Coreline Criteria for Inertial Particle Motion

Abstract

Dynamical systems, such as the second-order ODEs that govern the motion of finite-sized objects in fluids, describe the evolution of a state by a trajectory living in a high-dimensional phase space. The high dimensionality leads to visualization challenges and, for the case of inertial particles, multiple models exist that pose different assumptions. In this paper, we thoroughly address the extraction of a specific feature, namely the vortex corelines of inertial particles. Based on a general template model that comprises two of the most commonly used inertial particle ODEs, we first transform their high-dimensional tangent vector field into a Galilean reference frame in which the observed inertial particle flow becomes as steady as possible. In the optimal frame, we derive first-order and second-order vortex coreline criteria, allowing us to extract straight and bent inertial vortex corelines using 3D and 6D parallel vectors operators, respectively. With this, we generalize existing work in multiple ways: not only do we handle two inertial particle models at once, we extend the concept of second-order vortex corelines to the inertial case and make them Galilean-invariant by deriving the criteria from a steady reference frame, rather than from a geometric characterization.

Appendices

Appendix 1 - Derivation of First-Order 3D Criterion

Next, we show how the 6D parallel vectors condition of the first-order case:

$$\begin{aligned} \tilde{\mathbf{v}}^*\parallel \tilde{\mathbf{J}}^*\tilde{\mathbf{v}}^*\quad \Rightarrow \quad \begin{pmatrix} {\mathbf{v}}+ {\mathbf{d}}\\ {\mathbf{k}}- \frac{{\mathbf{v}}}{\kappa } \end{pmatrix} \parallel \begin{pmatrix} {\mathbf{k}}- \frac{{\mathbf{v}}}{\kappa } \\ \nabla {\mathbf{k}}({\mathbf{v}}+ {\mathbf{d}}) - \frac{{\mathbf{k}}- \frac{{\mathbf{v}}}{\kappa }}{\kappa } \end{pmatrix}. \end{aligned}$$

(46)

can be simplified to a 3D criterion. First, we look at the position subspace. Multiplying with $\kappa$ and adding ${\mathbf{v}}+ {\mathbf{d}}$ to the right hand side gives:

$$\begin{aligned} {\mathbf{v}}+ {\mathbf{d}}\parallel {\mathbf{k}}- \frac{{\mathbf{v}}}{\kappa }&\quad \Leftrightarrow \quad {\mathbf{v}}+ {\mathbf{d}}\parallel \kappa {\mathbf{k}}- {\mathbf{v}}\end{aligned}$$

(47)

$$\begin{aligned}&\quad \Leftrightarrow \quad {\mathbf{v}}+ {\mathbf{d}}\parallel \kappa {\mathbf{k}}+ {\mathbf{d}}, \end{aligned}$$

(48)

Equating Eq. (47) and Eq. (48) gives:

$$\begin{aligned} {\mathbf{k}}- \frac{{\mathbf{v}}}{\kappa } \parallel \kappa {\mathbf{k}}+ {\mathbf{d}}. \end{aligned}$$

(49)

Considering the velocity subspace of Eq. (46):

$$\begin{aligned} {\mathbf{k}}- \frac{{\mathbf{v}}}{\kappa } \parallel \nabla {\mathbf{k}}\, ({\mathbf{v}}+ {\mathbf{d}}) - \frac{{\mathbf{k}}- \frac{{\mathbf{v}}}{\kappa }}{\kappa }, \end{aligned}$$

(50)

and multiplying the right hand side with $\kappa$ gives:

$$\begin{aligned} {\mathbf{k}}- \frac{{\mathbf{v}}}{\kappa } \parallel \kappa \nabla {\mathbf{k}}\, ({\mathbf{v}}+ {\mathbf{d}}) - {\mathbf{k}}+ \frac{{\mathbf{v}}}{\kappa } \;. \end{aligned}$$

(51)

Adding the left hand side to the right hand side and dividing by $\kappa$:

$$\begin{aligned} {\mathbf{k}}- \frac{{\mathbf{v}}}{\kappa } \parallel \nabla {\mathbf{k}}\, ({\mathbf{v}}+ {\mathbf{d}}). \end{aligned}$$

(52)

Finally, substituting Eq. (49) on the left hand side of Eq. (52) and inserting Eq. (48) on the right hand side of Eq. (52) gives Eq. (25):

$$\begin{aligned} \kappa {\mathbf{k}}+ {\mathbf{d}}\parallel \nabla {\mathbf{k}}\, (\kappa {\mathbf{k}}+ {\mathbf{d}}) , \end{aligned}$$

(53)

which is a 3D condition, independent of the particle velocity ${\mathbf{v}}$.

