INDUCED DRAG

Is it possible to generate lift without Drag? Well, the answer is no, we can’t produce lift without drag. Here we will see how lift is related to drag.

Wingtip vortices tend to drag the surrounding air around that induces a small velocity component in the downwash direction at the wing this downward component is called as downwash. Denoted by symbol w.

As u can see in figure 1.a Relative wind and downwash add vectorially to produce a ‘’local’’ relative wind that is canted downward from the original direction of relative wind There is an increase in drag. This increase is called induced drag.

1.a

Calculation of induced drag

Consider a finite wing as shown in figure 1.b

1.b

Here,

Ø  R₁=Resultant aerodynamic force in an imaginary situation with no vortices

Ø  D₁= The component of R₁ parallel to v_∞

Ø  R= Resultant aerodynamic force in an actual situation with vortices

Ø  D= The component of R parallel to v_∞

Ø  D_i= The induced drag, difference between D and D1

·         D₁ is an imaginary case is due to skin friction and pressure drag due to flow separation.

·         D is an actual drag which includes the effect of the changed pressure distribution due to the wingtip vortices as well as skin friction and pressure drag due to flow separation.

As u can see in figure R is tiled backwards relatively to R₁, then D>D₁

To calculate the magnitude of D_i

We will consider finite wing as shown in figure 1.c

1.c

Here,

Geometric the angle of attack (a): The angle of attack defined between the mean chord of the wing and the direction of v∞ is called a geometric angle of attack

Induced angle of attack (a_i): The angle between the local flow direction and the free-stream direction is called the induced angle of attack

The airfoil section seeing an effective angle of attack

The local flow direction in the vicinity of the wing is inclined downward with respect to the free stream. The lift vector remains perpendicular to the relative wind, tilted back through angle ai , as shown in the figure, still considering drag to be parallel to be a free stream, tilted-lift vector contributes a certain amount of drag. This drag is called induced drag.

From figure 1.c

Value of a_i are generally small

Hence,

Note that in equation 3 ai should be in radians

ai for a given section of a finite wing depends on the distribution of downwash along of the span of the wing. To see this in more detail consider figure 1.d which is showing the front view of the finite wing.

1.d

The lift per span varies as the function of the distance along the wing

1.       The chord may vary in length along the wing

2.       The wing may be twisted in such way that each airfoil section of the wing is at a different geometric angle of attack

3.       The shape of the airfoil section may changes along the span.

In an elliptical lift distribution which in turn, produces a uniform downwash distribution

In the case of, incompressible flow theory predicts that

Here,

C_L is the lift coefficient of finite wing

AR is aspect ratio b²/S

Substitute the value of AR and ai into equation 3

Defining, the induced drag coefficient as

This result holds only for elliptical lift distribution, the wing no twist and same airfoil shape across the span.

For all wing in general, a span efficiency factor e can be defined such that for elliptical platform e=1 and for other e<1

From equation 5 is an important relation as it demonstrates

1.       To minimise induced drag we need to increase your AR

2.       Induced drag various as the square of the lift coefficient

 TOTAL DRAG
   

 Profile drag is the sum of skin friction drag and pressure drag (c_d).C_D,i for infinite wing (infinite aspect ratio) is zero.