Applied Mathematical Modelling 37 (2013) 6446–6457

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

A fast non-iterative numerical algorithm to predict unsteadypartial cavitation on hydrofoils

0307-904X/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.apm.2013.01.034

⇑ Corresponding author. Tel.: +98 9166320680.E-mail addresses: bnmorteza@scu.ac.ir (M. Behbahani-Nejad), maziar.ch@gmail.com, m-changizian@phdstu.scu.ac.ir (M. Changizian).

Morteza Behbahani-Nejad, Maziar Changizian ⇑Shahid Chamran University, Ahvaz 61357-83151, Iran

a r t i c l e i n f o

Article history:Received 30 May 2012Received in revised form 16 November 2012Accepted 10 January 2013Available online 1 February 2013

Keywords:Unsteady flowPartial cavityBoundary Element MethodHydrofoilFast algorithm

a b s t r a c t

A new algorithm to predict partial sheet cavity behavior on hydrofoils is proposed. The pro-posed algorithm models the unsteady partial cavitation using Boundary Element Method(BEM). In the proposed method the spatial iterative scheme is removed by means of anew approach determining the instantaneous cavity length. This iterative scheme isrequired in conventional algorithms to obtain the cavity length at each time step. Perfor-mance of the new algorithm for various unsteady cavitating flows with different reducedfrequencies, cavitation numbers, hydrofoil geometries and inflow conditions are investi-gated. Comparison between the obtained results using the proposed method and thoseof conventional ones indicates that the present algorithm works well with sufficient accu-racy. Moreover, it is shown that the proposed method is computationally more efficientthan the conventional one for unsteady sheet cavitation analysis on hydrofoils.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Cavitation is usually observed in high-speed liquid flows around many obstacles such as impellers and hydrofoils. It iswell understood that cavity flows will cause vibration, noise, damage and decline of efficiency in hydraulic systems. Partialsheet cavitation, cavitating tip vortex and supercavitating sheet are common types of cavitations that may occur when ahigh-speed liquid flows on a hydrofoil [1]. Today, there are several computational modeling approaches to simulate cavitat-ing flows around hydrofoils [2–9]. Many of these models assume potential flow because of its simplicity and its suitableaccuracy to analyze steady and unsteady cavitating flows around complex geometries [7–11]. Since BEM is known as a pow-erful computational method for potential problems, it is widely used by many researchers for cavitating flow analysis [7–11].

Analysis of cavitating hydrofoils using BEM have been published by, for instance, Pyo and Suh [2], Salvatore and Esposito[3] and Lee and Kinnas [8]. All these works are based on the 2-D closed partial cavity model of Kinnas and Fine [10], whichconsists of a so-called split panel approach and a pressure recovery region for the cavity closure. Vaz et al. [6] reviewed andcompared three different models for partial cavity flow modeling in the steady state. The above models are a fully non-linearmodel (FNL), a partially non-linear model with surface remeshing (PNL1), and a partially nonlinear model without surface re-meshing (PNL2). In FNL, the cavity surface is discretized using some boundary elements and the kinematic and dynamicboundary conditions are imposed on them [4,12,13]. But, in partially non-linear models one assumes a thin cavity andthe boundary elements on the hydrofoil beneath the cavity surface are considered as the cavity boundary elements. InPNL1 some of the boundary elements are allocated to the cavity surface. These elements should be resized in accordancewith the cavity length on each iteration step that results in resizing other boundary elements placed on the body. Therefore

M. Behbahani-Nejad, M. Changizian / Applied Mathematical Modelling 37 (2013) 6446–6457 6447

it is a surface remeshing procedure. On the other hand, in PNL2 the number of boundary elements assigned to the cavity ischanged according to the cavity length and it is not required to resize the initial mesh when the cavity length is changed.

