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Modelling and Simulation in Engineering
Volume 2012 (2012), Article ID 619419, 22 pages
http://dx.doi.org/10.1155/2012/619419
Research Article

New Theories on Boundary Layer Transition and Turbulence Formation

Department of Mathematics, The University of Texas at Arlington, Arlington, TX 76019, USA

Received 26 January 2012; Accepted 25 March 2012

Academic Editor: Guan Heng Yeoh

Copyright © 2012 Chaoqun Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is a short review of our recent DNS work on physics of late boundary layer transition and turbulence. Based on our DNS observation, we propose a new theory on boundary layer transition, which has five steps, that is, receptivity, linear instability, large vortex structure formation, small length scale generation, loss of symmetry and randomization to turbulence. For turbulence generation and sustenance, the classical theory, described with Richardson's energy cascade and Kolmogorov length scale, is not observed by our DNS. We proposed a new theory on turbulence generation that all small length scales are generated by “shear layer instability” through multiple level ejections and sweeps and consequent multiple level positive and negative spikes, but not by “vortex breakdown.” We believe “shear layer instability” is the “mother of turbulence.” The energy transferring from large vortices to small vortices is carried out by multiple level sweeps, but does not follow Kolmogorov's theory that large vortices pass energy to small ones through vortex stretch and breakdown. The loss of symmetry starts from the second level ring cycle in the middle of the flow field and spreads to the bottom of the boundary layer and then the whole flow field.

1. Introduction

These comments clearly show that the mechanism of turbulence formation and sustenance is still a mystery for research. Note that both Heisenberg and Lamb were not optimistic for the turbulence study.

1.1. Turbulence Is a Mystery

Turbulence is still covered by a mystical veil in nature after over a century of intensive study. Following comments are made by Wikipedia web page at http://en.wikipedia.org/wiki/Turbulence “Nobel Laureate Richard Feynman described turbulence as “the most important unsolved problem of classical physics” (USA Today 2006). According to an apocryphal story, Werner Heisenberg (another Nobel Prize winner) was asked what he would ask God, given the opportunity. His reply was “When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first.” [1]. HoraceLamb was quoted as saying in a speech to the British Association for the Advancement of Science, “I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic” [2, 3].

1.2. Richardson’s Vortex and Energy Cascade Theory (1928)

Richardson believed that a turbulent flow is composed by “eddies” of different sizes. The large eddies will be stretching, unstable, and breaking up to smaller eddies. These smaller eddies undergo the same process, giving rise to even smaller eddies. This process will continue until reaching a sufficiently small length scale such that the viscosity of the fluid can effectively dissipate the kinetic energy into internal energy. During the process of vortex breakdown, the kinetic energy of the initial large eddy is divided into the smaller eddies.

1.3. Kolmogorov Assumption (1941)

The classical theory on turbulence was given by Kolmogorov, a famous Russian mathematician. In general, the large scales of a flow are not isotropic, because they are determined by the particular boundary conditions. Agreeing with Richardson, Kolmogorov assumed that in Richardson's energy cascade, this geometrical and directional information is lost while the scale is reduced so that the statistics of the small scales have a universal character—they are statistically isotropic for all turbulent flows when the Reynolds number is sufficiently high. It was assumed that there is no dissipation during the energy transfer from large vortex to small vortex through “vortex breakdown.”

1.4. Kolmogorvo’s First and Second Hypotheses (1941)

Based on his assumption, Kolmogorov [6, 7] further gave very famous theories on the smallest length scale, which is later called Kolmogorov scale (first hypothesis), and turbulence energy spectrum (second hypothesis): 𝜈𝜂=3𝜀1/4,𝐸(𝑘)=𝐶𝜀2/3𝑘5/3,2𝜀=𝜈𝜕𝑢1𝜕𝑥12+2𝜕𝑢2𝜕𝑥22+2𝜕𝑢3𝜕𝑥32+𝜕𝑢2𝜕𝑥1+𝜕𝑢1𝜕𝑥22+𝜕𝑢3𝜕𝑥2+𝜕𝑢2𝜕𝑥32+𝜕𝑢1𝜕𝑥3+𝜕𝑢3𝜕𝑥12,(1) where, 𝜂 is Kolmogorov scale, 𝜈 is kinematic viscosity, 𝜀 is the rate of turbulence dissipation, 𝐸 is the energy spectrum function, 𝐶 is a constant and 𝜅 is the wave number. These formulas were obtained by Kolmogorov’s hypotheses that the small length scales are determined by 𝜈 and 𝜀, and 𝐸 is related to 𝜅 and 𝜀. These formulas are unique according to the dimensional analysis [8].

If all small vortices are generated by “shear layer instability” but not “vortex breakdown”, Kolmogorov’s first hypothesis will lose the background. The smallest length scales will be related to stability of the weakest unstable “shear layer”. This will require a deep study of “shear layer instability” not only “dimensional analysis”.

