What is wrong with my model of aerodynamic drag?

Question

I am creating a JavaScript program which simulates fireworks in a 2-dimentional context launching from the ground and exploding somewhere in mid-air.

The program takes into account gravity and drag, but something is wrong with the way I am calculating acceleration due to drag.

Observe these animations:

Without drag
With drag

My implementation of drag physics is producing an undesired, physically inaccurate "square" effect, where particles seem to have a different drag depending on the direction in which they are traveling (the more parallel they are to either the x or y axis, the greater the drag — with the least drag at a 45° angle from either axis). If I turn the drag "off", the firework particles explode evenly in all directions, as they should.

My calculations of acceleration due to drag are based on the following equations:

D = ½ρCAv² (Nasa)
a = D/m (Newton's Second Law of Motion)

And these are the snippets of code in the program that I believe are relevant to the problem (However, please feel free to audit the entire source code on GitHub: github.com/foxyjacob/vulcan):

function calculateAccelerationDueToDrag(fluidDensity, dragCoefficient, crossSectionalArea, speed, mass) {

    let acceleration = -(0.5 * fluidDensity * dragCoefficient * crossSectionalArea * (speed * speed)) / mass;

    if (speed < 0) {
        acceleration = -acceleration;
    }

    return acceleration;

}

...

let accelerationDueToDrag = [
    calculateAccelerationDueToDrag(
        AIR_DENSITY,
        this.dragCoefficient,
        this.crossSectionalArea,
        lastVelocity[0],
        this.mass
    ),

    calculateAccelerationDueToDrag(
        AIR_DENSITY,
        this.dragCoefficient,
        this.crossSectionalArea,
        lastVelocity[1],
        this.mass
    )
];

Essentially, I am calculating each particle's acceleration, velocity, and position on the x and y axes independently.

Why is my model of drag producing this strange "square" effect?

score 3 · Accepted Answer · answered Feb 08 '18 at 22:06

With a drag force proportional to velocity$^2$, you can't do the calculations for the $x$ and $y$ directions separately. $$\vec{D} = -k|\vec{v}|^2\hat{v} = -k|\vec{v}|\vec{v}$$ Therefore, $$D_{x} = -k\sqrt{v_x^2 + v_y^2}\ v_x$$ You can't just send the $x$-component of the velocity of the projectile to to the function to get the $x$-component of the drag force. Your calculateAccelerationDueToDrag() function needs to have inputs for both velocity components for both drag components.

What is wrong with my model of aerodynamic drag?

1 Answers1