navierStokesViscousOperator (1d, 2d, 3d)¶

The navierStokesViscousOperator computes the viscous stress and thermal conduction terms, with contributions from laminar (and optionally, turbulent) viscosity in the Navier Stokes equations using a conservative least squares interpolation scheme.

USim implements the viscous stress terms in the momentum and total energy equations in the Navier-Stokes operator using the form:

\[\notag \begin{align} \mathcal{S} \left(\mathbf{w}\right) = \left[ \begin{array}{c} \mathcal{S}_{\rho \mathbf{u}} \left(\mathbf{w}\right) \\ S_{E} \left(\mathbf{w}\right) \end{array} \right] = \left[ \begin{array}{c} c \nabla \cdot \mathbb{S} \left(\mathbf{w}\right) \\ c \nabla \cdot \left\{ \mathbb{S} \left(\mathbf{w}\right) \cdot \mathbf{u} \right\} \end{array} \right] \\ \mathbb{S} \left(\mathbf{w}\right) = 2 \left[\mu + \mu_{\mathrm{turb}} \right] \left[ \frac{\nabla \mathbf{u} + \left( \nabla \mathbf{u} \right)^{\mathrm{T}}}{2} - \frac{\mathbb{I}}{3} \nabla \cdot \mathbf{u} \right] - \frac{2 \mathbb{I}}{3} \mu_{\mathrm{turb}} \rho k \end{align}\]

Here, \(\mathbf{w}\) is the primitive state vector, \(\mathbf{u}\) is the fluid velocity, \(\mu\) is the laminar viscosity, \(\mu_{\mathrm{turb}}\) is the turbulent viscosity, \(\rho\) is the fluid density and \(k\) is the local local turbulent kinetic energy. The laminar viscosity can be computed through (e.g.) Sutherland’s law:

\[\notag \begin{align} \mu = \mu_0 \left( \frac{T}{T_0} \right)^{\frac{3}{2}} \left( \frac{T_0+S}{T_0+S} \right) \end{align}\]

where \(\mu_0 = 1.716 \times 10^{-5} \mathrm{kg} \; (\mathrm{ms} )^{-1}\), \(T_0 = 491.6R\), \(S = 198.6R\).

USim implements the thermal conduction terms in the total energy equation in the Navier-Stokes operator using the form:

\[\notag \begin{align} S_{E} \left(\mathbf{w}\right) = c \nabla \cdot \left[ -\left(\kappa + \kappa_{\mathrm{turb}}\right) \nabla T \right] \end{align}\]

where \(\kappa\) and \(\kappa_{\mathrm{turb}}\) are the laminar and turbulent heat conductivity, which can be modelled through:

\[\notag \begin{align} \kappa = \frac{c_p \mu}{\mathrm{Pr}}; \;\; \kappa_\mathrm{turb} = \frac{c_p \mu_\mathrm{turb}}{\mathrm{Pr}_\mathrm{turb}} \end{align}\]

where \(c_p\) is the heat capacity at constant pressure and \(\mathrm{Pr}\) and \(\mathrm{Pr}_\mathrm{turb}\) are the laminar and turbulent Prandtl numbers.

Data¶

in (string vector, required)

Defined by the choice of the enableThermal, enableViscous, and enableTurbulence flags defined below in the Parameters section. Input variables must be in the order listed below. The rules for the which arrays should be included in the vector are as follows:

fluid velocity: required if enableViscous = true or enableTurbulence = true
total viscosity: required if enableViscous = true or enableTurbulence = true
fluid temperature: required if enableThermal = true or enableTurbulence = true
total thermal conductivity: required if enableThermal = true or enableTurbulence = true
turbulence model: required if enableTurbulence = true
turbulence viscosity: required if enableTurbulence = true

Note

If enableTurbulence is true and enableViscous is false, then enableTurbulence has no effect.

If enableTurbulence and enableThermal don’t impact each other algorithmically.

If enableTurbulence is true, then the input variables are required to be present.

