User:Tohline/Appendix/Ramblings/Hybrid Scheme Implications
Implications of Hybrid Scheme
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Background
Key H_Book Chapters
[Ref01] Inertial-Frame Euler Equation
[Ref02] Traditional Description of Rotating Reference Frame
[Ref03] Hybrid Advection Scheme
[Ref04] Riemann S-type Ellipsoids
[Ref05] Korycansky and Papaloizou (1996)
Principal Governing Equations
Quoting from [Ref01] … Among the principal governing equations we have included the inertial-frame,
Lagrangian Representation
of the Euler Equation,
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<math>\frac{d\vec{v}}{dt} = - \frac{1}{\rho} \nabla P - \nabla \Phi</math> |
[EFE], Chap. 2, §11, p. 20, Eq. (38)
[BLRY07], p. 13, Eq. (1.55)
Shifting into a rotating frame characterized by the angular velocity vector,
<math>~\vec{\Omega}_f \equiv \hat\mathbf{k} \Omega_f \, ,</math>
and applying the operations that are specified in the first few subsections of [Ref02], we recognize the following relationships …
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<math>~\vec{v}_\mathrm{inertial}</math> |
<math>~=</math> |
<math>~\vec{v}_\mathrm{rot} + {\vec\Omega}_f \times \vec{x} \, ,</math> |
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<math>~\biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{inertial}</math> |
<math>~=</math> |
<math>~ \biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{rot} + 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot} + {\vec\Omega}_f \times ({\vec\Omega}_f \times \vec{x}) </math> |
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<math>~=</math> |
<math>~ \biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{rot} + 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot} - \frac{1}{2} \nabla | {\vec\Omega}_f \times \vec{x}|^2 </math> |
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<math>~=</math> |
<math>~ \biggl[ \frac{\partial \vec{v}}{\partial t} \biggr]_\mathrm{rot} + ({\vec{v}}_\mathrm{rot} \cdot \nabla){\vec{v}}_\mathrm{rot} + 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot} - \frac{1}{2} \nabla | {\vec\Omega}_f \times \vec{x}|^2 \, .</math> |
Making this substitution on the left-hand-side of the above-specified "Lagrangian Representation of the Euler Equation," we obtain what we have referred to also in [Ref02] as the,
Eulerian Representation
of the Euler Equation
as viewed from a Rotating Reference Frame
<math>\biggl[\frac{\partial\vec{v}}{\partial t}\biggr]_\mathrm{rot} + ({\vec{v}}_\mathrm{rot}\cdot \nabla) {\vec{v}}_\mathrm{rot}= - \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}}_\mathrm{rot} \, .</math>
This form of the Euler equation also appears early in [Ref05], where we set up a discussion of the paper by Korycansky & Papaloizou (1996, ApJS, 105, 181; hereafter KP96). But, for now, let's back up a couple of steps and retain the total time derivative on the left-hand-side. That is, let's select as the foundation expression the,
Lagrangian Representation
of the Euler Equation
as viewed from a Rotating Reference Frame
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<math>~\biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{rot} </math> |
<math>~=</math> |
<math>~- \frac{1}{\rho} \nabla P - \nabla \Phi - 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot} - {\vec\Omega}_f \times ({\vec\Omega}_f \times \vec{x}) \, ,</math> |
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[EFE], Chap. 2, §12, p. 25, Eq. (62) |
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which also serves as the foundation of most of our [Ref03] discussions.
Exercising the Hybrid Scheme
Focus on Tracking Angular Momentum
Let's begin by using <math>~\mathbf{u'}</math>, instead of <math>~{\vec{v}}_\mathrm{rot}</math>, to represent the fluid velocity vector as viewed from the rotating frame of reference. Our foundation expression becomes,
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<math>~\frac{d \bold{u'}}{dt} </math> |
<math>~=</math> |
<math>~- \frac{1}{\rho} \nabla P - \nabla \Phi - 2{\vec\Omega}_f \times \bold{u}' - {\vec\Omega}_f \times ({\vec\Omega}_f \times \vec{x}) \, ,</math> |
where we appreciate that we can move from the Lagrangian to an Eulerian representation by employing the operator substitution,
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<math>~\frac{d}{dt}</math> |
<math>~\rightarrow</math> |
<math>~\frac{\partial}{\partial t} + \mathbf{u'} \cdot \nabla </math> |
Next, using [Ref03] as a guide, let's focus on tracking angular momentum. We need to break the vector momentum equation, as well as the velocity vectors, into their <math>~(\bold{\hat{e}}_\varpi, \bold{\hat{e}}_\varphi, \bold{\hat{k}})</math> components.
