# Development of 1D Model of Constant-Volume Combustor and Numerical Analysis of the Exhaust Nozzle

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{x}emissions [12].

## 2. 1D Model of the Constant-Volume Combustion Test Rig

#### 2.1. Model Setup

#### 2.1.1. CVC Configuration

#### 2.1.2. Numerical Setup

#### 2.1.3. Parametrization of Valves

#### 2.1.4. Burning Rate

#### 2.1.5. Heat Loss Model

#### 2.2. Results of Model

#### 2.2.1. Validation of Model

#### 2.2.2. Transient Conditions of Exhaust System

## 3. 3D Transient Analysis of Exhaust System

#### 3.1. Governing Equations

#### 3.2. Numerical Setup

#### 3.3. Periodic Convergence Criteria

#### 3.4. Results of URANS CFD

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Greek Symbols | |

${\alpha}^{\ast}$ | Corrective Coefficient of viscosity $\phantom{\rule{4pt}{0ex}}[-]$ |

$\Delta P$ | Pressure Difference $\phantom{\rule{4pt}{0ex}}\left[\mathrm{Pa}\right]$ |

${\delta}_{i,j}$ | Kronecker Delta $\phantom{\rule{4pt}{0ex}}[-]$ |

$\u03f5$ | Relative Error $\phantom{\rule{4pt}{0ex}}[-]$ |

${\eta}_{D}$ | Damping Efficiency $\phantom{\rule{4pt}{0ex}}[-]$ |

${\mathsf{\Gamma}}_{\omega}$ | Effective Diffusivity for Specific Dissipation Rate $\phantom{\rule{4pt}{0ex}}[\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-3}]$ |

${\mathsf{\Gamma}}_{k}$ | Effective Diffusivity for Turbulence Kinetic Energy $\phantom{\rule{4pt}{0ex}}[\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}]$ |

$\lambda $ | Thermal Conductivity $\phantom{\rule{4pt}{0ex}}[\mathrm{W}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{-1}]$ |

$\mu $ | Dynamic Viscosity $\phantom{\rule{4pt}{0ex}}[\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}]$ |

${\mu}_{0}$ | Reference Dynamic Viscosity $\phantom{\rule{4pt}{0ex}}[\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}]$ |

${\mu}_{t}$ | Eddy Viscosity $\phantom{\rule{4pt}{0ex}}[\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}]$ |

$\omega $ | Specific Rate of Dissipation $\phantom{\rule{4pt}{0ex}}\left[{\mathrm{s}}^{-1}\right]$ |

${\overline{\zeta}}_{{P}_{t}}$ | Stagnation Pressure Losses $\phantom{\rule{4pt}{0ex}}[-]$ |

$\varphi $ | Equivalence Ratio $\phantom{\rule{4pt}{0ex}}[-]$ |

$\pi $ | Pressure Ratio $\phantom{\rule{4pt}{0ex}}[-]$ |

$\rho $ | Density $\phantom{\rule{4pt}{0ex}}[\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}]$ |

$\tau $ | Molecular Stress Tensor $\phantom{\rule{4pt}{0ex}}[\mathrm{N}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}]$ |

${\tau}_{cc}$ | Characteristic Time $\phantom{\rule{4pt}{0ex}}\left[\mathrm{s}\right]$ |

Roman Symbols | |

$\dot{m}$ | Mass Flow Rate $\phantom{\rule{4pt}{0ex}}[\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}]$ |

$\widehat{C}$ | Normalised Cross-Correlation $\phantom{\rule{4pt}{0ex}}[-]$ |

$\widehat{R}$ | Reduced Range $\phantom{\rule{4pt}{0ex}}[-]$ |

A | Area $\phantom{\rule{4pt}{0ex}}\left[\mathrm{m}\right]$ |

a | Specific Heat Capacity Coefficient $\phantom{\rule{4pt}{0ex}}[\mathrm{J}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{-1}]$ |

${c}_{p}$ | Specific Heat Capacity $\phantom{\rule{4pt}{0ex}}[\mathrm{J}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{-1}]$ |

