User:Tohline/PGE/Hybrid Scheme

Hybrid Scheme

Setting the Stage

Recognizing Statements of Conservation

When dealing with the compressible fluid equations, we will often encounter hyperbolic PDEs of the following form:

<math> \frac{d\psi}{dt} + \psi \nabla\cdot \vec{v} </math>

<math>~=~</math>

<math> S \, , </math>

where we are using <math>~\vec{v}</math> to represent the velocity field of the fluid as viewed from an inertial frame of reference, and the total (as opposed to partial) time derivative indicates the time-rate of change of <math>~\psi</math> is being measured in a so-called Lagrangian fashion, that is, at the location of some fluid element and moving along with that fluid element.

When we encounter a situation in which the "source" term, <math>~S</math>, on the right-hand side is zero, we will be able to identify the scalar variable, <math>~\psi</math>, as the volume density of some conserved quantity. For example, the continuity equation — which is a mathematical statement of mass conservation — has the form,

<math>\frac{d\rho}{dt} + \rho \nabla \cdot \vec{v} = 0</math>

or, equivalently,

<math> \frac{d\ln\rho}{dt} </math>

<math>~=~</math>

<math> ~- \nabla\cdot \vec{v} \, , </math>

where, <math>~\rho</math> is the mass per unit volume or, simply, the mass density of the fluid element. Clearly, when the mass of a Lagrangian fluid element is conserved, the fluid element's mass density changes only in accordance with the divergence of the local velocity field.

Similarly, if we are following the evolution of a fluid that expands and contracts adiabatically, we should expect to encounter an equation of the form,

<math> \frac{ds}{dt} + s\nabla\cdot \vec{v} </math>

<math>~=~</math>

<math> 0 \, , </math>

or, equivalently,

<math> \frac{d\ln s}{dt} </math>

<math>~=~</math>

<math> ~- \nabla\cdot \vec{v} \, , </math>

where, <math>~s</math> is the entropy density of a Lagrangian fluid element. Or, if an axisymmetric distribution of fluid is moving in an axisymmetric potential, we should expect the azimuthal component of the fluid's angular momentum to be conserved, in which case we should expect to encounter a dynamical equation of the form,

<math> \frac{d(\rho \varpi v_\phi)}{dt} + (\rho \varpi v_\phi) \nabla\cdot \vec{v} </math>

<math>~=~</math>

<math> 0 \, , </math>

where, <math>~\varpi</math> is the Lagrangian fluid element's (cylindrical radial) distance measured from the symmetry axis of the underlying potential and <math>~v_\phi = \varpi\dot\phi</math> is the azimuthal component of the inertial velocity field, <math>~\vec{v}</math>, at the location of the fluid element.

Alternative Reference Frames

Now, we might want to examine the time-dependent behavior of a distributed fluid system while viewing the flow from a reference frame that is more or less moving along with the fluid. This new frame of reference need not be an inertial frame; for example, when studying a rotating fluid, we may want to view the system's evolution from a rotating frame of reference. This will be accomplished mathematically by adjusting the dynamical equations so that the velocity that appears in the divergence term accounts for the new "frame" velocity field; specifically, we want to replace <math>~\vec{v}</math> with,

<math> \vec{v}_\mathrm{new} </math>

<math>~=~</math>

<math> \vec{v} + \vec{v}_\mathrm{frame} \, . </math>

(Here, we will consider only time-independent functional expressions for the frame velocity, <math>~\vec{v}_\mathrm{frame}</math>.) Of course, the introduction of <math>~\vec{v}_\mathrm{new}</math> must be done in such a way that the resulting, new PDE describes exactly the same physical behavior of the system as was described by the original equation; that is, the new equation must be derivable from the original one.

If <math>~\vec{v}_\mathrm{frame}</math> is a divergence-free velocity field, then the replacement is trivial. For example, if we choose a frame of reference that is rotating uniformly with angular velocity, <math>~\Omega_0</math>, then,

<math> \vec{v}_\mathrm{frame} </math>

<math>~=~</math>

<math> \boldsymbol{\hat{e}}_\phi (\varpi \Omega_0) \, , </math>

and, utilizing cylindrical coordinates,

<math> \nabla\cdot\vec{v}_\mathrm{frame} </math>

<math>~=~</math>

<math> \frac{\partial(0)}{\partial \varpi} + \frac{1}{\varpi}\frac{\partial(\varpi \Omega_0)}{\partial \phi} + \frac{\partial(0)}{\partial z} = 0 \, . </math>

Hence,

<math> \frac{d\psi}{dt} + \psi \nabla\cdot \vec{v}_\mathrm{new} </math>

<math>~=~</math>

<math> \frac{d\psi}{dt} + \psi \nabla\cdot [\vec{v} + \vec{v}_\mathrm{frame}] </math>

<math>~=~</math>

<math> \frac{d\psi}{dt} + \psi \nabla\cdot \vec{v} \, , </math>

while the generic hyperbolic PDE becomes,

<math> \frac{d\psi}{dt} + \psi \nabla\cdot \vec{v}_\mathrm{new} </math>

<math>~=~</math>

<math> S \, , </math>

and we can be confident that this new PDE represents the physics of the system just as well as the original PDE.

Eulerian Representation

We can shift any of the PDEs from a Lagrangian to an Eulerian representation — and thereby use them to follow the time-rate of change of physical variables at a point in space that is fixed with respect to the chosen frame of reference — by using the following transformation to replace each total time derivative with a partial time derivative:

<math> \frac{d\psi}{dt} </math>

<math>~~~\rightarrow~~~</math>

<math> \frac{\partial \psi}{\partial t} + \vec{v}_\mathrm{new} \cdot \nabla\psi \, . </math>

Hence, the "new" generic hyperbolic PDE can be rewritten as,

<math> \frac{\partial\psi}{\partial t} + \vec{v}_\mathrm{new}\cdot \nabla\psi + \psi \nabla\cdot \vec{v}_\mathrm{new} </math>

<math>~=~</math>

<math> S \, , </math>

or, more succinctly,

<math> \frac{\partial\psi}{\partial t} + \nabla\cdot (\psi \vec{v}_\mathrm{new}) </math>

<math>~=~</math>

<math> S \, . </math>

This equation also is broadly recognized as a conservation statement because, when <math>~S = 0</math>, the variable <math>~\psi</math> will represent the volume density of a conserved quantity.

We should emphasize that the inertial-frame version of this Eulerian conservation equation can be retrieved straightforwardly by setting <math>~\Omega_0 = 0</math>, which is equivalent to setting <math>~\vec{v}_\mathrm{new} = \vec{v}</math>. It is,

<math> \frac{\partial\psi}{\partial t} + \nabla\cdot (\psi \vec{v}) </math>

<math>~=~</math>

<math> S \, . </math>

The physics of the flow that is being described by this PDE is identical to the physics that is described by the preceding PDE. But an important distinction must be made between how the two equations are interpreted. The "inertial frame" version of the equation provides the time-rate of change of <math>~\psi</math> at a fixed point in inertial space, while the "new" version provides the time-rate of change of <math>~\psi</math> at a fixed point in the"new" moving — in this case, rotating — coordinate frame.

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