Vortex Simulation of the Bubbly Flow around a Hydrofoildownloads.hindawi.com/journals/ijrm/2007/072697.pdf · around a hydrofoil. Nishikawa et al. [8] performed the nu-merical simulation

Hindawi Publishing CorporationInternational Journal of Rotating MachineryVolume 2007, Article ID 72697, 9 pagesdoi:10.1155/2007/72697

Research ArticleVortex Simulation of the Bubbly Flow around a Hydrofoil

Tomomi Uchiyama1 and Tomohiro Degawa2

1 EcoTopia Science Institute, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan2 Graduate School of Information Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan

Received 18 March 2007; Revised 7 June 2007; Accepted 16 August 2007

Recommended by Eddie Y. K. Ng

This study is concerned with the two-dimensional simulation for an air-water bubbly flow around a hydrofoil. The vortex method,proposed by the authors for gas-liquid two-phase free turbulent flow in a prior paper, is applied for the simulation. The liquidvorticity field is discrerized by vortex elements, and the behavior of vortex element and the bubble motion are simultaneouslycomputed by the Lagrangian approach. The effect of bubble motion on the liquid flow is taken into account through the change inthe strength of vortex element. The bubbly flow around a hydrofoil of NACA4412 with a chord length 100 mm is simulated. TheReynolds number is 2.5 × 105, the bubble diameter is 1 mm, and the volumetric flow ratio of bubble to whole fluid is 0.048. It isconfirmed that the simulated distributions of air volume fraction and pressure agree well with the trend of the measurement andthat the effect of angle of attack on the flow is favorably analyzed. These results demonstrate that the vortex method is applicableto the bubbly flow analysis around a hydrofoil.

Copyright © 2007 T. Uchiyama and T. Degawa. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

1. INTRODUCTION

Turbo pumps are originally designed for handling single-phase liquid flows. But they are frequently operated undergas-liquid two-phase flow conditions because of their diverseuse. As the pump performance for two-phase flow generallydeteriorates, it is practically important to grasp and improvethe performance. A number of researchers [1–3] have inves-tigated the performance of pumps at various specific speeds,and they made clear the relation between the performanceand the flow in impeller. Some numerical simulations ontwo-phase flows in impeller have been carried out so as topredict the effect of the pump design parameters, such asthe impeller geometry and the rotating speed, on the pumpperformance [4, 5]. With the intention of clarifying the two-phase flow in impeller from a basic standpoint, the bubblyflow around a hydrofoil has also been studied. Ohashi et al.[6] and Ichikawa et al. [7] found from their measurementthat the gas volume fraction is distributed quite unevenlyaround a hydrofoil. Nishikawa et al. [8] performed the nu-merical simulation for a bubbly flow around a hydrofoil toexamine the slip motion of bubble relative to the water flow.Uchiyama [9, 10] analyzed the bubbly flow around a hydro-foil undergoing heaving and pitching motions to predict theunsteady characteristics of the hydrofoil.

The authors [11] proposed a vortex method for gas-liquid bubbly flow in a prior paper. The liquid vorticity fieldis discretized by vortex elements, and the behavior of vor-tex element and the bubble motion are simultaneously com-puted by the Lagrangian approach. The method can simu-late directly the development of vortical structure, such asthe formation and deformation of vortices. Therefore, it isusefully employed for the simulation of free turbulent flows,in which the large scale eddies play a dominant role. The au-thors applied the vortex method to simulate a bubbly flowaround rectangular cylinder [11], a bubble-laden plane mix-ing layer [12], and a bubble-driven plume [13]. The simula-tions demonstrated that the interaction between the large-scale eddies and the bubbles is successfully analyzed andthat the simulated two-phase flow field is favorably com-pared with the corresponding experimental results. Since thetwo-phase flow around a hydrofoil is chiefly governed by theliquid vortices and the bubble motion, the authors’ vortexmethod seems to be effectively employed for the simulation.

In this study, the vortex method is applied to the two-dimensional simulation for an air-water bubbly flow arounda hydrofoil. The flow was experimentally investigated byIchikawa et al. [7]. A hydrofoil of NACA4412 is mounted ina channel. The chord length is 100 mm, and the Reynoldsnumber is 2.5 × 105. The angle of attack is set at 0◦ and 10◦.

