Three-dimensional Incompressible Navier Research Articles

Monge–Ampère geometry and vortices

We introduce a new approach to Monge–Ampère geometry based on techniques from higher symplectic geometry. Our work is motivated by the application of Monge–Ampère geometry to the Poisson equation for the pressure that arises for incompressible Navier–Stokes flows. Whilst this equation constitutes an elliptic problem for the pressure, it can also be viewed as a non-linear partial differential equation connecting the pressure, the vorticity, and the rate-of-strain. As such, it is a key diagnostic relation in the quest to understand the formation of vortices in turbulent flows. We study this equation via an associated (higher) Lagrangian submanifold in the cotangent bundle to the configuration space of the fluid. Using our definition of a (higher) Monge–Ampère structure, we study an associated metric on the cotangent bundle together with its pull-back to the (higher) Lagrangian submanifold. The signatures of these metrics are dictated by the relationship between vorticity and rate-of-strain, and their scalar curvatures can be interpreted in a physical context in terms of the accumulation of vorticity, strain, and their gradients. We show explicity, in the case of two-dimensional flows, how topological information can be derived from the Monge–Ampère geometry of the Lagrangian submanifold. We also demonstrate how certain solutions to the three-dimensional incompressible Navier–Stokes equations, such as Hill’s spherical vortex and an integrable case of Arnol’d–Beltrami–Childress flow, have symmetries that facilitate a formulation of these solutions from the perspective of (higher) symplectic reduction.

A comprehensive approach to prediction of fractional flow reserve from deep-learning-augmented model

The underuse of invasive fractional flow reserve (FFR) in clinical practice has motivated research towards non-invasive prediction of FFR. Although the non-invasive derivation of FFR (FFRCT) using computational fluid dynamics (CFD) principles has become a common practice, its clinical application has been limited due to the considerable time required for computation of resulting changes in haemodynamic conditions. An alternative to CFD technology is incorporating a neural network into the computational process to reduce the time necessary for running an effective model.In this study we propose a cascade of data-driven and physic-based neural networks (DP-NN) for predicting FFR (DL-FFRCT). The first network of cascade network DP-NN includes geometric features, and the second network includes physical features. We compare the differences between data-driven neural network (D-NN) and DP-NN for predicting FFR. The training and testing datasets were obtained by solving the three-dimensional incompressible Navier–Stokes equations. Coronary flow and geometric features were used as inputs to train D-NN. In DP-NN the training process involves first training a D-NN to output resting ΔP as one input feature to the DP-NN. Secondly, the physics-based microcirculatory resistance as another input feature to the DP-NN.Using clinically measured FFR as the “gold standard", we validated the computational accuracy of DL-FFRCT in 77 patients. Compared to D-NN, DP-NN improved the prediction of ΔP (R2 = 0.87 vs. R2 = 0.92). Statistical analysis demonstrated that the diagnostic accuracy of DL-FFRCT was not inferior to FFRCT (85.71 % vs. 88.3 %) and the computational time was reduced by a factor of approximately 3000 (4.26 s vs. 3.5 h).DP-NN represents a near real-time, interpretable, and highly accurate deep-learning network, which contributes to the development of high-performance computational methods for haemodynamics. We anticipate that DP-NN will enable near real-time prediction of DL-FFRCT in personalized narrow blood vessels and provide guidance for cardiovascular disease treatments.

Assimilating experimental data of a mean three-dimensional separated flow using physics-informed neural networks

In this article, we address the capabilities of physics-informed neural networks (PINNs) in assimilating the experimentally acquired mean flow of a turbulent separation bubble occurring in a diffuser test section. The training database contains discrete mean pressure and wall shear-stress fields measured on the diffuser surface as well as three-component velocity vectors obtained with particle image velocimetry throughout the volumetric flow domain. Imperfections arise from the measurement uncertainty and the inability to acquire velocity data in the near-wall region. We show that the PINN methodology is suited to handle both of these issues thanks to the incorporation of the underlying physics that, in the present study, are taken into account by minimizing residuals of the three-dimensional incompressible Reynolds-averaged Navier–Stokes equations. As a result, measurement errors are rectified and near-wall velocity profiles are predicted reliably. The latter benefits from the incorporation of wall shear-stress data into the PINN training, which has not been attempted so far to the best of our knowledge. In addition to demonstrating the influence of this novel loss term, we provide a three-dimensional, highly resolved, and differentiable model of a separating and reattaching flow that can be readily used in future studies.

