Velocity Boundary Research Articles

Towards a theory of steady-state solidification process with a quasi-equilibrium two-phase region

AbstractThe process of directional crystallization in the presence of a quasi-equilibrium two-phase region located between the solid material and the liquid phase is studied theoretically. The mathematical model of the process is based on heat and mass transfer equations in the solid, liquid and two-phase regions, as well as boundary conditions at the phase interfaces “solid phase” – “two-phase region” and “two-phase region” – “liquid phase”, which are moving with a constant velocity. The process of directional crystallization is given by fixed temperature gradients in the solid and liquid phases, which determine a constant velocity of melt solidification. An exact analytical solution of the nonlinear problem with two moving boundaries of phase transformation is obtained, which is based on the transition to a new independent variable, the solid phase fraction, when integrating the nonlinear heat and mass transfer equations in the two-phase region. As a result of solving the problem, the distributions of temperature and concentration of dissolved impurity, the solid phase fraction in a two-phase region, the laws and velocities of motion of its interphase boundaries are determined. It is analytically shown that the impurity concentration and temperature in the two-phase region are only the functions of solid phase fraction, which, in turn, depends on the spatial coordinate. Analysis of the obtained solutions shows that the solid phase fraction in a two-phase region can be both a decreasing and increasing function of the spatial coordinate, which is directed from the solid material to the melt. This determines the internal structure of two-phase region, its permeability, average interdendritic spacing, distribution of dissolved impurity, crystallization velocity and laws of two-phase region boundaries.

Investigation of thermal properties of ethylene glycol-based Williamson hybrid-nanofluid over stretchable/shrinking flat plate and their effects on solar panels

The global requirement for sustainable energy supply to enhance industrial productivity and reduce production costs has focused researchers’ attention to renewable energy in recent years. Solar energy mitigates the dangers connected with the use of fossil fuels in electricity generation. This work is set to evaluate the heat transmission capacities of Williamson hybrid nanofluid flow across a flat plate with viscous dissipation and heat source. The mathematical model explaining the flow interaction of Williamson hybrid nanofluid, combining viscous dissipation, heat source, and temperature-variation thermal conductivity and viscosity, is created using conservation laws. The specified system of non-linear coupled partial differential equations undergoes non-similarity transformation. The resulting non-dimensional model is solved using the bivariate spectral weighted residual method. The accuracy of the method is proven by comparing obtained results with those in the literature, and a good agreement is observed. Graphs are utilized to explain the thermophysical properties that are being considered. The results show that the fluid temperature rises when there is a source of heating and viscous dissipation. The velocity and the fluid parameter (We) have an inverse connection, whereas the temperature of the fluid has the opposite impact. Moreover, when nanoparticles are present, the thermal boundary layer rises along with the nanoparticles, thickening the velocity boundary layer and decreasing fluid velocity. Findings also show that for the Vd∈[0.1,0.5], the skin drag force and Nusselt number retard by 0.64%,14.06% for the shrinking sheet and 0.21%,6.57% for the stretching sheet respectively. In the same vein, an 100% surge in the porosity parameter escalate the skin friction coefficient by 19.13% and 26.91% and the Nusselt number by 4.92% and 2.06% respectively for both the contracting and elastic sheet. The findings in the research will provide more insight in the design and improvement of solar panel plate efficiency.

An improved implicit direct-forcing immersed boundary method (DF-IBM) around arbitrarily moving rigid structures

An improved implicit direct-forcing immersed boundary method (DF-IBM) is presented for simulating incompressible flows around complex rigid structures undergoing arbitrary motion. The current approach harnesses the pressure implicit with splitting of operators algorithm to handle the fluid–solid system's dual constraints in a segregated manner. As a result, the divergence-free condition is preserved throughout the Eulerian domain, and the no-slip velocity boundary condition is exactly enforced on the immersed boundary. A new pressure Poisson equation (PPE) is derived, incorporating the boundary force where the no-slip condition is already met, enabling the use of fast iterative PPE solvers without modifications. The improvement involves integrating Lagrangian weight methods having better reciprocity over the IBM-related linear operators with the implicit formulation. An additional force initialization scheme is introduced to further boost the algorithm's performance. The method's accuracy, efficiency, and capability are verified through various stationary and moving immersed boundary cases. The results are validated against experimental and numerical data from the literature. The proposed improvements seamlessly integrate into existing incompressible fluid solvers with minimal adjustments to the original system equations, highlighting their ease of implementation.

