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Flow Deflection Angle due to Expansion Wave Formula

GoFlow Deflection Angle using Prandtl Meyer Function Formula

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Flow Deflection Angle is defined as the angle by which the flow turns through the expansion wave. Check FAQs

θ e = (\sqrt{\frac{γ e + 1}{γ e - 1}} \cdot a \tan (\sqrt{\frac{(γ e - 1) \cdot ({M e2}^{2} - 1)}{γ e + 1}}) - a \tan (\sqrt{{M e2}^{2} - 1})) - (\sqrt{\frac{γ e + 1}{γ e - 1}} \cdot a \tan (\sqrt{\frac{(γ e - 1) \cdot ({M e1}^{2} - 1)}{γ e + 1}}) - a \tan (\sqrt{{M e1}^{2} - 1}))

θ_e - Flow Deflection Angle?γ_e - Specific Heat Ratio Expansion Wave?M_e2 - Mach Number Behind Expansion Fan?M_e1 - Mach Number Ahead of Expansion Fan?

Flow Deflection Angle due to Expansion Wave Example

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Here is how the Flow Deflection Angle due to Expansion Wave equation looks like with Values.

Here is how the Flow Deflection Angle due to Expansion Wave equation looks like with Units.

Here is how the Flow Deflection Angle due to Expansion Wave equation looks like.

7.8669 = (\sqrt{\frac{1.41 + 1}{1.41 - 1}} \cdot a \tan (\sqrt{\frac{(1.41 - 1) \cdot (6^{2} - 1)}{1.41 + 1}}) - a \tan (\sqrt{6^{2} - 1})) - (\sqrt{\frac{1.41 + 1}{1.41 - 1}} \cdot a \tan (\sqrt{\frac{(1.41 - 1) \cdot (5^{2} - 1)}{1.41 + 1}}) - a \tan (\sqrt{5^{2} - 1}))

7.8669° = (\sqrt{\frac{1.41 + 1}{1.41 - 1}} \cdot a \tan (\sqrt{\frac{(1.41 - 1) \cdot (6^{2} - 1)}{1.41 + 1}}) - a \tan (\sqrt{6^{2} - 1})) - (\sqrt{\frac{1.41 + 1}{1.41 - 1}} \cdot a \tan (\sqrt{\frac{(1.41 - 1) \cdot (5^{2} - 1)}{1.41 + 1}}) - a \tan (\sqrt{5^{2} - 1}))

7.8669 = (\sqrt{\frac{1.41 + 1}{1.41 - 1}} \cdot a \tan (\sqrt{\frac{(1.41 - 1) \cdot (6^{2} - 1)}{1.41 + 1}}) - a \tan (\sqrt{6^{2} - 1})) - (\sqrt{\frac{1.41 + 1}{1.41 - 1}} \cdot a \tan (\sqrt{\frac{(1.41 - 1) \cdot (5^{2} - 1)}{1.41 + 1}}) - a \tan (\sqrt{5^{2} - 1}))

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Flow Deflection Angle due to Expansion Wave Solution

Follow our step by step solution on how to calculate Flow Deflection Angle due to Expansion Wave?

FIRST Step Consider the formula

θ e = (\sqrt{\frac{γ e + 1}{γ e - 1}} \cdot a \tan (\sqrt{\frac{(γ e - 1) \cdot ({M e2}^{2} - 1)}{γ e + 1}}) - a \tan (\sqrt{{M e2}^{2} - 1})) - (\sqrt{\frac{γ e + 1}{γ e - 1}} \cdot a \tan (\sqrt{\frac{(γ e - 1) \cdot ({M e1}^{2} - 1)}{γ e + 1}}) - a \tan (\sqrt{{M e1}^{2} - 1}))

Next Step Substitute values of Variables

∴

θ e = (\sqrt{\frac{1.41 + 1}{1.41 - 1}} \cdot a \tan (\sqrt{\frac{(1.41 - 1) \cdot (6^{2} - 1)}{1.41 + 1}}) - a \tan (\sqrt{6^{2} - 1})) - (\sqrt{\frac{1.41 + 1}{1.41 - 1}} \cdot a \tan (\sqrt{\frac{(1.41 - 1) \cdot (5^{2} - 1)}{1.41 + 1}}) - a \tan (\sqrt{5^{2} - 1}))

Next Step Prepare to Evaluate

∴

θ e = (\sqrt{\frac{1.41 + 1}{1.41 - 1}} \cdot a \tan (\sqrt{\frac{(1.41 - 1) \cdot (6^{2} - 1)}{1.41 + 1}}) - a \tan (\sqrt{6^{2} - 1})) - (\sqrt{\frac{1.41 + 1}{1.41 - 1}} \cdot a \tan (\sqrt{\frac{(1.41 - 1) \cdot (5^{2} - 1)}{1.41 + 1}}) - a \tan (\sqrt{5^{2} - 1}))

Next Step Evaluate

∴

θ e = 0.137303184990648rad

Next Step Convert to Output's Unit

∴

θ e = 7.86689301366959°

LAST Step Rounding Answer

∴

θ e = 7.8669°

Flow Deflection Angle due to Expansion Wave Formula Elements

Variables

Functions

Flow Deflection Angle

Flow Deflection Angle is defined as the angle by which the flow turns through the expansion wave.

