Γ Correlation Analysis on Volume Vorticity and Vortex in Late Boundary Layer Transition Xiangrui Dong Shuling Tian Chaoqun Liu Technical Report 2017-05 http://www.uta.edu/math/preprint/
Correlation Analysis on Volume Vorticity and Vortex in Late Boundary Layer Transition Xiangrui Dong 1, 3, Shuling Tian 2, 3, Chaoqun Liu 3, 1 National Key Laboratory of Transient Physics, Nanjing University of Science & Technology, Nanjing, Jiangsu, 210094, China 2 Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, 210016, China 3 Department of Mathematics, University of Texas at Arlington, Arlington, Texas, 76019, USA In this paper, two functions are introduced to describe turbulence generation in late flow transition. One is called the volume Omega Bar (volume Ω ), which represents the flow rotation or vortex strength. The other is called the volume vorticity, which shows the flow statistical fluctuations. Although they have very different definitions, one for fluctuation and the other for rotation, volume Ω and volume vorticity are found very highly correlated with a correlation factor greater than 0.9, which means there is a very close relation between flow fluctuation and flow rotation (vortex). While the vorticity flux keeps constant in the late flow transition through the integration over any sections either parallel or perpendicular to the flow direction, the volume Ω is fast increased along the flow direction and fast increased as the time marching during the flow transition process. This means the vortex structure is fast built up and rotation becomes more and more dominant. On the other hand, the total volume vorticity is also quickly increased, which shows the fluctuation is stronger. The flow transition is a process with significant volume vorticity increase, which was contributed by two Corresponding author. E-mail address: cliu@uta.edu
crucial factors: the length of the vorticity tubes from side boundary is quickly increased due to the vorticity line stretching, distortion and tangling and the generation of countless self-closed vorticity rings within the domain. Both the increase of the volume vorticity and the volume Ω can be a significant symbol of the flow transition from the laminar flow to turbulent flow. It also shows that vorticity (tubes or lines) cannot directly represent vortex, and should not be simply used as a signal of the turbulent transition process. Key words: DNS, vorticity, volume vorticity, volume Omega Bar, vortex, transition flow Nomenclature M = Mach number Re in = Reynolds number δ in = inflow displacement thickness µ = viscosity T = freestream temperature T w = wall temperature R = is the ideal gas constant γγ = the ratio of specific heats ρρ = freestream density UU = freestream velocity x, y, z streamwise, spanwise, normal directions Lz in = height at inflow boundary Lx Ly x in = length of computational domain along x direction = length of computational domain along y direction = distance between leading edge of flat plate and upstream boundary of computational domain 1 Introduction The significance of vorticity and vortex in fluid dynamics has been well noticed and recognized at both fundamental and applied levels in past decades. Vorticity has rigorous mathematical definition which is the curl of velocity field, while its physical interpretation is not as self-evident as its mathematical definition [1]. However, for vortex, the physical interpretation is some kind of clear, which can be defined as the rotation motion of fluid. The rotational flow of ideal fluid, first suggested by Helmholtz [2] in his three vorticity theorems, was named as vortex motions. More explanations of vortex like the sinews and muscles of the fluid and the sinews of turbulence respectively were respectively given by Kuchemann [3] and Moffatt et al. [4]. For a long time, many people track
vortex by vorticity or even directly regard vorticity magnitude as the vortex strength. Wu et al. [5] presented in his book that the name vortex tube is imprecise, since the rigorous definition of a vortex is still a controversial issue, and the side boundary of a vortex is not a vorticity surface. He also claimed a vortex as a connected fluid region with relatively high concentration of vorticity. However, Green [6] gave more detailed and different exposition, the early transition constitutes an event where vorticity is abruptly focused and ejected in a concentrated plume out of the surface layer; then it can be described as comprising a reorganization and local concentration of the existing vorticity field in a transition state. There are also other voices, vortex is not necessarily congregation of vorticity lines, but dispersion in most three dimensional cases [7], which means the vorticity in a vortex is not necessarily larger than the surrounding area in 3D cases. Transition from laminar flow to turbulent flow is always a complicated and controversial issue in fluid mechanics field. The vortices (vortex), a special existence form of fluid rotation motion, play a central role in the onset of turbulence in the late flow transition and fully developed turbulent boundary layers. In boundary layer transition, the turbulence occurs in a transitional zone following a laminar flow stretching along the wall surface. And the late transition are characterized by a rich and increasing number of complex tangle of vortices. However, the lack of a certain and unified definition of vortex has caused considerable confusions in identifying and investigating the vortex structures in transitional and turbulent flows. Lamb [8] defined vortices by using vortex tubes which has been proved to be inaccurate by many researchers [5, 9-11]. Therefore, due to the lack of the vortex definition, it is difficult to find a physical quantity (a scalar field or a vector field just like pressure or velocity vector) to be a signal directly representing the vortex increase in late flow transition or turbulence. Until