Membrane dynamic solution Analytical results • 3rd mode oscillations obtained

in unstable conditions • Membrane inertia does not

affect steady state solution in stable cases

Numerical results

• Strouhal number: 𝑆𝑡 =𝑓𝑐

𝑈∞

• Natural frequency of a string:

𝑆𝑡𝑛 =𝑛

2

𝐶𝑇

2𝜌ℎ

Rigid airfoil gust response in low Reynolds

• Lift drop is obtained when the gust front reaches the T.E. (at time 𝑠 = 2) due to a momentary failure to meet the Kutta condition

• A thinner airfoil presents a response more similar to the Kussner function • The airfoil camber does not affect the lift response

Membrane wing sharp-edge gust response

• Membrane presents no lift drop at time 𝑠 = 2 due to deformation of the T.E. area

• Membrane parameters: 𝐶𝑇0 = 1.5, 𝐸ℎ = 50, 𝜌ℎ = 0.5

• Flow parameters: 𝛼0 = 1∘, 𝛼𝑔 = 1∘, 𝑅𝑒 = 2500

Membrane Wing Gust Response

PhD Research Study by Sonya Tiomkin Under the Guidance of Prof. Daniella Raveh

Introduction

Mathematical Model

Results

Contact Information

Problem description • Equilibrium of forces between the aerodynamic load, the

membrane inertia and tension, where Shape is determined by loads and tension Loads vary with shape

Main assumptions

• 2D • Linearly elastic membrane • Potential flow/Low Reynolds flow

Objectives

• Establish knowledge of membrane wing dynamic response to gusts in terms of Shape adaptation Aerodynamic forces

c

. .L ET. .T ET

yF

𝑈∞

𝑤𝑔

Membrane model Membrane dynamic equation

𝜌ℎ𝜕2𝑦

𝜕𝑡2= 𝑇

𝜕2𝑦

𝜕𝑥21 +

𝜕𝑦

𝜕𝑥

2 −32

+ Δ𝑝

Constitutive relation

𝑇 = 𝑇0 + 𝐸ℎ𝑙 − 𝑙0𝑙0

Boundary conditions

𝑦 𝑥 = 0, 𝑡 = 𝑦 𝑥 = 𝑐, 𝑡 = 0

Initial conditions

𝑦 𝑥, 𝑡 = 0 = 𝑦0(𝑥) 𝜕𝑦

𝜕𝑡𝑥, 𝑡 = 0 = 0

Gust model Sharp-edge gust

𝑤𝑔 = 𝑤0 𝑥 < 𝑠0 𝑥 > 𝑠

𝑥 =𝑥

𝑏, 𝑠 =

𝑈∞𝑡

𝑏, 𝑏 =

𝑐

2

Lift build-up in response to sharp-edge gust

𝐿 = 2𝜋𝜌∞𝑈∞𝑏𝑤0𝜓(𝑠)

Kussner function approximation

𝜓 𝑠 ≅ 1 − 0.5𝑒−0.13𝑠 − 0.5𝑒−𝑠

Methods

Numerical solution • Iterative procedure • EZNSS - Elastic, Zonal, Navier-Stokes Solver (by ISCFDC) • O-type grid (401x86) • Laminar flow (Re=2500)

Analytical solution • Assumptions: small AoAs, small camber, potential flow • Membrane slope represented by a Fourier series

expansion with time dependent coefficients • Unsteady thin airfoil theory used • Added mass term omitted at first stage • Solved for elastic/constant tension cases

𝑦

𝑥

𝑤0 𝑈∞

Fig. 1 - Stability analysis of the analytical solution Fig. 2 – Membrane shape perturbation from the mean profile, obtained with the elastic analytical solution for 𝜌ℎ = 0.5, 𝐶𝑇0 = 1, 𝐸ℎ = 50, 𝛼 = 2∘

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

St

Am

pli

tud

e (C

L)

h=0.5

h=1

h=5

h=15

10-1

100

101

0.85

0.9

0.95

1

1.05

h

St E

/ S

t 2

(a) Frequency response (b) Dominant frequency response

Fig. 3 – Lift coefficient frequency response obtained with 𝐶𝑇0 = 1.5, 𝐸ℎ = 50, 𝛼 = 4∘ and 𝑅𝑒 = 2500 for various mass values

0 10 20 30 40 50 600

0.02

0.04

0.06

0.08

0.1

0.12

s

CL

0 1 2 30

0.035

0.07

Euler

Re=5000

Re=2500

0 10 20 30 40 50 600

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

s

CL

NACA0012

NACA0006

NACA0002

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

s

CL (

no

rmal

ized

)

Parabola 2% Re=2500

Parabola 4% Re=2500

Parabola 6% Re=2500

Kussner

Fig. 4 – Low Re effect on the lift response of a NACA0012 airfoil to a sharp-edge gust of 1∘

Fig. 5 – Lift response of 3 symmetric NACA airfoils of different thickness to a sharp-edge gust of 1∘ at 𝑅𝑒 = 2500

Fig. 6 – Lift response of parabolic airfoils of different camber to a sharp-edge gust of 1∘ at 𝑅𝑒 = 2500, compared to the Kussner function

(a) 𝑠 = 1.92 (b) 𝑠 = 2 (c) 𝑠 = 2.24

Fig. 7 – Velocity vector field during gust penetration for a NACA0012 airfoil at 𝑅𝑒 = 2500 (T.E. area)

• sonya@tx.technion.ac.il • Phone: 04-8293819

0 50 100 150 2000.2

0.25

0.3

0.35

0.4

s

CL

Membrane

Rigid

0 0.2 0.4 0.6 0.8 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

x/c

y/c

Initial

Final

0 5 10 15 200.2

0.25

0.3

0.35

0.4

s

CL

s=2

Fig. 8 – Comparison of the membrane response to a sharp-edge gust with the response of a rigid airfoil with the membrane initial profile

(a) Lift response (b) Membrane profile

Fig. 9 – Membrane response to a sharp-edge at initial transient

(a) Lift response (b) Membrane shape response

0 0.2 0.4 0.6 0.8 1

2

2.5

3

3.5

4

4.5

5

h

CT

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05max{real(

n)}

max{real(n)}=0

Scan to view a short movie of T.E. area velocity vectors