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EN
The paper studies the numerical solution of the inverse problem for a linearized two-dimensional system of Navier-Stokes equations in a circular cylinder with a final overdetermination condition. For a biharmonic operator in a circle, a generalized spectral problem has been posed. For the latter, a system of eigenfunctions and eigenvalues is constructed, which is used in the work for the numerical solution of the inverse problem in a circular cylinder with specific numerical data. Graphs illustrating the results of calculations are presented.
EN
Purpose: Mathematical modelling of the process of deformation of pipeline, transporting gasliquid mixtures with aggressive components and a comparative analysis of the value of the specified velocity depending on the dynamic viscosity of the multicomponent gas mixture is conducted. Design/methodology/approach: A mathematical model of the process of leakage of the transported product due to the loss of tightness of the pipe based on the system of Navier-Stokes equations with boundary conditions with considering the geometry of the leakage zones and the value of the leakage rate is implemented. Findings: Models of the process of deformation of the pipeline due to displacements of a certain set of surface points by specifying different types of functions, describing the geometry of deformed sections are constructed. The method of calculating the tensely deformed state based on the data on the movement of surface points by comparing different ways of setting functions, taking into account the actual configuration of sections and axes is improved. The change of flow characteristics in the pipeline when changing the structure of the mix, transported by studying of influence of change of dynamic viscosity is investigated; The method of calculating the rate of leakage of the mixture in case of loss of tightness due to the occurrence of critical stresses in the pipe material is improved. Research limitations/implications: Building a model of the deformation process, information about the nature, duration of forces and loads affecting the pipeline is not used. The law of the pipeline movement was constructed having taken into account the deformation of the sections in three directions. The necessity to take wind loads into account, estimating the real tensely deformed state was displayed. Practical implications: Using the method of calculating the tensely deformed state based on the data on the movement of surface points by comparing different ways of setting functions, taking into account the actual configuration of sections and axes. Originality/value: According to the computational algorithms created on the basis of the specified models, the calculations of the tense state of the pipelines and the flow rate of the mixture depending on its composition were performed. An analysis of the results of calculations - tense intensity and flow rate depending on the dynamic viscosity of the mixture is performed. The influence on the flow parameters - the flow rate of the mixture and the force of hydraulic resistance - changes in the dynamic viscosity of the mixture is analyzed.
EN
Knowledge about complex physical phenomena used in the casting process simulation requires continuous complementary research and improvement in mathematical modeling. The basic mathematical model taking into account only thermal phenomena often becomes insufficient to analyze the process of metal solidification, therefore more complex models are formulated, which include coupled heat-flow phenomena, mechanical or shrinkage phenomena. However, such models significantly complicate and lengthen numerical simulations; therefore the work is limited only to the analysis of coupled thermal and flow phenomena. The mathematical description consists then of a system of Navier-Stokes differential equations, flow continuity and energy. The finite element method was used to numerically modeling this problem. In computer simulations, the impact of liquid metal movements on the alloy solidification process in the casting-riser system was assessed, which was the purpose of this work, and the locations of possible shrinkage defects were pointed out, trying to ensure the right supply conditions for the casting to be free from these defects.
4
Content available Difference melt model
EN
The basic objective of the research is to construct a difference model of the melt motion. The existence of a solution to the problem is proven in the paper. It is also proven the convergence of the difference problem solution to the original problem solution of the melt motion. The Rothe method is implemented to study the Navier-Stokes equations, which provides the study of the boundary value problems correctness for a viscous incompressible flow both numerically and analytically.
EN
In this paper, an incompressible, two-dimensional (2D), time-dependent, Newtonian, laminar, and internal channel fluid flow over a skewed equilateral cavity is simulated using the finite difference method (FDM) and alternating direction implicit (ADI) technique. Navier-Stokes equations are solved numerically in stream function-vorticity formulation. The goal of tackling this problem depends on its academic significance by studying the difference between lid-driven and shear-driven cavity flows in terms of the formation of Moffatt eddies at the sharp corner, also to obtain the length and intensity ratios of these counter-rotating vortices. The value of velocity components along the centerlines of the skewed cavity was revealed at low and intermediate Reynolds numbers (Re), typically (Re = 200 and 2000) at two different skew angles of mainly 30° and 45°. Likewise, the blocked-off regions’ method is used to deal with the geometry of the skewed cavity especially the sharp corners. Furthermore, as Re increases, the main vortex approaches the skewed cavity center and the counter-rotating vortices get bigger in size and intensity, and their number increases.
EN
Heat transfer and fluid flow in the rectangular channel with an obstacle are considered. The problem is described by the Fourier-Kirchhoff equation, Navier-Stokes equations and continuity equation supplemented by appropriate boundary and initial conditions. To solve this system of equations the finite difference method with a staggered grid is used. The results of computations obtained using authorial computer program are compared with ANSYS Fluent simulation. Computations are carried out for obstacles of various sizes and positions, and on this basis the conclusions are formulated.
