Boundary-Layer Thickness - Flat Plate

This module contains a lot of functions to estimate the boundary-layer thicknesses of a incompressible/compressible, laminar/turbulent flat plate.

The four most important functions are: deltas_lam_ic, deltas_tur_ic, deltas_lam_c, deltas_tur_c as they are wrapper functions that allows to estimates all thicknesses with a single call.

pygasflow.atd.avf.thickness_fp.deltas_lam_ic(x, Re, to_dict=None)[source]

Compute different boundary-layer thicknesses for the incompressible laminar flat plate.

Parameters:

xfloat or array_like: Location where to compute the thickness.
Refloat or array_like: Free-stream Reynolds number computed at x.
to_dictbool, optional: If False, the function returns a list of results. If True, it returns a dictionary in which the keys are listed in the Returns section. Default to False (return a list of results).

Returns:

deltafloat or array_like: Boundary-layer thickness computed with the delta_lam_ic.
delta_1float or array_like: Boundary-layer displacement thickness computed with delta_1_lam_ic.
delta_2float or array_like: Momentum thickness computed with the delta_2_lam_ic.
H12float or array_like: Shape factor computed with the shape_factor_lam_ic.

See also

delta_lam_ic, delta_1_lam_ic, delta_2_lam_ic, shape_factor_lam_ic
deltas_lam_c, deltas_tur_ic, deltas_tur_c

Examples

The flow past a flat plate has a unit Reynolds number Re_u=10**6 m**-1. Assume incompressible laminar flow and determine on the plate at x=1.0 m the boundary-layer thicknesses:

>>> from pygasflow.atd.avf.thickness_fp import deltas_lam_ic
>>> import pint
>>> ureg = pint.UnitRegistry()
>>> Re_u = 1e06 / ureg.m
>>> x = 1 * ureg.m
>>> Re = Re_u * x
>>> res = deltas_lam_ic(x, Re, to_dict=True)
>>> res.show()
key      quantity     
----------------------
delta    δ [m]             0.00500000
delta_1  δ_1 [m]           0.00172080
delta_2  δ_2 [m]           0.00066410
H12      H12               2.59117603

pygasflow.atd.avf.thickness_fp.deltas_tur_ic(x, Re, to_dict=None)[source]

Compute different boundary-layer thicknesses for the incompressible turbulent flat plate.

Parameters:

xfloat or array_like: Location where to compute the thickness.
Refloat or array_like: Free-stream Reynolds number computed at x.
to_dictbool, optional: If False, the function returns a list of results. If True, it returns a dictionary in which the keys are listed in the Returns section. Default to False (return a list of results).

Returns:

deltafloat or array_like: Boundary-layer thickness computed with delta_tur_ic.
delta_vsfloat or array_like, optional: Viscous sub-layer thickness computed with delta_tur_ic_viscous.
delta_scfloat or array_like, optional: Scaling thickness computed with delta_tur_ic_scaling.
delta_1float or array_like: Boundary-layer displacement thickness computed with delta_1_tur_ic.
delta_2float or array_like: Momentum thickness computed with delta_2_tur_ic.
H12float or array_like: Shape factor computed with shape_factor_tur_ic.

See also

delta_tur_ic, delta_1_tur_ic, delta_2_tur_ic, shape_factor_tur_ic
deltas_tur_c, deltas_lam_ic, deltas_lam_c

Examples

The flow past a flat plate has a unit Reynolds number Re_u=10**6 m**-1. Assume incompressible turbulent flow and determine on the plate at x=1.0 m the boundary-layer thicknesses:

>>> from pygasflow.atd.avf.thickness_fp import deltas_tur_ic
>>> import pint
>>> ureg = pint.UnitRegistry()
>>> Re_u = 1e06 / ureg.m
>>> x = 1 * ureg.m
>>> Re = Re_u * x
>>> res = deltas_tur_ic(x, Re, to_dict=True)
>>> res.show()
key       quantity     
-----------------------
delta     δ [m]             0.02334542
delta_vs  δ_vs [m]          0.00011569
delta_sc  δ_sc [m]          0.00053538
delta_1   δ_1 [m]           0.00292133
delta_2   δ_2 [m]           0.00227145
H12       H12               1.28611111

pygasflow.atd.avf.thickness_fp.deltas_lam_c(x, Re, Tw_Tinf, Ts_Tinf, Minf, omega=0.65, gammainf=1.4, to_dict=None)[source]

Compute different boundary-layer thicknesses for the compressible laminar flat plate.

