Bubble Collapse Research Articles

The effect of temperature on the pressure energy distribution in the process of near-wall single bubble collapse

At the interface between rotating components and the working medium in fluid machinery, cavitation in the liquid is induced. The temperature of the working medium has a significant influence on the cavitation. However, the effect of temperature on the pressure and pressure energy in the collapse of bubbles cannot be explained accurately. This study established cavitation models containing insoluble gases at different temperatures and iron walls. The Transferable Intermolecular Potential 4 Points-Fluctuating Bonds water model and the Reax force field were selected to study the influence of temperature on pressure energy and pressure in the process of bubble collapse. Employing the model, we examined variations in release pressure and the pressure energy associated with bubble collapse within the system at diverse temperatures. This investigation was conducted by establishing 24 monitoring points. The findings revealed that the near-wall single-bubble collapse process could be delineated into four distinct stages: the precollapse stage, the inertia stage, the rebound stage, and the collapse stage. These four stages manifest on both the pressure curve in the simulation and the curve depicting changes in the cavity area during the experimental phase. The influence of temperature on bubble collapse primarily stemmed from its impact on saturation vapor pressure and viscosity. This influence was observed during the precollapse process, where saturation vapor pressure played a dominant role, and in the collapse stage, where viscosity was predominant.

The collapse of a sonoluminescent cavitation bubble imaged with X-ray free-electron laser pulses

Single bubble sonoluminescence (SBSL) is the phenomenon of synchronous light emission due to the violent collapse of a single spherical bubble in a liquid, driven by an ultrasonic field. During the bubble collapse, matter inside the bubble reaches extreme conditions of several gigapascals and temperatures on the order of 10000 K, leading to picosecond flashes of visible light. To this day, details regarding the energy focusing mechanism rely on simulations due to the fast dynamics of the bubble collapse and spatial scales below the optical resolution limit. In this work we present phase-contrast holographic imaging with single x-ray free-electron laser (XFEL) pulses of a SBSL cavitation bubble in water. X-rays probe the electron density structure and by that provide a uniquely new view on the bubble interior and its collapse dynamics. The involved fast time-scales are accessed by sub-100 fs XFEL pulses and a custom synchronization scheme for the bubble oscillator. We find that during the whole oscillation cycle the bubble’s density profile can be well described by a simple step-like structure, with the radius R following the dynamics of the Gilmore model. The quantitatively measured internal density and width of the boundary layer exhibit a large variance. Smallest reconstructed bubble sizes reach down to R≃0.8μm , and are consistent with spherical symmetry. While we here achieved a spatial resolution of a few 100 nm, the visibility of the bubble and its internal structure is limited by the total x-ray phase shift which can be scaled with experimental parameters.

An experimental investigation on the ice-breaking phenomenon induced by the spark-generated bubble near a vertical plate

The present work experimentally studied the coupled interaction between the fracture of an ice plate floating on the water surface, induced by the pulsation of a high-pressure bubble, and the associated deformation of a vertically submerged aluminum plate. The dynamic behaviors of bubbles and fractured ice were concurrently recorded using two high-speed cameras, and the plastic deformation of the aluminum plate was measured by an ultra-depth three-dimensional microscope. The results indicated that the jet direction of the collapsed bubble is heavily influenced by the position to generate the bubble due to the competing Bjerknes forces caused by various boundaries. There is also a significant discrepancy in the phenomena of bubble collapse near thin vs thick ice plates, attributed to the sudden alteration in boundary conditions caused by ice fracturing. Three distinct ice-breaking mechanisms, namely, the hogging moment, jet impact, and the secondary shock wave, were identified based on the types of loads, leading to the initial ice fractures. In general, it was observed that the efficiency of ice breaking improved with a decrease in the bubble–ice distance (γf) and an increase in the bubble–plate distance (γm). It was found that the contacting jet from an upwardly collapsed bubble is the most effective in breaking the thickest ice plates for γm>1.9. While the shock wave from a bubble collapsing near solid wall corners could fracture thick ice plates, the aluminum plate risked damage from jet impacts when γm was less than 0.6.

