Nonlinear Navier-Stokes Equations Research Articles

Computational fluid dynamics: Its carbon footprint and role in carbon reduction

Turbulent flow physics regulates the aerodynamic properties of lifting surfaces, the thermodynamic efficiency of vapor power systems, and exchanges of natural and anthropogenic quantities between the atmosphere and ocean, to name just a few applications of contemporary importance. The space-time dynamics of turbulent flows are described via numerical integration of the non-linear Navier–Stokes equation—a procedure known as computational fluid dynamics (CFD). At the dawn of scientific computing in the late 1950s, it would be many decades before terms such as “carbon footprint” or “sustainability” entered the lexicon, and longer still before these themes attained national priority throughout advanced economies. The environmental cost associated with CFD is seldom considered. Yet, large-scale scientific computing relies on intensive cooling realized via external power generation that is primarily accomplished through the combustion of fossil fuels, which leads to carbon emissions. This paper introduces a framework designed to calculate the carbon footprint of CFD and its contribution to carbon emission reduction strategies. We will distinguish between “hero” and “routine” calculations, noting that the carbon footprint of hero calculations—which demand significant computing resources at top-tier data centers—is largely determined by the energy source mix utilized. We will also review CFD of flows where turbulence effects are modeled, thus reducing the degrees of freedom. Estimates of the carbon footprint are presented for such fully and partially resolved simulations as functions of turbulence activity and calculation year, demonstrating a reduction in carbon emissions by two to five orders of magnitude at practical conditions. Besides generating a carbon footprint, the community's effort to avoid redundant calculations via turbulence databases merits particular attention, with estimates indicating that a single database could potentially reduce CO2 emissions by approximately O(1) × 106 metric tons.

Macroscopic finite-difference scheme and modified equationsof the general propagation multiple-relaxation-time lattice Boltzmann model.

In this paper we first present the general propagation multiple-relaxation-time lattice Boltzmann (GPMRT-LB) model and obtain the corresponding macroscopic finite-difference (GPMFD) scheme on conservative moments. Then based on the Maxwell iteration method, we conduct the analysis on the truncation errors and modified equations(MEs) of the GPMRT-LB model and GPMFD scheme at both diffusive and acoustic scalings. For the nonlinear anisotropic convection-diffusion equation(NACDE) and Navier-Stokes equations(NSEs), we also derive the first- and second-order MEs of the GPMRT-LB model and GPMFD scheme. In particular, for the one-dimensional convection-diffusion equation(CDE) with the constant velocity and diffusion coefficient, we can develop a fourth-order GPMRT-LB (F-GPMRT-LB) model and the corresponding fourth-order GPMFD (F-GPMFD) scheme at the diffusive scaling. Finally, three benchmark problems, the Gauss hill problem, the CDE with nonlinear convection and diffusion terms, and the Taylor-Green vortex flow in two-dimensional space, are used to test the GPMRT-LB model and GPMFD scheme, and it is found that the numerical results not only are in good agreement with corresponding analytical solutions, but also have a second-order convergence rate in space. Additionally, a numerical study on one-dimensional CDE also demonstrates that the F-GPMRT-LB model and F-GPMFD scheme can achieve a fourth-order accuracy in space, which is consistent with our theoretical analysis.

Spatially-Periodic Solutions for Evolution Anisotropic Variable-Coefficient Navier–Stokes Equations: I. Weak Solution Existence

We consider evolution (non-stationary) spatially-periodic solutions to the n-dimensional non-linear Navier–Stokes equations of anisotropic fluids with the viscosity coefficient tensor variable in spatial coordinates and time and satisfying the relaxed ellipticity condition. Employing the Galerkin algorithm with the basis constituted by the eigenfunctions of the periodic Bessel-potential operator, we prove the existence of a global weak solution.

