# Development and Validation of a Compressible Reacting Gas-Dynamic Flow Solver for Supersonic Combustion

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Governing Equations

#### 2.2. Subgrid Flow Equations

#### 2.3. Closure of Combustion Model

#### 2.4. Numerical Methods

## 3. Results

#### 3.1. Solver Validation

#### 3.1.1. Shock Tube Problem

- Inert Multicomponent Shock Tube

_{2}/10%O

_{2}/70%Ar (in mole%) is used in the simulation to verify the numerical accuracy of the model implementations for thermal and mass transport of a multicomponent mixture. The shock tube is 0.10 m in length, and it is discretized with 400 uniform elements. The time step is equal to $\u2206t={10}^{-7}\text{}s$. In the middle of the tube is a fixed diaphragm that separates the gas between the left side (${T}_{l}=400\text{}K$ and ${P}_{l}=8000\text{}Pa$) and the right side (${T}_{r}=1200\text{}K$ and ${P}_{r}=\mathrm{80,000}\text{}Pa$). At time $t=0,$ the diaphragm is eliminated, and the shock wave starts to propagate from right to left and the expansion wave moves in the opposite direction. The initial configuration of the gas mixture generates the propagation of shock and discontinuous waves to the left (low-pressure) and a rarefaction (expansion) wave to the right side (high-pressure) of the tube. Figure 2 shows the current simulation results (solid line) compared with the simulation results of Huang et al. [56]. The modeling results have good agreement. It is noteworthy that the compressibleCentralReactingFoam solver’s accuracy is 2nd order for both time and spatial convective terms.

- Reactive Multicomponent Shock Tube

_{2}/10% O

_{2}/70% Ar. The length of the tube is equal to $L=0.12\text{}m$, with the initial conditions corresponding to the left and right sides as follows: $\left({\rho}_{l},{u}_{l},{p}_{l}\right)=\left(0.072,\text{}0,\text{}7173\right)$ and $\left({\rho}_{r},{u}_{r},{p}_{r}\right)=\left(0.18075,-487.34,\text{}\mathrm{35,594}\right)$, where $\rho ,\text{}u\text{},\text{}\mathrm{a}\mathrm{n}\mathrm{d}\text{}p$ are measured in ($kg/{m}^{3}$), ($m/s$), and ($Pa$), respectively. The length of the tube is discretized by 400 uniform elements with 0.01 $\mu s$ timestep. The simulation used the reaction mechanism of Conaire et al. [57] for hydrogen, which consists of nine (9) chemical species (H

_{2}, O

_{2}, H, O, OH, HO

_{2}, H

_{2}O

_{2}, H

_{2}O, $Ar$), and 18 elementary reactions. A wall boundary condition is implemented on the left side of the tube, and on the right side, a non-reflected boundary condition is implemented. The initial configuration of non-stable discontinuity is decayed into a shock propagated to the tube’s left side. The solutions of spreading discontinuities in the shock tube for velocity, mass fraction of $H$, and temperature: ${U}_{x}$, ${Y}_{H}$, and $T$ are shown in Figure 3 for 230 $\mu s$ simulation time. Comparisons of the current solution (solid lines) in Figure 3 with computational data of Martínez-Ferrer et al. [34] (symbols) show good agreement. The visible discrepancies are seen in the vicinities of local maximums around x = 0.06 m at 230 $\mu s$ are explained that Martínez-Ferrer et al. [34] sed a seventh-order accurate Weighted Essentially Non-Oscillatory (WENO) scheme to discretize the non-linear advective terms.

_{2}/O

_{2}/Ar. Figure 4 shows the simulation results for the temperature profiles of inert and reactive mixtures at various simulation times.

#### 3.1.2. Simulation of Ladenburg Jet Problem

#### 3.1.3. Simulation of DLR Scramjet Combustor

- Simulation of Cold Flow DLR Combustor

- Simulation of Reacting Flow DLR Combustor

#### 3.2. Influence of Turbulence Models on Scramjet Combustion

#### 3.3. Analysis of Turbulence–Combustion Interaction in Scramjet Combustor

_{2}O correlates with the field of instantaneous temperature, while the field of HO

_{2}correlates with OH and Qdot on the shear layer.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Shock tube and spreading shock wave (red), the contact surface (black), and expansion fan (blue).

