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Helicity and geometric nature of particle trajectories in homogeneous isotropic turbulence
International Journal of Heat and Fluid Flow 31 (2010) 482–487
Contents lists available at ScienceDirect
International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
Helicity and geometric nature of particle trajectories in homogeneous isotropic turbulence Yeontaek Choi a, Yongnam Park b, Changhoon Lee b,c,* a
Division of Computational Sciences in Mathematics, National Institute for Mathematical Science, Daejeon 305-340, South Korea School of Mechanical Engineering, Yonsei University, Seoul 120-749, South Korea c Department of Computational Science and Engineering, Yonsei University, Seoul 120-749, South Korea b
a r t i c l e
i n f o
Article history: Received 9 October 2009 Received in revised form 27 November 2009 Accepted 2 December 2009 Available online 6 January 2010 Keywords: Lagrangian helicity Acceleration Curvature Torsion Particle trajectory
a b s t r a c t The role of helicity in Lagrangian turbulence is investigated using direct numerical simulation of homogeneous isotropic turbulence. Probability density functions and autocorrelations along a fluid particle trajectory associated with geometric quantities such as curvature and torsion of the Lagrangian trajectory are provided. The relationship between helicity and the ratio of torsion to curvature is investigated. We found that probability density functions of torsion and torsion normalized by curvature clearly show well-established slope in log–log plots. Finally, the relationship between helicity, acceleration and geometric quantities are discussed. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Intermittency of helicity, which is the inner product of velocity and vorticity vectors, in fluid turbulence is an unresolved phenomenon (Moffat and Tsinober, 1992). In a recent paper (Choi et al., 2009), it was observed that intermittency of helicity usually results from fluctuations, particularly rapidly rotating vortical structures. Furthermore, more prominent alignment between velocity and vorticity is found in the core of a coherent vortical structure. Therefore, the role of helicity in Lagrangian turbulence is expected to be intimately related to rotating structures as in Eulerian turbulence. In order to verify this, we investigate the geometric shape of the trajectory of a Lagrangian fluid particle in turbulent flows, because changes in trajectory are caused by background flow motions such as trapping and penetration. Similar observations have been presented in previous studies (Braun et al., 2006; Xu et al., 2007 and Scagliarini, 2009). However, the role of helicity, which measures the spiral motion of a fluid, in governing particle trajectory has not been previously investigated. Although Lagrangian helicity provides information on the spiral motions of fluids, knowledge of helicity does not directly lead to geometric interpretations of fluid particle trajectory. In this study, we demonstrate the relevance of Lagrangian helicity * Corresponding author. Address: Department of Computational Science and Engineering, Yonsei University, Seoul 120-749, South Korea. E-mail address: [email protected] (C. Lee). 0142-727X/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.ijheatfluidflow.2009.12.003
and behaviors to particle trajectory. In order to investigate this, we introduce the ratio of torsion to curvature of fluid particle trajectories. We use direct numerical simulations of homogeneous isotropic turbulence to calculate quantities in this paper. In order to simulate turbulent flow, the Navier–Stokes equations and continuity equation are solved using a spectral method for spatial discretization and the third-order Runge–Kutta scheme for time advance3 ment in a ½ð2pÞ cubic domain with periodicity conditions enforced in all three directions. Most calculations are carried out at 643 and 1283 resolutions. For the maintenance of stationarity, we use the forcing scheme proposed by Eswaran and Pope (1988) in which artificially forced low-wavenumber velocity components ðjkj < kf Þ generate statistically stationary turbulence fields. A four-point Hermite interpolation (Choi et al., 2004) is also used for the interpolation necessary for fluid particle tracking. The Reynolds number, based on the Taylor scale, Rk , and other simulation parameters are listed in Table 1. The remainder of the paper is organized as follows. In Section 2, definitions and fundamental concepts are introduced. Lagrangian helicity is defined and the geometric meaning of the ratio of torsion to curvature is given. In Section 3, we present observations to verify the relevance of helicity and geometric shapes to particle trajectory. First, we provide two statistics: probability distribution functions (PDF) and autocorrelations of velocity, vorticity, acceleration, and helicity. Next, we introduce statistics and correlations of helicity and changes of geometric shape of a fluid particle, with a
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Y. Choi et al. / International Journal of Heat and Fluid Flow 31 (2010) 482–487 Table 1 Simulation parameters. m; kmax ; g; T fL ; Dt and sg are viscosity, maximum wavenumber, Kolmogorov length scale, forcing time scale, time step and Kolmogorov time scale, respectively. N
Rk
m
kmax ; g
kf
T fL
Dt
pffiffiffi 2 2 2.514
0.4312
1=56sg
0.4312
1/80sg
64
47
0.03
1.24
128
70
0.015
1.5
