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1.
    
In this note, we show the link between the classical continuous surface stress and continuous surface force approaches together with special finite element method techniques toward a fully implicit level set method. Based on a modified surface stress formulation, neither normals nor curvature has to be explicitly calculated. The method is space‐dimension independent. Prototypical numerical tests of benchmarking character for a rising 2D bubble are provided for validating the accuracy of this new approach. We show additionally that the explicit redistancing can be avoided using a nonlinear PDE so that a fully implicit and even monolithic formulation of the corresponding multiphase problem gets feasible.  相似文献   

2.
    
In this paper, we present a numerical scheme for solving 2‐phase or free‐surface flows. Here, the interface/free surface is modeled using the level‐set formulation, and the underlying mesh is adapted at each iteration of the flow solver. This adaptation allows us to obtain a precise approximation for the interface/free‐surface location. In addition, it enables us to solve the time‐discretized fluid equation only in the fluid domain in the case of free‐surface problems. Fluids here are considered incompressible. Therefore, their motion is described by the incompressible Navier‐Stokes equation, which is temporally discretized using the method of characteristics and is solved at each time iteration by a first‐order Lagrange‐Galerkin method. The level‐set function representing the interface/free surface satisfies an advection equation that is also solved using the method of characteristics. The algorithm is completed by some intermediate steps like the construction of a convenient initial level‐set function (redistancing) as well as the construction of a convenient flow for the level‐set advection equation. Numerical results are presented for both bifluid and free‐surface problems.  相似文献   

3.
    
A three‐dimensional finite element method for incompressible multiphase flows with capillary interfaces is developed based on a (formally) second‐order projection scheme. The discretization is on a fixed (Eulerian) reference grid with an edge‐based local h‐refinement in the neighbourhood of the interfaces. The fluid phases are identified and advected using the level‐set function. The reference grid is then temporarily reconnected around the interface to maintain optimal interpolations accounting for the singularities of the primary variables. Using a time splitting procedure, the convection substep is integrated with an explicit scheme. The remaining generalized Stokes problem is solved by means of a pressure‐stabilized projection. This method is simple and efficient, as demonstrated by a wide range of difficult free‐surface validation problems, considered in the paper. Copyright © 2005 John Wiley & Sons, Ltd.  相似文献   

4.
    
Numerical issues arising in computations of viscous flows in corners formed by a liquid–fluid free surface and a solid boundary are considered. It is shown that on the solid a Dirichlet boundary condition, which removes multivaluedness of velocity in the ‘moving contact‐line problem’ and gives rise to a logarithmic singularity of pressure, requires a certain modification of the standard finite‐element method. This modification appears to be insufficient above a certain critical value of the corner angle where the numerical solution becomes mesh‐dependent. As shown, this is due to an eigensolution, which exists for all angles and becomes dominant for the supercritical ones. A method of incorporating the eigensolution into the numerical method is described that makes numerical results mesh‐independent again. Some implications of the unavoidable finiteness of the mesh size in practical applications of the finite‐element method in the context of the present problem are discussed. Copyright © 2009 John Wiley & Sons, Ltd.  相似文献   

5.
A fully discrete postprocessing mixed finite element scheme is considered for solving the time-dependent Navier–Stokes equations. In the PP method, we only consider a non-linear equation in the coarse-level subspace and a linear problem in the fine-level subspace. The analysis shows that the PP scheme can reach the same accuracy as the standard Galerkin method with a very fine mesh size h by an appropriate choice of H. Numerical examples are provided that confirm both the theoretical analysis and the corresponding improvement in computational efficiency.  相似文献   

6.
    
This works deals with sensitivity analysis (SA) for the Navier‐Stokes equations. The aim is to provide an estimate of the variance of the velocity field when some of the parameters are uncertain and then to use the variance to compute confidence intervals for the output of the model. First, we introduce the physical model and analyze its stability. The sensitivity equations are derived, and their stability analyzed as well. We propose a finite element‐volume numerical scheme for the state and the sensitivity, which is integrated into the open‐source industrial code TrioCFD. Finally, we present some numerical results: a steady and an unsteady test case for the channel flow problem are investigated. For the steady case, we compare the results to the Monte Carlo method and show how the SA technique succeeds in providing very accurate estimates of the variance. For the unsteady case, a new filtering procedure is proposed to deal with a sensitivity that grows in time. The filtered sensitivity is then used to compute the variance of the output and to provide confidence intervals.  相似文献   