Parameters:

xfloat or array_like: Location where to compute the thickness.
Refloat or array_like: Free-stream Reynolds number computed at x.
Tw_Tinffloat or array_like: Temperature Ratio Tw / Tinf between the wall temperature and the free stream temperature Tinf.
Ts_Tinffloat or array_like: Temperature ratio T* / Tinf between the reference temperature T* and the free stream temperature Tinf.
Minffloat or array_like: Free stream Mach number.
omegafloat, optional: Exponent of the viscosity power law. Default to 0.65, corresponding to T > 400K. Set omega=1 otherwise.
gammainffloat, optional: Free stream specific heats ratio. Default to 1.4. Must be > 1.
to_dictbool, optional: If False, the function returns a list of results. If True, it returns a dictionary in which the keys are listed in the Returns section. Default to False (return a list of results).

Returns:

deltafloat or array_like: Boundary-layer thickness computed with delta_lam_c.
delta_1float or array_like: Boundary-layer displacement thickness computed with delta_1_lam_c.
delta_2float or array_like: Momentum thickness computed with delta_2_lam_c.
H12float or array_like: Shape factor computed with shape_factor_lam_c.

See also

delta_lam_c, delta_1_lam_c, delta_2_lam_c, shape_factor_lam_c
deltas_lam_ic, deltas_tur_ic, deltas_tur_c

Examples

The flow past a flat plate has a unit Reynolds number Re_u=10**6 m**-1. Assume compressible laminar flow and determine on the plate at x=1.0 m the boundary-layer thicknesses:

>>> from pygasflow.atd.avf.thickness_fp import deltas_lam_c
>>> import pint
>>> ureg = pint.UnitRegistry()
>>> Re_u = 1e06 / ureg.m
>>> x = 1 * ureg.m
>>> Re = Re_u * x
>>> Tw_Tinf = 2.1
>>> Ts_Tinf = 1.2
>>> Minf = 6
>>> res = deltas_lam_c(x, Re, Tw_Tinf, Ts_Tinf, Minf, to_dict=True)
>>> res.show()
key      quantity     
----------------------
delta    δ [m]             0.00581158
delta_1  δ_1 [m]           0.00806313
delta_2  δ_2 [m]           0.00064325
H12      H12              12.53507315

pygasflow.atd.avf.thickness_fp.deltas_tur_c(x, Re, Tw_Tinf, Ts_Tinf, Minf, omega=0.65, gammainf=1.4, to_dict=None)[source]

Compute different boundary-layer thicknesses for the compressible turbulent flat plate.

Parameters:

xfloat or array_like: Location where to compute the thickness.
Refloat or array_like: Free-stream Reynolds number computed at x.
Tw_Tinffloat or array_like: Temperature Ratio Tw / Tinf between the wall temperature and the free stream temperature Tinf.
Ts_Tinffloat or array_like: Temperature ratio T* / Tinf between the reference temperature T* and the free stream temperature Tinf.
Minffloat or array_like: Free stream Mach number.
omegafloat, optional: Exponent of the viscosity power law. Default to 0.65, corresponding to T > 400K. Set omega=1 otherwise.
gammainffloat, optional: Free stream specific heats ratio. Default to 1.4. Must be > 1.
to_dictbool, optional: If False, the function returns a list of results. If True, it returns a dictionary in which the keys are listed in the Returns section. Default to False (return a list of results).

Returns:

deltafloat or array_like: Boundary-layer thickness computed with delta_tur_c.
delta_vsfloat or array_like, optional: Viscous sub-layer thickness computed with delta_tur_c_viscous.
delta_scfloat or array_like, optional: Scaling thickness computed with delta_tur_c_scaling.
delta_1float or array_like: Boundary-layer displacement thickness computed with delta_1_tur_ic.
delta_2float or array_like: Momentum thickness computed with delta_2_tur_c.
H12float or array_like: Shape factor computed with shape_factor_tur_c.