special focus on the ratio of torsion to curvature. Then statistics associated with acceleration are provided to investigate the relationship between geometry of the trajectory of a particle and the structure of the background flow. Finally, the conclusion is given in Section 4. 2. Definitions and basic concepts 2.1. Lagrangian helicity of a fluid particle Lagrangian helicity is defined as
HL ¼ uðxðtÞ; tÞ xðxðtÞ; tÞ;
ð1Þ
where time t 2 ½0; T; xðtÞ is the position of a fluid particle, uðxðtÞ; tÞ is the velocity of a fluid particle, and xðxðtÞ; tÞ is the vorticity of a fluid particle at time, t. Lagrangian variation of helicity of a fluid particle is calculated along the trajectory of a fluid particle, and indicates instantaneous interactions between velocity and vorticity vectors along the trajectory. Definition (1) corresponds to helicity density in Eulerian fields, which is the integrand of the total helicity R HE ¼ V v ðtÞ xðtÞdV (Choi et al., 2009), where v ðtÞ and xðtÞ are velocity and vorticity in an Eulerian field, respectively. 2.2. PDFs and autocorrelations We introduce the Lagrangian distributions of various quantities manifested along particle trajectories such as velocity, vorticity, acceleration, and helicity of a fluid particle. One- and two-particle Lagrangian acceleration correlations (Yeung, 1997) were investigated and heavy particle behaviors were also studied in Jung et al. (2008). We investigate autocorrelations for velocity u, vorticity x, acceleration a, and Lagrangian helicity HL of a fluid particle along its trajectory. The definition of an autocorrelation for variable x is as follows:
qxðtÞ ¼
hxðt0 Þxðt 0 þ tÞi : hx2 ðt 0 Þi
ð2Þ
2.3. Curvature and torsion of a Lagrangian trajectory It is natural to adopt torsion as well as curvature in to depict a particle trajectory in three dimensional space. The trajectory of a Lagrangian particle determined by dxðtÞ=dt ¼ uðxðtÞ; tÞ can be described in terms of a Frenet–Serret apparatus fjðsÞ; sðsÞ; TðsÞ; NðsÞ; BðsÞg, where s is a parameter of a curve, jðsÞ is curvature, sðsÞ is torsion, TðsÞ is a tangent vector, NðsÞ is a normal vector, and BðsÞ is a binormal vector (Millman and Parker, 1977). The Frenet apparatus has no dynamical information and contains pure geometric information. Reparametrizing the parameter s as time t, curvature and torsion for a curve that has non-unit speed vector, jdxðtÞ=dtj – 1, are written as follows:
a? ¼ 2; u juj3 _ u ða aÞ Torsion s ¼ ; u6 j2 Curvature
j¼
ju aj
where a? ; u and a_ are components of acceleration perpendicular to the velocity vector, the magnitude of velocity and material derivative of acceleration vector, respectively. There are several corollaries to characterize the properties of curvature and torsion for j – 0 (Millman and Parker, 1977):
j¼0 j – 0; s¼0 s ¼ 0; j ¼ constant > 0 s ¼ constant – 0; j ¼ constant > 0 s j
¼ constant
straight line plane curve circle circular helix helix
Here, a helix is defined as a regular curve such that for some fixed unit vector A, TA is constant. T is a unit tangent vector as defined above and A is the axis of the helix. In this context, for a fluid particle trajectory around a vortical structure, TA corresponds to the relative helicity which is defined by u=juj x=jxj ¼ cos h, where h is the angle between velocity and vorticity vectors. The theorem to characterize the helix was proven by Lancert in 1802. The unit speed curve aðsÞ; jdaðsÞ=dsj ¼ 1, with j – 0 is a helix if and only if there is a constant c such that s ¼ cj. If T A cos h, this leads to s=j ¼ cot h. We should, however, be cautious in dealing with Lancert’s theorem, for it is invalid for j ¼ 0 and jdaðsÞ=dsj – 1. The consequence of Lancert’s theorem is that the ratio s=j is explained by the angle between the tangent vector T which is parallel to a velocity vector, and the axis of helix A which is parallel to vorticity for a trajectory around a vortical structure. When Ak T; s=j ¼ 1, and when A ? T; c ¼ 0. Near a helix-like vortical structure, an investigation of s=j can reveal alignment trends between velocity and vorticity vectors. A circular helix is a special case. Circular refers to the fact that the projection in the ðx; yÞ plane is a circle. For example: a : R ! R3 be given by aðsÞ ¼ ðr cos s; r sin s; hsÞ, where h > 0 and r > 0. We expect that the ratio of torsion to curvature plays a crucial role in understanding the geometric shape of particle trajectory. In random flow, a trajectory is not a helix. 3. Lagrangian statistics 3.1. Lagrangian fluid particle trajectory Fig. 1 presents the sample trajectories of two Lagrangian particles at Rk ¼ 47. In this figure, the particles are shown to wrap around a vortical structure. Vorticity of the particles remains in the same direction as the axis of the vortical structure while accelerations are oriented toward the center of the vortical structure, clearly indicating that the vortical structure is responsible for converging distributions of acceleration. Investigations of torsion along the trajectories indicates that fluid particles closer to the vortical structure have much stronger torsion. Relative helicity, which is the cosine angle between velocity and vorticity vectors, is also closer to 1 near the vortical structure, implying that near the core of vortical structures, velocity and vorticity vectors tend to align with each other, consistent with our previous finding (Choi et al., 2009). Behaviors of acceleration, helicity and the geometric nature of trajectories around a vortical structure clearly suggest strong correlations between those quantities, particularly near vortical structures. In the rest of this paper, we suggest relationship between helicity, acceleration and particle trajectory shapes as depicted by curvature and torsion.
