Coanda Effect and relevant maths

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In summary, the Coanda Effect is the result of a fluid flowing around a curved surface. To understand it, you need to understand classical physics, calculus, and vector calculus. There are a few good resources for understanding the math, but the most difficult part will be proving the simulation is correct.
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Mathematical understanding of the Coanda Effect
Summary: Mathematical understanding of the Coanda Effect

Summary: Mathematical understanding of the Coanda Effect

Hi,

I am trying to understand how to understand the Coanda Effect in terms of mathematics. I am a high school student and have a basic understanding of calculus, and I do not fully understand the Navier-Stokes equations. I have a project where I have to investigate the parameters that affect whether a candle will be blown out if there is a bottle in front of the incoming airflow.

What are the relevant relationships for airflow around a convex shape (Coanda Effect). E.g. how is air-speed dependent on the radius of the shape, and how would you go about finding the minimum velocity for the Coanda Effect to be observed or for air to reach the other side (mathematical derivations).

Am I able to just equate forces to derive equations for velocity/pressure functions, and if so how would I go about doing that? E.g. centrifugal force = (some kind of force related to radial pressure).

Also, are there are any good resources for understanding fluid dynamics. I would really like to get to the stage where I can derive the navier-stokes equations so that I can use them for my project in MATLAB. Any good resources to help me understand the maths? Thank you.
 
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This is a very tough topic to cover:
1) You need to understand Calculus 1,2,3 to the point of understanding divergence, gradient and curl and the related line, surface and volume integrals
2) You need to understand classical physics of momentum, energy, stress and strain
3) You need to understand how to put these together in a Matlab numerical computation

Here's a video that explains the terms of the Navier Stokes equations:



To understand divergence, gradient and curl conceptually:

https://betterexplained.com/articles/vector-calculus-understanding-circulation-and-curl/https://betterexplained.com/articles/vector-calculus-understanding-the-gradient/https://betterexplained.com/articles/divergence/
https://betterexplained.com/articles/flux/
For understanding the vector calculus math Khan Academy or MathIsPower4U is a good place to begin.

For understanding Navier Stokes:

https://en.wikipedia.org/wiki/Navier–Stokes_equations
Numerical computing resources for Navier Stokes simulation:

https://www.mathworks.com/matlabcen...euler-and-navier-stokes-fluid-flow-simulation
https://math.mit.edu/~gs/cse/codes/mit18086_navierstokes.pdf
One of the hardest things will be proving your sim is correct which might mean physically testing it.

The sim problem boils down to either adding error or subtracting error as it runs and this manifests itself as energy coming in or energy being lost. THe easiest way o see that is a simple planetary simulation of the Earth going around the Sun as it runs the Earth will either spiral in from lost energy or spiral out energy gained from error in the simulation math (like rounding up or down too much)
 
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1. What is the Coanda Effect?

The Coanda Effect is a phenomenon in fluid dynamics where a fluid flowing close to a curved surface will follow the contour of the surface instead of continuing in a straight line. This effect is caused by the difference in pressure between the upper and lower surfaces of the curved surface.

2. How is the Coanda Effect related to aerodynamics?

The Coanda Effect has significant applications in aerodynamics, particularly in the design of aircraft wings and jet engines. By utilizing the Coanda Effect, engineers can manipulate the airflow around a curved surface to increase lift and improve the overall performance of an aircraft.

3. What are some real-world examples of the Coanda Effect?

Some common examples of the Coanda Effect in action include the lift generated by an airplane wing, the flow of exhaust gases in a jet engine, and the flow of water over a curved dam. It is also seen in everyday objects such as hair dryers and vacuum cleaners.

4. What mathematical equations are used to describe the Coanda Effect?

The Coanda Effect can be described by Bernoulli's principle, which states that an increase in the speed of a fluid results in a decrease in pressure. This is represented by the equation P + ½ρv² = constant, where P is pressure, ρ is density, and v is velocity. Other equations used to describe the Coanda Effect include the Navier-Stokes equations and the continuity equation.

5. How is the Coanda Effect relevant to the field of fluid mechanics?

The Coanda Effect is a fundamental principle in the study of fluid mechanics and has numerous practical applications. Understanding this phenomenon allows engineers to design more efficient and aerodynamic structures, as well as improve our understanding of fluid flow in various systems such as pipelines and turbines.

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