Non-Dimensional Numbers

pygasflow.atd.nd_numbers.Reynolds(rho, u, mu, L=1)[source]

Compute the Reynolds number, which is the ratio between the inertial forces to the viscous forces.

Parameters

rhofloat or array_like: Density.
ufloat or array_like: Velocity.
mufloat or array_like: Viscosity.
Lfloat or array_like, optional: Characteristic length. Default to L=1, which computes the unitary Reynolds number.

Returns

Refloat or array_like

Notes

Reynolds number is the principle similarity parameter governing viscous phenomena:

\(Re \rightarrow 0\): the molecular transport of momentum is much larger than the convective transport, the flow is the “creeping” or Stokes flow. The convective transport can be neglected.
\(Re \rightarrow \infty\): the convective transport of momentum is much larger than the molecular transport, the flow can be considered as inviscid, i. e. molecular transport can be neglected.
\(Re = O(1)\): the molecular transport of momentum has the same order of magnitude as the convective transport, the flow is viscous, i. e. it is boundary-layer, or in general, shear-layer flow.

pygasflow.atd.nd_numbers.Prandtl(*args, **kwargs)[source]

Compute the Prandtl number.

There are 5 modes of operation:

Prandtl(mu, cp, k)
Prandtl(gas) where gas is a Cantera’s Solution object.
Prandtl(gamma) which is a good approximation for both monoatomic and polyatomic gases. It is derived from Eucken’s formula for thermal conductivity.
Prandtl(Pe=Pe, Re=Re) by providing Peclét and Reynolds numbers.
Prandtl(Le=Le, Sc=Sc) by providing Lewis and Schmidt numbers.

Parameters

mufloat or array_like: Viscosity of the gas.
cpfloat or array_like: Specific heat at constant pressure.
kfloat or array_like: Thermal conductivity of the gas.
Pefloat or array_like: Peclét number.
Refloat or array_like: Reynolds number.
Lefloat or array_like: Lewis number.
Scfloat or array_like: Schmidt number.
gammafloat: Specific heats ratio. Default to None. Must be \(\gamma > 1\).
gasct.Solution: A Cantera’s Solution object.

Returns

Prfloat or array_like: Prandtl number.

See also

Peclet, Lewis, Reynolds, Schmidt

Notes

\(Pr \rightarrow 0\): the thermal boundary layer is much thicker than the flow boundary layer, which is typical for the flow of liquid metals.
\(Pr \rightarrow \infty\): the flow boundary layer is much thicker than the thermal boundary layer, which is typical for liquids.
\(Pr = O(1)\): the thermal boundary layer has a thickness of the order of that of the flow boundary layer. This is typical for gases.

Examples

Compute the Prandtl number of air with specific heat ratio of 1.4:

>>> from pygasflow.atd.nd_numbers import Prandtl
>>> Prandtl(1.4)
0.7368421052631579

Compute the Prandtl number of air at T=350K using a Cantera’s Solution object:

>>> import cantera as ct
>>> air = ct.Solution("gri30.yaml")
>>> air.TPX = 350, ct.one_atm, {"N2": 0.79, "O2": 0.21}
>>> Prandtl(air)
0.7139365242266411

Compute the Prandtl number by providing mu, cp, k:

>>> from pygasflow.atd.viscosity import viscosity_air_southerland
>>> from pygasflow.atd.thermal_conductivity import thermal_conductivity_hansen
>>> cp = 1004
>>> mu = viscosity_air_southerland(350)
>>> k = thermal_conductivity_hansen(350)
>>> Prandtl(mu, cp, k)
0.7370392202421769

pygasflow.atd.nd_numbers.Knudsen(*args)[source]

Compute the Knudsen number.

There are 2 modes of operation:

Knudsen(lambda, L)
Knudsen(Minf, Reinf_L, gamma)

Parameters

lambdafloat or array_like: Mean free path in the gas.
Lfloat or array_like: Characteristic length, which must be chosen according to the flow under consideration.For example, for boundary-layer flow it would be based on the boundary-layer thickness.
Minffloat or array_like: Free stream Mach number.
Reinf_Lfloat or array_like: Free stream Reynolds number computed at a characteristic length.
gammafloat: Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

Knfloat or array_like

See also

Reynolds

Notes

Knudsen number is employed to distinguish approximately between flow regimes:

\(Kn \lessapprox 0.01\): continuum flow
\(0.01 \lessapprox Kn \lessapprox 0.1\): continuum flow with slip effects (slip flow and temperature jumps at a body surface).
\(0.1 \lessapprox Kn \lessapprox 10\): disturbed free molecular flow (gas particles collide with the body surface and with each other).
\(Kn \gtrapprox 10\): free molecular flow (gas particles collide only with the body surface).

pygasflow.atd.nd_numbers.Stanton(*args)[source]

Compute the Stanton number, which represents a dimensionless form of the heat flux q_gw.

