User:Tohline/PGE/AStarScheme

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

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>~=~</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>~=~</math>

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

where,

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

<math>~=~</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>

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