Journal of Fluid Science and Technology

123456789 Bulletin of the JSME Vol.11, No.1, 2016 Journal of Fluid Science and Technology Relationships between local topology of vortical flow and pressure minimum feature derived from flow kinematics in isotropic homogeneous turbulence Katsuyuki NAKAYAMA, Yasumasa OHIRA and Hideki HASEGAWA Department of Mechanical Engineering, Aichi Institute of Technology Yakusa-cho, Toyota, Aichi 470-0392, JAPAN E-mail: nakayama@aitech.ac.jp Division of Mechanical Engineering, Graduate School of Engineering, Aichi Institute of Technology Yakusa-cho, Toyota, Aichi 470-0392, JAPAN Received 27 February 2015 Abstract Relationships between local flow topology (geometry) of vortical flow and pressure minimum feature derived from its flow kinematics are investigated in isotropic homogeneous turbulence, in a new topological point of view with respect to vortical flow symmetry in a swirl plane. Consistency of the pressure minimum plane and the swirl plane, and relationships between the pressure minimum and vortical flow symmetry in their development are also analyzed. The pressure minimum feature is specified by the λ 2 definition and also the ˇλ 2 definition that is the integrated definition of the, Q and λ 2 definitions and specifies the pressure minimum in the swirl plane. The swirlity ϕ that represents the geometrical average of the intensity of swirling flow is applied to the statistical analysis of the flow topology and pressure minimum. It shows that the pressure minimum requires a certain flow symmetry, and that the development or decay of the pressure minimum is associated with that of the vortical flow symmetry especially in the state (process) of a vortex attaining or losing the pressure minimum feature. The vortices with high intensity of ϕ have the feature that the effect of vorticity components parallel to the swirl plane decreases in specifying the pressure minimum plane by the λ 2 definition, and then this pressure minimum plane tends to approach the swirl plane. Key words : Vortical flow, Symmetry, Pressure minimum, Swirlity, Velocity gradient tensor 1. Introduction Vortices are associated with many fluid phenomena and engineering fields in various scales, such as fluid machinery, power plants, wind turbines, and bio-inspired flight objects. Elucidation of vortical flow phenomena in such fields is an important issue to control or suppress the vortices. An important characteristic of a vortex with swirling flow is its stability by its flow geometry (topology). The symmetry of the vortical flow is associated with the stability (Lundgren, 1982), and the vortical flow induces the pressure minimum feature. The pressure minimum supports an inflow motion that prevents the diffusion of the vorticity and induces vortex stretching (Tennekes and Lumley, 1972). Thus the relationships between the detail vortical flow geometry (symmetry) and the feature of the pressure minimum is of great interest and important in vortex dynamics and the flow control. In definition or identification of a vortex, many definitions have been proposed from various physical aspects of vortices (Chong et al., 1990,Hunt et al., 1988,Jeong and Hussain, 1995,Kida and Miura, 1998,Cucitore et al., 1999,Zhou et al., 1999, Chakraborty et al., 2005, Wu et al., 2005, Haller, 2005, Finn and Boghosian, 2006, Zhang and Choudhury, 2006, Kolář, 2007, Nakayama et al., 2007, Nakayama et al., 2014). Although no universal definition has been established, the (Chong et al., 1990), Q (Hunt et al., 1988), and λ 2 (Jeong and Hussain, 1995) definitions in local approach are Paper No.15-00139 1

