"""Dimensionless numbers: Reynolds, Froude, Mach, Strouhal, and Weber.
Fluid mechanics is governed by ratios, not absolute scales: two flows with
the same dimensionless numbers behave identically regardless of their size,
speed, or fluid, which is what makes wind-tunnel testing of a model
airplane meaningful for a full-size one. Each function here computes one
such ratio, and each result's magnitude is what determines which regime
(and which module of this package) actually applies -- a low Reynolds
number licenses :func:`physicskit.fluids.systems.viscous_flow.stokes_drag`,
a high one calls for
:class:`physicskit.fluids.systems.navier_stokes.NavierStokes2D` or the
turbulence tools in :mod:`physicskit.fluids.utils.spectral_analysis`.
"""
from __future__ import annotations
from physicskit.fluids.exceptions import InvalidParameterError
__all__ = [
"reynolds_number",
"froude_number",
"mach_number",
"strouhal_number",
"weber_number",
]
[docs]
def reynolds_number(velocity: float, length: float, nu: float) -> float:
"""The Reynolds number: inertial forces over viscous forces.
.. math::
Re = \\frac{U L}{\\nu}
Osborne Reynolds' 1883 pipe-flow experiments identified this ratio as
the single parameter controlling the transition from smooth (laminar)
to chaotic (turbulent) flow, regardless of the pipe's size or the
fluid's identity -- the founding result of dimensional-similarity
reasoning in fluid mechanics.
Parameters
----------
velocity : float
Characteristic flow speed `U`.
length : float
Characteristic length scale `L`.
nu : float
Kinematic viscosity.
Returns
-------
float
Reynolds number `Re`.
Raises
------
InvalidParameterError
If `length` or `nu` is not positive.
Examples
--------
>>> reynolds_number(velocity=2.0, length=0.5, nu=1e-6)
1000000.0
"""
if length <= 0:
raise InvalidParameterError(f"length must be positive, got {length}")
if nu <= 0:
raise InvalidParameterError(f"nu must be positive, got {nu}")
return velocity * length / nu
[docs]
def froude_number(velocity: float, length: float, g: float = 9.81) -> float:
"""The Froude number: inertial forces over gravitational forces.
.. math::
Fr = \\frac{U}{\\sqrt{gL}}
Governs free-surface (gravity wave) flows the way the Reynolds number
governs viscous ones: a ship model and its full-size counterpart make
the same waves, relative to their length, only if `Fr` matches. William
Froude's 1868 ship-hull towing-tank experiments established exactly
this scaling law.
Parameters
----------
velocity : float
Characteristic flow speed `U`.
length : float
Characteristic length scale `L`.
g : float, default 9.81
Gravitational acceleration.
Returns
-------
float
Froude number `Fr`.
Raises
------
InvalidParameterError
If `length` or `g` is not positive.
Examples
--------
>>> round(froude_number(velocity=3.0, length=9.81, g=9.81), 4)
0.3058
"""
if length <= 0:
raise InvalidParameterError(f"length must be positive, got {length}")
if g <= 0:
raise InvalidParameterError(f"g must be positive, got {g}")
return velocity / (g * length) ** 0.5
[docs]
def mach_number(velocity: float, speed_of_sound: float) -> float:
"""The Mach number: flow speed over the local speed of sound.
.. math::
M = \\frac{U}{c}
The single number that decides whether compressibility can be ignored
(:math:`M \\ll 1`, the regime of every other module in this package
except :mod:`physicskit.fluids.systems.compressible_flow`) or dominates
the flow (:math:`M \\gtrsim 1`, where :func:`~physicskit.fluids.systems.compressible_flow.normal_shock_relations`
applies).
Parameters
----------
velocity : float
Flow speed `U`.
speed_of_sound : float
Local speed of sound `c`.
Returns
-------
float
Mach number `M`.
Raises
------
InvalidParameterError
If `speed_of_sound` is not positive.
Examples
--------
>>> mach_number(velocity=340.0, speed_of_sound=340.0)
1.0
"""
if speed_of_sound <= 0:
raise InvalidParameterError(f"speed_of_sound must be positive, got {speed_of_sound}")
return velocity / speed_of_sound
[docs]
def strouhal_number(frequency: float, length: float, velocity: float) -> float:
"""The Strouhal number: a dimensionless vortex-shedding frequency.
.. math::
St = \\frac{f L}{U}
For flow past a bluff body shedding a
:func:`~physicskit.fluids.systems.vortex_dynamics.von_karman_vortex_street`
at frequency `f`, `St` stays remarkably constant (near 0.2 for a
circular cylinder) over a wide range of Reynolds number, letting a
single scaled model predict the shedding frequency at full scale.
Parameters
----------
frequency : float
Vortex-shedding frequency `f`.
length : float
Characteristic length scale `L` (e.g. the body's cross-stream width).
velocity : float
Characteristic flow speed `U`.
Returns
-------
float
Strouhal number `St`.
Raises
------
InvalidParameterError
If `velocity` is not positive.
Examples
--------
>>> round(strouhal_number(frequency=2.0, length=0.1, velocity=1.0), 4)
0.2
"""
if velocity <= 0:
raise InvalidParameterError(f"velocity must be positive, got {velocity}")
return frequency * length / velocity
[docs]
def weber_number(rho: float, velocity: float, length: float, surface_tension: float) -> float:
"""The Weber number: inertial forces over surface-tension forces.
.. math::
We = \\frac{\\rho U^2 L}{\\sigma}
Governs whether a free liquid surface (a droplet, a jet, a bubble)
deforms and breaks up under its own inertia (:math:`We \\gg 1`) or is
held together by surface tension (:math:`We \\lesssim 1`) -- the
parameter that decides, for instance, whether a jet of liquid breaks
into droplets.
Parameters
----------
rho : float
Fluid density.
velocity : float
Characteristic flow speed `U`.
length : float
Characteristic length scale `L` (e.g. a droplet diameter).
surface_tension : float
Surface tension coefficient :math:`\\sigma`.
Returns
-------
float
Weber number `We`.
Raises
------
InvalidParameterError
If `surface_tension` is not positive.
Examples
--------
>>> round(weber_number(rho=1000.0, velocity=1.0, length=0.001, surface_tension=0.072), 4)
13.8889
"""
if surface_tension <= 0:
raise InvalidParameterError(f"surface_tension must be positive, got {surface_tension}")
return rho * velocity**2 * length / surface_tension
