Problem Of Viscous Fluid Flow Research Articles

Multiscale model reduction technique for fluid flows with heterogeneous porous inclusions

Numerical treatment of the problem of two-dimensional viscous fluid flow in and around circular porous inclusions is considered. The mathematical model is described by Navier–Stokes equation in the free flow domain Ωf and nonlinear convective Darcy–Brinkman–Forchheimer equations in porous subdomains Ωp. It is well-known that numerical solutions of the problems in such heterogeneous domains require a very fine computational mesh that resolve inclusions on the grid level. The size alteration of the relevant system requires model reduction techniques. Here, we present a multiscale model reduction technique based on the Generalized Multiscale Finite Element Method (GMsFEM). We discuss construction of the multiscale basis functions for the velocity fields based on the solution of the local problems with and without oversampling strategy. Three test cases are considered for a given choice of the three key model parameters, namely, the Reynolds number (Re), the Forchheimer coefficient (C) and the Darcy number (Da). For the test runs, the Reynolds number values are taken to be Re=1,10,100 while the Forchheimer coefficient and Darcy number are chosen as C=1,10 and Da=10−5,10−4,10−3, respectively. We numerically study the convergence of the method as we increase the number of multiscale basis functions in each domain, and observe good performance of the multiscale method.

Flow Past a Pair of Separated Porous Spheres using Brinkman model

In this work, we obtained an analytical solution to the problem of viscous fluid flow past a pair of porous separated spheres. Stokesian approximation of the Navier-Stokes equation for the viscous fluid governs the flow in the region outside the two spheres, whereas Brinkman's model describes the flow in the porous region (within the spheres). Since the bipolar coordinate system is the most convenient system to represent separated spheres' geometry, we formulated this problem in the bipolar coordinate system. We then eliminated the pressure term from the equations governing the flow in the region outside the spheres, and they got reduced to a separable equation in terms of the stream function. Further, the flow governing equations inside the porous spheres gave rise to the Helmholtz equation. As the Helmholtz equation is not separable in the bipolar system, we used the spherical coordinate system to describe the fluid flow within each separated spheres and solved the resulting problem in the spherical coordinate system. Because of this, the flow variables on either side of the interface (spheres) are in different coordinate systems. To match the values of the field variables at the boundary, we used the transformation equations between the bipolar and spherical systems and transformed all the variables into the bipolar coordinate system. We then solved the governing equations with appropriate boundary conditions for the arbitrary constants and derived expressions for the stream and pressure functions. We plotted the respective functions for various values of the mathematical model's parameters to understand the flow pattern and pressure distribution in the flow domain and noted our observations. This study revealed an intuiting insight that the pressure distribution inside the porous spheres is independent of the non-dimensional parameter related to the medium's permeability and the fluid's viscosity.

A volume of solid implicit forcing immersed boundary method for solving incompressible Navier-Stokes equations in complex domain

In this study, a new implicit forcing immersed boundary (IFIB) method is proposed to solve incompressible viscous fluid flow problems involving complex domains. In the conventional immersed boundary (IB) method, a forcing term computed from the volume of solid (VOS) is added to the incompressible Navier-Stokes equations in order to satisfy the velocity condition within the embedded solid body. However, the velocity boundary condition and the divergence-free condition are enforced at different time levels, i.e. intermediate and new time levels. Penetration of streamlines into the stationary solid body is visible as the velocity boundary condition inside the solid body is not strictly enforced at the new time level. In the current work, the proposed IFIB method can ensure velocity field which satisfies both the velocity boundary condition and the divergence-free condition at the same (new) time level. This is accomplished by solving the pressure equation and calculating the forcing term simultaneously (implicitly). A modified pressure Poisson equation (MPPE) is derived in order to couple the pressure and the forcing terms by treating the forcing term as part of the source term of MPPE. Also, a new cell-based method is proposed to compute the VOS for getting a better parallel efficiency. The accuracy of the present IFIB method is then demonstrated by solving several benchmark problems. No penetration of streamlines has been found in the solid body.

