been calculated from the velocity profile assuming the turbulent intensity level Tuin and the turbulent characteristic length, λT,in , reported in Table 3; a subsonic outlet condition has been employed at the outflow boundary and the no-slip condition is imposed at the radial boundary; finally an axis boundary condition has been used along the centerline. Steady flamelet calculations have been performed using the FlameMaster code,13 with the chemistry described by the GRI-Mech 3.0,19 consisting of 325 elementary chemical reactions and 53 species involved. In the present simulation, the progress variable is defined as C = YCO2 + YH2 O + YH2 + YCO and the chemistry e and C e directions, and 25 point for the library is discretized with 125 uniformly distributed points in the Z ′′2 g Z direction. The main results measured along the flame centerline are reported in Figure 5. On the first row, the 8 of 15 American Institute of Aeronautics and Astronautics
Table 3. Conditions for SANDIA D flame experiment.
u (m/s) 49.6 ± 2
T (K) 294.0
e Z 1
e C 0
′′2 g Z 0
11.4 ± 0.2
∼1800
0.27
1
Co-flow 0.9 ± 0.05 Reynolds number Stoichiometric conditions Pressure (bar)
291.0
0
0
Main jet
Pilot stream
Tuin 5%
λT,in (mm) 0.5
Composition YCH4 = 0.156, YO2 = 0.196, YN2 = 0.648
0.0075
5%
0.5
YCO2 = 0.110, YO2 = 0.056, YN2 = 0.734, YOH = 0.002, YH2 O = 0.098
0
0.5%
YO2 = 0.233, YN2 = 0.767
1.0 22,400 Zst = 0.351 1.006
0.2
1.5 1.25
0.15
/
u" uref
2 ~
0.75
0.1
√
~ u/uref
1
0.5
0.05 0.25 0
0
1
0.2
0.8
0.15
0.1
√
~ Z
Z"
2 ~
0.6 0.4
0.05
0.2 0
0
2000
~ T, K
1500
1000
500
0 0
30
60
90
120
x/dref Figure 5. Sandia Flame D: comparison of measured (symbols) and numerical (lines) mean velocity, velocity rms, mean mixture fraction an its variance and mean temperature along the jet centerline. —— : fine grid; - - - - : coarse grid.
velocity and its rmsa are presented: the velocity profile is in quite good agreement with experimental data for both the computational grids while the velocity rms is under-predicted and its peak is shifted forward. ′′
a in
′′
′′
the present RANS simulations, under the hypothesis of isotropic turbulence, i.e., ui = uj = uk , the following relationship is valid: !2 ′′ u 2 k = = Tu2 , u 3 u2 where Tu is the local value of the turbulence intensity level.
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The calculated mean and rms mixture fraction, shown in the middle row of the figure, are found to be in overall good agreement with experimental data for the fine grid, whereas using the coarse grid, they are under-predicted for x/dref > 40. Both numerical results slightly over-predict the mixture fraction in the first part of the flame. Similar considerations are valid for the computed temperature profiles: both the numerical results under-predict the temperature for x/dref < 35, but they correctly capture the position of the peak with a flame length Lst = x(Zst ) = 50.07 dref (against an experimental value of Lst = 48 dref ) and the maximum temperature value. For x/dref > 45, the temperature profile computed on the fine grid results in better agreement with the experimental data, with respect to the solution on the coarse grid which predicts a thinner reaction zone. Figures 6, 7 and 8 show the computed and the measured Favre-averaged values of temperature, mixture fraction, its variance and the mass fraction of the principal species at six axial positions, x/dref = {1, 2, 3, 15, 30, 45}. The computed temperature and the mixture fraction, shown in Figure 6, are in overall good agreement with the experimental data for almost all the axial positions considered: some discrepancies occur at the axial position x/dref = 45 where the numerical solution predicts a narrower flame with respect to the experimental one. Also the predicted concentration of the principal species (reactants CH4 and O2 and the major products, CO2 and H2 O), in Figure 7, are in good agreement with the experimental data with the exception of the last axial position considered where a lower oxygen consumption results in a incomplete combustion and a lower prediction of the product concentrations. The intermediate species, in Figure 8, are predicted reasonably well with the exception of the nitrogen mono-oxide: The present results agree with one reported by Ihme20 who found that the FPV model is not able to correctly predict the concentration of the NOx, being their slow formation and strong dependence on the flame temperature a very tough problem to face. IV.C.