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NOTE: For the time being, we will write the velocity vector in terms of generic components, namely, <math>~\bold{u}' = \bold{\hat{e}}_\varpi u'_\varpi + \bold{\hat{e}}_\varphi u'_\varphi + \bold{\hat{k}}u'_z \, .</math> But, eventually, we want to explicitly insert the rotating-frame velocity that underpins the equilibrium properties of Riemann S-type ellipsoids. In Chap. 7, §47, Eq. 1 (p. 130) of [EFE], this is given in Cartesian coordinates, so we will need to convert his expressions to the equivalent cylindrical-coordinate components. |
The time-derivative on the left-hand-side of our foundation expression becomes,
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<math> \frac{d\mathbf{u'}}{dt} </math> |
<math>~=~</math> |
<math> \frac{d}{dt} [ \mathbf{\hat{e}}_\varpi u'_\varpi + \mathbf{\hat{e}}_\varphi u'_\varphi + \mathbf{\hat{k}} u'_z ] </math> |
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<math>~=~</math> |
<math> \mathbf{\hat{e}}_\varpi \frac{d u'_\varpi}{dt} + \mathbf{\hat{e}}_\varphi \frac{d u'_\varphi}{dt} + \mathbf{\hat{k}} \frac{d u'_z}{dt} + ( u'_\varpi) \frac{d}{dt}\mathbf{\hat{e}}_\varpi + ( u'_\varphi) \frac{d}{dt}\mathbf{\hat{e}}_\varphi </math> |
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<math>~=~</math> |
<math> \mathbf{\hat{e}}_\varpi \frac{d u'_\varpi}{dt} + \mathbf{\hat{e}}_\varphi \frac{d u'_\varphi}{dt} + \mathbf{\hat{k}} \frac{d u'_z}{dt} + \mathbf{\hat{e}}_\varphi(u'_\varpi) \frac{u'_\varphi}{\varpi} - \mathbf{\hat{e}}_\varpi(u'_\varphi) \frac{u'_\varphi}{\varpi} \, . </math> |
We also recognize that, when expressed in cylindrical coordinates,
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<math> ~{\vec{\Omega}}_f \times \vec{x} </math> |
<math>~=~</math> |
<math> {\hat\mathbf{k}} \Omega_f\times (\mathbf{\hat{e}}_\varpi \varpi + \mathbf{\hat{k}}z) = \mathbf{\hat{e}}_\varphi \Omega_f \varpi \, , </math> |
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<math> {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) </math> |
<math>~=~</math> |
<math> \hat{\mathbf{k}} \Omega_f \times ( \mathbf{\hat{e}}_\varphi \Omega_f \varpi ) = - \mathbf{\hat{e}}_\varpi \Omega_f^2 \varpi \, , </math> |
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<math> {\vec{\Omega}}_f \times {\mathbf{u'}} </math> |
<math>~=~</math> |
<math> {\hat\mathbf{k}} \Omega_f\times (\mathbf{\hat{e}}_\varpi u'_\varpi + \mathbf{\hat{e}}_\varphi u'_\varphi + \mathbf{\hat{k}}u'_z) = \mathbf{\hat{e}}_\varphi \Omega_f u'_\varpi - \mathbf{\hat{e}}_\varpi \Omega_f u'_\varphi \, , </math> |
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<math> {\vec{v}}_\mathrm{inertial} </math> |
<math>~=~</math> |
<math> \mathbf{u'} + \mathbf{\hat{e}}_\varphi \Omega_f \varpi \, . </math> |
The set of scalar momentum-component equations is obtained by "dotting" each unit vector into the vector equation.
