NOMENCLATURE

A

lasing gas

a

constant in gain equation

B

gas stores the vibrational energy

b

constant in gain equation

C

catalyst gas

c

mass fraction

e

specific internal energy

e_vib

specific vibrational energy

h

enthalpy

h

Planck's constant

k

Boltzmann constant

N

population

p

pressure

R

gas constant

T

temperature

t

time

u

axial velocity

v

lateral velocity

X

mole fraction

x

axial coordinate

y

lateral coordinate

λ

wavelength

ρ

density

τ _{I, II}

characteristic relaxation time of vibrational modes

τ₁₂

radiative lifetime

v _{1, 2, 3}

vibrational mode's frequency of gas A

v_c

collision frequency

Subscripts

I, II

vibrational modes

vib

vibrational

Superscripts

eq

equilibrium

2.0 GOVERNING EQUATIONS AND NUMERICAL PROCEDURE

In the present study, the laser mixture contains the A–B–C gas mixture in which A is the active lasing molecule, B stores the vibrational energy and C has the role of catalyst for augmentation of the population inversions. The 2D compressible equations governing the continuum inviscid flow representing the conservation of mass, momentum, and energy are used as the basic equations of the flow field. These equations are written in the following conservation form^{(Reference Kuo17)}.

(1) $$\begin{equation} \frac{{\partial U}}{{\partial t}} + \frac{{\partial F}}{{\partial x}} + \frac{{\partial G}}{{\partial y}} = 0, \end{equation}$$

where

(2) $$\begin{equation} U = \left[ \begin{array}{l} \rho \\ \rho u\\ \rho v\\ \rho e \end{array} \right]\,\,\,\,\,F = \left[ \begin{array}{l} \rho u\\ \rho uu + p\\ \rho uv\\ \rho uh \end{array} \right]\,\,\,\,\,G = \left[ \begin{array}{l} \rho v\\ \rho vu\\ \rho vv + p\\ \rho vh \end{array} \right] \end{equation}$$

The laser media under consideration has two modes of vibrational energy, so:

(3) $$\begin{equation} e = e\left( T \right) + \frac{{{V^2}}}{2} + {e_{vib}}_I + {e_{vib}}_{II} \end{equation}$$

The rate equations representing the relaxation of vibrational energies of these modes are:

(4) $$\begin{eqnarray} \frac{{\partial {e_{vib}}_I}}{{\partial t}} + u\frac{{\partial {e_{vib}}_I}}{{\partial x}} + v\frac{{\partial {e_{vib}}_I}}{{\partial y}} &=& \frac{1}{{{\tau _I}}}\left( {{e_{vib}}{{_I}^{eq}} - {e_{vib}}_I} \right)\nonumber\\ \frac{{\partial {e_{vib}}_{II}}}{{\partial t}} + u\frac{{\partial {e_{vib}}_{II}}}{{\partial x}} + v\frac{{\partial {e_{vib}}_{II}}}{{\partial y}} &=& \frac{1}{{{\tau _{II}}}}\left( {{e_{vib}}{{_{II}}^{eq}} - {e_{vib}}_{II}} \right) \end{eqnarray}$$

For the CO₂–N₂–H₂O or the N₂O–N₂–He mixtures as A–B–C system, the vibrational energies are defined as:

(5) $$\begin{eqnarray} {e_{vib}}_I &=& {c_A}{R_A}\left\{ {\left( {\frac{{{\rm{h}}{v_1}}}{k}} \right){{\left[ {exp\left( {\frac{{{\rm{h}}{v_1}}}{{k{T_{vib}}_I}}} \right) - 1} \right]}^{ - 1}} + \left( {\frac{{2{\rm{h}}{v_2}}}{k}} \right){{\left[ {exp\left( {\frac{{{\rm{h}}{v_2}}}{{k{T_{vib}}_I}}} \right) - 1} \right]}^{ - 1}}} \right\}\nonumber\\ {e_{vib}}_{II} &=& {c_A}{R_A}\left\{ {\left( {\frac{{{\rm{h}}{v_3}}}{k}} \right){{\left[ {exp\left( {\frac{{{\rm{h}}{v_3}}}{{k{T_{vib}}_{II}}}} \right) - 1} \right]}^{ - 1}}} \right\} \end{eqnarray}$$

The populations of two energy levels of the lasing molecule are:

(6) $$\begin{eqnarray} {N_{001}} &=& \frac{{{N_A}exp\left( { - \frac{{{\rm{h}}{v_3}}}{{k{T_{vib}}_{II}}}} \right)}}{{{{\left[ {1 - exp\left( { - \frac{{{\rm{h}}{v_1}}}{{k{T_{vib}}_I}}} \right)} \right]}^{ - 1}}{{\left[ {1 - exp\left( { - \frac{{{\rm{h}}{v_2}}}{{k{T_{vib}}_I}}} \right)} \right]}^{ - 2}}{{\left[ {1 - exp\left( { - \frac{{{\rm{h}}{v_3}}}{{k{T_{vib}}_{II}}}} \right)} \right]}^{ - 1}}}}\nonumber\\ {N_{100}} &=& \frac{{{N_A}exp\left( { - \frac{{{\rm{h}}{v_1}}}{{k{T_{vib}}_I}}} \right)}}{{{{\left[ {1 - exp\left( { - \frac{{{\rm{h}}{v_1}}}{{k{T_{vib}}_I}}} \right)} \right]}^{ - 1}}{{\left[ {1 - exp\left( { - \frac{{{\rm{h}}{v_2}}}{{k{T_{vib}}_I}}} \right)} \right]}^{ - 2}}{{\left[ {1 - exp\left( { - \frac{{{\rm{h}}{v_3}}}{{k{T_{vib}}_{II}}}} \right)} \right]}^{ - 1}}}}\qquad \end{eqnarray}$$