popular definitions which are used frequently (Chakraborty et al., 2005). The definition specifies the invariant swirling motion with eigenvalues of the velocity gradient tensor u. The Q definition is associated with the positive second invariant Q of u, and specifies a vortex where the vorticity exceeds the irrotational straining, which is equivalent to the positive pressure Laplacian. The λ 2 definition ascertains the signs of the pressure Hessians discarding the unsteady straining and viscous effects, which identifies the local pressure minimum region induced by vortical (swirling) motion. Because the pressure minimum feature induced by the vortical or swirling flow should be specified in its swirl plane, a vortex definition, say ˇλ 2 definition, has been proposed to specify this feature (Nakayama et al., 2014). The ˇλ 2 definition is a unified definition that integrates these three definitions, which specifies the invariant vortical (swirling) flow and resulting pressure minimum in the swirl plane, including the vorticity exceeding the strain in the plane. As for identification of flow geometry, because the eigenvalues of u can specify the local flow geometry (pattern) which is Galilei invariant, they not only derived the -definition where = (Q/3) 3 + (R/2) 2 (> 0) and R denotes the third invariant of u, but also have contributed in the classification of the local flow and thus the analysis of turbulent flow (Chong et al., 1990,Blackburn et al., 1996). Although the essential physical interpretation of the complex eigenvalues was lacked and the eigenvalues were difficult to identify the detail flow geometry, a mathematical analysis of the local velocity with its decomposition into the radial and azimuthal velocities has derived their interpretation and several properties specifying the detail geometry (Nakayama, 2014). It has been shown that the imaginary part of the complex eigenvalues represents the geometrical average of intensity of the azimuthal flow, and the real part does the arithmetic average of that of the radial flow. Consequently the properties to specify the symmetry of vortical flow have been derived from u, and one symmetry property is associated with the pressure minimum feature in the swirl plane specified by the ˇλ 2 definition (Nakayama et al., 2014). Then this definition and the symmetry property of the vortical flow enable to examine relationships between the flow geometry and the pressure minimum. The present paper investigates the relationships between the vortical flow geometry and the pressure minimum feature, in local flow kinematics. Vortices in isotropic homogeneous turbulence are our subject as a study of primary characteristics of a vortex. The pressure minimum is specified by the criteria in the λ 2 and ˇλ 2 definitions that are associated with the pressure minimum plane and pressure feature in the swirl plane induced by the vortical flow. The relation of these pressure minimums to the symmetry of vortical flow, and relationships between their development/decay are analyzed. The swirlity ϕ (Nakayama, 2014) representing the intensity of swirling flow of a vortex is used as a parameter to examine the features of the flow geometry and pressure minimum. It shows that the pressure minimum is associated with the symmetry of the vortical flow especially in the state or process of a vortex attaining or losing the pressure minimum, in which the pressure minimum feature and symmetry tend to develop or decay each other. The pressure minimum plane specified by the λ 2 definition differs from the swirl plane in general. However, vortices with high ϕ have the feature that these two planes approach. This characteristic of the vortices indicates that the vorticity components parallel to the swirl plane decrease, and the pressure minimum plane specified by the λ 2 definition approaches the swirl plane. 