Friction Drag Reduction of External Flows with Bubble and Gas Injectionby.genie.uottawa.ca/.../DragReductionBubbleGasInjection.pdf · 2012. 2. 2. · of) drag resulting from the gas

ANRV400-FL42-09 ARI 13 November 2009 13:46

Friction Drag Reduction ofExternal Flows with Bubbleand Gas InjectionSteven L. CeccioMechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2133;email: [email protected]

Annu. Rev. Fluid Mech. 2010. 42:183–203

First published online as a Review in Advance onAugust 18, 2009

The Annual Review of Fluid Mechanics is online atfluid.annualreviews.org

This article’s doi:10.1146/annurev-fluid-121108-145504

Copyright c© 2010 by Annual Reviews.All rights reserved

0066-4189/10/0115-0183$20.00

Key Words

cavitation, partial cavity, supercavity, bubble drag reduction, microbubble,ventilated cavity

AbstractThe lubrication of external liquid flow with a bubbly mixture or gas layerhas been the goal of engineers for many years, and this article presents theunderlying principles and recent advances of this technology. It reviews theuse of partial and supercavities for drag reduction of axisymmetric objectsmoving within a liquid. Partial cavity flows can also be used to reduce thefriction drag on the nominally two-dimensional portions of a horizontalsurface, and the basic flow features of two-dimensional cavities are presented.Injection of gas can lead to the creation of a bubbly mixture near the flowsurface that can significantly modify the flow within the turbulent boundarylayer, and there have been significant advances in the understanding of theunderlying physical process of drag reduction. Moreover, with sufficientgas flux, the bubbles flowing beneath a solid surface can coalesce to form athin drag-reducing air layer. The current applications of these techniques tounderwater vehicles and surface ships are discussed.

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1. INTRODUCTION

For objects moving through a liquid, the flow near the surface can be replaced by a layer of gas,creating a large reduction in near-wall density. This provides an attractive prospect for frictiondrag reduction on external flows. At issue are the conditions under which a stable gas or bubblylayer can be formed through the injection of gas at the surface, the amount of drag reduction thatcan be achieved, the required volume flux, and the possible increase in form (or other componentsof ) drag resulting from the gas injection that might outweigh the benefits of the gas-inducedskin-friction drag reduction.

2. AXISYMMETRIC CAVITY FLOWS

The formation of gas cavities behind canonical axisymmetric and two-dimensional objects hasbeen reviewed by several authors (including Brennen 1995, Knapp et al. 1970, May 1975, Wu1972) and is only briefly discussed here.

2.1. Basic Parameters

Figure 1 shows an axisymmetric underwater object that could employ a cavity for drag reduction.At the tip of the object is a cavitator with diameter d, which in this case is a round disk that isattached to a support and after-body of length L. The cavitator is immersed in a flow with free-stream velocity U and pressure PO, and as the flow moves over the cavitator, it separates and forms awake that may ultimately be filled with gas and vapor to form a cavity with gas contents at pressurePC . In classical cavity analysis, it is assumed that the cavity pressure is constant, although in practicethe flows of gas into, within, and out of a cavity may lead to minor pressure differences within thecavity. The minimum pressure that the gas in the cavity can achieve is the liquid vapor pressure,PV , and cavities that result from the flow-induced vaporization of the surrounding liquid are callednatural or vaporous cavities. If noncondensable gas is injected into the cavity, it is ventilated. Ifthe cavity length is shorter than the object (LC < L), the cavity is a partial cavity, whereas if thecavity extends beyond the test object (LC � L), it is a supercavity.

U

L

σ

dC

LC

Gas cavityCavitator with diameter d

Figure 1An example of canonical cavity flows for axisymmetric objects. The cavity separates from a cavitator and canreattach on the after-body of length L when LC < L (i.e., a partial cavity) or it may close downstream of theobject with LC � L (i.e., a supercavity).

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The cavitation number is defined as

σ = PO − PC12 ρU

2, (1)

where ρ is the fluid density. This is a basic parameter for cavity flows, along with the cavitationnumber based on the liquid vapor pressure, PV :

σV = PO − PV12 ρU

2. (2)

The density difference between the gas and the surrounding flow is often large, making buoyancyan important consideration. These effects can be scaled with the Froude number, which is definedas

Fr = U√gd

. (3)

(Care must be taken to determine if Fr is scaled on the size of the cavitator or a different cavitylength scale, such as its maximum diameter.) If gas is injected into the cavity, the nondimensionalflux is defined as

CQ = QUd 2 , (4)where Q is the volumetric gas flux at the injector exit. The drag coefficient for the cavitating flowis

CD = FD18 πd

2ρU 2. (5)

The geometry of the cavity and resulting drag coefficient is expected to be a function of thebasic nondimensional parameters σ , Fr, and CQ. If Fr � 1, the cavity is nominally axisymmetricabout an axis in the flow direction with a maximum diameter dC . Also, we assume for now thatCQ is such that the cavity pressure achieves a fixed value of σ . The drag coefficient increases withincreasing σ and scales with the value of CD at σ = 0. The approximate expressions for LC/d anddC/d for disk (and other sharp) cavitators take the form of a power law (Reichardt 1946):

LCd

= C1/2D

σ 3/2

[(σ + 0.008)

(1 − 0.132σ 1/7)1/2(1.7σ + 0.066)]

≈ C1/2D

σ

[ln

(1σ

)]1/2, (6)

dCd

= C1/2D

(σ − 0.132σ 8/7)1/2 ≈(

CDσ

)1/2, (7)

where the final approximation is from Garabedian (1956). The shape of the cavity is approximatelyellipsoidal.

