Laboratory Test Data

Two dimensional boundary layers are characterised by their velocity profile, as shown in Fig. 2.1. In the absence of flow control, the shape of the velocity profile depends solely on the streamwise pressure gradient. A favourable pressure gradient leads to a fuller velocity profile with a negative curvature (∂²u/∂y²). In contrast, an adverse pressure gradient leads to a less full profile with a region of positive curvature, and the consequent existence of an inflection point.

Figure 2.1: Stable and unstable velocity profiles in a flat-plate laminar boundary layer — flat plate corresponds to zero-pressure gradient. (Taken from Green (2006).)

The realm of stability theory is beyond the scope of this thesis; however, Green (2006) notes that for flows in which the velocity profile has an inflection point, transition occurs at much lower Reynolds numbers compared to the zero pressure-gradient case. In contrast, the opposite applies for a more convex velocity profile.

The change of boundary layer thickness and velocity profile due to suction is detailed schematically by Lachmann (1955) in Fig. 2.2. The effect of suction is two-fold: 1) the boundary layer profile is made more convex, and 2) the boundary layer thickness (and hence

2.1. LAMINAR FLOW CONTROL

local Reynolds number) is reduced. This combination leads to a more stable boundary layer.

Figure 2.2: Boundary layer development with suction (from Lachmann (1955)).

These observations can be understood through consideration of the incompressible bound-ary layer momentum equation with surface suction at the wall, which Green (2006) gives to be

µ∂²u

∂y² = dp dxưρV0

∂u

∂y

w

. (2.1)

This expression shows that, as expected, flow acceleration and suction through the wall both promote a convex velocity profile.

The boundary layer is characterised by two integral parameters: the displacement and momentum thickness. Displacement thickness (δ1) is a measure of the displacement of the free stream away from the surface due to boundary-layer growth, whilst momentum thickness (δ2) is a measure of the momentum deficit. These definitions are retained downstream of the aircraft and, for two-dimensional shock-free flow, δ₂ far downstream is directly proportional to the drag of the body.

δ₁ = Z ∞

0

1ư u

u_e

dy (2.2)

δ₂ = Z ∞ 0

u ue

1ư u

ue

dy (2.3)

On integration of Eq. 2.1, combined with the continuity equation and Eqs. 2.2 and 2.3, von K´arm´an derived the momentum integral equation

dδ2

dx + (2 +H12) 1 ue

due

dxδ2 =Cfư V0

ue

, (2.4)

where the shape factor

H12= δ1

δ2

. (2.5)

Equation 2.4 can be re-written in a form that shows the influence of suction on the rate of boundary layer growth

ρ d dx u²_eδ2

=δ1

dP

dx +τwư Vo

ue

. (2.6)

Green (2006) highlights that the term on the left is the rate of growth of momentum deficit in the boundary layer, whilst the first and second terms on the right are the contributions to local pressure and friction drag. He explains that when dP/dx is large in comparison to the skin friction term, the increase in momentum deficit, and hence drag, is proportional to the local thickness.

Drawing largely from a paper by Edwards (1961), Lee (1961) attempts to clarify some common misconceptions of how a reduction in profile drag through surface suction is actually manifested. He explains that, as expected of a laminar boundary layer, the pressure and skin friction drags are low. However, there is a limit to this: excessive suction will produce a boundary layer that is extremely thin, with an associated skin friction coefficient that can approach, or even exceed, that of a thick turbulent boundary layer. (Note, though, that a turbulent boundary layer would result in a larger wake with a larger amount of unrecovered energy left behind in the atmosphere.) Lee (1961) concludes that the application of suction should be such that the boundary layer remains on the verge of transition.

Lachmann (1955) highlights that the need to re-energise the sucked air was investigated by Pfenninger. Early experiments demonstrated an additional 40% reduction in effective drag if the (stagnant) sucked flow is accelerated to free-stream velocity. Therefore, the sucked low momentum fluid, which would otherwise need to be cancelled by an equal and opposite momentum increase in the propulsion wake, does not appear as a deficit in the wake, as illustrated in Fig. 2.3. To summarise, the actual drag reduction effects are: 1) a reduction of wake drag to a practically insignificant magnitude, and 2) a recovery with high mechanical efficiency of the kinetic energy of the air which has been sucked from the lower, almost stagnant, strata of the boundary layer.

In contrast to natural laminar flow, Lachmann (1955) observes that the critical parameter for laminar flow with distributed suction is Reynolds number per unit chord. Denning et al.

(1997) interpret this result as there being no upper limit to wing chord size because the boundary layer is continually ‘regenerating itself’.

The maximum possible unit Reynolds number depends on the surface finish and at-mospheric turbulence levels (Lachmann (1955)); these effects become less significant with

2.1. LAMINAR FLOW CONTROL

Figure 2.3: Wake profile for an integrated propulsion and suction system (from Lee (1961)).

increasing cruise altitude as a result of the increase in kinematic viscosity (Green (2006)).

The F94-A aeroplane maintained full-chord laminar flow for unit Reynolds numbers of over 13×10⁶ m⁻¹; therefore, based on the state-of-the-art at the time, Lachmann (1955) proposes that unit Reynolds numbers of up to 16×10⁶ m⁻¹ may realistically achievable.

With the growth rate of a boundary layer with suction independent of chord length, the boundary layer thickness relative to the wing chord is reduced. This result implies a reduction in drag coefficient with Re_c. This is shown in Fig. 2.4. Furthermore, at high Reynolds numbers, e.g. 100×10⁶, complete laminar flow aerofoils produce drag levels around one tenth of turbulent values.

There is a drawback with suction, however. Power is needed to energise the suction system, in order to develop a sufficient pressure ratio to maintain the desired flow rate. This is commonly referred to as an equivalent ‘pump drag’.

Green (2006) provides illustrative profile and pump drag coefficients taken from the upper surface suction glove, flight-tested by Northrop, on the F-94A; these are repeated in Tab. 2.1.

Green (2006) comments that the fully laminar values are conservative, as the suction flow rates are 10% higher than for minimum drag; added to this, they have been obtained by doubling the upper surface values (as no data for the lower surface was available). The profile drag for the fully laminar flow is approximately a fiftieth of the turbulent case, whilst the pump drag for the fully laminar aerofoil is more than 80% of the total drag. The total drag reduction is 89%: confirming the potential of laminarised surfaces. Green (2006) comments

Figure 2.4: Variation in wing drag coefficient with chord Reynolds number for laminar bound-ary layers (from Denning et al. (1997)).

that the same result is likely to apply for a fully laminar aircraft. Lachmann (1955) even goes to the extent of suggesting that, to a first approximation, the profile drag of laminarised surfaces may be neglected altogether.

Table 2.1: Aerofoil drag comparisons on the F-94A upper surface, all for Rec ≈ 30×10⁶ (from Green (2006)).

Fully turbulent Fully laminar Profile drag 0.00912 0.00018

Pump drag 0 0.00078

Total drag 0.00912 0.00096

2.1. LAMINAR FLOW CONTROL