THREE-DIMENSIONAL NUMERICAL SIMULATION OF A CONTINUOUSLY ROTATING DETONATION IN THE ANNULAR COMBUSTION CHAMBER WITH A WIDE GAP AND SEPARATE DELIVERY OF FUEL AND OXIDIZER

The possibility of integrating the Continuous Detonation Chamber (CDC) in a gas turbine engine (GTE) is demonstrated by means of three-dimensional (3D) numerical simulations, i. e., the feasibility of the operation process in the annular combustion chamber with a wide gap and with separate feeding of fuel (hydrogen) and oxidizer (air) is proved computationally. The CDC with an upstream isolator damping pressure disturbances propagating towards the compressor is shown to exhibit a gain in the total pressure of 15% as compared with the same combustion chamber operating in the de§agration mode.


INTRODUCTION
Modern power plants in aircraft are mainly GTEs utilizing Brayton thermodynamic cycle.A similar (constant-pressure combustion) cycle is widely used in liquid rocket engines (LREs).For many decades, GTEs and LREs were continuously improved and further improvement requires large capital investments.An alternative solution to signi¦cantly improve the thermodynamic e©ciency of modern GTEs and LREs is the use of the combustion chambers with the total pressure gain.Increasing the total pressure in the combustion chamber can be provided by an increase in the burning rate of a mixture of fuel with the oxidant and/or by changing the combustion mode.The most attractive combustion mode in terms of its thermodynamic e©ciency is the detonation [1,2].In a detonation wave, the chemical energy stored in the fuel is released with the extremely high rate in a very thin layer of shock-compressed mixture.There are two Figure 1 Architecture of the GTE operating on continuously rotating detonations basic schemes of detonation combustion: in periodic detonation waves traveling along the combustion chamber (pulse detonation chambers, PDCs [3,4]) and in a detonation wave continuously circulating in a tangential direction across the combustion chamber (CDCs [2,5,6]).These schemes are currently considered promising for both air-breathing and rocket engines [515].
In 2010, the Centre for Pulse Detonation Combustion at Semenov Institute of Chemical Physics of the Russian Academy of Sciences launched an ambitious project aimed at developing scienti¦c grounds for designing CDCs for power engineering and aerospace applications.The project implied the development of the computer code that allows full-scale 3D simulation of the operation process in CDCs of di¨erent design.
In [11,14], with this code the operation process in the annular CDC operating on the homogeneous stoichiometric hydrogenair mixture fed through the injector head with two annular openings of relatively small width (∼ 7 mm) was thoroughly investigated.It was shown that the CDC is the combustion chamber with the total-pressure gain: in the calculations, the total pressure in the chamber was increased by 10%14%.The calculations of [11,14] were mainly focused on the §ow characteristics in the input and output devices of the CDC referred to as upstream and downstream isolators separating the CDC from the compressor and turbine of the GTE, respectively (Fig. 1).It turned out that the chosen annular geometry of the upstream and downstream isolators did not provide proper damping of pressure disturbances generated by the detonation once they reach the GTE compressor and turbine.The estimated amplitudes of the pressure disturbances after passing the isolators attained very high values of 40%45% P in and 30%35% P in , respectively, where P in is the pressure behind the last stage of the GTE compressor.
In [15], a 3D numerical simulation of the operation process in an annular CDC with separate supply of fuel components ¡ hydrogen and air ¡ was performed.The design of the CDC and the main operational parameters were the same as in experiments [5,6].The air was fed to the chamber axially through the 2-millimeter wide annular gap, whereas hydrogen was fed through multiple radial holes of submillimeter size equally distributed along the circumference of the outer CDC wall.The calculations took into account the e¨ects associated with the ¦nite rates of molecular and turbulent mixing of fuel components with each other and with the detonation products, and with the ¦nite rates of chemical transformations.The calculation results were shown to be generally consistent with the experimental observations in terms of all the integral characteristics of the operation process (mean static pressure in the CDC, mass §ow rates of fuel components, the height of the detonating layer, etc.).
This article is the continuation of studies [11,14,15].The objective of the present work is to prove the feasibility of integrating the CDC into the GTE, i. e., to prove the possibility of arranging the operation process in an annular CDC with a wide gap (comparable to the height of the last stage of the GTE compressor blades) with separate supply of fuel and oxidizer.

