Revisiting turbulence small-scale behavior using velocity gradient triple decomposition

For the rotational case, we follow the VGT decomposition procedure outlined by Gao and Liu [12]. This method involves two coordinate-frame rotations to obtain the VGT in the desired lower block triangular form for decomposition. The steps are listed below:

• (a)  
  First we identify the rotation axis ($\to {\boldsymbol{r}}$), which is the eigenvector of the real eigenvalue of A. Then the coordinate frame is rotated such that the Z-axis of the new frame is aligned with $\to {\boldsymbol{r}}$. The VGT in this new coordinate frame is given by

  $$\begin{equation}{\boldsymbol{A}}^{\prime }=\boldsymbol{Q}\boldsymbol{A}{\boldsymbol{Q}}^{\mathrm{T}}\end{equation} \tag{ 3 }$$

  where, Q is a proper rotation matrix obtained from real Schur decomposition of A as shown by Liu et al [29].
• (b)  
  Next, the coordinate frame is further rotated about the Z-axis, i.e. in the XY-plane, by an azimuthal angle θ. This angle θ is chosen such that the angular velocity of the fluid is minimum. The VGT in this new coordinate frame (A*) is then given by

  $$\begin{equation}{\boldsymbol{A}}^{{\ast}}=\boldsymbol{P}{\boldsymbol{A}}^{\prime }{\boldsymbol{P}}^{\mathrm{T}}\quad \text{where}\quad \boldsymbol{P}=\left[\begin{bmatrix}{ccc}\mathrm{cos} \theta & \mathrm{sin} \theta & 0\\ -\mathrm{sin} \theta & \mathrm{cos} \theta & 0\\ 0& 0& 1\end{bmatrix}\right]\end{equation} \tag{ 4 }$$

  is also a proper rotation matrix.

These steps result in the VGT in a rotated coordinate frame such that it is of the form

$$\begin{equation}{\boldsymbol{A}}^{{\ast}}=\left[\begin{bmatrix}{ccc}{\lambda }_{\mathrm{c}\mathrm{r}}& -\phi & 0\\ \phi +{s}_{3}& {\lambda }_{\mathrm{c}\mathrm{r}}& 0\\ {s}_{2}& {s}_{1}& {\lambda }_{r}\end{bmatrix}\right]\end{equation} \tag{ 5 }$$

The VGT is then decomposed into the following constituent tensors,

$$\begin{equation*}{\boldsymbol{A}}^{{\ast}}=\boldsymbol{N}+\boldsymbol{H}+\boldsymbol{R}\quad \text{where}\end{equation*}$$

$$\begin{equation}\boldsymbol{N}=\left[\begin{bmatrix}{ccc}{\lambda }_{\mathrm{c}\mathrm{r}}& 0& 0\\ 0& {\lambda }_{\mathrm{c}\mathrm{r}}& 0\\ 0& 0& {\lambda }_{r}\end{bmatrix}\right],\quad \boldsymbol{H}=\left[\begin{bmatrix}{ccc}0& 0& 0\\ {s}_{3}& 0& 0\\ {s}_{2}& {s}_{1}& 0\end{bmatrix}\right],\quad \boldsymbol{R}=\left[\begin{bmatrix}{ccc}0& -\phi & 0\\ \phi & 0& 0\\ 0& 0& 0\end{bmatrix}\right]\end{equation} \tag{ 6 }$$

We define N as the normal-strain-rate tensor, which is a diagonal tensor containing the real parts of eigenvalues of VGT. Here, λ_cr is the real part of the complex conjugate eigenvalues of A and λ_r is the only real eigenvalue of A. The tensor N represents the normal compression and expansion of the fluid element along different directions. It must be noted that the volume of the fluid element is preserved. Incompressibility imposes the following condition:

$$\begin{equation}{\lambda }_{r}=-2{\lambda }_{\mathrm{c}\mathrm{r}}\end{equation} \tag{ 7 }$$

