User:Tohline/PGE/RotatingFrame

NOTE to Eric Hirschmann & David Neilsen... I have move the earlier contents of this page to a new Wiki location called Compressible Riemann Ellipsoids.

Rotating Reference Frame

At times, it can be useful to view the motion of a fluid from a frame of reference that is rotating with a uniform (i.e., time-independent) angular velocity <math>~\Omega_f</math>. In order to transform any one of the principal governing equations from the inertial reference frame to such a rotating reference frame, we must specify the orientation as well as the magnitude of the angular velocity vector about which the frame is spinning, <math>{\vec\Omega}_f</math>; and the <math>~d/dt</math> operator, which denotes Lagrangian time-differentiation in the inertial frame, must everywhere be replaced as follows:

<math> \biggl[\frac{d}{dt} \biggr]_{inertial} \rightarrow \biggl[\frac{d}{dt} \biggr]_{rot} + {\vec{\Omega}}_f \times . </math>

Performing this transformation implies, for example, that

<math> \vec{v}_{inertial} = \vec{v}_{rot} + {\vec{\Omega}}_f \times \vec{x} , </math>

and,

<math> \biggl[ \frac{d\vec{v}}{dt}\biggr]_{inertial} = \biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} + 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} + {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) </math>

<math> = \biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} + 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} - \frac{1}{2} \nabla \biggl[ |{\vec{\Omega}}_f \times \vec{x}|^2 \biggr] </math>

(If we were to allow <math>{\vec\Omega}_f</math> to be a function of time, an additional term involving the time-derivative of <math>{\vec\Omega}_f</math> also would appear on the right-hand-side of these last expressions; see, for example, Eq.~1D-42 in BT87.) Note as well that the relationship between the fluid vorticity in the two frames is,

<math> [\vec\zeta]_{inertial} = [\vec\zeta]_{rot} + 2{\vec\Omega}_f . </math>

Continuity Equation (rotating frame)

Applying these transformations to the standard, inertial-frame representations of the continuity equation presented elsewhere, we obtain the:

Lagrangian Representation
of the Continuity Equation
as viewed from a Rotating Reference Frame

<math>\biggl[ \frac{d\rho}{dt} \biggr]_{rot} + \rho \nabla \cdot {\vec{v}}_{rot} = 0</math> ;

Eulerian Representation
of the Continuity Equation
as viewed from a Rotating Reference Frame

<math>\biggl[ \frac{\partial\rho}{\partial t} \biggr]_{rot} + \nabla \cdot (\rho {\vec{v}}_{rot}) = 0</math> .

Euler Equation (rotating frame)

Applying these transformations to the standard, inertial-frame representations of the Euler equation presented elsewhere, we obtain the:

Lagrangian Representation
of the Euler Equation
as viewed from a Rotating Reference Frame

<math>\biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} = - \frac{1}{\rho} \nabla P - \nabla \Phi - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} - {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x})</math> ;

Eulerian Representation
of the Euler Equation
as viewed from a Rotating Reference Frame

<math>\biggl[\frac{\partial\vec{v}}{\partial t}\biggr]_{rot} + ({\vec{v}}_{rot}\cdot \nabla) {\vec{v}}_{rot}= - \frac{1}{\rho} \nabla P - \nabla \biggl[\Phi - \frac{1}{2}|{\vec{\Omega}}_f \times \vec{x}|^2 \biggr] - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} </math> ;

Euler Equation
written in terms of the Vorticity and
as viewed from a Rotating Reference Frame

<math>\biggl[\frac{\partial\vec{v}}{\partial t}\biggr]_{rot} + ({\vec{\zeta}}_{rot}+2{\vec\Omega}_f) \times {\vec{v}}_{rot}= - \frac{1}{\rho} \nabla P - \nabla \biggl[\Phi + \frac{1}{2}v_{rot}^2 - \frac{1}{2}|{\vec{\Omega}}_f \times \vec{x}|^2 \biggr]</math> .

Centrifugal and Coriolis Accelerations

Following along the lines of the discussion presented in Appendix 1.D, §3 of BT87, in a rotating reference frame the Lagrangian representation of the Euler equation may be written in the form,

<math>\biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} = - \frac{1}{\rho} \nabla P - \nabla \Phi + {\vec{a}}_{fict} </math>,

where,

<math> {\vec{a}}_{fict} \equiv - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} - {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) . </math>

So, as viewed from a rotating frame of reference, material moves as if it were subject to two fictitious accelerations which traditionally are referred to as the,

Coriolis Acceleration

<math> {\vec{a}}_{Coriolis} \equiv - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} , </math>

(see the related Wikipedia discussion) and the

Centrifugal Acceleration

<math> {\vec{a}}_{Centrifugal} \equiv - {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) = \frac{1}{2} \nabla\biggl[ |{\vec{\Omega}}_f \times \vec{x}|^2 \biggr] </math>

(see the related Wikipedia discussion).

