Examination of Turbulent Flow Effects in Rotating Detonation Engines

Transcription

1 Examination of Turbulent Flow Effects in Rotating Detonation Engines Colin A. Z. Towery, Katherine M. Smith, Prateek Shrestha, and Peter E. Hamlington Turbulence and Energy Systems Laboratory, Department of Mechanical Engineering, University of Colorado, Boulder, CO 839, USA Marthinus Van Schoor Midé Technology, Medford, MA 255, USA Subsonic and low-supersonic propulsion systems based on detonation waves have the potential to substantially improve efficiency and power density compared to traditional engines. Numerous technical challenges remain to be solved in such systems, however, including obtaining more efficient injection and mixing of air and fuels, more reliable detonation initiation, and better understanding of the flow leaving the detonation chamber. These challenges can be addressed using numerical simulations. Such simulations are enormously challenging, however, since accurate descriptions of highly unsteady flow fields are required in the presence of combustion, shock waves, fluid-structure interactions, and other complex physical processes. In this paper, we perform high-resolution two- and three-dimensional large eddy simulations of pulsed and rotating detonation engines and examine unsteady and turbulent flow effects on the operation, performance, and efficiency of the engine. These simulations are further used to test the accuracy of common Reynolds averaged turbulence models. I. Introduction Compared to other established (e.g., gas turbine) and advanced (e.g., scramjet) air-breathing propulsion systems, rotating and pulsed detonation engines (RDEs and PDEs, respectively) have the potential to provide higher energy efficiency, cleaner and more complete combustion, and higher power density. The concept underlying detonation engines has been in existence for many years (see [, 2] for reviews and historical perspectives), and recently renewed interest in these engines has included a focus on hybrid systems where exhaust gases from RDE or PDE detonation chambers are used in combination with traditional gas turbines. Numerous technical challenges remain to be solved in these systems, however, particularly in the transition region from the exit of the detonation combustor to the turbine stage (or turbocharger). The flow exiting the detonation chamber is expected to be turbulent and highly unsteady, with large localized pressures and temperatures. 3 Currently, there is inadequate understanding of how this unsteadiness affects turbine performance, efficiency, and fatigue. There are also a number of unresolved technical challenges in detonation combustion engines themselves, including obtaining more efficient injection and mixing of air and fuels, more reliable detonation initiation, and better understanding of the flow in the ejection nozzle., 4 Although it is possible to study each of these challenges using high-fidelity numerical simulations, major difficulties are posed by the wide range of spatial and temporal scales present in RDEs and PDEs. The range of spatial scales spans the width of the detonation wave, O( 5 m), to the size of the combustor, O( m), while the range of temporal scales extends from the reaction time scale, O( 6 s), to the time taken for several orbits of the detonation wave around the combustor in the case of RDEs, O( 3 s), or for several purge-refill cycles in the case of PDEs, O( 2 s). Although it is theoretically possible to resolve this full range of spatial and temporal scales using direct numerical simulations (DNS), the associated demand for computational resources is prohibitive in most practical applications. Consequently, approaches which model some, or all, of the relevant scales such as large eddy simulations (LES) and Reynolds Averaged Navier Stokes (RANS) approaches are the most practically useful. of 2

2 Past investigations have performed numerical simulations of both RDEs 5 3 and PDEs. 4 6 Although some PDE investigations have represented turbulence via RANS simulations, 7, 8 most simulations are shockcapturing approaches in the absence of an explicit treatment of turbulence, particularly simulations of RDEs. A more recent RDE investigation [3] used an explicit RANS treatment of turbulence; namely, using the standard κ ɛ model. We hypothesize, however, that turbulence will have a significant impact on the flow field and that standard models developed for incompressible, non-reacting flow will be insufficient to capture the full complexity of the flow. There is turbulence at the inflow to the domain, turbulence can broaden and disrupt the detonating flame front, and turbulence may even initiate auto-ignition at an undesirable time and location in the domain. Capturing these effects requires true representation of small-scale turbulence in simulations of high-speed combustion. In the present study, we perform simulations of an RDE using the ideal gas Euler equations with reaction source terms. We model a hydrogen-air RDE design, corresponding to prior numerical 6, and experimental 5, 9 studies of such systems. We address small-scale dissipation with numerical viscosity, thereby providing a representation of nonlinear, inertial turbulent flow effects without the uncertainty associated with most subgrid-scale models. We use these simulations to characterize turbulent flow effects in all regions of the RDE combustor, with a particular focus on the exhaust gases which would feed into a gas turbine stage in a hybrid pressure gain combustion system. We then evaulate existing RANS models on an a priori basis with these high-resolution datasets. II. Details of the Numerical Simulations The simulation code uses the Laboratory for Computational Physics Flux-Corrected Transport (LCPFCT) algorithm, which is described in detail in [2]. Schwer and Kailasanath [6 9] have used the same algorithm to simulate RDEs and is well-understood for simulations of high-speed reacting flows. We wrote the simulation code specifically for the RDE systems examined in this study, although we can also examine similar geometries (e.g., one dimensional detonation waves, shock tube problems, and pulsed detonation engines). The numerical simulations approximate the reactive Euler equations given by (ρe) t (ρu) t ρ t + (ρu) =, () + (ρu u) = p, (2) + (ρe)u = (pu) + qρ ω r, (3) ρ r t + (ρ ru) = ρ ω r, (4) where ρ is the density, u is the velocity vector, p is the pressure, E is the total energy, ρ r is the reactant density, and ω r is the reaction source term. The equations of state for a calorically-perfect gas are used to relate energy, temperature, and pressure, E = p ρ(γ ) + 2 (u u) + qy r, (5) T = p ρr, (6) where γ is the specific heat ratio, q is the total heat release, and Y r is the fuel mass fraction. The reaction source term ω r can be represented in the simulations using one-step Arrhenius kinetics or a two-step induction-reaction parameter model. II.A. One-step Reaction Kinetics Model For a one-step, Arrhenius reaction, ( ρ ω r = Aρ r exp E ) a, (7) RT 2 of 2