2. Representation of Flow topology and pressure minimum We summarize the symmetry property of the local vortical flow geometry and swirlity (Nakayama, 2014), and the λ 2 and ˇλ 2 definitions with the representation of u in a coordinate system associated with the swirl plane. We consider a local flow geometry around a point in an instantaneous velocity field. In the reference frame with respect to the point, the local velocity v i (i = 1, 2, 3) around the point can be expressed as: v i = dx i /dt = ( v i / x j )x j, where the summation convention is applied and v i / x i = 0 in incompressible fluids. Then the invariant local flow geometry is specified by the eigenvalues and eigenvectors of u (= [ v i / x j ]) (Chong et al., 1990). We assume that u has a pair of complex conjugate eigenvalues ε R ± iψ (ψ > 0) and their eigenvectors ξ pl ± iη pl, where i is the imaginary unit, and a real eigenvalue ε a and its eigenvector ξ axis. Then the flow trajectory can be represented as 2e ε Rt (ξ pl cos ψt η pl sin ψt) + e ε at ξ axis, where the flow swirls in the plane defined by ξ pl and η pl (Chong et al., 1990, Zhou et al., 1999), hereafter referred to as the swirl plane P. Swirlity ϕ represents the geometrical average of the intensity of the azimuthal velocity component in v i, i.e., the eigenvalues of the specific quadratic forms of the azimuthal velocity, and is specified as ϕ = Q + 3ε 2 a/4. The condition that 0 < ϕ is equivalent to the criterion that u has complex conjugate eigenvalues, i.e., > 0 (Chong et al., 1990), and ϕ = ψ in case of the vortical flow, i.e., 0 < ϕ (Nakayama, 2014). We note that both ξ pl and η pl are the degenerate eigenvectors of the matrix (A ε R E) 2 corresponding to the eigenvalue ψ 2, where A is a matrix of u in a coordinate system, i.e., A = [a i j ] = [ v i / x j ] (i, j = 1, 2, 3). The directions of ξ pl and η pl, and the relative length of these vectors, i.e., the ratio c = ξ pl / η pl, are specified by the eigenequations of A in terms 2

Fig. 1 flow geometries in swirl plane, with same complex eigenvalues (ε R = 0.5, ψ = 2) and different c. of the complex eigenvalues; A(ξ pl ± iη pl ) = (ε R ± iψ)(ξ pl ± iη pl ). Then we can specify that ξ pl and η pl are orthogonal, i.e., ξ pl η pl. The vortical flow geometry in P specified by the complex eigenvalues and eigenvectors of u, i.e., 2e ε Rt (ξ pl cos ψt η pl sin ψt), changes according to c. The geometry is elliptic in high c, and this symmetry increases as c approaches 1, as shown in Fig. 1. Then c is necessary to specify the geometry uniquely, and considered an invariant property of the symmetry of the swirling motion. c is also associated with the flow symmetry of the radial and azimuthal velocities in the plane, thus it represents the symmetry of whole vortical flow (Nakayama, 2014). We consider an orthonormal coordinate system ˇx i (i = 1, 2, 3) with bases ě i (i = 1, 2, 3) where ě 1 and ě 2 are parallel to ξ pl and η pl, respectively. Importantly, the ˇx 1 - ˇx 2 plane is P and the invariant subspace of u. u in this coordinate system, Ǎ = [ǎ i j ], can be expressed in the form Ǎ = ε R cψ ǎ 13 ψ/c ε R ǎ 23 0 0 ε a. It is noted that ǎ 12 ǎ 21 except when ξ pl = η pl (c = 1) and that ǎ 13 = ǎ 23 = 0 when ξ axis P. c specifies the symmetry of the rotational components of u in P. In the ˇx i coordinate system, we consider the formulation of the λ 2 and ˇλ 2 definitions with the above representation of u. The second derivative of the pressure obtained by differentiating the Navier-Stokes equation, neglecting the unsteady and viscous strain terms yields (1) 1 ρ p,i j = s iks k j + ω ik ω k j, (2) where the subscript after the comma denotes the derivative in the directions of indices, and ρ and p denote the density and the pressure discarding the above-mentioned terms, respectively (Jeong and Hussain, 1995). s i j and ω i j denote the rate-of-strain tensor and vorticity tensor, respectively. If two of the eigenvalues λ i (i = 1, 2, 3) of p,i j /ρ are