1.2 Numerical Solutions of Flow Problems

Transcription

1 1.2 Numerical Solutions of Flow Problems DIFFERENTIAL EQUATIONS OF MOTION FOR A SIMPLIFIED FLOW PROBLEM Continuity equation for incompressible flow: 0 Momentum (Navier-Stokes) equations for a Newtonian fluid with constant viscosity: Comments: The above system of four equations is closed, i.e., it has the same number of independent equations as the number of unknowns (u,v,w and p). Proper boundary and initial conditions must be specified. These equations are very difficult to solve because they are coupled and contain nonlinear terms. To simplify them, one may make new assumptions (e.g., to neglect friction), but this is possible only in certain cases. Analytical solutions are known for few simple cases only. Numerical solutions are possible for a wide range of problems.

2 TYPICAL STEPS FOR THE NUMERICAL SOLUTION OF A FLOW PROBLEM 1. Define the physical problem (i.e., the geometry and the required solution variables); if possible, make simplifying assumptions (e.g. incompressible flow or inviscid flow; justification must be given). 2. Formulate a mathematical model, including the governing equations (namely the mathematical expressions of the physical principles of conservation of mass, momentum and energy or other theoretical or empirical concepts) as well as proper boundary and/or initial conditions. The governing equations could be algebraic, differential (ordinary or partial), integral or integro-differential. The usual form in fluid mechanics is differential, which implies that the solution must specify the values of the required parameters at all points in the flow. 3. Transform differential equations into a form convenient for solution by digital computers. This means to discretize the differential equations by transforming them into a set of algebraic equations. Common discretization techniques are the finite difference technique and the finite element technique. The algebraic equations apply to selected positions of the flow field (e.g., at the nodes of a grid); values at other points can be found by interpolation. 4. Analyze the properties of the discretization scheme, namely check its stability, convergence and error bounds. Commercial packages have incorporated appropriate tests. 5. Solve the system of algebraic equations by iteration or with the use of matrix solvers. Commercial packages have incorporated properly optimized algorithms. Building block type subroutines are available for custom developed codes. 6. Present, interpret and discuss the results. Validate vs. experimental results and compare with other related studies, if available.

3 DISCRETIZATION IN CFD In fluid dynamics, space and time are in most cases assumed to be continuous. The differential equations of motion are also continuous, namely they apply to all mathematical points in space and time within the flow domain. Digital computers, however, cannot provide continuous solutions. Instead, they can estimate discrete values of the variables of interest at discrete locations. Therefore, a first step in all CFD formulations is to discretize both the space-time domain and the differential equations. Space discretization: This is achieved by defining a mesh or a grid that covers the flow domain and replacing the continuous domain by a finite number of discrete points. The part of a code that generates the discrete set of points is called the mesh generator. The solution will be then computed only at those points. Values at other points may be estimated by interpolation. Obviously, the higher the number of discrete points is, the more accurate the representation of the flow domain would be. One may also expect that the accuracy of the solution improves as the number of discrete points increases. On the other hand, as this number increases, the solution would require an increasing amount of computer resources (e.g., computer memory and execution time). Time discretization: This is usually achieved by dividing the time interval of interest into a number of equal discrete time steps and computing the solution at these steps only. Even steady flows are sometimes computed by using a time-dependent (pseudo-unsteady) formulation, in which the partial differential equations are replaced by ordinary differential equations with respect to time. Equation discretization: This requires the transformation of the continuous differential equations into a system of algebraic (i.e., without derivatives or integrals) equations of the variable values at the discrete space-time points mentioned earlier. The solution of this system of algebraic equations can be achieved either directly (i.e., in one step), or iteratively. The three most commonly used methods for the space-time discretization of differential operators are the finite difference method (FDM), the finite element method (FEM) and the finite volume method (FVM).

