User:Tohline/PGE/AStarScheme
From VisTrailsWiki
Contents 
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 this topic were included in my earliest version of this webbased 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 represent the content of these "year 2000" notes because the symbol fonts utilized throughout the original html page seem now not to be properly translated by some web browsers.
As my group discussed this proposed new algorithmic approach to advecting angular momentum over the first decade of the new millennium, it was often referred to as the "A* scheme," where, was the inertialframe angular momentum density as given by the sum of variables whose values were tracked in the rotating frame.
 Tiled Menu  Tables of Content  Banner Video  Tohline Home Page  
ASIDE: Relevant to hydrocode development
Curvature Terms
Converting from a Lagrangian to an Eulerian timederivative, Equation [I.A.5] becomes,



Now, if you're not working in Cartesian coordinates, care must be taken when dealing with the second term on the lefthandside of this equation because when the gradient operator acts on a vector quantity (in this case, ), 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 component of the equation of motion we may isolate these curvature terms as follows:



where,



So, for example, in cylindrical coordinates where and









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









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 timerateofchange of the azimuthal velocity, has been implemented in a different form, namely,



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






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" finitedifference scheme will guarantee that the physical quantity 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 htmlformatted notes, these variables are differentiated from inertialframe variables by color; specifically, rotatingframe variables are colored orange. Here we will abandon the color labeling and, instead, use to represent the velocity as measured in a frame rotating about the axis with angular frequency, . That is, is related to the inertialframe velocity through the expression,



The Socalled A* Scheme
Now, in a rotating frame of reference, this preferred form of the azimuthal component of the equation of motion takes the form,



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" finitedifference 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" finitedifference implementation, note that,






Hence, in a rotating frame of reference, the azimuthal component of the equation of motion can be written as,



or,



When advancing the angular momentum density (i.e., ) forward in time using a finitedifference scheme, I recommend that the "sourcing" step only include the terms on the righthandside of these last expressions (i.e., only the gradients in the pressure and gravitational potential), and the "fluxing" step should include the following terms:









This last expression is directly implementable using our standard fluxing scheme because all three terms have the form . (Note that the density that appears in the last two terms on the righthandside of this last expression must be taken from precisely the same point in time as the "" that appears in the first term on the righthandside.)
Whether you adopt precisely this final prescription or not, the primary point to keep in mind is that you want to advect the "intertial" [sic] angular momentum density using the "rotatingframe velocity at each grid cell face. Hence, you might prefer to use one slightly earlier relation to guide your design of the fluxing step. In the absence of "true" source terms (due to pressure and gravity), we have,



which is the same as,



But this may also be written as,



So, you can carry out the calculation by first adding the quantity to to get the value of the angular momentum density in the inertial frame; advect this "inertial" quantity; then subtract from this result the change in the quantity as determined from the updated value of the density as derived via the continuity equation.
Related Discussions
 Euler equation viewed from a rotating frame of reference or Main Page.
 An earlier draft of this "Euler equation" presentation.
© 2014  2019 by Joel E. Tohline 