### 4.1 VSF deviation

As discussed in Section 3.1, the tracking of vortex surfaces is perfect using Eq. (5) in ideal flows, whereas the assessment of the tracking becomes complex in non-ideal flows. On the accuracy of the VSF solution, Yang and Pullin [6] proposed a metric

$$ \epsilon_{\phi} \equiv \left \langle\left|\lambda_{ \boldsymbol{\omega} }\right|\right \rangle $$

(8)

to quantify the averaged deviation of a numerical VSF solution from an exact one with 0≤*ε*_{ϕ}≤1, where

$$ \lambda_{ \boldsymbol{\omega}} \equiv \frac { \boldsymbol{\omega} \cdot \nabla \phi_{ v} } { | \boldsymbol{\omega} | \left| \nabla \phi_{ v} \right| }, $$

(9)

the cosine of an angle between the VSF gradient and the vorticity, characterizes the local VSF deviation. Former VSF studies [5, 6] suggested that a satisfactory VSF solution should have *ε*_{ϕ}≤5*%*∼10*%*.

On the other hand, it is not sufficient to conclude that a temporally evolving VSF solution with a small *ε*_{ϕ} must be a good candidate for tracking vortex surfaces due to the nonuniqueness of the VSF solution [1]. An extreme example provided in a MHD flow (see in Fig. 8 in Ref. [3]) shows that the vortex surface should be time invariant, but the VSF isosurface evolves among nonunique VSF solutions with *ε*_{ϕ}=0.

We sketch this issue in Fig. 4. Two typical vortex surfaces represented by circular dashed lines at a time *t* evolve into the two represented by elliptical dashed lines at a later time *t*+*Δ**t*. An ideal vortex surface tracking should show the one-to-one map between the two inner loops or the two outer loops at the two times in a Lagrangian manner. There are two types of discrepancies in the tracking. For the inner vortex surface at *t* (thick solid line), two isosurfaces of numerical VSF solutions are sketched at *t*+*Δ**t*. The upper one is attached across the inner and outer vortex surfaces with satisfactory *ε*_{ϕ} and poor Lagrangian tracking performance, i.e., the solid loop obviously deviates from the inner dashed loop. On the contrary, the bottom one is oscillating around the inner vortex surface with large *ε*_{ϕ} and satisfactory Lagrangian tracking performance. Thus, besides *ε*_{ϕ}, we need another metric to quantify the Lagrangian tracking performance in the overall assessment of the VSF solution.

### 4.2 Volume overlap ratio of VSF isosurfaces

The evolution of vortex surfaces is characterized by the VSF isosurfaces of a particular contour value *ϕ*_{v}=*φ*. Although *φ* is expected to be constant, it is adjusted at different times due to the numerical dissipation. In this implementation, *φ* is implicitly determined from [11]

$$ V(\phi_{v} = \varphi, t) = V(\phi_{v} = C_{\phi}, t=0) $$

(10)

by searching the isocontour values *ϕ*_{v}=*φ* at a given time *t* and *ϕ*_{v}=*C*_{ϕ} at the initial time which correspond to the same fluid volume *V*_{ϕ}≡*V*(*ϕ*_{v}=*C*_{ϕ},*t*=0). Here, the fluid volume within an isosurface of *ϕ*_{v}=*φ* is calculated by the integration over the entire computational domain as

$$ V(\phi_{v}=\varphi)=\int_{\Omega} H_{\delta}(\phi_{v}=\varphi)d \boldsymbol{x}, $$

(11)

where

$$ H_{\delta}(\phi_{v}=\varphi)=\left\{ \begin{array}{ll} 0, & \qquad \phi_{v} - \varphi < - \delta,\\ \frac{1}{2\delta}(\phi_{v}-\varphi+\delta)+\frac{1}{2\pi}\sin\frac{\pi(\phi_{v}-\varphi)}{\delta}, & \qquad |\phi_{v} - \varphi|\leqslant\delta,\\ 1, & \qquad \phi_{v} - \varphi > \delta \end{array} \right. $$

(12)

is the discretized Heaviside function [28] with a smoothing parameter *δ*=1×10^{−6}. The previous VSF visualization shows that the isosurface of *ϕ*_{v}=*φ* can reasonably describe the continuous temporal evolution of a particular vortex surface, and this VSF isocontour selection is consistent with the volume conservation implied by the Helmholtz theorem in the limit of in ideal flows [11].

To quantify the tracking performance from numerical VSF solutions, we propose a volume overlap ratio

$$ R_{V} = \frac{V_{\bigcap}}{V_{\phi}}, $$

(13)

where the volume overlap is calculated by

$$ V_{\bigcap}=\int_{\Omega} H_{\delta}(\phi_{v}=\varphi)H_{\delta}(\phi_{v}^{\text{ref}}=\varphi^{\text{ref}})d\boldsymbol{x}. $$

(14)

Here, *ϕ*_{v} is the VSF solution being examined, and \(\phi _{v}^{\text {ref}}\) denotes an exact reference VSF solution. In the implementation, \(\phi _{v}^{\text {ref}}\) is obtained as a numerical VSF solution with high grid resolution, so it is presumed to be accurate enough with good convergence and low numerical dissipation. The isocontour values of *φ* and *φ*^{ref} satisfy

$$ \int_{\Omega} H_{\delta}(\phi_{v}=\varphi)d\boldsymbol{x}=\int_{\Omega} H_{\delta}(\phi_{v}^{\text{ref}}=\varphi^{\text{ref}})d\boldsymbol{x} = V_{\phi}. $$

(15)

Thus, *R*_{V} characterizes the normalized volume overlap ratio between the volumes within the isosurfaces of the candidate VSF solution *ϕ*_{v}=*φ* and the reference VSF solution \(\phi _{v}^{\text {ref}}=\varphi ^{\text {ref}}\).