The flow near the cavity closure requires special consideration. As the stream surface of thecavity converges downstream of the cavity (for σ > 0 in unbounded flow), a stagnation point formsin the wake of the cavity, forming a re-entrant jet. Re-entrant flows are often observed experi-mentally for both natural and ventilated partial cavities (Callenaere et al. 2001; Dang & Kuiper1999a,b; Knapp et al. 1970; Laberteaux & Ceccio 2001a,b), as well as supercavities (Campbell &Hilborne 1958, May 1975), which has important implications for the gas entrainment rate.

For Fr � 1, the re-entrant jet in the cavity closure flows upstream along the flow axis (e.g.,the re-entrant jet regime). In practice, this leads to the periodic filling of the cavity with liquid andsubsequent break-off. At moderate values of Fr, buoyancy can have a strong, stabilizing influenceon the cavity dynamics (Campbell & Hilborne 1958, Cox & Clayden 1955, Silbermann & Song1961, Semenenko 2001a, Stinebring et al. 1985). When gravity acts perpendicular to the cavity,the locus of the cavity centerline curves upward, and this curvature of the cavity produces lift

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and trailing vortices. Such a twin-vortex closure regime has profound implications for the cavitydynamics. The curvature of the cavity tends to orient the re-entrant jet downward and towardthe lower cavity surface, inhibiting the filling process and periodic shedding. The trailing vortexpair may also fill with vapor and extend far downstream, providing a sink for the cavity gas.Buyvol (1980) provides a transition criterion between the re-entrant and twin-vortex regimes thatis consistent with that of Campbell & Hilborne (1958):

Fr2σ 3/2 < 1.5. (8)

For Fr near the transition, there exists a toroidal shedding regime (Kirschner & Arzoumanian2008). It is also possible to produce a twin-vortex closure in cases where Fr � 1 when thecavitator’s axis of symmetry is tilted with respect to the direction of the incoming flow, producingsignificant camber in the cavity shape.

2.2. Gas Injection and Cavity Stability

The gas within the cavity can originate from three sources: vaporization when σ V ∼ σ , diffusion ofdissolved gas from the liquid into the cavity (Billet & Weir 1975, Brennen 1969, Yu & Ceccio 1997),and injection of noncondensable gas. The rate at which noncondensable gas must be injected intothe cavity strongly depends on the state of the cavity closure, Fr, and the ratio β = σ V /σ (Michel1984). Epshteyn (1961) used dimensional analysis to scale the gas entrainment rate for varying σand Fr:

CQσ 5/2 Fr

= 0.075(σ Fr4/3)/(1.52/3) − 1 . (9)

Note the rapid rise in CQ when (σ Fr4/3)/(1.52/3) approaches unity, which corresponds to thetransition to the twin-vortex regime. Kirschner & Arzoumanian (2008) report a relation for thegas-loss rate from a cavity with re-entrant closure to be

CQ ≈ k π4(1 + σ )

σCD(0)(β − 1), (10)

where 0.008 < k < 0.009. Moreover, for the case of the (more stable) twin-vortex closure,0.01 < k < 0.02, indicating that the gas flux is higher. β is referred to as the stability param-eter, and it is an important factor in the potential pulsation of the cavity. For natural cavities,β = 1, and with increasing β, the cavity contents contain more noncondensable gas, decreasingthe cavity compliance.

Cavity shape oscillations can occur in the cavity length and through the convection of waveson the cavity surface (Michel 1984, Semenenko 2001b, Tulin 1964, Woods 1966) (Figure 2).Figure 2b illustrates the process of wave formation and growth on an unstable axisymmetriccavity.

The relationships above reveal the strong coupling that exists among σ , CD, LC , dC , Fr, CQ, andβ. Hence, the cavity is part of a complex dynamical system that may be damped or exhibit self-sustained oscillations. Moreover, important dynamics may be coupled to the formation of localcavity disturbances that convect along the cavity surface, introducing a time delay in the systemdynamics (Silbermann & Song 1961). Paryshev (1978) analyzed cavity instability for axisymmetricflows using a linearized system of delayed differential equations, and he found that asymptoticallystable cavities occur when

PO(PO − PC ) < 2.65. (11)

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a

b

0 1 2 3 4 5 6 7 8 9 10

–0.5

0.0

0.5

Rad

ial

coo

rdin

ate

(m)

Axial coordinate (m)

Figure 2(a) Cavity vortex shedding and shape oscillation with visible re-entrant flow (Schauer 2003). Figure courtesyof R. Arndt. (b) The computed fifth-order mode of cavity oscillation for a cavity flow with β = 93.9. Thewaves intersect at the rear of the cavity, leading to the periodic pinch-off of gas bubbles (Kirschner &Arzoumanian 2008). Figure courtesy of I. Kirschner.

This value approximates β when PO � PV . Kirschner & Arzoumanian (2008) solved a similarsystem of delayed differential equations and found stable axisymmetric cavities for β < 2.70.

3. GAS-INJECTION DRAG REDUCTIONFOR UNDERWATER VEHICLES

The creation of a ventilated supercavity around an underwater vehicle affords the opportunityfor increased speed with drag reduced by as much as 90%. The efforts to realize this goal havebeen ongoing for the past several decades and have received renewed interest in the past severalyears. Much of this work has taken place in Russia and Ukraine, and researchers have ongoingefforts in supercavitating vehicle hydrodynamics, control, and propulsion at the Pennsylvania StateUniversity’s Applied Research Laboratory, the U.S. Navy’s Naval Undersea Warfare Center, AlionScience and Technology Corporation, and the University of Minnesota.