Appendix 2 - Tracer Particles as Limit Case

Next, we show that our inertial first-order and second-order criteria approach the massless case in the limit. The proofs of Model 1 and 2 are analogue. For brevity, we show the derivation for Model 1.

Inertial Motion. First, the motion of inertial particles is consistent with tracer particles for $r\rightarrow 0$, as shown by Günther and Theisel [6]: Rearranging the velocity subspace of $\tilde{\mathbf{v}}$ in Eq. (7) for ${\mathbf{v}}$ and substituting in the position subspace of $\tilde{\mathbf{v}}$ gives with Eq. (9):

$$\begin{aligned} \lim _{r\rightarrow 0} \frac{{\mathrm d}{\mathbf{x}}}{{\mathrm d}t} = {\mathbf{v}}= {\mathbf{u}}({\mathbf{x}},t) \underbrace{- r\frac{{\mathrm d}{\mathbf{v}}}{{\mathrm d}t} + r{\mathbf{g}}}_{0}. \end{aligned}$$

(54)

Vortex Centers in 2D. The motion of inertial particles is described with Eq. (9). We consider the limit $r\rightarrow 0$ for tracer particles:

$$\begin{aligned} \kappa = r \qquad \kappa {\mathbf{k}}&= {\mathbf{u}}({\mathbf{x}},t) + r {\mathbf{g}}\end{aligned}$$

(55)

$$\begin{aligned} \lim _{r \rightarrow 0} \kappa {\mathbf{k}}&= {\mathbf{u}}({\mathbf{x}},t) . \end{aligned}$$

(56)

With ${\mathbf{d}}= {\mathbf{J}}^{-1}{\mathbf{u}}_t = -{\mathbf{f}}$ from Eq. (15), we insert the limit in Eq. (56) into the general 2D vortex center condition in Eq. (17):

$$\begin{aligned} \kappa {\mathbf{k}}+ {\mathbf{d}}= {\mathbf{0}}\qquad \overset{r\rightarrow 0}{\Rightarrow }\qquad {\mathbf{u}}({\mathbf{x}},t) - {\mathbf{f}}= {\mathbf{0}}, \end{aligned}$$

(57)

which is the Galilean invariant 2D vortex coreline criterion for massless particles by Weinkauf et al. [27], cf. Eq. (55) in [23].

First-Order Corelines in 3D. For the first-order vortex corelines in 3D, we insert the limit in Eq. (56) into the general first-order vortex coreline condition in Eq. (25):

$$\begin{aligned} \kappa {\mathbf{k}}+ {\mathbf{d}}\parallel \nabla {\mathbf{k}}\, (\kappa {\mathbf{k}}+ {\mathbf{d}}) \qquad \overset{r\rightarrow 0}{\Rightarrow }\qquad {\mathbf{u}}- {\mathbf{f}}&\parallel \frac{{\mathbf{J}}}{r} \, ({\mathbf{u}}- {\mathbf{f}}) \end{aligned}$$

(58)

$$\begin{aligned} {\mathbf{u}}- {\mathbf{f}}&\parallel {\mathbf{J}}\, ({\mathbf{u}}- {\mathbf{f}}), \end{aligned}$$

(59)

which is the Galilean invariant first-order 3D vortex coreline criterion for massless particles by Weinkauf et al. [27], cf. Eq. (53) in [23].

Second-Order Corelines in 3D. We consider the transformed velocity $\tilde{\mathbf{v}}^*$ in Eq. (13) and the rate of acceleration $\tilde{\mathbf{b}}^*$:

$$\begin{aligned} \tilde{\mathbf{b}}^*= \begin{pmatrix} \nabla {\mathbf{k}}({\mathbf{v}}+{\mathbf{d}}) + \frac{{\mathbf{v}}}{\kappa ^2} - \frac{{\mathbf{k}}}{\kappa } \\ \nabla (\nabla {\mathbf{k}})({\mathbf{v}}+{\mathbf{d}})({\mathbf{v}}+{\mathbf{d}}) - \frac{\nabla {\mathbf{k}}}{\kappa }({\mathbf{v}}+{\mathbf{d}}) + \nabla {\mathbf{k}}({\mathbf{k}}-\frac{{\mathbf{v}}}{\kappa }) - \frac{{\mathbf{v}}}{\kappa ^3} + \frac{{\mathbf{k}}}{\kappa ^2} \end{pmatrix} \end{aligned}$$