Poisson bracket and integrals of motion in the problem of viscous fluid flow around an arbitrary flat contour

The Poisson bracket is used to find the integrals of motion. A numerical and analytical method is suggested to solve Navier-Stokes equations in Helmholtz form. The approximation of the sought solution by a linear combination of basis functions lies in this method’s core. The main idea behind this method is to select basis functions and generalized independent variables. A functional series is taken as the required solution, found from the equation’s solution obtained by equating the Poisson bracket to zero. The substitution of the selected solution into the Helmholtz equation reduces it to a nonlinear algebraic system. This system is solved analytically for a range of cases due to its structure. The authors present an algorithm for solving the boundary problem of viscous incompressible fluid flow around an arbitrary flat contour. In laminar flow around the body, the geometric dimension decreases, and the boundary problem is reduced to a linear system of algebraic equations. The solutions were tested on solving viscous fluid flow around a wedge and a thin plate.

Incompressible Viscous Flow Simulation Using the Quasi-Hydrodynamic Equations’ System

In this paper, we consider the problem of modeling viscous incompressible fluid flow using the quasi-hydrodynamic (QHD) system of equations. The use of this approach makes it possible to avoid instability in the pressure calculation when using the classical formulation of the Navier–Stokes equations, which is manifested when using cellular numerical schemes. For the numerical implementation of QHD equations, the finite volume method is used with cell-center approximation. In the case of a two-dimensional problem a square mesh is used. A cubic mesh is used in the case of a three-dimensional problem. Two series of calculations are performed for testing in areas of complex geometry for several Reynolds numbers. The results are compared with the ANSYS CFX software package. The comparison shows the high quality of the numerical simulation results using the QHD system of equations.

NUMERICAL REALIZATION OF THE ALGORITHM FOR SOLVING THE TASKS OF HYDROPHYSICS IN THE AREA WITH COMPLEX GEOMETRY

The work is devoted to the development and numerical implementation of an algorithm for solving the problem of modeling the process of hydrophysics, describing the transport of polluting biogenic substances in a complex shape. The velocity field of the water flow, calculated according to the hydrodynamic model of the Sea of Azov, is used in the model of transport of polluting nutrients as input data. To test the accuracy of the numerical solution of the hydrodynamic model problem, a test problem of viscous fluid flow between two coaxial semicylinders is used. To solve the interrelated problems of hydrophysics, including a model of hydrodynamic processes and a model of transport of polluting nutrients, rectangular grids are used, taking into account the “fullness” of the cells. The approximation of the problem in time is made on the basis of splitting schemes for physical processes. To assess the accuracy of the numerical solution of hydrodynamic problems, an analytical solution describing the Couette–Taylor flow is used as a reference. The simulation was carried out on a sequence of thickening computational grids with dimensions: 11×21, 21×41, 41×81 and 81×161 nodes in the cases of a smooth and stepped border. To increase the smoothness of the solution, grids were used that take into account the “fullness” of the cells. In the case of a stepwise approximation of the interface between two media, the calculation error reaches 70 % of the error solution of the problem. When using grids that take into account the “fullness” of cells, the error in the numerical solution of model problems of hydrodynamics, caused by the approximation of the boundary, does not exceed 6 % of the error solution of the problem. The example shows that the fragmentation of the grid by 8 times for each of the spatial directions does not lead to an increase in the accuracy that solutions obtained on the grids, which take into account the “fullness” of the cells, possess.

A Compressible Fluid Flow with Double-Deck Structure Inside an Axially Symmetric Wavy-Wall Pipe

The problem of viscous compressible fluid flow in an axially symmetric pipe with small periodic irregularities on the wall is considered for large Reynolds numbers. An asymptotic solution with double-deck structure of the boundary layer and unperturbed core flow is obtained. Numerical investigations of the influence of the density of the core flow on the flow behavior in the near-wall region are presented.

