Flow Deflection Angle due to Expansion Wave Calculator

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Compressible Flow

Fundamentals of Inviscid and Incompressible Flow

Three-Dimensional Incompressible Flow

Two-Dimensional Incompressible Flow

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Oblique Shock and Expansion Waves

Governing Equations and Sound Wave

Normal Shock Wave

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Expansion Waves

Oblique Shock

✖The Specific Heat Ratio Expansion Wave is the ratio of the heat capacity at constant pressure to heat capacity at constant volume.ⓘ Specific Heat Ratio Expansion Wave [γ_e]			+10% -10%
✖Mach Number Behind Expansion Fan is the Mach number of downstream flow across the expansion fan.ⓘ Mach Number Behind Expansion Fan [M_e2]			+10% -10%
✖Mach Number Ahead of Expansion Fan is the Mach number of the upstream flow.ⓘ Mach Number Ahead of Expansion Fan [M_e1]			+10% -10%

✖Flow Deflection Angle is defined as the angle by which the flow turns through the expansion wave.ⓘ Flow Deflection Angle due to Expansion Wave [θ_e]

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Formula

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Flow Deflection Angle due to Expansion Wave

Formula

`"θ"_{"e"} = (sqrt(("γ"_{"e"}+1)/("γ"_{"e"}-1))*atan(sqrt((("γ"_{"e"}-1)*("M"_{"e2"}^2-1))/("γ"_{"e"}+1)))-atan(sqrt("M"_{"e2"}^2-1)))-(sqrt(("γ"_{"e"}+1)/("γ"_{"e"}-1))*atan(sqrt((("γ"_{"e"}-1)*("M"_{"e1"}^2-1))/("γ"_{"e"}+1)))-atan(sqrt("M"_{"e1"}^2-1)))`

Example

`"7.866893°"=(sqrt(("1.41"+1)/("1.41"-1))*atan(sqrt((("1.41"-1)*(("6")^2-1))/("1.41"+1)))-atan(sqrt(("6")^2-1)))-(sqrt(("1.41"+1)/("1.41"-1))*atan(sqrt((("1.41"-1)*(("5")^2-1))/("1.41"+1)))-atan(sqrt(("5")^2-1)))`

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Flow Deflection Angle due to Expansion Wave Solution

STEP 0: Pre-Calculation Summary

Formula Used

Flow Deflection Angle = (sqrt((Specific Heat Ratio Expansion Wave+1)/(Specific Heat Ratio Expansion Wave-1))*atan(sqrt(((Specific Heat Ratio Expansion Wave-1)*(Mach Number Behind Expansion Fan^2-1))/(Specific Heat Ratio Expansion Wave+1)))-atan(sqrt(Mach Number Behind Expansion Fan^2-1)))-(sqrt((Specific Heat Ratio Expansion Wave+1)/(Specific Heat Ratio Expansion Wave-1))*atan(sqrt(((Specific Heat Ratio Expansion Wave-1)*(Mach Number Ahead of Expansion Fan^2-1))/(Specific Heat Ratio Expansion Wave+1)))-atan(sqrt(Mach Number Ahead of Expansion Fan^2-1)))
θ_e = (sqrt((γ_e+1)/(γ_e-1))*atan(sqrt(((γ_e-1)*(M_e2^2-1))/(γ_e+1)))-atan(sqrt(M_e2^2-1)))-(sqrt((γ_e+1)/(γ_e-1))*atan(sqrt(((γ_e-1)*(M_e1^2-1))/(γ_e+1)))-atan(sqrt(M_e1^2-1)))
This formula uses 3 Functions, 4 Variables

Functions Used

tan - The tangent of an angle is a trigonometric ratio of the length of the side opposite an angle to the length of the side adjacent to an angle in a right triangle., tan(Angle)
atan - Inverse tan is used to calculate the angle by applying the tangent ratio of the angle, which is the opposite side divided by the adjacent side of the right triangle., atan(Number)
sqrt - A square root function is a function that takes a non-negative number as an input and returns the square root of the given input number., sqrt(Number)

