A compressible-LEM turbulent combustion model for

A compressible-LEM turbulent combustion model for assessing gaseous explosion hazards Brian Maxwell a , Sam A.E.G. Falle b , Gary J. Sharpe c & Matei I. Radulescu a E-mail: [email protected] a

University of Ottawa, Ottawa, Canada University of Leeds, Leeds, United Kingdom c Blue Dog Scientific Ltd., Wakefield, United Kingdom b

Abstract A turbulent combustion model based on the Linear Eddy Model for Large Eddy Simulation (LEM-LES) is currently being developed to study highly compressible and reactive flows involving very rapid transients in pressure and energy. The model is a one-dimensional treatment of a diffusion-reaction system within each multi-dimensional LES cell. This reduces the expense of solving a complete multi-dimensional problem through Direct Numerical Simulation (DNS) while preserving micro-scale hotspots and their physical effects on ignition. The current approach features a Lagrangian description of fluid particles on the sub-grid for increased accuracy. Also, Adaptive Mesh Refinement (AMR) is implemented for increased computational efficiency. Recently, the model approach has been validated for various fundamental 1-D test cases, including inert mixing, constant volume ignition, shock tube problems and laminar flame propagation. In the current study, the model is extended to treat turbulent methane-air flames in one-dimension. Also, a 1-D piston-driven laminar shock-flame interaction and its subsequent transition to detonation is modelled. Results confirm that both laminar and turbulent flame speeds are reproduced when compared against DNS and experiments. Furthermore, the model offers a good prediction for detonation initiation in the presence of shock waves when compared to low-resolution DNS. Finally, the model is demonstrated to reproduce the correct detonation structures, velocities, and instability behaviour when compared to theory and DNS. Keywords: compressible flow, turbulent combustion, explosions, numerical simulation 1. Introduction A turbulent combustion model is currently being implemented in order to assess potential gaseous explosion hazards involving rapidly expanding jet fires from a compressed source (Maxwell et al., 2013b) and also scenarios where detonation initiation is possible via shock-flame interaction (Khokhlov et al., 1999). The model implemented here is based on the Linear Eddy Model for Large Eddy Simulation (LEM-LES) (Menon and Kerstein, 2011). In the current implementation, however, the model is adapted to treat highly compressible and reactive flows involving very rapid transients in pressure and energy. This compressible LEM-LES approach (henceforth referred to as CLEM-LES) also features Adaptive Mesh Refinement (AMR) for increased

Tenth International Symposium on Hazards, Prevention, and Mitigation of Industrial Explosions Bergen, Norway, 10-14 June 2014


computational efficiency. It is validated in this paper for laminar and turbulent one-dimensional flames involving premixed fuel-lean methane-air mixtures. Its performance is also assessed for combustion involving large pressure changes in laminar regimes. This is done through by modelling detonation initiation via shock-flame interaction. Finally, high resolution detonation propagation simulations are conducted to assess the model’s ability to capture detonation structure and propagation characteristics. The model itself is a 1-D treatment of a diffusion-reaction system within each multi-dimensional LES cell. This reduces the expense of solving a complete multi-dimensional problem through Direct Numerical Simulation (DNS) while preserving micro-scale hotspots and their physical effects on ignition. Furthermore, the model is appropriate for high speed flows and can treat flows where chemical reactions and micro-scale mixing occur on the same scales (Da ≈ 1). This method has been successfully applied to model turbulent premixed and non-premixed flames (Menon and Calhoon, 1996; Chakravarthy and Menon, 2000), and also supersonic inert mixing layers (Sankaran and Menon, 2005). While the method has not yet been applied to model turbulent compressible and reactive flows, the CLEM-LES formulation has recently been validated for various fundamental 1-D laminar test cases. These include inert mixing, constant volume ignition, shock tube problems, and laminar flame propagation (Maxwell et al., 2013a). In this paper, the method of Maxwell et al. (2013a) is extended to treat turbulent flows. In general, the model captures the correct laminar flame speeds and structure when compared to DNS. Furthermore, turbulent flame speeds are also reproduced when compared against experiments (Abdel-Gayed et al., 1984) and previous LEM implementations (Smith and Menon, 1997). For compressible regimes, comparison of CLEM-LES with high and low resolution DNS demonstrates the models ability to capture correctly detonations initiation events. Furthermore, the model captures correctly the theoretical detonation velocities, structure, and instability behaviour associated with high activation energies. 2. Model Formulation 2.1 Governing Equations for the Compressible and Reacting System In order to model gaseous explosions in compressible environments, the gas dynamic evolution is governed by the compressible Navier-Stokes equations. In order to simplify the analysis, reduce computational expense, and to isolate the roles of specific physical mechanisms that influence the overall fluid flow, a number of assumptions are made. First, a calorically perfect Newtonian fluid system is assumed. The chemistry is also simplified by considering only a single reactant species, with mass fraction Y , that undergoes chemical reaction forming products according to a single-step Arrhenius reaction rate law (Williams, 1985). Only premixed combustion is considered here, although extension of the model to non-premixed combustion is trivial. Changes in molecular weight from reactants to products are also neglected. The resulting conservation equations for mass, momentum, energy, and reactant mass are given below in Equations 1 - 4, respectively. The equations are given in non-dimensional form through Equations 9, where the various gas properties are normalized by a reference quiescent state. The nomenclature for the various symbols found throughout the paper are given in Table 1.