Physics-Informed Machine Learning of Argon Gas-Driven Melt Pool Dynamics

Abstract Melt pool dynamics in metal additive manufacturing (AM) is critical to process stability, microstructure formation, and final properties of the printed materials. Physics-based simulation, including computational fluid dynamics (CFD), is the dominant approach to predict melt pool dynamics. However, the physics-based simulation approaches suffer from the inherent issue of very high computational cost. This paper provides a physics-informed machine learning method by integrating the conventional neural networks with the governing physical laws to predict the melt pool dynamics, such as temperature, velocity, and pressure, without using any training data on velocity and pressure. This approach avoids solving the nonlinear Navier–Stokes equation numerically, which significantly reduces the computational cost (if including the cost of velocity data generation). The difficult-to-determine parameters' values of the governing equations can also be inferred through data-driven discovery. In addition, the physics-informed neural network (PINN) architecture has been optimized for efficient model training. The data-efficient PINN model is attributed to the extra penalty by incorporating governing PDEs, initial conditions, and boundary conditions in the PINN model.

Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid

Abstract This work is devoted to study the convection–reaction–diffusion behavior of contaminant in the recovered fracturing fluid which flows in the wellbore from shale gas reservoir. First, we apply various constitutive laws for generalized non-Newtonian fluids, diffusion principles, and friction relations to formulate the recovered fracturing fluid model. The latter is a partial differential system composed of a nonlinear and nonsmooth stationary incompressible Navier-Stokes equation with a multivalued friction boundary condition, and a nonlinear convection–reaction–diffusion equation with mixed Neumann boundary conditions. Then, we provide the weak formulation of the fluid model which is a hemivariational inequality driven by a nonlinear variational equation. We establish existence of solutions to the recovered fracturing fluid model via a surjectivity theorem for multivalued operators combined with an alternative iterative method and elements of nonsmooth analysis.

Stable production of fluid jets with vanishing diameters via tip streaming

We study numerically the microjetting mode obtained when a fluid is injected through a tube submerged in a uniaxial extensional flow. The steady solution to the full nonlinear Navier–Stokes equations is calculated. We obtain the linear global modes determining the linear stability of the steady solution. For sufficiently large outer viscosity, the flow remains stable for infinitely small values of the injected flow rate. This implies that jets with vanishing diameters can be produced regardless of the jet viscosity and outer flow strength. For a sufficiently small inner-to-outer viscosity ratio, the microjetting instability is associated only with the flow near the entrance of the jet. The tapering meniscus stretches and adopts a slender quasiconical shape. Consequently, the cone tip is exposed to an intense outer flow, which stabilizes the flow in the cone–jet transition region. This work presents the first evidence that fluid jets with arbitrarily small diameters can be stably produced via tip streaming. The results are related to those of a droplet in a uniaxial extensional flow with its equator pinned to an infinitely thin ring. The pinning of the equator drastically affects the droplet stability and breakup.

Accelerated simulation methodologies for computational vascular flow modelling.

Vascular flow modelling can improve our understanding of vascular pathologies and aid in developing safe and effective medical devices. Vascular flow models typically involve solving the nonlinear Navier-Stokes equations in complex anatomies and using physiological boundary conditions, often presenting a multi-physics and multi-scale computational problem to be solved. This leads to highly complex and expensive models that require excessive computational time. This review explores accelerated simulation methodologies, specifically focusing on computational vascular flow modelling. We review reduced order modelling (ROM) techniques like zero-/one-dimensional and modal decomposition-based ROMs and machine learning (ML) methods including ML-augmented ROMs, ML-based ROMs and physics-informed ML models. We discuss the applicability of each method to vascular flow acceleration and the effectiveness of the method in addressing domain-specific challenges. When available, we provide statistics on accuracy and speed-up factors for various applications related to vascular flow simulation acceleration. Our findings indicate that each type of model has strengths and limitations depending on the context. To accelerate real-world vascular flow problems, we propose future research on developing multi-scale acceleration methods capable of handling the significant geometric variability inherent to such problems.

