Modelling Lift of an Airfoil Using Bernoulli Principle

The Bernoulli equation can be used to create a simple model of lift generation of an airfoil as a function of the airfoil’s geometry, the pressure, density, and velocity of fluid around the airfoil, and other parameters such as elevation.

This is a simple C program that is run on the Signaloid Cloud Developer Platform, which incorporates uncertainty in some of the model parameters.

The Model, Parameters and Underlying Assumptions

For our design, we include the following parameters: air velocity ($v$), air density ($\rho$) and airfoil area ($A$). The lift force ($L$) generated by an airfoil is given by the expression

$\qquad$ $L = \Delta P \cdot A$

where $\Delta P$ is the pressure difference above and below the airfoil. The model is designed, with the following assumptions:

• We model the pressure difference using the Bernoulli's equation, given as:

$\qquad$ $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$.

• We assume that the height ($h$) is constant for our model, therefore

$\qquad$ $\Delta P = \frac{1}{2}\rho(v_2^2 - v_1^2)$,

$\qquad$ $v_1$ and $v_2$ is the velocity below and above the airfoil respectively.

• We assume standard atmospheric conditions for temperature and pressure, and the air density is measured using the device described in ¹, which has a tolerance of $8.5%$.

• We assume that the area of the airfoil designed by a manufacturer is $15\mathrm{m}^2$, and with an acceptable error margin of $0.01\mathrm{m}^2$.

• We assume that the measured velocity above the airfoil $v_2$ follows a normal distribution with a mean of $10\mathrm{ms}^{-1}$ and variance of $0.5\mathrm{ms}^{-1}$, and the measurements for $v_1$ is also normally distributed with mean of $6\mathrm{ms}^{-1}$ and variance of $0.3\mathrm{ms}^{-1}$.

Program Commandline Arguments

The C program can take in the following commandline arguments:

-v: velocity above airfoil

-s: variance of $v_2$

-u: velocity below airfoil

-r: variance of $v_1$

-A: area of airfoil

-t: tolerance for airfoil area

In the absence of any commandline argument, the default values are used.

Reference

Footnotes

1. Szymiczek, J. (2020). Air Density Measuring Device-Innovative Design, Calibration and Exemplary Results. Instrumentation, Mesures, Métrologies, 19(6). ↩