(60)

in the optimal reference frame, i.e., ${\mathbf{d}}$ is the translation rate:

$$\begin{aligned} \tilde{\mathbf{v}}^*= \begin{pmatrix} {\mathbf{v}}+{\mathbf{d}}\\ {\mathbf{a}}^*\end{pmatrix}. \qquad \tilde{\mathbf{b}}^*= \begin{pmatrix} {\mathbf{b}}^*\\ {\mathbf{c}}^*\end{pmatrix}. \end{aligned}$$

(61)

We rearrange the velocity subspace in Eq. (60), denoted as ${\mathbf{c}}^*$ with

$$\begin{aligned} {\mathbf{c}}^*= \nabla (\nabla {\mathbf{k}})({\mathbf{v}}+{\mathbf{d}})({\mathbf{v}}+{\mathbf{d}}) - \frac{\nabla {\mathbf{k}}}{\kappa }({\mathbf{v}}+{\mathbf{d}}) + \nabla {\mathbf{k}}({\mathbf{k}}-\frac{{\mathbf{v}}}{\kappa }) - \frac{{\mathbf{v}}}{\kappa ^3} + \frac{{\mathbf{k}}}{\kappa ^2} \end{aligned}$$

and with ${\mathbf{a}}^*= {\mathbf{k}}- \frac{{\mathbf{v}}}{\kappa } = {\mathbf{J}}({\mathbf{u}}-{\mathbf{f}})$ [23] and ${\mathbf{b}}^*$ in Eq. (60) into

$$\begin{aligned} {\mathbf{c}}^*= \nabla (\nabla {\mathbf{k}})({\mathbf{v}}+{\mathbf{d}})({\mathbf{v}}+{\mathbf{d}}) + \nabla {\mathbf{k}}\left( {\mathbf{J}}({\mathbf{u}}-{\mathbf{f}})\right) -\frac{{\mathbf{b}}^*}{\kappa }. \end{aligned}$$

(62)

Inserting Model 1 with Eqs. (56) and (57) gives for $r\rightarrow 0$:

$$\begin{aligned} {\mathbf{c}}^*&= \frac{1}{r}\nabla {\mathbf{J}}({\mathbf{u}}-{\mathbf{f}})({\mathbf{u}}-{\mathbf{f}}) + \frac{{\mathbf{J}}}{r}\left( {\mathbf{J}}({\mathbf{u}}-{\mathbf{f}})\right) -\frac{{\mathbf{b}}^*}{r} \end{aligned}$$

(63)

$$\begin{aligned} \Leftrightarrow \; r\cdot {\mathbf{c}}^*&= \nabla {\mathbf{J}}({\mathbf{u}}-{\mathbf{f}})({\mathbf{u}}-{\mathbf{f}}) + {\mathbf{J}}\left( {\mathbf{J}}({\mathbf{u}}-{\mathbf{f}})\right) -{\mathbf{b}}^*= {\mathbf{0}}. \end{aligned}$$

(64)

Rearranging for the rate of acceleration ${\mathbf{b}}^*$ gives the material derivative of the steady acceleration $\frac{D}{Dt}({\mathbf{J}}{\mathbf{u}}^*)$ in the optimal frame:

$$\begin{aligned} \lim _{r\rightarrow 0} {\mathbf{b}}^*= \left( \nabla {\mathbf{J}}{\mathbf{u}}^*+ {\mathbf{J}}{\mathbf{J}}\right) {\mathbf{u}}^*\quad \text {with}\quad {\mathbf{u}}^*={\mathbf{u}}-{\mathbf{f}}. \end{aligned}$$

(65)

Thus, the position subspace simplifies to ${\mathbf{u}}^*\parallel {\mathbf{b}}^*$, which is equivalent to the criterion of Roth and Peikert [9] in the optimal steady reference frame. Thus, all proposed inertial vortex criteria are consistent with the massless case for $r\rightarrow 0$.