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<math>\mathbf{\hat{e}}_\varpi:</math> |
<math>~\frac{d u'_\varpi}{dt} - \frac{(u'_\varphi)^2}{\varpi} </math> |
<math>~=</math> |
<math>~- \mathbf{\hat{e}}_\varpi \cdot \frac{\nabla P}{\rho} - \mathbf{\hat{e}}_\varpi \cdot \nabla \Phi + 2 \biggl[ \Omega_f u'_\varphi \biggr] + \Omega_f^2 \varpi </math> |
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<math>~\Rightarrow ~~~ \frac{d u'_\varpi}{dt} </math> |
<math>~=</math> |
<math>~- \mathbf{\hat{e}}_\varpi \cdot \frac{\nabla P}{\rho} - \mathbf{\hat{e}}_\varpi \cdot \nabla \Phi + \frac{1}{\varpi} \biggl[ (u'_\varphi)^2 + 2 \Omega_f u'_\varphi \varpi + \Omega_f^2 \varpi^2 \biggr]</math> |
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<math>~=</math> |
<math>~ - \mathbf{\hat{e}}_\varpi \cdot \frac{\nabla P}{\rho} - \mathbf{\hat{e}}_\varpi \cdot \nabla \Phi + \frac{1}{\varpi} (u'_\varphi + \Omega_f \varpi)^2 \, ; </math> |
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<math>\mathbf{\hat{e}}_\varphi:</math> |
<math>~\frac{d u'_\varphi}{dt} + \frac{u'_\varpi u'_\varphi}{\varpi} </math> |
<math>~=</math> |
<math>~- \mathbf{\hat{e}}_\varphi \cdot \frac{\nabla P}{\rho} - \mathbf{\hat{e}}_\varphi \cdot \nabla \Phi - 2\biggl[ \Omega_f u'_\varpi \biggr] </math> |
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(mult. thru by ϖ) <math>~\Rightarrow ~~~\frac{d (\varpi u'_\varphi )}{dt} </math> |
<math>~=</math> |
<math>~- \mathbf{\hat{e}}_\varphi \cdot \frac{\varpi \nabla P}{\rho} - \mathbf{\hat{e}}_\varphi \cdot \varpi \nabla \Phi - 2 \Omega_f \varpi u'_\varpi \, ; </math> |
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<math>\mathbf{\hat{k}}:</math> |
<math>~\frac{d u'_z}{dt} </math> |
<math>~=</math> |
<math>~- \mathbf{\hat{k}} \cdot \frac{\nabla P }{\rho} - \mathbf{\hat{k}} \cdot \nabla \Phi \, . </math> |
Now, recalling that <math>~\mathbf{u'} = (\mathbf{v} - \mathbf{\hat{e}}_\varphi \varpi \Omega_f)</math>, let's make the substitutions …
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<math>~u'_\varpi \rightarrow v_\varpi \, ,</math> |
<math>~u'_\varphi \rightarrow (v_\varphi - \varpi\Omega_f) \, ,</math> and, |
<math>~u'_z \rightarrow v_z \, .</math> |
This mapping gives,
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<math>\mathbf{\hat{e}}_\varphi:</math> |
<math>~\frac{d [\varpi v_\varphi - \varpi^2 \Omega_f]}{dt} </math> |
<math>~=</math> |
<math>~- \mathbf{\hat{e}}_\varphi \cdot \frac{\varpi \nabla P}{\rho} - \mathbf{\hat{e}}_\varphi \cdot \varpi \nabla \Phi - 2 \Omega_f \varpi v_\varpi \, ; </math> |
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<math>~\Rightarrow ~~~ \frac{d (\varpi v_\varphi )}{dt} </math> |
<math>~=</math> |
<math>~- \mathbf{\hat{e}}_\varphi \cdot \frac{\varpi \nabla P}{\rho} - \mathbf{\hat{e}}_\varphi \cdot \varpi \nabla \Phi \, ; </math> |
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<math>~\Rightarrow ~~~ \frac{1}{\varpi} ~\frac{d (\varpi v_\varphi )}{dt} </math> |
<math>~=</math> |
<math>~- \mathbf{\hat{e}}_\varphi \cdot \biggl[ \frac{\nabla P}{\rho} + \nabla \Phi \biggr] \, ; </math> |
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<math>\mathbf{\hat{k}}:</math> |
<math>~\frac{d v_z}{dt} </math> |
<math>~=</math> |
<math>~- \mathbf{\hat{k}} \cdot \biggl[ \frac{\nabla P }{\rho} + \nabla \Phi \biggr] \, . </math> |
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<math>\mathbf{\hat{e}}_\varpi:</math> |
<math>~\frac{d v_\varpi}{dt} </math> |
<math>~=</math> |
<math>~ - \mathbf{\hat{e}}_\varpi \cdot \biggl[ \frac{\nabla P}{\rho} + \nabla \Phi \biggr] + \frac{v_\varphi^2}{\varpi} \, ; </math> |
Steady-State Velocity Field for Jacobi Ellipsoids
In steady-state, the (Lagrangian time-derivative) operator on the left-hand-side of all three component equations maps to the following operator:
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<math>~\mathbf{u'} \cdot \nabla</math> |
<math>~=</math> |
<math>~\sum_{i=1}^3 u'_i \frac{\partial}{\partial x_i} \, ,</math> |
(in Cartesian coordinates); |
