Viscoelastic vapor bubble collapse near solid walls and corresponding shock wave formation

Lang, Christian; Adami, Stefan; Adams, Nikolaus A.

doi:10.1063/5.0175807

This study investigates the influence of viscoelasticity on the collapse of aspherical vapor bubbles near a solid boundary through numerical simulations. A fully compressible three-dimensional finite volume method is employed, incorporating a single-fluid homogeneous mixture cavitation model and the simplified linear Phan-Thien Tanner viscoelastic constitutive model. The collapse dynamics, liquid jetting, shock wave formation, and associated pressure impact are analyzed, and the viscous and viscoelastic stress fields are presented. A comparison of viscoelastic to Newtonian dynamics reveals significant differences in collapse behavior and shock wave formation due to viscoelasticity. Viscoelasticity can induce jet piercing, which is not observed in the Newtonian collapse, and increases vapor re-evaporation after the first collapse. The effect of changing the initial standoff distance is examined for both viscoelastic and Newtonian fluids, where a second jet formation is present only for the viscoelastic collapse, and the second collapse's intensity is increased due to increased vapor production during rebound. Additionally, the variation of elasticity in the viscoelastic case demonstrates a correlation between the amount of vapor produced during rebound and the relaxation time for the investigated cases.

I. INTRODUCTION

The dynamics of aspherical bubbles near rigid walls and their potential impact on the wall, including shock wave and jet formation, are crucial in various applications involving non-Newtonian fluids. Especially in biomedical applications, where the addressed materials often show a complex microstructure, cavitating bubble collapse is of significant importance. In sonoporation, the collapse of microbubbles close to a surface of a cell, the resulting liquid jet and shock wave emission are capable of producing large forces and perforating the cell membrane.^1,2 Drug-carrying microbubbles can be utilized together with focused ultrasound to interrupt the blood-brain barrier and deliver drugs targeted into brain tumors.³ Ultrasound and aspherical cavitating microbubbles can also be used for gene delivery by penetrating tissue.⁴ Shockwave lithotripsy also relies on aspherical bubble collapse to destroy kidney stones.⁵

Experiments proved that viscoelasticity has a significant impact on the collapse dynamics in the vicinity of a wall. Chahine and Fruman⁶ found with their experiments that polymer additives can inhibit liquid-jet formation during the collapse of spark-generated bubbles in the vicinity of a rigid wall. Brujan et al.⁷ examined the collapse of ultrasound-induced bubbles close to a rigid wall experimentally with the result that polymer additives can decrease the velocity of the liquid-jet. However, most numerical studies either neglect viscous forces^8–20 or consider Newtonian fluids.^21–32 Only a few numerical studies have considered non-Newtonian media for the wall-influenced aspherical bubble collapse.

One of the first numerical studies by Hara and Schowalter³³ revealed that viscoelastic effects are more substantial for the aspherical than for the spherical bubble collapse by applying a modified Rayleigh–Plesset equation. For the numerical simulation of bubble dynamics close to solid walls, the boundary element method (BEM) is frequently employed. The classical BEM assumes the flow field to be irrotational and incompressible to describe the flow by a potential function. Viscous forces, hence, are only incorporated through the dynamic boundary condition at the interface, and stresses in the remaining field are neglected. For high Reynolds number flows, where inertia is dominant and viscous forces are negligible, an irrotational flow field assumption can be appropriate. Furthermore, in the classical BEM two coinciding points or a doubly connected domain, respectively, would lead to a singularity. Thus, the simulation would fail as soon as a liquid jet pierces through the bubble, producing a toroid-shaped bubble. To overcome this limitation, methods have been developed to resolve the occurrence of fluid jets and associated toroidal bubbles. Different modifications, such as vortex-cut methods^10,34 and vortex-ring methods,^35,36 enable the BEM to resolve toroidal bubbles and doubly connected domains. A comprehensive comparison of the methods is provided in Han et al.³⁷ Lind and Phillips³⁸ use the boundary element method (BEM) to investigate the near-wall collapse of gas-filled bubbles. They consider incompressible and irrotational flow. Thus, stresses are only incorporated at the interface. The viscoelastic Maxwell model without a solvent contribution together with the material derivative as time derivative for the stress tensor is applied.^33,38 come to the conclusion that viscoelasticity can inhibit jet formation. However, it is found in Lind and Phillips,^23,38 that jet suppression also can be observed for highly viscous fluids with identical Newtonian viscosity. To consider viscous and viscoelastic stresses in the entire field Lind and Phillips³⁹ employ the more elaborate marker particle method together with spectral element discretization. In addition to not being restricted to irrotational flows, they account for compressibility and use a compressible Oldroyd-B model with objective rate and thus consider solvent contributions in the viscoelastic constitutive model. In both studies, the simulations are 2D, and a gas bubble is considered. Thus, evaporation and condensation are neglected. Walters and Phillips⁴⁰ introduce a singularity-free formulation of the BEM to simulate toroid-shaped bubbles and liquid jet formation. Due to the required velocity potential, incompressibility and an irrotational velocity field are assumed. Thus, stresses are likewise only introduced at the bubble interface and neglected in the remaining field. Viscoelasticity is described by the Oldroyd-B model, incorporating solvent contributions and the upper convected derivative for the viscoelastic stress tensor. The authors explain that the approach is restricted to moderate to high Reynolds number flows. Other recent numerical studies involving viscoelastic bubble dynamics investigate microbubbles in a corner geometry,⁴¹ and rising bubbles in viscoelastic fluid,⁴² respectively.

In the present study, we conduct fully compressible three-dimensional (3D) simulations for two-phase cavitating flows considering condensation and evaporation. The density-based approach⁴³ uses finite volume discretization and explicit time integration. The simplified linear Phan-Thien Tanner (LPTT) model is used to describe viscoelasticity. This viscoelasticity model is of the Maxwell-/Oldroyd-type, comprising a Newtonian (solvent) and a viscoelastic contribution. The method naturally considers viscoelastic stresses in the whole domain filled by the liquid and not only at the bubble interface. Opposed to the studies mentioned above, we consider vapor-filled bubbles in the present simulations, and jet formation and impingement on the rigid wall are resolved. Furthermore, the approach fully resolves wave dynamics throughout the collapse.

The article is structured as follows: In the subsequent Sec. II, the model and its governing equations and the numerical approach is introduced. Subsequently, Sec. III describes the setup, including boundary conditions and the non-dimensionalized numbers characterizing the problem. Simulation results are presented in Sec. IV, starting with comparing the general dynamics in Newtonian vs viscoelastic fluids. The dynamics for variations of initial standoff distance and elasticity are presented thereafter. Finally, the key findings are summarized in Sec. V.

II. PHYSICAL MODEL AND NUMERICS

A. Governing equations

We apply an Eulerian approach for three-dimensional (3D), compressible, cavitating flows in viscoelastic fluids. Phase change is assumed to be isentropic. Consequently, we assume the flow to be barotropic, and the energy conservation is not considered. Surface tension is neglected. The resulting governing equations consist of the mass and momentum conservation equations and an additional transport equation for the viscoelastic stresses. The conservative form of the governing equations for an arbitrary control volume V with surrounding surface

\partial V

is given by

\int_{V} \frac{\partial}{\partial t} Q d V = - \int_{\partial V} (F^{conv} + F^{diff}) d S + \int_{V} S d V,

(1)

where

Q = {[ρ, ρ u, ρ τ_{M}]}^{T}

represents the vector of variables consisting of the density ρ, the momentum

ρ u

⁠, and the additional viscoelastic stress contribution

ρ τ_{M}

⁠. Here,

τ_{M}

is the viscoelastic stress tensor, which we denote as Maxwell stress tensor referring to the series element of dashpot and spring, which is part of the 1D rheological representation⁴³ of Oldroyd-like viscoelastic constitutive models. The integral comprises convective

F^{conv}

and diffusive

F^{diff}

flux terms, and the source term S which is introduced by the viscoelastic constitutive equations. The convective and diffusive fluxes read

F^{conv} = [\begin{matrix} ρ \\ ρ u \\ ρ τ_{M} \end{matrix}] u \cdot n + [\begin{matrix} 0 \\ p \\ 0 \end{matrix}] \cdot n, F^{diff} = [\begin{matrix} 0 \\ - 2 μ_{S} d^{d} \cdot n - τ_{M} \cdot n \\ 0 \end{matrix}],

(2)

where u and p represent the velocity and the thermodynamic pressure.

d^{d} = d - \frac{1}{3} tr (d) I

is the deviatoric part of the symmetric shear rate tensor

d = \frac{1}{2} (l + l^{T})

⁠, where

l = \nabla u

is the velocity gradient and

tr (\dots)

represents the trace. The diffusive flux comprises Newtonian (solvent) stresses

τ_{S} = 2 μ_{S} d^{d}

and viscoelastic stresses

τ_{M}

⁠.

We apply the simplified linear Phan-Thien Tanner (LPTT) model,⁴⁴ which is written in differential formulation as follows:

τ_{M} (1 + ϵ \frac{λ}{μ_{M}} tr (τ_{M})) + λ {\overset{\nabla}{τ}}_{M} = 2 μ_{M} d^{d},

(3)

{\overset{\nabla}{τ}}_{M} = \frac{D τ_{M}}{D t} - l \cdot τ_{M} - τ_{M} \cdot l^{T} + tr (l) τ_{M},

(4)

where the Truesdell rate

{\overset{\nabla}{τ}}_{M}

is applied as objective rate for the viscoelastic stress tensor, which was identified in Lang et al.⁴³ as appropriate objective time derivative for compressible flows. μ_M represents the viscoelastic dynamic viscosity, denoted as Maxwell viscosity. λ is the relaxation time and

\frac{D τ_{M}}{D t}

is the material time derivative of the viscoelastic stress tensor. The extensibility parameter ϵ in Eq. is set to the commonly used value of

ϵ = 0.25

⁠. By using the mass conservation, the source term for the conservative formulation is obtained:

S = [\begin{matrix} 0 \\ 0 \\ ρ [l \cdot τ_{M} + τ_{M} \cdot l^{T} - (\nabla \cdot u) τ_{M} + \frac{1}{λ} (2 μ_{M} d^{d} - τ_{M})] \end{matrix}] .

