User:Tohline/PGE/Hybrid Scheme

Hybrid Scheme

Traditional Eulerian Representation (Review)

Here we review the traditional Eulerian representation of the Euler Equation, as has been discussed in detail earlier.

in terms of velocity:

The so-called "Eulerian form" of the Euler equation can be straightforwardly derived from the standard Lagrangian representation to obtain,

Eulerian Representation
of the Euler Equation,

<math>~\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla) \vec{v}= - \frac{1}{\rho} \nabla P - \nabla \Phi</math>

in terms of momentum density:

Also, we can multiply this expression through by <math>~\rho</math> and combine it with the continuity equation to derive what is commonly referred to as the,

Conservative Form
of the Euler Equation,

<math>~\frac{\partial(\rho\vec{v})}{\partial t} + \nabla\cdot [(\rho\vec{v})\vec{v}]= - \nabla P - \rho \nabla \Phi</math>

The second term on the left-hand-side of this last expression represents the divergence of the "dyadic product" of the vector momentum density (<math>~\rho</math><math>~\vec{v}</math>) and the velocity vector <math>~\vec{v}</math> and is sometimes written as, <math>\nabla\cdot [(\rho \vec{v}) \otimes \vec{v}]</math>.

Component Forms

Let's split the vector Euler equation into its three scalar components; various examples are identified in Table 1.

Example #	Grid		Momentum Vector
	Basis	Rotating?	Basis	Frame
1	Cartesian	No	Cartesian	Inertial
2	Cylindrical	Yes <math>~(\Omega_0)</math>	Cylindrical	Rotating <math>~(\Omega_0)</math>
3	Cylindrical	Yes <math>~(\Omega_0)</math>	Cylindrical	Rotating <math>~(\omega_0)</math>

In the following expressions, we will use <math>~\vec{v}</math> to denote the fluid velocity when it is associated with the rate of fluid transport across the coordinate grid, and we will use <math>~\vec{u}</math> to denote the fluid velocity when it is associated with the momentum density that is being advected. In all cases, it should be understood that <math>~\vec{v} = \vec{u}</math>, as both vectors refer to the same fluid velocity. In addition, we will use a "prime" notation to indicate when a velocity is being viewed from a rotating frame of reference; specifically, we will consider rotation about the <math>~z</math>-axis of the coordinate system, that is,

<math>~v'_\phi</math>

<math>~=~</math>

<math>~v_\phi - R\Omega_0 \, ,</math>

and,

<math>~u'_\phi</math>

<math>~=~</math>

<math>~u_\phi - R\omega_0 \, ,</math>

but we will not insist that the two rotation frequencies, <math>~\Omega_0</math> and <math>~\omega_0</math>, have the same value. Hence, in general, <math>~(\vec{u})' \ne (\vec{v})'</math>. It is worth emphasizing that, because we will only be considering frame rotation about the <math>z</math>-axis, the cylindrical <math>R</math> and <math>z</math> components of the velocity are interchangeable, that is: <math>~u'_R = v'_R = u_R = v_R</math>; and <math>~u'_z = v'_z = u_z = v_z</math>.

Example #1

This is certainly the most familiar component set.

<math>\boldsymbol{\hat{e}}_x: ~~~\frac{\partial (\rho v_x)}{\partial t} + \nabla\cdot[(\rho v_x) \vec{v}~]</math>	<math>~=~</math>	<math> -~\frac{\partial P}{\partial x} - \rho \frac{\partial \Phi}{\partial x} \, , </math>
<math>\boldsymbol{\hat{e}}_y: ~~~\frac{\partial (\rho v_y)}{\partial t} + \nabla\cdot[(\rho v_y) \vec{v}~]</math>	<math>~=~</math>	<math> -~\frac{\partial P}{\partial y} - \rho \frac{\partial \Phi}{\partial y} \, , </math>
<math>\boldsymbol{\hat{e}}_z: ~~~\frac{\partial (\rho v_z)}{\partial t} + \nabla\cdot[(\rho v_z) \vec{v}~]</math>	<math>~=~</math>	<math> -~\frac{\partial P}{\partial z} - \rho \frac{\partial \Phi}{\partial z} \, , </math>

where, for any one of the three scalar PDEs, advection of the relevant component of the momentum density, <math>~\psi_i</math>, is handled via the operation,

<math> \nabla\cdot[\psi_{i} \vec{v} ] </math>

<math>~=~</math>

<math> \frac{\partial (\psi_i v_x)}{\partial x} + \frac{\partial (\psi_i v_y)}{\partial y} + \frac{\partial (\psi_i v_z)}{\partial z} \, . </math>

Example #2

This component set has been spelled out in, for example, equations (5) - (7) of Norman & Wilson (1978) and equations (11), (12), & (3) of New & Tohline (1997).

