User:Tohline/Apps/Korycansky Papaloizou 1996 - Revision history

Tohline: /* Suggested Strategy by Joel */ Continue discussion

2010-03-31T22:49:35Z

Suggested Strategy by Joel: Continue discussion

← Older revision		Revision as of 22:49, 31 March 2010
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	</math>		</math>
	</div>		</div>
	Let me emphasize that, during this step of our SCF iteration cycle, we should assume that the functions <math>\rho(\vec{x})</math> and <math>f(\vec{x})</math> are known throughout the configuration.		Let me emphasize that, during this step of our SCF iteration cycle, we should assume that the functions <math>\rho(\vec{x})</math> and <math>f(\vec{x})</math> are known throughout the configuration. The continuity equation can therefore be written as,
			<div align="center">
			<math>
			\frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} = - \biggl[ v_x\frac{\partial}{\partial x}
			+ v_y \frac{\partial}{\partial y} \biggr] (\ln\rho)
			</math><br />

			<math>
			\Rightarrow ~~~~~
			f \frac{\partial (\ln v_0)}{\partial x} + \frac{\partial (\ln v_0)}{\partial y} =
			- ( 1 + f^2 )^{1/2} \biggl[ \frac{\partial [f ( 1 + f^2 )^{-1/2}]}{\partial x}
			+ \frac{\partial [( 1 + f^2 )^{-1/2}]}{\partial y} \biggr]
			- \biggl[ f \frac{\partial}{\partial x} + \frac{\partial}{\partial y} \biggr] (\ln\rho)
			</math>
			</div>



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Tohline: /* Suggested Strategy by Joel */ Discuss more about determining magnitude of velocity

2010-03-31T20:03:04Z

Suggested Strategy by Joel: Discuss more about determining magnitude of velocity

← Older revision		Revision as of 20:03, 31 March 2010
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	<div align="center">		<div align="center">
	<math>		<math>
	\nabla\cdot\vec{v} = -\vec{v}\cdot\nabla(\ln\rho) .		\nabla\cdot\vec{v} = -\vec{v}\cdot\nabla(\ln\rho) ,
	</math>		</math>
	</div>		</div>
			where, in terms of the function <math>f(\vec{x})</math>,
			<div align="center">
			<math>
			\vec{v} = v_0 \biggl[ \hat{i} f ( 1 + f^2 )^{-1/2} + \hat{j} ( 1 + f^2 )^{-1/2} \biggr] .
			</math>
			</div>
			Let me emphasize that, during this step of our SCF iteration cycle, we should assume that the functions <math>\rho(\vec{x})</math> and <math>f(\vec{x})</math> are known throughout the configuration.


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Tohline: /* Korycansky and Papaloizou (1996) */ Joel's suggested strategy

2010-03-27T03:26:02Z

Korycansky and Papaloizou (1996): Joel's suggested strategy

← Older revision		Revision as of 03:26, 27 March 2010
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	# They use the compressible version of the continuity equation;		# They use the compressible version of the continuity equation;
	# Instead of taking the divergence of the Euler equation to obtain a Poisson-like equation, they obtain an algebraic constraint on the Bernoulli function (as in our traditional SCF technique) by simply "dotting" <math>\vec{v}</math> into the Euler equation.		# Instead of taking the divergence of the Euler equation to obtain a Poisson-like equation, they obtain an algebraic constraint on the Bernoulli function (as in our traditional SCF technique) by simply "dotting" <math>\vec{v}</math> into the Euler equation.

			=Suggested Strategy by Joel=
			For the time being, I want to continue to discuss only planar flows. Aside from the Poisson equation and the equation of state, the three key scalar equations are:
			# Continuity equation;
			# Z-component of the curl of the Euler equation; and
			# Scalar product of <math>\vec{v}</math> and Euler.
			Let's take one of Shangli's compressible Riemann ellipsoids as our starting guess. We need to "tilt" each of the velocity vectors slightly so that they everywhere satisfy the compressible equation of state. By combining the first two of the three key scalar equations, the KP96 study tells us that we can do this '''not''' by orienting them tangent to isodensity contours but, rather, by orienting them tangent to iso-vortensity surfaces. That is, we need,
			<div align="center">
			<math>
			\vec{v} \cdot \nabla\Lambda = 0 .
			</math>
			</div>
			where,
			<div align="center">
			<math>
			\Lambda \equiv \ln\biggl[ \frac{(2\Omega + \zeta_z)}{\rho} \biggr] .
			</math>
			</div>
			Let's rely on this expression '''only''' to tell us how to orient each velocity vector, that is '''only''' to give us the ratio
			<div align="center">
			<math>
			f \equiv \frac{v_x}{v_y}
			</math>
			</div>
			everywhere. And let's assume that <math>\Omega</math>, <math>\rho(\vec{x})</math> and <math>\zeta_z(\vec{x})</math> are known from Shangli's model. Then we can straightforwardly conclude that, at all spatial locations within the configuration,
			<div align="center">
			<math>
			f(\vec{x}) = -\frac{\nabla_y(\Lambda)}{\nabla_x(\Lambda)} .
			</math>
			</div>
			Everywhere along the ''surface'', <math>\nabla\Lambda</math> may not be well-determined; in which case I suspect that we should align the velocity vectors tangent to the surface.

			Next, let's use ''either'' the continuity equation ''or'' the z-component of the curl of Euler to determine the updated '''magnitude''', <math>v_0(\vec{x})</math>, of each velocity vector. Using the continuity equation, for example, we need,
			<div align="center">
			<math>
			\nabla\cdot\vec{v} = -\vec{v}\cdot\nabla(\ln\rho) .
			</math>
			</div>



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Tohline: /* Two-Dimensional Planar Flow */ minor corrections

2010-03-27T02:40:44Z

Two-Dimensional Planar Flow: minor corrections

← Older revision		Revision as of 02:40, 27 March 2010
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	</div>		</div>

	Since, by design, streamlines defined by the momentum-density vector field must trace out lines of constant <math>\Psi</math> and, according to the conclusion drawn above, the momentum density vector must everywhere be tangent to contours of constant ''vortensity'', we can conclude — as did KP96 — that the ''vortensity'' <math>(\zeta_z + 2\~~omega~~)/\rho</math> must be constant along streamlines. The ''vortensity'' is therefore a function of <math>\Psi</math> alone, so we can write,		Since, by design, streamlines defined by the momentum-density vector field must trace out lines of constant <math>\Psi</math> and, according to the conclusion drawn above, the momentum density vector must everywhere be tangent to contours of constant ''vortensity'', we can conclude — as did KP96 — that the ''vortensity'' <math>(\zeta_z + 2\Omega)/\rho</math> must be constant along streamlines. The ''vortensity'' is therefore a function of <math>\Psi</math> alone, so we can write,
	<div align="center">		<div align="center">
	<math>		<math>
	\frac{\zeta_z + 2\~~omega~~}{\rho} = g(\Psi) .		\frac{\zeta_z + 2\Omega}{\rho} = g(\Psi) .
	</math>		</math>
	</div>		</div>

Tohline: Fix some key errors

2010-03-24T17:15:12Z

Fix some key errors

← Older revision		Revision as of 17:15, 24 March 2010
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	<div align="center">		<div align="center">
	<math>		<math>
	\nabla F_B - \vec{A} = 0 ,		\nabla F_B + \vec{A} = 0 ,
	</math>		</math>
	</div>		</div>

Tohline: Fix last expression for Bernoulli function

2010-03-22T20:58:59Z

Fix last expression for Bernoulli function

← Older revision		Revision as of 20:58, 22 March 2010
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	<div align="center">		<div align="center">
	<math>		<math>
	\frac{1}{2}v^2 + H + \Phi -\frac{1}{2}\~~omega^2 R~~^2 = F_B(\Psi) .		\frac{1}{2}v^2 + H + \Phi - \frac{1}{2}\|\Omega\hat{k} \times \vec{x}\|^2 = F_B(\Psi) .
	</math>		</math>
	</div>		</div>

Tohline: /* Governing Steady-State Equations */ Minor modifications; example, internal Wiki links

2010-03-22T20:56:41Z

Governing Steady-State Equations: Minor modifications; example, internal Wiki links

← Older revision		Revision as of 20:56, 22 March 2010
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	==Governing Steady-State Equations==		==Governing Steady-State Equations==

	KP96 begin with the standard set of principal governing equations, but choose to work from the set that is expressed in terms of a [~~http://www.vistrails.org/index.php?title=~~User:Tohline/PGE/RotatingFrame rotating frame of reference]. (Throughout the presentation on this page, it is to be understood that all variables are viewed from a rotating frame even though the subscript notation "rot" does not appear in the equations.) Their Eq. (1), for example, comes from the		KP96 begin with the standard set of principal governing equations, but choose to work from the set that is expressed in terms of a [[User:Tohline/PGE/RotatingFrame#Rotating_Reference_Frame\|rotating frame of reference]]. (Throughout the presentation on this page, it is to be understood that all variables are viewed from a rotating frame even though the subscript notation "rot" does not appear in the equations.) Their Eq. (1), for example, comes from the

	<div align="center">		<div align="center">
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	</div>		</div>

	where, in this last expression, we have ~~replace~~ the gradient of the pressure by the gradient of the [http://www.vistrails.org/index.php/User:Tohline/SR#Barotropic_Structure enthalpy] via the relation, <math>\nabla H = \nabla P/\rho </math>.		where, in this last expression, we have replaced the gradient of the pressure by the gradient of the [http://www.vistrails.org/index.php/User:Tohline/SR#Barotropic_Structure enthalpy] via the relation, <math>\nabla H = \nabla P/\rho </math>.
	Note that the KP96 notation is slightly different from ours:		Note that the KP96 notation is slightly different from ours:
	* <math>\Sigma</math> is used in place of <math>\rho</math> to denote a two-dimensional <i>surface</i> density;		* <math>\Sigma</math> is used in place of <math>\rho</math> to denote a two-dimensional <i>surface</i> density;

Tohline: Now contract earlier draft of discussion, getting rid of "solution strategy" subsection

2010-03-21T22:36:11Z

Now contract earlier draft of discussion, getting rid of "solution strategy" subsection

Show changes

Tohline: /* Korycansky and Papaloizou (1996) */ Streamline discussion of planar-flow constraints

2010-03-21T21:41:10Z

Korycansky and Papaloizou (1996): Streamline discussion of planar-flow constraints

← Older revision		Revision as of 21:41, 21 March 2010
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	===Scalar Product of Velocity and Euler===		===Scalar Product of Velocity and Euler===
	If we dot the vector <math>\vec{v}</math> into the Euler equation, we obtain the expression,		If we dot the vector <math>\vec{v}</math> into the steady-state Euler equation, we obtain the expression,
	<div align="center">		<div align="center">
	<math>		<math>
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	===Curl of Euler===		===Curl of Euler===

	If, on the other hand, we take the curl of the Euler equation, we obtain the expression,		If, on the other hand, we take the curl of the steady-state Euler equation, we obtain the expression,
	<div align="center">		<div align="center">
	<math>		<math>
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	In this case the gradient of the Bernoulli function disappears because the curl of any gradient is zero. This vector equation provides three independent physical constraints on our system, as all three Cartesian components of the curl of <math>\vec{A}</math> must independently be zero. Expressions for the three components of <math>\nabla\times\vec{A}</math> can be found in our accompanying [[User:Tohline/PGE/RotatingFrame#Nonlinear_Velocity_Cross-Product\|general discussion of the Euler equation]] as viewed from a rotating frame of reference.		In this case the gradient of the Bernoulli function disappears because the curl of any gradient is zero. This vector equation provides three independent physical constraints on our system, as all three Cartesian components of the curl of <math>\vec{A}</math> must independently be zero. Expressions for the three components of <math>\nabla\times\vec{A}</math> can be found in our accompanying [[User:Tohline/PGE/RotatingFrame#Nonlinear_Velocity_Cross-Product\|general discussion of the Euler equation]] as viewed from a rotating frame of reference.

	==~~No Vertical Motions~~==		==Two-Dimensional Planar Flow==

			In keeping with their objective to study steady-state flows in infinitesimally thin disks, KP96 imposed one additional important constraint on the set of governing equations.

			<font color="darkblue">Assumption #3:</font> KP96 set <math>v_z = 0</math> everywhere.

			Then, in order to determine the steady-state spatial distribution of the three principal physical variables <math>\rho(x,y)</math>, <math>v_x(x,y)</math>, and <math>v_y(x,y)</math>, they looked for solutions that would simultaneously satisfy the following three PDEs:
			* The z-component of the curl of the steady-state Euler equation, that is,
			<div align="center">
			<math>
			[\nabla\times\vec{A}]_z = 0 ;
			</math>
			</div>
			* The steady-state continuity equation, that is,
			<div align="center">
			<math>
			\vec{v}\cdot\nabla\rho + \rho\nabla\cdot\vec{v} = 0 ;
			</math>
			</div>
			and,
			* The scalar product of <math>\vec{v}</math> with the steady-state Euler equation, that is,
			<div align="center">
			<math>
			\vec{v} \cdot \nabla F_B = 0 .
			</math>
			</div>
			The algebraic equation of state that they used to supplement this coupled set of governing PDEs is identified in their paper, in the discussion associated with their Eq. (23).

			Drawing from our [[User:Tohline/PGE/RotatingFrame#Nonlinear_Velocity_Cross-Product\|accompanying discussion]] of how the curl of <math>\vec{A}</math> behaves when <math>v_z = 0</math>, the first of these PDEs takes the form,
			<div align="center">
			<math>
			[\nabla\times\vec{A}]_z = \frac{\partial}{\partial x}\biggl[ (\zeta_z + 2\Omega) v_x \biggr] + \frac{\partial}{\partial y}\biggl[ (\zeta_z + 2\Omega) v_y \biggr] = 0
			</math><br />
			<math>
			\Rightarrow ~~~~~\vec{v}\cdot\nabla(\zeta_z + 2\Omega) + (\zeta_z + 2\Omega)\nabla\cdot\vec{v} = 0 .
			</math>
			</div>
			This last expression appears as Eq. (2) in KP96. (For this last expression to be valid it must be understood that, for the inherently 2D problem under investigation by KP96, <math>nabla</math> is only operating in x and y.)

	Now we restrict the flow by setting <math>v_z = 0</math>, that is, from here on we will assume that all the motion is planar. Also, following the lead of KP96, we define the vorticity of the fluid,		Now we restrict the flow by setting <math>v_z = 0</math>, that is, from here on we will assume that all the motion is planar. Also, following the lead of KP96, we define the vorticity of the fluid,
	<div align="center">		<div align="center">

Tohline: /* Governing Steady-State Equations */ Add paragraphs about scalar product and curl of Euler equation

2010-03-21T20:07:57Z

Governing Steady-State Equations: Add paragraphs about scalar product and curl of Euler equation

← Older revision		Revision as of 20:07, 21 March 2010
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	* <math>\Phi_g</math> represents the combined, time-independent gravitational and centrifugal potential, that is, <math>\Phi_g = (\Phi - \|\Omega\hat{k} \times \vec{x}\|^2/2)</math>.		* <math>\Phi_g</math> represents the combined, time-independent gravitational and centrifugal potential, that is, <math>\Phi_g = (\Phi - \|\Omega\hat{k} \times \vec{x}\|^2/2)</math>.

	Up to this point, only the two assumptions itemized above have been imposed on the key governing equations. Hence, although KP96 apply these equations to the study of a two-dimensional flow problem, our derived forms for the equations can serve to describe a fully 3D problem.		Up to this point, only the two assumptions itemized above have been imposed on the key governing equations. Hence, although KP96 apply these equations to the study of a two-dimensional flow problem, our derived forms for the equations can serve to describe a fully 3D problem.

	~~Before proceeding further we should emphasize that~~, in the ~~context of~~ the Euler equation ~~written~~ in ~~this form (i.e.,~~ the form ~~preferred by KP96), the vector <math>\vec{A}</math> defined in the preamble, above, should be written~~,		Staying with this generalized approach, let's examine a few more aspects of these governing relations before focusing in on the more restrictive, 2D problem that has been tackled in KP96. First, let's rewrite the steady-state Euler equation in the form,
	<div align="center">		<div align="center">
	<math>		<math>
	\vec{A} = ~~2\omega\hat{k}\times\vec{v} +(\nabla\times\vec{v})\times\vec{v} + \nabla \biggl[\frac{1}{2}v^2 \biggr] .~~		\nabla F_B - \vec{A} = 0 ,
	</math>		</math>
	</div>		</div>
			where, the scalar "Bernoulli" function,
			<div align="center">
			<math>
			F_B \equiv \frac{1}{2}v^2 + H + \Phi - \frac{1}{2}\|\Omega\hat{k} \times \vec{x}\|^2 ;
			</math>
			</div>
			and,
			<div align="center">
			<math>
			\vec{A} \equiv ({\vec{\zeta}}+2{\vec\Omega}_f) \times {\vec{v}} ,
			</math>
			</div>
			is the vector involving a nonlinear cross-product of the velocity that has been introduced in our accompanying [[User:Tohline/PGE/RotatingFrame#Nonlinear_Velocity_Cross-Product\|general discussion of the Euler equation]] as viewed from a rotating frame of reference.

			===Scalar Product of Velocity and Euler===
			If we dot the vector <math>\vec{v}</math> into the Euler equation, we obtain the expression,
			<div align="center">
			<math>
			\vec{v} \cdot \nabla F_B = 0 .
			</math>
			</div>
			The vector <math>\vec{A}</math> disappears as a result of the dot product with <math>\vec{v}</math> because <math>\vec{A}</math> is necessarily everywhere perpendicular to <math>\vec{v}</math>.

			===Curl of Euler===

			If, on the other hand, we take the curl of the Euler equation, we obtain the expression,
			<div align="center">
			<math>
			\nabla\times\vec{A} = 0 .
			</math>
			</div>
			In this case the gradient of the Bernoulli function disappears because the curl of any gradient is zero. This vector equation provides three independent physical constraints on our system, as all three Cartesian components of the curl of <math>\vec{A}</math> must independently be zero. Expressions for the three components of <math>\nabla\times\vec{A}</math> can be found in our accompanying [[User:Tohline/PGE/RotatingFrame#Nonlinear_Velocity_Cross-Product\|general discussion of the Euler equation]] as viewed from a rotating frame of reference.

	==No Vertical Motions==		==No Vertical Motions==