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Prandtl Meyer function at upstream Mach number Solution

STEP 0: Pre-Calculation Summary
Formula Used
prandtl_meyer_function_at_upstream_mach = sqrt((Specific heat ratio+1)/(Specific heat ratio-1))*atan(sqrt(((Specific heat ratio-1)*((Mach Number ahead of shock^2)-1))/(Specific heat ratio+1)))-atan(sqrt(((Mach Number ahead of shock^2)-1)))
ν(M1) = sqrt((κ+1)/(κ-1))*atan(sqrt(((κ-1)*((M1^2)-1))/(κ+1)))-atan(sqrt(((M1^2)-1)))
This formula uses 3 Functions, 2 Variables
Functions Used
tan - Trigonometric tangent function, tan(Angle)
atan - Inverse trigonometric tangent function, atan(Number)
sqrt - Squre root function, sqrt(Number)
Variables Used
Specific heat ratio- The Specific heat ratio is the ratio of the heat capacity at constant pressure to heat capacity at constant volume.
Mach Number ahead of shock- Mach Number ahead of shock is the Mach number over the body before a shockwave has occurred
STEP 1: Convert Input(s) to Base Unit
Specific heat ratio: 1.392758 --> No Conversion Required
Mach Number ahead of shock: 1.5 --> No Conversion Required
STEP 2: Evaluate Formula
Substituting Input Values in Formula
ν(M1) = sqrt((κ+1)/(κ-1))*atan(sqrt(((κ-1)*((M1^2)-1))/(κ+1)))-atan(sqrt(((M1^2)-1))) --> sqrt((1.392758+1)/(1.392758-1))*atan(sqrt(((1.392758-1)*((1.5^2)-1))/(1.392758+1)))-atan(sqrt(((1.5^2)-1)))
Evaluating ... ...
ν(M1) = 0.208721685193954
STEP 3: Convert Result to Output's Unit
0.208721685193954 Radian --> No Conversion Required
FINAL ANSWER
0.208721685193954 Radian <-- Prandtl Meyer Function at upstream Mach no.
(Calculation completed in 00.002 seconds)

10+ Oblique Shock and Expansion Waves Calculators

Temperature behind oblique shock for given upstream temperature and normal upstream Mach number
temp_behind_shock = Temperature ahead of shock*((1+((2*Specific heat ratio)/(Specific heat ratio+1))*((Component of upstream mach normal to oblique shock^2)-1))/((Specific heat ratio+1)*(Component of upstream mach normal to oblique shock^2)/(2+((Specific heat ratio-1)*(Component of upstream mach normal to oblique shock^2))))) Go
Temperature ratio across the oblique shock
temperature_ratio_across_shock = (1+((2*Specific heat ratio)/(Specific heat ratio+1))*((Component of upstream mach normal to oblique shock^2)-1))/((Specific heat ratio+1)*(Component of upstream mach normal to oblique shock^2)/(2+((Specific heat ratio-1)*(Component of upstream mach normal to oblique shock^2)))) Go
Pressure behind expansion fan
pressure_behind_expansion_fan = Pressure ahead expansion fan*((1+0.5*(Specific heat ratio-1)*(Mach no. ahead expansion fan^2))/(1+0.5*(Specific heat ratio-1)*(Mach no. behind expansion fan^2)))^((Specific heat ratio)/(Specific heat ratio-1)) Go
Component of downstream Mach number normal to oblique shock for given normal upstream Mach number
component_of_downstream_mach_normal_to_obliqueshock = sqrt((1+0.5*((Specific heat ratio-1)*Component of upstream mach normal to oblique shock^2))/(Specific heat ratio*Component of upstream mach normal to oblique shock^2-0.5*(Specific heat ratio-1))) Go
Pressure ratio across the expansion fan
pressure_ratio_across_expansion_fan = ((1+0.5*(Specific heat ratio-1)*(Mach no. ahead expansion fan^2))/(1+0.5*(Specific heat ratio-1)*(Mach no. behind expansion fan^2)))^((Specific heat ratio)/(Specific heat ratio-1)) Go
Density behind oblique shock for given upstream density and normal upstream Mach number
density_behind_shock = Density ahead of shock*((Specific heat ratio+1)*(Component of upstream mach normal to oblique shock^2)/(2+((Specific heat ratio-1)*(Component of upstream mach normal to oblique shock^2)))) Go
Density ratio across the oblique shock
density_ratio_across_shock = (Specific heat ratio+1)*(Component of upstream mach normal to oblique shock^2)/(2+((Specific heat ratio-1)*(Component of upstream mach normal to oblique shock^2))) Go
Pressure behind oblique shock for given upstream pressure and normal upstream Mach number
static_pressure_behind_shock = Static pressure ahead of shock*(1+((2*Specific heat ratio)/(Specific heat ratio+1))*((Component of upstream mach normal to oblique shock^2)-1)) Go
Temperature ratio across the expansion fan
temp_ratio_across_expansion_fan = (1+0.5*(Specific heat ratio-1)*(Mach no. ahead expansion fan^2))/(1+0.5*(Specific heat ratio-1)*(Mach no. behind expansion fan^2)) Go
Pressure ratio across the oblique shock
pressure_ratio_across_shock = 1+((2*Specific heat ratio)/(Specific heat ratio+1))*((Component of upstream mach normal to oblique shock^2)-1) Go

