Solving Navier-Stokes Equations Research Articles

An efficient IMEX-DG solver for the compressible Navier-Stokes equations for non-ideal gases

We propose an efficient, accurate and robust IMEX solver for the compressible Navier-Stokes equations describing non-ideal gases with general cubic equation of state and Stiffened-Gas EOS. The method is based on an h-adaptive Discontinuous Galerkin spatial discretization and on an Additive Runge Kutta IMEX method for time discretization. It is specifically tailored for low Mach number applications and allows to simulate low Mach regimes at a significantly reduced computational cost, while maintaining full second order accuracy also for higher Mach number regimes. The method has been implemented in the framework of the deal.II numerical library, whose adaptive mesh refinement capabilities are employed to enhance efficiency. Refinement indicators appropriate for real gas phenomena have been introduced. A number of numerical experiments on classical benchmarks for compressible flows and their extension to real gases demonstrate the properties of the proposed method.

The acoustic streaming effects of sonic black hole

In this paper, the acoustic streaming effects of Sonic Black Hole (SBH) are deeply studied by using the numerical and simulation method. By solving Navier–Stokes equations of compressible fluid, we analyze the flow characteristics of acoustic medium in SBH excited by sound wave, and further discuss the acoustic streaming effects and sound wave capture mechanism of SBH. The phenomenon of simultaneous reduction of medium and high-frequency sound reflection and transmission is analyzed. Under the excitation of medium- and high-frequency sound waves, the interaction between fluid medium flow and SBH structure leads to the uneven internal velocity distribution in the sound propagation direction, which can lead to the phenomenon of sound wave capture and deceleration. For low-frequency sound waves, the velocity distribution of sound medium is uniform, and there is almost no change in velocity gradient, resulting in the uniform distribution of sound pressure without sound absorption. These theoretical and numerical results are in good agreement with the experimental results in the literature, and also verify our results. Finally, by improving the structure, increasing the complexity of sound medium flow and the gradient change in the direction of sound propagation, the sound absorption and insulation ability of the SBH can be further improved. This study reveals the sound transmission mechanism of SBH, which can provide a new idea for the suppression of low-frequency sound waves in SBH.

Geometry aware physics informed neural network surrogate for solving Navier–Stokes equation (GAPINN)

Many real world problems involve fluid flow phenomena, typically be described by the Navier–Stokes equations. The Navier–Stokes equations are partial differential equations (PDEs) with highly nonlinear properties. Currently mostly used methods solve this differential equation by discretizing geometries. In the field of fluid mechanics the finite volume method (FVM) is widely used for numerical flow simulation, so-called computational fluid dynamics (CFD). Due to high computational costs and cumbersome generation of the discretization they are not widely used in real time applications. Our presented work focuses on advancing PDE-constrained deep learning frameworks for more real-world applications with irregular geometries without parameterization. We present a Deep Neural Network framework that generate surrogates for non-geometric boundaries by data free solely physics driven training, by minimizing the residuals of the governing PDEs (i.e., conservation laws) so that no computationally expensive CFD simulation data is needed. We named this method geometry aware physics informed neural network—GAPINN. The framework involves three network types. The first network reduces the dimensions of the irregular geometries to a latent representation. In this work we used a Variational-Auto-Encoder (VAE) for this task. We proposed the concept of using this latent representation in combination with spatial coordinates as input for PINNs. Using PINNs we showed that it is possible to train a surrogate model purely driven on the reduction of the residuals of the underlying PDE for irregular non-parametric geometries. Furthermore, we showed the way of designing a boundary constraining network (BCN) to hardly enforce boundary conditions during training of the PINN. We evaluated this concept on test cases in the fields of biofluidmechanics. The experiments comprise laminar flow (Re = 500) in irregular shaped vessels. The main highlight of the presented GAPINN is the use of PINNs on irregular non-parameterized geometries. Despite that we showed the usage of this framework for Navier Stokes equations, it should be feasible to adapt this framework for other problems described by PDEs.