Vaz et al. [6] and Vaz [9] concluded that the overall performance of PNL1 is much better than PNL2 and fully nonlinearmodels. Moreover, Vaz et al. [6] concluded that PNL2 requires a larger number of panels than PNL1 to get numerical resultswith the same accuracy. However, robustness of PNL2 convinced Vaz [9] to use this model for three-dimensional steady cav-itation analysis. Regardless of which method (FNL, PNL1 or PNL2) is chosen, the cavity surface characteristics (cavity detach-ment point, cavity volume and length) are not known a priori and they should be obtained as part of the solution. Differentcriteria have been proposed in the literature to identify the location of cavity detachment point. The location of minimumpressure, the position where the surface pressure equals the fluid vapor pressure, the leading edge or the laminar separationpoint may be considered as the cavity detachment point. Moreover, a detachment condition known as the smooth separationcondition by Brioullin and Villat (see Franc and Michel [14]) may be used to locate the detachment point. In addition, a clo-sure condition is required in order to close the cavity surface. In steady state analysis a pressure recovery model has usuallybeen used [6–9] as a closure condition while a dynamic boundary condition without any pressure recovery at the cavity endis usually applied for unsteady problems [9].

One of the main difficulties in the analysis of cavitating flows is determination of the free streamline (the cavity surface)on which the pressure is prescribed. Since the cavity surface is unknown an iterative procedure is required to determine it,see e.g. [9–11]. These iterative procedures clearly increase CPU time and computational costs especially for unsteady anal-ysis. If another approach can be introduced so that eliminates time consuming iterative procedures, it will be more attractiveand efficient than the conventional ones. The aim of the present work is to introduce such an approach with acceptableaccuracy.

The current study is searching for a novel approach for analysis of unsteady sheet cavitation without any iterationrequirement. This new approach is based on the partially non-linear model without re-gridding (PNL2) previously developedfor steady and unsteady partial sheet cavitation [9]. First, the mathematical model of partial cavity is presented for a hydro-foil in an unsteady flow. Next, the numerical model based on BEM is presented and the proposed algorithm is introduced.Then, the proposed method is applied for different geometries, reduced frequencies, and inflow conditions. The obtained re-sults are discussed and it is demonstrated that the proposed non-iterative algorithm can analyze various unsteady problemswith sufficient accuracy. After that, the computational efficiency of the proposed method is investigated and shown that it ismore efficient than the previous iterative methods. Finally, the paper is concluded with the abilities of the proposed non-iterative method for unsteady partial cavitation analysis. Although the current approach is based on PNL2, one does not wor-ry about the large number of boundary elements and CPU times since there are no iteration steps.

2. Theoretical formulation

Consider a hydrofoil section with an attached cavity as shown in Fig. 1. If the hydrofoil moves with U1 (along-x direction)in a gusty flow described as W1ðx;tÞ ¼Ws þ DW f ðx; tÞ; the undisturbed flow velocity relative to a body fixed frame on thehydrofoil is

V!

0ðx;tÞ ¼ ðU1;Ws þ DWf ðx; tÞÞ; ð1Þ

where Ws, DW and f ðx; tÞ define the steady and unsteady parts of a possible gust in the flowfield. In a special case iff ðx; tÞ ¼ eiwt Eq. (1) describes a flowfield around an oscillating hydrofoil moving with a constant velocity of U1. Using Eq.(1) for low angles of attacks one obtains

W1ðx;tÞU1

¼ a0 þ Daf ðx; tÞ ð2Þ

ns

SC

SB

SWU

L

S oo

x

z

h

SBC

V0

Fig. 1. Two-dimensional partial sheet cavitation problem domain.

6448 M. Behbahani-Nejad, M. Changizian / Applied Mathematical Modelling 37 (2013) 6446–6457

where a0 is a steady state angle of attack and Da is an amplitude about it.Assuming potential flow, one can write

~rU ¼ ~V0 þ ~r/ ð3Þ

where U and / are total and perturbation velocity potentials, respectively. Outside the cavity, the perturbation potential sat-isfies Laplace’s equation, namely

r2/ ¼ 0 ð4Þ

along with the boundary conditions that will be later discussed in detail. Solving Eq. (4) subject to the corresponding bound-ary conditions, the perturbation potential is obtained and one can compute the pressure coefficient distribution using theunsteady Bernoulli equation that is [15]

