Two-dimensional Flow Of Viscous Fluid 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.

Numerical simulation of time-dependent two-dimensional viscous fluid flow with thermal radiation

Numerical simulation of time-dependent two-dimensional viscous fluid flow with thermal radiation

A New Technique to Solve Two-Dimensional Viscous Fluid Flow Among Slowly Expand or Contract Walls

In this research, we have proposed a new technique to solve two-dimensional (2D) viscous fluid flow among slowly expanding or contracting walls. The new technique depends on combining the algorithms of Yang transform and the homotopy perturbation methods. The results, obtained from the first iteration and by using the new method, show the accuracy and efficiency of this method compared to the other methods, used to find the analytical approximate solution for the problem caused by the 2D viscous fluid flow. Moreover, the graphs of the new solutions show the validity, usefulness and necessity of the new method.

Some geometrical properties of free boundaries in the Hele-Shaw flows

In this paper, we are concerned with certain geometric properties of the moving boundary in the case of two-dimensional viscous fluid flows in Hele-Shaw cells under injection. We study the invariance in time of free boundary for such a bounded flow domain under the assumption of zero surface tension. By applying various results in the theory of univalent functions, we consider the invariance in time of starlikeness of a complex order, almost starlikeness of order α ∈ [0, 1), and almost spirallikeness of type γ∈(−π/2,π/2) and order α ∈ (0, cos γ). This work complements recent work on planar Hele-Shaw flow problems in the case of zero surface tension.

On the Analysis of Laminar Flow of Viscous Fluid through a Porous Channel with Suction/Injection at Slowly Expanding or Contracting Walls

The vast biological and industrial applications of laminar flow of viscous fluid through a porous channel with contracting or expanding permeable wall have attest to the importance of studying the flow process. In this paper, two-dimensional flow of viscous fluid in a porous channel through slowly expanding or contracting walls with injection or suction is analyzed using variation parameter method. From the parametric studies using the developed approximate analytical solutions, it is shown that increase in the Reynolds number of the flow process leads to decrease in the axial velocity at the center of the channel during the expansion. The axial velocity increases slightly near the surface of the channel when the wall contracts at the same rate. Also, as the wall expansion ratio increases, the velocity at the center decreases but it increases near the wall. The results of the approximate analytical solution are verified by numerical solution using shooting method coupled with Runge-Kutta method. The results of the variation parameter method are in excellent agreement with the results obtained using numerical method.

INSTABILITY OF THE RAYLEIGH-BENARD CONVECTION FOR INCLINED LOWER WALL WITH TEMPERATURE VARIATION

This paper deals with an analysis of a two-dimensional viscous fluid flow between the two parallel plates inclined with respect to the horizontal plane, where the lower plate is heated and the upper one is cooled. The temperature difference between the plates is gradually increased during a certain time period after which it is temporarily constant. The temperature distribution on the lower plate is not constant in x-direction, there is a longitudinal sinusoidal temperature variation imposed on the mean temperature. We have investigated the wave number and amplitude influence of this variation on the subcritical stability and the onset of the Rayleigh-Bénard convective cells, by direct numerical simulation of 2D Navier-Stokes and energy equation.

High-Order Finite Difference Schemes for the First Derivative in Von Mises Coordinates

Third- and fourth-order accurate finite difference schemes for the first derivative of the square of the speed are developed, for both uniform and non-uniform grids, and applied in the study of a two-dimensional viscous fluid flow through an irregular domain. The von Mises transformation is used to transform the governing equations, and map the irregular domain onto a rectangular computational domain. Vorticity on the solid boundary is expressed in terms of the first partial derivative of the square of the speed of the flow in the computational domain, and the schemes are used to calculate the vorticity at the computational boundary grid points using up to five computational domain grid points. In all schemes developed, we study the effect of coordinate clustering on the computed results.

Rayleigh–Bénard convection instability in the presence of spatial temperature modulation on both plates

This paper is devoted to the analysis of two-dimensional viscous fluid flow between two parallel, horizontal plates, where the lower plate is heated and the upper one is cooled. The temperature difference between the plates increases gradually during the initial time period, and after that period it is temporarily constant. The temperature distribution on the lower and upper plates is not constant in the x-direction, there is longitudinal temperature modulation imposed on the mean temperature. Temperature modulation on the plates leads to periodic convection patterns for very small Rayleigh numbers. We investigate the instability of Rayleigh–Bénard convective cells by direct numerical simulation on the coarse grid, for the two-dimensional Navier–Stokes equation in the stream function–vorticity form and the thermal energy equation.

Rayleigh-Bénard convection instability in the presence of temperature variation at the lower wall

This paper analyzes the two-dimensional viscous fluid flow between two parallel plates, where the lower plate is heated and the upper one is cooled. The temperature difference between the plates is gradually increased during a certain time period, and afterwards it is temporarily constant. The temperature distribution on the lower plate is not constant in x-direction, and there is longitudinal sinusoidal temperature variation imposed on the mean temperature. We investigate the wave number and amplitude influence of this variation on the stability of Rayleigh-Benard convective cells, by direct numerical simulation of 2-D Navier-Stokes and energy equation.

Dynamically orthogonal field equations for continuous stochastic dynamical systems

In this work we derive an exact, closed set of evolution equations for general continuous stochastic fields described by a Stochastic Partial Differential Equation (SPDE). By hypothesizing a decomposition of the solution field into a mean and stochastic dynamical component, we derive a system of field equations consisting of a Partial Differential Equation (PDE) for the mean field, a family of PDEs for the orthonormal basis that describe the stochastic subspace where the stochasticity ‘lives’ as well as a system of Stochastic Differential Equations that defines how the stochasticity evolves in the time varying stochastic subspace. These new evolution equations are derived directly from the original SPDE, using nothing more than a dynamically orthogonal condition on the representation of the solution. If additional restrictions are assumed on the form of the representation, we recover both the Proper Orthogonal Decomposition equations and the generalized Polynomial Chaos equations. We apply this novel methodology to two cases of two-dimensional viscous fluid flows described by the Navier–Stokes equations and we compare our results with Monte Carlo simulations.

Onset of self-oscillations upon the loss of stability of spatially periodic two-dimensional viscous fluid flows relative to long-wave perturbations

The long-wave asymptotics of the secondary flow that arises after a steady, spatially periodic flow loses stability are studied when one of the periods tends to infinity and the rate of base flow along the longer period is equal to zero. It is shown that if certain non-degeneracy conditions are satisfied, then from the base solution a self-oscillatory regime branches off and both hard and soft stability loss is possible. For the leading terms of the asymptotics explicit formulas are obtained. Examples of self- oscillations calculated for specific flows are presented and the behavior of the fluid particle trajectories in the self-oscillatory regime branching off from the base flow is investigated.

309 二次元粘性流れ解析における格子ボルツマン法と従来型数値解法との比較に関する検討(O.S.3-2 熱流体解析における新潮流2)(オーガナイズドセッション3 : 熱流体解析における新潮流)

309 二次元粘性流れ解析における格子ボルツマン法と従来型数値解法との比較に関する検討(O.S.3-2 熱流体解析における新潮流2)(オーガナイズドセッション3 : 熱流体解析における新潮流)

K-0332 エレメント・フリー・ガレルキン法による二次元粘性流動問題の解析(第2報)(G01-3 メシュレス法・ボクセル法)(G01 計算力学部門一般講演)

The Element-Free Galerkin Method is applied to the numerical analysis for two-dimensional viscous fluid flows of high Reynolds numbers. An elliptic domain of influence is used for a node disposition of high aspect ratio. The driven cavity flow problems are solved up to the Reynolds number of 10,000. Steady solutions are obtained, which agree well with Ghia's.

A model for resolving the plankton paradox: coexistence in open flows

1. Recent developments in the field of chaotic advection in hydrodynamical/environmental flows encourage us to revisit the population dynamics of competing species in open aquatic systems. 2. We assume that these species are in competition for a common limiting resource in open flows with chaotic advection dynamics. As an illustrative example, we consider a time periodic two‐dimensional flow of viscous fluid (water) around a cylindrical obstacle. 3. Individuals accumulate along a fractal set in the wake of the cylinder, which acts as a catalyst for the biological reproduction process. While in homogeneous, well mixed environments only one species could survive this competition, coexistence of competitors is typical in our hydrodynamical system. 4. It is shown that a steady state sets in after sufficiently long times. In this state, the relative density of competitors is determined rather by the fractal nature of the spatial distribution of the advected species, and by their initial conditions, than by their competitive abilities. We argue that two factors, the strong chaotic mixing along a fractal set and the boundary layer around the obstacle, are responsible for the coexistence.

