How to Approximate Aerodynamic Coefficients? (Lift/Drag)

In summary: Shock-expansion theory is good when you have a symmetric body and can neglect the pressure on the opposite side of the body. The tangent-wedge/cone approximation is good for more complicated shapes where the pressure is not constant along the body.
  • #1
Dikuza
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TL;DR Summary
Looking for an example to calculate lift and drag coefficients for hypersonic flow over a blunt-body
Hello, this is a very specific question so any help is much appreciated!

GOAL: I'm trying to get a first-pass analytical approximation for the lift and drag coefficients for hypersonic flow over a blunt-body capsule spacecraft (similar to NASA's Apollo or SpaceX's Dragon) during atmospheric reentry from LEO.

METHOD: I understand that modified Newtonian fluid theory is the best (most accurate for its simplicity) approach before considering any computationally-demanding and time-consuming CFD simulations. The basis of this theory is to integrate the pressure coefficient over the portion of the 3D body that is exposed to air flow ("non-shadowed region") using the equation Cp = Cp(max)*sin^2(theta).

ISSUE: I've done a fair amount of research, however, not having an extensive background in aerodynamics am struggling with how exactly this is applied. For a given vehicle shape, angle of attack, and mach number, how exactly is this done? Every paper I read (3 of which I've referenced below as examples) seem to skip over the actual calculations and reference some code that's been written or use an existing program like CBAERO that I don't have access to. I've also found a couple of the original documents from the 60's ( https://ntrs.nasa.gov/api/citations/19660012440/downloads/19660012440.pdf) that go over this but are pretty hard to follow. Has anyone done this before and could walk me through the process or point me to an example calculation for my scenario or code? If this approach is reportedly simpler than CFD and used as a means of quick design iteration, I would think it's not super difficult but I'm lost with the complex integrations and limits.

SOURCES:
https://engineering.purdue.edu/~mjgrant/48th-aiaa-aerospace-science.pdf: "After analytic relations are developed, they are output to a Matlab-based aerodynamics module."
https://www.intechopen.com/chapters/21789]https://www.intechopen.com/chapters/21789: "A computer program is written to compute the aerodynamic coefficients using the Newtonian sine-squared law" https://www.researchgate.net/publication/269802955_Application_of_Modified_Newton_Flow_Model_to_Earth_Reentry_Capsules: "A Fortran code has been written, making benefit of existing in-house library"
 
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  • #2
You might take a look in newer resources like "Hypersonic and High-Temperature Gas Dynamics" by John Anderson. Since it's a textbook, it's goal is to inform a reader how to do things like this.

You may also need to employ something like shock-expansion theory or the tangent-wedge/cone approximation depending on your shape (they're also fairly easy to implement and are in the above book).
 
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FAQ: How to Approximate Aerodynamic Coefficients? (Lift/Drag)

What methods are commonly used to approximate aerodynamic coefficients for lift and drag?

Common methods include computational fluid dynamics (CFD), wind tunnel testing, analytical methods like potential flow theory, and empirical methods based on experimental data. Each method varies in complexity, cost, and accuracy, with CFD and wind tunnel testing providing more precise results but at a higher cost and computational effort.

How accurate are computational fluid dynamics (CFD) simulations in approximating lift and drag coefficients?

CFD simulations can be very accurate if properly set up and validated against experimental data. The accuracy depends on factors such as the quality of the mesh, turbulence models used, and boundary conditions. However, CFD requires significant computational resources and expertise to achieve high accuracy.

What are the limitations of using empirical methods to approximate aerodynamic coefficients?

Empirical methods rely on data from previous experiments and are typically limited to conditions similar to those of the original experiments. They may not accurately predict aerodynamic coefficients for novel designs or flow conditions outside the tested range. Additionally, they may not account for complex interactions in the flow field.

How can wind tunnel testing be used to approximate aerodynamic coefficients?

Wind tunnel testing involves placing a scale model of the aircraft or object in a controlled airflow environment and measuring the forces and moments acting on it. These measurements can then be used to calculate the lift and drag coefficients. Wind tunnel tests are highly accurate but can be expensive and time-consuming.

What role do analytical methods play in approximating aerodynamic coefficients?

Analytical methods, such as potential flow theory and thin airfoil theory, provide simplified mathematical models to estimate aerodynamic coefficients. These methods are useful for initial design and analysis but may not capture all the complexities of real-world fluid flow, such as turbulence and viscous effects. They are often used in conjunction with other methods for more comprehensive analysis.

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