PIV-based estimation of unsteady loads on a flat plate at high angle of attack using momentum equation approaches

PIV-based estimation of unsteady loads

on a flat plate at high angle of attack

using momentum equation approaches

A. Guissart1, L. P. Bernal2, G. Dimitriadis1 and V.E. Terrapon1

1_{Department of Aerospace and Mechanical Engineering, University of Liege} 2_{Department of Aerospace Engineering, University of Michigan}

Motivation

Forces measurement

using load sensor Not always possible

‚ Moving body with high inertia ‚ Small forces

Motivation

Forces measurement

using load sensor Not always possible

‚ Moving body with high inertia ‚ Small forces

Objective

From PIV experiment Indirect calculation of forces

t Fi

Integral momentum equation

Fi“ ´ ż S ρBtuidS ´ ż C ρui`ujnj˘ dC ´ ż C pnidC ` ż C τijnjdC

Noca’s flux equation

Fi “ ż C njγflux ji dC ´ d dt ż Cbptq ρnjujXi dC with γflux ji “ f`ui, Btui, Bjui, Xi ˘ calculate

S

C

C

C

_bptq

C

S

C

_bptq OR

Objective

From PIV experiment Indirect calculation of forces

t Fi

Integral momentum equation

Fi“ ´ ż S ρBtuidS ´ ż C ρui`ujnj˘ dC ´ ż C pnidC ` ż C τijnjdC

Noca’s flux equation

Fi “ ż C njγflux ji dC ´ d dt ż Cbptq ρnjujXi dC with γflux ji “ f`ui, Btui, Bjui, Xi ˘ calculate

S

C

C

C

_bptq

C

S

C

_bptq OR

Objective

From PIV experiment Indirect calculation of forces

t Fi

Integral momentum equation

Fi“ ´ ż S ρBtuidS ´ ż C ρui`ujnj˘ dC ´ ż C pnidC` ż C τijnjdC

Noca’s flux equation

Fi “ ż C njγflux ji dC ´ d dt ż Cbptq ρnjujXi dC with γflux ji “ f`ui, Btui, Bjui, Xi ˘ calculate

S

C

C

C

_bptq

C

S

C

_bptq OR

Objective

From PIV experiment Indirect calculation of forces

t Fi

Integral momentum equation

Fi“ ´ ż S ρBtuidS ´ ż C ρui ` ujnj˘ dC ´ ż C pnidC ` ż C τijnjdC

Noca’s flux equation

Fi “ ż C njγflux ji dC ´ d dt ż Cbptq ρnjujXi dC with γflux ji “ f`ui, Btui, Bjui, Xi ˘ calculate

S

C

C

C

_bptq

C

S

C

_bptq OR

Objective

From PIV experiment Indirect calculation of forces

t Fi

Integral momentum equation

Fi“ ´ ż S ρBtuidS ´ ż C ρui`ujnj˘ dC ´ ż C pnidC ` ż C τijnjdC

Noca’s flux equation

Fi “ ż C njγflux ji dC ´ d dt ż Cbptq ρnjujXi dC with γflux ji “ f`ui, Btui, Bjui, Xi ˘ calculate

S

C

C

C

_bptq

C

S

C

_bptq OR

Objective

From PIV experiment Indirect calculation of forces

t Fi

Integral momentum equation

Fi“ ´ ż S ρBtuidS ´ ż C ρui`ujnj˘ dC ´ ż C pnidC ` ż C τijnjdC

Noca’s flux equation

Fi “ ż C njγflux ji dC ´ d dt ż Cbptq ρnjujXi dC with γflux ji “ f`ui, Btui, Bjui, Xi ˘ calculate

S

C

C

C

_bptq

C

S

C

_bptq OR

Objective

From PIV experiment Indirect calculation of forces

t Fi

Integral momentum equation

Fi“ ´ ż S ρBtuidS ´ ż C ρui`ujnj˘ dC ´ ż C pnidC ` ż C τijnjdC

Noca’s flux equation

Fi “ ż C njγflux ji dC ´ d dt ż Cbptq ρnjujXi dC with γflux ji “ f`ui, Btui, Bjui, Xi ˘ calculate