2U21

@/@tþ jrUj2 � jV0j2

2¼ �Cp ð5Þ

where Cp is the pressure coefficient defined as

Cp ¼p� p112 qU2

1ð6Þ

In the above relation, p and p1 are the perturbed and undisturbed flow pressures, respectively, and q is the fluid density.Equation (4) is a boundary value problem and needs boundary conditions on the entire flow boundaries. Flow boundariescan be considered as illustrated in Fig. 1. They are the wet body surface SB (that part of the hydrofoil in contact with theliquid), the cavity surface SC, the wake surface SW (the wake sheet behind the lifting hydrofoil) and the far boundary denotedas S1. The flow disturbances via the body motion should be diminished on S1. In the other word

limr!1r/ ¼ 0 on S1 ð7Þ

where r is the distance from the origin of body’s frame of reference. On the wetted part of the body surface the fluid flow istangent to the hydrofoil and a kinematic Neumann boundary condition is imposed, i.e.

@/@n̂¼ �~V0:n̂ on SB ð8Þ

where n̂ is the unit normal to the boundary pointing into the flowfield.To impose the boundary condition on the wake surface, SW , one knows that the vorticity generated at the trailing edge is

shed into the wake and using Kelvin’s theorem it is convected along the wake surface with the free stream speed [15]. Thegenerated vorticity at the trailing edge can be described using the familiar Kutta condition, namely

ðD/ÞTE ¼ /UTE � /L

TE ð9Þ

where ()TE denotes trailing edge and ()U and ()L correspond the wake upper and lower surfaces, respectively. Kelvin’s theoremresults in,

D/W ðx; tÞ ¼ D/TE t � x� xTE

U1

� �onSW ð10Þ

The questionable boundary condition is the one that should be imposed on the cavity surface. Since the cavity surface isnot known a priori, two boundary conditions are considered; a dynamic boundary condition (DBC) and a kinematic boundarycondition (KBC). The dynamic boundary condition on the cavity surface confirms that the pressure everywhere on the cavitysurface is constant and equals to the vapor pressure. Using Eq. (5) it can be shown that this is equivalent to prescribingknown values of / on the cavity, which satisfies [9]

/ðs; tÞ ¼ /0ð0; tÞ þZ s

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU21rþ jV0j2 � 2

@/@t

r� ~V0 �~s

" #ds ð11Þ

where~s is a curvilinear coordinate tangent to the cavity surface as shown in Fig. 1, /0 is the perturbation potential at thecavity detachment point (s = 0), and r is the cavitation number defined as

r ¼ p1 � pv12 qU2

1ð12Þ

The unknown @/@t in Eq. (11) is evaluated using the earlier values of / from the previous time steps [11]. In addition, /0ð0; tÞ

is not known and in the present numerical scheme is expressed via a cubic extrapolation in terms of the unknown potentialson the wetted panels in front of the cavity [11]. As already discussed, boundary conditions are applied on the hydrofoil sur-face underneath the cavity (SBC in Fig. 1) when the partially non-linear model is used.

M. Behbahani-Nejad, M. Changizian / Applied Mathematical Modelling 37 (2013) 6446–6457 6449

The kinematic boundary condition guarantees the flow tangency on the cavity surface. It can be imposed using a partialdifferential equation for the cavity thickness, h, as follows [9]:

Us@h@s¼ Un �

@h@t

ð13Þ

where Us and Un are velocity components tangential and normal to the cavity surface, respectively.

3. Numerical model

In the present study BEM is used as the numerical model to analyze the governing equation along with the imposedboundary conditions for partial cavitation analysis around hydrofoils. Applying Green’s theorem and imposing the boundarycondition on S1 the equivalent boundary integral equation for Eq. (4) describing the perturbation potential at any point p onthe hydrofoil and cavity surfaces is [10]

m/pðtÞ ¼1

2p

ZSBðtÞ[SBC

ðtÞ/ðtÞ @

@n̂ðln rÞ � ln r

@/ðtÞ@n̂

� �dsþ

ZSW ðtÞ

D/wðtÞ@ðln rÞ@n̂

ds ð14Þ

where r is the distance from the point p to the boundary element ds, m ¼ 1=2 if p is on a smooth part of the surface and theintegrals on SBðtÞ [ SBC ðtÞ are in the sense of Cauchy principal value. In order to obtain an approximate solution for the bound-ary integral Eq. (14), the surfaces SB, SBC and SW are discretized using small straight line elements. The value of / and @/

