4.1.13. Vorticity Transport Equation What does irrotational mean?

\[\vec{\Omega} = \vec{0} \longrightarrow \nabla \times \vec{u} = \vec{0}\] What is an irrotational vortex?

Particles don’t rotate about their own axes, but about the axis of the vortex. Does irrotational and inviscid, irrotational and viscous, vortical and inviscid, vortical and viscous exist?

Type of flow Existing
Irrotational and inviscid Yes
Irrotational and viscous No
Vortical and inviscid Yes
Vortical and viscous Yes What is circulation?

The line integral around the contour of the tangential component of velocity \(\vec{u}\). Via Stokes theorem:

\[\Gamma = \oint_C \vec{u} \cdot d \vec{C} = \int_S \nabla \times \vec{u} \cdot d \vec{S} = \int_S \vec{\Omega} \cdot d \vec{S}\] What is the curl of the gradient of a scalar?

The curl of the gradient of a scalar is zero:

\[\nabla \times (\nabla \phi) = \vec{0}\]


\[\operatorname{curl} \left( \operatorname{grad} \phi \right) = \vec{0}\]

where \(\phi\) is a scalar

  • If the gradient points in the direction of increasing \(\phi\), then
  • Travelling in the direction of increasing \(\phi\) does not rotate. What is the vector identity for \(\vec{u} \cdot \nabla \vec{u}\)?

\[\vec{u} \cdot \nabla \vec{u} = \nabla \left( {1 \over 2} \vec{u} \cdot \vec{u} \right) - \vec{u} \times \left( \nabla \times \vec{u} \right)\] What is the curl of \(\vec{u} \times \vec{\Omega}\) for an incompressible fluid?

\[\nabla \times (\vec{u} \times \vec{\Omega}) = \vec{\Omega} \cdot \nabla \vec{u} - \vec{u} \cdot \nabla \vec{\Omega}\]

where: \(\nabla \cdot \vec{\Omega} = \nabla \cdot \vec{u} = 0\) How can the non-conservative form of the Navier-Stokes momentum equation use conservative body forces?


\[{{\partial \vec{u}} \over {\partial t}} + \vec{u} \cdot \nabla \vec{u} = - {1 \over \rho} \nabla p + \nu \nabla^2 \vec{u} + \vec{g}\]

Conservative body forces \(\longrightarrow \vec{g} = -\nabla \Phi\)

where: :math:`Phi = ` gravitational potential function

\[{{\partial \vec{u}} \over {\partial t}} + \vec{u} \cdot \nabla \vec{u} = - {1 \over \rho} \nabla p + \nu \nabla^2 \vec{u} -\nabla \Phi\] How can the Navier-Stokes momentum equation be arranged to include the potential function?

Navier-Stokes (with conservative body forces):

\[{{\partial \vec{u}} \over {\partial t}} + \vec{u} \cdot \nabla \vec{u} = - {1 \over \rho} \nabla p + \nu \nabla^2 \vec{u} -\nabla \Phi\]

By expansion:

\[\vec{u} \cdot \nabla \vec{u} = \nabla \left( {1 \over 2} \vec{u} \cdot \vec{u} \right) - \vec{u} \times \nabla \times \vec{u} = \nabla \left( {1 \over 2} \vec{u} \cdot \vec{u} \right) - \vec{u} \times \vec{\Omega}\]

By substitution:

\[{{\partial \vec{u}} \over {\partial t}} + \nabla( {1 \over 2} \vec{u} \cdot \vec{u}) - \vec{u} \times \vec{\Omega} = - {1 \over \rho} \nabla p + \nu \nabla^2 \vec{u} -\nabla \Phi\]


\[{{\partial \vec{u}} \over {\partial t}} - \vec{u} \times \vec{\Omega} = - \nabla \left( {1 \over \rho} p + {1 \over 2} \vec{u}^2 + \Phi \right) + \nu \nabla^2 \vec{u}\]

By re-definition:

\[{{\partial \vec{u}} \over {\partial t}} - \vec{u} \times \vec{\Omega} = - \nabla \Pi + \nu \nabla^2 \vec{u}\] How can the Navier-Stokes equations using the potential function be expressed as the vorticity transport equation?

Navier-Stokes (with potential function):

\[{{\partial \vec{u}} \over {\partial t}} - \vec{u} \times \vec{\Omega} = - \nabla \Pi + \nu \nabla^2 \vec{u}\]

Taking the curl of both sides:

\[\nabla \times \left( {{\partial \vec{u}} \over {\partial t}} - \vec{u} \times \vec{\Omega} \right) = \nabla \times \left( - \nabla \Pi + \nu \nabla^2 \vec{u} \right)\]

First part:

\[\nabla \times {{\partial \vec{u}} \over {\partial t}} = {{\partial \vec{\Omega}} \over {\partial t}}\]

Second part:

\[\nabla \times (- \vec{u} \times \vec{\Omega}) = -\nabla \times \vec{u} \times \vec{\Omega} = - \left( (\vec{\Omega} \cdot \nabla) \vec{u} - (\vec{u} \cdot \nabla) \vec{\Omega} \right) = -(\Omega \cdot \nabla) \vec{u} + (\vec{u} \cdot \nabla) \vec{\Omega}\]

Third part:

\[\nabla \times (\nabla \Pi) = 0\]