Inertial wave attractors in librating cuboids

Perturbed rapidly rotating flows are dominated by inertial oscillations, with restricted group velocity directions, due to the restorative nature of the Coriolis force. In containers with some boundaries oblique to the rotation axis, the inertial oscillations may focus upon reflections, whereby their energy increases whilst their wavelength decreases and their trajectories focus onto attractor regions. In a linear inviscid setting, these attractors are Delta-like distributions. The linear inviscid setting is obtained formally by setting both Ekman number ${E}$ (ratio of inertial to viscous time scales) and Rossby number ${Ro}$ (non-dimensional amplitude of the forcing that drives the inertial oscillations) to zero. These settings raise fundamental questions, in particular concerning the nature of energy dissipation in the vanishing Ekman number regime. Here, we consider a simple container geometry, a rectangular cuboid, in which the direction of the rotation axis is oblique to four of its walls, subject to librational forcing (small-amplitude harmonic oscillations of the rotation rate). This geometry allows for accurate and efficient direct numerical simulations of the three-dimensional incompressible Navier–Stokes equations with no-slip boundary conditions using a spectral-Galerkin spatial discretisation along with a third-order temporal discretisation. Solutions with Ekman and Rossby numbers as small as ${E}={Ro}=10^{-8}$ reveal many details of how the inertial oscillations focus, at the libration frequency considered, onto attractors, and how the focusing leads to increased localised nonlinear and dissipative processes as ${E}$ and ${Ro}$ are reduced. Even for extremely small forcing amplitudes, nonlinear effects have important dynamic consequences for the attractors.

Scalable methods for computing sharp extreme event probabilities in infinite-dimensional stochastic systems

We introduce and compare computational techniques for sharp extreme event probability estimates in stochastic differential equations with small additive Gaussian noise. In particular, we focus on strategies that are scalable, i.e. their efficiency does not degrade upon temporal and possibly spatial refinement. For that purpose, we extend algorithms based on the Laplace method for estimating the probability of an extreme event to infinite dimensional path space. The method estimates the limiting exponential scaling using a single realization of the random variable, the large deviation minimizer. Finding this minimizer amounts to solving an optimization problem governed by a differential equation. The probability estimate becomes sharp when it additionally includes prefactor information, which necessitates computing the determinant of a second derivative operator to evaluate a Gaussian integral around the minimizer. We present an approach in infinite dimensions based on Fredholm determinants, and develop numerical algorithms to compute these determinants efficiently for the high-dimensional systems that arise upon discretization. We also give an interpretation of this approach using Gaussian process covariances and transition tubes. An example model problem, for which we provide an open-source python implementation, is used throughout the paper to illustrate all methods discussed. To study the performance of the methods, we consider examples of stochastic differential and stochastic partial differential equations, including the randomly forced incompressible three-dimensional Navier–Stokes equations.

Global strong solutions to the Cauchy problem of the 3D heat-conducting fluids

In this paper, we study the global strong solutions to the three-dimensional (3D) heat-conducting incompressible Navier–Stokes equations with density-temperature-dependent viscosity and heat-conducting coefficients in . By using the t-weighted a priori estimates, we prove the global existence and exponential decay-in-time rates of strong solutions to the Cauchy problem when the -norm of the initial density is suitably small. It should be noted that the velocity and absolute temperature can be large initially, and the initial density contains vacuum case.