7.
Aerodynamic characteristics of various geometries are predicted using a finite element formulation coupled with several numerical techniques to ensure stability and accuracy of the method. First, an edge‐based error estimator and anisotropic mesh adaptation are used to detect automatically all flow features under the constraint of a fixed number of elements, thus controlling the computational cost. A variational multiscale‐stabilized finite element method is used to solve the incompressible Navier‐Stokes equations. Finally, the Spalart‐Allmaras turbulence model is solved using the streamline upwind Petrov‐Galerkin method. This paper is meant to show that the combination of anisotropic unsteady mesh adaptation with stabilized finite element methods provides an adequate framework for solving turbulent flows at high Reynolds numbers. The proposed method was validated on several test cases by confrontation with literature of both numerical and experimental results, in terms of accuracy on the prediction of the drag and lift coefficients as well as their evolution in time for unsteady cases.  相似文献   

8.
In this work, we consider a stabilised characteristic finite element method for the time-dependent Navier–Stokes equations based on the lowest equal-order finite element pairs. The diffusion term in these equations is discretised by using finite element method, the temporal differentiation and advection terms are treated by characteristic schemes. Unconditionally stable results and error estimates of optimal order for the velocity and pressure are established. Finally, some numerical results are provided to verify the performance of this method.  相似文献   

9.
    
This work presents a comprehensive framework for the sensitivity analysis of the Navier–Stokes equations, with an emphasis on the stability estimate of the discretized first-order sensitivity of the Navier–Stokes equations. The first-order sensitivity of the Navier–Stokes equations is defined using the polynomial chaos method, and a finite element-volume numerical scheme for the Navier–Stokes equations is suggested. This numerical method is integrated into the open-source industrial code TrioCFD developed by the CEA. The finite element-volume discretization is extended to the first-order sensitivity Navier–Stokes equations, and the most significant and original point is the discretization of the nonlinear term. A stability estimate for continuous and discrete Navier–Stokes equations is established. Finally, numerical tests are presented to evaluate the polynomial chaos method and to compare it to the Monte Carlo and Taylor expansion methods.  相似文献   

10.
    
Based on Navier‐Stokes stationary equations a mathematical model is constructed for liquid and gas media with funnel‐shaped rotation observed in atmospheric and sea tornados. Both compressible and incompressible media are considered. The differential equations corresponding to the mathematical model are integrated in elementary functions and the solutions are represented by seven rotationally symmetrical orthogonal curvilinear coordinates applicable to different shapes of funnels.  相似文献   

11.
The steady incompressible Navier–Stokes equations are coupled by a Poisson equation for the pressure from which the continuity equation is subtracted. The equivalence to the original N–S problem is proved. Fictitious time is added and vectorial operator-splitting is employed leaving the system coupled at each fractional-time step which allows satisfaction of the boundary conditions without introducing artificial conditions for the pressure. Conservative second-order approximations for the convective terms are employed on a staggered grid. The splitting algorithm for the 3D case is verified through an analytic solution test. The stability of the method at high values of Reynolds number is illustrated by accurate numerical solutions for the flow in a lid-driven rectangular cavity with aspect ratio 1 and 2, as well as for the flow after a back-facing step.  相似文献   

12.
    
In this paper, the finite element method with new spherical Hankel shape functions is developed for simulating 2‐dimensional incompressible viscous fluid problems. In order to approximate the hydrodynamic variables, the finite element method based on new shape functions is reformulated. The governing equations are the Navier‐Stokes equations solved by the finite element method with the classic Lagrange and spherical Hankel shape functions. The new shape functions are derived using the first and second kinds of Bessel functions. In addition, these functions have properties such as piecewise continuity. For the enrichment of Hankel radial basis functions, polynomial terms are added to the functional expansion that only employs spherical Hankel radial basis functions in the approximation. In addition, the participation of spherical Bessel function fields has enhanced the robustness and efficiency of the interpolation. To demonstrate the efficiency and accuracy of these shape functions, 4 benchmark tests in fluid mechanics are considered. Then, the present model results are compared with the classic finite element results and available analytical and numerical solutions. The results show that the proposed method, even with less number of elements, is more accurate than the classic finite element method.  相似文献   

13.
建立了不可压缩Navier-Stokes方程的Crank-Nicolson有限差分方法,数值模拟水槽晃动中流场及其涡流的数值变化规律。将数值解与解析解和前人的数值解进行比较,数值验证了不可压缩Naver-Stokes方程有限差分方法的有效性。通过数值模拟得到水槽在不同程度的倾斜激励晃动下流场及涡流的数值变换规律,当倾斜激励晃动的频率接近或远离共振频率时,水槽涡场的变化逐步由双涡变成单涡,再到不规则的涡场。当倾斜激励晃动的频率靠近共振频率ω_p=0.95ω_1附近时,水槽流场上部形成一个小涡,然后小涡扩大成整个水槽中的大涡,大涡下沉分裂成两个单涡,最后在底部消失;当倾斜激励晃动的频率在ω_p=0.75ω_1附近时,水槽底部形成一个小涡,然后扩大成大的单涡,最后在自由面消失;当倾斜激励晃动的频率在ω_p=0.55ω_1附近时,水槽底部出现小涡,然后扩大成大的单涡,大涡在自由面消失,继而出现不规则的大涡和不规则的小涡。  相似文献   

14.
    
