A leading edge vortex forms for pretty much the same reason as a tip vortex, at high AoA the boundary layer can't follow the sharp curve around the leading edge and separates from the surface, as it roles up into a vortex it produces extra lift.  On a wing without sweep the LEV can only stay on the wing for a few seconds before it expands to the point where it has to separate from the wing and then the wing stalls, this is what's called "dynamic stall".  As you increase the sweep of the wing the LEV can stay on the wing longer but it always leaves after a few seconds.  Until, that is, the sweep angle is greater than 55 degrees.  At 55 degrees sweep the spanwise component of flow is strong enough to push the axial component of the vortex toward the tip thereby creating a stabile, non expanding, cone shaped vortex.  Actually, on a delta wing, there is no distinct tip vortex, the LEV starts near the nose of the airplane and runs along the top surface to the tip, where it is dumped overboard just like a regular tip vortex.  The difference between a LEV and an ordinary tip vortex is that the LEV has a larger surface to act upon, thus it can turn some of the energy that went into creating it into lift so deltas have somewhat better lift to drag ratios than older 2D theory predicts.

    Many modern fighter planes use a structure called a "strake" (lately it's common to call it a LEX) to generate large vortex systems at high AoA (actually in this context "high AoA" is anything greater than 7 or 8 degrees).  These strakes are basically small deltas with sharp LEs attached to the LE of the main wing at the root.  The LEV that forms on the strake separates from its leading edge at the intersection with the main wing and continues down stream where it produces high velocity and low pressure on the upper surface of the main wing.  This widens the AoA range of the wing thereby allowing a small wing that's efficient at high speed to also give reasonably good handling at low speed.
Concorde in water tunnel Strake vortex on F-16B Strake vortices on F-16A
Shear layer instability on CF/A-18 F/A-18 in water tunnel

Strake vortices on F/A-18 Strake vortices on MIG-29

Strakes on cars SR-71 in water tunnel
Sab 35 Draken LE vortices on F-4

This drawing is fairly typical of older textbook illustrations of subsonic flow about deltas at high angle of attack.  It shows the starboard wing of a delta with a straight leading edge.  It's still completely valid but not many simple deltas are being built anymore.
Half wing with vortex
Drawing legend:
LE    Leading edge
PF     Potential Flow
PV     Primary Vortex
SV     Secondary Vortex
TE    Trailing Edge
RP     Reattachment Point
SF     Surface Flow (boundary layer)
These next two drawings are a bit harder to find, especially in older books.  They also show the starboard wing.
       Vortex "A" in the drawing of a double delta bellow  is the normal leading edge vortex.  Vortex "B" has the same direction of rotation as A and therefor when they come into contact they wrap around each other.  A tries to follow a line approximately parallel to the forward LE so when B wraps around A it is dragged farther inboard than it would go by itself, thus a larger area of the wing is affected by this type of vortex system than would have been affected by a single vortex.  Since the CLmax of a wing with vorticity can be expected to be 80% higher than with potential flow alone naturally the designer wants the vortices to cover as much area as possible.   In order to get this spiraling vortex pair the angle "K" must be greater that 10 degrees and the minimum sweep back of the aft portion of the wing should be more than 50 degrees. Half double delta with vortex
       The drawing below  depicts an unstable shear layer (as opposed to a stable shear layer, which is what the other drawings show).    It occurs at higher AoA than is needed to produce the type of LE vortices shown in the previous illustrations and with larger sweep angles. 
If you have any comments about this page pleas  let me know .
For a more detailed discussion of  the flow about a delta see "Aerodynamics of the Delta Wing".
Also see this historically interesting page  with an article about the AVRO 707A 
Picture credit:  Office National d'Etudes et Recherches Aeronautiques
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