Copyright © A. Filippone (1999-2003). All Rights Reserved.

Wings for All Speeds

Delta Wings


Certainly Concorde flies better than a lot of deltas, but one fact of life is that deltas come in on the back side of the drag curve, and you’ve got to remember it.

[B. Trubshaw, Director of Concorde Flight Test, 1969]

Delta wings (Deltas) are symmetrical triangular wings designed to fly at subsonic or supersonic speeds. At supersonic speeds the leading-edge can be subsonic, sonic or supersonic, depending on the relation between sweep angle and speed (see below).

Leading edges are generally linear, although there are cases of more complex geometries, such as the ogive delta (Concorde SST), the gothic delta, the cranked delta (Lockheed CL-823), the double delta (SAAB Viggen), delta + canards (North American XB-70 and others).

Almost all delta wings fall into the category of low aspect-ratio wings. Their aspect-ratio is defined by AR = 4/tan(D), where D is the leading edge sweep angle (this lead to AR less than 3 in most cases; about 1.8 in the case of Concorde). Wing thickness is generally small.

The problem is to find the aerodynamic properties of the wing (CL, Cd, Cm, Cp distribution, etc.), along with the lateral and longitudinal stability characteristics of the wings at different operation points.

The technical literature on deltas is huge, and it is safe to say that all speeds and sweep angles have been investigated (experimental, theoretical and computational research).

Delta Wing in Subsonic Flow

Flows past delta wings are severely compounded by the leading edge separation, by the roll-up structure of the concentrated vortices, and by the lateral and longitudinal instability that is consequent to large sweeps, high-angle of attack, and sharp manoeuvres.

Although the aerodynamics of the delta wing is non-linear, most of the reseach has relied for a long time on linearized small perturbation theory (shortly reviewed below).

Computational methods (ex. vortex lattice methods, panel codes) have proved tremendously effective at low speeds and unsteady flows (Katz, 1984, among others).

Linearized Theory

Linearized theory for the slender body with small angle of attack (Munk, 1924; RT Jones, 1945) leads to a very simple conclusion: lift is produced in the conical flow created by the streamwise variation of span b(x). There is one singularity at the apex of the delta, where the theoretical pressure would be infinite.

The expression for the lift coefficient is CL=2 , that is correct only for very low aspect-ratios (AR=1). The corresponding induced drag coefficient is , that is just half the value that is expected at angle of attack . The center of pressure is found at 2/3 chord from the pointed leading-edge, where the pressure is also singular.

Fuselage Effects

The effect of a fuselage can also be estimated by a more general formulation (Ashley-Landhal, 1965) that gives a lift coefficient in the wing-body configuration is lower that the wing-alone.

Lifting surface theory (for ex., vortex lattice method) is a better approach to the prediction of the basic coefficients. There are also methods for arrowhead wings (Mangler, 1955) and wings in yawed flow (Carafoli, 1969).

Delta Wings in Supersonic Flow

Delta wings are appropriate plan forms to fly at supersonic and hypersonic speeds, therefore there has been a long time interest in investigating the effects of high Mach numbers.

The principle of independence (Buseman, 1935) allows to investigate separately wings with subsonic and supersonic leading edges (e.g.for which the normal Mach number is below or above the speed of sound).

Separation Characteristics

In general, wings with subsonic leading edges are characterized by leading edge separation; wings with supersonic leading edge are characterized a Prandtl-Meyer expansion.

The main parameters of the wing problem are sweep, free stream Mach number, angle of attack and wing thickness. The effects of all these parameters can be collapsed in one single plane alfan-Machn, where the occurrence of subsonic or supersonic flow can be diagnosed as function of the parameters (Stanbrook- Squire, 1964).

Delta Wing with Subsonic Leading-Edge

The wing is inside the Mach cone if the sweep angle is greater than the Mach angle, thus yielding leading-edges that are fully subsonic, Fig. 1. Linearized theory leads again to simplified expressions for the main aerodynamic characterstics, which are quite powerful to describe the operation of the wing.