The tensor H represents the pure shearing deformation of the fluid element. The shear tensor is a lower triangular tensor with three elements—s₁, s₂ and s₃—that constitute the transverse gradients of velocity components. The s₃ component represents shearing within the plane of rigid-body-rotation and is non-negative by definition [12]. Moreover, the eigenvalues of such a tensor are zero. Finally, the rigid-body-rotation tensor R is an anti-symmetric tensor depending on only one unknown, ϕ. The tensor represents pure-rotation of the fluid element with an angular velocity given by ϕ. The rotational strength is defined as twice the value of this angular velocity of the fluid element ($\tilde {R}\equiv 2\phi$). The rotation axis, $\to {r}$, lies along the axial vector of the rigid-body-rotation tensor. Thus, the Rortex vector is given by [12],

$$\begin{equation}\to {\boldsymbol{R}}=\tilde {R}\to {\boldsymbol{r}}=2\phi \to {\boldsymbol{r}}\end{equation} \tag{ 8 }$$

The rigid-body-rotation tensor has one zero and two purely imaginary eigenvalues (±iϕ). Note that this additive triple decomposition of the VGT is in the principal frame of the normal-strain-rate tensor.

In this case, we use Schur decomposition to segregate the normal-strain-rate tensor from the shear tensor, as the rigid-body-rotation is identically zero. In a previous study, Keylock [22] segregated the effect of shear from the VGT by performing complex Schur decomposition, which however results in complex component tensors. In this study, we use real Schur decomposition for locally non-rotational points in the flow. Since A contains only real eigenvalues in this case, it can be transformed into an upper triangular tensor by real Schur decomposition:

$$\begin{equation}{\boldsymbol{A}}^{{\dagger}}={\boldsymbol{Q}}^{{\ast}\mathrm{T}}\boldsymbol{A}{\boldsymbol{Q}}^{{\ast}}\end{equation} \tag{ 9 }$$

such that A^† is a real upper triangular tensor which can now be decomposed into a diagonal (normal) tensor and a strictly upper-triangular (non-normal) tensor. Here, Q* is an orthogonal matrix responsible for coordinate transformation of the VGT from A to A^†. In order to be consistent with the lower triangular form of the shear tensor obtained in the triple decomposition method for the rotational case, we perform real Schur decomposition of the VGT transpose (A^T)

$$\begin{equation}{\boldsymbol{A}}^{{\ast}{\ast}}={\boldsymbol{Q}}^{\mathrm{T}}{\boldsymbol{A}}^{\mathrm{T}}\boldsymbol{Q}\end{equation} \tag{ 10 }$$

Then, the transpose of the resulting tensor yields

$$\begin{equation}{\boldsymbol{A}}^{{\ast}}={\boldsymbol{A}}^{{\ast}{\ast}\mathrm{T}}={\boldsymbol{Q}}^{\mathrm{T}}\boldsymbol{A}\boldsymbol{Q}\end{equation} \tag{ 11 }$$

where A* is the Schur decomposition of A in lower triangular form. Now, the VGT can be decomposed into the following normal and non-normal tensors

$$\begin{equation*}{\boldsymbol{A}}^{{\ast}}=\boldsymbol{N}+\boldsymbol{H}\quad \text{where}\end{equation*}$$

$$\begin{equation}\boldsymbol{N}=\left[\begin{bmatrix}{ccc}{\lambda }_{1}& 0& 0\\ 0& {\lambda }_{2}& 0\\ 0& 0& {\lambda }_{3}\end{bmatrix}\right]\quad \boldsymbol{H}=\left[\begin{bmatrix}{ccc}0& 0& 0\\ {s}_{3}& 0& 0\\ {s}_{2}& {s}_{1}& 0\end{bmatrix}\right]\end{equation} \tag{ 12 }$$