negative, i.e. λ 2 < 0 (here we set λ 1 λ 2 λ 3 ), then p has a local minimum characteristic, which yields the λ 2 definition. The ˇλ 2 definition focuses on the pressure minimum of p in P. The Hesse matrix of p, H = [h i j ] = [ p,i j /ρ] (i, j = 1, 2, 3), can be expressed as H = (AA + t A t A)/2, where the superscript t denotes the transpose of the matrix. Then, from Eq. (1), H in the ˇx i coordinate system, Ȟ, is derived as follows: ε 2 R ψ2 (c 1/c)ε R ψ ι 1 Ȟ = (c 1/c)ε R ψ ε 2 R ψ2 ι 2, (3) ι 1 ι 2 ε 2 a where ι 1 = ( ε R ǎ 13 + cψǎ 23 )/2 and ι 2 = ( ψǎ 13 /c ε R ǎ 23 )/2. The block matrix of Ȟ corresponding to P is expressed as Ȟ pl = [ȟ i j ] (i, j = 1, 2), which specifies the characteristics of p in P. Then the eigenvalues of Ȟ pl, ˇλ i (i = 1, 2; ˇλ 1 ˇλ 2 ), are given as: ˇλ 1, ˇλ 2 = (ε 2 R ψ2 ) (c 1 c )ε R ψ. (4) Because Ȟ pl is symmetric, ˇλ i (i = 1, 2) have the same sign if Ȟ pl > 0. The two conditions that Ȟ pl > 0 and ˇλ 1, ˇλ 2 < 0 can be integrated by the inequality with respect to ψ: c ε R < ψ, (5) 23

where c = c (1 c) or 1/c (c < 1). Then Eq. (5) is the condition that ˇλ 2 < 0, which represents the local minimum feature of p in P. Here we rotate the bases ě i (i = 1, 2) in P along ě 3 so that they coincide with the two orthonormal eigenvectors ˇζ i of ˇλ i (i = 1, 2), respectively. H in this orthonormal coordinate system ˇx i (i = 1, 2, 3), Ȟ is expressed as: ˇλ 1 0 ι 1 Ȟ = 0 ˇλ 2 ι 2 ι 1 ι 2 ε 2 a, where ι i (i = 1, 2) denotes the inner product of ˇζ i and ι (ι 1, ι 2 ), i.e., ι i = ˇζ i ι. Then p in P, ˇp, is expressed as ˇp/ρ = {ˇλ 1 ( ˇx 1 )2 + ˇλ 2 ( ˇx 2 )2 }/2. We examine the relationship among the λ 2 and ˇλ 2 definitions. Because p,ii /ρ = a i j a ji = Tr(H) (Jeong and Hussain, 1995), p,ii ( p) has the following relation: 1 ρ p = 3 i=1 λ i = 3 i=1 ˇλ i, where ˇλ 3 ε 2 a. The eigenequation of H, Φ λ2 (λ), can be expressed in terms of Ȟ in Eq. (6) as follows: Φ λ2 (λ) = Ψ(λ) ι 2 2 (ˇλ 1 λ) ι 2 1 (ˇλ 2 λ), (8) where Ψ(λ) (ˇλ 1 λ)(ˇλ 2 λ)(ˇλ 3 λ). Equation (8) is influenced by ι (ι 1, ι 2 ) which does not associate with P, and indicates Φ λ2 (ˇλ 1 ) < 0 and Φ λ2 (ˇλ 2 ), Φ λ2 (ˇλ 3 ) > 0. Then λ i ˇλ i (i = 1, 2) and ˇλ 3 λ 3, which indicates that the ˇλ 2 definition satisfies the λ 2 definition (λ 2 < 0) (Nakayama et al., 2014). ι is composed of ǎ 13 and ǎ 23 that are associated with the vorticity components parallel to P, i.e., ˇω 1 = ǎ 23 and ˇω 2 = ǎ 13, where ˇω i denotes the vorticity vector component in the ˇx i coordinate system. It indicates an important characteristic that λ i is influenced by ˇω 1 and ˇω 2. If ξ axis P, then ι = ι = o, i.e., ˇω 1 = ˇω 2 = 0. In this case, Φ λ2 (λ) is identical to Ψ(λ), which yields λ i = ˇλ i (i = 1, 2, 3). The difference between the λ 2 and ˇλ 2 definitions is the plane where the pressure minimum is specified. The ˇλ 2 definition ascertains the pressure minimum feature in P, on the other hand, λ 2 does the existence of a plane where the pressure has the minimum feature. This difference yields the characteristics of the two definitions that the ˇλ 2 definition is more severe and satisfies the λ 2 definition. 3. Numerical analysis We analyze the vortices in an isotropic homogeneous decaying turbulence by the pseudo-spectral method in the region (2π) 3, composed of 256 3 nodes (Jiménez et al., 1993,Kida and Miura, 1998,Ishihara et al., 2007). For the wavenumber vector k = (k 1, k 2, k 3 ), k < 121 where k = (k i k i ) 1/2, the phase shifting method is used for dealiasing. The time step is 0.001 in the fourth order Runge-Kutta method. An energy spectrum E(k) = (k/k p ) 4 exp{ 2(k/k p ) 2 } (k = k, k p = 4) (Kida and Miura, 1998) gives an initial velocity field with random phases of k, where the Taylor Reynolds number Re λ = 311, Taylor microscale λ T = 0.59, Kolmogorov length η = 0.015, and eddy turnover