Unsteady Incompressible Navier-Stokes Equations Research Articles

Prediction–correction projection immersed interface method for interfacial stress-induced flows and applications to electrohydrodynamics

We propose a robust, high-resolution prediction–correction projection immersed interface method (IIM) for solving the unsteady incompressible Navier–Stokes equations with traction boundary conditions, which arise from free surface flows driven by capillary, electric, and elastic forces. This method combines the advantages of traditional body-fitted moving mesh methods and immersed boundary methods (IBM), allowing for the accurate imposition of boundary conditions on free surfaces using Cartesian grids and providing detailed interface information that is typically smoothed out in traditional IBM. The irregular liquid-phase domain is embedded within a square region and discretized using a dynamically adaptive Cartesian mesh. The free surface is captured using a narrow-band level set with hybrid reinitialization. A prediction–correction projection scheme is constructed, incorporating additional pressure predictions and corrections to enhance robustness. The resulting Helmholtz/Poisson equations are solved using the augmented IIM for boundary value problems. Grid refinement analysis demonstrates second-order convergence of the L∞ error, even in challenging cases such as the oscillating drop test with low viscosity. We further apply this method to electrohydrodynamic (EHD) problems, constructing an implicit augmented IIM to solve the equations governing the electric field and charge conservation. Numerical experiments demonstrate that this method accurately addresses highly intense EHD phenomena, such as the formation of Taylor cones, highlighting its robustness.

A high-order upwind compact difference scheme for solving the streamfunction-velocity formulation of the unsteady incompressible Navier–Stokes equations

In this paper, we propose an upwind compact difference method with fourth-order spatial accuracy and second-order temporal accuracy for solving the streamfunction-velocity formulation of the two-dimensional unsteady incompressible Navier–Stokes equations. The streamfunction and its first-order partial derivatives (velocities) are treated as unknown variables. Three types of compact difference schemes are employed to discretize the first-order partial derivatives of the streamfunction. Specifically, these schemes include the fourth-order symmetric scheme, the fifth-order upwind scheme, and the sixth-order symmetric scheme derived by combining the two parts of the fifth-order upwind scheme. As a result, the fourth-order spatial discretization schemes are established for the Laplacian term, the biharmonic term, and the nonlinear convective term, along with the Crank–Nicolson scheme for the temporal discretization. The unconditional stability characteristic of the scheme for the linear model is proved by discrete von Neumann analysis. Moreover, six numerical experiments involving three test problems with the analytic solutions, and three flow problems including doubly periodic double shear layer, lid-driven cavity flow, and dipole-wall interaction are carried out to demonstrate the accuracy, robustness, and efficiency of the present method. The results indicate that the present method not only has good numerical performance but also exhibits quite efficiency.

Exact solutions of the Navier–Stokes equations generated by the eigenfunctions of the Stokes operator

We derive a class of exact solutions to the incompressible unsteady Navier–Stokes equations. These solutions are generated by the eigenfunctions of the Stokes operator and can be applied to studying boundary value problems for the Navier–Stokes equations related to pipe and channel flows. We explore the relation between the spectrum of the Stokes operator, generalized Beltrami flows, and exact solutions of the Navier–Stokes equations. Relevant properties of the Stokes eigenfunctions are also discussed in some detail.

Numerical solutions of boundary problems in partial differential equations: A deep learning framework with Green's function

