scaling

Definitions of dimensionless compressible flow equations.

This module defines the scaling for compressible flow equations, which are used to scale the governing equations of compressible flow to dimensionless form.

In the following, we derive the dimensionless compressible flow equations based on the choice of three different scalings: aerodynamic, acoustic, and aeroacoustic. We denote the dimensionful and dimensionless variables with and without overline e.g. the density \(\overline{\rho}, \rho\), respectively. Let us introduce the reference quantities used for the:

Reference quantities

Quantity	Aerodynamic	Acoustic	Aeroacoustic
\(\rho_{ref}\)	\(\overline{\rho}_\infty\)	\(\overline{\rho}_\infty\)	\(\overline{\rho}_\infty\)
\(u_{ref}\)	\(|\overline{\vec{u}}_\infty|\)	\(\overline{c}_\infty\)	\(|\overline{\vec{u}}_\infty| + \overline{c}_\infty\)
\(T_{ref}\)	\(\overline{T}_\infty \Ma_\infty^2 (\gamma - 1)\)	\(\overline{T}_\infty (\gamma - 1)\)	\(\overline{T}_\infty (1+\Ma_\infty)^2 (\gamma - 1)\)
\(L_{ref}\)	\(\overline{L}\)	\(\overline{L}\)	\(\overline{L}\)

From Buckingham \(\pi\) theorem, the remaining quantitites are all functions of the reference quantities above:

\(p_{ref}\)	\(\rho_{ref} u_{ref}^2\)	\(\rho_{ref} u_{ref}^2\)	\(\rho_{ref} u_{ref}^2\)
\(t_{ref}\)	\(\frac{L_{ref}}{u_{ref}}\)	\(\frac{L_{ref}}{u_{ref}}\)	\(\frac{L_{ref}}{u_{ref}}\)
\(\rho E_{ref}\)	\(\rho_{ref} u_{ref}^2\)	\(\rho_{ref} u_{ref}^2\)	\(\rho_{ref} u_{ref}^2\)

With these reference quantities, we can now start deriving the dimensionless compressible flow equations.

Continuity equation

\[\begin{split}\frac{\partial \overline{\rho}}{\partial \overline{t}} + \overline{\div}(\overline{\rho \vec{u}}) &= 0, \\ \frac{\rho_{ref}}{t_{ref}}\frac{\partial \rho}{\partial t} + \frac{\rho_{ref} u_{ref}}{L_{ref}} \div(\rho \vec{u}) &= 0, \\ \frac{\partial \rho}{\partial t} + \div(\rho \vec{u}) &= 0.\end{split}\]

Momentum equation

\[\begin{split}\frac{\partial \overline{\rho \vec{u}}}{\partial \overline{t}} + \overline{\div}(\overline{\rho \vec{u}} \otimes \overline{\vec{u}} + \overline{p} \mat{I} - \overline{\mat{\tau}})&= 0, \\ \frac{\rho_{ref} u_{ref}}{t_{ref}}\frac{\partial \rho \vec{u}}{\partial t} + \frac{\rho_{ref} u_{ref}^2}{L_{ref}} \div(\rho \vec{u} \otimes \vec{u} + p \mat{I} - \mat{\tau})&= 0, \\ \frac{\partial \rho \vec{u}}{\partial t} + \div(\rho \vec{u} \otimes \vec{u} + p \mat{I} - \mat{\tau})&= 0.\end{split}\]

Energy equation

\[\begin{split}\frac{\partial \overline{\rho E}}{\partial \overline{t}} + \overline{\div}((\overline{\rho E} + \overline{p}) \overline{\vec{u}} - \overline{\mat{\tau}}\overline{\vec{u}} + \overline{\vec{q}}) &= 0, \\ \frac{\rho E_{ref}}{t_{ref}}\frac{\partial \rho E}{\partial t} + \frac{\rho_{ref} u^3_{ref}}{L_{ref}} \div((\rho E + p) \vec{u} - \mat{\tau}\vec{u} + \vec{q}) &= 0, \\ \frac{\partial \rho E}{\partial t} + \div((\rho E + p) \vec{u} - \mat{\tau}\vec{u} + \vec{q}) &= 0.\end{split}\]

class Acoustic(mesh, root=None, **default)

property reference_velocity: CoefficientFunction

Returns the reference velocity.