Symbol: θ_e

Measurement: AngleUnit: °

Note: Value can be positive or negative.

Formulas to find Flow Deflection Angle

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.

Symbol: γ_e

Measurement: NAUnit: Unitless

Note: Value should be greater than 1.

Formulas to find Specific Heat Ratio Expansion Wave

Mach Number Behind Expansion Fan

Mach Number Behind Expansion Fan is the Mach number of downstream flow across the expansion fan.

Symbol: M_e2

Measurement: NAUnit: Unitless

Note: Value should be greater than 0.

Formulas to find Mach Number Behind Expansion Fan

Mach Number Ahead of Expansion Fan

Mach Number Ahead of Expansion Fan is the Mach number of the upstream flow.

Symbol: M_e1

Measurement: NAUnit: Unitless

Note: Value should be greater than 0.

Formulas to find Mach Number Ahead of Expansion Fan

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.

Syntax: 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.

Syntax: 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.

Syntax: sqrt(Number)

Other Formulas to find Flow Deflection Angle

Go Flow Deflection Angle using Prandtl Meyer Function

θ e = v M2 - v M1

Other formulas in Expansion Waves category

Go Pressure Ratio across Expansion Fan

P e,r = {(\frac{1 + 0.5 \cdot (γ e - 1) \cdot {M e1}^{2}}{1 + 0.5 \cdot (γ e - 1) \cdot {M e2}^{2}})}^{\frac{γ e}{γ e - 1}}

Go Temperature Ratio across Expansion Fan

T e,r = \frac{1 + 0.5 \cdot (γ e - 1) \cdot {M e1}^{2}}{1 + 0.5 \cdot (γ e - 1) \cdot {M e2}^{2}}

Go Pressure behind Expansion Fan

P 2 = P 1 \cdot {(\frac{1 + 0.5 \cdot (γ e - 1) \cdot {M e1}^{2}}{1 + 0.5 \cdot (γ e - 1) \cdot {M e2}^{2}})}^{\frac{γ e}{γ e - 1}}

Go Temperature behind Expansion Fan

T 2 = T 1 \cdot (\frac{1 + 0.5 \cdot (γ e - 1) \cdot {M e1}^{2}}{1 + 0.5 \cdot (γ e - 1) \cdot {M e2}^{2}})

How to Evaluate Flow Deflection Angle due to Expansion Wave?

Flow Deflection Angle due to Expansion Wave evaluator 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 evaluate 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 evaluate Flow Deflection Angle due to Expansion Wave using this online evaluator? To use this online evaluator 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.

FAQs on Flow Deflection Angle due to Expansion Wave

What is the formula to find Flow Deflection Angle due to Expansion Wave?

The formula of Flow Deflection Angle due to Expansion Wave is expressed as 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))). Here is an example- 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))).

How to calculate Flow Deflection Angle due to Expansion Wave?

With Specific Heat Ratio Expansion Wave (γ_e), Mach Number Behind Expansion Fan (M_e2) & Mach Number Ahead of Expansion Fan (M_e1) we can find Flow Deflection Angle due to Expansion Wave using the formula - 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))). This formula also uses Tangent (tan)Inverse Tan (atan), Square Root (sqrt) function(s).

What are the other ways to Calculate Flow Deflection Angle?

Here are the different ways to Calculate Flow Deflection Angle-

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

Can the Flow Deflection Angle due to Expansion Wave be negative?

Yes, the Flow Deflection Angle due to Expansion Wave, measured in Angle can be negative.

Which unit is used to measure Flow Deflection Angle due to Expansion Wave?

Flow Deflection Angle due to Expansion Wave is usually measured using the Degree[°] for Angle. Radian[°], Minute[°], Second[°] are the few other units in which Flow Deflection Angle due to Expansion Wave can be measured.

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Flow Deflection Angle due to Expansion Wave Formula

​GoFlow Deflection Angle using Prandtl Meyer Function Formula

Flow Deflection Angle due to Expansion Wave Example

Flow Deflection Angle due to Expansion Wave Solution

Flow Deflection Angle due to Expansion Wave Formula Elements

Other Formulas to find Flow Deflection Angle

Other formulas in Expansion Waves category

How to Evaluate Flow Deflection Angle due to Expansion Wave?

FAQs on Flow Deflection Angle due to Expansion Wave

Variable Name

GoFlow Deflection Angle using Prandtl Meyer Function Formula