3D Navier-Stokes Equations Research Articles

Effect of multiple cavities and tip injection on the aerothermal characteristics of the squealer tip in turbine stage

For modern gas turbines, the blades of the first-stage high-pressure turbines are subject to significant aerodynamic and thermal loads. As the region where the aerothermal environment is most severe, the aerothermal performance of the blade tip is crucial. Either the tip injection or squealer tip with ribs can effectively improve the tip leakage flow distribution, and thus reduce the tip thermal loads, while the effect of the combination of the two remains to be explored, especially when the number of ribs increases. To further understand the combined effects of the rib number and the tip injections on the aerothermal characteristics, steady-state numerical simulations of the first stage turbine aerothermal field are carried out with varied rib numbers and tip injection configurations in the current study. The kω SST turbulence model was used to solve the 3D Reynolds-Averaged Navier-Stokes (RANS) equations. The heat transfer coefficient distributions and the leakage flow rate distributions are compared with those of the conventional squealer tip case. The results show that adding a rib inside the tip cavity would reduce the size of the reflux vortex near the leading edge of the blade (LE), and effectively reduce the averaged heat transfer coefficient on the tip. In addition, an increase in the rib number would make the coolant accumulate in the cavities, thus increasing the cooling performance. The averaged heat transfer coefficient of the coolant injected squealer tip with three ribs (Rib3C) decreases by 19.55%, compared with the performance of the conventional squealer tip, due to the combined action of adding the multiple ribs and the coolant injection. Comprehensively considering both improving the aerodynamic performance and reducing the blade tip heat load using TOPSIS analysis, the coolant injected squealer tip with two ribs (Rib2C) achieves the best overall performance. In general, ribs arranged in the upstream region of the blade would improve the coolant coverage by blocking the separation region at the cavity bottom. To improve the harsh tip thermal environment and reduce the tip aerodynamic loss, optimizing the rib layout inside the cavity and the coolant injection configurations is a practical solution.

How and why non smooth solutions of the 3D Navier–Stokes equations could possibly develop

Fluid configurations in three-dimensions, displaying a plausible decay of regularity in a finite time, are suitably built and examined. Vortex rings are the primary ingredients in this study. The full Navier–Stokes system is converted into a 3D scalar problem, where appropriate numerical methods are implemented in order to figure out the behavior of the solutions. Further simplifications in 2D and 1D provide interesting toy problems, that may be used as a starting platform for a better understanding of blowup phenomena.

Uniqueness for active scalar equations in a Zygmund space

We consider a class of active scalar equations which includes, for example, the 2D Euler equations, the 2D Navier–Stokes equations, and various aggregation equations including the Keller–Segel model. For this class of equations, we establish uniqueness of solutions in the Zygmund space C ∗ 0 . This result improves upon that in (Trans. Amer. Math. Soc. 367 (2015) 3095–3118), where the authors show uniqueness of solutions in BMO. As a corollary of our methods, we establish the uniform in space vanishing viscosity limit of Hölder continuous solutions to the aggregation equation with Newtonian potential.

Analysis of fully discrete finite element methods for 2D Navier–Stokes equations with critical initial data

First-order convergence in time and space is proved for a fully discrete semi-implicit finite element method for the two-dimensional Navier–Stokes equations with L2 initial data in convex polygonal domains, without extra regularity assumptions or grid-ratio conditions. The proof utilises the smoothing properties of the Navier–Stokes equations in the analysis of the consistency errors, an appropriate duality argument, and the smallness of the numerical solution in the discrete L2(0, tm; H1) norm when tm is smaller than some constant. Numerical examples are provided to support the theoretical analysis.

Data assimilation with higher order finite element interpolants

AbstractThe efficiency of a nudging data assimilation algorithm using higher order finite element interpolating operators is studied. Numerical experiments are presented for the 2D Navier–Stokes equations in three cases: shear flow in an annulus, a forced flow in a disk with an off‐center cavity, and a forced flow in a box all satisfying Dirichlet boundary conditions. In all three cases, second order interpolation of coarse‐grain data is shown to outperform first order interpolation. Convergence of the nudged solution to that of a direct numerical reference solution is proved. The analysis points to a trade‐off in the estimates for higher order interpolating operators

Long time decay of solutions to the 3D incompressible Navier–Stokes equations with nonlinear damping

This paper is interested in the decay of solutions to 3D Navier–Stokes equations with a nonlinear damping term . We discuss the bounds of the global strong solutions in the negative Sobolev space and establish the optimal decay rates in of the global strong solutions. Moreover, the upper bound of the derivative is also obtained. Finally, we investigate the asymptotic stability of the solutions to the system.

