Solving Navier-Stokes Equations Research Articles

Applying Physics-Informed Neural Networks to Solve Navier–Stokes Equations for Laminar Flow around a Particle

In recent years, Physics-Informed Neural Networks (PINNs) have drawn great interest among researchers as a tool to solve computational physics problems. Unlike conventional neural networks, which are black-box models that “blindly” establish a correlation between input and output variables using a large quantity of labeled data, PINNs directly embed physical laws (primarily partial differential equations) within the loss function of neural networks. By minimizing the loss function, this approach allows the output variables to automatically satisfy physical equations without the need for labeled data. The Navier–Stokes equation is one of the most classic governing equations in thermal fluid engineering. This study constructs a PINN to solve the Navier–Stokes equations for a 2D incompressible laminar flow problem. Flows passing around a 2D circular particle are chosen as the benchmark case, and an elliptical particle is also examined to enrich the research. The velocity and pressure fields are predicted by the PINNs, and the results are compared with those derived from Computational Fluid Dynamics (CFD). Additionally, the particle drag force coefficient is calculated to quantify the discrepancy in the results of the PINNs as compared to CFD outcomes. The drag coefficient maintained an error within 10% across all test scenarios.

Effect of Gurney Flap on the Aerodynamic Performance of a Flapping Foil: Micro-Aerial Vehicle Application

Abstract Even with its small size, the Gurney flap (GF) can help considerably in increasing the lift of foils and wings. To exploit this feature, the objective of this research was to numerically study the effects of this flow control device on the aerodynamic performance of oscillating foils for micro-aerial vehicle (MAV) applications. Three sets of each important parameter were selected: the height (0.01c, 0.04c, and 0.16c), angle (45 deg, 90 deg, and 135 deg) and location from trailing edge (T.E, 0.05c and 0.1c). A two-dimensional laminar, incompressible Navier–Stokes equation solver was used to computationally investigate the effect of the Gurney flap on the aerodynamic performance of a flat plate (chord length = 10 mm and thickness = 0.03c). It was found that the best aerodynamic performance was obtained when the Gurney flap was installed at the trailing edge with a height of 0.04c and mount angle of 90 deg. The height of the Gurney flap had a major impact on aerodynamic performance. Results showed an increase of 23.5% in mean lift coefficient, 15.5% in maximum lift coefficient, and 5% in power economy as compared to flat plate, which is accredited to the increase in effective camber and the formation of counter-rotating vortices, decreasing the adverse pressure gradient. The weakening of counter-rotating vortices downstream of Gurney flap could also be the contributing factor to its good performance. The results suggest that the Gurney flap may be useful in enhancing the performance of wings for bio-inspired flapping wing MAVs.

Parameter Analysis of Active Flap Control for Rotor Aerodynamic Control and Design

Active rotor control of helicopters is the future development direction, and active flap control (AFC) is one of the most promising technologies. However, the numerical simulation of an AFC rotor is challenging. It is necessary to consider the fidelity of the local flow details while dealing with complex shapes and motions. Therefore, few simulations of the flow field and analyses of the influencing parameters have been conducted. In particular, there is a lack of aerodynamic design criteria and recommendations for the AFC rotor. Thus, a new overset assembly algorithm, an arbitrary multilevel moving grid transformation algorithm, and a solver for the unsteady Reynolds-averaged Navier-Stokes equations (URANS) are proposed to establish a suitable numerical method for AFC rotor simulation. The aerodynamic characteristics of the rotor and key influencing factors are systematically analyzed under different flow conditions and design and control parameters, and suggestions for the design of the AFC rotor are provided. The results show that the AFC significantly changes the load distribution of the rotor. The thrust loss of the rotor is approximately 1%, but the offset angle compensates for the loss. The control parameters show relatively consistent trends under different working conditions. The phase is the key control parameter, and the effect on the load is more pronounced when the control frequency is an integral multiple of the rotor’s natural load frequency. Increasing the chord length, span length, and deflection amplitude can also enhance the active control performance.

