The violent collapse of vapor bubbles in cryogenic liquids

Highlights

• •Vapor bubbles can collapse violently with temperatures of thousands of degrees.
• •Brenner’s thermodynamic parameter can partly predict these events.
• •Significant thermal effects emerge at the final stages.
• •The moment when the surrounding liquid layer enters supercritical state is critical.

Abstract

Vapor bubbles in cryogenic fluids may collapse violently under subcooled and pressurized conditions. Despite important implications for engineering applications such as cavitation erosion in liquid propellant rocket engines, these intense phenomena are still largely unexplored. In this paper, we systematically investigate the ambient conditions leading to the occurrence of violent collapses in liquid nitrogen and analyze their thermodynamic characteristics. Using Brenner’s time ratio χ, the regime of violent collapse is identified in the ambient pressure–temperature parameter space. Complete numerical simulations further refine the prediction and illustrate two classes of collapses. At 1 < χ < 10, the collapse is impacted by significant thermal effects and attains only moderate wall velocity. Only when χ > 10 does the collapse show more inertial features. A mechanism analysis pinpoints a critical time when the surrounding liquid enters supercritical state. The ultimate collapse intensity is shown to be closely associated with the dynamics at this moment. Our study provides a fresh perspective to the treatment of cavitation in cryogenic fluids. The findings can be instrumental in engineering design to mitigate adverse effects arising from intense cavitational activities.

1. Introduction

Due to thermal effects, the dynamics of cavitation bubbles in cryogenic fluids is usually benign. This feature originates from the distinctive physical properties of these mediums: small thermal conductivity (k_l), low ratio of liquid to vapor density (ρ_L/ρ_v), and rapid change in the saturated vapor pressure with temperature (dp_v/dT). Take liquid nitrogen at ambient temperature T_∞=77 K as an example, the values of these parameters are: k_L = 0.14 W/mK, ρ_L/ρ_v = 162.4, and dp_v/dT = 11.6 kPa/K. They are in sharp contrast to those of water at T_∞=300 K, for which k_L = 0.61 W/mK, ρ_L/ρ_v = 38941.4, and dp_v/dT = 0.2 kPa/K. The distinctions of cryogenic fluids dictate that when a bubble grows, vaporization at the liquid–gas interface substantially cools the surrounding medium and delays the development of cavitation [1]. On the other hand, when the bubble collapses, the vapor condensation results in a quick buildup of heat at the liquid–gas interface and a steep rise in vapor pressure, which cushions the inwards motion of the bubble wall and renders the collapse process mild [2]. Consequently, in hydraulic machinery using cryogenic fluids as the working medium, detrimental effects arising from violent bubble collapse such as cavitation erosion are conventionally disregarded.

However, an important but rarely-noticed study indicates that it may not be always the case. In 2007, Baghdassarian et al.[3] from UCLA reported their observation of laser-induced bubbles in liquid nitrogen. They found that when the fluids are pressurized to a few bars, the dynamics of these bubbles can be so violent that a short burst of light is emitted at the main collapse point. A spectrum analysis of the luminescence reveals that the peak temperature inside the cavity is of the order of 4500 K. In another study where the liquid nitrogen in an ultrasonic homogenizer is pressurized by 3 bars [4], noticeable cavitation erosion, albeit mild, is produced on the metallic specimen. We note that for many hydraulic machineries involving cryogenic fluids such as turbopumps used in rocket engines [5], [6], the operation pressure is much larger than the ambient pressures reported in these studies [3], [4]. Therefore, the treatment of cavitation in cryogenic fluids as mild and harmless should be called into question. If violent bubble collapses are frequently encountered in these equipment, adverse effects such as cavitation erosion, vibration, and noise should be seriously considered.

In this study, using Brennen’s thermodynamic parameter, we first clarify the parameter regime in which the collapse of vapor bubbles in cryogenic fluids is mild or becomes violent. For the latter case, a full numerical simulation accounting for the thermal effect is performed to examine the mechanical and thermal features in detail. These features are further highlighted and explained by comparing them with inertial collapse in water. The insights derived from this investigation offer a new perspective when analyzing intense cavitation events in cryogenic fluids and predicting their influences on engineering applications.

2. The numerical model

In the numerical simulation, we consider the collapse of a single vapor bubble in liquid nitrogen under static ambient pressure conditions, such as that of bubbles induced by laser irradiation [7], [8]. The simulation covers only the contraction phase of the bubble from the maximum size at R = R_max to the first rebound at R = R_reb. Thermodynamic equilibrium between the bubble and the surrounding liquid is assumed initially, which implies the internal temperature T_v(t = 0) = T_∞ and the pressure equals the corresponding saturated vapor pressure at T_∞, i.e., p(t = 0) = p_v(T_∞).

Our model relies heavily on the one developed by Prosperetti and his coworkers [9], [10], [11], but also comprises revisions that account for some features standing out in our studied case. The first and foremost is the significant thermal effects associated with phase change at the bubble interface. The second is the possibility that the vapor inside the bubble enters into supercritical state after the temperature and pressure therein pass the critical point (T = 126 K and p = 3.36 MPa for nitrogen). An immediate consequence in this scenario is the cessation of phase change. Thirdly, due to the rapid dynamics in violent collapse, thermodynamic equilibrium may break down, causing the pressure (p) inside the bubble to deviate from the saturated vapor pressure (p_v) corresponding to the temperature of the liquid at the bubble wall. This is especially evident when the heated liquid layer surrounding the bubble also becomes supercritical. In this situation, p_v is not well defined whereas p may continue to increase as the collapse proceeds.

In the numerical model, the evaporation and condensation are explicitly calculated. Together with the resolved temperature fields in both the gas and liquid phases, the thermal effect is therefore fully incorporated. This treatment also allows for enforcing the zero mass transfer at supercritical state. In addition, the pressure in the bubble is derived from the basic governing equations, rather than being assumed as the saturated vapor pressure. Next, we introduce the numerical model detailedly.