${C}_{v}$ | Coefficient of Velocity Chamber $\phantom{\rule{4pt}{0ex}}[-]$ |

$Cd$ | Discharge Coefficient $\phantom{\rule{4pt}{0ex}}[-]$ |

D | Diameter $\phantom{\rule{4pt}{0ex}}\left[\mathrm{m}\right]$ |

${D}_{\widehat{R}}$ | Damping Factor $\phantom{\rule{4pt}{0ex}}[-]$ |

$dt$ | Time Step $\phantom{\rule{4pt}{0ex}}\left[\mathrm{s}\right]$ |

e | Specific Energy $\phantom{\rule{4pt}{0ex}}[\mathrm{J}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}]$ |

F | Force $\phantom{\rule{4pt}{0ex}}\left[\mathrm{N}\right]$ |

f | Frequency $\phantom{\rule{4pt}{0ex}}\left[\mathrm{Hz}\right]$ |

g | Acceleration of Gravity $\phantom{\rule{4pt}{0ex}}[\mathrm{m}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-2}]$ |

${G}_{\omega}$ | Generation of Specific Dissipation Rate $\phantom{\rule{4pt}{0ex}}[\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-3}]$ |

${G}_{k}$ | Generation of Turbulence Kinetic Energy $\phantom{\rule{4pt}{0ex}}[\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-2}]$ |

H | Total Specific Enthalpy $\phantom{\rule{4pt}{0ex}}[\mathrm{J}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}]$ |

h | Specific Enthalpy $\phantom{\rule{4pt}{0ex}}[\mathrm{J}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}]$ |

${h}_{coef.}$ | Heat Transfer Coefficient $\phantom{\rule{4pt}{0ex}}[\mathrm{W}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{-1}]$ |

k | Turbulence Kinetic Energy $\phantom{\rule{4pt}{0ex}}[{\mathrm{m}}^{2}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-2}]$ |

L | Lift $\phantom{\rule{4pt}{0ex}}\left[\mathrm{m}\right]$ |

l | Length $\phantom{\rule{4pt}{0ex}}\left[\mathrm{m}\right]$ |

$Lash$ | Valve’s Clearance $\phantom{\rule{4pt}{0ex}}\left[\mathrm{m}\right]$ |

M | Mach Number $\phantom{\rule{4pt}{0ex}}[-]$ |

m | Mass $\phantom{\rule{4pt}{0ex}}\left[\mathrm{kg}\right]$ |

${M}_{w}$ | Molecular Weight $\phantom{\rule{4pt}{0ex}}[\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{Kmol}}^{-1}]$ |

$Nu$ | Nusselt Number $\phantom{\rule{4pt}{0ex}}[-]$ |

$P,p$ | Pressure $\phantom{\rule{4pt}{0ex}}\left[\mathrm{Pa}\right]$ |

${P}_{t}$ | Stagnation Pressure $\phantom{\rule{4pt}{0ex}}\left[\mathrm{Pa}\right]$ |

$Pr$ | Prandtl Number $\phantom{\rule{4pt}{0ex}}[-]$ |

${R}_{i,j}$ | Reynolds Stress $\phantom{\rule{4pt}{0ex}}[\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-2}]$ |

$Re$ | Reynolds Number $\phantom{\rule{4pt}{0ex}}[-]$ |

$RF$ | Rotating Factor $\phantom{\rule{4pt}{0ex}}[-]$ |

${S}_{0}$ | Temperature Term $\phantom{\rule{4pt}{0ex}}\left[\mathrm{K}\right]$ |

${S}_{\tau}$ | Shear Stress Term $\phantom{\rule{4pt}{0ex}}[\mathrm{N}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}]$ |

${S}_{e}$ | Energy Transfer Rate per Volume Term $\phantom{\rule{4pt}{0ex}}[\mathrm{W}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}]$ |

T | Temperature $\phantom{\rule{4pt}{0ex}}\left[\mathrm{K}\right]$ |

t | Time $\phantom{\rule{4pt}{0ex}}\left[\mathrm{s}\right]$ |

${T}_{0}$ | Reference Temperature $\phantom{\rule{4pt}{0ex}}\left[\mathrm{K}\right]$ |

