---
title: 'Liutex Formula: Vortex Rotation Analysis'
url: https://www.emergentmind.com/topics/liutex-formula
type: topic
---

# Liutex Formula: Vortex Rotation Analysis

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 [1812.10672].

## 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 [1812.10672].

A sign convention is imposed so that
$$
\vec\omega\cdot\vec r>0,
$$
which fixes the orientation of $\vec r$ and makes the Liutex direction unique up to that choice. With this convention, the Liutex magnitude is
$$
R
=
\vec\omega\cdot\vec r
-
\sqrt{(\vec\omega\cdot\vec r)^2-4\lambda_{ci}^2},
$$
and the Liutex vector is $\vec R=R\vec r$ [1812.10672].

| Quantity | Meaning | Role in Liutex |
|---|---|---|
| $\nabla\vec v$ | Velocity-gradient tensor | Local kinematic source tensor |
| $\vec r$ | Real eigenvector of $\nabla\vec v$ | Rotation-axis direction |
| $\lambda_r$ | Real eigenvalue | Axial eigenvalue |
| $\lambda_{cr}\pm i\lambda_{ci}$ | Complex eigenvalue pair | Swirling condition and $\lambda_{ci}$ |
| $\vec\omega$ | Vorticity vector | Supplies axial vorticity component |
| $R$ | Liutex magnitude | Rotational strength |
| $\vec R$ | 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 $Q$, $\lambda_2$, $\Delta$, or $\lambda_{ci}$ [2512.14953].

## 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 $\vec r$, then applied a rotation matrix $Q$ to align the local rotational axis with the $Z$-axis, and finally applied an in-plane rotation $P$ about that axis to locate the minimum angular velocity direction. In that formulation, the Liutex magnitude was defined by
$$
R=
\begin{cases}
2(\beta-\alpha),& \alpha-\beta<0,\\
0,& \alpha-\beta\ge 0,
\end{cases}
$$
with $\alpha$ and $\beta$ determined from the in-plane entries of the rotated velocity-gradient tensor [1812.10672].

The 2018 derivation replaced that construction by introducing the minimum and maximum angular velocities in the plane normal to $\vec r$, denoted $\dot\theta_{\min}$ and $\dot\theta_{\max}$. Two relations were then established:
$$
\dot\theta_{\min}+\dot\theta_{\max}=\omega_r,
\qquad
\dot\theta_{\min}\dot\theta_{\max}=\lambda_{ci}^2,
$$
where
$$
\omega_r=\vec\omega\cdot\vec r.
$$
Solving the quadratic
$$
x^2-\omega_r x+\lambda_{ci}^2=0
$$
for the smaller root gives
$$
\dot\theta_{\min}
=
\frac{\omega_r-\sqrt{\omega_r^2-4\lambda_{ci}^2}}{2}.
$$
Since Liutex magnitude is defined there as
$$
R=2\dot\theta_{\min},
$$
one obtains the explicit formula
$$
R
=
\omega_r-\sqrt{\omega_r^2-4\lambda_{ci}^2},
$$
and therefore
$$
\vec R
=
\left[
\omega_r-\sqrt{\omega_r^2-4\lambda_{ci}^2}
\right]\vec r
=
\left[
\vec\omega\cdot\vec r-\sqrt{(\vec\omega\cdot\vec r)^2-4\lambda_{ci}^2}
\right]\vec r.
$$
This is the formula generally referred to as the Liutex formula [1812.10672].

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 [1904.10094].

## 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,
$$
\omega_r=\vec\omega\cdot\vec r,
$$
is interpreted as twice the spatial mean angular velocity around that axis. The imaginary part of the complex eigenvalue, $\lambda_{ci}$, is interpreted as a pseudo-time average angular velocity associated with the local circular or spiral trajectory implied by the linearized dynamics. Liutex magnitude $R$ 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 [1812.10672].

This yields the relation
- $\omega_r$: twice the spatial mean angular velocity,
- $\lambda_{ci}$: pseudo-time mean angular velocity,
- $R$: 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
$$
\sqrt{\omega_r^2-4\lambda_{ci}^2}
$$
is the pure shear part, while
$$
R=\omega_r-\sqrt{\omega_r^2-4\lambda_{ci}^2}
$$
is the rotational part retained by Liutex [1812.10672].

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
$$
\lambda_{ci}^2=\frac{R}{2}\left(\frac{R}{2}+\varepsilon\right),
$$
so $\lambda_{ci}$ explicitly contains the in-plane shear parameter $\varepsilon$. In the same framework, $Q$ and $\Delta$ also contain shear and stretching terms, whereas Liutex is constructed to retain only the rigid-rotation block [2008.06779].

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
$$
R=2\min\left\{
\left|\frac{\partial u}{\partial y}\right|,
\left|\frac{\partial v}{\partial x}\right|
\right\},
$$
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 [1905.03228].

## 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 $z$-axis is parallel to the real eigenvector $\vec r$ and the in-plane orientation is fixed uniquely. In that frame, the velocity-gradient tensor takes a canonical form such as
$$
\nabla \mathbf V
=
\begin{bmatrix}
\lambda_{cr} & -R & 0\\
R+\varepsilon & \lambda_{cr} & 0\\
\xi & \eta & \lambda_r
\end{bmatrix},
$$
and is decomposed exactly into rigid rotation, shear, and stretching/compression [2008.06779].