2 International Journal of Rotating Machinery

The bubble diameter is 1 mm, while the volumetric flow ra-tio of bubble to whole fluid is 0.048. It is confirmed that thesimulated distributions of air volume fraction and pressureagree well with the trend of the measurement and that theeffect of angle of attack on the flow is favorably analyzed.These demonstrates that the vortex method is applicable tothe bubbly flow analysis around a hydrofoil.

2. BASIC EQUATIONS

2.1. Assumptions

The following assumptions are employed for the simulation.

(1) The mixture is a bubbly flow entraining small bubbles.(2) Both phases are incompressible and no phase changes

occur.(3) The mass and momentum of the gas phase are very

small and negligible compared with those of the liquidphase.

(4) The bubbles maintain their spherical shape, and nei-ther fragmentation nor coalescence occurs.

2.2. Conservation equations for bubbly flow

The conservation equations for the mass and momentum ofthe bubbly flow are expressed by the following equations un-der the assumptions (1)–(3):

∂αl∂t

+∇·(αlul) = 0, (1)

αlDulDt

= − 1ρl∇p + νl∇2ul + αlg, (2)

where the third term on the right-hand side of (2) expressesthe buoyancy effect, and the volume fraction satisfies the fol-lowing relation:

αg + αl = 1. (3)

2.3. Equation of motion of bubble

It is postulated that the virtual mass force, the drag force,the gravitational force, and the lift force act on the bubble.The equation of motion for the bubble is expressed by thefollowing equation under the assumption (4):

dugdt

= 1 + CVβ + CV

DulDt

− 1β + CV

3CD4d

ur∣∣ur∣∣

+β − 1β + CV

g− CLβ + CV

ur ×(∇× ul

),

(4)

where d is the bubble diameter, ur = ug − ul and β = ρg/ρl.CV , CD, and CL are the virtual mass coefficient, the drag co-efficient and the lift coefficient, respectively. CD is given as[14],

CD = 24Reb

(1 + 0.15Re0.687

b

), (5)

where Reb = d|ur|/vl.

2.4. Decomposition of liquid velocity field

According to the Helmholtz’ theorem, any vector field canbe represented as the summation of the gradient for a scalarpotential φ and the rotation for a vector potential ψ . Thus,the liquid velocity ul is written as

ul = ∇φ +∇× ψ . (6)

The velocity calculated from (6) remains unaltered evenwhen any gradient of scalar function is added to ψ . To re-move this arbitrariness, a solenoidal condition is imposed onψ :

∇·ψ = 0. (7)

When substituting (6) into (1) and rewriting the resul-tant equation by using a relation ∇·(∇× ψ) = 0, the follow-ing equation is obtained:

∂αl∂t

+∇·(αl∇φ)

+∇× ψ·∇αl = 0. (8)

Taking the rotation of (6) and substituting (7) into theresultant equation, the vector Poisson equation is derived:

∇2ψ = −ω, (9)

where ω is the vorticity for the liquid phase.

2.5. Lagrange analysis for vortex element and bubble

When taking the rotation of (2), the vorticity equation forthe bubbly flow is derived. For the two-dimensional calcula-tion, it is expressed by the following equation:

∂ω

∂t+∇·(ωul

) = νlαl∇2ω +

1αl∇αl ×

(g− Dul

Dt

). (10)

The vorticity field is discretized by vortex elements. It ispostulated that the vortex element has a viscous core and thatthe vorticity distribution around the element is representedwith the Gaussian curve. When the vortex element γ at xγ

has a circulation Γγ and a core radius σγ, the vorticity at xinduced by the element is expressed as

ωγ(x) = Γγπσ2

γexp

[

−(∣∣x − xγ

∣∣

σγ

)2]

. (11)

The convection of the vortex element γ is estimated by theLagrangian calculation of the following equation:

dxγ

dt= ul

(xγ). (12)

Equations (8) and (9) are solved by a finite elementmethod. The two-dimensional computational domain in thex-y plane is resolved into quadrilateral elements. An elementis shown in Figure 1. The scalar potential φ, the z-componentfor vector potential ψz, the vorticity ω, and the liquid veloc-ity ul are defined on the nodes, while the volume fraction isdefined at the center of element. When solving (9) for ψz, the

T. Uchiyama and T. Degawa 3

NodeCenter

φ,ψz ,ω, ulαg

Figure 1: Quadrilateral element.