(5)

The viscoelastic implementation has been validated and grid convergence studies have been performed for single-phase channel flows in Lang et al.⁴³

B. Single-fluid homogeneous mixture cavitation model

Cavitating two-phase flow is modeled by a single-fluid homogeneous mixture approach,⁴⁵ assuming that a computational cell contains either pure liquid or a homogeneous water–vapor mixture. For mixtures, we assume that both phases are in mechanical and thermodynamic equilibrium with infinitely fast and isentropic phase change without time delay. The model inherently considers condensation and evaporation since the generated phase is directly reproduced according to the thermodynamic equilibrium state through the equation of state. An additional mass-transfer rate equation, such as the Schnerr–Sauer cavitation model,⁴⁶ is not required. The cavitation model was extensively used and validated for various applications such as bubble dynamics,^31,45,47,48 fuel injectors,⁴⁹ and condensation shocks.⁵⁰ The governing Eq. apply to homogeneous mixtures of multiple components without limitation. Consequently, we can assume that all quantities are represented by volume- and phase-averaged mixtures via

\bar{◻} : = \frac{1}{V_{Ω}} \int_{V_{Ω}} ◻ d V

(6)

for each computational cell

V_{Ω}

⁠, describing the homogeneous mixture. Within a discrete finite volume, the vapor volume fraction is calculated from

α = \frac{V_{v, Ω}}{V_{Ω}} = {\begin{matrix} 0, & if \bar{ρ} \geq ρ_{l, sat} (pure liquid), \\ \frac{ρ_{l, sat} - \bar{ρ}}{ρ_{l, sat} - ρ_{v, sat}}, & else (liquid - vapor mixture) \end{matrix}

(7)

with the vapor volume

V_{v, Ω}

in a computational cell and the saturation densities of vapor

ρ_{l, sat}

and liquid

ρ_{l, sat}

⁠. For pure liquids, a modified Tait equation of state by Saurel et al.⁵¹ is applied,

\bar{p} = B [{(\frac{\bar{ρ}}{ρ_{l, sat}})}^{N} - 1] + p_{sat},

(8)

where N = 7.15 and

B = 3.3 \times 10^{8}

Pa at reference temperature

T_{ref} = 293.15 K

⁠. The speed of sound in computational cells of pure liquid is calculated by

c = {(\frac{N (\bar{p} + B)}{\bar{ρ}})}^{1 / 2} .

(9)

For mixtures of saturated liquid and vapor, the definition of the isentropic speed of sound

c^{2} = {\frac{\partial p}{\partial ρ} |}_{s = const .} \Rightarrow p (ρ) - p_{sat} = \int_{ρ_{l, sat}}^{ρ} c^{2} d \tilde{ρ}

(10)

allows to obtain the pressure of the mixture by integration. The speed of sound is given by

\frac{1}{\bar{ρ} c^{2}} = \frac{α}{ρ_{v, sat} c_{v, sat}^{2}} + \frac{1 - α}{ρ_{l, sat} c_{l, sat}^{2}} + \frac{((1 - α) ρ_{l, sat} c_{p, l, sat} - α ρ_{v, sat} c_{p, v, sat}) T_{ref}}{{(ρ_{v, sat} l_{v})}^{2}}

(11)

containing the latent heat of vaporization l_v.⁵² A detailed derivation of the pressure calculation is given in Lang et al.⁴³ The solvent viscosity is calculated similarly for pure liquid and mixture regions following the approach of Beattie and Whalley,⁵³

μ_{S} = (1 - α) (1 + \frac{5}{2} α) μ_{l, sat} + α μ_{v, sat} .

(12)

The fluid properties applied in the present study are summarized in Table I.

TABLE I.

Fluid properties of the barotropic model at $T_{ref} = 293.15 K$ ⁠.

Property	Symbol	Value
Reference temperature	T_ref	293.15 K
Density of saturated liquid	$ρ_{l, sat}$	998.16 kg/m³
Density of saturated vapor	$ρ_{v, sat}$	0.017 21 kg/m³
Saturation pressure	p_sat	2339.3 Pa
Dynamic viscosity of saturated liquid	$μ_{l, sat}$	$1.0014 \times 10^{- 3} Pa s$
Dynamic viscosity of saturated vapor	$μ_{v, sat}$	$9.7275 \times 10^{- 6} Pa s$
Speed of sound of saturated liquid	$c_{l, sat}$	1482.2 m/s
Speed of sound of saturated vapor	$c_{v, sat}$	423.18 m/s
Specific heat capacity of saturated liquid	$c_{p, l, sat}$	4184.4 J/(kg K)
Specific heat capacity of saturated vapor	$c_{p, v, sat}$	1905.9 J/(kg K)
Latent heat of vaporization	l_v	$2453.5 \times 10^{3} J / kg$

Property	Symbol	Value
Reference temperature	T_ref	293.15 K
Density of saturated liquid	$ρ_{l, sat}$	998.16 kg/m³
Density of saturated vapor	$ρ_{v, sat}$	0.017 21 kg/m³
Saturation pressure	p_sat	2339.3 Pa
Dynamic viscosity of saturated liquid	$μ_{l, sat}$	$1.0014 \times 10^{- 3} Pa s$
Dynamic viscosity of saturated vapor	$μ_{v, sat}$	$9.7275 \times 10^{- 6} Pa s$
Speed of sound of saturated liquid	$c_{l, sat}$	1482.2 m/s
Speed of sound of saturated vapor	$c_{v, sat}$	423.18 m/s
Specific heat capacity of saturated liquid	$c_{p, l, sat}$	4184.4 J/(kg K)
Specific heat capacity of saturated vapor	$c_{p, v, sat}$	1905.9 J/(kg K)
Latent heat of vaporization	l_v	$2453.5 \times 10^{3} J / kg$

C. Numerical approach

The finite volume discretization is applied using body-fitted, hexahedral cells with non-staggered cell-centered variables, and the governing equations are formulated in Cartesian coordinates. The cell-face fluxes are calculated separately for convective fluxes and diffusive fluxes, respectively. Convective fluxes are calculated by an upwind-biased low-Mach number consistent AUSM-type (advection upstream splitting method) approximate Riemann solver. The cell-face values for density, pressure, velocities, and viscoelastic stresses are calculated by higher-order MUSCL (monotone upstream-centered schemes for conservation laws) reconstruction on a four-point stencil with the Min-Mod⁵⁴ limiter. The convective flux calculation is described in more detail as baseline finite-volume scheme in Egerer et al.⁴⁷ The diffusive flux and the source term are calculated by a second order central reconstruction. The time discretization is realized by an explicit four-step Runge–Kutta method with a modified time step criterion considering the viscoelastic transport equation. Further information on the numerical approach, including the discretization, numerical flux term approximation, and time integration, is given in Lang et al.⁴³

III. NUMERICAL SETUP

The collapse of a vapor cavity with initial radius

R_{O} = 1 \times 10^{- 4} m

surrounded by (viscoelastic) fluid is simulated. The pressure inside the bubble is initialized with the constant saturated vapor pressure p_sat, and the pressure in the surrounding is initialized with a distribution,

p (r, t = 0) = p_{\infty} - \frac{R_{0}}{r} (p_{\infty} - p_{sat}), r > R_{0}

(13)

and driving pressure of

p_{\infty} = 10 \times 10^{5} Pa

⁠. The bubble and the surrounding fluid are initially at rest. The pressure field represents the solution of the Rayleigh–Plesset equation for the Besant problem,⁵⁵ and has been used to simulate similar problems³¹ to suppress artificial pressure waves due to pressure jumps at the interface. The computational mesh is depicted in Fig. 1. The initial standoff distance h₀ represents the initial distance of the bubble center from the solid wall. We exploit two symmetry planes and hence simulate only one quarter of the collapsing vapor bubble. The bubble is located inside a refined rectangular subdomain with the size of

1.5 R_{0} \times 3.5 R_{0} \times 1.5 R_{0}

and an equally spaced grid. The grid resolution of the refined zone is

100 cells / R_{0}

⁠. A coarsened region is attached to the refined zone to dissipate outgoing waves and minimize wave reflections toward the refined zone. The computational domain has a total size of

30 R_{0} \times 30 R_{0} \times 30 R_{0}

⁠. As boundary conditions, two symmetry boundary conditions in the

x_{1} / x_{2}

- and the

x_{2} / x_{3}

-plane are applied, as well as a no-slip boundary condition in the

x_{1} / x_{3}

-plane. Outlet boundary conditions apply for the remaining planes.

FIG. 1.

$Initialization of the vapor bubble and the imposed boundary conditions. (a) Entire computational domain with refined zone and attached coarse mesh [for the refined zone in (a) only every 10th grid line is shown]; (b) isosurface ( α = 0.01) visualizing initial bubble surface and adjacent boundaries; and (c) closeup of the initial vapor volume fraction α ( − ) and computational grid in the refined region.$

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Initialization of the vapor bubble and the imposed boundary conditions. (a) Entire computational domain with refined zone and attached coarse mesh [for the refined zone in (a) only every 10th grid line is shown]; (b) isosurface (⁠ $α = 0.01$ ⁠) visualizing initial bubble surface and adjacent boundaries; and (c) closeup of the initial vapor volume fraction $α (-)$ and computational grid in the refined region.