This work introduces Green Networks (GreenNets), a novel class of neural network architectures designed to address a wide spectrum of boundary problems in partial differential equations (PDEs). Demonstrated versatility includes solving the Poisson equation, convection-diffusion equation, and unsteady incompressible Navier-Stokes equations. GreenNets comprises two specialized variants: Boundary Transformation Green Network (BT-GreenNet), focusing on transforming Dirichlet boundary conditions into source terms, and Boundary Integral Green Network (BI-GreenNet), considering boundary integral terms with shared weights between the boundary and source integrals. Comparisons with existing deep learning and traditional numerical methods underscore their accuracy and efficiency. BT-GreenNet excels in efficiency, and BI-GreenNet offers enhanced flexibility. The research pinnacle, Boundary Unified Green Network (BU-GreenNet), merges BT-GreenNet and BI-GreenNet to address coupled PDEs, specifically unsteady incompressible Navier-Stokes equations. BU-GreenNet stands out by surpassing traditional numerical approaches, offering remarkable computational efficiency while faithfully capturing the intricate evolution of fluid dynamics phenomena. This research redefines the means of solution to boundary problems in PDEs through innovative neural network architectures, emphasizing practical utility and computational efficiency.

Convergence analysis of high-order IMEX-BDF schemes for the incompressible Navier–Stokes equations

In this paper, we consider developing high-order temporal integration schemes for the unsteady incompressible Navier–Stokes equations in bounded two-dimensional domain subjected to the periodic boundary conditions. Utilizing the k-step (k=3,4,5) backward differentiation formula (BDF) coupled with the implicit–explicit (IMEX) treatment of the nonlinear convective term in an anti-symmetry form, a class of IMEX-BDFk schemes up to fifth-order in time are constructed and analyzed. By imposing a zero-mean constrain on the finite-dimensional space for the pressure, the proposed numerical schemes are proven to be uniquely solvable. Based on the recent theoretical framework consisting of a class of discrete orthogonal convolution kernels, rigorous L2 norm error estimates for both the velocity and the pressure are established by using a novel divergence free projection system. The proposed schemes are then implemented in two benchmark experiments, including a Taylor–Green vortex problem and a double shear layer flow at various high Reynolds numbers. Numerical results demonstrate the expected solution accuracy and the computational effectiveness in simulating the realistic flow dynamics.

An aneurysm-specific preconditioning technique for the acceleration of Newton-Krylov method with application in the simulation of blood flows.

In this paper, we develop an algorithm to simulate blood flows in aneurysmal arteries and focus on the construction of robust and efficient multilevel preconditioners to speed up the convergence of both linear and nonlinear solvers. The work is motivated by the observation that in the local aneurysmal region, the flow is often quite complicated with one or more vortices, but in the healthy section of the artery, the principal component of blood flows along the centerline of the artery. Based on this observation, we introduce a novel two-level additive Schwarz method with a mixed-dimensional coarse preconditioner. The key components of the preconditioner include (1) a three-dimensional coarse preconditioner covering the aneurysm; (2) a one-dimensional coarse preconditioner covering the central line of the healthy section of the artery; (3) a collection of three-dimensional overlapping subdomain preconditioners covering the fine meshes of the entire artery; (4) extension/restriction operators constructed by radial basis functions. The blood flow is modeled by the unsteady incompressible Navier-Stokes equations with resistance outflow boundary conditions discretized by a stabilized finite element method on fully unstructured meshes and the second-order backward differentiation formula in time. The resulting large nonlinear algebraic systems are solved by a Newton-Krylov algorithm accelerated by the new preconditioner in two ways: (1) the initial guess of Newton is obtained by solving a linear system defined by the coarse preconditioner; (2) the Krylov solver of the Jacobian system is preconditioned by the new preconditioner. Numerical experiments indicate that the proposed preconditioner is highly effective and robust for complex flows in a patient-specific artery with aneurysm, and it significantly reduces the numbers of linear and nonlinear iterations.

Simulation of Drag Crisis in Flow Past a Circular Cylinder using 2D Computations

The unsteady incompressible 2D Navier-Stokes equations are solved using a finite element method with Streamline-Upwind/Petrov-Galerkin (SUPG) and Pressure-Stabilizing/Petrov- Galerkin (PSPG) stabilization terms for high Reynolds number (105 to I06) flow past a cylinder to stimulate drag-crisis. The mean drag coefficient for various Re are compared with experimental data. It is observed that the low at Re =106 separates much later, from the cylinder surface as compared to that at Re =105. The wake at Re =106 is narrower than that at Re = 105. The time-averaged flow field shows a standing bubble just downstream of the flow separation point for the Re = 105 flow. The bubble is not observed for the flow at Re = 106. The energy spectrum of the flow exhibits the characteristics of 2D isotropic turbulence.

