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Lagrangian-based investigation of multiphase flows by finite-time Lyapunov exponents

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Abstract

Multiphase flows are ubiquitous in our daily life and engineering applications. It is important to investigate the flow structures to predict their dynamical behaviors effectively. Lagrangian coherent structures (LCS) defined by the ridges of the finite-time Lyapunov exponent (FTLE) is utilized in this study to elucidate the multiphase interactions in gaseous jets injected into water and time-dependent turbulent cavitation under the framework of Navier-Stokes flow computations.

For the gaseous jets injected into water, the highlighted phenomena of the jet transportation can be observed by the LCS method, including expansion, bulge, necking/breaking, and back-attack. Besides, the observation of the LCS reveals that the back-attack phenomenon arises from the fact that the injected gas has difficulties to move toward downstream region after the necking/breaking. For the turbulent cavitating flow, the ridge of the FTLE field can form a LCS to capture the front and boundary of the re-entraint jet when the adverse pressure gradient is strong enough. It represents a barrier between particles trapped inside the circulation region and those moving downstream. The results indicate that the FTLE field has the potential to identify the structures of multiphase flows, and the LCS can capture the interface/barrier or the vortex/circulation region.

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Abbreviations

d e :

Diameter of nozzle exit

d s :

Diameter of cross-section of propulsion system

d t :

Diameter of nozzle throat

E g :

Total energy of gas

E w :

Total energy of water

k eff :

Effective thermal conductivity, k eff = k + k t

k g :

Thermal conductivity of gas

k t :

Turbulent thermal conductivity

k w :

Thermal conductivity of water

m +, m − :

Source and sink terms in the cavitation model

p 0 :

Stagnation pressure

p a :

Ambient pressure

p e :

Pressure at nozzle exit

Q :

Vortex definition criterion, Q = 1/2(|Ω|2 − |S|2)

S :

The rate-of strain tensor

t :

Time

t 0 :

Initial time

t ∞ :

Reference time scale

t*:

Normalized time scale, t* = t/t ∞

T :

Temperature

T LE :

Finite integration time interval

υ :

Velocity field

υ 0 :

Initial velocity

x :

Particle trajectory

x 0 :

Initial position of the particles

α g :

Volume fraction of gas

α 1 :

Volume fraction of liquid

α w :

Volume fraction of water

Δ :

Cauchy-Green deformation tensor

λ :

Eigenvalue of the Cauchy-Green deformation tensor

µg :

Dynamic viscosity of gas

µm :

Dynamic viscosity of mixture

µw :

Dynamic viscosity of water

ρ g :

Density of gas

ρ 1 :

Density of liquid

ρ m :

Density of mixture

ρ w :

Density of water

σ :

Lyapunov exponent

Ω :

Vorticity tensor

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Authors and Affiliations

  1. School of Aerospace Engineering, Bei**g Institute of Technology, 100081, Bei**g, China

    Jia-Ning Tang & Ning-Fei Wang

  2. Department of Mechanical and Electro-Mechanical Engineering, National Taiwan Sun Yat-Sen University, Kaohsiung, Taiwan, China

    Chien-Chou Tseng

  3. Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI, 48109, USA

    Jia-Ning Tang

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  1. Jia-Ning Tang
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Correspondence to Chien-Chou Tseng.

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Tang, JN., Tseng, CC. & Wang, NF. Lagrangian-based investigation of multiphase flows by finite-time Lyapunov exponents. Acta Mech Sin 28, 612–624 (2012). https://doi.org/10.1007/s10409-012-0037-3

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