Module 2: Introduction to Finite Volume Method Lecture 14: The Lecture deals with: The Basic Technique. Objectives_template

Size: px

Start display at page:

Download "Module 2: Introduction to Finite Volume Method Lecture 14: The Lecture deals with: The Basic Technique. Objectives_template"

William Milton Gilbert
6 years ago
Views:

1 The Lecture deals with: The Basic Technique file:///d /chitra/nptel_phase2/mechanical/cfd/lecture14/14_1.htm[6/20/2012 4:40:30 PM]

2 The Basic Technique We have introduced the finite difference method. In the context of the method of weighted residuals, it can be said that the Finite Difference procedure is a collection method with piecewise definition of the field variable in the neighborhood of chosen grid points (or collection points). In a similar fashion the Finite Volume Method is a subdomain method with piecewise definition of the field variable in the neighborhood of chosen control volumes. The total solution domain is divided into many small control volumes which are usually rectangular (or arbitrary quadrilateral in shape. Nodal points are used within these control volumes for interpolating the field variable and usually, a single node at the centre of the control volume is used for each control volume. This method was developed by Patankar and Spalding (1972) and they proposed the use of the physical approach (where possible) for deriving the nodal equations. We shall illustrate the technique with the help of the 2-D heat conduction problem in rectangular geometry. Figure 14.1: Grid Arrangement for the Finite Volume Method Consider 2-D, steady heat conduction in rectangular geometry (Figure 14.1). The 2-D heat conduction equation is (14.1) where is the temperature field, is the thermal conductivity and Q is the heat generation per unit volume. At present we shall not consider any specific set of boundary conditions for the problem, but we shall discuss the handling of various type of boundary condition in due course. file:///d /chitra/nptel_phase2/mechanical/cfd/lecture14/14_2.htm[6/20/2012 4:40:30 PM]

3 The Basic Technique The two alternative ways of setting up the nodal equations are the weighted residual approach and the physical approach. Using the weighted residual approach, the 2-D heat conduction equation can be approximately satisfied by: (14.2) where the weight within the control volume. outside the control volume. Thus, we get, for each i = 1,...n (14.3) Interesting equation (14.3) by parts, we get: where the Gauss divergence theorem has been used to convert the volume integral to a surface integral. (14.4) The meaning of Eqn. (14.4) is that the net heat generation rate in the control volume is equal to the net sum of the rate of heat energy going out of the control volume where is the boundary of the control volume Equation (14.4) can be taken as an energy balance equation for the control volume. This balance equation can also be obtained physically, considering the balance of heat flux in Figure file:///d /chitra/nptel_phase2/mechanical/cfd/lecture14/14_3.htm[6/20/2012 4:40:30 PM]

4 Figure 14.2: Balance of Heat Flux in a Control Volume. For a typical node P with neighbors E,N,W,S (standing for east, north, west and south etc.) and corresponding control volume boundaries in those directions denoted by e,n,w,s etc., the heat balance for the control volume can be written as follows (for unit depth in z- direction): when is the heat flux (per unit area) on the east face, is the heat flux on the west face etc., and the faces are taken to be one unit deep perpendicular to the plane of the figure. Thus, is the total heat flux through the east face. The fluxes are taken to be positive in the directions indicated by the arrows. Physically, the above equation is equivalent to saying : Net rate of heat energy leaving the control volume through the boundary = Rate of heat generation within the control volume (CV) at steady state Thus, file:///d /chitra/nptel_phase2/mechanical/cfd/lecture14/14_3.htm[6/20/2012 4:40:30 PM]

5 (14.5) Which is the same statement as equation (14.4). In the implementation of the FVM procedure, the heat fluxes are expressed in terms of the nodal temperatures (T E, etc. at the CV centers) using piecewise interpolation around the control volume for the field variable (temperature in this case). Thus, assuming temperature to have linear variation between points E and P, the heat flux can be evaluated as follows: (14.6) while deriving (14.6) it has been assumed that the cell size is, constant in x-direction (equal to ). Similarly, is given by (14.7) Using similar expression for and also, the nodal equation for point P becomes: (14.8) This equation can be rewritten in the familiar form used in finite difference as: (14.9) where During numerical implementation, the subscripts E, W, etc. will be changed to numerical indices of i, j and solved in the same way (using point-by-point or line-by-line procedure etc.) as mentioned in previous lectures on finite differences. file:///d /chitra/nptel_phase2/mechanical/cfd/lecture14/14_3.htm[6/20/2012 4:40:30 PM]

6 The boundary conditions of a typical heat transfer problem can be handled in the following way. When the heat flux at the boundary is prescribed, say the corresponding heat flux term in the balance Equation (14.5) is set equal to the applied heat flux. For instance, for the control volumes adjacent to the x = 0 boundary as shown in Fig. 14.3, the term will be substituted by in equation (14.3). thus, (14.10) Equation (14.10) will be the nodal equation for such nodes. Figure 14.3: Prescribed Heat Flux at the Boundary. file:///d /chitra/nptel_phase2/mechanical/cfd/lecture14/14_4.htm[6/20/2012 4:40:31 PM]

7 When the boundary temperature is specified, the control volume shapes near the boundary can be changed to facilitate the implementation of the boundary conditions. For instance, consider the condition T = T L on the x = L boundary (see Fig. 14.4). For the nodes on the boundary, an imaginary extension of the control volumes outside the actual domain can be considered in line with the finite difference methodology described earlier. The physical boundary is taken to be at the center of boundary cell of width (see Figure 14.4), while the widths if the adjacent cells are thus reduced to Consider a typical control volume i near the x = L boundary as shown in Fig Figure 14.4: Boundary Condition, at x = L, T = T L. The boundary cells will need no nodal equation as the T = T L will be applied. The nodal equation for the adjacent cell P will be written considering a shortened control volume: (14.11) file:///d /chitra/nptel_phase2/mechanical/cfd/lecture14/14_5.htm[6/20/2012 4:40:31 PM]

8 where Note that T L has been used instead of T E in the above equation. So, the boundary condition is being directly applied. In this fashion, by adjusting the control volume spacing and the placement of nodes, nodal equations can be obtained at all nodes and these can be solved simultaneously by the matrix inversion technique, line-byline technique or point-by-point technique as discussed earlier. Having done the above exercise, we may like to look at a more generalized description of the finite volume method. Congratulations, you have finished Lecture 14. To view the next lecture select it from the left hand side menu of the page or click the next button. file:///d /chitra/nptel_phase2/mechanical/cfd/lecture14/14_5.htm[6/20/2012 4:40:31 PM]

FINITE VOLUME METHOD: BASIC PRINCIPLES AND EXAMPLES

FINITE VOLUME METHOD: BASIC PRINCIPLES AND EXAMPLES SHRUTI JAIN B.Tech III Year, Electronics and Communication IIT Roorkee Tutors: Professor G. Biswas Professor S. Chakraborty ACKNOWLEDGMENTS I would like