1.5. A short Review of Study on Late Boundary Layer Transition

The transition process from laminar to turbulent flow in boundary layers is a basic scientific problem in modern fluid mechanics. After over a hundred years of study on flow transition, the linear and weakly non-linear stages of flow transition are pretty well understood [10, 11]. However, for late non-linear transition stages, there are still many questions remaining for research [4, 1216]. Adrian [17] described hairpin vortex organization in wall turbulence, but did not discuss the sweep and ejection events and the role of the shear layer instability. Wu and Moin [18] reported a new DNS for flow transition on a flat plate. They did obtain fully developed turbulent flow with structure of forest of ring-like vortices by flow transition at zero pressure gradients. However, they did not give the mechanism of the late flow transition. The important mechanism of boundary layer transition such as sweeps, ejections, and positive spikes cannot be found from that paper. Recently, Guo et al. [19] conducted an experimental study for late boundary layer transition in more details. They concluded that the U-shaped vortex is a barrel-shaped head wave, secondary vortex, and is induced by second sweeps and positive spikes.

In order to get deep understanding the mechanism of the late flow transition in a boundary layer and physics of turbulence, we recently conducted a high order direct numerical simulation (DNS) with 1920 × 241 × 128 gird points and about 600,000 time steps to study the mechanism of the late stages of flow transition in a boundary layer at a free stream Mach number 0.5 [2032]. The work was supported by AFOSR, UTA, TACC and NSF Teragrid. A number of new observations are made and new mechanisms are revealed in late boundary layer transition (LBLT) including:(i)mechanism on secondary and tertiary vortex formation;(ii)mechanism on first ring-like vortex formation;(iii)mechanism on second sweep formation;(iv)mechanism on high share layer formation;(v)mechanism on positive spike formation;(vi)mechanism on multiple ring formation;(vii)mechanism on U-shaped vortex formation;(viii)mechanism on small length vortices generation; (ix)mechanism on multiple level high shear layer formation;(x)mechanism of energy transfer paths;(xi)mechanism on symmetry loss or so called “flow randomization;” (xii)mechanism on thickening of turbulence boundary layer;(xiii)mechanism of high surface friction of turbulent flow.

A 𝜆2 technology developed by Jeong and Hussain [33] is used for visualization.

1.6. Different Physics Observed by Our High-Order DNS

According to our recent DNS, “vortex breakdown” is not observed [24, 29, 30]. Liu gave a new theory that “turbulence is not generated by vortex breakdown but shear layer instability” [24, 30, 32]. Liu believes that “shear layer instability” is the “mother of turbulence.”

As Richardson’s energy cascade and Kolmogorov’s assumption about “vortex breakdown” are challenged, Kolmogorov’s assumption that the smallest vortices are statistically isotropic becomes questionable. Since the smallest vortices are generated by the “shear layer instability,” which is closely related to the shape of body configuration, they cannot be isotropic.

1.7. Questions to Classical Theory on Boundary Layer Transition

The classical theory, which considers “vortex breakdown” as the last stage of boundary layer transition on a flat plate, is challenged and the phenomenon of “hairpin vortex breakdown to smaller structures” is not observed by our new DNS [24, 27, 28]. The so-called “spikes” are actually a process of multibridge or multiring formation, which is a rather stable large vortex structure and can travel for a long distance.

As experiment is quite expensive and has very limited power in date acquisition, direct numerical simulation (DNS) becomes a more and more important tool to discover physics. The purpose of this work is to find physics of turbulence by direct numerical simulation. The paper is organized as follows: Section 1 is a background review; Section 2 shows the case set up and code validation; Section 3 is our observation and analysis; Section 4 is a summary of our new finding; Section 5 is the conclusions which are made based on our recent DNS.

1.8. New Theory on Boundary Layer Transition Given by Liu

Classical theory on boundary layer transition can be described four stages: (1) boundary layer receptivity; (2) linear instability; (3) non-linear growth; (4) vortex breakdown to turbulence. Apparently, we disagree with the classical theory on “vortex breakdown to turbulence.” The new theory of boundary layer transition can be described by five stages: (1) boundary layer receptivity; (2) linear instability; (3) large vortex structure formation; (4) small vortices generation; (5) symmetry loss and turbulence formation. By the way, the vortex cascade in turbulence given by Richardson, Kolmogorov and others is not observed.

1.9. New Theory on Turbulence Generation and Sustenance Given by Liu

The new theory on turbulence formation and sustenance can be described as follows. All small length scales (turbulence) are generated by shear layer instability which is produced by large vortex structure with multiple level vortex rings, multiple level sweeps and ejections, and multiple level negative and positive spikes near the laminar sublayers. Therefore, “turbulence” is not generated by “vortex breakdown” but rather positive and negative spikes and consequent high shear layers. “Shear layer instability” is considered as the “mother of turbulence.” This new theory may give a universal mechanism for turbulence generation and sustenance—the energy is brought by large vortex structure through multiple level sweeps in a boundary layer.