In that same paper, the principal decomposition is written as
$$
\nabla \mathbf V=A+B+C,
$$
where $A$ is the rigid-rotation block, $B$ contains shear terms, and $C$ 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 [2008.06779].

A mathematical study of the local fluid rotation axis gave a more axiomatic statement. It defines a local rotation axis as a direction $\vec d$ satisfying
$$
\nabla\vec v\cdot\vec d=\alpha\vec d,
$$
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 $\nabla\vec v$, 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,
$$
\nabla\vec v=
\begin{bmatrix}
\lambda_{cr} & -R/2 & 0\\
R/2+\varepsilon & \lambda_{cr} & 0\\
\xi & \eta & \lambda_r
\end{bmatrix},
$$
and the rigid-rotation tensor has local angular speed $\Omega=R/2$, yielding the statement that Liutex magnitude equals twice the local angular speed [2103.07525].

The principal decomposition was later transferred back to the original Cartesian coordinates. For a Liutex vector $\mathbf R=(R_x,R_y,R_z)$ and unit direction $\mathbf r$, the rigid-rotation tensor becomes
$$
\tilde R
=
\frac12
\begin{bmatrix}
0 & -R_z & R_y\\
R_z & 0 & -R_x\\
-R_y & R_x & 0
\end{bmatrix},
$$
while the stretching/compression tensor is
$$
\widetilde{SC}
=
\lambda_{cr}I+(\lambda_r-\lambda_{cr})\,\mathbf r\mathbf r^T.
$$
The shear tensor is then the remainder
$$
\tilde S=\nabla\mathbf v-\tilde R-\widetilde{SC}.
$$
This Cartesian principal decomposition was proposed as a direct alternative to the Cauchy–Stokes split into symmetric and antisymmetric parts [2106.10564].

## 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 $Q$, $\lambda_2$, $\Delta$, and $\lambda_{ci}$. 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 [2512.14953].

The relation to $\lambda_{ci}$ is especially important. In principal-coordinate analysis,
$$
\lambda_{ci}^{\,2}=\frac{R}{2}\left(\frac{R}{2}+\varepsilon\right),
$$
so $\lambda_{ci}$ 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 [2008.06779]. 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 $\lambda_{ci}$, $Q$, and $\lambda_2$ correlate better with Liutex but are still not exact substitutes [2008.06781].

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:
$$
\nabla R\times\mathbf r=0,
$$
with $R>0$. 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
$$
\nabla R=0.
$$
The same paper notes an important degeneracy: in pure rigid-body rotation, every point satisfies the condition because $\nabla R=0$ everywhere, so the axis is not unique under that criterion [1904.10094].

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 [1904.10094].

A later “relative Liutex” formulation introduced a normalized measure
$$
R^*=\frac{G_R G_R}{G_{ij}G_{ij}},
$$
where $G_R$ 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 $R$, because it normalizes the rigid-rotation strength by the total local velocity-gradient strength [2512.23857].

## 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 $2.76\,\text{s}$ to $1.75\,\text{s}$, an improvement of about $37\%$, while the overall workflow including file I/O and velocity-gradient calculation accelerated by about $7.9\%$ [1812.10672].

Subsequent work embedded the same explicit formula into turbulence modeling. A 2025 LES study used Liutex magnitude as the SGS velocity scale in
$$
\nu_t=(C_R\Delta_R)^2R,
$$
with a Liutex-based length scale
$$
\Delta_R=2\max_{m=1,\ldots,6}\left|\mathbf n_R\times \mathbf r_{cf}^{(m)}\right|,
$$
and a dynamic coefficient based on triple decomposition,
$$
C_R=\frac{G_R G_R}{G_{ij}G_{ij}}.
$$
That model was tested for turbulent channel flow at friction Reynolds number $544$ and reported improved near-wall Reynolds-stress prediction with only $4.21\%$ computational overhead [2512.14953].

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
$$
\tau_{xy}=\tau_{yx}+2\gamma\frac{\partial R_z}{\partial t},
$$
with analogous formulas for the other cyclic pairs. The paper treats Liutex magnitude as twice the angular speed and uses $\partial_t R_i$ as the rotational-acceleration contribution in the constitutive corrections [2008.06784].

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
$$
\bm R_{\text{Liutex}}=\bm R_N^+=2\psi^+\bm e_3
=
\left(\omega_3-\sqrt{\omega_3^2-4\lambda_{ci}^2}\right)\bm e_3,
$$
while an alternative invariant branch
$$
2\psi^-=\omega_3+\sqrt{\omega_3^2-4\lambda_{ci}^2}
$$
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 [2509.00372].

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 [1812.10672]. Direct numerical solution of the axis-line condition $\nabla R\times\mathbf r=0$ is difficult because of computational error, and the proposed extraction method in the 2019 axis-line paper is explicitly preliminary and manual [1904.10094]. 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 [2509.00372].

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 [1812.10672].

Source: https://www.emergentmind.com/topics/liutex-formula