mathematical modeling and decomposition of hydrodynamics...

Scientia Iranica B (2014) 21(1), 213{221

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringwww.scientiairanica.com

Mathematical modeling and decomposition ofhydrodynamics-acoustic �elds using perturbationmethods

M. Riahi�, N.M. Nouri and A. Valipour

Department of Mechanical Engineering, Iran University of Science and Technology, Tehran, P.O. Box 1684613114, Iran.

Received 11 December 2012; received in revised form 17 June 2013; accepted 16 July 2013

KEYWORDSMathematicsmodeling;Cavitation;Supercavitation;Perturbation method;Scale analysis.

Abstract. Conservation of mass, momentum, energy and state equations are recognizedas basic mathematical models in analysis of the acoustic behavior of cavitation, as wellas supercavitation. Also, it is known that the order of acoustic e�ects is not as high asthat of hydrodynamics. Therefore, in this paper, initially, for comparing di�erent terms ofequations, using scale analysis, conservation equations are converted into dimensionlessones. Then, by comparing all conditions, coupled with weighting terms available inthose equations, groups of parameters most appropriate with the hydrodynamics andhydroacoustics of the cavitating ow, are selected. By regarding acoustics as lowerorder phenomena, compared to the hydrodynamics of ow, and simultaneously using theperturbation method, two equations containing leading and �rst orders and di�erent termscan be attained. Obtained results indicate that leading order equations represent thehydrodynamics of the cavitating ow, and �rst order equations indicate the acoustics ofcavitation or supercavitation. Acoustic equations of the present study contain terms relatedto uid viscosity, density and pressure changes, and background ow velocity. As acousticequations are coupled with leading order equations, in order to �nd the noise of cavitation,equations of uid ow for compressible ow should be resolved.© 2014 Sharif University of Technology. All rights reserved.

1. Introduction

Evaporation of liquid due to pressure decrease down toless than the pressure of the saturated vapor of thatliquid, is called cavitation. Cavitation occurs whenthe local pressure of a liquid uid suddenly reachesbelow a critical value. This critical value is relatedto the pressure of the vapor of the liquid uid. Byplacing uid in this area, small bubbles made of vaporand other gases begin to form. These bubbles movewith the current of uid and when they reach areas

*. Corresponding author. Tel.: +98 21 77240540E-mail addresses: [email protected] (M. Riahi),[email protected] (M. Nouri), [email protected] (A.Valipour)

with higher pressure, will dissolve and dissipate. Ingeneral, there are two methods to realize cavitationphenomenon. First is visual observation, and second isacoustic observation [1]. As a result, noise generated bycavitation, although recognized as an undesired e�ect,can be viewed as a means for recognition of cavitationoccurrence. Noise is a mechanical perturbation thatpropagates in an elastic region. In uids, noise isgenerated when there is a relative motion between two uids and/or between a uid and a surface. Noise isalways regarded as a sound or an undesired sound thathas an e�ect on the normal performance of a system.Noise can appear in a uid ow due to turbulence,because of chaos in the uid or on the basis of cavitationoccurrence. Turbulence and vertex usually lead to lowfrequency pressure uctuations, whereas, noise caused