The flow of a viscous fluid with a predetermined pressure gradient through periodic structures

In the Stokes approximation, the problem of viscous fluid flow through two-dimensional and three-dimensional periodic structures is solved. A system of thin plates of a finite width is considered as a two-dimensional structure, and a system of thin rods of finite length is considered as a three-dimensional structure. Plates and rods are periodically located in space with certain translation steps along mutually perpendicular axes. On the basis of the procedure developed earlier, the authors constructed an approximate solution of the equations for fluid flow with an arbitrary orientation of structures relative to a given vector of pressure gradient. The solution is sought in a finite region (cells) around inclusions in the class of piecewise smooth functions that are infinitely differentiable in the cell, and at the cell boundaries they satisfy the continuity conditions for velocity, normal and tangential stresses. Since the boundary value problem for the Laplace equation is solved, it is assumed that the solution found is unique. The type of functions allows us to separate the variables and to reduce the problem's solution to the solution of ordinary differential equations. It is found that the change in the flow rate of a fluid through a characteristic cross section is determined mainly by the geometric dimensions of the cells of the free liquid in such structures and is practically independent of the size of the plates or rods.

Numerical Solution of the Problem of Incompressible Fluid Flow in a Plane Channel with a Backward-Facing Step at High Reynolds Numbers

Numerical solutions of the problem of steady incompressible viscous fluid flow in a plane channel with a backward-facing step have been obtained by the grid method. The fluid motion is described by the Navier–Stokes equations in velocity-pressure variables. The main computations were performed on a uniform 6001 × 301 grid. The control-volume method of the second order in space was used for the difference approximation of the original equations. The results were validated for the range of Reynolds numbers (100 ≤ Re ≤ 3000) by comparing them with the experimental and theoretical data found in the literature. The stability of the computational algorithm at high Reynolds numbers was achieved by using a fine difference grid (a small grid step). The study has been carried out for a short channel at Reynolds numbers from 1000 to 10 000 with a step of 1000. A nonstandard structure of the primary vortex behind the step—the presence of numerous centers of rotation both inside the vortex and in the near-wall region under it—has been revealed. The number of centers of rotation in the primary recirculation zone is shown to grow with increasing Reynolds number. The profiles of the coefficients of friction and hydrodynamic resistance to the flow as a function of Reynolds number have also been analyzed. The results obtained can be useful for comparison and validation of the solutions of problems of such a type.

Unsteady self-similar viscous flow near a stagnation point

The problem of unsteady viscous incompressible fluid flow near a stagnation point is considered. Self-similar solutions describing plane and axisymmetric flows are constructed.

On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface

In this paper, we prove a local in time unique existence theorem for some two phase problem of compressible and compressible barotropic viscous fluid flow without surface tension in the $L_p$ in time and the $L_q$ in space framework with $2< p <\infty$ and $N< q <\infty$ under the assumption that the initial domain is a uniform $W^{2-1/q}_q$ domain in $\mathbb{R}^N (N\ge 2)$. After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve the problem by the contraction mapping principle with the maximal $L_p$-$L_q$ regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key step of our method is to prove the existence of $\mathcal{R}$-bounded solution operator to resolvent problem corresponding to linearized problem. The $\mathcal{R}$-boundedness combined with Weis's operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal $L_p$-$L_q$ regularity theorem.

