Cadot, O

Jimenez-Gonzalez, JI

Lorite-Diez, M

Martinez-Bazan, C

Pastur, L

This study reveals that injecting a light fluid of density ρ_{b} in the recirculating bubble of a bluff body at Re≈6.4×10^{4} has a greater drag reduction potential than blowing fluid of a density greater than or equal to that of the free stream ρ. It is found that the maximum drag reduction scales as (ρ_{b}/ρ)^{-1/6}. This power law combines the ability of the recirculating bubble to diffuse the injected momentum and the effectiveness of the injection to increase the recirculating bubble length

University of Liverpool Repository

Drag reduction of three-dimensional bodies by base-blowing with various gas densitiesM. Lorite-Dı́ez,1 J. I. Jiménez-González,1 L. Pastur,2 O. Cadot,3 and C. Mart́ınez-Bazán11Departamento de Ingenieŕıa Mecánica y Minera. Universidad de Jaén. Campus de las Lagunillas, 23071, Jaén, Spain.2IMSIA-ENSTA ParisTech, 828 Bd des Maréchaux F-91762 Palaiseau, France.3School of Engineering, University of Liverpool, Liverpool L69 3GH, UK.(Dated: June 16, 2020)The study reveals that injecting a light fluid of density ρb in the recirculating bubble of a bluffbody at Re ≈ 6.4 × 104 has a greater drag reduction potential than blowing fluid of density greaterthan or equal to that of the free-stream, ρ. It is found that the maximum drag reduction scales as(ρb/ρ)− 16 . This power law combines the ability of the recirculating bubble to diffuse the injectedmomentum and the effectiveness of the injection to increase the recirculating bubble length.Since the seminal work of Wood [1], it is well knownthat injecting fluid in the recirculating bubble at the rearof two dimensional bluff bodies reduces the aerodynamicdrag. The drag reduction mechanism, later confirmed byBearman [2], is associated with a growth of the recircu-lating flow known to be intimately related to the basesuction CB (Roshko [3]) defined as minus the mean pres-sure at the base of the body. The technique is efficientto reduce drag of high speed projectiles and most of thefundamental research has focused on compressible flowswith transonic and supersonic regimes of axisymmetricbodies [4, 5]. Although the technique is also consideredto reduce aerodynamics drag in all transportation indus-tries implying blunt base bluff bodies, or to suppress bluffbodies vortex-induced vibrations [6], there is only fewfundamental investigations for three-dimensional bodiesin incompressible flows in the literature [7–9].The authors recently proposed a budget model [9]that establishes the relationship between the recirculat-ing bubble length and the volumetric injection flow rate.The model is inspired by the mechanics of the forma-tion region of vortices behind bluff bodies due to Gerrard[10]. Results are obtained from an experiment with slitsinjection at the body base, likely to be implemented inpractice since they also work as passive drag reductiondevices [11]. Lorite-Dı́ez et al. [9] distinguish two dif-ferent regimes, namely a mass regime and a momentumregime. The mass regime takes place at small injectionflow rate, in which the added fluid is considered as a pas-sive scalar, i.e. as a diffusive species that is present insuch low concentration that it has no dynamical effect onthe fluid motion itself [12, 13]. In this case, the injectedfluid inflates the recirculating bubble, whose equilibriumsize results from the extra leakages through the mixinglayers introduced by the growth of the recirculating bub-ble. These leakages are shown to be in agreement withthe entrainment velocity measured for isolated turbulentmixing layers [9]. This mass regime is associated withdrag reduction and usually referred to as base bleed ef-fects in the literature [1, 2]. Besides, at large injectionflow rate, a new regime, called the momentum regimein Lorite-Dı́ez et al. [9], is characterized by a strong flowmodification associated with a bubble shortening. In thiscase, the injected fluid is directly evacuated from the re-U∞p∞cµ, ρlO xz6-axis Load cellPositioning systemqbGas supply ValvePressure scanner piwhyzOUb(b)(a)sFigure 1. (a) Sketch of the experimental set-up. (b) Baseof the bluff body depicting the pressure taps location (withblack dots) and the blowing slit of surface s placed and thebottom of the body base.circulating bubble by its own momentum and the bubblesize equilibrium results from additional leakages in themixing layers. The momentum regime is associated witha drag increase, and the transition between the mass andthe momentum regimes is then easily defined as the in-jection flow rate producing the largest bubble length orequivalently the lowest drag.The aim of the present