2. Case Setup and Code Validation

2.1. Case Setup

The computational domain is displayed in Figure 1. The grid level is 1920 × 128 × 241, representing the number of grids in streamwise (𝑥), spanwise (𝑦), and wall normal (𝑧) 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 (𝑌+=0.43). The parallel computation is accomplished through the message passing interface (MPI) together with domain decomposition in the streamwise direction (Figure 2). The flow parameters, including Mach number and Reynolds number are listed in Table 1. Here, 𝑥in represents the distance between leading edge and inlet, 𝐿𝑥, 𝐿𝑦, 𝐿𝑧in are the lengths of the computational domain in 𝑥-, 𝑦-, and 𝑧-directions, respectively, and 𝑇𝑤 is the wall temperature.

tab1
Table 1: Flow parameters.
619419.fig.001
Figure 1: Computation domain.
619419.fig.002
Figure 2: Domain decomposition along the streamwise direction in the computational space.
2.2. Code Validation

The DNS code—“DNSUTA” has been validated by NASA Langley and UTA researchers [24, 32, 34] carefully to make sure the DNS results are correct.

2.2.1. Comparison with Linear Theory

Figure 3 compares the velocity profile of the 𝑇-𝑆 wave given by our DNS results to linear theory. Figure 4 is a comparison of the perturbation amplification rate between DNS and LST. The agreement between linear theory and our numerical results is quite good.

fig3
Figure 3: Comparison of the numerical and LST velocity profiles at Rex = 394300.
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Figure 4: Comparison of the perturbation amplification rate between DNS and LST.
2.2.2. Grid Convergence

The skin friction coefficient calculated from the time-averaged and spanwise-averaged profile on a coarse and fine grid is displayed in Figure 5. The spatial evolution of skin friction coefficients of laminar flow is also plotted out for comparison. It is observed from these figures that the sharp growth of the skin-friction coefficient occurs after 𝑥450𝛿in, which is defined as the “onset point.” The skin friction coefficient after transition is in good agreement with the flat-plate theory of turbulent boundary layer by Cousteix in 1989 [35]. Figures 5(a) and 5(b) also show that we get grid convergence in skin friction coefficients.

fig5
Figure 5: Streamwise evolutions of the time- and spanwise-averaged skin-friction coefficient.
2.2.3. Comparison with Log Law

Time-averaged and spanwise-averaged streamwise velocity profiles for various streamwise locations in two different grid levels are shown in Figure 6. The inflow velocity profiles at 𝑥=300.79𝛿in is a typical laminar flow velocity profile. At 𝑥=632.33𝛿in, the mean velocity profile approaches a turbulent flow velocity profile (log law). This comparison shows that the velocity profile from the DNS results is a turbulent flow velocity profile and the grid convergence has been realized.

fig6
Figure 6: Log-linear plots of the time- and spanwise-averaged velocity profile in wall unit.
2.2.4. Spectra and Reynolds Stress (Velocity) Statistics

Figure 7 shows the spectra in 𝑥- and 𝑦-directions. The spectra are normalized by 𝑧 at location of Re𝑥=1.07×106 and 𝑦+=100.250. In general, the turbulent region is approximately defined by 𝑦+>100 and 𝑦/𝛿<0.15. In our case, The location of 𝑦/𝛿=0.15 for Re𝑥=1.07×106 is corresponding to 𝑦+350, so the points at 𝑦+=100 and 250 should be in the turbulent region. A straight line with slope of 3/5 is also shown for comparison. The spectra tend to tangent to the 𝑘5/3 law.

fig7
Figure 7: (a) Spectra in 𝑥-direction (b) Spectra in 𝑦-direction.

Figure 8 shows Reynolds shear stress profiles at various streamwise locations, normalized by square of wall shear velocity. There are 10 streamwise locations which are selected starting from leading edge to trailing edge. As expected, close to the inlet at Re𝑥=326.8×103 where the flow is laminar, the values of the Reynolds stress is much smaller than those in the turbulent region. The peak value increases with the increase of 𝑥. At around Re𝑥=432.9×103, a big jump is observed, which indicates the flow is in transition. After Re𝑥=485.9×103, the Reynolds stress profile becomes close to each other in the turbulent region. So for this case, we can consider that the flow transition starts after Re𝑥=490×103.