time T eddy = 1.12. The kinematic viscosity is 0.002. Hereafter ϕ, λ 2 and ˇλ 2 are nondimensionalized by their root mean square values in the corresponding time, and c is expressed as c (0 < c 1). Figure 2 shows contours of ˇλ 2 and c in vortical regions conditioned on 0 < ϕ, in an instantaneous velocity field where Re λ = 35. The regions occupied only by the contours of c (c = 0.75, 0.85) where ˇλ 2 contours (ˇλ 2 = 3, 2) do not overlap with are vortical regions where negative ˇλ 2 (pressure minimum) is not as high as those of the contours. Figure 3 shows a zoomed vortical region indicated by the arrow in Fig. 2, where the contours of λ 2 = 3, 2 are also shown. In these vortical regions, both symmetry and pressure minimum are higher in the core region. The contours of λ 2, ˇλ 2 = 2 cover those of λ 2, ˇλ 2 = 3 respectively, and of c = 0.75, 0.85. Then the contours of λ 2, ˇλ 2 = 3 and c = 0.75 overlap in part, and the contour of c = 0.85 is almost inside those of λ 2, ˇλ 2 = 3. Figure 4 shows characteristics of joint probability density functions (JPDFs) of (c, λ 2 ) and (c, ˇλ 2 ) in the vortical region (0 < ϕ) where Re λ = 35, 50 and 104, respectively. It shows that λ 2 and ˇλ 2 have wide ranges in their values for an arbitrary value of c. However, both λ 2 and ˇλ 2 require high c in the high pressure minimum. These characteristics are similar irrespective of Re λ described above, although the distribution of the JPDF becomes diffusive and has a higher correlation coefficient cor in Re λ = 35. As for the development or decay of the pressure minimum, Fig. 5 shows JPDFs of c/ t and λ 2 / t as well as ˇλ 2 / t, where Re λ = 35, 50 and 104, and the time t is nondimensionalized by Kolmogorov time at the moment. cor between c/ t (6) (7) 24

Nakayama, Ohira and Hasegawa, Journal of Fluid Science and Technology, Vol.11, No.1 (2016) Fig. 2 Contours of λ 2 and c where c = 0.75, 0.85 and λ 2 = 2, 3 in an instantaneous velocity field where Reλ = 35. Fig. 3 Contours of c = 0.75, 0.85 and λ2, λ 2 = 3, 2 in a zoomed vortical region indicated by the arrow in Fig. 2. (a) λ2 and (b) λ 2. and λ2 / t or λ 2 / t is low in the whole vortical region. We focus on their relationships in the vortical region with low λ2 and λ 2, where the probability density seems to be concentrated as shown in Fig. 4. Figure 6 shows JPDFs of (ϕ, λ2 ) and (ϕ, λ 2 ) in the vortical region (0 < ϕ) where Reλ = 35. The distributions of the JPDFs have a shape of a quadratic curve, and, in the JPDF of (ϕ, λ 2 ), the region where ϕ 1 is the most concentrated region. Thus we extract the vortical region in terms of 0.7 < ϕ < 1.2 where the probability is approximately 0.24 (24%) and the rate of vortices satisfying λ 2 < 0 is approximately 61%. Figure 7 shows JPDFs of ( c/ t, λ2 / t) and ( c/ t, λ 2 / t) in this region. Then a feature is clearly exhibited that both λ2 and λ 2 develop or decay with c. 4. Discussion 4.1. Pressure minimum feature and flow symmetry In Fig. 4, the very weak vortices that do not have the pressure minimum feature, i.e., 0 < λ2, λ 2, have a wide range of the symmetry c. However, as positive λ2 and λ 2 is higher, the range of c becomes narrow and c is lower. Thus vortices with high positive λ2 or λ 2 have a tendency to have low symmetry. In the comparison of the regions where 0 < λ2 and 0 < λ 2, the 0 < λ 2 region (area) is larger, which indicates that vortices where 0 < λ 2 may have a pressure minimum plane except P, i.e., λ2 < 0. Even though a vortex does not have the pressure minimum feature in P, it may have that feature in the other plane, i.e., the eigenplane specified by the λ2 definition, by the effect of the vorticity components parallel to P. Then some vortices that satisfy the λ2 definition do not satisfy the λ 2 definition. As λ2 or λ 2 becomes negative and the vortices have the pressure minimum feature, vortices have a certain symmetry and seem to have a lower limit of c according to λ2 or λ 2. Furthermore, if the vortices have higher pressure minimum (higher negative λ2 or λ 2 ), then the symmetry increases. This reason is associated with the relationship between ϕ and c, as shown in Fig. 8 that is a JPDF of ϕ and c where Reλ = 35. In Fig. 8, ϕ and c have a high correlation, and high ϕ requires 25