ð3Þ
3.2. Probability density functions and autocorrelations
ð4Þ
Lagrangian statistics were observed either numerically or experimentally in several previous papers (Vedula and Yeung,
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Y. Choi et al. / International Journal of Heat and Fluid Flow 31 (2010) 482–487
1X
X X XX XX XX XX
0.6
ρt
ux ax ωx HL
XX
0.8
X XX
XX
0.4
XX
XX
XX
XX
XX
0.2
XX
XX
XX
XX XX XX
0 0 Fig. 1. (a) Trajectories of two sample Lagrangian particles moving around a vortical structure (green structure). Red vectors represent vorticity and black vectors represent acceleration. Color of particles represents the magnitude of torsion, (b) the same trajectories viewed from a different angle with distribution of the magnitude of vorticity and (c) the color of particles denotes the magnitudes of relative helicity, i.e., cosine values of the angle between velocity and vorticity vectors. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
1999; Mordant et al., 2004). Here, we observed statistics of several Lagrangian quantities for a fluid particle moving along its trajectory, such as velocity, vorticity, acceleration and helicity (Fig. 2). The PDF of velocity, whose distribution is nearly Gaussian, is compared to PDFs of acceleration, helicity and vorticity, which show wider tails. The statistics are also compared to the PDF of the ratio s=j. Compared to other variables, s=j shows significantly intermittent distribution. Local values of that are s=j 1000 times larger than its mean value are frequently observed. Fig. 3 presents autocorrelations of Lagrangian velocity, vorticity, acceleration and helicity in the cases of two different Reynolds numbers Rk ¼ 47 and 70. The Lagrangian trajectory for each fluid particle is tracked by employing the 3rd order Runge–Kutta scheme, and flow quantities at each time are obtained by a 4th order Hermite interpolation scheme. In order to obtain a fully converged distribution of the correlation function, data are collected
10
-2
-3
10
1.5 t / TL
2
2.5
in the period of 270T L and 54T L for Rek ¼ 47 and 70, respectively, where T L is the velocity integral time scale. The number of particles used in the calculation are 3 108 and 4:5 107 for Rek ¼ 47 and 70, respectively. Helicity becomes decorrelated faster than velocity and vorticity, as shown in Fig. 3. Sensitivity to the Reynolds number is found to be relatively weak. 3.3. Statistics of curvature and torsion Braun et al. (2006) numerically observed the exponential behavior for PDFs of curvature, i.e., the 2/5 exponent of PDF for high values of curvature, in their simulation of a turbulent flow. In Fig. 4, PDFs for j; s and s=j obtained from our simulations are illustrated to clearly show the 3 exponent for high values of torsion and the 2 exponent for s=j. Xu et al. (2007) experimentally measured the curvature of particles moving along Lagrangian trajectories and PDF of curvature of robust power-law tails. According to Xu et al. (2007), there is a strong correlation between
100 X X
10 10
-4
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X X
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-2
P(x)
PDF(x)
10
1
Fig. 3. Autocorrelations of velocity, acceleration, vorticity and helicity of a fluid particle along a trajectory for Rk ¼ 47 (black lines) and Rk ¼ 70 (red lines). The horizontal axis represents time normalized by the velocity integral time scale R1 T L ¼ 0 qu ðtÞdt. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
acceleration velocity helicity vorticity τ/κ
-1
0.5
-2.5
X
X X
X X
X X
X X
X
X
10
-7
0
X
X
10
20 x/σx
30
40
Fig. 2. The PDF of Lagrangian velocity, acceleration, vorticity, helicity and s=j at Rk ¼ 70. rx is the standard deviation of x. The PDF of velocity (dotted line) is nearly Gaussian.
10
X
-10
10
X
-2
10
-1
10
0
1
10
x = κ , τ , τ/κ
10
2
Fig. 4. PDFs of curvature, torsion and s=j in a log–log plot. The Reynolds number effect is shown to be small in the range of Rk ¼ 47 and 70.
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high curvature and low velocity in addition to the fact that curvature is associated with small, intense vortex tubes (Moisy and Jimenez, 2004; Biferale et al., 2005). Xu and colleagues found that high curvature is closely related to low velocity, where the direction of velocity reverses, rather than to high normal acceleration. Due to experimental limitations they were unable to measure
(a) 101
the material derivatives of acceleration required to calculate torsion. Scagliarini (2009) also reported the 2/5 exponent of PDF for curvature in their turbulence simulation and attributed it to large scale flow reversal by conditioning on events with not-sosmall velocity. However, it is not clear whether such conditioning really excludes large scale motions only.
(a) 3 10-4 10
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2
(b) 3 10
10
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10
-2
10
-1
1
-2
10
0
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τ/κ
10
2
10
4
Fig. 5. Correlations between magnitudes of acceleration, torsion, curvature, and s=j. (a) Magnitude of acceleration vs. curvature, (b) magnitude of acceleration vs. torsion, and (c) magnitude of acceleration vs. s=j. ra is the standard deviation of acceleration and the brackets denote an ensemble average.
0
-2
10
0
10
2
τ/κ
10
10
4
Fig. 6. Correlations between magnitudes of velocity, torsion, curvature, and s=j. (a) Magnitude of velocity vs. curvature, (b) magnitude of velocity vs. torsion and (c) magnitude of velocity vs. s=j. ru is the standard deviation of velocity fluctuation.
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While correlations between curvature, acceleration and velocity show results similar to those of Xu et al. (Figs. 5a and 6a), correlations between torsion, acceleration, and velocity (Figs. 5b and 6b) show a slightly different trend. High torsion tends to be correlated with low acceleration and low velocity. Torsion itself seems not to provide a clear geometrical picture of fluid particle trajectory. We claim that a better quantity, which has more distinct features, is torsion normalized by curvature, s=j. It is quite interesting to find that PDF of s=j clearly shows the 2 exponent as shown in Fig. 4. Furthermore, PDFs of s=j for two different Reynolds numbers completely collapse to each other. It is not so interesting to find that high values of s=j are correlated with low acceleration and high velocity. More meaningful features can be found near s=j ’ 1, where both acceleration and velocity show peak values as shown in Figs. 5c and 6c. We carefully interpret this as evidence supporting that a fluid particle trajectory around a vortical structure that is usually accompanied by high acceleration and high velocity tends to possess a unique geometrical nature such that s ’ j.
in Section 2.3. Figs. 7a and b deliver a unique feature in that the high Lagrangian helicity and high relative Lagrangian helicity are found at s ’ j. This is consistent with our previous observation of correlations between acceleration, velocity and s=j in Figs. 5c and 6c in that large magnitudes of acceleration and velocity naturally lead to large helicity, since high acceleration is always found near vortical structures that have high vorticity. Furthermore, events in which the angle between velocity and vorticity vectors, h, tends to approach zero are always found at s ’ j as shown in Fig. 7b, which contributes much to high helicity at that location. In their simulation of isotropic turbulence, Choi et al. (2009) showed that velocity and vorticity vectors tend to be aligned in the core region of vortical structures. Indeed, all of these observations confirm that trajectories around a vortical structure have the unique property that s ’ j. For the helix geometry of a trajectory around a vortical structure, s=j ¼ cot h as explained in Section 2. It is interesting to note that a strong alignment between velocity and vorticity vectors ðh ! 0; cot h ! 1Þ is found to be relatively well correlated with the curve, s=j ¼ cot h as shown in Fig. 7b.
3.4. Geometry of particle trajectory and coherent structures In Fig. 7, correlations between magnitudes of helicity and s=j, and between cot hð¼ cos h=ð1 cos2 hÞ1=2 Þ with cos h denoting relative helicity and s=j, are presented since for a helix-like trajectory s=j and cot h are expected to have strong correlations as explained
1
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8 10-4
HL/<|HL|>
10
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-1
10
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10-4
-1
10
10 -3
4
-2
3×10
10 0
-1
2
4
6
8
10
HL/<|HL|>
2
Fig. 8. Correlations between magnitudes of Lagrangian helicity and acceleration.
