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Study on multiple ring-like vortex formation and small vortex generation in late flow transition on a flat plate

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Abstract

A high-order direct numerical simulation (DNS) of flow transition over a flat plate at a free stream Mach number 0.5 has been carried out. During the simulation, we cannot find, according to the classical theory of boundary layer transition, the “hairpin vortex breakdown to smaller structures” in the last stage of flow transition on a flat plate. However, we did discover the so-called spikes as a result of a multibridge or multiring formation. This indicated a large and stable vortex structure which can travel for a long distance. We believe that this is a result of ring heads that are located in an inviscid region. These heads of the “turbulence spot” never seem to break down and persist as a stable structure. In addition, we discovered that the U-shaped vortex is a part of an existing coherent structure instead of a newly generated one. The U-shaped vortex also provides an additional channel to transfer vorticity to the ring from the wall. During travel, the leading primary ring in the front of the spot is sloped and skewed, causing disappearance of the second sweep. As a consequence, no energy is brought down to the lower boundary layer near the vortex ring head of the spot. Thus, the small length scale structures become damped and the existing U-shaped vortex structure becomes distinguishable. So, the question is where do the turbulent small length scale vortices come from? We will address this with a new theory which states that all of small length scales (turbulence) are generated by high-shear (HS) layers rather than being produced by “vortex breakdown.” The new DNS shows that the HS layers are produced by strong positive spikes surrounded by low-speed fluids and by the interaction between the secondary and higher-level vortices and the wall surface especially near the ring neck. This multiple ring-like vortex generation also follows the first Helmholtz vortex conservation law. Furthermore, the Λ-shaped vortex is formed and rolling up, and the Λ-vortex is stretched and narrowed, and a new bridge is generated after the neck. The bridge will further become a second ring and so on. Besides the original vortex legs, there are also U-shaped vortex tubes. Finally, the multiple ring vortex structure is formed. From this process, no evidence is found to support that two consequent multiring circles are mixed to generate small vortices. The connection of downdraft/updraft motions is also studied.

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Abbreviations

M ∞ :

Mach number

\({\delta _{in}}\) :

Inflow displacement thickness

T ∞ :

Free stream temperature

Lx :

Length of computational domain along x direction

Ly :

Length of computational domain along y direction

x in :

Distance between leading edge of flat plate and upstream boundary of computational domain

A 2d :

Amplitude of 2D inlet disturbance

ω :

Frequency of inlet disturbance

β :

Spanwise wave number of inlet disturbance

γ :

Ratio of specific heats

ω x :

Streamwise vorticity

λ 2 :

Const representing vortex tube surface

Re :

Reynolds number

T w :

Wall temperature

Lz in :

Height at inflow boundary

A 3d :

Amplitude of 3D inlet disturbance

α 2d :

Streamwise wave number of inlet disturbance

R :

Ideal gas constant

μ ∞ :

Viscosity

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Authors and Affiliations

  1. Department of Mathematics, University of Texas at Arlington, Arlington, TX, 76019, USA

    Chaoqun Liu, Lin Chen, Ping Lu & Xiaobing Liu

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  1. Chaoqun Liu
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  2. Lin Chen
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  4. Xiaobing Liu
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Correspondence to Chaoqun Liu.

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Liu, C., Chen, L., Lu, P. et al. Study on multiple ring-like vortex formation and small vortex generation in late flow transition on a flat plate. Theor. Comput. Fluid Dyn. 27, 41–70 (2013). https://doi.org/10.1007/s00162-011-0247-5

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  • DOI: https://doi.org/10.1007/s00162-011-0247-5

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