Linear Instability Mechanisms on Airfoils at Low Reynolds Number

152 Pages·2016·13.58 MB·English

´
UNIVERSIDAD POLITECNICA DE MADRID
ESCUELA TE´CNICA SUPERIOR DE INGENIEROS AERONA´UTICOS
Linear Instability Mechanisms on Airfoils at Low Reynolds
Number: Massive Separation, Wingtip Vortex Formation
and the Trailing Vortex System
Doctoral Thesis
By
Wei He
MEng Aeronautical Engineering
Madrid, July 2016

´ ´
ESCUELA TECNICA SUPERIOR DE INGENIEROS AERONAUTICOS
´
UNIVERSIDAD POLITECNICA DE MADRID
Doctoral Thesis
Linear Instability Mechanisms on Airfoils at Low Reynolds
Number: Massive Separation, Wingtip Vortex Formation
and the Trailing Vortex System
by
Wei He
MEng Aeronautical Engineering
directed by
Vassilis Theofilis
Jos´e Miguel P´erez
Ph.D. in Aeronautical Engineering
Madrid, July 2016
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Piano, piano, si va molto lontano!
– from an Italian proverb
ii
Abstract
The global linear instability analysis of separated flow on homogeneous infinite and finite
wings are numerically investigated, including analysis of the wingtip vortex.
Two- and three-dimensional modal and non-modal instability mechanisms of steady
spanwise homogeneous laminar separated flows over airfoil profiles, placed at large angles
of attack against the oncoming flow, have been investigated using linear global theory.
Three NACA profiles of distinct thickness and camber were considered, in order to assess
geometry effects on the laminar-turbulent transition paths discussed. At the conditions
investigatedlarge-scalesteadyseparationoccurs,suchthatTollmien-Schlichtingandcross-
flow mechanisms were not considered. It is found that the leading modal instability on all
three airfoils is that associated with the Kelvin-Helmholtz (KH) mechanism, taking the
formoftheeigenmodesknownfromanalysisofgenericbluffbodies. Thethree-dimensional
stationaryeigenmodeofthetwo-dimensionallaminarseparationbubble, associatedinear-
lier analyses with the formation on the airfoil surface of large-scale separation patterns
akin to stall-cell, is shown to be stronger damped than the KH mode. Non-modal insta-
bility analysis reveals the potential of the flows to sustain transient growth which becomes
stronger with increasing angle of attack and Reynolds number. Optimal initial conditions
were computed and were found to be analogous to those on a cascade of Low Pressure
Turbine blades. By changing the time-horizon of the analysis these linear optimal initial
conditions are found to evolve into the KH mode. Steady Navier-Stokes equations for a
high angle of attack and Reynolds number flow are achieved through selective damping
frequency method, and its modal analysis is performed. The most unstable mode is os-
cillating after the airfoil and dominates about O(10) chord length. The sub-leading mode
is a KH type as appeared in the low Reynolds number steady flow. The stationary mode
starts immediately behind the airfoil and then decays into the wake.
Thetime-periodicbaseflowsensuinglinearamplificationoftheKHmodeareanalyzed
here via temporal Floquet theory. Two amplified modes are discovered, having character-
istic spanwise wavelengths of approximately 0.6 and 2 chord lengths, respectively. Unlike
secondary instabilities on the circular cylinder, three-dimensional short-wavelength per-
turbations are the first to become linearly unstable on all airfoils. Long-wavelength per-
turbations are quasi-periodic, standing or traveling-wave perturbation that also become
unstable as the Reynolds number is increased further. The dominant short-wavelength
instability gives rise to spanwise periodic wall-shear patterns, akin to the separation cells
encountered on airfoils at low angles of attack and the stall cells found in flight at condi-
tions close to stall. Thickness and camber have quantitative but not qualitative effect on
the secondary instability analysis results obtained.
The previous analysis assums an idealistic wing flow which has a homogeneous bound-
ary conditions in the spanwise direction. A generalized wingtip developed downstream
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Abstract
should be taken into account. To this purpose, a finite wing laminar flow has been con-
sidered. Instability analysis of flow in the wake of a low aspect ratio three-dimensional
wing of elliptic platform, constructed with appropriately scaled Eppler E387 airfoils, has
been performed. The flow field over the airfoil and in its wake was computed by full three-
dimensional direct numerical simulation at a chord Reynolds number of Re = uc = 1750
c ν
and two angles of attack, AoA = 0◦ and 5◦. The spatial eigenvalue problem govern-
ing linear global small-amplitude perturbations superposed upon this base flow has been
solved and results were used to initialize a linear PSE-3D marching procedure without
any simplifying assumptions regarding the form of the trailing vortex system, other than
weak dependence of all flow quantities on the axial spatial direction. Two classes of lin-
early unstable perturbations were identified, namely stronger-amplified symmetric modes
and weaker-amplified antisymmetric disturbances, both peaking at the vortex sheet which
connects the trailing vortices. The amplitude functions of both classes of modes were doc-
umented and N-factor predictions for potential laminar breakdown have been computed.
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