PROBLEM FORMULATION
Figure 2 shows the schematic of the CDC with an annular gap of width δ = 23 mm equipped with the upstream isolator in the form of the expansion chamber with the maximum width -.The outer diameter of the CDC is 306 mm.The detailed design of the isolator is a subject of patent [16] and is not discussed here.Oxidant (air) under static pressure P in and temperature T in is fed in the CDC through the isolator axially.Fuel (hydrogen) is supplied to the CDC from the fuel reservoir (not shown in Fig. 2) through 80 radial injector bores: 40 in the outer wall and 40 in the inner wall.Pressure and temperature of hydrogen in the fuel reservoir are kept constant and equal to P f and T f , respectively.Downstream of the CDC, a provision is made for a divergent nozzle formed by a conical central body (cone half-angle is 10 • ) and a large bu¨er volume (not shown in Fig. 2) exceeding considerably the volume of the chamber and used for avoiding parasitic numerical re §ections.
The main tasks attacked in this work are: (i) to design the CDC ensuring continuous detonative combustion with an appreciable gain in total pressure and (ii) to design the upstream isolator of the CDC which could provide e¨ective damping of pressure disturbances propagating towards the GTE compressor.
The physicochemical processes in the CDC are simulated using the mathematical model described in detail in [11].Here, the authors limit themselves to a brief description of its main features.The §ow of a viscous compressible gas in the CDC is described using the 3D unsteady Reynolds-averaged NavierStokes (URANS), energy, and species conservation equations for a multicomponent mixture.The turbulent §uxes of species, momentum, and energy are modeled within the framework of the standard kε turbulence model for a compressible §ow.Given that all physicochemical processes in the CDC occur in a very short time, the contribution from the frontal (laminar and/or turbulent) combustion to the chemical sources in the equations of conservation of energy and components of the mixture is neglected.The contributions of these reactions to the bulk (volumetric) chemical sources are determined using the particle method (PM) [11].The most important advantage of the PM is its ability to accurately determine the rates of chemical reactions in a turbulent §ow without invoking any hypothesis about the in §uence of turbulent §uctuations of the temperature and concentrations of the reactants on the mean rate of the reaction.In the PM algorithm, the instantaneous local states of a turbulent reacting §ow are represented as a set of interacting (Lagrangian) particles.Each particle has its individual properties: the position in space, three velocity components, volume, density, temperature, mass fractions of chemical species, and statistical weight, which is used to determine the mean values of the variables over the ensemble of particles.For each particle, the system of equations of conservation of mass of the species, momentum, and energy is solved; the §ux (transport) terms are calculated using the classic models of linear relaxation to the mean [17].The equations of the model are closed by the caloric and thermal equations of state of a mixture of ideal gases with variable speci¦c heats, as well as by the initial and boundary conditions.All the thermophysical parameters of the gas are considered variable.
Numerical solution of the governing equations of the problem is carried out using the coupled algorithm ¤Semiimplicit method for pressure linked equations (SIMPLE) Monte Carlo method.¥The chemical sources are calculated by an implicit scheme with internal time stepping.The coupled algorithm was previously used to simulate §ame acceleration and de §agration-to-detonation transition in smooth tubes and in tubes with obstacles [18,19], as well as to solve the problems of shock-initiated autoignition and pre §ame ignition in con¦ned spaces [20].In all cases, satisfactory agreement between the results of calculations and experiments were observed.In addition, this algorithm was used to solve the problem of the limits of existence of detonation in the CDC operating on homogeneous hydrogenair mixture.As in [11,14,15], the oxidation of hydrogen was described by a single-step reaction scheme: The rate of hydrogen oxidation at elevated pressures P (5 to 40 atm) and temperatures T (11002000 K) was calculated by the formula: where P is the pressure and [S] denotes the concentration of species S. Equation (2) was obtained by ¦tting the dependences of the induction period on the pressure and temperature obtained for reaction (1) to