Corotational Hookean Models of Dilute Polymeric Fluids: Existence of Global Weak Solutions, Weak-Strong Uniqueness, Equilibration, and Macroscopic Closure

We prove the existence of global weak solutions to the corotational Hookean dumbbell model, a system of PDEs arising in the kinetic theory of dilute polymers, involving the unsteady incompressible Navier–Stokes equations in a bounded domain coupled to a Fokker–Planck type parabolic equation including a center-of-mass diffusion term, satisfied by the probability density function, modeling the evolution of the configuration of noninteracting polymer molecules in a viscous incompressible solvent. The micro-macro interaction is manifested by the presence of a corotational drag term in the Fokker–Planck equation and the divergence of a polymeric extra-stress tensor on the right-hand side of the Navier–Stokes momentum equation. We also analyze certain properties of weak solutions to this system of PDEs: we use the relative energy method to deduce a weak-strong uniqueness type result, and derive the macroscopic closure of the kinetic model; a corotational Oldroyd-B model with stress-diffusion. Finally, we discuss the existence and uniqueness of global weak solutions to this class of corotational Oldroyd-B models with stress-diffusion.

A less time-consuming upwind compact difference method with adjusted dissipation property for solving the unsteady incompressible Navier-Stokes equations

In this paper, a new strategy to establish less time-consuming upwind compact difference method with adjusted dissipation is introduced for solving the incompressible Navier-Stokes (N-S) equations in the streamfunction-velocity form efficiently. By weighted combination of the numerical solutions calculated using the upwind term and the downwind term of the general third-order upwind compact scheme (UCD3), a new fourth-order compact formulation and a third-order upwind compact formulation with adjusted dissipation nature are proposed for computing the first derivatives. Further, they are used to approximate the biharmonic term and the convective terms in the streamfunction-velocity formulation of the N-S equations, respectively. Meanwhile, the first derivatives of the streamfunction (velocities) in the coefficients in the convective terms are solved by the newly proposed fourth-order compact formulation. Temporal discretization for the streamfunction-velocity formulation is addressed with the help of the second-order Crank-Nicolson scheme. Moreover, the newly proposed scheme for the linear models is proved to be unconditionally stable by virtue of the discrete Fourier analysis. Finally, five numerical problems, viz. the analytic solution, Taylor-Green vortex problem, doubly periodic double shear layer flow problem, lid-driven square cavity flow problem and two-sided square cavity flow problem are solved numerically to verify the efficiency and accuracy of the present method. Results solved by the present method match well with the analytic solutions and the existing results proving the accuracy of it. What is more, it is less time-consuming and has lower dissipation than the existing method [20] .

Details study on the kinematic characteristics of manta ray section in flapping motion and exploring its application in wave glider propulsion system

It has always been a human challenge to inspire natural configurations and phenomena and benefit from their merits in improving the performances of man-made proposed aero/hydro vehicles. For example, the manta rays are known for their great swimming performances. To design and fabricate an underwater robot based on the manta ray geometry and its kinematic characteristics, it is important to initially study its hydrodynamic behavior and possibly arrive at some key design parameters, which can remarkably help to figure out an optimum geometry with high swimming performances. The main objective of this study is to focus on the merits of gliding motion inspired by the manta ray fish considering the real sea conditions. In this regard, two wave-glider configurations with specific submersible geometries are selected to practice the bases of this objective. They are the manta-glider, which has a complicated 3D configuration representing the real manta ray swimming behavior, and the wing-glider, which has much simpler configuration constructing from a flat and uniform wing span shape. The 3D unsteady incompressible Navier-Stokes equations are numerically solved to simulate the gliding motion performances of these two gliders. The results show that the maximum efficiency of manta-glider model is ηmax = 50 %, which is much higher than that of the wing-glider with ηmax = 36 %, within an operational range with a relatively constant significant wave height (SWH). Alternatively, the wing-glider efficiency is about 5 % higher than that of the manta-glider in cases with non-constant SWH values. Irrespective of their efficiencies, the interesting outcome of this study is that the manta-glider model shows fewer fluctuations and performs more stable conditions than a simplified glider model in energy-absorbing applications. Considering the achieved outcomes, this work also suggests a simple guideline, which is so beneficial to arrive at a suitable wave glider propulsion design.

