User:Tohline/PGE/RotatingFrame
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Compressible Analogs of Riemann S-Type Ellipsoids
Here we attempt to develop a self-consistent-field type, iterative technique that will permit the construction of steady-state structures that are compressible analogs of Riemann S-Type (incompressible) ellipsoids. We will build upon the recent work of Ou (2006).
Standard Steady-State Governing Relations
As viewed from a rotating frame of reference and written in Eulerian form, the steady-state version of the three-dimensional principal governing equations are:
<math> \nabla\cdot(\rho \vec{v}) = 0 </math>
<math> (\vec{v}\cdot \nabla)\vec{v} = -\nabla \biggl[H + \Phi -\frac{1}{2}\omega^2 R^2 \biggr] -2\vec{\omega}\times\vec{v} </math>
<math> \nabla^2 \Phi = 4\pi G \rho </math>
Proposed Solution Strategy
Preamble:
Specify the three "polar" boundary locations, <math>a, b,</math> and <math>c</math>; specify the direction but not the magnitude of the rotating frame's angular velocity, for example, <math>(\vec{\omega}/\omega) = \hat{k}</math>; pin the central density to the value <math>\rho_c = 1</math>. Define the following dimensionless density, velocity vector, angular velocity, enthalpy, gravitational potential, and position vector:
<math> \rho^* \equiv \frac{\rho}{\rho_c} ; ~~~~~{\vec{v}}^* \equiv \frac{\vec{v}}{[a^2G\rho_c]^{1/2}} ; ~~~~~\omega^* \equiv \frac{\omega}{[G\rho_c]^{1/2}} ; </math>
<math> H^* \equiv \frac{H}{[a^2G\rho_c]} ; ~~~~~\Phi^* \equiv \frac{\Phi}{[a^2G\rho_c]} ; ~~~~~{\vec{x}}^* \equiv \frac{\vec{x}}{a} . </math>
From here, on, spatial operators are assumed to be in terms of the dimensionless coordinates.
Step #1:
Guess a 3D density distribution — such as a uniform-density ellipsoid, or one of the converged models from Ou (2006) — that conforms to a selected set of positional boundary conditions, that is, where the density goes to zero along the three principal axes at <math>x=a</math>, <math>y = b</math>, and <math>z = c</math>. Solve the Poisson equation in order to derive values for <math>\Phi</math> everywhere inside and on the surface of the 3D configuration:
<math> \nabla^2 \Phi^* = 4\pi \rho^* . </math>
Step #2:
Use the continuity equation and the curl of the Euler equation to numerically derive the structure but not the overall magnitude of the velocity flow-field throughout the 3D configuration. Take advantage of the fact that the direction, <math>(\vec{\omega}/\omega)</math>, has been specified; and assume that the 3D density distribution is known. Here are the relevant equations:
<math> \nabla\cdot(\rho^* {\vec{v}}^*) = 0 </math>
<math> \nabla\times [({\vec{v}}^*\cdot \nabla){\vec{v}}^*] = -2 \nabla\times \biggl[\hat{k}\times {\vec{v}}^* \biggr] </math>
The first of these is a scalar equation; the second is a vector equation and it will presumably provide two useful scalar equations (constraining the two components of <math>\vec{\zeta}</math> that are perpendicular to <math>\hat{k}</math> ?). Since the left-hand-side of the second equation is obviously nonlinear in the velocity, we may have to linearize this set of equations and look for small "corrections" <math>\delta\vec{v}</math> to an initial "guess" for the velocity field, such as the flow field in Riemann S-type ellipsoids, which is also the flow-field adopted by Ou (2006).
Step #3:
Take the divergence of the Euler equation and use it to solve for <math>H</math> throughout the configuration, given the structure of the flow-field obtained in Step #2. Boundary conditions at the three "poles" of the configuration may suffice to uniquely determine <math>\omega</math>, the overall normalization factor for the flow-field <math>\vec\zeta</math> — hopefully this is analogous to solving for the vorticity parameter <math>\lambda</math> in Ou (2006) — and the Bernoulli constant (or something equivalent). The relevant "Poisson"-like equation is:
<math> \nabla^2 \biggl[H^* + \Phi^* -\frac{1}{2}(\omega^*)^2 \biggl(\frac{R}{a}\biggr)^2 \biggr] = - \nabla\cdot [({\vec{v}}^*\cdot \nabla){\vec{v}}^* + 2 {\vec{\omega}}^*\times {\vec{v}}^* ] . </math>
Detailed Example
Preamble:
Adopt Cartesian coordinates, write vector expression in component form, and linearize individual equations, as necessary.
Step #2:
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© 2014 - 2021 by Joel E. Tohline |