Hybrid numerical methods for modelling multi-physics mass transport in coal

It is a challenging task to predict and analyse the transient flow physics in unconventional complex geological formations such as coal seams. This is due to the multi-scale multi-physics nature of processes taking place in coal seam gas reservoirs. Therefore, numerical tools allowing the incorporation of various flow physics inherent to coal into the simulation routine are of particular importance. At the pore-scale, the rock images obtained by micro-computed tomography are frequently used for flow modelling in porous or fractured media by direct numerical simulations (Lattice Boltzman method or by solving Navier Stokes equations) or by utilising so-called pore-network modelling. Both methods have advantages and drawbacks, while their hybridisation may allow us avoid some of the major pitfalls. Due to a limited number of research dedicated to image-based pore-scale modelling in coal, we develop a hybrid numerical model that combines the Hagen-Poiseuille analytical solution and the volume of fluid advection scheme for fluid flow in fractures with the continuum Darcy flow in the coal matrix. The explicit coupling of fluid flow across the fracture-matrix domains is implemented via corresponding source terms in the governing equations. Besides, additional physics including sorption and matrix swelling-shrinkage phenomena are introduced into the numerical model. The hybrid solver is validated against analytical solutions and a previously developed Darcy-Brinkman-Stokes framework, which has similar functionality for modelling fluid flow in coal seams. To validate the developed model and show its capability to capture multi-scale multi-physics flow in coal, several illustrative simulations are conducted using synthetic and realistic digital images of coal. As a result, the accuracy of the hybrid solver was within 4% when compared to the Darcy-Brinkman-Stokes framework for flow simulation on realistic geometry. Besides, the non-parallelised hybrid model outperformed the parallelised direct numerical simulation, being more than 90 times faster in terms of computational time. The findings are beneficial for conducting larger core-scale simulations of multi-scale multi-physics flow in coal seams with potential applications to natural gas production, hydrogen storage, and carbon dioxide geosequestration.

Experimental and numerical investigation of different wing profiles

Computational Fluid Dynamics (CFD) techniques are widely used in product development processes due to efficient algorithms to solve Navier Stokes equations. In this study, the coordinates of the NACA 1412 and NACA 2415 sections selected from the standard wing sections of the National Aviation Advisory Committee (NACA) have been implemented with specific procedures to create new geometries and a new wing profile design using the coordinate method. The wing profile geometry is created from the coordinates available in the literature. Fixed two-dimensional viscous flows around the generated wing profiles are simulated using commercially available ANSYS-Fluent software. The mesh structure used was created using the C type and rectangular structure. CFD and wind tunnel analysis of attack angles ranging from certain degrees. The aerodynamic coefficients of the related statuses, i.e. the lift and drag coefficients obtained from the analysis, have been compared with the wind tunnel test results. Aerodynamic research in recent years focuses on active flow control concepts, laminar flow control, fuel economy and higher transport/drag force ratio. This is designed to understand and analyze flow behavior around airfoils using the CFD method.

Effects of intersection geometry on flow velocity interference and flow rate redistribution in furcating fractures

Fracture intersections are an essential element of fracture networks in energy extraction. The accurate evaluation of the flow behavior at intersections in the furcating fracture is the key to describing the flow behavior of fractured reservoirs, yet fracture intersections are often oversimplified or even ignored. To fill this knowledge gap, the effect of geometric characteristics internal to the intersection was investigated based on direct numerical simulation by solving Navier-Stokes equations. Two parameters, θ1 and θ2, which are the angles between the normal inlet branch direction and the two sides of a closed triangle, are introduced to characterize the geometry of the intersection in a furcating fracture, and the role of geometrical parameters θ1 and θ2 in the interference effect and flow rate distribution is investigated. The results demonstrate that intersection geometry interferes with the fluid flow passing through the intersection to a certain range. With an increasing hydraulic gradient J and θ, the flow interference range increases. Under the conditions of these simulations (J = 10−3 – 10−2, Re = 101–102), J is linearly correlated with flow rate q, but with an increasing J and θ, the area and percentage of the low-velocity zone caused by an intersection increases leading to the enhancement of nonlinear flow. Furthermore, based on the simulation results, an empirical formula for quantifying the flow redistribution at intersections is proposed, contributing to the effectiveness and accuracy of modeling seepage in the fracture networks. Finally, the physical meanings of geometric parameters θ1 and θ2 are determined. This study advances the understanding of flow behavior at intersections commonly observed in discrete fracture networks from a fresh perspective, despite the difficulty of monitoring.