Finally, these populations are used to compute the small signal gain using the relation:

(7) $$\begin{equation} {G_o} = \left( {\frac{{{\lambda ^2}}}{{4\pi {\tau _{12}}{v_c}}}} \right)\left( {{N_{001}} - {N_{100}}} \right)\left( {\frac{a}{T}} \right)exp\left( { - \frac{b}{T}} \right) \end{equation}$$

The parameters in these equations are defined in the nomenclature. Here, the cell-centred finite volume method used to discretise the six governing equations is defined. All equations are coupled and are discretized on structured grids. The time integration is accomplished by an explicit time stepping scheme^{(Reference Hirsch18)}. The flux terms are treated using an AUSM⁺ method (Advection Upstream Splitting Method) at cell faces^{(Reference Liou19)}. The local time steps are used to speed up the computations and obtain the final steady state results.

The developed numerical procedure has been validated and used in different flow fields^{(Reference Tahsini20-Reference Tahsini22)}. In addition, to show the validity of this code for the considered problem, two different GDLs are considered. At first, the CO₂–N₂–H₂O flow of Reference AndersonRef. 5 is solved in convergent-divergent nozzle geometry to compare the distribution of the vibrational temperature as shown in Fig. 1. Then, the N₂O–N₂–He gas dynamic laser^{(Reference Reddy10)} is calculated to compare the small signal gain distribution as illustrated in Fig. 2. The results show the accuracy of the present numerical simulation program.

Figure 1. Vibrational and translational temperature distribution.

Figure 2. Small signal gain distribution.

Since the results of the mentioned references have been calculated using quasi-1D method, for comparison, the present 2D code is reduced and converted to 1D form. To do this, the lateral momentum equation is omitted and the effect of area change is entered as a source term in an axial momentum equation^{(Reference Tahsini23)}.

3.0 RESULTS AND DISCUSSION

The gas dynamic laser is the converging-diverging nozzle which is continued by the constant area chamber as an optical cavity (laser resonator). The gas dynamics of such supersonic flow field in this cavity may contain some expansion and compression waves, which make the diamond pattern there. This pattern causes the flow's inhomogeneities, which are improper for the laser beam. The presence of the diamond pattern and its influence must be studied using 2D simulations. The GDL considered here is N₂O–N₂–He laser with geometry (not scale) and flow properties as shown schematically in Fig. 3. The inlet condition is T_o = 1,200K, p_o = 7 bar, X_N2O = 0.25, X_N2 = 0.45 and X_He = 0.3. The outlet flow is supersonic and the outlet boundary condition is extrapolated from the inside. The divergent part has three different geometries: linear, quarter of cosine function, and half of cosine-function. The grid resolution of 150 × 50 is chosen here after the grid independency studies. The 2D simulations last about 2 hours on the core-i7 2.8 GHz CPU PC.

Figure 3. Geometry and flow properties of the considered GDL.

The results of the flow field simulation (without considering the vibrational energies) are shown in Fig. 4, which illustrates the diamond pattern in the supersonic region. The geometry of the divergent nozzle has some impacts on flow field but apart from that, there are many differences between the 2D and quasi-1D simulations as shown in Figs 5 and 6. Temperature and Mach number distributions along the cavity's centreline (in a straight channel after divergent nozzle) indicate the effect of the diamond pattern as about 50% deviation in Mach number and 140% deviation in temperature at some places in the optical cavity. Such 2D phenomena alter the performance of the GDL, as will be illustrated.

Figure 4. Lateral velocity contours for different geometries of divergent nozzle.

Figure 5. Mach number distribution along the cavity's centreline.

Figure 6. Temperature distribution along the cavity's centreline.

Taking into account the equations for vibrational energy modes helps to compute the population inversions and small signal gain of the GDL. The distribution of the population inversion is shown in Fig. 7 using the quasi-1D simulation, which indicates that the maximum inversion occurs at the initial sections of the cavity and is about 0.011. The 2D result for the first configuration is shown in Fig. 8. The maximum inversion decreases slightly due to the 2D effects and occurs at a position downstream of the 1D simulation.

Figure 7. Distribution of population inversion from quasi 1D simulation.

Figure 8. Population inversion contours for the first geometry.

The small signal gain and the second mode's vibrational temperature are also compared in Figs 9 and 10, respectively. The results indicate that the maximum gain depends to the geometry of the divergent nozzle, and its position is downstream of the results of the 1D computation. The maximum gain growth is about 40% between different geometries. The diamond pattern drastically decreases the population inversions and the small signal gain along the cavity. Compression waves (oblique shock waves) slow down the supersonic flow in the cavity and increase the resident time, so the nonequilibrium effects are weakened.

Figure 9. Distribution of small signal gain along the centreline.

Figure 10. Distribution of the second mode's vibrational temperature along the centreline.