contribution is to includedensity effect for this drag reduction mechanism, knownto modify the stability properties of wake [14]. We willshow in this rapid communication that the transitionfrom the mass to the momentum regime, where thelowest drag is observed, is governed by the ability of therecirculating bubble to diffuse the injected momentum.The key result is that a lighter gas allows to increase theinjection velocity for a given momentum, extending themass regime to a larger maximum injection rate, andthus producing a longer bubble along with a lower drag.However, the bubble growth efficiency with the injectionrate is found to be reduced with a lighter gas.The experimental set-up is identical to that of Lorite-2CB0.20Cq Cq(ρb/ρ)1/30 0.015 0.02Cq(ρb/ρ)1/2(a) (b) (c)0.160.170.180.190.215×10−3 0.010.015 0.020  5×10−3 0.01 0.015 0.020 5×10−3 0.01AirHeCO2momentummassFigure 2. Base suction coefficient CB versus the volumetric flow rate coefficient Cq (a), and the scaled coefficients Cq(ρb/ρ)α (band c) as power laws of the density ratio. Vertical dashed lines in (a) indicate the transition from the mass to the momentumregimes for each gas as described in Lorite-Dı́ez et al. [9]Dı́ez et al. [9]. Briefly, the body shown in Fig. 1 hasdimensions l = 291 mm, w = 97.25 mm and h = 72 mm.It is supported by four cylindrical rods of 7.5 mm in di-ameter with a ground clearance c = 20 mm = 0.278h.It is placed inside a 390 mm × 390 mm test section ofan Eiffel-type wind tunnel. The velocity of the uniformstream is U∞ = 13.3 m.s−1 giving a Reynolds number Re= ρU∞h/µ ≈ 6.4×104, where µ is the dynamic viscosityand ρ the density of air. The turbulent intensity is below0.5% and the velocity homogeneity over the test sectionbetter than 0.3% [see 15, for further details]. Blowing ofAir, Carbon Dioxide and Helium with density ρb is car-ried out through a single narrow horizontal slit placed atthe bottom of the body base, having dimensions 87.25mm × 2 mm and a surface s = 0.025hw (see Fig. 1b).Their density ratios are ρb/ρ =1, 1.648 and 0.148 forAir, CO2 and He respectively. As sketched in Fig. 1(a),the slit is connected to an internal cavity, pressurised byfour pneumatic tubes at each corner which are joined toa single pneumatic injection tube. This injection tubeincludes an Aalborg digital flow meter specifically cali-brated for each gas, a precision valve and a pressure reg-ulator to ensure a constant volumetric flow rate qb [see 9,for details]. The volumetric flow rate blowing coefficientis defined asCq = qb/(hwU∞). (1)In the following, quantities with an asterisk superscriptare made dimensionless using h and U∞.Pressure measurements are performed through fourdifferent pressure taps, placed at (y∗, z∗) = (±0.3,±0.2)as indicated in Fig. 1, and connected to a ScanivalveZOC22 pressure scanner working at a sampling rate of100 Hz per channel. As established by Grandemangeet al. [16], such an arrangement can be considered as therelevant minimal set of measurements to assess the meanbase suction coefficient [3, 17] defined asCB = −14Σ4i=1pi − p∞ρU2∞/2, (2)where pi denotes the mean pressure at the location i andp∞ is the reference static pressure at the inlet of the testsection. The duration for the averaging is 60 s and theuncertainty in the base suction coefficient is estimated tobe ±0.002.The drag force exerted on the body is measured witha multi-axial load cell (model AMTI-MC3A-100lb) asshown in Fig. 1(a) and acquired at a sampling rate of1 kHz. The uncertainty of the measurements is estimatedto be 0.002 N. Since we are interested in the mean dragwithout any contribution produced by the blowing sys-tem itself, we subtract to the mean force obtained fora given base blowing and wind speed, fd(Cq, U∞), themean force produced by the blowing alone (without windflow), fd(Cq, U∞ = 0). The mean value then readsC̃d =fd(Cq, U∞) − fd(Cq, U∞ = 0)hwρU2∞/2. (3)Hence, C̃d is the aerodynamic drag coefficient whenthe blowing jet effects are removed. The duration forthe averaging is 30 s, being the experimental protocol toobtain accurate values fully described in [9].The near wake velocity fields are measured using Par-ticle Image Velocimetry (PIV) in the two perpendicularplanes y∗ = 0 and z∗ = 0. The PIV system uses a dualpulse laser (Nd:YAG, 2 x 135mJ, 4ns) synchronizedwith a FlowSense EO, 4 Mpx, CCD camera. Statisticsare performed over 500 velocity fields sampled at 10Hz. Velocities were computed from an interrogationwindow of 16 pixels × 16 pixels with an overlap of 50%,resulting in a spatial resolution of 1% of the body’sheight. Further details about the PIV procedure can be30-0.510.50z*x*0-0.510.5z*0 1 2 0 1 2x*x*00.51Ux(a)(b) (c)*1 