those calculated within the framework of an extensively tested detailed kinetic mechanism of hydrogen oxidation [21,22].Note that the heat of reaction (1) was modi¦ed to make the calculated ChapmanJouguet detonation velocity D CJ for the stoichiometric hydrogenair mixture be consistent with its thermodynamic value (D CJ ≈ 1970 m/s).The boundary conditions for the average §ow velocity, pressure, temperature, turbulent kinetic energy and its dissipation rate, and mean concentrations of chemical species on the solid walls of the CDC are set using the formalism of wall functions on the assumption that the walls are isothermal (T w = 293.15K), impermeable, and noncatalytic, with noslip properties.
The inlet boundary conditions in air and hydrogen reservoirs are taken in the form of ¦xed values of the pressure, temperature, turbulent kinetic energy and its dissipation rate, as well as ¦xed values of the average concentrations of oxidant and fuel, respectively.
At the boundaries of the bu¨er volume attached to the CDC nozzle, the von Neumann condition, grad(P ) = 0, is set.The rest of the variables (velocity, temperature, turbulent kinetic energy and its dissipation rate, and the concentrations of components) are extrapolated to these boundaries from the computational domain.Special calculations demonstrated that the speci¦ed boundary conditions at the bu¨er volume boundaries produce no e¨ect whatsoever on the solution.
The boundary conditions for the particles (the components of the velocity vector and scalar variables) on the solid walls of the CDC and the open boundaries of the computational domain are formulated in such a way that they are consistent with the boundary conditions for the mean values of the relevant variables.This consistency is continuously monitored by comparing the values of the variables obtained by averaging over the ensemble of particles in the computational mesh with the average values of the same variables obtained by solving the averaged §ow equations.
The initial conditions for the average parameters of the §ow are formulated as follows.It is assumed that at initial time t = 0, the air and hydrogen reservoirs are ¦lled with air and hydrogen under static pressures P in and P f , respectively, and the rest of the region is ¦lled with quiescent air at atmospheric pressure.
The initial positions of the particles in the computational domain are selected using a random number generator capable of providing an on-average uniform distribution over a unit-length interval.At the initial time, each particle is characterized by a set of speci¦c values of all relevant variables consistent with the initial distributions of the corresponding average values.The nominal number of particles in the computational mesh is speci¦ed before simulation, N p = 10.Note that in the process of computations, the actual number of particles in the meshes can change (particles migrate over the computational domain).To keep the number of particles unchanged, special procedures of cloning and clustering are applied.The pattern of the §ow in the CDC is generally dependent on the chosen value of N p and on the computational grid.However, previous calculations in [11,14,15] showed that at N p > 1015, the dependence of the §ow pattern on N p becomes weak.The in §uence of the computational grid was investigated by comparing the results of calculations on di¨erent grids.
The calculation procedure is started with purging the CDC for 0.4 ms, a time long enough to form a 10-centimeter thick active layer of hydrogenair mixture over the plane of hydrogen injectors.Then, the procedure of detonation initiation is performed.This procedure amounts to a rapid burning of particles located in the initiator region, a limited-size area in the active layer.The combustion of the particles rapidly raises the pressure in the initiator region, thereby forming a shock wave.To ensure the propagation of the detonation wave in the desired direction, for example, counterclockwise, the initial distribution of particles in the CDC in the clockwise direction from the initiator region contains a layer of temporarily inert particles.Immediately after the initiation of detonation, these particles become active.
The calculations are performed on structured grids with hexagonal cells ranging in size from 0.5 to 2 mm.Despite the fact that such computational grids do not resolve the internal structure of the detonation front, the use of the PM takes into account the in §uence of longitudinal and transverse acoustic waves on chemical reactions inside the computational cell.The integration time step does not exceed 1 µs.

