# Passive scalar mixing studies to identify the mixing length in a supersonic confined jet

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## Abstract

Supersonic jet with a co-flow, closely bounded by walls is known as supersonic confined jet. Supersonic confined jet is encountered in practical devices like the supersonic ejector. Mixing of the primary and the secondary fluid inside the confined passage is complex. From a design perspective, it is necessary to have an accurate knowledge of the mixing length (*L* _{MIX}). Tracers that do not actively participate in the flow behavior but rather mark the fluids such that they faithfully follow the fluid motion are known as passive scalars. Passive scalars help in the understanding the progression of mixing amidst interacting flows. In this work, we have performed passive scalar mixing studies in a supersonic confined jet for different operating conditions using an existing low area ratio (AR = 3.7) rectangular supersonic gaseous ejector. Air is used as the working fluid in both the primary and the secondary flow. The design Mach number of the primary flow nozzle (*M* _{PD} = 1.5–3.0) and the total pressure of the primary flow (*P* _{OP} = 4.89–9.89 bar) are varied during the experiments. Using the planar laser-induced fluorescence (PLIF) technique and acetone as the passive scalar, *L* _{MIX} is determined. A 266 nm Nd-YAG laser with a repetition rate of 8 Hz is used to excite the acetone molecules in the flow field, and the emitted fluorescence is captured by an ICCD camera. A new method is proposed to study the passive scalar distribution from the acetone PLIF images through digital image processing. Spatial Scalar Fluctuations Intensity (SSFI or *ψ*) is a parameter defined at every transverse section along the flow direction. Based on the variation of *ψ* along the jet, the location of *L* _{MIX} can be identified. *L* _{MIX} is defined as the length from the supersonic nozzle exit where *ψ* first attains a value of 0.05. For the first time, *L* _{MIX} is quantified in a supersonic confined jet. *L* _{MIX} values are observed to be in the range of 3H to 6H for the cases under study, where H is the height of the confined passage. The behavior of *L* _{MIX} is closely dependent on the nozzle operating conditions. The values of *L* _{MIX} are found to be reduced by 17.67% for the over-expanded flows and increased by 15.76% for the under-expanded flows from the perfectly expanded condition. This study also provides other supersonic confined jet characteristics like the potential core length (*L* _{PC}) and the shock cell spacing (*S* _{x}) of the primary supersonic jet. Compared to the supersonic free jet, values of *L* _{PC} and *S* _{x} are found to be different in the supersonic confined jet.

## Keywords

Secondary Flow Passive Scalar Primary Flow Planar Laser Induce Fluorescence Nozzle Pressure Ratio## Abbreviations

- PLIF
Planar laser induced fluorescence

- SNR
Signal to noise ratio

- ICCD
Intensified charge-coupled device

- SSFI
Spatial scalar fluctuations intensity

## List of symbols

*P*_{OP}Primary flow stagnation pressure (bar)

*P*_{OS}Secondary flow stagnation pressure (bar)

*T*_{OP}Primary flow stagnation temperature (K)

*T*_{OS}Secondary flow stagnation temperature (K)

*P*_{e}Mixed flow exit pressure (bar)

*P*_{min}Minimum pressure encountered at the top wall of the mixing duct (bar)

*P*_{ne}Pressure near the primary flow nozzle exit (bar)

*P*_{a}Pressure in the ambient conditions (bar)

- \( \dot{m}_{p} \)
Primary mass flow rate (kg/s)

- \( \dot{m}_{s} \)
Secondary flow mass flow rate (kg/s)

- SPR
Stagnation pressure ratio (

*P*_{OP}/*P*_{OS})- CR
Compression ratio (

*P*_{e}/*P*_{OS})- NPR
Nozzle pressure ratio (

*P*_{OP}/*P*_{ne})- ER or ω
Entrainment ratio \( \left( {\dot{m}_{\text{s}} /\dot{m}_{\text{p}} } \right) \)

*L*_{PC}Length of the primary flow potential core (mm)

*L*_{NM}Non-mixed length from PLMS experiments (mm)

*L*_{MIX}Mixed length from acetone PLIF experiments (mm)

*M*_{PD}Design Mach number of the primary nozzle

*M*_{PJ}Fully expanded jet Mach number of the primary nozzle

*M*_{PD}/*M*_{PJ}Mach number ratio

*M*_{c}Convective Mach number

*M*_{s}Secondary flow Mach number

- AR
Area ratio of the supersonic confined jet \( \left[ {\left( {W/w} \right) \cdot \left( {H/h} \right)} \right] \)

*W*Width of the mixing duct (mm)

*w*Width of the primary flow convergent-divergent (CD) nozzle (mm)

*H*Height of the constant area mixing duct (mm)

*h*Height of the primary flow CD nozzle (mm)

*µ*Dynamic viscosity of the fluid (kg/m s)

- Re
_{J} Reynolds number corresponding to fully expanded primary jet conditions (

*ρ*_{J}*u*_{J}*h*_{J}/*μ*_{J})*h*_{J}Fully-expanded primary jet height (mm)

*D*_{J}Equivalent hydraulic jet diameter (mm) [(4

*Wh*_{J})/(2*W*+*h*_{J})]*u*_{J}Fully-expanded primary jet velocity (m/s)

*ρ*_{J}Fully-expanded primary jet density (kg/m

^{3})*β*_{mix}Mixing parameter used in the definition of

*L*_{NM}and*L*_{PC}- [
*S*_{x}]_{n} Successive shock cell spacing along x-direction (mm)

*n*Shock cell number

*ψ*Spatial scalar fluctuations intensity

*Θ*Normalized local scalar quantity (

*I*/*I*_{max})- \( \bar{\varTheta } \)
Time-averaged scalar quantity

*Θ*^{′}Fluctuating scalar quantity

## Notes

### Acknowledgements

The first author is grateful for the scholarship provided by the Ministry of Human Resource Development (MHRD)–Govt. of India, during the research period. The authors would like to sincerely acknowledge the research grants (Grant No. DRDO 0626) from the Defence Research and Development Organization (DRDO), India. The authors would like to thank Dr. Vikas M. Shelar, Dr. Bindu, and Dr. Yedhu for their valuable inputs in the analysis of experimental results. The authors would also like to thank the research students in LHSR, especially Mr. Kannan Munusamy, Mr. Sneh Deep and Mr. G. Yogeshwaran for their timely help in setting up and conducting the experiments. Technical assistance offered by the LaVision support team specifically Mr. Arun and the expertise from Tesscorn Aerofluid Inc., chiefly Mr. Satyanarayana are duly acknowledged.

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