# What aerodynamic equation should be used to determine the drop in air pressure over an airfoil embedded within a vertical pipe?

I have a conceptual idea for a thruster for a VTOL aircraft/drone which should produce lift by utilizing a upward-flowing airflow to produce low air pressure over the top surface of an airfoil that is embedded within a vertical pipe.

I would like to be able to calculate what the drop in air pressure will be, but being that I am neither an aerospace or aeronautical engineer, I do not know which aerodynamic equation to use to determine this.

I have made some CAD drawings of this conceptual VTOL aircraft/drone thruster to help illustrate and explain how the thruster should produce lift/thrust. (Note: The embedded airfoil is fastened to the vertical pipe with screws and it is colored orange to make it stand out from the rest of the thruster.)

I am thinking of using a Drag Equation that I found on a NASA website to calculate the drop in air pressure, but I'm not sure if it is the right one to use. Reference https://www.grc.nasa.gov/www/k-12/airplane/drageq.html

So, my question is what aerodynamic equation should be used to determine the drop in air pressure over an airfoil embedded within a vertical pipe?

• Why not have 100% of the flow exit from the bottom and keep the top completley closed ?
– AJN
May 1 at 5:01
• @AJN, I have thought of that and I agree that it would be a more ideal thruster to use, yet I am very curious to see how much thrust can be created using this thruster design. May 1 at 12:39
• I'm not sure the upper port will produce any thrust.
– Drew
May 2 at 5:46
• There isn't an equation. You'd have to solve the entire flow field of the jet and entrained air. This is done with real fluid equations and a nonlinear solver. Even a decent RANS solver with basic jet boundary heuristics isn't going to be very good at this. This is basically operating on the Coandra effect. There's a fabulous amount of garbage out there about it. Don't believe any of it. You'd be better off turning the lip of the pipe outward and putting a lid over it. Then you can approximate the flow exit conditions and use momentum theory. May 2 at 20:38
• many equations and ways to go about this exist. conservation of momentum is the most basic one since you are already sizing for 50% of air to go one way and another. is that 50% by mass? what are the the areas? if you have those uou can solve. The other thing to consider is where this thrust is transmitted to your system - draw your boundaries in a smart way and solve for all flows in and out of it. the ambient air part would only apply if pressures were not cancelled elsewhere at the top- they will be unless you are blasting the ground to have some upward flow- how far is the ground?
– Abel
May 3 at 13:02

The equation you quote isn't a physical law, it's a definition of $$C_{\textrm{d}}$$. You will need it, but for finding the value of $$C_{\textrm{d}}$$, there's no substitute for doing a scale model experiment, geometrically similar to and at the same Reynolds number as (and unless the Mach number is very small, also at the same Mach number as) the situation about which you want to make predictions.