optimal disturbances and receptivity of 3d boundary layers · david tempelmann ardeshir hanifi, dan...

10/23/08

1

1

Ercoftac SIG 33 October 2008

David Tempelmann

Linné FLOW Centre KTH Mechanics

Optimal Disturbances and Receptivity

of 3D Boundary Layers

David Tempelmann Ardeshir Hanifi, Dan Henningson

Ercoftac SIG 33 Meeting, 16-18th October 2008

The work described in this presentation is supported by the European Commission through the FP6 project “Telfona” (Contract No AST4-CT-2005-516109)

2


David Tempelmann


Outline

•  Background –  Motivation –  Previous Work

•  Method –  Governing Equations

•  Results –  Verification –  Parametric Study

•  Conclusions

10/23/08

2

3


David Tempelmann


Motivation

•  Work is part of EU-project TELFONA –  Objective to demonstrate ability to predict aircraft

performance based on wind tunnel tests and CFD-results

•  Investigate receptivity in 3D-boundary layers –  Different levels of free stream turbulence

in wind tunnel compared to free flight –  How do external perturbations enter

the boundary layer

•  Compute optimal disturbances –  First step towards a receptivity model –  Can later be used to compute receptivity coefficients

Initial energy

4


David Tempelmann


Previous Work

•  Optimal disturbances in 2D boundary layer –  Andersson et. al. (1999) and Luchini (2000) –  Corbett & Bottaro (2000) and Levin & Henningson (2003) –  Streamwise vortices that develop into streamwise streaks –  Very different from Tollmien-Schlichting waves

•  Optimal growth in swept boundary layers –  Corbett & Bottaro (2001) –  Temporal approach using Orr-Sommerfeld eq. –  Transient and exponential growth are complementary

•  Receptivity in 3D boundary layers –  Pralits & Luchini (Ercoftac 2007) –  Compute amplification factor of crossflow modes –  Projection of disturbances on adjoint modes

10/23/08

3

5


David Tempelmann


Governing Equations

•  Objective: Describe algebraically and exponentially growing 3D-disturbances using parabolic equations

•  Why not use PSE?

–  PSE follows one exponentially growing mode –  Transient growth results from superposition of modes

•  PSE can not describe transient growth

6


David Tempelmann


•  Disturbance Ansatz

•  Optimal disturbances almost aligned with the external streamline (Corbett & Bottaro 2001)

Governing Equations

→ Prescribing α removes ambiguity

→ Oscillations absorbed by exp-function

→ Growth absorbed by shape function

→

10/23/08

4

7


David Tempelmann


Governing Equations •  Assume slow variation of shape function along x

–  Restriction in terms of exponential growth

•  How should terms be scaled? –  Transient growth associated with boundary layer scaling –  PSE-scaling used for exponential growth

•  Apply both scalings, keep terms of order lower than –  One extra term resulting from B.L.-scaling

compared to PSE:

– 

•  Optimise using method of Lagrangian multipliers

8


David Tempelmann


Verification

•  Does optimal disturbance develop into crossflow mode?

–  Compare downstream development of optimal disturbance with that of corresponding cross flow mode (same β)

–  Compare solution of adjoint equation at initial position to that of the corresponding adjoint PSE solution

10/23/08

5

9


David Tempelmann


Results •  Baseflow: Falkner-Skan-Cooke boundary layer

–  Sweep angle: 45°

–  Adverse (βH = -0.05) and favourable (βH = 0.1) pressure gradient

– 

•  Optimal Growth for βH = 0.1, x0/L = 0.005 and ω = 0 N-Factor of maximum growth - 0.5*ln(E/E0)

10


David Tempelmann


Results •  Compare optimal disturbances and their downstream

response for selected points in x-β plane

Optimal init. disturbance (subcritical)

Response

Optimal init. disturbance (supercritical)

Response

Optimal init. disturbance Response

vs

us

w

us

w, vs

10/23/08

6

11


David Tempelmann


•  Adverse pressure gradient case (βH=-0.05) Results

Optimal init. disturbance (supercritical)

Response

Optimal init. disturbance Response vs

us

w

us

w, vs

N-Factor of maximum growth - 0.5*ln(E/E0)

12


David Tempelmann


Results

•  Dependence of the optimal growth at xopt = 1 on the initial position x0

•  Leading edge is not the optimal initial position

βH = 0.1 βH = -0.05

N-Factor of maximum growth - 0.5*ln(E/E0) N-Factor of maximum growth - 0.5*ln(E/E0)

10/23/08

7

13


David Tempelmann


Results •  Dependence of optimal growth on frequency •  N-Factors of maximum growth, x0 = 0.005

βH = 0.1 βH = -0.05 su

per

critic

al

subcr

itic

al

14


David Tempelmann


Conclusions / Outlook •  Set of parabolic equations describes algebraically and

exponentially growing disturbances

•  Restricted in terms of exponential growth

•  Transient and exponential growth generate disturbances of similar structure

•  Use optimal disturbances for receptivity calculations –  Project onto free stream turbulence

optimal disturbances and receptivity of 3d boundary layers · david tempelmann ardeshir hanifi, dan...

Documents