On the use of fast projection methods with unsteady velocity boundary conditions

This article aims at clarifying and amending a previously published result on the use of pseudo-pressure approximations for fast incompressible Navier-Stokes solvers with time-dependent boundary conditions. Fast projection methods were proposed to speed up high-order incompressible Navier-Stokes solvers by replacing pressure projections with simple approximations. A generalized approximation theory was proposed in [1], in which the time-dependent boundary conditions were ignored. In this comment, we investigate the effect of unsteady boundary conditions on the order of accuracy and demonstrate that the pressure approximations derived in [1] hold for all RK2 integrators, while RK3 approximations are only third order for certain sets of coefficients. We show that third order is lost for other cases and shed light on why order is broken. This work serves as a reference for the treatment of unsteady boundary conditions for fast-projection methods. Numerical validation is performed on a well established channel flow problem with a variety of unsteady inlet conditions for a wide range of explicit RK2 and RK3 integrators.

Lattice Boltzmann Modeling of Forced Imbibition Dynamics in Dual-Wetted Porous Media

Forced imbibition dynamics are critical for enhancing recovery rates in reservoirs, as efficient fluid displacement directly impacts resource extraction. This study employs the Zou-He velocity boundary condition and a modified convective boundary condition (mCBC) within the multi-component Shan-Chen lattice Boltzmann method (LBM), thus facilitating unobstructed flow of multi-component fluids at the outlet while maintaining constant outlet pressure. The model's validity was established through outflow tests of immiscible droplets in channels. Its accuracy was further confirmed by comparing results from forced imbibition dynamics tests in dual-wetted pores with theoretical predictions. A symmetrical porous medium was constructed using the stacking method, achieving dual wettability by fixing the contact angle in the upper region and varying it in the lower region. We performed 20 simulation sets under unfavorable viscosity ratios and varying capillary numbers, focusing on overall displacement efficiency before and after breakthrough, and employing energy balance equations to evaluate the dominant forces. Results reveal that capillary forces predominantly dictate forced imbibition dynamics in low capillary number scenarios. In strongly wetted regions, the invading fluid fully occupies pore spaces; however, in weakly wetted regions, displacement efficiency significantly declines as wettability approaches neutrality, even nearing zero. As capillary numbers increase, viscous forces become more prominent, controlling dynamics and leading to fingering and trapping of defending fluids. In weakly wetted areas, the increasing influence of viscous forces enhances fluid displacement, resulting in significant improvements in overall displacement efficiency compared to conditions dominated by capillary forces.

An equivalent source method for acoustic problems with thermoviscous effects.

This paper presents an equivalent source method (ESM) for analyzing sound propagation in small-scale acoustic structures with thermoviscous effects. The formulations that describe the thermal, viscous, and acoustic modes for thermoviscous acoustic problems are introduced. The concept of ESM is then applied to solve these formulations, resulting in an efficient numerical computation and implementation procedure. Based on two different strategies, the obtained ESM formulations are coupled at the boundary using the isothermal, non-slip, and null-divergence conditions. The coupling based on the first strategy is efficient for solving thermoviscous acoustic problems with few matrices required. However, this procedure faces the evaluation of the tangential derivatives of the boundary velocity. Coupling the ESM formulations directly for each component of the total particle velocity at the boundary has no such problem, which leads to the second strategy. However, it entails a larger memory usage compared to the former. Additionally, the coupled finite element method (FEM)-ESM formulations based on the above strategies are developed for acoustic-structural interaction. The validity of the presented ESM formulations is demonstrated through benchmark examples, and that of the coupled FEM-ESM formulation is illustrated by the numerical analysis of a simplified microphone.