RCM-3 2001: Mascotte single injector Supercritical Combustion
The MASCOTTE cryogenic combustion test facility was developed by ONERA to study fundamental processes which are involved in the combustion of cryogenic propellants, namely, liquid oxygen (LOx) and gaseous hydrogen (GH2 ). In particular, the test case RCM-3 2001,21 presented at the 2nd International Workshop on Rocket Combustion Modeling in 2001, deals with the super-critical liquid oxygen/hydrogen combustion problem. The injector consists of an inner duct for the oxygen with a diameter of 3.6 mm at the inlet, diverging to a diameter of 5 mm at the orifice. Hydrogen is injected coaxially, through an annular duct with an inner diameter of 5.6 mm and an outer diameter of 10 mm. The length of the injector is 50 mm in order to get
~ T, K
2200
x/dref=1.0
x/dref=2.0
x/dref=3.0
x/dref=15.0
x/dref=30.0
x/dref=45.0
1650 1100 550 0
1.0000
~ Z
0.7500 0.5000 0.2500
0.2250 0.1500
√
Z"
2 ~
0.0000
0.0750 0.0000
0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 6
r/dref
r/dref
r/dref
r/dref
r/dref
r/dref
Figure 6. Sandia Flame D: comparison of measured (symbols) and numerical (lines) mean temperature, mixture fraction and its variance at several stream-wise locations in the flame. —— : fine grid; - - - - : coarse grid.
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x/dref=1.0
x/dref=2.0
x/dref=3.0
x/dref=15.0
x/Dref=30.0
x/Dref=45.0
~ YCH
4
0.2000 0.1500 0.1000 0.0500 0.0000
~ YO
2
0.3000 0.2250 0.1500 0.0750 0.0000
0.1125
2
~ YH O
0.1500
0.0750 0.0375 0.0000
~ YCO
2
0.1500 0.1125 0.0750 0.0375 0.0000
0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 6
r/d
r/d
r/d
r/d
r/d
r/d
ref ref ref ref ref ref Figure 7. Sandia Flame D: comparison of measured (symbols) and numerical (lines) mean mass fraction for the reactants and the principal products of the combustion at several stream-wise locations in the flame. —— : fine grid; - - - - : coarse grid.
x/dref=1.0
x/dref=2.0
x/dref=3.0
x/dref=15.0
x/dref=30.0
x/dref=45.0
~ YH
2
0.0030 0.0023 0.0015 0.0008 0.0000
~ YCO
0.0400 0.0300 0.0200 0.0100 0.0000
~ YOH
0.0030 0.0023 0.0015 0.0008 0.0000 -4
4.0×10
~ YNO
-4
3.0×10
-4
2.0×10
-4
1.0×10
0.0
0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 6
r/d
r/d
r/d
r/d
r/d
r/d
ref ref ref ref ref ref Figure 8. Sandia Flame D: comparison of measured (symbols) and numerical (lines) mean mass fraction for some intermediate products of combustion at several stream-wise locations in the flame. —— : fine grid; - - - - : coarse grid.
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Table 4. Conditions for RCM-3 2001 test case.
LOx GH2 Pressure (bar)
Mass flow rate m ˙ (kg/s) 0.1 0.07
T (K) 85 287
ρ (kg/m3 ) 1177.8 5.51 60
u (m/s) 4.35 236
Tuin (-) 5% 5%
λT,in , mm 4 4
Figure 9. Reference geometry for Test RCM-3 2001 and a detail of the computational grid for the injectors.