@n̂ areassumed to be constant within each element. Therefore, the collocation method yields the following relation for each collo-cation point on the body [9]:

/p ¼XNBþNC

j¼1

Apj/j þXNBþNC

j¼1

Bpj@/@n̂

� �j

þXNW

j¼1

ApjD/j; p ¼ 1;2; . . . ;NBþ NC ð15Þ

where NB, NC and NW are the number of elements on the wet body, cavity and the wake of the hydrofoil, respectively.Moreover,

Apj ¼1

2p

ZSj

@

@n̂ðln rÞdsj ð16Þ

Bpj ¼ �1

2p

ZSj

ðln rÞdsj ð17Þ

are influence coefficients and sj is the surface of jth element.For the first term on the right hand side of Eq. (15) /j is known on the cavity surface from boundary condition described

by Eq. (11). On the other hand @/@n̂ on the wet body is determined using the tangency boundary condition represented by Eq.

(8). The third term on the right hand side of Eq. (15) is also known because D/ can be obtained using the Kutta condition andKelvin’s theorem denoted by Eq. (10). Therefore, there are NB unknowns for / on the wet body and NC unknowns for @/

@n̂ onthe cavity surface that are obtained using Eq. (15).

Because the cavity surface is not known a priori, imposing the boundary condition on the cavity surface is not a clear andstraightforward procedure. One does not know where the start and the end points of cavity are. Therefore, the partial cavityproblems are usually analyzed using iterative algorithms. For example let hðsÞ denotes the cavity thickness. It should beequal to zero at s ¼ lc where lc is the cavity length. In a conventional unsteady algorithm a spatial iterative scheme is usedat each time step to find the correct value of lc . The spatial iterative scheme is begun with an initial guess for the cavitylength. Next, Eq. (15) with corresponding boundary conditions is solved and / and @/

@n̂ distribution on the wet body and cavitysurface are determined, respectively. Then the cavity thickness, hðsÞ;is calculated using Eq. (13). If the cavity thickness ats ¼ lc converges to zero [9,11], namely

jhðlcÞj 6 ea ð18Þ

where ea is an accepted small error, the solution is converged and the guessed length is the correct one. Of coursejhðlcÞj ¼ e > ea at each iteration step if the cavity length is smaller or larger than the converged value.

In the present work, a non-iterative algorithm is proposed which eliminates the above time consuming spatial iterativeprocedure. The idea of developing the current algorithm is based on the behavior of cavity thickness, hðlcÞ; versus the guessedcavity length in a steady flow. Let’s for example consider an unsteady partial cavitation flow with r ¼ 1 over a NACA 16-006hydrofoil with four degrees average angle of attack. Of course the cavity length is not known a priori. For initial guesses ofcavity lengths (lc), the corresponding cavity thicknesses at s ¼ lc are calculated using steady flow assumption. They are illus-trated in Fig. 2 for various angles of attacks around the average one. The cavity detachment point is considered at the leadingedge. Clearly the correct cavity length is the one for which e is zero. As is shown in the figure the error variation near thesolution (e ffi 0) is quite linear and its slop is nearly the same for all angles of attack. Thus, for each problem the error slopcan be considered as the one that obtained at the average angle of attack.

Cavity guessed length

ε/c

0 0.2 0.4 0.6 0.8-0.06

-0.04

-0.02

0

0.02

0.04

AOA=3.5AOA=4AOA=4.5

lc/c

Fig. 2. Cavity heights (e) vs. different guesses of cavity lengths for NACA 16-006 with r ¼ 1.

6450 M. Behbahani-Nejad, M. Changizian / Applied Mathematical Modelling 37 (2013) 6446–6457

Now, let’s see how the proposed algorithm is applied for unsteady partial cavitation analysis. In the current algorithm acavity length and an error slop are obtained for the corresponding steady flow with an average loading condition (averageangles of attacks of the corresponding unsteady problem). Next, considering the obtained cavity length as an initial guess Eq.(15) with boundary conditions 7, 8, 10, and 11, is solved that results in the unknown distribution of sources and dipolesstrengths on the cavity and wet body of the hydrofoil. Consequently Eq. (13) gives e and Eq. (18) may not be satisfied. Then,using e and the error slop already obtained, one can predict the correct cavity length without any iteration process. Finally,based on the new cavity length hydrodynamic characteristics of the hydrofoil in unsteady partial cavitation are obtained ateach time step.