Variables Used

Flow Deflection Angle - (Measured in Radian) - Flow Deflection Angle is defined as the angle by which the flow turns through the expansion wave.
Specific Heat Ratio Expansion Wave - The Specific Heat Ratio Expansion Wave is the ratio of the heat capacity at constant pressure to heat capacity at constant volume.
Mach Number Behind Expansion Fan - Mach Number Behind Expansion Fan is the Mach number of downstream flow across the expansion fan.
Mach Number Ahead of Expansion Fan - Mach Number Ahead of Expansion Fan is the Mach number of the upstream flow.

STEP 1: Convert Input(s) to Base Unit

Specific Heat Ratio Expansion Wave: 1.41 --> No Conversion Required
Mach Number Behind Expansion Fan: 6 --> No Conversion Required
Mach Number Ahead of Expansion Fan: 5 --> No Conversion Required

STEP 2: Evaluate Formula

Substituting Input Values in Formula

θ_e = (sqrt((γ_e+1)/(γ_e-1))*atan(sqrt(((γ_e-1)*(M_e2^2-1))/(γ_e+1)))-atan(sqrt(M_e2^2-1)))-(sqrt((γ_e+1)/(γ_e-1))*atan(sqrt(((γ_e-1)*(M_e1^2-1))/(γ_e+1)))-atan(sqrt(M_e1^2-1))) --> (sqrt((1.41+1)/(1.41-1))*atan(sqrt(((1.41-1)*(6^2-1))/(1.41+1)))-atan(sqrt(6^2-1)))-(sqrt((1.41+1)/(1.41-1))*atan(sqrt(((1.41-1)*(5^2-1))/(1.41+1)))-atan(sqrt(5^2-1)))

Evaluating ... ...

θ_e = 0.137303184990648

STEP 3: Convert Result to Output's Unit

0.137303184990648 Radian -->7.86689301366959 Degree (Check conversion here)

FINAL ANSWER

7.86689301366959 ≈ 7.866893 Degree <-- Flow Deflection Angle

(Calculation completed in 00.004 seconds)

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Home » Physics » Aerodynamics » Compressible Flow » Oblique Shock and Expansion Waves » Expansion Waves » Flow Deflection Angle due to Expansion Wave

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Created by Harsh Raj

Indian Institute of Technology, Kharagpur (IIT KGP), West Bengal

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< 10+ Expansion Waves Calculators

Flow Deflection Angle due to Expansion Wave

Go Flow Deflection Angle = (sqrt((Specific Heat Ratio Expansion Wave+1)/(Specific Heat Ratio Expansion Wave-1))*atan(sqrt(((Specific Heat Ratio Expansion Wave-1)*(Mach Number Behind Expansion Fan^2-1))/(Specific Heat Ratio Expansion Wave+1)))-atan(sqrt(Mach Number Behind Expansion Fan^2-1)))-(sqrt((Specific Heat Ratio Expansion Wave+1)/(Specific Heat Ratio Expansion Wave-1))*atan(sqrt(((Specific Heat Ratio Expansion Wave-1)*(Mach Number Ahead of Expansion Fan^2-1))/(Specific Heat Ratio Expansion Wave+1)))-atan(sqrt(Mach Number Ahead of Expansion Fan^2-1)))

Prandtl Meyer Function at Upstream Mach Number

Go Prandtl Meyer Function at Upstream Mach no. = sqrt((Specific Heat Ratio Expansion Wave+1)/(Specific Heat Ratio Expansion Wave-1))*atan(sqrt(((Specific Heat Ratio Expansion Wave-1)*(Mach Number Ahead of Expansion Fan^2-1))/(Specific Heat Ratio Expansion Wave+1)))-atan(sqrt(Mach Number Ahead of Expansion Fan^2-1))

