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Advances in Mechanical Engineering
Volume 2011 (2011), Article ID 743975, 10 pages
The Combination of Polymer, Compliant Wall, and Microbubble Drag Reduction Schemes
Institute of Thermophysics, Siberian Branch of Russian Academy of Sciences, Prospekt Ac. Lavrentyev, 1, Novosibirsk 630090, Russia
Received 9 March 2011; Accepted 30 April 2011
Academic Editor: Jinjia Wei
Copyright © 2011 Boris N. Semenov. 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.
The promising study of turbulence management by joint use of compliant coatings with other drag reduction means is proposed. Its outlooks are conditioned by different considered factors and confirmed by the first experimental and theoretical results.
The combined use of different means is one of the main principles of nature development. The study of hydrodynamic problems of bionics (Aleyev , Bushnell and Moore ) also convinces us of the correctness of this statement. Bionics is the way from observations and astonishment at making the first estimations (the conclusion about the paradox existence) to the explanation for the phenomenon.
The characteristic “nature” example of the study of bodies with low drag is the investigation of dolphins, the search of reasons of well-known paradox of Gray . These investigations showed that in consequence of long evolution dolphins possess different variants of adaptation to different, rapidly changing conditions of their inhabitation in sea (Woodcock , Focke , Semenov , Alekseeva and Semenov , Wu and Chwang ). Here, excellent variants of economical swimming of dolphins were discovered and described. For example, Woodcock  described the “motionless” swimming of dolphins near the ship nosing. Focke  investigated this fact. He showed by calculations that dolphins (using pressure distribution near the ship nosing) can swim with any ship velocity and without essential energy losses (as “external passengers of ship-travellers without tickets”). The other example: Wu and Chwang  show by theoretical calculations that dolphins can obtain an energy for their swimming from a wavy stream. So, they can swim in sea waves with minimum energy losses (quoted work permits to explain the physical essence of surf boards too). Above-mentioned results requested to introduce new, additional conditions for selection of dolphin speed observations (used for analysis of Gray’s paradox). But note: they cannot explain Gray’s paradox for observations of high-speed swimming of dolphins under conditions of the absolute calm, far from ships. And here, the other conclusion is important. As the result of long evolution, dolphins enjoyed different variants of adaption to very different and often changed residing conditions in sea. So, our aim is to search and study many dolphins “secrets”. Here, the analysis of the dolphin body shape (Young , Hertel ) was the important step to explain the observed low drag. The other important step was made by Kramer [11–13], who simulated the dolphin skin compliance in delaying the transition to turbulence. Semenov  gave the additional explanation for low drag (of dolphin Tursiops Tursio Ponticus) taking into account also the possibilities of joint use of compliant dolphin skin, water-soluble secretions decreasing drag, and gas microbubbles observed in experiments.
Technical progress is connected with this main principle of nature development (the combined use of different means) too. There are a lot of possible variants of the combined use of different (and numerous) methods of drag reduction for different hydrodynamic conditions. Two passive means (compliant coatings and riblets) and two active means (polymeric additives and gas microbubbles) are considered here in order to estimate outlooks for their joint action investigations.
2. Some Notes on Investigation Outlooks
These notes can be interesting to both research of near-wall turbulence and representatives of industry who use scientific successes. So, first of all, it is important to note that all considered methods of turbulence management (compliant coatings, riblets, air microbubbles, and PEO additives) satisfy the ecology requirements.
Motivations of fine outlooks on joint use of the considered methods of drag reduction can be divided into four groups.
2.1. Initial Approach
The initial approach to joint use of different drag reducing means took into account only the simplified dependence of possible drag reduction efficiency for their joint action on their individual efficiencies This expression is correct if all considered drag reducing means act independently and do not change the action conditions for the others (here and further, drag reduction efficiency is considered concerning turbulent friction coefficient for smooth hard surface: ).
In this case, the possible drag reduction efficiency for joint action of different drag reducing means must be less than the sum of their individual drag reducing possibilities The prognosticated negative deviation from the sum of individual efficiencies depends on their values and number of means used jointly for turbulence management.
These dependences can be analysed at ease for the variant of equal individual efficiencies: . So, the deviation from the sum of individual efficiencies is calculated as This deviation increases for increasing .