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<math>~\mathbf{u'} \cdot \nabla</math> |
<math>~=</math> |
<math>~ u'_\varpi \frac{\partial}{\partial \varpi} + \frac{u'_\varphi}{\varpi} \frac{\partial}{\partial \varphi} + u'_z \frac{\partial}{\partial z} \, ,</math> |
(in cylindrical coordinates); |
We know, as well, that,
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<math>~u'_\varpi = u'_x \cos\varphi + u'_y \sin\varphi \, ,</math> |
and, |
<math>~u'_\varphi = u'_y \cos\varphi - u'_x \sin\varphi \, .</math> |
Hence, the cylindrical-coordinate-based operator may be rewritten as,
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<math>~\mathbf{u'} \cdot \nabla</math> |
<math>~=</math> |
<math>~ ( u'_x \cos\varphi + u'_y \sin\varphi ) \frac{\partial}{\partial \varpi} + ( u'_y \cos\varphi - u'_x \sin\varphi )\frac{1}{\varpi} \frac{\partial}{\partial \varphi} + u'_z \frac{\partial}{\partial z} \, .</math> |
Drawing from [ Ref04 ] … As Ou(2006) has pointed out, the velocity field of a Riemann S-type ellipsoid as viewed from a frame rotating with angular velocity <math>~{\vec{\Omega}}_f = \boldsymbol{\hat{k}} \Omega_f</math> takes the following form:
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<math>~{\mathbf{u'}}</math> |
<math>~=</math> |
<math>~\lambda \biggl[ \boldsymbol{\hat{\imath}} \biggl(\frac{a}{b}\biggr)y - \boldsymbol{\hat{\jmath}} \biggl(\frac{b}{a}\biggr)x \biggr] \, ,</math> |
Ou(2006), p. 550, §2, Eq. (3)
where <math>~\lambda</math> is a constant that determines the magnitude of the internal motion of the fluid, and the origin of the x-y coordinate system is at the center of the ellipsoid. This velocity field, <math>~\mathbf{u'}</math>, is designed so that velocity vectors everywhere are always aligned with elliptical stream lines by demanding that they be tangent to the equi-effective-potential contours, which are concentric ellipses. Hence, for Riemann S-type ellipsoids, we have,
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<math>~u'_x = \lambda\biggl(\frac{a}{b}\biggr)y = \lambda\biggl(\frac{a}{b}\biggr)\varpi \sin\varphi \, ;</math> |
<math>~u'_y = -\lambda\biggl(\frac{b}{a}\biggr)x = -\lambda\biggl(\frac{b}{a}\biggr)\varpi \cos\varphi \, ;</math> |
<math>~u'_z = 0 \, .</math> |
So, for the velocity flow that underpins Riemann S-type ellipsoids, the cylindrical-coordinate-based operator is
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<math>~\mathbf{u'} \cdot \nabla</math> |
<math>~=</math> |
<math>~ \biggl[ \lambda\biggl(\frac{a}{b}\biggr)\varpi \sin\varphi \cos\varphi -\lambda\biggl(\frac{b}{a}\biggr)\varpi \cos\varphi \sin\varphi \biggr] \frac{\partial}{\partial \varpi} + \biggl[ -\lambda\biggl(\frac{b}{a}\biggr)\varpi \cos\varphi \cos\varphi - \lambda\biggl(\frac{a}{b}\biggr)\varpi \sin\varphi \sin\varphi \biggr] \frac{1}{\varpi} \frac{\partial}{\partial \varphi} </math> |
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<math>~=</math> |
<math>~ \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\lambda \varpi \sin\varphi \cos\varphi \frac{\partial}{\partial \varpi} - \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr]\lambda \frac{\partial}{\partial \varphi} \, .</math> |
And, given that,
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<math>~\mathbf{\hat{e}}\Omega_f \varpi</math> |
<math>~=</math> |
<math>~ \Omega_f \varpi \biggl[ \boldsymbol{\hat{\jmath}} \cos\varphi - \boldsymbol{\hat{\imath}} \sin\varphi \biggr] \, , </math> |
the inertial-frame velocity components are,
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<math>~ v_x = \lambda\biggl(\frac{a}{b}\biggr)\varpi \sin\varphi - \Omega_f \varpi \sin\varphi = \biggl[ \lambda\biggl(\frac{a}{b}\biggr) - \Omega_f \biggr]\varpi\sin\varphi \, ;</math> |
<math>~ v_y = -\lambda\biggl(\frac{b}{a}\biggr) \varpi\cos\varphi + \Omega_f\varpi \cos\varphi = \biggl[\Omega_f -\lambda\biggl(\frac{b}{a}\biggr) \biggr]\varpi\cos\varphi \, ;</math> |
<math>~v_z = 0 \, .</math> |
That is,
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<math>~v_\varpi = v_x \cos\varphi + v_y \sin\varphi</math> |