See also

delta_tur_c, delta_1_tur_c, delta_2_tur_c, shape_factor_tur_c
deltas_tur_ic, deltas_lam_ic, deltas_lam_c

Examples

The flow past a flat plate has a unit Reynolds number Re_u=10**6 m**-1. Assume compressible turbulent flow and determine on the plate at x=1.0 m the boundary-layer thicknesses:

>>> from pygasflow.atd.avf.thickness_fp import deltas_tur_c
>>> import pint
>>> ureg = pint.UnitRegistry()
>>> Re_u = 1e06 / ureg.m
>>> x = 1 * ureg.m
>>> Re = Re_u * x
>>> Tw_Tinf = 2.1
>>> Ts_Tinf = 1.2
>>> Minf = 6
>>> res = deltas_tur_c(x, Re, Tw_Tinf, Ts_Tinf, Minf, to_dict=True)
>>> res.show()
key       quantity     
-----------------------
delta     δ [m]             0.02479314
delta_vs  δ_vs [m]          0.00015166
delta_sc  δ_sc [m]          0.00068105
delta_1   δ_1 [m]           0.01864145
delta_2   δ_2 [m]           0.00201025
H12       H12               9.27318000

Laminar Incompressible

pygasflow.atd.avf.thickness_fp.delta_lam_ic(x, Re, c=5)[source]

Boundary-layer thickness of the incompressible laminar flat plate.

Parameters:

xfloat or array_like: Location where to compute the thickness.
Refloat or array_like: Free-stream Reynolds number computed at x.
cfloat, optional: Proportionality constant from Blasius theory. Default to 5.

Returns:

outfloat or array_like

Notes

delta_lam_ic is the distance at which locally the tangential velocity component u(y) has approached the inviscid external velocity u_e by eps * u_e:

u_e - u(y) <= eps * u_e

Usually, the boundary-layer thickness is defined with eps=0.01, which gives c=5. If eps=0.001 is taken, then c=6.

References

“Basic of aerothermodynamics” by Ernst Heinrich

pygasflow.atd.avf.thickness_fp.delta_1_lam_ic(x, Re)[source]

Compute the integral parameter delta_1, also known as the boundary-layer displacement thickness, for the laminar incompressible case.

Parameters:

xfloat or array_like: Location where to compute the thickness.
Refloat or array_like: Free-stream Reynolds number computed at x.

Returns:

outfloat or array_like

See also

delta_2_lam_ic

References

“Basic of aerothermodynamics” by Ernst Heinrich

pygasflow.atd.avf.thickness_fp.delta_2_lam_ic(x, Re)[source]

Compute the integral parameter delta_2, also known as the momentum thickness, for the laminar incompressible case.

Parameters:

xfloat or array_like: Location where to compute the thickness.
Refloat or array_like: Free-stream Reynolds number computed at x.

Returns:

outfloat or array_like

See also

delta_1_lam_ic

References

“Basic of aerothermodynamics” by Ernst Heinrich

pygasflow.atd.avf.thickness_fp.shape_factor_lam_ic()[source]

Compute the shape factor, H12, for the laminar incrompressible case.

Returns:

outfloat

Laminar Compressible

pygasflow.atd.avf.thickness_fp.delta_lam_c(x, Re, Ts_Tinf, omega=0.65, c=5)[source]

Boundary-layer thickness of the compressible laminar flat plate.

Parameters:

xfloat or array_like: Location where to compute the thickness.
Refloat or array_like: Free-stream Reynolds number computed at x.
Ts_Tinffloat or array_like: Temperature ratio T* / Tinf between the reference temperature T* and the free stream temperature Tinf.
omegafloat: Exponent of the viscosity power law. Default to 0.65, corresponding to T > 400K. Set omega=1 otherwise.
cfloat, optional: Proportionality constant from Blasius theory. Default to 5.

Returns:

outfloat or array_like

See also

delta_lam_ic

References

“Basic of aerothermodynamics” by Ernst Heinrich

pygasflow.atd.avf.thickness_fp.delta_1_lam_c(x, Re, Tw_Tinf, Ts_Tinf, Minf, omega=0.65, gammainf=1.4)[source]

Compute the integral parameter delta_1, also known as the boundary-layer displacement thickness, for the laminar compressible case.

Parameters:

xfloat or array_like: Location where to compute the thickness.
Refloat or array_like: Free-stream Reynolds number computed at x.
Tw_Tinffloat or array_like: Temperature Ratio Tw / Tinf between the wall temperature and the free stream temperature Tinf.
Ts_Tinffloat or array_like: Temperature ratio T* / Tinf between the reference temperature T* and the free stream temperature Tinf.
Minffloat or array_like: Free stream Mach number.
omegafloat, optional: Exponent of the viscosity power law. Default to 0.65, corresponding to T > 400K. Set omega=1 otherwise.
gammainffloat, optional: Free stream specific heats ratio. Default to 1.4. Must be > 1.