619419.fig.008
Figure 8: Reynolds stress.
2.2.5. Comparison with Other DNS

Although we cannot compare our DNS results with those given by Borodulin et al. [4] quantitatively, we still can find that the shear layer structure are very similar in two DNS computations in Figure 9.

fig9
Figure 9: Qualitative comparison of contours of streamwise velocity disturbance u in the (𝑥,𝑧)-plane (light shades of gray correspond to high values).
2.2.6. Comparison with Experiment

By this 𝜆2-eigenvalue visualization method, the vortex structures shaped by the nonlinear evolution of 𝑇-𝑆 waves in the transition process are shown in Figure 10. The evolution details are briefly studied in our previous paper [20] and the formation of ring-like vortices chains is consistent with the experimental work ([5], Figure 11) and previous numerical simulation by Bake and his coauthors [15].

fig10
Figure 10: Evolution of vortex structure at the late-stage of transition (Where 𝑇 is the period of 𝑇-𝑆 wave).
fig11
Figure 11: Evolution of the ring-like vortex chain by experiment [5].
2.2.7. U-Shaped Vortex in Comparison with Experimental Results

Figure 12(a) [19] represents an experimental investigation of the vortex structure including ring-like vortex and barrel-shaped head (U-shaped vortex). The vortex structures of the nonlinear evolution of 𝑇-𝑆 waves in the transition process are given by DNS in Figure 12(b). By careful comparison between the experimental work and DNS, we note that the experiment and DNS agree with each other in a detailed flow structure comparison. This cannot be obtained by accident, but provides the following clues: (1) both DNS and experiment are correct, (2) disregarding the differences in inflow boundary conditions (random noises versus enforced 𝑇-𝑆 waves) and spanwise boundary conditions (nonperiodic versus periodic) between experiment and DNS, the vortex structures are same, (3) no matter K-, H- or other types of transition, the final vortex structures are same, (4) there is a universal structure for late boundary layer transition; (5) turbulence has certain coherent structures (CS) for generation and sustenance.

619419.fig.0012
Figure 12: Qualitative vortex structure comparison with experiment.

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

3. Observation and Analysis done by Our DNS

Following observations have been made and reported by our previous publications [2028, 3032, 36, 37].

3.1. Mechanism of Large Coherent Vortex Structure

Late boundary transition starts from the formation of the first vortex ring.

3.1.1. Vortex Rings in Flow Field [23]

According to Helmholtz vorticity conservation law, the vortex tube cannot have ends inside the flow field. The only form of vortex tube existing inside the flow filed must be ring with or without legs (Figure 13).

fig13
Figure 13: Vortex rings with or without legs.
3.1.2. Mechanism of First Ring-Like Vortex Formation [23]

The ring shape of first ring-like vortex is caused by the interaction between the prime streamwise vortices and secondary streamwise vortices (Figure 14), which is quite different from the one given by Moin et al. [38]. The ring-like vortex is located at the edge of the boundary layer or in other words located in the inviscid zone (𝑧=3.56,𝑈=0.99𝑈𝑒). The vortex ring is perpendicularly standing and the shape of the ring is almost perfectly circular, which will generate the strongest downwash sweeps (Figures 14 and 15). More important, no vortex “pinch off” is observed.

fig14
Figure 14: Formation processes of first ring-like vortex.
619419.fig.0015
Figure 15: The shape and position of ring-like vortices in boundary layer (𝑧=3.56,𝑈=0.99𝑈𝑒).
3.1.3. Mechanism of Multiple Ring Formation [37]

The ejection by rotation of the primary vortex legs brings low speed flow from lower boundary layer up and forms a cylinder-like momentum deficit zone in the middle of the two legs (green in Figure 16). The momentum deficit zone is also located above the legs and forms shear layers due to the stream-wise velocity difference between surrounding high-speed flow and low-speed flow in the deficit zone. These shear layers are not stable and multiple ring-like vortices (yellow in Figure 16) are generated by the shear layers one by one (Figure 17). This process must satisfy the Helmholtz vorticity conservation and the primary streamwise vorticity must be reduced when a new spanwise-oriented ring is formed (Figure 18). Different from Borodulin et al. [4], neither “vortex breakdown and reconnection” nor “Crow theory” is observed by our new DNS.

619419.fig.0016
Figure 16: Three-dimensional momentum deficit and multiple ring-like vortex formation.
fig17
Figure 17: (a) Sketch for mechanism of multiring generation, (b) multiple ring formation.
619419.fig.0018
Figure 18: Maximum streamwise and spanwise vorticity along 𝑥-coordinate (Helmoholtz vorticity conservation).
3.1.4. Primary, Secondary, and U-Shaped Vortices [31]

U-shaped vortex is part of large coherent vortex structure and is generated by secondary vortices which are induced by the primary vortex. Being different from Guo et el. [19], the U-shaped vortex is not a wave and is not induced by second sweeps and positive spikes. Actually, the U-shaped vortex is a tertiary (not secondary) vortex with same vorticity sign as the original ring legs (Figures 19 and 20). In addition, the U-shaped vortices serve as additional channels to supply vorticity to the multiple ring structure [32].