214 M. Riahi et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 213{221

by cavitation typically has an acoustic frequency rangefrom 1 to 100 kHz [2]. Brennen's investigation [3] indi-cates that bursting bubbles at high frequencies occursmore often than similar occurrences without cavitation.The acoustic e�ect of cavitation is more in the rangeof ultrasonic wide-band waves (in excess of 20 kHz) [3].In general, sound is a perturbation term in steady con-dition [4]. Pierce [5] and Goldstein [6] expressed meansof �nding general forms of wave equations in detail.Morch [7] and Chahine [8] concentrated their workon the dynamics of some �xed cavitation known as�xed unsteady cavitation, otherwise, recognized cloudcavitation. Experimental studies of Reisman et al. [9]showed that noise caused by bursting cloud cavitationexceeds by far that of collective noise generated bysummation of all single bubbles existing in a cloudcavitation. Wang [10] investigated generated shockwaves in cloud cavitation in his doctoral dissertation.The growth and dissolving of a spherical cloud madefrom cavitation bubbles were modeled in a nonlinearform. Because of that, he was able to �nd pressurepulses, as well as produced momentum, causing noiseand erosion. Wang combined a continuum mixturemodel and Rayleigh-Plesset equation and solved itusing the Lagrangian integration method. In followingyears, Brennen et al. [11,12] developed their studies inrecognizing acoustic noise caused by cloud cavitationin numerical form. Reisman studied the acoustics ofcloud cavitation, numerically and experimentally. Byprocessing received signals from his experiments, heshowed that the frequency range of signals belongingto noise of this type of cavitation wave was almost be-tween 10 Hz to 100 kHz [13]. Levy et al. [14] developedhis research on noise generated by cloud cavitationin numerical as well as experimental form in a watertunnel with high speed and on a NACA0015 hydrofoil.Seo et al. [15] estimated cavitation ow noise arounda two-dimensional cylinder with circular cross-sectionand by using Direct Numerical Simulation (DNS). Inrecent years; extensive studies on supercavitation havetaken place in the hydrodynamics laboratory of IranUniversity of Science and Technology. Nouri et al.[16] and Moghimi [17] studied the steady conditionof a cavity boundary while supercavitation. Howeand Foley [18] and Foley et al. [19,20] have conductedvast investigations into �nding propagated sounds fromventilated supercavitation. Formation of ventilatedsupercavitation around underwater vehicles causes ve-hicles to reach high speed and generates low and highfrequency sound noises [21]. High frequency noisescan also interfere with the guiding and controllingsystem of the vehicle, whereas, low frequency noiseshave a tendency to propagate to the �eld far fromthe supercavity [22]. As acoustic propeller noise isthe most important noise source of underwater vehi-cles, some numerical [23,24] and experimental [25,26]

investigations have been studied recently. Salvatoret al. [27] used the Ffowcs Williams and Hawkings(FWH) model in investigation of underwater propellernoise. This type of noise estimation methodologypresents many assumptions, such as those consideredin linear acoustics, low Mach number and compressedsound sources [28]. Figure 1 depicts formation ofsupercavitation in a water tunnel.

In light of host applications concerning noisefrom cavitation and supercavitation, mainly from en-vironmental and marine aspects, the degree of itsimportance becomes obvious. In this research, in orderto estimate noise caused by the occurrence of cavitationand supercavitation in uid ow, governing conserva-tion equations of uid ow need to be derived. So, inthis paper, using scale analysis, conservation equationsincluding di�erential terms and weighting terms, areconverted into dimensionless form. Hydrodynamicstudies of cavitating ow are not in the same order asa hydroacoustic study of it. So, for decomposing thesestudies into two di�erent orders, a perturbation methodshould be used. By selecting a suitable weighting termas a perturbed term and applying the perturbationmethod, conservation equations are decomposed intotwo orders containing leading and �rst orders. Leadingorder equations display the hydrodynamics of uid owwith cavitation (or supercavitation), and �rst orderequations explain the acoustics of cavitation. Finally,the validity of these two di�erent models is investigatedby comparing them with other study results.

2. Scale analysis

Scale analysis is one of the most useful methods inclassical uid mechanics. Using scale analysis andmaking equations non-dimensional causes all terms ofequations to have appropriate weight. Consequently,comparing the weight of each term with each otherspeci�es the importance of each term or variableparameter, and the lower weight can be neglected. So,

Figure 1. Ventilated supercavitation produced inhydrodynamic laboratory at Iran University of Scienceand Technology.

