Incompressible Navier Research Articles

Numerical Investigation of Gas Bubble Interaction in a Circular Cross-Section Channel in Shear Flow

This work’s goal is to numerically investigate the interactions between two gas bubbles in a fluid flow in a circular cross-section channel, both in the presence and in the absence of gravitational forces, with several Reynolds and Weber numbers. The first bubble is placed at the center of the channel, while the second is near the wall. Their positions are set in such a way that a dynamic interaction is expected to occur due to their velocity differences. A finite element numerical tool is utilized to solve the incompressible Navier–Stokes equations and simulate two-phase flow using an unfitted mesh to represent the fluid interface, akin to the front-tracking method. The results show that the velocity gradient influences bubble shapes near the wall. Moreover, lower viscosity and surface tension force account for more significant interactions, both in the bubble shape and in the trajectory. In this scenario, it can be observed that one bubble is trapped in the other’s wake, with the proximity possibly allowing the onset of coalescence. The results obtained contribute to a deeper understanding of two-phase inner flows.

Weak adversarial networks for solving forward and inverse problems involving 2D incompressible Navier–Stokes equations

Weak adversarial networks for solving forward and inverse problems involving 2D incompressible Navier–Stokes equations

Kolmogorov’s dissipation number and determining wavenumber for dyadic models

We study some dyadic models for incompressible magnetohydrodynamics and Navier–Stokes equation. The existence of fixed point and stability of the fixed point are established. The scaling law of Kolmogorov’s dissipation wavenumber arises from heuristic analysis. In addition, a time-dependent determining wavenumber is shown to exist; moreover, the time average of the determining wavenumber is proved to be bounded above by Kolmogorov’s dissipation wavenumber. Additionally, based on the knowledge of the fixed point and stability of the fixed point, numerical simulations are performed to illustrate the energy spectrum in the inertial range below Kolmogorov’s dissipation wavenumber.

Linear and nonlinear receptivity mechanisms in boundary layers subject to free-stream turbulence

Large-eddy simulations of a flat-plate boundary layer, without a leading edge, subject to multiple levels of incoming free-stream turbulence are considered in the present work. Within an input–output model, where nonlinear terms of the incompressible Navier–Stokes equations are treated as an external forcing, we manage to separate inputs related to perturbations coming through the intake of the numerical domain, whose evolution represents a linear mechanism, and the volumetric nonlinear forcing due to triadic interactions. With these, we perform the full reconstruction of the statistics of the flow, as measured in the simulations, to quantify pairs of wavenumbers and frequencies more affected by either linear or nonlinear receptivity mechanisms. Inside the boundary layer, different wavenumbers at near-zero frequency reveal streaky structures. Those that are amplified predominantly via linear interactions with the incoming vorticity occur upstream and display transient growth, while those generated by the nonlinear forcing are the most energetic and appear in more downstream positions. The latter feature vortices growing proportionally to the laminar boundary layer thickness, along with a velocity profile that agrees with the optimal amplification obtained by linear transient growth theory. The numerical approach presented is general and could potentially be extended to any simulation for which receptivity to incoming perturbations needs to be assessed.

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.

The Double‐Logarithmic Regularity Criterion of Pressure for the 3D Navier–Stokes Equations in the Besov Space

In the paper, we consider the regularity of the weak solutions for the incompressible 3D Navier–Stokes (N–S) equations. Our main result is the double‐logarithmic regularity criterion of pressure for the 3D Navier–Stokes equations in the Besov space. Our results extend and generalize previous works.

Advancing Finite Difference Solutions for Two‐Dimensional Incompressible Navier–Stokes Equations Using Artificial Compressibility Method and Sparse Matrix Computation

In this article, a numerical scheme for solving two‐dimensional (2D) time‐dependent incompressible Navier–Stokes equations is presented. The artificial compressibility technique is used to incorporate a time derivative of pressure term to the continuity equation. It is employed for pressure–velocity coupling. The scheme consists of backward difference approximation for time derivatives and central difference approximation for spatial derivatives, implemented on a collocated grid. The discretization of the differential equations yields a system of algebraic equations with a block coefficient matrix. To solve this system efficiently, matrix inversion with sparse matrix computation is employed. The proposed numerical scheme is applied to solve three flow problems (lid‐driven cavity flow, rectangular channel flow, and Taylor–Green vortex problem) to validate the accuracy and applicability of the scheme. The results affirm the scheme’s capability to provide precise approximations for solutions to the Navier–Stokes equations. With slight modifications, the scheme can be applied to solve various flow problems with high accuracy, less memory usage, and reduced computational time.