Gas kinetic principles in Navier-Stokes finite-volume solvers

We present a new approach for introducing more physics into the flux function of a finite-volume solver for compressible Navier-Stokes equations. This approach is conceived as an alternative solution to classical approximate Riemann solvers.The peculiarity of the solution we designed comes from the use of the gas distribution function, which is employed in the kinetic theory of gases; once computed from the Boltzmann equation with a simplified collision source term (BGK model), this distribution function is used, then, for the computation of the cell interface flux.Two solutions are investigated for computing the distribution function. In the first solution, this function is approximated as a first-order approximation of the BGK equation; in that case the resulting numerical flux smoothly evolves between a Godunov-like flux and a flux vector splitting: this first choice generates the simple gas kinetic scheme (SCG scheme). In the second solution, we use the kinetic meaning of the local sound speed at cell interfaces, to introduce macroscopic wave informations into the gas distribution function; this way, the resulting numerical flux is a convex combination between a HLLE-type flux and a modified flux-vector-splitting: this is what we call the acoustic gas kinetic scheme (ASCGE scheme). Both solutions are inserted into a space-time separation algorithm that is solved by a positive Runge-Kutta (SSPRK) method; in addition, a MUSCL procedure for generating a high-order positivity-preserving scheme, is selected.Viscous fluxes are computed by using kinetic informations coming from both sides of the cell face of the finite-volume method. Lastly, both schemes (SCG and ASCGE schemes) are investigated and analyzed over a variety of test-cases including inviscid/viscous and steady/unsteady problems.

A variable time-step IMEX-BDF2 SAV scheme and its sharp error estimate for the Navier–Stokes equations

We generalize the implicit-explicit (IMEX) second-order backward difference (BDF2) scalar auxiliary variable (SAV) scheme for Navier–Stokes equation with periodic boundary conditions (Huang and Shen, SIAM J. Numer. Anal. 59 (2021) 2926–2954) to a variable time-step IMEX-BDF2 SAV scheme, and carry out a rigorous stability and convergence analysis. The key ingredients of our analysis are a new modified discrete Grönwall inequality, exploration of the discrete orthogonal convolution (DOC) kernels, and the unconditional stability of the proposed scheme. We derive global and local optimal H1 error estimates in 2D and 3D, respectively. Our analysis provides a theoretical support for solving Navier–Stokes equations using variable time-step IMEX-BDF2 SAV schemes. We also design an adaptive time-stepping strategy, and provide ample numerical examples to confirm the effectiveness and efficiency of our proposed methods.

Simple and robust h-adaptive shock-capturing method for flux reconstruction framework

In this paper, a simple and robust shock-capturing method is developed for the Flux Reconstruction (FR) framework by combining the Adaptive Mesh Refinement (AMR) technique with the positivity-preserving property. The adaptive technique avoids the use of redundant meshes in smooth regions, while the positivity-preserving property makes the solver capable of providing numerical solutions with physical meaning. The compatibility of these two significant features relies on a novel limiter designed for mesh refinements. It ensures the positivity of solutions on all newly created cells. Therefore, the proposed method is completely positivity-preserving and thus highly robust. It performs well in solving challenging problems on highly refined meshes and allows the transition of cells at different levels to be completed within a very short distance. The performance of the proposed method is examined in various numerical experiments. When solving Euler equations, the technique of Local Artificial Diffusivity (LAD) is additionally coupled to damp oscillations. More importantly, when solving Navier-Stokes equations, the proposed method requires no auxiliaries and can provide satisfying numerical solutions directly. The implementation of the method becomes rather simple.