Recent progress of lattice Boltzmann method and its applications in fluid-structure interaction

Fluid-structure interaction (FSI) is a very common physical phenomenon which extensively exists in nature, human daily life and many engineering applications. The lattice Boltzmann method (LBM) is an alternative of solving Navier–Stokes equations to obtain complex fluid dynamics. Since the proposal of lattice Bhatnagar–Gross–Krookmodel, the LBM has been improved and applied to various complex flows ranging from laminar flow to turbulent flow and Newtonian flow to non-Newtonian flow. To handle the associated FSI, the bounce-back scheme was proposed and then the immersed boundary method (IBM) was incorporated into the LBM for the simplicity and robustness of IBM in handling complex and moving geometries and the significant efficiency of LBM in solving fluid dynamics. In recent years, the combined frameworks of LBM, that is, bounce-back–lattice Boltzmann method and immersed boundary–lattice Boltzmann method have witnessed a significant development and the successful applications can be found in a range of physical problems such as flow around bluff bodies, flapping wings, wind turbines, aeroacoustics, sea wave modelling and et al. In this paper, the recent progress of LBM and some typical applications involving FSI are reviewed.

CFD simulation of heat transfer and phase change characteristics of the cryogenic liquid hydrogen tank under microgravity conditions

The heat transfer and phase change processes of cryogenic liquid hydrogen (LH2) in the tank have an important influence on the working performance of the liquid hydrogen-liquid oxygen storage and supply system of rockets and spacecrafts. In this study, we use the RANS method coupled with Lee model and VOF (volume of fraction) method to solve Navier-stokes equations. The Lee model is adopted to describe the phase change process of liquid hydrogen, and the VOF method is utilized to calculate free surface by solving the advection equation of volume fraction. The model is used to simulate the heat transfer and phase change processes of the cryogenic liquid hydrogen in the storage tank with the different gravitational accelerations, initial temperature, and liquid fill ratios of liquid hydrogen. Numerical results indicate greater gravitational acceleration enhances buoyancy and convection, enhancing convective heat transfer and evaporation processes in the tank. When the acceleration of gravity increases from 10−2 g0 to 10−5 g0, gaseous hydrogen mass increases from 0.0157 kg to 0.0244 kg at 200s. With the increase of initial liquid hydrogen temperature, the heat required to raise the liquid hydrogen to saturation temperature decreases and causes more liquid hydrogen to evaporate and cools the gas hydrogen temperature. More cryogenic liquid hydrogen (i.e., larger the fill ratio) makes the average fluid temperature in the tank lower. A 12.5% reduction in the fill ratio resulted in a decrease in fluid temperature from 20.35 K to 20.15 K (a reduction of about 0.1%, at 200s).

Deep learning method for fast prediction of film cooling performance

This study examines the predictive capability of deep learning method for adiabatic film cooling effectiveness distribution with variable operating conditions and geometric layouts. A conditional generative adversarial network is trained to establish nonlinear mapping from input to output. We embed the boundary condition information directly into the input tensor, thereby imparting the capability to address variable operating conditions. The processed input tensors include different blowing ratios M, incoming turbulence intensity Tu, geometry profile controlled by the inclination angle ϕ, compound angle θ, number of hole rows n, pitch between hole rows Px, and span-wise hole pitch Py. The output data are adiabatic film cooling effectiveness fields generated by Reynolds-averaged Navier–Stokes simulations. The prediction results are in good agreement with the computational fluid dynamics results in terms of various statistical assessments. Furthermore, compared with conventional methods of solving Navier–Stokes equations, predictions based on deep learning result in better response times. Therefore, the method proposed in this study is of high significance in the early design of cooling structures for turbine blades.

