Solution Of Flow Problems Research Articles

Modification of the Euler equations for ‘‘vorticity confinement’’: Application to the computation of interacting vortex rings

A new ‘‘vorticity confinement’’ method is described which involves adding a term to the momentum conservation equations of fluid dynamics. This term depends only on local variables and is zero outside vortical regions. The partial differential equations with this extra term admit solutions that consist of Lagrangian-like confined vortical regions, or covons, in the shape of two-dimensional (2-D) vortex ‘‘blobs’’ and three-dimensional (3-D) vortex filaments, which convect in a constant external velocity field with a fixed internal structure, without spreading, even if the equations contain diffusive terms. Solutions of the discretized equations on a fixed Eulerian grid show the same behavior, in spite of numerical diffusion. Effectively, the new term, together with diffusive terms, constitute a new type of regularization of the inviscid equations which appears to be very useful in the numerical solution of flow problems involving thin vortical regions. The discretized Euler equations with the extra term can be solved on fairly coarse, Eulerian computational grids with simple low-order (first- or second-) accurate numerical methods, but will still yield concentrated vortices which convect without spreading due to numerical diffusion. Since only a fixed grid is used with local variables, the vorticity confinement method is quite general and can automatically accommodate changes in vortex topology, such as merging. Applications are presented for incompressible flow in 3D, where pairs of thin vortex rings interact and, in some cases, merge.

Two‐dimensional multilayer coextrusion flow in a flat coat‐hanger die. Part I: Modeling

AbstractThe multilayer stratified flow of several polymers in a flat coat‐hanger die was modeled using a finite element method. The problem was formulated using the lubrication approximation theory. A solution procedure in decoupling pressure and streamlines was developed. This new method is very powerful in comparison with more classical approaches, permitting the solution of flow problems involving a great number of layers in a complex industrial geometry. It allows us to obtain, among other things, pressure field, streamlines, residence times, and the values of the interface positions in the whole die.

Finite-element solution of flow problems with trailing conditions

The paper deals with the finite-element solution of stream function problems describing nonviscous subsonic irrotational flows past profiles. The main emphasis is laid on the treatment of the nonstandard trailing stagnation conditions which lead to physically admissible solutions. The paper presents a general conception of stream function finite-element modelling of complicated flow problems and a complete theory of the finite-element approximations, including the investigation of the existence and uniqueness of the solution of the nonsymmetric discrete problem and the convergence of approximate solutions to the exact solution.

A new computational method for the solution of flow problems of microstructured fluids. Part 1. Theory

The theoretical basis for a new computational method is presented for the solution of flow problems of microstructured fluids: examples include suspensions of rigid particles and polymeric liquids ranging from liquid crystals to concentrated solutions or melts of flexible chains. The method is based on a Lagrangian conservation statement for the distribution function of the conformation of the local structure, which can be derived from the conventional, Eulerian conservation statement and is exactly equivalent. The major difference, which is reflected in the numerical technique, is that the Lagrangian representation of the distribution function allows for computation of the Brownian contribution and of moments of distribution functions in ways that do not require explicit knowledge of the distribution function, and involve no approximation whatsoever. This suggests a new type of efficient, self-adaptive numerical scheme that is suitable for the solution of flow problems of microstructured fluids, in which macroscopic properties depend on the state of the microstructure.

Flow separation in the vicinity of a moving boundary

THE effect of boundary motion on the laminar, incompressible flow in the entry region of a two-dimensional channel is considered in this study. The formation of a separated region during the upstream motion of a section of the lower boundary is of particular interest. The size of the separated region depends on the Reynolds number and the velocity of the moving boundary. The solution is obtained numerically by considering the continuity and the Navier-Stokes equations using the vorticity-stream function formulation. A computational scheme, which consists of the alternating-direction implicit method for the vorticity equation and the successive over-relaxation method for the stream function equation, is used to solve the finite difference equations. Contents Specifying solutions for flow problems comprising moving boundaries has potential importance in aerospace applications such as oscillating airfoils, impellers, helicopter rotors, and compressor blades. The primary objective of this work is to numerically simulate the steady, laminar, incompressible, confined flow in the presence of boundary motion.1 As a preliminary configuration, the problem is modeled as viscous flow past a moving boundary section placed within the entry region of a two-dimensional channel, with uniform velocity at the inlet and Poiseuille flow at the downstream end. Interest is focused on the development of a separated region in the vicinity of a lower boundary section permitted to move with a constant velocity in its own plane in the direction opposite to the inlet velocity. Laminar separation and reattachement occur in regimes of low Reynolds number flow around airfoils. Also, separated regions can be generated by the flow near the sharp leading edge of a thin airfoil. Unpredictable rapid growth of such separated regions can cause stall. Viscous, incompressible flows are governed by the NavierStokes equations. For two-dimensional flows, it is often desirable to introduce the vorticity and the stream function as dependent variables. The x- and j>-component Navier-Stokes equations yield the z-component vorticity equation: a2*

Numerical Solution of Incompressible Flow of Power-Law Fluid Using Boundary-Fitted Curvilinear Coordinates.