Prandtl Meyer function at upstream Mach number Formula

prandtl_meyer_function_at_upstream_mach = sqrt((Specific heat ratio+1)/(Specific heat ratio-1))*atan(sqrt(((Specific heat ratio-1)*((Mach Number ahead of shock^2)-1))/(Specific heat ratio+1)))-atan(sqrt(((Mach Number ahead of shock^2)-1)))
ν(M1) = sqrt((κ+1)/(κ-1))*atan(sqrt(((κ-1)*((M1^2)-1))/(κ+1)))-atan(sqrt(((M1^2)-1)))

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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 = sqrt((Specific heat ratio+1)/(Specific heat ratio-1))*atan(sqrt(((Specific heat ratio-1)*((Mach Number ahead of shock^2)-1))/(Specific heat ratio+1)))-atan(sqrt(((Mach Number ahead of shock^2)-1))) to calculate the Prandtl Meyer Function at upstream Mach no., The Prandtl Meyer function at upstream Mach number formula is obtained by substituting the upstream Mach number in Prandtl Meyer function. Prandtl Meyer Function at upstream Mach no. and is denoted by ν(M1) 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 (κ) and Mach Number ahead of shock (M1) and hit the calculate button. Here is how the Prandtl Meyer function at upstream Mach number calculation can be explained with given input values -> 0.208722 = sqrt((1.392758+1)/(1.392758-1))*atan(sqrt(((1.392758-1)*((1.5^2)-1))/(1.392758+1)))-atan(sqrt(((1.5^2)-1))).