Prandtl Meyer Function

Go Prandtl Meyer Function = sqrt((Specific Heat Ratio Expansion Wave+1)/(Specific Heat Ratio Expansion Wave-1))*atan(sqrt(((Specific Heat Ratio Expansion Wave-1)*(Mach Number^2-1))/(Specific Heat Ratio Expansion Wave+1)))-atan(sqrt(Mach Number^2-1))

Pressure behind Expansion Fan

Go Pressure Behind Expansion Fan = Pressure Ahead of Expansion Fan*((1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Ahead of Expansion Fan^2)/(1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Behind Expansion Fan^2))^((Specific Heat Ratio Expansion Wave)/(Specific Heat Ratio Expansion Wave-1))

Pressure Ratio across Expansion Fan

Go Pressure Ratio Across Expansion Fan = ((1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Ahead of Expansion Fan^2)/(1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Behind Expansion Fan^2))^((Specific Heat Ratio Expansion Wave)/(Specific Heat Ratio Expansion Wave-1))

Temperature behind Expansion Fan

Go Temperature Behind Expansion Fan = Temperature Ahead of Expansion Fan*((1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Ahead of Expansion Fan^2)/(1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Behind Expansion Fan^2))

Temperature Ratio across Expansion Fan

Go Temperature Ratio Across Expansion Fan = (1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Ahead of Expansion Fan^2)/(1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Behind Expansion Fan^2)

Flow Deflection Angle using Prandtl Meyer Function

Go Flow Deflection Angle = Prandtl Meyer Function at Downstream Mach no.-Prandtl Meyer Function at Upstream Mach no.

Forward Mach Angle of Expansion Fan

Go Forward Mach Angle = arsin(1/Mach Number Ahead of Expansion Fan)

Rearward Mach Angle of Expansion Fan

Go Rearward Mach Angle = arsin(1/Mach Number Behind Expansion Fan)

Flow Deflection Angle due to Expansion Wave Formula

What is expansion wave?

When a supersonic flow is turned away from itself an expansion wave is formed. The fan continuously opens in the direction away from the corner. The expansion wave is in the shape of a fan centered at the corner.

How to Calculate Flow Deflection Angle due to Expansion Wave?

Flow Deflection Angle due to Expansion Wave calculator uses Flow Deflection Angle = (sqrt((Specific Heat Ratio Expansion Wave+1)/(Specific Heat Ratio Expansion Wave-1))*atan(sqrt(((Specific Heat Ratio Expansion Wave-1)*(Mach Number Behind Expansion Fan^2-1))/(Specific Heat Ratio Expansion Wave+1)))-atan(sqrt(Mach Number Behind Expansion Fan^2-1)))-(sqrt((Specific Heat Ratio Expansion Wave+1)/(Specific Heat Ratio Expansion Wave-1))*atan(sqrt(((Specific Heat Ratio Expansion Wave-1)*(Mach Number Ahead of Expansion Fan^2-1))/(Specific Heat Ratio Expansion Wave+1)))-atan(sqrt(Mach Number Ahead of Expansion Fan^2-1))) to calculate the Flow Deflection Angle, The Flow Deflection Angle due to Expansion Wave formula is defined as difference of Prandtl Meyer Functional values at upstream and downstream of expansions wave. Flow Deflection Angle is denoted by θ_e symbol.

How to calculate Flow Deflection Angle due to Expansion Wave using this online calculator? To use this online calculator for Flow Deflection Angle due to Expansion Wave, enter Specific Heat Ratio Expansion Wave (γ_e), Mach Number Behind Expansion Fan (M_e2) & Mach Number Ahead of Expansion Fan (M_e1) and hit the calculate button. Here is how the Flow Deflection Angle due to Expansion Wave calculation can be explained with given input values -> 450.7398 = (sqrt((1.41+1)/(1.41-1))*atan(sqrt(((1.41-1)*(6^2-1))/(1.41+1)))-atan(sqrt(6^2-1)))-(sqrt((1.41+1)/(1.41-1))*atan(sqrt(((1.41-1)*(5^2-1))/(1.41+1)))-atan(sqrt(5^2-1))).