ω value on the nodes is determined by taking the summa-tion of the vorticity produced by every vortex element. Theαl value in (8) is estimated from (3) after computing the gasvolume fraction αg from the Lagrangian calculation of (4).When the bubbles are assumed to distribute uniformly withan interval Δz in the z-direction normal to the x-y plane, αgfor a quadrilateral element with an area A is expressed as

αg =(π/6)d3ngAΔz

, (13)

where ng is the number of bubbles in the element.It is found from (10) that the vorticity varies with the

lapse of time due to the viscous diffusion and the gradient ofthe volume fraction. These variations are considered throughthe changes in σ and Γ for the vortex element, and they arecomputed simultaneously with the Lagrangian calculation of(12).

2.6. Change in core radius due to viscous diffusion

The viscous diffusion is simulated by applying a core spread-ing method, which was used for single-phase flow simula-tions [15],

dσdt= 2νl

σ. (14)

2.7. Change in circulation due to bubble motion

When employing the Reynolds transport theorem and (10),the time rate of change in the circulation Γ along any closedcurve (area element dS) is expressed as

DΓDt

=∫

S

[1αl∇αl ×

(g− Dul

Dt

)]

dS, (15)

where the viscous diffusion term is omitted because it is al-ready considered by (14).

The application of (15) to a quadrilateral element(Figure 1) yields the time rate of change in Γ, ΔΓ/Δt, in theelement. In the case where the number of vortex elements ina quadrilateral element is nv, the change in the circulation foreach vortex element duringΔt is supposed to beΔΓ/nv. In thecase where there are no vortex elements in the quadrilateralelement, a vortex element with a circulation ΔΓ is generatedat the center of the quadrilateral element.

−1

0

1

y/l Flow

B0

B1

B2

B3B4

l

θ

NACA4412

−1 0 1 2 3

x/l

Computational domain

Figure 2: Computational domain.

2.8. Numerical procedure

When the flow field at t = t is known, the flow at t = t + Δt isestimated by the following procedures.

(1) Simulate the bubble motion from (4).(2) Calculate αg from (13) to estimate αl from (3).(3) Calculate σ from (14).(4) Calculate ΔΓ from (15).(5) Simulate the convection of vortex element from (12)

to calculate ω from (11).(6) Calculate ψ from (9).(7) Calculate φ from (8).(8) Calculate ul from (6).

3. SIMULATION CONDITIONS

The vortex method is applied to the two-dimensional sim-ulation for an air-water bubbly flow around a hydrofoil.The hydrofoil of NACA4412 is mounted in a channel with200 mm × 100 mm cross-sectional area. The chord length lis 100 mm. The Reynolds number, based on the water ve-locity upstream of the hydrofoil ul0 and l, is 2.5 × 105. Thevolumetric flow ratio of bubble to whole fluid αg0 is 0.048.

Figure 2 shows the computational domain. The inlet andoutlet boundaries, B0 and B3, are located: l upstream and 3ldownstream of the leading edge of the hydrofoil. The simu-lations are performed at the angles of attack of 0◦ and 10◦.

The vorticity field is generated by the bubble motion andthe velocity shear layer originating from the hydrofoil. Tosimulate the vorticity field due to the velocity shear layer, thevorticity layer on the hydrofoil is divided into segments witha length s, and a vortex element is released from each seg-ment into the flow field at a time interval Δt. In this simula-tion, the number of segments is 50, the core radius σ0 of thevortex element at the release is 0.5s, and the height of vortic-ity layer is set at 10/

√Re. The σ0 value for a vortex element

generated from a quadrilateral element by the bubble mo-tion is determined so that πσ2

0 coincides with the area for thequadrilateral element.