Here, N is the normal tensor, i.e. a diagonal tensor containing the eigenvalues of A. It is therefore referred to as the normal-strain-rate tensor and reflects the compression and expansion that the fluid element undergoes along different directions. In order to ensure that the VGT decomposition is unique, the ordering of diagonal elements of the normal-strain-rate tensor is fixed to λ₁ ⩾ λ₂ ⩾ λ₃. This tensor consists of only two unknowns due to the incompressibility condition,

$$\begin{equation}{\lambda }_{1}+{\lambda }_{2}+{\lambda }_{3}=0\end{equation} \tag{ 13 }$$

The lower triangular tensor H is the non-normal tensor containing information about the eigenvectors of A. This tensor consists of three independent elements representing shearing of the fluid element in three orthogonal planes and is therefore, referred to as the shear tensor.

The VGT decomposition in both rotational and non-rotational cases are considered in the principal frame of N. In the remaining sections of this work, the VGT will be used in this coordinate frame for convenience and will be represented by A. The results presented in this study are frame invariant and do not depend on the coordinate frame of reference.

In terms of strain-rate and rotation-rate, velocity gradient magnitude (Frobenius norm squared: A² = A_ijA_ij) can be written as:

$$\begin{equation}{A}^{2}={S}^{2}+{W}^{2}\end{equation} \tag{ 16 }$$

where S² = S_ijS_ij is strain-rate magnitude related to dissipation (νS²) and W² = W_ijW_ij is vorticity magnitude or enstrophy. The triple decomposition of VGT results in the following expression for magnitude:

$$\begin{equation}{A}_{ij}{A}_{ij}=\;{N}_{ij}{N}_{ij}+{H}_{ij}{H}_{ij}+{R}_{ij}{R}_{ij}+2{R}_{ij}{H}_{ij}\end{equation} \tag{ 17 }$$

which can be restated as

$$\begin{equation}{A}^{2}=\;{N}^{2}+{H}^{2}+{R}^{2}+2RH\end{equation} \tag{ 18 }$$

where N² = N_ijN_ij, H² = H_ijH_ij and R² = R_ijR_ij are defined as the magnitudes (Frobenius norm squared) of normal-strain-rate, shear and rigid-body-rotation tensors, respectively. For locally fluid rotational case (equations (6) and (7)), these are of the form

$$\begin{equation}{N}^{2}=6{\lambda }_{\mathrm{c}\mathrm{r}}^{2},\quad {H}^{2}={s}_{1}^{2}+{s}_{2}^{2}+{s}_{3}^{2},\quad {R}^{2}=2{\phi }^{2}\end{equation} \tag{ 19 }$$

and for the fluid non-rotational case (equations (12) and (13)), these constituent magnitudes are of the form

$$\begin{equation}{N}^{2}=2\left({\lambda }_{1}^{2}+{\lambda }_{2}^{2}+{\lambda }_{1}{\lambda }_{2}\right),\quad {H}^{2}={s}_{1}^{2}+{s}_{2}^{2}+{s}_{3}^{2},\quad {R}^{2}=0\end{equation} \tag{ 20 }$$

The term RH = R_ijH_ij is defined as the correlation term between the R and H tensors. The other possible correlation terms such as N_ijH_ij and N_ijR_ij are identically zero since N is a diagonal matrix while both H and R are hollow matrices (all diagonal elements are zero). Using equation (6) it can be shown that the shear-rotation correlation term, which exists only if the flow is locally rotational, is given by,

$$\begin{equation}2RH=2{R}_{ij}{H}_{ij}=2\phi {s}_{3}\end{equation} \tag{ 21 }$$

Therefore, the shear-rotation correlation term contributing to velocity gradient (VG) magnitude depends only on the rigid-body-rotation strength (2ϕ) and the component of shear that is in the plane of rigid-body-rotation (s₃). Since both ϕ and s₃ are non-negative [12] by definition, 2RH ⩾ 0.