RESULTS
Table 1 lists the calculation conditions used in this study and indicates the resultant combustion modes.
As mentioned above, initiation of detonation in the CDC is preceded by purging the chamber with fuel components.Figure 3 shows the successive distributions of hydrogen mass fraction in the CDC cross section located at an axial distance of 50 mm downstream from the hydrogen injectors during ¦rst 180 µs of purging in Run 1 (see Table 1).It can be seen that by about 120 µs, the mean hydrogen mass fraction in this cross section is close to stoichiometric (0.028), but the mixture is periodically strati¦ed: mixture layers enriched with fuel are interspersed with fuel-lean layers.
About 34 ms after the detonation initiation, a periodic operation mode with a transverse detonation wave propagating continuously in the annular gap at the average velocity of ∼ 1850 m/s is established in the CDC in Run 1 (referred to as ¤continuous detonation¥ mode in Table 1).The settled frequency of detonation rotation in the chamber is approximately equal to 2 kHz.  Figure 4 shows the calculated instantaneous static pressure (Fig. 4a) and temperature (Fig. 4b) distributions in the vicinity of the CDC outer wall in the periodic operation mode with a single detonation wave (Run 1).Detonation propagates from right to left.It is seen that the hydrogenair mixture ahead of the detonation wave is periodically strati¦ed not only in terms of the composition (see Fig. 3), but also in terms of temperature (see Fig. 4b), as air enters the upstream isolator with temperature 550 K and hydrogen, expanding into the injector holes, is cooled to a temperature of ∼ 200 K.In addition to the periodic strati¦cation of the mixture in terms of temperature and composition in the CDC cross section, the calculations show a substantial variation in the temperature and composition of the hydrogenair mixture in the axial direction: along the mixture layer ahead of the detonation wave.For this reason, the detonation wave possesses a highly curved front with its upper part displaced ahead of its lower part by about 2δ and with the apparent front height of approximately 4.5δ (∼ 100 mm).A very similar continuous detonation mode is obtained in Run 2 (see Table 1) but a large amount of unburned hydrogen exists at the CDC outlet due to a highly fuel-rich overall mixture composition.
For the sake of comparison, Figs. 5 and 6 show the calculated instantaneous static pressure (see Fig. 5a) and temperature (see Figs. 5b and 6) distributions in the vicinity of the outer wall under conditions of stabilized de §agration (diffusion combustion) in the same annular chamber (Runs 3 and 4 in Table 1).In Run 3 (see Fig. 5), with the same calculation conditions as in Run 1, the stabilized de §agration rather than continuous detonation mode is readily obtained by igniting the mixture evenly by a hot ignition source all throughout the area over hydrogen injectors.In Run 4 (see Fig. 6), the de §agration mode is obtained in the course of process evolution as a result of detonation failure and gradual establishment of the stabilized de §agration mode.A similar system evolution was observed in Run 5.In the stabilized de §agration mode, combustion covers the entire cross section of the chamber and the static pressure and temperature decrease monotonically toward the nozzle.
Runs 6 and 7 in Table 1 represent the limiting conditions of CDC operation.In these Runs, the detonation wave once initiated is blown o¨from the CDC without further establishment of the stabilized de §agration mode.
Among all the combustion modes observed in the calculations, the continuous detonation mode of Run 1 is of most interest.Therefore, this mode is considered here in more detail.Figure 7 shows the calculated time histories of the static pressure at two points in the CDC: at a point located 50 mm above hydrogen injectors (solid curve) and at a point in the upstream isolator with -= 90 mm (dashed curve).It is seen that the static pressure in the CDC is pulsating with the maximum and minimum values attaining 3035 and 34 atm, respectively.Taking the average of the solid curve in Fig. 7 gives the average static pressure in the CDC of ∼ 8.6 atm, which is somewhat lower (by 4.4%) than the inlet air pressure P in .Nevertheless, as shown below, the CDC exhibits a gain in the total pressure.
One of the most important results of this work is the proof that using the upstream isolator of speci¦c design, one can ensure almost complete damping of pressure disturbances propagating upstream the CDC towards the last stage of the GTE compressor.Indeed, the maximum deviation of the dashed curve from the straight line in Fig. 7 is only 3% P in .
Runs 8 and 9 in Table 1 demonstrate the e¨ect of the isolator width -on its damping e©ciency.As is seen in Fig. 8, the decrease in the isolator widthfrom 90 to 45 mm results in deterioration of the isolator damping e©ciency: at P in = 7 atm, the amplitude of pressure disturbances in the isolator changes from 11.5 atm (14%21% P in ) at -= 90 mm to 26 atm (29%86% P in ) at -= 45 mm.
Figure 9 compares the calculated distributions of the cross-section-averaged total pressure along the CDC operating in the continuous detonation mode of Run 1 (solid curve) and that in the stabilized de §agration mode of Run 3 (dashed curve).The cross-section averaged total pressure p * for the quasi-stationary periodic §ow was determined as where p is the static pressure; ρ is the density; U is the length of the velocity vector; V is the volume; -t is the detonation rotation time period; and index i denotes the computational cell number in the chosen cross section.It is seen from Fig. 9 that unlike the de §agration mode in which the total pressure gradually decreases along the axis of the chamber, the detonation mode exhibits the increase in the total pressure to 10.3 atm, i. e., is higher than P in by about 15%.This is another important result of the present study.One more interesting result is depicted in Fig. 10 demonstrating the velocity ¦eld at the CDC exit.It follows from Fig. 10 that the transverse §ow direction at the CDC exit is opposite to the direction of detonation propagation due to speci¦c pressure distribution in the annular combustion chamber with a continuously rotating detonation wave.As a matter of fact, in Fig. 10, the detonation wave propagates in the clockwise direction (for the observer at the CDC top), whereas the §ow ahead of the detonation wave is directed counterclockwise.
It is also seen that the transverse velocity component is considerably smaller than the axial one with the latter exceeding the local sound velocity.