The viscoelastic flow can be characterized by the non-dimensional Reynolds- and Deborah number. The Reynolds number relates the timescale for diffusive momentum transport and the Deborah number the relaxation timescale to the characteristic inertia-related timescales of the flow. The inertia-related timescale is defined from the characteristic velocity for the spherical Rayleigh collapse

u_{0} : = \sqrt{Δ p / ρ_{l, sat}}

⁠. For the viscoelastic bubble collapse, Re and De are defined as follows:

R e = \frac{R_{0} \sqrt{ρ_{l, sat} Δ p}}{μ_{0}}, D e = \frac{λ}{R_{0}} \sqrt{\frac{Δ p}{ρ_{l, sat}}},

(14)

where

Δ p = p_{\infty} - p_{sat}

is the pressure difference of driving and saturation pressure of vapor inside the bubble and

μ_{0} = μ_{S} + μ_{M}

is the sum of solvent and Maxwell viscosity. The non-dimensional numbers are calculated with respect to the spherical bubble collapse (Rayleigh collapse) in an infinite domain without solid wall influence. Furthermore, β describes the ratio of solvent to total viscosity,

β = \frac{μ_{S}}{μ_{0}},

(15)

which is chosen to

β = 0.1

⁠. The non-dimensional initial standoff distance

h^{*}

⁠, the non-dimensional radial distance from the center in the plane of the solid wall

r^{*}

⁠, and the non-dimensional x₂-direction

x_{2}^{*}

are defined as

h^{*} : = \frac{h_{0}}{R_{0}}, r^{*} : = {\frac{x_{1}^{2} + x_{3}^{2}}{R_{0}} |}_{x_{2} = 0}, x_{2}^{*} : = \frac{x_{2}}{R_{0}} .

IV. RESULTS AND DISCUSSION

A. General dynamics of the aspherical collapse

We simulate the vapor bubble collapse with a standoff distance of $h^{*} = 1.1$ for highly viscous flow at Re = 40. This Reynolds number is chosen since rheological effects become significant for Re < 100 (Ref. 56), but jet formation can still be expected for such a Reynolds number.³⁸ The parameters as mentioned above represent the reference for which Newtonian and viscoelastic collapse with an elasticity of De = 2 are compared. Figure 2 shows the total vapor content over time for the Newtonian opposed to the LPTT case. In Figs. 3 and 4, the collapse behavior for both fluids is illustrated qualitatively. During the initial collapse, the bubble in the Newtonian fluid collapses before the water-hammer impinges onto the opposite side of the vapor bubble. Contrarily, the LPTT fluid exhibits larger bubble deformation and elongation in wall normal direction. Also, the resulting radial water-hammer leads to an initial pressure wave already before the first collapse (cf. Fig. 6). The pressure wave emitted after the first collapse results from a superposition of the water-hammer and the collapse of the vapor cavity in the LPTT fluid as opposed to the Newtonian collapse, where only collapse is observed. Subsequently, the LPTT collapse yields larger amounts of vapor during rebound than the Newtonian fluid, where only a tiny toroidal vapor region is produced. The vapor after rebound occupies a larger coherent area in LPTT, especially during later stages of the collapse, enabling jet formation during the second collapse (cf. Fig. 5). Additionally, the vapor region in LPTT is squeezed and dragged along the solid wall after the rebound, and a splashing effect is observed. The vapor cavity in the center leads to a complex second vapor structure. In the Newtonian case, the small toroidal vapor region after rebound does not lead to jet formation during the second collapse.

FIG. 2.

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Non-dimensional vapor content in the domain over time for $R e = 40, D e = 2, h^{*} = 1.1$ with close-up of the rebound. Blue line–Newtonian fluid; black line–LPTT fluid.

FIG. 3.

$Collapse in Newtonian fluid for initial standoff distance h * = 1.1 and Re = 40. First and third column: vapor volume fraction α ( − ). Second and fourth columns: Wall normal velocity in x2-direction u 2 ( m / s ) through the x1/x2-midplane and velocity vectors scaled by the velocity magnitude. Pressure distribution p ( Pa ) through the x2/x3-midplane and at the wall. Isosurface shows constant vapor content of α = 0.01.$

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Collapse in Newtonian fluid for initial standoff distance $h^{*} = 1.1$ and Re = 40. First and third column: vapor volume fraction $α (-)$ ⁠. Second and fourth columns: Wall normal velocity in x₂-direction $u_{2} (m / s)$ through the x₁/x₂-midplane and velocity vectors scaled by the velocity magnitude. Pressure distribution $p (Pa)$ through the x₂/x₃-midplane and at the wall. Isosurface shows constant vapor content of $α = 0.01$ ⁠.

FIG. 4.

$Collapse in LPTT fluid for initial standoff distance h * = 1.1 and R e = 40 , D e = 2. First and third column: vapor volume fraction α ( − ). Second and fourth columns: Velocity in x2-direction u 2 ( m / s ) through the x1/x2-midplane and velocity vectors scaled by the velocity magnitude. Pressure distribution p ( Pa ) through the x2/x3-midplane and at the wall. Isosurface shows constant vapor content of α = 0.01.$

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Collapse in LPTT fluid for initial standoff distance $h^{*} = 1.1$ and $R e = 40, D e = 2$ ⁠. First and third column: vapor volume fraction $α (-)$ ⁠. Second and fourth columns: Velocity in x₂-direction $u_{2} (m / s)$ through the x₁/x₂-midplane and velocity vectors scaled by the velocity magnitude. Pressure distribution $p (Pa)$ through the x₂/x₃-midplane and at the wall. Isosurface shows constant vapor content of $α = 0.01$ ⁠.

FIG. 5.

Jet velocity u 2 , jet ( m / s ) in x2-direction measured at centerline over time. Blue line—Newtonian fluid; black line—LPTT fluid.

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Jet velocity $u_{2, jet} (m / s)$ in x₂-direction measured at centerline over time. Blue line—Newtonian fluid; black line—LPTT fluid.

In the following, the jet evolution during collapse and the pressure wave emission for the first collapse are investigated in detail. Figure 5 depicts the jet velocity during collapse for the Newtonian and the LPTT fluid. The jet velocity is identified by the maximum wall normal velocity magnitude along the centerline. Note that a finite jet velocity does not necessarily indicate that the flow from the top of the bubble actually pierces through the bubble deforming it to a toroid. We observe that during the first collapse, the jet velocity evolution for both cases agrees qualitatively, and that the maximum absolute jet velocities are larger for the Newtonian case compared to the LPTT fluid. For LPTT, a second jet forms during the second collapse, which cannot be found for the Newtonian case. The second jet results from the collapse of the vapor cavity formed after rebound.

In Fig. 6, the pressure distribution during the first collapse is examined in detail. For the collapse in a Newtonian fluid (upper row), no water penetration and corresponding piercing of the vapor cavity can be observed. The bubble sphericity increases during collapse, and pressure increase is observed at the upper side of the bubble due to the decelerated water column. A shock wave is emitted after the full collapse. The LPTT case (lower row) shows a radial water-hammer impinging the opposite side of the vapor cavity leading to shock wave emission before condensation and separation of the remaining vapor cavities into a toroidal upper part and a droplet-like lower part. The pressure wave emission is followed by the complete collapse of the two remaining vapor cavities. Shock wave interference results in a complex pressure field.

FIG. 6.

Pressure distribution p ( Pa ) and illustration of shock formation during first collapse through the x1/x2-midplane for R e = 40 , D e = 2 , h * = 1.1. Top: Newtonian fluid; bottom: LPTT fluid. Black isoline shows vapor content of α = 0.01.

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Pressure distribution $p (Pa)$ and illustration of shock formation during first collapse through the x₁/x₂-midplane for $R e = 40, D e = 2, h^{*} = 1.1$ ⁠. Top: Newtonian fluid; bottom: LPTT fluid. Black isoline shows vapor content of $α = 0.01$ ⁠.

In Fig. 7, we show pressure and vapor distribution in x₂-direction along the centerline for both fluids. First, it can be seen that the pressure magnitudes in the vicinity of the collapsing vapor cavities are significantly larger for the Newtonian case, which can be attributed to the more focused Newtonian collapse as compared to the LPTT case (cf. Fig. 6), where two vapor cavities collapse separately. Moreover, in the LPTT fluid, a distinct pressure rise is observable right before full condensation (⁠ $t = 3.47 \times 10^{- 6} s$ ⁠) caused by the pressure wave emitted from the radial water impingement. Subsequently, the pressure wave from water impingement and the collapse of the remaining lower vapor cavity interfere. For the Newtonian fluid, however, the pressure rises not before collapse.

FIG. 7.

$Distribution of vapor volume fraction α ( − ) (top) and pressure p ( Pa ) (bottom) along the centerline in x2-direction for R e = 40 , D e = 2 , h * = 1.1. (a) Newtonian fluid: Black line— t = 3.546 × 10 − 6 s, blue line— t = 3.547 × 10 − 6 s, red line— t = 3.548 × 10 − 6 s. (b) LPTT fluid: Black line— t = 3.466 × 10 − 6 s, blue line— t = 3.47 × 10 − 6 s, red line— t = 3.471 × 10 − 6 s.$

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Distribution of vapor volume fraction $α (-)$ (top) and pressure $p (Pa)$ (bottom) along the centerline in x₂-direction for $R e = 40, D e = 2, h^{*} = 1.1$ ⁠. (a) Newtonian fluid: Black line— $t = 3.546 \times 10^{- 6} s$ ⁠, blue line— $t = 3.547 \times 10^{- 6} s$ ⁠, red line— $t = 3.548 \times 10^{- 6} s$ ⁠. (b) LPTT fluid: Black line— $t = 3.466 \times 10^{- 6} s$ ⁠, blue line— $t = 3.47 \times 10^{- 6} s$ ⁠, red line— $t = 3.471 \times 10^{- 6} s$ ⁠.