We begin with Keller’s equation describing the radial dynamics of the bubble [12]

$$\left(1-\frac{\dot{R}}{\mathit{\text{c}}}\right)R\dot{R}+\frac{3}{2}\left(1-\frac{\dot{R}}{3c}\right){\dot{R}}^2=\frac{1}{{\rho }_L}\left[\left(1+\frac{\dot{R}}{\mathit{\text{c}}}\right)\left(p_B-p_{\infty }\right)+\frac{R}{c}{\dot{p}}_B\right],$$ (1)

where R is the bubble radius as a function of the time t, ρ_L the liquid density, c the sound speed, p_∞ the ambient pressure, and p_B the pressure on the liquid side of the interface. All the overdots here and thereafter represent time derivatives.

The Keller’s equation is accurate to the first order of Mach number $\dot{R}/c$ when accounting for the influence of liquid compressibility [13]. Strictly speaking, more advanced models such as the Gilmore model can provide better results when simulating violent collapses and investigating the emitted shock waves. However, since our focus is the general features of bubble collapse in cryogenic liquids and the variation trend of collapse intensity under different thermal effects, we didn’t attempt those models.

The liquid pressure p_B is related to gas pressure p by the balance of normal stress across the bubble interface

$$\mathit{\text{p}}=p_B+\frac{2S}{R}+\frac{4{\mu }_L\dot{R}}{R},$$ (2)

in which S is the surface tension and μ_L the liquid viscosity.

The mass conservation at the bubble interface stipulates [11]

$$\dot{m}={\rho }_v\left(\dot{R}-v_R\right)={\rho }_L\left(\dot{R}-u_R\right),$$ (3)

where $\dot{m}$ is the mass flux due to the phase change (positive for evaporation and negative for condensation), and v_R and u_R the velocity of vapor and liquid at the interface, respectively. From this equation, we get

$$v_R=\dot{R}-\frac{R}{3}\frac{\dot{n}}{n},$$ (4)

in which we use n, the number of moles, to refer to the amount of vapor contained in the bubble. Obviously, m = nM, with the molecular weight M = 28 g/mol for nitrogen.

On the other hand, v_R itself should follow the velocity distribution in the vapor phase. Assuming uniform pressure inside the bubble and perfect gas behavior for the vapor, the conservation of mass and energy leads to the expression of the velocity field [9]

$$v=\frac{1}{\gamma p}\left[\left(\gamma -1\right)k_v\frac{\partial T_v}{\partial r}-\frac{1}{3}r\dot{p}\right],$$ (5)

where r is the radial distance from the bubble center at r = 0, γ the ratio of specific heats, k_v the thermal conductivity, and T_v the vapor temperature. Applying this relation at the bubble wall (r = R) and equating it with Eq. (4), we arrive at

$$\dot{\mathit{\text{p}}}=\frac{3}{R}\left[\left(\gamma -1\right)k_v{\left(\frac{\partial T_v}{\partial r}\right|}_{r=R}-\gamma p\left(\dot{R}-\frac{R}{3}\frac{\dot{n}}{n}\right)\right].$$ (6)

Note that the last term in the parenthesis represents the influence of phase change, which differs our equation for p from that in Ref [11], which assumes p = p_v since weakly oscillating bubbles were studied and thermodynamic equilibrium prevails.

Following the same assumption of uniform pressure, the energy equation inside the bubble becomes [9]:

$${\rho }_{\mathit{\text{v}}}{c}_{\mathit{pv}}\left(\frac{\partial T_v}{\partial t}+v\frac{\partial T_v}{\partial r}\right)-\dot{p}=\frac{1}{r^2}\frac{\partial }{\partial r}\left(k_v{r}^2\frac{\partial T_v}{\partial r}\right),0\le r\le R\left(\mathit{\text{t}}\right),$$ (7)

where c_pv is the specific heat of vapor at constant pressure. Substituting Eq. (5) and (6) for v and p into Eq. (7) yields an equation for the gas temperature that is solved in our simulation.

The temperature field in the outside liquid can be described in the same manner as

$$\frac{\partial T_L}{\partial t}+u\frac{\partial T_L}{\partial r}=\frac{D_L}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial T_L}{\partial r}\right),r\ge R\left(\text{t}\right),$$ (8)

where u is the velocity of the liquid at radial distance r and D_L is the thermal diffusivity. Assuming the liquid as incompressible, u can be determined from Eq. (3) as

$$u=\frac{R^2}{{\mathit{\text{r}}}^2}\left(\dot{R}-\frac{\dot{m}}{{\rho }_L}\right).$$ (9)

To close the energy equations, the continuity of temperature and heat flux at the gas–liquid interface is imposed:

$${\left(T_L\right|}_{r=R}={\left(T_v\right|}_{r=R},$$ (10)

$$k_L{\left(\frac{\partial T_L}{\partial \mathit{\text{r}}}\right|}_{r=R}=k_v{\left(\frac{\partial T_v}{\partial \mathit{\text{r}}}\right|}_{r=R}+\dot{m}\mathcal{L},$$ (11)

where $\mathcal{L}$ is the latent heat.

The rate of phase change is determined by the thermodynamic state at the bubble interface and can be described by Hertz–Knudsen–Langmuir formula [14]

$$\dot{n}=4\pi R^2\frac{{\alpha }_M\left(p_v-p\right)}{\sqrt{2\pi M\mathcal{R}{T}_w}},$$ (12)

where α_M is the accommodation coefficient and T_w the temperature at the bubble wall. While there is no consensus on the exact value of α_M, we found the impact of this parameter is not great when it is in the conventional range of 0.1–0.5. Therefore, α_M = 0.4 is selected out of our previous modeling experiences [15], [16], [17], [18] and studies by others [14], [19], [20]. To reflect the fact that phase change is halted at supercritical state, $\dot{n}$ is set as zero when T_w > 126 K.