<math>~=</math> |
<math>~ \biggl[ \lambda\biggl(\frac{a}{b}\biggr) - \Omega_f \biggr]\varpi\sin\varphi \cos\varphi + \biggl[\Omega_f -\lambda\biggl(\frac{b}{a}\biggr) \biggr]\varpi\cos\varphi \sin\varphi </math> |
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<math>~=</math> |
<math>~ \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\lambda \varpi\sin\varphi \cos\varphi \, ; </math> |
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<math>~v_\varphi = v_y \cos\varphi - v_x \sin\varphi</math> |
<math>~=</math> |
<math>~ \biggl[\Omega_f -\lambda\biggl(\frac{b}{a}\biggr) \biggr]\varpi\cos^2\varphi - \biggl[ \lambda\biggl(\frac{a}{b}\biggr) - \Omega_f \biggr]\varpi\sin^2\varphi </math> |
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<math>~=</math> |
<math>~ \Omega_f \varpi -\lambda \varpi \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi +\biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr] \, . </math> |
Note, as well, that,
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<math>~\frac{v_\varphi^2}{\varpi}</math> |
<math>~=</math> |
<math>~ \frac{1}{\varpi} \biggl\{ \Omega_f \varpi -\lambda \varpi \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi +\biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr] \biggr\}^2 </math> |
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<math>~=</math> |
<math>~ \varpi \biggl\{ \Omega_f^2 - 2\lambda \Omega_f \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi +\biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr] + \lambda^2 \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi +\biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr]^2 \biggr\} \, . </math> |
Finally, then, we find that the left-hand-side of the momentum-component expressions are,
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<math>\mathbf{\hat{k}}:</math> |
<math>~\frac{d v_z}{dt} </math> |
<math>~=</math> |
<math>~0 \, ; </math> |
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<math>\mathbf{\hat{e}}_\varpi:</math> |
<math>~\frac{d v_\varpi}{dt} </math> |
<math>~=</math> |
<math>~ \biggl\{ \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\lambda \varpi \sin\varphi \cos\varphi \frac{\partial}{\partial \varpi} - \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr]\lambda \frac{\partial}{\partial \varphi} \biggr\}\biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\lambda \varpi\sin\varphi \cos\varphi </math> |
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<math>~=</math> |
<math>~ \lambda \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\biggl\{ \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\lambda \varpi \sin\varphi \cos\varphi \frac{\partial}{\partial \varpi} \biggl[ \varpi\sin\varphi \cos\varphi \biggr] - \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr]\lambda \frac{\partial}{\partial \varphi}\biggl[ \varpi\sin\varphi \cos\varphi \biggr] \biggr\} </math> |
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<math>~=</math> |
<math>~ \lambda \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\biggl\{ \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\lambda \varpi \sin^2\varphi \cos^2\varphi + \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr] \lambda\varpi \biggl[ \sin^2\varphi - \cos^2\varphi \biggr] \biggr\} </math> |
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<math>~=</math> |
<math>~ \lambda^2 \varpi \biggl( \frac{a}{b} - \frac{b}{a} \biggr) \biggl[ \biggl( \frac{a}{b}\biggr) \sin^4\varphi - \biggl(\frac{b}{a}\biggr) \cos^4\varphi \biggr] \, ; </math> |
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<math>\mathbf{\hat{e}}_\varphi:</math> |
<math>~\frac{1}{\varpi} ~\frac{d (\varpi v_\varphi )}{dt} </math> |
<math>~=</math> |
<math>~ \frac{1}{\varpi}\biggl\{ \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\lambda \varpi \sin\varphi \cos\varphi \frac{\partial}{\partial \varpi} - \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr]\lambda \frac{\partial}{\partial \varphi} \biggr\} \biggl\{ \biggl[\Omega_f -\lambda\biggl(\frac{b}{a}\biggr) \biggr]\varpi^2\cos^2\varphi - \biggl[ \lambda\biggl(\frac{a}{b}\biggr) - \Omega_f \biggr]\varpi^2\sin^2\varphi \biggr\} </math> |