A new finite volume multi-resolution central WENO scheme for Navier–Stokes equations on staggered meshes

In this paper, a new fifth-order finite volume multi-resolution central WENO (MRC-WENO) scheme is proposed for solving Navier–Stokes equations on staggered meshes. A series of high-order spatial reconstruction procedures using unequal-sized stencils and a time discretization method with the natural continuous extension (NCE) of the Runge–Kutta method are applied to design this scheme. By gradually deteriorating from the fifth-order to the third-order, and eventually to the first-order in non-smooth areas, this new MRC-WENO method may preserve the original fifth-order accuracy in smooth regions while maintaining sharp shock transitions without solving any local Riemann problems on staggered meshes. Unlike many traditional central WENO schemes which employ many equal-sized stencils on staggered meshes, the new MRC-WENO scheme employs smaller unequal-sized stencils, easily makes the choice of the stencils, and uses arbitrary positive linear weights provided their summation is one. This is the first time that only three unequal-sized hierarchical central spatial stencils are used in multi-dimensional spatial reconstruction procedures on staggered meshes. This MRC-WENO scheme is simple to construct and can be easily implemented to arbitrary high-order accuracy in higher dimensions. And it is suitable for large scale engineering applications. Some benchmark one-dimensional and two-dimensional viscous examples are supplied to demonstrate the good performance of this new finite volume MRC-WENO scheme on staggered meshes.

Effects of Ambient Temperature and Solar Radiation on Solar Chimney Power Plants (SCPP) Performances

In this paper, the air flow inside the Solar Chimney Power Plants (SCPP) of Manzanares (Spain) has been investigated through numerical simulation, aiming to study the influence of environmental parameters on the SCPP performances. The electrical power produced by the SCPP of different regions in Algeria has been also compared in order to determine the best location for the installation of SCPP in Algeria. The resolution of Navier-Stokes equations that govern the air flow through the SCPP has been conducted using Ansys Fluent software. The k–ε RNG model and the standard wall mode have been selected to simulate the turbulent flow conditions, while the discrete ordinate model has been chosen for the thermal radiation modulation. The solar radiation had a direct effect on the output power while it had no effect on the effectiveness of the SCPP. The Ambient temperature has negatively affected the SCPP efficiency and the output power. Based on the results of this study, solar radiation and ambient temperature are the two main factors that could allow the possibility of adopting the SCPP system in a specific region. The global solar radiation is the dominant factor in the allocation of SCPP output power, relatively to the ambient temperature. It has been concluded that areas with high solar radiation and low ambient temperature are more appropriate for SCPP installation to increase electrical energy production.

Numerical Solutions of Two-Dimensional Navier–Stokes Equations Based on a Generalized Harmonic Polynomial Cell Method With Non-Uniform Grid

Abstract It is essential for a Navier–Stokes equations solver based on a projection method to be able to solve the resulting Poisson equation accurately and efficiently. In this paper, we present numerical solutions of the 2D Navier–Stokes equations using the fourth-order generalized harmonic polynomial cell (GHPC) method as the Poisson equation solver. Particular focus is on the local and global accuracy of the GHPC method on non-uniform grids. Our study reveals that the GHPC method enables the use of more stretched grids than the original HPC method. Compared with a second-order central finite difference method (FDM), global accuracy analysis also demonstrates the advantage of applying the GHPC method on stretched non-uniform grids. An immersed-boundary method is used to deal with general geometries involving the fluid–structure interaction problems. The Taylor–Green vortex and flow around a smooth circular cylinder and square are studied for the purpose of verification and validation. Good agreement with reference results in the literature confirms the accuracy and efficiency of the new 2D Navier–Stokes equation solver based on the present immersed-boundary GHPC method utilizing non-uniform grids. The present Navier–Stokes equations solver uses second-order central FDM and Quadratic Upstream Interpolation for Convective Kinematics scheme for the discretization of the diffusion term and advection term, respectively, which may be replaced by other higher-order schemes to further improve the accuracy.