3. Numerical method

The equations introduced above are fully coupled. To solve these equations, the second-order ordinary differential equation (Eq. (1)) for bubble radius R is first rewritten as a system of two first-order equations. The partial differential equations governing the temperature fields (Eqs. (7), (8)) are reformulated into a fixed domain [0, 1]. For Eq. (7), a variable y, y = r/R(t), is introduced [10]. After substituting Eq. (5) for the velocity v, it becomes [11]

$$\frac{\gamma }{\gamma -1}\frac{\mathit{\text{p}}}{T_v}\left\{\frac{\partial T_v}{\partial t}-y\frac{\dot{R}}{R}\frac{\partial T_v}{\partial \mathit{\text{y}}}+\frac{1}{\gamma p{R}^2}\left[\left(\gamma -1\right)k_v\frac{\partial T_v}{\partial \mathit{\text{y}}}-\frac{1}{3}y{R}^2\dot{p}\right]\frac{\partial T_v}{\partial y}\right\}-\dot{p}=\frac{1}{R^2{y}^2}\frac{\partial }{\partial y}\left(k_v{y}^2\frac{\partial T_v}{\partial y}\right)$$ (13)

For Eq. (8), the introduced auxiliary variable for transformation is ζ, ζ = L/(L + y-1) [21], where $L=B{\left(D_L{t}_c/R_{\max }^2\right)}^{\text{1/2}}$ with B the constant and t_c the Rayleigh collapse time. Physically, L represents a multiple B of the thermal penetration depth normalized by R_max [11]. After substituting Eq. (9) for the velocity u, the equation becomes

$$\frac{\partial T_L}{\partial t}={\left(\frac{\zeta }{L}\right)}^2\left[\frac{D_L}{R^2}\left({\zeta }^2\frac{{\partial }^2{T}_L}{\partial {\zeta }^2}+2\zeta \frac{\partial T_L}{\partial \zeta }\right)\right]-\frac{{\zeta }^2}{L}\left[\frac{2D_L}{y{R}^2}+\frac{\dot{R}}{R}\left(y-\frac{1-\dot{m}/{\rho }_L\dot{R}}{y^2}\right)\right]\frac{\partial T_L}{\partial \zeta }$$ (14)

Both Eqs. (13) and (14) are solved by the Chebyshev collocation method. Specifically, the temperature field is projected to the N-dimensional manifold spanned by Chebyshev polynomials. Considering the zero temperature gradient at the bubble center (y = 0) and the infinity (ζ = 0), only the even polynomials are used. For the temperature field inside the bubble, it is expanded using M Chebyshev polynomials

$$T_v=\sum _{n=0}^M{a}_n{T}_{2n}\left(y\right),$$ (15)

where ${\mathit{\text{T}}}_{\text{2}\mathit{\text{n}}}\left(\mathit{\text{y}}\right)\text{=cos}\left(\text{2}\mathit{\text{n}}{\text{cos}}^{\text{-1}}\mathit{\text{y}}\right)$ and a_n are the expansion coefficients. Substituting the new expression of temperature into Eq. (13) and enforcing the equations at the Gauss-Lobatto collocation points

$$\mathit{\text{y}}=\cos \left(i\pi /2M\right),i=1,2,\cdots M,$$ (16)

M equations with a_n as the unknown parameters are derived. A similar treatment is applied to the temperature equation in the liquid (Eq. (14)) using N Chebyshev polynomials. Combined with the continuity equations at the bubble interface (Eqs. (10), (11)), the two first-order equations for bubble radius and wall velocity, the pressure equation (Eq. (6)), and the equation for mass transfer (Eq. (12)), a system of M + N + 6 differential–algebraic equations is obtained, which is integrated using the stiff solver ode15s in MATLAB. In this way, the parameters R, $\dot{R}$, p, n, and the expansion coefficients a_n and b_n are obtained at each time step. The physical properties, i.e., p_v, k_v, k_L, D_L, S, and μ, are functions of temperature and pressure. Their real-time values in the simulation are supplied from the NIST database using a MATLAB script [22].

4. Prediction of violent collapse

4.1. The Brennen’s method

Before delving into the details of violent bubble collapse in cryogenic fluids, we first need to identify when those events will occur. This problem is examined within the parameter space of ambient pressure, p_∞, and temperature, T_∞, since in practice they are the main adjustable variables in tests such as those involving laser-induced bubbles [3], [7], [8]. The bubble size is also at one’s disposal, but is only weakly associated with the intensity of collapse as will be shown. In addition, since the focus of this study is the dynamics of bubble collapse, we concentrate the discussion on the phase of bubble contraction unless otherwise specified.

In Brennen’s theory [23], thermodynamic equilibrium is assumed and the pressure inside the bubble p equals the saturated vapor pressure p_v. Neglecting the influence of surface tension and viscosity, the bubble’s radial dynamics is governed by the Rayleigh-Plesset equation

$${\rho }_L\left[R\frac{d^{2}R}{d{t}^2}+\frac{3}{2}{\left(\frac{\mathit{dR}}{\mathit{dt}}\right)}^2\right]=p_v\left(T_w\right)-p_{\infty },$$ (17)

where T_w is the liquid temperature at the bubble wall. It is seen from the above equation that thermal effects enter into the bubble dynamics by changing the instant p_v through T_w. By limiting the variation of T_w close to T_∞, p_v can be expanded by the first-order Taylor series [23]

$$p_v\left(T_w\right)\text{-}{p}_v\left(T_{\infty }\right)=\frac{\partial p_v}{\partial T}\left(T_w-T_{\infty }\right),$$ (18)

where the derivative of p_v with respect to T can be obtained by differentiating the Clausius-Clapeyron relation and reads

$$\frac{\partial p_v}{\partial T}=\frac{{\rho }_v\left(T_{\infty }\right)\mathcal{L}\left(T_{\infty }\right)}{T_{\infty }},$$ (19)

in which the physical properties on the right-hand side are also evaluated at T_∞. Substituting Eqs. (18), (19) into Eq. (17) leads to

$${\rho }_L\left[R\frac{d^{2}R}{d{t}^2}+\frac{3}{2}{\left(\frac{\mathit{dR}}{\mathit{dt}}\right)}^2\right]=\frac{{\rho }_v\left(T_{\infty }\right)\mathcal{L}\left(T_{\infty }\right)}{T_{\infty }}\left(T_w-T_{\infty }\right)+p_v\left(T_{\infty }\right)-p_{\infty }.$$ (20)