On the higher derivatives estimate of the surface growth equation

In this paper, we are concerned with the higher derivatives estimate of the surface growth equation which has similar features to the 3D Navier–Stokes equations. We extend the well-known apriori estimates for the Navier–Stokes equations due to Foias–Guillope–Temam in C. Foias et al. (1981) to this model. Several new sufficient conditions that guarantee regularity of solutions to the generalized surface growth equation are also presented. As a byproduct, the global well-posedness of the generalized surface growth equation in the sense of Lions’s classical work in J.L. Lions (1969) is obtained.

Added Mass and Damping of an Hexagonal Rod Vibrating in Highly Confined Viscous Fluids

Abstract This paper deals with fluid–structure interaction analysis of an hexagonal rod enclosed in a narrow viscous gap. A new analytical solution for a two-dimensional (2D) cylindrical case is derived and described. A numerical solution of 2D Navier–Stokes equations coupled with a harmonic structure model is applied to both cylindrical and prism geometries. The comparison between the numerical tool and the analytical solution is discussed and a method to apply the analytical solution to the hexagonal case is proposed. An original definition of the added mass and damping based on an energetic approach is provided avoiding the dependence from the geometry and the type of forcing (free or forced vibration). An experimental facility is provided accounting for an hexagonal prism vibrating within a 7 mm enclosure. Free vibration experiments in water allow assessing the added mass and added damping effect on the modal parameters. The fluid flow is affected by a three-dimensional (3D) effect—named down-strokes flow—at the top and the base of the assembly because of free surface and stocky geometry. This produces a higher frequency than the 2D theoretical value given both by the analytical solution and the numerical simulation. A geometry-based correction factor is suggested to taken into account in the 2D numerical simulation the 3D effect. Velocity measured within the gap provides further insight on this phenomenon and agrees well with the prediction of the transposed cylindrical analytical model.

Numerical Analysis for Supercavitation Characteristics around Underwater Vehicle according to Ventilated Gas Temperature

Supercavitation is a phenomenon in which the cavity covers the entire underwater vehicle. The purpose of this paper is to compare and analyze the thermal effect on the cavity characteristics by changing the ventilated gas temperature through computational analysis. For this study, a homogeneous mixture model based on the 3D Navier-Stokes equation was used. As a phase change model, it is its own code considering both pressure change and temperature change. A dimensionless number <i>T<sub>m</sub></i> was presented to analyze the numerical results, and as the <i>T<sub>m</sub></i> increased, the cavity length increased by about 3.6 times and the cavity width by about 3.3 times at 393.15 K compared to room temperature. Analyzing these thermal effects, it was confirmed that rapid heat exchange and heat transfer between the gas phase and the liquid phase occurred at the location where the ventilated gas was sprayed, affecting the cavity characteristics. In addition, it can be confirmed that the initial cavity surface becomes unstable as the ventilated gas temperature increases, and it can be confirmed based on the numerical analysis results that the critical temperature at which the cavity surface becomes unstable is 373.15 K.

Eilmer: An open-source multi-physics hypersonic flow solver

This paper introduces Eilmer, a general-purpose open-source compressible flow solver developed at the University of Queensland, designed to support research calculations in hypersonics and high-speed aerothermodynamics. Eilmer has a broad userbase in several university research groups and a wide range of capabilities, which are documented on the project's website, in the accompanying reference manuals, and in an extensive catalogue of example simulations. The first part of this paper describes the formulation of the code: the equations, physical models, and numerical methods that are used in a basic fluid dynamics simulation, as well as a handful of optional multi-physics models that are commonly added on to do calculations of hypersonic flow. The second section describes the processes used to develop and maintain the code, documenting our adherence to good programming practice and endorsing certain techniques that seem to be particularly helpful for scientific codes. The final section describes a half-dozen example simulations that span the range of Eilmer's capabilities, each consisting of some sample results and a short explanation of the problem being solved, which together will hopefully assist new users in beginning to use Eilmer in their own research projects. Program summaryProgram Title: EilmerCPC Library link to program files:https://doi.org/10.17632/gy2ds2fyxm.1Developer's repository link:https://github.com/gdtk-uq/gdtkCode Ocean capsule:https://codeocean.com/capsule/7226427Licensing provisions: GPLv3Programming language: D, LuaSupplementary material:https://gdtk.uqcloud.netNature of problem: Eilmer solves the compressible Navier-Stokes equations with a particular emphasis on flows at hypersonic speeds. The code includes modelling for high-temperature gas effects such as chemical and vibrational nonequilibrium. Eilmer can be used for the simulation for unsteady and steady flows.Solution method: The code is implemented in D [1] and built on a finite-volume formulation that is capable of solving the Navier-Stokes equations in 2D and 3D computational domains, discretised with structured or unstructured grids. Grids may be generated using a built-in parametric scripting tool or imported from commercial gridding software. The inviscid fluxes are computed using the reconstruction-evolution approach. In structured-grid mode, reconstruction stencils up to fourth-order spatial accuracy are available. In unstructured-grid mode, least-squares reconstruction provides second-order spatial accuracy. A variety of flux calculators are available in the code. Viscous fluxes are computed with compact stencils with second-order spatial accuracy. For unsteady flows, explicit time-stepping with low-order RK-family schemes are available, along with a point-implicit Backward-Euler update scheme for stiff systems of equations. For steady flows, convergence can be greatly accelerated using a Jacobian-free Newton-Krylov update scheme, which seeks a global minimum in the residuals using a series of large pseudo-timesteps. Domain decomposition is used for parallel execution using both shared memory and distributed memory programming techniques.Additional comments including restrictions and unusual features: Eilmer provides a programmable interface for pre-processing, post-processing and user run-time customisations. The programmable interface is enabled using a built-in embedded interpreter for the Lua programming language [2]. Run-time customisations include used-defined boundary conditions, source terms and grid motion.

