http://www.uta.edu/math/preprint/

Technical Report 2016-08

Observation Of The Development Of Lambda-Vortex

To Hairpin Vortex Packet

Yong Yang Chaoqun Liu

Γ

American Institute of Aeronautics and Astronautics

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Observation of the development of 𝚲𝚲-vortex to hairpin vortex packet

Yong Yang1 and Chaoqun Liu2 University of Texas at Arlington, Arlington, Texas, USA 76019

A DNS simulation is carried out to reveal the development of a Λ-vortex to a hairpin vortex packet. To visualize the vortices, the new omega vortex identification method is utilized. According to the observation, the couter rotation of two legs of 𝚲𝚲-vortex sweeps the low speed zone up to form the high strain rate region; the moving up of the high strain rate region is along with the generation of vortex rings; the separation of the first hairpin vortex is related to the development of the proturberance of the second vortex ring; the separation of the second and third hairpin vortices is similar with the first one, but the fourth and fifth hairpin generate in a different way from the first three hairpins.

Nomenclature

∞M = Mach number Re = Reynolds number

inδ = inflow displacement thickness wT = wall temperature

∞T = free stream temperature inLz = height at inflow boundary

outLz = height at outflow boundary Lx = length of computational domain along x direction Ly = length of computational domain along y direction

inx = distance between leading edge of flat plate and upstream boundary of computational domain x, y, z – stremwise, spanwise, normal directions 𝑽𝑽 = velocity vector 𝝎𝝎 = vorticity vector 𝜔𝜔𝑥𝑥 = streamwise vorticity 𝜔𝜔𝑦𝑦 = spanwise vorticity 𝜔𝜔𝑧𝑧 = normal vorticity

I. Introduction ORTEX plays a significant role in turbulence, which is called “the sinews of turbulence” [1]. The transition process from laminar to turbulent flow in boundary layers with the generation and development of vortices is a

basic scientific problem in fluid mechanics. After over a hundred year of study, researchers have figured out the linear and weakly non-linear stage of flow transition pretty well [2], [3]. Many reseach worker studied the late non-linear trasitional stage numerically and experimentally. Moin et al [4] gave a explaination of the evolution of a curved vortex filament into a vortex ring in 1986. Liu et at [5], [6] reported the whole process of K- and H-type transition in 1995 and 1996 and Rist [7] gave a quantitative comparison of experiment and direct numerical simulation in 2002. Adrian [8] gave a review about the hairpin vortex organization in boundary layer in 2007. Moin et al [9] reported a new direct numerical simulation (DNS) for flow transition on a flat plate in 2009. However there are still many questions to solve in late non-linear stage of flow transition. Some misunderstandings are still polular. Even though the explaination of the evolution of a vortex filament into a vortex ring is given [4], the development of Λ-vortex to hairpin vortex can still not be explained since the difference of vortex filament and vortex [10] , see Figure 1. Rigorously, the vortex

1 PhD student, Department of Mathematics, AIAA Member. 2 Professor, Department of Mathematics, AIAA Associate Fellow.

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filament which is defined as the a line everywhere tangent to the local vorticity vector 𝝎𝝎 = 𝛁𝛁 × 𝐕𝐕 should be called a vorticity line, to disambigue the misunderstanding.

Figure 1. Vortex filaments (black lines) with 𝚲𝚲-vortex (green surface). (Wang et al. 2016)

Some researchers believe the multiple hairpin vortices in a vortex packet is produced by auto-generation [8], however, recent researches show that the development of vortex packets is related to the high shear layer instability [10]. To study the process of Λ-vortex developing to vortex packet carefully, a detailed observation based on DNS data will be given in this paper. The case setup and code validation will be given in section II. A revisit of the new omega vortex visualization method will be given in section III. Then a detailed observation of the development of a Λ-vortex to a vortex packet will be given in section IV. And section V will summarize the conclutions.

II. Case setup and Code Validation

A. Case setup The physical domain is displayed in Figure 2, where 𝑥𝑥𝑖𝑖𝑖𝑖 represents the distance between leading edge and inlet,

𝐿𝐿𝑥𝑥 and 𝐿𝐿𝐿𝐿 are the lengths of the computational domain in 𝑥𝑥 and 𝐿𝐿 directions respectively, and 𝐿𝐿𝑧𝑧𝑖𝑖𝑖𝑖 is the length of the inlet in 𝑧𝑧 direction. The details are listed in Table 1. The grid level is 1920 × 128 × 241, representing the number of grids in streamwise (𝑥𝑥), spanwise (𝐿𝐿), and wall normal (𝑧𝑧) directions. The grid is stretched in the normal direction and uniform in the streamwise and spanwise directions. The length of the first grid interval in the normal direction at the entrance is found to be 0.43 in wall units (𝑍𝑍+ = 0.43).

