Influence of heat transfer on the dynamic response of a spherical gas/vapor bubble

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Conference on Modelling Fluid Flow (CMFF’09) The 14^thInternational Conference on Fluid Flow Technologies Budapest, Hungary, September 9-12, 2009

I NFLUENCE OF HEAT TRANSFER ON THE DYNAMIC RESPONSE OF A SPHERICAL GAS / VAPOR BUBBLE

Ferenc HEGED ˝ US

¹

, Csaba H ˝ OS

²

, L ´aszl ´o KULLMANN

³

1Department of Hydrodynamic Systems, Budapest University of Technology and Economics. E-mail: hegedusf@hds.bme.hu

2Department of Hydrodynamic Systems, Budapest University of Technology and Economics. E-mail: csaba.hos@hds.bme.hu

3Department of Hydrodynamic Systems, Budapest University of Technology and Economics. E-mail: kullmann.laszlo@hds.bme.hu

ABSTRACT

The starting point of studying cavitating flows is the well-known Rayleigh-Plesset equation (RPE), which neglects heat transfer in the fluid. In this paper an extended model is presented which couples the RPE and the heat equation. The extended model is solved by a Galerkin-based numerical procedure.

Vapor bubble growth velocity, free and forced oscillation studies are presented which highlight the im- portance of heat transfer on bubble dynamics. It is also shown that the presence of heat transfer results in nonlinear effects such as hysteresis of the bubble oscillation modes.

Keywords: Rayleigh-Plesset equation, heat trans- fer, bubble dynamics, cavitation

NOMENCLATURE

t [s] time

r [m] radial coordinate

ξ [−] transformed coordinate

R [m] bubble radius

p [P a] pressure

T [K] temperature

ρ [kg/m³] density

u [m/s] velocity in the liquid domain µ [kg/m/s] dynamic viscosity

C [N/m] surface tension

α [m²/s] thermal diffusivity L [J/kg] thermal diffusivity

y [K] basis function

a [−] basis function coefficient m [−] derivativ of the temperature

field with respect to the transformed coordinate at1 N [−] number of the basis functions

A [m] amplitude

ω [rad/s] angular frequency ωN [rad/s] natural angular frequency ωp [rad/s] pressure excitation frequency Erel [−] relative error of the

temperature field

Subscripts and Superscripts b inside the bubble g gas property o reference property L liquid property V vapor property

∞ far from bubble

Γ differential operator of the heat equation

- average

· differentiation with respect to time

′ differentiation with respect to transformed coordinate

˜ shifted quantity

∗ decomposed quantity ref reference calculation G Galerkin based calculation 1. INTRODUCTION

The Rayleigh-Plesset equation has been exten- sively studied in bubble dynamics since despite several severe simplifications employed in its derivation it still gives a qualitative picture of the dynamics of a single bubble. The first analysis performed by Lord Rayleigh [1] assumed incompressible liquid and neglected the thermal effects, surface tension and viscosity. This model was extended by Plesset [2] to include viscosity and surface tension, resulting in the classic form of the Rayleigh-Plesset equation (RPE). The RPE still assumes incompressible liquid, no thermal energy flow at the bubble wall and the physical properties of the mixture inside the bubble are constant both in space and time. Moreover, although several studies (e.g. [3], [4]) on the RPE pre- dict extremely high temperatures and pressure (notably during the collapse), the real behavior of the water and the vapour phase has not been taken into account (e.g. via the Haar-Gallagher-Kell equation of state [5]).

The simplicity of the RPE allows analytical cal- culations on the transient behavior of a single bubble subjected to periodic pressure excitation. In the 80’s many studies examined this simple model in the light of modern nonlinear and chaos theory, e.g. Chang

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and Chen [6], Feng and Leal [7] or Lauteborn and Parlitz [8].

