Homework 1_R1

Homework 1 AAE 334 Assigned: Friday, January 12, 2024 Due: Friday, January 19, 2024 (11:59 p.m. ET) -10% Monday, January 22, 2024 (11:59 p.m. ET) These problems review material from AAE 333 or other similar courses in fluid dynamics. 1. [10 points] Typically inviscid flow solutions for flow around bodies indicate that the fluid flows smoothly around the body, even for blunt bodies like a cylinder. However, experience reveals that due to the presence of viscosity, the main flow may actually separate from the body creating a wake behind the body. As was discussed in AAE 333, whether or not separation takes place depends on the pressure gradient along the surface of the body, as calculated by inviscid flow theory. For a circular cylinder placed in a uniform stream with velocity, U , determine a. [5 pts] an expression for the pressure gradient in the direction of flow on the surface of the cylinder in terms of the free stream velocity, U, the cylinder radius, a, and the angle 𝜃𝜃 , and b. [ 5 pts] the range of values for the angle θ where an adverse pressure gradient occurs. That is where 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 > 0 .

2. [10 pts] It is hard to find the origin of the Cow versus Jeep aerodynamics meme, but I had a group of students do a series of wind tunnel tests to confirm that a Jeep with the top and sides open does have a higher drag coefficient than a cow. (https://www.instagram.com/engineeringprinciples/p/CVpEk0JszzO/) Performing a series of wind tunnel tests, the students determined the drag coefficient based on the projected frontal area of the Jeep is about 0.5. According to Edmunds.com a 2022 Jeep Wrangler has a maximum power of 285 hp and a top speed of 110 mph. The Jeep is 188.4 inches long by 73.8 inches wide by 73.6 inches high. a. [2 pts] Estimate the power (in Watts) from the engine to overcome the drag when the Jeep is at top speed. What percentage of the maximum engine power is this? b. [6 pts] Openvsp is a program that allows you to rapidly draw an airplane. One of the tools in Openvsp allows you to determine the parasitic drag of an airplane. You can read more about this tool at https://openvsp.org/wiki/doku.php?id=parasitedrag . Part of determining the parasitic drag is determining the skin friction coefficient. We are going to use the explicit fit of Spalding and Chi to model average skin friction coefficient on one side of a flat plate with a turbulent boundary layer. The Spalding and Chi model is 𝐶𝐶 𝐷𝐷𝐷𝐷 = 0.43 [log 10 ( 𝑅𝑅𝑒𝑒 𝐿𝐿 )] 2 . 32 where the drag coefficient is based on the wetted area and the Reynolds number, 𝑅𝑅 𝑒𝑒 𝐿𝐿 , is based on the plate length. Modeling the car as a flat rectangular box, use the flat plate formula shown above to estimate the drag (in Newtons) due to skin friction at top speed. Omit the front and back surfaces from the calculation. The OpenVSP link above has an FF in the equation for drag that you can assume is equal to one. FF is a form factor which corrects the flat plate skin friction coefficients due to geometry or other effects. They are commonly used in Drag Build Up or Component Buildup methods. You can find more information about the drag build up process in “Aircraft Design” by Raymer in chapter 12, but this is beyond the scope of our problem.