Returns:

outfloat or array_like

pygasflow.atd.avf.thickness_fp.delta_2_lam_c(x, Re, Ts_Tinf, omega=0.65)[source]

Compute the integral parameter delta_2, also known as the momentum thickness, for the laminar compressible case.

Parameters:

xfloat or array_like: Location where to compute the thickness.
Refloat or array_like: Free-stream Reynolds number computed at x.
Ts_Tinffloat or array_like: Temperature ratio T* / Tinf between the reference temperature T* and the free stream temperature Tinf.
omegafloat, optional: Exponent of the viscosity power law. Default to 0.65, corresponding to T > 400K. Set omega=1 otherwise.

Returns:

outfloat or array_like

pygasflow.atd.avf.thickness_fp.shape_factor_lam_c(x, Re, Tw_Tinf, Minf, gammainf=1.4)[source]

Compute the shape factor, H12, for the laminar crompressible case.

Returns:

outfloat or array_like

Turbulent Incompressible

pygasflow.atd.avf.thickness_fp.delta_tur_ic(x, Re)[source]

Boundary-layer thickness of the incompressible turbulent flat plate. This is valid for a low Reynolds number. It comes from the (1/7) power velocity distribution law.

Parameters:

xfloat or array_like: Location where to compute the thickness.
Refloat or array_like: Free-stream Reynolds number computed at x.

Returns:

outfloat or array_like

References

“Basic of aerothermodynamics” by Ernst Heinrich

pygasflow.atd.avf.thickness_fp.delta_tur_ic_viscous(x, Re)[source]

Viscous sub-layer thickness of the incompressible turbulent flat plate.

Parameters:

xfloat or array_like: Location where to compute the thickness.
Refloat or array_like: Free-stream Reynolds number computed at x.

Returns:

outfloat or array_like

References

“Basic of aerothermodynamics” by Ernst Heinrich

pygasflow.atd.avf.thickness_fp.viscous_edge_velocity(ue, Re)[source]

Viscous sub-layer edge velocity. It is the velocity corresponding to delta_tur_ic_viscous.

Parameters:

uefloat or array_like: Edge velocity at the boundary layer.
Refloat or array_like: Free-stream Reynolds number computed at x.

Returns:

outfloat or array_like

References

“Basic of aerothermodynamics” by Ernst Heinrich

pygasflow.atd.avf.thickness_fp.delta_tur_ic_scaling(x, Re)[source]

Scaling thickness for the incompressible turbulent flat plate, where the non-dimensional velocity u+ and the wall distance y+ are equal. It is somewhat similar to the viscous sub-layer thickness.

Parameters:

xfloat or array_like: Location where to compute the thickness.
Refloat or array_like: Free-stream Reynolds number computed at x.

Returns:

outfloat or array_like

References

“Basic of aerothermodynamics” by Ernst Heinrich

pygasflow.atd.avf.thickness_fp.delta_1_tur_ic(x, Re)[source]

Compute the integral parameter delta_1, also known as the boundary-layer displacement thickness, for the turbulent incompressible case.

Parameters:

xfloat or array_like: Location where to compute the thickness.
Refloat or array_like: Free-stream Reynolds number computed at x.

Returns:

outfloat or array_like

See also

delta_2_tur_ic

References

“Basic of aerothermodynamics” by Ernst Heinrich

pygasflow.atd.avf.thickness_fp.delta_2_tur_ic(x, Re)[source]

Compute the integral parameter delta_1, also known as the momentum thickness, for the turbulent incompressible case.

Parameters:

xfloat or array_like: Location where to compute the thickness.
Refloat or array_like: Free-stream Reynolds number computed at x.

Returns:

outfloat or array_like

See also

delta_1_tur_ic

References

“Basic of aerothermodynamics” by Ernst Heinrich

pygasflow.atd.avf.thickness_fp.shape_factor_tur_ic()[source]

Compute the shape factor, H12, for the turbulent incrompressible case.

Returns:

outfloat

Turbulent Compressible

pygasflow.atd.avf.thickness_fp.delta_tur_c(x, Re, Ts_Tinf, omega=0.65)[source]

Boundary-layer thickness of the compressible turbulent flat plate. This is valid for a low Reynolds number. It comes from the (1/7) power velocity distribution law.

