TA6: 3D Shock Control Bump Optimisation
When flying at transonic speeds, shock waves, terminating rejoins of supersonic flow on the upper surface of an aircraft’s wings, generate drag. This drag component can represent a large proportion of overall drag for flight at high chord-wise Mach numbers and lift coefficients, and for airfoil profiles vulnerable to strong shocks such as natural laminar flow wings.
Shock control bumps are capable of significantly reducing wave drag by splitting a single shock wave into a number of weaker oblique compressions. Two dimensional bumps extending along a large proportion of a wing’s span, although capable of good ‘on-design’ performance, are ineffective at reducing drag across a range of typical flight conditions.
Dividing 2D shock control bumps into a number of discrete 3D bumps can reduce the sensitivity of the device, whilst retaining the benefits of shock control.
The design and placement of 3D bumps on a transonic wing is however far from trivial. To limit the computational resource necessary, the optimisation of a 3D shock control device on a simplified geometry, representative of an infinite wing with a constant chord and fixed bump spacing, is proposed.
The aim of current optimisation problem is a multipoint minimisation of overall drag using 3D shock bump control applied to the natural laminar flow RAE5243 airfoil. Each case is given equal weighting, with a successful design aiming to minimise average overall drag. A typical flight condition range is simulated here using three flight Mach numbers, found to have a strong influence on shock Mach number and position, with all other flight variables held constant, as summarised in Table 1.
Reynolds averaged Navier-Stokes solutions using a turbulence model.
Solutions should be carried out on high quality grids with: a far field boundary greater than 20 chord lengths away from the airfoil in all directions and a value of $y^+ \sim 1$ at the aerofoil surface.
Figure 1: Illustration of geometric parameters describing the 3D shock control bump.
Figure 2: RAE5243 wing with 3D shock control bump.
| Case | $M_\infty$ | $\textrm{Re}_{c,\infty}$ | $C_l$ | Flow condition |
|---|---|---|---|---|
| A | 0.74 | $10 \times 10^6$ | 0.45 | 40% transition |
| B | 0.75 | $10 \times 10^6$ | 0.45 | 40% transition |
| C | 0.76 | $10 \times 10^6$ | 0.45 | 40% transition |
- Steady state.
- Non-slip at airfoil surface and far-feild at boundary edges.
A single design minimising average overall drag for the three design cases, $\frac{1}{3} (C_{d\ A} + C_{d\ B} + C_{d\ C} )$, as outlined in Table 1.
Specification of the 3D control bump geometry is achieved using the following six geometric parameters (Fig. 1):
- ramp angle, $θ$
- control position, $cp$
- bump length, $bl$
- crest position, $crest \equiv \frac{\mathrm{control\ length,}\ cl}{\mathrm{bump\ length,}\ bl}$
- aspect ratio, $AR \equiv \frac{\mathrm{bump\ length,}\ bl}{\mathrm{bump\ width,}\ bw}$
- crest ratio, $CR \equiv \frac{\mathrm{crest\ width,}\ cw}{\mathrm{bump\ width,}\ bw}$
Cubic splines define surfaces between the control points listed above, with tangential conditions applied between: the bumps edges and the aerofoil surface and the bump crest and the airfoil surface. Geometories generated on a flat surface, as illustrated in Fig. 1(b), are ‘wrapped’ onto the upper surface of the curved RAE5243 profile, as illustrated in Fig. 2. Span-wise, centre to centre, control bump spacing is fixed at 30% chord. It is suggested that, to take advantage of the problem’s geometrical symmetry, solutions only be carried out over half the domain.
Bounds on Design Parameters
Optimisation is subject to the following six geometric constraints:
$0 \geq \theta \leq 10$
$cp + (1 − crest) bl \leq 0.9$
$cp − crest\ bl \geq 0.3$
$\frac{bl}{AR} \le 0.05$
$0.1 \geq crest \leq 0.9$
$0.1 \geq CR \leq 0.9$
$\min \frac{1}{3} (C_{d\ A} + C_{d\ B} + C_{d\ C} )$
- Drag polars of optimum design illustrating force coefficients at five lift coefficients spaced evenly between $0.4 < C_l < 0.5$ for each design Mach number including comparison with datum ‘no bump’ case.
- Average overall drag of optimum design, including comparison with the datum ‘no bump’ overall average.
