optimal disturbances and receptivity of 3d boundary layers · david tempelmann ardeshir hanifi, dan...
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10/23/08
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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)
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Ercoftac SIG 33 October 2008
David Tempelmann
Linné FLOW Centre KTH Mechanics
Outline
• Background – Motivation – Previous Work
• Method – Governing Equations
• Results – Verification – Parametric Study
• Conclusions
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Ercoftac SIG 33 October 2008
David Tempelmann
Linné FLOW Centre KTH Mechanics
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
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Ercoftac SIG 33 October 2008
David Tempelmann
Linné FLOW Centre KTH Mechanics
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
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Ercoftac SIG 33 October 2008
David Tempelmann
Linné FLOW Centre KTH Mechanics
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
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Ercoftac SIG 33 October 2008
David Tempelmann
Linné FLOW Centre KTH Mechanics
• 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
→
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Ercoftac SIG 33 October 2008
David Tempelmann
Linné FLOW Centre KTH Mechanics
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
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Ercoftac SIG 33 October 2008
David Tempelmann
Linné FLOW Centre KTH Mechanics
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
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Ercoftac SIG 33 October 2008
David Tempelmann
Linné FLOW Centre KTH Mechanics
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)
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Ercoftac SIG 33 October 2008
David Tempelmann
Linné FLOW Centre KTH Mechanics
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
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Ercoftac SIG 33 October 2008
David Tempelmann
Linné FLOW Centre KTH Mechanics
• 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)
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Ercoftac SIG 33 October 2008
David Tempelmann
Linné FLOW Centre KTH Mechanics
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)
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Ercoftac SIG 33 October 2008
David Tempelmann
Linné FLOW Centre KTH Mechanics
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
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Ercoftac SIG 33 October 2008
David Tempelmann
Linné FLOW Centre KTH Mechanics
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