${T}_{cycle}$ | Period of Cycle $\phantom{\rule{4pt}{0ex}}\left[\mathrm{s}\right]$ |

${T}_{t}$ | Stagnation Temperature $\phantom{\rule{4pt}{0ex}}\left[\mathrm{K}\right]$ |

$U,u$ | Velocity $\phantom{\rule{4pt}{0ex}}[\mathrm{m}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}]$ |

V | Volume $\phantom{\rule{4pt}{0ex}}\left[{\mathrm{m}}^{3}\right]$ |

x | Length $\phantom{\rule{4pt}{0ex}}\left[\mathrm{m}\right]$ |

${X}_{b}$ | Non-Dimensional Burning Rate $\phantom{\rule{4pt}{0ex}}[-]$ |

${Y}_{\omega}$ | Dissipation of Specific Rate of Dissipation $\phantom{\rule{4pt}{0ex}}[\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-3}]$ |

${Y}_{k}$ | Dissipation of Turbulent Kinetic Energy $\phantom{\rule{4pt}{0ex}}[\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-2}]$ |

Subscripts | |

0 | Closing of Intake Valves Moment |

$av$ | Average Time Moment |

$bends$ | Bends and Taps Contribution |

$cc$ | Combustion Chamber |

$e,\phantom{\rule{4pt}{0ex}}add.$ | Additional Source Term |

$e,\phantom{\rule{4pt}{0ex}}loss$ | Loss Source Term |

$ex$ | Exhaust |

f | Fuel |

$friction$ | Friction Contribution |

i | Average Time Moment |

$in$ | Intake |

K | Middle Cross-Section of Chamber |

$Lash$ | Valve’s Clearance |

$op$ | Operation |

t | Total |

Abbreviations | |

PDC | Pulse Detonation Combustor |

RDC | Rotating Detonation Combustor |

HPT | High-Pressure Turbine |

IGV | Inlet Guide Vane |

PGC | Pressure Gain Combustion |

CVC | Constant Volume Combustor |

GCI | Grid Convergence Index |

RANS | Reynolds-Averaged Navier–Stokes |

URANS | Unsteady Reynolds-Averaged Navier–Stokes |

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**Figure 2.**Operating description of model and calibration of a. discharge coefficients and b. burning rate.

**Figure 5.**Validation of 1D model with experimental chamber’s pressure: (

**a**) Comparison of experimental and model pressure, (

**b**) Comparison of model with experimental limits and (

**c**) Comparison of 9

^{th}experimental cycle with the model.

**Figure 6.**Stagnation Properties after the exhaust valves (

**a**) and cross-area evolution of the exhaust system (

**b**).

**Figure 7.**Flow domain and simulation mesh: (

**a**) Exhaust system ensemble flow domain—view 1, (

**b**) Exhaust system ensemble flow domain—view 2, (

**c**) Exhaust system grid—view 1, (

**d**) Exhaust system grid—view 2, (

**e**) Exhaust system with artificial plenum grid—view 1 and (

**f**) Exhaust system with artificial plenum grid—view 2. For (

**b**) the symbols specify the locations of Inlet of Domain (i), Inlet of transition part (ii), Inlet of nozzle (iii), Throat of nozzle (iv) and Outlet of nozzle (v).

**Figure 11.**Cross-correlation of transient CFD signals for mass flow rate (

**a**), mass flow—weighted Mach number (

**b**) and mass flow—weighted total temperature (

**c**).

**Figure 12.**Variable operating conditions of exhaust system: (

**a**) Transient inlet boundary conditions of flow domain, (

**b**) Center-line profile of pressure ratio of domain and (

**c**) Mach number contour of exhaust nozzle for time moments ${t}_{1}$, ${t}_{2}$, ${t}_{3}$, ${t}_{4}$ and ${t}_{5}$.

**Figure 13.**Losses and oscillations’ characterisation of exhaust system: (

**a**) Damping factor, (

**b**) Cumulative damping factor, (

**c**) Total pressure losses and (

**d**) Cumulative total pressure losses.

Valve | n | ${\mathit{Cd}}_{\mathit{max}}$ | ${\mathit{Cd}}_{\mathit{Lash}}$ |
---|---|---|---|