2Figure 3. Averaged streamwise velocity U∗x contours at y∗ = 0plane along with averaged flow streamlines (grey lines). Theblack continuous line denotes the values U∗x = 0. Case with noinjection (a), cases with air (b) at COptq = 0.008 and Helium(c) at COptq = 0.021. The blowing slit is indicated with thewhite band.also found in [9].The evolution of the base suction CB with the volu-metric flow rate coefficient is shown in Fig. 2(a) for thethree injected gases. They all present a constant slope atsmall Cq over the mass regime range. The effect of thegas density is clearly observable from these curves: theheavier the gas, the steeper the slope and the shorter themass regime. Figure 2(b) suggests that a simple powerlaw scaling of density ratio to describe the mass regimeisCB = CB(Cq = 0) − a (ρb/ρ)13 Cq, (4)with a ≈ 1.67. Similarly, Fig. 2(c) suggests that a powerlaw scaling to describe the optimal blowing coefficient forwhich the minimum base suction is reached isCOptq = b (ρb/ρ)− 12 , (5)where b ≈ 8 × 10−3. Note that, the lightest gas leadsto the best reduction in base suction (with 10.6% forHelium, 6.3% for Air and 5.9% for CO2) but with less ef-ficiency since the optimal drag reduction is reached witha slope that decreases with the gas density as (ρb/ρ)13(see Eq. 4) and, thus, it is achieved at higher values ofCq. The main change in the flow with base blowing isthe growth of the recirculating bubble [1, 2, 9], as illus-trated with the mean velocity fields shown in Fig. 3. Thebubble is clearly identifiable with the streamlines patternand corresponds to the low speed region in blue contain-ing two counter rotating vortices. The black line repre-sents the loci of U∗x = 0 and encloses by definition thewhole feedback flow (i.e. U∗x < 0). The characteristic size(a)Cq = 0AirHeCO2CB0.160.180.20Lr1.3 1.4 1.5 1.6(b)* Cd0.38 0.4 0.42~Figure 4. Recirculation length L∗r (a) and drag coefficient C̃d(b) vs. base suction CB for the 3 gases and the case with noblowing.of the bubble is defined as the distance from the base tothe furthest location where the mean streamwise velocityU∗x is equal to zero. We can see that bubbles with opti-mal base blowing in Fig. 3(b,c) are elongated comparedto Fig. 3(a) and the longest bubble is achieved with He-lium which also corresponds to the largest base suctionreduction. As in Lorite-Dı́ez et al. [9] we define the bub-ble size L∗r as the average of the characteristic sizes (asdefined from above) measured in the two perpendicularplanes y∗ = 0 and z∗ = 0. It is plotted versus the basesuction in Fig. 4(a) and shows a very good correlationindependently of the gas used. It reminds the work ofBearman [2] about a 2D blunt cylinder who achieved aunique relationship using either iso-density base bleed-ing or splitter plate. This universal behavior confirmsthe relevance of the inviscid cavity models (see the re-view by [18]) for blunt bodies at large Reynolds number.These models establish a general behavior of base suc-tion evolving as positive power laws of the inverse of thecavity length. Physically, the low base pressure is pro-duced by the inviscid flow curvature around the cavity.For viscous wake flows at large Reynolds number, thistrend still remains with the recirculating bubble playingthe role of the cavity [3]. In our case, the variations ofthe bubble length L∗r are small compared to the referencevalue of L∗r = 1.45 which linearizes the power law to theaffine law as observed in Fig. 4(a).The relationship between the drag coefficient and thebase suction is straightforward since the base suction CBis the rear part contribution of the pressure drag. Thedrag coefficient thus reads C̃d = CB + CF + Cf , whereCF accounts for the forebody pressure integration andCf for the wall friction. With base blowing, neither theforebody nor the friction contributions are supposed tochange significantly, thus leading to identical variationsfor base suction and drag coefficient, ∆C̃d ≈ ∆CB . Thediscrepancy to the slope 1 observed for larger drag (i.e.smaller bubble length obtained with large blowing) inFig. 