Fourth part:

\[\nabla \times ( \nu \nabla^2 \vec{u} ) = \nu \nabla^2 \vec{\Omega}\]


\[{{\partial \vec{\Omega}} \over {\partial t}} + (\vec{u} \cdot \nabla) \vec{\Omega} = (\vec{\Omega} \cdot \nabla) \vec{u} + \nu \nabla^2 \vec{\Omega}\] What is the physical meaning of the terms in the vorticity transport equation?

\[{{\partial \vec{\Omega}} \over {\partial t}} + (\vec{u} \cdot \nabla) \vec{\Omega} = (\vec{\Omega} \cdot \nabla) \vec{u} + \nu \nabla^2 \vec{\Omega}\]
  • Temporal change of vorticity / unit volume:
\[{{\partial \vec{\Omega}} \over {\partial t}}\]
  • Spatial change of vorticity / unit volume:
\[(\vec{u} \cdot \nabla) \vec{\Omega}\]
  • Vortex stretching term:
\[(\vec{\Omega} \cdot \nabla) \vec{u}\]
  • Diffusion of vorticity / unit volume:
\[\nu \nabla^2 \vec{\Omega}\] What is the vortex stretching phenomenon?

  • The vortex stretching term is \((\vec{\Omega} \cdot \nabla) \vec{u}\)
  • Conservation of angular momentum:
\[{{D(I \vec{\Omega})} \over {Dt}} = I {{D \vec{\Omega}} \over {Dt}} + \vec{\Omega} {{DI} \over {Dt}}\]
  • For constant angular momentum:
\[{{D(I \vec{\Omega})} \over {Dt}} = 0\]
  • Hence:
\[{{D \vec{\Omega}} \over {Dt}} = - {{\vec{\Omega}} \over {I}} {{DI} \over {Dt}}\]
  • Vortex stretching is an increase in vorticity caused by reduced momentum intertia of vortex elements in 3D What is the vorticity transport equation in 2D?

  • In 2D:
\[(\vec{\Omega} \cdot \nabla) \vec{u} = \left( \Omega_x {\partial \over {\partial x}} + \Omega_y {\partial \over {\partial y}} + \Omega_z {\partial \over {\partial z}} \right) \vec{u} = 0\]

as \(\Omega_x = \Omega_y = {\partial \over {\partial z}} = 0\)

  • Hence:
\[{{\partial \vec{\Omega}} \over {\partial t}} + (\vec{u} \cdot \nabla) \vec{\Omega} = \nu \nabla^2 \vec{\Omega}\]
  • For \(\Omega_z\):
\[{{\partial \Omega_z} \over {\partial t}} + (\vec{u} \cdot \nabla) \Omega_z = \nu \nabla^2 \Omega_z\]
  • By expansion:
\[{{\partial \Omega_z} \over {\partial t}} + u {{\partial \Omega_z} \over {\partial x}} + v {{\partial \Omega_z} \over {\partial y}} = \nu \left( {{\partial^2 \Omega_z} \over {\partial x^2}} + {{\partial^2 \Omega_z} \over {\partial y^2}} \right)\]
  • Also:
\[\Omega_x = \Omega_y = 0\] How are the terms in the 3D vorticity transport equation expanded?

  • \(\Omega_x\) by expansion:
\[{{\partial \Omega_x} \over {\partial t}} + u {{\partial \Omega_x} \over {\partial x}} + v {{\partial \Omega_x} \over {\partial y}} + w {{\partial \Omega_x} \over {\partial z}} = \Omega_x {{\partial u} \over {\partial x}} + \Omega_y {{\partial u} \over {\partial y}} + \Omega_z {{\partial u} \over {\partial z}} + \nu \left( {{\partial^2 \Omega_x} \over {\partial x^2}} + {{\partial^2 \Omega_x} \over {\partial y^2}} + {{\partial^2 \Omega_x} \over {\partial z^2}} \right)\]
  • \(\Omega_y\) by expansion:
\[{{\partial \Omega_y} \over {\partial t}} + u {{\partial \Omega_y} \over {\partial x}} + v {{\partial \Omega_y} \over {\partial y}} + w {{\partial \Omega_y} \over {\partial z}} = \Omega_x {{\partial v} \over {\partial x}} + \Omega_y {{\partial v} \over {\partial y}} + \Omega_z {{\partial v} \over {\partial z}} + \nu \left( {{\partial^2 \Omega_y} \over {\partial x^2}} + {{\partial^2 \Omega_y} \over {\partial y^2}} + {{\partial^2 \Omega_y} \over {\partial z^2}} \right)\]
  • \(\Omega_z\) by expansion:
\[{{\partial \Omega_z} \over {\partial t}} + u {{\partial \Omega_z} \over {\partial x}} + v {{\partial \Omega_z} \over {\partial y}} + w {{\partial \Omega_z} \over {\partial z}} = \Omega_x {{\partial w} \over {\partial x}} + \Omega_y {{\partial w} \over {\partial y}} + \Omega_z {{\partial w} \over {\partial z}} + \nu \left( {{\partial^2 \Omega_z} \over {\partial x^2}} + {{\partial^2 \Omega_z} \over {\partial y^2}} + {{\partial^2 \Omega_z} \over {\partial z^2}} \right)\]