An important drawback of the RPE is that it does not consider that the heat needed for evaporation (on the bubble surface) is not available instantly but has to be transported to the bubble wall from the fluid domain via conduction and convection. One of the first models including thermal effects was developed by Plesset and Zwick [9], which was confined to cases in which the thickness of the thermal boundary layer surrounding the bubble is small compared with the radius of the bubble. In [10] Plesset and Prosperetti derived a model for purely gas filled bubbles allowing the dissolution, neglecting the convective term in the diffusion equation. Kamath and Prosperetti in [11] present a model, in which the heat equation is solved in the bubble interior. The damping effect of heat transfer can also be modeled by a (virtual) effec- tive liquid viscosity as described in Prosperetti [12], Kameda and Matsumoto [13] or Storey and Szeri [14].

Due to the expensive computational require- ments of the correct treatment of the heat transport, researchers have tended to develop reduced order models recently which are capable of capturing the damping effect of heat transfer with reasonable computational effort, e.g. Preston, Colonius and Bren- nen [15], [16] or [17]. These works are based on the proper orthogonal decomposition (POD) method and consider many physical effects, such as heat and diffusion equation for the bubble interior and exterior and show that these effects have great influence on the bubble motion especially at high bubble wall velocities.

This paper focuses on the heat transport phenomena. In Section 2. the mathematical modeling is presented, i.e. the RPE and the heat equation with their boundary conditions are described. Section 3.

presents the numerical solution technique including the Galerkin projection of the heat equation. The projection results in a system of ordinary differential equations (ODEs), which are coupled with the RPE and integrated numerically. Section 4. includes the results. Section 4.1. studies the influence of heat transfer on the growth of a purely vapor bubble, section 4.2. investigates the free oscillations of a gas/vapor bubble and section 4.3. compares the frequency response curves with and without heat transfer. Finally Section 5. concludes the paper.

2. MATHEMATICAL MODELING

Let’s consider a spherical bubble of radiusR(t) (where t stands for time) in an infinite domain of liquid (e.g. a vapor bubble in water), see Figure 1.

The bubble includes either vapor or gas or their mixture. The pressure, temperature and density inside the bubble ispb(t),Tb(t)andρb(t), respectively. The velocity, pressure and temperature field in the liquid domain areu(r, t),p(r, t)andT(r, t). Far from the bubble the pressure and temperature arep∞(t)and

T∞. The liquid density and viscosity are ρL and µL, the surface tension coefficient isC. The material properties of the liquid and vapor are considered to be constant and are evaluated atT∞andp∞with the help of Haar-Gallagher-Kell equation of state [5], wherep∞is the average pressure far from the bubble.

R(t) r p (t)b

T (t)b

u(r,t) p(r,t) T(r,t) p (t)_∞

T∞

liquid

vapour/gas far from

bubble

r_b(t) r_L

mL

C

Figure 1. Vapor bubble in an infinite liquid do- main

In what follows, the following assumptions are considered.

A.1 Spatially uniform temperature, pressure and density field inside the bubble.

A.2 The liquid is incompressible with constant material properties as described above. (In- compressibility also assumes low bubble wall velocities compared to the sonic velocity.) A.3 AsρV ≪ ρL, the water/vapour mass trans-

fer across the bubble wall is neglected, thus the water velocity at the wall and the bubble wall velocity equals, i.e. u(R(t), t) = ˙R(t) (dot stands for differentiation with respect to time).

A.4 If gas content is present in the bubble, no dissolution into the liquid is considered. (In other words the gas content of the bubble re- mains constant.)

A.5 Gas is handled as ideal gas, while the partial pressure of the vapour is the saturated pressure atTb.

Using the above assumptions, the RPE can be written as (for details of the derivation see [18], [2], [10]

3

2R˙²+RR¨+4µL

ρL

R˙ R = 1

ρL

pV(Tb)−p∞(t) +pg−2C R

. (1) The gas is assumed to be ideal, hence

pg=pgo

Tb

T∞

Ro

R 3

, (2)

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wherepgo,T∞andRoare the properties of the reference state. Vapor pressure is the saturated pressure corresponding toTb and the bubble pressure is the sum of these two partial pressures, i.e. pb = pV(Tb) +pg.

Next, we consider two different models. The first model is the classical RPE and assumes that the bubble temperature equals the ambient temperature, i.e.Tb=T∞, this case will be referred as RPE. The second model employs the heat equation to compute the bubble temperature and will be referred as ERPE (extended Rayleigh-Plesset equation).

