1-Plot θ-β-M∞ for flow over a cone using singularity...

1-Plot $\theta-\beta-M_{\infty}$ for flow over a cone using singularity distribution method 2-Cosider an axisymmetric aerodynamic body with profile of ($R(x) = \frac{x}{5} - \frac{x^2}{50}$ 0<x<5) exposed to a supersonic flow with $M_{\infty} = 2$. Consider the control points at x=1, 2,3,4,5 and solve the flow field over the body using Karman-Moor method 1- Plot Cp, vs. x 2- Calculate drag coefficient 3-Show that the solution for flow over a slender axially symmetric body of profile section Gives: $R = \frac{2t}{L^2}x(L-x)$ $u = \frac{\partial \phi}{\partial x} = -\frac{4Ut^2}{L^2} \left[ 2x(L-3x) + (L^2 - 6Lx + 6x^2)ln\frac{2x^2}{R_A} \right]$

Transcript

00:02 For this question, we are given a vector field f, which is x, cubed, y, cubed, and z cubed.

00:07 We're told that a region w will lie between spheres of radius 1 and 2 and within the positive cone.

00:14 And this region w has a boundary, which is the surface s.

00:20 This is a closed surface, and our goal is to find the flux of our vector field through the surface.

00:25 Now, because the surface is closed, we can use divergence theorem, which states that our flux instead is equal to the triple integral over the region w of our divergence of the vector field.

00:49 Now, the reason this is preferable to calculating the surface integral directly is because with the way they're describing the surface, you would need to do three separate flux integrals, which can be a pain.

01:03 Instead, we can do a single triple integral.

01:06 And not only that, but the way that the surface is defined really lends itself over to spherical coordinates, which should make this divergence integral easy to calculate.

01:18 So let's give it a shot.

01:20 First things first, let's go ahead and convert into spherical coordinates for our surfaces.

01:29 Now, with that said, spherical coordinates, the way that it works, in particular, x squared plus y squared plus z squared, is equal to our variable row, the radius squared.

01:44 And also, the differential of volume is going to be rewritten as row squared sine fee.

01:56 And then we'll have our d theta, d fee, and d row.

02:01 Okay, so this is our jacobian that needs to be inserted into our integral.

02:07 All right.

02:08 So with that said, let's also calculate the divergence of our vector field.

02:18 Notice that we'll be adding the x, y, and z partials of the x, y, and z coordinates, which turns out to be 3x squared plus 3y squared plus 3 z squared.

02:34 And again, you can see why spherical coordinates are beneficial in this case, because when you pull out the three, you end up getting this sum of squares, which perfectly equals row squared based on our conversion to spherical coordinates.

02:57 All right, so let's give this a shot.

02:59 So we will have our triple integral over w of the divergence times the jacobian of row squared, sign fee, and then we will have our three differentials.

03:25 So this is the integral that we need to solve for.

03:27 The one thing we're missing are going to be the integration bounds.

03:31 So those we can pull relatively quickly from the description.

03:36 We are lying between two spheres with radius one and two.

03:40 So there will be the radius bounds.

03:43 We'll say the radius row is between one and two.

03:49 Let's talk about our theta angle, the rotation about the positive z axis.

03:55 There is nothing that is restricting the bounds.

03:58 For theta.

03:59 So you should expect theta should be between 0 and 2 pi.

04:07 And then we have the fee angle.

04:10 The fee angle is our angle between the surface and the positive z axis.

04:17 So this angle here is fee...

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