∂ρ + ∇ · (ρu) = 0 ∂t

(1)

∂(ρu) + ∇ · (ρuu) + ∇p − ∇ · τ = 0 ∂t

(2)

∂E γ + ∇ · (E + p)u − u · τ − ∇ · (α∇T ) = −Qω˙ ∂t γ−1

(3)

∂(ρY ) + ∇ · (ρuY ) − ∇ · ∂t

α ∇Y Le

= ω˙

(4)

where: ω˙ = −ρn AY e(−Ea /T )

(5)

1 p + ρuu (γ − 1) 2

(6)

p ρ

(7)

E=

T =

2 T τ = µ ∇u + (∇u) − (∇ · u)Iˆ 3

(8)

The various parameters and variables are given in non-dimensional form according to

ρ=

ρˆ , ρô µ=

u=

uˆ , cô

1 µ ˆ = , ˆ Re ρô cô L Ea =

p=

pˆ pˆ , 2 = γ pô ρô cô

α= Eâ , cô 2

ˆ cˆp µ k/ = , ˆ Pr ρô cô L Q=

ˆ Q , cô 2

T =

Tˆ , γ Tô

Le =

A=

x=

xˆ , ˆ L

ˆ cˆp Sc k/ = , ˆ Pr ρˆD

Aˆ ˆ ρô n−1 ) cô /(L

t=

γ=

tˆ ˆ cô L/

cˆp cˆv

,

(9)


Table 1: Nomenclature. Variables A Pre-exponential factor Cλ LEM model constant c Speed of sound cp Specific heat capacity at constant pressure cv Specific heat capacity at constant volume D Mass diffusivity Da Damkohler number DCJ Detonation Mach number DT Turbulent diffusivity E Total energy Ea Activation energy F˙ Turbulent stirring source term k Heat conductivity L Reference length scale Le Lewis number l Eddy size Ms Shock Mach number m Mass coordinate N Number of subgrid nodes Nη Kolmogorov scale constant n Reaction order Pr Prandtl number Subscripts o Quiescent reference state i Subgrid node number ∗ Pre-compression reference state

p Q Re Sc SL T t u V x Y α γ ¯ ∆ δ η λ λ1/2 ρ µ ν ω˙

Pressure Heat release Reynolds number Schmidt number Laminar flame speed Temperature time Velocity Volume Cartesian distance coordinate Mass fraction of reactant Heat diffusivity Ratio of specific heats LES cell width Laminar flame width Kolmogorov scale Stirring event frequency per unit length Detonation half reaction length Density Dynamic viscosity Kinematic viscosity Reaction rate

Superscripts ˆ Dimensional quantity 0 Fluctuating quantity

2.2 1-D Subgrid model for the CLEM-LES Strategy For the CLEM-LES strategy, the large scale pressure evolution is solved separately from the small scale mixing and reactions. This is achieved through appropriate coupling of the pressure and energy fields. Thus, Equations 1 through 3 are solved on the large scales without the chemical reaction terms. A model is then applied to describe the small scale molecular mixing and chemical reactions. In this particular LES approach, the sub-grid model is a one-dimensional representation of the flow field within each LES cell whose orientation is aligned in the direction of local flow. A good general summary of the model formulation for weakly compressible flows (LEM-LES) is found in Menon and Kerstein (2011) and a more comprehensive description of the model is found in Sankaran (2003). In order to derive the sub-grid model formulation, pressure gradients are locally neglected within the small scales of each LES cell. Thus it is assumed that pressure waves travel much faster than the physical expansion or contraction of the local fluid relative to it’s convective motion. The result is that low-Mach number approximations are applied to treat the small scales while the pressure on the large scales is treated using piece-wise


constant discretization of the flow field. Therefore, the pressure field evolution is solved entirely on the LES. Also, the pressure changes determined from the LES are prescribed to the sub-grid, as a source term, in order to account for energy changes due to rapid compression or expansions. This formulation was previously applied to determine ignition limits of idealized non-turbulent and rapidly expanding hydrogen jets (Maxwell and Radulescu, 2011). The previous model formulation, however, contained only pressure and energy coupling in one-direction; from the large scales to the small scales. Thus, the model could only be used to determine the onset of ignition, and not the subsequent influence of ignition on the fluid evolution. Therefore, this current model serves as an extension to the previous work (Maxwell and Radulescu, 2011) by providing twoway coupling between the large scale pressure evolution and small scale mixing and reactions. The system of equations that is solved on the sub-grid is the conservation of energy, Equation 10, and conservation of reactant mass, Equation 11: DT − ρ Dt

γ − 1 ∂p ∂ ∂T γ−1 −ρ ρα =− Qω˙ + F˙T γ ∂t ∂m ∂m γ DY ∂ α ∂Y −ρ ρ ρ = ω˙ + F˙Y Dt ∂m Le ∂m