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<math>~=</math> |
<math>~ \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr] 2\varpi \lambda \sin\varphi \cos\varphi \biggl\{ \biggl[\Omega_f -\lambda\biggl(\frac{b}{a}\biggr) \biggr]\cos^2\varphi - \biggl[ \lambda\biggl(\frac{a}{b}\biggr) - \Omega_f \biggr]\sin^2\varphi \biggr\} </math> |
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<math>~+ \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr] \biggl[ \frac{a}{b} - \frac{b}{a} \biggr] 2\varpi \lambda^2 \sin\varphi \cos\varphi </math> |
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<math>~=</math> |
<math>~ \biggl( \frac{a}{b} - \frac{b}{a} \biggr) 2\varpi \lambda \sin\varphi \cos\varphi \biggl\{ \biggl[\Omega_f -\lambda\biggl(\frac{b}{a}\biggr) \biggr]\cos^2\varphi - \biggl[ \lambda\biggl(\frac{a}{b}\biggr) - \Omega_f \biggr]\sin^2\varphi +2\lambda\biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr] \biggr\} </math> |
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<math>~=</math> |
<math>~ \biggl( \frac{a}{b} - \frac{b}{a} \biggr) 2\varpi \lambda \sin\varphi \cos\varphi \biggl\{ \biggl[\Omega_f +\lambda\biggl(\frac{b}{a}\biggr)\biggr]\cos^2\varphi + \biggl[ \Omega_f + \lambda\biggl( \frac{a}{b}\biggr) \biggr]\sin^2\varphi \biggr\} \, . </math> |
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Try Again
As viewed from a cylindrical-coordinate-based <math>~(\varpi, \varphi, z)</math> inertial reference frame, we are interested in specifying the location,
<math>~\bold{x} = \mathbf{\hat{e}}_\varpi \varpi + \bold{\hat{k}} z \, ,</math>
[BT87], p. 646, Appendix §1.B.2, Eq. (1B-18)
of a Lagrangian fluid element at time <math>~t = 0</math> — hereafter denoted by the subscript, <math>~0</math> — as well as at later times. Although the position vector, <math>~\bold{x}</math>, does not explicitly display a dependence on the azimuthal coordinate angle, <math>~\varphi</math>, it is important to realize that the orientation in space of the unit vector, <math>~\bold{\hat{e}}_\varpi</math>, does depend on the value of this coordinate angle.
At any point in time, the instantaneous velocity of this Lagrangian fluid element will correspond precisely with the (total) time-derivative of its instantaneous position vector, that is,
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<math>~\bold{v}</math> |
<math>~\equiv</math> |
<math>~\frac{d\bold{x}}{dt}</math> |
<math>~=</math> |
<math>~\bold{\hat{e}}_\varpi \frac{d\varpi}{dt} + \bold{\hat{k}} \frac{dz}{dt} + \varpi \frac{d \bold{\hat{e}}_\varpi}{dt}</math> |
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<math>~=</math> |
<math>~\bold{\hat{e}}_\varpi \frac{d\varpi}{dt} + \bold{\hat{k}} \frac{dz}{dt} + \varpi \biggl[ \bold{\hat{e}}_\varphi \frac{d\varphi}{dt} \bigg] \, .</math> |
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[BT87], p. 647, Appendix §1.B.2, Eq. (1B-23) | ||||
In carrying out this time differentiation, the last term on the right-hand-side accounts for the aforementioned dependence of <math>~\bold{\hat{e}}_\varpi</math> on <math>~\varphi</math>. Similarly, the following component breakdown of the Lagrangian fluid element's acceleration takes into account the dependence of <math>~\bold{\hat{e}}_\varphi</math> on <math>~\varphi</math>:
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<math>~\bold{a}</math> |
<math>~\equiv</math> |
<math>~\frac{d\bold{v}}{dt}</math> |
<math>~=</math> |
<math>~ \bold{\hat{e}}_\varpi \frac{d^2\varpi}{dt^2} + \bold{\hat{k}} \frac{d^2z}{dt^2} + \bold{\hat{e}}_\varphi \biggl[\frac{d\varpi}{dt} \cdot \frac{d\varphi}{dt} + \varpi \frac{d^2\varphi}{dt^2}\biggr] + \varpi \frac{d\varphi}{dt} \biggl[ \frac{d\bold{\hat{e}}_\varphi}{dt} \biggr] + \frac{d\varpi}{dt} \biggl[ \frac{d\bold{\hat{e}}_\varpi}{dt} \biggr] </math> |
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<math>~=</math> |