Existence and regularity of weak solutions for a fluid interacting with a non-linear shell in three dimensions

We study the unsteady incompressible Navier–Stokes equations in three dimensions interacting with a non-linear flexible shell of Koiter type. This leads to a coupled system of nonlinear PDEs where the moving part of the boundary is an unknown of the problem. The known existence theory for weak solutions is extended to non-linear Koiter shell models. We introduce a priori estimates that reveal higher regularity of the shell displacement beyond energy estimates. These are essential for non-linear Koiter shell models, since such shell models are non-convex (with respect to terms of highest order). The estimates are obtained by introducing new analytical tools that allow dissipative effects of the fluid to be exploited for the (non-dissipative) solid. The regularity result depends on the geometric constitution alone and is independent of the approximation procedure; hence it holds for arbitrary weak solutions. The developed tools are further used to introduce a generalized Aubin–Lions-type compactness result suitable for fluid–structure interactions.

Propulsive performance of plunging airfoils in biplane configuration

Biplane configuration of pure plunging airfoils is investigated in terms of vortex dynamics both experimentally and numerically by utilizing particle image velocimetry and unstructured finite volume solver of incompressible unsteady Navier–Stokes equations. Experiments are carried out to disclose the vortex shedding and interaction mechanisms for various values of frequency and amplitude of the plunging motion. For the non-dimensional plunge amplitude with respect to the chord of airfoil h = 0.2, the effect of the reduced circular frequency based on chord length and the free stream velocity k = 1 and 10 are considered, whereas for h = 0.3, k = 2, 4, 8, and 10 cases are examined. Influence of the plunge amplitude is studied for h = 0.25 at k = 2.5 and for h = 0.0875, 0.15, and 0.3 at k = 4. Numerical simulations are performed to investigate the effect of phase difference on vortex structures and propulsive characteristics, such as thrust and Froude efficiency. Two cases having the highest thrust and efficiency values k = 2.5, h = 0.25, and k = 4, h = 0.15 value couples are selected for the phase angle of ϕ=0°, ϕ=90°, ϕ=180°, and ϕ=270°. Opposed plunge, ϕ=180°, was found as the most efficient amongst all phase angles that were investigated, where ϕ=90° is beneficial in lift production. Additionally, three-dimensional simulations indicate no significant three dimensionalities for the parameters used herein.

Efficient parallel simulation of hemodynamics in patient-specific abdominal aorta with aneurysm

Surgical planning for aortic aneurysm repair is a difficult task. In addition to the morphological features obtained from medical imaging, alternative features obtained with computational modeling may provide additional useful information. Though numerical studies are noninvasive, they are often time-consuming, especially when we need to study and compare multiple repair scenarios, because of the high computational complexity. In this paper, we present a highly parallel algorithm for the numerical simulation of unsteady blood flows in the patient-specific abdominal aorta before and after the aneurysmic repair. We model the blood flow with the unsteady incompressible Navier-Stokes equations with different outlet boundary conditions, and solve the discretized system with a highly scalable domain decomposition method. With this approach, a high resolution simulation of a full-size adult aorta can be obtained in less than an hour, instead of days with older methods and software. In addition, we show that the parallel efficiency of the proposed method is near 70% on a parallel computer with 2, 880 processor cores.