EN
The free convection heat transfer from an isothermal vertical plate in open space is investigated theoretically. In contrast to conventional approaches we use neither boundary layer nor self-similarity concepts. We base on expansion of the fields of velocity and temperature in a Taylor Series in x coordinate with coefficients being functions of the vertical coordinate (y). In the minimal version of the theory we restrict ourselves by cubic approximation for both functions. The Navier-Stokes and Fourier-Kirchhoff equations that describe the phenomenon give links between coefficient functions of y that after exclusion leads to the ordinary differential equation of forth order (of the Mittag-Leffleur type). Such construction implies four boundary conditions for a solution of this equation while the links between the coefficients need two extra conditions. All the conditions are chosen on the basis of the experience usual for free convection. The choice allows us to express all the theory parameters as functions of the Rayleigh number and the temperature difference. To support the conformity of the theory we derive the Nusselt-Rayleigh numbers relation that has the traditional form. The solution in the form of velocity and temperature profiles is evaluated and illustrated for air by examples of plots of data.
EN
The article presents the study of the flow of non-Newtonian liquid on the hydrodynamic initial section in doubly connected region. A method which allows solving this problem is to use the principle of conservation of momentum for a moving fluid. The study used the equation Navier- Stokesa for in viscid fluid moving in a two dimensional space. For the solution of the Navier-Stokes equations it is necessary to know the function of cross linking for the velocity distribution in hydrodynamic initial section in doubly connected region. This function was obtained by integrating the Navier-Stokes equations and after substitution in the equation for the distribution of local velocities allows calculating the local velocity in the liquid flow on the hydrodynamic initial section in doubly connected region. In addition to the analysis of the function of cross linking was discussed.
PL
W artykule przedstawiono analizę przepływu cieczy nienewtonowskiej na styku początkowym podwójnie połączonego odcinka hydraulicznego. Sposób, który pozwala na rozwiązanie tego problemu to jest zastosowanie zasady zachowania pędu do poruszającego się płynu. W badaniu analitycznym zastosowano równanie Naviera-Stokesa dla lepkiego płynu poruszającego się w przestrzeni dwuwymiarowej. Do układu równań Naviera-Stokesa należy wprowadzić funkcję sieciowania dla rozkładu prędkości na początku przepływu hydrodynamicznego w dwuwymiarowym obszarze. Funkcję otrzymano przez całkowanie równań Naviera-Stokesa. Po podstawieniu do równań dla rozkładu lokalnych prędkości umożliwiło to obliczenie lokalnej prędkości w cieczy w dwuwymiarowym przepływie hydrodynamicznym.
EN
The Yosida methods for incompressible viscous flows are investigated numerically in the aspect of local and global errors of volume conservation. Unsteady Stokes and Navier–Stokes flows past an obstacle inserted into 2D channel are used as the test cases. Open boundary conditions are imposed at the channel’s inlet and outlet. The results obtained by the Yosida-based Spectral Element Method (SEM) solvers are compared to the results obtained by the SEM solver using exact factorization of the Uzawa system. Analysis of parametric variation of the velocity divergence and the flow rate errors is presented. It is concluded that switching to higher-order Uzawa methods reduces substantially volume conservation errors and removes numerical artifacts observed at the channel’s inlet when the basic Yosida method is used.
EN
We examine the conditional regularity of the solutions of Navier–Stokes equations in the entire three-dimensional space under the assumption that the data are axially symmetric. We show that if positive part of the radial component of velocity satisfies a weighted Serrin type condition and in addition angular component satisfies some condition, then the solution is regular.
EN
A novel solution of the free convection boundary problem is represent ed in analytical form for velocity and temperature for an isothermal vertical plate, as an examp le. These fields are built as a Taylor Series in the x coordinate with coefficients as functions of the vertical coordinate ( y ). We restrict ourselves by cubic approximation for both functions. T he basic Navier-Stokes and Fourier-Kirchhoff equations and boundary conditions give links between coefficients and connected with free convection heat transfer phenomen on which define the analytical form of the solution as a function of the Grashof number only. In t he solution the non zero velocity of a fluid flow through a leading edge of the plate is take n into account. The solution in the form of velocity and temperature profiles is numerically evaluated and illustrated for air.
EN
The main aim of this article is numerical solution to the Navier–Stokes equations for incompressible, non-turbulent and subsonic fluid flows with Gaussian physical random parameters. It is done with the use of the specially adopted Finite Volume Method extended towards probabilistic analysis by the generalized stochastic perturbation technique. The key feature of this approach is the weighted version of the Least Squares Method implemented symbolically in the system MAPLE to recover nodal polynomial response functions of the velocities, pressures and temperatures versus chosen input random variable(s). Such an implementation of the Stochastic Finite Volume Method is applied to model 3D flow problem in the statistically homogeneous fluid with uncertainty in its viscosity and, separately, coefficient of the heat conduction. Probabilistic central moments of up to the fourth order and the additional characteristics are determined and visualized for the cavity lid driven flow owing to the specially adopted graphical environment FEPlot. Further numerical extension of this technique is seen in an application of the Taylor–Newton–Gauss approximation technique, where polynomial approximation may be replaced with the exponential or hyperbolic ones.