And for , it has the limit
Results of this prognosis are shown in Figure 1. The prognosticated negative deviations are small when the sum of individual efficiencies is less than 20%. But they are very considerable for 80% sum: for example, for two combined drag reducing means and for .
This approach was used for our initial estimations. Viscoelastic coatings, riblets, gas bubbles, and polymer additives are four well-known means for the action on near-wall turbulence. Their actions for the decrease of the turbulence production are very different.
Compliant surface reacts on long-wave disturbances. According to the estimation of the interference theory of Semenov  and experimental data of Kulik et al.  the real viscoelastic coating is deformed by the pressure wave with length more than one thousand viscous scales. Viscous scale is , where friction velocity is , and are density and viscosity of flow, and is friction stress on a wall (Hinze ). Small additives in a flow put out the microeddy turbulence for the turbulence linear scales less than one hundred viscous scales (Greshilov et al. ). Riblets manage microeddy structures too (Choi ). The flowing screen of gas bubbles can destroy long-wave powerful fluctuations going to the wall from the turbulent core and background flow (Bogdevich et al. ).
It is known (Hinze , Cantwell ) that in the main both microeddies of viscous sublayer and long waves of turbulent core generate a new turbulence. So, the joint use of considered methods of drag reduction gives possibility to wait for new qualities of turbulence production decrease. Therefore, the combined use of these four methods permits to obtain the best results in turbulent drag reduction as compared with above described prognosis.
2.2. Association of Useful Qualities
A study of joint use of different methods of drag reduction is promising because of a number of other reasons too. It is attractive as a possible origin of other useful properties (in addition to drag reduction possibility) which are inherent in separate methods.
For example, drag reducing compliant coatings can have high anticorrosion properties. One-layer coatings created in Institute of Thermophysics of Russian Academy of Sciences (Kulik et al. ) have the excellent immunity to a damage by acids and alkalis.
Another example: tests carried out by Russian and Bulgarian scientists (Malyuga et al. ) show that creation of an air-bubble layer in a near-wall region is a sufficiently effective method for reducing the amplitudes of the propelled-induced pressures and the plate vibrations for ships.
And thirdly, for joint use of compliant coatings, air microbubbles, and polymeric additives, it is possible to suppress turbulent wall-pressure fluctuations in the very wide frequency band that is impossible for any method used separately. So, it is possible to believe that these combinations will lead to the strong decrease of the hydrodynamic noise in the very wide frequency band too.
2.3. Here, It Is Important to Take into Account the Economic Factor
The turbulence management by compliant coatings and riblets is particularly useful due to their passive nature. As a result, additional energy is not required for the turbulence control. The injection of gas microbubbles and polymer additives is connected with consumption of some energy and materials. Although drag reduction by the high-molecular polymer additive use is realized for its very small concentration in a flow, expenses for its use may be higher than the economy (for example) of expenses for fuel. Therefore, Berman  suggested to estimate the specific efficiency , determining the expediency of drag reduction. He showed for a flow in a pipe that decreased as the concentration increased (for a flow with constant concentration of polymer additives) and was significantly less at the friction minimization than the specific efficiency at moderate values of drag reduction . It is connected with nonlinear form of dependence of on and asymptotic achievement of maximum value of drag reduction. Semenov [25, 26] carried out analogous analysis for a flow with variable concentration of polymer additives in a flow (for turbulent boundary layer on a plate) and showed that from the point of view of profit, it is worthwhile not to tend to the drag minimization but to restrict drag reduction nearly twice (). So, the combined investigations must be carried out for variants of small consumptions of PEO too. And only the joint use of the considered methods can permit to achieve maximum and profitable efficiency of drag reduction.
The similar situation is realized for drag reduction using gas bubbles. However, in this case, it is possible even to achieve drag reduction “free of charge” by the use of engine exhaust.
2.4. “Mutual Aid” of Different Drag Reducing Means
And after all, here, it is necessary to enumerate to some other factors of an interaction between jointly used methods of turbulence management. They are subject to a study as proposed factors of “a mutual aid” promoting to an appearance of new qualities.
The flowing screen of gas bubbles destroys powerful fluctuations going to a wall from the turbulent core and background flow. So, the bubble screen defends polymer additives acting with high efficiency just in a near-wall region. It decreases their ousting from this region.