M. Riahi et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 213{221 215

applying scale analysis can be helpful for using simplermodels for investigation of a problem. Acoustic noiseis part of uid dynamics phenomena which occurs ina cavitating ow. Hence, using governing conservationlaws of mass, momentum and energy for a fundamentalelement of that uid, in addition to the hydrodynamicsof uid ow, acoustics generated through cavitationor supercavitation can also be described and modeled.Moreover, since the objective of this investigation isto present a model for an unknown phenomenon, itis necessary to consider the governing equations beanalyzed in a dimensionless condition. Using scaleanalysis to produce governing dimensionless equationsleads to the comparison among the weight of presentterms, which yields recognition of the importance ofthose terms. Also, gaining more knowledge aboutthe importance of unknown terms causes the solutionprocess to be simple and, consequently, reduces thedependability of the solution on physical quantities forpresenting a dimensionless model. Eqs. ((1) to (3))represent a dimension form of continuity (conservationof mass), momentum and combined equations of stateand energy for compressible liquid uid in a reversibleprocess [29,30].

@�@t

+r:(�~u) = �Q; (1)

@(�~u)@t

+ (~u:r)(�~u) = �rp+ ��r2~u

+��

13

+�v�

�r(div(~u))

��+ �~uQ; (2)

[@p@t

] + [(~u:r)p] = c2[��div(~u)]

+�c2�cp

��v(r:~u)2 + 2�feijeji � 1

3e2iig�: (3)

In these equations �(X; t) is density, ~u(X; t) is velocity,Q(X; t)) is the volume source of the expansion of uidmass, p(X; t) is uid pressure, � is viscosity, �v is bulkviscosity, eij is the tensor of the strain rate, c is soundspeed, cp is heat capacity at constant pressure and � =1�

� @�@T

�p0

is constant. In order to dimensionally analyzethe above equations, the following presumptions areconsidered:

p = ~Pp� for example ) ~P = P1 = �0U20 ;

u = ~Uu� for example ) ~U = U0;

� = ~�� for example ) ~� = �0 = �water;

Q = ~QQ�;

t =t�~!

) ~! =~U�

) � =~U~!;

X; Y; Z = Lx�; Ly�; Lz� ) r =@@xi

=@

L@x�i:(4)

In the above relations, terms that have the upper caseindex (*) are dimensionless in order of one [O(1)]:So, �� is dimensionless density, ~u� is dimensionlessvelocity, Q� is the dimensionless volume source forexpansion in the mass of uid, p� is the dimensionlesspressure of the uid and t� is the dimensionless time([O(1)]). Also, terms containing upper case index(�) are scale parameters that are related to the samephysical parameters in the problem. In case of pressureand density, examples are mentioned therein. Byplacing the above relations in Eqs. (1) to (3), thefollowing equations will be produced:

f~�~!g@��@t� +

� ~� ~ULr(��u�) = f~� ~Qg��Q�; (5)

f~� ~U ~!g�@(��~u�)@t�

�+�

~� ~U2)L

�[(~u�:r)(��~u�)]

=�

~pL

�[�rp�] +

� ~UL2

��r2~u�

+�

13

+�v�

�(rdiv(~u�))

�+ f~� ~U ~Qg��~u�Q�;

(6)

f~p~!g[@p�@t�]+

�~U ~pL

�[(~u�:r)p�]=

�~� ~UL

�c2[��div(~u�)]

+�c2�cp

��v� ~U2

L2

�(r:~u�)2

+ �� ~U2

L2

�2fe�ije�ji � 1

3e�2ii g

�: (7)

As evident, the presence of terms with (*) index, hascaused di�erential terms to be in one order [O(1)].Also, the presence of parameters having sign (�) asthe coe�cients of di�erential terms causes the weight ofdi�erential terms to become comparable, despite beingdimensional. This weight, in fact, could represent thedegree importance of di�erent terms compared witheach other. In order to better compare, the weightof di�erential terms, coe�cients also need to becomedimensionless. Eventually, by multiplying suitableparameters, and simpli�cation, along with the de�ni-tion of dimensionless numbers, in sequence, the non-dimensional form of continuity equations (conservation


of mass) and momentum, as well as combined equationsof state and energy, are found in the following forms:

St@��@t� +r:(��~u�) =

�L ~Q~U

��Q�; (8)

St@(��~u�)@t� + (~u�:r)(��~u�) = Eu[�rp�]+

1Re

�r2~u� +

�13

+�v�

�(rdiv(~u�)

�+�L ~Q~U

��~u�Q�; (9)

(St)(Eu)[@p�@t� ] + Eu[(~u�:r)(p�)] = (

c~U

)2[��div(~u�)]

+�c2�cp

��l

Rev(r:~u�)2+

lRe

2fe�ije�ji� 13e�2ii g

�:

(10)

In the above equations, dimensionless parameters ofReynolds number, Strouhal number and Euler numberare de�ned as follows. Also, it is noteworthy that inthis condition, di�erential terms, as well as coe�cients,have become dimensionless.