The bubbles are released into the water flow field witha velocity ug0 from the inlet boundary B0 at a time inter-val Δt. The diameter is set at the measured value 1 mm, andug0 is estimated from a drift-flux model [16]. To reproducethe bubbly flow condition, the bubble releasing positions are


−1

0

1

y/l

−0.5 0 1 2

x/l

(a) θ = 0◦

−1

0

1

y/l

−0.5 0 1 2

x/l

(b) θ = 10◦

Figure 3: Distribution of water velocity.

Figure 4: Closeup of velocity field near the separated region.

determined by using random numbers. The second-orderAdams-Bashforth method is used for the Lagrangian calcu-lation of vortex element and bubble. The time increment Δtis set at ul0Δt/s = 0.025.

Equations (8) and (9) are solved by using the followingboundary conditions: the uniform flow on the inlet bound-ary B0; the convective outflow on the outlet boundary B3; theslip on the channel walls B1 and B2, and the nonslip on thehydrofoil surface B4. The nonslip condition is also applicableto B1 and B2 by representing the vorticity layer with the samemethod applied on the hydrofoil surface. Since the Reynoldsnumber is so high, this study assumes that the vorticity layer

−1

0

1

y/l

−0.5 0 1 2

x/l

(a) θ = 0◦

−1

0

1

y/l

−0.5 0 1 2

x/l

(b) θ = 10◦

Figure 5: Distribution of vortex element.

scarcely affects the flow around the hydrofoil and imposes theslip condition.

The above-mentioned boundary conditions are summa-rized as follows:

∂φ

∂n= −ul0, ψz = 0 on B0,

∂φ

∂n= 0, ψz = 0 on B1, B2, and B4,

∂φ

∂n= ul0 on B3,

(16)

where n is the outward unit vector normal to the boundary.The Sommerfeld radiation condition is imposed for ψz onthe boundary B3.

The pressure is calculated from the Poisson equation de-rived by taking the divergence of (2).


−1

0

1

y/l

−0.5 0 1 2

x/l

0 αg /αg0 1.4

(a) θ = 0◦

−1

0

1

y/l

−0.5 0 1 2

x/l

0 αg /αg0 1.4

(b) θ = 10◦

Figure 6: Distribution of air volume fraction.

4. RESULTS AND DISCUSSIONS

4.1. Distribution of water velocity

The distribution of water velocity around the hydrofoil isshown in Figure 3. When the angle of attack θ is 0◦, the flowparallels the hydrofoil surface and remains steady. But theflow at θ = 10◦ separates from the upper surface of the rearhalf for the hydrofoil.

The closeup of the velocity field near the separated re-gion is shown in Figure 4. The flow exhibits an unsteadybehavior.

The vortex elements introduced from the hydrofoil sur-face distribute as plotted in Figure 5, where the distributionat the same instant as Figure 3 is indicated. As the vortexelements generated by the bubble motion distribute in thewhole region, their depiction is omitted. When θ = 0◦, thevortex elements flow along the hydrofoil surface as shown inFigure 5(a), since the water flow is in parallel with the sur-face. They flow almost straight in the wake of the hydro-

l

0.6l

0.4l

0.2l

0.1l

L1

L2

L3

A B C

D E F

G

L4

L5

L6

L0

Figure 7: Lines and points for air volume fraction around hydrofoil.

foil. The distribution of vortex element at θ = 10◦ is shownin Figure 5(b). The flow separation from the upper surfaceof the hydrofoil is confirmed. The vortex elements meandermarkedly in the wake, and they form clusters. The clusters,corresponding to the large-scale eddies, flow downstream atalmost even interval. The authors [12] analyzed such large-scale eddies in a bubble-laden plane mixing layer by the vor-tex method. From the present simulation, one can grasp thegeneration and development for a velocity shear layer down-stream of the hydrofoil.