Assume the basketball is a Spalding TF-150 size 7 basketball that has a circumference of 29.5 inches and weighs 1.41 pounds. The acceleration due to gravity is g = 9.81 m/s 2 . The spin rate, s, is specified in parts (c) and (d) below. Consider a coordinate system where the basketball is released at the origin, y is the vertical direction (positive upward, negative downward) and x is the horizontal direction (positive in the direction of the ball flight in the video, away from the dam). Write out equations for the x and y components of the forces acting on the basketball in terms of the x and y components of the velocity, V x and V y , and the other parameters of the problem, accounting for gravity, drag and lift. (Note: it is useful to define an angle that describes the direction of the velocity vector.) (b) [6 pts.] Write a Matlab program [ see note at end ] (or a program in another language of your choice that performs the required computations) to solve the equations of motion of the basketball (assuming the spin rate remains constant), 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 = 𝑉𝑉 𝑥𝑥 , 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 = 𝑉𝑉 𝑦𝑦 , 𝑑𝑑𝑉𝑉 𝑥𝑥 𝑑𝑑𝑑𝑑 = 1 𝑀𝑀 𝐹𝐹 𝑥𝑥 , 𝑑𝑑𝑉𝑉 𝑦𝑦 𝑑𝑑𝑑𝑑 = 1 𝑀𝑀 𝐹𝐹 𝑦𝑦 . We will do this using an explicit Euler method, where we approximate the time derivatives using 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 ≈ 𝑑𝑑 𝑛𝑛 − 𝑑𝑑 𝑛𝑛−1 𝑑𝑑 𝑛𝑛 − 𝑑𝑑 𝑛𝑛−1 = 𝑑𝑑 𝑛𝑛 − 𝑑𝑑 𝑛𝑛−1 𝑑𝑑𝑑𝑑 = 𝑅𝑅𝑅𝑅𝑅𝑅 . This allows us to advance the solution of the variable 𝑑𝑑 in time from step 𝑖𝑖 − 1 to step 𝑖𝑖 using 𝑑𝑑 𝑛𝑛 = 𝑑𝑑 𝑛𝑛−1 + 𝑑𝑑𝑑𝑑 ∗ 𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛−1 , where RHS is the right hand side of the differential equation. The initial conditions (t = 0) are given by 𝑑𝑑 1 = 𝑑𝑑 (0) = 0 , 𝑑𝑑 1 = 𝑑𝑑 (0) = 0 , 𝑉𝑉 𝑥𝑥1 = 𝑉𝑉 𝑥𝑥 (0) = 0 , 𝑉𝑉 𝑦𝑦1 = 𝑉𝑉 𝑦𝑦 (0) = 0 . The Matlab program should loop through time steps and advance the position (measured in meters) and velocity (measured in

meters per second) of the basketball. Keep the x and y positions and Vx and Vy velocity components in arrays that can be plotted. Use a while loop (see “help while” in the Matlab command window) to stop the time advancement after the basketball hits the water (when y < -126.5 m). Since the computation of the ball position will not stop precisely when the ball hits the water, include in your program the use of linear interpolation between the last two time steps to determine the values of time, the x position, and the velocity component values when the basketball hits the water. (We will count that as when the center of the basketball reaches the height y = -126.5 m, and we will not account for the non-zero radius of the basketball.) Use a constant value of the time step dt in your program. We will use values that are small enough that the accuracy constraints in parts (c) and (d) are met. Print a listing of your program. (Do this after you have used it to solve parts (c) and (d) and you are confident it works correctly – make sure you turn in the latest version of your program). (c) [3 pts.] Run your program with no spin (s = 0) and a time step dt = 1 sec. To check that the computation is accurate, rerun your program using time steps of dt = 0.1 and 0.01 sec. As you do this tabulate the values of the time step, dt , the time when the ball hits the ground, 𝑑𝑑 𝐷𝐷 , and the corresponding velocity magnitude, 𝑉𝑉 𝐷𝐷 . We want to determine the time of impact, 𝑑𝑑 𝐷𝐷 , to within 0.01 sec and the velocity at impact, 𝑉𝑉 𝐷𝐷 , to within 0.1 m/s. Based on the change in the solution as the time step is refined, do we meet the accuracy constraints? If not, then reduce the value of dt further to 0.001 sec and include these values in the table and your analysis. Make the following plots (using the value of dt needed to meet the accuracy constraints): • Height of the basketball, y, as a function of time, t. • Velocity magnitude as a function of time. Describe the variation of the height with time. Describe variation of the velocity with time. Does the velocity approach a constant value? Does it behave monotonically? (d) [7 pts.] Run your program using a rotation rate of s = 2 rev/sec and time steps dt of 1.0, 0.1 and 0.01 sec, as in part (c). As you do this tabulate the values of the time step, dt , the time when the ball hits the ground, 𝑑𝑑 𝐷𝐷 , and the corresponding velocity magnitude, 𝑉𝑉 𝐷𝐷 and x location, 𝑑𝑑 𝐷𝐷 . We want to determine the time of impact, 𝑑𝑑 𝐷𝐷 , to within 0.01 sec, the velocity at impact, 𝑉𝑉 𝐷𝐷 , to within 0.1 m/s, and the x location, 𝑑𝑑 𝐷𝐷 , to within 0.5 m. Based on the change in the solution as the time step is refined, do we meet the accuracy constraints? If not, then reduce the value of dt further to 0.001 sec and include these values in the table and your analysis. Make the following plots (using the value of dt needed to meet the accuracy constraints): • Height of the basketball, y, as a function of time, t. • The horizontal position of the basketball, x, as a function of time, t. • Trajectory of the basketball, y as a function of x. • The vertical component of velocity, 𝑉𝑉 𝑦𝑦 , as a function of time, t. • The horizontal component of velocity, 𝑉𝑉 𝑥𝑥 , as a function of time, t. • Velocity magnitude as a function of time.