Nonlinear Velocity Cross-Product

In some contexts — for example, our discussion of Riemann ellipsoids or the analysis by Korycansky & Papaloizou (1996) of nonaxisymmetric disk structures — it proves useful to isolate and analyze the term in the "vorticity formulation" of the Euler equation that involves a nonlinear cross-product of the rotating-frame velocity vector, namely,

<math> \vec{A} \equiv ({\vec{\zeta}}_{rot}+2{\vec\Omega}_f) \times {\vec{v}}_{rot} . </math>

NOTE: To simplify notation, for most of the remainder of this subsection we will drop the subscript "rot" on both the velocity and vorticity vectors.

Align <math>{\vec\Omega}_f</math> with z-axis

Without loss of generality we can set <math>{\vec\Omega}_f = \hat{k}\Omega_f</math>, that is, we can align the frame rotation axis with the z-axis of a Cartesian coordinate system. The Cartesian components of <math>{\vec{A}}</math> are then,

<math> \hat{i}: ~~~~~~ A_x = \zeta_y v_z - (\zeta_z + 2\Omega) v_y , </math>

<math> \hat{j}: ~~~~~~ A_y = (\zeta_z + 2\Omega) v_x - \zeta_x v_z , </math>

<math> \hat{k}: ~~~~~~ A_z = \zeta_x v_y - \zeta_y v_x , </math>

where it is understood that the three Cartesian components of the vorticity vector are,

<math> \zeta_x = \biggl[\frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z} \biggr] ,

~~~~~~ \zeta_y = \biggl[ \frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x} \biggr] ,
~~~~~~ \zeta_z = \biggl[ \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} \biggr] .

</math>

In turn, the curl of <math>\vec{A}</math> has the following three Cartesian components:

<math> \hat{i}: ~~~~~~ [\nabla\times\vec{A}]_x = \frac{\partial}{\partial y}\biggl[ \zeta_x v_y - \zeta_y v_x \biggr] - \frac{\partial}{\partial z}\biggl[ (\zeta_z + 2\Omega) v_x - \zeta_x v_z \biggr], </math>

<math> \hat{j}: ~~~~~~ [\nabla\times\vec{A}]_y = \frac{\partial}{\partial z}\biggl[ \zeta_y v_z - (\zeta_z + 2\Omega) v_y \biggr] - \frac{\partial}{\partial x}\biggl[ \zeta_x v_y - \zeta_y v_x \biggr] , </math>

<math> \hat{k}: ~~~~~~ [\nabla\times\vec{A}]_z = \frac{\partial}{\partial x}\biggl[ (\zeta_z + 2\Omega) v_x - \zeta_x v_z \biggr] - \frac{\partial}{\partial y}\biggl[ \zeta_y v_z - (\zeta_z + 2\Omega) v_y \biggr] . </math>

When <math>v_z = 0</math>

If we restrict our discussion to configurations that exhibit only planar flows — that is, systems in which <math>v_z = 0</math> — then the Cartesian components of <math>{\vec{A}}</math> and <math>\nabla\times\vec{A}</math> simplify somewhat to give, respectively,

<math> \hat{i}: ~~~~~~ A_x = - (\zeta_z + 2\Omega) v_y , </math>

<math> \hat{j}: ~~~~~~ A_y = (\zeta_z + 2\Omega) v_x , </math>

<math> \hat{k}: ~~~~~~ A_z = \zeta_x v_y - \zeta_y v_x , </math>

and,

<math> \hat{i}: ~~~~~~ [\nabla\times\vec{A}]_x = \frac{\partial}{\partial y}\biggl[ \zeta_x v_y - \zeta_y v_x \biggr] - \frac{\partial}{\partial z}\biggl[ (\zeta_z + 2\Omega) v_x \biggr], </math>

<math> \hat{j}: ~~~~~~ [\nabla\times\vec{A}]_y = - \frac{\partial}{\partial z}\biggl[(\zeta_z + 2\Omega) v_y \biggr] - \frac{\partial}{\partial x}\biggl[ \zeta_x v_y - \zeta_y v_x \biggr] , </math>

<math> \hat{k}: ~~~~~~ [\nabla\times\vec{A}]_z = \frac{\partial}{\partial x}\biggl[ (\zeta_z + 2\Omega) v_x \biggr] + \frac{\partial}{\partial y}\biggl[ (\zeta_z + 2\Omega) v_y \biggr] , </math>

where, in this case, the three Cartesian components of the vorticity vector are,

<math> \zeta_x = - \frac{\partial v_y}{\partial z} ,

~~~~~~ \zeta_y = \frac{\partial v_x}{\partial z} ,
~~~~~~ \zeta_z = \biggl[ \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} \biggr] .

</math>

Related Discussions

• Wikipedia discussion of vorticity.
• Wikipedia discussion of Coriolis Effect.
• Wikipedia discussion of Centrifugal acceleration.

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation