Korycansky and Papaloizou (1996)

Korycansky & Papaloizou (1996, ApJS, 105, 181; hereafter KP96) developed a method to find nontrivial, nonaxisymmetric steady-state flows in a two-dimensional setting. Specifically, they constructed infinitesimally thin steady-state disk structures in the presence of a time-independent, nonaxisymmetric perturbing potential. While their problem was only two-dimensional and they did not seek a self-consistent solution of the gravitational Poisson equation, the approach they took to solving the 2D Euler equation in tandem with the continuity equation for a compressible fluid is instructive. What follows is a summary of their approach.

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 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

Eulerian Representation
of the Continuity Equation
as viewed from a Rotating Reference Frame

<math>\frac{\partial\rho}{\partial t} + \nabla \cdot (\rho {\vec{v}}) = 0</math> ,

and the,

Eulerian Representation
of the Euler Equation
as viewed from a Rotating Reference Frame

<math>\frac{\partial\vec{v}}{\partial t} + ({\vec{v}}\cdot \nabla) {\vec{v}} = - \frac{1}{\rho} \nabla P - \nabla \biggl[\Phi - \frac{1}{2}|{\vec{\Omega}}_f \times \vec{x}|^2 \biggr] - 2{\vec{\Omega}}_f \times {\vec{v}} </math> .

And their Eq. (7) can be derived straightforwardly from the

Euler Equation
written in terms of the Vorticity and
as viewed from a Rotating Reference Frame

<math>\biggl[\frac{\partial\vec{v}}{\partial t}\biggr] + ({\vec{\zeta}}+2{\vec\Omega}_f) \times {\vec{v}}= - \frac{1}{\rho} \nabla P - \nabla \biggl[\Phi + \frac{1}{2}v^2 - \frac{1}{2}|{\vec{\Omega}}_f \times \vec{x}|^2 \biggr]</math> .

Assumption #1: KP96 align the angular velocity vector of the rotating frame of reference with the z-axis of a Cartesian coordinate system. Specifically, they set

<math>{\vec{\Omega}}_f = \hat{k}\Omega</math>.

Assumption #2:Because KP96 are seeking steady-state solutions, they set all Eulerian time-derivatives to zero.

Hence, the steady-state versions of the Euler and continuity equations shown above give rise to Eq. (1) of KP96, namely,

<math> (\vec{v}\cdot \nabla)\vec{v} + 2\Omega\hat{k}\times\vec{v} + \frac{1}{\rho}\nabla P + \nabla \biggl[\Phi -\frac{1}{2}\omega^2 \varpi^2 \biggr] = 0 , </math>

<math> \nabla\cdot(\rho \vec{v}) = \vec{v}\cdot\nabla\rho + \rho\nabla\cdot\vec{v} = 0 . </math>

And, if written in terms of the vorticity, our steady-state Euler equation becomes essentially Eq. (7) of KP96, namely,

<math> 0 = ({\vec{\zeta}}+2\Omega{\hat{k}}) \times {\vec{v}} + \frac{1}{\rho} \nabla P + \nabla \biggl[\Phi + \frac{1}{2}v^2 - \frac{1}{2}|\Omega\hat{k} \times \vec{x}|^2 \biggr] </math>

<math> = - {\vec{v}}\times({\vec{\zeta}}+2\Omega{\hat{k}}) + \nabla \biggl[\frac{1}{2}v^2 + H + \Phi - \frac{1}{2}|\Omega\hat{k} \times \vec{x}|^2 \biggr] , </math>

where, in this last expression, we have replace the gradient of the pressure by the gradient of the enthalpy via the relation, <math>\nabla H = \nabla P/\rho </math>. 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 surface density;
<math>\hat{z}</math> is used instead of <math>\hat{k}</math> to denote a unit vector in the z-coordinate direction;
the vorticity vector is written as <math>\hat{z}\omega</math> instead of <math>\vec\zeta</math>;
<math>W</math> is used instead of <math>~H</math> to denote the enthalpy; and
<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.

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,

<math> \nabla F_B - \vec{A} = 0 , </math>

where, the scalar "Bernoulli" function,

<math> F_B \equiv \frac{1}{2}v^2 + H + \Phi - \frac{1}{2}|\Omega\hat{k} \times \vec{x}|^2 ; </math>

and,

<math> \vec{A} \equiv ({\vec{\zeta}}+2{\vec\Omega}_f) \times {\vec{v}} , </math>

is the vector involving a nonlinear cross-product of the velocity that has been introduced in our accompanying 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 steady-state Euler equation, we obtain the expression,

<math> \vec{v} \cdot \nabla F_B = 0 . </math>

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 steady-state Euler equation, we obtain the expression,

<math> \nabla\times\vec{A} = 0 . </math>

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 general discussion of the Euler equation as viewed from a rotating frame of reference.

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.

Assumption #3: 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,

<math> [\nabla\times\vec{A}]_z = 0 ; </math>

The steady-state continuity equation, that is,

<math> \vec{v}\cdot\nabla\rho + \rho\nabla\cdot\vec{v} = 0 ; </math>

and,

The scalar product of <math>\vec{v}</math> with the steady-state Euler equation, that is,

<math> \vec{v} \cdot \nabla F_B = 0 . </math>

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 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,

<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>
<math> \Rightarrow ~~~~~\vec{v}\cdot\nabla(\zeta_z + 2\Omega) + (\zeta_z + 2\Omega)\nabla\cdot\vec{v} = 0 . </math>

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,

<math> \vec{\zeta} \equiv \nabla\times\vec{v} = \hat{i}\zeta_x + \hat{j}\zeta_y + \hat{k}\zeta_z . </math>

[Note that (unfortunately) KP96 use <math>\omega</math> instead of <math>\zeta</math> to represent the rotating-frame vorticity.] In terms of the components of the vorticity vector, the steady-state Euler equation therefore becomes,

<math> (2\omega + \zeta_z)\hat{k}\times\vec{v} + (\hat{i}\zeta_x + \hat{j}\zeta_y)\times\vec{v} + \nabla \biggl[\frac{1}{2}v^2 + H + \Phi -\frac{1}{2}\omega^2 R^2 \biggr] = 0 . </math>

Continuing to follow the lead KP96, we next take the curl of this Euler equation. Because the curl of a gradient is always zero, this leads us to the same condition discussed above — but this time written in terms of the components of the vorticity — namely,

<math> \nabla\times\vec{A} = 0 = \nabla\times [(2\omega + \zeta_z)\hat{k}\times\vec{v} + (\hat{i}\zeta_x + \hat{j}\zeta_y)\times\vec{v}] . </math>

Using another vector identity, namely,

<math> \nabla\times(\vec{C} \times \vec{B}) = (\vec{B}\cdot\nabla)\vec{C} - (\vec{C}\cdot\nabla)\vec{B} + \vec{C}(\nabla\cdot\vec{B}) - \vec{B}(\nabla\cdot\vec{C}), </math>

and remembering that we are assuming <math>v_z = 0</math>, we see in this case that the vector condition <math>\nabla\times\vec{A}=0</math> leads to the following three independent scalar constraints:

<math> ~~~~~\hat{i}:~~~~~ [\nabla\times\vec{A}]_x = - \frac{\partial }{\partial z}\biggl[ (2\omega + \zeta)v_x \biggr] + \frac{\partial}{\partial y} \biggl[ \zeta_x v_y - \zeta_y v_x \biggr] = 0 ; </math>
<math> ~~~~~\hat{j}:~~~~~ [\nabla\times\vec{A}]_y = - \frac{\partial }{\partial z} \biggl[ (2\omega + \zeta)v_y \biggr] - \frac{\partial}{\partial x} \biggl[ \zeta_x v_y - \zeta_y v_x \biggr] = 0 ; </math>
<math> ~~~~~\hat{k}:~~~~~ [\nabla\times\vec{A}]_z = (2\omega + \zeta)\nabla\cdot\vec{v} + \biggl[ v_x \frac{\partial}{\partial x} + v_y \frac{\partial}{\partial y} \biggr](2\omega + \zeta) = 0 . </math>

With the understanding that, by definition,

<math> \zeta_x \equiv - \frac{\partial v_y}{\partial z} , ~~~~~ \zeta_y \equiv + \frac{\partial v_x}{\partial z} , ~~~~~ \mathrm{and} ~~~~~ \zeta_z \equiv \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} , </math>

it can be shown that these three constraints are identical to the ones presented in the preamble, above.