a fully developed turbulent flow profiles at the injector exit. The combustion chamber has a cuboid shape with 50 mm × 50 mm section, but in the present study it is modeled by a cylindrically shaped chamber with a radius of 28.21 mm to conserve the chamber volume. The chamber pressure was held at 6 MPa, above the critical pressure for both oxygen and hydrogen (50.43 and 13.13 bar, respectively). Oxygen is injected as a liquid at the temperature of 85 K, which corresponds to a density of 1177.8 kg/m3 , while hydrogen is injected at the ambient temperature of 287 K. A 2D axisymmetric computations have been performed on a computational grid of about 18,000 cells distributed between 24 structured blocks. A partial view of the computational grid is reported in Figure 9. At the inlet points, the boundary conditions have been imposed as summarized in Table 4; subsonic outflow conditions have been assigned at the exit of the combustion chamber; whereas the walls have been considered no-slip adiabatic surfaces. Finally, the axis conditions have been used for the centerline boundary. Steady flamelet calculations have been performed with the chemistry described by the Warnatz scheme,22 consisting of 38 elementary chemical reactions and 8 species. In the present simulation, the progress variable is defined as C = YH2 O and the chemistry library is discretized with 125 uniformly distributed points in the ′′2 g e and C e directions, and 25 point for the Z Z direction. Two different simulations have been performed using both the ideal- and the real-gas equation of state. As argued in Section II.A, the ideal-gas equation of state completely fails in the prediction of the density of oxygen at the injection temperature of 85 K and a chamber pressure of 60 bar: it provides a value of 271.7 kg/m3 against an experimental value23 of 1177.8 kg/m3 . The PR equation of state, with a predicted value of 1310 kg/m3 , slightly over-predicts the density of oxygen. It is an expected result since, as described by Poschner and Pfitzner24 and Cutrone et al.,8 the PR equation of state shows the largest deviation from experimental data for reduced temperature Tr = T /Tcr < 0.7 and in the present simulation the value of such parameter is about 0.55. On one hand, keeping constant the oxygen injection velocity, the mass flow rate at the inflow boundary is equal to 0.023 and 0.11 kg/s when using the ideal-gas and the Peng–Robinson equation of state, respectively, against a nominal value of 0.1 kg/s; on the other hand, keeping constant the mass flow rate, for the case of the ideal-gas equation of state, the injection velocity is 18.8 m/s, which is very large with respect to the nominal injection velocity: for this reason such a test is not considered in the present work. 12 of 15 American Institute of Aeronautics and Astronautics
YOH:
0
0.02
YOH:
0.001 0.007 0.013 0.019 0.025 0.031
0.04
0
0.02
0.001 0.007 0.013 0.019 0.025 0.031
0.04
X
X
(a)
(b)
Figure 10. Comparisons of OH mass fractions. (a) Abel transformed emission (on the top) and ideal gas equation of state result (on the bottom); (b) Abel transformed emission (on the top) and Peng Robinson equation of state result (on the bottom).
For RCM-3 2001 test case, only the LIF (Laser Induced Fluorescence) image of the hydroxyl radical from the experiment is available for validation.25 In Figures 10 the contours of the predicted OH mass fraction are provided for both the numerical test and compared with the experimental Abel transformed emission image. It should be noticed that a scale is not given for the LIF data. Qualitatively, the predicted flame zone and spreading angle of the shear layer, using the PR equation of state, correspond very closely to those of the experimental images. Since the flame location and the spreading angle of the jet are determined essentially by mixing and combustion of propellants, the qualitative similarity indicates that the model is able to reasonably provide a correct description of the turbulent mixing and the thermal properties. The ideal-gas computation provides a smaller flame, essentially due to the reduced oxygen mass flow rate, and consequently a different O/F ratio. Moreover, one characteristic feature of the flame, that is clearly seen in the real-gas computation, reported in Figure 11, and that is absent in the ideal-gas simulation, is the sudden apparent expansion of the LOx core several injector diameter downstream the injection plate caused by a recirculation region that is formed by the expanding LOx core.
V.
Conclusions
A self-consistent model is presented for the simulation of a generic mixture of reacting real gases. The model is able to describe the correct behavior of the thermodynamic properties of a generic gas mixture for a wide range of temperatures and pressures, and in particular in the range of interest for the simulation of liquid rocket combustion chambers. A non-reacting-flow test case has been computed for validating the thermodynamic model at supercritical conditions. The good agreement between computed and experimental data demonstrates that the Eulerian single-phase approach is suitable for the simulation of supercritical flows. Then, the flamelet progress-variable model has been validated versus two reacting-flow test cases: 1) the combustion of air and methane at atmospheric pressure, known as Sandia D flame test; 2) the combustion of liquid oxygen and hydrogen at high pressure, presented at the 2nd International Workshop of Rocket Combustion Modeling. Such test cases allowed for a detailed quantitative and qualitative validation of the proposed approach. In conclusion, the overall proposed numerical method has been proven to be accurate for simulating the combustion process of mixtures of real gases for which a single-phase model is suitable.
References 1 Branam, R. and Mayer, W., “Characterization of Cryogenic Injection at Supercritical Pressure,” Journal of Propulsion and Power , Vol. 19, No. 3, May–June 2003, pp. 342–354.
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Figure 11. Hydroxyl radical fraction (top) and temperature (bottom) contours for the real gas simulation.