<math>~ \bold{\hat{e}}_\varpi \frac{d^2\varpi}{dt^2} + \bold{\hat{k}} \frac{d^2z}{dt^2} + \bold{\hat{e}}_\varphi \biggl[\frac{d\varpi}{dt} \cdot \frac{d\varphi}{dt} + \varpi \frac{d^2\varphi}{dt^2}\biggr] + \varpi \frac{d\varphi}{dt} \biggl[- \bold{\hat{e}}_\varpi \frac{d\varphi}{dt} \biggr] + \frac{d\varpi}{dt} \biggl[ \bold{\hat{e}}_\varphi \frac{d\varphi}{dt} \biggr] </math> |
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<math>~=</math> |
<math>~ \bold{\hat{e}}_\varpi \biggl[\frac{d^2\varpi}{dt^2} - \varpi \biggl(\frac{d\varphi}{dt}\biggr)^2 \biggr] + \bold{\hat{e}}_\varphi \biggl[ 2 \biggl( \frac{d\varpi}{dt} \cdot \frac{d\varphi}{dt} \biggr) + \varpi \frac{d^2\varphi}{dt^2}\biggr] + \bold{\hat{k}} \frac{d^2z}{dt^2} \, . </math> |
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[BT87], p. 647, Appendix §1.B.2, Eq. (1B-24) | ||||
Let's rewrite the velocity vector as,
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<math>~\bold{v}</math> |
<math>~=</math> |
<math>~\bold{\hat{e}}_\varpi \dot\varpi + \bold{\hat{e}}_\varphi \varpi \dot\varphi + \bold{\hat{k}} \dot{z} \, ,</math> |
and (the second line of) this acceleration expression as,
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<math>~~\bold{a} \equiv \frac{d\bold{v}}{dt} = \bold{\hat{e}}_\varpi \frac{d \dot\varpi}{dt} + \bold{\hat{e}}_\varphi \frac{d}{dt}\biggl[\varpi \dot\varphi \biggr] + \bold{\hat{k}} \frac{d \dot{z}}{dt} + \underbrace{ \dot\varpi \biggl[ \bold{\hat{e}}_\varphi \frac{d\varphi}{dt} \biggr] - \varpi \dot\varphi \biggl[\bold{\hat{e}}_\varpi \frac{d\varphi}{dt} \biggr] }_\text{curvature terms}\, . </math> |
Now, if <math>~\bold{B}</math> is a vector quantity that characterizes some property of a fluid element — such as momentum density, velocity, or vorticity — the difference between the Lagrangian and Eulerian time-derivatives of that vector quantity is given by the expression,
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<math>~\frac{d\bold{B}}{dt} - \frac{\partial \bold{B}}{\partial t}</math> |
<math>~=</math> |
<math>~(\bold{v} \cdot \bold\nabla)\bold{B} \, ,</math> |
where the various elements of this right-hand-side mathematical operator can be obtained by replacing <math>~\bold{A}</math> with <math>~\bold{v}</math> in the so-called convective operator.
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Convective Operator in Cylindrical Coordinates
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In particular, if we are examining the behavior of the fluid velocity <math>~(\bold{B} \rightarrow \bold{v} )</math>, we find that,
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<math>~\frac{d\bold{v}}{dt} - \frac{\partial \bold{v}}{\partial t}</math> |
<math>~=</math> |
<math>~(\bold{v} \cdot \bold\nabla)\bold{v} </math> |
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<math>~=</math> |
<math>~ \bold{\hat{e}}_\varpi \biggl[ v_\varpi \frac{\partial v_\varpi}{\partial \varpi} + \frac{v_\varphi }{\varpi}\frac{\partial v_\varpi}{\partial \varphi} + v_z \frac{\partial v_\varpi}{\partial z} - \frac{v_\varphi v_\varphi}{\varpi} \biggr] + \bold{\hat{e}}_\varphi \biggl[ v_\varpi \frac{\partial v_\varphi}{\partial \varpi} + \frac{v_\varphi }{\varpi}\frac{\partial v_\varphi}{\partial \varphi} + v_z \frac{\partial v_\varphi}{\partial z} + \frac{v_\varphi v_\varpi}{\varpi} \biggr] + \bold{\hat{e}}_z \biggl[ v_\varpi \frac{\partial v_z}{\partial \varpi} + \frac{v_\varphi }{\varpi}\frac{\partial v_z}{\partial \varphi} + v_z \frac{\partial v_z}{\partial z} \biggr] </math> |
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<math>~=</math> |
<math>~ \bold{\hat{e}}_\varpi \biggl[ \dot\varpi \frac{\partial \dot\varpi}{\partial \varpi} + \frac{\varpi \dot\varphi }{\varpi}\frac{\partial \dot\varpi }{\partial \varphi} + \dot{z} \frac{\partial \dot\varpi}{\partial z} \biggr] + \bold{\hat{e}}_\varphi \biggl[ \dot\varpi \frac{\partial (\varpi \dot\varphi) }{\partial \varpi} + \frac{(\varpi \dot\varphi) }{\varpi}\frac{\partial (\varpi \dot\varphi) }{\partial \varphi} + \dot{z} \frac{\partial (\varpi \dot\varphi) }{\partial z} \biggr] + \bold{\hat{e}}_z \biggl[ \dot\varpi \frac{\partial \dot{z}}{\partial \varpi} + \frac{\varpi \dot\varphi }{\varpi}\frac{\partial \dot{z}}{\partial \varphi} + \dot{z} \frac{\partial \dot{z}}{\partial z} \biggr] </math> |