By comparing the Newtonian and the LPTT collapse at the identical Reynolds number, we show that jet formation, or more precisely, jet piercing through the vapor bubble, is enabled by viscoelasticity for the investigated parameters. However, Lind and Phillips^38,39 assert that jet formation is suppressed by viscoelasticity, although they also found that jet formation is likewise suppressed for highly viscous fluids.^23,38 Karri et al.⁵⁷ ascertained in their experimental study for Newtonian fluid, that high viscosity can yield bubble oscillations and jet suppression. Their viscoelastic simulations are performed for the same or even lower Reynolds numbers as in the Newtonian case. For this comparison, we cannot see clear evidence that jet suppression is purely caused by viscoelasticity. For the identical or further decreased Reynolds numbers in the viscoelastic case, jet formation is expectedly suppressed, but it is not evident whether the mitigating effect is unambiguously caused by viscoelasticity or if viscous and viscoelastic stresses similarly can inhibit jet formation. Moreover, in Lind and Phillips,³⁸ a parameter study for the variation of Reynolds and Deborah number, associated with the variation of viscous and elastic influence, was conducted and determined that for given Reynolds number increasing elasticity in a viscoelastic fluid can lead to jet formation in the first place.

Figure 8 compares the pressure evolution at the position of maximum wall pressure (for $h^{*} = 1.1$ at the focus point $r^{*} = 0$ ⁠) and at the maximum pressure position along the centerline for both fluids. The maximum pressures at the wall during first collapse are comparable. Opposed to the pressures at the wall, the pressure along the centerline during the first collapse is larger for the Newtonian case. Furthermore, the maximum pressure from the first collapse occurs at a larger distance from the wall for the Newtonian collapse. The overall pressure evolutions for the first collapse are qualitatively similar. A substantial pressure rise caused by the collapse (Newtonian case) or the superposition of the pressure waves from the water-hammer and collapse (LPTT case), respectively, is immediately followed by the jet impinging the rigid wall. Gonzalez-Avila et al.⁵⁸ observed a similar pressure evolution at the wall for the same initial standoff distance and gas-filled bubbles. The subsequent pressure evolution shows an additional pressure peak related to the second collapse after the rebound. The second pressure peak is much stronger for the viscoelastic fluid, caused by the larger vapor cavity formed during rebound. Moreover, the second collapse for LPTT is delayed compared to the Newtonian case due to the larger vapor cavity after the rebound and the increased total stresses. Between the first and second collapse, the pressure at the wall decreases to vapor pressure in the LPTT case, since the vapor region is pushed to the center. In the Newtonian fluid, vapor is not observable at the wall before the second collapse, and hence the pressure drop between the two collapses is not as pronounced as in LPTT.

FIG. 8.

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Pressure $p (Pa)$ over time for $R e = 40, D e = 2, h^{*} = 1.1$ (logarithmic scale). Top: at the wall (⁠ $x_{2}^{*} = 0$ ⁠) at the location of maximum pressure: (a) Newtonian fluid at $r^{*} = 0.02$ and (b) LPTT fluid at $r^{*} = 0$ ⁠. Bottom: Along the centerline at the position of maximum occurring pressure for (a) Newtonian fluid at $x_{2}^{*} = 0.22$ ⁠, and (b) LPTT fluid at $x_{2}^{*} = 0.16$ ⁠.

The differences in the collapse dynamics and shock wave formation are consequence of the different constitutive relations. In the following, different Newtonian and LPTT stress components are compared. Figures 9 and 10 show τ₁₁, the component which differs most distinctly when comparing the aspherical collapse of Newtonian and viscoelastic fluid.³⁹ An additional τ₁₁-stress-layer can be observed for the collapse in LPTT, a unique feature of viscoelastic fluids also observed by Lind and Phillips,³⁹ which is caused by normal stress effects. These positive stresses accelerate fluid along the wall away from the center and lead to splashing (⁠ $t = 5.8 \times 10^{- 6} s$ ⁠) in the later instants of the collapse. By comparing solvent (⁠ $τ_{S, 11}$ ⁠) and viscoelastic (⁠ $τ_{M, 11}$ ⁠) stresses in general, we can observe that solvent stresses exhibit more oscillations as compared to the smoother viscoelastic stresses. This behavior can be explained by the delayed viscoelastic stress build-up, whereas solvent stresses are directly related to the occurring deformation rate. Furthermore, solvent stresses in the Newtonian case are higher than in the LPTT case since the total stress is divided into additional viscoelastic stresses, and the solvent stresses themselves. Additionally, it is observable that in the LPTT case, large viscoelastic stresses occur at the two upper tips of the bubble and a narrow region of negative stresses at the centerline (⁠ $t = 3.46 \times 10^{- 6} s$ ⁠).

FIG. 9.

Solvent stress τ S , 11 ( Pa ) through x2/x3-midplane for R e = 40 , h * = 1.1 in Newtonian fluid. Black isoline shows vapor content of α = 0.01.

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Solvent stress $τ_{S, 11} (Pa)$ through x₂/x₃-midplane for $R e = 40, h^{*} = 1.1$ in Newtonian fluid. Black isoline shows vapor content of $α = 0.01$ ⁠.

FIG. 10.

Stress through x2/x3-midplane for R e = 40 , D e = 2 , h * = 1.1 in LPTT fluid. Black isoline shows constant vapor content of α = 0.01. Top: solvent stress τ S , 11 ( Pa ); bottom: viscoelastic stress τ M , 11 ( Pa ).

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Stress through x₂/x₃-midplane for $R e = 40, D e = 2, h^{*} = 1.1$ in LPTT fluid. Black isoline shows constant vapor content of $α = 0.01$ ⁠. Top: solvent stress $τ_{S, 11} (Pa)$ ⁠; bottom: viscoelastic stress $τ_{M, 11} (Pa)$ ⁠.

For the shear stress component τ₁₂ in Figs. 11 and 12, significant negative viscoelastic stresses occur in the lower part and large positive stresses in the upper part of the tips in LPTT case. Positive viscoelastic stresses cover a much larger region than the solvent stresses.

FIG. 11.

Solvent stress τ S , 12 ( Pa ) through x2/x3-midplane for R e = 40 , h * = 1.1 in Newtonian fluid. Black isoline shows vapor content of α = 0.01.

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Solvent stress $τ_{S, 12} (Pa)$ through x₂/x₃-midplane for $R e = 40, h^{*} = 1.1$ in Newtonian fluid. Black isoline shows vapor content of $α = 0.01$ ⁠.

FIG. 12.

Stress through x2/x3-midplane for R e = 40 , D e = 2 , h * = 1.1 in LPTT fluid. Black isoline shows constant vapor content of α = 0.01. Top: solvent stress τ S , 12 ( Pa ); bottom: viscoelastic stress τ M , 12 ( Pa ).

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Stress through x₂/x₃-midplane for $R e = 40, D e = 2, h^{*} = 1.1$ in LPTT fluid. Black isoline shows constant vapor content of $α = 0.01$ ⁠. Top: solvent stress $τ_{S, 12} (Pa)$ ⁠; bottom: viscoelastic stress $τ_{M, 12} (Pa)$ ⁠.

The τ₂₂ component, visualized in Figs. 13 and 14, likewise shows more delayed evolution and smoother distribution of viscoelastic stresses as compared to solvent stresses. The viscoelastic stress component $τ_{M, 22}$ shows a large region of positive stresses at the centerline. This component represents the normal stress in x₂-direction, pulling fluid in this direction, and is responsible for the re-evaporation of vapor in this region (⁠ $t = 5.8 \times 10^{- 6} s$ ⁠).

FIG. 13.

Solvent stress τ S , 22 ( Pa ) through x2/x3-midplane for R e = 40 , h * = 1.1 in Newtonian fluid. Black isoline shows vapor content of α = 0.01.

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Solvent stress $τ_{S, 22} (Pa)$ through x₂/x₃-midplane for $R e = 40, h^{*} = 1.1$ in Newtonian fluid. Black isoline shows vapor content of $α = 0.01$ ⁠.

FIG. 14.

Stress through x2/x3-midplane for R e = 40 , D e = 2 , h * = 1.1 in LPTT fluid. Black isoline shows constant vapor content of α = 0.01. Top: solvent stress τ S , 22 ( Pa ); bottom: viscoelastic stress τ M , 22 ( Pa ).

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Stress through x₂/x₃-midplane for $R e = 40, D e = 2, h^{*} = 1.1$ in LPTT fluid. Black isoline shows constant vapor content of $α = 0.01$ ⁠. Top: solvent stress $τ_{S, 22} (Pa)$ ⁠; bottom: viscoelastic stress $τ_{M, 22} (Pa)$ ⁠.

B. Variation of initial standoff distance

The following simulations examine the variation of initial standoff distances (⁠ $h^{*} = {0.35, 1.5}$ ⁠). Figure 15 shows the total vapor content for the modified initial distances compared to the reference case with $h^{*} = 1.1$ while Re and De remain unchanged. For the Newtonian fluid, collapse time decreases for both standoff distances $h^{*} = 0.35$ and $h^{*} = 1.5$ ⁠, with the shortest collapse time for $h^{*} = 1.5$ ⁠. Furthermore, the largest vapor cavity formed during rebound is observed for $h^{*} = 1.1$ ⁠. The general behavior for LPTT is similar. The largest cavity during rebound is formed for $h^{*} = 1.1$ ⁠, the smallest for $h^{*} = 0.35$ ⁠, although the formed cavities are conspicuously larger as for the Newtonian case. The shortest collapse time in LPTT is observed for $h^{*} = 1.5$ ⁠, and the longest time can be appreciated for $h^{*} = 1.1$ ⁠.

FIG. 15.

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Non-dimensional vapor content in the domain over time for $R e = 40, D e = 2$ and different initial standoff distances with close-up of the rebound: (a) Newtonian fluid: blue line— $h^{*} = 0.35$ ⁠, black line— $h^{*} = 1.1$ ⁠, red line— $h^{*} = 1.5$ ⁠; (b) LPTT fluid: blue line— $h^{*} = 0.35$ ⁠, black line— $h^{*} = 1.1$ ⁠, red line— $h^{*} = 1.5$ ⁠.