The marker‐density‐function (MDF) method has been developed to conduct direct numerical simulation (DNS) for bubbly flows. The method is applied to turbulent bubbly channel flows to elucidate the interaction between bubbles and wall turbulence. The simulation is designed to clarify the structure of the turbulent boundary layer containing microbubbles and the mechanism of frictional drag reduction. It is deduced from the numerical tests that the interaction between bubbles and wall turbulence depends on the Weber and Froude numbers. The reduction of the frictional resistance on the wall is attained and its mechanism is explained from the modulation of the three‐dimensional structure of the turbulent flow. Copyright © 2001 John Wiley & Sons, Ltd.  相似文献   

15.
    
A liquid at rest in a container will show a contact angle at the wall depending on material properties. If the liquid, or the boundary walls, are moving with constant speed, this angle will change with velocity. We perform numerical experiments for a two‐dimensional model free boundary value problem that has been proved to be well‐posed and show that the dependence of the contact angle on the velocity is qualitatively correct.  相似文献   

16.
    
In this article, the aim is to study the incompressible Navier‐Stokes equations with a permeable wall and derive the complete asymptotic expansion with respect to the kinematic viscosity ν. More precisely, we derive the physically uniform (in time and space) estimates of the boundary layers and prove that the asymptotic expansion is reasonable in uniform norm. This improves the results already obtained by R. Temam and X. Wang (2000, ZAMM, Volume 80 Issue 11–12, Pages 835–843).  相似文献   

17.
    
The purpose of this research is to analyze the application of neural networks and specific features of training radial basis functions for solving 2‐dimensional Navier‐Stokes equations. The authors developed an algorithm for solving hydrodynamic equations with representation of their solution by the method of weighted residuals upon the general neural network approximation throughout the entire computational domain. The article deals with testing of the developed algorithm through solving the 2‐dimensional Navier‐Stokes equations. Artificial neural networks are widely used for solving problems of mathematical physics; however, their use for modeling of hydrodynamic problems is very limited. At the same time, the problem of hydrodynamic modeling can be solved through neural network modeling, and our study demonstrates an example of its solution. The choice of neural networks based on radial basis functions is due to the ease of implementation and organization of the training process, the accuracy of the approximations, and smoothness of solutions. Radial basis neural networks in the solution of differential equations in partial derivatives allow obtaining a sufficiently accurate solution with a relatively small size of the neural network model. The authors propose to consider the neural network as an approximation of the unknown solution of the equation. The Gaussian distribution is used as the activation function.  相似文献   

18.
    
We study both, by experimental and numerical means the fluid dynamical phenomenon of edge tones. Of particular interest is the verification of scaling laws relating the frequency f to given quantities, namely d, the height of the jet, w, the stand–off distance and the velocity of the jet. We conclude that the Strouhal number Sd is related to the geometrical quantities through Sd = C ⋅ (d / w)n with n ≈ 1, in contrast to some analytical treatments of the problem. The constant C of the experiment agrees within 13–15% with the result of the numerical treatment. Only a weak dependence on the Reynolds number with respect to d is observed. In general, a very good agreement of the experimental and the numerical simulations is found.  相似文献   

19.
    
Let Ω ⊆ ℝ3 be a uniformly regular domain of the class C3 or Ω = ℝ3. Let A denote the Stokes operator and {Eλ; λ > 0} be the resolution of identity of A. We show as the main result of the paper that if w is a nonzero global weak solution to the Navier‐Stokes equations in Ω satisfying the strong energy inequality, then there exists a nonnegative finite number a = a(w) such that for every ε > 0 [lim_{t rightarrow infty} frac {||(E_{a+varepsilon}‐E_{a‐varepsilon}) w(t)||} {||w(t)||} = 1, ] where we put Ea‐ε = 0 if a‐ε < 0. Thus, every nonzero global weak solution satisfying the strong energy inequality exhibits large‐time energy concentration in a particular frequency. Moreover, the solutions with the exponentially decreasing energy are characterized by the positivity of a. In Appendix, we present some further results describing in detail the large‐time behavior of w.  相似文献   

20.
The time-dependent Navier–Stokes equations and the energy balance equation for an incompressible, constant property fluid in the Boussinesq approximation are solved by a least-squares finite element method based on a velocity–pressure–vorticity–temperature–heat-flux ( u –P–ω–T– q ) formulation discretized by backward finite differencing in time. The discretization scheme leads to the minimization of the residual in the l2-norm for each time step. Isoparametric bilinear quadrilateral elements and reduced integration are employed. Three examples, thermally driven cavity flow at Rayleigh numbers up to 106, lid-driven cavity flow at Reynolds numbers up to 104 and flow over a square obstacle at Reynolds number 200, are presented to validate the method.  相似文献   

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