If the bubble temperature Tb changes in time, one has to solve the heat equation:

∂T

∂t + ˙R R

r 2

∂T

∂r =αL

r²

∂

∂r

r²∂T

∂r

, (3) whereαLis the thermal diffusivity of the liquid. The domain extends from the bubble radiusR(t)to infin- ity. The boundary conditions are:

T(∞, t) =T∞, and (4) R˙ = λL

ρVL

∂T

∂r

_r=R(t), (5)

whereλL is the heat conductivity of the liquid and Lis the latent heat. Boundary condition (5) does not allow the heating or cooling of the bubble contents thus we have an extra assumption corresponding to the ERPE model.

A.6 The heat flow through the wall covers only the heat needed for condensation/evaporation.

Therefore only small-amplitude oscillations around the equilibrium can be studied with this model.

Instead of solving (3) over the domain r ∈ [R(t),∞), a new spatial coordinateξis defined as

ξ= R(t)

r , where ξ∈[0,1]. (6) Applying this new coordinate and shifting the temperature field by the constantT∞, equations (3), (4) and (5) can be written as

Γ( ˜T)^def.= ∂T˜

∂t −R˙ Rξ⁴∂T˜

∂ξ −αL

R²ξ⁴∂²T˜

∂ξ² = 0, (7)

T˜(0, t) = 0, (8)

−RR˙ = λL

ρVL

∂T˜

∂ξ

_ξ=1. (9) whereT˜ =T−T∞. We immediately drop the tilde overT˜.

3. SOLUTION TECHNIQUE

In this section an efficient Galerkin-based solution technique of the heat equation is presented. The main idea is to decompose the temperature field with the help of a set of spatial basis functions with purely time-dependent coefficients and then minimize the error of the approximation, resulting in a set of ordinary differential equations for the time-dependent coefficients. In order to satisfy boundary conditions (8) and (9), the temperature field is generated in the following special form:

T^∗=

N

X

i=1

ai(t)yi(ξ) +m(t)yo(ξ), with (10) m(t) = ∂T^∗

∂ξ

_ξ=1=−ρVL λL

RR,˙ (11) y^′_o(1) = 1, y^′_i(1) = 0, and (12) yo(0) = 0, yi(0) = 0. (13) Substituting (10) into (7) results in an approximation errorΓ(T^∗)6= 0asN is a finite number. To minimize this error, we force this residual to be orthogonal to the subspace spanned by the basis func- tionsyi:

0 =hΓ(T^∗), yji= Z 1

0

Γ(T^∗)yjdξ, j= 1. . . N, (14) whereh., .istands for scalar product. The integration can be evaluated term by term, for example the first term turns into

∂T^∗

∂t , yj

=

N

X

i=1

˙ ai

Z 1

0

yiyjdξ

+ ˙m Z 1

0

yoyjdξ ^def.= Aija˙i+ ˙mBj. (15) The resulting system of ordinary differential equations can be written as

˙

a=A⁻¹ R˙

RC+αL

R²E

! a+

A⁻¹ Rm˙

R D+αLm

R² F−mB˙

! , (16) where:

Aij = Z 1

0

yiyjdξ, Bi = Z 1

0

yoyjdξ, Cij =

Z 1

0

ξ⁴yi^′yjdξ, Di= Z 1

0

ξ⁴y^′oyjdξ, Eij =

Z 1

0

ξ⁴y^′′iyjdξ, Fi= Z 1

0

ξ⁴yo^′′yjdξ. (17)

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System (16) is integrated together with (1) (in a monolithic way), the coupling between the two sets of equations is that Tb = T^∗(1) in the Rayleigh- Plesset equation and thatRandR˙ appears explicitly in (16).

In our examination two sets of basis functions (yi) are constructed. The first system is polynomial and is built up with the Gram-Schmidt orthogonalization method. This saves the need of computing and inverting matrixAin (16). As the shifted temperature field tends to zero asξ → 0(r → ∞) the basis function’s derivatives should tend to zero (although technically this is not a boundary condition).