Jet velocities, calculated as described in Sec. IV A, are illustrated in Fig. 16. For $h^{*} = 0.35$ ⁠, the Newtonian and the LPTT show an oscillatory behavior of the jet, with a negative and ensuing positive jet velocity. The jet velocities are smallest for an initial distance of $h^{*} = 0.35$ for Newtonian and viscoelastic fluid compared to the remaining standoff distances. The largest absolute jet velocities are produced for $h^{*} = 1.1$ for both fluids. Comparing Newtonian to viscoelastic fluid, the jet velocities for LPTT are larger than for the Newtonian fluid for $h^{*} = 0.35$ ⁠. For $h^{*} = 1.1, h^{*} = 1.5$ ⁠, the jet around the first collapse is faster for the Newtonian fluid. For $h^{*} = 1.1$ and the increased standoff distance $h^{*} = 1.5$ ⁠, the jet velocity exhibits a second jet formation only for the LPTT fluid.

FIG. 16.

Jet velocity u 2 , jet ( m / s ) in x2-direction over time measured at centerline for R e = 40 , D e = 2 and different initial standoff distances: (a) Newtonian fluid: blue line— h * = 0.35, black line— h * = 1.1, red line— h * = 1.5; (b) LPTT fluid: blue line— h * = 0.35, black line— h * = 1.1, red line— h * = 1.5.

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Jet velocity $u_{2, jet} (m / s)$ in x₂-direction over time measured at centerline for $R e = 40, D e = 2$ and different initial standoff distances: (a) Newtonian fluid: blue line— $h^{*} = 0.35$ ⁠, black line— $h^{*} = 1.1$ ⁠, red line— $h^{*} = 1.5$ ⁠; (b) LPTT fluid: blue line— $h^{*} = 0.35$ ⁠, black line— $h^{*} = 1.1$ ⁠, red line— $h^{*} = 1.5$ ⁠.

In the following, the collapse for decreased initial standoff distance $h^{*} = 0.35$ is examined in detail. For all subsequent variations, the evolution of the collapse is only shown for the relevant stages starting right before the first collapse. Figures 17 and 18 illustrate the collapse dynamics for selected time instants for Newtonian and LPTT fluid. We observe that the bubble is more elongated in x₂-direction during the first collapse in the viscoelastic case (Newtonian: $t = 3.41 \times 10^{- 6} s$ ⁠, LPTT: $t = 3.26 \times 10^{- 6} s$ ⁠). Furthermore, while the jet penetrates the bubble and impinges the wall, more fluid is pushed below the vapor cavity in LPTT (Newtonian: $t = 3.44 \times 10^{- 6} s$ ⁠, LPTT: $t = 3.3 \times 10^{- 6} s$ ⁠). After the first collapse, a toroidal vapor cavity forms in the viscoelastic case, which is not observed for the Newtonian fluid. This vapor torus in LPTT yields a second collapse, less intense than the first.

FIG. 17.

$Collapse in Newtonian fluid for initial standoff distance h * = 0.35 and Re = 40. First and third column: vapor volume fraction α ( − ). Second and fourth columns: Velocity in x2-direction u 2 ( m / s ) through the x1/x2-midplane and velocity vectors scaled by the velocity magnitude. Pressure distribution p ( Pa ) through the x2/x3-midplane and at the wall. Isosurface shows constant vapor content of α = 0.01.$

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Collapse in Newtonian fluid for initial standoff distance $h^{*} = 0.35$ and Re = 40. First and third column: vapor volume fraction $α (-)$ ⁠. Second and fourth columns: Velocity in x₂-direction $u_{2} (m / s)$ through the x₁/x₂-midplane and velocity vectors scaled by the velocity magnitude. Pressure distribution $p (Pa)$ through the x₂/x₃-midplane and at the wall. Isosurface shows constant vapor content of $α = 0.01$ ⁠.

FIG. 18.

$Collapse in LPTT fluid for initial standoff distance h * = 0.35 and R e = 40 , D e = 2. First and third column: vapor volume fraction α ( − ). Second and fourth columns: Velocity in x2-direction u 2 ( m / s ) through the x1/x2-midplane and velocity vectors scaled by the velocity magnitude. Pressure distribution p ( Pa ) through the x2/x3-midplane and at the wall. Isosurface shows constant vapor content of α = 0.01.$

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Collapse in LPTT fluid for initial standoff distance $h^{*} = 0.35$ and $R e = 40, D e = 2$ ⁠. First and third column: vapor volume fraction $α (-)$ ⁠. Second and fourth columns: Velocity in x₂-direction $u_{2} (m / s)$ through the x₁/x₂-midplane and velocity vectors scaled by the velocity magnitude. Pressure distribution $p (Pa)$ through the x₂/x₃-midplane and at the wall. Isosurface shows constant vapor content of $α = 0.01$ ⁠.

Figure 19 compares the shock wave inception in the Newtonian and the viscoelastic bubble collapse for reduced standoff distance. The initial time steps look similar, despite that the vapor cavity is more elongated in wall normal direction for the viscoelastic fluid. During jet impingement on the solid wall, the liquid jet penetrates below the vapor cavity for the LPTT case. Due to the absence of liquid penetrating below the vapor cavity in the Newtonian case, a narrower and more focused jet is observed. The subsequent high-pressure region exhibits higher pressures for the Newtonian collapse.

FIG. 19.

Pressure distribution p ( Pa ) and illustration of shock formation during first collapse through the x1/x2-midplane for R e = 40 , D e = 2 , h * = 0.35. Top: Newtonian fluid; bottom: LPTT fluid. Black isoline shows vapor content of α = 0.01.

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Pressure distribution $p (Pa)$ and illustration of shock formation during first collapse through the x₁/x₂-midplane for $R e = 40, D e = 2, h^{*} = 0.35$ ⁠. Top: Newtonian fluid; bottom: LPTT fluid. Black isoline shows vapor content of $α = 0.01$ ⁠.

Figure 20 shows the pressure evolution at the position of maximum occurring pressure at the wall and along the centerline, respectively. The reference case (⁠ $h^{*} = 1.1$ ⁠) is shown for comparison. For the Newtonian case, the maximum pressure at the wall does not occur at the center. We also show the pressure evolution at the center to compare the pressures induced by jet impingement. We observe that the maximum pressures at the wall are larger for decreased standoff distance compared to the reference case for both fluids. Pressure maxima at the wall are not caused by jet impingement but by the subsequent collapse for both fluids. The pressure curves at the wall center position reveal a pre-compression due to jet impingement, which is followed by a higher pressure peak originating from the actual collapse. Furthermore, an additional prominent pressure peak induced by the second collapse (⁠ $t \approx 3.9 \times 10^{- 6} s$ ⁠) of the re-evaporation vapor can be observed for the LPTT fluid. By looking at the positions of maximum pressure along the centerline, we observe that the maximum pressures in the field are smaller than for the reference case for both fluids.

FIG. 20.

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Pressure $p (Pa)$ for reduced initial standoff distance $h^{*} = 0.35$ vs reference case $h^{*} = 1.1$ at $R e = 40, D e = 2$ (logarithmic scale). Top: at the wall (⁠ $x_{2}^{*} = 0$ ⁠): (a) Newtonian fluid: blue line—at $r^{*} = 0.25$ (location of maximum pressure), light blue line— $r^{*} = 0$ ⁠, black line—reference case (⁠ $h^{*} = 1.1$ ⁠) at $r^{*} = 0.02$ ⁠; (b) LPTT: blue line—at $r^{*} = 0$ ⁠, black line—reference case (⁠ $h^{*} = 1.1$ ⁠) at $r^{*} = 0$ ⁠. Bottom: Along the centerline at the position of maximum pressure: (a) Newtonian fluid: blue line—at $x_{2}^{*} = 0$ ⁠, black line—reference case (⁠ $h^{*} = 1.1$ ⁠) at $x_{2}^{*} = 0.22$ ⁠; (b) LPTT: blue line—LPTT fluid at $x_{2}^{*} = 0.06$ ⁠, black line—reference case (⁠ $h^{*} = 1.1$ ⁠) at $x_{2}^{*} = 0.16$ ⁠.

To explain the different collapse dynamics in Newtonian and viscoelastic fluid, stress evolution is examined in more detail. In the following, we only show the development of the most relevant stress components τ₁₁ in Figs. 21 and 22. For the interested reader, the remaining components are included in Appendix A. We again observe a distinct viscoelastic stress layer $τ_{M, 11}$ in the LPTT fluid right above the solid wall, pulling vapor in opposite x₁-directions and leading to the larger vapor cavity after the rebound than for the Newtonian case.

FIG. 21.

Solvent stress τ S , 11 ( Pa ) through x2/x3-midplane for R e = 40 , h * = 0.35 in Newtonian fluid. Black isoline shows vapor content of α = 0.01.

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Solvent stress $τ_{S, 11} (Pa)$ through x₂/x₃-midplane for $R e = 40, h^{*} = 0.35$ in Newtonian fluid. Black isoline shows vapor content of $α = 0.01$ ⁠.

FIG. 22.

Stress through x2/x3-midplane for R e = 40 , D e = 2 , h * = 0.35 in LPTT fluid. Black isoline shows constant vapor content of α = 0.01. Top: solvent stress τ S , 11 ( Pa ); bottom: viscoelastic stress τ M , 11 ( Pa ).

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Stress through x₂/x₃-midplane for $R e = 40, D e = 2, h^{*} = 0.35$ in LPTT fluid. Black isoline shows constant vapor content of $α = 0.01$ ⁠. Top: solvent stress $τ_{S, 11} (Pa)$ ⁠; bottom: viscoelastic stress $τ_{M, 11} (Pa)$ ⁠.

Next, the collapse dynamics for the increased initial distance $h^{*} = 1.5$ is investigated. Figures 23 and 24 show the evolution of the collapse in Newtonian and LPTT fluid. In the Newtonian fluid, the bubble entirely collapses before the liquid jet pierces through the bubble. After a violent collapse, a toroidal vapor cavity re-evaporates before the second collapse. In the viscoelastic fluid, the liquid jet vertically pierces through the bubble before the collapse. During rebound, a large heart-shaped vapor cavity is formed (⁠ $t = 4 \times 10^{- 6} s$ ⁠). Due to the larger vapor cavity, the second collapse is more violent than in the Newtonian fluid.

FIG. 23.

