Aerodynamics Example #1: Finding the Critical Mach Number of an Airfoil


Some Background

In the field of aerodynamics, airspeeds are usually stated in terms of Mach number, which is the local airspeed divided by the local speed of sound.
For example, assume we are considering airflow at a point, say, "A" on the surface of an airfoil.
Assume we know the local airspeed is vA.

The local speed of sound, a, depends upon the local temperature according to the relation

a   =   √( γ   R   T )

γ is defined as

γ   ≡ Cp/Cv

where Cp and Cv are the specific heats of a gas at constant pressure and volume, respectively. For most calculations, standard air conditions are assumed, and a value of γ = 1.4 is used.

R is the specific gas constant. For air at standard conditions, R = 287 J/(kg K).

T is the temperature, in degrees Kelvin.

Assume we know the speed of sound at "A" is aA.

The Mach number at "A" would be calculated as

MA   =   vA/aA.

Further analysis would continue using MA rather than vA or aA.

Why is the critical Mach number important?
For values of M slightly greater than Mcr, the airfoil experiences a dramatic increase in drag coefficient. As a result, for more detailed analysis, it is important to know when this condition occurs.

On With the Present Example

Consider an airfoil in sub-sonic airflow. The undisturbed airflow far ahead of the airfoil is called the freestream airflow, whose airspeed is usually denoted M.

As the airflow passes over the airfoil, its speed increases and, somewhere on the surface of the airfoil, its speed reaches a maximum value before completely passing over the airfoil and returning to its freestream airspeed. Now increase the the speed of the freestream airflow until the maximum airspeed achieved over the airfoil is M   =   1. This condition defines the critical Mach number. By definition, the critical Mach number, Mcr, is the freestream Mach number at which sonic flow is first achieved on the airfoil.

Calculating the critical Mach number
The critical Mach number is determined by equating two pressure coefficients: (Cp)min and Cp,cr.

(Cp)min is given by

(Cp)min   =   (Cp,0)min/√(1 - (M)2)

where (Cp,0)min is the minimum pressure coefficient for the airfoil at a 0 degree angle of attack. It is usually supplied for a particular airfoil, determined experimentally (in a wind tunnel).

In turn, Cp,cr is calculated from an equation:

Cp,cr   =   2/(γ Mcr2)   {[(1 + ((γ - 1)/2)Mcr2)/(1 + (γ - 1)/2)]γ/(γ - 1) - 1}

The critical Mach number is then calculated by setting

(Cp)min   =   Cp,cr

which can be rearranged as

(Cp)min   -   Cp,cr   =   0

Expanding the terms yields:

(Cp,0)min/√(1 - (M)2)   -   2/(γ M2)   {[(1 + ((γ - 1)/2)M2)/(1 + (γ - 1)/2)]γ/(γ - 1) - 1}   =   0

Note that all instances of Mcr were replaced by M; for the desired condition, they are the same.

A Numerical Example
Assume you are trying to determine the critical Mach number of the NACA 0012 airfoil.
For the given conditions, you are given that (Cp,0)min   =   -0.43.
You plan to use the equation just above to determine the critical Mach number:

This equation is solved by using Root-Finder Utility #2; however, the data must be altered a little to get it in the correct form for use with the utility.
For Root-Finder Utility #2,
constant a1 is entered as a1   =   (Cp,0)min   (γ/2)   =   (-0.43)(1.4/2)   =   -0.301.
constant a2 is entered as a2   =   (γ - 1)/2   =   (1.4 - 1)/2   =   0.2.
constant a3 is entered as a3   =   γ/(γ - 1)   =   1.4/(1.4 - 1)  =   3.5.
The endpoints b and c can be suggested by graphing the above equation, or simply trial-and-error. The upper endpoint can be set to 1, because we know M MUST BE LESS THAN 1. Let us set b = 0.5 and c = 1 for the present example.
Plugging these numbers into Root-Finder Utility #2 yields Mcr = 0.7371059

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