S

C

C

C

_bptq

C

S

C

_bptq OR

Objective

From PIV experiment Indirect calculation of forces

t Fi

Integral momentum equation

Fi“ ´ ż S ρBtuidS ´ ż C ρui`ujnj˘ dC ´ ż C pnidC ` ż C τijnjdC

Noca’s flux equation

Fi “ ż C njγflux ji dC ´ d dt ż Cbptq ρnjujXi dC with γflux ji “ f`ui, Btui, Bjui, Xi ˘ calculate

S

C

C

C

_bptq

C

S

C

_bptq OR

Objective

From PIV experiment Indirect calculation of forces

t Fi

Integral momentum equation

Fi“ ´ ż S ρBtuidS ´ ż C ρui`ujnj˘ dC ´ ż C pnidC ` ż C τijnjdC

Noca’s flux equation

Fi “ ż C njγflux ji dC ´ d dt ż Cbptq ρnjujXi dC with γflux ji “ f`ui, Btui, Bjui, Xi ˘ calculate

S

C

C

C

_bptq

C

S

C

_bptq OR

Test cases

Flat plate c t ‚ c “ 7.6cm ‚ c_{{t“ 16} ‚ Rounded edges

1. Large amplitude pitching

∆α LE center ‚ α “ 0˝ ‚ ∆α “ 30˝ ‚ f based on k “ 0.2 ‚ 2 pivot axis

2. Small amplitude pitching

α ∆α ‚ α “ α `∆α sin p2πftq ‚ α “ 45˝ ‚ f based on St “ 0.155 ‚ ∆α « 1˝

Test cases

Flat plate c t ‚ c “ 7.6cm ‚ c_{{t“ 16} ‚ Rounded edges

1. Large amplitude pitching

∆α LE center ‚ α “ 0˝ ‚ ∆α “ 30˝ ‚ f based on k “ 0.2 ‚ 2 pivot axis

2. Small amplitude pitching

α ∆α ‚ α “ α `∆α sin p2πftq ‚ α “ 45˝ ‚ f based on St “ 0.155 ‚ ∆α « 1˝

Test cases

Flat plate c t ‚ c “ 7.6cm ‚ c_{{t“ 16} ‚ Rounded edges

1. Large amplitude pitching

∆α LE center ‚ α “ 0˝ ‚ ∆α “ 30˝ ‚ f based on k “ 0.2 ‚ 2 pivot axis

2. Small amplitude pitching

α ∆α ‚ α “ α `∆α sin p2πftq ‚ α “ 45˝ ‚ f based on St “ 0.155 ‚ ∆α « 1˝

Methodology

Data collection

Data collection Pre-processing

Pre-processing

Forces calculation

Forces calculation

‚ Water channel at the University of Michigan

Rotary

stage traverseLinear Plate Sensor

ñ Synchronized PIV

ñ Direct force measurements

shadow

laser laser

shadow

overlap overlap

‚ Shadow due to mounting ‚ Use of symmetry ‚ Stitching of two images

ñ Overlap used for stitching ñ But may introduce noise

t Fi

S

C

Fi“ ´ ż S ρBtuidS ´ ż C ρui`ujnj˘ dC ´ ż C pnidC ` ż C τijnjdC Fi“ ż C njγjiflux dC´ d dt ż Cbptq ρnjujXi dC

Methodology

Data collection

Data collection Pre-processing

Pre-processing

Forces calculation

Forces calculation

‚ Water channel at the University of Michigan

Rotary

stage traverseLinear Plate Sensor

ñ Synchronized PIV

ñ Direct force measurements

shadow

laser laser

shadow

overlap overlap

‚ Shadow due to mounting ‚ Use of symmetry ‚ Stitching of two images

ñ Overlap used for stitching ñ But may introduce noise

t Fi

S

C

Fi“ ´ ż S ρBtuidS ´ ż C ρui`ujnj˘ dC ´ ż C pnidC ` ż C τijnjdC Fi“ ż C njγjiflux dC´ d dt ż Cbptq ρnjujXi dC

Methodology

Data collection

Data collection

Pre-processing

Pre-processing Forces calculation

Forces calculation ‚ Water channel at the

University of Michigan

Rotary

stage traverseLinear Plate Sensor

ñ Synchronized PIV

ñ Direct force measurements

shadow

laser

laser

shadow

overlap overlap

‚ Shadow due to mounting ‚ Use of symmetry ‚ Stitching of two images

ñ Overlap used for stitching ñ But may introduce noise

t Fi

S

C

Fi“ ´ ż S ρBtuidS ´ ż C ρui`ujnj˘ dC ´ ż C pnidC ` ż C τijnjdC Fi“ ż C njγjiflux dC´ d dt ż Cbptq ρnjujXi dC

Methodology

Data collection

Data collection

Pre-processing

Pre-processing Forces calculation

Forces calculation ‚ Water channel at the

University of Michigan

Rotary

stage traverseLinear Plate Sensor

ñ Synchronized PIV

ñ Direct force measurements

shadow

laser

laser

shadow

overlap overlap

‚ Shadow due to mounting ‚ Use of symmetry ‚ Stitching of two images

ñ Overlap used for stitching ñ But may introduce noise

t Fi

S

C

Fi“ ´ ż S ρBtuidS ´ ż C ρui`ujnj˘ dC ´ ż C pnidC ` ż C τijnjdC Fi“ ż C njγjiflux dC´ d dt ż Cbptq ρnjujXi dC

Methodology

Data collection

Data collection

Pre-processing

Pre-processing Forces calculation

Forces calculation ‚ Water channel at the

University of Michigan

Rotary

stage traverseLinear Plate Sensor

ñ Synchronized PIV

ñ Direct force measurements

shadow

laser laser

shadow

overlap overlap

‚ Shadow due to mounting ‚ Use of symmetry ‚ Stitching of two images

ñ Overlap used for stitching ñ But may introduce noise

t Fi

S

C

Fi“ ´ ż S ρBtuidS ´ ż C ρui`ujnj˘ dC ´ ż C pnidC ` ż C τijnjdC Fi“ ż C njγjiflux dC´ d dt ż Cbptq ρnjujXi dC

Methodology

Data collection Data collection Pre-processing Pre-processing Forces calculation Forces calculation

‚ Water channel at the University of Michigan

Rotary

stage traverseLinear Plate Sensor

ñ Synchronized PIV

ñ Direct force measurements

shadow

laser laser

shadow

overlap overlap

‚ Shadow due to mounting ‚ Use of symmetry ‚ Stitching of two images

ñ Overlap used for stitching ñ But may introduce noise

t Fi

S

C

Fi“ ´ ż S ρBtuidS ´ ż C ρui`ujnj˘ dC ´ ż C pnidC ` ż C τijnjdC Fi“ ż C njγjiflux dC´ d dt ż Cbptq ρnjujXi dC

Methodology

Data collection Data collection Pre-processing Pre-processing Forces calculation Forces calculation

‚ Water channel at the University of Michigan

Rotary

stage traverseLinear Plate Sensor

ñ Synchronized PIV

ñ Direct force measurements

shadow

laser laser

shadow

overlap overlap

‚ Shadow due to mounting ‚ Use of symmetry ‚ Stitching of two images

ñ Overlap used for stitching ñ But may introduce noise

t Fi

S

C

Fi“ ´ ż S ρBtuidS ´ ż C ρui`ujnj˘ dC ´ ż C pnidC ` ż C τijnjdC Fi“ ż C njγjifluxdC´ d dt ż Cbptq ρnjujXi dC