The algorithm was applied for an unsteady partial cavitation flow around NACA16-006 hydrofoil withaðTÞ ¼ 4

�þ 0:5

�sinðTÞ where T ¼ 2U1t=c is the non-dimensional time. Fig. 3 shows the hydrofoil lift coefficient time vari-

ations obtained using the current algorithm and the conventional iterative approach (the conventional iterative approachis based on PNL2). The same panel configurations are used for both the conventional and current methods.

U t/c

CL

0 10 20 30 40 500.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

Conventional methodCurrent method

oo

Fig. 3. Time changes of lift coefficient for NACA 16-006 at r ¼ 1.

x/c

CP

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

Conventional methodCurrent method

Fig. 4. Current and conventional methods results for pressure coefficient distribution on NACA 16-006.

M. Behbahani-Nejad, M. Changizian / Applied Mathematical Modelling 37 (2013) 6446–6457 6451

As is illustrated in Fig. 3, relatively large differences exist between the results of the current and conventional algorithms.However, the results of current algorithm have nearly the same amplitude but with some shift in comparison with those ofthe conventional iterative approach. With more assessment it was found out that the main reason for aforementioned shift isthe pressure distribution behavior adjacent the reattachment region of cavity. The pressure coefficient distribution at theinstant of fourth cycle of hydrofoil oscillation is shown in Fig. 4.

As is depicted in Fig. 4 one can observe that there is a large difference between the pressure coefficients computed by thetwo methods near the cavity end. Because the end point of cavity should be stagnation point (see Fig. 5), the pressure coef-

Reattachment Point

End Points of ElementsCollocation Points

Fig. 6. Graphical representation of srf and s⁄.

Stagnation Point

Fig. 5. Stagnation point at the end of cavity.

x/c

CP

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

Conventional methodProposed method

Fig. 7. Proposed and conventional methods results for pressure coefficient distribution on NACA 16-006.

6452 M. Behbahani-Nejad, M. Changizian / Applied Mathematical Modelling 37 (2013) 6446–6457

ficient should be near one there. Thus, the results obtained by the conventional iterative method are more consistent withthe physics of the flow.

At the moment, an ad hoc procedure is introduced to fix that problem. The procedure modifies the flow velocity adjacentthe cavity end so that the zero velocity condition is fulfilled there. Accordingly, it is expected that the current approach isrelatively improved. Let gðs�Þ denotes a pressure recovery function by which the flow velocity is corrected as

V� ¼ 1� gðs�ÞV ð19Þ

where V is the total velocity, V⁄ is the corrected velocity and

gðs�Þ ¼srf

s��srflc 6 s� 6 lc þ Dl;

0 otherwise

(ð20Þ

in which srf is the distance between the reattachment point and the collocation point on its neighbor element (see Fig. 6).Moreover, s⁄ is the distance between the reattachment point and the end point of next elements and Dl is a suitable regionnear the reattachment point. Based on the authors’ numerical experiences Dl ¼ 0:3 lc is an appropriate choice for variousproblems.

If the algorithm is modified based on the above procedure, the corresponding pressure coefficient distribution of Fig. 4will be changed as shown in Fig. 7. As is illustrated in Fig. 7 the modified algorithm results approach the ones of conventionaliterative method with small acceptable error. Therefore, the modified algorithm is proposed as the final approach denoted bythe proposed method.

4. Results and discussions

To verify the generality of the proposed method it is applied for various unsteady partial cavitation flows with differentcavitation numbers, reduced frequencies, and inflow conditions flows. NACA16-006 and NACA16-009 hydrofoils are consid-ered as they are more conventional for partial cavitation flows.