**Figure 2.**Multicomponent inert shock tube profiles of H

_{2}/O

_{2}/Ar mixture at 40 μs. Keys: solid line—current simulation; symbols—simulation of Huang et al. [56].

**Figure 3.**Multicomponent reactive shock tube profiles of H

_{2}/O

_{2}/Ar mixture at 230 μs. Keys: solid line—current simulation; symbols—Martínez-Ferrer et al. [34].

**Figure 4.**Temperature profiles of shocks/detonation waves for (

**a**) inert and (

**b**) reactive mixture at various times.

**Figure 5.**Computational 3D mesh for simulation of underexpanded jet of Ladenburg experiment. Inflow is shown by the yellow surface, the free stream inlet surface is violet, and the free stream surface is red.

**Figure 6.**(

**a**) Schematics of an underexpanded air jet. Keys: 1 expansion waves, 2 oblique shock, 3 normal shock, 4 reflected shock, 5 reflected expansion waves. (

**b**) Contours of Mach numbers at the plane through the center of computational domain (see Figure 5).

**Figure 7.**Density distribution of the supersonic jet measured (reprinted with permission from Ladenburg [58]) (

**lower panel**), compared with current computational results (

**upper panel**). The values of the density contours are provided in kg/m

^{3}.

**Figure 8.**Geometry of DLR experiment [60]. The 12° wedge provides a stagnation zone for flame holding.

**Figure 14.**Mean axial velocity profile along the middle of the channel at $y=25mm$. Keys: symbols—experimental data [60]; line—current simulation.

**Figure 15.**Contours of mean temperature (upper) and mean temperature at the three cross-sections with different turbulence models (bottom). Keys: symbols—experimental data [60]; lines—simulation. Color keys: black—WALE; red—SMG; blue—LDkEqn.

**Figure 16.**Mean temperature at three cross-sections compared with simulation data in the literature. Keys: symbols—experimental data [60]; lines—simulation. Color keys: black—current simulation with WALE model; red—Berglund and Fureby [63]; green—Genin and Menon [12]; blue—Zhang et al. [30]; purple—Oevermann [61].

**Figure 17.**Instantaneous distribution of: (

**a**) Mach number; (

**b**) normalized heat release rate $Qdo{t}_{Norm}=Qdot/{10}^{10}$; (

**c**) temperature; (

**d**) normalized contour lines of density gradient.

**Figure 18.**Takeno Flame Index (TFI) [59].

**Figure 19.**Instantaneous distribution of species mass fraction. (

**a**) H; (

**b**) OH; (

**c**) HO

_{2}; (

**d**) H

_{2}O.

**Figure 20.**Instantaneous distribution of: (

**a**) mixture fraction; (

**b**) scalar dissipation rate $\chi $ (in the log scale).

**Figure 21.**(

**a**) Borghi diagram showing different turbulent combustion regimes; (

**b**) Damköhler number (Da in logarithmic scale); (

**c**) fraction of reactive cells (${\gamma}^{*}$). The white lines indicate locations where $Da=1.$

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**MDPI and ACS Style**

Gilmanov, A.; Gokulakrishnan, P.; Klassen, M.S.
Development and Validation of a Compressible Reacting Gas-Dynamic Flow Solver for Supersonic Combustion. *Dynamics* **2024**, *4*, 135-156.
https://doi.org/10.3390/dynamics4010008

**AMA Style**

Gilmanov A, Gokulakrishnan P, Klassen MS.
Development and Validation of a Compressible Reacting Gas-Dynamic Flow Solver for Supersonic Combustion. *Dynamics*. 2024; 4(1):135-156.
https://doi.org/10.3390/dynamics4010008

**Chicago/Turabian Style**

Gilmanov, Anvar, Ponnuthurai Gokulakrishnan, and Michael S. Klassen.
2024. "Development and Validation of a Compressible Reacting Gas-Dynamic Flow Solver for Supersonic Combustion" *Dynamics* 4, no. 1: 135-156.
https://doi.org/10.3390/dynamics4010008