DNS Study of Turbulence Structure in a Boundary Layer

This DNS study is focused on the structure of turbulence in a boundary layer. First, it is not appropriate to call the turbulent flow as a “random” motion. Any mean flow like Blasius solution, which is a Navier-Stokes solution, added by white noise or random perturbation would be rejected by Navier-Stokes equations and the perturbation will be damped quickly. The governing Navier-Stokes equations only accept certain structured flows and turbulent flow is a solution that the Navier-Stokes equations can accept. Turbulent flow is not random solutions because the governing Navier-Stokes equations do not allow. The DNS finding shows turbulence is built up by organized “vortex packages” which can be accepted by Navier-Stokes equations. Since these packages keep moving around, it would make a fake impression that the flow is “random”. The so-called “intermittence” really represents the motion of the “vortex packages”, self-motion and relative-motion, and the packages never disappear or destroyed periodically in the turbulent flow. The package itself must have certain structure to generate a variety of different size vortices and keep the energy transport from high energy inviscid flow to the viscous boundary layer bottom. The small size vortices need energy to survive to balance the viscous dissipation. This paper will provide a detailed observation and analysis on the formation of these vortex packages. Unfortunately, the “vortex breakdown” and “energy cascade” proposed by Richardson and accepted by Kolmogorof are never observed by any DNS or experiment. Following observations have been made by our recent DNS (Chen et al. 2011; Liu et al, 2010 a, 2010b, 2010d, 2011a, 2011b; Lu et al, 2011a, 2011b 2012). First, the large multiple vortex structure is studied including the vortex legs (rotation cores), first vortex ring, multiple vortex rings, stretch of the rings, secondary vortex and tertiary vortex. Here, a 2 λ visualization method developed by Jeong et al (1995) is used. Second, the sweeps and ejections induced by the large vortex rings are found very important. They produce low speed zones (negative spikes) in the upper boundary layer and high speed zones (positive spikes) near the bottom and further generate high shear layers. These multiple level high shear layers will further generate smaller vortices of different size at different level due to the shear layer instability. Therefore, the small vortices (and turbulence) are not generated by “vortex breakdown” which can never happen, but by multiple level shear layers near the solid wall surface. Therefore, these vortex packages have certain structure and are pretty stable with the energy transport channel to bring the energy down from the high energy inviscid area to the viscous boundary layer bottom. In this way, the vortex packages can be self-supported. The reason why the packages are sometime clear and sometime look like break or disappear is that the package is filled and surrounded by countless small vortices. Actually, these large vortex structures in the package are stable and must be kept in order to survive in a viscous flow. Of course, the structure of the package keeps changing inside and moving around. The packages can never stop and the relative motion between these packages can never stop either if the mean flow is not zero. This is the reason why the turbulent flow look likes “random”. The reason why turbulent flow consists of these packages is because only these packages can satisfy the Navier-Stokes equations. The mean flow with random perturbations will be rejected by Navier-Stokes equations. Apparently, these turbulence packages satisfy mass, momentum and energy conservation law and have the channel to supply enough energy for survival of the dissipative small vortices, but not the base flow like Blasius solution with random perturbations. In addition, we can find four vortices merge to two and two to one, but did not see one vortex break to two vortices and two break to four vortices. This may be restrained by the second thermodynamics law. 1. Sketch of turbulence structure Turbulence consists of many vortex packages (Figure1). Each vortex package has certain large vortex structure including vortex legs, multiple rings, secondary and tertiary vortices. The small vortex structure is generated by shear layers which are produced by sweeps and ejections due to large vortex rotation and consequent positive (high speed zone) and negative (low speed zone) spikes. These spikes will generate high shear layers and further multilevel vortex rings due to the shear layer instability (Figure 2.) Therefore, small vortices (turbulence) are not generated by “vortex breakdown” (impossible) but multiple level shear layer instability (Kelvin-Helmohotz type). Each vortex package has multiple level sweeps to bring the energy from high energy inviscid zone to the low level boundary layers to support those dissipative small vortices to survive (Figure 3.) These large and small vortex structures will further lose the symmetry and become disordered due to the internal instability of these multiple level vortex structure (Figure 4). Some people may argue that there is no accurate definition of vortex which could be vortex tube as most people think, blobs of vorticity, rotation centers or even vortex sheets (Davidson, 2004). However, small vortices (turbulence) can never be generated by ‘vortex breakdown” in any sense and the Richardson eddy cascade is not observed by any DNS or experiments. (1) Front view (2) Side view (3) Top view Figure 1 Turbulence consists of vortex packages Figure 2 Sketch of vortex package Figure 3 Energy brought down from inviscid zone to bottom through multiple level sweeps (a) Section view in y-z plane (b) Bottom view of positive spike Figure 4 The Flow lost symmetry in second level rings at t=15.0T 2. Late flow transition control Turbulence onset starts at the location of the first vortex ring formation. In order to delay the flow transition, we must delay the first ring formation. Whenever the first vortex ring is formed, turbulence generation is not avoidable. However, the small vortex generation is directly caused by strong sweeps which is generated by the first vortex ring (Figure 5). Changing the layout direction of large vortex rings could weaken the sweep and, therefore, to delay the small vortex formation. For example, an ejection of counter vortex may delay the first vortex ring formation and then delay the transition. The perfect circular shape and perpendicularly standing vortex rings would generate strongest sweeps and should be controlled (Figure 6.) Figure 5 Sweeps and ejections by vortex ring Figure 6 Perfectly circular shape and perpendicularly standing of the first vortex ring 3. Richardson vortex cascade revisit Figure 7 Sketch of Richardson’s cascade process (Frisch et al, 1978) Classical turbulence theory about vortex chains was given by Richardson (1924). He has a famous poem that “ big whirls have little whirls, which feed on their velocity; And little whirls have lesser whirls, and so on to viscosity in the molecular sense.” However, the vortex chain generated by large vortex breakdown is never observed. It is also hard to explain why one vortex breaks two and then four, etc. It is also hard to believe why viscosity plays role when the vortex size is equivalent to Re=1. How about eddy Re=2, 4, 6 etc. where viscosity does not play any role at all? As shown by our DNS, turbulence has different size of vortices from the large to the small. However, they are all generated by shear layer instability (K-H type) without exception and no vortex breakdown is observed. 4. Kolmogorov’s hypothesis revisit 4.1 Kolmogorov scale: The famous Kolmogorof scale is given by Russian Mathematician Kolmogorof in 1941. The scale is obtained by dimensional analysis. Assume the velocity and length related to the largest eddy are L and U , ν is the viscosity, ε is the kinetic energy, the velocity and length related to the smallest eddy are V and η , we will have the energy relation: