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Sigma Relative Velocity Ao2008

Particula Index / Particula / Dynamics / Coagulation / Turbulent Dns Kernel / Sigma Relative Velocity Ao2008

Auto-generated documentation for particula.dynamics.coagulation.turbulent_dns_kernel.sigma_relative_velocity_ao2008 module.

VelocityCorrelationTerms

Show source in sigma_relative_velocity_ao2008.py:35

Parameters from computing velocity correlation terms.

Signature

class VelocityCorrelationTerms(NamedTuple): ...

_compute_cross_correlation_velocity

Show source in sigma_relative_velocity_ao2008.py:214

Compute the cross-correlation of the fluctuating velocities of droplets

The function is given by:

⟨ v'^(1) v'^(2) ⟩ = (u'² f₂(R) / (τ_p1 τ_p2)) * [b₁ d₁ Φ(c₁, e₁) - b₁ d₂ Φ(c₁, e₂) - b₂ d₁ Φ(c₂, e₁) + b₂ d₂ Φ(c₂, e₂)]

  • u' (turbulence_intensity) : Fluid RMS fluctuation velocity [m/s].
  • τ_p1, τ_p2 : Inertia timescales of droplets 1 and 2 [s].
  • f₂(R) : Longitudinal velocity correlation function.
  • Φ(c, e) : Function Φ computed using get_phi_ao2008.

Arguments

- turbulence_intensity : Fluid RMS fluctuation velocity [m/s].
- collisional_radius : Distance between two colliding droplets [m].
- `-` *particle_inertia_time* - Inertia timescale of droplet 1 [s].
- `-` *particle_velocity* - Droplet velocity [m/s].
- taylor_microscale : Taylor microscale [m].
- eulerian_integral_length : Eulerian integral length scale [m].
- velocity_correlation_terms : Velocity correlation coefficients [-].

Returns

- Cross-correlation velocity ⟨ v'^(1) v'^(2) ⟩ [m²/s²].

References

  • Ayala, O., Rosa, B., & Wang, L. P. (2008). Effects of turbulence on the geometric collision rate of sedimenting droplets. Part 2. Theory and parameterization. New Journal of Physics, 10. https://doi.org/10.1088/1367-2630/10/7/075016

Signature

@validate_inputs(
    {
        "turbulence_intensity": "positive",
        "collisional_radius": "positive",
        "particle_inertia_time": "positive",
        "particle_velocity": "positive",
        "taylor_microscale": "positive",
        "eulerian_integral_length": "positive",
    }
)
def _compute_cross_correlation_velocity(
    turbulence_intensity: float,
    collisional_radius: Union[float, NDArray[np.float64]],
    particle_inertia_time: Union[float, NDArray[np.float64]],
    particle_velocity: Union[float, NDArray[np.float64]],
    taylor_microscale: float,
    eulerian_integral_length: float,
    velocity_correlation_terms: VelocityCorrelationTerms,
) -> Union[float, NDArray[np.float64]]: ...

See also

_compute_rms_fluctuation_velocity

Show source in sigma_relative_velocity_ao2008.py:132

Compute the square of the RMS fluctuation velocity for the k-th droplet.

The function is given by:

⟨ (v'^(k))² ⟩ = (u'² / τ_pk) * [b₁ d₁ Ψ(c₁, e₁) - b₁ d₂ Ψ(c₁, e₂) - b₂ d₁ Ψ(c₂, e₁) + b₂ d₂ Ψ(c₂, e₂)]

  • u' (turbulence_intensity) : Fluid RMS fluctuation velocity [m/s].
  • τ_pk (particle_inertia_time) : Inertia timescale of the droplet k [s].
  • Ψ(c, e) : Function Ψ computed using get_psi_ao2008.

Arguments

- turbulence_intensity : Fluid RMS fluctuation velocity [m/s].
- particle_inertia_time : Inertia timescale of the droplet k [s].
- velocity_correlation_terms : Velocity correlation coefficients [-].

Returns

- RMS fluctuation velocity squared ⟨ (v'^(k))² ⟩ [m²/s²].

References

  • Ayala, O., Rosa, B., & Wang, L. P. (2008). Effects of turbulence on the geometric collision rate of sedimenting droplets. Part 2. Theory and parameterization. New Journal of Physics, 10. https://doi.org/10.1088/1367-2630/10/7/075016

Signature

@validate_inputs(
    {"turbulence_intensity": "positive", "particle_inertia_time": "positive"}
)
def _compute_rms_fluctuation_velocity(
    turbulence_intensity: float,
    particle_inertia_time: Union[float, NDArray[np.float64]],
    velocity_correlation_terms: VelocityCorrelationTerms,
) -> Union[float, NDArray[np.float64]]: ...

See also

get_relative_velocity_variance

Show source in sigma_relative_velocity_ao2008.py:48

Compute the variance of particle relative-velocity fluctuation.

The function is given by:

σ² = ⟨ (v'^(2))² ⟩ + ⟨ (v'^(1))² ⟩ - 2 ⟨ v'^(1) v'^(2) ⟩

  • ⟨ (v'^(k))² ⟩ is the square of the RMS fluctuation velocity for droplet k
  • ⟨ v'^(1) v'^(2) ⟩ is the cross-correlation of the fluctuating velocities.

Arguments

- turbulence_intensity : Fluid RMS fluctuation velocity [m/s].
- collisional_radius : Distance between two colliding droplets [m].
- particle_inertia_time : Inertia timescale of droplet 1 [s].
- particle_velocity : Droplet velocity [m/s].
- taylor_microscale : Taylor microscale [m].
- eulerian_integral_length : Eulerian integral length scale [m].
- lagrangian_integral_time : Lagrangian integral time scale [s].

Returns

- σ² : Variance of particle relative-velocity fluctuation,
    (n, n) matrix where n is number of particles [m²/s²],

References

  • Ayala, O., Rosa, B., & Wang, L. P. (2008). Effects of turbulence on the geometric collision rate of sedimenting droplets. Part 2. Theory and parameterization. New Journal of Physics, 10. https://doi.org/10.1088/1367-2630/10/7/075016

Signature

@validate_inputs(
    {
        "turbulence_intensity": "positive",
        "collisional_radius": "positive",
        "particle_inertia_time": "positive",
        "particle_velocity": "positive",
    }
)
def get_relative_velocity_variance(
    turbulence_intensity: float,
    collisional_radius: NDArray[np.float64],
    particle_inertia_time: NDArray[np.float64],
    particle_velocity: NDArray[np.float64],
    taylor_microscale: float,
    eulerian_integral_length: float,
    lagrangian_integral_time: float,
) -> Union[float, NDArray[np.float64]]: ...