3 where A is the pre-exponential factor, E a is the activation energy, R is the specific gas constant, and T is the temperature. The parameters A, E a, and R depend on the fuel mixture, in this case stoichiometric, 2 hydrogren-air. The model uses single-component values of the reactant properties in Eq. (7), namely A = s, E a = J/kg, (8) and the heat release parameter, q, ratio of specific heats, γ, and specific gas constant, R, are q = J/kg, γ =.29, R = J/kg K. (9) It will be shown in section III that the model used here agrees satisfactorily with theoretical predictions from the Chapman-Jouguet (CJ) relations for a detonation wave. II.B. Two-Step Reaction Kinetics Model In the compressible gas detonations studied there are two distinct periods involved in the chemical kinetics of combustion: a chain-branching period of reactant dissociation into free radicals in which essentially no heat is released; and then a chain-terminating period of product formation with heat release. 22 This two-period description of detonation corresponds closely to the shockwave-deflegration zone description of Zel Dovichvon Neumann-Doring (ZND) theory and a two-step induction-reaction model applies this description in practice. II.B.. Induction Parameter Model If we consider the Lagrangian motion of a fluid droplet moving through a flow field with varying temperature and pressure, the ratio dt/t ind is the infinitesimal fraction of the induction period which has elapsed during dt. The induction time, t ind, is a measure of the time required for the reactants to dissociate into a critical density of free radicals, at which point the free radicals react to form products and release heat. As such, t ind is a function of the local temperature, pressure, and reactant equivalence ratio, t ind (T, P, φ). Therefore, we can relate the induction process to a progress variable, τ i, where τ i = t dt t ind (T, P, φ), () dτ i dt = t ind (T, P, φ). () Since the induction parameter is Lagrangian, the code must transport it along with the reacting fluid, and since T and P are functions of both space and time, dτ i / dt is a material derivative with the form (ρτ i ) t + (ρτ i )u = ρ t ind (T, P, φ). (2) Once τ i =, the reaction period begins, and we use a reaction parameter to describe the time-dependent transformation of reactants into products and release of heat. The function for induction time can be interpolated from either controlled experimental results or numerical simulations with detailed kinetic mechanisms. We chose to fit a function to the numerical results obtained in [23] for a 2::4 mixture of H 2 :O 2 :N 2, which were also used in [6 9]. II.B.2. Reaction Parameter Model We can relate the chemical reaction of any fluid particle to a progress variable, τ r, which measures the extent of the chemical transformation of the fluid particle. Assuming all chemical species are consumed or created at the same rate, τ r = Y i Y ir Y ir Y ir, (3) dτ r dt = dy i = ω r, (4) Y ip Y ir dt 3 of 2

4 where Y i is the mass fraction of the ith chemical species, Y ir and Y ip are the mass fractions before and after the reaction is complete, and ω r is the chemical transformation rate. 22 As with the induction parameter, the code must transport the reaction parameter with the flow: (ρτ r ) t + (ρτ r )u = ρ ω r. (5) Once τ r =, the reaction is complete. Without putting any restrictions on the form of ω r, or the numerical solver, one must specify { (ρτ r ) (τ + (ρτ r )u = i < τ r > ). (6) t ρ ω r (τ i & τ r < ) The parameterization of the reaction period is independent of the choice of modeled chemical species. Therefore, the two-step induction-reaction model only requires that two species are tracked: a pre-mixed, stoichiometric reactant and its product. In this simplified case from which we obtain Y i = [Y r, Y p ], Y r = Y p, (7) τ r = Y r = Y r = Y p, (8) dτ r dt = dy p = dy r = ω r, dt dt (9) [ ] (ρτr ) + (ρτ r )u = (ρy r) + (ρy r )u = ρ ω r. (2) t t We can then finally simplify this relation to obtain a continuity equation for reactant density as ρ r t + (ρ ru) = ρ ω r. (2) As a result, the only additional computation the model requires compared to a one-step reaction is the addition of Eq. (2) and logic which suppresses ω r while τ i < and while ρ r =. We can represent the transformation rate, ω r, by either an Arrhenius or linear rate form 22 ( ρ r a exp b ) ρ ω r = RT. (22) ρc The simplest case, ω r = c, describes a constant-rate reaction. Several previous studies have used this method, including [6 9] where they specify the reaction rate, c as the duration of heat release, t react, and implicitly specify the simulation time step such that the heat release is spread out over n steps: ρ ω r = ρ, and t = t react /n. (23) t react Other studies use a very similar approach where ρ ρ ω r = t react (T, P ), (24) as in [24], and t react is no longer constant. This requires that we finds a function for t react by the same process used to elicit t ind. If we instead choose an Arrhenius rate, ( ρ ω r = ρ R a exp b ), (25) RT we can tune the activation energy, b, so that simulations produce the correct detonation cell size for the reactants, and we can tune the pre-exponential factor, a, to produce the correct reaction time for a given detonation, as in [25]. 4 of 2