EN
An efficient method for simulating laminar flows in complex geometries is presented. The artificial compressibility method was applied to solve two- and three-dimensional Navier-Stokes equations in primitive variables on Cartesian grids. Two numerical approaches were proposed in this work, which are based on the method of lines process in conjunction with transfer of all the variables from the boundaries to the nearest uniform grid knots. Initial value problems for the systems of ordinary differential equations for pressure and velocity components were computed using the one-step backward-differentiation predictor-corrector method or the Galerkin-Runge-Kutta method of third order. Some test calculations for laminar flows in square, half-square, triangular, semicircular, cubic, half-cubic, half-cylinder and hemisphere cavities with one uniform moving wall were reported. The present results were compared with the available data in the literature and the Fluent solver numerical simulations.
14
Content available remote A mixed, scalable domain decomposition method for incompressible flow
EN
This work deals with the construction of a mixed and extensible domain decomposition method for incompressible flows. In the scheme proposed here, the solution is sought at the intersection of two spaces, one containing the solution of the Navier–Stokes equations considered separately in each subdomain, and theother one containing the solutions of the compatibility equations on the interfaces. A solution verifying all the equations of the two spaces is achieved iteratively. One di?culty is that the interface problem is large and dense. In order to reduce its cost while maintaining the extensibility of the method, we defined an interface macroproblem in terms of a few interface macro unknowns. The best option takes advantage of the incompressibility condition to prescribe an interface macroproblem which propagates the information to the whole domain by making use of only two unknowns per interface. Several examples are used to illustrate the main properties of the method.
EN
The objective of this work is the developement and assessment of a fourth-order compact scheme for the three-dimensional unsteady incompressible viscous flows. The equations of the flow are discretized on a staggered grid and using fourth-order compact scheme for the three directions. The accuracy of the method is demonstrated in the Taylor-Green vortex problem. Finally, the turbulent natural convection in a vertical channel is investigated to validate the numerical methods.
EN
For the determination of viscous incompressible flows a pure stream-function formulation for the fourth-order equation, the artificial compressibility method, and velocity correction method are employed. Test calculations are performed for various flows inside square, triangular, semicircular and cubic cavities with one uniform wall, the backward-facing step, double bent channels, the flow around an aerofoil at large angle of attack and for flows over models of buildings. Some complex geometrical configurations can be decomposed into a set of simpler subdomains. A practical methodology for the computation of the Navier-Stokes equations in arbitrarily complex geometries is also considered. The simplest approach for specifying boundary conditions near curved or irregular boundaries is to transfer all the variables from the boundaries to the nearest grid knots.
EN
We study the shape differentiability of a cost function for the flow of an incompressible viscous fluid of power-law type. The fluid is confined to a bounded planar domain surrounding an obstacle. For smooth perturbations of the shape of the obstacle we express the shape gradient of the cost function which can be subsequently used to improve the initial design.
EN
A general framework for calculating shape derivatives for domain optimization problems with partial differential equations as constraints is presented. The first order approximation of the cost with respect to the geometry perturbation is arranged in an efficient manner that allows the computation of the shape derivative of the cost without the necessity to involve the shape derivative of the state variable. In doing so, the state variable is only required to be Lipschitz continuous with respect to geometry perturbations. Application to shape optimization with the Navier-Stokes equations as PDE constraint is given.
PL
W pracy omówiono wybrane aspekty matematycznego opisu turbulentnych przepływów cieczy. W szczególności, odniesiono się do szóstego Problemu Milenijnego dotyczącego istnienia, jednoznaczności i regularności rozwiązań zagadnienia Cauchy'ego dla równań Naviera-Stokesa. Rozważono rozwiązania klasyczne, słabe w sensie Leray'a oraz - krótko - podejście półgrupowe Kato-Fujity. Zwrócono również uwagę na recepcję tego problemu wśród fizyków teoretyków i przedstawicieli dyscyplin technicznych.
EN
This paper reviews the selected aspects of mathematical description of turbulent fluid flows. In particular, the basic results concerning existence, uniqueness and regularity of the Cauchy problem for the Navier-Stokes equations (NSE) are described (the sixth problem of the Millenium). The classical solutions, the weak formulation of the NSE and semi-group approach of Kato-Fujita are considered. Some remarks about the significance of these problems for theoretical physicists and engineers are also briefly presented.
PL
Na podstawie równań Naviera-Stokesa dla przepływu jednowymiarowego opracowano równania modelu przewodu pneumatycznego o parametrach skupionych. Model matematyczny oparty na równaniach różniczkowych zwyczajnych może być bez trudności zaimplementowany w większości programów komputerowych do symulacji złożonych systemów inżynierskich. Opracowany model przewodu wykorzystano do symulacji dynamiki prostego obwodu pneumatycznego z siłownikiem tłokowym jednostronnego działania w programie Matlab-Simulink.
EN
The govering Navier-Stokes equations for quasi-one-dimensional flow arę presented and from them the lumped govering equations for modeling pneumatic pipelines are derivated.The mathematical model based entirely on ODEs can readily be implemented in most ODE-based simulator for modelling complex heterogeneous engineering systems. The pipeline model was used to simułate dynamics of simple pneumatic circuit with one-acting cylinder in Matlab-Simulink.
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