In order to measure the contribution of each component toward VG magnitude, we normalize these magnitudes by the local VG magnitude

$$\begin{equation}{n}^{2}\equiv \frac{{N}^{2}}{{A}^{2}},\quad {h}^{2}\equiv \frac{{H}^{2}}{{A}^{2}},\quad {r}^{2}\equiv \frac{{R}^{2}}{{A}^{2}},\quad 2rh\equiv \frac{2RH}{{A}^{2}}\end{equation} \tag{ 22 }$$

Then, the normalized normal-strain, shear and rigid-body-rotation magnitudes have the following bounds,

$$\begin{equation}0{\leqslant}{n}^{2}{\leqslant}1,\quad 0{\leqslant}{h}^{2}{\leqslant}1,\quad 0{\leqslant}{r}^{2}{\leqslant}1\end{equation} \tag{ 23 }$$

It can be shown using equation (6) that the normalized correlation term 2rh have more restricted bounds, i.e.

$$\begin{equation}0\;{\leqslant}\;2rh\;{\leqslant}\;\frac{1}{\sqrt{2}+1}\;\;\approx 0.41\end{equation} \tag{ 24 }$$

A detailed proof of the above result is included in appendix A.

As shown previously in equation (15), the shear tensor contributes to both S as well as W in the form of its symmetric (H_S) and anti-symmetric (H_W) counterparts, respectively. Since the shear tensor is a lower triangular tensor, the symmetric-shear and anti-symmetric-shear tensors are of the form

$$\begin{equation}{\mathbf{H}}_{\mathbf{S}}=\left[\begin{bmatrix}{ccc}0& {s}_{3}/2& {s}_{2}/2\\ {s}_{3}/2& 0& {s}_{1}/2\\ {s}_{2}/2& {s}_{1}/2& 0\end{bmatrix}\right]\quad \text{and}\quad {\mathbf{H}}_{\mathbf{W}}=\left[\begin{bmatrix}{ccc}0& -{s}_{3}/2& -{s}_{2}/2\\ {s}_{3}/2& 0& -{s}_{1}/2\\ {s}_{2}/2& {s}_{1}/2& 0\end{bmatrix}\right]\end{equation} \tag{ 25 }$$

The magnitudes of H_S and H_W are equal since,

$$\begin{equation}{H}_{\mathrm{S}}^{2}={H}_{\mathrm{W}}^{2}=\frac{1}{2}\left({s}_{1}^{2}+{s}_{2}^{2}+{s}_{3}^{2}\right)=\frac{{H}^{2}}{2}\end{equation} \tag{ 26 }$$

Therefore, the shear-magnitude H² is divided equally between strain-rate and vorticity magnitudes. From equations (15) and (26), we then obtain

$$\begin{equation}{S}^{2}={N}^{2}+\frac{{H}^{2}}{2},\quad {W}^{2}={R}^{2}+2RH+\frac{{H}^{2}}{2}\end{equation} \tag{ 27 }$$

It may be noted that vorticity has an additional dependence on shear via the shear-rotation correlation term.

The occurrence of extreme values of A² is critical in the investigation of turbulence intermittency. The primary goal of this study is to examine the contribution of normal-strain-rate, shear and rigid-body-rotation towards the overall VG magnitude and its relation with the pressure field.

In this subsection, we revisit local streamline topology and geometry using triple decomposition of VGT. The topology of local streamlines in incompressible turbulent flows is defined by the second and third invariants of VGT as proposed by Chong et al [30],

$$\begin{equation}{Q}_{A}=-\frac{1}{2}{A}_{ij}{A}_{ji},\quad {R}_{A}=-\frac{1}{3}{A}_{ij}{A}_{jk}{A}_{ki}\end{equation} \tag{ 28 }$$

