Prandtl Meyer Function at Upstream Mach Number Solution

STEP 0: Pre-Calculation Summary
Formula Used
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))
vM1 = sqrt((γe+1)/(γe-1))*atan(sqrt(((γe-1)*(Me1^2-1))/(γe+1)))-atan(sqrt(Me1^2-1))
This formula uses 3 Functions, 3 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
Prandtl Meyer Function at Upstream Mach no. - (Measured in Radian) - Prandtl Meyer Function at Upstream Mach no. is the Prandtl Meyer Functional value at upstream of 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 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 Ahead of Expansion Fan: 5 --> No Conversion Required
STEP 2: Evaluate Formula
Substituting Input Values in Formula
vM1 = sqrt((γe+1)/(γe-1))*atan(sqrt(((γe-1)*(Me1^2-1))/(γe+1)))-atan(sqrt(Me1^2-1)) --> sqrt((1.41+1)/(1.41-1))*atan(sqrt(((1.41-1)*(5^2-1))/(1.41+1)))-atan(sqrt(5^2-1))
Evaluating ... ...
vM1 = 1.32473545821219
STEP 3: Convert Result to Output's Unit
1.32473545821219 Radian -->75.9017507269022 Degree (Check conversion here)
FINAL ANSWER
75.9017507269022 75.90175 Degree <-- Prandtl Meyer Function at Upstream Mach no.
(Calculation completed in 00.004 seconds)

Credits

Created by Shikha Maurya
Indian Institute of Technology (IIT), Bombay
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Indian Institute for Aeronautical Engineering and Information Technology (IIAEIT), Pune
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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)

Prandtl Meyer Function at Upstream Mach Number Formula

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))
vM1 = sqrt((γe+1)/(γe-1))*atan(sqrt(((γe-1)*(Me1^2-1))/(γe+1)))-atan(sqrt(Me1^2-1))

Which law is implemented for flow visualization by optical system?

Snell's law is implemented for flow visualization by an optical system. According to Snell's law, a light ray, passing through a nonhomogeneous refracted field, is deflected from its original direction and a light path is different from that of an undisturbed ray. If a recording plane is placed in front of the light ray, after disturbing media, three quantities can be measured: the vertical displacement of the disturbed ray, the angular deflection of the disturbed ray with respect to the undisturbed one and the phase shift between both rays, owning to their different optical path lengths.

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Flow visualization is essential for exploring, and understanding fluid behaviour and can be both qualitative and quantitative. The main methods for visualization of these flows are optical methods. The three principal optical methods are shadow, schlieren and interferometry.

How to Calculate Prandtl Meyer Function at Upstream Mach Number?

Prandtl Meyer Function at Upstream Mach Number calculator uses 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)) to calculate the Prandtl Meyer Function at Upstream Mach no., Prandtl Meyer Function at Upstream Mach Number formula describes the relationship between the Mach number of the flow and the corresponding Prandtl-Meyer angle, which characterizes the amount of expansion or turning required to achieve the specified Mach number. Prandtl Meyer Function at Upstream Mach no. is denoted by vM1 symbol.

How to calculate Prandtl Meyer Function at Upstream Mach Number using this online calculator? To use this online calculator for Prandtl Meyer Function at Upstream Mach Number, enter Specific Heat Ratio Expansion Wave e) & Mach Number Ahead of Expansion Fan (Me1) and hit the calculate button. Here is how the Prandtl Meyer Function at Upstream Mach Number calculation can be explained with given input values -> 4407.204 = 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 Prandtl Meyer Function at Upstream Mach Number?
Prandtl Meyer Function at Upstream Mach Number formula describes the relationship between the Mach number of the flow and the corresponding Prandtl-Meyer angle, which characterizes the amount of expansion or turning required to achieve the specified Mach number and is represented as vM1 = sqrt((γe+1)/(γe-1))*atan(sqrt(((γe-1)*(Me1^2-1))/(γe+1)))-atan(sqrt(Me1^2-1)) or 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)). The Specific Heat Ratio Expansion Wave is the ratio of the heat capacity at constant pressure to heat capacity at constant volume & Mach Number Ahead of Expansion Fan is the Mach number of the upstream flow.
How to calculate Prandtl Meyer Function at Upstream Mach Number?
Prandtl Meyer Function at Upstream Mach Number formula describes the relationship between the Mach number of the flow and the corresponding Prandtl-Meyer angle, which characterizes the amount of expansion or turning required to achieve the specified Mach number is calculated using 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)). To calculate Prandtl Meyer Function at Upstream Mach Number, you need Specific Heat Ratio Expansion Wave e) & Mach Number Ahead of Expansion Fan (Me1). With our tool, you need to enter the respective value for Specific Heat Ratio Expansion Wave & 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.
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