4 EQUATION DISCRETIZATION METHODS Finite difference method: This is the simplest and oldest method, and it is best applied to relatively regular meshes. The flow domain is fitted by a grid consisting of intersecting families of lines, straight or curved, and the solution is obtained at the intersection points (nodes). The various derivatives in the equations are approximated as finite differences, by using the Taylor series expansion. Thus, each differential equation is replaced by a set of algebraic, finite difference equations. Finite element method: The flow domain is divided into a number of finite elements, namely subdomains with various shapes (e.g., rectangular, triangular, tetrahedral etc.). These elements are defined by their vertices. They should not overlap and should cover the entire flow domain. A number of discrete points (nodes) are defined for each finite element, either at its interior or at its boundary. The discrete values of the unknown parameters and their derivatives are expressed at these nodes. The differential equations are first expressed in an integral form over the entire flow domain. Then, the approximate solutions, valid at the nodes, are expressed as linear combinations of certain functions, multiplied by coefficients which are selected such that the integral relationships are satisfied. One approach to find these coefficients is to use a variational method (Ritz method) that minimizes some appropriate function. Another method is the weighted residual method or Galerkin method. The finite element method was first developed in structural dynamics and it took some time before it could be applied to fluid mechanics. This method is suitable for coupling fluid and solid motions (fluid-structure interaction). Finite volume method: In this approach, the integral forms of the basic principles (i.e., continuity, momentum and energy principles) are discretized directly in the physical space. Although conceptually this method resembles the finite difference method, it is suitable for complex geometries, as it utilizes arbitrary mesh shapes. Thus, the finite volume method is similar to the finite element method as far as the subdivision of space is concerned.

5 STRUCTURED AND UNSTRUCTURED MESHES We wish to solve numerically the flow in a rectangular channel containing a solid cylinder. We need to fit a mesh in the space between the cylinder and plane wall surfaces. Two choices are possible: a structured mesh, which follows the boundaries in some regular fashion and an unstructured mesh, which has randomly connected elements. Structured mesh would generally provide higher accuracy of the solution but they may be difficult or impossible to generate for complex geometries. Structured mesh with detail near the gap Unstructured mesh with detail near the gap

6 FINITE DIFFERENCE APPROXIMATION OF LAPLACE'S EQUATION USING A RECTANGULAR GRID N PN W y PW x PE E P 2 Δ y PS 2 Δ x S 2-D Laplace s Equation: 0 In the following, we shall use Taylor series expansions to approximate derivatives by finite differences. As a first step, the first-order derivatives at mid-points PE and PW are approximated as Δ Δ Then, the second derivative at the central point P is approximated as Δ Δ Δ 2 Δ 2

7 Similarly, following the same procedure along the y-axis, one can get 2 Δy 2 Then the 2-D Laplace's equation in finite difference form becomes Δ 2 Δy 2 Further solving for the function at the central point, one gets Δy 2 Δ 2 2 Δ 2 2 Δ 2 For a square grid ( x = y), this is simplified to the central average value 4 1

8 N b y y W P E S x a x a<1, b<1, x= y If the grid has uneven spacing (e.g., near a boundary, as shown in the sketch), the above relation has to be modified as where the coefficients are given by the expressions 2 1, 2 1,

9 NUMERICAL SOLUTION OF LAPLACE'S EQUATION Boundary condition: Laplace's equation can be solved if and only if the values of the unknown function are specified over the entire boundary (Note: other types of equations may require the specification of boundary conditions in part of the boundary only as well as the specification of initial conditions). For simplicity, assume that the flow domain is a square (left side of the figure). First, assume that a uniform grid with only 3 rows and three columns is fitted to this domain. If is specified at the four boundary nodes, then eq. (1) can be used to compute the value of at the only existing interior node 2,2 and the problem is solved, within this level of approximation. Next, assume that a uniform grid with M rows and N columns is fitted to the domain (right side of the figure). must be specified at all boundary nodes. There will be (N-2)(M-2) interior nodes, for each of which one can write a linear algebraic equation like eq. (1). Then, one can obtain a system of (N-2)(M-2) linear algebraic equations with an equal number of unknowns, namely the values of at all interior nodes. This is a well known problem and can be solved exactly, e.g., by matrix inversion methods. However, when N and M are large, this approach is extremely tedious and even unnecessary because most of the coefficients of these equations are zero. A more practical approach is to use an iterative, approximate method of solution, such as the Gauss-Seidel iteration method.

10 GAUSS-SEIDEL ITERATION METHOD FOR SOLVING A SYSTEM OF LINEAR ALGEBRAIC EQUATIONS This method consists of the following steps: a) as the boundary condition, specify the values of at all exterior nodes b) prescribe starting values of (e.g., zero values) at all interior nodes c) compute new values of at all interior nodes using equations like (1) d) compare the new and old values e) if the mean square difference between new and all values is smaller than a prescribed error limit, stop the iteration; otherwise, go back to step c) and repeat steps d) and e)