Effects of coflow-jet active flow control on airfoil stall

Purpose Active flow control on the NACA 0024 airfoil defined as suction-injection jet at the chord-based Reynolds number of 1.5 × 1e + 5 is studied. Design/methodology/approach The three-dimensional incompressible unsteady Reynolds-averaged Navier–Stokes equations with the SST k-ω turbulence model are used to study the effects of coflow-jet (CFJ) on the dynamic and static stall phenomena. CFJ implementation is conducted with several momentum coefficients to investigate their turnover. Furthermore, the current work intends to analyze the CFJ performance by varying the Reynolds number and jet momentum coefficient and comparing all states to the baseline airfoil, which has not been studied in prior research investigations. Findings It is observed that at the momentum coefficient (Cµ) of 0.06, the lift coefficients at low attack angles (up to a = 15) dramatically increase. Furthermore, the dynamic stall at the given Reynolds number and with the lowered frequency of 0.15 is explored. In the instance of Cµ = 0.07, the lift coefficient curve does not show a noticeable stall feature compared to Cµ = 0.05, suggesting that a more powerful stronger jet can entirely control the dynamic stall. Originality/value Furthermore, the current work intends to analyze the CFJ performance by varying the jet momentum coefficient and comparing all states to the baseline airfoil, which has not been studied in prior research investigations.

Global-in-time probabilistically strong and Markov solutions to stochastic 3D Navier–Stokes equations: Existence and nonuniqueness

We are concerned with the three-dimensional incompressible Navier–Stokes equations driven by an additive stochastic forcing of trace class. First, for every divergence free initial condition in L2 we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions, solving one of the open problems in the field. This result, in particular, implies nonuniqueness in law. Second, we prove nonuniqueness of the associated Markov processes in a suitably chosen class of analytically weak solutions satisfying a relaxed form of an energy inequality. Translated to the deterministic setting, we obtain nonuniqueness of the associated semiflows.

Numerical investigation of distinct flow modes in a wide-gap spherical annulus using OpenMP

Spherical Couette flow (SCF) between two rotating concentric spheres has many applications in different fields of Mathematics, Physics, and Engineering. The transitions occurring in this flow phenomena have the relevancy with geophysical motions. In this paper, we use the time-dependent three-dimensional incompressible Navier–Stokes equations (3D-INSEs) to investigate distinct flow modes for a wide-gap clearance ratio (CR) . An Artificial Compressibility method has been used along with three implicit solvers, i.e. alternating direction implicit (ADI), line Gauss–Seidel (LGS), and point Gauss–Seidel (PGS). We apply the basic techniques of OpenMP to our Fortran code. First, we make a performance comparison of the three implicit solvers running on a single central processing unit (CPU) with that of the multi-core CPUs using Open Multi-Processing (OpenMP) to choose the best solver for the simulation of the SCF problem. The comparison shows that the ADI solver is the most efficient solver compared to LGS and PGS solvers. Using the most efficient parallel ADI solver, we compute distinct flow modes for this wide gap in a range of Reynolds number between . For this wide gap, first we obtain the basic 0-vortex flow (0-VF) at . By increasing the the flow becomes symmetric 1-vortex flow (1-VF) and asymmetric spiral 1-VF at and respectively. The flow returns back to super critical basic flow at . Further increasing the the flow becomes spiral 0-VF with m = 4 and m = 5 spiral waves at and respectively. Increasing the further the flow finally becomes turbulent. For this CR, we have first time numerically simulate these flow transitions from basic flow to super critical spiral basic flow with increasing . Abbreviations: ACM: artificial compressibility method; ADI: alternating directionimplicit; CPU: central processing unit; CR: clearance ratio; CVD: circumferentialvelocity distribution; INSEs: incompressible Navier–Stokes equations; LGS: lineGauss–Seidel; OpenMP: open multi-processing; PGS: point Gauss–Seidel; Re: Reynoldsnumber; SCF: spherical Couette flow; SG: spherical-gaps; TG: Taylor–Gortler; TVF:Taylor vortex flow; 0-VF: 0-vortex flow; 1-VF: 1-vortex flow; WENO: weightedessentially non-oscillatory.