Gaining two derivatives on a singular force in the 2D Navier-Stokes equations

It has long been known, for the autonomous 2D Navier-Stokes equations with singular forcing in H−1, that there exists a unique solution, globally bounded in L2, i.e., with a gain of one derivative on the force in terms of regularity. These classical techniques also show us that the solution is almost everywhere in H1. On the other hand, if the forcing term is in L2, it is known that the solution is bounded in H2 globally – a gain of two derivatives. In this paper we prove new maximal regularity results for the 2D Navier-Stokes equations. First, we explore classical techniques to show that if the force is in Hα for α∈(−1,0), then the solution gains two derivatives globally. These methods break down for forces in H−1. In this scenario, we use the Littlewood-Paley decomposition in Fourier space, and an appropriate frequency splitting, to show that for any initial data in H1 there exists a bounded in H1 solutions for a short time interval.

Stabilization of 3D Navier–Stokes Equations to Trajectories by Finite-Dimensional RHC

Local exponential stabilization of the three-dimensional Navier–Stokes system to a given reference trajectory via receding horizon control (RHC) is investigated. The RHC enters as the linear combinations of a finite number of actuators. The actuators are spatial functions and can be chosen in particular as indicator functions whose supports cover only a part of the spatial domain.

An optimal regularity criterion for 3D Navier-Stokes equations involving the gradient of one velocity component

In this paper, we provide an optimal regularity criterion for 3D Navier-Stokes equations involving the gradient of one velocity component in the framework of anisotropic Lebesgue spaces. More precisely, employing the anisotropic Littlewood-Paley theory, we prove that a weak solution u is regular if ∇u3 belongs to scaling invariant space L2(0,T;Lv∞Lh2), where h and v denote the horizontal and vertical components, respectively. This result verifies the limiting case of a previous result established by Guo, Caggio and Skalák (2017).

Global existence of the strong solution to the 3D incompressible micropolar equations with fractional partial dissipation

Abstract In this paper, we considered the global strong solution to the 3D incompressible micropolar equations with fractional partial dissipation. Whether or not the classical solution to the 3D Navier–Stokes equations can develop finite-time singularity remains an outstanding open problem, so does the same issue on the 3D incompressible micropolar equations. We establish the global-in-time existence and uniqueness strong solutions to the 3D incompressible micropolar equations with fractional partial velocity dissipation and microrotation diffusion with the initial data $(\mathbf {u}_0,\ \mathbf {w}_0)\in H^1(\mathbb {R}^3)$ .

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.