Sr =~!L~U;

1Re

=�

~� ~UL; Eu =

~p~� ~U2

: (11)

Eqs. (8) to (10) are dimensionless forms of equationsdescribing the dynamics of uid ow. Since termshaving an asterisk as their upper-case are in the order ofone, it could be noticed that terms of di�erential orderare also in the order of one. Moreover, dimensionlessnumbers are, in fact, the weight of di�erential terms,depicting the importance of each term. By comparingthe weight of di�erential terms (dimensionless num-bers), being 3 in number, it becomes evident that forthe above three equations, 27 di�erent conditions arefeasible. This is re ective of the fact that there are3 di�erent conditions for every dimensionless number:A) The order of the dimensionless number is far greaterthan the order of one; B) It is equal to the orderof one; and C) It is much smaller than the order ofone. Di�erent conditions of governing equations areindicative of di�erent problems. Favorable conditionsfor the occurrence of cavitation (or supercavitation) ina ow is when Euler and Strouhal numbers are in theorder of one and the Reynolds number is not muchsmaller than the order of one. The reason for this isthat under other conditions, the needed terms for themodeling of cavitation ow, as well as supercavitation,are eliminated. As an example, in Reynolds numbersmuch smaller than the order of one, the e�ects ofthe convective term are eliminated. This is where, inthe modeling of bubble movement by uid ow, the

calculation of this term is necessary. Also, in conditionsrelated to very small Strouhal number, steady ow willform, which is not an objective of this study.

3. Utilization of perturbation method

In general sense, in a physical problem when valueof a term is extremely minute, whereas its e�ect issigni�cant, the perturbation method becomes ratherimportant. As an example, viscosity in a viscose ow is considered a turbulent term in a non-viscous ow. This fact indicates that although viscosity isa small parameter, its e�ect in a ow sometimes isvery noticeable. As mentioned earlier, acoustic noiseis a part of uid ow dynamics. However, the entityof acoustic noise and its behavior in propagation aredi�erent with uid ow. This could be justi�ed bythe positioning of noise from ow and ow dynamicsin di�erent orders of basic equations. In cases wherecavitation is observed through perturbation, equationsused in the modeling of ow will be placed in an orderhigher than the acoustic equations of the ow. It isbecause, when the sound waves propagate through a uid, such small changes occur in pressure, density andvelocity vector components, that no one can see them.On the other hand, the occurrence of hydrodynamic uctuations of pressure, density and velocity vectorcomponents during cavitation (or supercavitation) ismuch higher than for acoustic ones.

In the light of everything mentioned, it will be at-tempted to separate continuity equations (conservationof mass), momentum and the combined equations ofstate and energy from the point of view of order. Thisway, the necessary equations for the recognition andsimulation of the acoustics generated by cavitation andsupercavitation, will be presented. Also, the relation ofthis type of equation with the dynamics of uid owcould be shown.

