User:Tohline/ThreeDimensionalConfigurations/Challenges
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(→Trial #3) 
(→Exponent q = 3) 

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\biggl( \frac{\rho}{\rho_c}\biggr) \biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr)  \biggl( \frac{\rho}{\rho_c}\biggr) \biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr)  
\biggr]  \biggr]  
  \biggr\}  +  \biggr\} \, . 
+  </math>  
+  </td>  
+  </tr>  
+  </table>  
+  
+  If we choose a function,  
+  <table border="0" cellpadding="5" align="center">  
+  
+  <tr>  
+  <td align="right">  
+  <math>~F_B(\Psi)</math>  
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~  
+  \frac{D_1}{2} \Psi^{1 / 2} + D_2 \Psi  
+  </math>  
+  </td>  
+  </tr>  
+  
+  <tr>  
+  <td align="right">  
+   
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~  
+  \frac{D_1}{2} \Psi_0^{1 / 2} \biggl(\frac{\rho}{\rho_c}\biggr) + D_2 \Psi_0 \biggl( \frac{\rho}{\rho_c}\biggr)  
+  </math>  
+  </td>  
+  </tr>  
+  </table>  
+  we obtain,  
+  <table border="0" cellpadding="5" align="center">  
+  
+  <tr>  
+  <td align="right">  
+  <math>~\nabla F_B(\Psi)</math>  
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~  
+  ~\frac{D_1}{2} \Psi_0^{1 / 2} \cdot \nabla \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr)  
+  </math>  
+  </td>  
+  </tr>  
+  
+  <tr>  
+  <td align="right">  
+   
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~  
+  ~D_1 \Psi_0^{1 / 2}  
+  \biggl[ \boldsymbol{\hat\imath} \biggl(\frac{x}{a^2} \biggr) + \boldsymbol{\hat\jmath} \biggl(\frac{y}{b^2} \biggr) \biggr]  
+  \, .  
</math>  </math>  
</td>  </td> 
Revision as of 11:32, 26 September 2020
Contents 
Challenges Constructing EllipsoidalLike Configurations
First, let's review the three different approaches that we have described for constructing Riemann Stype ellipsoids. Then let's see how these relate to the technique that has been used to construct infinitesimally thin, nonaxisymmetric disks.
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Riemann Stype Ellipsoids
Usually, the density, , and the pair of axis ratios, and , are specified. Then, the Poisson equation is solved to obtain in terms of , , and . The aim, then, is to determine the value of the central enthalpy, — alternatively, the thermal energy density, — and the two parameters, and , that determine the magnitude of the velocity flowfield. Keep in mind that, as viewed from a frame of reference that is spinning with the ellipsoid (at angular frequency, ), the adopted (rotatingframe) velocity field is,



Hence, the inertialframe velocity is given by the expression,



While we will fundamentally rely on the parameter pair to define the velocity flowfield, in discussions of Riemann Stype ellipsoids it is customary to also refer to the following two additional parameters: The (rotatingframe) vorticity,






and the dimensionless frequency ratio,



2^{nd}Order TVE Expressions
As we have discussed in detail in an accompanying chapter, the three diagonal elements of the 2^{nd}order tensor virial equation are sufficient to determine the equilibrium values of , , and .
Indices  2^{nd}Order TVE Expressions that are Relevant to Riemann SType Ellipsoids  





The element gives directly in terms of known parameters. The and elements can then be combined in a couple of different ways to obtain a coupled set of expressions that define and .
and,

Ou's (2006) Detailed Force Balance
In a separate accompanying chapter, we have described in detail how Ou(2006) used, essentially, the HSCF technique to solve the detailed forcebalance equations. Beginning with the,
Eulerian Representation
of the Euler Equation
as viewed from a Rotating Reference Frame
it can be shown that, for the velocity fields associated with all Riemann Stype ellipsoids,









Hence, within each steadystate configuration the following Bernoulli's function must be uniform in space:



Ou(2006), p. 550, §2, Eq. (6)
where is a constant. So, at the surface of the ellipsoid (where the enthalpy H = 0) on each of its three principal axes, the equilibrium conditions demanded by the expression for detailed force balance become, respectively:
 On the xaxis, where (x, y, z) = (a, 0, 0):
 On the yaxis, where (x, y, z) = (0, b, 0):
 On the zaxis, where (x, y, z) = (0, 0, c):
This third expression can be used to replace the lefthandside of the first and second expressions. Then via some additional algebraic manipulation, the first and second expressions can be combined to provide the desired solutions for the parameter pair, , namely,



and 



Ou(2006), p. 551, §2, Eqs. (15) & (16)
where,


and, 



Hybrid Scheme
In a separate chapter we have detailed the hybrid scheme. For steadystate configurations, the three components of the combined Euler + Continuity equations give,
Hybrid Scheme Summary for SteadyState Configurations

In this context, the vector acceleration that drives the fluid flow is, simply,



Then, for the velocity flowpatterns in Riemann Stype ellipsoids, we have,


(because ); 









Vertical Component: Given that , we deduce that,



Azimuthal Component: Irrespective of the location of each fluid element, this component requires,



Radial Component: After inserting the "azimuthal component" relation and marching through a fair amount of algebraic manipulation, we find that Irrespective of the location of each fluid element, this component requires,



Compressible Structures
Here we draw heavily on the published work of Korycansky & Papaloizou (1996, ApJS, 105, 181; hereafter KP96) that we have reviewed in a separate chapter, and on the doctoral dissertation of Saied W. Andalib (1998).
Part I
Returning to the abovementioned,
Eulerian Representation
of the Euler Equation
as viewed from a Rotating Reference Frame
we next note — as we have done in our broader discussion of the Euler equation — that,
where, as above, is the vorticity. Making this substitution, we obtain what is essentially equation (7) of KP96, that is, the,
Euler Equation
written in terms of the Vorticity and
as viewed from a Rotating Reference Frame
.
Hence, in steadystate, the Euler equation becomes,
where, the scalar "Bernoulli" function,
and,
For later use …

We will leave discussion of the Euler equation, for the moment, and instead look at the continuity equation. As viewed from the rotating frame of reference,



If we are able to write the momentum density (in the rotating frame) in terms of a streamfunction, , such that,



then satisfying the steadystate continuity equation is guaranteed because the divergence of a curl is always zero. Note, as well, that when written in terms of this streamfunction, the zcomponent of the vorticity will be,






Note that the steadystate continuity equation may be rewritten in the form,



It can also be shown that the condition, can be rewritten as,



By combining these last two expressions, we appreciate that,



This means that, in the steadystate flow whose spatial structure we are seeking, the velocity vector, (and also the momentum density vector, ) must everywhere be tangent to contours of constant vortensity, .
We need a function such that,






Let's try, , and




and, 

Then,









Hence,



Next, given that,



we conclude that, to within an additive constant,









where,



Here's what to do:
Given , write out the functional forms of and . Then see if .
Part II
Consider a steadystate configuration that is the compressible analog of a Riemann Stype ellipsoid; even better, give the configuration a "peanut" shape rather than the shape of an ellipsoid. As viewed from a frame that is spinning with the configuration's overall angular velocity, , generally we expect the configuration's internal (and surface) flow to be represented by a set of nested streamlines and at every location the fluid's velocity (and its momentumdensity vector) will be tangent to the streamline that runs through that point. It is customary to represent the streamfunction by a scalar quantity, , appreciating that each streamline will be defined by a curve, ; also, the local spatial gradient of will be perpendicular to the local streamline, hence it will be perpendicular to the local velocity vector as well. If we specifically introduce the streamfunction via the relation,



it will display all of the justdescribed attributes and we are also guaranteed that the steadystate continuity equation will be satisfied everywhere, because the divergence of a curl is always zero.
We also have demonstrated that the vector, , has the right properties if,



This means that, at every location in the plane of the fluid flow, the gradient of the vortensity also must be perpendicular to the velocity vector. This constraint can be immediately satisfied if we simply demand that the vortensity be a function of the streamfunction, , that is, we need,



In other words, the scalar vortensity is constant along each streamline. And, once we have determined the mathematical expression for this function, we will know that,



Furthermore, we should be able to determine the mathematical expression for because,



As a check, we should find that,



Part III
Here we closely follow Chapter 4 of Saied W. Andalib (1998).
From §4.1 (p. 80): "Euler's equation, the equation of continuity, the Poisson equation and the equation of state … govern the dynamics and evolution of these equilibrium configurations."
Equation of Continuity
In steady state, . Hence the rotatingframebased continuity equation becomes,