Topological sensitivity analysis for the 3D nonlinear Navier–Stokes equations

This work is devoted to a topological asymptotic expansion for the nonlinear Navier–Stokes operator. We consider the 3D Navier–Stokes equations as a model problem and we derive a topological sensitivity analysis for a design function with respect to the insertion of a small obstacle inside the fluid flow domain. The asymptotic behavior of the perturbed velocity field with respect to the obstacle size is examined. The performed mathematical framework can be applied for a large class of design functions and arbitrarily shaped geometric perturbations. The obtained asymptotic formula can serve as a useful tool for solving a variety of topology optimization problems in fluid mechanics.

Flow of Newtonian Incompressible Fluids in Square Media: Isogeometric vs. Standard Finite Element Method

This article highlights a study focused on resolving a nonlinear problem in fluid dynamics using the Navier–Stokes equations as a mathematical model. The study focuses on comparing the isogeometric analysis (IGA) B-spline method with the traditional finite element method (FEM) in a two-dimensional context. The objective is to showcase the superior performance of the IGA method in terms of result quality and computational efficiency. The study employs GEOPDE’s MATLAB code for implementing and computing the NURBS method and COMSOL Software’s FEM code for comparison. The advantages of the IGA B-spline method are highlighted, including its ability to accurately capture complex flow behavior and its reduced computation time compared to FEM. The study aims to establish the superiority of the IGA method in solving nonlinear Navier–Stokes equations, providing valuable insights for fluid dynamics and practical implications for engineering simulations.

On Exact Solutions to Problems of Steady Flow of Incompressible Viscous Fluid in a Channel and in a Circular Pipe

Turning to the well-known Couette flow, we again, but now in a new formulation, consider the steady motion of an incompressible viscous fluid in a channel with two parallel flat walls, one of which is stationary while the other moves at a constant speed in its plane. In the classical formulation of this problem on a layered (laminar) fluid flow in a channel, the wall velocity and the constant pressure drop along the channel are independent quantities and are considered predetermined. However, when a solid body moves along the fluid surface, the fluid particles are set in motion. This occurs due to the arising pressure generated in the fluid caused by the movement of the solid body. Therefore, here the pressure or pressure drop is not predetermined but is determined as a result of solving the corresponding two-dimensional boundary value problem of a steady flow of a viscous fluid in a channel. For this, the corresponding boundary value problem of a fluid flow in the channel is formulated based on the linearized Navier-Stokes equations obtained from the general nonlinear Navier-Stokes equations for a steady flow of a viscous fluid neglecting the convective terms, which is true for small Reynolds numbers. Using the Fourier integral transform method, an exact (closed) solution to this boundary value problem is constructed; the velocity and pressure components are determined. Then it is shown that the resulting solution, different from the known Couette flow solution, also satisfies the original nonlinear Navier-Stokes equations, the continuity equation, and boundary conditions. A comparative analysis of the new solution with the Couette flow solution is carried out. The well-known Hagen-Poiseuille problem is also discussed from the point of view of establishing the conditions for the laminar axisymmetric and steady flow of an incompressible viscous fluid in a straight circular pipe According to the study's results, to implement laminar flow under the indicated conditions, the pressure drop along the pipe must be constant. The study demonstrates that this condition is simultaneously sufficient, and therefore, the necessary and sufficient condition is established for a laminar flow in a straight circular pipe.

On the Design of Global-in-Time Newton-Multigrid-Pressure Schur Complement Solvers for Incompressible Flow Problems