On efficient modelling of radical production in cavitation assisted reactors

Process intensification by cavitation is gaining widespread attention due to the benefits that the intense bubble collapse conditions can provide, yet, several knowledge gaps exist in the modelling of such systems. This work studies the numerical prediction of single bubble dynamics and the various approaches that can be employed to estimate the changes in the chemical composition of cavitating bubbles. Specific emphasis is placed on the prediction of the radical production rates during bubble collapse and the computational performance, with the aim of coupling the single bubble dynamics to flow models for reactor hydrodynamics. The results reveal that the choice of chemical reaction approach has virtually no effect on the bubble dynamics, whereas the predicted radical production rates can differ substantially. It is found that evaluating the radical production only on temperature peaks, an approach commonly followed in literature, may result in the most erroneous estimations (on average 12.8 times larger than those of the full kinetic model), while a simplified kinetic model yields more accurate predictions (2.3 times larger) at the expense of increased computational times. Continuous evaluation of the bubble content by assuming equilibrium when the bubble temperature is above a certain threshold (≈1500K) is shown to be capable of predicting total radical production values close to those estimated by solving the kinetics of a detailed reaction model (19.8% difference), as well as requiring only 22.2% more computational costs compared to simulations without chemical reaction modelling. Such an equilibrium approach is therefore recommended for future studies aiming to couple flow simulations with single bubble dynamics to accurately predict radical production rates in cavitation devices, involving numerous bubbles following different flow trajectories. Furthermore, an algebraic expression that successfully approximates the full kinetic simulation results is proposed as a function of the initial nucleus size and the time integral of the liquid pressure when it is under vapor pressure. Such a model can be applied in modelling efforts that do not require local instantaneous radical concentrations, and paves the way for efficient closure modelling of radical production in CFD simulations of hydrodynamic reactors.

Influence of a hemispherical bulge on a flat wall upon the collapse jet of cavitation bubbles

Uneven sections along a wall cause local flow deterioration and can result in severe cavitation erosion. This paper investigates the influence of a hemispherical bulge on a flat wall upon the collapse dynamics of a cavitation bubble in terms of the Kelvin impulse theory and high-speed photography based on the Weiss theorem and the image method. The evolution and characteristics of the bubble collapse morphology, the flow field distribution, and the Kelvin impulse (in terms of strength, direction, and directional sensitivity) are analyzed for symmetric and asymmetric configurations. The results show that the bubble collapse jet can be divided into three scenarios: mainly induced by the hemispherical bulge, mainly induced by the flat wall, and broadly equivalent effects of both. Additionally, as the dimensionless distance between the bubble and the flat wall increases, the range of the jet attraction zone induced by the bulge initially increases, subsequently decreases, and ultimately converges to the diameter of the bulge. The maximum width of the attraction zone can reach 3.45 times the radius of the bulge. Finally, the spatial sensitivity of the jet direction is found to be significant near the junction between the flat wall and the bulge.

Vapor compression and energy dissipation in a collapsing laser-induced bubble

The composition of the gaseous phase of cavitation bubbles and its role on the collapse remains to date poorly understood. In this work, experiments of single cavitation bubbles in aqueous ammonia serve as a novel approach to investigate the effect of the vapor contained in a bubble on its collapse. We find that the higher vapor pressure of more concentrated aqueous ammonia acts as a resistance to the collapse, reducing the total energy dissipation. In line with visual observation, acoustic measurements, and luminescence recordings, it is also observed that higher vapor pressures contribute to a more spherical collapse, likely hindering the growth of interface instabilities by decreasing the collapse velocities and accelerations. Remarkably, we evidence a strong difference between the effective damping and the energy of the shock emission, suggesting that the latter is not the dominant dissipation mechanism at collapse as predicted from classical correction models accounting for slightly compressible liquids. Furthermore, our results suggest that the vapor inside collapsing bubbles gets compressed, consistently with previous studies performed in the context of single bubble sonoluminescence, addressing the question about the ability of vapors to readily condense during a bubble collapse in similar regimes. These findings provide insight into the identification of the influence of the bubble content and the energy exchanges of the bubble with its surrounding media, eventually paving the way to a more efficient use of cavitation in engineering and biomedical applications.