619419.fig.0019
Figure 19: U-shaped vortex with streamwise velocity perturbation contour.
619419.fig.0020
Figure 20: Isosurface of 𝜆2 and stream trace at 𝑥=530.348𝛿in.
3.2. Mechanism of Small Length Vortices Generation

Turbulence has two futures: (1) small length vortices; (2) nonsymmetric structure (Randomization).

3.2.1. “Vortex Breakdown to Turbulence” is Challenged [29, 30]

In the new DNS [27, 28], it has been observed that the multiple vortex structure is a quite stable structure and never breaks down. Previously reported “vortex breakdown” is either based on 2-D visualization or made by using low pressure center as the vortex center (Figure 21(a)). We can use a different 𝜆2 value to get similar “vortex breakdown” (Figure 21(b)), which is faked. Here, we define vortex as a tube with a rotated core and isosurfaces of constant vorticity flux. However, there is no evidence that the vortex breaks down. Let us look at the head of the so-called “turbulence spot” from different directions of view (Figure 22). Due to the increase of the ring (bridge) number and vorticity conservation, the leading rings will become weaker and weaker until they cannot be detected, but they never break down (see Figure 23).

fig21
Figure 21: (a) U-shaped vortex [9], (b) DNS with a different 𝜆-value (fake).
fig22
Figure 22: View of young turbulence spot head from different directions (𝑡=8.8𝑇).
fig23
Figure 23: 3D angle view and 2D top view of the young turbulence spot (𝑡=8.8𝑇).
3.2.2. Mechanism of Small-Length Scale (Turbulence) Generation—by “Shear Layers” [30, 32]

The question will be raised that where do the small length scales come from if the small vortices are not generated by “vortex breakdown”? Since we believe that the small vortices are generated by shear layers near the wall surface, we take snap shots in the direction of view from the bottom to top (Figure 24). The evidence provided by our new DNS confirms that the small length scale vortices are generated near the bottom of the boundary layer. Being different from classical theory which believes turbulence is generated by “vortex breakdown”, our new DNS found that all small length scales (turbulence) are generated by high-shear layers (HS) near the bottom of the boundary layers. There is no exception. When we look at the later stage of flow transition at 𝑡=15𝑇, we can see that all small length scales are located around the high-shear layers, especially at those near the wall surface (Figures 25 and 26).

fig24
Figure 24: Small length scale vortex generation at different time steps (view up from bottom): small length scale vortices are generated by the solid wall near the ring necks from the beginning to the end.
fig25
Figure 25: Visualization of isosurface 𝜆2 and velocity perturbation at 𝑥=508.633 for (𝑌, 𝑍)-Plane 𝑡=15𝑇.
619419.fig.0026
Figure 26: Visualization of isosurface 𝜆2 and velocity perturbation at 𝑦=4 for (𝑋, 𝑍)-Plane 𝑡=15𝑇.
3.2.3. High-Shear Layer Formation by Sweeps [30]

Experiment and our numerical results have confirmed that there is a second sweep excited by every ring-like vortex (Figure 27). Combined with the first sweep generated by the original 𝜆 vortex legs, it forms strong positive spikes which generate strong high shears at the bottom of the boundary layer (red in Figure 28). Figures 29 and 30 are contours of isosurface of 𝜆2 and velocity perturbation at 𝑥=508.66. The second sweep movement induced by ring-like vortices working together with first sweep will lead to huge energy and momentum transformation from high-energy inviscid zones to low-energy zones near the bottom of the boundary layers and we can observe that all small length scales are generated around high shear layers between the positive spike (momentum increment) and solid wall surface.

619419.fig.0027
Figure 27: Velocity vector field.
619419.fig.0028
Figure 28: Vorticity distribution at 𝑥=508.633.
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Figure 29: :Isosurface of 𝜆2 and velocity at 𝑥=508.66.
619419.fig.0030
Figure 30: Isosurface of 𝜆2 and stream traces at 𝑥=508.66.
3.2.4. Multiple-Level Sweeps and Multiple-Level Negative and Positive Spikes [32]

The positive spikes (momentum increment) could generate new ring-like vortices. The new ring-like vortices can further generate new sweeps and form new positive spikes (Figure 31) at the location which are very close to the bottom of the boundary layer (laminar sublayer). The new positive spike could induce new smaller vortex rings by unstable shear layers. These multilevel sweeps and multilevel negative and positive spikes provide channels for energy transfer from the inviscid area (high energy area) down to the bottom (low energy area). This is the mechanism why turbulence can be generated and sustained. This mechanism can be interpreted as a universal mechanism for both transitional and turbulent flow in a boundary layer.

fig31
Figure 31: (a) Multilevel sweeps; (b) multilevel positive spikes.
3.2.5. Energy Transfer Paths and Universal Turbulence Spot Structure [32]

Figures 32 and 33 are sketches describing energy transfer and likely universal turbulence structure.