4.2. Distribution of air volume fraction

Figure 6 shows the distribution of air volume fraction αg atthe same instant as Figures 3 and 5. When θ = 0◦, αg islow near the hydrofoil surface, as found from Figure 6(a).This is because the bubbles are prevented from approach-ing the leading edge due to the stagnation pressure, and ac-cordingly they flow downstream in parallel with the hydro-foil surface. The αg value is also low in the wake. This iscaused by the above-mentioned sparse distribution of bub-ble near the hydrofoil surface. Near the rear half of theupper-surface, αg takes its maximum value. The distribu-tion for θ = 10◦ is shown in Figure 6(b). The αg value re-duces near the hydrofoil surface. The reduction near thelower surface is larger than that for θ = 0◦. Because the pres-sure rises in the downstream direction near the leading edgeon the lower surface, accordingly such positive pressure gra-dient prevents the bubbles from approaching the hydrofoilsurface. It should be noted that αg is higher near the rearhalf of the upper-surface. Since the bubbles are entrainedinto the large-scale eddies shed from the trailing edge, αgis also higher in the wake. It reaches its maximum value atthe center of every eddy. Rotating blades are used in engi-neering devices handling the chemical reaction of gas-liquidtwo-phase mixtures. It is considered that the shed of large-scale eddies and the bubble entrainment into the eddiescan attain the active contact and mixing between the twophases.

Murakami and Minemura [17] performed the experi-mental investigation on an axial-flow pump operated underair-water two-phase flow conditions. Their flow observation


0.4

0.3

0.2

0.1

0

y′/l

0 0.04 0.08

αg

L1

θ = 0◦

θ = 10◦

(a)

0.4

0.3

0.2

0.1

0

y′/l

0 0.04 0.08

αg

L2

θ = 0◦

θ = 10◦

(b)

0.4

0.3

0.2

0.1

0

y′/l

0 0.04 0.08

αg

L3

θ = 0◦

θ = 10◦

(c)

0

0.1

0.2

0.3

0.4

y′/l

0 0.04 0.08

αg

L4

θ = 0◦

θ = 10◦

(d)

0

0.1

0.2

0.3

0.4

y′/l

0 0.04 0.08

αg

L5

θ = 0◦

θ = 10◦

(e)

0

0.1

0.2

0.3

0.4

y′/l

0 0.04 0.08

αg

L6

θ = 0◦

θ = 10◦

(f)

Figure 8: Time-averaged air volume fraction around hydrofoil.

inside the impeller made clear that the entrained bubbles ac-cumulate on the rear half of the suction surface for the blade.The present vortex method simulates such bubble accumula-tion near the rear half of the upper surface (suction surface)for a hydrofoil, as demonstrated in Figure 6. Consequently,it seems to be usefully employed for the simulation of two-phase flow in rotating impeller.

Calculating the time-averaged air volume fraction αg onthe six lines L1–L6 indicated in Figure 7, it distributes asshown in Figure 8, where αg is plotted against the distancey′ from the hydrofoil surface. The result at θ = 0◦ is shownby the broken lines. In the upper-surface regions (L1, L2, L3),αg takes its maximum value near the surface. The maximumvalue increases in the downstream direction. In both sides ofthe hydrofoil, αg reduces with approaching the hydrofoil sur-face. The distributions of αg at θ = 10◦ are plotted by the solidlines. Comparing with the results at θ = 0◦, αg is lower nearthe point C which locates in the separated region. It is alsolower near the lower surface.

Figure 9 shows the distribution of αg on the line L0 pass-ing through a point G downstream of the trailing edge. Whenθ = 0◦, αg reaches its minimum value at the center of wake(y′/l = 0), indicating quite uneven distribution. It is foundthat the wake is markedly affected by the bubble distributionnear the hydrofoil surface. Similarly, uneven distribution isalso observed when θ = 10◦.

The above-mentioned distribution of αg agrees well withthe trend of the measured result [7]. But the maximum valuefor αg is lower than the measurement. This may be becausethe present simulation ignores the bubble deformation.

4.3. Pressure distribution

Figure 10 shows the distribution of pressure coefficient Cp.The result at θ = 0◦ is plotted in Figure 10(a). In the re-gion of 0.3 ≤ x/l ≤ 1 near the upper-surface, the pressurerises in the downstream direction. Such positive pressuregradient lowers the velocity of the bubble flowing along the


0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

y′/l

0 0.04 0.08

αg

L0

G

(a) θ = 0◦

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

y′/l

0 0.04 0.08

αg

L0

G

(b) θ = 10◦

Figure 9: Time-averaged air volume fraction in wake.

−1

0

1

y/l

−0.5 0 1 2

x/l

−0.83 Cp 0.67

(a) θ = 0◦

−1

0

1

y/l

−0.5 0 1 2

x/l

−2.66 Cp 0.55

(b) θ = 10◦

Figure 10: Distribution of pressure coefficient.

hydrofoil surface and causes the bubble accumulation there,accordingly αg heightens as found in Figure 6(a). The bub-ble accumulation does not occur near the lower-surface, be-cause the pressure remains unaltered in the flow direction.Figure 10(b) shows the result at θ = 10◦. The pressure is lo-cally high near the leading edge of the lower-surface. As thepressure gradient prevents the bubble from approaching thehydrofoil surface, it causes the marked decrement in αg nearthe lower-surface as shown in Figures 6 and 8.

The time-averaged value for Cp, Cp distributes on the hy-drofoil surface as shown in Figure 11. On the lower-surface,the distribution for the two-phase flow (αg0 = 0.048) is al-most the same as that for the single-phase flow (αg0 = 0). Thisis because the air volume fraction is lower near the lower-surface, as found from Figure 6. On the upper-surface, how-ever, the absolute value |Cp| decreases at the two-phase flowcondition. According to the experiment at Re = 4 × 105 [7],|Cp| on the upper-surface reduces with an increment in αg0.Therefore, the present simulation seems to be reasonable.


−1

0

1

Cp

0 0.2 0.4 0.6 0.8 1

x′/l

αg0 = 0αg0 = 0.048

Upper surface

Lower surface

(a) θ = 0◦

−3

−2

−1

0

1

Cp

0 0.2 0.4 0.6 0.8 1

x′/l

αg0 = 0αg0 = 0.048

Upper surface

Lower surface

(b) θ = 10◦

Figure 11: Pressure distribution on hydrofoil.

5. CONCLUSIONS

The air-water bubbly flow around a hydrofoil is simulatedwith the vortex method proposed by the authors in a priorstudy. A hydrofoil of NACA4412 with a chord length 100 mmis mounted in a channel. The water Reynolds number is2.5 × 105, the bubble diameter is 1 mm, and the volumetricflow ratio of bubble to whole fluid is 0.048. The results aresummarized as follows.

(1) When the angle of attack θ is 0◦, the flow parallels thehydrofoil surface, and the air volume fraction αg takesits maximum value near the rear half of the upper-surface. When θ = 10◦, the flow separates from theupper-surface of the rear half, and large-scale eddiesare shed from the trailing edge. The bubbles are en-

trained into the eddies, and αg reaches its maximumvalue at the center of each eddy.

(2) The time-averaged value for αg , αg takes its maxi-mum value near the hydrofoil surface, and it reduceswith approaching the surface. The maximum value in-creases in the downstream direction. The αg value atθ = 10◦ is lower than that at θ = 0◦ in the separatedflow region and near the lower-surface.

(3) The absolute value of the pressure on the upper-surface for the bubbly flow is lower than that for thesingle-phase flow.

(4) The simulated results are in good agreement with thetrend of the measurement. It is confirmed that the au-thors’ vortex method is indeed applicable to the bubblyflow analysis around a hydrofoil.

NOMENCLATURE

A: area of elementCp: pressure coefficient = (p − p0)/(ρmu

2l0/2)

d: bubble diameterl: chord lengthg: gravitational constantng : number of bubbles in quadrilateral elementp: pressureRe: Reynolds number = ul0l/νlt: timeu: velocityα: volume fractionαg0: volumetric flow ratio of bubble to whole fluidθ: angle of attackΓ: circulationν: kinematic viscosityρ: densityρm: mean density at inlet boundary = αg0ρg + αl0ρlσ : core radiusφ: scalar potentialψ : vector potentialω: vorticity =∇× ul

Subscripts0: inlet boundary or initial valueg: gas phasel: liquid phasez: direction normal to x-y-plane

REFERENCES

[1] J. Mikielewicz, D. G. Wilson, T. Chan, and A. Goldfinch, “Amethod for correlating the characteristics of centrifugal pumpperformance in two-phase flow,” ASME Journal of Fluids Engi-neering, vol. 100, no. 4, pp. 395–409, 1978.