Fluid velocity (nodalArray, 3-components, optional): Vector of fluid velocities, required if enableViscous = true: 0. \(u_{\hat{\mathbf{i}}} = \mathbf{u} \cdot \hat{\mathbf{i}}\): velocity in the \(\hat{\mathbf{i}}\) direction 1. \(u_{\hat{\mathbf{j}}} = \mathbf{u} \cdot \hat{\mathbf{j}}\): velocity in the \(\hat{\mathbf{j}}\) direction 2. \(u_{\hat{\mathbf{k}}} = \mathbf{u} \cdot \hat{\mathbf{k}}\): velocity in the \(\hat{\mathbf{k}}\) direction
Total viscosity (nodalArray, 1-components, optional): Sum of the laminar and turbulent viscosities, \(\mu + \mu_{\mathrm{turb}}\) (if turbulence is not modelled, the latter can be ommitted). Required if enableViscous = true:
Fluid Temperature (nodalArray, 1-components, optional): Required if enableThermal = true.
Total thermal conductivity (nodalArray, 1-components, optional): Sum of the laminar and turbulent thermal conductivities, \(\mu + \mu_{\mathrm{turb}}\) (if turbulence is not modelled, the latter can be ommitted). Required if enableThermal = true.
Turbulence model (nodalArray, 1-components, optional): Required if enableTurbulence = true. One of:

Turbulent kinetic energy density \(\rho k - \rho \epsilon\)

Turbulent kinetic energy density dissipation rate \(\rho k - \rho \omega\)
Turbulent viscosity (nodalArray, 1-components, optional): Turbulent viscosities, \(\mu_{\mathrm{turb}}\). Required if enableTurbulence = true:

out (string vector, required)

Vector of Fluxes (nodalArray, 4-components)

\(\mathcal{S}\left( \rho\,u_{\hat{\mathbf{i}}} \right)\): \(\hat{\mathbf{i}}\) momentum flux
\(\mathcal{S}\left( \rho\,u_{\hat{\mathbf{j}}} \right)\): \(\hat{\mathbf{j}}\) momentum flux
\(\mathcal{S}\left( \rho\,u_{\hat{\mathbf{k}}} \right)\): \(\hat{\mathbf{k}}\) momentum flux
\(\mathcal{S}\left( E \right)\): total energy flux

Parameters¶

The navierStokesViscousOperator updater accepts the parameters below, in addition to those required by Updater:

orderAccuracy (integer, required)

Order of the polynomial that is used to form the operator. Choice of 1, 2 or 3 corresponding, respectively to first, second and third order accuracy. The appropriate choice of order varies on the problem type and the mesh used.

numberOfInterpolationPoints (integer, required)

Number of points to be considerd for the least squares fit. This parameter varies from mesh to mesh and should be determined by computing a known function on the mesh.

The numberOfInterpolationPoints must be greater than (or equal to) the number of coefficients in the polynomial approximation. This means that in 1d the value is 4, in 2D the value is at least 6 and in 3D the value is at least 10.

These choices do not guarantee that a matrix inverse will be found. The following values though appear to be adequate in general: in 1D 4; in 2D 8 and in 3D 20.

coefficient (float, required)

Constant floating point value, c that multiplies the output of the diffusion updater.

enableThermal (boolean, optional)

Tell USim whether to include the contribution from heat conduction. Default true.

enableViscous (boolean, optional)

Tell USim whether to inclue the contribution from viscous stress. Default true.

enableTurbulence (boolean, optional)

Tell USim whether to include the turbulence. Default false.

Examples¶

<Updater computeViscousSource>
  kind = navierStokesViscousOperator2d
  onGrid = domain

  coefficient = 1.0

  numberOfInterpolationPoints = 8
  orderAccuracy = 2

  enableThermal = false
  enableViscous = true

  in = [velocity, dynamicViscosity, temperature, thermalCoefficient]
  out = [viscousSource]
</Updater>

<Updater computeViscousSource>
  kind = navierStokesViscousOperator2d
  onGrid = domain
  isRadial = true
  coefficient = 1.0

  numberOfInterpolationPoints = 8

  enableThermal = false
  enableViscous = true
  enableTurbulence = true
  temperatureIndex = 0

  in = [velocity, totalVisc, avgTemp, totalCond, kEpsilon, turbulentViscosity]
  out = [viscousSource]
</Updater>

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Contents

navierStokesViscousOperator (1d, 2d, 3d)¶

Data¶

Parameters¶

Examples¶