(10)

(11)

where the source terms, F˙T and F˙Y account for the effect of turbulence on the subgrid in the form of random ”stirring” events (Kerstein, 1991) and m is a one-dimensional mass weighted coordinate whose transformation to Cartesian spatial coordinates is given by Z

x

m(x, t) =

ρ(x, t)dx

(12)

xo

The sub-grid model formulation in this study differs from previous LEM-LES implementations (Smith and Menon, 1997; Menon and Kerstein, 2011; Sankaran, 2003) for two reasons. First, the pressure term in Equation 10 accounts for the energy changes associated with rapid changes in pressure. Second, the 1-D sub-grid domains are formulated with Lagrangian mass-weighted coordinates, m, as given by Equation 12. This is done in order to account for not only the expansion or contraction of the fluid along particle paths, but also changes in spatial distance between computational nodes on the subgrid (dx). 2.3 LEM-LES coupling through source terms The two-way coupling between the large scale hydrodynamics (the LES) and the small scale mixing and reaction (the subgrid) is achieved through source terms in the energy equations, 3 and 10. Specifically, this is done by maintaining consistent pressure changes between the two energy fields. First, the large scale fluid flow, the LES, provides the local pressure changes in each cell at each time step to the subgrid model. For isentropic flow in the form of expansions and weak compressions, the pressure term in Equation 10 is solved exactly by considering its contribution to the enthalpy change separately from the diffusion/reaction terms via operator splitting. Thus the enthalpy change of each subgrid node due to isentropic pressure changes is accounted for through


T2 = T1

p2 p2

(γ−1)/γ (13)

The diffusion terms of Equation 10 are then solved separately at constant pressure. To treat irreversible shock-waves, however, the post-shock temperature of each subgrid node is determined according to the Rankine-Hugoniot jump conditions through T − T∗ =

1 1 γ−1 (P + P∗ ) − 2 ρ∗ ρ

(14)

where the * denotes a pre-compression reference state. In this approach, if the local flow experiences more than 0.1% compression, Equation 14 is solved in order to update the temperature and density of each subgrid node according to the pressure increase. If the local flow does not compress sufficiently, isentropy is assumed and Equation 13 is solved. Also, the reference state (denoted by *) is reset accordingly. In order to implement this procedure, however, subgrid diffusion and chemical reactions are frozen during the irreversible compression process since a pre-compression reference state is needed. At the end of the subgrid simulation, the model provides the energy contribution due to chemical reactions to the LES. This is done by prescribing the reaction rate exactly in Equation 3 according to the precise amount of reactant mass consumed during the subgrid simulation. Thus, ω˙ LES =

∆(ρY )subgrid ∆tLES

(15)

where for each LES cell: PN (ρY )subgrid =

i=1

mi Yi

Vcell

(16)

A key feature of this two-way energy-coupling is that the pressures (and consequently, the internal energies per unit cell volume) of both the sub-grid and LES cell are always returned to the same value. This ensures that energy, along with mass, is always conserved. 2.4 Turbulent stirring (LEM) For the Linear Eddy Model (LEM), the turbulent ”stirring” (Kerstein, 1991) on the subgrid is represented by the source terms, F˙T and F˙Y in Equations 10 and 11. The actual ”stirring” events, however, are implemented as a series of random instantaneous re-mapping procedures. The remapping procedure is designed to simulate the effect that a multi-dimensional eddy would have on a one-dimensional sample of the flow field. In order to achieve this, a triplet map can implemented as detailed by Kerstein (1991). In this procedure, fluid properties are re-mapped in in such a way that mass is conserved. The effect of this re-mapping procedure on a 1-D sample of a flow field is illustrated in Figure 1.


Effect of eddies on flow field (l≫δ example): flame

flame

1D sample of flow field (LEM subgrid domain)

encounter with eddy

unburned gas

burned gas

unburned gas

burned gas

Equivalent 1-D domain temperature profile: burned gas Apply triplet map to 1D sample: unburned pocket

unburned gas

Figure 1: Representation of LEM triplet map to model eddies on a 1D sample of the flow field In order to implement the mapping procedure, three random variables are needed; the size of an eddy, its location, and the time of appearance. The location of each eddy is simply chosen randomly from a uniform distribution. However, its size, l, is randomly determined according to a Probability Density Function (PDF) which is derived from Komogorov scaling laws associated with turbulence dissipation. For eddies ranging in size from the integral scale of a ¯ to the Kolmogorov scale, η, the PDF of eddy sizes is given by Kerstein LES cell width, ∆, (1991):

f (l) =

(5/3)l−8/3 ¯ −5/3 ) (η −5/3 − ∆

(17)

Finally, the frequency of eddy stirring events is determined by considering a turbulent diffusivity resulting from a random walk of re-mapping events (Kerstein, 1991). By considering this diffusivity as an analogue to the diffusivity resulting from random collisions of molecules in a gas (Hinze, 1975; Jeans, 1940), a turbulent diffusivity of ”stirring” events can derived: 2 DT = λ 27