Optimum Size and Heat Transfer and Velocity Correlations of the Outer and Inner Surfaces of Vertical Hollow Polygonal Cylinders in Natural Convection

Abstract Laminar natural convection heat transfer from vertical hollow polygonal cylinders with a wide range of cross-sectional areas is investigated. The buoyancy-driven three-dimensional (3D) flow around hollow polygonal cylinders immersed in quiescent ambient air with equal outer and inner surface temperatures is analyzed. The governing equations are numerically solved in nondimensional variables using the finite volume method. The numerical solution is validated using available experimental and numerical data. Results of the mean Nusselt number for the outer (Nu¯ho) and inner (Nu¯hi) surfaces are obtained by varying a number of key parameters. These parameters are the Rayleigh number based on the cylinder height (Rah) in the range 103≤ Rah≤ 107, the nondimensional cross-sectional area (AC) in the range 0.006 ≤ AC≤ 0.5, and the number of sides of the polygon (N) in the range 6 ≤ N ≤∞. In all cases, a Prandtl number (Pr) of 0.7 has been assumed. The study shows that at a certain Rayleigh number and a certain number of sides, the heat transfer rate from the inner surface decreases (by as much as 79.8%) as the polygon area decreases (by as much as 83.32%), whereas the heat transfer rate on the outer surface increases (by as much as 133.3%) as the polygon area decreases (by as much as 83.32%). It has also been found that the behavior of the buoyancy-driven flow in the vicinity of the outer surface is fundamentally different than that near the inner surface. Additional details about this fundamental difference are presented in the Results and Discussion section of the paper. New correlations to calculate the average velocity at the exit surface of the cylinder inner core and the mean Nusselt number for both the outer and inner surfaces have also been developed. Also, correlations have been developed for selecting the optimal cross-sectional area for purposes of identifying the regions where the thermal and velocity boundary layers overlap within the inner core of the cylinder.

Three-dimensional numerical investigation on flow and heat transfer characteristics and multi-objective optimization of interconnected microchannels

The interconnected microchannel demonstrates excellent performance in enhancing flow heat transfer. However, the optimal width of the interconnecting slot has not been thoroughly investigated. To address this gap, this paper numerically simulates the three-dimensional flow and heat transfer of interconnected microchannels featuring varied slot widths and channel depths. The investigated slot widths range from 50 to 1000 μm and depths of 100, 200, and 300 μm. Evaluation of overall performance encompasses heat transfer and flow characteristics such as average wall temperature, heat transfer coefficient, thermal resistance as well as pressure drop. It is the interconnected slots that interrupt the velocity and thermal boundary layers, contributing to enhanced heat transfer. Furthermore, the back propagation neural network and non-dominated sorting genetic algorithm-II are utilized for multi-objective optimization of thermal resistance and pressure drop. The technique for order preference by similarity to an ideal solution (TOPSIS) method combined with the entropy weight method is employed to select the compromise optimal solution from the Pareto optimal solutions. The optimized slot widths are 369, 424, and 167 μm for channel depths of 100, 200, and 300 μm, respectively. This research provides data support for the optimization design and engineering manufacturing of interconnected microchannels.

Kinetics of the bcc to hcp phase transition in [formula omitted]-oriented single crystal iron under ramp compression

Non-Equilibrium Molecular Dynamics simulations have been used to investigate the kinetics of the bcc to hcp phase transition in ramp-compressed [001]-oriented single crystal iron in the presence of defects consisting in micro-voids. Depending on the ramp maximum pressure, the microstructure of the daughter hcp phase is observed to be strongly affected by the plastic activity in the parent bcc phase. The hcp phase is first nucleated around the pre-existing microvoids, then it grows along preferential crystal directions and the subsequent volume reduction induces a local relaxation of the pressure. The velocity of the bcc-hcp phase boundary is shown to increase gradually with maximum pressure up to supersonic values. For high over-compression beyond the transition onset pressure, the phase transformation becomes almost instantaneous and essentially independent of the initial defects. Meanwhile, the ramp wave steepens with propagation distance and evolves into a shock front, up to the over-driven regime classically observed in both experiments and hydrodynamic, macro-scale simulations.

An improved numerical model of high-speed friction stir welding using a new tool-workpiece slip ratio

The interface slip state of high-speed friction stir welding (HSFSW) is challenging to determine, and accurate boundary conditions cannot be obtained. First, this paper studies the effects of velocity and temperature on the interface slip state, and establishes a new slip coefficient equation. Then, this equation was used to modify the velocity boundary conditions and improve the numerical calculation method of HSFSW. A numerical model of HSFSW was established and validated through experimental data. Finally, the effects of traverse speed and rotational speed on the temperature and void defects of HSFSW were elaborated, and the applicable process parameter range was determined. Specifically, void defects were observed at a rotational speed of 6000 rpm with traverse speeds of 1000 mm/min and 2000 mm/min, while surface peeling occurred at a traverse speed of 3000 mm/min with a rotational speed of 7000 rpm. This study establishes the optimal parameter range for aluminum alloy HSFSW, providing valuable guidance for future industrial applications.