Unconditional energy conservation and conditional regularity for the incompressible Navier-Stokes Maxwell system

In this paper, we study a hydrodynamic system modeling the evolution of a plasma subject to a self-induced electromagnetic Lorentz force in incompressible viscous fluids. The system consists of the Navier–Stokes equations coupled with a Maxwell equation. In the three dimensional case, we show that every weak solution verifies the energy equality for the incompressible Navier–Stokes–Maxwell equations with damping. We also establish some non-explosion criteria in terms of the velocity and magnetic of local strong solution for standard Navier–Stokes–Maxwell system.

Effect of Oldroyd-B fluid on thermal convection stability in a bidispersive porous media with relatively large macropores

This study investigates the effect of Oldroyd-B fluids on the thermal convection instability in a bi-disperse porous medium with relatively large macropores. The modified momentum equation of Oldroyd-B fluid in the macro- and micropores is first derived, assuming local thermal equilibrium. The Orr–Sommerfeld (O–S) equation is derived using the new modified incompressible Navier–Stokes equations of Oldroyd-B fluid. The Chebyshev collocation method is employed to solve the generalized eigenvalue problem, and the neutral stability curves and temporal growth rates are obtained. The results demonstrate that the relaxation time enhances the convective instability, while the retardation time suppresses it. Furthermore, the effect of relaxation time is generally small for smaller wave number, but the effect of relaxation time is significant for large wave number, and this effect becomes accentuated as the wave number increases. The retardation time does not have any influence when the wavenumber is small, and the influence of retardation time becomes small with the large relaxation time. Moreover, the convection instability is enhanced with the higher micropermeability and a higher ratio of momentum transfer coefficient between the macro- and microphases will suppress the instability.

Data assimilation finite element method for the linearized Navier-Stokes equations with higher order polynomial approximation

In this article, we design and analyze an arbitrary-order stabilized finite element method to approximate the unique continuation problem for laminar steady flow described by the linearized incompressible Navier–Stokes equation. We derive quantitative local error estimates for the velocity, which account for noise level and polynomial degree, using the stability of the continuous problem in the form of a conditional stability estimate. Numerical examples illustrate the performances of the method with respect to the polynomial order and perturbations in the data. We observe that the higher order polynomials may be efficient for ill-posed problems, but are also more sensitive for problems with poor stability due to the ill-conditioning of the system.

On global solutions to the inhomogeneous, incompressible Navier–Stokes equations with temperature-dependent coefficients

On global solutions to the inhomogeneous, incompressible Navier–Stokes equations with temperature-dependent coefficients

Global well-posedness of 2D incompressible Navier–Stokes–Darcy flow in a type of generalized time-dependent porosity media

Global well-posedness of 2D incompressible Navier–Stokes–Darcy flow in a type of generalized time-dependent porosity media

Designing the edges of holes (with bias flow) to maximise acoustic damping

Circular holes with a mean bias flow passing through them can amplify or damp acoustic energy and this property is relevant for many industrial applications. In this work, we propose a methodology to design the edges of such holes so that the acoustic damping is maximised. The approach relies on a Bayesian optimisation framework and is illustrated in a short circular hole with a mean laminar bias flow. The acoustic response of the perforation is characterised numerically using a two-step approach where, first, a steady mean flow is computed as the solution of the incompressible Navier–Stokes equations. Second, small-amplitude acoustic perturbations are superimposed on this mean flow and their dynamics are obtained as the solution of the linearised compressible Navier–Stokes equations. Both the upstream and downstream edges of the hole are modified with 45°chamfers. The sizes of these two chamfers are the control parameters optimised to maximise the acoustic absorption coefficient. The results of this letter show that the careful design of the edges can dramatically increase the acoustic energy that holes can damp: the hole investigated here goes from one generating 55% more acoustic energy than incident upon it at a given frequency to one that damps 46% of this acoustic energy.

Optimal decay rate of the incompressible Navier–Stokes–Maxwell system with Ohm’s law

This work is to establish optimal time decay rate of global classical solutions to the two-fluid incompressible Navier–Stokes–Maxwell system with Ohm’s law and the incompressible Navier–Stokes–Maxwell system with Ohm’s law in R3. The results show that the optimal time decay rate for the velocity of these two systems achieves the same as that of the incompressible Navier–Stokes equation. In other words, Ohm’s law can offset the negative effects brought by the Maxwell equation on the decay rate of solutions. This is interesting compared to the compressible Navier–Stokes system, whose time decay rate decreases when it is coupled with the self-consistent Maxwell system.

Derivation and simulation of a two-phase fluid deformable surface model

To explore the impact of surface viscosity on coexisting fluid domains in biomembranes we consider two-phase fluid deformable surfaces as model systems for biomembranes. Such surfaces are modelled by incompressible surface Navier–Stokes–Cahn–Hilliard-like equations with bending forces. We derive this model using the Lagrange–d’Alembert principle considering various dissipation mechanisms. The highly nonlinear model is solved numerically to explore the tight interplay between surface evolution, surface phase composition, surface curvature and surface hydrodynamics. It is demonstrated that hydrodynamics can enhance bulging and furrow formation, which both can further develop to pinch-offs. The numerical approach builds on a Taylor–Hood element for the surface Navier–Stokes part, a semi-implicit approach for the Cahn–Hilliard part, higher-order surface parametrizations, appropriate approximations of the geometric quantities, and mesh redistribution. We demonstrate convergence properties that are known to be optimal for simplified subproblems.