Numerical predictions of underwater radiated noise from a non-cavitating model-scale propeller

Propeller-induced acoustic noise from marine vessels is the largest source of anthropogenic underwater radiated noise (URN) and a significant threat to marine ecosystems. Under typical operation, cavitation dominates the URN emissions. Cavitation, which is a pressure-driven phase-change process that results in the violent formation and collapse of vapor bubbles in the wake of the propeller, is often unavoidable during realistic, full-scale operating conditions. However, at a model scale, inducing cavitation requires a depressurized flow facility that makes acoustic measurements difficult due to confinement effects. Numerical simulation is, therefore, appealing as a tool for predicting URN, but the simulation of the cavitation phenomenon and the associated acoustics involves considerable uncertainty and a range of potential sources of error. In propeller operation, cavitation frequently occurs in the core of the vortices shed from the tips of propeller blades. In the present work, we developed a delayed detached-eddy simulation (DDES) of a model-scale ship with the focus on predicting fluctuating pressure due to shed vorticity. The solution is compared to hydrophone measurements from non-cavitating tow-tank experiments. Finally, we numerically introduce cavitation at model scale in the numerical solution and examine its effects.

The violent collapse of vapor bubbles in cryogenic liquids

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.

Sono-catalytic syngas production by low-temperature biomass gasification

Syngas holds a crucial role in diverse industrial applications, including chemical synthesis and power generation. It has been traditionally generated through high-temperature processes, such as steam reforming or biomass gasification, demanding temperatures of up to 1200 °C, which results in substantial energy consumption. Consequently, there is a growing interest in exploring alternative, low-temperature biomass gasification processes. This paper introduces an innovative method for syngas production through sono-catalytic biomass conversion at significantly lower temperatures. The approach employs a sonochemical reactor that concentrates acoustic waves to activate inertial cavitation. Furfural, a biomass-derived platform chemical, was chosen as the primary chemical for the study, showcasing the potential of sonochemical syngas production. The reaction converted furfural into various gas products, particularly a mixture of hydrogen and carbon monoxide, with the chemical reaction hypothesized to be triggered by pyrolysis during bubble collapse hot spots. The incorporation of heterogeneous metal foams into the system further enhanced furfural conversion rates. This improvement was attributed to the preferential nucleation of cavitation bubbles on the porous metal foam surface, bringing the hot spots into proximity with furfural molecules. This application of sonochemistry in furfural pyrolysis at low temperatures holds promise for a more economical approach to biomass valorization.

Laser direct overall water splitting for H2 and H2O2 production

Hydrogen (H2) and hydrogen peroxide (H2O2) play crucial roles as energy carriers and raw materials for industrial production. However, the current techniques for H2 and H2O2 production rely on complex catalysts and involve multiple intermediate steps. In this study, we present a straightforward, environmentally friendly, and highly efficient laser-induced conversion method for overall water splitting to simultaneously generate H2 and H2O2 at ambient conditions without any catalysts. The laser direct overall water splitting approach achieves an impressive light-to-hydrogen energy conversion efficiency of 2.1%, with H2 production rates of 2.2 mmol/h and H2O2 production rates of 65 µM/h in a limited reaction area (1 mm2) within a short real reaction time (0.36 ms/h). Furthermore, we elucidate the underlying physics and chemistry behind the laser-induced water splitting to produce H2 and H2O2. The laser-induced cavitation bubbles create an optimal microenvironment for water-splitting reactions because of the transient high temperatures (104 K) surpassing the chemical barrier required. Additionally, their rapid cooling rate (1010 K/s) hinders reverse reactions and facilitates H2O2 retention. Finally, upon bubble collapse, H2 is released while H2O2 remains dissolved in the water. Moreover, a preliminary amplification experiment demonstrates the potential industrial applications of this laser chemistry. These findings highlight that laser-based production of H2 and H2O2 from water holds promise as a straightforward, environmentally friendly, and efficient approach on an industrial scale beyond conventional chemical catalysis.

