In this paper, the Navier-Stokes equations describing the motion of viscous incompressible fluid in a bounded domain is considered. Method of fictitious domains is applied for approximate solution of the problem taking into account boundary conditions with uniform accuracy.

The Boussinesq equations are widely used to model incompressible thermally driven flows, but their numerical simulation becomes more challenging in domains containing fixed or moving rigid obstacles, particularly when the prescribed obstacle motion is uncertain. In this work, we develop a Brinkman-type fictitious-domain formulation for the incompressible Boussinesq system on a fixed computational domain. Brinkman penalization is used to impose the prescribed solid velocity, while a thermal penalty term enforces the temperature inside the immersed bodies. The resulting problem is discretized by a finite element method using skew-symmetric convective forms and grad-div stabilization. The uncertainty in the obstacle motion is introduced through a random oscillation amplitude and treated by a non-intrusive stochastic collocation method based on Gauss–Legendre quadrature. Stability and convergence of the fully discrete scheme are established. Numerical experiments for moving heated and hot–cold obstacles demonstrate the influence of the uncertain oscillation amplitude on the Nusselt number, kinetic energy, mean vorticity, and the spatial distributions of the mean and standard deviation of the solution fields. The results indicate that the proposed method provides a stable numerical framework for simulating incompressible thermally coupled flows with moving immersed obstacles under uncertainty in the prescribed obstacle motion.

Several numerical methods can be used to approximate the solution of the problem. In order to determine the most effective of them, it is necessary to carefully study each method. The most efficient approximation method is characterized by properties such as high accuracy of the solution, fewer iterations and parameters in the calculation, calculation speed, etc. In this paper we consider the Dirichlet problem for the Poisson equation described by the initial-boundary value problem for the elliptic type of the second order. As an effective iterative method for its approximate solution, variational methods for constructing difference equations and variational methods for constructing iterative algorithms were used. The article presents the results of calculations developed using the variational method for the selected model problem. Examples of calculations for model problems are given. The results of the computational experiment demonstrate the high efficiency of the proposed iterative method. Several numerical methods can be used to approximate the solution of the problem. In order to determine the most effective of them, it is necessary to carefully study each method. The most efficient approximation method is characterized by properties such as high accuracy of the solution, fewer iterations and parameters in the calculation, calculation speed, etc. In this paper we consider the Dirichlet problem for the Poisson equation described by the initial-boundary value problem for the elliptic type of the second order. As an effective iterative method for its approximate solution, variational methods for constructing difference equations and variational methods for constructing iterative algorithms were used. The article presents the results of calculations developed using the variational method for the selected model problem. Examples of calculations for model problems are given. The results of the computational experiment demonstrate the high efficiency of the proposed iterative method.

Moving-boundary problems for incompressible flows often require repeated mesh generation when body-fitted finite element methods are used. This may become costly for flows with moving rigid obstacles, since the numerical solution from the previous time level also has to be transferred to the new finite element space. In this paper, we study a fictitious-domain finite element method for the two-dimensional incompressible Navier–Stokes equations with rigid obstacles. The aim is to analyze a penalized formulation on a fixed background domain and to compare it with a classical body-fitted finite element approach. The flow problem is extended to a larger fixed domain, and the effect of the solid domain is imposed by a nonlinear penalty term depending on the parameters $\epsilon$ and $\beta$. A fully discrete finite element scheme is constructed, and stability and convergence estimates are obtained for the fixed-obstacle auxiliary penalized problem. The moving obstacle case is examined numerically. Several numerical tests are presented. A manufactured auxiliary problem is used to check the spatial convergence of the method, the influence of the penalty parameter, and the behavior of the solution inside the fictitious domain. We also consider two moving obstacle examples: a rotating disk moving near a no-slip wall and two rotating disks moving along symmetric trajectories. The fictitious-domain solutions are compared with body-fitted finite element solutions, and the main flow patterns are reproduced without remeshing. The results indicate that the fictitious-domain finite element method is a practical alternative for moving obstacle flow problems. It is not universally faster than body-fitted methods, but it becomes advantageous when the geometry changes in time, since the fixed background mesh avoids repeated remeshing and the transfer of the previous solution between nonmatching meshes.

In this paper, we consider an initial-boundary value problem for an unsteady flow of a viscous incompressible fluid in a bounded region, solved using a system of nonlinear Navier-Stokes equations. The equations describe the motion of the fluid considering viscosity, pressure, and mass force, as well as the solenoidality condition of the velocity field. In the general case, finding an analytical solution to the system of equations presents significant difficulties, and it has not yet been proven whether there is always a smooth solution for all possible conditions. In this regard, the fictitious domain method is used to solve the problem, which allows us to reduce the problem by solving a system of differential equations with appropriate boundary conditions. Particular attention is paid to introducing the concept of twice the mean curvature of the surface, which is necessary for the correct application of the fictitious domain method. For this purpose, the article provides a detailed calculation of the mean curvature using surface parameterization and matrices of the first and second forms. Proof of a lemma related to the calculation of twice the mean curvature is also given, which is of great importance for further numerical methods for solving the Navier-Stokes system of equations. The obtained results expand the scope of application of the fictitious domain method in solving hydrodynamic problems, especially in complex geometries, and can be used to develop more efficient numerical algorithms.

The object of this study is steady two-dimensional incompressible viscous flow in L-shaped channels with different bend angles. The problem to be solved is to determine how the bend angle affects the flow pattern and hydraulic losses. Two geometries are compared: channels with 45° and 90° bends. The computations are carried out for Re = 500, 1000, and 2000. The flow is described by the stationary Navier-Stokes equations, which are solved by the finite element method. Newton's method is used to solve the nonlinear system. The comparison is performed under the same boundary conditions and computational parameters. The results are analyzed using velocity and pressure fields, stream-function distributions, pressure drop, loss coefficient, and Euler number. The calculations show that the 90° bend causes a stronger change in the flow after the corner than the 45° bend. In the 90° channel, the direction of motion changes more sharply. Therefore, the velocity field downstream of the bend is more distorted, the gradients are higher, and recirculation zones become more visible as Re increases. In the 45° channel, the turn is smoother, so the velocity and pressure fields change more regularly. The integral characteristics confirm this result. For all Re considered, the 90° bend gives a larger pressure drop than the 45° bend. The loss coefficient and Euler number vary with Re, but the 90° geometry remains less favorable. This is explained by stronger rearrangement of the flow caused by the sharper turn. The influence of the bend angle is shown both by field plots and by pressure-loss indicators. The results may be used in design of ducts, cooling channels, and pipelines when pressure losses in curved parts should be reduced