Solution Strategy

Constraint #1: For their two-dimensional disk problem, KP96 focused on the constraint provided by the z-component of the curl of the Euler equation, which can be rewritten as (see above derivation, or Eq. 2 of KP96),

<math> \nabla\cdot\vec{v} =-\vec{v} \cdot \biggl[ \frac{\nabla(2\omega + \zeta_z)}{(2\omega + \zeta_z)} \biggr] = -\vec{v} \cdot \nabla[\ln(2\omega + \zeta_z)]. </math>

Constraint #2: But from the continuity equation they also know that,

<math> \nabla\cdot\vec{v} = -\vec{v}\cdot\biggl[\frac{\nabla\rho}{\rho} \biggr] = -\vec{v} \cdot \nabla[\ln\rho] . </math>

Hence,

<math> \vec{v} \cdot \nabla[\ln(2\omega + \zeta_z)] = \vec{v} \cdot \nabla[\ln\rho] , </math>

that is,

<math> \vec{v} \cdot \nabla\ln\biggl[ \frac{(2\omega + \zeta_z)}{\rho} \biggr] = 0 . </math>

This is essentially KP96's Eq. (3).

Introduce stream function: The constraint implied by the continuity equation also suggests that it might be useful to define a stream function in terms of the momentum density — instead of in terms of just the velocity, which is the natural treatment in the context of incompressible fluid flows. KP96 do this. They define the stream function, <math>\Psi</math>, such that (see their Eq. 4),

<math> \rho\vec{v} = \nabla\times(\hat{k}\Psi) . </math>

in which case,

<math> v_x = \frac{1}{\rho} \frac{\partial \Psi}{\partial y} ~~~~~\mathrm{and}~~~~~ v_y = - \frac{1}{\rho} \frac{\partial \Psi}{\partial x} . </math>

This implies as well that the z-component of the fluid vorticity can be expressed in terms of the stream function as follows (see Eq. 5 of KP96):

<math> \zeta_z = - \nabla\cdot \biggl( \frac{\nabla\Psi}{\rho} \biggr) = - \frac{\partial}{\partial x} \biggl[ \frac{1}{\rho} \frac{\partial\Psi}{\partial x} \biggr] - \frac{\partial}{\partial y} \biggl[ \frac{1}{\rho} \frac{\partial\Psi}{\partial y} \biggr]. </math>

According to KP96, this expression, taken in combination with the conclusion drawn above from the second constraint — that is, Eq. (3) taken in combination with Eq. (4) from KP96 — "tell us that the 'vortensity' <math>(\zeta_z + 2\omega)/\rho</math> is constant along streamlines which are lines of constant <math>\Psi</math>." The vortensity is therefore a function of <math>\Psi</math> alone, so we can write,

<math> \frac{\zeta_z + 2\omega}{\rho} = g(\Psi) . </math>

Constraint #3:

Taking the scalar product of <math>\vec{v}</math> and the following form of the steady-state Euler equation,

<math> 2\omega\hat{k}\times\vec{v} - \vec{v}\times(\nabla\times\vec{v}) + \nabla \biggl[\frac{1}{2}v^2 + H + \Phi -\frac{1}{2}\omega^2 R^2 \biggr] = 0 , </math>

we obtain the constraint,

<math> \vec{v}\cdot\nabla \biggl[\frac{1}{2}v^2 + H + \Phi -\frac{1}{2}\omega^2 R^2 \biggr] = 0 . </math>

When tied with our earlier discussion, this means that the Bernoulli function also must be constant along streamlines. Hence, we can write,

<math> \frac{1}{2}v^2 + H + \Phi -\frac{1}{2}\omega^2 R^2 = F(\Psi) . </math>

KP96 then go on to demonstrate that the relationship between the functions <math>g(\Psi)</math> and <math>F(\Psi)</math> is,

which allows the determination of <math>F</math> up to a constant of integration.

Summary

In summary, KP96 constrain their flow as follows:

They use the z-component of the curl of the Euler 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.

User:Tohline/Apps/Korycansky Papaloizou 1996

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