The lift coefficient depends on the aspect-ratio, according to en expression that is fairly approximate for incidences less than 5 degrees (Ashley-Landhal, 1965). According to the theory, the strong leading edge suction gives rise to a leading edge thrust that decreases the amount of drag (in practice only a small amount of this suction can be realised.)

Subsonic Leading Edge

Figure 1: Delta wing with subsonic leading-edge

Delta Wing with Supersonic Leading-Edge

The leading-edge is inside the Mach cone, by virtue of the comparatively larger sweep angle (Fig. 2). In such a case there is no interaction of flows between upper and lower surface.

The pressure jump at any given point on the wind surface has a definite expression, which is constant along lines through the wing vertex (conical flow). By integration of the pressure jump one finds that the lift coefficient is independent from the sweep angle, and the lift-curve slope is also independent from the angle of attack for as long as the leading edge is supersonic.

Supersonic Leading Edge

Figure 2: Delta wing with supersonic leading-edge

Carafoli (1969) report analytical studies of a wide array of delta wings, polygonal wings, and T-wings, also in yawed flow.

Flow Separation on Highly Swept Wings

Real cases of flow past slender delta wings (wings of small aspect-ratios) are almost certainly separated, and to a great extent. Separation starts from the leading-edge and produces a series of vortical regions that have a conical shape growing streamwise. The angle of attack at which these vortices appear depends on the slenderness.

Separation is at the leading-edge when the leading edge is sharp, and leads to performances largely independent from the Reynolds number. The presence of the leading-edge vortices is the cause of a number of phenomena:

  • The lift coefficient is larger than that predicted with linearized theory (see below). This is due to the suction effect of the separation vortices. The difference between the linear value of the lift and its actual value is called vortex lift.

  • The leading-edge vortices induce a field of low pressure on the suction side of the wing. The increased suction is a reason for increased lift (point above).

  • Stall occurs at a large angle of attack, because of the vortex instability, leading to vortex burst. When the vortex core bursts the suction effect disappears. Aa vortex burst far behind the trailing edge, the burst has little or no effect; vortex burst on the wing itself will reduce the vortex lift.

  • The vortex pattern behind the delta wing depends on the slenderness, because slenderness, together with angle of attack, is what decides the vortex burst.

  • Vortex aysmmetry appears on very slender wings at lower and lower angles of attack, because the vortex finds less physical limits for development, therefore becoming soon unstable.

Flow separation characteristics depend on speed (Mach number), wing sweep, angle of attack and wing thickness.

Wings with subsonic leading edge are dominated by leading edge separation. Secondary separation appears at moderate to high angles of attack, Fig. 3.

Wings with supersonic leading edges are characterized by a Prandtl-Meyer expansion behind the bow shock and by an attached leading edge flow Fig. 4.

Flow Separation on Delta Wing

Figure 3: Flow separation on delta wing with subsonic leading edge. A = attachment; S = separation; V = vortex.

Flow Separation on Delta Wing

Figure 4: Flow separation on delta wing with supersonic leading edge. SW = shock wave

Related Material

Other Wings

Selected References

  • Ashley H, Landahl M. Aerodynamics of Wings and Bodies, Addison-Wesley Publ. Company, Reading, Mass. 1965.

  • Katz J Plotkin J. Low Speed Aerodynamics, McGraw-Hill, Inc., New York, 1991.

  • Carafoli E. Wing Theory in Supersonic Flow, Pergamon Press, 1969.

  • Nickel K, Wohlfahrt M. Tailless Aircraft in Theory and Practice, Edward Arnold, London 1994 (also available from AIAA).

  • Peake DJ, Tobak M. Three-Dimensional Interactions and Vortical Flows with Emphasis on High Speeds, AGARDograph AG-252, July 1980.

  • Riebe, JM, William C. Low-Speed Stability Characteristics of a Cambered-Delta-Wing Model, NACA RM-L55L21a, 1956.

  • Henderson A. Supersonic Wave Drag of Nonlifting Delta Wings with Linearly Varying Thickness Ratio, NACA TN 2858, 1952.

The literature on this subject is staggering. For details please inquire.

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Copyright © A. Filippone (1999-2003). All Rights Reserved.