[2] K. Minemura and M. Murakami, “Developments in the re-search of air-water two-phase flows in turbomachinery,” JSMEInternational Journal Series B, vol. 31, no. 4, pp. 615–622, 1988.

[3] A. Furukawa, S.-I. Shirasu, and S. Sato, “Experiments on air-water two-phase flow pump impeller with rotating-stationarycircular cascades and recirculating flow holes,” JSME Interna-tional Journal Series B, vol. 39, no. 3, pp. 575–582, 1996.

[4] K. Minemura and T. Uchiyama, “Three-dimensional calcula-tion of air-water two-phase flow in centrifugal pump impeller


based on a bubbly flow model,” Journal of Fluids Engineering,vol. 115, no. 4, pp. 766–771, 1993.

[5] A. Tremante, N. Moreno, R. Rey, and R. Noguera, “Numer-ical turbulent simulation of the two-phase flow (liquid/gas)through a cascade of an axial pump,” Journal of Fluids Engi-neering, vol. 124, no. 2, pp. 371–376, 2002.

[6] H. Ohashi, Y. Matsumoto, Y. Ichikawa, and T. Tsukiyama,“Air/water two-phase flow test tunnel for airfoil studies,” Ex-periments in Fluids, vol. 8, no. 5, pp. 249–256, 1990.

[7] Y. Ichikawa, H. Ohashi, and Y. Matsumoto, “Bubble motronaround an airfoil in bubbly flow,” in Proceedings of ASME Cav-itation and Multiphase Flow Forum, vol. FED-109, pp. 155–158, Portland, Ore, USA, June 1991.

[8] H. Nishikawa, Y. Matsumoto, and H. Ohashi, “Numericalcalculation of the bubbly two-phase flow around an airfoil,”Computers & Fluids, vol. 19, no. 3-4, pp. 453–460, 1991.

[9] T. Uchiyama, “Numerical prediction of bubbly flow aroundan oscillating hydrofoil by incompressible two-fluid model,”International Journal of Rotating Machinery, vol. 8, no. 4, pp.295–304, 2002.

[10] T. Uchiyama, “Numerical study on the bubbly flow around ahydrofoil in pitching and heaving motions,” Proceedings of theInstitution of Mechanical Engineers, Part C, vol. 217, no. 7, pp.811–820, 2003.

[11] T. Uchiyama and T. Degawa, “Numerical simulation for gas-liquid two-phase free turbulent flow based on vortex in cellmethod,” JSME International Journal Series B, vol. 49, no. 4,pp. 1008–1015, 2006.

[12] T. Uchiyama and T. Degawa, “Numerical simulation forbubble-laden plane mixing layer by vortex in cell method,” inProceedings of the 3rd International Symposium on Two-PhaseFlow Modelling and Experimentation, Pisa, Italy, September2004, CD-ROM.

[13] T. Uchiyama and T. Degawa, “Numerical simulation ofbubble-driven plume by vortex method,” in Proceedings of the3rd International Conference on Vortex Flows and Vortex Mod-els, pp. 350–355, Yokohama, Japan, November 2005.

[14] L. Schiller and A. Z. Naumann, “Uber die grundlegendenBerechnungen bei der Schwerkraftaufbereitung,” Zeitschriftdes Vereines Deutscher Ingenieure, vol. 77, no. 12, pp. 318–320,1933.

[15] A. Leonard, “Vortex methods for flow simulation,” Journal ofComputational Physics, vol. 37, no. 3, pp. 289–335, 1980.

[16] M. Ishii, “One-dimensional drift-flux model and constitutiveequations for relative motions between phases in various twophase flow regimes,” ANL Report ANL-77-47, Argonne Na-tional Laboratory, Argonne, Ill, USA, 1977.

[17] M. Murakami and K. Minemura, “Effects of entrained air onthe performance of a horizontal axial-flow pump,” Journal ofFluids Engineering, vol. 105, no. 4, pp. 382–388, 1983.

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Vortex Simulation of the Bubbly Flow around a Hydrofoildownloads.hindawi.com/journals/ijrm/2007/072697.pdf · around a hydrofoil. Nishikawa et al. [8] performed the nu-merical simulation