There are 3 modes of operation:

Stanton(q_gw, q_inf)
Stanton(q_gw, rho_inf, v_inf)
Stanton(q_gw, rho_inf, v_inf, delta_h)

Parameters

q_gwfloat or array_like: Heat flux in the gas at the wall. It is the heat transported towards the surface of the vehicle by diffusion mechanisms.
q_inffloat or array_like: Heat transported towards a flight vehicle: q_inf = rho_inf * v_inf * h_t
rho_inffloat or array_like: Free stream density.
v_inffloat or array_like: Free stream velocity.
delta_hfloat or array_like: Difference between the enthalpy related to the recovery temperature and the enthalpy related to the wall temperature, hr - hw.

Returns

Snfloat or array_like

pygasflow.atd.nd_numbers.Strouhal(*args)[source]

Compute the Strouhal number.

There are 2 modes of operation:

Strouhal(t_res, t_ref)
Strouhal(L_ref, t_ref, v_ref)

Parameters

L_reffloat or array_like: Reference length (for example, the body vehicle length).
t_reffloat or array_like: Reference time.
v_reffloat or array_like: Reference velocity (for example, free stream velocity).
t_resfloat or array_like: Residence time, defined as t_res = L_ref / v_ref.

Returns

Srfloat or array_like

Notes

In our applications we speak about steady, quasi-steady, and unsteady flow problems. The measure for the distinction of these three flow modes is the Strouhal number, Sr:

\(Sr = 0\): steady flow.
\(Sr \rightarrow 0\): quasi-steady flow. The residence time is small compared to the reference time, in which a change of flow parameters happens. For practical purposes, quasi-steady flow is permitted for Sr <= 0.2.
\(Sr = O(1)\): unsteady flow.

Note: the movement of a flight vehicle may be permitted to be considered as at least quasi-steady, while at the same time truly unsteady movements of a control surface may occur. In addition there might be configuration details, where highly unsteady vortex shedding is present.

pygasflow.atd.nd_numbers.Peclet(rho, mu, cp, L, k)[source]

Compute the Peclét number.

Parameters

rhofloat or array_like: Density.
mufloat or array_like: Viscosity.
cpfloat or array_like: Specific heat at constant pressure.
Lfloat or array_like: Characteristic length.
kfloat or array_like: Thermal conductivity.

Returns

Pefloat or array_like

Notes

Peclét number relates the molecular transport of heat to the convective transport. In particular:

\(Pe \rightarrow 0\): the molecular transport of heat is much larger than the convective transport.
\(Pe \rightarrow \infty\): the convective transport of heat is much larger than the molecular transport.
\(Pe = O(1)\): the molecular transport of heat has the same order of magnitude as the convective transport.

pygasflow.atd.nd_numbers.Lewis(rho, D, cp, k)[source]

Compute the Lewis number, which is interpreted as the ratio of ‘heat transport by mass diffusion’ to ‘heat transport by conduction’ in a flow with chemical non-equilibrium.

Parameters

rhofloat or array_like: Density.
Dfloat or array_like: Mass diffusivity.
cpfloat or array_like: Specific heat at constant pressure.
kfloat or array_like: Thermal conductivity

Returns

Lwfloat or array_like

pygasflow.atd.nd_numbers.Eckert(M, gamma=1.4)[source]

Compute the Eckert number, which can be interpreted as ratio of kinetic energy to thermal energy of the flow.

Parameters

Mfloat or array_like: Mach number.
gammafloat: Specific heats ratio. Default to None. Must be \(\gamma > 1\).

Returns

Efloat or array_like

pygasflow.atd.nd_numbers.Schmidt(*args)[source]

Compute the Schmidt number.

There are 2 modes of operation:

Schmidt(Pr, Le)
Schmidt(rho, mu, D)

Parameters

Prfloat or array_like: Prandtl number.
Lefloat or array_like: Lewis number.
rhofloat or array_like: Density.
mufloat or array_like: Viscosity.
Dfloat or array_like: Mass diffusivity.

Returns

Scfloat or array_like