Numerical study on adjustment of the main flow field with guide vanes in transition zone for an S-duct

A 30°/30° S-duct is often employed by a modern civil and military aircraft. Due to the vortex generators have poor control effect on the flow separation in the large curvature S-duct, so a new method adjusting the direction of the main flow in the transition zone for S-duct is proposed in the present study. These studies are carried out by solving Navier-Stokes equations with SST k-ω turbulence model. In order to explore the control mechanism of guide vanes on the flow field for S-duct, the internal flow field of S-duct with/without guide vanes is analyzed and compared in detail, including the velocity distribution on the symmetry plane, the secondary vortex intensity at different cross sections, turbulent kinetic energy on different cross sections, total pressure distribution on different cross sections, three-dimensional streamline, and so on. The results show that guide vanes can not only effectively adjust the direction of the main flow, but also restrain the flow separation, reduce the total pressure loss, and improve the uniformity of flow field.

A Direct-Forcing Immersed Boundary Method for Incompressible Flows Based on Physics-Informed Neural Network

The application of physics-informed neural networks (PINNs) to computational fluid dynamics simulations has recently attracted tremendous attention. In the simulations of PINNs, the collocation points are required to conform to the fluid–solid interface on which no-slip boundary condition is enforced. Here, a novel PINN that incorporates the direct-forcing immersed boundary (IB) method is developed. In the proposed IB-PINN, the boundary conforming requirement in arranging the collocation points is eliminated. Instead, velocity penalties at some marker points are added to the loss function to enforce no-slip condition at the fluid–solid interface. In addition, force penalties at some collocation points are also added to the loss function to ensure compact distribution of the volume force. The effectiveness of IB-PINN in solving incompressible Navier–Stokes equations is demonstrated through the simulation of laminar flow past a circular cylinder that is placed in a channel. The solution obtained using the IB-PINN is compared with two reference solutions obtained using a conventional mesh-based IB method and an ordinary body-fitted grid method. The comparison indicates that the three solutions are in excellent agreement with each other. The influences of some parameters, such as weights for different loss components, numbers of collocation and marker points, hyperparameters in the neural network, etc., on the performance of IB-PINN are also studied. In addition, a transfer learning experiment is conducted on solving Navier–Stokes equations with different Reynolds numbers.

A Conservative and Monotone Characteristic Finite Element Solver for Three-Dimensional Transport and Incompressible Navier-Stokes Equations on Unstructured Grids

A Conservative and Monotone Characteristic Finite Element Solver for Three-Dimensional Transport and Incompressible Navier-Stokes Equations on Unstructured Grids

비압축성 입자법 내 압력해법에 관한 GPU 가속화에 대한 연구

In the Moving Particle Simulation (MPS) method, which is a gridless particle method with simple algorithm for solving Navier-Stokes equation, insufficient number of particles in the vicinity of boundary has a considerable influence on the numerical calculation, which causes directly the flow field to become unstable and inaccurate. In particular, the Poisson Equation for pressure is solved iteratively at every time step to obtain an approximate solution. This gratly incereases the amount of computation when performing large-scale particle simulations, which requires high-performance computing. The main objective of this study is to develop a GPU(Graphics Processing Unit)-accelerated MPS code using CUDA(Compute Unified Device Architecture) language to solve the Poisson equation. Furthermore in order to compare the performance of the computational speed, a 3-dimensional calculation of broken-dam break problem has been carried out using the GPU-accelerated MPS method. It can be seen that the performance of the speed ratio in the GPU-based code is improved by about 10 times compared to that of the CPU-based code.

TLBfind: a Thermal Lattice Boltzmann code for concentrated emulsions with FINite-size Droplets