The initial set of basis functions (shown in Figure 2.) are

yo=1

2ξ² (18)

yi= (i+ 2)ξⁱ⁺¹−(i+ 1)ξⁱ⁺². (19)

0 0.2 0.4 0.6 0.8 1

yo y1

y2 y3

ξ[-]

Figure 2. The inital set of polynomial basis func- tions

0 0.2 0.4 0.6 0.8 1

−1.5

−1

−0.5 0 0.5 1 1.5 2

yo y1

y2 y3

ξ[-]

Figure 3. The orthonormal polynomial basis functions

However, this system yields poorly conditioned matrices, which cause numerical problems when calcu- lating the inverse of the matrixA. Therefore we con- struct an orthonormal system by applying the Gram- Smith orthogonalization method. A drawback of this system is that the polynomial coefficients are of very different orders of magnitude and this might lead to

problems when evaluating the functions. As the coefficient matrices are dense lower-triangle matrices it is time-consuming to evaluate them. The resulting polynomial basis is not given explicitly here but shown in Figure 3. Another common choice of con- structing a basis system is the use of trigonometric functions. The following orthonormal basis functions were chosen (see Figure 4):

yo=−sin(πξ)

π (20)

yi=√ 2 sin

(i−1)πξ+π 2ξ

. (21)

0 0.2 0.4 0.6 0.8 1

−1.5

−1

−0.5 0 0.5 1 1.5

yo y1

y2 y3

ξ[-]

Figure 4. The trigonometric basis functions A test case was defined to compare the perfor- mance of the polynomial and trigonometric basis functions. The heat equation was not coupled to (1) instead the bubble radius time history was defined as (see Figure 5)

R(t) =Ro+At²e^−tcos(ωt), (22) withRo= 0.1m,A= 0.001m,ω= 20π/s.

0 1 2 3 4 5 6

0.0994 0.0996 0.0998 0.1 0.1002 0.1004 0.1006

R[m]

t[s]

Figure 5. The bubble radius time history of the test case (22)

As a reference computation, the heat equation was discretized with a central difference scheme in space and a first-order forward difference scheme in time. A non-uniform spatial grid of 1000 points was used with semi-Chebyshev distribution (i.e. the pro- jections onto theξ-axis of equally spaced points on

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the unit circle but only in the interval[0,1]). Sev- eral consecutive computations were performed with halved time steps and the Richardson extrapolation technique was finally employed to improve the accu- racy of the results. The relative error is defined as

Erel(t) =

ξ∈[0,1]max |Tref.(ξ, t)−TG(ξ, t)|

ξ∈[0,1]maxTref.(ξ, t) , (23) whereTref stands for the solution of the reference (central difference-based) technique andTGdenotes the Galerkin solution. Figure 6 and Figure 7 de- pict the convergence history for the polynomial and trigonometric systems for various numbers of basis functions.

0 2 4 6 8 10

10⁻⁴ 10⁻² 10⁰

N10 N16

N20 N24

N28

Erel[-]

t[s]

Figure 6. The relative error of the polynomial system

0 2 4 6 8 10

10⁻⁴ 10⁻²

10⁰ N10

N20 N30 N40

E[-]rel N50

t[s]

Figure 7. The relative error of the trigonometric system

It is clear that for this test case the polynomial system gives the more accurate results with the same number of functions. However, in the case of the polynomial system we experienced Gibbs-like phenomena close to the bubble wall when the Galerkin system was coupled with the RPE. These heavy oscillations disappeared when the trigonometric system was employed, thus in what follows the trigonometric system will be used.

4. RESULTS

4.1. Vapour bubble growth velocity In this section the growth velocity of a pure vapor bubble (i.e. without gas content, pg = 0) is studied with and without heat transfer. During the evaporation process the required heat is extracted from the fluid thus its temperature decreases at the bubble wall. The equilibrium of this heat extrac- tion and the amount of heat arriving form the liquid domain via conduction and convection results in a lower bubble growth velocity. This damping effect is clearly seen in Figure 8. The parameter values were T∞= 106^oC,p∞= 1barandR(0) = 10⁻⁵m(the first two values are taken from [9]). The Galerkin solution of the ERPE model was also verified with the finite difference solver (FDS), the computational time was 73 seconds for the Galerkin-based solver and 500 seconds for the FD solver (on a standard desktop PC). Since Plesset and Zwick in [9] found approximately seven times larger maximum bubble growth velocity, we speculate that this discrepancy is due to the different thermal boundary condition (in [9] the heat equation is not solved but only a thermal boundary layer model is applied).