The shape of the vehicle is often set to fill the majority of the cavity interior, leaving onlya small gap (usually a few centimeters) between the vehicle surface and the outer flow. For fastmoving objects (typically with U > 70 m s−1) the cavity extends beyond the stern of the vehicle.Slower vehicles develop a partial cavity that closes near the stern of the object and can result in afully cavitating wake (i.e., a base cavity). The cavity must separate cleanly from the vehicle, whichnecessitates a sharp edge, and the disk is the most basic shape employed. The cavitator can becanted to generate lift and side forces on the nose of the vehicle (both to overcome the weight ofthe vehicle and to steer it). More complex cavitator shapes have been explored, including cones,


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disks with circumferential patterns, and cones with variable area (Ashley 2001). Noncondensablegas is supplied to the cavity through one or more vent ports, and care must be taken to preventthe gas injection from disturbing the cavity interface. The ventilation gas may be stored on boardor generated as a product of a chemical reaction.

It is critical that the vehicle remains within the cavity during changes in speed and whilemaneuvering, and this presents a challenging control problem. The vehicle may partially rest onthe bottom cavity surface or periodically touch the cavity surface as it translates (and possiblyrotates) within the cavity. Planing surfaces at the aft of the vehicle can be used to prevent such tailslap. Lifting surfaces with supercavitating sections can be employed that pierce the cavity surfacefor vehicle control and maneuvering. Because the center of pressure on the body is located at thecavitator, it is far ahead of the vehicle’s center of gravity, and this reduces the vehicle’s directionalstability. Dzielski & Kurdila (2003), Kirschner et al. (2002), Balas et al. (2006), and Vanek et al.(2007) discuss control strategies for the six degree-of-freedom motion of supercavitating objects.

Savchenko (2001) discusses different methods of vehicle propulsion. Solid rockets or ram jetsusing the surrounding flow (e.g., water) as the oxidizer and a metal fuel (e.g., aluminum) canproduce thrust directly or create steam to drive a turbine. In the latter case, a supercavitating orsurface-piercing propulsor can then be used to drive the vehicle forward (Kinnas 2001; Young &Kinnas 2003, 2004). The supercavitating propulsor must be in contact with the liquid flow outsidethe cavity to develop thrust, so the cavity should terminate upstream of the propulsor (i.e., it mustbe a partial cavity). This, in turn, can lead to complex, coupled interactions between the propulsorand the cavity flow as the rate of gas entrainment and re-entrant flow will be influenced by thepropulsor hydrodynamics.

4. PARTIAL CAVITY DRAG REDUCTION ON SURFACES

Gas cavities can also be formed on the surfaces of nonaxisymmetric objects. The drag on stationarylifting surfaces and struts can be reduced through cavitation or ventilation, as discussed by Acosta(1973). Cavities may also be formed beneath a flat surface with the goal of reducing the net dragon the hulls of ships, and both passive ventilation (e.g., the use of stepped hulls on planing crafts)and gas injection have been employed. Figure 3 presents examples of two canonical partial cavityflows that can be formed on two-dimensional surfaces. Here we consider flows in which gravityacts transverse to the flow direction. As in the case of axisymmetric cavity flows, the flow in thecavity closure is of critical importance.

4.1. Basic Features of Two-Dimensional Cavities at High Froude Number

As for axisymmetric cavities, approximate expressions for LC/h and dC/h formed from a protrudingtwo-dimensional cavitator take the form of a power law for small σ :

LCh

≈ 16CD(0)πσ 2

, (12)

hCh

≈ 4CD(0)πσ

, (13)

where h is the length scale of the cavitator (e.g., its height above the surface) (Tulin 1955). Theshapes of the cavities are also approximately elliptic.

The potential flow solution of a two-dimensional cavity forming downstream of a backward-facing step for Fr � 1 is discussed by Callenaere et al. (2001). The length of the cavity normalized

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U

LC

Gascavity

g

h

h

Figure 3Two canonical partial cavity flows formed on two-dimensional surfaces. The cavity can separate from aprotruding cavitator and then reattach to the surface, or the cavity can form downstream of abackward-facing step.

with the step height, h, is given by

LCh

= λπ

[(1 − λ)2

λ2+ ln

(λ2

1 − 2λ + 2λ2)]

(14)

with the nondimensional re-entrant jet thickness λ = hJ/h given by

λ = hJh

= 12

[1 − 1√

1 + σ

]. (15)

The cavity shape is given by

yC (r)h

= −(1 − λ)[

1 − 1π

cos−1(2r − 1) − 2π

(r(1 − r))1/2]

, (16)

xC (r)h

= λπ

[1 − 2λ + 2λ2

λ2r + ln(1 − r)

], (17)

where r is a parameter equal to zero at the cavity detachment and unity at the infinite extentof the re-entrant jet. For high-speed flow, the partial cavity profile generally takes this shape.Two-dimensional solutions capture much of the partial cavity flows, but spanwise variation of theflow can lead to significant changes in the closure region of the cavity and offers the possibility ofmanaging the re-entrant flow to prevent cavity break-off (De Lange & De Bruin 1997).