Intake | $2.5$ | $0.29$ | 13% of $C{d}_{max}^{in}$ |

Exhaust | 1 | $0.81$ | 6% of $C{d}_{max}^{ex}$ |

$\mathit{RF}$ | ${\mathit{t}}_{\mathit{start}}/{\mathit{T}}_{\mathit{cycle}}$ | ${\mathit{t}}_{\mathit{final}}/{\mathit{T}}_{\mathit{cycle}}$ |
---|---|---|

$0.055$ | 0 | $0.2569$ |

${\mathbf{\left(}{\mathit{P}}_{\mathit{cc}}\mathbf{\right)}}^{\mathit{Model}}\phantom{\rule{4pt}{0ex}}$ vs. | ${\overline{\mathbf{\Delta}\mathit{P}}}_{\mathit{cc}}$ | $\mathit{Max}\mathbf{\left[}\mathbf{\Delta}{\mathit{P}}_{\mathit{cc}}\mathbf{\right]}$ | $\mathsf{\sigma}$$\mathbf{\left[}\mathbf{\Delta}{\mathit{P}}_{\mathit{cc}}\mathbf{\right]}$ |
---|---|---|---|

${\overline{P}}_{cc}^{Exp.}$ | −3.0564% −0.048936 [bar] | 21.1084% 1.232 [bar] | 10.1056% 0.37494 [bar] |

${\left[{P}_{cc}\right]}_{no:\phantom{\rule{4pt}{0ex}}9}$ | 0.90957% 0.020247 [bar] | 28.9296% 0.51703 [bar] | 9.392% 0.2258 [bar] |

**Table 4.**Boundary conditions of domain. Colours correspond to Figure 7c–f.

Boundary Conditions | |
---|---|

Type | Properties |

Inlet | ${P}_{t,ex.}\left(t\right)$ and ${T}_{t,ex.}\left(t\right)$ |

Outlet | ${P}_{amb.}=1\phantom{\rule{4pt}{0ex}}\left[atm\right]$ |

Symmetry | - |

No - Slip Wall | Adiabatic |

Free - Slip Wall | Adiabatic |

Property | Grid | Refinement Ratio | GCI | Asymptotic Range of Convergence |
---|---|---|---|---|

$\dot{m}$ | Coarse–Medium Medium–Fine | $1.261$ $1.260$ | $0.745\%$ $0.474\%$ | $1.0049$ |

M | Coarse–Medium Medium–Fine | $1.261$ $1.260$ | $0.716\%$ $0.490\%$ | $1.0037$ |

${T}_{t}$ | Coarse–Medium Medium–Fine | $1.261$ $1.260$ | 1.2 × ${10}^{-5}\%$ 6.2 × ${10}^{-6}\%$ | $1.002$ |

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## Share and Cite

**MDPI and ACS Style**

Gallis, P.; Misul, D.A.; Boust, B.; Bellenoue, M.; Salvadori, S.
Development of 1D Model of Constant-Volume Combustor and Numerical Analysis of the Exhaust Nozzle. *Energies* **2024**, *17*, 1191.
https://doi.org/10.3390/en17051191

**AMA Style**

Gallis P, Misul DA, Boust B, Bellenoue M, Salvadori S.
Development of 1D Model of Constant-Volume Combustor and Numerical Analysis of the Exhaust Nozzle. *Energies*. 2024; 17(5):1191.
https://doi.org/10.3390/en17051191

**Chicago/Turabian Style**

Gallis, Panagiotis, Daniela Anna Misul, Bastien Boust, Marc Bellenoue, and Simone Salvadori.
2024. "Development of 1D Model of Constant-Volume Combustor and Numerical Analysis of the Exhaust Nozzle" *Energies* 17, no. 5: 1191.
https://doi.org/10.3390/en17051191