A numerical method using boundary-fitted curvilinear coordinates (BFC) is applied to compute non-Newtonian flow equations in primitive variable formulations for an incompressible inelastic power-law fluid. The flow equations are solved on a special staggered grid by the method of lines using the rational Runge-Kutta time integration scheme and the central finite difference method for spatial discretization, while the Poisson equation is solved by means of the successive overrelaxation (SOR) method using the modified simplified marker and cell (MSMAC) method. It is concluded that the results using BFC are comparable to those using Cartesian coordinates, and the proposed method can be applied to the numerical solution of flow problems of a power-law fluid in arbitrary computational domains.

Finite Element Solution of Flow Problems with Mixed‐Time Integration

A mixed‐time integration method that had been developed for the finite element analysis of structural and thermal dynamics has been implemented for the study of both steady and unsteady fluid mechanics problems. The method to be discussed is capable of partitioning the domain into implicit and explicit regions in an attempt to capitalize on the desirable properties of each method, namely the stability and accuracy of the implicit method, and the manageable computational resource demands of an explicit method. In addition, the explicit region is further divided into subregions, each of which may have a different time step that is governed by the local stability criterion of an explicit method. To demonstrate the applicability of these methods to equation systems that govern fluid flow, several examples are presented. These include one‐ and two‐dimensional advection of a cosine hill, as well as two‐dimensional steady and unsteady inviscid, compressible flow problems. These examples will be used to show the ...

Chebyshev collocation solutions of flow problems

Chebyshev collocation methods are reviewed. For general second-order elliptic equations, the algebraic system obtained through the collocation process is ill-conditioned. Recent work on finite element preconditioning shows that bilinear elements give full satisfaction. For Stokes problems, as interpolants of different degree are used for the velocity and the pressure, the classical nine-node Lagrangian element with biquadratic velocities and bilinear pressures constitutes the best choice. In order to treat the non-linear terms, a Newton's method is designed. Domain decomposition is set up with the jumps of the stress vector across interfaces between adjacent subdomains. Two-dimensional curvy geometries are handled by a coordinate mapping. Problems with singularities are treated by a mixed finite element and spectral approximation. Finally, current developments for Non-Newtonian fluids are evoked.

PCG solutions of flow problems in random porous media using mixed finite elements

The behavior of preconditioned conjugate gradient (PCG) solutions of flow problems in random porous media is examined. The flow problems are discretized with the mixed finite element method, the resulting matrix is reduced in size by block Gauss elimination, and PCG is applied to the resulting problem. As a result of the matrix reduction and the need to store the resultant matrix implicitly because of storage limitations for large problems, the selection of a preconditioner is not obvious. Four heuristic preconditioners, selected because of the simplicity with which they can be constructed from the implicit matrix representation and because of their potential for high performance computers, are tested: diagonal, tridiagonal, scaled diagonal, and scaled tridiagonal. Tests on fields with random hydraulic conductivity, with both isotropic and anisotropic correlation functions and with a large range of standard deviations, are performed. The scaled tridiagonal and diagonal preconditioners prove very effective and efficient for a broad range of problems, even for conductivity variations of 25 orders of magnitude.

From molecular models to the solution of flow problems

From molecular models to the solution of flow problems

Authors

Authors

Orthogonal and conformal mapping with application to MHD equilibrium and stability calculations

There is a growing interest in the use of orthogonal and conformal mapping methods for the solution of flow problems in two and three dimensions. Here, the non-linear integral equations of Symm and...

Model polymer fluid systems

Abstract

Exact solutions of boundary-value problems for nonlinear flow in porous media

Exact solutions for flow problems in porous media with a limiting gradient in the case when the flow region in the hodograph plane is a half-strip with a longitudinal cut [1] are known only for two models of the resistance law [2–6]. The present study gives a one-parameter family of flow laws, and argues the possibility of effective determination of exact and approximate analytical solutions on the basis of successive reduction to boundary-value problems for the Laplace equation or for the equation studied in detail in [1]. It should be noted that the characteristics of the flow are determined without additional quadratures.

Numerical prediction of a laminar boundary layer flow on a rotating sphere

Numerical solutions to a laminar boundary layer flow past a sphere are considered. The solutions are presented using the procedure of Gosman et al. [1] with appropriate modifications. Successful numerical solution procedures have been devised for the solution of flow problems, see [5]. The SOR method is chosen as a method of solution. Although it looks like a simple method, application of such a method to nonlinear Navier-Stokes equations is highly nontrivial. The matrix method is not used because convergence was not a problem for the type of flow considered in this paper. The governing nonlinear differential equations are converted into finite difference equations by integrating the equations over a control volume and are then solved by an iterative procedure. The numerical results predict that the transverse velocity v θ is positive in the upper hemisphere, goes to zero in the equitorial plane and becomes negative in the lower hemisphere.