5 II.C. Solution Procedure We initially simulated the RDE by unrolling the computational domain and solving Eqs. () (6) in two 6, spatial dimensions on a Cartesian grid. Previous studies have used this approach to simulate RDEs and it is especially accurate as the annular combustion chamber becomes infinitely thin. As a result of this unrolling, the azimuthal direction (θ) becomes the horizontal direction with coordinate x and the axial direction (z) becomes the vertical direction with coordinate y. For a detonation chamber of height h and radius r, the length and height of the corresponding two-dimensional Cartesian grid are L x = 2πr and L y = h, where x = [, L x ] and y = [, L y ]. The x-boundaries are periodic to close the geometry. We also simulated the RDE with a three-dimensional Cartesian grid in order to develop fully three-dimensional turbulence while still isolating the model from centrifugal effects. The code integrates Eqs. () (6) in time using a split-step approach. LCPFCT solves the advection terms in a directionally-split fashion wherein the advection equations are first integrated from time t to t + t/2 and the resulting pressure gradient terms are used to re-integrate from time t to t + t. This is a recommended time-stepping procedure for the LCPFCT algorithm, as discussed in [2]. Since we perform the simulations in two dimensions, the solution variables are ρ, ρu (horizontal momentum), ρv (vertical momentum), E, and ρ R. Our code solves the reaction terms as ordinary differential equations with no spatial dependence. For Arrhenius reaction rates in either the one or two-step kinetics models ( dρ r dt = ρ ω r = ρ r a exp b ). (26) RT Assuming that T is constant, we can solve this equation analytically to give [ ( ρ r (t + t) = ρ r (t + t) exp a t exp b ) ], (27) RT where the code obtains ρ r (t + t) from the solution of the advection terms. For two-step induction-reaction kinetics, we can also solve Eqs. (), (23), and (24) analytically since T and P are held constant over a time step, ρ(t + t) t τ(t + t) = τ(t + t) + t ind ( T (t + t), p(t + t)), (28) ρ r (t + t) = ρ r (t + t) + cρ(t + t) t. (29), Yi, et al, used this approach for handling the reaction term and found it substantially relaxes the requirements on the spatial and temporal resolution of the simulations. II.D. Initial Conditions Initially, the code fills the detonation chamber with quiescent fluid at ambient pressure and temperature, denoted p and T, respectively. The code then obtains the ambient density from the equation of state as ρ = p RT. (3) We assume that ρ r = everywhere in the domain at t =, except for a thin band of height y r along the bottom of the chamber where ρ r = ρ (corresponding to pure reactants in the mixture and Y r = ). The code introduces a detonation wave to the chamber at t = by specifying a driving pressure and temperature, p d and T d, in a small region at the bottom left corner of the simulation domain of size x = [, x d ] and y = [, y d ]. Initially, the code sets the vertical boundaries at x = and x = L x as reflective walls in order to ensure that the detonation wave moves in only one direction, from x = to x = L x. After a sufficient time, the the code removes the walls and replaces them with a periodic boundary condition, thereby allowing the detonation wave to travel around the detonation chamber. 5 of 2

6 II.E. Boundary Conditions The code sets the upper boundary of the detonation chamber as an extrapolative outflow for all times. The bottom boundary is more complicated, however, and must account for the inflow of fresh reactant mixture into the domain. In practice, experiments have provided this inflow with a ring of injectors along the bottom boundary. In the present study, we explore the effects of discrete nozzle injection and control for nozzle injection effects using an idealized inflow by averaging the discrete inflow over the entire inlet boundary, following the example of prior studies. 6, The LCPFCT algorithm allows for relatively simple treatment of boundary conditions. Integration of the governing equations requires one guard cell at either end of the one-dimensional variable arrays passed as input to LCPFCT. We write the variable values in these guard cells, denoted ϕ g, as ϕ g = S ϕ ϕ I + V ϕ, (3) where ϕ I is the value of ϕ in the solution domain immediately adjacent to the guard cell and S ϕ and V ϕ depend on the type of boundary condition. The code must include two guard-cells for each integration using LCPFCT; one at the beginning of the one-dimensional array and one at the end of the array. II.E.. Physical Boundaries We use three types of physical boundary conditions in the simulations: reflective walls, extrapolative outflow, and prescribed inflow. Reflective walls: For a wall, V ϕ = for all ϕ. For the component of the momentum perpendicular to the wall (either ρu for a vertical wall or ρv for a horizontal wall), S ϕ = such that there is no flow through the wall. For the tangential momentum and all other solution variables, S ϕ =. Extrapolative outflow: Although modifications can be made depending on whether the outflow is subsonic or supersonic, here we model the entire outflow boundary in the same manner. For all ϕ, S ϕ = η and V ϕ = ηϕ, where η is a relaxation parameter that is typically small (we use η =. in these simulations) and ϕ represents the ambient value of ϕ. Prescribed inflow: For a prescribed inflow, S ϕ = for all ϕ. We give the values of V ϕ as V ρ = ρ in, V ρu =, V ρv = ρ in v in, V ρe = ρ in E in, V ρr = ρ in, (32) where ρ in, v in, E in and p in are the inflow values of the density, perpendicular velocity, total energy, and pressure, respectively, and we assume that the inflow boundary is horizontal (extending from x = to x = L x along the bottom of the simulation domain). Following [6, ], we determine the inflow values based on the value of p in the first grid cell: If p p, where p is the stagnation pressure in the nozzles, then p in = p and there is no flow, v in =, and S ϕ and V ϕ are the same as for the reflective boundary. If p > p > p cr, where p cr is the critical pressure given by ( ) γ/(γ ) 2 p cr = p, (33) γ + then p in = p and ( ) γ p γ T in = T p, v in = 2γRT ( p γ p ) γ γ, (34) where T is the stagnation temperature. If p p cr, then the flow is choked with p in = p cr and ( ) γ pcr γ T in = T, v in = 2γRT p γ ( pcr p ) γ γ. (35) 6 of 2