Always, in the perturbation method, the perturb-ing parameter is depicted by ("). By glancing throughpresumed equation systems in Eqs. (8) to (10), it isevident that �ve dimensionless terms (coe�cients ofdi�erential terms) can be very small. In other words,they could be "1 � "5. In this type of problem, aterm is selected from all the perturbing parameters.Then, separation of the di�erent order of equationstakes place on the basis of that parameter. The point isthat selection of that parameter has to be based on thetype of investigated problem. As volume uctuationsin uid lead to the propagation of acoustic noise, theorder of governing equations in solving the acoustics of uid ow could be chosen equal to the order of thesevolume uctuations. In light of the points stated above,for the present problem in this study, in identi�cationof acoustics stemming from supercavitation, the per-turbing term will be " = ~QL= ~U . The reason for this


selection is that, in supercavitation ow, the existenceof very small bubbles inside the cavity is regarded asperturbation in a ow without cavitation. Hence, (Q)has to be viewed as a perturbing term in the equations.An interesting point is that one of the major sourcesof acoustic noise generation is uctuations of bubblesinside the cavity. Results presented at the end of thissection in the form of obtained acoustical equations,are indicative of the point stated above. Therefore,initially, the following assumptions will be entered inthe equations:

p�(X; t; ") =h0(")p0(X; t) + h1(")p1(X; t)

+ o(h1("));

~u�(X; t; ") =f0(")~u0(X; t) + f1(")~u1(X; t)

+ o(f1("));

��(X; t; ") =g0(")�0(X; t) + g1(")�1(X; t)

+ o(g1("));

Q�(X; t; ") =m0(")Q0(X; t)+m1(")Q1(X; t)

+ o(m1(")): (12)

In these relations, parameters which have zero indexes,include ~u0; �0; p0 and Q0, which are dimensionlessvelocity vector, dimensionless density, dimensionlesspressure and dimensionless volume source uctuations,respectively, in the order of hydrodynamic. Also,parameters which have one index, include ~u1; �1; p1 andQ1, which are dimensionless velocity vector, dimension-less density, dimensionless pressure and dimensionlessvolume uctuations, respectively, in the order of hy-droacoustics. Terms h0; h1; :::;m0;m1 are the indicatedweight of dimensionless parameters.

xn+1(") = o(xn(")); " =~QL~U:

After placing Eq. (12) in the dimensionless Eqs. (8)to (10), they need to be separated into two di�erentorders. This requires that the relation of weighingterms with perturbation parameter ("), in terms oforder, is being determined. Also, according to thede�nition applied in this study, leading order equationsmust be satis�ed conditions, which govern the hydro-dynamics of uid ow, and �rst order equations mustbe satis�ed conditions governing hydroacoustics. Inother words, the relation between weighing terms andperturbation terms has to be found, such that, afterdecomposition of equations into di�erent orders, theobtained equations in the leading order must be able todisplay the ow �eld precisely. Moreover, the obtained

equations of the �rst order have to be indicative oflinear wave equations after applying linear acousticsassumptions (non-viscous and stationary background uid ow). Beside these assumptions, the de�nitionof xn+1(") = o(xn(")), and the principle of minimumpossible conditions have been used. By doing so, inEqs. (8) to (10), and then simplifying, by using theprinciple of minimum possible conditions, it becomesevident that the above parameters need to be presentedin the following form:

p� = p0 + "p1; ~u� = ~u0 + "~u1;

�� = �0 + "�1; Q� = Q0 + "Q1: (13)

Placing the parameters as Eq. (13) causes di�erentorders of equations to be formed. By separationof all terms having one order, O(1) as the termsof leading order equations and all terms having (")order 0("), as �rst order equations, it will be possibleto recognize equations related to the dynamics ofsupercavitation ow (hydrodynamics of ow), as well asacoustics generated by it. The leading order of formedequations represents the modeling of cavitation andsupercavitation ow, and �rst order equations presenttheir acoustics.

As the work is preceded, initially, results obtainedfor di�erent orders are presented and, then, whilethe validation of leading and �rst order equations isinspected, the interpretation of terms related to theseequations is developed.

4. Results

Results indicate that acoustical equations are at alower order compared to the dynamics of cavitation(or supercavitation) ow. In other words, acousticsas a \perturbation phenomena" have been imposedon the cavitation (or supercavitation) ow. Hence,in light of obtained results, two groups of leadingand �rst order equations could be designated as amathematical model of the uid dynamics of cavitationand supercavitation, and as a mathematical model forits acoustics.