The drag reducing polymers (polyethylene oxide, polyacril amide, etc.) are surface-active substances which decrease the surface tension and so the separation diameter of a bubble at its generation on the porous injecting insert. Besides, polymer additives in flow prevent the bubble coalescence and also impede bubble rising. Note that it is very important for drag reduction to have microbubbles with diameter less than 0.2 mm. The decrease of microbubble diameter leads to an improvement of screening properties of a bubble layer, to a displacement of the peak concentration of gas bubbles in a water flow to a wall, and to a decrease of the bubble buoyancy velocity. Hence, one can expect that the flow of high-polymer solutions aerated by gas bubbles will result in mutual increase of effects of drag reduction on a streamlined surface (Malyuga et al. [27, 28]).
Waves and eddies are responsible for the near-wall turbulence production near a smooth surface. The wave action role is decreased as a result of the surface roughness increase. Compliant coatings respond to the pressure fluctuation waves. So, the viscoelastic boundary action losses a physical sense as a result of high roughness of a surface (Semenov and Semenova [29, 30]). The increase of the viscous sublayer thickness by polymer additives increases the permissible roughness of a compliant surface that simplifies and cheapens the coatings preparation technology.
Semenov and Semenova [29–31] have carried out the first calculations for joint action of compliant boundary and polymer additives in the turbulent boundary layer in order to explain the obtained experimental results (Semenov et al. [32, 33] and Kulik et al. [34, 35]). One of possible factors of an interaction between two considered methods of turbulence management is the action of compliant boundary on mass transfer in a near-wall region. Carried-out calculations show that the decrease (increase) of mass transfer, achieved by the use of a viscoelastic coating, decreases (increases) the polymer consumption a little. The other factor is the influence of polymer additives in a flow on the interference action of viscoelastic boundary on near-wall turbulence. The calculations show that injected polymer additives extend the phase-frequency region of positive action of compliant boundary; that is, they extend possibilities of drag (and noise) reduction by compliant coatings. These two problems are described in Section 4 in details.
Semenov and Semenova [29, 30] considered the action of drag reducing riblets for joint use with compliant coating and concluded that it extends the phase-frequency region of positive action of compliant boundary too.
The viscoelastic coating for drag reduction is the mechanical vibrational system with amplitude-phase-frequency characteristic chosen for action on the near-wall turbulence spectrum band responsible for the main production of new turbulence. And, of course, this choice must take into account the existence of natural turbulence background conditions. However, both for different usual experimental hydrodynamic installations and for practical objects (ships and pipe-lines) the existence of additional strong pressure fluctuations in a flow is quite possible. These additional pressure fluctuations can swing the compliant coating in the frequency region of its negative action very essentially. So, the total production of new turbulence (for all frequency regions) can be even increased. The important factor of the gas bubble layer action is the defence of a near-wall region of the turbulent boundary layer. So, the injection of gas bubbles into a near-wall flow will ensure the stable drag reduction action of viscoelastic coating for different exploitation conditions.
Further, the following indexes are used for meaning: compliant surface—, polymer additives—, air microbubbles—, riblets—, and joint use—their combinations.
3. Experimental Investigations
The quantity of experimental investigations is small still. Only some variants of joint use of different drag reducing means were considered.
Already the first experiments (carried out at the Institute of Thermophysics RAS) for joint use of compliant coatings and polymer additives (Semenov et al. [32, 33]) showed fine outlooks of this study. There was obtained that the total effectiveness of turbulent drag reduction is equal to the algebraic sum of individual small efficiencies of these methods of turbulence management. These successes initiated new investigations.
3.1. Experimental Conditions
The experiments were carried out in the saline lake Issyk-Kool, where 2.1 m—long, 0.175 m—diameter streamline body of revolution was towed by the tow boat with speed m/s.
This model (see Figure 2) was described in details formerly by Kulik et al. [16, 36]. It was equipped (in the middle of its length) with 0.66 m—long “floating” surface element for measuring the skin-friction drag. There were tested different variants of these cylindrical elements. One had a solid smooth surface, and the others were mounted with compliant coatings. Careful measurement of friction coefficient for the case of hard polished surface in water flow was used for comparison as a standard.
The model nosing had a ring slot for polymeric solution injection. The model was equipped with the 35-mm long insert made from porous metal for air injection. Sizes of injected microbubbles varied from 0.07 mm to 0.2 mm.