7 II.E.2. In the case of idealized inflow, we assign prescribed inflow to the entire boundary and scale v in by the area ratio, A in /A wall, providing an averaging effect. We model discrete injection nozzles by assigning alternating boundary conditions to the inflow at the bottom of the RDE chamber; that is, the prescribed inflow without scaling where an injector is present and a reflective wall where one is not. With these values of p in, T in, and v in, the corresponding inflow density, ρ in, and energy, E in, are ρ in = Parallel Domain Decomposition p in p in, E in = RT in ρ in (γ ) + 2 v2 in + q. (36) We parallelized the code with MPI using two-dimensional domain decomposition. Where an MPI task subdomain falls along the model domain, we assign boundary conditions as above. Where a subdomain borders a neighboring subdomain, MPI tasks communicate a number of halo cells, extending and overlapping the two subdomains. We found a two-cell halo was sufficient for the simulations reported here. The parallel code scales very well down to, cells per subdomain (52 processors in Figure ) and simulations reported here were executed with anywhere from processor for D simulations to 44 processors for 2D RDE simulations and up to 32 processors for 3D RDE simulations. Walltime (s) rde2d.794 unitary Number of Processors Figure : Strong scaling study of the TESLa RDE model. III. Pulsed Detonation Engine Simulations III.A. One-Dimensional Detonation We examined a one-dimensional detonation wave in order to verify that the advection-reaction equations implemented in the code give correct detonation wave solutions. Comparisons with theoretical CJ solutions and prior studies of one-dimensional detonation waves at similar conditions enable the validation. The code initializes the D domain with reactant mixture at ambient conditions, p = atm and T = 3 K, with ρ, E, and ρ r all determined by the corresponding equations of state. The code then initiates a detonation wave on the left wall of the chamber (at x = ) by introducing reactant at p = 2 atm and T = 3 K. The resulting wave propagates to the right in the chamber (towards x = L x ). we show pressure, temperature, and horizontal velocity profiles for a one-step reaction detonation wave at t =. ms in Figure 2. The figure shows results for different spatial resolutions, corresponding to grid sizes N x = [2, 4,, 2] with x = L x /N x at temporal resolution t = 8 s. The figure also 7 of 2

8 PDE 2D RDE-A 3D RDE-A 3D RDE-B Simulation parameters Domain size, (Lx, Ly, Lz) (.5 m,.4 m, n/a) (.4398 m,.77 m, n/a) (.4398 m,.77 m,. m) (.299 m,.885 m,.5 m) Grid dimensions, (nx, ny, nz) various (6, 24) (24,96,4) (24,96,4) (run 4), (2,48,2) (runs 5, 6) Time step, t various 5 9 s, 9 s 5 9 s 5 9 s Total simulation time, T various 5.89 ms,.57 ms 2.23 ms (run ), 2.98 ms (runs 4-6) 4.55 ms (run 2),.3 ms (run 3) Ambient conditions Ambient velocity, v m/s Ambient pressure, P atm Ambient temperature, T 3 K ρ Ambient density,.95 kg/m 3 Initial conditions Initial fuel height, Lr,y n/a. m restart from idealized restart from idealized Initial fuel density, ρr, ρ ρ (runs, 2) (runs 4, 5) Det. initi. size, (Ld,x, Ld,y) (. m,.4 m) (. m,. m) initialized with nozzles initialized with nozzles Det. initi. pressure, Pd 2P 5P (run 3) (run 6) Det. initi. temperature, Td T T see 2D RDE-A for conditions see 2D RDE-A for conditions Boundary conditions Injection area ratio, Ain/Awall n/a Number of nozzles, Ninj n/a idealized injection 5 (run ), (runs 2, 3) (runs 4-6) Inflow stag. pressure, P n/a P P P Inflow stag. temperature, T n/a T T T Outflow pressure, Pb n/a P P P Table : Summary of parameters used to perform the simulations described in this report. The RDE-A case corresponds to the studies conducted by Schwer and Kailasanath 7, 9 8 of 2

9 shows results for N x = 2 and t = 9 s, which allows the effects of the temporal resolution to be examined. Figure 2 shows that for N x 5, the simulations provide good agreement with CJ theory, which predicts the CJ temperature T CJ = 2942 K, CJ pressure p CJ = 5.56 atm, left wall pressure p w = 5.9 atm, and detonation velocity S CJ = m/s. For N x = 2, however, the solution begins to oscillate for t = 8 s; decreasing this time-step to t = 9 s reduces these oscillations. The profiles shown in Figure 2 are in good agreement with results presented in [] for a one-dimensional detonation wave under similar conditions. Pressure x Tem p er at u r e x Velocity x x Figure 2: One-step kinetics profiles of pressure (left), temperature (center), and horizontal (x) velocity (right) for a onedimensional detonation wave at t =. ms. The horizontal dashed lines in the pressure panel show p w = 5.9 atm and p CJ = 5.56 atm. Lines correspond to N x = 2, t = 8 s (cyan), N x = 4, t = 8 s (green), N x =, t = 8 s (red), N x = 2, t = 8 s (blue), and N x = 2, t = 9 s (black). We show profiles for a two-step reaction detonation wave at t =.2 ms in Figure 3. We fit the induction parameter model from the numerical results of [23] and tuned a constant-rate reaction parameter model to best match CJ theory, as in [7]. The parameters for this reaction model are c = t react = 6 s, q = J/kg. (37) Figure 3 shows results for a single spatial resolution, corresponding to grid size N x = 5 and x = 4 m, roughly one order of magnitude greater than the thickness of a real shockwave. We ran simulations at other spatial resolutions and temporal resolutions, with very close agreement between the simulations. The profiles shown in Figure 3 are in good agreement with results presented in [7], including the oscillation observed near the head of the detonation tube. Note that while the two-step induction-reaction model adds a great deal of computational cost compared to the one-step Arrhenius model, larger time steps are possible for a given resolution, recovering the imposed cost Pressure 3 2 Temperature 3 2 Velocity Figure 3: Two-step kinetics profiles of pressure (left), temperature (center), and horizontal (x) velocity (right) for a onedimensional detonation wave at t =.2 ms. The horizontal dashed lines in the panels show p w = 5.9 atm, p CJ = 5.56 atm, and T CJ = 2942 K as predicted by CJ theory. Solid lines correspond to the simulation with x = 4 m and t = 2. 9 s. The resulting detonation wave-speed was S det = m/s. III.B. Two-Dimensional Detonation We examined a two-dimensional detonation chamber (2D PDE) to verify and validate both reaction models in two dimensions, and with the MPI parallelism, using well-established experimental results regarding cellular 9 of 2