4.2. Cavity Profiles at Moderate Froude Number

In many applications, buoyancy dominates the shape and closure of the cavity. Butuzov (1967) andMatveev (2003) provide a potential flow solution of the cavity flow produced by cavitating wedgesbeneath a surface with finite Fr. In these solutions, the fictitious surface used to close the cavity hasa prescribed height, but its angle is determined as part of the cavity solution such that the cavitycloses smoothly. The cavity rises toward the surface, and the profile approaches that of a gravitywave that is similar to the flow at the transom of a ship (Maki et al. 2008, Schmidt 1981). Matveev(2007) used this similarity for a case in which multiple waves occur in a single partial cavity. Thehalf-wavelength of a deep-water gravity wave, LG, is given by

LG = πU2

g, (18)


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and LG would be the limiting length of the partial cavity with a single wave. Schmidt (1981)presents the asymptotic cavity profile far from cavity detachment:

yC (x) = σ LG2π[

1 −√

2 cos(

π

(x

LG+ 1

8

))]= hG

[1 −

√2 cos

(π

(x

LG+ 1

8

))], (19)

where hG = (PO − PC )/ρg. The first wave crest occurs when xC1 = 7/8 LG and has a height abovethe cavity detachment point of

yC (xC1) = σ LG2π (1 +√

2) = hG(1 +√

2). (20)

Hence, the physical dimensions of the partial cavity for moderate Fr depend on length scales LGand yC(xC1).

4.3. Ventilation of Partial Cavities and Management of the Cavity Closure

Gas injection is usually required to create a partial cavity with length LC/d � 1 or LC/h � 1,and few data have been published on the required ventilation rates for two-dimensional partialcavities. Ideally, the closure of the cavity would occur with a near-zero angle to prevent strongre-entrant flows, cavity shedding, and increased gas entrainment.

Amromin et al. (2006), Kopriva et al. (2007), and Kopriva et al. (2008) discuss the use ofpartial cavitation to reduce the drag on a specially designed hydrofoil that has a contour designedto produce a leading-edge cavity that would smoothly reattach near the cavity mid-chord. Bothnatural and ventilated cavities were examined for a variety of foil shapes at chord-based Reynoldsnumbers of order 105, with reported improvements in the foil lift-to-drag ratio of 30% to 50%.The design cavitation number for natural cavities was σ = σ V = 0.95, and mild ventilation wasdetermined to stabilize the cavity in steady and gusting flow. The flow coefficients reported wereCQ on the order of 0.005, where CQ/(hwU ), h is the maximum depth of the cavity depression,and w is the foil span. [It is unclear if the gas flux reported by Kopriva et al. (2008) is at normalconditions. If so, the volume flux into the cavity would be nearly three times larger because POwas approximately 1/3 of an atmosphere during the experiment.]

Lay et al. (2009) reported on a large-scale experiment that examined the ventilated cavity flowformed downstream of a backward-facing step. The experiment was conducted to determine thelevel of drag reduction that could be achieved through the creation of a stable partial cavity viagas injection. Figure 4 presents a view of the test model, which is 12.9 m long and 3.01 m in span,also showing the 0.187-m step and the cavity-arresting beach geometry. The maximum height ofthe beach was 0.088 m, or one-half the step height, and a section of the beach was instrumentedto measure the total flow-induced force in the streamwise direction. The model was tested ina recirculating water channel, the U.S. Navy’s William B. Morgan Large Cavitation Channel,described by Etter et al. (2005). The Reynolds number based on the step height was order 106

and order 107 based on the cavity length.The maximum partial cavity length is nearly LG = 10 m based on the geometry of the model,

implying a design flow speed of U ∼ 5 m s−1 to produce the desired cavity if the confinementeffects of the test section can be neglected (which was the experimentally observed result). Partialcavities were created over a range of speeds between 3 m s−1 < U < 7 m s−1 for varying gasflow coefficients, CQ = Q/(hwU ), where w is the span of the test model and h the step height,with stable partial cavities produced at speeds in a band around U = 5 m s−1, for 0.012 < CQ <0.020. At higher speeds, a supercavity was created, and at lower speeds, the cavity was unstable.Figure 5 shows the closure region of the cavity on the beach. Cloud cavities were created as a resultof local re-entrant flow, and their aspect ratio and shedding rate were similar to those formed from

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Flow

4.46 m (injector to mid-cavity tap)

0.48 m (injector to near injector cap)

1.47 m

3.01 mPressure taps

Injection slot

Floating plates

Floating beach

Beach flat

2.77 m

0.42 m 1.10 m 1.04 m

7-inch step

Gravity

7.14 m

Figure 4The HIPLATE test model configured for partial cavity flows (Lay et al. 2009). Gray areas represent floating plates that wereinstrumented with load cells to measure the streamwise component of drag. The gray area of the beach was anchored to two floatingplates.

natural cavities (Kawanami et al. 1998). The cavitation numbers for stable cavities were slightlypositive and on the order of σ = 0.1. However, if the injection rate exceeded the entrainmentrate at the cavity closure, the cavity pressure would rise significantly and lead to cavity instability.The stable cavity led to drag reduction of roughly 80% relative to the drag on a flat surface.

Cloud cavitation

Surface instabilitypropagates upstream

Sheet cavity is cutand detached here

0 ms 6 ms 12 ms 18 ms

24 ms 30 ms 36 ms 42 ms

Beach flat

Flow

Beach

Figure 5A time series of images showing local cavity break-off in the closure region on the partial cavity (Lay et al.2009).


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5. NUMERICAL COMPUTATION OF CAVITY FLOW

Analysis of complex, three-dimensional cavity shapes requires the use of numerical solutions.Uhlman (1987, 1989, 2006) and Varghese et al. (2005) discuss the use of boundary element methodsfor the computation of partially cavitating and supercavitating flows, and Kirschner et al. (2001)review the use of slender body theory and boundary element methods. Cavity flow calculationsusing these methods have successfully captured the steady and dynamic behavior of ventilatedcavity flows.

Reynolds-averaged Navier-Stokes (RANS) and large eddy simulation solutions of the cavityflows are also possible, where the flow is modeled as homogeneous fluid with a variable mixturedensity. A single set of mass and momentum equations is solved for the mixture with a variablemixture density or void fraction, including turbulent transport. In the most basic formulation,the mixture density is governed by an equation of state that forces an abrupt change in densitybetween the liquid and vapor phases as the local pressure crosses the vapor pressure (Song & Qin2001, Wang & Ostoja-Starzewski 2007). These methods, however, are limited to natural cavityflows.