2 Mayer et al., W., “Atomization and Breakup of Cryogenic Propellants Under High-Pressure Subcritical and Supercritical Conditions,” Journal of Propulsion and Power , Vol. 14, No. 5, September 1998, pp. 835–842. 3 Peng, D. and Robinson, D., “A New Two-Constant Equation of State,” Ind. Eng. Chem. Fundam., Vol. 15, 1976, pp. 58–64. 4 Pierce, C. D. and Moin, P., “Progress-variable approach for large-eddy simulation of non-premixed turbulent combustion,” Journal of fluid Mechanics, Vol. 504, 2004, pp. 73–97. 5 Schwer, D. A., Numerical study of unsteadiness in non-reacting and reacting mixing layers, PhD Thesis, The Pennsylvania State University, 1999. 6 Harstad, K., Miller, R., and Bellan, J., “Efficient high pressure state equations,” American Institute of Chemical Engineers Journal , Vol. 43, 1997, pp. 1675–1706. 7 Twu, C., Bluck, D., Cunningham, J., and Coon, J., “A Cubic Equation of state with a New Alpha Function and a New Mixing Rule,” Fluid Phase Equilibria, Vol. 69, 1991, pp. 33–50. 8 Cutrone, L., Ihme, M., and Herrmann, M., “Modeling of high-pressure mixing and combustion in liquid rocket injectors,” Studying Turbulence Using Numerical Simulation Databases (Proceeding of the CTR Summer Program 2006), edited by CTR, Vol. XI, July 2006, pp. 269–281. 9 Prausnitz, D., Lichtenthaler, R., and Azevedo, E. D., Molecular Thermodynamics for Fluid Phase Equilibrium, PrenticeHall, 1986. 10 Chung, T., Lee, L., and Starling, K. E., “Applications of the kinetic gas theories and multi-parameter correlation for prediction of diluite gas viscosity and thermal-conductivity.” Ind. Eng. Chem. Fund., Vol. 23, 1984, pp. 8–13. 11 Ely, J. and Hanley, H., “Prediction of transport properties: thermal conductivity of pure fluids and mixtures.” Ind. Eng. Chem. Fund, Vol. 22, 1983, pp. 90–97. 12 Reid, R., Prausnitz, J., and Poling, B., The Properties of Gases and Liquids, MGraw-Hill Inc., 4th ed., 1987. 13 Pitsch, H., “FlameMaster V3.3. A C++ Computer Program for 0D Combustion and 1D Laminar Flame Calculations,” 1998a, Available at http://www.stanford.edu/∼ hpitsch. 14 Pitsch, H. and Peters, N., “Unsteady Flamelet Modeling of Differential Diffusion in Turbulent Jet Diffusion Flames,” Combustion and Flame, Vol. 123, No. 3, 2000, pp. 358–374. 15 Peters, N., “Laminar diffusion flamelet models in non-premixed turbulent combustion,” Prog. Energy Combustion Sciences, Vol. 10, 1984, pp. 319–339. 16 Barlow, R. and Frank, J., “Effects of turbulence on species mass fraction in methane/air jet flames,” Proceedings of the Combustion Institute, Vol. 27, 1998, pp. 1087–1095. 17 Pierce, C., Progress-Variable Approach for Large-Eddy Simulation of Turbulent Combustion, PhD Thesis, Stanford University, 2001. 18 Sandia National Laboratories, TNF Workshop, Boulder, Colorado, USA, 1998, http://www.ca.sandia.gov/TNF. 19 Smith, G. P., Golden, D. M., Frenklach, M., Moriarty, N. W., Eiteneer, B., Goldenberg, M., Bowman, C. T., Hanson, R. K., Song, S., Gardiner, W. C., Lissianski, V. V., and Qin, Z., 2000, http://www.me.berkeley.edu/gri mech/. 20 Ihme, M., Pollutant Formation and Noise Emission in Turbulent Non-Premixed Flames, Phd thesis, Stanford University, 2007.
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21 Thomas, J. and Zurbach, S., “Test case RCM-3: supercritical spray combustion at 60 bar at Mascotte,” Lampoldshausen, Germany, March 25-27 2001. 22 Maas, U. and Warnatz, J., “Ignition Processes in Hydrogen-Oxygen Mixtures,” Combustion and Flame, Vol. 74, 1988, pp. 53–69. 23 http://webbook.nist.gov/chemistry/fluid. 24 Poschner, M., P. M., “Real gas CFD simulation of supercritical H2-LOX combustion in the Mascotte single-injector combustor using a commercial CFD code,” 46th AIAA Aerospace Sciences Meeting and Exhibit, AIAA, Reno, Nevada, January 2008. 25 Vingert, L., Habiballah, M., Vuillermoz, P., and S., Z., “Mascotte, a test facility for cryogenic combustion research at high pressure,” 51th International Astronautical Congress, International Astronautical Federation, Rio de Janeiro, October 2000.
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