HW4Fall2023

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University of Illinois, Urbana Champaign *

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311

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Aerospace Engineering

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Oct 30, 2023

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pdf

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Fall 2023 AE 311 Homework 4 Due Friday October 20, by the end of the day For all problems in this Homework except for Problem 1 sketches are not only suggested but required. You do not need to re-derive all the expressions seen in class, you can simply use them. This is, you know everything about the elementary flow solutions and some of specific superposition cases that we have seen/introduced in class. 1. Rankine Oval A line source and sink with strengths of equal magnitude | Λ | in m 2 /s (volumetric flow rate per unit depth), are placed on the x-axis at x = -a and x = +a, respectively. A uniform flow, with speed V ∞ in the positive x-direction is added to obtain the flow past a Rankine oval. (a) Obtain the stream function, velocity potential and velocity field (velocity components indicating their direction, conceptual sketch, and coordinate system is a must, as usual) for the combined flow. Sketch conceptually the lines of constant streamfunction. Note: While you do not need to re-derive all theory seen in class, you need to show you understand how it is done. You can consider the elementary flow solutions known and show the derivation from there on. (b) Find the value (mathematically) of ψ = constant on the stagnation streamline. Identify the stagnation streamline in your sketch, the stagnation points and provide and algebraic solution for the x sp , y sp , coordinates of the stagnation point/s. (c) Assign two sets of values for | Λ | , V ∞ , and a of your choice and using Matlab or Python plot the streamlines or lines of constant streamfunction. The expected result for this section are two separate plots for different values of the constants indicated above. As usual, provide your code and the chosen values for the constants with corresponding units. (d) For | Λ | = 4 π , V ∞ = 30 m/s, and a = 0.3 m, obtain the equation that provides the geometry of the resulting body. Note: this is an equation that relates x and y along the dividing streamline. (e) Obtain an expression for the pressure distribution along the surface of the body, this is, valid for all points along the surface. Using that expression, plot the pressure profile qualitatively along the x b coordinate for the upper part of the oval, where x b is defined parallel to the x axis with x b = 0 at the forward stagnation point. ”Qualitatively” (different from quantitatively, see next) means that you are not asked to plot the exact p ( x b ) distribution, simply by hand sketch its trend from the forward stagnation point to the backward stagnation point, this is from the Leading Edge to the Trailing Edge in this case. (f) ( For extra points :) Solve for the pressure at the different points along the surface and plot the quantitative distribution as a function of the x b coordinate using Matlab or Python. Search online or on books for how the pressure distribution around a symmetric airfoil at 0deg angle of attack looks like and comment on similarities and/or differences. 1

2. 3-hole probe Air at standard atmospheric conditions and a velocity of 65 m/s approaches a cylinder (say, the streamlined strut of a general aviation aircraft). (a) Using potential flow theory, find the velocity magnitude and pressure at the following points at the cylinder surface: a) the leading and trailing stagnation points, b) the cylinder shoulder, this is half way between the leading and trailing stagnation points, c) and halfway between the shoulder and the front and back trailing stagnation points. (b) Three orifices drilled normal to the surface in the cylindrical body of the previous section are used as a flow direction indicator, as shown in the picture below. The flow direction indicator works in the following way: when the pressures on the two side holes are equal, the center hole (half way between the side holes) will point directly into the flow, and the pressure at that central hole will be the stagnation pressure. This is indeed the principle of operation of an instrument called a cylindrical yaw probe, as it serves to measure the yaw angle. If the side orifices of a yaw probe were located such that they measure the free-stream static pres- sure, at what angle ( α in the sketch) would they have to be located? (demonstrate mathematically that the pressure at those ports is exactly p ∞ ). Assume incompressible, potential flow. (c) Find the sensitivity of the surface pressure with respect to change in angle, ∂p/∂θ , at the locations of the side orifices if they are at the position calculated in section (b). Note: this is very important to know which precission you need when drilling the holes. 2

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