1. The gure below illustrates a thermocline, or region of strong temperature gradient in a strati ed uid. Oceans exhibit this type of behavior. The depth of the thermocline is de ned as the location at which the curvature of the depth-temperature curve goes to zero (d2T=dz2 = 0, an in ection point). The temperature gradient at this location determines the heat transfered between the ocean surface and the ocean oor according to Fourier s law, q = k dT=dz where q is the heat ux density in W/m 2 and k is the thermal conductivity in W/mK. Note that the data should be plotted as shown in the gure with depth on the vertical axis, but that the curve t should be done for temperature as a function of depth (not depth as a function of temperature). For this problem, assume these data are exact. Depth, m 0 100 300 400 500 700 1000 1250 1500 1750 Temperature, C 22 21.9 20.8 19.1 16.4 10 6.2 5.1 4.6 4.4 (a) By solving a linear system (Vandermonde matrix), nd the polynomial that passes through all data points. Plot the polynomial curve. On the same graph, plot the data points. Predict the thermocline depth using the interpolation results. You may want to use the MATLAB function roots for nding the in ection point, or you may wish to use fzero. Note any di culties you have with this calculation. Estimate the temperature gradient at this location. (b) What is the minimum-order polynomial that could reasonably be used to predict the thermocline depth? (Hint: you are looking for an in ection point.) For this order polynomial, nd the least-squares curve t using the MATLAB function polyfit. Plot the polynomial curve and the data points. Predict the thermocline location. Estimate the temperature gradient at this location. (c) Use the MATLAB function spline to interpolate the data as cubic splines. You can call the function by typing >>f = spline(x,y,xi); where x is the vector of depth locations, y is the vector of temperatures and xi is a vector of depths at which the interpolation is to be completed. f will be a vector of interpolated temperatures. Plot the curve and the data points. To estimate the second derivative, you need to unmake the partialpolynomial form. To do this, type >>pp = spline(x,y); The resulting vector can be unmade by typing >>[breaks,coef,L,K] = unmkpp(pp); The matrix, coef, will be an N 4 matrix where N is the number of intervals. The rows of the matrix are the cubic polynomial coe cients for the corresponding interval. For example, the cubic describing the second interval is given by f2(x) = coef(2,1)(x x2)3 + coef(2,2)(x x2)2 + coef(2,3)(x x2) + coef(2,4) Since you ought to be able to determine in which interval the in ection point will occur, you should be able to extract the depth where the second derivative reaches zero. (d) Compare the thermocline depth predictions and the temperature gradient estimates from eachof the three methods. Discuss, including other observations about the three approximations. Which do you think is likely to give the most physically realistic result? (e) With the help of the MATLAB function csape, repeat parts (c) and (d) for end conditions corresponding to the natural ( variational ) spline and the default ( Lagrange ) spline and compare with results for the MATLAB default "not-a-knot" end condition. Comment on the dierences. Which do you think is likely more correct and why? 2. The gure shows the forces acting on an aircraf ying in steady ight (no acceleration). The steady- ight (or trim) equations can be expressed in a coordinate system aligned with the ight path as Fx = 0 = T cos W sin XX D (1) XMy = 0 = Fz = 0 = T sin 'WLW + W cos 'HLH (3) hTT LW LH (2) Note that hT is not shown in the gure, but it represents the vertical distance between the center of gravity (the origin of the coordinates) and the thrust vector. In these equations, T thrust W aircraf weight D aircraf drag LW wing lif LH = horizontal tail lif ight-path angle aircraf angle of attack First, we must determine what is unknown. The unknown quantities depend on the problem to be solved, but a typical situation may be that the aircraf properties are known and a certain ight path and velocity are desired. For this case, W and are known, but the rest of the values are not. We therefore have three equations with ve unknowns. To complete our set of equations, we introduce expressions for the lif and drag: (4) (5) where is in radians. We may assume that all the new variables introduced in these two equations are known: air density V aircraf velocity S aircraf wing area aW wing lif-curve slope CD0 parasite drag coe cient K = induced drag parameter We are now faced with a set of ve nonlinear equations and ve unknowns, T, LW, LH, D, and . The following table gives values for the aircraf and operating parameters typical of a 180-passenger transport aircraf (Boeing 737-900) ying at an altitude of 31,000 f. W (N) V (m/s) (kg/m3) CD0 aW S (m2) K 'W (m) 'H (m) hT (m) 1:3 735,00 235 0.44 0.02 5.2 125 1.06 1.0 21.5 0 (a) Solve the above ve equations for the ve unknowns. You may write your own Newton-Raphsonsolver or use MATLAB s built-in solver, fsolve.Use three dierent ight-path angles ( = 0; 10 , 4 ). For initial guesses, keep in mind that the total lif approximately balances the weight, and that the lif-to-drag ratio for an aircraf like this one runs around 15. The angle of attack usually remains below about 12 (0.21 radians). (b) Do your answers for part a make sense? Explain why you think so or think not. Note especiallythe solution for = 4 . Note that the relative values of the unknowns will vary greatly in order of magnitude (check the dierence between the expected wing lif LW vs. the angle of attack, ). This large variation in variable magnitude will likely lead to a large variation in variable error, even if the relative errors are similar in magnitude. This could mean that fsolve gives an unusual error message. Solution to these equations will tell a designer how much thrust is required for the mission (and therefore how big an engine is needed). Knowing the required thrust can also be used to determine the amount of fuel used