A mechanistic model for estimating bacteria levels in stormwater ponds

This paper presents a three-dimensional CFD based hydro-environmental model that simulates fate and transport of bacteria in water bodies. The model numerically solves unsteady incompressible Reynolds-Averaged Navier-Stokes equations on a structured grid. Free-floating and particle-attached bacteria were modelled separately regarding both fate and transport. Therefore, a sediment transport model was integrated into the main model in order to model particle-attached bacteria transport. In addition, Volume of Fluid approach was implemented to capture the water surface movements. Wind effect was also considered in the modelling using shear stress on the water surface. Since stormwater reuse is the source of some public health concerns, a stormwater pond was chosen as the test case for the model. The model was applied to simulate the distribution of bacterial indicator organisms in the Inverness Stormwater Pond in Calgary, Alberta, which is a large T-shaped pond with several inlets and outlets. The bacteria distribution in the pond was simulated for three rain events that occurred in the area. In six locations of the pond the modelled bacteria distribution was compared to collected data using non-dimensional bacteria concentrations. The comparison showed good agreement and indicated that the middle of the pond, close to the surface had the lowest levels of bacteria and thus, was considered the optimal location for withdrawal for reusing pond water. Furthermore, planting a tree barrier on the north bank of the West wing of the pond was shown to mitigate bacteria transport away from the inlets into the pond body and substantially decrease the risk of contamination at the optimal water withdrawal location.

Assessment of three-dimensional effects on slamming load predictions using OpenFoam

Three-dimensional effects on slamming loads predictions are investigated numerically using the unsteady incompressible Reynolds-Average Navier-Stokes (RANS) equations and volume of fluid (VOF) method, which are implemented in the open-source library OpenFoam. Three cases of free-falling water entry problems are considered. The numerical solutions are validated through the comparisons of slamming loads and motions between the CFD simulations and the available experimental values. The total slamming force and slamming pressures on the 3D wedge cylinder, ship section and slender circular cylinder are compared with the results on the 2D sections, respectively. The results show that the 3D predictions on the maximum slamming force coefficient are around 21%∼23% lower for the wedge with a deadrise angle of 30º and are around 16% lower for the slender circular cylinder, relative to the 2D calculations.

Multirate timestepping for the incompressible Navier-Stokes equations in overlapping grids

We develop a multirate timestepper for semi-implicit solutions of the unsteady incompressible Navier-Stokes equations (INSE) based on a recently-developed multidomain spectral element method (SEM) [1]. For incompressible flows, multirate timestepping (MTS) is particularly challenging because of the tight coupling implied by the incompressibility constraint, which manifests as an elliptic subproblem for the pressure at each timestep. The novelty of our approach stems from the development of a stable overlapping Schwarz method applied directly to the Navier-Stokes equations, rather than to the convective, viscous, and pressure substeps that are at the heart of most INSE solvers. Our MTS approach is based on a predictor-corrector (PC) strategy that preserves the temporal convergence of the underlying semi-implicit timestepper. We present numerical results demonstrating that this approach scales to an arbitrary number of overlapping grids, accurately models complex turbulent flow phenomenon, and improves computational efficiency in comparison to singlerate timestepping-based calculations.

Evaluation of ANN and ANFIS Methods in Study of the Motion of a Bubble in A Combined Couette-Poiseuille Flow

The equilibrium position of a deformable bubble in a combined Couette-Poiseuille flow is investigated numerically by solving the full Navier-Stokes equations using a finite-difference/front-tracking method. The present approach is examined to predict the migration of a bubble in a combined Couette-Poiseuille flow at finite Reynolds numbers of 5, 10, and 15. The related unsteady incompressible full Navier-Stokes equations are solved using a conventional finite-difference method with a structured staggered grid. The purpose of this study is to evaluate ANN and ANFIS methods in study of the lateral migration of the bubble. Evaluation criteria of accuracy in test set derived from ANFIS demonstrates that estimated values of correlation coefficient (r), Mean Absolute Error (MAE), and Root Mean Square Error (RMSE) are 0.97, 0.001, and 0.0014, respectively. The ANN model with RMSE of 0.0007, MAE of 0.0004 and r of 0.99, is better than ANFIS model. It is also demonstrated that the bubble position estimated by the ANN and ANFIS models closely follows the one achieved from front tracking method.