4.1. Mathematical hydrodynamic model forsimulating of cavitation orsupercavitation ow

On the basis of obtained results and by separatingterms having the order of one, a group of leading orderequations for simulation of cavitation (or supercavita-tion) ow is used. The leading order system related tocontinuity and momentum equations, as well as those ofcombined equations of state and energy, are presentedin Eqs. (14) to (16):


(St)@�0

@t+r:(�0~u0) = 0; (14)

(St)@(�0~u0)@t

+ (~u0:r)(�0~u0) = (Eu)[�rp0]

+1

Re

�r2~u0 +

�13

+�v�

�(rdiv(~u0))

�; (15)

(St)(Eu)@p0

@t)+ (Eu)(~u0:r)p0 = (

c~U

)2�0(r:~u0)

+�c2�cp

��1

Rev(r:~u0)2 +

1Re

2fe0ije

0ji

� 13

(r:~u0)2g�: (16)

Consequently, it becomes evident that obtained resultsfor a leading order system, in fact, indicating thatused equations for the simulation of cavitation orsupercavitation ow are true, by the assumption of uid ow being viscous and compressible.

4.2. Mathematical acoustics model forsimulating of cavitation orsupercavitation ow

Since acoustical equations are in a lower order com-pared with hydrodynamic equations, the order of theseequations is assumed equal to that of the perturba-tion parameter mentioned in Eq. (12). Similarly, byreordering the available terms in the equations on thebasis of a perturbation term, otherwise known as termswith an order of ("), equations of the �rst order couldbe obtained. In this regard, systems of governingequations are presented in the following form:

(St)@�1

@t+r:(�0~u1) +r:(�1~u0) = �0Q0; (17)

(St)�@(�1~u0)@t

+@(�0~u1)@t

�+ (~u0:r)(�1~u0)

+ (~u0:r)(�0~u1) + (~u1:r)(�0~u0) = (Eu)[�rp1]

+1

Re

�r2~u1+

�13

+�v�

�(rdiv(~u1))

�+�0u0Q0;

(18)

(St)(Eu)@p1

@t+ (Eu)(~u0:r)p1 + (Eu)(~u1:r)p0

= �� c~U �2�0(r:~u1)� � c~U �2�1(r:~u0) +�c2�cp

�"

2Rev (r:~u0)(r:~u1)+ 2

Ref 14@uoi@xj

@u1j@xi + 1

4@uoi@xj

@u1i@xj

+ 14@u1i@xj

@u0j@xi + 1

4@uoj@xi

@u1j@xi � 2

3 (r:~u0)(r:~u1)g#:

(19)

These equations are indicative of the fact that a groupof acoustic equations are coupled with equations ofleading order and are a�ected by them. In other words,although acoustic equations of ow have no e�ecton the hydrodynamic equations of cavitation ow, aspresented by Eqs. (14) to (16), the hydrodynamicsequations of ow a�ect the acoustic equations. Then,the validity of the system of leading and �rst orderequations is inspected regarding certain assumptions.After obtaining the equations from di�erent orders, thevalidity of these equations needs to be veri�ed.

4.3. Validity inspection of hydrodynamicmodel of supercavitation ow

In order to inspect the validity of equations of leadingorder, they should be compared with equations ofmomentum, continuity, energy and state used in thesimulation of supercavitation ow. However, sinceequations were presented in previous studies of thesimulation of supercavitation as incompressible, in thisstudy, also, that assumption was held. Then, theequation systems of leading order were obtained and,subsequently, incompressible forms of the equationswere compared with the results of other studies. There-fore, incompressible forms of governing equations insupercavitation ow are presented. Also, it is assumedin these equations that the Strouhal and Euler numberare in the order of one. Initially, it is assumed that the ow is incompressible. So, in that case:

r:(~u0) = 0: (20)

By considering this assumption in the leading orderequations, continuity equations will be presented in thefollowing form:

@�0

@t+ (~u0:r)�0 = 0;

D�0

Dt= 0: (21)

And, similarly, for equations of momentum and com-bined state, as well as energy, Eqs. (22) and (23) willbe:�

�0@~u0

@t+~u0

@�0

@t

�+ (~u0:r)(�0~u0)

= �r�0 +1Re

[r2~u0];(22)

@p0

@t+ (~u0:r)p0 =

�c2�cp

��1Re

2fe0ije

0jig�: (23)

Obtained results are the same equations as the gov-erning equation of incompressible uid ow in theOpen Foam [16]. Also, these equations are those


used in Moghimi's PhD dissertation [16] for simulationof arti�cial supercavitation assuming incompressible uid. Consequently, in light of points discussed, thevalidity of leading order equations for compressible owis realized.

