Steady-state Navier-Stokes Equations Research Articles

Stacked Deep Learning Models for Fast Approximations of Steady-State Navier–Stokes Equations for Low Re Flow

Computational fluid dynamics (CFD) simulations are broadly used in many engineering and physics fields. CFD requires the solution of the Navier–Stokes (N-S) equations under complex flow and boundary conditions. However, applications of CFD simulations are computationally limited by the availability, speed, and parallelism of high-performance computing. To address this, machine learning techniques have been employed to create data-driven approximations for CFD to accelerate computational efficiency. Unfortunately, these methods predominantly depend on large labeled CFD datasets, which are costly to procure at the scale required for robust model development. In response, we introduce a weakly supervised approach that, through a multichannel input capturing boundary and geometric conditions, solves steady-state N-S equations. Our method achieves state-of-the-art results without relying on labeled simulation data, instead using a custom data-driven and physics-informed loss function and small-scale solutions to prime the model for solving the N-S equations. By training stacked models, we enhance resolution and predictability, yielding high-quality numerical solutions to N-S equations without hefty computational demands. Remarkably, our model, being highly adaptable, produces solutions on a 512 × 512 domain in a swift 7 ms, outpacing traditional CFD solvers by a factor of 1,000. This paves the way for real-time predictions on consumer hardware and Internet of Things devices, thereby boosting the scope, speed, and cost-efficiency of solving boundary-value fluid problems.

Data-physics driven multiscale approach for high-pressure resin transfer molding (HP-RTM)

We present a multiscale computational framework for high-pressure resin transfer molding of fiber-reinforced composites. Due to the relatively rapid speed of resin flow and the significant convective effects, this process is governed by the nonlinear steady-state Navier–Stokes equations, as opposed to the linear Stokes equations commonly adopted for the simulation of classical resin transfer molding processes. To overcome the computational challenge of directly solving the high-pressure resin filling problem through the fiber preform, we developed a data-driven nonlinear homogenization approach where the average velocity and instantaneous permeability computed from the microscale representative volume element problem provide solution-dependent coefficients for the macroscale problem. The proposed data-physics driven computational framework has been validated against the direct numerical solution of the steady-state Navier–Stokes equations.

Homogenization of the steady-state Navier-Stokes equations with prescribed flux rate or pressure drop in a perforated pipe

The steady motion of a viscous incompressible fluid in a pipe (perforated with a large number of small holes) is modeled through the Navier-Stokes equations with mixed boundary conditions involving the Bernoulli pressure and the tangential velocity on the inlet and outlet of the tube, while either the transversal flux rate or the pressure drop is prescribed along the pipe. Applying the classical energy method in homogenization theory, we study the asymptotic behavior of the solutions to these systems, without any restriction on the magnitude of the data, as the size of the perforations goes to zero and show that the effective equations remain unmodified in the limit. The main novelty of the present work lies in the obtainment of the required uniform bounds, which are achieved (in the case of the prescribed flux problem) by a contradiction argument based on Bernoulli's law for solutions of the stationary Euler equations.

Role of preferential hydration on diffusiophoresis of globular proteins

Diffusiophoresis is the migration of a colloidal particle through a fluid, caused by a cosolute concentration gradient. Preferential hydration is a thermodynamic phenomenon responsible for particle diffusiophoresis from high to low cosolute concentration in water. Here, steady-state Navier-Stokes equation for incompressible fluids, with particle modeled as an inpenetrable rigid sphere, is employed for deriving a mathematical expression of the diffusiophoresis coefficient. Mean-force potentials with arbitrary interaction range but with interaction energies small or comparable with thermal energy are employed to describe water-mediated particle-cosolute interactions. This choice of potential is movitated by previous work showing that only fluid particles weakly interacting with the colloildal particle can cause diffusiophoresis. Preferential hydration has been extensively investigated for proteins. Thus, the proposed model is applied to available experimental diffusiophoresis coefficients of lysozyme in aqueous NaCl and KCl at pH 4.5 and 25 °C. This analysis also includes data on salt osmotic diffusion, describing salt diffusion caused by protein concentration gradients. The preferential-hydration contribution of experimental lysozyme diffusiophoresis is obtained after removing electrophoretic diffusiophoresis. Examination of these experimental data shows that the mean-force potential describing preferential hydration is indeed weak, with an interaction range that is an appreciable fraction of particle size.

