# Modeling Lecture 1 - imtek.de Aerospace Modeling Tutorial Lecture 2 Basic Aerodynamics Greg and Mario February 2, 2015 Our system dynamics: ( 11 21

31 12 22 32 13 0

23 = 33 )( 0

0 )( 11 21 31 12 22 32 =

( )+ 13 23 33 )

= =

Our model Navier Stokes Equations Solving Navier Stokes - CFD Computationally demanding Not suitable for real time simulation Not suitable for dynamic optimization How to simplify things?

Thin airfoil theory Assumptions: 2-dimensional flow Inviscid flow Incompressible flow Solve simplified NS (just Laplaces equation) with flow tangency condition Thin airfoil theory Results: (Lift)

Advantages: Easy to compute Fits well to data Drawbacks: Predicts 0 drag Real wings arent 2-dimensional xfoil viscous solution in the boundary layer Inviscid outside gives parasitic drag still 2d

Prandtl lifting line theory Still inviscid, incompressible Model flow field as a sum of horseshoe vortices Solve for circulation of each 2-d section 2 = = +2

Still need to account for wingtail interaction Ignores spanwise viscous flow Vortex lattice Model the wing as a panel of ring vortices Can handle arbitrary shapes Disadvantage: intrinsically computational, no handy formulas AVL Athena Vortex Lattice (Mark Drela)

popular code, includes parasitic drag Inputs: geometry, alpha/beta/airspeed Outputs: force/moment vectors + derivatives w.r.t. omega Strategy: sweep alpha/beta, fit curves for all coefficients Our model Homework 1: 2-dimensional model 2 State:

= + 0.01 =2 Control input: Mass 2 Aspect ratio 10 Sref 0.5 Gravity 9.8 1. Starting from [0,-10,10,0], fly as far as possible in 10

seconds, in the x direction 2. Starting from the same place, fly as long as possible (maximum time) Altitude must always be positive!! Homework 2 (optional): 3 dimensional model Implement the full aerodynamic model, using coefficients from https:// github.com/ghorn/rawesome/blob/master/rawe/models/betty_conf.py (There is also a reference model there) R(0) = eye(3)

p(0) = [0,0,0] v(0) = [15, 0, 0] (0) = [1, 0, 0] Do something like, R(5.0)=eye(3), w(5.0) = [0,0,0], vy(5.0) = 0, minimize u^2 Probably best to simulate first to validate model