Z

¯ ∆

l3 f (l)dl

(18)

η

where the Kolmogorov scale is estimated from η=

¯ Nη ∆ (Re,∆¯ )3/4

(19)


Also, an equivalent expression of DT (Kerstein, 1991; Smith and Menon, 1997) is related to the a turbulent Reynolds number, (Re,∆¯ ), through DT ≈

νRe,∆¯ Cλ

(20)

¯ u0 ∆ ν

(21)

where Re,∆¯ =

Equations 18 and 20 are then be equated in order to determine the frequency of eddy stirring events per unit space, λ. The time between stirring events is then simply ¯ ∆tstir = 1/(λ∆)

(22)

Thus, the only required inputs for the re-mapping procedure are two empirical model constants, Cλ and Nη , and also the local velocity fluctuation of the turbulent flow, u0 . Specifically, Nη is calibrated to provide the Kolmogorov scale and typically has a value between 1.28 to 13 (Smith and Menon, 1997). The constant Cλ is calibrated to experiments. Finally, the velocity fluctuation, u0 , is normally obtained according to local flow conditions though closure of the momentum Equation 2 with a turbulent viscosity model. However, for the one-dimensional simulations conducted here, turbulent contributions to the conservation of momentum, Equation 2, are neglected. Thus, the velocity fluctuation, u0 , is only considered for turbulent stirring of Equations 10 and 11 and is provided as a constant. This is consistent with one-dimensional LEM simulations of Smith and Menon (1997). 2.5 Numerical Implementation In order to solve the system of Equations 1 through 4, for DNS or LES, a numerical framework developed by Mantis Numerics Ltd. is employed. The compressible flow solver features a second order accurate exact Godunov solver to treat the convection terms (Falle, 1991) and the diffusive terms are handled explicitly. Adaptive Mesh Refinement (AMR) and is also implemented for increased computational efficiency (Falle and Giddings, 1996). For the sub-grid model, the system of Equations 10 and 11 is solved across each LES time step using operator splitting to treat the various terms. The diffusion terms are solved explicitly in time and spatially discretized using central differences. The reaction terms are solved implicitly using the Backward Euler method with variable-time stepping. For implementation of AMR for the LES approach, the sub-grid model is only solved on the finest grid level of the LES. Finally, it should be noted that the reactant evolution does not need to be solved on the LES scale since all of the reactant information is stored on the sub-grid. Equations 5 and 11 are completely decoupled. A step-by-step summary of the algorithm for implementing the LES model (for 1-D laminar flows) across a single LES time step is found in Maxwell et al. (2013a).


2.6 Model Parameters For the validation experiments below, the model parameters are calibrated for premixed fuel-lean methane-air combustion at Tˆ = 328K and Pˆ = 1atm, with an equivalence ratio of φ = 0.717. This is consistent with the experiments of Abdel-Gayed et al. (1984) and numerical simulations of Smith and Menon (1997). The activation energy, heat release, and Lewis number are all taken from the Abdel-Gayed et al. (1984) experiments. The ratio of specific heats, sound speed, fluid viscosities, and heat conductivity are computed for the unburned gas composition using Cantera (Goodwin, 2013) and the GRI-3.0 mechanism (Smith et al., 2013). The pre-exponential factor was chosen such that the laminar flame speed matches the experiments of Abdel-Gayed et al. ˆ was chosen to be the (1984), which was given as SˆL = 0.36m/s. Finally, the length scale L ˆ laminar flame thickness, δ, as given by Abdel-Gayed et al. (1984). Dimensional quantities for the methane-air mixture properties and the corresponding non-dimensional model parameters are given in Table 2. Table 2: Fluid properties and model parameters for methane combustion. Dimensional properties Tô 328 K ρô 1.038 kg/m3 Pô 1 atm cô 368 m/s SˆL 0.36 m/s δˆ 5.21x10−5 m Non-dimensional model parameters α/ρ 0.00137 Pr 0.716 Le 0.973 A 1.0x104 Cλ 1.0 Nη 10.76

νˆ ˆ ρcˆp ) k/(ˆ

1.88 m2 /s 2.62 m2 /s

Eâ ˆ Q

5452.1 kJ/kg 1671.8 kJ/K

Ea Q SL

40.3 12.4 0.001

γ n

1.39 1.0

3. Model Validation for 1-D Flows 3.1 Freely Propagating Laminar and Turbulent Flames The first series of experiments conducted was to validate the current CLEM-LES model with laminar flame speed predictions of high resolution DNS and also the turbulent methane-air flame speed experiments of (Abdel-Gayed et al., 1984). The current model is also compared against a previously developed constant pressure LEM strategy (Smith and Menon, 1997) which also uses the Abdel-Gayed et al. (1984) experiments as validation. In order to initiate the experiments, the unburned and burned fluid properties are initiated in a domain with the flame located at (x = 0), according to: ρ(x, 0) = u(x, 0) =

1 (1 + (γ − 1)Q)−1

0.5 0.5 + (SL /ρ)(1 − ρ) p(x, 0) = 1/γ

if x < 0 otherwise if x < 0 otherwise

(23)