To investigate the performance of the proposed method for different cavitation numbers, it is applied forr ¼ 1:0; 1:1and1:2 for unsteady partial cavitation flows over a NACA16-006 hydrofoil. The unsteady flow described byEq. (2) is considered as

aðTÞ ¼ 4�þ 0:5

�sinðkTÞ ð21Þ

where k ¼ xc2U1

is the reduced frequency. For a time interval equivalent to 16 cycles of oscillations, spatial and temporal dis-cretizations are considered as those of Vaz [9], i.e., 400 panels on hydrofoil with cosine distribution and 10 time steps pereach cycle. The first step is to calculate cavity length and error slope in the corresponding steady flow for average loading(a0 ¼ 4

�) and related cavitation number. Table 1 depicts the required cavity lengths and error slops obtained using the steady

analysis.

Table 1Cavity length and error slope for average loading (a0 ¼ 4

�).

Cavity length (lc) Error slope

r ¼ 1:0 0.40 �0.1r ¼ 1:1 0.32 �0.11r ¼ 1:2 0.26 �0.11

M. Behbahani-Nejad, M. Changizian / Applied Mathematical Modelling 37 (2013) 6446–6457 6453

The next step is applying the proposed method for unsteady analysis based on the data depicted in Table 1. Fig. 8 illus-trates time changes of cavity lengths that are obtained with the conventional (iterative PNL2) and the proposed methods. Asis shown in the figure the results of present approach are in good agreement with those of the conventional method (themaximum relative difference is less than 5%). However, there are relatively large differences for a few initial time stepswhere nonphysical peaks exist due to the time derivative term in relation (11) [9]. Fortunately, these transients decayquickly.

Fig. 9 shows time variations for the lift coefficient which are obtained using the conventional and the proposed algo-rithms. As is shown in this figure, good agreements are observed between the results of conventional and proposed methods.Excluding the first transient, the maximum relative difference is less than 4%.

Next, the ability of the proposed method for analysis of unsteady cavitating flows with various reduced frequencies isinvestigated. Time changes of cavity length and lift coefficients for different reduced frequencies are obtained and discussed.Figs. 10 and 11 illustrate time changes of cavity length and lift coefficient, respectively for NACA16-006 hydrofoil. In the cor-responding unsteady partial cavitation flows r ¼ 1:0 and reduced frequencies of 0.5, 1 and p=2 are considered. It is observedthat the results of proposed method are in accord with the conventional ones. The maximum relative error is about 6% forboth cavity length and lift coefficient predictions. In the other words the proposed method can analyze unsteady flows withsufficient accuracy in a wide range of reduced frequencies.

Now, let’s study the performance of the present approach in gusty flows. If

Len

gth

ofC

avit

y/c

CL

W1 ¼ Vgust ¼ 0:07þ 0:0175 sin 2kxc�wt

� �h ik ð22Þ

then the flow velocity in the body fixed frame is

~V0ðx; tÞ ¼ 1;0:07þ 0:0175 sin 2kxc�wt

� �h ið23Þ

U t/c0 10 20 30 40 50

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Conventional methodProposed method

oo U t/c

Len

gth

ofC

avit

y/c

0 10 20 30 40 500.25

0.3

0.35

0.4

0.45

Conventional methodProposed method

oo U t/c

Len

gth

ofC

avit

y/c

0 10 20 30 40 500.15

0.2

0.25

0.3

0.35

0.4

Conventional methodProposed method

oo

Fig. 8. Cavity lengths time changes of NACA 16-006 for various cavitation numbers and k ¼ 1.

U t/c0 10 20 30 40 50

0.4

0.45

0.5

0.55

0.6

0.65

0.7

Conventional methodProposed method

oo U t/c

CL

0 10 20 30 40 500.4

0.45

0.5

0.55

0.6

0.65

0.7

Conventional methodProposed method

oo U t/c

CL

0 10 20 30 40 500.4

0.45

0.5

0.55

0.6

0.65

0.7

Conventional methodProposed method

oo

Fig. 9. Lift coefficient time changes of NACA 16-006 for various cavitation numbers and k ¼ 1.

