Nightingale College Therapy Solve Using Bracketing Method Discussion

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King Abdulaziz University
Aero. Eng. Dept.
AE-419 Computational Fluid Dynamics
Summer 2022
First Assignment: Root-Finding
Figure 1. Shock angle, β, as function of the deflection angle, θ, and upstream Mach
number M1 .
Problem Statement:
Reproduce figure 1 for oblique shock waves by solving the following relations:


M12 sin2 β − 1
tan θ = 2 cot (β) 2
M1 (γ + cos 2β) + 2
1
M2 =
sin (β − θ)
v
t
1 + γ−1
M12 sin2 β
2
γ M12 sin2 β − γ−1
2
θ : defliction angle
β : shock angle
M1 : upstream Mach number
M2 : down stream Mach number
Figure 2. Schematic for inward deflected supersonic flow
Tasks:
1. Methods’ Performance:
Compare the rate of convergence of Bisection method and the Secant method. To do so:
1. Select a point (M1 ,θ) and compute β using both methods.
2. Record and plot the convergence history for both methods. Use a tolerance  = 1 × 10−7
2. Generating the Chart:
Select the method with faster convergence and produce a chart similar to figure 1 for the following range of values:
1. The upstream Mach number, M1 , varies from 1.1 to 3.0 with step 0.2.
2. The deflection angle, θ, starts from 0.0◦ with step 0.1◦