The Jocobian coordinate transformation is employed from physical domain to computational domain, see Figure 3(a) and the Message Passing Interface (MPI), together with domain decomposition in the ξ-direction, is utilized to accomplish the parallel computation, see Figure 3(b). The flow parameters, including Mach number, Reynolds number, etc. are listed in Table 2. Here, 𝑇𝑇𝑤𝑤 = 273.15𝐾𝐾 is the wall temperature.

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Figure 2. Physical domain

𝑥𝑥𝑖𝑖𝑖𝑖 𝐿𝐿𝑥𝑥 𝐿𝐿𝐿𝐿 𝐿𝐿𝑧𝑧𝑖𝑖𝑖𝑖

300.79𝛿𝛿𝑖𝑖𝑖𝑖 798.03𝛿𝛿𝑖𝑖𝑖𝑖 22𝛿𝛿𝑖𝑖𝑖𝑖 40𝛿𝛿𝑖𝑖𝑖𝑖

Table 1. Geometry parameters

(a)

(b)

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Figure 3. Schematics of (a) coordinate transformation and (b) domain decomposition.

𝑀𝑀∞ 𝑅𝑅𝑅𝑅 𝑇𝑇𝑤𝑤 𝑇𝑇∞

0.5 1000 273.15K 273.15K

Table 2. Flow paramaters

B. Code validation The DNS code – ‘‘DNSUTA’’ has been validated by NASA Langley and UTA researchers [11]–[13] carefully to

make sure that the DNS results are correct. Since the detailed code validation has been reported by Liu and Chen [14] we only give a short description here.

1. Comparison with Log Law and grid convergence

Time and spanwise-averaged streamwise velocity profiles for various streamwise locations in two different grid levels are shown in Figure 4. The inflow velocity profiles at 𝑥𝑥 = 300.79𝛿𝛿𝑖𝑖𝑖𝑖 is a typical laminar flow velocity profile. At 𝑥𝑥 = 632.33𝛿𝛿𝑖𝑖𝑖𝑖, the mean velocity profile approaches a turbulent flow velocity profile (Log law). This comparison shows that the velocity profile from the DNS results is turbulent flow velocity profile and the grid convergence has been realized.

(a) Coarse Grids (960x64x121) (b) Fine Grids (1920x128x241)

Figure 4. Log-linear plots of the time-and spanwise-averaged velocity profile in wall unit

2. Comparison with experiment

By using Ω criterion method [15], the vortex structures shaped by the nonlinear evolution of T-S waves in the transition process are shown in Figure 5. The evolution details are studied in our previous paper [10], [16] and the formation of ring-like vortices chains is consistent with the experimental work [17], see Figure 6.

z+

U+

100 101 102 1030

10

20

30

40

50x=300.79x=632.33Linear LawLog Law

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(a) t=6.0T (b) t=6.2T

(c) t=6.4T (d) t=7.0T

Figure 5. Evolution of vortex structure at the late-stage of transition (Where T is the period of T-S wave)

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Figure 6. Evolution of the ring-like vortex chain by experiment (Lee et al, 2007)

3. Comparison with Rist’s DNS data

Figure 7 shows a comparison of our DNS results with the data set provided by Rist as his personal kindness. The comparison shows both DNS have same vortex structure.

(a) Our DNS (b) Rist’s DNS data

Figure 7. Comparison of our DNS results with Rist’s DNS data.

All these verifications and validations above show that our code is correct and our DNS results are reliable.

III. Revisit of the new omega vortex identification method Liu et al [15] proposed a new vortex identification method, Ω method, recently. The basic idea of this method is

that vorticity can be decomposed into a vortical part and a non-vortical part. One conspicuous evidence of this idea is that vorticity exists in both rotating flow like vortex and flow without rotation like Blasius solution.

Ω is defined as following:

Ω =‖𝐵𝐵‖𝐹𝐹2

‖𝐴𝐴‖𝐹𝐹2 + ‖𝐵𝐵‖𝐹𝐹2 + 𝜖𝜖 (1)

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where ‖ ∙ ‖𝐹𝐹 is the Frobenius normand 𝐴𝐴 is the symmetric part of velocity gradient tensor ∇𝑽𝑽, 𝐵𝐵 is the anti-symmetric

part and ε is a small positive number introduced to avoid division by zero:

𝐴𝐴 =12

(∇𝑽𝑽 + ∇𝑽𝑽𝑇𝑇) (2)

𝐵𝐵 =12

(∇𝑽𝑽 − ∇𝑽𝑽𝑇𝑇) (3)

According to Liu et al. [15], a vortex can be identified as a connected region where Ω > 0.5 and the iso-surface of Ω = 0.52 can be utilized to indicate the structures of vortices.