On the Unsteady Rotationally Symmetric Flow Between a Stationary and a Finite Rotating Disk With a Given Change in the Axial Velocity

Abstract We consider the scenario of an unsteady viscous flow between two coaxial finite disks, one stationary and the other rotating with an axial velocity changing impulsively from zero to a constant value. The three-dimensional (3D) incompressible Navier–Stokes equations are analytically solved by postulating the polynomial profiles for the axial and circumferential velocity components and by employing the open-end condition of zero pressure difference and an integral approach. It is shown that the time-dependent squeezing of the fluid between the disks and the edge effects of the finite open-ended disks in the flow domain are determined by the compressing Reynolds number and the rotating Reynolds number for the case of laminar flow at low Reynolds numbers and small aspect ratios. The general explicit formulae are derived for the velocity and pressure distributions as a function of the compressing and rotating Reynolds numbers in this unsteady flow process (and the steady-state solutions are then obtainable as the compressing Reynolds number vanishes). A simple theoretical relationship between the radial and axial pressure gradients is deduced to hinge the radial and circumferential velocity components together. The values of the compression and rotation Reynolds numbers suitable to this theory are also suggested for the problem of rotating disk flows at low Reynolds numbers. The validity of the theoretical predictions for the circumferential, radial, and axial velocity components is partially verified through comparison with previous steady experimental and numerical results. These analytical results have the immediate engineering applications of fluid flows with varying gap widths, including wet brakes, wet clutches, hydrostatic bearings, face seals, and rotating heat exchangers.

A forced convection of water aluminum oxide nanofluid flow and heat transfer study for a three dimensional annular with inner rotated cylinder

The article examines a water alumina nanofluid and heat transfer through the three-dimensional annular. The annular is constructed by the two concentric cylinders in which the inner cylinder can rotate along the tangential direction at a constant speed. A slip boundary condition will be imposed to vanish the viscous effect in the vicinity of the outer cylinder wall. Moreover, the rotating cylinder is kept at a hot temperature, and the outer one is at a cold temperature. A three-dimensional incompressible Navier Stokes and energy equations were carried in cylindrical coordinates. The simulation was observed using the emerging computational tool of COMSOL Multiphysics 5.6, which implements Least Square Galerkin's scheme of finite element method. The parametric study will be done by altering the speed of rotation of the inner cylinder from 1 to 4, volume fraction from 0.001 to 0.9, and the aspect ratio from 0.4 to 0.6 for a fixed Reynolds number of 35,000. The results will be displayed with graphs and tables for average values of the Nusselt number, the percentage change in the temperature, and the skin friction at the middle plan. It was found that the average Nusselt number at the middle of the annular increases before the volume fraction of 0.2 and then decreases for all values of the volume fraction for a fixed rotation of the inner cylinder. The average percentage change relative to the inner cylinder's hot temperature decreases with the volume fraction increase for the fixed rotation. Also, it was found that the quantity of nanoparticles in the domain is improving the average skin friction in the middle of the channel, and it can be reduced by improving the rotation of the inner cylinder by about 10–23% strictly depending upon the aspect ratio for a particular case.

The Impact of Floating Raft Aquaculture on the Hydrodynamic Environment of an Open Sea Area in Liaoning Province, China

The sea area of Changhai County in Dalian City is a typical floating raft aquaculture area, located in Liaoning Province, China, where a key issue in determining the scale and spatial layout of the floating raft aquaculture is the assessment of the impact of aquaculture activities on the hydrodynamic environment. To address this issue, we established depth-averaged two-dimensional shallow water equations and three-dimensional incompressible Reynolds-averaged Navier–Stokes equations for the open sea area described in this paper. The impact of floating rafts for aquaculture on hydrodynamic force was reflected in the numerical model by changing the Manning number, where scenarios with different aquaculture densities were taken into account. Finally, the water exchange rate of the floating raft aquaculture area in the study area was calculated. It was found, through a comparison between the simulated value and the measured value obtained via layered observation, that the two values were in good agreement with each other, indicating that the model exhibits great accuracy. In addition, the calculation results for scenarios before and after aquaculture were compared and analyzed, showing that from low-density to high-density aquaculture zones, the variation in flow rate was greater than 80% at the peak of a flood tide. The water exchange rates of the water body after 1 day, 4 days, and 8 days of water exchange were also calculated, and the results show that they had been reduced by 17.92%, 13.59%, and 1.63%, respectively, indicating that the existence of floating rafts for aquaculture indeed reduced the water exchange capacity of the water body. The model described in this paper can serve as a foundation for other studies on aquaculture in open sea areas, and it provides a theoretical basis for the scientific formulation of marine aquaculture plans and the rational optimization of the spatial layout.

