In two-dimensional boundary layers, Tollmien-Schlichting (T-S) waves are the primary insta-bility mechanism. Saric and Reed (2006) describe these as essentially two-dimensional convec-tive travelling wave disturbances generating streamwise instabilities that are slow growing.
This instability is influenced by the shape of the non-dimensional boundary-layer velocity profile, which Hall et al. (1984) explain is due to the associated effects on the distribution of vorticity within the layer and viscous effects due to the change in Reynolds number. Green (2006) explains that, for high Reynolds number flows, these disturbances can be amplified even for profiles which do not have an inflection point; whilst for low Reynolds number flows small disturbances can be damped out.
2.1.3.2 Taylor-G¨ortler Vortices
Over the concave section of the lower surface of a cambered aerofoil, Taylor-G¨ortler vortex-induced instabilities dominate. These are stationary, streamwise oriented, counter-rotating vortices, as shown schematically in Fig. 2.6. Saric and Reed (2006) comment that this mechanism can easily be controlled through the use of convex curvature, whereby vortices of the opposite sign are generated, and are more stabilising than those produced by the concave curvature are destabilising; furthermore, they become less significant the larger the sweep angle is.
Figure 2.6: Schematic diagram of Taylor-G¨ortler vortices (from Joslin (1998b)).
2.1.3.3 Crossflow Instabilities
The crossflow (CF) instability appears as co-rotating vortices, which all rotate in the same direction and whose axes are aligned to within a few degrees of the inviscid streamlines, as shown in Fig. 2.7(a). They dominate in regions where the flow is either accelerating or
2.1. LAMINAR FLOW CONTROL
decelerating, as both the surface and flow streamlines are highly curved. The combination of wing sweep and pressure gradient deflect the inviscid-streamlines inboard (in a manner shown in Fig. 2.7(a)), which, when combined with the low momentum fluid in the boundary layer, makes the deflection even larger (Reed and Saric (1989)).
Figure 2.7(b) shows that there is an inflection point in the crossflow velocity profile: at the wall the no-slip condition is satisfied, whilst the crossflow velocity component in the free stream is zero. This causes a strong primary instability which depends on the crossflow Reynolds number, defined as ReCF =Wmaxδ10/ν — the boundary layer thicknessδ10 corre-sponds to the point where the the crossflow velocity is at 10% of its maximum value (Reed and Saric (1989)).
(a) Crossflow instabilities (from Joslin (1998a))
(b) Velocity components within a swept boundary layer (from Reed and Saric (1989)).
Figure 2.7: Schematic diagram of the crossflow instability and velocity profile.
2.1.3.4 Attachment-Line Instabilities
A further issue in swept-wing transition is the stability of the ‘attachment-line’ boundary layer. This instability develops along the ‘stagnation line’ and causes transition in the leading edge region. It is shown schematically in Fig. 2.8.
Figure 2.8: Schematic diagram of attachment-line flow (from Joslin (1998b)).
By considering an infinite swept-wing attachment-line boundary layer in a low turbulence free stream, Poll (1979) derived a governing Reynolds number dependent upon edge conditions and a characteristic length (which in turn depends on the chordwise velocity gradient), given by
Rea.l =
Ve2c νeU∞U⊥
12
, (2.7)
whereU⊥= Uc
∞
due
ds
s=0. The critical value of this attachment-line Reynolds number beyond which transition occurs was experimentally determined by Poll (1979) to be in the range 600 – 700.
Taking Eq. 2.7, and considering the potential-flow solution for a cylinder, Poll (1979) showed that, for this case,
Rea.l= U∞r
ν ×sin Λ tan Λ 2
12
. (2.8)
In Eq. 2.8 we see that there is an explicit dependence on unit Reynolds number, leading-edge radius and wing sweep. Saric and Reed (2006) add that, for most applications, reduction of the nose radius for a specific sweep angle is a practical form of control. (This also has the compounded benefit of decreasing the chordwise extent of the crossflow region and providing a more rapid acceleration of the flow over the wing.)
2.1. LAMINAR FLOW CONTROL
2.1.3.5 Shock Wave / Laminar Boundary Layer Interactions
During flight testing of the F-94A a new potential problem appeared. As the aircraft speed was increased to the point where the local Mach number on the wing surface exceeded about 1.10, full-chord laminar flow was lost with the slot configuration tested (Braslow (1999)).
The problem was attributed to the steep pressure rise through the shock waves that formed.
In more recent experiments performed by Harris et al. (1988), much effort was put into obtaining a shock-free supercritical aerofoil with a large supersonic zone of over 80% of the upper surface. The maximum local Mach number was recorded to have reached about 1.11
— greater in both extent and magnitude than those measured on the F-94A. A problem with the aerofoil, however, was that it had a very narrow off-design operating range.
Harris et al. (1988) reviewed some of the earlier work on this and found that suction laminarisation only appears practicable in regions of weak shocks on transonic aerofoils;
moreover, the pressure rise that a laminar boundary layer can negotiate in the region of an incident shock-wave decreases with increasing Reynolds number, unless the thickness of the upstream boundary layer can be reduced in some manner, for example, by suction. However, Harris et al. (1988) conclude that quantification for boundary layer control in such shock-interaction regions still requires experimentation.
2.1.3.6 Example Suction Distributions
As we have seen, the presence of T-S and CF disturbances in the boundary layer are primarily dependent on the pressure gradient and wing sweep angle. However, Joslin (1998b) comments that for wings with sweeps between 0◦ and 25◦, T-S wave instabilities dominate; whilst for wing sweeps greater than 25◦ both T-S and CF disturbances are present; and for even greater wing sweeps of above 30◦ to 35◦, CF disturbances alone dominate near the wing leading edge.
An example chordwise suction distribution necessary for laminarisation of the 30◦ swept X-21A aircraft is shown in Fig. 2.9. (The stability calculations were performed assuming incompressible laminar boundary layers for flow over an infinitely swept wing. The calculated suction distribution is compared against that measured during flight tests in the figure, and agrees reasonably well.) The pronounced suction over the leading edge is due to a combination of strong crossflow instabilities in the favourable pressure gradient region and attachment-line instabilities. (In order to prevent attachment-line contamination from the turbulent fuselage, a number of control techniques were employed: a fence, vertical slots, and a gutter.) Over the mid-chord, where there is a mild adverse pressure gradient, T-S waves dominate and these require a lower degree of surface suction for stabilisation. Toward the trailing edge, the increased adverse pressure gradient results in a higher requirement, due to a combination of
T-S and CF instabilities.
Figure 2.9: Example suction flow velocity distribution for the 30◦ (quarter-chord) sweep X-21A aircraft (Joslin (1998a)).
In contrast, we can see in Fig. 2.10 that the shape of the suction distribution calculated for the Vampire, with a quater-chord sweep of 10◦, differs considerably. Suction stabilisation against crossflow instabilities over the front of the aerofoil is absent given the reduced wing sweep, whilst the requirement increase over the rear of the aerofoil, due to the adverse pressure gradient, is much further downstream.
Figure 2.10: Example suction flow velocity distribution for the 10◦ (quarter-chord) sweep Vampire aircraft (from Head (1955)).
In a review of the historical aspects of LFC, Denning et al. (1997) quote Lachmann, who comments that only 5 – 10% extra suction flow is required on a swept planform relative to an unswept one. Quantification for a specific case confirming this does not seem to be available in the open literature.
2.1. LAMINAR FLOW CONTROL