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<math>~ + \bold{\hat{e}}_\varphi \biggl[ \frac{(\varpi \dot\varphi)\dot\varpi }{\varpi}
\biggr] - \bold{\hat{e}}_\varpi \biggl[ \frac{\varpi^2 {\dot\varphi}^2}{\varpi} \biggr] </math> |
It can readily be appreciated that the first few terms in our next-to-last acceleration expression may be grouped together as the partial time-derivative of the velocity, that is, we can write,
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<math>~\frac{d\bold{v}}{dt}</math> |
<math>~=</math> |
<math>~\frac{\partial \bold{v}}{\partial t} - \bold{\hat{e}}_\varpi \biggl[\varpi \biggl(\frac{d\varphi}{dt}\biggr)^2 \biggr] + \bold{\hat{e}}_\varphi \biggl[ 2 \biggl( \frac{d\varpi}{dt} \cdot \frac{d\varphi}{dt} \biggr) + \varpi \frac{d^2\varphi}{dt^2}\biggr] </math> |
the azimuthal coordinate location it is critically important to recognize that the orientation in space of the pair of cylindrical-coordinate-based unit vectors, <math>~\bold{\hat{e}}_\varpi</math> and <math>~\bold{\hat{e}}_\varphi</math>, depend on the fluid element's coordinate location. Specifically,
Let's begin by building a model in a cylindrical coordinate system that is spinning about its <math>~\bold{\hat{k}}</math>-axis with an angular velocity, <math>~{\vec{\Omega}}_f = {\hat\mathbf{k}} \Omega_f</math>. Furthermore, let's use <math>~\bold{v}</math> to represent the inertial-frame velocity and use <math>~\bold{u}</math> to represent the velocity as viewed in the rotating frame. We know that,
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<math>~\frac{d}{dt} \bold{\hat{e}}_\varpi</math> |
<math>~=</math> |
<math>~\mathbf{\hat{e}}_\varphi \frac{u_\varphi}{\varpi} \, ,</math> |
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<math>~\frac{d}{dt} \bold{\hat{e}}_\varphi</math> |
<math>~=</math> |
<math>~ - \mathbf{\hat{e}}_\varpi \frac{u_\varphi}{\varpi} \, ,</math> and, |
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<math> \bold{v} - \bold{u} = {\vec{\Omega}}_f \times \vec{x} </math> |
<math>~=~</math> |
<math> {\hat\mathbf{k}} \Omega_f\times (\mathbf{\hat{e}}_\varpi \varpi + \mathbf{\hat{k}}z) = \mathbf{\hat{e}}_\varphi \Omega_f \varpi \, . </math> |
Let's take the total time-derivative of an arbitrary vector, <math>~\bold{A}</math>, which when written in cylindrical coordinates takes the form,
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<math>~\bold{A}</math> |
<math>~=</math> |
<math>~\bold{\hat{e}}_\varpi A_\varpi + \bold{\hat{e}}_\varphi A_\varphi + \bold{\hat{k}}A_z \, .</math> |
Associated with this Lagrangian fluid element will be various (generally, time-dependent) scalar attributes, <math>~q(\bold{x}_0, t)</math>, such as mass-density or pressure, and various vector attributes, <math>~\bold{A}(\bold{x}_0,t)</math>, such as velocity or vorticity.
One Approach
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<math> \frac{d\mathbf{A}}{dt} </math> |
<math>~=~</math> |
<math> \frac{d}{dt} [ \mathbf{\hat{e}}_\varpi A_\varpi + \mathbf{\hat{e}}_\varphi A_\varphi + \mathbf{\hat{k}} A_z ] </math> |
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<math>~=~</math> |
<math> \mathbf{\hat{e}}_\varpi \frac{d A_\varpi}{dt} + \mathbf{\hat{e}}_\varphi \frac{d A_\varphi}{dt} + \mathbf{\hat{k}} \frac{d A_z}{dt} + ( A_\varpi) \frac{d}{dt}\mathbf{\hat{e}}_\varpi + ( A_\varphi) \frac{d}{dt}\mathbf{\hat{e}}_\varphi </math> |
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<math>~=~</math> |
<math> \mathbf{\hat{e}}_\varpi \frac{d u'_\varpi}{dt} + \mathbf{\hat{e}}_\varphi \frac{d u'_\varphi}{dt} + \mathbf{\hat{k}} \frac{d u'_z}{dt} + \mathbf{\hat{e}}_\varphi(u'_\varpi) \frac{u'_\varphi}{\varpi} - \mathbf{\hat{e}}_\varpi(u'_\varphi) \frac{u'_\varphi}{\varpi} \, . </math> |
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© 2014 - 2021 by Joel E. Tohline |