A Stabilized Finite Element Framework for Anisotropic Adaptive Topology Optimization of Incompressible Fluid Flows

This paper assesses the feasibility of performing topology optimization of laminar incompressible flows governed by the steady-state Navier–Stokes equations using anisotropic mesh adaptation to achieve a high-fidelity description of all fluid–solid interfaces. The present implementation combines an immersed volume method solving stabilized finite element formulations cast in the variational multiscale (VMS) framework and level-set representations of the fluid–solid interfaces, which are used as an a posteriori anisotropic error estimator to minimize interpolation errors under the constraint of a prescribed number of nodes in the mesh. Numerical results obtained for several two-dimensional problems of power dissipation minimization show that the optimal designs are mesh-independent (although the convergence rate does decreases as the number of nodes increases), agree well with reference results from the literature, and provide superior accuracy over prior studies solved on isotropic meshes (fixed or adaptively refined).

Large-scale parallel topology optimization of three-dimensional incompressible fluid flows in a level set, anisotropic mesh adaptation framework

This papers considers the topology optimization of duct flows governed by the three-dimensional steady state Navier–Stokes equations, using anisotropic mesh adaptation to achieve a high-fidelity description of all fluid–solid interfaces. The numerical framework combines an immersed volume method solving stabilized, linear equal-order finite element formulations cast in the Variational Multiscale (VMS) framework, and level set representations of the interface, used as a posteriori anisotropic error estimator to minimize the interpolation error under the constraint of a prescribed number of nodes in the mesh. Both the resolution and remeshing steps are performed in a massively parallel framework allowing for the optimization of large-scale systems. In particular, an original parallelization strategy is used for mesh adaptation, that combines local remeshing performed sequentially and independently on each subdomain with blocked interfaces, and constrained repartitioning to optimally move the interfaces between subdomains in an optimal way, both iterated until a satisfying mesh and partition are obtained. The proposed approach reduces the computational burden related to the call of the finite element solver, compared to classical optimization schemes working on uniform grids with similar mesh refinement. For a given number of nodes, it improves the accuracy in the geometric description of all layouts. Finally, it has the potential to alleviates the end user from most of the post-processing step aiming at extracting the final layout, due to ability of anisotropic adapted meshes to generate intrinsically smooth designs. Numerical results are provided for several three-dimensional problems of power dissipation minimization involving several dozen million state degrees of freedom, for which the optimal designs agree well with reference results from the literature, while providing superior accuracy over prior studies solved on isotropic meshes (in the sense that the flow is better resolved, especially in the near-wall regions, and the layouts are more smooth). The potential of the method for engineering problems of practical interest is eventually exposed by optimizing the distributor section conveying the cold fluid within the plates of a plate fin heat exchanger.

A short note on the Liouville problem for the steady-state Navier–Stokes equations

A short note on the Liouville problem for the steady-state Navier–Stokes equations

An Improved Arrow–Hurwicz Method for the Steady-State Navier–Stokes Equations

An Improved Arrow–Hurwicz Method for the Steady-State Navier–Stokes Equations

Multi-objective topology optimization and thermal performance of liquid-cooled microchannel heat sinks with pin fins