For example, Fig. 5 plots cross-sections of VSF solutions on the *y*–*z* slices at *x*=3 and *t*_{TG}=5.0 in the TG flow for *R**e*=400 with *N*=128 (red for the VSF solution to be tested) and *N*=512 (blue for the reference VSF solution) along with their overlap region (shaded), where Fig. 5c combines Fig. 5a and b. The isocontour value corresponds to *V*_{Ω}/8 and the volume overlap ratio is *R*_{V}=0.44, where *V*_{Ω} denotes the volume of the whole computational domain.

The volume overlap ratio characterizes the time coherence of the vortex surface tracking based on VSF solutions. The best tracking performance has *R*_{V}=1. To balance reasonably good time coherence and moderate computational cost, we propose a criterion *R*_{V}≥0.5 for satisfactory performance of vortex surface tracking. Note that the VSF solution is closer to the reference one while the VSF grid resolution is more demanding for larger *R*_{V}.

### 4.3 Conservation of numerical VSF solutions

In practice, *R*_{V} may be not available due to the lack of the reference VSF solution with high grid resolution, so we need to find a surrogate metric to assess the Lagrangian tracking performance of numerical VSF solutions. In the present VSF evolution, the VSF isosurface is volume-preserving under the selection of isocontour levels in Eq. (10), but it can be artificially distorted by the dissipative numerical regularization.

The distortion is related to the numerical dissipation, which can be quantified by

$$ R_{\phi}(t) \equiv \frac{\sqrt{\left \langle\phi_{v}^{2} (t)\right \rangle }}{\sqrt{\left \langle \phi_{v}^{2} (t=0)\right \rangle}}. $$

(16)

Note that *R*_{ϕ}=1 for the ideal vortex surface tracking based on Eq. (5). In similar front tracking problems, such as in the level set method [29, 30], the conservation degree of the numerical solution of the tracer scalar governed by the equation in the form of Eq. (5) is an important metric to assess the tracking performance. Thus, we examine the correlation of *R*_{ϕ} and *R*_{V} to demonstrate that whether *R*_{ϕ} is suitable to characterize the vortex surface tracking.

In the VSF calculations for the TG flow with *R**e*=400, we use the VSF solution with *N*=512 as the reference one, and collect *R*_{V} and *R*_{ϕ} at different times and with different grid resolutions. Note that in order to avoid the grid resolution effect of velocity field, we set the same number of grid points 512^{3} for DNS but different *N* for VSF calculations. In Fig. 6, the data points on the same curve from right to left are obtained at *t*_{TG}=4.0,4.5,5.0,5.5,6.0,6.5,7.0. In this time period, the TG flow has a transition to the turbulent-like state. The isocontour values correspond to *V*_{ϕ}=*V*_{Ω}/16 and *V*_{Ω}/4.

We find that *R*_{ϕ} and *R*_{V} are positively correlated, indicating the effectiveness of *R*_{ϕ} for characterizing the surface tracking performance. Moreover, the correlation depends on the grid resolution and the isocontour level. It is noted that the volume *V*_{ϕ} within the isosurface of *ϕ*_{v}=*C*_{ϕ} decreases with the increasing *C*_{ϕ}. For the same *R*_{ϕ},*R*_{V} increases with *N* and *V*_{ϕ}. This suggests that the isosurface at a large isocontour level with small volume tends to be significantly distorted on a coarse grid, so we still need to use a reasonably high grid resolution and a suitable *C*_{ϕ} corresponding to a large enough volume for the assessment. For *N*=256, we observe that the criterion *R*_{V}>0.5 is satisfied for *R*_{ϕ}>0.05, i.e., the corresponding data points are within the shaded region in Fig. 6.

As an illustrative example, we show the close-up view of VSF isosurfaces in the TG flow for *R**e*=400 at *t*_{TG}=7.0 with *N*=512,*N*=128, and *N*=64 in Fig. 7. We observe two clear twisted vortex tubes from the reference numerical VSF solution with *N*=512 and *R*_{ϕ}=0.5624 in Fig. 7a. By contrast, the vortex tube is barely resolved for the low-resolution VSF solution with *N*=128 and *R*_{ϕ}=0.0508 in Fig. 7b. Although some parts are under-resolved, e.g., the very twisted part at the right end of the vortex tube in Fig. 7b, the essential tube structure is still preserved with *R*_{ϕ}=0.0508. On the other hand, the vortex tube is not well resolved in Fig. 7c with *N*=64 and *R*_{ϕ}=0.0212.

Therefore, we propose the criteria

$$ \epsilon_{\phi} <0.05~~~\text{and}~~~R_{\phi}>0.05 $$

(17)

to quantify both the VSF deviation and Lagrangian tracking performance. The threshold values are selected for the perturbed TG flow which has features of both turbulent and transitional flows with large-scale coherent and small-scale turbulent flow structures.