Modeling Subduction With Extremely Fast Trench Retreat

AbstractThe Tonga‐Kermadec subduction zone exhibits the fastest observed trench retreat and convergence near its northern end. However, a paradox exists: despite the rapid trench retreat, the Tonga slab maintains a relatively steep dip angle above 400 km depth. The slab turns flat around 400 km, then steepening again until encountering a stagnant segment near 670 km. Despite its significance for understanding slab dynamics, no existing numerical model has successfully demonstrated how such a distinct slab morphology can be generated under the fast convergence. Here we run subduction models that successfully reproduce the slab geometries while incorporating the observed subduction rate. We use a hybrid velocity boundary condition, imposing velocities on the arc and subducting plate while allowing the overriding plate to respond freely. This approach is crucial for achieving a good match between the modeled and observed Tonga slab. The results explain how the detailed slab structure is highly sensitive to physical parameters including the seafloor age and the mantle viscosity. Notably, a nonlinear rheology, where dislocation creep reduces upper mantle viscosity under strong mantle flow, is essential. The weakened upper mantle allows for a faster slab sinking rate, which explains the large dip angle. Our findings highlight the utilizing rheological parameters that lead to extreme viscosity variations within numerical models to achieve an accurate representation of complex subduction systems like the Tonga‐Kermadec zone. Our study opens new avenues for further study of ocean‐ocean subduction systems, advancing our understanding of their role in shaping regional and global tectonics.

Numerical study on the motion of two parallel spherical particles with different diameters in upward flow

The settling of particles is related to many industrial processes and research fields. However, due to the complex particle–particle and particle–fluid interactions, the settling mechanism of particles in flowing fluids is not fully understood. This article conducts numerical research on the settling process of two particles with different diameters in parallel in upward flow using the immersion boundary method. The numerical method was validated against experimental results including one particle settling, two parallel particles settling, and two series particles settling. The effects of large particle diameter, upward flow velocity, and initial particle spacing on the settling process were explored. The results indicate that the two particles with same diameter will repel each other when settling in upward flow. Moreover, when the diameters differ, the two particles can experience both attractive and repulsive interactions. The larger the diameter of the large particle, the stronger its attractive influence on the small particle. When the diameter of large particle d2 = 3.0d1, large particle only has an attractive effect on small particle. The wake of each particle forms a distinct velocity boundary with the upward fluid. As the upward flow velocity increases, the interactions between the two particles become increasingly intense. With increasing initial spacing between the particles, their mutual interactions gradually weaken.

Spontaneous ignition and flame propagation in hydrogen/methane wrinkled laminar flames at reheat conditions: Effect of pressure and hydrogen fraction

Direct numerical simulations (DNS) with detailed chemical kinetics and chemical explosive mode analysis (CEMA) are performed to investigate the effect of operating pressure and hydrogen–methane blending on laminar wrinkled flames at reheat combustion conditions. Building upon previous three-dimensional DNS datasets of reheat hydrogen flames, the present work considers two-dimensional configurations that render computationally-feasible simulations of high-pressure combustion and significantly more complex hydrocarbon chemistry. A geometrically-simplified representation of the reheat combustor is adopted and the thermodynamic states considered in the simulations are carefully chosen to mimic gas turbine operation conditions, which are constrained to achieve a target flame position and flame temperature. The two-dimensional DNS results are first assessed against the available three-dimensional data and then analyzed to extract the main trends exhibited by the reheat combustion process with respect to the fraction of the fuel consumed by spontaneous ignition and flame propagation, for increasing pressures and hydrogen fractions. Due to significant differences in the characteristic features of the velocity and thermal boundary layers between the three-dimensional and the two-dimensional configurations, a different global flame shape is observed (V-shaped flame vs W-shape flame) together with a ∼10% bias in the fraction of fuel consumed by spontaneous ignition. Crucially, results from the scaling studies reveal very clear trends with a significant decrease in the fuel consumption by spontaneous ignition for increasing pressures and hydrogen fractions, at the conditions investigated. Finally, this study highlights the important role that first-principle direct numerical simulations, even when conducted in computationally affordable two-dimensional simplified geometrical configurations, can play in obtaining detailed insights and quantitative trends useful for the characterization of complex reactive flows.Novelty and significanceThe reheat burners of the two-stage sequential gas turbine combustors are designed to stabilize flames primarily due to spontaneous ignition. However, prior numerical simulations revealed the presence of an additional assisted-ignition mode aided by recirculation zones. In this study, we used practically feasible two-dimensional direct numerical simulations of premixed hydrogen–air mixtures under lean conditions to quantify the roles of the two flame stabilization modes at different operating pressures. Achieving fundamental understanding of the effect of pressure and hydrogen fraction on flame stabilization is crucial to the development of two-stage sequential combustion and the present two-dimensional calculations provide important new insights. We show that at high pressures, both modes consume fuel to a similar extent. We also investigated the effect of methane blending on flame stabilization. This study also discusses how two-dimensional direct numerical simulations can be leveraged in the design optimization of reheat burners at low technology readiness levels.