\[u_{ref} = \overline{c}_\infty\]

property velocity

Returns the dimensionless farfield velocity.

\[|\vec{u}_\infty| = \Ma_\infty\]

property reference_temperature: CoefficientFunction

Returns the reference temperature.

\[T_{ref} = \overline{T}_\infty (\gamma - 1)\]

property temperature: CoefficientFunction

Returns the dimensionless farfield temperature.

\[T_\infty = \frac{1}{(\gamma - 1)}\]

property pressure: CoefficientFunction

Returns the dimensionless farfield pressure.

\[p_\infty = \frac{1}{\gamma}\]

property reference_reynolds_number: CoefficientFunction

Returns the reference Reynolds number.

\[\Re_{ref} = \frac{\Re_\infty}{\Ma_\infty}\]

property speed_of_sound: float

Returns the dimensionless farfield speed of sound.

\[c_\infty = 1\]

class Aeroacoustic(mesh, root=None, **default)

property reference_velocity: CoefficientFunction

Returns the reference velocity.

\[u_{ref} = |\overline{\vec{u}}_\infty| + \overline{c}_\infty\]

property velocity

Returns the dimensionless farfield velocity.

\[|\vec{u}_\infty| = \frac{\Ma_\infty}{1 + \Ma_\infty}\]

property reference_temperature: CoefficientFunction

Returns the reference temperature.

\[T_{ref} = \overline{T}_\infty (1+ \Ma_\infty)^2 (\gamma - 1)\]

property temperature: CoefficientFunction

Returns the dimensionless farfield temperature.

\[T_\infty = \frac{1}{(1+\Ma_\infty)^2 (\gamma - 1)}\]

property pressure: CoefficientFunction

Returns the dimensionless farfield pressure.

\[p_\infty = \frac{1}{(1+\Ma_\infty)^2 \gamma}\]

property reference_reynolds_number: CoefficientFunction

Returns the reference Reynolds number.

\[\Re_{ref} = \frac{\Re_\infty (1+\Ma_\infty)}{\Ma_\infty}\]

property speed_of_sound: float

Returns the dimensionless farfield speed of sound.

\[c_\infty = \frac{1}{1 + \Ma_\infty}\]

class Aerodynamic(mesh, root=None, **default)

property reference_velocity: CoefficientFunction

Returns the reference velocity.

\[u_{ref} = |\overline{\vec{u}}_\infty|\]

property velocity

Returns the dimensionless farfield velocity.

\[|\vec{u}_\infty| = 1\]

property reference_temperature: CoefficientFunction

Returns the reference temperature.

\[T_{ref} = \overline{T}_\infty \Ma^2_\infty (\gamma - 1)\]

property temperature: CoefficientFunction

Returns the dimensionless farfield temperature.

\[T_\infty = \frac{1}{\Ma^2_\infty (\gamma - 1)}\]

property pressure: CoefficientFunction

Returns the dimensionless farfield pressure.

\[p_\infty = \frac{1}{\Ma^2_\infty \gamma}\]

property reference_reynolds_number: CoefficientFunction

Returns the reference Reynolds number.

\[\Re_{ref} = \Re_\infty\]

property speed_of_sound: float

Returns the dimensionless farfield speed of sound.

\[c_\infty = \frac{1}{\Ma_\infty}\]

class Scaling(mesh, root=None, **default)

Base class for compressible flow scalings.

property farfield: dict[str, CoefficientFunction]

Returns the dimensionful farfield values \(\overline{\cdot}_\infty\) for the scaling.

Getter:

Returns the farfield values as dictionary.

Setter:

Sets the farfield values, defaults to

• \(\overline{L} = 1 \, \mathrm{m}\)

• \(\overline{\rho}_\infty = 1.293 \, \mathrm{kg/m^3}\)

• \(|\overline{\vec{u}}_\infty| = 1 \, \mathrm{m/s}\)

• \(\overline{c}_\infty = 343 \, \mathrm{m/s}\)

• \(\overline{T}_\infty = 293.15 \, \mathrm{K}\)

• \(\overline{p}_\infty = 101325 \, \mathrm{Pa}\)

property reference_density: CoefficientFunction

Returns the reference density.

\[\rho_{ref} = \overline{\rho}_\infty\]

property density: float

Returns the dimensionless farfield density.

\[\rho_\infty = 1\]