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Linear stability analysis of surface waves of liquid jet injected in transverse gas flow with different angles

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Abstract

A theoretical and experimental study was conducted to investigate the effect of injection angle on surface waves. Linear stability theory was utilized to obtain the analytical relation. In the experimental study, high-speed photography and shadowgraph techniques were used. Image processing codes were developed to extract information from photos. The results obtained from the theoretical relation were validated with the experimental results at different injection angles. In addition, at the injection angle of 90\({^\circ }\), the theoretical results were evaluated with the experimental results of other researchers. This evaluation showed that the theory results were in good agreement with the experimental data. The proper orthogonal decomposition (POD) and the power spectra density (PSD) analysis were also used to investigate the effect of the injection angle on the flow structures. The results obtained from the linear stability were used to determine the maximum waves’ growth rate, and a relation was presented for the breakup length of the liquid jet at different injection angles. The breakup length results were compared with theory and published experimental data. The presented relation is more consistent with experimental data than other theories due to considering the nature of waves. The results showed that the instability of the liquid jet is influenced by three forces: inertial, surface tension, and aerodynamic. Therefore, Rayleigh–Taylor, Kelvin–Helmholtz, Rayleigh–Plateau, and azimuthal instabilities occur in the process. Decreasing the injection angle changes the nature of waves and shifts from Rayleigh–Taylor to Kelvin–Helmholtz. That reduces the wavelength and increases the growth rate of the waves. Axial waves have a significant impact on the physics of the waves and influence parameters. If axial waves are not formed, the growth rate of the waves is independent of the injection angle. An increase in the gas Weber number causes a change in the type of dominant waves and a greater instability of the liquid jet. In contrast, an increase in the liquid Weber number causes an enhancement in the resistance of the liquid jet against the transverse flow without changing the type of the dominant waves. Decreasing the density ratio reduces the effect of Rayleigh–Taylor waves and strengthens the Kelvin–Helmholtz waves. It causes two trends to be observed for the growth rate of waves at low spray angles, while one trend occurs at high spray angles.

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Abbreviations

\(\textrm{At}\) :

Atwood number, \(\frac{\rho _{l}-\rho _{g}}{\rho _{l}+\rho _{g}}\)

c :

Constant

\({\hat{e}}\) :

Unit vector

f :

Body force

g :

Gravity acceleration

I :

First kinds modified Bessel function

K :

Second kinds modified Bessel function

k :

Axial wavenumber

L :

Breakup length

m :

Azimuthal wavenumber

p :

Pressure

R :

Radius

t :

Time

V :

Velocity vector

v :

Velocity

\(v^{*}\) :

Air/liquid velocity ratio, \(\frac{v_{g}}{v_{l}}\)

We:

Weber number, \(\frac{\rho v^{2}R}{\sigma }\)

N, S :

Local tangential-vertical coordinate axes

\(r,\theta ,z\) :

Reference cylindrical coordinate axes

x, z :

Cartesian coordinate axes

\(\beta \) :

Circumferential angle

\(\eta \) :

Interface displacement

\(\mu \) :

Dynamic viscosity

\(\upsilon \) :

Kinematic viscosity

\(\xi \) :

Effective thickness

\(\rho \) :

Density

\(\rho ^{*}\) :

Air/liquid density ratio, \(\frac{\rho _{g}}{\rho _{l}}\)

\(\sigma \) :

Surface tension

\(\varphi \) :

Velocity potential function

\(\psi _{0}\) :

Injection angle

\({\Gamma }\) :

Gamma function

\(\omega \) :

Complex frequency

I :

Imaginary part of complex number

R :

Real part of complex number

l :

Liquid

g :

Gas

0:

Initial point

opt:

Optimum point

ref:

Reference point

i, j :

Tensor index

\(\infty \) :

Free stream condition

‘:

Perturbation quantity

\(*\) :

Dimensionless quantity

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  1. Turbulence and Multiphase Flow Laboratory, Department of Aerospace Engineering, Amirkabir University of Technology, Tehran, Iran

    Si. Kasmaiee, M. Tadjfar & Sa. Kasmaiee

  2. Department of Mechanical and Aerospace Engineering, Clarkson University, Potsdam, NY, 13699, USA

    G. Ahmadi

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All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by SK, MT, SK and GA. The first draft of the manuscript was written by SK and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Si. Kasmaiee.

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Kasmaiee, S., Tadjfar, M., Kasmaiee, S. et al. Linear stability analysis of surface waves of liquid jet injected in transverse gas flow with different angles. Theor. Comput. Fluid Dyn. 38, 107–138 (2024). https://doi.org/10.1007/s00162-024-00685-2

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