Exploring viscosity influence mechanisms on coating removal: Insights from single cavitation bubble behaviours in low-frequency ultrasonic settings

The role of acoustic cavitation in various surface cleaning disciplines is important. However, the physical mechanisms underlying acoustic cavitation-induced surface cleansing are poorly understood. This is due to the combination of microscopic and ultrashort timescales associated with the dynamics of acoustic cavitation bubbles. Here, we have precisely controlled single-bubble cavitation in both space and time. Ultrasonic excitation leads to the cavitation of generated single bubbles. A synchronous ultrafast photomicrographic system simultaneously records the dynamics of single acoustic cavitation bubbles (SACBs) and the cleaning process of the nearby surface in liquids with varying viscosities. Finally, we analysed the correlation between bubble dynamics and surface cleaning situations. The differences in the typical dynamic characteristics of the bubbles during collapse in liquids with varying viscosities reveal two main mechanisms underlying surface cleaning by acoustic cavitation, which are respective the Laplace pressure during the bubble’s movement and liquid jets during bubble collapse. Our study provides a better physical understanding of the ultrasonic cleaning process based on acoustic cavitation, and will help to optimize and facilitate the applications of surface cleaning, especially for the cleaning of substrates with tightly attached dirt.

Repeatable Acoustic Vaporization of Coated Perfluorocarbon Bubbles for Micro-Actuation Inspired by Polypodium aureum.

Polypodium aureum, a fern, possesses a specialized spore-releasing mechanism like a catapult induced by the quick expansion of vaporized bubbles. This study introduces lipid-coated perfluorocarbon droplets to enable repeatable vaporization-condensation cycles, inspired by the repeatable vaporization of Polypodium aureum. Lipid-perfluorocarbon droplets have been considered not to exhibit repeatable oscillations due to bubble collapse of the low surface tension of lipid layers. However, a single lipid-dodecafluoropentane droplet with a diameter of 9.17 µm shows expansion-contraction oscillations over 4000 cycles by changing lipid composition and applying a low-power 1.7 MHz ultrasound to induce the partial vaporization of the droplets. The optimal combinations of shell composition, droplet fabrication, and acoustic conditions can minimize the damage on shell structure and promote a quick recovery of damaged shell layers. The highly expanding oscillatory microbubbles provide a new direction for fuel-free micro- or nanobots, as well as biomedical applications of contrast agents and drug delivery.

Research on Erosion Effect of Various Submerged Cavitating Jet Nozzles and Design of Self-Rotating Cleaning Device

Submerged cavitating jets can effectively remove marine organisms from ship hulls without damaging the surface paint. To enhance the cleaning efficiency of cavitating jets, the selection of an appropriate nozzle structure and the design of an efficient cleaning device are crucial. In this study, the submerged cavitation effect of different nozzles was analyzed by numerical simulation. The actual cleaning efficacy of the nozzles was confirmed through erosion experiments as well. The simulation and experiment showed that the shear nozzle, absent of a pre-shrinking section and featuring a spherical outlet connected to a diffusion cylindrical section, maintained stable erosion performance at a standoff distance of 30–50 mm. This erosion was primarily attributed to denudation caused by bubble collapse. Based on this shear nozzle, a self-rotating cleaning device was designed and manufactured. A test rig was also established to test the cleaning effect and some parameters of the cleaning device.

Cavitation bubble collapse near a rigid wall with an oil layer

This paper reports on the study of a laser-generated cavitation bubble in water collapsing near a rigid wall that is covered with a layer of silicone oil. The oil viscosity is between 5 and 5000 cSt. It is known that a collapsing cavitation bubble near a rigid boundary develops a jet. In this study, we show that the viscous oil layer competes with the flow generated by the bubble, and affects the short- and long-term bubble dynamics. For very low viscosity silicone oil (5, 10 and 20 cSt) the vortices formed when the water jet hits the wall result in a gas–water mixture in the oil layer. With increasing viscosity, i.e. 50 and 100 cSt, the formation of water droplets with gas bubbles inside the oil layer is observed. The bubble dynamics and flow in the oil layer are nicely reproduced with VoF simulations up to the second bubble collapse. In higher viscosity silicone oils (350, 500 and 1000 cSt), the water jet typically retracts completely during the re-expansion of the bubble. When a very high viscosity oil (5000 cSt) is used, a ‘mushroom’-shaped bubble is formed in water which splits in two as it collapses. To separate the regimes, we introduce a non-dimensional stress ratio relating the inertia and viscosity. This inverse Reynolds number Π becomes approximately 1 in the jet retraction regime. When Π≥ 1.0, the jet impact and vortices are effectively hindered on the rigid wall.