Parameters:

xfloat or array_like: Location where to compute the thickness.
Refloat or array_like: Free-stream Reynolds number computed at x.
Ts_Tinffloat or array_like: Temperature ratio T* / Tinf between the reference temperature T* and the free stream temperature Tinf.
omegafloat: Exponent of the viscosity power law. Default to 0.65, corresponding to T > 400K. Set omega=1 otherwise.

Returns:

outfloat or array_like

See also

delta_tur_ic

References

“Basic of aerothermodynamics” by Ernst Heinrich

pygasflow.atd.avf.thickness_fp.delta_tur_c_viscous(x, Re, Ts_Tinf, omega=0.65)[source]

Viscous sub-layer thickness of the compressible turbulent flat plate.

Parameters:

xfloat or array_like: Location where to compute the thickness.
Refloat or array_like: Free-stream Reynolds number computed at x.
Ts_Tinffloat or array_like: Temperature ratio T* / Tinf between the reference temperature T* and the free stream temperature Tinf.
omegafloat: Exponent of the viscosity power law. Default to 0.65, corresponding to T > 400K. Set omega=1 otherwise.

Returns:

outfloat or array_like

See also

delta_tur_ic_viscous

References

“Basic of aerothermodynamics” by Ernst Heinrich

pygasflow.atd.avf.thickness_fp.delta_tur_c_scaling(x, Re, Ts_Tinf, omega=0.65)[source]

Scaling thickness for the compressible turbulent flat plate, where the non-dimensional velocity u+ and the wall distance y+ are equal. It is somewhat similar to the viscous sub-layer thickness .

Parameters:

xfloat or array_like: Location where to compute the thickness.
Refloat or array_like: Free-stream Reynolds number computed at x.
Ts_Tinffloat or array_like: Temperature ratio T* / Tinf between the reference temperature T* and the free stream temperature Tinf.
omegafloat: Exponent of the viscosity power law. Default to 0.65, corresponding to T > 400K. Set omega=1 otherwise.

Returns:

outfloat or array_like

References

“Basic of aerothermodynamics” by Ernst Heinrich

pygasflow.atd.avf.thickness_fp.delta_1_tur_c(x, Re, Tw_Tinf, Ts_Tinf, Minf, omega=0.65, gammainf=1.4)[source]

Compute the integral parameter delta_1, also known as the boundary-layer displacement thickness, for the turbulent compressible case.

Parameters:

xfloat or array_like: Location where to compute the thickness.
Refloat or array_like: Free-stream Reynolds number computed at x.
Tw_Tinffloat or array_like: Temperature Ratio Tw / Tinf between the wall temperature and the free stream temperature Tinf.
Ts_Tinffloat or array_like: Temperature ratio T* / Tinf between the reference temperature T* and the free stream temperature Tinf.
Minffloat or array_like: Free stream Mach number.
omegafloat, optional: Exponent of the viscosity power law. Default to 0.65, corresponding to T > 400K. Set omega=1 otherwise.
gammainffloat, optional: Free stream specific heats ratio. Default to 1.4. Must be > 1.

Returns:

outfloat or array_like

pygasflow.atd.avf.thickness_fp.delta_2_tur_c(x, Re, Ts_Tinf, omega=0.65)[source]

Compute the integral parameter delta_2, also known as the momentum thickness, for the turbulent compressible case.

Parameters:

xfloat or array_like: Location where to compute the thickness.
Refloat or array_like: Free-stream Reynolds number computed at x.
Tw_Tinffloat or array_like: Temperature Ratio Tw / Tinf between the wall temperature and the free stream temperature Tinf.
Ts_Tinffloat or array_like: Temperature ratio T* / Tinf between the reference temperature T* and the free stream temperature Tinf.
Minffloat or array_like: Free stream Mach number.
omegafloat, optional: Exponent of the viscosity power law. Default to 0.65, corresponding to T > 400K. Set omega=1 otherwise.
gammainffloat, optional: Free stream specific heats ratio. Default to 1.4. Must be > 1.

Returns:

outfloat or array_like

pygasflow.atd.avf.thickness_fp.shape_factor_tur_c(Tw_Tinf, gammainf, Minf)[source]

Compute the shape factor, H12, for the turbulent incrompressible case.

Returns:

outfloat or array_like