Fundamental solutions and numerical methods for flow problems

AbstractAn approach for the numerical solution of flow problems based on the concept of fundamental solutions of differential equations is described. This approach uses the finite element methodology but does not rely on the concept of variational principle or that of residuals. The approach is shown to be well‐suited for many types of flow problems. Various applications of this approach are discussed in this paper, with particular emphasis placed on the solution of potential flows and viscous flows containing appreciable regions of separation.

Long range memory effects in flows involving abrupt changes in geometry: Part 2: the expansion/contraction/expansion problem

We give further theoretical consideration to the solution of flow problems involving abrupt changes in geometry using implicit rheological equations of state, so that the stress components are dependent variables along with the velocity components and the pressure. Attention is confined to the expansion/contraction/expansion problems and main interest centres on the effect of long range memory on flow characteristics, particularly the recirculating vortices arising from the presence of the abrupt changes in geometry.

Solution of Two-Phase Flow Problems Using Implicit Difference Equations

Abstract Many difference equations used to approximate reservoir flow problems treat the phase pressures implicitly but not the mobility-density coefficients. Such difference equations are neither wholly explicit nor implicit, but might be described as mixed. Mixed equations are relatively easy to apply. But the associated time truncation error is relatively large, and when used to solve problems characterized by high flow rates, these equations may be unstable for practical size time steps. This paper outlines the development of a completely implicit difference analogue for reservoir simulation, along with a Newtonian iterative method for solving the resulting nonlinear set of algebraic equations that arise at each time step. While this implicit equation requires two to three times more work than does the mixed equation, it is shown to markedly decrease the time truncation error and to yield a stable solution for much larger time steps than does the mixed equation. Introduction Calculation of multiphase, multidimensional flow in porous media is generally accomplished by numerical methods that involve approximating systems of particle differential equations by systems of partial difference equations. The early development of difference equation techniques was directed toward the solution of linear differential systems. However, the equations of multiphase fluid flow through porous media are highly nonlinear in that mobility and density often are strong functions of pressure and saturation. Thus, solution of the flow pressure and saturation. Thus, solution of the flow problems by difference equations involves solving problems by difference equations involves solving sets of algebraic equations whose coefficients change from step to step of the calculations. Current practice is to evaluate most coefficients at the beginning of a time step and then to apply difference methods well suited for solving linear problems. At least two major difficulties arise from problems. At least two major difficulties arise from the use of this practice. First, evaluation of coefficients at the old time level and evaluation of pressures at the new time level results in larger pressures at the new time level results in larger time truncation errors near displacement fronts than if all quantities in the distance derivative were evaluated at the same time level. Second, evaluation of mobility coefficients at old time levels results in an unstable difference equation in regions of high flow rate. The first of these difficulties often results in optimistic recovery behind the displacement front, and the second requires the use of small time steps or special techniques in solving coning problems or problems of gas percolation. Evaluation of all quantities in the distance differences at the new time level results in a completely implicit difference system. Such a difference system has a lower time truncation error than equations in which mobilities are evaluated at the old level and pressures evaluated at the net, level. The algebraic equations resulting from the fully implicit difference equation are, in this case, nonlinear and require some iterative method for their solution. This paper develops fully implicit difference equations for two-phase flow in porous media, describes their solution by Newtonian iteration, and gives examples of problems more easily solved by the new method than by previous methods.

Wells in homogeneous anisotropic aquifers

When the hydraulic conductivity of an aquifer is a function of direction only, the aquifer is referred to as homogeneous and anisotropic. Anisotropic aquifers are a common occurrence in nature. The effect of anisotropy on the flow of groundwater has been of concern to many hydrologists. Nearly all solutions of flow problems now available do not take into consideration such effects. The process of obtaining such solutions, however, can be used to obtain solutions for problems in homogeneous and anisotropic aquifers. Boundary‐value problems describing the flow in homogeneous anisotropic mediums may be changed by using a simple transformation of coordinates to problems whose solutions are either available or can be obtained through the usual processes of mathematical analysis. Although the process of carrying out the transformation to completion is routine to hydrologists who are versed in mathematical analysis, it is not routine to many active and experienced hydrologists. Solutions are presented herein for the drawdown around wells in homogeneous anisotropic aquifers which, during the period of pumping and the subsequent period of recovery, may be assumed for all practical purposes to be infinite in areal extent. (Key words: Groundwater; hydraulics; wells.)

The relationship between solutions of flow problems in isotropic and anisotropic soils

Problems of flow through a given region of nonisotropic porous medium may be solved by means of an equivalent isotropic flow region obtained from the given region by an appropriate scale change. The derivation of these scales is based on the similarity between the equations describing the flow in both systems. Several special cases of flow of a single liquid and two liquids under confined and phreatic conditions are considered. The scales derived in this way are most useful in model investigations.