0 0.2 0.4 0.6 0.8 1

x 10⁻³ 0

0.2 0.4 0.6 0.8

1x 10⁻⁴

Rayleigh−Plesset (RPE)

Thermal effect (ERPE)

R[m]

t[s]

Figure 8. Typical time history of bubble growth without (RPE) and with (ERPE) heat transfer

100 102 104 106 108 110

10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹

RPE

˙R[m/s]max ERPE

T∞[^oC]

Figure 9. Maximum bubble wall velocity as a function of ambient temperature without (RPE) and with (ERPE) heat transfer

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The bubble growth velocity tends to an asymp- totic maximum valueR˙max as t → ∞. In Figure 9 these maximum bubble wall velocities are compared for different ambient temperaturesT∞. Note that there is a striking difference of three orders of magnitude.

4.2. Free oscillations of vapor-gas bub- bles

System (1) has only one stable equilibrium point if the ambient pressure is larger than the saturation pressure (ifpV > p∞ we have either two or zero equilibria). In this section free oscillations around this equilibrium are studied.

0.004

0.003 0.005

I. II. III.

3.46 3.48 3.5 3.52 3.54x 10⁻⁴

R[m]

t[s]

Figure 10. Free oscillations of a slightly dis- turbed gas/vapor bubble without (solid line, RPE model) and with (dashed line, ERPE model) heat transfer. Temperature distributions attI.,tII.

andtIII.are depicted in Figure 11

1 1.1 1.2

−0.01

−0.005 0 0.005 0.01

T−T∞[oC]

r/R(t)[-]

tI.= 0.0003s tII.= 0.0009s

tIII.= 0.0015s

Figure 11. Tipical temperature distributions at tI. =T /4,tII. = 3T /4,tIII. = 5T /4, whereT stands for the period of the first oscillation

Figure 10 depicts the time history of the free oscillation of a vapor-gas bubble without (solid line) and with (dashed line) heat transfer. There is a significant difference in the decay rate of the two cases as in the RPE model the damping effect stems only from the viscous term, consequently the decay rate is small especially if compared to the effect of heat transfer. The simulation parameters were

T∞ = 30^oC, p∞ = 5000 P a, the initial bubble radius was perturbed by one percent around the equilibrium value ofReq. = 0.35mm. Again the Galerkin-based results were verified with the central difference solver, the computation time was 8.6 seconds for the Galerkin-type solver and 123.3 seconds for the finite difference solver.

Figure 11 presents typical temperature distributions at different instants (see Figure 10). Note that in this case the thickness of the thermal boundary layer is smaller than 20% of the bubble radius. It can be clearly seen that during bubble growth (at tII), evaporation extracts heat from the fluid, thus the gradient of the temperature distribution at the wall is positive. On the other hand, during condensation (decreasing bubble radius -tI.andtIII.) heat is con- ducted towards the fluid, thus the temperature gradient on the bubble wall is negative. The temperature gradient grows up to170^oC/mstrongly influencing heat transfer.

4.3. Forced oscillations and amplifica- tion

In this section forced bubble oscillations and amplification are studied. In what follows the ambient pressure is a harmonic function of time written as

p∞(t) = ¯p∞+Apsin(ωpt). (24) whereAp = 100P a,p¯∞ = 50000P a, the ambient temperature isT∞= 30^oCandωpis the excitation frequency. The natural frequency of the bubble is (see [18])

ωN =

s3(¯p∞−pV) ρLR²eq

+ 4C

ρLReq³ −8ν_L² R⁴eq

. (25)

0.4 0.6 0.8 1 1.2

10⁻³ 10⁻² 10⁻¹

*

orbit (a)

orbit (b)

(Rmax−RE)/RE[-]

ωp/ωN [-]

Figure 12. Frequency response diagram of a harmonically excited gas/vapor bubble without (solid line) and with (dashed line) heat transfer.