In this paper, we present TLBfind, a GPU code for simulating the hydrodynamics of droplets along with a dynamic temperature field. TLBfind hinges on a two-dimensional multi-component lattice Boltzmann (LB) model simulating a concentrated emulsion with finite-size droplets evolving in a thermal convective state, just above the transition from conduction to convection. The droplet concentration of the emulsion system is tunable and at the core of the code lies the possibility to measure a large number of physical observables characterising the flow and droplets. Furthermore, TLBfind includes a parallel implementation on GPU of the Delaunay triangulation useful for the detection of droplets' plastic rearrangements, and several types of boundary conditions, supporting simulations of channels with structured rough walls. Program summaryProgram Title: TLBfindCPC Library link to program files:https://doi.org/10.17632/hbk45696nf.1Developer's repository link:https://github.com/FrancescaPelusi/TLBfindLicensing provisions: MITProgramming language: CUDA-CNature of problem: Hydrodynamics of concentrated emulsions with finite-size droplets in a thermal convective state.Solution method: Single relaxation time Lattice Boltzmann method to solve Navier-Stokes equations for fluids, coupled with the temperature field dynamics. The output describes the dynamics of finite-size droplets of concentrated emulsions in presence of a temperature field. The temperature field obeys the advection-diffusion equation.Additional comments including restrictions and unusual features: Plastic rearrangements of droplets are detected via the parallel implementation of the Delaunay triangulation, and boundary conditions are tunable.

Fast Fluid Thermodynamics Simulation by Solving Heat Diffusion Equation

In mechanical engineering educations, simulating fluid thermodynamics is rather helpful for students to understand the fluid’s natural behaviors. However, rendering both high-quality and realtime simulations for fluid dynamics are rather challenging tasks due to their intensive computations. So, in order to speed up the simulations, we have taken advantage of GPU acceleration techniques to simulate interactive fluid thermodynamics in real-time. In this paper, we present an elegant, basic, but practical OpenGL/SL framework for fluid simulation with a heat map rendering. By solving Navier-Stokes equations coupled with the heat diffusion equation, we validate our framework through some real-case studies of the smoke-like fluid rendering such as their interactions with moving obstacles and their heat diffusion effects. As shown in Fig. 1, a group of experimental results demonstrates that our GPU-accelerated solver of Navier-Stokes equations with heat transfer could give the observers impressive real-time and realistic rendering results.

Modelling of Eulerian incompressible fluid flows by using peridynamic differential operator

This study presents a novel method for modelling of Eulerian incompressible fluid flow by using peridynamic differential operator. The peridynamic differential operator is used to calculate partial derivatives in the Navier-Stokes equations. The pressure Poisson equation is used to obtain the pressure field whereas the velocity field is obtained by solving momentum equations. The numerical procedure to solve Navier-Stokes equations in peridynamics for the incompressible fluid is also provided. The capability of the proposed peridynamic incompressible fluid model is demonstrated by considering problems of two-dimensional cavitation, a flow inside an open channel, and a flow over a cylinder. Moreover, to demonstrate the capabilities of the proposed model, the problems of two-dimensional cavitation and a flow over a cylinder are investigated for different Reynolds numbers. The vortex shedding is also captured for the problem of a flow over a cylinder at Reynolds number of Re=100. For verification purposes, the peridynamic results are compared with the results obtained by ANSYS Fluent, a commercial fluid dynamics software.

Numerical simulations of sintering coupled with moisture transfer

In many applications, amorphous particles bond together through a phenomenon known as sintering to minimize their surface energy. Water is a plasticizer for many food and pharmaceutical powders and the strong reduction in viscosity induced by moisture absorption can accelerate strongly particle sintering [1]. Numerical simulations of particle sintering usually neglect the coupling with moisture transfer, considering a uniform viscosity throughout the particle.In this study, a novel approach based on solving Navier-Stokes equation using an Arbitrary Lagrangian-Eulerian (ALE) approach is proposed to model the dynamics of particle sintering coupled with moisture transfer. Maltodextrin DE21 is considered as an industrially relevant example of amorphous particles. Due to moisture uptake, strong gradients of viscosity can exist in the particles undergoing sintering.FEM simulations consider accurately the forces acting on the contact area between the particles, leading to slower dynamics than commonly used approximate analytical models. This study highlights that FEM simulations considering a homogeneous moisture and viscosity within the particles are in many cases sufficiently accurate and identifies the limits of validity of this assumption. In the conditions considered in this study, the intraparticle gradients were found to condition significantly the sintering dynamics only when particle diameter is above 1.5 mm. The particle size affects strongly both the dynamics of sintering and of moisture transfer. Moreover, higher external relative humidity leads to a lower viscosity and a faster sintering kinetics. The initial water content was found to have a lower impact in the conditions studied.This coupled simulation approach can be used to identify conditions reducing the risk of caking during the storage of amorphous powders or to master sintering during powder structuration processes. Furthermore this study helps identifying when simpler simulation approaches considering homogeneous particles can be safely used and shows the limitations of simplified analytical models.