Methodology: forces calculation

t Fi PIV data ‚ Phase averaging ‚ Definition of C Gradients calculation ‚ Finite difference Calculation of p ‚ Using NS equations Bip “ ´ρBtui´ ρujBjui `µB2_jjui´ Bju1 iu1j ‚ Integration along C Calculation of ´d dt ż Cbptq ρnjujXi dC ‚ Analytical formulation

or

Fi calculation ‚ Integration on C and S

C

S

A “ E

B

C

D

Methodology: forces calculation

t Fi PIV data ‚ Phase averaging ‚ Definition of C Gradients calculation ‚ Finite difference Calculation of p ‚ Using NS equations Bip “ ´ρBtui´ ρujBjui `µB2_jjui´ Bju1 iu1j ‚ Integration along C Calculation of ´d dt ż Cbptq ρnjujXi dC ‚ Analytical formulation

or

Fi calculation ‚ Integration on C and S

C

S

A “ E

B

C

D

Methodology: forces calculation

t Fi PIV data ‚ Phase averaging ‚ Definition of C Gradients calculation ‚ Finite difference Calculation of p ‚ Using NS equations Bip “ ´ρBtui´ ρujBjui `µB2_jjui´ Bju1 iu1j ‚ Integration along C Calculation of ´d dt ż Cbptq ρnjujXi dC ‚ Analytical formulation

or

Fi calculation ‚ Integration on C and S

C

S

A “ E

B

C

D

Methodology: forces calculation

t Fi PIV data ‚ Phase averaging ‚ Definition of C Gradients calculation ‚ Finite difference Calculation of p ‚ Using NS equations Bip “ ´ρBtui´ ρujBjui `µB2_jjui´ Bju1 iu1j ‚ Integration along C Calculation of ´d dt ż Cbptq ρnjujXi dC ‚ Analytical formulation

or

Fi calculation ‚ Integration on C and S

C

S

A “ E

B

C

D

Methodology: forces calculation

t Fi PIV data ‚ Phase averaging ‚ Definition of C Gradients calculation ‚ Finite difference Calculation of p ‚ Using NS equations Bip “ ´ρBtui´ ρujBjui `µB2_jjui´ Bju1 iu1j ‚ Integration along C Calculation of ´d dt ż Cbptq ρnjujXi dC ‚ Analytical formulation

or

Fi calculation ‚ Integration on C and S

C

S

A “ E

B

C

D

Methodology: forces calculation

t Fi PIV data ‚ Phase averaging ‚ Definition of C Gradients calculation ‚ Finite difference Calculation of p ‚ Using NS equations Bip “ ´ρBtui´ ρujBjui `µB2_jjui´ Bju1 iu1j ‚ Integration along C Calculation of ´d dt ż Cbptq ρnjujXi dC ‚ Analytical formulation

or

Fi calculation ‚ Integration on C and S

C

S

A “ E

B

C

D

Methodology: forces calculation

t Fi PIV data ‚ Phase averaging ‚ Definition of C Gradients calculation ‚ Finite difference Calculation of p ‚ Using NS equations Bip “ ´ρBtui´ ρujBjui `µB2_jjui´ Bju1 iu1j ‚ Integration along C Calculation of ´d dt ż Cbptq ρnjujXi dC ‚ Analytical formulation