Topological derivatives applied to fluid flow channel design optimization problems

Topology optimization methods application for viscous flow problems is currently an active area of research. A general approach to deal with shape and topology optimization design is based on the topological derivative. This relatively new concept represents the first term of the asymptotic expansion of a given shape functional with respect to the small parameter which measures the size of singular domain perturbations, such as holes and inclusions. In previous topological derivative-based formulations for viscous fluid flow problems, the topology is obtained by nucleating and removing holes in the fluid domain which creates numerical difficulties to deal with the boundary conditions for these holes. Thus, we propose a topological derivative formulation for fluid flow channel design based on the concept of traditional topology optimization formulations in which solid or fluid material is distributed at each point of the domain to optimize the cost function subjected to some constraints. By using this idea, the problem of dealing with the hole boundary conditions during the optimization process is solved because the asymptotic expansion is performed with respect to the nucleation of inclusions --- which mimic solid or fluid phases --- instead of inserting or removing holes in the fluid domain, which allows for working in a fixed computational domain. To evaluate the formulation, an optimization problem which consists in minimizing the energy dissipation in fluid flow channels is implemented. Results from considering Stokes and Navier-Stokes are presented and compared, as well as two- (2D) and three-dimensional (3D) designs. The topologies can be obtained in a few iterations with well defined boundaries.

An Efficient Implicit Compact Streamfunction Velocity Formulation of Two Dimensional Flows

In this paper, we develop a class of implicit, unconditionally stable compact schemes for solving coupled fourth order and second order semilinear streamfunction and temperature equations respectively representing unsteady Navier---Stokes (N---S) equations for incompressible viscous fluid flows. The governing equations are presented in a generic form from which specific equations are obtained for several two dimensional (2D) incompressible viscous fluid flow problems. The streamfunction and the temperature equations are all solved iteratively in a coupled system of equations for the four field variables consisting of streamfunction, two velocities and temperature. The discretized form of biharmonic equation in streamfunction consists on 5-point stencil of streamfunction and is found to be second-order accurate both for spatially and temporally. In addition, we have considered two strategies for the discretization of the energy equation, which are second order accurate 5-point and fourth order accurate 9-point compact scheme. The formulation is made efficient by decreasing the computational complexity and is used to solve several 2D time dependent well studied benchmark problems. It is seen to efficiently capture both time wise and steady-state solutions of the N---S equations with Dirichlet as well as Neumann boundary conditions. The results obtained using our proposed scheme are in excellent agreement with the results available in the literature and they clearly establish the efficiency and the accuracy of the proposed scheme.

Meshless analysis and applications of a symmetric improved Galerkin boundary node method using the improved moving least-square approximation

This paper combines variational formulations of boundary integral equations (BIEs) and the improved moving least-square (IMLS) approximation to develop a symmetric meshless method, the improved Galerkin boundary node method (IGBNM) for boundary-only analysis of boundary value problems in potential theory and viscous fluid flow. The IMLS approximation is used in the IGBNM to construct meshless shape functions. In the IMLS approximation, the system of algebra equations can be solved without the inverse matrix. Compared with other MLS-based meshless methods, the IGBNM is a direct numerical method in which the basic unknown quantities are the actual nodal values. Besides, boundary conditions in the IGBNM are implemented directly and easily, and the resulting ‘stiffness’ matrices conserve the symmetry and positive definiteness of the variational formulations. Thus, it gives a higher computational efficiency. Total details of numerical implementation and error analysis of the IGBNM are first given for general BIEs. Then, taking potential problems and viscous fluid flow problems as examples, we set up a framework for numerical implementation and asymptotic error estimates of the IGBNM. Finally, some numerical tests are presented to demonstrate the efficiency of the method.

Remark on classical Crane’s solution of viscous flow past a stretching plate

An efficient compatibility criterion is proposed to solve the nonlinear boundary problem arising in the study of the classical problem of viscous fluid flow due to a stretching sheet due to Crane (Crane, 1970). The compatibility and generalized group analysis make it simple to obtain the exact solution of the classical boundary layer problem.

