Compressible Flow Calculator
High Speed Aerodynamics
Name:
Prajapati Neel
Roll No.: 23B0002
Instructor: Professor Vineeth Nair
Indian Institute of Technology Bombay
Aerospace Engineering Department
September 29, 2025
Compressible Flow Report
High Speed Aerodynamics
Contents
1 Isentropic Flow Calculator
2
2 Normal Shock Flow Calculator
3
3 Oblique Shock Flow Calculator
4
Abstract
The primary goal of this project is to apply the concepts of isentropic flows, normal
shocks, and oblique shocks to develop a functional Gas Table Calculator. This tool aims
to simplify the calculation of flow parameters, facilitating quick and accurate engineering
analysis.
Objectives
• Formulate equations to determine various flow parameters given a single known
quantity in isentropic flow, normal shock, and oblique shock scenarios.
• Implement these equations in code to create a practical and user-friendly Gas Table
Calculator.
Libraries Used
1. math: Provides basic mathematical functions and constants. Used for trigonometric
conversions, square roots, and powers in compressible flow and shock calculations.
2. numpy: Handles numerical computations and arrays efficiently, such as vectorized operations, linearly spaced arrays, and calculations in θ–β–M relations and Mach numbers.
3. ipywidgets: Creates interactive GUI elements in Jupyter notebooks, enabling input
of Mach number, turn angle, wave angle, etc., and dynamic display of computed results.
4. scipy.optimize: Solves non-linear equations numerically. Functions like fsolve find
Mach numbers from area ratios and Prandtl-Meyer angles, while brentq and bisect find
weak and strong oblique shock angles.
5. IPython.display: Renders widgets and outputs cleanly in Jupyter notebooks.
These libraries together enable computation, numerical solving, and interactive visualization for compressible flow, normal shock, and oblique shock analysis.
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Compressible Flow Report
1
High Speed Aerodynamics
Isentropic Flow Calculator
The following relations are used to determine isentropic flow properties:
p
=
pt
ρ
ρt
γ
=
T
Tt
γ
γ−1
(1)
− γ
γ − 1 2 γ−1
1+
M
2
−1
γ−1 2
T
= 1+
M
Tt
2
− 1
ρ
γ − 1 2 γ−1
= 1+
M
ρt
2
γ+1
2(γ−1)
A
2
γ−1 2
1
1
+
M
=
A∗
M γ+1
2
p
= 0.528
p∗
ρ
= 0.634
ρ∗
T
= 0.833
T∗
p
=
pt
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Here, M is the Mach number, γ is the ratio of specific heats, p, ρ, T are static properties,
pt , ρt , Tt are stagnation conditions where v = 0 and starred quantities(∗ ) denote throat
conditions where M = 1.
Key Observations
• ppt < 1: Static pressure is always less than stagnation pressure.
• ρρt < 1: Static density is always less than stagnation density.
• TTt < 1: Static temperature is always less than stagnation temperature.
• AA∗ :
– > 1 for M < 1 (subsonic regime),
– = 1 for M = 1 (sonic condition),
– < 1 for M > 1 (supersonic regime).
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Compressible Flow Report
2
High Speed Aerodynamics
Normal Shock Flow Calculator
For a normal shock, the governing relations are:
s
M2 =
M12 (γ − 1) + 2
2γM12 − (γ − 1)
(9)
p2
2γM12 − (γ − 1)
=
p1
γ+1
ρ2
(γ + 1)M12
=
ρ1
(γ − 1)M12 + 2
2γ 2
γ−1
2
1
+
−
1
M
M
1
1
2
γ−1
T2
=
T1
M 2 2γ + γ−1
1
p02
=
p01
γ−1
(γ + 1)M12
2 + (γ − 1)M12
(10)
(11)
(12)
2
γ γ−1
γ+1
2
2γM1 − (γ − 1)
1
γ−1
(13)
Here M1 is the upstream Mach number, M2 is the downstream Mach number, p1 ,ρ1
and T1 are the upstream conditions, while p2 ,ρ2 and T2 are the downstream conditions
and p01 ,p02 are the stagnation pressures.
Key Observations
• M1 > 1: Shock occurs only in supersonic flow.
• M2 < 1: Post-shock Mach number is subsonic.
• pp12 > 1: Pressure increases across the shock.
• ρρ12 > 1: Density increases across the shock.
• TT12 > 1: Temperature increases across the shock.
02
• pp01
< 1: Stagnation pressure decreases across the shock.
3
Compressible Flow Report
3
High Speed Aerodynamics
Oblique Shock Flow Calculator
In the case of an oblique shock, the following relations apply:
Mn1 = M sin β
(14)
2
2
+ 2)
− (γ − 1))((γ − 1)Mn1
T2
(2γMn1
=
2
T1
((γ + 1)Mn1 )
2
p2
2γMn1 − (γ − 1)
=
p1
γ+1
2
(γ + 1)Mn1
ρ2
=
2
+2
ρ1
(γ − 1)Mn1
γ 1
2
γ−1
γ−1
pt2
(γ + 1)Mn1
γ+1
=
2
2
pt1
(γ − 1)Mn1
+2
2γMn1
− (γ − 1)
Mn1
β = arcsin
M
(15)
(16)
(17)
(18)
(19)
Where M is the upstream Mach number, p1 ,ρ1 and T1 are the upstream conditions,
p2 ,ρ2 and T2 are the downstream conditions, pt1 , pt2 are the stagnation pressures, Mn1 is
the normal upstream Mach number and β is the shock wave angle
Key Observations
• M > 1: Oblique shocks occur only in supersonic flows.
• Mn1 ≤ M : Normal Mach number is always ≤ upstream Mach number.
• pp21 > 1: Pressure rises after the shock.
• ρρ12 > 1: Density rises after the shock.
• TT12 > 1: Temperature rises after the shock.
t2
• ppt1
< 1: Stagnation pressure decreases after the shock.
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Compressible Flow Report
High Speed Aerodynamics
Conclusion
The compressible flow relations summarized above are central in analyzing high-speed
aerodynamic phenomena. By comparing isentropic, normal shock, and oblique shock
behaviors, one can predict pressure, density, and temperature variations, which are crucial
in the design of nozzles, diffusers, and supersonic aircraft components.
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