The temperature rise, T_w-T_∞, can be approximated by the Plesset- Zwick relation [24]. By further assuming that the variation of bubble radius with time follows the power relation, R = R*tⁿ, where R* is a constant and n the power index, the first term (thermal term) on the right side of Eq. (20) becomes ρ_L∑CR*t^n−0.5, in which C is another constant of order unity and ∑ the well-known thermodynamic parameter

$$\sum (T_{\infty })=\frac{{\mathcal{L}}^2\left(T_{\infty }\right){\rho }_{\text{v}}^2\left(T_{\infty }\right)}{{\rho }_L^2{c}_{\mathit{pL}}{T}_{\infty }{D}_L^{0.5}\left(T_{\infty }\right)}.$$ (21)

It is thus clear that the thermal term in Eq. (20) is a function of time and its influence increases with the development of collapse. Therefore, there exists a critical time, t_c4, when the order of magnitude of the thermal term equals that of the dynamic terms on the left side. Brennen gave this time as [23]

$${\mathit{\text{t}}}_{c4}={\left[R_{\max }/\sum \left(T_{\infty }\right)\right]}^{2/3},$$ (22)

where R_max is the maximum bubble radius at the t = 0. Physically, t_c4 marks the moment when the elevated vapor pressure begins to cushion the collapse significantly.

On the other hand, if the critical time emerges after the collapse has been completed, the process can be considered inertial and would be violent in nature. In the absence of thermal damping, the time duration for a pure inertial collapse is commonly known as the Rayleigh collapse time t_c [25],

$$t_c=0.915R_{\max }\sqrt{\frac{{\rho }_L}{p_{\infty }-p_v\left(T_{\infty }\right)}}.$$ (23)

Therefore, the relative magnitude between t_c and t_c4 provides an indicator of the intensity of bubble collapse. Violent collapses will be expected in cases where t_c $\ll$ t_c4. On the contrary, if t_c $\gg$ t_c4, the early occurrence of cushioning effect brought up by the vapor pressure would render the collapse benign. In this regard, the ratio of t_c4 to t_c is defined as χ,

$$\chi ={\mathit{\text{t}}}_{c4}/t_c=\frac{1}{0.915}{\left(\frac{1}{R_{\max }{\sum }^2}\right)}^{\frac{1}{3}}\sqrt{\frac{p_{\infty }-p_v\left(T_{\infty }\right)}{{\rho }_L}},$$ (24)

and will be used in this study to predict the regime of violent collapse. It is seen that χ mainly depends on the ambient pressure p_∞ and the physical properties of the liquid, which are determined by the ambient temperature T_∞. A large χ is achieved with a high p_∞ and a small T_∞, and vice versa.

The main issue of Brennen’s theory as illustrated above is the assumption of small temperature change in T_w and the associated thermodynamic equilibrium, which admits the evaluation of the thermodynamic parameter ∑ at T_∞. In the case of violent collapse in cryogenic liquids, this assumption can be easily violated. As will be seen in our investigated cases, the liquid layer near the collapsing bubble can be heated dramatically with T_w increasing by hundreds of degrees. Then the real ∑ would increase significantly over the course of the collapse. In this regard, ∑ should be treated as a process variable. An evaluation based on the initial temperature ∑(T_∞) and the resultant χ may underestimate the extent of true thermal effects.

Inspired by Florschuetz and Chao [26] with their treatment of p_v, a possible remedy for this issue is to replace ∑(T_∞) with an integrated average value over the variation range of T_w

$${\sum }^{\text{'}}=\frac{1}{T_{w,\max }-T_{\infty }}{\int }_{T_{\infty }}^{T_{w,\max }}\sum \left(T\right)dT.$$ (25)

However, the maximum wall temperature, T_w,max, required in the integral is not a priori known, and a complex iterative approach is needed for the calculation. Therefore, we don’t attempt this method, but instead resort to the complete numerical simulation to refine the prediction.

4.2. The regime of violent collapse in liquid nitrogen

In Fig. 1(a) and (b), the value of χ is displayed in the p_∞-T_∞ parameter space for bubbles in liquid nitrogen with R_max = 0.75 mm and 17 μm, respectively. We stress again that due to the issue of Brennen’s theory, the condition for violent collapse is $\chi \gg 1$, rather than the weak form $\chi >1$. The bubble sizes are chosen according to the previous studies [3], [4], [7], [8], [27], [28], which will be referenced when discussing the numerical results. More generally, they represent two categories of bubbles that are investigated extensively: bubbles of millimeters in size induced by laser irradiation, and those of micrometers existing in ultrasonic reactors. Although their sizes differ by three orders of magnitude, Fig. 1 shows that for both kinds of bubbles, only a small region of high pressure and low temperature in the parameter space satisfies the requirement of violent collapse, i.e., $\chi \gg 1$. Note the dash-dotted line in the figure denoting χ = 1.

Fig. 1.

The distribution of χ in the p_∞-T_∞ parameter phase for bubbles in liquid nitrogen with R_max = 0.75 mm (a) and 17 μm (b). The dash-dot line denotes χ = 1. The points indicated by P1 (p_∞=6.8 bar, T_∞=66 K, χ = 3.6), P2 (p_∞=1.0 bar, T_∞=77.23 K, χ = 0.02), P3 (p_∞=1.1 bar, T_∞=78 K, χ = 0.03), P4 (p_∞=1.2 bar, T_∞=66.5 K, χ = 5.5), and P5 (p_∞=4 bar, T_∞=77.4 K, χ = 1.6) denote cases studied by previous tests [3], [4], [7], [8], [27], [28] and will be discussed in the text. Note that in some of the referenced tests, the ambient pressure p_∞ is reported as gauge pressure, which is zero under atmospheric conditions. In this study, they are all converted to absolute values. Also, the exact maximum bubble radii in some studies are only near, but not necessarily equal to, the chosen values for the figure. The influence of this deviation is negligible since χ is weakly dependent on R_max.