In this work, a new global-in-time solution strategy for incompressible flow problems is presented, which highly exploits the pressure Schur complement (PSC) approach for the construction of a space–time multigrid algorithm. For linear problems like the incompressible Stokes equations discretized in space using an inf-sup-stable finite element pair, the fundamental idea is to block the linear systems of equations associated with individual time steps into a single all-at-once saddle point problem for all velocity and pressure unknowns. Then the pressure Schur complement can be used to eliminate the velocity fields and set up an implicitly defined linear system for all pressure variables only. This algebraic manipulation allows the construction of parallel-in-time preconditioners for the corresponding all-at-once Picard iteration by extending frequently used sequential PSC preconditioners in a straightforward manner. For the construction of efficient solution strategies, the so defined preconditioners are employed in a GMRES method and then embedded as a smoother into a space–time multigrid algorithm, where the computational complexity of the coarse grid problem highly depends on the coarsening strategy in space and/or time. While commonly used finite element intergrid transfer operators are used in space, tailor-made prolongation and restriction matrices in time are required due to a special treatment of the pressure variable in the underlying time discretization. The so defined all-at-once multigrid solver is extended to the solution of the nonlinear Navier–Stokes equations by using Newton’s method for linearization of the global-in-time problem. In summary, the presented multigrid solution strategy only requires the efficient solution of time-dependent linear convection–diffusion–reaction equations and several independent Poisson-like problems. In order to demonstrate the potential of the proposed solution strategy for viscous fluid simulations with large time intervals, the convergence behavior is examined for various linear and nonlinear test cases.

Numerical treatment of MHD Al2O3–Cu/engine oil-based nanofluid flow in a Darcy–Forchheimer medium: Application of radiative heat and mass transfer laws

This computational study was designed to examine the effect of alumina and copper on the flow of a nanofluid based on engine oil through a Darcy–Forchheimer porous media. The flow model incorporates the generalized radiative heat and mass transport rules. The Darcy–Forchheimer terms in the momentum equation and radiation term in the energy equation are integral parts of the governing nonlinear Navier–Stokes equations, which are two-dimensional partial differential equations. Under the constraint of the convective boundary, the stated PDEs are transformed into highly nonlinear versions of the ordinary differential equations. In order to solve the final ODEs, the numerical RK-45 approach is combined with the shooting methodology. Important aspects of the governing model include porosity, magnetohydrodynamics (MHD), a convective boundary, thermal radiation, and viscous dissipation. The final findings show that when the Forchheimer number increases, the velocity decreases because of the inertial effect included in the flow model. In addition, the velocity profile is improved due to the increased volume percentage of both kinds of nanoparticles. The temperature varies greatly depending on the volume fraction. A higher Biot number and the resulting convective border cause a greater heat flow than a non-convective barrier. For two particular examples, with and without MHD influence, interesting streamlines and contour graphs are produced.

Differentiability Properties for Boundary Control of Fluid-Structure Interactions of Linear Elasticity with Navier-Stokes Equations with Mixed-Boundary Conditions in a Channel

In this paper we consider a fluid-structure interaction problem given by the steady Navier Stokes equations coupled with linear elasticity taken from (Lasiecka et al. in Nonlinear Anal 44:54–85, 2018). An elastic body surrounded by a liquid in a rectangular domain is deformed by the flow which can be controlled by the Dirichlet boundary condition at the inlet. On the walls along the channel homogeneous Dirichlet boundary conditions and on the outflow boundary do-nothing conditions are prescribed. We recall existence results for the nonlinear system from that reference and analyze the control to state mapping generalizing the results of (Wollner and Wick in J Math Fluid Mech 21:34, 2019) to the setting of the nonlinear Navier-Stokes equation for the fluid and the situation of mixed boundary conditions in a domain with corners.

Fictitious domain method with the idea of conjugate optimization for non-linear Navier-Stokes equations

Fictitious domain method with the idea of conjugate optimization for non-linear Navier-Stokes equations

Numerical simulation of nonlinear phenomena in a standing-wave thermoacoustic engine with gas-liquid coupling oscillation

