Liutex Formula: Vortex Rotation Analysis

• Liutex formula is defined as a vector quantity that encapsulates the local rotation axis and rotational strength derived from the velocity-gradient tensor.
• It is computed using the real eigenvector and the imaginary part of the complex eigenvalue, thereby isolating rigid-body-like rotation from shear effects.
• The explicit formula improves computational efficiency and accuracy in vortex detection and turbulence modeling applications.

The Liutex formula is the closed-form expression for the Liutex vector, a vortex quantity introduced in the Liutex/Rortex framework to encode both the local rotation axis and the local rotational strength. In its standard explicit form,

$$\vec R = R\vec r = \left[ \vec\omega\cdot\vec r - \sqrt{(\vec\omega\cdot\vec r)^2-4\lambda_{ci}^2} \right]\vec r,$$

where $\vec\omega=\nabla\times\vec v$ is the vorticity vector, $\vec r$ is the real eigenvector of the velocity-gradient tensor $\nabla\vec v$ associated with its real eigenvalue, and $\lambda_{ci}$ is the imaginary part of the complex-conjugate eigenvalue pair. The formula was derived explicitly in 2018 to replace an earlier geometric two-rotation construction and to clarify how Liutex isolates rigid-body-like rotation from shear contamination (Wang et al., 2018).

1. Definition and mathematical setting

Liutex, previously named Rortex, is defined as a vector quantity whose direction is the local rotational axis and whose magnitude is the local rotational strength in the Liutex sense. Its construction begins from the velocity-gradient tensor $\nabla\vec v$. If $\nabla\vec v$ has one real eigenvalue $\lambda_r$ and one complex-conjugate pair $\lambda_{cr}\pm i\lambda_{ci}$, local swirling motion is taken to exist; the corresponding real eigenvector $\vec r$ gives the local rotation axis. If all three eigenvalues are real, the Liutex vector is set to zero (Wang et al., 2018).

A sign convention is imposed so that

$\vec\omega=\nabla\times\vec v$0

which fixes the orientation of $\vec\omega=\nabla\times\vec v$1 and makes the Liutex direction unique up to that choice. With this convention, the Liutex magnitude is

$\vec\omega=\nabla\times\vec v$2

and the Liutex vector is $\vec\omega=\nabla\times\vec v$3 (Wang et al., 2018).

Quantity	Meaning	Role in Liutex
$\vec\omega=\nabla\times\vec v$4	Velocity-gradient tensor	Local kinematic source tensor
$\vec\omega=\nabla\times\vec v$5	Real eigenvector of $\vec\omega=\nabla\times\vec v$6	Rotation-axis direction
$\vec\omega=\nabla\times\vec v$7	Real eigenvalue	Axial eigenvalue
$\vec\omega=\nabla\times\vec v$8	Complex eigenvalue pair	Swirling condition and $\vec\omega=\nabla\times\vec v$9
$\vec r$0	Vorticity vector	Supplies axial vorticity component
$\vec r$1	Liutex magnitude	Rotational strength
$\vec r$2	Liutex vector	Axis plus strength

Later Liutex papers retained the same core formula and summarized Liutex as a vector quantity whose direction gives the local rotation axis and whose magnitude gives twice the local angular velocity of rigid rotation, in contrast with scalar criteria such as $\vec r$3, $\vec r$4, $\vec r$5, or $\vec r$6 (Chen et al., 16 Dec 2025).

2. Derivation of the explicit formula

Before the explicit formula was obtained, Liutex magnitude was computed by a geometric algorithm. One first found the real eigenvector $\vec r$7, then applied a rotation matrix $\vec r$8 to align the local rotational axis with the $\vec r$9-axis, and finally applied an in-plane rotation $\nabla\vec v$0 about that axis to locate the minimum angular velocity direction. In that formulation, the Liutex magnitude was defined by

$\nabla\vec v$1

with $\nabla\vec v$2 and $\nabla\vec v$3 determined from the in-plane entries of the rotated velocity-gradient tensor (Wang et al., 2018).

The 2018 derivation replaced that construction by introducing the minimum and maximum angular velocities in the plane normal to $\nabla\vec v$4, denoted $\nabla\vec v$5 and $\nabla\vec v$6. Two relations were then established:

$\nabla\vec v$7

where

$\nabla\vec v$8

Solving the quadratic

$\nabla\vec v$9

for the smaller root gives

$\lambda_{ci}$0

Since Liutex magnitude is defined there as

$\lambda_{ci}$1

one obtains the explicit formula

$\lambda_{ci}$2

and therefore

$\lambda_{ci}$3

This is the formula generally referred to as the Liutex formula (Wang et al., 2018).

The same explicit form was later used as the operational definition in multiple follow-up papers, including vortex-axis extraction, contamination analysis, Cartesian principal decomposition, and SGS modeling (Liu et al., 2019).