Other homogenous flow models have been developed, including multiple phases with masstransport. A separate volume fraction transport equation is solved that includes source terms forthe dynamic exchange of mass between the phases (possibly as a result of phase change) usinga transport equation–based model. Senocak & Shyy (2002; 2004a,b) and Wu et al. (2005) use apressure-based algorithm to solve the Reynolds-averaged Navier-Stokes equations for each phase,simulating both steady and unsteady natural cavitation. Kunz et al. (2000) and Venkateswaran et al.(2002) developed a preconditioned time-marching algorithm for the computation of both com-pressible and incompressible multiphase mixtures. These methods have been used to successfullycompute a variety of canonical and practical, three-dimensional, unsteady, cavitating flows (Kunzet al. 2000, 2001; Lindau et al. 2002). Figure 6 illustrates sample results from Kunz et al. (2001)using this method.

6. BUBBLE AND AIR-LAYER DRAG REDUCTION

If bubbles of sufficient volume fraction are injected into a liquid turbulent boundary layer (TBL),they can significantly influence the turbulent transport of momentum and the resulting frictiondrag. And, if enough gas is injected beneath a flat surface, the bubbles may coalesce to form a gaslayer (e.g., a thin drag-reducing cavity).

A number of basic flow parameters are used to scale bubble injection and the resulting dragreduction. It is customary to characterize the TBL with the parameters of the baseline flow,such as the thickness, δ, and the displacement thickness, δ∗. The local shear stress is τW , andthe viscous length is lν = ν/u∗, where ν is the liquid kinematic viscosity, and friction velocityu∗ = √τW/ρ. The volume flux of gas, Q, is normalized with the volume of liquid flowing throughthe boundary layer upstream of the injector, QW = Uw(δ−δ∗), or for the flux of water in a channelflow,

ᾱ = QQ + QW =

CQ1 + CQ , (21)

to define the void fraction, ᾱ. The span is w, and CQ = Q/QW. Two other important parametersare the size of the bubbles compared to the viscous length, dB/lν , and the Weber number We =ρ dB (u∗)2/λ, where λ is the surface tension.

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a

b

cVapor

Noncondendablegas

Figure 6The results of calculations reported by Kunz et al. (2001): (a) the gas and vapor flow near a cavitator withmultiple injection ports, (b) two realizations of an unsteady natural cavity on an ogive shape with theisosurface of 90% void fraction highlighted for σ = 0.30 and Reynolds number of 1.46 × 105 based on thediameter, and (c) the flow around a supercavitating object moving at a 5◦ attack angle with the 90% voidfraction isosurface and the magnitude of the velocity in a plane. Figure courtesy of R. Kunz.

6.1. Experimental Observation of Bubble Drag Reduction with Gas Injection

Numerous investigators have achieved bubble drag reduction (BDR) of external flows on a varietyof body test models, including flat plates, axisymmetric objects, and ship hull forms. Merkle &Deutsch (1992) provide a review of the earliest work. Bogdevich & Evseev (1976) reported BDRusing gas injection for the flows over and under a 1-m-long flat surface at ReL ∼ 106. Maximumdrag reduction levels of ∼80% were reported for ᾱ ∼ 0.2 for the plate-on-top configuration.This, and other, work from the former Soviet Union represents some of the first detailed studiesof BDR.

BDR has been extensively studied by researchers at the Pennsylvania State University’s AppliedResearch Laboratory. Madavan et al. (1984, 1985a) examined gas injection into the TBL of a water-tunnel test section for ReL ∼ 106, reporting 0 < ᾱ < 0.4 with integral drag reductions of over 80%for the plate-on-top configuration. Their data showed that BDR improved with decreasing flowspeed, increasing gas injection rates, and when buoyancy assisted in retaining the bubbles in theboundary layer.

Deutsch & Castano (1986) studied BDR on an axisymmetric body that was 0.089 m in diameterand 0.62 m in total length for ReL < 107 and gas injection rates corresponding to 0 < ᾱ < 1. Theyshowed that drag reduction improved with increasing Fr as the influence of buoyancy was reducedand the bubbles stayed closer to the surface of the body. Clark & Deutsch (1991) examined theeffects of pressure gradients on this model and found that a weak favorable pressure gradientsignificantly inhibited drag reduction, whereas a weak adverse pressure gradient caused separationand large reductions in the skin friction drag at low gas injection rates.


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Guin et al. (1996), Kato et al. (1998), Kodama et al. (2000), Moriguchi & Kato (2002), andMurai et al. (2007) report a series of BDR experiments undertaken by researchers in Japan. In mostof these experiments, bubbles were injected beneath the top and bottom flow boundaries of a TBLformed by a channel flow (ReL < 106 for most experiments). The levels of friction drag reductionreported by Guin et al. (1996) and Kato et al. (1998) were similar to those of Madavan et al. (1984,1985a) at equivalent values of near-wall void fraction. Takahashi et al. (2001, 2003) examined BDRon a flat surface that was a 50-m-long flat bottom of a towed model. The experiment achieveda maximum speed of U = 7 m s−1 or ReL ∼ 108. A maximum total drag reduction of 13% wasreported, presumably resulting from a significant reduction in friction drag beneath the hull.