now, many researchers track vortex by high concentration of vorticity. That is why some researchers hold the view that the transition is a process with vorticity increase while others believe the total vorticity is constant during the transition process [6, 12, 13]. So what is the true symbol of vortex generation and growth in the transition from laminar flow to turbulence? Several vortex identification criteria, such as method, Q and λ 2 criteria [14, 15, 9], have achieved some success and have been widely used in capturing vortex structures when applied to DNS data of a transitional boundary layer [16]. However, a case related threshold is required case by case and time by time for these vortex identification methods, and all of them cannot give a quantitative definition of vortex. In addition, zhang et al. [17] give the detailed discussion on different vortex identification methods in a review, which clearly indicates that these identification methods are sensitive to the chosen values for the vortex analysis, and quantitative analysis of the vortex will be badly
needed. Recently, a new vortex identification method called Omega method which was proposed by Liu et al. [18] appears to fix these problems. The Omega criterion is a normalized and a case independent function from 0 to 1, and has a clear physical meaning which is that vortices are located where vorticity overtake deformation. For deeper investigating the difference and correlation between vortex, vorticity and volume vorticity as well as finding the true symbol of the vortex generation and growth in the transition from laminar flow to turbulence, a boundary layer transition flow on a flat plate at a freestream Mach number of 0.5, which was simulated based on highorder direct numerical simulation (DNS), is utilized, and some view points towards vortex and vorticity are obtained in this paper. The paper is organized as follows: Section 2 presents the numerical methods and the case description; Section 3 shows the vorticity expression and new vortex definition; Section 4 provides our DNS results and addresses the detailed analyses on the comparison of vorticity, vorticity flux, volume vorticity and volume Ω and on the correlation between volume vorticity and volume Ω in the late boundary layer transition. Finally, we give some conclusions. 4 Conclusions In this paper, both vortex and vorticity characteristics in the process of boundary layer transition on a flat plate are analyzed based on our high-order direct numerical simulation (DNS). Two functions are introduced to describe turbulence generation and increase in late flow transition. Several conclusions based on our DNS results analysis are obtained as follows: (1) According to the law of vorticity flux conservation, the vorticity flux over any spanwise cross-section remains the same as the one over the side boundary. Since all vorticity tubes have constant integral of vorticity flux on any cross-section along the tube, including the self-closed vorticity tubes which come from a plane and must go back through this plane. The integral of vorticity vector over whole domain always keeps conserved although more and more vortical structures appear during the transition process. The vorticity should not be simply treated as a signal of the turbulent transition process as the vorticity flux keeps constant in all cross sections parallel to the streamwise direction. (2) The volume vorticity shows the flow statistical fluctuations. The flow transition is a process with significant volume vorticity increase, which is contributed by two crucial factors: the elongation of the vorticity tubes from side
boundary due to the vorticity line stretching, distortion and tangling and the numerous generation of the self-closed vorticity rings within the domain. (3) The volume Omega Bar (volume Ω ), which represents the rotation of vortex, equals zero in the laminar area but quickly increased during the flow transition process. Both the increase of volume vorticity and the volume Ω can be a significant symbol of the flow transition from the laminar flow to turbulent flow. (4) There is a very high correlation between the volume vorticity and the volume Ω, with the coefficient at least 0.9, that means there is a close relation between the flow fluctuation and the flow rotation (vortex), in other word, the fluctuation of the flow in transition is mainly caused by the flow rotation. Acknowledgments The work was supported by Department of Mathematics at University of Texas at Arlington. The authors are grateful to Texas Advanced Computing Center (TACC) for the computation hours provided. This work is accomplished by using Code DNSUTA released by Dr. Chaoqun Liu at University of Texas at Arlington in 2009. Xiangrui Dong also would like to acknowledge the China Scholarship Council (CSC) for financial support. References [1] Lugt, H.J. Introduction to vortex theory. Vortex Flow Press (Potomac, Maryland, 1996). [2] Helmholtz, H., About integrals of hydrodynamic equations related with vortical motions, J. für die reine Angewandte Mathematik, 55 (1858). [3] Küchemann, D., Report on the IUTAM symposium on concentrated vortex motions in fluids, Journal of Fluid Mechanics, 21(1), 1-20 (1965). [4] Moffatt, H. K., Kida, S., & Ohkitani, K., Stretched vortices the sinews of turbulence; large-reynolds-number asymptotics, Journal of Fluid Mechanics, 259, 241-264 (1994). [5] Wu, J. Z., Ma, H. Y., & Zhou, M. D. Vorticity and vortex dynamics. Springer Science & Business Media (Berlin New York, 2007). [6] Green, B. Fluid vortices. Springer Science & Business Media (Dordrecht, Boston, 2012). [7] Wang, Y., Yang, Y., Yang, G., & Liu, C., DNS Study on vortex and vorticity in late boundary layer transition, Communications in Computational Physics, 22(2), 441-459 (2017). [8] Lamb, H. Hydrodynamics. Cambridge At the university Press (New York, 1932).
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