Dynamics of collapse of free-surface bubbles: effects of gravity and viscosity

The rupture of the thin film at the top of a bubble at a liquid–gas interface leads to an axisymmetric collapse of the bubble cavity. We present scaling laws for such a cavity collapse, established from experiments conducted with bubbles spanning a wide range of Bond ( ${10^{-3}&lt; Bo\leq 1}$ ) and Ohnesorge numbers ( ${10^{-3}&lt; Oh&lt;10^{-1}}$ ), defined with the bubble radius $R$ . The cavity collapse is a capillary-driven process, with a dependency on viscosity and gravity, affecting respectively, precursory capillary waves on the cavity boundary and the static bubble shape. The collapse is characterised by the normal interface velocity ( $U_n$ ) and by the tangential wave propagation velocity of the kink ( $U_t$ ), defined by the intersection of the concave cavity boundary formed after the rupture of the thin film with the convex boundary of the bubble cavity. During the collapse, $U_t$ remains constant and is shown to be $U_t=4.5U_c{\mathcal {W}}_R$ , where $U_c$ is the capillary velocity and ${\mathcal {W}}_R(Oh,Bo)={(1-\sqrt {Oh {\mathscr {L}}} )^{-1/2}}$ is the wave resistance factor due to the precursory capillary waves, with $\mathscr {L}(Bo)$ being the path correction of the kink motion. The movement of the kink in the normal direction is part of the inward shrinkage of the whole cavity due to the sudden reduction of gas pressure inside the bubble cavity after the thin film rupture. This normal velocity is shown to scale as $U_c$ in the equatorial plane, while at the bottom of the cavity $\bar {U}_{nb}=U_c(Z_c/R)({\mathcal {W}_R}/ {\mathscr {L}})$ , where $Z_c(Bo)$ is the static cavity depth. The filling rate of the cavity, which remains a constant throughout the collapse, is shown to be entirely determined by the shrinking velocity and scales as ${Q_T\simeq 2{\rm \pi} R Z_c U_c}$ . From $Q_T$ we recover the jet velocity scaling, thereby relating the cavity collapse with the jet velocity scaling.

Experimental study on cavitation inhibition in a butterfly valve with different plate shapes

A valve is a kind of mechanical device used to control the flow of fluid in pipelines or systems, which has a wide range of uses in various industrial, commercial, and household applications. When the fluid passes through the valve, cavitation often occurs in the valve, resulting in the valve not working properly and even damaging the valve and related systems. In this paper, experimental visualization was used to investigate cavitation evolution through a butterfly valve with different plate shapes under different pressure conditions. The results show that with the increase in the curvature of both sides of the butterfly plate and the increase in the total pressure condition, the cavitation type inside the butterfly valve will change from non-attached cavitation to attached cavitation. In addition, the maximum length and thickness of the attached cavitation bubbles gradually increase as well. However, when the shape of both ends of the butterfly plate is triangular, the adhesion ability to cavitation bubbles is greatly reduced. As the curvature increases, the frequency of the shedding of cavitation bubble gradually decreases, the volume of cavitation bubble increases, and the shedding of the cavitation bubble caused by the shock wave is more obvious. When the shape of both sides of the butterfly plate is triangular, the position where the bubble collapses is closer to the front end of the butterfly plate, and the shock wave generated by the bubble collapse will lead to the collapse of the attached cavitation bubble.

