The Finite Difference Method

Transcription

1 Chapter 5. The Finite Difference Method This chapter derives the finite difference equations that are used in the conduction analyses in the next chapter and the techniques that are used to overcome computational instabilities encountered when using the algorithm. A two-dimensional heat-conduction problem at steady state is governed by the following partial differential equation. T T + = 0 (5.1.1) This is the Laplace equation, and this type of problem is classified as an elliptic system. The finite difference equations and solution algorithms necessary to solve a simple elliptic system can be found in the literature. Because of the way that the present problem is defined two boundary conditions specified in one of the two dimensions, a new solution algorithm becomes necessary. The following steps explain how the equations are derived, and the algorithm is formulated. 5.1 Derivation of the Finite Difference Equations Interior nodes A finite difference equation (FDE) presentation of the first derivative can be derived in the following manner. In a descritized domain, if the temperature at the node i is T(i), the temperature at the node i+1, spatially separated from node i by in the x-direction (fictitious direction in space), can be approximated as the Taylor series expansion. 60

2 T T T T T (i + 1) = T(i) (5.1.)!!! Similarly at the node i-1, T T T T T (i 1) = T(i) + + (5.1.)!!! Here, represents the distance between centers of adjacent nodes. Adding Eqns. (5.1.) and (5.1.) results in the following form. T T T (i + 1) + T(i 1) = T(i) (5.1.)!! Solving for the second derivative term, T T(i + 1) T(i) + T(i 1) =! T (5.1.5) The nature of the Taylor series expansion lets higher order terms be truncated because magnitudes of the higher order terms are smaller. If one truncated all the right-hand-side terms, except the first, in Eqn. (5), it would look like the following. T T(i + 1) T(i) + T(i 1) = + O( ) (5.1.5 ) Here, O( ) contains all the truncated terms, or is regarded as the truncation error due to the approximation of the original derivative. This finite difference approximation of the x-derivative may be classified as central difference and of second order accuracy with truncation of the order of magnitude. The central difference indicates that the derivative approximation is centered at node i. As partial derivatives are used thoroughly in the above derivation, the same procedure applies in the y-direction. Hence, Equation (5.1.5 ) and the corresponding expression in the y-direction can be substituted into Eqn. (5.1.1). It yields the following, while dropping the higher order terms. T(i + 1, j) T(i, j) + T(i 1, j) T(i, j + 1) T(i, j) + T(i, j 1) + = 0 (5.1.6) 61

3 j+1 Direction of Parabolicity j j 1 i 1 i i+1 Figure 5.1 Neighboring nodes for the finite difference presentation of the Laplace equation about the node (i,j). indicates the node at which the temperature is to be calculated based on those at the nodes indicated with. Figure 5.1 shows the relative positioning of the points involved in Eqn. (5.1.6) to approximate the Laplace equation with the order and about the node (i,j). Because of the way the problem is defined, the calculation proceeds from one side of the domain where j=1 to the other where j=nj. Therefore, Equation (5.1.6) must be solved for T(i,j+1). (i, j + 1) = T(i, j) T(i, j 1) T { T(i + 1, j) T(i, j) + T(i 1, j) } (5.1.7) This, then, is the form in which the Laplace equation is implemented in the computational code. 6

4 5.1. Nodes at boundaries Equation (5.1.7) cannot be applied to nodes at boundaries. Therefore, modified equations suitable for capturing the conditions at the boundaries must be derived. In the problem, there are three types of boundaries and four boundaries. The first type is a boundary at which both temperature and heat flux are specified. The calculation algorithm starts with this boundary and proceeds in the direction of parabolicity that is indicated in Fig The second type applies to the side boundaries, i = 1 and NI. At the side boundaries, doubly connected or zero-flux boundary conditions can be specified. The doubly connected boundaries mean that both temperature and heat flux are specified. Detailed procedures can be found in Hoffmann and Chiang (1998). The last type is at the end of the calculation, j = NJ. There is no need to derive a finite difference equation at this level since each value will be computed from Eqn. (5.1.7). First, at the starting level, both temperature and heat flux are prescribed, and the to-be-derived finite difference equation must reflect that. Considering very small area elements indicated by the dashed lines in Fig. 5.1, transport of heat can be looked at in terms of Fourier s law, which states: T T q x " = ka ; q y " = ka (5.1.8) and (5.1.9) Heat transfer crossing the dashed line between the nodes (i,j) and (i,j+1) can be approximated as, assuming a linear temperature distribution: T(i, JN 1) T(i, JN) q S = k (5.1.10) Here, the S (South) indicates the relative location from node (i,j) of the surface through which the heat flows. Positive values indicate northerly (increasing j) flow. Similar expressions can be derived for all four directions. However, the prescribed temperature at level j=jn is uniform; therefore, q W and q E are both zero. The heat flow on the north boundary, from the gas to the wall, is: q N [ T T(i, JN) ] = h (i) (5.1.11) o g 6