All experiments were carried out for low background turbulence conditions. The spectrum analysis of measured wall-pressure fluctuations (see the example in Figure 3) in frequency band from 10 Hz to 10 kHz revealed strong peaked deviation from smooth distribution in frequency only for low frequencies (below 20 Hz), that is inessential for these investigations.
All experimental conditions were described in details by Semenov et al. .
3.2. Joint Action of Compliant Coatings and Polymer Additives
New results of these investigations were described by Kulik et al. [34, 35] and Semenov et al. . There the mass consumption of polyethylene oxide (PEO of different molecular mass ) was varied. The corresponding dimensionless parameter is , where is diameter of the measured “floating” element, : density of PEO, : thickness of turbulent boundary layer calculated for water flow (with temperature ) without polymer additives for the middle abscissa of the “floating” element (with solid smooth surface). According to Kutateladze and Leontyev , the thicknesses of diffusion and dynamic turbulent layers near this “floating” element are approximately equal. So, is like to the near-wall concentration of PEO for the middle abscissa of the “floating” element.
The first experimental results of Semenov et al. [32, 33] showed that is shifted concerning so as ; that is, the summarizing property was discovered for the joint of compliant coating and polymer additives that confirmed our initial prognosis for small individual effectivenesses. However, contrary to initial estimations, it was noted that for the case of the increase of separate effects the magnitude of combined drag reduction exceeded their sum. So, further, it is considered the deviation of the drag reduction efficiency for joint action from the sum of the drag reduction efficiencies for separate actions in order to investigate this summarizing property.
Shown in Figure 4 are data (from Semenov et al. ) for the joint use of different compliant coatings (both decreasing and increasing the turbulent friction) and polymer additives (for the great variation of polymer consumption and, accordingly, ). These results witness the existence of three zones:(1)the zone of the exact sum of individual efficiencies ;(2)the zone of positive deviation ;(3)the zone of negative deviation .
Here, the zone of the exact sum is observed for all tested variants till . Zones of positive and negative deviations follow the zone of the exact sum when polymer consumption increases. But here, we see considerable differences for different tested variants.
Experimental results from Figure 4 are shown in Figure 5 again for their comparison with initial prognosis. Here, these results are considered in dependence on drag reduction of hard surface by polymer additives, that is, on individual efficiency of polymer additives .
According to (1), . The prognosticated deviation must be So, in this case, the deviation must be negative for a “positive coating” () and positive for a “negative coating” ().
The deviations prognosticated according to (5) (shown in Figure 5 by lines) are contrary to experimental data for the second and third zones. Thus, these results show the presence of an interaction of compliant coating and polymeric additives. So, above-mentioned zones can be termed as(2) the zone of positive interaction of two considered methods of drag reduction (with ),(3) the zone of negative interaction of two considered methods of drag reduction (with ).
3.3. Joint Action of Air-Microbubbles and Polymer Additives
Malyuga et al. [27, 28] carried out first experiments on drag reduction using the injection of PEO (WSR-301)—solutions aerated by air bubbles. They measured the friction in 3 points of the hard flat plate from distance 0.25 m (N1), 0.99 m (N2) and 2.23 m (N3) behind the slot for –10 m/s. They determined that an aeration of injected PEO solutions can lead to an increase of their efficiency of drag reduction. The maximum additional increase of their efficiency was measured: 36% in point N2 and 16% in point N3 but in point N2 were measured both an increase and a decrease of drag reduction efficiency. And here, the results were worse for an increase of PEO consumption. It is important to note that highly large consumption of injected air and polymer was used in this experiment. The corresponding dimensionless parameters were
Here, is the surface of studied plate part, and is the volumetric consumption of injected air.
Some above-mentioned results and new data (Semenov et al. ) obtained in experiments (described in Section “Experimental conditions”) for very small consumption of air and polymer are shown in Figure 6. Here, we can see the same three zones: the zone of the exact sum, zones of positive and negative interaction.
Note that the negative interaction zone corresponds to very high consumption of PEO and air.
3.4. Joint Action of Compliant Coating and Air Microbubbles
The first experiment is described by Semenov et al. . One compliant coating was tested for very small consumption of injected air . m/s, °C. Drag reduction of hard surface by air-microbubbles varied from 7% to 14%. There was obtained that the total efficiency of turbulent drag reduction is equal to the sum of individual efficiencies .