Computational study of heat transfers and hydrodynamic of nanofluid flow inside double pipe heat exchanger in presence of non-uniform magnetic field

The improvement of heat exchangers is important for energy savings and environmental protection. In this work, the application of the non-uniform magnetic source for the enhancement of the hydrodynamic in double-pipe heat exchangers is comprehensively studied. The hydrodynamics of nanofluid flow is also investigated in our research by performing computational modeling. In addition, the thermal efficiency of a selected system in different conditions of magnetic field and nanofluid velocity are studied. In addition, role of wall profile of inner tube on hydrodynamic is also disclosed in this work. The simulation of the nanofluid flow is done by computational fluid dynamics and solving Navier-stokes equations. The usage of non-uniform magnetic field would raise average skin friction about 230 %, 400 % and 500 % on sinusoidal, triangular and rectangular profiles, respectively.

Fast aerodynamics prediction of laminar airfoils based on deep attention network

The traditional method for obtaining aerodynamic parameters of airfoils by solving Navier–Stokes equations is a time-consuming computing task. In this article, a novel data-driven deep attention network (DAN) is proposed for reconstruction of incompressible steady flow fields around airfoils. To extract the geometric representation of the input airfoils, the grayscale image of the airfoil is divided into a set of patches, and these are input into the transformer encoder by embedding. The geometric parameters extracted from the transformer encoder, together with the Reynolds number, angle of attack, flow field coordinates, and distance field, are input into a multilayer perceptron to predict the flow field of the airfoil. Through analysis of a large number of qualitative and quantitative experimental results, it is concluded that the proposed DAN can improve the interpretability of the model while obtaining good prediction accuracy and generalization capability for different airfoils and flow-field states.

Solving Navier–Stokes with mimetic operators

We present a new scheme for solving Navier–Stokes equations using mimetic difference operators. These operators can be constructed to high orders of accuracy and maintain the physical properties of the problem under consideration. We demonstrate the effectiveness of our scheme by modeling a lock release in 3D Cartesian coordinates, then extend our techniques to 3D curvilinear grids. The resulting scheme allows for simple and efficient computation of fluid processes on curvilinear grids, which allows us to solve problems in more complex regions while minimizing the restrictions of finite difference methods.

Numerical analysis on the high-speed impulsive noise propagation characteristic of helicopter rotor in the presence of strong shear flow

Based on the Linearized Euler Equations (LEE) and the Computational Fluid Dynamics (CFD) technique, the propagation characteristics of helicopter rotor High-Speed Impulsive (HSI) noise in non-uniform flowfield are simulated and analyzed. A Hybrid Computational Aeroacoustics (HCAA) method for solving the sound field of rotor is first developed. The acoustic source region is simulated by solving Navier-Stokes equations, and the acoustic near-field is simulated by solving LEE based on the Runge-Kutta Discontinuous Galerkin (RKDG) method. The numerical analysis method is then validated through comparisons with experimental data, and the aeroacoustics characteristics of the UH-1H rotor are calculated. Afterwards, the propagation characteristics of HSI noise in the wind tunnel are calculated, and the “trap” effect of the shear layer on HSI noise is determined. Finally, the peak negative pressure, time-averaged acoustic energy, and instantaneous perturbed pressure field are analyzed in detail, and some conclusions are drawn. In jet flow, the HSI noise energy is reinforced, and the time of the peak negative pressure is significantly advanced. In still air, the HSI noise energy is attenuated, and the time of the peak negative pressure is slightly delayed. These investigations offer valuable data for developing advanced theoretical correction method to consider the effect of shear layer on rotating acoustic source.

Physically plausible and conservative solutions to Navier–Stokes equations using physics-informed CNNs

The physics-informed neural network (PINN) is an emerging approach for efficiently solving partial differential equations (PDEs) using neural networks. The physics-informed convolutional neural network (PICNN), a variant of PINN enhanced by convolutional neural networks (CNNs), has achieved better results on a series of PDEs since the parameter-sharing property of CNNs is effective in learning spatial dependencies. However, applying existing PICNN-based methods to solve Navier–Stokes equations can generate oscillating predictions, which are inconsistent with the laws of physics and the conservation properties. To address this issue, we propose a novel method that combines PICNN with the finite volume method to obtain physically plausible and conservative solutions to Navier–Stokes equations. We derive the second-order upwind difference scheme of Navier–Stokes equations using the finite volume method. Then we use the derived scheme to calculate the partial derivatives and construct the physics-informed loss function. The proposed method is assessed by experiments on steady-state Navier–Stokes equations under different scenarios, including convective heat transfer and lid-driven cavity flow. The experimental results demonstrate that our method can effectively improve the plausibility and accuracy of the predicted solutions from PICNN.