Now we discuss the implications of Fig. 1 for laser-induced bubbles and ultrasound-driven bubbles, respectively. Previous experiments investigating the former class of bubbles in liquid nitrogen were usually conducted under atmospheric pressure conditions (p_∞=1 bar) [7], [8]. Nevertheless, Fig. 1(a) suggests that this is not a good choice if the purpose is to study the rapid bubble dynamics. With $\chi <1$, most of these collapses would be mild, including the ones in Sato et al. [8] (p_∞=1.0 bar, T_∞=77.23 K, χ = 0.02) and Tomita et al.[7] (p_∞=1.1 bar, T_∞=78 K, χ = 0.03). They are marked with P2 and P3 in Fig. 1(a), respectively. Even for tests with ambient temperatures falling within the region on the left side of the demarcating line, it is seen that χ is only slightly higher than 1 since T_∞ should be larger than the freezing temperature to maintain the liquid state of the medium. As a result, no violent collapse would occur.

A feasible path towards realizing violent collapse is increasing the ambient pressure. Baghdassarian et al. [3] adopted such a strategy and observed a hallmark of rapid bubble dynamics, i.e., luminescence, at collapse. The ambient condition of their test is indicated by P1 (p_∞=6.8 bar, T_∞=66 K) in Fig. 1(a). With the thermodynamic parameter χ of 3.6, the reported strong collapse is expected.

For micrometer-sized bubbles produced in an ultrasonic sound field, the region of violent collapse in the parameter space is enlarged as shown in Fig. 1(b). Also, the maximum χ increases to 38, showing the benefits brought from the decreased size as $\chi \propto R_{\mathrm{m}\mathrm{a}\mathrm{x}}^{-1/3}$. The obstacle to achieving intense collapse in this case, however, is the phase mismatch between the applied sound pressure and bubble evolution. Take sinusoidal acoustic pressure p_s, p_s(t) = p_asin(ωt) where p_a is the amplitude and ω = 2πf the angular frequency, the bubble expands to the maximum size and begins to collapse at the end of the rarefication phase of the sound wave when p_s < 0. This means the ambient pressure is negated by the sound pressure and the total external pressure (p_total = p_∞+p_s) compressing the bubble in the subsequent collapse is weakened. For a bubble oscillating in liquid nitrogen under ultrasonic driving with p_a = 1.0 bar and f = 28.5 kHz [27], numerical simulation [28] shows that p_s maintains negative throughout the main collapse process as displayed in Fig. 2. In consequence, with p_total $\ll$ p_∞, only weak bubble oscillation was observed [27], [28]. Hypothetically, nevertheless, if the sound pressure disappears during the collapse (p_s = 0), intense collapse will be anticipated since χ increases to 5.5 in this circumstance. The assumptive ambient condition is indicated by P4 in Fig. 1(b).

Fig. 2.

Simulation of bubble dynamics in liquid nitrogen under ultrasonic pressure p_s(t) = p_asin(2πft) with the p_a = 1.0 bar and frequency f = 28.5 kHz. The ambient conditions are p_∞=1.2 bar and T_∞=66.5 K. The figure is adapted from Ref. [28] with the permission of the American Physical Society.

As in the case of laser-induced bubbles, the collapse intensity in ultrasonic fields can be strengthened by enhancing the ambient pressure. Dular and Petkovšek [4] increased p_∞ to 3 bar and observed weak cavitation erosion in liquid nitrogen. Ignoring the sound pressure, the ambient condition in their test is indicated by P5 in Fig. 1(b). Still, the moderate χ of 1.6 in this hypothetical condition is insufficient to induce violent collapses as inferred from the test. We suspect that heat convection due to the relative movement between the bubble and liquid plays an important role in this case. Discussion on this aspect will be deferred till the mechanism of thermal effects is revealed later. The drawback of this strategy is the increased threshold of initiating cavitation in the first place, which requires high-intensity ultrasound and large power output from the ultrasonic transducer.

5. Simulation results

5.1. Comparing bubble collapse intensity in liquid nitrogen

In the above section, we identified the regime of bubble collapse in liquid nitrogen of different intensities in the parameter space by using Brenner’s thermodynamic parameter χ. Now, we check some representative cases in each regime using the full simulation model introduced in Section 2. The investigated bubbles are that of Baghdassarian et al. [3] (Case B, p_∞=6.8 bar, T_∞=66 K, χ = 3.6) and Sato et al. [8] (Case C, p_∞=1.0 bar, T_∞=77.23 K, χ = 0.02). Also, we add two cases in the simulation to compare more strongly bubble collapses. The first is a hypothetical one derived from Baghdassarian et al.[3] where the ambient pressure is increased to 55 bar while other parameters are kept unchanged. It should be noted that no tests with such a high ambient pressure have been reported for liquid nitrogen. We devise this case purely out of curiosity and aim to examine the violent collapse. The second is from Obreschkow et al. [29] where the bubble collapses in water under microgravity conditions but with normal ambient temperatures, where p_∞=28.6 kPa, T_∞=26 ℃, and R_max = 2.786 mm. Despite different hosting mediums, these two cases have the same χ, χ = 10.4, and are termed Case A and D, respectively. The simulation of the water vapor bubble (Case D) is conducted with the same numerical model as in the case of nitrogen, except with the physical properties of water. To be consistent with the assumption of pure vapor bubbles, no noncondensable gas exists inside the cavity for all the cases.