Y (x, 0) =

1 0

if x < 0 otherwise

In the following simulations, the boundaries of the domain are located sufficiently far such that acoustic signals reflecting from the boundaries do not interfere with the flame itself. Also, a uniform flow velocity, u = 0.5, is imposed in the flow field in order to ensure that diffusion occurs between all sub-grid nodes. This is necessary since the sub-grid diffusion of Equations 10 and 11 is neglected across the LES cell faces (Menon and Kerstein, 2011; Sankaran, 2003). Therefore, in this configuration, the flame is propagating to the left, but advected to the right. First, laminar flame experiments are conducted in order to verify the LES strategy. The turbulent LEM component of the CLEM-LES is then validated following the laminar cases. 3.1.1 The Laminar Flame For the laminar case, the resolved flame speed of the CLEM-LES model was found to converge to the same value as that found through Direct Numerical Simulation (DNS) of Equations 1 - 4. For the LES model, the number of subgrid nodes per LES cell was varied from N = 8 to N = 32 and the recorded steady flame speeds are shown below in Figure 2. The recorded flame speeds in this paper are determined according to the rate of reactant mass consumption at each time step. Two observations are drawn from this; 1) increasing the number of subgrid nodes per cell allows for a more coarse grid to be used on the LES in order to capture laminar flame speeds, and 2) providing there is fluid motion, neglecting subgrid diffusion between neighbouring LES cells has a negligible effect on the laminar flame speed. Finally, the flame structures of the resolved DNS ¯ = δ/104 and LES flames are presented in Figure 3. The resolution of the DNS in this figure is ∆ ¯ = δ/13 with N = 16 subgrid nodes per cell. Observation and the resolution for the LES is ∆ of Figure 3 reveals that the laminar flame structure is maintained by the LES strategy. Also, the flame width, δ, indicated by the arrows in the Figure, is consistent with Abdel-Gayed et al. (1984). The flame width indicated by the arrows in the Figure is measured by the distance from Y = 0.1 to Y = 0.9. 3.1.2 The Turbulent Flame To validate the LEM portion of the CLEM-LES formulation, the numerical turbulent flame experiment of Smith and Menon (1997) was repeated for the flame parameters described in Section ¯ = 740, and the Kolmogorov scale was set by letting 2.6. The size of each LES cell was set to ∆ Nη = 10.76, both of which are consistent with flame ”A1” in Smith and Menon (1997). These choices effectively set the range of eddy sizes from the order of δ to 740δ. In the numerical experiments, N = 1480 subgrid nodes per cell was computed as the minimum resolution, which was sufficiently small enough to resolve the smallest eddy stirring event. (u0 /SL ) was varied from 1 to 30 and the numerical experiments were all run up to t = 500, 000. This corresponds to eddy turn over times, defined in Equation 24, which range from 0.75 to 20, depending on u0 . ¯ 0 τ = ∆/u

(24)


Flamespeed (SL) vs Grid Resolution (7% CH4 + Air) DNS LES (N=8) LES (N=16) LES (N=32) Steady Solution

0.014 0.012

Note: δ = ν/SL

SL

0.01 0.008 0.006 0.004 0.002 0 0.1

1 10 Resolution (cells/δ)

100

1000

Figure 2: Laminar flame speeds vs. Grid Resolution Laminar Flame Structure (temperature profile)

Laminar Flame Structure (reactant profile)

6

1.2 DNS (∆=δ/104) LES (∆=δ/13, N=16)

5

0.8

3

Y

4 T

DNS (∆=δ/104) LES (∆=δ/13, N=16)

1

δ

0.6 δ

0.4

2

0.2 1 0 0 2495

2496

2497

2498 x

2499

2500

2494

2495

2496

2497 x

2498

2499

2500

Figure 3: Laminar flame structure: Temperature (left) and reactant (right) The results of the experiment are presented in Figure 4, where Cλ = 1.0 was found to provide good agreement between the CLEM-LES model and the validating experiments (Smith and Menon, 1997; Abdel-Gayed et al., 1984). One notable difference between the current CLEMLES formulation and Smith and Menon (1997) is the choice of Cλ . Smith and Menon (1997) had found that Cλ = 15.0 correlates well with the experiments of Abdel-Gayed et al. (1984). It should be noted that this difference can be attributed to differences in the model parameters used. Q and A, for example, are calibrated differently in Smith and Menon (1997). Furthermore Smith and Menon (1997) is formulated in a purely constant pressure system, unlike the current model. The flame speeds given in Figure 4 for the CLEM-LES are taken as the average speed recorded throughout the duration of each numerical experiment. Finally, a typical turbulent flame structure is shown for u0 /SL = 30 in Figure 5. The action of the eddy stirring events under high turbulence causes a one-dimensional ”Flame brush” to form. This leads to pockets of unreacted gas to form behind the leading flame front. This effectively increases the total surface area of the flame, which increases the burning rate, or flame speed due to the large-scale turbulence. This was also observed in Smith and Menon (1997).