FAQ

What is Flow Deflection Angle due to Expansion Wave?

The Flow Deflection Angle due to Expansion Wave formula is defined as difference of Prandtl Meyer Functional values at upstream and downstream of expansions wave and is represented as θ_e = (sqrt((γ_e+1)/(γ_e-1))*atan(sqrt(((γ_e-1)*(M_e2^2-1))/(γ_e+1)))-atan(sqrt(M_e2^2-1)))-(sqrt((γ_e+1)/(γ_e-1))*atan(sqrt(((γ_e-1)*(M_e1^2-1))/(γ_e+1)))-atan(sqrt(M_e1^2-1))) or Flow Deflection Angle = (sqrt((Specific Heat Ratio Expansion Wave+1)/(Specific Heat Ratio Expansion Wave-1))*atan(sqrt(((Specific Heat Ratio Expansion Wave-1)*(Mach Number Behind Expansion Fan^2-1))/(Specific Heat Ratio Expansion Wave+1)))-atan(sqrt(Mach Number Behind Expansion Fan^2-1)))-(sqrt((Specific Heat Ratio Expansion Wave+1)/(Specific Heat Ratio Expansion Wave-1))*atan(sqrt(((Specific Heat Ratio Expansion Wave-1)*(Mach Number Ahead of Expansion Fan^2-1))/(Specific Heat Ratio Expansion Wave+1)))-atan(sqrt(Mach Number Ahead of Expansion Fan^2-1))). The Specific Heat Ratio Expansion Wave is the ratio of the heat capacity at constant pressure to heat capacity at constant volume, Mach Number Behind Expansion Fan is the Mach number of downstream flow across the expansion fan & Mach Number Ahead of Expansion Fan is the Mach number of the upstream flow.

How to calculate Flow Deflection Angle due to Expansion Wave?

The Flow Deflection Angle due to Expansion Wave formula is defined as difference of Prandtl Meyer Functional values at upstream and downstream of expansions wave is calculated using Flow Deflection Angle = (sqrt((Specific Heat Ratio Expansion Wave+1)/(Specific Heat Ratio Expansion Wave-1))*atan(sqrt(((Specific Heat Ratio Expansion Wave-1)*(Mach Number Behind Expansion Fan^2-1))/(Specific Heat Ratio Expansion Wave+1)))-atan(sqrt(Mach Number Behind Expansion Fan^2-1)))-(sqrt((Specific Heat Ratio Expansion Wave+1)/(Specific Heat Ratio Expansion Wave-1))*atan(sqrt(((Specific Heat Ratio Expansion Wave-1)*(Mach Number Ahead of Expansion Fan^2-1))/(Specific Heat Ratio Expansion Wave+1)))-atan(sqrt(Mach Number Ahead of Expansion Fan^2-1))). To calculate Flow Deflection Angle due to Expansion Wave, you need Specific Heat Ratio Expansion Wave (γ_e), Mach Number Behind Expansion Fan (M_e2) & Mach Number Ahead of Expansion Fan (M_e1). With our tool, you need to enter the respective value for Specific Heat Ratio Expansion Wave, Mach Number Behind Expansion Fan & Mach Number Ahead of Expansion Fan and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.

How many ways are there to calculate Flow Deflection Angle?

In this formula, Flow Deflection Angle uses Specific Heat Ratio Expansion Wave, Mach Number Behind Expansion Fan & Mach Number Ahead of Expansion Fan. We can use 1 other way(s) to calculate the same, which is/are as follows -

Flow Deflection Angle = Prandtl Meyer Function at Downstream Mach no.-Prandtl Meyer Function at Upstream Mach no.

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