Modified Galerkin Analysis for the Unsteady Three-Dimensional Viscous Fluid Flow (Approach by the Error Analysis Technique).

In our previous research, the modified Galerkin method was proposed as one of the most efficient methods for the convection-diffusion problems and the two-dimensional viscous fluid flows. In the modified Galerkin method, since the inertia term is considered explicitly, only the symmetrical matrixes appear. Then an artificial viscosity is introduced through an error analysis approach to improve its accuracy and stability. In this paper, this modified Galerkin method is applied to the three-dimensional incompressible viscous fluid flow analysis coupled with the Semi-Implicit Method for Pressure-Linked Equations formulation. The cubic cavity flow problem sare investigated for the Reynolds number of 400, 1000, 2000 and 3200. Finally, we confirmed the effectiveness of our proposed method.

Finite Element Error Analysis Approach for Three-Dimensional Incompressible Viscous Fluid Flow Analysis

In our previous research, the modified Galerkin method was proposed as one of the most efficient methods for the analyses of convection-diffusion problems and two-dimensional viscous fluid flow problems. In this modified Galerkin method, the inertia term is considered explicitly, so only the symmetrical matrixes appear. Then an artificial viscosity is introduced through an error analysis approach to improve its accuracy and stability. In this paper, we proposed a new finite element formulation for three-dimensional incompressible viscous fluid flow analysis. This formulation (‘MS’ algorithm and ‘MSR’ algorithm) is based on the modified Galerkin method coupled with the Semi-Implicit Method for Pressure-Linked Equations. The cubic cavity flow problems were investigated for the Reynolds number of 400, 1,000, 2,000 and 3,200 using non-uniform meshes. Finally, we confirmed the effectiveness of our proposed method through the comparison with other research works.

Modified Galerkin Method for Unsteady Two-Dimensional Viscous Fluid Flow Analysis

The Modified Galerkin Method (MGM) has been proposed as one of the most efficient methods for two-dimensional convection-diffusion equations. In the MGM, the non-symmetric matrices, which are derived from the convection term in the Galerkin formulation, are not used, and an artificial diffusion is introduced through an error analysis approach to improve its discretization accuracy in both time and space directions. In this study, the MGM is applied for two-dimensional viscous fluid flow analysis, and the driven cavity flow problems are solved up to Reynolds number of 10,000 using the vorticity-stream function formulation and non-uniform meshes. The results show the effectiveness of MGM.

Wrinkling of stable fronts in viscous flow

Stable wrinkled fronts were often observed in experiments on fluid invasion into porous media, when the invaded phase is less viscous than the invading one. Recent experiments by a group at the Weizmann institute, show that capillary wetting of paper may also lead to rough self-affine fronts. A dominant factor controlling the interface roughening in these systems is the nature of the two-dimensional viscous fluid flow in the medium behind it. Since the disorder in the medium modifies this flow, it also determines the roughness of the interface. We present a theoretical analysis of the connection between the nature of randomness in the medium and the resulting roughness of the fronts.

Modified Galerkin Analysis for Unsteady Two-Dimensional Viscous Fluid Flow. Approach by the Error Analysis Technique.

In our previous research, the modified Galerkin method (MGM) was proposed as one of the most efficient methods for one-dimensional, two-dimensional and three-dimensional convection-diffusion equations. In the MGM, the non-symmetric matrixes, which are derived from the convection term in the Galerkin formulation, are not used, and an artificial viscosity is introduced through the error analysis approach to improve its discretization accuracy in both time and space directions. In this study, this MGM is applied for two-dimensional viscous fluid flow analysis, and the driven cavity flow problems are solved up to the Reynolds number of 10 000 using the vorticity-stream function formulation for non-uniform meshes. The results show the effectiveness of our modified Galerkin method.

Justification of the linear long-wave approximation to viscous fluid flow down an inclined plane

The objective of this paper is to justify the linear long-wave approximation used in the derivation of approximate equations for long waves on the free surface of a two-dimensional viscous fluid flow down an inclined plane. To the first order of a small parameter, the approximate equation is a heat equation, which becomes ill-posed if a Reynolds number R R is greater than some critical value R c {R_c} . To overcome this difficulty we consider a higher-order approximate equation, which is well-posed even if R > R c R > {R_c} , and show that the solution of the higher-order equation is an approximation to the solution of the linearized Navier-Stokes equations. The justification is based upon a set of long-wave initial conditions, and the error bounds can also be expressed in terms of pointwise estimates.