5 h o (i) is a local value of heat transfer coefficient, and T g is the freestream temperature of gas. The positive sign indicates northerly flow when T g > T(I,JN). Now, conservation of energy can be applied, considering positive heat flow to be in +x and +y directions, i.e. Substituting, q N + q E + qs q W = 0 [ ] T(i, JN 1) T(i, JN) T T(i, JN) + 0 k h(i) g = Solving for T(i,), T(i,JN [ T T(i, JN) ] (5.1.1) (5.1.1) h o (i) 1) = T(i, JN) g (5.1.1) k The side boundaries are doubly connected, which means that both temperature and heat flux match between the side boundaries. In other words, the node (1, j) is identical to node (NI, j). To achieve this condition, Eqn. (5.1.7) can be modified as follows. At i=1, for example, there is no such node as i-1, so this is replaced with IN-1. (1, j + 1) = T(1, j) T(1, j 1) T Similarly at i=in, { T(, j) T(1, j) + T(IN 1, j) } (5.1.15) (IN, j + 1) = T(IN, j) T(IN, j 1) T { T(, j) T(IN, j) + T(IN 1, j) } (5.1.16) Now that all boundaries seem to have been addressed, the question remains, How to deal with the corner nodes. At j = JN, Eqns. (5.1.15) and (5.1.16) are applicable as is Eqn. (5.1.7) to the nodes at the j = JN level. At j = 1 then, as seen from the comparison between Eqns. (5.1.15) and (5.1.16), the corresponding equations do not lead to having the corner nodes being identical since the heat transfer coefficients may be different at I=1 and IN. Therefore, for the present study, one of the two possible values of the heat transfer coefficients either h(1) or h(in) is used. 6

6 5. Pre-Computation Analyses Before the above finite difference equations are implemented into the computational code, there are a few analyses one can do to check whether the equations are consistent with the original partial differential equation (consistency analysis) and to derive stability criteria for the computation (perturbation analysis) Consistency analysis This analysis confirms that the derived finite difference equation is consistent with the original governing partial differential equation the Laplace equation. Equation (5.1.7) is going to be applied to most of the nodes in the computation. Thus, this analysis looks at Eqn. (5.1.7) only. Equations (5.1.1), (5.1.15) and (5.1.16), are merely modified cases of Eqn. (5.1.7). To test the consistency, the Taylor series expansions of T(i+1, j), T(i-1, j), T(i, j+1) and T(i, j-1) are substituted into Eqn. (7). The resulting form is: T (i, j) + +! T +! T T T(i, j) +! T + = T(i, j) T! T + T T T T(i, j) T(i, j)!! Rearranging to get, T T(i, j) +! T! T (5..1)! T +! T! T +! = T (5..) 65

7 Multiplying the both sides by and dividing by, T + 1 T T T + = 0 (5..) Consistency is confirmed if the original partial differential equation is recovered when and go to zero. This is true for Eqn. (5..), and the consistency is confirmed. 5.. Perturbation analysis A perturbation analysis is a way to determine a criterion for computational stability. This is achieved by introducing a small perturbation into the finite difference equation, in this case Eqn. (5.1.7). Looking at propagation of the perturbation in the direction of computation, a stability criterion can be derived in terms of combinations of and. This procedure is based on Chapter in Hoffmann and Chiang (1998) where they derive stability criteria for one-dimensional, transient heat conduction computations. They introduce different types of perturbations at a particular time and examine how they grow or decay. Certainly, the larger the decay, the more stable is the algorithm. They find that the criterion that they derive for spatially oscillating perturbations is the strictest. Following their procedure, such perturbations are introduced into level j of Eqn. (5.1.7). A small perturbation is denoted by ε, and +ε is introduced at node (i, j), -ε at (i-1, j) and (i+1, j). The values before the level j are zero; therefore, T(i, j-1) is equal to 0. Equation (5.1.7) then becomes: (i, j + 1) = ( + ε) 0 {( ε) ( +ε) + ( ε) } (5..) T Rearranging to obtain the ratio of T(i, j+1) to ε, T(i, j + 1) ε = + (5..5) This ratio represents a propagation of the error. If the ratio is less than unity, the error decays with continued calculation. If the ratio is greater than unity, the error grows, and 66