The time histories of the two responses at∗are presented in Figure 13

Figure 12 depicts the frequency response curve for the RPE model (solid line) and the ERPE model

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(dashed line). Both curves were computed numerically by integrating the corresponding system of ODEs with a fourth-order adaptive step-size Runge- Kutta solver. The natural frequency of the bubble wasωN = 116kHz,RE stands for the equilibrium bubble radius (0.102mm) corresponding top¯∞.

As expected, the RPE model for small- amplitude oscillations (Ap/p¯∞ = 0.2%) behaves similarly to a linear system with damping (solid line), i.e. there is a large but finite peak atωp/ωN = 1. However, even for this small-amplitude excitation several interesting phenomena emerge in the case of the ERPE model (dashed line). The peak is slightly shifted fromωp/ωN = 1to approx.1.14. The peak is less ’sharp’, i.e. large-amplitude response occurs in a wide range of peak. The third and most interesting issue is that in a wide interval ofωp/ωN two possible responses co-exist and hysteresis occurs.

0 0.02 0.04 0.06 0.08 0.1

−1 0 1

0 0.02 0.04 0.06 0.08 0.1

−1 0 1

0 0.02 0.04 0.06 0.08 0.1

−1 0 1

ˆR[-]ˆTb[-]

ˆ ˙Q[-]

t[ms]

Figure 13. Time simulations atωp/ωN = 1.1 large and small amplitude oscillation

Table 1. Scaling values in (26) for Figure 13

orbit (a) Φ ∆

Φ =R [mm] 9.95·10⁻⁵ 8.42·10⁻⁶ Φ =Tb [^oC] 289.8 2.24 Φ = ˙Q [J/s] −3.51·10⁻¹⁰ 1.24·10⁻⁸

orbit (b) Φ ∆

Φ =R [mm] 1.02·10⁻⁴ 3.72·10⁻⁷ Φ =Tb [^oC] 303.1 0.005 Φ = ˙Q [J/s] −2.62·10⁻¹¹ 4.95·10⁻¹⁰

The two possible bubble oscillation modes are compared in Figure 13 in such a way that all quan- tities are scaled as given by (26). The actual scaling values are given in Table 1. Although the oscillation amplitudes∆(before the scaling, see Table 1) are different by two orders of magnitude, the shapes are similar except the small phase difference inTˆb. The analysis of these interesting co-existing vibration modes is beyond the scope of this paper.

Φ(t) =ˆ Φ(t)−Φ maxt (Φ(t)−Φ)

def.= Φ(t)−Φ

∆ . (26) 5. CONCLUSION

The classical Rayleigh-Plesset equation in its original form neglects the heat transfer effects. In this paper it was shown that heat transfer has a significant impact even in the case of small-amplitude oscillations around the equilibrium. A Galerkin-based numerical technique was presented which allows the study of these effects with reasonable computational effort compared to finite-difference based methods.

It was shown that heat conduction in the fluid is a severely limiting issue for vapor bubble growth velocity. If we neglect it the bubble growth velocity increases by three orders of magnitude. It was also reported that heat transfer introduces an important damping effect both in the case of free and forced oscillations. Moreover, heat transfer results in addi- tional nonlinear effects, notably hysteresis. The authors are not aware of any previous publication re- porting on the co-existence of such bubble vibration modes.

It is important to improve the thermal boundary condition allowing the simulation of large-amplitude oscillations. Qualitative information could be gained on the nonlinear nature of bubble response with the help of the mathematical tools of nonlinear dynamics. Furthermore we intend to extend our physical model such as application of the real equation of state [5] and mass transfer effect for the gas content.

ACKNOWLEDGMENT

The authors wish to acknowledge the kind help of Dr. Tam´as K¨ornyey.

The research described in this paper was sup- ported by a grant from National Scientific Research Fund (OTKA), Hungary, project No: 061460.

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