Hybrid Advection Scheme (Background)

March 1, 2014 by Joel E. Tohline

As is mentioned in the preface to our primary discussion of the Hybrid Advection Scheme, my early notes on the topic have been preserved, as they were included in my earliest version of this web-based H_Book. The relevant page can be accessed via this link; it is an html file whose linux time stamp is August 27, 2000. Here we re-present the content of those "year 2000" notes because the "symbol" fonts utilized throughout the original html page seem now not to be properly translated by some web browsers.

ASIDE: Relevant to hydrocode development

Curvature Terms

Coverting [sic] from a Lagrangian to an Eulerian time-derivative, Equation [I.A.5] becomes,

<math> \partial_t(\rho\boldsymbol{v}) + (\boldsymbol{v}\cdot\nabla)(\rho \boldsymbol{v}) + (\rho \boldsymbol{v})\nabla\cdot \boldsymbol{v} </math>

<math> -\nabla P - \rho \nabla\Phi \, . </math>

Now, if you're not working in Cartesian coordinates, care must be taken when dealing with the second term on the left-hand-side of this equaiton because when the gradient operator acts on a vector quantity (in this case, <math>~\rho\boldsymbol{v}</math>), various curvature terms will arise reflecting the fact that, in general, the unit vectors of your curvilinear coordinate system point in different directions as the fluid moves to different locations in space. Quite generally, though, for the <math>~j^\mathrm{th}</math> component of the equation of motion we may isolate these curvature terms as follows:

<math> \partial_t(\rho v_j) + \nabla\cdot (\rho v_j \boldsymbol{v}) + (\mathrm{curvature})_j </math>

<math> -\nabla_j P - \rho \nabla_j \Phi \, , </math>

where,

<math> ~(\mathrm{curvature})_j </math>

<math> \Sigma_{i=1,2,3} \{ [ (\rho v_i)/(h_i h_j) ] [ v_j \partial_{\xi_i} h_j - v_i \partial_{\xi_j} h_i ] \} \, . </math>

So, for example, in cylindrical coordinates where <math>~h_1 = h_\varpi = 1, h_2 = h_\theta = \varpi,</math> and <math>~h_3 = h_z = 1,</math>

<math> ~(\mathrm{curvature})_\varpi </math>	<math>~=~</math>	<math> [ (\rho v_\theta)/\varpi ] [ -v_\theta ] = - \rho v_\theta^2/\varpi \, ; </math>
<math> ~(\mathrm{curvature})_\theta </math>	<math>~=~</math>	<math> [ (\rho v_\varpi)/\varpi ] [ v_\theta ] = \rho v_\varpi v_\theta/\varpi \, ; </math>
<math> ~(\mathrm{curvature})_z </math>	<math>~=~</math>	<math> 0 \, ; </math>

Thus, in cylindrical coordinates the three components of the equation of motion become,

<math> \partial_t(\rho v_\varpi) + \nabla\cdot (\rho v_\varpi \boldsymbol{v}) </math>	<math>~=~</math>	<math> -\nabla_\varpi P - \rho \nabla_\varpi \Phi + \rho v_\theta^2/\varpi \, ; </math>
<math> \partial_t(\rho v_\theta) + \nabla\cdot (\rho v_\theta \boldsymbol{v}) </math>	<math>~=~</math>	<math> -\nabla_\theta P - \rho \nabla_\theta \Phi - \rho v_\varpi v_\theta/\varpi \, ; </math>
<math> \partial_t(\rho v_z) + \nabla\cdot (\rho v_z \boldsymbol{v}) </math>	<math>~=~</math>	<math> -\nabla_z P - \rho \nabla_z \Phi \, . </math>

Conservation of Angular Momentum

Now, the first and third of these expressions are indeed the ones we are utilizing in our hydrocode. but the middle one, expressing the time-rate-of-change of the azimuthal velocity, has been implemented in a different form, namely,

<math> \partial_t(\rho \varpi v_\theta) + \nabla\cdot(\rho\varpi v_\theta \boldsymbol{v} ) </math>

<math> ~ - \varpi \nabla_\theta P - \varpi \rho \nabla_\theta \Phi \, . </math>

The $64,000 question is, "Are these equivalent expressions?" Well, let's play with the left-hand-side of this last expression.

<math> \partial_t(\rho \varpi v_\theta) + \nabla\cdot(\rho\varpi v_\theta \boldsymbol{v} ) </math>	<math>~=~</math>	<math> \varpi \partial_t(\rho v_\theta) + (\rho v_\theta) \partial_t(\varpi) + \varpi \nabla\cdot(\rho v_\theta \boldsymbol{v}) + (\rho v_\theta \boldsymbol{v}\cdot\nabla)\varpi </math>
	<math>~=~</math>	<math> \varpi \{ \partial_t(\rho v_\theta) + \nabla\cdot(\rho v_\theta \boldsymbol{v}) \} + (\rho v_\theta v_\varpi) </math>

It is easy to see, therefore, that the two expressions are equivalent; but the latter one is used in preference to the former because it provides a direct statement of conservation of angular momentum. Specifically, when the external forces (due to gradients in the gravitational potential and pressure) balance, our selected "conservative" finite-difference scheme will guarantee that the physical quantity <math>~A = \rho\varpi v_\theta</math> is conserved globally to precisely the same degree of accuracy as mass is conserved.

Notation Adjustment

In what follows, the governing PDEs will be expressed in terms of velocities that are measured in the context of a rotating reference frame. In our original html-formatted notes, these variables are differentiated from inertial-frame variables by color; specifically, rotating-frame variables are colored orange. Here we will abandon the color labeling and, instead, use <math>~\boldsymbol{u}</math> to represent the velocity as measured in a frame rotating about the <math>~z</math>-axis with angular frequency, <math>~\Omega_f</math>. That is, <math>~\boldsymbol{u}</math> is related to the inertial-frame velocity <math>~\boldsymbol{v}</math> through the expression,

<math> ~\boldsymbol{u} </math>

<math> ~\boldsymbol{v} - \boldsymbol{\hat{e}}_\theta \varpi\Omega_f \, . </math>

The So-called A* Scheme

Now, in a rotating frame of reference, this preferred form of the azimuthal component of the equation of motion takes the form,

<math> \partial_t(\rho \varpi u_\theta) + \nabla\cdot(\rho\varpi u_\theta \boldsymbol{u} ) </math>

<math> ~ - \varpi \nabla_\theta P - \varpi \rho \nabla_\theta \Phi -2\rho\varpi \Omega_f u_\varpi \, . </math>

Notice that a new term has appeared due to the coriolis force. Traditionally, in numerical hydrodynamics, this new term has been treated explicitly as a source term. hence, this component of the equation of motion no longer takes on a strictly conservative form, and the adopted "conservative" finite-difference is no longer a particularly useful tool to guarantee that the angular momentum is globally conserved even when the external forces due to pressure and gravity balance one another.

To derive a form of this equation that is a lot more suited to a "conservative" finite-difference implementation, note that,

Related Discussions

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

User:Tohline/PGE/AStarScheme

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