4.4. Validity inspection of acoustic modelgenerated by supercavitation ow.

In order to inspect the validity of �rst order equations,these equations are compared with those of momentum,continuity, energy and state, all presenting simulationof supercavitation ow acoustics. Since obtaining lin-ear equations of acoustics is also done using equationsof continuity, momentum, energy and state, utilizingthis method to inspect the validity of obtained acous-tics equations in the �rst order is suitable. To do this,initially, assumptions relating to linear acoustics areentered into the equations and, then, obtained resultsare compared with the results of linear acoustics. Also,it is assumed that Euler and Strouhal numbers are inthe order of one. Assumptions used in obtaining linearacoustic equations are as follows:

�0 = cons; u0 = 0;

� = 0; Q = 0: (24)

By exerting these assumptions in the equations ofcontinuity, momentum, energy and state of �rst order,the following equations are obtained:

@�1

@t+ �0(r:~u1) = 0; (25)

�0@(~u1)@t

= [�r(p1)]; (26)

@p1

@t= �� c~U )2�0(r:~u1): (27)

It is evident that these equations are equal to those oflinear acoustics equations [20]. Therefore, in the lightof points presented in the preceding two sections, thevalidity of the obtained equations is realized from bothleading order equations, as well as �rst order.

5. Discussions

As mentioned earlier, equations of leading order areindicative of a system of equations for determinationof uid ow conditions. These equations introducea general form of principle equations in uids. Lackof thermal charge existence, otherwise known as non-occurrence of work on the basis of heat transfer, is theonly assumption in this form of equations. On the otherhand, this assumption, in light of the problem underconsideration, meaning simulation of supercavitation ow, is a suitable and logical point. Considering the

obtained leading order Eqs. (14) to (16), as well asthe dimensionless weight of their di�erent terms, itbecomes possible to decide about problem formationin di�erent conditions. In simple terms, this meansthat in a given problem, the e�ect of a term with lowerweight could be neglected against other terms andvice versa. At any rate, having this type of equation(leading order) at hand enables researchers to simulatesupercavitation ow in a compressible viscous ow ina designated time.

Regarding acoustic equations of the �rst order, itis also fair to state that these obtained equations are ageneral form of those equations used in the analysis oflinear acoustics. Di�erences between the obtained �rstorder equations and linear acoustics equations are:

a) Equations of linear acoustic are connected to sta-tionary uid ow. Whereas, obtained equations inthis study include the velocity of uid ow as well.The term expressing the uids velocity is depictedby ~u0.

b) In equations of linear acoustics, the uid is assumedto be incompressible with �xed density. In condi-tions where obtained equations from a perturbationmethod have regarded the uid as compressible,then, density in these acoustic equations could notbe constant.

c) Contrary to linear acoustic equations, in theseequations, the term of the acoustics source existsand is shown by (Q), which is present in bothcontinuity equations as well as momentum. Never-theless, the importance of this term becomes evidentwhen large supercavity occurs in the uid owcomprised of many smaller bubbles. These smallbubbles are transferred from low pressure areasto high pressure areas via the movement of ow.This causes considerable changes in the volume andradii of each bubble. Fluctuations of each bubble,in addition to being observed experimentally, aremonitored through numerical analysis by utilizingthe equation of Rayleigh{Plesset [13]. Since noiseis generated by the uctuation of each bubble, itcan be regarded as the source of acoustic noiseproduction. Therefore, this term (Q) has to beconsidered. Another interesting point is that theterm of the volume source in equations of leadingorder is not evident. The reason for this is the smallorder of these uctuations compared with those ofleading order, despite being in equations of the �rstorder. So, even though leading order equations areproducers of supercavitation ow, it is expected thatby using equations of leading order, the uctuationsof each bubble become recognized. Then, by usingequations of the �rst order, the acoustic noise of each uctuation could be obtained.