0
10 -2
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-2
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-2
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10-2 -2
10
-2
-1
10 10
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τ/κ
10 10
2
10
4
Fig. 7. Correlations between magnitudes of helicity, cot h, and s=j. The dashed line denotes s=j ¼ cot h.
-1
10
0
10
1
10
an/<|a|> Fig. 9. Correlations between magnitudes of acceleration and the normal component of acceleration. Ninty precent of the data is found in the region between the red line and 45° line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Y. Choi et al. / International Journal of Heat and Fluid Flow 31 (2010) 482–487
Finally, the correlation between acceleration and helicity is investigated. Previous studies showed that large accelerations are related to rotational structures of turbulence (Lee et al., 2004; Lee and Lee, 2005). Lee et al. clearly showed that large acceleration and dissipation on the edge of a vortex filament are strongly correlated, whereas enstrophy shows a weak correlation with strong dissipation. Fig. 8 presents the correlation between the magnitudes of Lagrangian helicity and acceleration. High acceleration always accompanies high helicity, but not vice versa. This reflects the relevance of the spiral curve of a particle trajectory to acceleration. Acceleration of particle motion decomposes into the tangential component, ðu aÞ=u, and the centripetal component, ja uj=u. The latter comprises most of the acceleration as shown in Fig. 9, and thus is responsible for the intermittency of acceleration since high centripetal acceleration is always found near vortical structures, and high acceleration is strongly correlated with high helicity or high relative helicity. 4. Conclusions In this paper, helicity is investigated in a Lagrangian flow field to study turbulent transport and diffusion problems. The goal of this paper is to explain the relationship between geometric parameters such as curvature and torsion, and Lagrangian helicity which represents the spiral motion of a particle trajectory. First, the Lagrangian helicity is defined and the ratio of torsion to curvature is proposed, with this ratio being the most unique quantity presented in our study. Through our observations of various geometric quantities, we find that: (1) The Lagrangian helicity is intermittently distributed on particle trajectories. (2) The ratio of torsion to curvature reveals the unique geometric nature of trajectories around vortical structures. (3) Coherent structures can be understood in terms of geometric shapes and their changes along a particle trajectory, which are determined by curvature, torsion, and the ratio between the two. (4) High Lagrangian helicity is always found, where high acceleration is observed. More studies of the dynamic role of torsion, or the ratio of torsion to curvature, are necessary for a complete description of Lagrangian turbulence.
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Acknowledgements This research was supported by the National Research Foundation of Korea through Grants R01-2008-000-10664-0, R31-2008000-10049-0 (WCU Program), and 20090093134 (ERC Program). Most computations were carried out at KISTI Supercomputing Center, Daejeon, Korea. References Biferale, L., Boffeta, G., Celani, A., Devenish, B.J., Lanotte, A., Toschi, F., 2005. Multifractal statistics of Lagrangian velocity and acceleration in turbulence. Phys. Rev. Lett. 93, 064502. Braun, W., De Lillo, F., Eckhardt, B., 2006. Geometry of particle paths in turbulent flows. J. Turbul. 7 (62). Choi, J.-I., Yeo, K., Lee, C., 2004. Lagrangian statistics in turbulent channel flow. Phys. Fluids 16, 779–793. Choi, Y., Kim, B.-K., Lee, C., 2009. Alignment of velocity and vorticity and intermittent distribution of helicity in isotropic turbulence. Phy. Rev. E 80, 017301. Eswaran, V., Pope, S.B., 1988. One- and two-particle Lagrangian acceleration correlation in numerically simulated homogeneous turbulence. Phys. Fluids 9, 2983–2990. Jung, J., Yeo, K., Lee, C., 2008. Behavior of heavy particles in isotropic turbulence. Phys. Rev. E 77, 016307. Lee, C., Yeo, K., Choi, J.-I., 2004. Intermittent nature of acceleration in near wall turbulence. Phys. Rev. Lett. 92, 144502. Lee, S., Lee, C., 2005. Intermittency of acceleration in isotropic turbulence. Phys. Rev. E 71, 056310. Millman, R., Parker, G., 1977. Elements of Differential Geometry. Prentice Hall. Moffat, H.K., Tsinober, 1992. Helicity in laminar and turbulent flow. Ann. Rev. Fluid Mech. 24, 281–312. Moisy, F., Jimenez, J., 2004. Geometry and clustering of intense structures in isotropic turbulence. J. Fluid Mech. 512, 111–133. Mordant, N., Crawford, A.M., Bodenschatz, E., 2004. Three-dimensional structure of the Lagrangian acceleration in turbulent flows. Phys. Rev. Lett. 93, 214501. Scagliarini, A., 2009. Geometric properties of particle trajectories in turbulent flows. J. Turbul. arXiv:0901.3521v1 (nlin.CD, 22 January 2009). Vedula, P., Yeung, P.K., 1999. Similarity scaling of acceleration and pressure statistics in numerical simulations of isotropic turbulence. Phys. Fluids 11, 1208–1220. Xu, H., Ouellette, N., Bodenschatz, E., 2007. Curvature of Lagrangian trajectories in turbulence. Phys. Rev. Lett. 98, 050201. Yeung, P.K., 1997. An examination of forcing in direct numerical simulations of turbulence. Comput. Fluids 16, 257–278.