Sanders et al. (2006) and Elbing et al. (2008) discuss the results of a high–Reynolds numberexperiment examining BDR for a near-zero pressure gradient TBL. The HIPLATE test modeldescribed above was used for these experiments, but, unlike the configuration shown in Figure 5,the working surface of the test model was flat with flush-mounted gas injectors beneath the model.The free-stream speed range was 6 m s−1 < U < 20 m s−1, yielding ReL ∼ 108. Peak levels of localfriction drag reduction of ∼80% were achieved near the injector with 0 < ᾱ < 0.4. However, thepersistence of the drag reduction was significantly reduced with increasing U.

Figure 7 shows a composite of BDR results from different experiments on plates and on theboundary of channel flows. It is clear that ᾱ can be used to scale the results of a given experiment,but that it fails to collapse results between experiments. This is not surprising given the differencein plate configuration, Reynolds number, flow speed, and injection methods. Another limitationis the use of the average void fraction across the TBL, not the near-wall value. Finally, the scalingdoes not include the change in drag reduction as with downstream distance from the injector.

α

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Sanders et al. 2006Madavan et al. 1985aDeutsch et al. 2003Elbing et al. 2008

Sanders et al. 2006Channel flow data, Kato et al. 1998Murai et al. 2007Kodama et al. 2000

Figure 7A composite of bubble–drag reduction (DR) results from different experiments on plates and on theboundary of channel flows. Flat plate data are from Bogdevich et al.

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6.2. Mechanisms and Persistence of Bubble Drag Reduction

The level of drag reduction achieved with BDR is strongly related to the near-wall void fraction(Elbing et al. 2008, Guin et al. 1996, Madavan et al. 1985a, Sanders et al. 2006). This is consistentwith our understanding that much of the turbulent transport processes important to the productionof friction drag take place within a few tens of wall units of the surface (Pope 2000).

The underlying mechanism of BDR continues to be an active topic of study, but several pro-cesses are at work that can reduce friction drag. First, the presence of the gas changes the averagedensity and viscosity of the fluid. Second, the bubbles may interact with the turbulent flow ofthe liquid to modify the turbulent transport. Third, the bubbles introduce compressibility to theflow (Lo et al. 2006). And, fourth, the dynamics and interactions of the bubbles (e.g., splitting andmerger) may lead to changes in the flow (Meng & Uhlman 1998). Legner (1984) discusses howthe fluid near the wall may be considered as a homogeneous mixture of varying void fraction, andthe change in bulk properties and turbulent transport can then lead to the reduction in frictiondrag. Madavan et al. (1985b) present a similar result, and both analyses demonstrate how thepresence of finite void fraction near the wall can lead to a thickening of the sublayer and reducedfriction drag, concluding that the bubbles are most effective when they reside in the buffer layerof the TBL. Kunz et al. (2007) employ an Eulerian two-fluid calculation for BDR that capturesthe evolution of the near-wall bubble populations that are coupled to the development and loss ofdrag reduction.

Advances in high-fidelity computation of multiphase flows now permit the detailed investiga-tion of bubble-flow interactions in the context of BDR. Recent work includes that of Xu et al.(2002) and Ferrante & Elghobashi (2004, 2005), who employ Eulerian-Lagrangian models withpoint-force coupling. These simulations reveal detailed interactions between bubbles and turbu-lent flow in the near-wall region. Bubbles or bubble clusters can modify near-wall streamwisevortices, leading to a reduction in friction drag, even at relatively small void fractions. The mi-crobubbles employed in these simulations typically have sizes on the order of the viscous wallunit with We � 1. Lu et al. (2005) employed a direct numerical simulation of the bubbly flow toreveal how large (dB/lν ∼ 50) deformable (We ∼ 0.5) bubbles can strongly interact with streamwisevortices when they are present in the buffer region (Figure 8). Experimental evidence for suchdrag-reducing bubble-flow interactions has been reported by Murai et al. (2006) and Gutierrez-Torres et al. (2008) for flows at relatively low Reynolds number and void fractions.

Investigators have used the Couette-Taylor flow as a model to study bubble-vortex interactionsthat may be responsible for BDR (Murai et al. 2008, van den Berg et al. 2005). Sugiyama et al.(2008) show that the buoyant motions of microbubbles can disrupt coherent vortices of low–Reynolds number flows, reducing the friction drag. However, as the Reynolds number increasesand the coherence of the vortical flow is reduced, the effect of the bubbles on the friction dragis less pronounced. The authors suggest that, as the Weber and Reynolds numbers of the flowincrease, other flow processes (such as bubble splitting, deformation, and compressibility) maybecome more important.

The influence of bubble size on BDR has been difficult to ascertain for high–Reynolds numberflows. Moriguchi & Kato (2002) and Kawamura et al. (2003) reported no change in drag reductionover a range of injected bubble sizes, all of which were much larger than the viscous length. Winkelet al. (2004) and Elbing et al. (2008) used saltwater and surfactants to reduce the average bubblesize with little change far from the injector. Shen et al. (2006) injected bubbles with size rangingfrom 5 < dB/lν < 100 without a significant change in observed drag reduction for a given ᾱ. Sanderset al. (2006) showed that the TBL flow will split the injected gas into bubbles with size the orderof dB/lν ∼ 102 and We ∼ 1 due to interactions with the TBL, and this result is consistent with the


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Z

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Wall shear

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Figure 8Simulation of bubble-flow interactions in a channel leading to friction drag reduction: (a) the baseline flow,(b) bubbles with We = 0.203, and (c) We = 0.405. The left column is for early time, and the right is for latertime. Isocontours of streamwise vorticity are shown along with shear on the bottom wall. Figure taken fromLu et al. 2005, courtesy of G. Tryggvason.

observed insensitivity of the bubble formation to the method of gas injection and the resultingdrag reduction (Madavan et al. 1985a, Pal et al. 1988). Larger bubbles may form for slower flows,but the mechanical energy of the TBL cannot be used to produce bubbles on the order of theviscous length for liquids with finite surface tension. Therefore, BDR employing large numbers ofmicrobubbles (dB/lν < 10) requires a separate process to create such small bubbles before injection.