Three-dimensional flows of incompressible Navier–Stokes fluids in tubes containing a sinus, with varying slip conditions at the wall

The objective of this study is to understand the formation of vortices and other flow characteristics associated with the three-dimensional motions of an incompressible Navier–Stokes fluid in tubes containing a sinusoidal extension. The study has some bearing on two conjectures that da Vinci made concerning the flow of blood through the aortic root. We investigate how the flow attributes change with increasing sinus radius and with the nature of the slip at the tube wall characterized by the parameter θ , 0 ≤ θ ≤ 1 , with θ values being 0 for free slip and 1 for the “no-slip” (adherence) condition. Two time-dependent solvers – one fully three-dimensional and the other based on the assumption that admissible flows are axially symmetric – are used to solve the equations governing the flow in a geometry associated with the inflow velocities and the other conditions closely related to flows of blood in a blood vessel containing the aortic root. Both these solvers passed a benchmark test based on steady flow in a cylinder with different slip conditions. Computing the flows systematically using both solvers, focusing first on problems with constant-in-time inflow, we have found that for the circular cylinder with constant cross-section, serving as the reference domain, the flow is steady and unidirectional for all boundary conditions that are considered, with maximal vorticity and dissipation for no-slip. For tubes with the sinus radius up to 16 mm the flow remains steady and axially symmetric, but the vorticity and the bulk dissipation in the sinus are maximal for θ between 0.6 and 0.7. For a tube with the sinus radius of 20 mm the flow remains steady and axially symmetric only for θ greater or equal than 0.9 including the case of no-slip. For θ below this value, not only does the solution become unsteady (oscillatory or even chaotic), it does not converge to the steady state and is thus different from the corresponding axially symmetric flows, and also the total dissipation and the vorticity depend on θ in a non-monotone manner. In particular, for a tube with sinus radius of 20 mm, the bulk dissipation and the vorticity are the highest for θ = 0 , i.e. for (free) slip of the fluid at the wall. For tubes with the sinus radius of 20 mm and for θ below 0.9, we computed two different solutions (one steady, axially symmetric, the other unsteady, fully three-dimensional) to three-dimensional evolutionary incompressible Navier–Stokes equations for the same set of the initial and boundary data. This last result supports the idea that, under the conditions considered, the axially symmetric solution looses its stability. Finally, we computed the problems with a pulsatile inflow. For the tube with the largest sinus radius of 20 mm, we observed that the full problem is axially symmetric for the slip parameter θ ≥ 0 . 9 , while for more significant slip, i.e. for θ < 0 . 9 , we again obtained two different solutions under the same initial and boundary conditions.

Stabilized reduced-order models for unsteady incompressible flows in three-dimensional parametrized domains

In this work we derive a parametric reduced-order model (ROM) for the unsteady three-dimensional incompressible Navier–Stokes equations without additional pre-processing on the reduced-order subspaces. Concerning the high-fidelity, full-order model, we start from a streamline-upwind Petrov–Galerkin stabilized finite element discretization of the equations using P1−P1 elements for velocity and pressure, respectively. We rely on Galerkin projection of the discretized equations onto reduced basis subspaces for the velocity and the pressure, respectively, obtained through Proper Orthogonal Decomposition on a dataset of snapshots of the full-order model. Both nonlinear and nonaffinely parametrized algebraic operators of the reduced-order system of nonlinear equations, including the projection of the stabilization terms, are efficiently assembled exploiting the Discrete Empirical Interpolation Method (DEIM), and its matrix version (MDEIM), thus obtaining an efficient offline–online computational splitting. We apply the proposed method to (i) a two-dimensional lid-driven cavity flow problem, considering the Reynolds number as parameter, and (ii) a three-dimensional pulsatile flow in stenotic vessels characterized by geometric and physiological parameter variations. We numerically show that the projection of the stabilization terms on the reduced basis subspace and their reconstruction using (M)DEIM allows to obtain a stable ROM with coupled velocity and pressure solutions, without any need for enriching the reduced velocity space, or further stabilizing the ROM. Additionally, we demonstrate that our implementation allows to compute the ROM solution about 20 times faster than the full order model.