Contents lists available at ScienceDirect
International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
Helicity and geometric nature of particle trajectories in homogeneous isotropic turbulence Yeontaek Choi a, Yongnam Park b, Changhoon Lee b,c,* a
Division of Computational Sciences in Mathematics, National Institute for Mathematical Science, Daejeon 305-340, South Korea School of Mechanical Engineering, Yonsei University, Seoul 120-749, South Korea c Department of Computational Science and Engineering, Yonsei University, Seoul 120-749, South Korea b
a r t i c l e
i n f o
Article history: Received 9 October 2009 Received in revised form 27 November 2009 Accepted 2 December 2009 Available online 6 January 2010 Keywords: Lagrangian helicity Acceleration Curvature Torsion Particle trajectory
a b s t r a c t The role of helicity in Lagrangian turbulence is investigated using direct numerical simulation of homogeneous isotropic turbulence. Probability density functions and autocorrelations along a fluid particle trajectory associated with geometric quantities such as curvature and torsion of the Lagrangian trajectory are provided. The relationship between helicity and the ratio of torsion to curvature is investigated. We found that probability density functions of torsion and torsion normalized by curvature clearly show well-established slope in log–log plots. Finally, the relationship between helicity, acceleration and geometric quantities are discussed. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Intermittency of helicity, which is the inner product of velocity and vorticity vectors, in fluid turbulence is an unresolved phenomenon (Moffat and Tsinober, 1992). In a recent paper (Choi et al., 2009), it was observed that intermittency of helicity usually results from fluctuations, particularly rapidly rotating vortical structures. Furthermore, more prominent alignment between velocity and vorticity is found in the core of a coherent vortical structure. Therefore, the role of helicity in Lagrangian turbulence is expected to be intimately related to rotating structures as in Eulerian turbulence. In order to verify this, we investigate the geometric shape of the trajectory of a Lagrangian fluid particle in turbulent flows, because changes in trajectory are caused by background flow motions such as trapping and penetration. Similar observations have been presented in previous studies (Braun et al., 2006; Xu et al., 2007 and Scagliarini, 2009). However, the role of helicity, which measures the spiral motion of a fluid, in governing particle trajectory has not been previously investigated. Although Lagrangian helicity provides information on the spiral motions of fluids, knowledge of helicity does not directly lead to geometric interpretations of fluid particle trajectory. In this study, we demonstrate the relevance of Lagrangian helicity * Corresponding author. Address: Department of Computational Science and Engineering, Yonsei University, Seoul 120-749, South Korea. E-mail address: [email protected] (C. Lee). 0142-727X/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.ijheatfluidflow.2009.12.003
and behaviors to particle trajectory. In order to investigate this, we introduce the ratio of torsion to curvature of fluid particle trajectories. We use direct numerical simulations of homogeneous isotropic turbulence to calculate quantities in this paper. In order to simulate turbulent flow, the Navier–Stokes equations and continuity equation are solved using a spectral method for spatial discretization and the third-order Runge–Kutta scheme for time advance3 ment in a ½ð2pÞ cubic domain with periodicity conditions enforced in all three directions. Most calculations are carried out at 643 and 1283 resolutions. For the maintenance of stationarity, we use the forcing scheme proposed by Eswaran and Pope (1988) in which artificially forced low-wavenumber velocity components ðjkj < kf Þ generate statistically stationary turbulence fields. A four-point Hermite interpolation (Choi et al., 2004) is also used for the interpolation necessary for fluid particle tracking. The Reynolds number, based on the Taylor scale, Rk , and other simulation parameters are listed in Table 1. The remainder of the paper is organized as follows. In Section 2, definitions and fundamental concepts are introduced. Lagrangian helicity is defined and the geometric meaning of the ratio of torsion to curvature is given. In Section 3, we present observations to verify the relevance of helicity and geometric shapes to particle trajectory. First, we provide two statistics: probability distribution functions (PDF) and autocorrelations of velocity, vorticity, acceleration, and helicity. Next, we introduce statistics and correlations of helicity and changes of geometric shape of a fluid particle, with a
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Y. Choi et al. / International Journal of Heat and Fluid Flow 31 (2010) 482–487 Table 1 Simulation parameters. m; kmax ; g; T fL ; Dt and sg are viscosity, maximum wavenumber, Kolmogorov length scale, forcing time scale, time step and Kolmogorov time scale, respectively. N
Rk
m
kmax ; g
kf
T fL
Dt
pffiffiffi 2 2 2.514
0.4312
1=56sg
0.4312
1/80sg
64
47
0.03
1.24
128
70
0.015
1.5
special focus on the ratio of torsion to curvature. Then statistics associated with acceleration are provided to investigate the relationship between geometry of the trajectory of a particle and the structure of the background flow. Finally, the conclusion is given in Section 4. 2. Definitions and basic concepts 2.1. Lagrangian helicity of a fluid particle Lagrangian helicity is defined as
HL ¼ uðxðtÞ; tÞ xðxðtÞ; tÞ;
ð1Þ
where time t 2 ½0; T; xðtÞ is the position of a fluid particle, uðxðtÞ; tÞ is the velocity of a fluid particle, and xðxðtÞ; tÞ is the vorticity of a fluid particle at time, t. Lagrangian variation of helicity of a fluid particle is calculated along the trajectory of a fluid particle, and indicates instantaneous interactions between velocity and vorticity vectors along the trajectory. Definition (1) corresponds to helicity density in Eulerian fields, which is the integrand of the total helicity R HE ¼ V v ðtÞ xðtÞdV (Choi et al., 2009), where v ðtÞ and xðtÞ are velocity and vorticity in an Eulerian field, respectively. 2.2. PDFs and autocorrelations We introduce the Lagrangian distributions of various quantities manifested along particle trajectories such as velocity, vorticity, acceleration, and helicity of a fluid particle. One- and two-particle Lagrangian acceleration correlations (Yeung, 1997) were investigated and heavy particle behaviors were also studied in Jung et al. (2008). We investigate autocorrelations for velocity u, vorticity x, acceleration a, and Lagrangian helicity HL of a fluid particle along its trajectory. The definition of an autocorrelation for variable x is as follows:
qxðtÞ ¼
hxðt0 Þxðt 0 þ tÞi : hx2 ðt 0 Þi
ð2Þ
2.3. Curvature and torsion of a Lagrangian trajectory It is natural to adopt torsion as well as curvature in to depict a particle trajectory in three dimensional space. The trajectory of a Lagrangian particle determined by dxðtÞ=dt ¼ uðxðtÞ; tÞ can be described in terms of a Frenet–Serret apparatus fjðsÞ; sðsÞ; TðsÞ; NðsÞ; BðsÞg, where s is a parameter of a curve, jðsÞ is curvature, sðsÞ is torsion, TðsÞ is a tangent vector, NðsÞ is a normal vector, and BðsÞ is a binormal vector (Millman and Parker, 1977). The Frenet apparatus has no dynamical information and contains pure geometric information. Reparametrizing the parameter s as time t, curvature and torsion for a curve that has non-unit speed vector, jdxðtÞ=dtj – 1, are written as follows:
a? ¼ 2; u juj3 _ u ða aÞ Torsion s ¼ ; u6 j2 Curvature
j¼
ju aj
where a? ; u and a_ are components of acceleration perpendicular to the velocity vector, the magnitude of velocity and material derivative of acceleration vector, respectively. There are several corollaries to characterize the properties of curvature and torsion for j – 0 (Millman and Parker, 1977):
j¼0 j – 0; s¼0 s ¼ 0; j ¼ constant > 0 s ¼ constant – 0; j ¼ constant > 0 s j
¼ constant
straight line plane curve circle circular helix helix
Here, a helix is defined as a regular curve such that for some fixed unit vector A, TA is constant. T is a unit tangent vector as defined above and A is the axis of the helix. In this context, for a fluid particle trajectory around a vortical structure, TA corresponds to the relative helicity which is defined by u=juj x=jxj ¼ cos h, where h is the angle between velocity and vorticity vectors. The theorem to characterize the helix was proven by Lancert in 1802. The unit speed curve aðsÞ; jdaðsÞ=dsj ¼ 1, with j – 0 is a helix if and only if there is a constant c such that s ¼ cj. If T A cos h, this leads to s=j ¼ cot h. We should, however, be cautious in dealing with Lancert’s theorem, for it is invalid for j ¼ 0 and jdaðsÞ=dsj – 1. The consequence of Lancert’s theorem is that the ratio s=j is explained by the angle between the tangent vector T which is parallel to a velocity vector, and the axis of helix A which is parallel to vorticity for a trajectory around a vortical structure. When Ak T; s=j ¼ 1, and when A ? T; c ¼ 0. Near a helix-like vortical structure, an investigation of s=j can reveal alignment trends between velocity and vorticity vectors. A circular helix is a special case. Circular refers to the fact that the projection in the ðx; yÞ plane is a circle. For example: a : R ! R3 be given by aðsÞ ¼ ðr cos s; r sin s; hsÞ, where h > 0 and r > 0. We expect that the ratio of torsion to curvature plays a crucial role in understanding the geometric shape of particle trajectory. In random flow, a trajectory is not a helix. 3. Lagrangian statistics 3.1. Lagrangian fluid particle trajectory Fig. 1 presents the sample trajectories of two Lagrangian particles at Rk ¼ 47. In this figure, the particles are shown to wrap around a vortical structure. Vorticity of the particles remains in the same direction as the axis of the vortical structure while accelerations are oriented toward the center of the vortical structure, clearly indicating that the vortical structure is responsible for converging distributions of acceleration. Investigations of torsion along the trajectories indicates that fluid particles closer to the vortical structure have much stronger torsion. Relative helicity, which is the cosine angle between velocity and vorticity vectors, is also closer to 1 near the vortical structure, implying that near the core of vortical structures, velocity and vorticity vectors tend to align with each other, consistent with our previous finding (Choi et al., 2009). Behaviors of acceleration, helicity and the geometric nature of trajectories around a vortical structure clearly suggest strong correlations between those quantities, particularly near vortical structures. In the rest of this paper, we suggest relationship between helicity, acceleration and particle trajectory shapes as depicted by curvature and torsion.
ð3Þ
3.2. Probability density functions and autocorrelations
ð4Þ
Lagrangian statistics were observed either numerically or experimentally in several previous papers (Vedula and Yeung,
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1X
X X XX XX XX XX
0.6
ρt
ux ax ωx HL
XX
0.8
X XX
XX
0.4
XX
XX
XX
XX
XX
0.2
XX
XX
XX
XX XX XX
0 0 Fig. 1. (a) Trajectories of two sample Lagrangian particles moving around a vortical structure (green structure). Red vectors represent vorticity and black vectors represent acceleration. Color of particles represents the magnitude of torsion, (b) the same trajectories viewed from a different angle with distribution of the magnitude of vorticity and (c) the color of particles denotes the magnitudes of relative helicity, i.e., cosine values of the angle between velocity and vorticity vectors. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
1999; Mordant et al., 2004). Here, we observed statistics of several Lagrangian quantities for a fluid particle moving along its trajectory, such as velocity, vorticity, acceleration and helicity (Fig. 2). The PDF of velocity, whose distribution is nearly Gaussian, is compared to PDFs of acceleration, helicity and vorticity, which show wider tails. The statistics are also compared to the PDF of the ratio s=j. Compared to other variables, s=j shows significantly intermittent distribution. Local values of that are s=j 1000 times larger than its mean value are frequently observed. Fig. 3 presents autocorrelations of Lagrangian velocity, vorticity, acceleration and helicity in the cases of two different Reynolds numbers Rk ¼ 47 and 70. The Lagrangian trajectory for each fluid particle is tracked by employing the 3rd order Runge–Kutta scheme, and flow quantities at each time are obtained by a 4th order Hermite interpolation scheme. In order to obtain a fully converged distribution of the correlation function, data are collected
10
-2
-3
10
1.5 t / TL
2
2.5
in the period of 270T L and 54T L for Rek ¼ 47 and 70, respectively, where T L is the velocity integral time scale. The number of particles used in the calculation are 3 108 and 4:5 107 for Rek ¼ 47 and 70, respectively. Helicity becomes decorrelated faster than velocity and vorticity, as shown in Fig. 3. Sensitivity to the Reynolds number is found to be relatively weak. 3.3. Statistics of curvature and torsion Braun et al. (2006) numerically observed the exponential behavior for PDFs of curvature, i.e., the 2/5 exponent of PDF for high values of curvature, in their simulation of a turbulent flow. In Fig. 4, PDFs for j; s and s=j obtained from our simulations are illustrated to clearly show the 3 exponent for high values of torsion and the 2 exponent for s=j. Xu et al. (2007) experimentally measured the curvature of particles moving along Lagrangian trajectories and PDF of curvature of robust power-law tails. According to Xu et al. (2007), there is a strong correlation between
100 X X
10 10
-4
10
X X
X X
X
X
X
-4
X
X
X
X
X
X
X
X
κ
x
X
X
X
X
X
X
X
τ/κ
X X
X
X X
X X
X X
X
X
X X
X
X X
X
X X
X X
X
X X
10
x
X X
X
-6
Rλ=70
X X
X
X
X
X
X
X
x
X
X
X
-8
-3
X
X
Rλ=47
x-2
X
X
X
X
10
X
X
τ
10
-6
X
X
X
10
-5
X X
-2
P(x)
PDF(x)
10
1
Fig. 3. Autocorrelations of velocity, acceleration, vorticity and helicity of a fluid particle along a trajectory for Rk ¼ 47 (black lines) and Rk ¼ 70 (red lines). The horizontal axis represents time normalized by the velocity integral time scale R1 T L ¼ 0 qu ðtÞdt. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
acceleration velocity helicity vorticity τ/κ
-1
0.5
-2.5
X
X X
X X
X X
X X
X
X
10
-7
0
X
X
10
20 x/σx
30
40
Fig. 2. The PDF of Lagrangian velocity, acceleration, vorticity, helicity and s=j at Rk ¼ 70. rx is the standard deviation of x. The PDF of velocity (dotted line) is nearly Gaussian.