If we insist that the momentumdensity vector be expressible in terms of the curl of a vector — for example,



Saied W. Andalib (1998), §4.1, p. 80, Eq. (4.1) 
then satisfying this steadystate continuity equation is guaranteed because the divergence of a curl is always zero. "The task of satisfying the steadystate equation of continuity then shifts to identifying an appropriate expression for the vector potential, " In the most general case, in terms of this vector potential the three Cartesian components of the momentumdensity vector are,









Here, we will follow Andalib's lead and only look for fluid flows with no vertical motions. That is to say, we will set , in which case this last expression establishes the constraint,



"A general solution to this equation can be found only if there exists a scalar function such that …"

and, 

note that this adopted functional behavior works because the constraint becomes,



Hence, the expressions for the x and ycomponents of the momentumdensity vector may be rewritten, respectively, as,






If we again follow Andalib's lead and only look for models in which the xyplane flow is independent of the vertical coordinate, z, then, and must be functions of x and y only. Therefore, is independent of z and is at most linear in z. Now, rather than focusing on the determination of , we can just as well define the scalar function,



in which case "… the components of the momentum density may be written as:"



It is straightforward to demonstrate that this expression for the momentumdensity vector does satisfy the steadystate continuity equation. "The function will serve a similar role as the velocity potential for incompressible fluids."
Related Useful Expressions
Given that, by our design, the fluid motion will be confined to the xyplane, the fluid vorticity will have only a zcomponent; that is,
,
where,



And when it is written in terms of , this zcomponent of the vorticity will be obtained from the expression,



This is useful to know because, in the Euler equation (see immediately below) we will encounter a term that involves the cross product of the vector, , with the rotatingframebased velocity vector. Appreciating as well that the vector, , only has a nonzero zcomponent, we recognize that this term may be written as,









Saied W. Andalib (1998), §4.2, p. 83, Eq. (4.13) 
Euler Equation
We begin with the,
Euler Equation
written in terms of the Vorticity and
as viewed from a Rotating Reference Frame
.
Next, we rewrite this expression to incorporate the following three realizations:
 For a barotropic fluid, the term involving the pressure gradient can be replaced with a term involving the enthalpy via the relation, .
 The expression for the centrifugal potential can be rewritten as, .
 In steady state, .
This means that,



If the term on the lefthandside of this equation can be expressed in terms of the gradient of a scalar function, then it can be readily grouped with all the other terms on the righthandside, which already are in the gradient form.
Striving for Gradient Form As we have already demonstrated above, the term on the lefthandside of the Euler equation can be rewritten as,
If the term inside the square brackets on the righthandside were a constant — that is, independent of position — then it could immediately be moved inside the gradient operator and we will have accomplished our objective. But, while is constant "… generally the vorticity and density are both functions of and ." As Andalib has explained, "The expression … can be cast in the form of a gradient only if
where is an arbitrary function." Specifically in this case, the term on the lefthandside of the Euler equation may be written as,
That is, we accomplish our objective by recognizing that the soughtafter function, , is obtained from via the relation,
For example, try …
This means that,

Having transformed the lefthandside term into the gradient of the scalar function, , the Euler equation can now be written as,






where we will refer to as the Bernoulli constant.
Strategy
STEP 0: Choose the pair of modelsequence parameters, , that are associated with the function, . Hold these fixed during iterations.
STEP 1: Guess a density distribution, . For example, pick the equatorialplane (uniform) density distribution of a Riemann Stype ellipsoid with an equatorialplane axisratio, and meridionalplane axisratio, ; use the same ratio to define two points on the configuration's surface throughout the iteration cycle.
STEP 2: Given , solve the Poisson equation to obtain, . In the first iteration, this should exactly match the values associated with the chosen Riemann Stype ellipsoid.
STEP 3: Guess a value of — perhaps the spinfrequency associated with your "initial guess" Riemann ellipsoid — then solve the following twodimensional, elliptic PDE to obtain …



Boundary Condition
Moving along various rays from the center of the configuration, outward, the surface is determined by the location along each ray where goes to zero for the first time. We set at these various surface locations. At each of these locations, the velocity vector must be tangent to the surface. This requirement also, then, sets the value of and at each location. 
STEP 4: Determine (rotatingframe) velocity from knowledge of and .






