Keywords

1 Introduction

A vortex is recognized as the rotational motion of fluids. Many vortex identification methods have been developed within the last several decades to track the vortical structure in a fluid flow; however, we still lacked unambiguous and universally accepted vortex identification criteria. This obstacle caused a lot of confusion and misunderstandings in turbulence research [1]. These methods are characterized into three generations starting with vorticity-based methods as the first generation, eigenvalue-based such as \(\Delta\) [2, 3], Q [4], \({\lambda }_{2}\) [5], \({\lambda }_{ci}\) [6], and \(\Omega\) [7, 8], as the second generation, and the recently developed Liutex methods as the third generation of vortex identification methods [9]. The Liutex method is a novel eigenvector-based method that is local, accurate, unique, and systematic [10].

In Computational fluid dynamics, many researchers and textbooks accept that vorticity is vortex. This is due to a misunderstanding from the Cauchy-Stokes decomposition of the velocity gradient tensor. It was understood that the symmetric tensor \({\varvec{A}}\) represented stretching/compression, and the antisymmetric tensor \({\varvec{B}}\) (vorticity tensor) represented rotation [11]. In this paper, we will show that antisymmetric tensor \({\varvec{B}}\) (vorticity tensor) does not represent only rotation and investigate the behavior of shear, Liutex, and vorticity in the boundary layer from laminar flow to turbulent flow.

This manuscript is split into sections, where Sect. 11.2 is a review of vorticity and Liutex, Sect. 11.3 shows that vorticity is not strictly rotation, Sect. 11.4 shows the data structure for the DNS study, and Sect. 11.5 is the results of the investigation of the behavior of shear Liutex, and vorticity.

2 Review of Related Vortex Identification Methods

The vortex identification methods that we will analyze in this paper are vorticity and Liutex. We will review these methods in this section.

2.1 Vorticity

In 1858, Helmholtz introduced the concept of the vorticity tube/filament [12]. Since then, many researchers have believed that vortices consist of small vorticity tubes called vortex filaments, and the magnitude of vorticity gives the vortex strength.

2.1.1 Vorticity Vector

The vorticity vector is mathematically defined as the curl of the velocity. i.e.,

$${\text{vortivity}} = {\mathbf{\nabla }} \times {\mathbf{\mathop{v}\limits^{\rightharpoonup} }} = \left| {\begin{array}{*{20}c} {\mathbf{i}} & {\mathbf{j}} & {\mathbf{k}} \\ {\frac{\partial }{\partial x}} & {\frac{\partial }{\partial y}} & {\frac{\partial }{\partial z}} \\ u & v & w \\ \end{array} } \right|$$
$$=\mathbf{i}\left(\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z}\right)-\mathbf{j}\left(\frac{\partial w}{\partial x}-\frac{\partial u}{\partial z}\right)+\mathbf{k}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)$$
(11.1)

The vorticity vector is also derived as follows [5]:

$$\mathop{\omega }\limits^{\rightharpoonup} = {\mathbf{\nabla }} \times {\mathbf{\mathop{v}\limits^{\rightharpoonup} }} = \left( {\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}} \right)^{{\varvec{T}}} \times ({\text{u}},{\text{v}},{\text{w}})^{{\varvec{T}}}$$
$$= {\left(\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z}, \frac{\partial u}{\partial z}-\frac{\partial w}{\partial x}, \frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)}^{{\varvec{T}}}.$$
(11.2)

2.1.2 Vorticity Magnitude

The vorticity magnitude is defined as

$$\Vert \mathop{\omega }\limits^{\rightharpoonup} \Vert =\sqrt{{\left(\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z}\right)}^{2}+{\left(\frac{\partial u}{\partial z}-\frac{\partial w}{\partial x}\right)}^{2}+{\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)}^{2}}.$$
(11.3)

2.1.3 Vorticity Tensor

The vorticity tensor is the antisymmetric tensor \({\varvec{B}}\) from the traditional Cauchy-Stokes decomposition of the velocity gradient tensor \({\mathbf{\nabla \mathop{v}\limits^{\rightharpoonup} }}\) [13]:

$${\varvec{B}}=\frac{1}{2}\left({\mathbf{\nabla }} {\mathbf{\mathop{v}\limits^{\rightharpoonup} }} - {\mathbf{\nabla }} {\mathbf{\mathop{v}\limits^{\rightharpoonup} }}^{\bf{T}} \right)=\left[\begin{array}{ccc}0& \frac{1}{2}\left(\frac{\partial u}{\partial y} - \frac{\partial v}{\partial x}\right)& \frac{1}{2}\left(\frac{\partial u}{\partial z}-\frac{\partial w}{\partial x}\right)\\ \frac{1}{2}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)& 0& \frac{1}{2}\left(\frac{\partial v}{\partial z}-\frac{\partial w}{\partial y}\right)\\ \frac{1}{2}\left(\frac{\partial w}{\partial x}-\frac{\partial u}{\partial z}\right)& \frac{1}{2}\left(\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z}\right)& 0\end{array}\right].$$
(11.4)