4(b) is likely to be due to the blowing jet effectsthat cannot be completely removed using Eq. (3) becauseof the interaction of the blowing jet with the flow. In40 0.5 10 0.5 10 0.5 10.30.20.10y*−0.5−0.4−0.3−0.6Cq = 00 0.5 1Ux*AirHex*Figure 5. Mean velocity profiles of the mixing layer that de-velops from the bottom edge of the base. Time averagedvelocity profiles, U∗x at several streamwise locations for thenon-blowing case and two blowing flow rates employing Heand Air which produce the same recirculation region length,L∗r = 1.54.addition, the low velocity of the incoming flow U∞ =13.3 m.s−1 makes our data more scattered due to theforce balance limitation than those presented in Lorite-Dı́ez et al. [9], working at U∞ = 20 m.s−1. Overall,the drag coefficient evolves monotonously with the basesuction as expected.The bubble growth model in the mass regime [9] resultsfrom the equilibrium between mass fluxes feeding andemptying the recirculating region as introduced by [10].Based on simple arguments they obtainL∗r = L∗r0 +wV ∗E`Cq, (6)where ` = 2h+ 2w is the peripheral length of the mixinglayers bounding the recirculation bubble.Figure 5 shows the mixing layer that develops at thebottom of the base, from the z∗ = −0.5, x∗ = 0 edge.The case with no injection is showed as a reference, thecase with air injection at Cq = 0.008 corresponds tothe end of the mass regime and with Helium injectionat Cq = 0.014 approximately corresponds to the mid-dle of the mass regime. These two cases of injection havebeen chosen because they lead to the same bubble length,L∗r = 1.54 and then to the same base suction. The ve-locity profiles indicate that the Helium injection has lesseffect on the mixing layer development than Air injec-tion has. The weak tendency is a slightly steeper gradi-ent with Helium than with Air that is consistent with adecrease in the entrainment velocity. This observation isin agreement with the density effects reported on turbu-lent mixing layers by Brown and Roshko [19] and it isalso expected with linear stability analysis [7]. Withinthe framework of the model in Eq. (6), this flow modi-fication fails to explain the weaker efficiency using lowerdensity fluid (i.e. slope approximated by Eq. 4) since theweaker the efficiency means the larger the entrainmentvelocity. It is then better to consider the injected fluidas a passive scalar and we suggest that the pressure fielddynamics inside and at the closure of the recirculationbubble is responsible for the density effect through thehorsehoe vortical structures [20–22] connecting the inte-rior of the separation and the wake. As low density fluidis attracted and high density fluid repelled by low pres-sure associated with the vortical structure we speculatethat low density fluid is preferentially evacuated from thebubble while high density fluid is likely to remain.The transition between the mass and momentumregime leading to optimal drag reduction was found inLorite-Dı́ez et al. [9], by changing the area of injection,to depend on the global flow rate injected in the bubble,and not on the local momentum injection. This resultis generalised including density effect with Eq. (5). Itis interpreted as the maximum injected momentum flux,ρbuOptb suOptb , which can be efficiently diffused by the flowin the separated area such that the injected fluid behavesas a passive scalar (the mass regime condition). The mo-mentum diffusion is taken into account with a dilutionfactor s/(hw) such that the maximum injected momen-tum flux readsΠ = ρbuOptb suOptb ×shw=ρbhw(qOptb)2. (7)In dimensionless unit, Π/(ρhwU2∞) = (ρb/ρ)(COptq )2 =b2 as found experimentally in Eq. (5). Hence, usingthe measured value of b, the maximum momentum fluxborn by the bubble in the mass regime is Π ≈ 6.4 ×10−5ρhwU2∞. Finally, taking into account the scalinglaws in Eqs.(4) and (5), the maximum drag variationthat can be achieved is given by∆COptB = −a b (ρb/ρ)− 16 ≈ −1.33 × 10−2(ρb/ρ)−16 . (8)ACKNOWLEDGMENTSThis work has been partially supported by the SpanishMINECO and European Funds under projects DPI2017-89746- R and DPI2017-88201-C3-2-R. Moreover, J.I.J.G.and M.L.D want to thank the Spanish MECD for thefinancial support provided under José Castillejo grantCAS18/00379 and Fellowship FPU 014/02945 respec-tively.[1] C. J. Wood, Aeronaut. J. 68, 477 (1964).[2] P. W. Bearman, Aeronaut. Quart. 18, 207 (1967).[3] A. Roshko, J. Wind Eng. Ind. Aerodyn. 49, 79 (1993).[4] M. Tanner, Prog. Aerosp. Sci. 16, 369 (1975).5[5] P. R. Viswanath, Prog. Aerosp. Sci. 32, 79 (1996).[6] S. Dong, G. S. 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Fluid Mech. 4, 243 (1972).[19] G. L. Brown and A. Roshko, J. Fluid Mech. 64, 775(1974).[20] A. Evrard, O. Cadot, V. Herbert, D. Ricot, R. Vigneron,and J. Délery, J. Fluids Struct. 61, 99 (2016).[21] H. J. Schmidt, R. Woszidlo, C. N. Nayeri, andO. Paschereit, Exp. Fluids 59, 107 (2018).[22] G. Pavia, M. A. Passmore, M. Varney, and G. Hodgson,J. Fluid Mech. 888, A33 (2020).

Drag reduction of three-dimensional bodies by base blowing with various gas densities

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Drag reduction of three-dimensional bodies by base blowing with various gas densities

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