Design and fabrication of a bioreactor for the proliferation of mesenchymal stem cells

Currently, the number of organ donors is low and there are different methods to deal with this situation, including cell cultures, decellularization and bioreactors, which play an important role in helping the required processes to be carried out in a faster and more optimized way. The objective of this work is to make the cell proliferation process efficient using a bioreactor as an alternative, in which flow simulations can be made to previously analyze how it works. The radial flow hydraulic circuit was designed and simulated in SOLIDWORKS solving Navier-Stokes equations to verify that we have a non-turbulent laminar flow and simulate cell detachment without damaging the cell matrices. The design was printed with additive printing techniques, using photosensitive resin. Bioreactors also play a part in tissue engineering, as they allow for changes that can be monitored and reproduced in particular environmental factors. This article shows the successful design of a bioreactor to proliferate mesenchymal stem cells printed and subjected to simulations, whose results lead to the conclusion that there is non-turbulent laminar flow in the behavior of the system without subjecting the cells to a possible stress that causes damage.

Double-precision hardware accelerator for incompressible Navier–Stokes equations solver based on discontinuous Galerkin method

We present the first known hardware accelerator for a complete solver based on the modal discontinuous Galerkin method (DGM) of full incompressible Navier–Stokes equations. The solver is designed for hexahedral finite elements and, at the current stage, the modal basis is limited to second-order polynomials giving 27 degrees of freedom per finite element. The solution algorithm employs the projection method for the treatment of the incompressibility constraint, but the Poisson pressure equation is replaced with a parabolic equation solved in pseudo-time. Our implementation requires about 105 floating-point operations (FLOP) for a single finite element per time step, and its arithmetic intensity is 1.8 FLOP/byte on average. We also designed the hardware based on a set of simple, simultaneously working microprogrammable computing units that contain only a floating-point arithmometer and a small memory. The design fits a medium-size field-programmable gate array (FPGA, we used Xilinx XCZU15EG) and, for all tested cases excluding one, achieves performance over 100 000 mesh elements per second (122 000 in peak) with a reasonable hardware utilisation: memory bandwidth between 8 and 9 GB/s (over 0.4 of the machine peak) and computational performance between 13 and 15 GFLOPS (almost 0.7 of the peak). The main reasons limiting hardware utilisation are: internal hardware limits in FPGA that prevent allocation of all DDR memory bandwidth for computations; and the differences in arithmetic intensities of different parts of the algorithm — the arithmetic intensities vary between 1.1 and 3.3 FLOP/byte.

A lattice Boltzmann exploration of two-phase displacement in 2D porous media under various pressure boundary conditions

While experimental designs developed in recent decades have contributed to research on dynamic nonequilibrium effects in transient two-phase flow in porous media, this problem has been seldom investigated using direct numerical simulation (DNS). Only a few studies have sought to numerically solve Navier–Stokes equations with level-set (LS) or volume-of-fluid (VoF) methods, each of which has constraints in terms of meniscus dynamics for various flow velocities in the control volume (CV) domain. The Shan–Chen multiphase multicomponent lattice Boltzmann method (SC-LBM) has a fundamental mechanism to separate immiscible fluid phases in the density domain without these limitations. Therefore, this study applied it to explore two-phase displacement in a single representative elementary volume (REV) of two-dimensional (2D) porous media. As a continuation of a previous investigation into one-step inflow/outflow in 2D porous media, this work seeks to identify dynamic nonequilibrium effects on capillary pressure–saturation relationship ( P c – S ) for quasi-steady-state flow and multistep inflow/outflow under various pressure boundary conditions. The simulation outcomes show that P c , S and specific interfacial area ( a nw ) had multistep-wise dynamic effects corresponding to the multistep-wise pressure boundary conditions. With finer adjustments to the increase in pressure over more steps, dynamic nonequilibrium effects were significantly alleviated and even finally disappeared to achieve quasi-steady-state inflow/outflow conditions. Furthermore, triangular wave-formed pressure boundary conditions were applied in different periods to investigate dynamic nonequilibrium effects for hysteretical P c – S . The results showed overshoot and undershoot of P c to S in loops of the nonequilibrium hysteresis. In addition, the flow regimes of multistep-wise dynamic effects were analyzed in terms of Reynolds and capillary numbers ( Re and Ca ). The analysis of REV-scale flow regimes showed higher Re (1 < Re < 10) for more significant dynamic nonequilibrium effects. This indicates that inertia is critical for transient two-phase flow in porous media under dynamic nonequilibrium conditions.