2.2 Liutex

Liutex [9, 10] is a vector defined as \(\mathop{R}\limits^{\rightharpoonup} = R\mathop{r}\limits^{\rightharpoonup}\). \(R\) represents the Liutex magnitude defined as twice the angular velocity, and \(\mathop{r}\limits^{\rightharpoonup}\) represents the directional unit vector of Liutex. According to Wang [14], \(\mathop{r}\limits^{\rightharpoonup}\) is the real eigenvector of the velocity gradient tensor, and the explicit formula of \(R\) is

$$R = \mathop{\omega }\limits^{\rightharpoonup} \cdot \mathop{r}\limits^{\rightharpoonup} - \sqrt {(\mathop{\omega }\limits^{\rightharpoonup} \cdot \mathop{r}\limits^{\rightharpoonup} )^{2} - 4\lambda_{ci}^{2} }$$
(11.5)

Liutex, as a vector, overcomes the drawbacks of the scalar methods, e.g., the threshold requirement when creating and analyzing graphics.

3 Vorticity Versus Rotation

In this section, we will decompose vorticity in the principal coordinate to show that vorticity is not strictly rotational.

3.1 Vorticity Tensor in the Principal Coordinate

The Vorticity tensor in the Principal Coordinate has the following form:

$$\begin{array}{c}{{\varvec{B}}}_{{\varvec{P}}{\varvec{C}}}=\left[\begin{array}{ccc}0& \frac{1}{2}\left(\frac{\partial U}{\partial Y} - \frac{\partial V}{\partial X}\right)& \frac{1}{2}\left(\frac{\partial U}{\partial Z}-\frac{\partial W}{\partial X}\right)\\ \frac{1}{2}\left(\frac{\partial V}{\partial X}-\frac{\partial U}{\partial Y}\right)& 0& \frac{1}{2}\left(\frac{\partial V}{\partial Z}-\frac{\partial W}{\partial Y}\right)\\ \frac{1}{2}\left(\frac{\partial W}{\partial X}-\frac{\partial U}{\partial Z}\right)& \frac{1}{2}\left(\frac{\partial W}{\partial Y}-\frac{\partial V}{\partial Z}\right)& 0\end{array}\right]\end{array}$$
$${{\varvec{B}}}_{{\varvec{P}}{\varvec{C}}}=\left[\begin{array}{ccc}0& -\frac{R+\varepsilon }{2}& -\frac{\xi }{2}\\ \frac{R+\varepsilon }{2}& 0& -\frac{\eta }{2}\\ \frac{\xi }{2}& \frac{\eta }{2}& 0\end{array}\right]$$
(11.6)

and can be decomposed further into this form:

$${{\varvec{B}}}_{{\varvec{P}}{\varvec{C}}}= \left[\begin{array}{ccc}0& -\frac{R}{2}& 0\\ \frac{R}{2}& 0& 0\\ 0& 0& 0\end{array}\right]+\left[\begin{array}{ccc}0& -\frac{\varepsilon }{2}& -\frac{\xi }{2}\\ \frac{\varepsilon }{2}& 0& -\frac{\eta }{2}\\ \frac{\xi }{2}& \frac{\eta }{2}& 0\end{array}\right]={\varvec{R}}+{\varvec{A}}{\varvec{S}}.$$
(11.7)

where \({\varvec{R}}\) \(\& {\varvec{A}}{\varvec{S}}\) represent the rotation and antisymmetric shear deformation part, respectively. This implies that the vorticity tensor is not strictly rotation [15].

3.2 Vorticity Vector in the Principal Coordinate

Using Eq. 11.2 from Sect. 11.2.1.1 and applying the Principal Coordinate yields:

$$\mathop{\omega }\limits^{\rightharpoonup} = {\mathbf{\nabla }} \times \mathop{{\mathbf{v}}_{\theta } }\limits^{ \rightharpoonup} ={\left(\frac{\partial }{\partial X}, \frac{\partial }{\partial Y}, \frac{\partial }{\partial Z}\right)}^{{\varvec{T}}}\times {\left(U, V, W\right)}^{{\varvec{T}}}$$
$$={\left(\frac{\partial W}{\partial Y}-\frac{\partial V}{\partial Z}, \frac{\partial U}{\partial Z}-\frac{\partial W}{\partial X}, \frac{\partial V}{\partial X}-\frac{\partial U}{\partial Y}\right)}^{{\varvec{T}}}$$
$$={\left(\upeta , -\upxi , \frac{R}{2}+\varepsilon -\left(-\frac{R}{2}\right)\right)}^{{\varvec{T}}}= {\left(\upeta , -\upxi , R+\varepsilon \right)}^{{\varvec{T}}}$$
$$={\left(\upeta , -\upxi , \varepsilon \right)}^{{\varvec{T}}}+{\left(0, 0, R\right)}^{{\varvec{T}}}=\mathop{S}\limits^{\rightharpoonup}+\mathop{R}\limits^{\rightharpoonup}$$
(11.8)