U t/c

CL

0 20 40 60 80 1000.4

0.45

0.5

0.55

0.6

0.65

0.7

Conventional methodProposed method

k = 0.5

oo U t/c

CL

0 10 20 30 40 500.4

0.45

0.5

0.55

0.6

0.65

0.7

Conventional methodProposed method

k = 1

oo U t/c

CL

0 5 10 15 20 25 300.4

0.45

0.5

0.55

0.6

0.65

0.7

Conventional methodProposed method

k = π/2

oo

Fig. 11. Lift coefficient time changes of NACA 16-006 for various reduced frequencies and r ¼ 1.

U t/c

Len

gth

ofC

avit

y/c

0 20 40 60 80 1000.3

0.35

0.4

0.45

0.5

0.55

0.6

Conventional methodProposed method

0.3

0.35

0.6

Len

gth

ofC

avit

y/c

0.4

0.45

0.5

0.55

U t/c0 10 20 30 40 50

Conventional methodProposed method

U t/c

Len

gth

ofC

avit

y/c

0 5 10 15 20 25 300.3

0.35

0.4

0.45

0.5

0.55

0.6

Conventional methodProposed method

Fig. 10. Cavity lengths time changes of NACA 16-006 for various reduced frequencies and r ¼ 1.

6454 M. Behbahani-Nejad, M. Changizian / Applied Mathematical Modelling 37 (2013) 6446–6457

For the above flowfield, the unsteady partial cavitation analysis results in time variations of lift coefficient and cavity lengthas illustrated in Figs. 12 and 13, respectively. In the present study it is assumed that r = 1and k ¼ p=2. It is observed that thecurrent non-iterative approach beautifully works for gusty flows as well. The maximum difference between the results ofpresent and conventional methods is less than 7% for both cavity length and lift coefficient calculations.

Next, the capability of the proposed method is discussed for other geometries. A NACA16-009 hydrofoil is considered asanother geometry and the obtained results of the present method are compared with those of the conventional one. The

U t/c

Len

gth

ofC

avit

y/c

0 5 10 15 20 25 300.3

0.35

0.4

0.45

0.5

0.55

0.6

Conventional methodProposed method

oo

Fig. 12. Cavity lengths time changes of NACA 16-006 in the gusty flow (r ¼ 1 k ¼ p=2).

U t/c

CL

0 5 10 15 20 25 300.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

Conventional methodProposed method

oo

Fig. 13. Lift coefficient time changes of NACA 16-006 in the gusty flow (r = 1, k ¼ p=2).

M. Behbahani-Nejad, M. Changizian / Applied Mathematical Modelling 37 (2013) 6446–6457 6455

unsteady flow is considered as Eq. (21) with k ¼ p=2 and r ¼ 1. Moreover, steady state analysis results in the error slope of�0.085 for this hydrofoil. Figs. 14 and 15 show time changes of cavity length and lift coefficient, respectively. As are illus-trated in the figures good agreement exists between the results of proposed and conventional ones. Maximum relative errorsare 6% and 7% for cavity length and lift coefficient calculations. Based on the obtained results one concludes that the pro-posed method can be applied for other hydrofoil geometries with sufficient accuracy. In other words the proposed methodis a sufficiently accurate and general approach for unsteady partial cavitation analysis on hydrofoils.

Finally, the computational efficiency of the proposed method is discussed through comparison between the CPU times ofthe proposed method and the conventional iterative one. Let’s define time efficiency of the present method as

g ¼ jTc � TpjTc

� 100 ð24Þ

where Tc and Tp are CPU times of the conventional and present methods, respectively. Table 2 depicts CPU times in secondsfor analysis of unsteady cavitating flows with various reduced frequencies on NACA 16-006 section via the conventional and

U t/c

Len

gth

ofC

avit

y/c

0 5 10 15 20 25 300.25

0.3

0.35

0.4

0.45

0.5

0.55

Conventional methodProposed method

oo

Fig. 14. Cavity lengths time changes of NACA16-009 (r = 1, k ¼ p=2).

U t/c

CL

0 5 10 15 20 25 300.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

Conventional methodProposed method

oo

Fig. 15. Lift coefficient time changes of NACA16-009 (r = 1, k ¼ p=2).