$Collapse in Newtonian fluid for initial standoff distance h * = 1.5 and Re = 40. First and third column: vapor volume fraction α ( − ). Second and fourth columns: Velocity in x2-direction u 2 ( m / s ) through the x1/x2-midplane and velocity vectors scaled by the velocity magnitude. Pressure distribution p ( Pa ) through the x2/x3-midplane and at the wall. Isosurface shows constant vapor content of α = 0.01.$

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Collapse in Newtonian fluid for initial standoff distance $h^{*} = 1.5$ and Re = 40. First and third column: vapor volume fraction $α (-)$ ⁠. Second and fourth columns: Velocity in x₂-direction $u_{2} (m / s)$ through the x₁/x₂-midplane and velocity vectors scaled by the velocity magnitude. Pressure distribution $p (Pa)$ through the x₂/x₃-midplane and at the wall. Isosurface shows constant vapor content of $α = 0.01$ ⁠.

FIG. 24.

$Collapse in LPTT fluid for initial standoff distance h * = 1.5 and R e = 40 , D e = 2. First and third column: vapor volume fraction α ( − ). Second and fourth columns: Velocity in x2-direction u 2 ( m / s ) through the x1/x2-midplane and velocity vectors scaled by the velocity magnitude. Pressure distribution p ( Pa ) through the x2/x3-midplane and at the wall. Isosurface shows constant vapor content of α = 0.01.$

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Collapse in LPTT fluid for initial standoff distance $h^{*} = 1.5$ and $R e = 40, D e = 2$ ⁠. First and third column: vapor volume fraction $α (-)$ ⁠. Second and fourth columns: Velocity in x₂-direction $u_{2} (m / s)$ through the x₁/x₂-midplane and velocity vectors scaled by the velocity magnitude. Pressure distribution $p (Pa)$ through the x₂/x₃-midplane and at the wall. Isosurface shows constant vapor content of $α = 0.01$ ⁠.

Figure 25 compares the pressure wave formation during the first collapse between Newtonian and LPTT fluid. As mentioned, we can only observe jet piercing in LPTT fluid. The fundamental collapse is more focused for the Newtonian fluid since there is no separation of high-pressure regions due to piercing.

FIG. 25.

Pressure distribution p ( Pa ) and illustration of shock formation during first collapse through the x1/x2-midplane for R e = 40 , D e = 2 , h * = 1.5. Top: Newtonian fluid; bottom: LPTT fluid. Black isoline shows vapor content of α = 0.01.

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Pressure distribution $p (Pa)$ and illustration of shock formation during first collapse through the x₁/x₂-midplane for $R e = 40, D e = 2, h^{*} = 1.5$ ⁠. Top: Newtonian fluid; bottom: LPTT fluid. Black isoline shows vapor content of $α = 0.01$ ⁠.

The pressure evolutions for increased standoff distance are shown in Fig. 26 and compared to the reference case. The top row of Fig. 26 shows the pressure evolution over time at the location where the maximum wall pressure occurs. For both the Newtonian and the LPTT case, the pressures are smaller than in the reference case. Yang et al.²⁰ also observed decreasing wall pressures for increasing standoff distances. The pressure peak during the first collapse is slightly higher for the LPTT fluid. The second collapse leads to much stronger pressure impact at the wall in LPTT as compared to the Newtonian case. We attribute this observation to the larger re-evaporated cavity during rebound in LPTT. Furthermore, a pressure rise around $t \approx 4.5 \times 10^{- 6} s$ is present in LPTT, which is caused by jet impact (cf. Fig. 4) and cannot be detected in the Newtonian case. The pressures at the position of maximum pressure along the centerline are illustrated in the lower part of Fig. 26. The peak pressures are again smaller than in the reference case for both fluids. The maximum pressure in the LPTT is associated with the second collapse of the re-evaporated vapor. Additionally, the pressure evolution for the position of the maximum pressure during the first collapse is shown. The maximum pressure during the first collapse is larger in the Newtonian case due to the more focused collapse (cf. Fig. 25). However, the second collapse in LPTT is distinctly stronger than for the Newtonian fluid due to the mentioned extensive re-evaporation area.

FIG. 26.

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Pressure $p (Pa)$ for increased initial standoff distance $h^{*} = 1.5$ vs reference case $h^{*} = 1.1$ at $R e = 40, D e = 2$ (logarithmic scale). Top: at the wall (⁠ $x_{2}^{*} = 0$ ⁠) at the location of maximum pressure: (a) Newtonian fluid: red line—at $r^{*} = 0.02$ ⁠, black line—reference case (⁠ $h^{*} = 1.1$ ⁠) at $r^{*} = 0.02$ ⁠; (b) LPTT: red line— $r^{*} = 0$ ⁠, black line—reference case (⁠ $h^{*} = 1.1$ ⁠) at $r^{*} = 0$ ⁠. Bottom: Along the centerline at the position of maximum pressure: (a) Newtonian fluid: red line—at $x_{2}^{*} = 1.13$ ⁠, black line—reference case (⁠ $h^{*} = 1.1$ ⁠)—at $x_{2}^{*} = 0.22$ ⁠; (b) LPTT: red line at $x_{2}^{*} = 0.29$ (location of maximum pressure), light blue line—at $x_{2}^{*} = 1.19$ ⁠, black line—reference—case (⁠ $h^{*} = 1.1$ ⁠) at $x_{2}^{*} = 0.16$ ⁠.

Subsequently, we examine the relevant stress components. τ₁₁ is illustrated in Figs. 27 and 28. A large region of positive $τ_{M, 11}$ stresses formed along the solid wall can be seen for the viscoelastic stress. These stresses are responsible for the cusped concave shape (⁠ $t = 4.6 \times 10^{- 6} s$ ⁠) of the bubble during the collapse, an effect also described in Lind and Phillips.^39,59 The viscoelastic stresses along the solid wall pull fluid radially away, creating suction that accelerates fluid from the bulk of the bubble. Furthermore, the Newtonian case exhibits larger solvent stresses than the LPTT fluid, again due to the additional viscoelastic stresses emerging in the LPTT case.

FIG. 27.

Solvent stress τ S , 11 ( Pa ) through the x2/x3-midplane for different time instants during the collapse for Re = 40 and initial standoff distance h * = 1.5 in Newtonian fluid. Black isoline shows constant vapor content of α = 0.01.

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Solvent stress $τ_{S, 11} (Pa)$ through the x₂/x₃-midplane for different time instants during the collapse for Re = 40 and initial standoff distance $h^{*} = 1.5$ in Newtonian fluid. Black isoline shows constant vapor content of $α = 0.01$ ⁠.

FIG. 28.

Stress through x2/x3-midplane for R e = 40 , D e = 2 , h * = 1.5 in LPTT fluid. Black isoline shows constant vapor content of α = 0.01. Top: solvent stress τ S , 11 ( Pa ); bottom: viscoelastic stress τ M , 11 ( Pa ).

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Stress through x₂/x₃-midplane for $R e = 40, D e = 2, h^{*} = 1.5$ in LPTT fluid. Black isoline shows constant vapor content of $α = 0.01$ ⁠. Top: solvent stress $τ_{S, 11} (Pa)$ ⁠; bottom: viscoelastic stress $τ_{M, 11} (Pa)$ ⁠.

By looking at the τ₂₂ stress components in Figs. 29 and 30, it can be appreciated that significant viscoelastic stresses $τ_{M, 22}$ covering two lengthy regions above the vapor cavity during rebound for the LPTT case (⁠ $t = 3.4 \times 10^{- 6} s, t = 4.6 \times 10^{- 6} s$ ⁠). These stresses yield positive forces in x₂-direction, resulting in the observed heart-shaped vapor cavity before the second collapse. For completeness, the remaining stress distributions are included in Appendix B.

FIG. 29.

Solvent stress τ S , 22 ( Pa ) through the x2/x3-midplane for different time instants during the collapse for Re = 40 and initial standoff distance h * = 1.5 in Newtonian fluid. Black isoline shows constant vapor content of α = 0.01.

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Solvent stress $τ_{S, 22} (Pa)$ through the x₂/x₃-midplane for different time instants during the collapse for Re = 40 and initial standoff distance $h^{*} = 1.5$ in Newtonian fluid. Black isoline shows constant vapor content of $α = 0.01$ ⁠.

FIG. 30.

Stress through x2/x3-midplane for R e = 40 , D e = 2 , h * = 1.5 in LPTT fluid. Black isoline shows constant vapor content of α = 0.01. Top: solvent stress τ S , 22 ( Pa ); bottom: viscoelastic stress τ M , 22 ( Pa ).

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Stress through x₂/x₃-midplane for $R e = 40, D e = 2, h^{*} = 1.5$ in LPTT fluid. Black isoline shows constant vapor content of $α = 0.01$ ⁠. Top: solvent stress $τ_{S, 22} (Pa)$ ⁠; bottom: viscoelastic stress $τ_{M, 22} (Pa)$ ⁠.

C. Variation of elasticity

The following section investigates the influence of elasticity, respectively, the variation of relaxation time, for the LPTT case. The Reynolds number Re = 40 remains unchanged. Figure 31 shows the vapor content over time for the three considered Deborah numbers $D e = {1, 2, 4}$ ⁠. For all cases, we observe a distinct rebound after the initial complete collapse. The largest vapor content during re-evaporation is observed for the largest elasticity De = 4, whereas the smallest vapor content is produced for the smallest elasticity De = 1. A comparison of the jet velocities in Fig. 32 reveals that the largest jet velocity magnitudes occur for the smallest elasticity De = 1 during the first and second collapse. The jet velocities are smallest for the largest elasticity De = 4. In Fig. 33, the collapse behavior for decreased elasticity (De = 1) is depicted. It appears generally similar to the reference case, including splashing. The collapse appears to be more focused with decreasing elasticity. Figure 34 shows the formation of the emitted shock waves during the first collapse. The overall phenomenology is comparable to the reference case. Radial impingement of the water-hammer is followed by the collapse of the remaining vapor cavities, and the pressure waves of the water-hammer and collapse (complete condensation) overlap.