619419.fig.0032
Figure 32: Energy transfer paths and universal structure for turbulence.
619419.fig.0033
Figure 33: Sketch of mechanism of multiple rings formation and small vortices formation.
3.2.6. Surface Friction

Figure 34 is a time and spanwise averaged surface friction coefficient (CF). From the figure, it is clearly seen that there is a jump starting at 𝑥=430 which indicates an onset of flow transition and reach a maximum value at 𝑥=508.663. It is conventionally believed that the CF is large in the turbulent area and small in the laminar area due to the strong mixing in turbulent flow. That is the reason why most turbulence models are formulated based on change of turbulent viscosity. However, from Figure 34, it is easy to find that the CF reaches maximum in a laminar area and is not directly related to mixing. Further analysis found the viscosity is not changed for incompressible flow, but the shear layer is changed sharply when the first and second sweeps caused positive spikes (momentum increment), HS layers and consequent small vortices generations. The velocity gradients suddenly jump to a very large level in the laminar sub-layer and then the CF becomes very large. Therefore, the CF jump is not caused by mixing or viscosity coefficients increase, but is, pretty much, caused by velocity gradient jump due to small vortices formation.

619419.fig.0034
Figure 34: Maximum CF at 𝑥=508.633 (Laminar).
3.3. Loss of Symmetry (Randomization)

The existing theory [41] believes that the flow randomization in DNS is caused by the background noises and removal of periodic boundary conditions. They believe that randomization starts from loss of symmetry, which is caused by background noise. According to their theory, the ring tip is affected first, which rapidly influences the small length scale structure. This will lead to loss of symmetry and randomization for the whole flow field quickly.

However, what we observed in our new DNS is that the loss of symmetry starts from second level rings (Figure 35) while the top and bottom rings are still symmetric (Figure 36). The nonsymmetric structure of second level rings will influence the small length scale at the bottom quickly. The change of symmetry in the bottom of the boundary layer is quickly spread to the up level through ejections. This will lead to randomization of the whole flow field. Therefore, the internal instability of multiple level ring spot overlapping structure, especially the middle ring cycles, is a critical reason for flow randomization, but mainly not the background noise. In addition, the loss of symmetry starts in the middle of the flow domain, not the inflow and not the outflow (Figure 37(a)). As mentioned, we did not change the periodic boundary conditions and the solution is still periodic in the spanwise direction (period=2𝜋 is enforced). In addition, we did not add any additional background noise or inflow perturbation. However, we found that the flow lost symmetry first and then was randomized step by step (see Figures 38(a)38(d)). The flow was first periodic (period=𝜋) and symmetric in the spanwise direction, 𝑛𝑘=0𝑎𝑘cos(2𝑘𝑦), but then lost symmetry in some areas, and finally everywhere. It was periodic with a period of 𝜋, same as the inflow, but changed with a period of 2𝜋. The flow is still periodic because we enforced the periodic boundary condition in the spanwise direction with a period of 2𝜋. This means the flow does not only have 𝑛𝑘=0𝑎𝑘cos(2𝑘𝑦) but also has 𝑛𝑘=0𝑏𝑘sin(2𝑘𝑦) which is newly generated. Meanwhile, flow lost periodicity with period = 𝜋, but has to be periodic with period = 2𝜋 (Figures 38(c) and 38(d)), which we enforced. Since the DNS study is focused on the mechanism of randomization and the DNS computation only allows use of two periods in the spanwise direction, we consider that the flow is randomized when the symmetry is lost and period is changed from 𝜋 to 2𝜋 (Figures 38(a) and 38(b)): 𝑓(𝑦)=𝑛𝑘=0𝑎𝑘cos(2𝑘𝑦)+𝑛1𝑘=1𝑏𝑘+sin(2𝑘𝑦)𝑛1𝑘=0𝑐𝑘cos(2𝑘𝑦+𝑦)+𝑛1𝑘=0𝑑𝑘sin(2𝑘𝑦+𝑦).(2) In real flow, there is no such a restriction of periodic boundary condition in the spanwise direction. A more detailed discussion about the mechanism of flow randomization will be given by another AIAA 2012 paper [36].