IV. Development of a 𝚲𝚲-vortex to a vortex packet Based on the DNS data and Ω vortex identification method, a detailed observation is carried out to figure out the

process of a Λ-vortex developing to a vortex packet.

A. 𝚲𝚲-vortex structure Figure 8 gives the structures of Λ-vortex and strain rate in different views at 𝑡𝑡 = 6.0𝑇𝑇, where 𝑇𝑇 is T-S wave period.

The green surface is the iso-surface of Ω = 0.52 which indicates the structure of Λ-vortex while the yellow surface is the iso-surface of ‖𝐴𝐴‖𝐹𝐹 = 0.6 which show the position of shear layer. According to Figure 8, the shear layer concentrates beyond the head of the Λ-vortex.

(a) Glabal view

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(b) Y-Z view

(c) X-Z view

(d) X-Y view

Figure 8. The iso-surfaces of 𝛀𝛀 = 𝟎𝟎.𝟓𝟓𝟓𝟓 (green) and ‖𝑨𝑨‖𝑭𝑭 = 𝟎𝟎.𝟔𝟔 (yellow) at 𝒕𝒕 = 𝟔𝟔.𝟎𝟎𝟎𝟎, where 𝟎𝟎 is the period of T-S wave.

When we increase the iso-value of Ω, see Figure 9, the shear layer places at the same position along streamwise direction with the iso-surface of Ω = 0.65. It indicates the shear layer is high near the strong vortex. Figure 10 gives a more clear exlaination. The distributions of 𝜔𝜔𝑥𝑥 and 𝑢𝑢 on the slice 𝑥𝑥 = 443 are given in Figure 10(b) and (c). We can find a strong rotation exist at this position and the low speed zone is swept to higer postion, then 𝜕𝜕𝜕𝜕

𝜕𝜕𝑧𝑧 becomes

greater around the position 𝐿𝐿 = 5.5, 𝑧𝑧 = 2, where the shear layer places.

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Figure 9. The iso-surfaces of 𝛀𝛀 = 𝟎𝟎.𝟔𝟔𝟓𝟓 (green) and ‖𝑨𝑨‖𝑭𝑭 = 𝟎𝟎.𝟔𝟔 (yellow) at 𝒕𝒕 = 𝟔𝟔.𝟎𝟎𝟎𝟎, where 𝟎𝟎 is the period of T-S wave.

(a) Position of the slice of 𝒙𝒙 = 𝟒𝟒𝟒𝟒𝟒𝟒.

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(b) 𝝎𝝎𝒙𝒙 with stream traces.

(c) 𝒖𝒖 with stream traces

Figure 10. The distribution of 𝝎𝝎𝒙𝒙 and 𝒖𝒖 on the slice of 𝒙𝒙 = 𝟒𝟒𝟒𝟒𝟒𝟒.

B. The ring of hainpin vortex generation Figure 11 shows the iso-surface of Ω = 0.52 and ‖𝐴𝐴‖𝐹𝐹 = 0.6 at 𝑡𝑡 = 6.16𝑇𝑇, where 𝑇𝑇 is the period of T-S wave.

From Figure 11(b), we find the first ring generates at the top of the high strain rate region. Comparing Figure 11(b) with Figure 8(c), the high strain rate region moves up. Liu et al [18] reported this phenomenon that the generation of vortex ring is due to the moving up of the shear layer. This also can checked by Figure 12, which shows the iso-surface of Ω = 0.52 and ‖𝐴𝐴‖𝐹𝐹 = 0.6 at 𝑡𝑡 = 6.30𝑇𝑇. Comparing Figure 12 with Figure 11(b), the second high strain rate region moves upper at 𝑡𝑡 = 6.30𝑇𝑇. Then the second ring generates following as a result.

(a) Global view

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(b) X-Z view

Figure 11. The iso-surface of 𝛀𝛀 = 𝟎𝟎.𝟓𝟓𝟓𝟓 and ‖𝑨𝑨‖𝑭𝑭 = 𝟎𝟎.𝟔𝟔 at 𝒕𝒕 = 𝟔𝟔.𝟏𝟏𝟔𝟔𝟎𝟎, where 𝟎𝟎 is the period of T-S wave.

Figure 12. The iso-surface of 𝛀𝛀 = 𝟎𝟎.𝟓𝟓𝟓𝟓 and ‖𝑨𝑨‖𝑭𝑭 = 𝟎𝟎.𝟔𝟔 at 𝒕𝒕 = 𝟔𝟔.𝟒𝟒𝟎𝟎𝟎𝟎, where 𝟎𝟎 is the period of T-S wave.

C. The first three hairpins separating from 𝚲𝚲-vortex Note that in Figure 12, there is a protuberance at the “mandible” of the second ring. This protuberance grows along

lower front direction to enfold the legs of the first vortex ring. It is more clear in Figure 13. Figure 13 gives the vortex structures at 𝑡𝑡 = 6.43𝑇𝑇 in global, X-Z and X-Y view. The red arrows indicate the direction of the protuberance developing.

(a) Global view

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(b) X-Z view

(c) X-Y view

Figure 13. The iso-surface of 𝛀𝛀 = 𝟎𝟎.𝟓𝟓𝟓𝟓 at 𝒕𝒕 = 𝟔𝟔.𝟒𝟒𝟒𝟒𝟎𝟎, where 𝟎𝟎 is the period of T-S wave.

Figure 14 shows the the vortex structures at 𝑡𝑡 = 6.43𝑇𝑇 in X-Z view, X-Y top view and bottom view. The protuberance extends to the bottom and is parallel to the leg of the first hairpin vortex, see Figure 14(a). And the first hairpin vortex has been separated from the Λ-vortex, see Figure 14(c).

(a) X-Z view

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(b) X-Y view (Top)

(c) X-Y view (Bottom)

Figure 14. The iso-surface of 𝛀𝛀 = 𝟎𝟎.𝟓𝟓𝟓𝟓 at 𝒕𝒕 = 𝟔𝟔.𝟔𝟔𝟎𝟎𝟎𝟎, where 𝟎𝟎 is the period of T-S wave.

The distribution of 𝜔𝜔𝑥𝑥 on the slice of 𝑥𝑥 = 475 with stream traces (black lines) at t=6.60T is shown in Figure 15. We can find the protuberance and the leg of the first hairpin are counter-rotating, see Figure 15(b). This is the possible reason that the first hairpin vortex separate from the Λ-vortex. Also from Figure 15(a), we can find the separations of the first three hairpin vortex from Λ-vortex have the similar process, that is, the separations are all with the extension of the protuberance of latter one.

(a) The position of slice of 𝒙𝒙 = 𝟒𝟒𝟒𝟒𝟓𝟓

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(b)

Figure 15. The distribution of 𝝎𝝎𝒙𝒙 on the slice of 𝒙𝒙 = 𝟒𝟒𝟒𝟒𝟓𝟓 with stream traces (black lines) at 𝒕𝒕 = 𝟔𝟔.𝟗𝟗𝟎𝟎𝟎𝟎, where 𝟎𝟎 is the period of T-S wave.

D. The fouth and fifth hairpin vortices generation The process of the generation of the fourth and fifth hairpin vortices is different from the first three hairpins, see

Figure 16. The details will be developed in the complete paper.

(a) 𝒕𝒕 = 𝟒𝟒.𝟓𝟓𝟎𝟎𝟎𝟎

(b) 𝒕𝒕 = 𝟒𝟒.𝟒𝟒𝟎𝟎𝟎𝟎

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(c) 𝒕𝒕 = 𝟒𝟒.𝟔𝟔𝟎𝟎𝟎𝟎

Figure 16. The process of the fourth and fifth hairpin vortices generation in top view.

V. Conclusion A DNS simulation is carried out to reveal the development of a Λ-vortex to a hairpin vortex packet. The new

omega vortex identification method is utilized to visualize the vortices. According to the observation, the high strain rate region on the top of Λ-vortex is caused by the sweep due to the counter rotation of two legs of Λ-vortex; the generation of vortex rings is related to the moving up of high strain rate region; the separation of the first hairpin vortex is related to the development of the proturberance of the second vortex ring; the separation of the second and third hairpin vortices is similar with the first one, but the fourth and fifth hairpin generate in a different way from the first three hairpins, which will be studies in future and concluded in the complete paper.

Acknowledgments This work was supported by the Department of Mathematics at University of Texas at Arlington. The authors are

grateful to Texas Advanced Computing Center (TACC) for providing computation hours. This work is accomplished by using Code DNSUTA which was released by Dr. Chaoqun Liu at University of Texas at Arlington in 2009.

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