User:Tohline/PGE/Hybrid Scheme
Hybrid Scheme
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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.
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Example # |
Grid |
Momentum Vector |
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Basis |
Rotating? |
Basis |
Frame |
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1 |
Cartesian |
No |
Cartesian |
Inertial |
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2 |
Cylindrical |
Yes <math>~(\Omega_0)</math> |
Cylindrical |
Rotating <math>~(\Omega_0)</math> |
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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,
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<math>~v'_\phi</math> |
<math>~=~</math> |
<math>~v_\phi - R\Omega_0 \, ,</math> |
and,
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<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>.
Example #1
This is certainly the most familiar component set.
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<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> |
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<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> |
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<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,
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<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).
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<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> |
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<math>~=~</math> |
<math> -~\frac{\partial P}{\partial R} - \rho \frac{\partial \Phi}{\partial R} + \frac{\rho}{R} (v_\phi + R\Omega_0)^2 \, , </math> |
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<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> |
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<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,
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<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
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<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} (u'_\phi + R\omega_0)(v'_\phi + R\Omega_0) </math> |
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<math>~=~</math> |
<math> -~\frac{\partial P}{\partial R} - \rho \frac{\partial \Phi}{\partial R} + \frac{\rho u'_\phi v'_\phi}{R} + \frac{\rho[u'_\phi R\Omega_0 + v'_\phi R \omega_0]}{R} + \rho\omega_0\Omega_0 R \, , </math> |
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<math>~\boldsymbol{\hat{e}}_\phi:</math> |
<math>~\frac{\partial [\rho R (u_\phi - R\omega_0)]}{\partial t} + \nabla\cdot[\rho R (u_\phi - R\omega_0) (\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> |
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<math>~\boldsymbol{\hat{e}}_z:</math> |
<math>~\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,
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<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 - 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.
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© 2014 - 2021 by Joel E. Tohline |