Delta Wings
Summary
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 leadingedge 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 CL823), the double delta (SAAB Viggen), delta + canards (North
American XB70 and others).
Almost all delta wings fall into the category of low aspectratio wings. Their
aspectratio 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).
Flows past delta wings are severely compounded by the leading edge separation, by the
rollup structure of the concentrated vortices, and by the lateral and longitudinal
instability that is consequent to large sweeps, highangle of attack, and sharp
manoeuvres.
Although the aerodynamics of the delta wing is nonlinear, 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 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 aspectratios (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 leadingedge, where the pressure is also
singular.
The effect of a fuselage can also be estimated by a more general formulation
(AshleyLandhal, 1965) that gives a lift coefficient in the wingbody configuration
is lower that the wingalone.
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 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).
In general, wings with subsonic leading edges are characterized by leading edge
separation; wings with supersonic leading edge are characterized a PrandtlMeyer
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
alfanMachn, where the occurrence of subsonic or supersonic flow can be diagnosed as
function of the parameters (Stanbrook Squire, 1964).
The wing is inside the Mach cone if the sweep angle is greater than the Mach angle,
thus yielding leadingedges 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 aspectratio, according to en expression that is
fairly approximate for incidences less than 5 degrees (AshleyLandhal, 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.)
Figure 1: Delta wing with subsonic leadingedge
Delta Wing with Supersonic LeadingEdge
The leadingedge 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 liftcurve slope is also independent from the angle of
attack for as long as the leading edge is supersonic.
Figure 2: Delta wing with supersonic leadingedge
Carafoli (1969) report analytical studies of a wide array of delta wings,
polygonal wings, and Twings, also in yawed flow.
Real cases of flow past slender delta wings (wings of small aspectratios) are almost
certainly separated, and to a great extent. Separation starts from the leadingedge
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 leadingedge when the leading edge is sharp,
and leads to performances largely independent from the Reynolds number.
The presence of the leadingedge 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 leadingedge 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 PrandtlMeyer expansion behind the bow shock and by an attached leading edge flow
Fig. 4.
Figure 3: Flow separation on delta wing with subsonic leading edge.
A = attachment;
S = separation;
V = vortex.
Figure 4: Flow separation on delta wing with supersonic leading edge.
SW = shock wave
Related Material
Other Wings
 Ashley H, Landahl M. Aerodynamics of Wings and Bodies,
AddisonWesley Publ. Company, Reading, Mass. 1965.
 Katz J Plotkin J. Low Speed Aerodynamics, McGrawHill, 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.
ThreeDimensional Interactions and Vortical Flows with Emphasis on High
Speeds, AGARDograph AG252, July 1980.
 Riebe, JM, William C. LowSpeed Stability Characteristics of a
CamberedDeltaWing Model, NACA RML55L21a, 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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