The optimized structures for the liquid-cooled microchannel heat sinks with different pin-fin arrays were designed via using the topology optimization method for the better performance, including minimization of the flow energy dissipation and the average temperature of the bottom surface, aiming to facilitate the more efficient design of microchannel heat sink for electronic chips’ cooling. The heat transfer and flowing performance was simulated with steady-state incompressible Navier-Stokes equation and the energy equation. Based on the rational approximation of material properties (RAMP) method, the mathematical optimization model for heat sink was established according to the multi-objective function. The Galerkin finite element method (GFEM) was employed to calculate the flow and heat transfer system. Furthermore, the globally convergent method of moving asymptotes (GCMMA) was adopted to deal with the mathematical optimization problem as the optimization method. The entropy generation analysis and comprehensive heat transfer factor were employed to evaluate the performance of the conjugate heat transfer problem of 3D microchannel heat sinks with various pin-fin structures. The calculation results reveal that the multi-objective optimal structure has good heat transfer performance, and the temperature distribution of the optimized structure is more uniform. As the same pumping power, the comprehensive heat transfer efficiency of the optimized inline and staggered pin-fin arrays increases by 31.20% and 32.18% compared to the smooth rectangular channel. Compared with the cylindrical pin-fin configuration with the same heat transfer area, the entropy generation rates of the optimized inline and staggered pin-fin configurations decrease by 4.67% and 4.25%, respectively. The results can provide theoretical basis and application support for the electronic chip cooling technology.

Numerical Study of Film-Cooling and Aerodynamic Performance of Contoured Cratered Holes

Film cooling with discrete cratered holes has been proven to be one of the effective ways to increase the cooling effectiveness for flat-plate conditions. Up to now, there has been a lack of relevant research on various discrete cratered holes under cascade conditions. In this paper, the cooling and aerodynamic performances of a reference contoured cratered hole and two optimized contoured cratered holes are studied employing the numerical analysis method. The velocity and temperature fields for the film cooling are obtained by solving the steady-state Reynolds-averaged Navier-Stokes equations with the Shear Stress Transport turbulence closure. The numerical results show that these two optimized holes significantly enhance the cooling performance with an ignorable increment of pressure loss compared to the reference hole. When the cooling hole is placed at the pressure or suction side, the local pressure gradient plays a key role in developing the downstream flow field for either cooling hole. Furthermore, the pressure loss is mainly affected by the blowing ratio and the film cooling hole location. The pressure loss in the hole and its contribution to the total pressure loss increase with the increase of the blowing ratio.

Numerical Investigation of Flow and Dispersion over Two-Dimensional Semi-Open Street Canyon

A semi-open street canyon is able to protect pedestrians from unpleasant situations such as direct sunlight and rain. However, the protruding elements of the two opposite building facades that form the semi-open configuration can affect the air quality of the urban canopy layer (UCL). Therefore, this paper investigated the influence of the eave structures on the flow and pollutant dispersion over an idealized 2D street canyon with a unity aspect ratio. The length of the eaves was varied into 0.25H and 0.5H (H is the building height) and placed either on the leeward wall, the windward wall, or on both building facades located at the same elevation as the street canyon. Numerical simulations were performed using the steady-state Reynolds-averaged Navier-Stokes (RANS) equations in conjunction with Re-Normalization Group (RNG) k-ε as the turbulence closure model. The pollutant was released from a line source in the center of the bottom of the target canyon with uniform flow rate. Six different eave configurations were simulated in the wind direction perpendicular to the canyon axis, representing the worst condition of canyon ventilation. The evolution of the primary vortex, which occupied the entire canyon with the characteristic of skimming flow, showed less dependence on the length and position of the eave, except for the longest eave on the windward wall. However, the position of the vortex center depicted opposite results. The pollutant concentration is always higher near the leeward wall, but for the eave that protrudes from the windward wall with a length of 0.5H, the pollutant accumulates near the windward region. The ratio of pollutant concentration showed higher concentration in the semi-open configurations compared to the fully open layout as a result of limited penetration of shear flow into the canyon, which leads to deterioration of pollutant removal.

