Solving for shock wave angle using Mach and deflection angle

In summary, the conversation revolves around trying to determine the shock wave angle, Beta, while keeping the deflection angle constant and increasing the Mach number. The equation being used is implicit and an approximate relationship between Beta and theta is provided for large Mach numbers. The use of graphical solutions is suggested, but the focus is on finding a method for solving the equation for different Mach numbers.
  • #1

twmggc

5
0
I am trying to show how the shock wave angle varies as I hold the deflection angle constant and increase the Mach number. I am trying to solve for Beta (shock wave angle) using the theta-beta-mach relation:

tan(θ)= 2cot(β) * (M2sin2(β)-1) / (M2(1.4+cos(2β))+2)

This seems like it should be a simple problem, but I can't seem to figure it out.
Note, I am using constant specific heats hence the 1.4 in the equation.

Any hints?
 
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  • #2
If I remember my gas dynamics correctly that equation is implicit with respect to beta.
However, "Elements of Gas Dynamics", by Liepmann and Rosko give an approximate relationship between β and θ when the mach number is large:

[itex]\beta ≈ (\gamma +1)/2 * \theta [/itex]

L&R provide a graphical solution to the weak and strong shocks on page 87. This is probably the best method to show the relationship between deflection angle and shock angle. Also, they note that for a specific mach number there is a maximum possible deflection angle, θ.
 
  • #3
Thank you for the response, but since I am attempting to model the change in Beta with constant deflection angle the Liepmann and Rosko equation won't help me too much. As far as the graphical solutions go, I have an anderson book with theta-beta-mach relations plotted but they don't go to into detail for higher mach numbers.
 
  • #4
It's just an implicit equation. Plug it into fsolve or something similar in Matlab. You can't solve it explicitly exactly.
 
  • #5
Yep, gogo Newton's method. Gotta love Matlab.
 
Last edited:

What is the formula for calculating shock wave angle using Mach and deflection angle?

The formula for calculating shock wave angle is given by sin(β) = (M^2 sin(α - β))/[1 + (M^2 (γ-1))/2], where β is the shock wave angle, α is the deflection angle, M is the Mach number, and γ is the ratio of specific heats.

How do Mach and deflection angle affect the shock wave angle?

The shock wave angle is directly proportional to the deflection angle and inversely proportional to the Mach number. This means that as the deflection angle increases, the shock wave angle also increases, and as the Mach number increases, the shock wave angle decreases.

What is the significance of calculating shock wave angle?

Calculating shock wave angle is important in understanding and predicting the behavior of shock waves in supersonic flow. It helps in designing and analyzing aircraft and other high-speed vehicles, as well as in studying the effects of shock waves on structures and materials.

Can the formula for shock wave angle be used for all types of shock waves?

The formula for shock wave angle is specifically designed for calculating the oblique shock wave angle in a compressible flow field. It cannot be used for other types of shock waves, such as normal shock waves, which have their own unique equations.

What are some real-world applications of solving for shock wave angle using Mach and deflection angle?

The calculation of shock wave angle is used in various fields, such as aerodynamics, aerospace engineering, and high-speed vehicle design. It is also important in understanding the behavior of shock waves in natural phenomena, such as thunder and lightning, and in industrial processes involving supersonic flows.

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