Table 2Comparison of CPU time (s) in different situations.

k ¼ 0:5 k ¼ 1 k ¼ p=2

Tc 49.3 42 62.4Tp 8 7.83 7.55g 84% 82% 88%

6456 M. Behbahani-Nejad, M. Changizian / Applied Mathematical Modelling 37 (2013) 6446–6457

proposed methods. The presented results are based on numerical computations using a dual core-2100 MHz with 2-GB RAMcomputer. As is depicted in the Table 2, the proposed method is much more efficient than the conventional iterative one. It isobserved that the present method has a time efficiency of more than 80% which is an excellent efficiency for a computationalapproach. Thus, one can consider the present method as a fast non-iterative algorithm.

5. Conclusions

The proposed numerical algorithm can accurately and efficiently predict partial cavity effects on hydrofoils in unsteadyflows. Using the proposed method the cavity length and lift coefficient can be accurately predicted without any iterationrequirement at each time step. An algebraic and simple relation is proposed to modify the pressure distribution close tothe cavity reattachment point. However, the proposed method prettily works for several cavitation numbers, reduced fre-quencies, inflow conditions and different geometries. Comparison between the results of conventional and proposed meth-ods shows that the maximum relative difference is less than 7%. Moreover, the obtained results show that the proposedmethod has excellent time efficiency greater than 80%. Having the above characteristics, one concludes that the proposedmethod is a powerful numerical approach for analysis of unsteady partial cavitation problems.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.apm.2013.01.034.

References

[1] J.S. Carlton, Marine Propellers and Propulsion, Butterworth Heinemann, 1994.[2] S.W. Pyo, J.C. Suh, Modified split panel method applied to the analysis of cavitating propellers, J. Ship Ocean Eng. 4 (2000) 1–10.[3] F. Salvatore, P.G. Esposito, Numerical analysis of cavitating propellers including viscous effects. in: Proceedings of the 8th of International Symposium

on Practical Design of Ships and Mobile Units (PRADS 2001), 2001.[4] F. Salvatore, C. Testa, L.A Greco, Viscous-inviscid coupled formulation for unsteady sheet cavitation modelling of marine propellers. in: Cav2003

Proceedings, November, Osaka, Japan, 2003.

M. Behbahani-Nejad, M. Changizian / Applied Mathematical Modelling 37 (2013) 6446–6457 6457

[5] Y.L. Young, Fluid and structural modelling of cavitating propeller flows. in: Cav2003 Proceedings, November, Osaka, Japan, 2003.[6] G. Vaz, J. Bosschers, J. Compo, Two-dimensional modelling of partial cavitation with BEM. in: Analysis of Several Models, Proceedings of Cav 2003,

Osaka, Japan, 2003.[7] L. Greco, F. Salvatore, F. Felice, Validation of a quasi-potential flow model for the analysis of marine flow propellers. in: Proceedings of the 25th

Symposium on Naval Hydrodynamics, August, St.John’s, Canada, 2004.[8] H. Lee, S.A. Kinnas, Application of a BEM in the prediction of unsteady blade sheet and developed tip vortex cavitation on marine propellers, Journal of

Ship Research 48 (2004) 15–30.[9] G. Vaz, Modelling of Sheet Cavitation on Hydrofoils and Marine Propellers using Boundary Element Methods, PhD thesis, IST, UTL, Lisbon, Portugal;

2005.[10] S.A. Kinnas, N.E. Fine, A numerical nonlinear analysis of the flow around two- and three-dimensional partially cavitating hydrofoils, J. Fluid Mech. 254

(1993) 151–181.[11] N.E. Fine, Non-linear Analysis of Cavitating Propellers in Nonuniform Flow, PhD thesis, Massachusetts Institute of Technology – MIT, October 1992.[12] J. Dang, Numerical Simulation of Unsteady Partial Cavity Flows, PhD thesis, Delft University of Technology, February 2001.[13] J. Dang, G. Kuiper, Re-entrant jet modeling of partial cavity flow on two-dimensional hydrofoils, J. Fluids Eng. 121 (1999) 773–780.[14] J.P. Franc, J.M. Michel, Attached cavitation and the boundary layer: experimental investigation and numerical treatment, J. Fluid Mech. 154 (1985) 63–

90.[15] J. Katz, A. Plotkin, Low-Speed Aerodynamics, 2nd ed., Cambridge University Press, 2001.