In the fluid rotational case, the invariants can be expressed in terms of VGT constituents (applying equations (6) and (7)) as follows,

$$\begin{equation}\begin{aligned}\hfill & {Q}_{A}=-3{\lambda }_{\mathrm{c}\mathrm{r}}^{2}+{\phi }^{2}+\phi {s}_{3}=\frac{1}{2}\left({R}^{2}+2RH-{N}^{2}\right)\hfill \\ \hfill & {R}_{A}=2{\lambda }_{\mathrm{c}\mathrm{r}}\left({\lambda }_{\mathrm{c}\mathrm{r}}^{2}+{\phi }^{2}+\phi {s}_{3}\right)={\lambda }_{\mathrm{c}\mathrm{r}}\left(\frac{{N}^{2}}{3}+{R}^{2}+2RH\right)\hfill \end{aligned}\end{equation} \tag{ 29 }$$

It is important to note here that the invariants do not depend on the components of shear outside the plane of rigid body rotation, i.e. s₁ and s₂. Therefore, the topology of locally rotational streamlines can be expressed as a function of normal-strain-rate eigenvalues, rigid-body-rotation strength and the component of shear within the plane of rigid-body-rotation. The invariants of each of the constituent tensor (N, R and H) are given by,

$$\begin{equation}\begin{aligned}\hfill & {Q}_{N}=-3{\lambda }_{\mathrm{c}\mathrm{r}}^{2},\quad {Q}_{R}={\phi }^{2},\quad {Q}_{H}=0\hfill \\ \hfill & {R}_{N}=-2{\lambda }_{\mathrm{c}\mathrm{r}}^{3},\quad {R}_{R}=0,\quad {R}_{H}=0\hfill \end{aligned}\end{equation} \tag{ 30 }$$

In the fluid non-rotational case (applying equations (12) and (13)), the VGT invariants are given by

$$\begin{equation}\begin{aligned}\hfill & {Q}_{A}=-\left({\lambda }_{1}^{2}+{\lambda }_{2}^{2}+{\lambda }_{1}{\lambda }_{2}\right)=-\frac{{N}^{2}}{2}\hfill \\ \hfill & {R}_{A}={\lambda }_{1}{\lambda }_{2}\left({\lambda }_{1}+{\lambda }_{2}\right)=-{\lambda }_{1}\left(\frac{{N}^{2}}{2}-{\lambda }_{1}^{2}\right)\hfill \end{aligned}\end{equation} \tag{ 31 }$$

In this case, both Q_A and R_A are functions of only the normal-strain-rate tensor and do not depend on the shear tensor at all. In fact, invariants of the consituent tensors N and H are given by

$$\begin{equation}\begin{aligned}\hfill & {Q}_{N}={Q}_{A},\quad {Q}_{H}=0\hfill \\ \hfill & {R}_{N}={R}_{A},\quad {R}_{H}=0\hfill \end{aligned}\end{equation} \tag{ 32 }$$

Therefore, the topology of locally non-rotational streamlines depends only on the normal strain-rate tensor.

Topology, however, only provides information about the connectivity of a geometric shape. The complete geometric shape, as shown by Das and Girimaji [31, 32], is better represented in the bounded phase space of normalized VGT invariants given by,

$$\begin{equation}q=\frac{{Q}_{A}}{{A}^{2}},\quad r=\frac{{R}_{A}}{{\left({A}^{2}\right)}^{\frac{3}{2}}}\end{equation} \tag{ 33 }$$

Note that these normalized invariants depend on all the shear components due to the normalization. Thus, shear might not be as critical in determining the topology of the local flow but it is important in determining its complete geometric shape. We now examine the shape of the local streamline geometry associated with the different velocity gradient tensor components N, H and R in the q–r plane.