Oscillatory flow around a vertical wall-mounted cylinder: Flow pattern details

The interaction between surface gravity waves and a vertical wall-mounted rigid cylindrical structure such as a pier or vegetation involves various interdependent physical phenomena including the scouring process and the vertical mixing and horizontal dispersion of materials. As a step toward understanding this interaction, detailed flow fields of sinusoidal oscillatory flow passing a vertical wall-mounted cylinder are numerically investigated for three different wave conditions. With a moderately wide range of the Keulegan–Carpenter number from 6 to 20, numerical simulations are systematically performed, and the data are extensively investigated to determine the dynamic characteristics of the oscillatory flow past a wall-mounted cylinder. In this study, three-dimensional unsteady incompressible Navier–Stokes equations are solved using the open-source software, OpenFOAM®. Grid-convergence tests are conducted, and the undisturbed oscillating Stokes boundary-layer determined by numerical simulations is validated by a good agreement with the analytical solution. Flow details in the form of profiles, streamlines, and contours of calculated turbulence fields are presented. Coherent structure dynamics is illustrated using iso-surfaces of the Q-criterion. The synthesis of various flow variables presents a mechanistic view of the bed shear and processes responsible for scour near the cylinder-wall junction.

A parallel high-order compact scheme for the pure streamfunction formulation of the 3D unsteady incompressible Navier–Stokes equation

The high-order compact scheme for the pure streamfunction formulation of the Navier-Stokes equation (Yu and Tian, 2019) is extended to unsteady problems. The unsteady term is discretized by the Crank-Nicolson scheme. A new boundary scheme is also established for symmetrical boundaries to reduce the computational cost of symmetrical flow problems. The parallelization of the present scheme is realized through a boundary approximation approach which maintains the accuracy of calculations for the first- and second-order derivatives on the interior boundaries. The multigrid method for solving the governing equations is parallelized as well. These means are essential to feasible large-scale simulations. Test on a problem with an analytical solution proves that the proposed algorithm is fourth-order accurate in space and second-order in time. Several numerical experiments on lid-driven cavity problems with various configurations and Reynolds numbers up to 3200 are carried out. Results show that the present scheme is efficient and robust enough to resolve fine flow structures and temporal characteristics with moderate grid sizes and time steps.

A parallel non-nested two-level domain decomposition method for simulating blood flows in cerebral artery of stroke patient.

Numerical simulation of blood flows in patient-specific arteries can be useful for the understanding of vascular diseases, as well as for surgery planning. In this paper, we simulate blood flows in the full cerebral artery of stroke patients. To accurately resolve the flow in this rather complex geometry with stenosis is challenging and it is also important to obtain the results in a short amount of computing time so that the simulation can be used in pre- and/or post-surgery planning. For this purpose, we introduce a highly scalable, parallel non-nested two-level domain decomposition method for the three-dimensional unsteady incompressible Navier-Stokes equations with an impedance outlet boundary condition. The problem is discretized with a stabilized finite element method on unstructured meshes in space and a fully implicit method in time, and the large nonlinear systems are solved by a preconditioned parallel Newton-Krylov method with a two-level Schwarz method. The key component of the method is a non-nested coarse problem solved using a subset of processor cores and its solution is interpolated to the fine space using radial basis functions. To validate and verify the proposed algorithm and its highly parallel implementation, we consider a case with available clinical data and show that the computed result matches with the measured data. Further numerical experiments indicate that the proposed method works well for realistic geometry and parameters of a full size cerebral artery of an adult stroke patient on a supercomputers with thousands of processor cores.