fig35
Figure 35: The flow lost symmetry in second level rings and bottom structure at 𝑡=15.0𝑇.
619419.fig.0036
Figure 36: The top ring structure is still symmetric at 𝑡=15.0𝑇.
fig37
Figure 37: The whole flow field lost symmetry at 𝑡=21.25𝑇.
fig38
Figure 38: (a) Whole domain is symmetric and periodic ---𝑎×cos2𝑦 (the period = 𝜋; spanwise domain is 𝜋<𝑦<𝜋), (b) symmetric and asymmetric—redrectangular: periodic and symmetric at 𝑦=𝜋/2, 0, 𝜋/2, that is, 𝑓(𝜋/2𝑦) = 𝑓(𝜋/2+𝑦), 𝑓(𝑦) = 𝑓(𝑦), 𝑓(𝜋/2𝑦) = 𝑓(𝜋/2+𝑦); yellow diamond: periodic, 𝑓(𝑦+𝜋) = 𝑓(𝑦), period = 𝜋; but asymmetric 𝑓(𝑦)𝑓(𝑦); the spanwise domain is 𝜋<𝑦<𝜋, (c) periodic but asymmetric—yellow diamond: periodic, period = 𝜋; black box: periodic but period = 2𝜋; the spanwise domain is 𝜋<𝑦<𝜋. (d) Periodic but asymmetric—all black boxes: periodic but asymmetric (period=2𝜋); the spanwise domain is 𝜋<𝑦<𝜋.

4. Summary of Our DNS Findings

4.1. Our New Observations

Since turbulence generation and sustenance are one of the top secret in nature, this research will bring significant impact on fundamental fluid mechanics as the classical theory and dominant concepts on late flow transition and turbulence structure are challenged. Table 2 gives a comparison of classical theory and currently dominant conclusions with the observation by our DNS.

tab2
Table 2: Comparison of classical theory and our DNS observation.
4.2. New Theory on Turbulence Formation and Sustenance by Liu
4.2.1. Classical Theory on Turbulence

A turbulent flow is characterized by a hierarchy of scales through which the energy cascade takes place. According to Richardson, turbulence is random interactions of “eddies” as “big whorls have little whorls, little whorls have smaller whorls, that feed on their velocity, and so on to viscosity” (Figure 39 by [39]). On the other hand, vortex stretching is the core mechanism on which the turbulence energy cascade relies to establish the structure function. According to Richardson (1928) and Kolmogorov [6, 7], the radial length scale of the vortices decreases and the larger flow structures breakdown into smaller structures (Figure 40 by [40]). The process continues until the small-scale structures are small enough to the extent where their kinetic energy is overwhelmed by the fluid's molecular viscosity and dissipated into heat.

619419.fig.0039
Figure 39: Sketch of vortex breakdown [39] (Tsubota et al, 2009). A vortex ring (a) can break up into smaller rigs if the transition between states (b) and (c) allowed when the separation of vortex lines becomes of atomic dimensions. The eventual small rings (d) may be identical to rotons.
619419.fig.0040
Figure 40: Sketch of Richardson cascade process [40].

Richardson’s energy cascade theory has several important points of view which can be summarized as follows:(1)stremwise vortex is stretching;(2)vortex must break down to smaller structure due to the stretching;(3)energy transfers from large vortex to smaller one through “vortex breakdown;”(4)these smaller vortices continue the same process until the smallest one appears (Kolmgorov’s length scale) in which viscosity is dominant;(5)the micro-scale 𝜂 (Kolmogorov scale) is several order smaller than the macro-scale 𝐿 (main flowscale) and the middle length scales 𝑟 (𝜂𝑟𝐿) are basically inertial.

4.2.2. New Observations and Analyses

On the other hand, our new observations can be described as follows:(1)stremwise vortex is stretched but never breaks down;(2)vortex has strong rotation which causes strong mixing;(3)in the middle of two vortex legs, there is a low-speed zone formed by ejection from boundary layer bottom to bring momentum deficit up to this area. This momentum deficit forms a long-shear layer;(4)this shear layer in the middle of two legs is unstable and multiple rings with spanwise vorticity will form one by one to generate multiple rings near the inviscid region due to the shear layer instability especially near the narrowed vortex neck where rotation is very strong and the low speed momentum deficit is large. In this process, Helmholtz vorticity conservation must be satisfied;(5)in the two sides of the vortex legs, high-speed zones are formed by sweeps which bring high energy (energy increment) from inviscid area to the near wall region. The energy increment will form shear layers near the wall; (6)the momentum increment shear layers near the wall are unstable (shear layer instability) and will form multiple smaller ring-like vortices;(7)these newly generated large and smaller vortices will continue a similar process to generate smaller ring-like vortices until the vortex size is too small and the new shear layer becomes stable. The shear layer stability analysis should be further studied; (8)energy is transferred from the large size vortices to small size vortices through multiple level sweeps. Dissipation is inevitable during the energy transfer; (9)all smaller vortices are newly generated by shear layer instability, but not by original vortex “breakdown;”(10)smallest length scales are all generated near the wall surface which must be related to the surface configuration and cannot be “isotropic;” (11)surface drag is determined by small length scale generation, which must be closely related to turbulence modeling; (12)“shear layer” control is the key issue for flow transition control, turbulent flow control, and drag reduction.