• A CFD model of gas-liquid standing-wave thermoacoustic engine is developed and experimentally verified. • Time-domain impedance boundary condition is employed to truncate the modelling of liquid column. • Entire transient evolution of self-excited thermoacoustic oscillation from onset to saturation is captured. • Three different acoustic streamings are observed through inspecting the time-averaged mass flux field. A computational fluid dynamics (CFD) model based on the compressible, nonlinear Navier-Stokes equations is developed to simulate the nonlinear phenomena in a gas-liquid standing-wave thermoacoustic engine, which employs gas as the working fluid and liquid as the phase-matching element. The CFD model is well validated when comparing the simulated onset and steady-state performances with those measured from a real experimental system. Based on the CFD model, the nonlinear phenomena of the engine are analyzed from three aspects. Firstly, the nonlinear dynamic phenomena of self-excited thermoacoustic oscillation from onset to saturation are investigated, focusing on onset temperature difference and steady-state pressure amplitude. Then, the nonlinear acoustic phenomena of the engine operating at steady state are analyzed. The steady-state fluctuating pressure is decomposed into a superimposition of the acoustic modes with different frequencies, which reveals the existence of high-order harmonics. Finally, the nonlinear hydrodynamic phenomena including multi-dimensional flow effect and mass streaming are observed by examining the steady-state flow fields in the engine. The proposed CFD model provides an effective approach to comprehending the mechanism of nonlinear phenomena in gas-liquid coupling thermoacoustic engines.

Analysis of Pseudo-spectral Methods Used for Numerical Simulations of Turbulence

Events in turbulent flows computed by direct numerical simulation (DNS) are often calibrated with properties based on homogeneous isotropic turbulence, advanced by Kolmogorov, and given in Turbulence, U. Frisch, Cambridge Univ. Press, UK (1995). However, these computational procedures are not calibrated using numerical analyses in order to assess their strengths and weaknesses for DNS. This is with the exception in "A critical assessment of simulations for transitional and turbulence flows- Sengupta, T.K., In Proc. of IUTAM Symp. on Advances in Computation, Modeling and Control of Transitional and Turbulent Flows, 491-532, (eds. Sengupta, Lele, Sreenivasan, Davidson), WSPC, Singapore (2016)", where such a calibration has been advocated for numerical schemes using global spectral analysis (GSA) for the convection equation. In recent times, due to growing computing power, simulations have been reported using pseudo-spectral methods, with spatial discretization performed in Fourier spectral space and time-integration by multi-stage Runge-Kutta (RK) methods. Here, we perform GSA of Fourier spectral methods for the first time with RK2 and other multistage Runge-Kutta methods using the model linear convection and convection-diffusion equations. With the help of GSA, various sources of numerical errors are quantified. The major limitations of the RK2-Fourier spectral method are demonstrated for DNS and alternate choices are presented. We specify optimal parameters to achieve the best possible accuracy for simulations. There is a one-to-one correspondence of numerical solutions obtained by linear convection-diffusion equation and nonlinear Navier-Stokes equation with respect to numerical parameters. This enables us to investigate the capabilities of the numerical methods for DNS.

A numerical approach for solving one nonlinear problem of hydrodynamics in a non-convex polygonal domain

The paper considers the two-dimensional stationary problem obtained as a result of time discretization of the nonlinear Navier-Stokes equations in rotation form, which describe the incompressible viscous fluid flow in a non-convex polygonal domain. In order to solve a nonlinear problem, it is necessary to find solutions to a sequence of approximate linear problems. For linear problems, the concept of an Rv-generalized solution in weighted sets obeying the weighted analogue of the Ladyzhenskaya-Babuška-Brezzi condition is introduced. A scheme of the weighted finite element method is constructed which suppresses the error arising in the neighborhood of the vertex of the reentrant angle and does not allow it to propagate into the inner part of the computational domain. The mass conservation law is valid directly at the grid nodes, and not only in the (weak) integral sense. Numerical experiments are carried out and a comparative analysis is made in polygonal domains with different values of reentrant angles. The advantage of the proposed approach over the classical finite element method is the order of convergence with respect to the grid step. A set of optimal parameters of the method is determined to obtain the required order of convergence. In this case, the exponent of the weight function depends on the value of the reentrant angle, and the parameters determining the Rv-generalized solution are independent on its value. In contrast to the classical approach, the order of convergence of the proposed approximate method to the exact solution of a nonlinear problem does not depend on the value of the reentrant angle. The constructed approximate method can be used without geometric mesh refinement in the vicinity of the singularity point.