10 detonation structure. We stimulated the cellular detonation structure by applying a random noise field to the initial conditions of the hot driver gas. The code generates, scales, and subtracts a pseudo-random number field from the pressure and density fields. Figure 4 shows results for the one-step reaction model with the noise varied from -2% in 5% intervals. In the case of hydrogen-air detonations, we expect an average cell size of 8 mm, with a high-degree of irregularity due to the Nitrogen diluent, as seen in experiment smoke foils. The detonation chamber in Figure 4 is 3 cm wide, and therefore we expect 4 cells to span the field. The simulations in Figure 4 used a spatial resolution of 5 5 m and one-step kinetics. Figure 5 shows the results from two-step reaction simulations, wherein we kept the level of noise constant at 2% and used two spatial resolutions, x = 2 4 and 4 m. The higher-resolution simulation shows all the hallmarks of experimental studies and other high-resolution studies, namely, the cellular structure begins uniformly and small, with a fish-scale shape, and transitions to an irregular structure after a short period. Figure 4: Fields of pressure (left column) and x-momentum (right column) in a one-step reaction PDE model for no heterogeneity in the fuel mixture (top), 5% heterogeneity in the fuel (second row), % heterogeneity in the fuel (third row), and 2% heterogeneity in the fuel (bottom row)..4 Max Pressure Max Pressure Figure 5: Traces of maximum pressure in a two-step reaction PDE model with 2% heterogeneity in the initiation mixture for x = 2. 4 m (top) and x =. 4 m (bottom). IV. Rotating Detonation Engine Simulations We list the physical parameters of the rotating detonation engine examined in Table, which correspond to the cases studied in [6 8]. All simulations reported here used Arrhenius reaction kinetics. of 2

11 IV.A. Idealized Fuel Injection We simulated the idealized injection case with a spatial resolution of m and with two different temporal resolutions, t = 9 and 5 9 s. We ran the case for roughly 25 revolutions of the detonation wave, resulting in the unrolled density, momentum, pressure, and temperature fields shown in Figure 6. The fields have reached an approximate steady state (in the phase-averaged sense), although some local and instantaneous variations in the fields occur due to the Kelvin-Helmholtz eddies shed from the detonation wave..8 Density ( 2 kg/m 3 ) 8.8 Density gradient x-momentum (-6,5 kg/m 2 s) Pressure ( 7 atm) y-momentum (-3 28 kg/m 2 s) Temperature (2 35 K) Figure 6: Fields of density, x-momentum, pressure, and temperature for the idealized injection 2D RDE-A simulation (see Table ) at T = 5.3 ms Figure 7 shows time-series of the density, x-velocity, pressure, and temperature for the exit plane of the idealized RDE chamber at location x = L x /2. Despite some variability in the peak values, we obtained a fairly stable periodic cycle after approximately ms. The spectra for these time series in Figure 8 reveal strong peaks near 4 khz, corresponding to the time taken for the detonation wave to make one complete cycle of the cylindrical chamber. The harmonics of this peak are also evident in each of the panels in Figure 8. Both the density and temperature spectra in Figure 8 have relatively strong low frequency modes compared to the peak at the detonation wave frequency. IV.B. Discrete Fuel Injection We modeled the discrete fuel injection in 3D for the RDE-A and RDE-B conditions outlined in Table. We simulated these cases with a spatial resolution of.83 4 m and a temporal resolution of t = 5 9 s. Verification simulations in 2D indicated that significant flame-holding might be present in the 3D model with Arrhenius kinetics. In order to determine the effect of inter-nozzle spacing on the flame-holding, we restarted three runs from ideal injection conditions: the RDE-A case with 5 nozzles (herein run ) and nozzles (run 2), and the RDE-B case corresponding to an engine at :2 linear scale with nozzles (run 4). Run 4 doubled both the number of nozzles per unit length as well as the linear mesh resolution. Therefore, to control for the effect of doubling the linear mesh resolution, we started another run with the original mesh resolution restored (run 5). Furthermore, in order to determine if the restart from ideal conditions has a of 2

12 Density (kg/m 3 ) Pressure (atm) x-velocity (m/s) Temperature (K) Figure 7: Time series of the density, x-velocity, pressure, and temperature during the idealized injection RDE simulation at the exit plane for x = L x /2 m.. Fluid De nsity 5 x-velocity Spectral Density (f/hz) Frequency (/s) Spectral Density (f/hz) Frequency (/s) Spectral Density (f/hz).5.5 Pre ssure Frequency (/s) Spectral Density (f/hz) Temperature Frequency (/s) Figure 8: Frequency spectra of the density, x-velocity, pressure, and temperature during the idealized injection RDE simulation at the exit plane for x = L x /2 m, 2 khz sampling frequency. 2 of 2

13 significant effect on engine operation, we started two runs from scratch with discrete injection: an RDE-A case with nozzles (run 3), and an RDE-B case with nozzles and the lower resolution (run 6). We show spatial fields of instantaneous temperature at the centerline of the RDE chamber near the end of runs -6 in Figure 9. Subfigure 9(a) shows that run, with the largest separation between nozzles, produces extensive flame-holding and an extremely weak detonation wave which often breaks down into a broad deflagration wave. Subfigures 9(b), 9(c), and 9(d) show that runs 2-4 all converge to steady state solutions with multiple detonation waves, which form spontaneously, while Subfigures 9(e) and 9(f) demonstrate how runs 5 and 6 maintain a tenuously stable solution of four and three waves, respectively, which do not form until the last.5 ms of the simulations. Prior to this pseudo-stable period, runs 5 and 6 are characterized by the continual extinction and ignition of detonation waves, traveling in both directions, with several detonation wave collisions causing periods of very little combustion, a condition Schwer and Kailasanath have seen previously..8 Temperature (5 275 K).8 Temperature (5 275 K) (a) Run (b) Run 2.8 Temperature (5 275 K).8 Temperature (5 275 K) (c) Run 3 (d) Run 4 Temperature (5 275 K) Temperature (5 275 K).. Figure 9: (e) Run 5 (f) Run 6 Centerline instantaneous fields of temperature for runs -6 of the 3D RDEs with discrete injection. Formally, when the characteristic length-scale of the nozzle-generated flow is the same order of magnitude as the characteristic length-scale of numerical diffusion, which is resolution-dependent, the inflow becomes highly non-uniform (runs 5-6). This non-uniformity in turn causes auto-ignition of new detonation waves in regions of high pressure and fuel density, and extinction of existing detonation waves in regions of low pressure and fuel density. When the length-scale of the nozzle-generated flow is much larger than the length-scale of numerical diffusion (run ), then deflagration becomes the dominant combustion mode, extinguishing detonation waves by depriving them of necessary fuel. Stable operation is achieved when inter-wave deflagration (flame-holding) consumes fuel more slowly than the mass inflow rate. This appears to favor the creation of multiple detonation waves, which increases the frequency of detonation, thereby decreasing the build-up of 3 of 2