Potentially Singular Behavior of the 3D Navier–Stokes Equations

Whether the 3D incompressible Navier–Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the incompressible axisymmetric Navier–Stokes equations with smooth initial data of finite energy seem to develop potentially singular behavior at the origin. This potentially singular behavior is induced by a potential finite time singularity of the 3D Euler equations that we reported in a companion paper published in the same issue, see also Hou (Potential singularity of the 3D Euler equations in the interior domain. arXiv:2107.05870 [math.AP], 2021). We present numerical evidence that the 3D Navier–Stokes equations develop nearly self-similar singular scaling properties with maximum vorticity increased by a factor of \(10^7\). We have applied several blow-up criteria to study the potentially singular behavior of the Navier–Stokes equations. The Beale–Kato–Majda blow-up criterion and the blow-up criteria based on the growth of enstrophy and negative pressure seem to imply that the Navier–Stokes equations using our initial data develop a potential finite time singularity. We have also examined the Ladyzhenskaya–Prodi–Serrin regularity criteria (Kiselev and Ladyzhenskaya in Izv Akad Nauk SSSR Ser Mat 21(5):655–690, 1957; Prodi in Ann Math Pura Appl 4(48):173–182, 1959; Serrin in Arch Ration Mech Anal 9:187–191, 1962) that are based on the growth rate of \(L_t^q L_x^p\) norm of the velocity with \(3/p + 2/q \le 1\). Our numerical results for the cases of \((p,q) = (4,8),\; (6,4),\; (9,3)\) and \((p,q)=(\infty ,2)\) provide strong evidence for the potentially singular behavior of the Navier–Stokes equations. The critical case of \((p,q)=(3,\infty )\) is more difficult to verify numerically due to the extremely slow growth rate in the \(L^3\) norm of the velocity field and the significant contribution from the far field where we have a relatively coarse grid. Our numerical study shows that while the global \(L^3\) norm of the velocity grows very slowly, the localized version of the \(L^3\) norm of the velocity experiences rapid dynamic growth relative to the localized \(L^3\) norm of the initial velocity. This provides further evidence for the potentially singular behavior of the Navier–Stokes equations.

Potential Singularity of the 3D Euler Equations in the Interior Domain

Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D axisymmetric incompressible Euler equations with smooth initial data of finite energy develop a potential finite time singularity at the origin. This potential singularity is different from the blow-up scenario revealed by Luo and Hou (111:12968–12973, 2014) and (12:1722–1776, 2014), which occurs on the boundary. Our initial condition has a simple form and shares several attractive features of a more sophisticated initial condition constructed by Hou and Huang in ( arXiv:2102.06663 , 2021) and (435:133257, 2022). One important difference between these two blow-up scenarios is that the solution for our initial data has a one-scale structure instead of a two-scale structure reported in Hou and Huang ( arXiv:2102.06663 , 2021) and (435:133257, 2022). More importantly, the solution seems to develop nearly self-similar scaling properties that are compatible with those of the 3D Navier–Stokes equations. We will present numerical evidence that the 3D Euler equations seem to develop a potential finite time singularity. Moreover, the nearly self-similar profile seems to be very stable to the small perturbation of the initial data.

Dynamic Weight Strategy of Physics-Informed Neural Networks for the 2D Navier-Stokes Equations.

When PINNs solve the Navier–Stokes equations, the loss function has a gradient imbalance problem during training. It is one of the reasons why the efficiency of PINNs is limited. This paper proposes a novel method of adaptively adjusting the weights of loss terms, which can balance the gradients of each loss term during training. The weight is updated by the idea of the minmax algorithm. The neural network identifies which types of training data are harder to train and forces it to focus on those data before training the next step. Specifically, it adjusts the weight of the data that are difficult to train to maximize the objective function. On this basis, one can adjust the network parameters to minimize the objective function and do this alternately until the objective function converges. We demonstrate that the dynamic weights are monotonically non-decreasing and convergent during training. This method can not only accelerate the convergence of the loss, but also reduce the generalization error, and the computational efficiency outperformed other state-of-the-art PINNs algorithms. The validity of the method is verified by solving the forward and inverse problems of the Navier–Stokes equation.

Theoretical and experimental research on the coating process of ceramic stereolithography

The coating process in ceramic stereolithography is one of the principal sources of inaccuracy in the height dimension. The coating in stereolithography is a layer-by-layer build-up process, so the thickness of one layer coated is influenced by the previous layer and affects the thickness of the next layer. If inconsistent layer thicknesses exist, they will result in the poor quality of the final product. This paper analyzes the coating process of the slurry-separated system based on a simple 2D model and Navier-Stokes equations, presenting the theoretical equations of layer thickness for the commonly used blade types. Given the characteristics of layer-by-layer coating in stereolithography, the theoretical equations should be constantly iterated to obtain the thickness of each layer. This paper discovers the initial blade gap has a significant impact on the layer thicknesses by theoretical analyses. With the verification of experiments, the results shows that the presence of the unreasonable initial blade gap may lead to large fluctuations in layer thicknesses of the first few layers, and the coating in stereolithography is a "self-regulating" process with a tendency to the desired thickness. The larger the desired layer thickness, the worse this "self-regulating ability" is. This paper defines a coefficient of blade gap and proposes that using the coefficient of blade gap to determine a reasonable initial blade gap based on different working conditions can effectively control the consistency of each layer thickness.

Remarks on regularity criteria for the 3d Navier–Stokes equations

Remarks on regularity criteria for the 3d Navier–Stokes equations