150USRT Class R-513A Refrigerant Two-Stage Centrifugal Compressor Design Point and Separation Point Flow Field Simulation Analysis

This study used computational fluid dynamics for low greenhouse effect refrigerant (R-513A) simulation analysis in the two-stage 150 USRT class oil-free centrifugal refrigerant compressor using integrated part load value (IPLV) and internal flow field. The compressor rotor speed and mass flow rate for the IPLV working conditions with various loads were planned using Stage 1 and Stage 2 simulations, respectively. The performance and flow field numerical simulation analyses for the two-stage centrifugal compressor is discussed. This study used Ansys-CFX software for numerical simulation analysis and the conservation form of a 3D steady-state Navier–Stokes equation set with the finite volume discretization method for computation. The computing mode produced better computing efficiency and flexible mesh setting using the k-omega (k-ω) model, which has better computational performance in the near wall boundary layer and low Reynolds number flow field (used as the turbulence model) for simulation. The R-513A refrigerant property setting was calculated using the Soave–Redlich–Kwong gas equation. This study discusses the shaft work, pressure ratio, and isentropic efficiency and also describes the main research findings with the meridional pressure, isentropic efficiency contour maps, and flow field velocity vector diagrams. According to the numerical simulation results, in Stage 1 and Stage 2 simulations, the isentropic efficiency produced the highest numerical results in the 75% load case, which are 88.19% and 89.06%, respectively. The isentropic efficiency decreased to 75.93% and 82.26%, respectively, in the 25% load case. The flow field velocity vector diagram shows that in the 25% load case, there was back-flow field distribution near the impeller shroud. The compressor performance was also analyzed.

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.

Структура течения и теплоперенос в газокапельном течении за внезапным сужением канала

The numerical study of the local flow structure and heat transfer in a gas-droplet turbulent flow behind a forward-facing step in the two-phase gas-droplet flow is carried out. Two-dimensional steady-state Reynolds-averaged Navier-Stokes (RANS) equations are used for the numerical solution. They were written taking into account the presence of a dispersed phase. The Eulerian two-fluid approach is used to describe the flow dynamics and heat and mass transfer in the gaseous and dispersed phases. The turbulence of the carrier phase was described using an elliptical model for the transport of Reynolds stress components, taking into account the presence of droplets. The effect of evaporating droplets flowing after the forward-facing step on the local flow structure, turbulence, distribution of the dispersed phase, and heat transfer inetensification is analyzed.

CFD INVESTIGATION OF HEAT TRANSFER ON FOUR CYLINDERS IN IN-LINE SQUARE ARRANGEMENT EQUIPPED WITH SPLITTER PLATES

A numerical study was carried out for heat transfer in flow past four cylinders in in-line square arrangement with a pitch-to-diameter ratio <i>L/D</i> = 4.2. Computational fluid dynamics (CFD) computations were performed for Reynolds numbers of 9300 and 20,000 to investigate passive control using a rectangular splitter plate placed on the rear side of the cylinders. Three different configurations were considered: the first one contains only smooth cylinders, the second uses cylinders equipped with splitter plate in the upstream row, and the third configuration where all cylinders have a splitter plate. The steady-state Reynolds-averaged Navier-Stokes equations were solved using a finite volume method, where the k-ω SST turbulence model was used to produce a closed system of solvable equations. Results of the simulations in terms of temperature and Nusselt number distributions for all cylinders are presented and compared. The effect of new configurations on the heat transfer is well demonstrated, where a drop of about 7.92% in Nusselt number is reported using the splitter plate.

