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properties of the conical flow generated by a right circular cone flying at a supersonic Mach number at zero pitch and zero yaw through an inviscid perfect gas. The cone has a sharp tip (vertex). Numerical Approach: To accomplish this, numerically integrate the Taylor-Maccoll (T-M) equation, which is a second order and very nonlinear ODE. Do so by first converting this second order equation into two first-order ODEs. Solve these two by the use of the powerful and popular algorithm: 4th-order (Classical) Runge-Kutta method. Use the "inverse" approach described in Anderson (Modern Compressible Flow, 2021, Section 10.4, pp. 373-374) by starting at the given conical shock wave angle 6, and marching in the negative -direction, in small increments of A from the shock towards the cone. O is the spherical angular variable, measured from the cone axis up to an arbitrary ray emanating from the cone vertex. Recall that "conical flow" is just the flow between the shock and the cone body, is fully isentropic, and is characterized by the remarkable characteristic that all flow properties are constant along each ray, though they vary from ray to ray. Of course, the free stream flow is isentropic ahead of the shock. The only region of non-isentropic flow is across the conical shock wave, which like all other shocks is adiabatic. Thus, (using Anderson's notation) T = constant everywhere, but po will drop across the shock, and will remain constant behind the shock. Also, note that, although, the shock wave is 3-D, the shock is locally planar, and thus locally can be treated by the use of 2-D oblique shock theory. Governing Equations: In Anderson (2021), T-M equation is Eq. 10.13, but instead work with the more convenient non-dimensional version of it in Eq. 10.15. You will also be using Eq. 10.14 to find the 0-component of velocity V. The latter will look the same in non-dimensional form, except for adding a prime on each side. Once you have found the two components of velocity, V, and V, you can combine them to find the magnitude of the local velocity vector V. From the latter, you can now find the local Mach number M from Eq. 10.16, which you will need for isentropic flow calculations in the conical flow field. For convenience, drop the primes from these three equations, i.e., Eqs. 10.14, 10.15 and 10.16. Notice a small misprint in Anderson p. 374, Fig 10.4: V-component of velocity just downstream of the shock should be parallel to and just downstream of the shock, not as shown. Notation: Use that of Anderson (2021): • Station 1 or ∞ is the free stream 8 • Station 2 is right behind the shock • Station c is at the surface of the cone • Subscript o is stagnation values • Cone angle is • Conical shock angle is ẞ or • Initial turning of the streamline as the flow crosses the shock from 1 to 2 is 8 which is <0 Case Study: 1. Run your program for: M₁ = 6 y=1.4 2 0. = 12° A0=-0.1° (this is small enough to give you good accuracy, we are marching in the negative direction, thus a negative sign on the step size). Find that produces this shock wave and the corresponding M. 2. Once your program is working, we will calculate and extract more data from it: calculate non-dimensional flow properties on each ray and plot as a function of Programming: See the attachment for the governing equations and the 4th-order Runge-Kutta (R-K) algorithm. In your computer program, use all the same notations as in the attachment. Structure your computer program with separate modules and each one should start with one or more comment statement(s) that describe what is being done in that module and also define your notation for the module. Note that you will need the free stream velocity nondimensionalized by Vmax, which is obtained from Eq. 10.16, applied to the free stream. You will see that the answer is not 1, but less than 1. You will need this value to compute your initial conditions at Station 2 to start your R-K computations. Hint: The final answers for 0, Cpc, and M can be read from Charts 5, 6, and 7 in NACA 1135, good checks on the veracity of the core of your program. 3 ARO 3111 Computer Assignment Part II Conical Flow: Supersonic Flow Past Un-Yawed Cone Now that your computer program is working, we will add some gas dynamics to it and extract some of the flow properties in the conical flow region, between the shock and the cone. Again, our Case Study is: M₁ = 6 y=1.4 0. = 12° A0=-0.1° You already found the values of 0 and M. for this flow from your two plots. Now, continue your work as follows: 1. Use locally planar oblique shock relations to obtain from flow properties at Station 1 (the free stream), flow properties at Station 2 (right behind the shock), as you did in Problem 10.1 of Anderson (2021) in PS9. Do your work in non-dimensional form, by hand calculation, with 4 significant figures, as well as by incorporating into your program, to find: M2, P2 P1, P2 P₁, T₁/T₁. They should agree completely. This also helps debug the early part of your program. 2. Now, recalling that flow from Station 2 to Station c is isentropic, use your previously developed program to compute and plot, in one color graph, local values of these properties as a function of the spherical variable 0, i.e., M, p/p₁, p/p₁, T/T. Theta will now go from the above to O, that is, an inverted scale. Add the results in (1) to complete your plot for theta between and 45°, to fully see what happens to the fluid particle as it approaches the cone. 3. Add one table of all three plots vs 0 in increments of one degree. 4. You will present your work in a report.