Simulation of drop impact on substrate with micro-wells

In this paper, we numerically investigate drop impact on a micro-well substrate to understand the phenomena of non-wettability. The simulation is carried out by solving three-dimensional incompressible Navier–Stokes equations using a density projection method and an adaptive grid refinement algorithm. A very sharp interface reconstruction algorithm, known as the moment-of-fluid method, is utilized to identify the multi-materials and multi-phases present in the computation domain. Our simulations predicted that a micro-well with a deep cavity can significantly reduce a solid–liquid contact in the event of drop impact. The results from the drop impact on the micro-well substrate are compared with results from drop impact on a flat substrate. Significant differences are observed between these two cases in terms of wetted area, spreading ratio, and kinetic energy. Our simulation shows that under the same conditions, a drop is more apt to jump from a micro-well substrate than from a flat surface, resulting in smaller wetted area and shorter contact time. Based on the simulation results, we draw a drop jumping region map. The micro-well substrate has a larger region than the flat surface substrate. Finally, we present a comparative analysis between a flat substrate and a substrate constructed with a dense array of micro-wells and, therefore, show that the array of micro-wells outperforms the smooth substrate with regard to non-wettability and drop wicking capability.

Numerical Simulation of the Force Acting on the Riser by Two Internal Solitary Waves

An internal wave is a typical dynamic process. As an internal wave, an internal solitary wave usually occurs between two layers of fluids with different densities. Compared with general internal waves, internal solitary waves have large amplitudes, fast propagation speeds, short-wave periods, and often have tremendous energy. The propagation causes strong convergence and divergence of seawater and generates a sudden strong current. Due to its various characteristics, the propagation of internal solitary waves can cause serious harm to offshore engineering structures. Therefore, studying the effects of internal solitary waves on risers is vital in preventing environmental pollution caused by riser damage. Although the research on internal solitary waves has achieved very fruitful results, the research on structures is mostly focused on a single condition, and the occurrence of internal solitary wave, as a complex ocean phenomenon, is often accompanied by many situations. Therefore, this paper constructs a numerical simulation of the interaction between two columns of internal solitary waves and risers. This study explores the force and flow field changes of the riser under the condition of multiple internal solitary waves using the Star-CCM+ software in the simulation. The improved K-epsilon turbulence model was adopted to close the three-dimensional incompressible Navier–Stokes equation, and the solitary wave solution of the eKdV equation was used as the initial and boundary conditions. The interaction between single and double internal solitary waves and a riser was calculated, compared, and analyzed using numerical analysis. The experiment results indicate that the conditions of two internal solitary waves differ from those of a single internal solitary wave. After colliding at the riser, the waves gradually merge into a single wave, and the flow field reaches its minimum velocity. Under the two-wave condition, the horizontal force on the riser as a whole is less than the single-wave condition. As the amplitude difference between the two internal solitary waves gradually decreases, the horizontal opposing force received by the riser first increases and then decreases, while the horizontal positive force gradually decreases.

Spontaneous symmetry breaking for extreme vorticity and strain in the three-dimensional Navier-Stokes equations.

We investigate the spatio-temporal structure of the most likely configurations realizing extremely high vorticity or strain in the stochastically forced three-dimensional incompressible Navier–Stokes equations. Most likely configurations are computed by numerically finding the highest probability velocity field realizing an extreme constraint as solution of a large optimization problem. High-vorticity configurations are identified as pinched vortex filaments with swirl, while high-strain configurations correspond to counter-rotating vortex rings. We additionally observe that the most likely configurations for vorticity and strain spontaneously break their rotational symmetry for extremely high observable values. Instanton calculus and large deviation theory allow us to show that these maximum likelihood realizations determine the tail probabilities of the observed quantities. In particular, we are able to demonstrate that artificially enforcing rotational symmetry for large strain configurations leads to a severe underestimate of their probability, as it is dominated in likelihood by an exponentially more likely symmetry-broken vortex-sheet configuration.This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.