Nakayama, Ohira and Hasegawa, Journal of Fluid Science and Technology, Vol.11, No.1 (2016) Fig. 4 JPDFs of (c, λ2 ) ((a), (b) and (c)) and (c, λ 2 ) ((d), (e) and (f)) in terms of 0 < ϕ, where Reλ = 35, 50, 104. (note: cor denotes the correlation coefficient.) high c to some extent. Because the high pressure minimum is derived from the high intensity of the swirling motion, i.e., high ϕ, higher pressure minimum needs higher c. In high pressure minimum vortices, because λ2 and λ 2 have a feature that λ2 < λ 2 (< 0), λ2 may seem to indicate higher pressure minimum than λ 2. However, their features of the distributions in JPDFs of (c, λ2 ) and (c, λ 2 ) are similar in these regions. We examine the difference between the pressure minimum plane Λ specified by the λ2 definition and P. Because the normal vector of Λ can be expressed by the eigenvector of λ3, say ζ 3, the angle θ between ζ 3 and the normal vector of P denoted as e 3 ξ pl η pl are analyzed in terms of λ2 and λ 2. Figure 9 shows JPDFs of (θ, λ2 ) and (θ, λ 2 ). It indicates that θ approaches 0, i.e., Λ approaches P, as λ2 and λ 2 become high negative. Then the vorticity components parallel to P become less in the high pressure minimum vortices. This feature is the reason that λ2 and λ 2 have a similar feature in their high negative regions. 4.2. Relationships between flow symmetry and pressure minimum in dynamics Figure 5 shows the feature that, in distributions of the JPDFs of ( c/ t, λ2 / t) and ( c/ t, λ 2 / t), high concentrated regions have downward (negative) slope shapes, although each cor is low. This feature is exhibited clearer in Fig. 7, where λ2 and λ 2 are low. This reason is also associated with the relationship between ϕ and c, that is, high ϕ requires high c. In 26

Nakayama, Ohira and Hasegawa, Journal of Fluid Science and Technology, Vol.11, No.1 (2016) Fig. 5 JPDFs of ( c/ t, λ2 / t) ((a), (b) and (c)) and ( c/ t, λ 2 / t ) ((d), (e) and (f)) in terms of 0 < ϕ, where Reλ = 35, 50, 104. Fig. 6 Fig. 7 JPDFs of (a) (ϕ, λ2 ) and (b) (ϕ, λ 2 ) where Reλ = 35. JPDFs of (a) ( c/ t, λ2 / t) and (b) ( c/ t, λ 2 / t ) in terms of 0.7 < ϕ < 1.2, where Reλ = 35 (cor: (a) 0.631 and (b) 0.648). 27

Fig. 8 JPDFs of (c, ϕ) (Re λ = 35, cor = 0.838). Fig. 9 JPDFs of (a) (θ, λ 2 ) and (b) (θ, ˇλ 2 ) (Re λ = 35). addition, Fig. 8 shows that such tendency between ϕ and c is prominent in low ϕ region. Figure 10 shows the correlation coefficients of ( c/ t, λ 2 / t) and ( c/ t, ˇλ 2 / t) in terms of ϕ < 0.4, 0.8, and 1.5 < ϕ, in Re λ = 35. It is clear that the low ϕ region has high correlation with respect to ( c/ t, λ 2 / t) and ( c/ t, ˇλ 2 / t), whereas high ϕ region has lower correlation. In particular, the most concentrated probability density region in JPDF of (ϕ, λ 2 ) or (ϕ, ˇλ 2 ) in Fig. 6 has higher correlation as shown in Fig. 7, where vortices with/without the pressure minimum feature are mixed. Thus, in the state or process of a vortex attaining or losing the pressure minimum feature, c and λ 2 or ˇλ 2 have a tendency to codevelop or codecay. In Fig. 10, each range of ϕ has a higher cor than that in whole vortical region. Thus the reason of the lower cor in the whole vortical region can be derived from the different characteristics of the relationships of ( c/ t, λ 2 / t) or ( c/ t, ˇλ 2 / t) in terms of the range of λ 2 or ˇλ 2. On the other hand, it is clear that ˇλ 2 has the higher correlation with c than λ 2 in Figs. 5, 7, and 10, because the pressure minimum in P, not in other plane searched by the λ 2 definition, is directly associated with c. 5. Conclusion The relationships between the vortical flow symmetry and the pressure minimum feature in terms of the flow kinematics specified by the λ 2 and ˇλ 2 definitions were investigated in the vortices in isotropic homogeneous decaying turbulence. 28