CONCLUDING REMARKS
The possibility of arranging a cyclic operation in the detonation mode of the CDC with a wide annular gap (23 mm) comparable to the height of the blades of the last stage GTE compressor and with separate delivery of fuel and oxidizer has been demonstrated computationally based on 3D URANS simulations coupled with the Particle method modeling micromixing and turbulencechemistry interaction.A 15 percent in total pressure has been obtained in the CDC.A proposed design of the upstream isolator was shown to provide almost complete damping of pressure disturbances propagating upstream from the CDC towards the compressor.
Further e¨orts will be directed towards transition from hydrogen to liquid hydrocarbon fuel and the design of a downstream isolator protecting the turbine from pressure disturbances generated in the CDC.

Figure 2
Figure 2 Schematic of the annular CDC with the upstream isolator, radial fuel injectors, and a nozzle.Dimensions are in millimeters

Figure 3
Figure 3 Instantaneous distributions of hydrogen mass fraction in the CDC cross section at the stage of chamber ¦lling with fuel components before detonation initiation in Run 1

Figure 4
Figure 4 Instantaneous distributions of static pressure (a) and temperature (b) near the outer wall of the CDC operating in the continuous detonation mode in Run 1. Detonation propagates from right to left

Figure 5
Figure 5 Instantaneous distributions of static pressure (a) and temperature (b) near the outer wall of the CDC operating in the stabilized de §agration mode in Run 3

Figure 6
Figure 6 Transition from detonation to stabilized de §agration mode in Run 4

Figure 7 Figure 8
Figure 7  Predicted time histories of static pressure in the middle of the CDC (solid curve) and in the upstream isolator with -= 90 mm (dashed curve)

Figure 9 Figure 10
Figure 9 distribution of the total pressure averaged over the cross section in the annular combustion chamber with detonation (solid curve, Run 1) and de §agration (dashed curve, Run 3)

Table 1
Calculation conditions and the resultant combustion modes a Mixture is ignited by a hot ignition source all throughout the area over hydrogen injectors.