A reduced order method for nonlinear parameterized partial differential equations using dynamic mode decomposition coupled with k-nearest-neighbors regression

Accurately constructing a reduced order model (ROM) of nonlinear parameterized partial differential equations (PDEs) has always been a challenging problem in engineering and applied sciences. Dynamic mode decomposition (DMD) is a popular and efficient data-driven method for ROM, however, it is proposed for the model order reduction of time-dependent problems that it is inapplicable for the parameterized problems. In this paper, a new ROM is proposed based on the k-nearest-neighbors (KNN) regression and DMD, namely, KNN-DMD. The KNN can approximate the solution at any given parameter value by choosing and averaging the nearest k DMD solutions based on the distance between the given parameter value and other parameter values, leading to the applicability of DMD to parameterized problems. We apply the proposed method to various nonlinear parameterized PDEs, i.e., heat equations, reaction-diffusion equations, Burgers equations, and Navier-Stokes equations for the two-dimensional fluid flow over a cylinder. The results demonstrate the applicability and efficiency of the proposed KNN-DMD as a real-time ROM for parameterized PDEs. Furthermore, KNN-DMD shows better predictive ability than the POD-based ROMs outside of the training time region.

Comparative analysis of physical methods of transport disinfection

Transport is an object of mass concentration of people where workers and passengers are likely to be exposed to pathogens of infectious diseases. 95% are diseases that do not respond effectively to the effects of specific prophylaxis, influenza and acute respiratory viral infections. Currently, measures to contain the infection have led to a restriction of transport activity. In such conditions, the development of effective ways to prevent the occurrence and spread of infectious diseases transmitted by airborne droplets, not controlled by means of specific prevention, as well as the preservation of the health of transport industry workers, is particularly relevant. The aim of the study is to determine the location and type of air disinfection device in the driver's cabin of a railway vehicle and the passenger compartment of a sitting car with various breathing options (natural exhalation, cough, sneezing). To determine the location and capacity of the decontamination plant, the mathematical method of gas-dynamic calculations using solid-state three-dimensional models was used. The study was performed in the SolidWorks software package, using the FlowSimulation module to solve the system of nonlinear Navier-Stokes equations. The trajectories of air flows from human breathing in the cab of the vehicle are calculated taking into account the influence of ventilation. Research on the effectiveness of disinfection was carried out by a certified laboratory in the operating conditions of the vehicle. The results of the calculation allowed us to establish that it is advisable to place devices for physical disinfection of air in the cab of the vehicle at the level of the information panel on the left side or directly in the center. The basis for choosing the technical characteristics of the device for air disinfection should be information about the maximum speed of particles approaching the information panel, which is equal to 2.2-3 m/s. The safest method of disinfection of the air environment of vehicles is the use of closed UV radiation installations, the effectiveness of which is confirmed in the course of a full-scale experiment. It is concluded that the formation of an infection zone when an infected person is in the cab or cabin of a vehicle largely depends on the duration of his stay in the cabin, the intensity of breathing and the use of personal protective equipment. The advantage of closed UV radiation installations is the simultaneous provision of high disinfection efficiency and satisfaction of all safety requirements (chemical, environmental, fire, electrical, etc.), as well as the possibility of using the method in the presence of people. The analysis of the regulatory and technical documentation showed that UV irradiators are the only type of disinfection equipment that meets the requirements of domestic legislation in the field of sanitary and epidemiological welfare.

New Necessary Conditions for the Well-Posedness of Steady Bioconvective Flows and Their Small Perturbations

We introduce new necessary conditions for the existence and uniqueness of stationary weak solutions and the existence of the weak solutions for the evolution problem in the system arising from the modeling of the bioconvective flow problem. Our analysis is based on the application of the Galerkin method, and the system considered consists of three equations: the nonlinear Navier–Stokes equation, the incompressibility equation, and a parabolic conservation equation, where the unknowns are the fluid velocity, the hydrostatic pressure, and the concentration of microorganisms. The boundary conditions are homogeneous and of zero-flux-type, for the cases of fluid velocity and microorganism concentration, respectively.