Fig. 10 Correlation coefficients of ( c/ t, λ 2 / t) and ( c/ t, ˇλ 2 / t) in terms of ϕ < 0.4, ϕ < 0.8, and 1.5 < ϕ (Re λ = 35). The pressure minimum feature requires a certain flow symmetry, and the symmetry increases according to the degree of the pressure minimum. The development or decay of the pressure minimum is associated with that of the vortical flow symmetry especially in the state or process of attaining or losing the pressure minimum feature. The pressure minimum plane specified by the λ 2 definition approaches the swirl plane as the swirlity increases. It yields a characteristic that the λ 2 definition tends to be similar to the ˇλ 2 definition. References Blackburn, H. M., Mansour, N. N., and Cantwell, B. J., Topology of fine-scale motions in turbulent channel flow, J. Fluid Mech., Vol.310 (1996), pp.269-292. Chong, M. S., Perry, A. E., and B. J. Cantwell, A general classification of three-dimensional flow fields, Phys. Fluids, Vol.A2 No.5 (1990), pp.765-777. Cucitore, R., Quadrio, M., and Baron, A., On the effectiveness and limitations of local criteria for the identification of a vortex, E. J. Mech. B/Fluids Vol.18, No.2 (1999), pp.261-282. Chakraborty, P., Balachandar, S., and Adrian, R. J., On the relationships between local vortex identification schemes, J. Fluid Mech. Vol.535 (2005), No 189-214. Finn, L. I., and Boghosian, B. M., A global variational approach to vortex core identification, Physica A Vol. 362 (2006), pp.11-16. Haller, G., An objective definition of a vortex, J. Fluid Mech. Vol.525 (2005), pp.1-26. Hunt, J. C. R., Wray, A. A., and Moin, P., Eddies, streams, and convergence zones in turbulent flows, Center for Turbulence Research CTR-S88 (1988), pp.193 208. Ishihara, T., Kaneda, Y., Yokokawa, M., Itakura, K. and Uno. A., Small-scale statistics in high-resolution direct numerical simulation of turbulence: Reynolds number dependence of one-point velocity gradient statistics, J. Fluid Mech., Vol.592 (2007), pp.335 366. Jeong, J., and Hussain, F., On the identification of a vortex, J. Fluid Mech. Vol.285 (1995), pp.69-94. Jiménez, J., Moin, P., Moser, R., and Keefe, L., Ejection mechanisms in the sublayer of a turbulent channel, Phys. Fluids Vol.31, No.6 (1988), pp.1311-1313. Jiménez, J., Wray, A. A., Saffman, P. G., and Rogallo, R. S., The structure of intense vorticity in isotropic turbulence, J. Fluid Mech. Vol.255 (1993), pp.65-90. Kida, S., and Miura, H., Identification and analysis of vortical structures, E. J. Mech. B/Fluids Vol.17 No.4 (1998), pp.471-488. Kolář, V., Vortex identification: new requirements and limitations, Int. J. Heat Fluid Flow, Vol.28 (2007), pp.638-652. Lundgren, T. S., Strained spiral vortex model for turbulent fine structure, Phys. Fluids, Vol.25 No.12 (1982), pp.2193-2203. Nakayama, K., Physical properties corresponding to vortical flow geometry, Fluid Dyn. Res., Vol.46 (2014), 055502. Nakayama, K., Sugiyama, K., and Takagi, S., A unified definition of a vortex derived from vortical flow and the resulting pressure minimum, Fluid Dyn. Res. Vol.46 (2014), 055511. Nakayama, K., Umeda, K., Ichikawa, T., Nishihara, Y., and Takagi, S., Definition of swirl function and its application to identification method of swirling motion, Transactions of the Japan Society of Mechanical Engineers, Series B, Vol.73, No.725 (2007), pp.22-29 (in Japanese). 29

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