8 the calculation becomes unstable. Thus, a proper combination of and must be sought from Eqn. (5..5). The requirement then is to limit the magnitude of Eqn. (5..5) to less than unity: T(i, j_ + 1) < 1 (5..6) ε In other words, + < 1 (5..7) There are two possible conditions: + < 1 and < 1 + x (5..8) From the second condition, y 1 > (5..9) This is always satisfied since both length scales are positive values. However, from the first condition, y 1 < (5..10) This cannot be satisfied by any combination of and. In conclusion, this perturbation analysis did not lead to a criterion for computational stability; rather, it is anticipated that the computation will be unstable since Eqn. (5..10) cannot be satisfied. At this point, it has become necessary to derive a technique to overcome the anticipated instability. The next section deals with this issue. 67

9 5. Smoothing the Temperature Profile Solutions, or temperature fields, calculated with the above finite difference equation will surely produce errors. The growth of the errors is similar to that of a system governed by a parabolic partial differential equation such as a one-dimensional, transient heat conduction system. In both cases, computational instability produces oscillating errors. They may not be visible during an earlier stage when a profile is plotted. When the stability criterion is not met, these errors grow as the calculation progresses, and the solutions diverge. An idea to improve the situation is to smooth the temperature profile over each layer when the calculation on each layer is done. The oscillations are purely computational. By smoothing the temperature profile on each layer of the calculations, the oscillating errors are prevented from growing Smoothing parameter MATLAB is used for the computations, and the smoothing function chosen for this conduction problem within MATLAB has the function name CSAPS.M. The function returns smoothed new values of dependent variables based on a vector of independent variables, a vector of old dependent variables (or data) and a smoothing parameter. The smoothing parameter can be any value between 0.0 and 1.0. The new values of dependent variables fully depend of a choice of the parameter. This section reviews how the smoothing parameter affects the resulting profiles of dependent variables. The smoothing function has two extreme modifying routines built in. One is to take the least-square straight line. This routine can be selected by setting the smoothing parameter to zero. With this parameter value, the new dependent variable will be a straight line through the range of independent variables. Therefore, this would not be accurate unless the dependent variables are expected to be linear functions. The other extreme is to follow all the old values of the dependent variable. This can be selected by setting the parameter to unity. However, the vector of the new dependent variable would 68

10 be identical to one of the old values, and there would be no smoothing. Any value between results in a weighed average of the two extreme cases. In Fig. 5., observations are made of two different values of the smoothing parameters using simple functions. In (a) and (b), smoothing parameters of 0.1 and 0.9 are respectively applied to a linear function with oscillating errors. In (c) and (d), the same pair of the smoothing parameters, 0.1 and 0.9, is applied to a combination of linear functions peaking at the middle point. In any case, the data points are indicated with, and they deviate by 0.06 from the true functions, which are plotted as the solid lines. The smoothing function generates the dashed lines based only on the data points given by. Therefore, performance of smoothing function is measured by how close the smoothed lines are to the true profile. 69

11 1 (a) 1 (b) Node No Node No. 1 (c) 1 (d) Node No Node No. Figure 5. Smoothing parameters and their effects on the smoothing. The solid line is the true profile, the crosses the original data points with oscillating errors from the true profile and the dashed line the smoothed profile intended to recover the true profile. (a) and (c) use a smoothing parameter of 0.1 and (b) and (d) 0.9. In (a) and (b), the effects of the different smoothing parameters are minor while they are quite significant in (c) and (d). It seems that the smoothing function is better at dealing with simply increasing or decreasing functions than it is at dealing with peaks in the functions. Properly recovering a peak is an important issue. The actual conduction problem in the main project has much higher heat fluxes around the leading edge of the airfoil. The peak is expected to give rise to strong temperature variations in that region. If there is such a phenomenon, it is important to properly capture it. 70

12 By comparing (c) and (d), it is obvious that the higher value of the smoothing parameter, 0.9, does a much better job. However, the true value will not be recovered even with the value 1.0 since the smoothed line with the value 1.0 merely goes through all the data points. Therefore, in the main project, it is important to have many data points within a peak so that such a peak can be recovered. In other words, the goal of implementing this smoothing function is to eliminate locally oscillating profiles, or local peaks; therefore, any true peak with one data point, such as the one in (d) of Fig. 5., can be strongly smoothed. Since at least one peak is expected in the main conduction problem, a value of the smoothing parameter that will be used is expected to be close to 1.0. In fact, the values used in the curved wall analysis (see Sections 6. and 6.) are on the order of Example This section considers a classical one-dimensional, transient heat conduction problem that can be analytically solved. The comparisons are made between the analytical solution and numerical solutions with and without smoothing. This example will show that the smoothing process does not cause any unphysical effects on the resulting solution. This is Problem.1 from Hoffmann and Chiang (1998). The situation is given in Fig. 5., and the system is governed by a parabolic PDE given in Eqn. (5..1). 71