14 deflagration pressure and temperature. When the time between passing detonation waves at a point is too large, deflagration will build up local pressure at that point until a new detonation wave is ignited. If this occurs directly in front of an existing detonation wave, coupled extinction and ignition occurs, making it appear as if the existing detonation wave has jumped forward. By these two processes of wave creation and wave-jumping, the detonation waves self-regulate and form a stable state of detonation (runs 2-4). We chose run 2 as the best case for further study, and continued the run through roughly seven stable revolutions of its three detonation waves. Figure presents spatial fields of instantaneous density, momentum, pressure, and temperature at the end of the simulation. D e n s i ty ( 2 k g/ m 3 ) D e n s i ty g r a d i e n t y (m ) y (m ) x (m ) x - M o m e n t u m ( k g / m 2s ).22 x (m ) y - M om e n t u m ( k g /m 2 s ) y (m ) y (m ) x (m ) P r e s s u r e ( 7 at m ) Te m p e r at u r e ( K ) y (m ).8 y (m ).22 x (m ) x (m ) x (m ) Figure : Fields of density, x-momentum, pressure, and temperature for the RDE-A simulation with discrete injection (see Table ) at T = 4.55 ms. For the purposes of quantifying performance and unsteadiness, we wrote the two-dimensional inflow and outflow planes of run 2 to file at a rate of 2 MHz and output single-point data to file at a very high frequency of 4 MHz. The detonation wave speed, Sdet, is approximately 8 m/s, corresponding to a Courant number, Co, of.49, a cell-crossing time of.8 7 s, and a Nyquist frequency of approximately MHz for the spatial and temporal discretization listed in Table. The single-point data is sampled at 4 times the Nyquist frequency and should therefore be free of aliasing. We show the time series and frequency spectrum for density, velocity, pressure, and temperature in Figures and 2, calculated from the singlepoint data. We used the boundary plane data to calculate engine performance. The first three peaks in Figure 2 correspond to the fundamental engine cycle ( 4 Hz), and one-half ( 82 Hz) and one-third cycle periods ( 23 Hz), corresponding to signals generated by two detonation waves and three detonation waves, respectively. Since the 4 MHz data is free of aliasing, we can be sure the higher frequency peaks are a property of the flow, and not ringing from the fundamental detonation frequencies. IV.C. Engine Performance We show the hydrogen mass flow rate, thrust, and specific impulse in Figures 3 and 4. The primary performance metric of interest is the specific impulse, Isp (t), for an air-breathing engine. We calculated the 4 of 2

15 .2 6 Density (kg/m 3 ) x-velocity (m/s) Pressure (atm) Temperature (K) Figure : Time series of the density, x-velocity, pressure, and temperature at the exit plane for x = L x/2 during the discrete-injection RDE simulation. Spectral Density (f/hz) Fluid Density Frequency (/s) Spectral Density (f/hz) Velocity x Frequency (/s) Spectral Density (f/hz) Pressure Frequency (/s) Spectral Density (f/hz) Temperature Frequency (/s) Figure 2: Frequency spectra of the density, x-velocity, pressure, and temperature at the exit plane for x = L x /2 during the discrete-injection RDE simulation, 4 MHz sampling frequency. 5 of 2

16 specific impulse for an air-breathing engine from the mass flow rate of fuel into the engine, ṁ f (t), in this case hydrogen, and the propulsive thrust, F (t), at the exit of the engine. For the purposes of comparing performance to the investigation in [7], we assumed the two-dimensional inflow and outflow boundaries extend. cm in the z-direction. The hydrogen mass flow rate is defined as ṁ H2 (t) = ρ H2 v da. (38) Ω We calculated the density of hydrogen, ρ H2, from the mass fraction of hydrogen in the stoichiometric hydrogen-air reactant mixture: We calculated the propulsive thrust as F (t) = 2H 2 + O N 2, ρ H2 =.28326ρ r. (39) Ω [ ] ρv 2 + (P P b ) da, (4) where P b is the back pressure at the outflow. Once we knew the mass flow and thrust, we calculated the specific impulse: I sp (t) = F (t) g o ṁ H2 (t). (4) Mass Flow Rate (g/s) Thrust Force (N) Specific Impulse (s) Figure 3: Hydrogen mass flow rate, thrust, and specific impulse for the RDE with idealized fuel injection for the full duration of the simulation with t = 5 9 s and output at MHz. Mass Flow Rate (g/s) Thrust Force (N) Specific Impulse (s) Figure 4: Hydrogen mass flow rate, thrust, and specific impulse for the RDE with discrete fuel injectors for the last seven engine cycles, with boundary plane output at 2 MHz. V. Turbulence Characterization In order to characterize the mean and turbulent flow fields, we temporally phase-averaged the simulation outputs. That is, for each 2D domain snapshot (or 3D domain pre-averaged in the z-direction), the postprocessing code determined the location of the detonation wave(s) and shifted the domain to center the wave in each snapshot. We obtained the mean fields by time-averaging the phase-shifted snapshots. We then obtained the turbulent fields by subtracting the mean field from each instantaneous, phase-shifted field. We calculated the Reynolds stress tensor, u i u j, from the turbulent velocity fields, u i, and then time-averaged. We calculated the mean rate of strain tensor from the gradients of the mean velocity fields, u i, ( ) S ij u i + u j. (42) 2 x j x i 6 of 2