4.3. Possible Impact on Fluid Mechanics
4.3.1. Impact on Fundamental Fluid Mechanics

Since turbulence generation and sustenance are one of the top mysteries in nature, this research will bring significant impact on fundamental fluid mechanics as the classical theories on late flow transition and turbulence structure are challenged. LBLT classical theory, Richardson eddy cascade, Kolmogorov first and second hypotheses should be all revisited.

4.3.2. Impact on DNS and LES

As shown by the new DNS, it is not observed that “Kolmogorov scale” is the smallest turbulence scale but is found that the smallest length scale is determined by “shear layer instability”. The estimate of energy containing scale, 𝜂min/𝜂ref=(1/Re)3/4, is too conservative. For example, an engineering hydrodynamics problem has commonly a Reynolds number, Re=108. Considering 3D in space and 1D in time, the requirement for computation would be 𝑄=Re4×3/4=(108)3=1024. This is apparently impossible for any currently available computer or any near future computers. However, researchers in US National Research Labs and many others around the world do “implicit LES” which is really “coarse grid DNS”. They do DNS, but are afraid to call DNS but “ILES”. The reason is that they are afraid of “Kolmogorov scale.” Since we could not find that energy is transformed from large vortices to small vortices by “vortex breakdown”, “Kolmogorov scale” would lose solid scientific foundation and must be revisited. This will provide guidance to DNS which is supposed to resolve all energy-contained small length scales and LES which is expected to resolve the large eddies and model the subgird scales.

4.3.3. Impact on Turbulence Modeling

The eddy viscosity turbulence model is mainly based on “Boussinesq Assumption” (1872) that the nonlinear perturbation terms can be modeled by resolved mean scales with new turbulent coefficients of viscosity. However, most turbulence models are still empirical and case-related. The right turbulence models can be developed with deep understanding of turbulence generation and sustenance. As shown by our DNS, the perturbation and small length scales are not isotropic but configuration related. Yes, Prandtl may be right that the turbulence can be described by “mixing lengths” which represent the length of different size vortices. Although Prandtl's “mixing length” model is an old concept [42], turbulence is really generated by mixing of variety of different size vortices and the “mixing length” concept coincides with physics. The serious question is how to find the right size of “mixing length”. We should develop a new "mixing length" model based on DNS data and "shear layer stability” analysis.

4.3.4. Impact on Flow Control of Flow Transition and Turbulence

Flow transition and turbulence control should be based on right understanding on flow transition and turbulence structure. Based on our observation, it is proposed to cut the energy transfer channels for high shear layer generation to stop or delay turbulence generation. It is also found that the “shear layer instability” is the “mother of turbulence.” Apparently, control of shear layer formation and structure is the key issue to the control of the transitional flow and fully developed turbulent flow. It is also an important way for drag production of turbulent flow, for example, why oscillating wall and rough wall with riblets can reduce the surface drag.

5. Conclusions

In our recent DNS work [2032, 36, 37] we have made a number of new findings on late flow transition in a boundary layer including the mechanism of large vortex structure formation, small-length scale generation and flow randomization (loss of symmetry).

New theory on flow transition and turbulence generation and sustenance has been given by Liu. Instead of four steps, that is, receptivity, linear instability, non-linear growth, vortex breakdown to turbulence, described by classical theory, Liu proposed a new theory with five steps, that is, receptivity, linear instability, large vortex formation, small length scale generation, loss of symmetry and randomization to turbulence. For turbulence generation and sustenance, the classical theory, described with Richardson’s energy cascade and Kolmogorov length scale, is not observed by our DNS. Liu proposed a new theory on turbulence generation that all small length scales are generated by “shear layer instability” through multiple level ejections and sweeps and consequent multiple level positive and negative spikes, but not by “vortex breakdown” which can never happen. We believe “shear layer instability” is the “mother of turbulence.” Flow randomization is caused by the internal property of instability of the structure of multiple vortex-ring-spot overlapping.

Nomenclature

𝑀:Mach number
Re: Reynolds number
𝛿in:Inflow displacement thickness
𝑇𝑤:Wall temperature
𝑇:Free stream temperature
𝐿𝑧in:Height at inflow boundary
𝐿𝑧out:Height at outflow boundary
𝐿𝑥:Length of computational domain along x direction
𝐿𝑦:Length of computational domain along y direction
𝑥in:Distance between leading edge of flat plate and upstream boundary of computational domain
𝐴2𝑑:Amplitude of 2D inlet disturbance
𝐴3𝑑:Amplitude of 3D inlet disturbance
𝜔:Frequency of inlet disturbance
𝛼2𝑑,𝛼3𝑑:Two and three dimensional streamwise wave number of inlet disturbance
𝛽:Spanwise wave number of inlet disturbance
𝑅:Ideal gas constant
𝛾:Ratio of specific heats
𝜇:Viscosity.

Acknowledgments

This work was supported by AFOSR Grant FA9550-08-1-0201 supervised by Dr. John Schmisseur and 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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