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As a result, most of the experimental results above are for bubbles that are very large comparedto the inner scales of the TBL. Moreover, the level of drag reduction typically achieved is on theorder of the near-wall void fraction, suggesting that a basic mechanism of drag reduction is densityreduction near the wall as well as bubble interactions with the near-wall vortical flow.

As the bubbly flow convects away from the gas source, the near-wall void fraction within theTBL evolves as a result of liquid entrainment, turbulent mixing, and buoyancy. Sanders et al.(2006) and Elbing et al. (2008) showed that the level of drag reduction can decrease rapidly fromits peak level, as a nearly bubble-free layer forms near the wall. At high shear rates, bubble liftforces are sufficient to overcome buoyancy and move bubbles away from the wall a distance on theorder of the bubble diameter. Because dB/lν ∼ 102 in these flows, the turbulent transport acrossthis thin liquid layer is largely uninterrupted, and the level of BDR is significantly reduced.

6.3. Transition from Bubble to Air-Layer Drag Reduction

Bubbles injected into the TBL can coalesce to form macrobubbles with size on the order of δ, andthese bubbles can also reduce drag (Murai et al. 2007). Such large, nonspherical bubbles persistwhen the Weber number of the flow is low, usually because of slow free-stream speeds. Reed(1994) and Fukuda et al. (2000) discuss the use of surface grooves and hydrophobic coatings toenhance the formation of macrobubbles. However, even at higher speeds, smaller bubbles cancoalesce beneath the flow surface given a sufficiently high gas flux. Elbing et al. (2008) examinedthe transition from BDR to air-layer drag reduction (ALDR) for flow beneath the HIPLATEmodel. Three flow regimes were identified: BDR, transitional, and ALDR. Figure 9a shows thelevel of drag reduction at a measurement location 6 m from the injector for varying gas fluxes. Thesteep rise in the reduction of friction drag occurs when the air layer forms. Figure 9b shows howthe transitional and critical gas fluxes vary with U. The air layer is essentially a thin supercavity. Itcan form from a flush line source of gas, but the layer is more robust when it forms downstreamof a spanwise step with height that is on the order of δ. The physics and scaling of the BDR toALDR transition, along with the stability of the air layer, are current topics of study.

7. APPLICATION OF GAS-INJECTION FOR DRAG REDUCTIONON SURFACE SHIPS

The concept of using air lubrication to reduce the drag of ship hulls has been present for manyyears, and Latorre (1997) reports that patents on this subject were filed as early as 1880. He providesa brief review of the earliest developments in this field, including pioneering work originating fromthe former Soviet Union. Researchers there developed and fielded displacement and planing ships,employing a variety of drag reduction methods, air layers, and partial cavities. The typical reductionin total resistance was between 10% and 20%. Efforts are ongoing in Japan and Europe to developair-lubricated ships, and researchers have been conducting both model- and prototype-scale trials.Nagamatsu et al. (2002) and Kodama et al. (2006) review a series of efforts coordinated by theNational Maritime Research Institute of Japan. The DK Group has been conducting both model-and full-scale tests of an air-cavity ship that is a 1:4 scale model of a very large container carrier.The ship uses a series of parallel cavities along its length that are fed with actively controlledair injectors. They project a net fuel savings of up to 15%, with a corresponding reduction inemissions ( J.P. Winkler, private communication). Cavity flows on surface ships are typically forlow Fr with the streamwise length of a single cavity insufficient to extend the length of the ship.For a large, moderate-speed ship, it is likely that several streamwise cavities will need to be created(Amromin & Mizine 2003). Cavities formed beneath surfaces with finite span will also encounterend effects, possibly producing transverse wave patterns on the cavity interface (Matveev 2007).


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Critical flux

Transitional flux

Figure 9The gas flux required to transition from bubble drag reduction to air-layer drag reduction on the HIPLATEtest model. The percentage of drag reduction (DR) is measured 6 m from the injector. Figure taken fromElbing et al. 2008.

Important practical considerations arise when air lubrication is considered for drag reductionon ships. Foremost, the power required to pump the air beneath the hull must not be larger thanthe propulsive power saved. Additionally, the presence of the gas around the hull cannot degradethe propulsion of the ship. The proportion of friction drag that a hull experiences is highest atlower speeds (compared to form and wave drag), which makes friction drag reduction attractive.With increasing speed, the proportion of resistance from the friction drag typically decreases.Because the power loss due to friction rises as U 3, whereas the supply rate of the injected gasusually increases at a slower rate, friction drag reduction with gas injection may become morecost-beneficial at higher speeds. The lines of the ship are designed to reduce the total resistance,so care must be taken not to increase the other components of drag in an effort to contain thelubricating layers.

The most significant challenge is the unsteady flows around the hull that result from shipmotions and maneuvers while traveling in a seaway that would break up and dissipate cavities or

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air layers. The loss of a partial cavity can lead to tremendous increases in drag. Also, the presenceof the cavities cannot degrade the sea keeping of the ship.

8. CONCLUSIONS

Scientists and engineers have made considerable progress in our understanding and applicationof gas injection for the reduction of drag on external liquid flows. These technologies are beingactively developed for both underwater and surface craft. As our understanding of these multiphaseflows increases, gas injection and cavity management processes may be further optimized. Giventhe complex dynamics of the air layers, and the need of vehicles for maneuvering and sea keeping,it is likely that active monitoring and control of the gas injection and cavity formation processwill be required for crafts operating in the open water. Design of such hull forms will rely on acombination of modeling and experiments using the most advanced tools we have developed.