STEP 5: Using the "scalar Euler equation,"



Saied W. Andalib (1998), §4.3, p. 85, Eq. (4.23) 
 Set at two different points on the surface of the configuration — usually at and — to determine values of the two constants, and .
 At all points inside the configuration, determine .
STEP 6: Use the barotropic equation of state to determine the "new" massdensity distribution from the knowledge of the enthalpy, .
Compare
Incompressible


















Returning to the abovementioned,
Eulerian Representation
of the Euler Equation
as viewed from a Rotating Reference Frame
we next note — as we have done in our broader discussion of the Euler equation — that,
Making this substitution, we obtain



it can be shown that, for the velocity fields associated with all Riemann Stype ellipsoids,









As a check,








Yes! This expression matches the one that appears just a few lines earlier in this discussion.
Now, let's switch the order of terms in the steadystate Euler equation to permit an easier comparison with our attempt to develop the compressible models.






Compressible
Now in our above discussion of Andalib's work, the steadystate form of the Euler equation was formulated as,



It is easy to appreciate that,



As we have shown, in the case of the incompressible (Riemann Stype ellipsoid) models,



If we attempt to directly relate these two expressions, we must acknowledge that,






As we have discussed above, Andalib (1998) found that some interesting model sequences could be constructed if he adopted the functional form,



Evidently, the incompressible (uniformdensity) Riemann Stype ellisoids can be retrieved from our derived compressiblemodel formalism if we set, , and,



with,





Afterthought: Because we want to go to zero at the surface, it likely will be better to set,
then adjust the sign of and add a constant (zerothorder term) to the definition of .
Trial #1
Restricting our discussion to nonaxisymmetric, thin disks, let's assume is uniform throughout the configuration and that,



This means that,

and, 

The momentum density vector is governed by the relation,






[ has units of "density × length^{2} per time"] 
First of all, let's see if the steadystate continuity equation is satisfied:











Q.E.D. 
Next, let's determine the zcomponent of the vorticity and the vortensity:












This means that,









But this entire process was designed to ensure that,



where, . Let's see if we get the same expression …









This is indeed the same expression as above if we set,



Hooray!!
Finally, let's make sure that the elliptic PDE "constraint" equation is satisfied.














Yes! 
Trial #2
Still restricting our discussion to nonaxisymmetric, thin disks, let's try, , and




and, 

IMPORTANT NOTE (by Tohline on 22 September 2020): As I have come to appreciate today — after studying the relevant sections of both EFE and BT87 — this is an example of a heterogeneous density distribution whose gravitational potential has an analytic prescription. As is discussed in a separate chapter, the potential that it generates is sometimes referred to as a Ferrers potential, for the exponent, n = 1. In our accompanying discussion we find that,
where,
More specifically, in the three cases where the indices, ,

This means that,

and, 

The Elliptic PDE Constraint Equation

The momentum density vector is governed by the relation,









[ has units of "density × length^{2} per time"] 
As above, let's see if the steadystate continuity equation is satisfied:














Yes! 
Next, as above, let's determine the zcomponent of the vorticity and the vortensity:












This means that,









Now, let's examine the gradient of .












which is identical to the immediately preceding expression if we set,



Continuing with the above examination of the elliptic PDE "constraint" equation, we find that,








Hooray! 
Trial #3
If the density distribution has been specified, then is the "streamfunction" from which all rotatingframe velocities are determined. Specifically,



Most importantly, as has been detailed above, the term on the lefthandside of the steadystate Euler equation becomes,



Saied W. Andalib (1998), §4.2, p. 83, Eq. (4.13) 
where,



Still restricting our discussion to infinitesimally thin, nonaxisymmetric disks, let's assume that,




and, 


and, 

And, let's assume that,






and, similarly, 


This means that,


















Hence,















Exponent q = 2
Notice that, if ,



Now, if we choose a function,



we obtain,






This is consistent with the elliptic PDE constraint if,



Also if , we have,















Keep in mind that, as discussed above, we are trying to satisfy the scalar Bernoulli relation,






The righthandside of this expression does not appear to be rich enough to balance the gravitational potential (on the lefthandside) which, as detailed above, contains and and terms.
Exponent q = 3
Alternatively, if ,






If we choose a function,






we obtain,






Exponent q = 4



See Also
© 2014  2020 by Joel E. Tohline 