FIG. 31.

View large Download slide

Non-dimensional vapor content in the domain over time for $R e = 40, h^{*} = 1.1$ and different relaxation times in LPTT fluid; close-up of the rebound. Blue line—De = 1, black line—De = 2, red line—De = 4.

FIG. 32.

Jet velocity u 2 , jet ( m / s ) in x2-direction over time measured at centerline for R e = 40 , h * = 1.1 and different relaxation times in LPTT fluid. Blue line—De = 1, black line—De = 2, red line—De = 4.

View large Download slide

Jet velocity $u_{2, jet} (m / s)$ in x₂-direction over time measured at centerline for $R e = 40, h^{*} = 1.1$ and different relaxation times in LPTT fluid. Blue line—De = 1, black line—De = 2, red line—De = 4.

FIG. 33.

$Collapse in LPTT fluid for R e = 40 , D e = 1 , h * = 1.1 (increased relaxation time). First and third column: vapor volume fraction α ( − ). Second and fourth columns: Velocity in x2-direction u 2 ( m / s ) through the x1/x2-midplane and velocity vectors scaled by the velocity magnitude. Pressure distribution p ( Pa ) through the x2/x3-midplane and at the wall. Isosurface shows constant vapor content of α = 0.01.$

View large Download slide

Collapse in LPTT fluid for $R e = 40, D e = 1, h^{*} = 1.1$ (increased relaxation time). First and third column: vapor volume fraction $α (-)$ ⁠. Second and fourth columns: Velocity in x₂-direction $u_{2} (m / s)$ through the x₁/x₂-midplane and velocity vectors scaled by the velocity magnitude. Pressure distribution $p (Pa)$ through the x₂/x₃-midplane and at the wall. Isosurface shows constant vapor content of $α = 0.01$ ⁠.

FIG. 34.

Pressure distribution p ( Pa ) and illustration of shock formation during first collapse through the x1/x2-midplane for R e = 40 , D e = 1 , h * = 1.1 in LPTT fluid. Black isoline shows vapor content of α = 0.01.

View large Download slide

Pressure distribution $p (Pa)$ and illustration of shock formation during first collapse through the x₁/x₂-midplane for $R e = 40, D e = 1, h^{*} = 1.1$ in LPTT fluid. Black isoline shows vapor content of $α = 0.01$ ⁠.

For the pressure evolutions at the position of maximum pressure at the wall (top of Fig. 35) and at the centerline (bottom of Fig. 35), we can observe that the maximum emitted pressure at the wall is marginally increased for De = 1 as compared to the reference case. The pressure at the wall due to the second collapse is larger for the reference case, which can be associated with the increased water cavity produced during rebound. For the location of the maximum pressure along the centerline, the first collapse of the reference case is more intense, while the pressure generated by the second collapse is almost similar. All stress distributions for the case with decreased and increased elasticity can be found in Appendixes C and D.

FIG. 35.

View large Download slide

Pressure $p (Pa)$ for decreased elasticity De = 1 vs reference case De = 2 at $R e = 40, h^{*} = 1.1$ (logarithmic scale) in LPTT fluid. Top: at the wall (⁠ $x_{2}^{*} = 0$ ⁠) at the location of maximum pressure: Blue line—De = 1 at $r^{*} = 0.05$ and black line—the reference case (De = 2) at $r^{*} = 0$ ⁠. Bottom: Along the centerline at the position of maximum pressure blue line—De = 1 at $x_{2}^{*} = 0.195$ and black line—the reference case at $x_{2}^{*} = 0.155$ ⁠.

Figure 36 depicts different stages of the collapse for increased relaxation (De = 4). It is observed that more vapor is produced during rebound than for the other elasticity. However, the qualitative collapse evolution is similar for all cases with identical initial standoff distance $h^{*} = 1.1$ ⁠.

FIG. 36.

$Collapse in LPTT fluid for R e = 40 , D e = 4 , h * = 1.1 (increased relaxation time). First and third column: vapor volume fraction α ( − ). Second and fourth columns: Velocity in x2-direction u 2 ( m / s ) through the x1/x2-midplane and velocity vectors scaled by the velocity magnitude. Pressure distribution p ( Pa ) through the x2/x3-midplane and at the wall. Isosurface shows constant vapor content of α = 0.01.$

View large Download slide

Collapse in LPTT fluid for $R e = 40, D e = 4, h^{*} = 1.1$ (increased relaxation time). First and third column: vapor volume fraction $α (-)$ ⁠. Second and fourth columns: Velocity in x₂-direction $u_{2} (m / s)$ through the x₁/x₂-midplane and velocity vectors scaled by the velocity magnitude. Pressure distribution $p (Pa)$ through the x₂/x₃-midplane and at the wall. Isosurface shows constant vapor content of $α = 0.01$ ⁠.

The pressure distribution over time depicted in Fig. 37 shows a radial impingement of the water-hammer toward the other side of the vapor cavity, followed by the complete collapse of the bubble. The pressures over time at the locations of maximum occurring pressures in Fig. 38 reveal that the pressure at the wall is smaller for the first and second collapse compared to the reference. At the location of maximum pressure along the centerline, the pressure of the first collapse is also smaller for increased elasticity. The pressure during the second collapse is of comparable magnitude for the increased and the reference elasticity.

FIG. 37.

Pressure distribution p ( Pa ) and illustration of shock formation during first collapse through the x1/x2-midplane for R e = 40 , D e = 4 , h * = 1.1 in LPTT fluid. Black isoline shows vapor content of α = 0.01.

View large Download slide

Pressure distribution $p (Pa)$ and illustration of shock formation during first collapse through the x₁/x₂-midplane for $R e = 40, D e = 4, h^{*} = 1.1$ in LPTT fluid. Black isoline shows vapor content of $α = 0.01$ ⁠.

FIG. 38.

View large Download slide

Pressure $p (Pa)$ for decreased elasticity De = 4 vs reference case De = 2 at $R e = 40, h^{*} = 1.1$ (logarithmic scale) in LPTT fluid. (a) At the wall (⁠ $x_{2}^{*} = 0$ ⁠) at the location of maximum pressure: red line—De = 4 at $r^{*} = 0.035$ and black line—the reference case (De = 2) at $r^{*} = 0$ ⁠. (b) Along the centerline at the position of maximum pressure: red line—De = 4 at $x_{2}^{*} = 0.175$ and black line—the reference case (De = 2) at $x_{2}^{*} = 0.155$ ⁠.

V. CONCLUSION

Viscoelastic vapor bubble collapse near solid walls is investigated conducting fully compressible 3D simulations using the linear Phan-Thien Tanner viscoelastic model. The following novelties apply to the investigation:

Usage of viscoelastic constitutive model including solvent contribution and shear-thinning behavior (LPTT).
Application of a fully 3D-approach.
Fully compressible method enables resolution of shock wave formation and of emitted pressure waves.
Simulation of a cavitating vapor bubble considering condensation and evaporation in contrast to gas-filled bubbles neglecting condensation and evaporation.^38–40

The near-wall collapse for different initial wall-distances by comparing the results of viscoelastic and Newtonian fluid reveals that viscoelasticity can strongly affect collapse dynamics. For wall-detached bubbles, viscoelasticity changes the shock wave formation mechanism by introducing liquid-jet-piercing, which cannot be observed for Newtonian collapse. Furthermore, viscoelasticity significantly increases the amount of re-evaporated fluid during the rebound for the considered standoff distances. While the pressure emitted during the first collapse is larger or comparable for the Newtonian fluid, the pressure peak during the second collapse is considerably larger in the viscoelastic liquid for all cases. Furthermore, the re-evaporated fluid generates a second jet only for the viscoelastic collapse. For initially wall-attached bubbles, re-evaporation and subsequent second collapse is only observed for the viscoelastic fluid. The different collapse dynamics are caused by the additional viscoelastic stresses. Analyzing solvent and viscoelastic stress distributions shows a smoother distribution of the viscoelastic compared to the solvent stresses due to relaxation. We observe that viscoelastic stresses influence the amount of re-evaporation during rebound. The variation of elasticity shows that the vapor produced during rebound is correlated with the relaxation time. Increasing the relaxation time yields significantly more vapor, at least for the range of investigated parameters. The jet velocity is largest for the lowest elasticity.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the Leibniz Supercomputing Centre for funding this project by providing computing time and support on its Linux-Cluster. The first author is member of the Technical University of Munich (TUM) Graduate School.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Christian Lang: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Stefan Adami: Conceptualization (supporting); Formal analysis (supporting); Investigation (supporting); Supervision (equal); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (equal). Nikolaus A. Adams: Funding acquisition (lead); Supervision (lead); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

APPENDIX A: VISCOUS AND VISCOELASTIC STRESS DISTRIBUTIONS FOR Re = 40, De = 2, $h^{*} = 0.35$

The following figures show the stress distributions for an initial standoff distance h^* = 0.35 and Re = 40, De = 2 for different time instants. Figures 39 and 41 illustrate the solvent stress distributions for the Newtonian collapse and Figs. 40 and 42 depict the solvent and viscoelastic stress distributions for the collapse in LPTT fluid.

FIG. 39.

Solvent stress τ S , 12 ( Pa ) through x2/x3-midplane for R e = 40 , h * = 0.35 in Newtonian fluid. Black isoline shows vapor content of α = 0.01.

View large Download slide

Solvent stress $τ_{S, 12} (Pa)$ through x₂/x₃-midplane for $R e = 40, h^{*} = 0.35$ in Newtonian fluid. Black isoline shows vapor content of $α = 0.01$ ⁠.

FIG. 40.

Stress through x2/x3-midplane for R e = 40 , D e = 2 , h * = 0.35 in LPTT fluid. Black isoline shows constant vapor content of α = 0.01. Top: solvent stress τ S , 12 ( Pa ); bottom: viscoelastic stress τ M , 12 ( Pa ).