3.5. Joint Action of Riblets and Surface Compliance
According to theoretical estimations of Semenov and Semenova , this combination must be the fine variant of passive (without energy expenditure) methods of turbulent drag reduction.
But experimental data are still absent.
3.6. Joint Action of Riblets and Polymer Additives
The first experimental results were described by Reidy and Anderson  and Choi et al. . They found out that individual efficiencies of two methods of drag reduction are summed up for their joint use. Note that they considered very small consumption of polymers.
Koury and Virk  and Virk and Koury  investigated this problem in detail: for two polyethyleneoxides (and ) and one polyacrylamide (), in two hydraulically smooth pipes of 7.82 mm and 10.2 mm i.d. and in four riblets pipes formed by, respectively, lining each of the smooth pipes with 0.11 mm and 0.15 mm V—groove riblets of equal height and spacing. Within the polymeric regime, at moderate drag reductions of order 50%, drag reduction in the riblet walled pipe significantly exceeded that in the smooth pipe, by as much as 15%. But the greatest drag reduction by riblets in water was measured ~10%. So, the positive deviation from the exact sum of individual efficiencies is observed here. At conditions of asymptotic maximum drag reduction, of order 80%, friction factors in the present riblet-walled pipe were identical to smooth for but departed off the smooth asymptote in the direction of lesser drag reduction for . And here, the negative interaction is observed.
3.7. Joint Action of Riblets and Air Microbubbles
The opinion about the promising study of this combination is based on an expectation that riblets and air microbubbles manage with very differed structures of turbulence. But both experimental and theoretical investigations were not carried out still.
3.8. Joint Action of Compliant Coating, Air Microbubbles, and Polymeric Additives
The first experiment is described by Semenov et al. . Russian scientists measured the friction of a floating cylindrical element (see “Experimental Conditions” here). They carried out tests for very small consumption of air and PEO. They used the one-layer compliant coating tested also by Choi et al.  after this experiment. Results are shown in Figure 7. Here, the positive deviation increases monotonously with increasing consumption of air and PEO. It shows the presence of an interaction of compliant coating, air microbubbles, and polymer additives in the whole region of this investigation.
Note that the effectiveness of drag reduction for joint use of compliant coating, air microbubbles, and PEO additives exceeded the sum of individual efficiencies by as much as 11% (for ).
4. Theoretical Analysis of Interaction between Compliant Boundary and Polymer Additives
The discovered peculiarities of drag reduction using a complex of different methods of turbulence management require theoretical explanations.
Compliant coatings and polymer additives manage with very differed structures of near-wall turbulence. So, both methods of drag reduction are independent according to this point of view.
But the other factor of an interaction between compliant boundary and polymer additives is a possible reason of observed contradictions between experimental data and initial prognosis: a change of action conditions of one method by other method of drag reduction.
4.1. The Considered Influence of the Viscoelastic Boundary on the Turbulent Diffusion of Polymer Additives
One possible factor of an interaction between two considered methods of turbulence management is the action of compliant boundary on mass transfer in a near-wall region. Here, the integral approach was used. The calculation analysis was carried out on the base of approximate model  for a flat plate analogous to the construction scheme tested in quoted experiments [32–35] described here in Section “Experimental Conditions”.
It is supposed that the slot injection of PEO solutions at satisfies the conditions of pulseless injection of polymeric additives into a near-wall flow . Here, the constant efficiency of drag variation using compliant coating (independent on polymer additives in flow) is considered from to . is the body length. For this part of the body, it was calculated The local friction reduction by PEO additives is determined according to the formula grounded in 
The near-wall concentration of PEO may be determined according to the experimental data of Fabula and Burns  as
The thickness of turbulent boundary layer is determined as where , is the kinematic coefficient of water viscosity, for and . Here, the existence of laminar boundary layer from to is proposed. In the point of transition from laminar form a flow to a turbulent one (at ) the condition of continuity of momentum thickness is written. On its base, the initial thickness of turbulent boundary layer at is determined. Here, the power form of the velocity profile with index 1/11 was taken.
So, the friction coefficient (without polymer injection) is calculated according to the Falkner's formula 
The system of (8), (9), (10) is solved for given molecular , dimensionless coefficient of PEO consumption , Reynolds number . After its solution, the drag variation (for ) and drag reduction (for ) are calculated according to (7). On the base of these calculations, the deviation of drag reduction for joint use of compliant surface and polymer additives from the sum of efficiencies for separate actions is determined.