Research on cavitation bubble behaviors between a dual-particle pair

The interactions between particle clusters and cavitation bubbles play an important role in several industrial applications, e.g., the abrasive erosion of hydraulic machinery, the efficiency improvement of ultrasonic cleaning, and the mineral flotation. This paper explores the influence of a dual-particle pair of unequal sizes on the bubble collapse dynamics based on a Kelvin impulse model and high-speed photography. The impact of the radius ratio of the dual-particle pair (ranging from 1.0 to 3.0), defined as the ratio of the large-particle radius to the small-particle radius, on the bubble interface deformation, collapse jet, and the translational movement of the bubble is analyzed. The main findings are as follows: (1) the Kelvin impulse based on the Weiss theorem can effectively predict the asymmetric bubble behaviors in terms of the moving distance and the direction of the bubble centroid during the bubble collapse; (2) the particle radius ratio is a key parameter affecting the intensity and direction of the Kelvin impulse, as well as the bubble centroid moving distance; and (3) the zero-Kelvin-impulse point is always biased toward the smaller particle in the dual-particle pair.

Modeling of vapor bubble dynamics considering the metastable fluid state

This study investigates the dynamic modeling of a spherical vapor bubble in an infinite liquid. Initially, the liquid is in a state of saturation, but then, there is a change in ambient pressure. Unlike previous studies that assumed vapor saturation, our model fully considers the metastable fluid state of subcooled vapor and superheated liquid. The theoretical model is based on the state equation of vapor and the kinetic theory of gases, allowing the visualization of the effects of superheating and subcooling on the bubble pressure and the phase transition rate. The accuracy of the numerical solutions for the bubble collapse and growth is confirmed by experimental results using water as the working fluid. Two approaches of heat transfer simplification are validated in the numerical solution, and an adhesion coefficient within the range of 0.1–1 is recommended for the calculation. Additionally, this study provides insights into how the metastable fluid state affects the bubble pressure and phase transition, especially in the early stages of bubble collapse. Furthermore, the conservation of vapor mass inside the bubble is also demonstrated.

Dynamic behaviors of a bubble near a rectangular wall with a bulge

In this paper, the cavitation bubble dynamics near a rectangular wall with a bulge are theoretically investigated. High-speed photography is employed to provide experimental verification of the theoretical results. Through a series of conformal transformations and the image method, the analytical description of how this complex wall configuration affects the bubble is shown to be equivalent to the superposition of eight virtual bubbles. The physical meaning of the eight virtual bubbles can be divided into four groups, corresponding to the influence of the left wall, the bottom wall, the angle formed by the two flat walls, and the bulge. The influence of the bulge on the liquid velocity distribution, as well as the intensity and direction of the Kelvin impulse exerted on the bubble, is explored for cases in which the bubble is located at symmetric and asymmetric positions. The main findings are given as follows: During the bubble collapse, a high-velocity area of the liquid exists to the side of the bubble farthest from the bulge, and three stagnation points with three low-velocity areas appear on the bulge surface. The bulge mainly influences the impulse intensity when the bubble is located near the symmetric position. The existence of the bulge causes the impulse angle to attain a minimum as the distance between the bubble and the bulge increases. For a larger bulge radius, the changes in the impulse angle become more complicated as the bubble position angle increases.

Numerical simulation of water jet load induced by spherical bubble collapse under underwater explosion

In this study, a numerical method combining the HLLC solver and FV-WENO scheme is proposed to accurately simulate the behavior of multiphase compressible fluids and investigate the formation process of water jets. The HLLC solver approximates the conservation equations of the fluid by using numerical fluxes from the approximate Riemann problem, while the FV-WENO scheme improves the accuracy of the numerical solution by employing the weighted essentially non-oscillatory (WENO) reconstruction method. The combination of these methods allows for a more accurate simulation of the behavior of multiphase compressible fluids, leading to an accurate depiction of the formation process of water jets. By simulating the interaction between underwater shock waves from explosions and single or multiple spherical bubbles, we study the generation, development, and collapse of water jets during the bubble collapse process. By revealing the characteristics of the load during the formation of water jets and the interaction mechanisms between bubbles, our study provides support for further research on water jet phenomena. The formation process of water jets is of great significance for understanding the propagation of underwater shock waves and structural damage. Through this study, we delve into the mechanism of water jet formation, providing a new perspective on the propagation of underwater shock waves and structural damage. Our research results demonstrate that the application of the HLLC solver and FVWENO scheme in water jet simulation effectively improves the accuracy and precision of the numerical solution.