where \(\mathop{R}\limits^{\rightharpoonup}\) is a rotational vector and \(\mathop{S}\limits^{\rightharpoonup}\) is a non-rotational shear vector. This implies that \(\mathop{\omega }\limits^{\rightharpoonup}\) contains shearing and rotation components. Therefore, the vorticity vector is not strictly rotational [5, 11, 16].

3.3 Vorticity Magnitude in the Principal Coordinate

The vorticity magnitude in the Principal Coordinate is

$$\Vert \mathop{\omega }\limits^{\rightharpoonup} \Vert =\sqrt{{\left(\eta \right)}^{2}+{\left(\xi \right)}^{2}+{\left(R+\varepsilon \right)}^{2}}.$$
(11.9)

The vorticity magnitude contains rotation \(R\) and shearing components \(\eta\), \(\xi\), \(\varepsilon\); therefore, the vorticity magnitude does not only represent rotational strength [11].

4 Data Structure

The computational domain has the grid number 1920 \(\times 128\times 241,\) representing the number of grids in streamwise (x), spanwise (y), and wall-normal (z) directions. In normal direction, these grids are stretched, while in streamwise and spanwise directions, they are uniform. The length of the first grid interval in the normal direction at the entrance is 0.43 in wall units (Z+ = 0.43) (see Fig. 11.1).

Fig. 11.1
A 3 D rectangle diagram, with a length of L subscript x and a height of L subscript Z in, in the x, y, and z planes is above another rectangle with a width of L subscript y.

Computation domain

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The flow parameters are listed in Table 11.1. Here, \({M}_{\infty }\) is Mach number, \(Re\) is Reynolds number and, \({T}_{w}\,and\,{T}_{\infty }\) are wall and free stream temperature, respectively. Likewise, \({x}_{in}\) represents the distance between the leading edge and the inlet of the flat plate. \(Lx, Ly\) and \({Lz}_{in}\) are the lengths of computational domain in x, y, and z directions. \({\delta }_{in}\) is the inflow displacement thickness [1, 17].

Table 11.1 DNS parameters
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Figure 11.2 shows the formation of the vortex structures from the Y direction view in laminar, transitional, and turbulent flow. In laminar flow, little to no vortex activity is detected. In transitional flow, the formation of hairpin vortex rings begins, and this is where weak to strong vortex activity starts to be detected. Then in turbulent flow, many vortex rings have formed. This area is more chaotic and complicated. This means the strength of the vortex should increase from laminar flow to turbulent flow. According to Dong et al., the spanwise Y direction is the most prominent since it contributes the most to the value of the magnitudes [8].

Fig. 11.2
An illustration represents the formation of the vortex structures in Y versus X directions on a scale, ranging from 380 to 520, that starts with 4 vortex rings.

Vortex structures by modified Liutex-Omega with \({\stackrel{\sim}{{\varvec{\Omega}}}}_{{\varvec{L}}}=0.52\) at t = 13.00 T

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5 Results

There are two objectives of this study; by using the DNS simulation of boundary layer transition we:

  1. 1.

    Investigate the behavior of shear, Liutex, and Vorticity from laminar flow to turbulent flow.

  2. 2.

    Analyze the effect of shear on vorticity.

Statistical analysis is performed over the whole grid domain and across time domain T. The Statistical integration formula is

$$\sum_{i}\sum_{j}\sum_{k}{\tau }_{ijkt}{vol}_{ijk}={\iota }_{t},$$
(11.10)

where \({\tau }_{ijkt}\) is a point in the grid, i = 1 to 1920, j = 1 to 128, k = 1 to 241 and t represents the step-in time. \({vol}_{ijk}\) is the volume of the space around the point \({\tau }_{ijkt}\). \({\iota }_{t}\) is the integration output value at time step t.