d) Equations of linear acoustics are for non-viscous uid ow, whereas the acoustic equations of thepresent study contain terms related to uid viscosity.Viscosity terms could be observed in the form ofReynolds number in the equations. The left sideof the combined equation of state and energy arerepresentative of acoustic pressure distribution, andits right side presents two e�ective mechanismson the distribution of acoustic pressure. Thoseterms are like dilatation terms, having Mach numberas their coe�cient, and another term related toviscosity could be obtained via equations, in which,by comparing the weight of Reynolds number andMach number, each mechanism's contribution toacoustic pressure could be determined.

6. Conclusions

As the acoustic modeling of cavitation and super-cavitation are instrumental in recognition of thesehydrodynamic phenomena, presenting a precise andvalid model for estimating propagated acoustic noise isof great importance. Consequently, as presented earlierin this study,

1. In order to provide a more precise numerical andmathematical model of the noise produced bysupercavitation, initially, governing equations ofthe uid containing equations of state, continuity,momentum and energy were divided into di�erentorders. This was done by dimensional analysis andthe perturbation method. Obtained results fromdi�erent orders of the equations were indicativeof acoustic noise present as a perturbation termin the governing equations of uid hydrodynamics.It is noteworthy that the terms of the equationswere scaled and had become unity ordered usingdimensional analysis and gauge functions. Thedegree of scaled leading equations is in the order ofthe unit. Also, the order of the acoustic equations isfrom the order of volume source uctuations. Thesevolume sources in supercavitation ow are bubblesinside a large cavity.

2. According to the conducted analysis, in order toanalyze the acoustic behavior of supercavitation,�rst, equations of the leading order need to besolved. Under most general conditions, in order toestimate propagated noise as precisely as possiblefrom cavitation and supercavitation, �rst, equa-tions of continuity, Navier stokes, energy and statefor compressible uid (with or without viscosity)have to be solved.

3. By solving this system of equations, a supercavityand bubbles inside it will be formed. Then, byconsidering the conditions of ow (location of each

bubble, pressure and pressure gradient distribution,velocity distribution, density and gradient of veloc-ity and density) as boundary conditions, as wellas initial conditions, equations of �rst order, oth-erwise known as acoustic equations (coupled withequations of hydrodynamic of ow), will be solved.Through solving these equations, acoustic pressuredistribution, velocity and the density induced bynoise can be calculated.

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Biographies

Mohammad Riahi received BS and MS degreesin Mechanics from Illinois, USA, in 1981 and 1986,respectively, and a PhD degree in Manufacturing fromIowa, USA, in 1991. He is currently Associate Professorof Mechanical Engineering at Iran University of Scienceand Technology (IUST), Tehran, Iran. Dr. Ri-ahi's research activities include nondestructive testing(NDT), acoustics and AE, measurements and conditionmonitoring in industrial maintenance.

Norooz M. Nouri graduated in Mechanical Engi-neering from the University of Zahedan, Iran, withemphasis on uid mechanics, and obtained a PhDdegree in 1993 on Vortex Phenomenon from EcolePolytechnique, France. Dr. Nouri has completedmany industrial and scienti�c projects and is currentlyAssociate Professor of Mechanical Engineering at IranUniversity of Science and Technology (IUST), Tehran,Iran.

Ali Valipour received his BS degree from AzadUniversity of Ahvaz, Iran, in 2005 and his MS degreein Mechanical Engineering from Iran University ofScience and Technology (IUST), Tehran, Iran, wherehe is currently pursuing his PhD degree. His researchactivities include acoustics, �nite element simulation,o�shore structures, nondestructive testing and noisemeasurement.

mathematical modeling and decomposition of hydrodynamics...

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