New numerical method for solving optimal control problem for the stationary Navier-Stokes equations

<abstract><p>The purpose of this article is to propose a new numerical method to solve the problem of optimal control of the steady-state Navier-Stokes equations via the velocity-pressure formulation. We apply a new technique to derive a system of equations from which the solution can be calculated. The spectral method is used to discretize the problem. An extended relaxation method is proposed in the numerical part to ensure the correct convergence of the system. Finally, numerical results are provided to confirm the effectiveness of this approach.</p></abstract>

Development of a 3D Eulerian/Lagrangian Aircraft Icing Simulation Solver Based on OpenFOAM

A 3D icing simulation code is developed in the open-source CFD toolbox OpenFOAM. A hybrid Cartesian/body-fitted meshing method is used to generate high-quality meshes around complex ice shapes. Steady-state 3D Reynolds-averaged Navier-Stokes (RANS) equations are solved to provide the ensemble-averaged flow around the airfoil. Considering the multi-scale nature of droplet size distribution, and more importantly, to represent the less uniform nature of the Super-cooled Large Droplets (SLD), two droplet tracking methods are realized: the Eulerian method is used to track the small-size droplets (below 50 m) for the sake of efficiency; the Lagrangian method with random sampling is used to track the large droplets (above 50 m); the heat transfer of the surface overflow is solved on a virtual surface mesh; the ice accumulation is estimated via the Myers model; finally, the final ice shape is predicted by time marching. Limited by the availability of experimental data, validations are performed on 3D simulations of 2D geometries using the Eulerian and Lagrangian methods, respectively. The code proves to be feasible and accurate enough in predicting ice shapes. Finally, an icing simulation result of the M6 wing is presented to illustrate the full 3D capability.

New Numerical Approach for the Steady-State Navier–Stokes Equations with Corner Singularity

The steady-state nonlinear problem governing the Newtonian flow of an incompressible viscous fluid in convection form in L-shaped domain is considered. The solution as [Formula: see text]-generalized one in weighted sets is determined. Unlike the classical definition of the generalized solution in a weak formulation, which has a symmetric structure, our setting has an asymmetric one so that a special inf-sup condition in weighted sets is proved. A new weighted finite element scheme for an approximate [Formula: see text]-generalized solution is constructed. A converging iterative method to solve the sequence of linear problems with block preconditioning of their matrices is introduced. The results of numerical simulations of several tests have shown an advantage over the standard methods so that an approximate [Formula: see text]-generalized solution tends to the exact one with a rate almost twice in order exceed, relative to the grid step, than classical approaches. The result is achieved without using mesh refined in the neighborhood of a reentrant corner.

English

We study linear stability of solutions to the Navier\textendash Stokes equations with stochastic viscosity. Specifically, we assume that the viscosity is given in the form of a~stochastic expansion. Stability analysis requires a solution of the steady-state Navier-Stokes equation and then leads to a generalized eigenvalue problem, from which we wish to characterize the real part of the rightmost eigenvalue. While this can be achieved by Monte Carlo simulation, due to its computational cost we study three surrogates based on generalized polynomial chaos, Gaussian process regression and a shallow neural network. The results of linear stability analysis assessment obtained by the surrogates are compared to that of Monte Carlo simulation using a set of numerical experiments.

Hyperbolic Method to Explore Multiplicity Flow Solutions in a Four-Sided Lid-Driven Cavity

In this study, the hyperbolic method is adopted to explore the flow field states of incompressible flow in a four-sided lid-driven square cavity. In particular, we focus on the flow bifurcation obtained at the critical Reynolds number . In the hyperbolic method, the diffusive term is transformed into a hyperbolic one by introducing a diffusion flux term, which is the solution of an additional equation. A classical Riemann-like solver with a finite-volume discretization is thus employed for the full flux (split into advective and diffusive parts), in order to solve the steady-state incompressible Navier-Stokes equation. The incompressibility of the flow is treated via the artificial pseudo-compressibility method. It is shown that our numerical code is able to detect the bifurcation by the analysis of the residual term relaxation during the pseudo-time iteration procedure. Moreover, depending on the combination choice of slope limiters for the two spatial directions, our method is able to select the first or the second stable solution among the double flow field state obtained when the Reynolds number is higher than the critical value that is estimated to be 129.4 in our study.