Carried out calculations show that the mass transfer decrease (increase) by the use of viscoelastic coating decreases (increases) the polymer consumption a little. So, it is unlikely that it is the main factor of the interaction between these two methods of turbulence management. However, this approach can and must be taken into account for future investigations and accurate analysis.
One example is shown in Figure 8. We see that in both considered cases ( and ), the calculated deviations (points) differ from the initial prognosis (lines) inessentially.
4.2. The Interference Action of Viscoelastic Boundary on Near-Wall Turbulence in Flow with Polymer Additives
Here, the other factor of interaction between two methods of drag reduction (the influence of polymer additives in a flow on the interference action of viscoelastic boundary on near-wall turbulence) is considered.
Formerly, the interference form of a compliant boundary action was analysed by Semenov [15, 46] for a turbulent near-wall flow of Newtonian fluids. He used the near-wall turbulence model of Sternberg . The main modeling parameter (written by Semenov for solution of the problem ) is the complex dimensionless compliance of a boundary. He determined the region of this parameter values for drag reduction [48–50]. This theoretical model was used for modeling and choice of one-layer compliant drag reducing coatings. These coatings provided up to 20% drag reduction in experiments [16, 36]. They were used in above-written experimental combined investigations of different methods of turbulence management too.
Here, the interference approach is used for a compliant boundary of a water flow with PEO additives. In this case is suitable the former solution  of the problem on an interaction between a viscoelastic boundary and the viscous sublayer of a turbulent boundary layer. Here, we take into account that PEO additives in a flow do not change long-wave structures, the ratio of wave numbers for transverse () and main () directions.
Drag reduction by polymer additives, a change of the velocity profile , viscosity and wave velocity are taken into account in calculations. It is important to note that the increase of the viscous sublayer thickness by polymer additives increases the region of permissible use of the linear theory near a wall.
The complex compliance of the boundary (the modeling parameter) is characterised by amplitude and phase of the boundary displacement relative to the turbulent pressure fluctuation. This parameter must be determined for the frequency band of the main production of turbulence. The increase of permissible amplitudes of viscoelastic boundary oscillations follows the increase of thickness of a viscous sublayer.
The obtained solution  shows the restriction of the phase region for positive action of a viscoelastic boundary (for drag reduction). This positive action is connected with the decrease of near-wall turbulence production. For fixed frequency (, where is cyclic frequency) the production change of the turbulence energy should be Index “” corresponds to a compliant boundary. The interference action of a compliant boundary for fixed frequency is neutral if this integral is equal to zero. According to the near-wall turbulence model of Sternberg , the calculated viscous sublayer thickness is connected with the fluctuation frequency as .
For the neutral action variant, the mean velocity profile is written according to the experimental data for a hard wall.
The improved interference theory (presented by Semenov and Semenova ) was used for first calculations of joint action of a compliant boundary and polymer additives.
Neutral phase-frequency lines (calculated according to condition (12)) restrict (from below) a region of for positive action of compliant boundary (). One example for is shown in Figure 9 (for two variants of the abscissa). The phase shift of the compliant boundary displacement relative to acting fluctuating pressure is on the ordinate. The dimensionless frequency is on the abscissa. In Figure 9(a), it is made dimensionless by the use of real flow viscosity near a wall and real friction velocity . In Figure 9(b), it is made dimensionless by the use of kinematic viscosity of water and friction velocity without drag reduction in order to compare the different influences of drag reducing polymer additives for identical conditions of a water flow.
We see that injected polymer additives extend the phase-frequency region of positive action (PFRPA) of compliant boundary. This extension of PFRPA is maximum at .
The injection of drag reducing polymeric additives into a flow leads to a displacement of PFRPA to the left that can lead even to the change of the action sign of compliant boundary (from “+” to “−” and on the contrary).
We see that from the right branch of the neutral line is displaced distinctly to the left. So, minimum velocity of possible drag reduction using compliant coating must increase with the increasing individual efficiency of drag reducing polymeric additives. For example, it must increase to two times at .
The used theoretical approach does not permit still to carry out a quantitative comparison. It is a problem for future investigations.
So, this exposition shows fine outlooks of further study of turbulence management by joint use of compliant coatings, riblets, polymer additives, and microbubbles.
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