17 .8 Phase- averaged pressure ( 7 atm) 6.8 Phase- averaged temperature (2 35 K) Phase- averaged x-velocity, (-5 5 m/s) 5.8 Phase- averaged y-velocity, (- 5 m/s) Figure 5: Phase-averaged pressure, temperature, x-velocity, and y-velocity for the 2D RDE-A case (see Table )..8 Reynolds Stress u u ( 7. m 2 /s 2 5 ).8 Reynolds Stress v v ( 3.8 m 2 /s 2 5 ) Reynolds Stress u v ( m 2 /s 2 5 ) Turbulence Kinetic Energy k ( 5. m 2 /s 2 5 ) Figure 6: Phase-averaged Reynolds stresses u u, v v, u v, and turbulence kinetic energy k for the 2D RDE-A case (see Table ) Figure 5 shows the phase averaged pressure, temperature and velocities for the 2D RDE-A case outlined in Table. Figure ( 6 shows the turbulence statistics, with increased Reynolds stresses and turbulence kinetic energy k 2 u 2 + v 2 + w 2) at the shock wave and in the vicinity of the Kelvin-Helmholtz eddies that trail the shock and detonation waves. We show similar mean flow and turbulence fields in Figures 7 and 8 for the discrete injection case (see Table ), where the averaging is additionally carried out over the z direction. The presence of discrete inflow nozzles in this case results in substantially greater unsteadiness in the flow and hence somewhat less crisp phase-averaged quantities. A particularly interesting aspect of the unsteadiness generated by the discrete injection nozzles is that the detonation waves oscillate back-and-forth in relation to each other, similar to a mass-spring system, and wax and wane in strength as they recede and approach the wave ahead. This oscillation is best seen in the short-duration, phase-averaged pressure and temperature fields as a blurring of the detonation fronts of the outer waves (Figure 7). Furthermore, strong phase-averaging artifacts can be seen in Figure 8, which occur because the phase-averaging algorithm was tasked with centering the strongest wave, leading the algorithm to pick different waves at different instances. These relative wave oscillations 7 of 2

18 complicate the process of calculating the turbulence statistics of the entire domain, as they introduce a time dependence to the phase-averaging process, which should ideally be time-independent. P h as e - A v e r ag e d P r e s s u r e ( a t m ) P h as e - A v e r ag e d Te m p e r a t u r e ( K ) P h a s e - av e r a g e d x - v e l o c i ty ( m /s ) P h as e - av e r ag e d y - v e l o c i ty ( m /s ) Figure 7: Phase- and z-averaged pressure, temperature, x-velocity, and y-velocity for the 3D RDE-A case (see Table ). R e y n o l d s S t r e s s u u ( 6. 8 m 2/ s 2 5) R e y n ol d s S t r e s s v v (. 8 m 2 / s 2 5 ) Tu r b u l e n c e K i n e t i c E n e r gy k ( 3. 6 m 2 / s 2 5 ) R e y n o l d s S t r e s s u v ( m 2 / s 2 5 ) Figure 8: Phase-averaged Reynolds stresses u u, v v, u v, and turbulence kinetic energy k for the 3D RDE-A case (see Table ). We used the phase-averaged turbulence statistics to test a common RANS modeling assumption, the gradient transport hypothesis, also known as the Boussinesq hypothesis! 2 u u 2 i j ui uj = kδij νt + = kδij 2νT S ij. (43) 3 xj xi 3 The hypothesis states that the sub-grid scale Reynolds stresses are proportional to the mean rates of strain, with the non-linear coefficient of proportionality, νt, known as the eddy viscosity. Popular RANS models, such as the k ε and k ω models, assume a form for the eddy viscosity and then calculate the sub-grid scale stresses using the Boussinesq hypothesis. Since the Boussinesq hypothesis is not an assumption of implicit large eddy simulations as conducted in this report, we can test the hypothesis a priori by calculating the eddy 8 of 2

19 viscosity field from the turbulence statistics. We calculated a scalar eddy viscosity field for both injection cases from the phase-averaged data using the definition ν T = k 2 where a ij is the Reynolds stress anisotropy tensor given by a ij S ij S ij S ij, (44) a ij u i u j k 2 3 δ ij. (45) We show spatial fields for the xy-component of anisotropy and mean rate of strain in Figures 9 and 2, and the computed eddy viscosities are shown in Figures 2 and 22. We can see from comparison of a 2 and S 2 in these figures that there is an approximate correspondence between large magnitudes of a 2 and large (oppositely signed) magnitudes of S 2, particularly near the shock wave. For the 2D case in Figure 9, ν T has its largest non-zero values in the Kelvin-Helmholtz instability region, although there is a significant probability of negative ν T. These negative values are indicative of a break-down of the Boussinesq hypothesis due to a non-proportionality of a ij and S ij. Negative values of ν T are even more prominent for the 3D case shown in Figure 2, where ν T < at many of the locations behind the secondary shock and detonation waves. In the future, specific models for ν T based on variables such as k, ε, and ω will be additionally tested..8.9 Anisotropy a Figure 9: Rate of Strain, S 2, ( s 6 ) xy-components of the anisotropy and mean rate of strain tensors for the RDE with idealized fuel injection Eddy Viscosity, νt = k(aijsij)/(2sijsij) Figure 2: Eddy viscosity calculated according to the gradient transport hypothesis for the RDE with idealized fuel injection. ν T =.5665, σ =.86, max(ν T ) = 53.88, min(ν T ) = Anisotropy a2.8 Rate of Strain, S 2, ( s 5 ) Figure 2: xy-components of the anisotropy and mean rate of strain tensors for the RDE simulation with discrete fuel injection. 9 of 2