Investigation on the microscale combustion characteristics of AP/HTPB propellant under wide pressure range

This study developed a combustion model for ammonium perchlorate (AP)/hydroxyl–terminated polybutadiene (HTPB) composite propellant to capture the flame structure and elucidate the mechanism governing burning rate at different pressures, AP sizes and surface morphologies. A numerical model was proposed to characterize gas domain by solving Navier-Stokes equations of reacting flow field with 3-step and 12-species global kinetics; and the condensed domain was coupled by considering interfacial energy balance. The present model was validated with experimental results of both pure AP and AP/HTPB combustion, in which the mean absolute percentage error of burning rate for all cases was less than 5%. The standoff distances of AP monopropellant, premixed binder and primary diffusion flames were determined as 14.2, 12.1 and 25.8 μm at 2 MPa, respectively, reducing with pressure. The front and height of final diffusion flame were determined by axially varied radial gradient of temperature and its height was obtained as 152 μm at 2 MPa, which rises as the pressure or AP particle size increases. The increase of burning rate with pressure is attributed to the raise of surface heat flux dominated by standoff distances of AP monopropellant, premixed binder and primary diffusion flames, in which the effect of primary diffusion flame enhances as the AP size decreases resulting from enlarged lateral diffusion of species. Finally, the effects of protrusive, recessive AP surfaces and their combination with planar AP surface on the flow and flame properties were examined.

A Study of the Impact of Methanol, Ethanol and the Miller Cycle on a Gasoline Engine

This paper focuses on the investigation and optimisation of the Miller cycle, methanol, ethanol and turbocharging when applied to a high-performance gasoline engine. These technologies have been applied both individually and concurrently to test for potential compounding effects. Improvements have been targeted with regards to both emission output and performance. Also assessed is the capability of the engine to operate when exclusively powered by biofuels. This has been carried out numerically using the 1D gas dynamics tool ‘WAVE’, a 1D Navier–Stokes equation solver. These technologies have been implemented within the McLaren M838T 3.8L twin-turbo engine. The Miller cycle early intake valve close (EIVC) improved peak efficiency by 0.17% and increased power output at low and medium loads by 11%. Reductions of 6% for both NOx and CO were also found at rated speed. The biofuels achieved NOx and CO reductions of 60% and 96% respectively, alongside an efficiency increase of 2.5%. Exclusive biofuel use was found to be feasible with a minimum 35% power penalty. Applied cooperatively, the Miller cycle and biofuels were not detrimental to each other, compounding effects of a further 0.05% efficiency and 2% NOx improvements were achieved.

A deep learning based prediction approach for the supercritical airfoil at transonic speeds

In traditional ways, the aerodynamic property of the aircraft is obtained by solving Navier-Stokes equations or performing tunnel experiments. However, these methods are time consuming for the aircraft design and optimization. In comparison, the deep learning technique is capable of handling high dimensional parameters and can describe compressible flow structures clearly and efficiently. For these, an efficient and accurate prediction approach based on the deep neural network is proposed for the compressible flows over the transonic airfoils in this study. By investigating the effects of the input coordinate features of the deep learning method on the prediction accuracy and robustness, the aerodynamic characteristics, such as lift, drag, and pitch coefficients, are obtained from the predicted flow fields. Results indicate that the proposed deep learning prediction method is with a high resolution and efficiency. It is promising to be extended to the optimization and design process of the supercritical airfoil.