The results of the Direct Numerical Simulation are recorded, Statistical Analysis is performed over the whole grid domain, and the data is plotted across time domain T for:

  • \({{\varvec{l}}}_{{\varvec{m}}{\varvec{a}}{\varvec{g}}}\) = Liutex magnitude component

  • \({{\varvec{\omega}}}_{{\varvec{m}}{\varvec{a}}{\varvec{g}}}\) = Vorticity magnitude component

  • \({{\varvec{s}}}_{{\varvec{m}}{\varvec{a}}{\varvec{g}}}\) = Shear magnitude component

  • \({{\varvec{l}}}_{{\varvec{y}}}\) = Liutex component in the y direction

  • \({{\varvec{\omega}}}_{{\varvec{y}}}\) = Vorticity component in the y direction

  • \({{\varvec{s}}}_{{\varvec{y}}}\) = Shear component in the y direction

The difference in the values of vorticity and Liutex is significantly high, so the relative values are used to compare the change in the values over period T.

Figures 11.3 and 11.4 show the behavior of vorticity, shear, and Liutex from the y direction. It is observed that the relative \({{\varvec{l}}}_{{\varvec{y}}}\) values increased significantly over period T, which is the T-S wave period. While \({{\varvec{\omega}}}_{{\varvec{y}}}\) showed no significant change throughout period T. The \({{\varvec{s}}}_{{\varvec{y}}}\) values were observed to be decreasing. This change is negative in nature and coincides with the increase in \({{\varvec{l}}}_{{\varvec{y}}}\) and the behavior of \({{\varvec{\omega}}}_{{\varvec{y}}}\). The period T travels from laminar flow to turbulent flow. There should be minuscule rotation or vortex activity in laminar flow and an increasing trend of vortex activity as we move into transitional flow, where hairpin vortex rings are formed, and on to turbulent flow. This behavior only coincides with the behavior of the \({{\varvec{l}}}_{{\varvec{y}}}\) relative values.

Fig. 11.3
A line graph of statistical integration versus period T plots an increasing curve for I subscript y and 2 horizontal merged lines for omega subscript y and s subscript y.

Relative integration values for \({{\varvec{\omega}}}_{{\varvec{y}}}\), \({{\varvec{l}}}_{{\varvec{y}}}\) & \({{\varvec{s}}}_{{\varvec{y}}}\) plotted over time domain T

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Fig. 11.4
3 line graphs for whole domains plot a curve in a linear trend of omega subscript y, an increasing curve of I subscript y, and a decreasing curve of s subscript y.

Relative integration values for \({{\varvec{\omega}}}_{{\varvec{y}}}\), \({{\varvec{l}}}_{{\varvec{y}}}\) & \({{\varvec{s}}}_{{\varvec{y}}}\) plotted over time domain T - individual graphs

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It can be observed from Figs. 11.5 and 11.6 that the relative change in \({{\varvec{l}}}_{{\varvec{m}}{\varvec{a}}{\varvec{g}}}\) across time is much greater than \({{\varvec{\omega}}}_{{\varvec{m}}{\varvec{a}}{\varvec{g}}}\). As we moved from laminar flow to turbulent flow in time, the values of \({{\varvec{l}}}_{{\varvec{m}}{\varvec{a}}{\varvec{g}}}\) continually increased, showing that \({{\varvec{l}}}_{{\varvec{m}}{\varvec{a}}{\varvec{g}}}\) picked up the formation of vortex rings, whereas \({{\varvec{\omega}}}_{{\varvec{m}}{\varvec{a}}{\varvec{g}}}\) barely changed.

Fig. 11.5
A line graph of statistical integration versus period T plots an increasing curve for I subscript mag and 2 horizontal lines with increasing trends toward the end for omega subscript mag and s subscript mag.

Relative integration values for \({{\varvec{\omega}}}_{{\varvec{m}}{\varvec{a}}{\varvec{g}}}\), \({{\varvec{l}}}_{{\varvec{m}}{\varvec{a}}{\varvec{g}}}\) & \({{\varvec{s}}}_{{\varvec{m}}{\varvec{a}}{\varvec{g}}}\) plotted over time domain T

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Fig. 11.6
3 line graphs for whole domains plot a curve in a linear trend of omega subscript mag, an increasing curve of I subscript mag, and an increasing curve of s subscript mag.

Relative integration values for \({{\varvec{\omega}}}_{{\varvec{m}}{\varvec{a}}{\varvec{g}}}\), \({{\varvec{l}}}_{{\varvec{m}}{\varvec{a}}{\varvec{g}}}\) & \({{\varvec{s}}}_{{\varvec{m}}{\varvec{a}}{\varvec{g}}}\) plotted over time domain T - individual graphs 

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6 Conclusion

The effect of shear on vorticity can be substantial, leading to a misrepresentation of vortex indication in the laminar flow where there is basically no rotation. The Y direction graphs show that vorticity had little to no change in value over time, while Liutex increased as time progressed. The shear was observed to be decreasing as Liutex increased. This shows that Liutex has a negative relation with Shear deformation. The magnitude graphs show that the increase in Liutex as time progressed is more significant than the increase in vorticity and shear. Therefore, vorticity should not be considered as Vortex.