Точные решения задачи о течении вязкой жидкости в цилиндрической области с меняющимся радиусом

Точные решения задачи о течении вязкой жидкости в цилиндрической области с меняющимся радиусом

Numerical analysis of the external slow flows of a viscous fluid using the R-function method

In this paper the problem of calculating the external slow flow of viscous incompressible fluid past bodies of revolution is considered. A numerical method, based on the joint use of the R-function structural method of V. L. Rvachev for building the structure of the boundary-value problem solution and the Galerkin projection method for approximating the indeterminate components of the structure, is proposed for solving the problem. The problem of slow viscous incompressible fluid flow past an ellipsoid of revolution is numerically solved using the proposed approach to test the developed method. The solution obtained is compared with the known exact solution of the problem in question. In addition, the problems of flow past two touching spheres and past two joined ellipsoids are solved.

An application of networked EWS to fluid flow analysis

An effective use of networked engineering work stations is considered for the finite–element solution of two–dimensional viscous fluid flow problems. Unknowns involved in the governing equations are distributed to each work station, so that the finite–element equations can be solved in parallel on the network. Communication between the work stations is achieved by a mail system. The overhead time for the mail communication is small as compared to the time required for the numerical solution of the equations. Illustrative examples of a viscous fluid flow around a cylinder, a viscous flow over a back step, and a natural convection in a rectangular cavity are presented.

A corrected smoothed particle hydrodynamics approach to solve the non-isothermal non-Newtonian viscous fluid flow problems

In this paper, a corrected smoothed particle hydrodynamics (SPH) method is proposed to solve the problems of non-isothermal non-Newtonian viscous fluid. The proposed particle method is based on the corrected kernel derivative scheme under no kernel derivative and incompressible conditions, which possesses higher accuracy and better stability than the traditional SPH method. Meanwhile, a temperature-discretization scheme is deduced by the concept of SPH method for the purpose of precisely describing the evolutionary process of the temperature field. Reliability of the corrected SPH method for simulating the non-Newtonian viscous fluid flow is demonstrated by simulating the isothermal Poiseuille flow and the jet fluid of filling process; and the validity and accuracy of the proposed SPH discrete scheme in a temperature model for solving the non-isothermal fluid flow are tested by solving the non-isothermal Couette flow and 4:1 contraction flow. Subsequently, the proposed corrected SPH method combined with the SPH temperature-discretization scheme is tentatively extended to include the simulation of the non-isothermal non-Newtonian viscous free-surface flows in the ring-shaped and C-shaped cavities. Especially, the convergence of numerical simulations is analyzed, and the influences of heat flow parameters on the temperature and fluid flow at different positions are discussed.

On some free boundary problem for a compressible barotropic viscous fluid flow

In this paper, we prove a local in time unique existence theorem for the free boundary problem of a compressible barotropic viscous fluid flow without surface tension in the \(L_p\) in time and \(L_q\) in space framework with \(2 < p < \infty \) and \(N < q < \infty \) under the assumption that the initial domain is a uniform \(W^{2-1/q}_q\) one in \({\mathbb {R}}^{N}\, (N \ge 2\)). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve problem by the Banach contraction mapping principle based on the maximal \(L_p\)–\(L_q\) regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key issue for the linear theorem is the existence of \({\mathcal {R}}\)-bounded solution operator in a sector, which combined with Weis’s operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal \(L_p\)–\(L_q\) regularity theorem. The nonlinear problem we studied here was already investigated by several authors (Denisova and Solonnikov, St. Petersburg Math J 14:1–22, 2003; J Math Sci 115:2753–2765, 2003; Secchi, Commun PDE 1:185–204, 1990; Math Method Appl Sci 13:391–404, 1990; Secchi and Valli, J Reine Angew Math 341:1–31, 1983; Solonnikov and Tani, Constantin caratheodory: an international tribute, vols 1, 2, pp 1270–1303, World Scientific Publishing, Teaneck, 1991; Lecture notes in mathematics, vol 1530, Springer, Berlin, 1992; Tani, J Math Kyoto Univ 21:839–859, 1981; Zajaczkowski, SIAM J Math Anal 25:1–84, 1994) in the \(L_2\) framework and Holder spaces, but our approach is different from them.