13 T s T i T s 1-D x Figure 5. Example problem T T = (5..1, 6) t The temperatures at the surfaces is raised to T s at time zero, and initially the slab has a temperature of T i. The analytical solution can be obtained by the Fourier series, and is given as: π π = + ( ) m m 1 ( 1) m x T(x, t) Ts Ti Ts exp αt sin (5.., 7) m= 1 L mπ L The physical parameters used are the domain length, L, of 1 [ft], the initial temperature, T i, of 100 [ F], the surface temperature, T s, of 00 [ F] and the thermal diffusivity, α, of 0.1 [ft /hr]. A computational method, similar to the one derived earlier in this chapter for the main conduction problem, is the explicit method called Forward Time Central Space (FTCS) method. The finite difference equation states: T n+ 1 i n n n [ T T + T ] n α t = Ti + i+ 1 i i 1 (5.., 6) 7

14 The index i indicates x-position and n time. The steps, t and, are in time and space, respectively. A perturbation analysis can be done in the same way as done in the earlier case. It results in the criterion: α t 1 (5.., 7) By restricting the spatial step,, to 0.05 [ft], the critical value of time step, t, can be calculated since the diffusivity is given; it is [hr]. Therefore, if the time step is set to a value which is less than 0.015, the calculation will be stable. The time step of 0.01 [hr] is used for the stable calculation, and 0.0 [hr] for the smoothed calculation. To be more precise, three calculations are compared in the following discussions. One is the analytical solution given by Eqn. (5..). The second is a numerical calculation with the stability criterion met. In this case, the time step of 0.01 [hr] is used, and there is no smoothing process. The last is yet another numerical calculation with the stability criterion NOT met. In this case, the time step of 0.0 [hr] is used. Since the calculation is unstable, the smoothing process is implemented every time step. The results are shown in the following figures. The smoothing parameter used is In the analytical solution, the series terms up to the fifth are included. Figures 5. (a) and (b) present the entire domain at times 0.1 and 1.0 hour, respectively. The deviations of the numerical methods from the analytical are barely visible in these scales. These figures are magnified in (c) and (d), centering the locations where deviations are seemingly the greatest. In (c) and (d), besides the magnitudes of the deviations, the comparisons evidently show what the smoothing process is least good at capturing precipitous changes in gradients. While the analytical solution draws the outermost curvature, the stable solution misses capturing the peak values, and each of the smoothed calculations has even less curvature. 7

15 T [ o F] (a) smoothed stable analytical T [ o F] (b) 100 t=0.1 hr X [ft] 00 t=1.0 hr X [ft] (c) (d) T [ o F] T [ o F] t=0.1 hr X [ft] t=1.0 hr X [ft] Figure 5. Results of example problem testing the smoothing parameter Figure 5.5 shows distributions of the deviations from the analytical solution. The errors are the greatest where the analytical solution has the highest rate of changes in gradients. The trends of the error distributions are similar between the stable solution and the smoothed solution, except for the magnitudes. The most remarkable advantage of using the smoothing technique is that the instabilities in the algorithm have been overcome. The time step of 0.0 [hr] exceeds the stability criterion given by Eqn. (5..) and has a value If an ordinary calculation were begun with this condition, the solution would diverge; however, with the use of the smoothing function, the errors are of the same magnitude as those of the stable calculations. 7

16 1 t=0.1 hr (a) (b) Error [%] 0 1 stable smoothed X [ft] Error [%] t=1.0 hr X [ft] Figure 5.5 Error distributions of the example problem, (a) at 0.1 hour and (b) at 1.0 hour 75

17 5. Conclusions Finite difference equations have been developed for the two-dimensional heat conduction computations to be performed in the next Chapter. The equations allow parabolic solution algorithms that become necessary because of the way that the problem is proposed. Equation (5.1.7) is used for interior nodes with uniform grids; Eqns. (5.1.15) and (5.1.16) are used for nodes at side boundaries. Consistency analysis verifies that the newly developed FDE, Eqn. (5.1.7), is consistent with the governing Laplace equation. However, perturbation analysis concludes that the algorithm is not stable, i.e. errors growing as the computation proceeds in the parabolic direction. To overcome the instability, a profile-smoothing technique is developed and tested in a simple, one-dimensional, transient, heat conduction example. A temperature profile is smoothed by a smoothing function to eliminate oscillating errors. The example has yielded a solution that is almost identical to the analytical solution, verifying that the smoothing function helps overcome instability. Still, the smoothing procedure may reduce a true peak value if the profile is kinky at the peak. 76