DISCLOSURE STATEMENT

The author is not aware of any affiliations, memberships, funding, or financial holdings that mightbe perceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS

The author would like to thank Mr. Simo Mäkiharju, Dr. Brian Elbing, Dr. Ivan Kirschner, andDr. Keary Lay for their assistance in preparing this manuscript and Prof. Roger Arndt, Dr. RobertKunz, and Prof. Gretar Trggvason for the use of several figures. He would also like to recognizehis ongoing collaboration with Prof. David Dowling and Prof. Marc Perlin on the general topicof drag reduction, and the support he has received from the Office of Naval Research and theDefense Advanced Research Projects Agency.

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AR400-FM ARI 13 November 2009 15:33

Annual Review ofFluid Mechanics

Volume 42, 2010Contents

Singular Perturbation Theory: A Viscous Flow out of GöttingenRobert E. O’Malley Jr. � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 1

Dynamics of Winds and Currents Coupled to Surface WavesPeter P. Sullivan and James C. McWilliams � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �19

Fluvial Sedimentary PatternsG. Seminara � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �43

Shear Bands in Matter with GranularityPeter Schall and Martin van Hecke � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �67

Slip on Superhydrophobic SurfacesJonathan P. Rothstein � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �89

Turbulent Dispersed Multiphase FlowS. Balachandar and John K. Eaton � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 111

Turbidity Currents and Their DepositsEckart Meiburg and Ben Kneller � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 135

Measurement of the Velocity Gradient Tensor in Turbulent FlowsJames M. Wallace and Petar V. Vukoslavčević � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 157

Friction Drag Reduction of External Flows with Bubble andGas InjectionSteven L. Ceccio � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 183

Wave–Vortex Interactions in Fluids and SuperfluidsOliver Bühler � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 205

Laminar, Transitional, and Turbulent Flows in Rotor-Stator CavitiesBrian Launder, Sébastien Poncet, and Eric Serre � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 229

Scale-Dependent Models for Atmospheric FlowsRupert Klein � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 249

Spike-Type Compressor Stall Inception, Detection, and ControlC.S. Tan, I. Day, S. Morris, and A. Wadia � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 275

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AR400-FM ARI 13 November 2009 15:33

Airflow and Particle Transport in the Human Respiratory SystemC. Kleinstreuer and Z. Zhang � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 301

Small-Scale Properties of Turbulent Rayleigh-Bénard ConvectionDetlef Lohse and Ke-Qing Xia � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 335

Fluid Dynamics of Urban Atmospheres in Complex TerrainH.J.S. Fernando � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 365

Turbulent Plumes in NatureAndrew W. Woods � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 391

Fluid Mechanics of MicrorheologyTodd M. Squires and Thomas G. Mason � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 413

Lattice-Boltzmann Method for Complex FlowsCyrus K. Aidun and Jonathan R. Clausen � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 439

Wavelet Methods in Computational Fluid DynamicsKai Schneider and Oleg V. Vasilyev � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 473

Dielectric Barrier Discharge Plasma Actuators for Flow ControlThomas C. Corke, C. Lon Enloe, and Stephen P. Wilkinson � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 505

Applications of Holography in Fluid Mechanics and Particle DynamicsJoseph Katz and Jian Sheng � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 531

Recent Advances in Micro-Particle Image VelocimetrySteven T. Wereley and Carl D. Meinhart � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 557

Indexes

Cumulative Index of Contributing Authors, Volumes 1–42 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 577

Cumulative Index of Chapter Titles, Volumes 1–42 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 585

Errata

An online log of corrections to Annual Review of Fluid Mechanics articles may be foundat http://fluid.annualreviews.org/errata.shtml

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Annual Reviews OnlineSearch Annual ReviewsAnnual Review of Fluid MechanicsOnlineMost Downloaded Fluid Mechanics ReviewsMost Cited Fluid MechanicsReviewsAnnual Review of Fluid MechanicsErrataView Current Editorial Committee

All Articles in the Annual Review of Fluid Mechanics Vol. 42 Singular Perturbation Theory: A Viscous Flow out of GottingenDynamics of Winds and Currents Coupled to Surface WavesFluvial Sedimentary PatternsShear Bands in Matter with GranularitySlip on Superhydrophobic SurfacesTurbulent Dispersed Multiphase FlowTurbidity Currents and Their DepositsMeasurement of the Velocity Gradient Tensor in Turbulent FlowsFriction Drag Reduction of External Flows with Bubble and Gas InjectionWave–Vortex Interactions in Fluids and SuperfluidsLaminar, Transitional, and Turbulent Flows in Rotor-Stator CavitiesScale-Dependent Models for Atmospheric FlowsSpike-Type Compressor Stall Inception, Detection, and ControlAirflow and Particle Transport in the Human Respiratory SystemSmall-Scale Properties of Turbulent Rayleigh-BenardConvectionFluid Dynamics of Urban Atmospheres in Complex TerrainTurbulent Plumes in NatureFluid Mechanics of MicrorheologyLattice-Boltzmann Method for Complex FlowsWavelet Methods in Computational Fluid DynamicsDielectric Barrier Discharge Plasma Actuators for Flow ControlApplications of Holography in Fluid Mechanics and Particle DynamicsRecent Advances in Micro-Particle Image Velocimetry

Documents

Friction Drag Reduction of External Flows with Bubble and Gas Injectionby.genie.uottawa.ca/.../DragReductionBubbleGasInjection.pdf · 2012. 2. 2. · of) drag resulting from the gas