Fig. 3 displays the bubble radius, temperature fields inside and outside of the bubble, and the bubble wall velocity for each case. It is observed that the simulation reproduced well the bubble radius from the referenced tests [3], [8], [29] in Case B-D. In addition, in the experiment of Baghdassarian et al.[3], the collapsing temperature is estimated to be 4500 K from the spectra analysis of the emitted light. Our simulation gives the peak temperature as 4242 K, again showing excellent agreement with the test.

Fig. 3.

Comparison of bubble radius (top row), temperature field (middle row), and wall velocity (bottom row) during bubble collapse in liquid nitrogen (Case A-C) and water (Case D). The ambient conditions are p_∞=55 bar, T_∞=66 K, χ = 10.4 for Case A, p_∞=6.8 bar, T_∞=66 K, χ = 3.6 for Case B, p_∞=1.0 bar, T_∞=77.23 K, χ = 0.02 for Case C, and p_∞=28.6 kPa, T_∞=26 ℃, χ = 10.4 for Case D. Case B and C are indicated by P1 and P3 in Fig. 1(a), respectively. “LN₂” stands for liquid nitrogen. The figure displaying the temperature field for Case D uses logarithmic scale due to the large variation range. The abscissa of all the figures displays the time normalized by Rayleigh collapse time t_c.

A comparison of Case A-C shows the reduced collapse intensity in liquid nitrogen caused by a more significant thermal effect as χ decreases. The violence of collapse is quantified by several key parameters, including the compression ratio, the maximum wall velocity, and the collapsing temperature, and the differences among the cases are displayed in Table 1. For Case C, the small χ dictates that no collapse occurs as expected. The bubble only oscillates weakly around the equilibrium size. The collapses in Cases A and B result in the strong heating of the vapor inside the bubble with a collapsing temperature higher than 4000 K, which well explains the luminescence observed at the collapsing point [3].

Table 1.

Comparison of the main parameters for bubble collapse in liquid nitrogen and water.

Studied case	Rmax/Rmin	${\left|\dot{R}\right|}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, m/s	$T_{\mathrm{m}\mathrm{a}\mathrm{x}}$, K
Case A, nitrogen bubble p∞=55 bar, T∞=66 K, χ = 10.4	15	844	5782
Case B, nitrogen bubble p∞=6.8 bar, T∞=66 K, χ = 3.6	10	193	4242
Case C, nitrogen bubble p∞=1.0 bar, T∞=77.23 K, χ = 0.02	1.3	3.5	110
Case D, water vapor bubble p∞=28.6 kPa, T∞=26 ℃, χ = 10.4	125	1254	54,584

It is interesting to note that for Case B, while a high collapsing temperature is generated, the velocity of bubble wall motion is moderate with ${\left|\dot{R}\right|}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ ∼ 200 m/s, indicating the relatively slow dynamics. The associated shock waves, if any, would be rather weak in this case. The relative slow bubble motion is made clear when compared with the collapse of the water vapor bubble in Case D, where ${\left|\dot{R}\right|}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is almost 100 times larger. It has been reported that the pulse of cavitation luminescence in liquid nitrogen can be 100 times longer than in water [3]. The difference in bubble dynamics revealed from the simulation agrees well with the experimental observation.

The slow dynamics in Case B indicates that the collapse is still cushioned significantly by thermal effect, even though χ is larger than 1 and such effect is predicted to be absent. This contradiction corroborates our earlier conclusion regarding the issue of Brennen’s theory. The inadequacy of the thermodynamic parameter ∑ in characterizing the true thermal effect is also noticed by Dular and Petkovšek [4], who observed the difference in cavitation erosion in hot water and liquid nitrogen under similar ∑.

Besides the selected cases displayed in Fig. 3, we also simulated bubble collapses in liquid nitrogen under different ambient pressures while other parameters were kept unchanged. The purpose is to elucidate the complete change pattern of collapse dynamics in response to varying χ. The results for T_∞=66 K and R_max = 0.75 mm are shown in Fig. 4 where p_∞ varies between 0.5 bar and 55 bar.

Fig. 4.

The change in the maximum temperature and bubble wall velocity under different ambient pressures for collapsing bubbles in liquid nitrogen. The ambient temperature and initial bubble radius are kept unchanged with T_∞=66 K and R_max = 0.75 mm.

The figure clearly displays the steep rise in T_max when p_∞<7 bar before leveling off towards higher ambient pressures. Unlike T_max, however, the maximum collapsing velocity increases continuously with higher p_∞. The diverging trends illustrate that while the collapse is strongly influenced by the thermal effect when χ is small, it becomes increasingly inertial at larger χ.

5.2. Comparing collapse intensity in liquid nitrogen and water

The difference in the collapse dynamics is also reflected in the energy distribution. Following the energy partition scheme proposed by Tinguely et al. [30], we calculate the fractions of different energies among the total energy at R = R_max. The results are displayed in Table 2 with the definitions for each energy listed in Table 3.

Table 2.

Comparison of energy partition in the collapse under different ambient conditions. η_i denotes the fraction of energy in the total energy, η_i = E_i/E₀, where E_i is defined in Table 2 for the respective energy i.

ηs, %	ηreb, %	ηheat, %	ηU, %	Parameters	Reference
93.8	4.0	0.6	1.6	Medium: liquid nitrogen, p∞=55 bar, T∞=66 K, Rmax = 0.75 mm	Case A, present study
58.8	34.4	6.7	0.06	Medium: liquid nitrogen, p∞=6.8 bar, T∞=66 K, Rmax = 0.75 mm	Case B, present study
93.7	6.2	0.07	0.03	Medium: water, p∞=0.286 bar, T∞=26 K, Rmax = 2.786 mm	Case D, Present study
80–99	1–15	–	–	Medium: water, p∞=0.1–0.8 bar, T∞=290–294 K, Rmax = 2–5.6 mm	Tinguely et al., 2012 [30]
89.7	6.2	–	–	Medium: water, p∞=1 bar, T∞=293 K, Rmax = 1.2 mm	Wen et al., 2023 [33]

Table 3.

Definition of various energies in the bubble collapse.