Degenerate geometries: these are the limiting cases, illustrated in figure 2(a), that represent a point or a line in the q–r plane. These degenerate shapes are discussed below:

• (a)  
  Pure-rotation (A = R; N = H = 0): the top-most point in the plane (r = 0, q = 1/2) represents an anti-symmetric VGT with one zero and two purely imaginary eigenvalues. This results in pure rigid-body-rotation of the fluid element or locally planar circular streamlines.
• (b)  
  Pure-shear (A = H; N = R = 0): the lower triangular tensor H has zero second and third invariants by its definition. Therefore, the origin of the q–r plane represents pure shearing of the fluid element.
• (c)  
  Normal-straining (A = N; R = H = 0): The q = −1/2 line or bottom-most boundary of the plane constitutes the case when the VGT itself is a normal matrix. This represents pure stretching/compression of the fluid element in three orthogonal directions.
• (d)  
  Rotation and shear (A = R + H; N = 0): the upper half (q > 0) of the r = 0 line represents VGT with zero normal-strain-rate tensor. The VGT here has one zero and two purely imaginary eigenvalues, resulting in planar closed circulating streamlines that are elliptic in shape due to the presence of shear. In the absence of shear, the streamlines are purely circular (at q = 1/2).
• (e)  
  Normal-strain and rotation (A = N + R; H = 0): the shear tensor is zero at the left and right boundaries of the plane given by [32]

  $$\begin{equation}r={\pm}\left(\frac{1+q}{3}\right){\left(\frac{1-2q}{3}\right)}^{1/2}\end{equation} \tag{ 34 }$$

  The left boundary represents rigid-body-rotation about the expansive normal-strain-rate eigenvector while the right boundary represents rigid-body-rotation about the compressive normal-strain-rate eigenvector, both in the absence of any shear. This implies locally stable/unstable spiraling streamlines stretching/compressing perpendicular to its focal plane.

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Non-degenerate geometries: figure 2(b) illustrates the four non-degenerate geometries covering the area of the q–r plane and the corresponding VGT decomposition. These are discussed below:

• (a)  
  Rotational geometries (A = N + R + H): above the zero discriminant (d = q³ + (27/4)r² = 0) lines the VGT has complex eigenvalues and the flow is locally rotational. All three constituent tensors are in general non-zero. In the stable-focus-stretching (SFS) topology, the unique eigenvalue of N (λ_r: real eigenvalue of A) is positive, i.e. λ_r > 0, representing stretching of the fluid element. The other two equal eigenvalues of N are negative, i.e. λ_cr < 0, representing convergence of the local streamlines towards a stable focus. Similarly in unstable-focus-compression (UFC) topology, λ_r < 0 represents compression of the fluid element and λ_cr > 0 represents diverging streamlines from an unstable focus. The non-zero eigenvalues of tensor R (±iϕ) denote the angular velocity of rigid-body-rotation of the fluid element. The tensor H controls the shearing of the fluid element, resulting in varied orientations of the spiraling with respect to the direction of stretching/compression. The angle of alignment between the vorticity vector ($\to {\boldsymbol{\omega }}\equiv$ dual vector of W) and the eigenvectors of S can take any value. On the contrary, in this decomposition the rotation vector ($\to {\boldsymbol{r}}\equiv$ dual vector of R) is always aligned along the unique eigenvector of N.
• (b)  
  Non-rotational geometries (A = N + H; R = 0): below the zero discriminant line, VGT has only real eigenvalues. Stable-node-saddle-saddle (SNSS) topology region represents compression in two directions and expansion in one, i.e. λ₁ > 0, λ_2,3 < 0, while unstable-node-saddle-saddle (UNSS) topology implies λ_1,2 > 0, λ₃ < 0. However, these directions are in general oblique with respect to each other. The information about the magnitude of stretching/compression is contained in N but the orientation of these directions, designated by the λ_i-eigenvectors, are contained in H.

In summary, pure shear occurs at the origin of the q–r plane while normal-straining and rigid-body-rotation occur at the boundaries of the plane. The entire area inside the plane is populated in a turbulent flow field as a result of the combination of all three constituents. The distribution asymptotes to a nearly universal teardrop-like shape around the origin following the right discriminant line in fully-developed turbulent flows [31].