Turbulent Flame Speeds for 7% CH4 + Air 20 Abdel-Gayed et al. (1984) Smith and Menon (1997) - LEM A1 CLEM-LES (Nη=10.76, Cλ=1.0)

ut/SL

15

10

5

0 0

5

10

15 u’/SL

20

25

30

Figure 4: Turbulent flame speeds vs. intensity (u0 /SL ) Turbulent Flame Structure (temperature profile)

Turbulent Flame structure (reactant profile) 1.2

CLEM-LES (u’/SL = 30)

5

1

4

0.8

3

Y

T

6

CLEM-LES (u’/SL = 30)

0.6 0.4

2

0.2 1 0 0 230140

230880 x

231620

230140

230880 x

231620

Figure 5: Turbulent flame structure: Temperature (left) and reactant (right) 3.2 Detonation Initiation via Shock-Flame Interaction In this numerical experiment, laminar detonation initiation through a shock-flame interaction was modelled using the LES strategy and compared against DNS at two different resolutions. The LEM turbulence in 1-D was not considered in this experiment. The simulation set-up, illustrated in Figure 6, was conducted in the frame of reference of a piston, whose surface is initially located 4000 laminar flame thickness widths (4000δ) downstream from the flame in the burned products (on the left of the domain). An inflow boundary of unburned gas is specified 1000δ upstream from the flame (on the right of the domain). The flame modelled is consistent with the parameters in 2.6 and the speed at which the unburned gas flows into the domain is sufficient to drive a shock through the products with a strength of Ms = 2.2 relative to the burned gas. In this context, the flame propagates towards the right, however it is convected to the left (towards the piston). Mathematically, the domain is initialized according to


ρ(x, 0) = u(x, 0) =

(1 + (γ − 1)Q)−1 1 −up (SL /ρ)(1 − ρ) − up

if x < 4000 otherwise if x < 4000 otherwise

(25)

p(x, 0) = 1/γ Y (x, 0) =

0 1

if x < 4000 otherwise

where the piston velocity in the absolute frame of reference is given by 1/2 (Ms − 1/Ms ) up = 2 1 + (γ − 1)Q (γ + 1) piston

flame

inflow boundary unburned gas

burned gas

x=0

(26)

x=4000

x=5000

Figure 6: Setup for the shock-flame interaction simulation. ¯ = δ/64 and ∆ ¯ = 10δ, corresponding to high resoThe DNS is conducted for resolutions of ∆ ¯ = 10δ lution and low resolution respectively. The LES is conducted for a base resolution of ∆ containing N = 640 subgrid nodes per cell, thus with diffusion and chemical reaction resolved to the same scale as the high resolution DNS case. The density and reactant mass fraction evolutions for all three cases is shown in Figure 7. For the high resolution DNS, a detonation wave was found to initiate ahead of the flame, in the unreacted gas, at approximately (x, t) = (1000, 1500). Specifically, the location where detonation initiation occurred was found to coincide with the shock-shock interaction involving the initially transmitted shock into the unreacted gas, and a second transmitted shock originating a shock reflection at the piston wall. In this case, detonation initiation occurs due to sufficient shock-compression. In the low resolution DNS, however, a detonation wave was found to initiate almost immediately after the incident shock from the piston reached the flame, at approximately (x, t) = (1400, 900). In this case, increased reaction rates owing to numerical diffusion (and a faster moving flame) at the flame surface are sufficient to initiate the detonation wave prematurely. For the LES, on the other hand, detonation initiation occurs at approximately the same instant and position as the DNS. Finally, differences between the LES and high resolution DNS, however, are observed where subsonic burning of reactant gas occurs in the near zero velocity region between the original flame and newly formed detonation wave. This error in the LES prediction is due to insufficient diffusive mixing across cell faces since the model relies on sufficient flow velocity across cell boundaries.


Density evolution for high resolution DNS (∆=δ/64)

Reactant mass fraction evolution for high resolution DNS (∆=δ/64)

2500

2000

10

2500

1

8

2000

0.8

6

1500

0.6

4

1000

0.4

2

500

0.2

0

0

D

t

s s 1000

detonation initiation

F s

t

1500

F

s

s unburned gas

500

s

F

burned gas 0 0

1000

2000

3000

4000

5000

0 0

1000

2000

x

3000

4000

5000

x

Density evolution for low resolution DNS (∆=10δ)

Reactant mass fraction evolution for low resolution DNS (∆=10δ)

2500

10

2500

1

2000

8

2000

0.8

6

1500

0.6

4

1000

0.4

2

500

0.2

0

0

1500

t

t

D

1000


s

unburned gas 500

s

F

burned gas 0 0

1000

2000

3000

4000

5000

0 0

1000

2000

x

3000

4000

5000

x

Density evolution for laminar LES (∆=10δ, N=640)

Reactant mass fraction evolution for laminar LES (∆=10δ, N=640)