View large Download slide

Stress through x₂/x₃-midplane for $R e = 40, D e = 2, h^{*} = 0.35$ in LPTT fluid. Black isoline shows constant vapor content of $α = 0.01$ ⁠. Top: solvent stress $τ_{S, 12} (Pa)$ ⁠; bottom: viscoelastic stress $τ_{M, 12} (Pa)$ ⁠.

FIG. 41.

Solvent stress τ S , 22 ( Pa ) through x2/x3-midplane for R e = 40 , h * = 0.35 in Newtonian fluid. Black isoline shows vapor content of α = 0.01.

View large Download slide

Solvent stress $τ_{S, 22} (Pa)$ through x₂/x₃-midplane for $R e = 40, h^{*} = 0.35$ in Newtonian fluid. Black isoline shows vapor content of $α = 0.01$ ⁠.

FIG. 42.

Stress through x2/x3-midplane for R e = 40 , D e = 2 , h * = 0.35 in LPTT fluid. Black isoline shows constant vapor content of α = 0.01. Top: solvent stress τ S , 22 ( Pa ); bottom: viscoelastic stress τ M , 22 ( Pa ).

View large Download slide

Stress through x₂/x₃-midplane for $R e = 40, D e = 2, h^{*} = 0.35$ in LPTT fluid. Black isoline shows constant vapor content of $α = 0.01$ ⁠. Top: solvent stress $τ_{S, 22} (Pa)$ ⁠; bottom: viscoelastic stress $τ_{M, 22} (Pa)$ ⁠.

APPENDIX B: VISCOUS AND VISCOELASTIC STRESS DISTRIBUTIONS FOR Re = 40, De = 2, $h^{*} = 1.5$

The following figures show the stress distributions for an initial standoff distance h^* = 1.5 and Re = 40, De = 2 for different time instants. Figure 43 illustrates the solvent stress distributions for the Newtonian collapse and Fig. 44 shows the solvent and viscoelastic stress distributions for the collapse in LPTT fluid.

FIG. 43.

Solvent stress τ S , 12 ( Pa ) through the x2/x3-midplane for different time instants during the collapse for Re = 40 and initial standoff distance h * = 1.5 in Newtonian fluid. Black isoline shows constant vapor content of α = 0.01.

View large Download slide

Solvent stress $τ_{S, 12} (Pa)$ through the x₂/x₃-midplane for different time instants during the collapse for Re = 40 and initial standoff distance $h^{*} = 1.5$ in Newtonian fluid. Black isoline shows constant vapor content of $α = 0.01$ ⁠.

FIG. 44.

Stress through x2/x3-midplane for R e = 40 , D e = 2 , h * = 1.5 in LPTT fluid. Black isoline shows constant vapor content of α = 0.01. Top: solvent stress τ S , 12 ( Pa ); bottom: viscoelastic stress τ M , 12 ( Pa ).

View large Download slide

Stress through x₂/x₃-midplane for $R e = 40, D e = 2, h^{*} = 1.5$ in LPTT fluid. Black isoline shows constant vapor content of $α = 0.01$ ⁠. Top: solvent stress $τ_{S, 12} (Pa)$ ⁠; bottom: viscoelastic stress $τ_{M, 12} (Pa)$ ⁠.

APPENDIX C: VISCOUS AND VISCOELASTIC STRESS DISTRIBUTIONS FOR Re = 40, De = 1, $h^{*} = 1.1$

The following figures show stress distributions for an initial standoff distance h^* = 1.1 and Re = 40, De = 1 for different time instants in LPTT fluid. Figures 45 to 47 illustrate the solvent and viscoelastic stress distributions during the collapse.

FIG. 45.

Stress through x2/x3-midplane for R e = 40 , D e = 1 , h * = 1.1 (decreased relaxation time) in LPTT fluid. Black isoline shows constant vapor content of α = 0.01. Top: solvent stress τ S , 11 ( Pa ); bottom: viscoelastic stress τ M , 11 ( Pa ).

View large Download slide

Stress through x₂/x₃-midplane for $R e = 40, D e = 1, h^{*} = 1.1$ (decreased relaxation time) in LPTT fluid. Black isoline shows constant vapor content of $α = 0.01$ ⁠. Top: solvent stress $τ_{S, 11} (Pa)$ ⁠; bottom: viscoelastic stress $τ_{M, 11} (Pa)$ ⁠.

FIG. 46.

View large Download slide

Stress through x₂/x₃-midplane for $R e = 40, D e = 1, h^{*} = 1.1$ (decreased relaxation time) in LPTT fluid. Black isoline shows constant vapor content of $α = 0.01$ ⁠. Top: solvent stress $τ_{S, 12} (Pa)$ ⁠; bottom: viscoelastic stress $τ_{M, 12} (Pa)$ ⁠.

FIG. 47.

View large Download slide

Stress through x₂/x₃-midplane for $R e = 40, D e = 1, h^{*} = 1.1$ (decreased relaxation time) in LPTT fluid. Black isoline shows constant vapor content of $α = 0.01$ ⁠. Top: solvent stress $τ_{S, 22} (Pa)$ ⁠; bottom: viscoelastic stress $τ_{M, 22} (Pa)$ ⁠.

APPENDIX D: VISCOUS AND VISCOELASTIC STRESS DISTRIBUTIONS FOR Re = 40, De = 4, $h^{*} = 1.1$

The following figures show stress distributions for an initial standoff distance h^* = 1.1 and Re = 40, De = 4 for different time instants in LPTT fluid. Figures 48 to 50 illustrate the solvent and viscoelastic stress distributions during the collapse.

FIG. 48.

Stress through x2/x3-midplane for R e = 40 , D e = 4 , h * = 1.1 (increased relaxation time) in LPTT fluid. Black isoline shows constant vapor content of α = 0.01. Top: solvent stress τ S , 11 ( Pa ); bottom: viscoelastic stress τ M , 11 ( Pa ).

View large Download slide

Stress through x₂/x₃-midplane for $R e = 40, D e = 4, h^{*} = 1.1$ (increased relaxation time) in LPTT fluid. Black isoline shows constant vapor content of $α = 0.01$ ⁠. Top: solvent stress $τ_{S, 11} (Pa)$ ⁠; bottom: viscoelastic stress $τ_{M, 11} (Pa)$ ⁠.

FIG. 49.

View large Download slide

Stress through x₂/x₃-midplane for $R e = 40, D e = 4, h^{*} = 1.1$ (increased relaxation time) in LPTT fluid. Black isoline shows constant vapor content of $α = 0.01$ ⁠. Top: solvent stress $τ_{S, 12} (Pa)$ ⁠; bottom: viscoelastic stress $τ_{M, 12} (Pa)$ ⁠.

FIG. 50.

View large Download slide

Stress through x₂/x₃-midplane for $R e = 40, D e = 4, h^{*} = 1.1$ (increased relaxation time) in LPTT fluid. Black isoline shows constant vapor content of $α = 0.01$ ⁠. Top: solvent stress $τ_{S, 22} (Pa)$ ⁠; bottom: viscoelastic stress $τ_{M, 22} (Pa)$ ⁠.

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Viscoelastic vapor bubble collapse near solid walls and corresponding shock wave formation

I. INTRODUCTION

II. PHYSICAL MODEL AND NUMERICS

A. Governing equations

B. Single-fluid homogeneous mixture cavitation model

C. Numerical approach

III. NUMERICAL SETUP

IV. RESULTS AND DISCUSSION

A. General dynamics of the aspherical collapse

B. Variation of initial standoff distance

C. Variation of elasticity

V. CONCLUSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX A: VISCOUS AND VISCOELASTIC STRESS DISTRIBUTIONS FOR Re = 40, De = 2, $h^{*} = 0.35$

APPENDIX B: VISCOUS AND VISCOELASTIC STRESS DISTRIBUTIONS FOR Re = 40, De = 2, $h^{*} = 1.5$

APPENDIX C: VISCOUS AND VISCOELASTIC STRESS DISTRIBUTIONS FOR Re = 40, De = 1, $h^{*} = 1.1$

APPENDIX D: VISCOUS AND VISCOELASTIC STRESS DISTRIBUTIONS FOR Re = 40, De = 4, $h^{*} = 1.1$

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Viscoelastic vapor bubble collapse near solid walls and corresponding shock wave formation Open Access

I. INTRODUCTION

II. PHYSICAL MODEL AND NUMERICS

A. Governing equations

B. Single-fluid homogeneous mixture cavitation model

C. Numerical approach

III. NUMERICAL SETUP

IV. RESULTS AND DISCUSSION

A. General dynamics of the aspherical collapse

B. Variation of initial standoff distance

C. Variation of elasticity

V. CONCLUSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX A: VISCOUS AND VISCOELASTIC STRESS DISTRIBUTIONS FOR Re = 40, De = 2, h * = 0.35

APPENDIX B: VISCOUS AND VISCOELASTIC STRESS DISTRIBUTIONS FOR Re = 40, De = 2, h * = 1.5

APPENDIX C: VISCOUS AND VISCOELASTIC STRESS DISTRIBUTIONS FOR Re = 40, De = 1, h * = 1.1

APPENDIX D: VISCOUS AND VISCOELASTIC STRESS DISTRIBUTIONS FOR Re = 40, De = 4, h * = 1.1

REFERENCES

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Viscoelastic vapor bubble collapse near solid walls and corresponding shock wave formation

APPENDIX A: VISCOUS AND VISCOELASTIC STRESS DISTRIBUTIONS FOR Re = 40, De = 2, $h^{*} = 0.35$

APPENDIX B: VISCOUS AND VISCOELASTIC STRESS DISTRIBUTIONS FOR Re = 40, De = 2, $h^{*} = 1.5$

APPENDIX C: VISCOUS AND VISCOELASTIC STRESS DISTRIBUTIONS FOR Re = 40, De = 1, $h^{*} = 1.1$

APPENDIX D: VISCOUS AND VISCOELASTIC STRESS DISTRIBUTIONS FOR Re = 40, De = 4, $h^{*} = 1.1$