3. Physical interpretation

The explicit formula is tied to three distinct angular-velocity notions. In the 2018 derivation, the vorticity component along the local rotation axis,

$\lambda_{ci}$4

is interpreted as twice the spatial mean angular velocity around that axis. The imaginary part of the complex eigenvalue, $\lambda_{ci}$5, is interpreted as a pseudo-time average angular velocity associated with the local circular or spiral trajectory implied by the linearized dynamics. Liutex magnitude $\lambda_{ci}$6 is then twice the minimum angular velocity in the plane normal to the axis, and that minimum is identified with the rigid-body-like rotational core of the local motion (Wang et al., 2018).

This yields the relation

• $\lambda_{ci}$7: twice the spatial mean angular velocity,
• $\lambda_{ci}$8: pseudo-time mean angular velocity,
• $\lambda_{ci}$9: twice the minimum angular velocity.

Within the RS decomposition used by the original Liutex literature, the in-plane local motion can be written as the sum of a rigid-rotation part and a shear part. In that reading, the quantity

$\nabla\vec v$0

is the pure shear part, while

$\nabla\vec v$1

is the rotational part retained by Liutex (Wang et al., 2018).

A later analysis of stretching and shearing contamination reformulated the same idea in a principal coordinate. There, the imaginary part of the complex eigenvalue satisfies

$\nabla\vec v$2

so $\nabla\vec v$3 explicitly contains the in-plane shear parameter $\nabla\vec v$4. In the same framework, $\nabla\vec v$5 and $\nabla\vec v$6 also contain shear and stretching terms, whereas Liutex is constructed to retain only the rigid-rotation block (Shrestha et al., 2020).

A local-plane interpretation of the same rigid-rotation selection principle was emphasized in the symmetry-to-asymmetry analysis of Liutex/Rortex. In that paper, for a two-dimensional local rotational plane, the Liutex magnitude was written as

$\nabla\vec v$7

so the smaller member of the off-diagonal pair represents the common matched rigid-rotation content, while the excess in the larger member is interpreted as shear. In that interpretation, same-sense added shear does not increase Liutex, whereas opposite-sense shear can reduce it (Liu et al., 2019).

4. Principal-coordinate and tensor decompositions

The Liutex formula is closely connected with a canonical local frame, usually called the principal coordinate. In the contamination-analysis literature, this coordinate system is defined so that its $\nabla\vec v$8-axis is parallel to the real eigenvector $\nabla\vec v$9 and the in-plane orientation is fixed uniquely. In that frame, the velocity-gradient tensor takes a canonical form such as

$\nabla\vec v$0

and is decomposed exactly into rigid rotation, shear, and stretching/compression (Shrestha et al., 2020).

In that same paper, the principal decomposition is written as

$\nabla\vec v$1

where $\nabla\vec v$2 is the rigid-rotation block, $\nabla\vec v$3 contains shear terms, and $\nabla\vec v$4 contains stretching/compression terms. The paper also proves that for an arbitrary velocity-gradient tensor there exists one and only one principal coordinate, and argues that the decomposition is unique only in that coordinate (Shrestha et al., 2020).

A mathematical study of the local fluid rotation axis gave a more axiomatic statement. It defines a local rotation axis as a direction $\nabla\vec v$5 satisfying

$\nabla\vec v$6

so that the velocity increment along the axis is purely axial stretching or compression. Under that definition, the local rotation axis must be a real eigenvector of $\nabla\vec v$7, and Liutex direction is therefore identified with the local fluid rotation axis. In the same work, the principal-frame velocity-gradient tensor is written with a half-strength convention,

$\nabla\vec v$8

and the rigid-rotation tensor has local angular speed $\nabla\vec v$9, yielding the statement that Liutex magnitude equals twice the local angular speed (Nottage et al., 2021).

The principal decomposition was later transferred back to the original Cartesian coordinates. For a Liutex vector $\lambda_r$0 and unit direction $\lambda_r$1, the rigid-rotation tensor becomes

$\lambda_r$2

while the stretching/compression tensor is

$\lambda_r$3

The shear tensor is then the remainder

$\lambda_r$4

This Cartesian principal decomposition was proposed as a direct alternative to the Cauchy–Stokes split into symmetric and antisymmetric parts (Liu et al., 2021).

5. Relations to other vortex measures and axis-line extraction

Liutex was introduced partly in response to the limitations of vorticity and second-generation scalar criteria. The Liutex literature repeatedly contrasts it with $\lambda_r$5, $\lambda_r$6, $\lambda_r$7, and $\lambda_r$8. The stated differences are that Liutex is vector-valued rather than scalar-valued, provides both axis direction and rotational strength, and is intended to remove contamination by shear and, in some formulations, stretching (Chen et al., 16 Dec 2025).