New finite difference unequal-sized Hermite WENO scheme for Navier-Stokes equations

In this paper, a new finite difference unequal-sized Hermite weighted essentially non-oscillatory (US-HWENO) scheme is proposed for solving multi-dimensional Navier-Stokes equations on structured meshes, which could achieve sixth-order accuracy in one dimension and fifth-order accuracy in two and three dimensions, respectively. The high-order spatial reconstruction procedure uses a convex combination of one quintic polynomial and two linear polynomials defined on three unequal-sized central or biased spatial stencils. Compared with the classical finite difference, finite volume, or mixed finite difference and finite volume HWENO schemes [32] , [33] , the new scheme uses same largest stencil and only three stencils in spatial reconstruction procedures, and the first-order derivative values in auxiliary equations can be obtained directly rather than using complex HWENO-type reconstruction methodologies [32] , [33] . Moreover, the associated viscosity terms could be directly computed from the function values and first-order derivative values by using the characteristic of the WENO-type spatial reconstructions. Meanwhile, the linear weights can be any positive numbers on condition that their summation equals one, which avoids the problem of negative linear weights in classical HWENO schemes [24] , [25] . So the new finite difference US-HWENO scheme is more suitable for solving Navier-Stokes equations, and can be easily implemented to multi-dimensions on structured meshes. Some benchmark viscous examples are illustrated to verify the good performance of this new finite difference US-HWENO scheme.

Parallel Computation Using Non-Overlapping Domain Decomposition Coupled with Compact Local Integrated RBF for Navier–Stokes Equations

A non-overlapping domain decomposition-based parallel algorithm coupled with a compact local integrated radial basis function (CLIRBF) method is developed for solving Navier-Stokes equations. For this approach, a problem is divided into subdomains. In each sub-domain, a CLIRBF scheme is applied to solve the Navier-Stokes equations of flows. A relaxation factor is used at the interface between sub-domains to ensure the quick convergence of the present method. The Bitmap termination detection technique is introduced to complete the global termination. The present approach is verified using two fluid flow problems: the lid-driven cavity and the natural convection in concentric annuli flow. The numerical results have demonstrated the efficiency of the present parallel method compared with the corresponding sequential one and other published methods. Especially, super-linear speed-up was achieved for several CPUs. In terms of accuracy, the obtained results are in very good agreement with benchmark results.

DRVN (deep random vortex network): A new physics-informed machine learning method for simulating and inferring incompressible fluid flows

We present the deep random vortex network (DRVN), a novel physics-informed framework for simulating and inferring the fluid dynamics governed by the incompressible Navier–Stokes equations. Unlike the existing physics-informed neural network (PINN), which embeds physical and geometry information through the residual of equations and boundary data, DRVN automatically embeds this information into neural networks through neural random vortex dynamics equivalent to the Navier–Stokes equation. Specifically, the neural random vortex dynamics motivates a Monte Carlo-based loss function for training neural networks, which avoids the calculation of derivatives through auto-differentiation. Therefore, DRVN can efficiently solve Navier–Stokes equations with non-differentiable initial conditions and fractional operators. Furthermore, DRVN naturally embeds the boundary conditions into the kernel function of the neural random vortex dynamics and, thus, does not need additional data to obtain boundary information. We conduct experiments on forward and inverse problems with incompressible Navier–Stokes equations. The proposed method achieves accurate results when simulating and when inferring Navier–Stokes equations. For situations that include singular initial conditions and agnostic boundary data, DRVN significantly outperforms the existing PINN method. Furthermore, compared with the conventional adjoint method when solving inverse problems, DRVN achieves a 2 orders of magnitude improvement for the training time with significantly precise estimates.