2500

2000

10

2500

1

8

2000

0.8

6

1500

0.6

4

1000

0.4

2

500

0.2

0

0

D

t

s s 1000


F s

t

1500

F

s

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Figure 7: Shock-flame interaction for DNS (at high and low resolution) and LES (laminar). The left frames show density (ρ) while the right frames show the reactant mass fraction (Y ). The diagrams show the various shock, flame, and detonation features in the frame of reference of a piston (the left boundary). The right boundary is an inflow boundary. (s=shock, F=flame, D=detonation)


3.3 Detonation Propagation and Structure The final experiment conducted here assesses the performance of the model in highly compressible and reactive flows. Specifically, a highly resolved detonation was simulated and the resulting structure and propagation characteristics were analysed. In order to initiate the detonation, the Zel’dovich - von Neumann - Doring (ZND) structure (Fickett and Davis, 1979) is imposed with ˆ is the leading shockwave located at (x = 0), as illustrated in Figure 8. The length scale, L, given in terms of the half-reaction length, (λ1/2 ) and is obtained from the steady ZND solution. The non-dimensional distance, time, and pre-exponential factor are scaled accordingly. The left boundary, located at x = −90, is fixed at the Chapman-Jouguet (CJ) conditions (Fickett and Davis, 1979), while the right boundary, located at x = 1200, is an outflow boundary with zero gradients. The initial quiescent fluid, located at x > 0, has the properties ρ(x, 0) = 1, u(x, 0) = 0, p(x, 0) = 2, and Y (x, 0) = 1. The simulation is run up to t = 180 for a base res¯ = λ1/2 /128. The results are compared against the theoretical ZND model and also olution of ∆ DNS with the same base resolution. For the LES model, only a few subgrid nodes are included (N = 4). boundary fixed at CJ properties

ZND solution for x0 x=0

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Figure 8: Initial conditions for detonation propagation experiment. Figure 9 shows the velocity of the wave as a function of time for both the DNS and the LES model. For both simulations at sufficient resolution, the initial disturbance (numerical errors) at (x, t) = (0, 0) is sufficient to induce the oscillatory unstable (pulsating) behaviour associated with high activation numbers and low reaction orders (Ng et al., 2005; Short, 2005). For the LES model, this oscillatory behaviour is triggered faster than the DNS. Despite this transient difference at the simulation start up, the amplitude, period, and mode of velocity fluctuations are the same for both simulations despite being out of phase. Furthermore, the detonation velocities recorded in both simulations was found to oscillate about the Chapman-Jouguet (CJ) velocity of DCJ = 5.33. The difference in behaviour at start up is believed to be influenced by differences in resolution of the diffusive scales between the two simulations, but is not further investigated here. Finally, comparison of the detonation structure of the LES model and the DNS with the steady ZND solution is shown in Figure 10. Excellent agreement is observed between the LES model, DNS, and ZND solutions for the reactant and density profiles. 4. Conclusions In this paper an LES strategy (CLEM-LES), based on LEM-LES (Menon and Kerstein, 2011), has been proposed for modelling gaseous explosion hazards which are highly compressible and

Tenth International Symposium on Hazards, Prevention, and Mitigation of Industrial Explosions Bergen, Norway, 10-14 June 2014 Detonation Speeds for ∆=(λ1/2/128) 12

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Figure 9: Comparison of detonation velocities for DNS and LES. Detonation (density profile)

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Figure 10: Detonation structure: density(left) and reactant(right) profiles. involve rapid transients in pressure and energy. The model has been validated against various one-dimensional laminar and turbulent combustion scenarios. The model has been demonstrated to resolve and capture laminar flame speeds and structure when compared against DNS. Also, the simulations validate assumptions made by Menon and Kerstein (2011) regarding the negligible effect that subgrid diffusion between neighbouring LES cells has on the laminar flame speed, providing there is sufficient fluid motion. Furthermore, the model is able to reproduce the turbulent flame speed trends observed experimentally (Abdel-Gayed et al., 1984) and also numerically (Smith and Menon, 1997). It is worthwhile to note that for the 1-D turbulent flame calculations presented here, the turbulent production on the subgrid scale is prescribed as a constant. In a multi-dimensional simulation, however, adequate closure to the momentum equation would be required. In this case, multi-dimensional fluid motion would generate turbulence, which would then be reflected on the subgrid through the stirring events. These stirring events would then influence the energy and reactant evolution, through source terms, which would then further influence fluid motion and turbulence. Piston-driven shock-flame interaction experiments showcase the ability of the model to simulate detonation initiation resulting from complex gas-dynamics and rapid chemical reactions in the presence of shock waves. The model has been found to address difficulties with asso-