10
X
-10
10
X
-2
10
-1
10
0
1
10
x = κ , τ , τ/κ
10
2
Fig. 4. PDFs of curvature, torsion and s=j in a log–log plot. The Reynolds number effect is shown to be small in the range of Rk ¼ 47 and 70.
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high curvature and low velocity in addition to the fact that curvature is associated with small, intense vortex tubes (Moisy and Jimenez, 2004; Biferale et al., 2005). Xu and colleagues found that high curvature is closely related to low velocity, where the direction of velocity reverses, rather than to high normal acceleration. Due to experimental limitations they were unable to measure
(a) 101
the material derivatives of acceleration required to calculate torsion. Scagliarini (2009) also reported the 2/5 exponent of PDF for curvature in their turbulence simulation and attributed it to large scale flow reversal by conditioning on events with not-sosmall velocity. However, it is not clear whether such conditioning really excludes large scale motions only.
(a) 3 10-4 10
0
10
10
2
-1
10
-3
10
-2
10
-1
1
-2
10
-4
|u|/σu
|a|/σa
10
10-2
-1
10
-3
-2
-1
10
10
10
0
1
2
10
κ/<κ>
0
10
1
(b) 10
-2
-1
0
1
10
10
10
κ/<κ>
-4
10
0
10
10
10
-1
10
10-1
-2
10
|u|/σu
|a|/σa
2 10-3 -4
-3
-2
10-1
1
-2
10
-2
10
10
0
2
0
10
τ/<|τ|>
(c) 101
-2
10
-1
10
0
1
10
10
τ/<|τ|>
10
2
(c) 3 10
0
10
10
-3
-1
10
-4 -4
10 -3 10
-2
2
10
|u|/σu
|a|/σa
10
-1
10
-2
2
(b) 3 10
10
10
10
-2
10
-1
1
-2
10
0
10
τ/κ
10
2
10
4
Fig. 5. Correlations between magnitudes of acceleration, torsion, curvature, and s=j. (a) Magnitude of acceleration vs. curvature, (b) magnitude of acceleration vs. torsion, and (c) magnitude of acceleration vs. s=j. ra is the standard deviation of acceleration and the brackets denote an ensemble average.
0
-2
10
0
10
2
τ/κ
10
10
4
Fig. 6. Correlations between magnitudes of velocity, torsion, curvature, and s=j. (a) Magnitude of velocity vs. curvature, (b) magnitude of velocity vs. torsion and (c) magnitude of velocity vs. s=j. ru is the standard deviation of velocity fluctuation.
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While correlations between curvature, acceleration and velocity show results similar to those of Xu et al. (Figs. 5a and 6a), correlations between torsion, acceleration, and velocity (Figs. 5b and 6b) show a slightly different trend. High torsion tends to be correlated with low acceleration and low velocity. Torsion itself seems not to provide a clear geometrical picture of fluid particle trajectory. We claim that a better quantity, which has more distinct features, is torsion normalized by curvature, s=j. It is quite interesting to find that PDF of s=j clearly shows the 2 exponent as shown in Fig. 4. Furthermore, PDFs of s=j for two different Reynolds numbers completely collapse to each other. It is not so interesting to find that high values of s=j are correlated with low acceleration and high velocity. More meaningful features can be found near s=j ’ 1, where both acceleration and velocity show peak values as shown in Figs. 5c and 6c. We carefully interpret this as evidence supporting that a fluid particle trajectory around a vortical structure that is usually accompanied by high acceleration and high velocity tends to possess a unique geometrical nature such that s ’ j.
in Section 2.3. Figs. 7a and b deliver a unique feature in that the high Lagrangian helicity and high relative Lagrangian helicity are found at s ’ j. This is consistent with our previous observation of correlations between acceleration, velocity and s=j in Figs. 5c and 6c in that large magnitudes of acceleration and velocity naturally lead to large helicity, since high acceleration is always found near vortical structures that have high vorticity. Furthermore, events in which the angle between velocity and vorticity vectors, h, tends to approach zero are always found at s ’ j as shown in Fig. 7b, which contributes much to high helicity at that location. In their simulation of isotropic turbulence, Choi et al. (2009) showed that velocity and vorticity vectors tend to be aligned in the core region of vortical structures. Indeed, all of these observations confirm that trajectories around a vortical structure have the unique property that s ’ j. For the helix geometry of a trajectory around a vortical structure, s=j ¼ cot h as explained in Section 2. It is interesting to note that a strong alignment between velocity and vorticity vectors ðh ! 0; cot h ! 1Þ is found to be relatively well correlated with the curve, s=j ¼ cot h as shown in Fig. 7b.