Determination of Local Heat Transfer Coefficients and Friction Factors at Variable Temperature and Velocity Boundary Conditions for Complex Flows

Transient conjugate heat transfer measurements under varying temperature and velocity inlet boundary conditions at incompressible flow conditions were performed for flat plate and ribbed channel geometries. Therefrom, local adiabatic wall temperatures and heat transfer coefficients were determined. The data were analyzed using typical heat transfer correlations, e.g., Nu=CRemPrn, determining the local distributions of C and m. It is shown that they are closely linked. A relationship lnC=A−mB is observed, with A and B as modeling parameters. They could be related to parameters in log-law or power-law representations for turbulent boundary layer flows. The parameter m is shown to have a close link to local pressure gradients and, therewith, near wall streamlines as well as friction factor distributions. A normalization of the C parameter allows one to derive a Reynolds analogy factor and, therefrom, local wall shear stresses.

Cattaneo-Christov and Darcy-Forchheimer heat flux on Reiner-Philippoff fluid with Velocity and Thermal Slip Boundary Condition under heat Sink/Source

Cattaneo-Christov and Darcy-Forchheimer heat flux on Reiner-Philippoff fluid with Velocity and Thermal Slip Boundary Condition under heat Sink/Source

Multi-cycle Direct Numerical Simulations of a Laboratory Scale Engine: Evolution of Boundary Layers and Wall Heat Flux

AbstractMulti-cycle direct numerical simulations (DNS) of a laboratory-scale engine at technically relevant engine speeds (1500 and 2500 rpm) are performed to investigate the transient velocity and thermal boundary layers (BL) as well as the wall heat flux during the compression stroke under motored operation. The time-varying wall-bounded flow is characterized by a large-scale tumble vortex, which generates vortical structures as the flow rolls off the cylinder wall. The bulk flow is found to strongly affect the development of the BL profiles, especially at higher engine speeds. As a result, the large-scale flow structures lead to alternating pressure gradients near the wall, invalidating the flow equilibrium assumptions used in typical wall modeling approaches. The thickness of the velocity BL and of the viscous sublayer was found to scale inversely with engine speed and crank angle. The thermal BL thickness also scales inversely with engine speed but increases with in-cylinder temperature. In contrast, thermal displacement thickness, which is sometimes used as a proxy for thermal BL thickness, was found to decrease with increasing temperature in the bulk. Examination of the heat flux distribution revealed areas of increased heat flux, particularly at places characterized by strong flow directed towards the wall. In addition, significant cyclic variations in the surface-averaged wall heat flux were observed for both engine speeds. An analysis of the cyclic tumble ratio revealed that the cycles with lower tumble ratio values near top dead center (TDC), indicative of an earlier tumble breakdown, also exhibit higher surface averaged wall heat fluxes. These findings extend previous numerical and experimental results for the evolution of BL structure during the compression stroke and serve as an important step for future engine simulations under realistic operating conditions.

