Linearization

Let us begin the linearization process with the boundary conditions. The small perturbations approach will be used here. Accordingly, let U be the free stream speed in positive x direction, Fig. 2.1.

Let “ be the perturbation velocity component in x direction which makes the total velocity component in x direction: “ = U + “’.In addition, defining function ф’ as the perturbation potential gives us the relation between the two potentials as follows: ф = ф’ + Ux. As a result, we can write the relation between the pertur­bation potential and the velocity components in following form

The small perturbation method is based on the assumption that the perturbation speeds are quite small compared to the free stream speed, i. e. “’, v, w ^ U. In addition, because of thin wing theory the slopes of the body surface are small therefore we can write

which gives the approximate expression for the boundary condition

w = 0za + U0za (2.20)

0t 0x v ;

Equation 2.20 is valid at angles of attack less than 12° for thin airfoils whose thickness ratio is less than 12%. For the upper and lower surfaces, the linearized downwash expression will be denoted as follows.

Upper surface (u) : w = + U0"; z = 0+

0t 0x

Lower surface (l) : w = + U; z = 0-.

ot ox

Now, let us obtain an expression for the linearized surface pressure coefficient. For this purpose we are going to utilize the linearized version of Eq. 2.8. The second term of the equation is linearized as follows

q2 U2 .

ffi + 2U"

22

For the right hand side of Eq. 2.8 if we arbitrarily choose F(t) = U2/2 then the term with the integral reads as

dP 0/ ,

2U " .

P 0t

The relation between the velocity potential and the perturbation potential gives: = °0І/-. If we now evaluate the integral from the free stream pressure value p? to any value P and omit the small perturbations in pressure and in density we obtain

p

dP ^ P – Pi

P P1

Pi

Using the definition of pressure coefficient

Here, the pressure coefficient is expressed in terms of the perturbation potential only.

Example Let the equation of the surface of a body immersed in a free stream U be

Zu, i = ±a i(0 < x < l)

If this body pitches about its nose simple harmonically with a small amplitude, find the downwash at the upper and the lower surfaces of the body in terms of a, l and the amplitude and the frequency of the oscillatory motion.

Answer Let a = a sin xt (a: small amplitude and x: angular frequency) be the pitching motion, let x, z be the stationary coordinate and x’, z’ be the moving coordinate system attached to the body. The relation between the fixed and the moving coordinate system is given by Fig. 2.2 in terms of a.

The coordinate transformation gives

x’ = x cos a — z sin a

Z = x sin a + z cos a

In body fixed coordinates the surface equations zUl = iay^jr(0 < x < l)

In terms of the stationary coordinate system B(x, z, t) = z’ — zu l (x) = x sin a +

z cos a T a(x cos a—z sin a)1/2 for small a sin a ffi a and cos a ffi 1. Then

B(x, z, t)= xa + z T a(x~lza)1=2- Equation 2.17 gives

. a a z x — za —1/2 a x — za —1/2 aa x — za —1/2

wu ’l = —{xa±-2r(—) +U[aT2^(-^) 1 **(—)

Here a = xa cos xt. Now, let us express the downwash for t = 0 wu ; l

axx ± aa. x z(f)—1/2T § ® —1/2

z with l the non dimensional form of the downwash expression becomes

Fig. 2.2 a pitch angle and the coordinate systems

If we write the reduced frequency: k = U and the nondimensional coordinates a* = у : x* = у ve z* = f, new form of the downwash becomes

akx* ± a*akz*(x*)~1,/2^—(x*)~1,/2 .

In the last expression, the first two terms are time dependent and the last term is the term due to the steady flow.

Now, we can linearize Eq. 2.15 for the scalar potential with small perturbation approach. The nonlinear terms are the second and third terms in parentheses. The velocity vector in the second term is q = Ui + V /’ = Ui + u’i

(2’4b)