FAQ

What is Prandtl Meyer function at upstream Mach number?
The Prandtl Meyer function at upstream Mach number formula is obtained by substituting the upstream Mach number in Prandtl Meyer function and is represented as ν(M1) = sqrt((κ+1)/(κ-1))*atan(sqrt(((κ-1)*((M1^2)-1))/(κ+1)))-atan(sqrt(((M1^2)-1))) or prandtl_meyer_function_at_upstream_mach = sqrt((Specific heat ratio+1)/(Specific heat ratio-1))*atan(sqrt(((Specific heat ratio-1)*((Mach Number ahead of shock^2)-1))/(Specific heat ratio+1)))-atan(sqrt(((Mach Number ahead of shock^2)-1))). The Specific heat ratio is the ratio of the heat capacity at constant pressure to heat capacity at constant volume and Mach Number ahead of shock is the Mach number over the body before a shockwave has occurred.
How to calculate Prandtl Meyer function at upstream Mach number?
The Prandtl Meyer function at upstream Mach number formula is obtained by substituting the upstream Mach number in Prandtl Meyer function is calculated using prandtl_meyer_function_at_upstream_mach = sqrt((Specific heat ratio+1)/(Specific heat ratio-1))*atan(sqrt(((Specific heat ratio-1)*((Mach Number ahead of shock^2)-1))/(Specific heat ratio+1)))-atan(sqrt(((Mach Number ahead of shock^2)-1))). To calculate Prandtl Meyer function at upstream Mach number, you need Specific heat ratio (κ) and Mach Number ahead of shock (M1). With our tool, you need to enter the respective value for Specific heat ratio and Mach Number ahead of shock 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 Prandtl Meyer Function at upstream Mach no.?
In this formula, Prandtl Meyer Function at upstream Mach no. uses Specific heat ratio and Mach Number ahead of shock. We can use 10 other way(s) to calculate the same, which is/are as follows -
  • density_ratio_across_shock = (Specific heat ratio+1)*(Component of upstream mach normal to oblique shock^2)/(2+((Specific heat ratio-1)*(Component of upstream mach normal to oblique shock^2)))
  • component_of_downstream_mach_normal_to_obliqueshock = sqrt((1+0.5*((Specific heat ratio-1)*Component of upstream mach normal to oblique shock^2))/(Specific heat ratio*Component of upstream mach normal to oblique shock^2-0.5*(Specific heat ratio-1)))
  • density_behind_shock = Density ahead of shock*((Specific heat ratio+1)*(Component of upstream mach normal to oblique shock^2)/(2+((Specific heat ratio-1)*(Component of upstream mach normal to oblique shock^2))))
  • static_pressure_behind_shock = Static pressure ahead of shock*(1+((2*Specific heat ratio)/(Specific heat ratio+1))*((Component of upstream mach normal to oblique shock^2)-1))
  • temp_behind_shock = Temperature ahead of shock*((1+((2*Specific heat ratio)/(Specific heat ratio+1))*((Component of upstream mach normal to oblique shock^2)-1))/((Specific heat ratio+1)*(Component of upstream mach normal to oblique shock^2)/(2+((Specific heat ratio-1)*(Component of upstream mach normal to oblique shock^2)))))
  • pressure_ratio_across_shock = 1+((2*Specific heat ratio)/(Specific heat ratio+1))*((Component of upstream mach normal to oblique shock^2)-1)
  • temperature_ratio_across_shock = (1+((2*Specific heat ratio)/(Specific heat ratio+1))*((Component of upstream mach normal to oblique shock^2)-1))/((Specific heat ratio+1)*(Component of upstream mach normal to oblique shock^2)/(2+((Specific heat ratio-1)*(Component of upstream mach normal to oblique shock^2))))
  • pressure_ratio_across_expansion_fan = ((1+0.5*(Specific heat ratio-1)*(Mach no. ahead expansion fan^2))/(1+0.5*(Specific heat ratio-1)*(Mach no. behind expansion fan^2)))^((Specific heat ratio)/(Specific heat ratio-1))
  • temp_ratio_across_expansion_fan = (1+0.5*(Specific heat ratio-1)*(Mach no. ahead expansion fan^2))/(1+0.5*(Specific heat ratio-1)*(Mach no. behind expansion fan^2))
  • pressure_behind_expansion_fan = Pressure ahead expansion fan*((1+0.5*(Specific heat ratio-1)*(Mach no. ahead expansion fan^2))/(1+0.5*(Specific heat ratio-1)*(Mach no. behind expansion fan^2)))^((Specific heat ratio)/(Specific heat ratio-1))
Where is the Prandtl Meyer function at upstream Mach number calculator used?
Among many, Prandtl Meyer function at upstream Mach number calculator is widely used in real life applications like {FormulaUses}. Here are few more real life examples -
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