<math>\boldsymbol{\hat{e}}_R: ~~~~~~~\frac{\partial (\rho v_R)}{\partial t} + \nabla\cdot[(\rho v_R) \vec{v}~]</math>	<math>~=~</math>	<math> -~\frac{\partial P}{\partial R} - \rho \frac{\partial \Phi}{\partial R} + \frac{(\rho R v_\phi)^2}{\rho R^3} + \rho\Omega_0^2 R + \frac{2\Omega_0 (\rho R v_\phi)}{R} \, , </math>
	<math>~=~</math>	<math> -~\frac{\partial P}{\partial R} - \rho \frac{\partial \Phi}{\partial R} + \frac{\rho}{R} (v_\phi + R\Omega_0)^2 \, , </math>
<math>\boldsymbol{\hat{e}}_\phi: ~~~\frac{\partial (\rho R v_\phi)}{\partial t} + \nabla\cdot[(\rho R v_\phi) \vec{v}~]</math>	<math>~=~</math>	<math> -~\frac{\partial P}{\partial \phi} - \rho \frac{\partial \Phi}{\partial \phi} - 2\rho (\Omega_0 R )v_R \, , </math>
<math>\boldsymbol{\hat{e}}_z: ~~~~~~~~\frac{\partial (\rho v_z)}{\partial t} + \nabla\cdot[(\rho v_z) \vec{v}~]</math>	<math>~=~</math>	<math> -~\frac{\partial P}{\partial z} - \rho \frac{\partial \Phi}{\partial z} \, , </math>

where, as noted above,

<math> \nabla\cdot[\psi_{i} \vec{v} ] </math>

<math>~=~</math>

<math> \frac{\partial (\psi_i v_R)}{\partial R} + \frac{1}{R} \frac{\partial (\psi_i v_\phi)}{\partial\phi} + \frac{\partial (\psi_i v_z)}{\partial z} \, . </math>

Example #3

<math>~\boldsymbol{\hat{e}}_R:</math>	<math>~\frac{\partial (\rho u'_R)}{\partial t} + \nabla\cdot[\rho u'_R (\vec{v})'~]</math>	<math>~=~</math>	<math> -~\frac{\partial P}{\partial R} - \rho \frac{\partial \Phi}{\partial R} + \frac{\rho}{R} (v'_\phi + R\Omega_0)^2 </math>
		<math>~=~</math>	<math> -~\frac{\partial P}{\partial R} - \rho \frac{\partial \Phi}{\partial R} + \frac{\rho (v'_\phi)^2}{R} + 2\rho \Omega_0 v'_\phi + \rho \Omega_0^2 R \, , </math>
<math>~\boldsymbol{\hat{e}}_\phi:</math>	<math>~\frac{\partial \{\rho R [u'_\phi + R(\Omega_0 - \omega_0)]\} }{\partial t} + \nabla\cdot[ \{ \rho R [u'_\phi + R(\Omega_0 - \omega_0)] \} (\vec{v})'~]</math>	<math>~=~</math>	<math> -~\frac{\partial P}{\partial \phi} - \rho \frac{\partial \Phi}{\partial \phi} - 2\rho R\omega_0 v'_R \, , </math>
<math>~\boldsymbol{\hat{e}}_z:</math>	<math>~\frac{\partial (\rho u'_z)}{\partial t} + \nabla\cdot[\rho u'_z (\vec{v})'~]</math>	<math>~=~</math>	<math> -~\frac{\partial P}{\partial z} - \rho \frac{\partial \Phi}{\partial z} \, , </math>

where, as noted above,

<math>~u'_\phi</math>

<math>~=~</math>

<math>~u_\phi - R\omega_0 \, ,</math>

and, for any one of the three scalar PDEs, advection of the relevant component of the momentum density, <math>~\psi_i</math>, is handled via the operation,

<math> \nabla\cdot[\psi_{i} (\vec{v})' ] </math>	<math>~=~</math>	<math> \frac{\partial (\psi_i v'_R)}{\partial R} + \frac{1}{R} \frac{\partial (\psi_i v'_\phi)}{\partial\phi} + \frac{\partial (\psi_i v'_z)}{\partial z} </math>
	<math>~=~</math>	<math> \frac{\partial (\psi_i v_R)}{\partial R} + \frac{1}{R} \frac{\partial [\psi_i (v_\phi - R\Omega_0)]}{\partial\phi} + \frac{\partial (\psi_i v_z)}{\partial z} \, . </math>

Related Discussions

• Euler equation viewed from a rotating frame of reference or Main Page.
• An earlier draft of this "Euler equation" presentation.

© 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