3.4. Geometry of particle trajectory and coherent structures In Fig. 7, correlations between magnitudes of helicity and s=j, and between cot hð¼ cos h=ð1 cos2 hÞ1=2 Þ with cos h denoting relative helicity and s=j, are presented since for a helix-like trajectory s=j and cot h are expected to have strong correlations as explained
1
10
0
10
8 10-4
HL/<|HL|>
10
10-2
-1
10
|a|/σa
(a) 10
6
-3
10-4
-1
10
10 -3
4
-2
3×10
10 0
-1
2
4
6
8
10
HL/<|HL|>
2
Fig. 8. Correlations between magnitudes of Lagrangian helicity and acceleration.
0
10 -2
0
10
10
2
10
τ/κ
-2
10
4
10
(b) 10
0
6
a/<|a|>
cotθ
8
1
10-4
10
10
-4
10
-3
-1 10-2 10
-1
10
4
10-3
0-4 10
-2
10
2
10-2 -2
10
-2
-1
10 10
0
τ/κ
10 10
2
10
4
Fig. 7. Correlations between magnitudes of helicity, cot h, and s=j. The dashed line denotes s=j ¼ cot h.
-1
10
0
10
1
10
an/<|a|> Fig. 9. Correlations between magnitudes of acceleration and the normal component of acceleration. Ninty precent of the data is found in the region between the red line and 45° line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Y. Choi et al. / International Journal of Heat and Fluid Flow 31 (2010) 482–487
Finally, the correlation between acceleration and helicity is investigated. Previous studies showed that large accelerations are related to rotational structures of turbulence (Lee et al., 2004; Lee and Lee, 2005). Lee et al. clearly showed that large acceleration and dissipation on the edge of a vortex filament are strongly correlated, whereas enstrophy shows a weak correlation with strong dissipation. Fig. 8 presents the correlation between the magnitudes of Lagrangian helicity and acceleration. High acceleration always accompanies high helicity, but not vice versa. This reflects the relevance of the spiral curve of a particle trajectory to acceleration. Acceleration of particle motion decomposes into the tangential component, ðu aÞ=u, and the centripetal component, ja uj=u. The latter comprises most of the acceleration as shown in Fig. 9, and thus is responsible for the intermittency of acceleration since high centripetal acceleration is always found near vortical structures, and high acceleration is strongly correlated with high helicity or high relative helicity. 4. Conclusions In this paper, helicity is investigated in a Lagrangian flow field to study turbulent transport and diffusion problems. The goal of this paper is to explain the relationship between geometric parameters such as curvature and torsion, and Lagrangian helicity which represents the spiral motion of a particle trajectory. First, the Lagrangian helicity is defined and the ratio of torsion to curvature is proposed, with this ratio being the most unique quantity presented in our study. Through our observations of various geometric quantities, we find that: (1) The Lagrangian helicity is intermittently distributed on particle trajectories. (2) The ratio of torsion to curvature reveals the unique geometric nature of trajectories around vortical structures. (3) Coherent structures can be understood in terms of geometric shapes and their changes along a particle trajectory, which are determined by curvature, torsion, and the ratio between the two. (4) High Lagrangian helicity is always found, where high acceleration is observed. More studies of the dynamic role of torsion, or the ratio of torsion to curvature, are necessary for a complete description of Lagrangian turbulence.
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Acknowledgements This research was supported by the National Research Foundation of Korea through Grants R01-2008-000-10664-0, R31-2008000-10049-0 (WCU Program), and 20090093134 (ERC Program). Most computations were carried out at KISTI Supercomputing Center, Daejeon, Korea. References Biferale, L., Boffeta, G., Celani, A., Devenish, B.J., Lanotte, A., Toschi, F., 2005. Multifractal statistics of Lagrangian velocity and acceleration in turbulence. Phys. Rev. Lett. 93, 064502. Braun, W., De Lillo, F., Eckhardt, B., 2006. Geometry of particle paths in turbulent flows. J. Turbul. 7 (62). Choi, J.-I., Yeo, K., Lee, C., 2004. Lagrangian statistics in turbulent channel flow. Phys. Fluids 16, 779–793. Choi, Y., Kim, B.-K., Lee, C., 2009. Alignment of velocity and vorticity and intermittent distribution of helicity in isotropic turbulence. Phy. Rev. E 80, 017301. Eswaran, V., Pope, S.B., 1988. One- and two-particle Lagrangian acceleration correlation in numerically simulated homogeneous turbulence. Phys. Fluids 9, 2983–2990. Jung, J., Yeo, K., Lee, C., 2008. Behavior of heavy particles in isotropic turbulence. Phys. Rev. E 77, 016307. Lee, C., Yeo, K., Choi, J.-I., 2004. Intermittent nature of acceleration in near wall turbulence. Phys. Rev. Lett. 92, 144502. Lee, S., Lee, C., 2005. Intermittency of acceleration in isotropic turbulence. Phys. Rev. E 71, 056310. Millman, R., Parker, G., 1977. Elements of Differential Geometry. Prentice Hall. Moffat, H.K., Tsinober, 1992. Helicity in laminar and turbulent flow. Ann. Rev. Fluid Mech. 24, 281–312. Moisy, F., Jimenez, J., 2004. Geometry and clustering of intense structures in isotropic turbulence. J. Fluid Mech. 512, 111–133. Mordant, N., Crawford, A.M., Bodenschatz, E., 2004. Three-dimensional structure of the Lagrangian acceleration in turbulent flows. Phys. Rev. Lett. 93, 214501. Scagliarini, A., 2009. Geometric properties of particle trajectories in turbulent flows. J. Turbul. arXiv:0901.3521v1 (nlin.CD, 22 January 2009). Vedula, P., Yeung, P.K., 1999. Similarity scaling of acceleration and pressure statistics in numerical simulations of isotropic turbulence. Phys. Fluids 11, 1208–1220. Xu, H., Ouellette, N., Bodenschatz, E., 2007. Curvature of Lagrangian trajectories in turbulence. Phys. Rev. Lett. 98, 050201. Yeung, P.K., 1997. An examination of forcing in direct numerical simulations of turbulence. Comput. Fluids 16, 257–278.