Non-Invasive Quantification of Fraction Flow Reserve Based on Steady-State Geometric Multiscale Models.

Background: The underuse of invasive fraction flow reserve (FFR) in clinical practice has motivated research towards its non-invasive prediction. The early attempts relied on solving the incompressible three-dimensional Navier–Stokes equations in segmented coronary arteries. However, transient boundary condition has a high resource intensity in terms of computational time. Herein, a method for calculating FFR based on steady-state geometric multiscale (FFRSS) is proposed. Methods: A total of 154 moderately stenotic vessels (40–80% diameter stenosis) from 136 patients with stable angina were included in this study to validate the clinical diagnostic performance of FFRSS. The method was based on the coronary artery model segmented from the patient’s coronary CTA image. The average pressure was used as the boundary condition for the inlet, and the microcirculation resistance calculated by the coronary flow was used as the boundary condition for the outlet to calculate the patient-specific coronary hyperemia. Then, the flow velocity and pressure distribution and the FFRss of each coronary artery branch were calculated to evaluate the degree of myocardial ischemia caused by coronary stenosis. Also, the FFRSS and FFRCT of all patients were calculated, and the clinically measured FFR was used as the “gold standard” to verify the diagnostic performance of FFRSS and to compare the correlation between FFRSS and FFRCT. Results: According to the FFRSS calculation results of all patients, FFRSS and FFR have a good correlation (r = 0.68, p < 0.001). Similarly, the correlation of FFRSS and FFRCT demonstrated an r of 0.75 (95%CI: 0.67–0.72) (p < 0.001). On receiver-operating characteristic analysis, the optimal FFRSS cut point for FFR≤0.80 was 0.80 (AUC:0.85 [95% confidence interval: 0.79 to 0.90]; overall accuracy:88.3%). The overall sensitivity, specificity, PPV, and NPV for FFRSS ≤0.80 versus FFR ≤0.80 was 68.18% (95% CI: 52.4–81.4), 93.64% (95% CI: 87.3–97.4), 82.9%, and 91.1%, respectively. Conclusion: FFRSS is a reliable diagnostic index for myocardial ischemia. This method was similar to the closed-loop geometric multiscale calculation of FFR accuracy but improved the calculation efficiency. It also improved the clinical applicability of the non-invasive computational FFR model, helped the clinicians diagnose myocardial ischemia, and guided percutaneous coronary intervention.

Deformation and acceleration of water droplet in continuous airflow

The present work investigates the deformation and acceleration of water droplets in continuous airflow. The numerical approach is based on the level set method for capturing the liquid–gas interface and the projection method for solving the three-dimensional incompressible Navier–Stokes equations. The effects of the incoming airflow velocity (10–100 m/s), initial droplet diameter (20–100 μm), and supercooling on water droplet deformation are investigated. The results indicate that the droplet enters the breakup regime at a critical Weber number of 10, which agrees with the published literature. A dimensionless deformation factor L is defined to describe the droplet deformation. The results confirm that the extreme values of L increase with increasing Weber number during droplet movement. Two models are proposed to predict the minimum deformation factor and the corresponding dimensionless time. The effect of supercooling on water droplet deformation is analyzed, and it is found that the deformation factor of supercooled droplets is lower than that of room-temperature droplets, while supercooled water droplets exhibit greater acceleration. Furthermore, based on the numerical results, a model governed by the Weber number and the Ohnesorge number is proposed for predicting droplet acceleration.

Global solutions to 3D incompressible Navier–Stokes equations with some large initial data

In this paper, we derive a new smallness hypothesis of initial data for the three-dimensional incompressible Navier–Stokes equations. More precisely, we prove that if ‖u01+u02‖Ḃp,13p−1+‖u03‖Ḃp,13p−1‖u01‖Ḃp,13p−1+‖u02‖Ḃp,13p−1×expC(‖u0‖Ḃ∞,2−12+‖u0‖Ḃ∞,1−1) is small enough, the Navier–Stokes equations have a unique global solution. As an application, we construct two examples of initial data satisfying the smallness condition, but whose Ḃ∞,∞−1(R3) norm can be arbitrarily large.