ciated with unresolved (low resolution) DNS in terms of detonation initiation resulting from increased reaction rates and faster flame speeds that are driven by numerical diffusion. It was found that the CLEM-LES model captures correctly detonation initiation events when compared to high resolution DNS. Errors in the burning rates of reactant gas in near-zero velocity regions are observed, however, which further confirms the model requirement of sufficient flow velocity to ensure adequate diffusive mixing across cell boundaries. Finally, detonation structure and stability characteristics arising from perturbations in the flow field are captured by the LES model. As a final note, it was found that computational expense of the model was comparable to DNS for 1-D simulations. The model, however, has the potential to reduce computation expense by a square-root or cubic-root factor in two and three-dimensional simulations, respectively. This is attributed to the fact that multi-dimensional problems would effectively be reduced to one-dimensional problems on the finest scales. In conclusion, the CLEM-LES formulation has been demonstrated to improve on previous LEM-LES formulations (Menon and Calhoon, 1996; Chakravarthy and Menon, 2000; Menon and Kerstein, 2011; Sankaran, 2003) by simultaneously treating rapid pressure changes and chemical reactions in turbulent environments. Acknowledgements The Authors would like to Acknowledge sponsorship for the project via the NSERC Hydrogen Canada (H2CAN) Strategic Research Network and by an NSERC Discovery Grant. B.M.M. also acknowledges financial support through the NSERC Alexander Graham Bell Scholarship, and has also previously received financial support from the Ontario Ministry of Training, Colleges and Universities via an Ontario Graduate Scholarship. Additional funding was also previously provided through a CFD Society of Canada Graduate Scholarship and the NSERC Michael Smith Foreign Study Supplement. References Abdel-Gayed, R. G., Al-Khishali, K. J., and Bradley, D. (1984). Turbulent burning velocities and flame straining in explosions. Proceedings of the Royal Society of London A, 391:393. Chakravarthy, V. K. and Menon, S. (2000). Subgrid modeling of turbulent premixed flames in the flamelet regime. Flow, Turbulence and Combustion, 65:133–161. Falle, S. A. E. G. (1991). Self-similar jets. Monthly Notices of the Royal Astronomical Society, 250:581–596. Falle, S. A. E. G. and Giddings, J. R. (1996). Monthly Notices of the Royal Astronomical Society, 278:586–602. Fickett, W. and Davis, W. C. (1979). Detonation Theory and Experiment. Dover. Goodwin, D. G. (Accessed 2013). Cantera. http://code.google.com/p/cantera/. Hinze, J. (1975). Turbulence. McGraw-Hill, 2nd edition.


Jeans, J. H. (1940). An Introduction to the Kinetic Theory of Gases. Cambridge University Press. Kerstein, A. R. (1991). Linear-eddy modeling of turbulent transport. part 6: Microstructure of diffusive scalar mixing fields. Journal of Fluid Mechanics, 231:361–394. Khokhlov, A. M., Oran, E. S., and Thomas, G. O. (1999). Numerical simulations of deflagrationto-detonation transition: The role of shock-flame interactions in turbulent flames. Combustion and Flame, 117:323–339. Maxwell, B. M., Falle, S. A. E. G., Sharpe, G. J., and Radulescu, M. I. (2013a). A Turbulent Combustion Model for Ignition of Rapidly Expanding Hydrogen Jets: Validation for 1-D Laminar Flows. 5th International Conference on Hydrogen Safety, Brussels, Belgium. Maxwell, B. M. and Radulescu, M. I. (2011). Ignition limits of rapidly expanding diffusion layers: Application to unsteady hydrogen jets. Combustion and Flame, 158 No. 10:1946– 1959. Maxwell, B. M., Tawagi, P., and Radulescu, M. I. (2013b). The role of instabilities on ignition of unsteady hydrogen jets flowing into an oxidizer. International Journal of Hydrogen Energy, 38:2908–2918. Menon, S. and Calhoon, W. H. (1996). Subgrid mixing and molecular transport modeling in a reacting shear layer. 26th International Symposium on Combustion, Naples, Italy. Menon, S. and Kerstein, A. R. (2011). Turbulent Combustion Modeling: Advances, New Trends and Perspectives, chapter 10: The Linear-Eddy Model, pages 221–247. Springer. Ng, H., Higgins, A., Kiyanda, C., Radulescu, M., Lee, J., Bates, K., and Nikiforakis, N. (2005). Nonlinear dynamics and chaos analysis of one-dimensional pulsating detonations. Combustion Theory and Modeling, 9:1:159–170. Sankaran, V. (2003). Sub-grid Combustion Modeling for Compressible Two-Phase Reacting Flows. PhD thesis, School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, Georgia. Sankaran, V. and Menon, S. (2005). LES of scalar mixing in supersonic mixing layers. Proc. Combust. Inst., 30:2835–2842. Short, M. (2005). Theory and modeling of detonation wave stability: A brief look at the past and toward the future. 20th International Colloquium on the Dynamics of Explosions and Reactive Systems, Montreal, Canada. Smith, G. P., Golden, D. M., Frenklach, M., Moriarty, N. W., Eiteneer, B. E., Goldenberg, M., Bowman, C. T., Hanson, R. K., Song, S., Gardiner, W. C., Lissianski, V. V., and Qin, Z. (Accessed 2013). GRI-Mech 3.0. http://www.me.berkeley.edu/gri mech/. Smith, T. and Menon, S. (1997). One-dimensional simulations of freely propagating turbulent premixed flames. Combustion Science and Technology, 128:99–130. Williams, F. A. (1985). Combustion Theory. Benjamin/Cummings Publishing Company Inc., Menlo Park, CA, 2nd edition.