Numerical simulation of the underexpanded plume spectral radiance using Monte-Carlo method

Numerical simulation of the tactical rocket plumes spectral radiance is performed. An influence of the chemical composition and temperature of the plume exhausts as well as the angle of sight and scattering on the solid particles is discussed. Numerical simulation results for homogeneous plume of tactical rocket are in a good agreement with published data.


Introduction
Most of the modern subsonic and supersonic vehicles produces plumes which radiate in the infrared region of the electromagnetic spectrum. Due to partial atmospheric opacity in this spectral region the radiance reaches the detector with some distortion. Particular shape of the spectrum depends on the type of the engine (solid or liquid), chemical composition of the engine fuel, radius of the nozzle exit, presence of solid particles, its phase and radius, altitude of the flight.
For this reason, numerical simulation of the plume spectral radiance is complex problem which requires the knowledge of large amount of data such as: spatial distribution of temperature and pressure in the plume, chemical composition, absorbance coefficients for different chemical species, complex refractive index of solid particles as well as physical and mathematical models [1][2][3][4][5] included different nonequilibrium processes proceeded in the plume. It is well known that the most universal algorithm to calculate spectral radiance is the Monte-Carlo method [1,6,7]. The main purpose of this work is to calculate spectral signature of the underexpanded plumes with the use of Monte-Carlo method and investigate its dependence on the plume characteristics: temperature, emitted volume, number density of solid particles and its size, imaginary part of the refractive index. 23 Al O particles The plume flowfield was modelled as a homogeneous cylinder. The cylinder size, temperature and pressure, as well as chemical composition was assumed to be the same as in [8]. It's length is equal to L = 600 cm and radius r = 10 cm . The cylinder is isothermal with temperature T = 1000 K and pressure p = 1 atm . Gas and solid particle concentrations were uniform. All the particles are spherical and have the same radius. Chemical composition of the plume and its particle properties are given in table 1. The aspect angle was assumed to be equal to 90°. The absorption and scattering coefficients of the solid particles are calculated with the use of the Mie theory. So the spectral signature of the plume with scattering particles is significantly depends on the imaginary part of the refractive index which in its turn depends on the wavelength, temperature and phase of solid particles. Comparison of the values of imaginary part of the refractive index for 23 Al O particles given by different authors [9 -15] is presented in figure 1. It can be seen that its values differ by several orders of magnitude for different temperature and phase of particles.

Spectral radiance of the plume with
The influence of the imaginary part of the refractive index on the spectral radiance of the plume is given in figure 2. Numerical simulation of the signature performed for the particle radius r = 1 mcm and different concentrations of solid 23 Al O . Two approximations of the imaginary part of the refractive index were used provided by Dombrovskii [9] and Toon [12]. With the increase of the solid particle concentration difference in the spectral radiance of the plume for data [12] and data [9]   Numerical simulation results for the matrix of solid particle concentrations and radiuses and its comparison to the data [8] are given in figure 3.
The are two sets of numerical simulation results. The first set (left column) is the total volume emission of the homogeneous plume and the second set (right column) is the emission only for a path through the diameter of the radiative cylinder. The numerical simulation results with the use the values of the refraction index provided by Toon [12] give satisfactory agreement with the data [8]. 23 BO particles As in previous section the plume flowfield was modelled as a homogeneous cylinder. The cylinder size, temperature and pressure, as well as chemical composition was assumed to be the same as in [16]. It's length is equal to L = 1500 cm and radius r = 75 cm . The cylinder is isothermal with temperature and pressure given in table 2. The temperature is assumed to be equal to T = 1000 K (variant 1) and T = 1600 K ( variant 2) and pressure p = 1 atm (both cases). Its gas and solid particle concentrations were uniform. All the particles are spherical and have the same radius. It was assumed that 23 BO are the scattering solid particles in this case. Values of the complex refractive index for 23 BO particles were taken from [16]. Chemical composition of the plume and its particle properties are given in table 2. The aspect angle was assumed to be equal to 90°.  Figure 3. Total volume emission of the plume (a) and the emission for a path through the diameter of the cylinder (b) and its comparison to data [8].

Spectral radiance of the plume with
Numerical simulation results for the matrix of solid particle concentrations and radiuses and its comparison to the data [16] are given in figure 4.
There are two sets (variant 1 and variant 2) of numerical simulation results. The conditions for any case are given in table 2. A good agreement with the data [16] is obtained for all considered cases.  Figure 4. Spectral radiance of the plume with 23 BO particles and its comparison with data [16].
In figure 5 comparison of the spectral signature for different angles of observation is given (

Spectral radiance of the nonhomegeneous plume of tactical rocket
The plume flowfield in this section is calculated from a system of a Navier-Stokes equations for compressible chemically reacting gas [2,17,18]. The system of equations is formulated in twodimensional axissimmetric geometry: where Ф  is dissipative function; t -is time; z, r -orthogonal cylindrical coordinates;  The algebraic Penner-Haselman-Edwards model of turbulent transfer [19] (PHE), were used to consider the effects of turbulent mixing in the model: where K is an empirical constant with recommended value of 0.125 [19]. Its value is varied in calculations.
For the numerical simulation of the afterburning processes in the plume the following kinetic scheme was used: Rate coefficients for the processes (1)  (8) were taken from the [20]. Rate coefficients for the reaction (9)  (10) were taken from the [21].
The initial data for the numerical simulation were assumed to be the same as in [22]. The temperature is equal to T = 2070 K and pressure is equal to p = 1.069 K . Chemical composition is given in table 3. Table 3. Chemical composition.
Chemical component: Mass fraction The rocket radius is equal to r = 8 cm . The nozzle exit radius is assumed to be equal to r = 3.81 cm. The nozzle exit velocity isn't given in the paper [22] so it has been varied. Numerical simulation results for the plume flowfield are presented in figure 6. Spatial two-dimensional distribution in the plane trough the axis of the plume for the density, temperature, and mass fraction of chemical species are presented in this figure. This results were obtained for the nozzle exit velocity  Spectral signature of the plume for the different angles of observation (broadside and nose on) are given in figure 8. It is considered that in the case of the nose on signature ( ο 0  ) the rocket body partially obscure the radiative plume. Numerical simulation results for the plume flowfields for different altitudes are given in figure 9. With the increase of the altitude plume significantly expands and the absolute value of temperature on the plume axis is decreased. Comparison of the plume radiance for different altitudes are presented in figure 10.  23 BO particles and its comparison with data [5].

Spectral radiance of the Atlas plume
Numerical simulation of the Atlas rocket plume flowfield was performed with the use of the technique described in the previous section. The boundary conditions for the freestream and nozzle exit regions were taken from the [23]. The specified boundary conditions are given in the table 4. Spatial distributions of temperature and density in the Atlas plume are presented in figure 11. Comparison of the axial and spatial distributions calculated with those obtained in the [23]. for perfect and chemically reacting gas is given in figure 12. Numerical simulation results presented in this figure were obtained for the laminar case. There is a satisfactory agreement for both (perfect gas and chemically reacting gas) cases.
Comparison of the spectral signature with those obtained by [24] is given in figure 13.

Conclusion
Numerical simulation of the plume flowfields and spectral signatures for different altitudes, chemical composition, altitudes and solid particle properties were performed. A satisfactory agreement with published results are obtained. An influence of the altitude and imaginary part of the refractive index on the plume spectral signature was shown.
The present study was supported by the Russian Science Foundation project № 161110275.   Figure 13. Spectral signature of the Atlas plume and its comparison to data [24].