viscosity

Definitions of viscous constitutive relations for compressible flow

This module defines the dynamic viscosity and heat conductivity for the compressible flow solvers, which are used to compute the viscous stress tensor and the heat flux in the Navier-Stokes equations.

We derive the dimensionless viscous stress tensor \(\mat{\tau}\) and heat flux \(\vec{q}\) from the dimensional one, namely:

\[\begin{split}\overline{\mat{\tau}} &= 2 \overline{\mu} \left[ \frac{\overline{\nabla} \overline{\vec{u}} + (\overline{\nabla} \overline{\vec{u}})^\T}{2} - \frac{1}{3} \overline{\div}(\overline{\vec{u}}) \right], \\ \rho_{ref} u^2_{ref} \mat{\tau} &= 2\mu \overline{\mu}_\infty \frac{u_{ref}}{L_{ref}} \left[ \frac{\nabla \vec{u} + (\nabla \vec{u})^\T}{2} - \frac{1}{3} \div(\vec{u}) \right], \\ \mat{\tau} &= 2 \mu \frac{\overline{\mu}_\infty}{\rho_{ref} u_{ref} L_{ref}} \left[ \frac{\nabla \vec{u} + (\nabla \vec{u})^\T}{2} - \frac{1}{3} \div(\vec{u}) \right], \\ \mat{\tau} &= \frac{2 \mu}{\Re_{ref}} \left[ \frac{\nabla \vec{u} + (\nabla \vec{u})^\T}{2} - \frac{1}{3} \div(\vec{u}) \right],\end{split}\]

and

\[\begin{split}\overline{\vec{q}} &= -\overline{k} \overline{\nabla} \overline{T}, \\ \rho_{ref} u^3_{ref} \vec{q} &= -\overline{k}_\infty \frac{T_{ref}}{L_{ref}} k \nabla T, \\ \vec{q} &= -\frac{\overline{k}_\infty}{\overline{\mu}_\infty} \frac{T_{ref}}{u^2_{ref}} \frac{\overline{\mu}_\infty}{\rho_{ref} u_{ref}L_{ref}} k \nabla T, \\ \vec{q} &= -\frac{\overline{k}_\infty}{\overline{\mu}_\infty c_p} \frac{\overline{\mu}_\infty}{\rho_{ref} u_{ref}L_{ref}} k \nabla T, \\ \vec{q} &= -\frac{k}{\Pr_\infty \Re_{ref}} \nabla T.\end{split}\]

To set the dimensionless dynamic viscosity \(\mu\) and heat conductivity \(k\), the following definitions are available:

Inviscid

\[\mu := 0, \quad k := 0.\]

Constant viscosity and heat conductivity

\[\mu := 1, \quad k:= 1.\]

Sutherland viscosity

The dimensionless dynamic viscosity \(\mu\) is derived from the dimensional one as follows:

\[\begin{split}\overline{\mu}(\overline{T}) &= \overline{\mu}_\infty \left( \frac{\overline{T}}{\overline{T}_\infty} \right)^{3/2} \frac{\overline{T}_\infty + \overline{S}}{\overline{T} + \overline{S}}, \\ \overline{\mu}_\infty \mu(\overline{T}) &= \overline{\mu}_\infty \left( \frac{\overline{T}}{\overline{T}_\infty} \right)^{3/2} \frac{T_{ref}}{T_{ref}} \frac{\overline{T}_\infty + \overline{S}}{\overline{T} + \overline{S}}, \\ \mu(T) &= \left( \frac{T}{T_\infty} \right)^{3/2} \frac{T_\infty + S}{T + S}.\end{split}\]

with \(S = \overline{S}/T_{ref}\).

For the heat conductivity \(k\) the same relation holds:

\[k(T) = \left( \frac{T}{T_\infty} \right)^{3/2} \frac{T_\infty + S}{T + S}.\]

See scaling for the definition of the reference Reynolds number \(\Re_{ref}\).

class Constant(mesh: Mesh, root: Configuration = None, **default)

viscosity(U: flowfields): Returns the dynamic viscosity and heat conductivity for a constant viscosity flow.

\[\mu = k = 1\]

class DynamicViscosity(mesh: Mesh, root: Configuration = None, **default): Base class for dynamic viscosity and heat conductivity definitions.

class Inviscid(mesh: Mesh, root: Configuration = None, **default)

viscosity(U: flowfields)

Returns the dynamic viscosity and heat conductivity for an inviscid flow.

\[\mu = k = 0\]

Raise:: TypeError If the dynamic viscosity is requested in an inviscid setting.

class Sutherland(mesh, root=None, **default)

property sutherland_temperature: float

The Sutherland temperature \(\overline{S}\) in Kelvin.

Getter:: Returns the Sutherland temperature.
Setter:: Sets the Sutherland temperature. Defaults to \(110.4 \, K\)

viscosity(U: flowfields): Returns the dynamic viscosity and heat conductivity for a Sutherland flow.

\[\mu(T) = k(T) = \left( \frac{T}{T_\infty} \right)^{3/2} \frac{T_\infty + S}{T + S}\]