20 .8 Eddy Viscosity, νt = k(aijs ij)/(2s ijs ij) Figure 22: Eddy viscosity calculated according to the gradient transport hypothesis for the RDE with discrete fuel injectors. ν T =.366, σ =.249, max(ν T ) = 97.49, min(ν T ) = VI. Conclusions We performed a preliminary characterization of turbulence in an RDE, including time series and spectra at the outflow of the detonation chamber. Future work will further characterize the turbulent flow fields in the RDE, particularly for discrete nozzle designs where there is substantial spatial and temporal unsteadiness. Accurate consideration of this unsteadiness requires three-dimensional simulations, more advanced chemical reaction mechanisms, inclusion of centrifugal forces, and more complex geometries. A particular area of interest is in understanding how and when additional detonation waves auto-ignite in the nozzle cases, and what effects these multiple waves have on RDE performance. Finally, future work will also evaluate the accuracy of additional RANS and LES subgrid-scale models. Acknowledgments The authors gratefully acknowledge support from the Defense Advanced Research Projects Agency (DARPA). This work utilized the Janus supercomputer, which is supported by the National Science Foundation (award number CNS-82794), the University of Colorado Boulder, the University of Colorado Denver, and the National Center for Atmospheric Research (NCAR). The Janus supercomputer is operated by the University of Colorado Boulder. We would also like to acknowledge high-performance computing support from Yellowstone (ark:/8565/d7wd3xhc) provided by NCAR s Computational and Information Systems Laboratory, sponsored by the National Science Foundation. 2 of 2

21 References G. D. Roy, S. M. Frolov, A. A. Borisov, and D. W. Netzer. Pulse detonation propulsion: challenges, current status, and future perspective. Prog. Energy. Combust. Sci., 3: , K. Kailasanath. The Rotating-Detonation-Wave Engine Concept: A Brief Status report. AIAA Paper, 2-58, 2. 3 K. P. Rouser. Unsteady specific work and isentropic efficiency of a radial turbine driven by pulsed detonations. PhD Dissertation, E. M. Braun, F. K. Lu, D. R. Wilson, and J. A. Camberos. Airbreathing rotating detonation wave engine cycle analysis. Aerospace Science and Technology, M. Hishida, T. Fujiwara, and P. Wolanski. Fundamentals of rotating detonations. Shock Waves, 9:, D. A. Schwer and K. Kailasanath. Numerical Investigation of Rotating Detonation Engines. AIAA Paper, 2-688, 2. 7 D. A. Schwer and K. Kailasanath. Numerical investigation of the physics of rotating-detonation-engines. Proc. Comb. Inst., 33: , 2. 8 D. A. Schwer and K. Kailasanath. Effect of Inlet on Fill Region and Performance of Rotating Detonation Engines. AIAA Paper, 2-644, 2. 9 D. A. Schwer and K. Kailasanath. Fluid dynamics of rotating detonation engines with hydrogen and hydrocarbon fuels. Proc. Comb. Inst., 34:99 998, 23. T.-H. Yi, C. Turangan, J. Lou, P. Wolanski, and J. Kindracki. A Three-Dimensional Numerical Study of Rotational Detonation in an Annular Chamber. AIAA Paper, , 29. T.-H. Yi, J. Lou, C. Turangan, J.-Y. Choi, and P. Wolanski. Propulsive Performance of a Continuously Rotating Detonation Engine. J. Propulsion Power, 27:7 8, 2. 2 S. M. Frolov, A. V. Dubrovskii, and V. S. Ivanov. Three-dimensional numerical simulation of the operation of the rotating-detonation chamber. Russian Journal of Physical Chemistry B, 6(2): , S. M. Frolov, A. V. Dubrovskii, and V. S. Ivanov. Three-dimensional numerical simulation of the operation of a rotatingdetonation chamber with separate supply of fuel and oxidizer. Russian Journal of Physical Chemistry B, 7():35 43, K. Kailasanath and G. Patnaik. Performance Estimates of Pulsed Detonation Engines. Proc. Comb. Inst., 28:595 6, 2. 5 K. Kailasanath, G. Patnaik, and C. Li. The Flowfield and Performance of Pulse Detonation Engines. Proc. Comb. Inst., 29: , X. He and A. R. Karagozian. Numerical Simulation of Pulse Detonation Engine Phenomena. J. Sci. Comp., 9:2 224, V. F. Nikitin, V. R. Dushin, Y. G. Phylippov, and J. C. Legros. Pulse detonation engines: Technical approaches. Acta Astro., 64:28 287, Y. G. Phylippov, V. R. Dushin, V. F. Nikitin, V. A. Nerchenko, N. V. Korolkova, and V. M. Guendugov. Fluid mechanics of pulse detonation thrusters. Acta Astro., 76:5 26, J. Kindracki, P. Wolanski, and Z. Gut. Experimental research on the rotating detonation in gaseous fuels oxygen mixtures. Shock Waves, 2:75 84, 2. 2 J. P. Boris, A. M. Landsberg, E. S. Oran, and J. H. Gardner. LCPFCT-A Flux-Corrected Transport Algorithm for Solving Generalized Continuity Equations. NRL Tech. Report, NRL/MR/ , F. Ma, J.-Y. Choi, and V. Yang. Thrust Chamber Dynamics and Propulsive Performance of Single-Tube Pulse Detonation Engines. J. Propulsion Power, 2:52 526, M. Sichel, N. A. Tonello, E. S. Oran, and D. A. Jones. A two step kinetics model for numerical simulation of explosions and detonations in H 2 -O 2 mixtures. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 458(27):49 82, T.L. Burks and E. S. Oran. A computational study of the chemical kinetics of hydrogen combustion. memorandum report. Technical report, Naval Research Lab., Washington, DC (USA), M. H. Lefebvre, E. S. Oran, K. Kailasanath, and P. J. Van Tiggelen. The influence of the heat capacity and diluent on detonation structure. Combustion and flame, 95():26 28, D. A. Jones, G. Kemister, N. A. Tonello, E. S. Oran, and M. Sichel. Numerical simulation of detonation reignition in H 2 -O 2 mixtures in area expansions. Shock Waves, ():33 4, 2. 2 of 2