Energy	Definition
Potential energy	$E_0=\frac{4\pi }{3}{R}_{\max }^3\left(p_{\infty }-p_{R=R_{\max }}\right)\mathrm{at}R=R_{\max }$$E_{\mathrm{reb}}=\frac{4\pi }{3}{R}_{\mathrm{reb}}^3\left(p_{\infty }-p_{R=R_{\mathrm{reb}}}\right)\mathrm{at}R=R_{\mathrm{reb}}$
Internal energy	$U_1=\frac{4\pi }{3\left(\gamma -1\right)}{R}_{\max }^3{p}_{R=R_{\max }}\mathrm{at}R=R_{\max }$$U_2=\frac{4\pi }{3\left(\gamma -1\right)}{R}_{\mathrm{reb}}^3{p}_{R=R_{\mathrm{reb}}}\mathrm{at}R=R_{\mathrm{reb}}$
Energy for heating liquid	$E_{\mathrm{heat}}={\int }_{R_{\mathrm{reb}}}^{\infty }{c}_p{\rho }_{L}4\pi r^2{T}_L\left(r\right)dr-{\int }_{R_{\max }}^{\infty }{c}_p{\rho }_{L}4\pi r^2{T}_{\infty }dr$
Energy in emitted sound	$E_{\text{s}}=E_0-U_1-E_{\mathrm{reb}}+U_2-E_{\mathrm{heat}}$

For inertial collapse in water, the peak velocity of the bubble wall can reach thousands of meters per second, and shock waves are emitted into the surrounding liquid [31], [32]. The energy lost in this process accounts for the highest proportion of the total energy, with a value near or exceeding 90 % [30], [33]. For bubble collapses in liquid nitrogen in Case B, due to the mild dynamics, the energy associated with the emitted sound constitutes a moderate fraction, 58.8 %, of the total energy. On the other hand, 34.4 % of the energy is conserved in the rebounded bubble as the potential energy. With less energy lost in the emitted sound, more is used to heat the vapor inside the bubble during the collapse. This well explains the high collapsing temperatures relative to the small wall velocity for this case as observed in Fig. 3. When the collapse is strengthened as in Case A, the energy distribution resembles those for inertial collapses more closely.

5.3. Mechanism of the slow bubble dynamics in liquid nitrogen

Now we explore the root cause for the relative slow collapse dynamics in liquid nitrogen even with χ > 1. At first sight, the lower wall velocity observed in Case A and Case B compared with inertial collapse in Case D is very puzzling, since the ambient pressure exerted on the nitrogen bubble is much higher than on the water vapor bubble (p_∞=55 and 6.8 bar in Case A and B, versus 28.6 kPa in Case D). Even though the vapor pressure is larger in liquid nitrogen than in water (21145 Pa versus 2894 Pa), the net compression pressure is still an order of magnitude larger, which should induce a faster bubble motion.

An inspection of the detailed collapse process confirms our suspicion. Fig. 5 displays the bubble wall velocity as a function of time before the collapse point t*, which refers to the moment when$\dot{R}\left(t^{\ast }\right)$ = 0. The results show that the nitrogen bubble contracts more rapidly than the water vapor bubble over the major portion of the collapse process. Only at $\left(t^{\ast }-t\right)=2.5\times {10}^{-3}{t}_c$ and $\left(t^{\ast }-t\right)=7.0\times {10}^{-4}{t}_c$ does the collapsing velocity of the water vapor bubble surpass that of the nitrogen bubble in Case B and A, respectively. Note these instants are very close to the collapse point.

Fig. 5.

Evolution of bubble wall velocity in liquid nitrogen and water. The simulated bubble is the nitrogen bubble in Case A (p_∞=55 bar, T_∞=66 K, χ = 10.4) and Case B (p_∞=6.8 bar, T_∞=66 K, χ = 3.6), as well as the water vapor bubble in Case D (p_∞=28.6 kPa, T_∞=26 ℃, χ = 10.4). The normalized time on the abscissa is relative to the collapse point where$\dot{R}\left(t^{\ast }\right)$ = 0. t_c1, t_c2 and t_c3 denote the time when the heated liquid at the bubble interface becomes supercritical.

Since the bubble collapse is essentially governed by the same dynamic equation (Eq. (1)), the evolution of the various pressure terms therein should explain the difference in the intensity. Following Hilgenfeldt et al. [34], the investigated pressures include the acceleration term, $p_{\mathit{acc}}={\rho }_{L}R\ddot{R}$, the velocity term, $p_{\mathit{vel}}=\frac{3}{2}{\rho }_L{\dot{R}}^2$, the gas pressure term, $p_{\mathit{gas}}=p$, and the emitted sound term, $p_{\mathit{snd}}=\frac{R}{c_L}\frac{\mathit{dp}}{\mathit{dt}}$. Their contributions to the collapse during the collapse are displayed in Fig. 6 for the three cases.

Fig. 6.

Evolution of pressure contributions during the collapse for the nitrogen bubbles in Case A (a) and Case B (b), as well as for the water vapor bubble in Case D (c). The pressure terms are normalized by the ambient pressure and are defined in the text.

For all the bubbles, Fig. 6 shows that the dynamic terms $p_{\mathit{acc}}$ and $p_{\mathit{vel}}$ are the dominating contributions for a large part of the collapse phase. Dynamics governed by the two terms is inertial and can be described by the classical Rayleigh equation [25]

$$R\ddot{R}+\frac{3}{2}{\dot{R}}^2=p_{\infty }.$$ (26)

As $t$ approaches $t^{\ast }$, the inertial regime breaks down in all cases, but due to different reasons. For the nitrogen bubble in Case B, the gas pressure term $p_{\mathit{gas}}$ increases rapidly in magnitude and begins to balance $p_{\mathit{acc}}$ at $\left(t^{\ast }-t\right)=1.4\times {10}^{-2}{t}_c$ as shown in Fig. 6(b). For the nitrogen bubble in Case A and water vapor bubble in Case D, however, it is the elevated sound term, $p_{\mathit{snd}}$, that counteracts $p_{\mathit{acc}}$ and changes the dynamics, which agrees with the finding from Hilgenfeldt et al.[34]. Also, the inertial breakdown for the latter two cases occurs later.