A multiblock (MIB) finite element method for accurate and efficient blood flow simulation

We present an efficient and accurate immersed boundary (IB) finite element (FE) method for internal flow problems with complex geometries (e.g. blood flow in the vascular system). In this study, we use a voxelized flow domain (discretized with hexahedral and tetrahedral elements) instead of a box domain, which is frequently used in IB methods. We highlight the applicability and efficiency of the proposed method in generating the flow domain mesh and removing unnecessary portions of the domain, compared to the standard IB methods that rely on a full Cartesian mesh. Our approach is attractive for internal flow problems that use IB methods. The proposed method accurately satisfies the boundary conditions on the immersed object using velocity and pressure correction procedures. The velocity correction applies implicitly such that the velocity on the immersed boundary (Lagrangian points) interpolated from the corrected velocity values computed on the mesh nodes (Eulerian nodes) accurately satisfies the prescribed velocity boundary conditions. The proposed method utilizes the well-established incremental pressure correction scheme (IPCS) FE solver, and the boundary condition-enforced IB (BCE-IB) method to numerically solve the transient, incompressible Navier-Stokes (NS) flow equations. We verify the accuracy of our numerical method using the analytical solution for the Poiseuille flow in a cylinder, and the available experimental data (laser Doppler velocimetry) for the flow in a three-dimensional 90° angle tube bend. We further examine the accuracy and applicability of the proposed method by considering flow within complex geometries, such as blood flow in aneurysmal vessels and the aorta, flow configurations that would otherwise be difficult to solve by most IB methods. Our method offers high accuracy, as demonstrated by the verification examples, and high applicability, as demonstrated through the solution of blood flow within complex geometry. The proposed method is efficient, since it is as fast as the traditional finite element method used to solve the Navier–Stokes flow equations, with a small overhead (not more than 5%) due to the numerical solution of a linear system formulated for the IB method.

A hybrid FEM-IBM-level set algorithm for water entry of deformable body

Water entry of deformable bodies has been investigated extensively despite the major challenges in precise simulations of the whole physical process. Inspired by the basic idea of immersed boundary method (IBM), a hybrid FEM-IBM-level set algorithm is proposed where structural compliance is considered. The flow is described by Navier-Stokes equations with an added body force while the drastically moving object of arbitrary geometry is represented by several Lagrangian points on solid boundary. The direct forcing IBM is enhanced and coupled with finite element method (FEM) to facilitate simulation of severe fluid-structure interactions. Consequently, the added body force is calculated implicitly based on the exact velocity boundary condition with divergence-free condition ensured simultaneously. The improved conservative level set technique is adopted for accurate depiction of instantaneously distorted water surface, and the strongly coupled system is resolved iteratively through a partitioned scheme. Examples including sedimentation of a disk, water entry of deformable wedge and cylindrical shell are performed for validation. Specifically, hydroelastic impact of a three-dimensional conical shell is investigated, and the influence factors of entry dynamics are discussed. The results demonstrate the reliability and great potential of present algorithm for practical applications in ocean engineering, naval architecture and military engineering, etc.

Analytical method for studying acoustic fields radiated by general spherical sound sources in thermoviscous fluids.

The study of acoustic radiation from spherical sound sources plays a crucial role in understanding the thermoviscous effects in practical acoustic problems. However, finding a general solution of acoustic radiation from spherical sound sources in thermoviscous fluids remains a formidable challenge. To advance this issue, an analytical method is developed in this paper to calculate the acoustic field radiated by spherical sound sources with the isothermal boundary condition and arbitrary velocity boundary condition. The developed method is utilized to present the solutions of the acoustic field generated by an oscillating sphere and a general spherical sound source, and the accuracy and validity of these solutions are verified through analytical and numerical methods.

Rate-independent mechanism of deformation twinning in single crystal magnesium

The dynamics of {101‾2}〈101‾1〉 extension twins in a single crystal of pure magnesium are studied through uniaxial compression experiments coupled with direct optical imaging. The experiments covered a seven orders of magnitude strain rate, between 10−5−102s−1, covering all rates of engineering applications of magnesium and its alloys. Under uniaxial compression along a direction perpendicular to prismatic planes, only two variants of extension twins are activated, facilitating in-situ tracking of individual twin boundaries throughout the deformation process. Across the entire strain rate range, the twinning transformation evolves in a self-similar pattern of twin nucleation and thickening. Deformation fronts initiate at the edges of the sample and propagate toward the center, creating parallel twins with characteristic thicknesses and distances from each other. The velocity of the deformation fronts scales linearly with the imposed external strain rate and a nearly uniform resolved shear stress is measured over the entire strain rate range. We show that this unique process is attributed to the constant ratio between the nucleation rate and sideways velocity of individual twin boundaries, where both processes are proportional to the imposed strain rate. These findings indicate that, within the large strain rate range investigated in this study, the evolution mechanism of extension twinning is practically rate-independent.