or

Fi calculation ‚ Integration on C and S

C

S

A “ E

B

C

D

Large amplitude pitching

center LE center LE

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces Momentum

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces Momentum Noca

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces Momentum Noca

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d Lift Drag

0

T

_{

₄ T

_{

₂

´1

0

1

2

c

l Center Leading edge

0

T

_{

₄ T

_{

₂

´1

0

1

2

c

l from Noca from momentum Lift

‚ Mean, standard deviation and time evolution from both

‚ Noca’s equation more noise sensitive

Large amplitude pitching

center LE center LE

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces Momentum

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces Momentum Noca

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces Momentum Noca

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d Lift Drag

0

T

_{

₄ T

_{

₂

´1

0

1

2

c

l Center Leading edge

0

T

_{

₄ T

_{

₂

´1

0

1

2

c

l from Noca from momentum Lift

‚ Mean, standard deviation and time evolution from both

‚ Noca’s equation more noise sensitive

Large amplitude pitching

center LE center LE

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces Momentum

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces Momentum Noca

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces Momentum Noca

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d Lift Drag

0

T

_{

₄ T

_{

₂

´1

0

1

2

c

l Center Leading edge

0

T

_{

₄ T

_{

₂

´1

0

1

2

c

l from Noca from momentum Lift

‚ Mean, standard deviation and time evolution from both

‚ Noca’s equation more noise sensitive

Large amplitude pitching

center LE center LE

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces Momentum

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces Momentum Noca

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces Momentum Noca

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d Lift Drag

0

T

_{

₄ T

_{

₂

´1

0

1

2

c

l Center Leading edge

0

T

_{

₄ T

_{

₂

´1

0

1

2

c

l from Noca from momentum Lift

‚ Mean, standard deviation and time evolution from both

‚ Noca’s equation more noise sensitive

Large amplitude pitching

center LE center LE

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces Momentum

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces Momentum Noca

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces Momentum Noca

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d Lift Drag

0

T

_{

₄ T

_{

₂

´1

0

1

2

c

l Center Leading edge

0

T

_{

₄ T

_{

₂

´1

0

1

2

c

l from Noca from momentum Lift

‚ Mean, standard deviation and time evolution from both

‚ Noca’s equation more noise sensitive

Large amplitude pitching

center LE center LE

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces Momentum

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces Momentum Noca

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´2

´1

0

1

2

c

l Direct forces Momentum Noca

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

´0.5

0

0.5

1

1.5

c

d Lift Drag

0

T

_{

₄ T

_{

₂

´1

0

1

2

c

l Center Leading edge

0

T

_{

₄ T

_{

₂

´1

0

1

2

c

l from Noca from momentum Lift

‚ Mean, standard deviation and time evolution from both

‚ Noca’s equation more noise sensitive ‚ Momentum able to compute phase-lag

Small amplitude pitching

45◦ ∆α

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

l Direct forces

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

l Direct forces Momentum

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

l Direct forces Momentum Noca

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

d

‚ Only mean from momentum ‚ Unusable results from Noca’s

equation

ñ Why a such big difference?

Small amplitude pitching

45◦ ∆α

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

l Direct forces

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

l Direct forces Momentum

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

l Direct forces Momentum Noca

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

d

‚ Only mean from momentum ‚ Unusable results from Noca’s

equation

ñ Why a such big difference?

Small amplitude pitching

45◦ ∆α

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

l Direct forces

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

l Direct forces Momentum

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

l Direct forces Momentum Noca

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

d

‚ Only mean from momentum ‚ Unusable results from Noca’s

equation

ñ Why a such big difference?

Small amplitude pitching

45◦ ∆α

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

l Direct forces

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

l Direct forces Momentum

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

d

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

l Direct forces Momentum Noca

0

T

_{

₄ T

_{

₂ 3T

_{

₄

_T

0.8

1

1.2

1.4

c

d

‚ Only mean from momentum ‚ Unusable results from Noca’s

equation

ñ Why a such big difference?

Small amplitude pitching

45◦

∆α

center

‚ Small amplitude pitching ñ flow is 3D ñ Impact on stitching and noise

Small amplitude pitching

45◦

∆α

center

‚ Small amplitude pitching ñ flow is 3D ñ Impact on stitching and noise

Conclusion and future work

Indirect methods are able to estimate forces ‚ Good estimation of mean coefficients

‚ Good estimation of temporal evolution for large amplitude

‚ Methods are noise sensitive

ñ Integral momentum equation seems to be more robust

‚ Impact of 3rddimension on p