The fact that the rising pressure term balances the dynamic term in Case B indicates a profound thermal effect emerging in the final stage of bubble collapse in liquid nitrogen. This demonstrates again the deficiency of Brenner’s theory since such thermal effect should be absent with χ = 3.6.

The rapid increase of internal pressure in the collapsing nitrogen bubble is well expected due to the physical properties of liquid nitrogen as we explained before. Besides this well-known mechanism, we propose another factor at the latter stage that ultimately retards collapse. In introducing the numerical model in Section 2, we highlighted a critical point in the collapse process when both the vapor inside the bubble and the liquid at the interface enter into supercritical state. After that, vapor condensation ceases since there is no distinction between phases in supercritical state. In consequence, the remaining vapor is effectively trapped inside the bubble and behaves as noncondensable gas. The compression of these supercritical substances as the collapse proceeds would steepen the pressure rise and cushion the collapse more strongly.

The critical temperature and pressure are 126 K and 3.3 MPa for liquid nitrogen, while they are 647 K and 22 MPa for water. Combined with a more rapid rise of vapor pressure, the supercritical state is achieved much earlier in liquid nitrogen than in water as observed in Fig. 7. The elevated pressure after this point quickly counteracts the compression and decelerates the collapse as can be seen from the evolution of wall velocity shown in Fig. 5. Therefore, it is the very final stage that ultimately determines the collapse intensity. The early deceleration of wall motion for the nitrogen bubble caused by the trapping of vapor leads to a lower collapse intensity.

Fig. 7.

Comparison of pressure rises inside the collapsing bubble in liquid nitrogen and water. t_c1, t_c2, and t_c3 denote the time when the heated liquid at the bubble interface becomes supercritical.

For nitrogen bubbles in Cases A and B, the fluid enters the supercritical state at identical times. In this regard, the effect of imposing larger ambient pressure on the nitrogen bubble is to cause a larger collapse velocity by the moment when the supercritical state is achieved, which in turn would lead to a stronger ultimate collapse as seen in Case A.

5.4. Implication for cavitation erosion in cryogenic liquids

From the simulation results, it is seen that for vapor bubbles in liquid nitrogen, ambient pressure of dozens of bars is required for the collapse to transition into inertial. In practice, however, applying such high pressures is a huge challenge for laboratory tests involving cryogenic fluids. Most experiment allows for much lower pressures (a few bars) like that in the study of Baghdassarian et al. [3] and Dular and Petkovšek [4]. Thermal effect in this circumstance would be significant, which in turn leads to mild bubble collapse. As a result, side effects such as cavitation erosion should be absent.

Nevertheless, there is another route through which the thermal effect in cryogenic fluids can be mitigated. In both Brenner’s theory and our simulation, the bubble is assumed to be at rest in a liquid and the heat transfer between the bubble and the surrounding liquid is accomplished by conduction. The extremely small thermal diffusivity makes the dissipation of energy in the liquid highly inefficient, which increases the temperature at the interface of the collapsing bubble and raises the vapor pressure quickly. In a real fluid environment, the heat transfer at the interface can be enhanced by convection due to the relative movement between the vapor bubble and the liquid [35]. The enhancement may be significant as the associated Nusselt number can be as large as 10⁶ [36]. With more efficient heat dissipation and lower temperature at the bubble wall, the pressure rise inside the collapsing bubble may be slower and the cushioning effect can be weakened. Violent collapse is therefore attained and shock waves can be resulted. This effect probably explains the occurrence of cavitation erosion observed in liquid nitrogen [4] even though the estimated χ in that case is slightly larger than 1.

In the last, we point out that even if the collapse in cryogenic fluids is not violent enough to generate shock waves, cavitation erosion is still possible. The simulation results displayed in Fig. 4 indicate that collapsing temperatures of thousands of Kelvins can be easily achieved inside the bubble by slightly pressurizing the liquid. When bubbles under such ambient conditions are attached to a solid surface and collapse, direct contact between the heated contents and the material can inflict damage to the surface. A similar scenario has been observed in a prior test [37], where hemispherical bubbles collapsing on a boundary were seen to emit light and damage the surface.

6. Conclusion

In this paper, we focused on the class of violent collapse of vapor bubbles in liquid nitrogen and investigated its thermodynamic features. Brenner’s dimensionless time ratio χ was introduced and adopted to predict when those intense events can occur. In addition, a complete numerical simulation was performed to examine the detailed collapse process. The collapse of the nitrogen bubble was further compared with that of water vapor bubbles to illustrate the extent of thermal impact in different ambient conditions. The main conclusions are summarized as follows.

(1) The dimensionless time ratio χ is instrumental in identifying the parameter regime for violent collapse, but has to be extended due to its inherent issue.

(2) For vapor bubbles in liquid nitrogen, collapses with 1 < χ < 10 are still strongly influenced by thermal effect and characterized by relatively high collapsing temperature and moderate wall velocity. The collapse is strengthened with increasing χ and becomes inertial when χ > 10.

(3) For the moderately violent collapse of nitrogen bubbles, the inertial breaks down due to the rapid rising of internal pressure, which differs from that of a water vapor bubble in the absence of thermal effect.

(4) In addition to well-known physical properties, the pronounced thermal effect is also attributed to the lower threshold of supercritical state for cryogenic liquids. The ultimate collapse intensity is closely associated with the dynamics at the time when the surrounding liquid layer becomes supercritical.

CRediT authorship contribution statement

Kewen Peng: Writing – review & editing, Writing – original draft, Software, Investigation, Formal analysis, Conceptualization. Shouceng Tian: Investigation, Funding acquisition. Yiqun Zhang: Investigation. Jingbin Li: Investigation. Wanjun Qu: Investigation, Funding acquisition. Chao Li: Investigation, Funding acquisition.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.