The relation to $\lambda_r$9 is especially important. In principal-coordinate analysis,

$\lambda_{cr}\pm i\lambda_{ci}$0

so $\lambda_{cr}\pm i\lambda_{ci}$1 depends on both rigid rotation and in-plane shear. This is why Liutex papers treat swirling strength as an incomplete measure of true local rotation (Shrestha et al., 2020). Correlation analysis in a DNS of boundary-layer transition further reported that the correlation between vorticity and Liutex is minimal in strong-shear regions, while $\lambda_{cr}\pm i\lambda_{ci}$2, $\lambda_{cr}\pm i\lambda_{ci}$3, and $\lambda_{cr}\pm i\lambda_{ci}$4 correlate better with Liutex but are still not exact substitutes (Yu1 et al., 2020).

Because Liutex is a vector field, it also supports line-type vortex definitions. A Liutex-based definition of vortex axis line was proposed in 2019:

$\lambda_{cr}\pm i\lambda_{ci}$5

with $\lambda_{cr}\pm i\lambda_{ci}$6. The geometric meaning is that the gradient of Liutex magnitude is aligned with the Liutex direction, so the point is a transverse extremum of rotational strength in the plane normal to the local axis. In two-dimensional flow this reduces to

$\lambda_{cr}\pm i\lambda_{ci}$7

The same paper notes an important degeneracy: in pure rigid-body rotation, every point satisfies the condition because $\lambda_{cr}\pm i\lambda_{ci}$8 everywhere, so the axis is not unique under that criterion (Liu et al., 2019).

That vortex-axis paper also presents a preliminary manual extraction procedure: draw Liutex or Omega isosurfaces, identify the gathering line of Liutex gradient lines, choose a point on that gathering structure, and then trace the Liutex line through the point. The method was tested on Burgers vortex and a hairpin-vortex DNS case (Liu et al., 2019).

A later “relative Liutex” formulation introduced a normalized measure

$\lambda_{cr}\pm i\lambda_{ci}$9

where $\vec r$0 is the rigid-rotation part of the triple decomposition of the velocity-gradient tensor. That paper emphasizes that relative Liutex is not equivalent to merely lowering the threshold on $\vec r$1, because it normalizes the rigid-rotation strength by the total local velocity-gradient strength (Chen et al., 29 Dec 2025).

6. Later developments, applications, and ongoing reinterpretations

The immediate practical consequence of the explicit 2018 Liutex formula was computational. By eliminating the second in-plane rotation needed for the earlier geometric definition, the authors reported that the Liutex-computation portion of their Fortran code decreased from $\vec r$2 to $\vec r$3, an improvement of about $\vec r$4, while the overall workflow including file I/O and velocity-gradient calculation accelerated by about $\vec r$5 (Wang et al., 2018).

Subsequent work embedded the same explicit formula into turbulence modeling. A 2025 LES study used Liutex magnitude as the SGS velocity scale in

$\vec r$6

with a Liutex-based length scale

$\vec r$7

and a dynamic coefficient based on triple decomposition,

$\vec r$8

That model was tested for turbulent channel flow at friction Reynolds number $\vec r$9 and reported improved near-wall Reynolds-stress prediction with only $\vec\omega=\nabla\times\vec v$00 computational overhead (Chen et al., 16 Dec 2025).

Liutex was also inserted into a modified Navier–Stokes framework derived from angular-momentum balance on a finite control volume. In that formulation, reciprocal shear stresses differ by terms involving time derivatives of Liutex components, for example

$\vec\omega=\nabla\times\vec v$01

with analogous formulas for the other cyclic pairs. The paper treats Liutex magnitude as twice the angular speed and uses $\vec\omega=\nabla\times\vec v$02 as the rotational-acceleration contribution in the constitutive corrections (Yu et al., 2020).

A broader kinematic reinterpretation appeared in 2025 through a comparison between direction-dependent vorticity decompositions and invariant vorticity decompositions. In that framework, the Liutex branch is written as

$\vec\omega=\nabla\times\vec v$03

while an alternative invariant branch

$\vec\omega=\nabla\times\vec v$04

is also algebraically admissible. That paper interprets Liutex as the minimum admissible rigid-rotation branch under its phase-space bounds and therefore as embodying a minimization principle rather than the only possible invariant branch (Chen et al., 30 Aug 2025).

Several recurring limitations remain explicit in the literature. The basic Liutex definition requires a point where the velocity-gradient tensor has one real eigenvalue and one complex-conjugate pair; if all three eigenvalues are real, Liutex is set to zero (Wang et al., 2018). Direct numerical solution of the axis-line condition $\vec\omega=\nabla\times\vec v$05 is difficult because of computational error, and the proposed extraction method in the 2019 axis-line paper is explicitly preliminary and manual (Liu et al., 2019). More broadly, the general kinematic theory suggests that Liutex is one physically selected branch inside a richer invariant decomposition landscape, which indicates that its interpretation continues to be refined rather than closed (Chen et al., 30 Aug 2025).

Within the Liutex literature, the formula has therefore come to occupy two roles simultaneously: a practical computational expression for a vector-valued vortex quantity, and a theoretical statement that the rigid-body-like rotational content of local motion can be extracted from the axial vorticity and complex eigenvalue structure of the velocity-gradient tensor by removing the part attributed to shear (Wang et al., 2018).