Global solutions to the rotating Navier–Stokes equations with large data in the critical Fourier–Besov spaces

AbstractWe consider the initial value problem for the 3D incompressible Navier–Stokes equations with the Coriolis force. The aim of this paper is to prove the existence of a unique global solution with arbitrarily large initial data in the scaling critical Fourier–Besov spaces (, ), provided that the size of the Coriolis parameter is sufficiently large. Moreover, if the initial data additionally belong to the scaling sub‐critical spaces, we obtain an explicit relationship between the initial data and the Coriolis force, which ensures the existence of a unique global solution.

The production of uncertainty in three-dimensional Navier–Stokes turbulence

We derive the evolution equation of the average uncertainty energy for periodic/homogeneous incompressible Navier–Stokes turbulence and show that uncertainty is increased by strain rate compression and decreased by strain rate stretching. We use three different direct numerical simulations (DNS) of non-decaying periodic turbulence and identify a similarity regime where (a) the production and dissipation rates of uncertainty grow together in time, (b) the parts of the uncertainty production rate accountable to average strain rate properties on the one hand and fluctuating strain rate properties on the other also grow together in time, (c) the average uncertainty energies along the three different strain rate principal axes remain constant as a ratio of the total average uncertainty energy, (d) the uncertainty energy spectrum's evolution is self-similar if normalised by the uncertainty's average uncertainty energy and characteristic length and (e) the uncertainty production rate is extremely intermittent and skewed towards extreme compression events even though the most likely uncertainty production rate is zero. Properties (a), (b) and (c) imply that the average uncertainty energy grows exponentially in this similarity time range. The Lyapunov exponent depends on both the Kolmogorov time scale and the smallest Eulerian time scale, indicating a dependence on random large-scale sweeping of dissipative eddies. In the two DNS cases of statistically stationary turbulence, this exponential growth is followed by an exponential of exponential growth, which is, in turn, followed by a linear growth in the one DNS case where the Navier–Stokes forcing also produces uncertainty.

Local volume-conservation-improved diffuse interface model for simulation of Rayleigh–Plateau fluid instability

The Rayleigh–Plateau instability is a typical multi-phase fluid interface problem which is driven by the surface tension. After the break-up of a liquid thread, the satellites droplets will appear. To prevent the disappearance of small droplets due to interfacial length minimization, we present a new Cahn–Hilliard-type diffuse interface model to improve the conservation of local volume. By coupling with the incompressible Navier–Stokes equations, the fluid flow-coupled binary system and its axisymmetric form are derived. The pressure projection method and the semi-implicit time-marching method are used to discretize the fluid equations and the diffuse interface equations in time. In the axisymmetric domain, the finite difference method is used to perform the spatial discretization. The fully discrete solution algorithm is introduced in detail. The numerical experiments show that the proposed model has better local volume conservation than the classical Cahn–Hilliard model. The qualitative comparison with a real experiment indicates that the proposed model has a good potential to simulate the coating problem of a cylindrical fibre with a liquid film.

Active-learning-driven surrogate modeling for efficient simulation of parametric nonlinear systems

When repeated evaluations for varying parameter configurations of a high-fidelity physical model are required, surrogate modeling techniques based on model order reduction are desirable. In absence of the governing equations describing the dynamics, we need to construct the parametric reduced-order surrogate model in a non-intrusive fashion. In this setting, the usual residual-based error estimate for optimal parameter sampling associated with the reduced basis method is not directly available. Our work provides a non-intrusive error-estimator-based optimality criterion to efficiently populate the parameter snapshots, thereby, enabling us to effectively construct a parametric surrogate model. We consider parameter-specific proper orthogonal decomposition subspaces and propose an active-learning-driven surrogate model using kernel-based shallow neural networks (KSNNs), abbreviated as ActLearn-POD-KSNN surrogate model. The center location for each kernel, along with center-dependent kernel widths, can be learned for the KSNN by using an alternating dual-staged iterative training procedure. To demonstrate the efficiency of our proposed ideas, we present numerical experiments using four physical models, including incompressible Navier–Stokes equations. The ActLearn-POD-KSNN surrogate model efficiently predicts the solution at new parameter locations, even for a setting with multiple interacting shock profiles and a fluid flow scenario with Hopf bifurcation. We also provide an investigation of the surrogate’s performance when the available data is noisy.