Funciones a cumplir para el área de crédito y cobranza

The response of the wall to the boundary layer disturbances and the effect of the compliant wall on the disturbance amplitudes can be determined to some degree by considering the tions which can be obtained from them are invaluable in providing some understanding of the physical processes which occur when instability arises and how they change when wall compliance is introduced. The following subsections describe how the eigenfunctions are calculated for this problem and observations are then discussed.

4 .1 3 .1 Flow Eigenfunctions.

As stated previously a consequence of the Gram-Schmidt process required to numerically integrate the Orr-Sommerfeld equation is that the eigenfunctions cannot be obtained as a direct result of the eigenvalue calculation. However, there does exist a set of intermediate solutions which are valid between the orthonormalisation points y\ < < ■ ‘ < VK = Ve and these are required to match at each of the orthonormalisation points in order to generate the eigenfunctions subject to some normalisation condition.

The intermediate solutions take the form

constants to be determined.

From the requirement of continuity across an orthonormalisation point the solutions must satisfy

wall and flow eigenfunctions. The flow eigenfunctions and corresponding energy distribu­

Vm(y) = +»m)(ÿ) for y € [ÿm,ÿm + l] (4.71)

where t>m\ » = 1,2, are the two fundamental solutions and am, for m = 1 , . . . , A', are the

v m — 1 (l/m ) — (Vm ) (4.72)

or + = <Jm ) • (4.73)

However, from the Gram-Schmidt orthonormalisation process itself

(4.74)

l l « & ! l ( i * n ) |l

and

(4.75) 102

Substitution of the expression in Eq.( 4.75) into Eq.( 4.73) gives

+ V(^ h ( y m ) - < > ®m-l- Using the form in Eq.( 4.74) then determines an iterative formula for the constants, a„

am-lV^LAym) ~

Um— 1 (4.76)

At the edge of the boundary layer, yx, a suitable value for ax is chosen and in this case the initial value is determined by normalising the eigenfunctions to unity at the edge of the boundary layer. Eigenfunctions obtained for the Blasius profile were found to be in good agreement with those published in Yeo (1988).

A comparison of the eigenfunctions for the Falkner-Skan profile at /2J = 5000 for the rigid wall and the compliant wall which gave the maximum reduction in growth rate is made in Figures 4.9 and 4.10. These show similar characteristics to those which have been observed for Blasius flow in that wall compliance brings about a reduction in amplitude of the normal perturbation velocity and generates non-zero values at the wall in accordance with the changes in wall boundary conditions. The point of maximum amplitude is still found to occur some way into the boundary layer region near rj = 2. The rigid wall eigenfunction for the streamwise component of the perturbation velocity shows a double peak in the real part near the wall which becomes smoothed out when the compliant wall is introduced.

Figure 4.9: Eigenfunctions for rigid wall at u = 0.125 and Rj. = 5000 corre­ sponding to position of maximum growth rate, a = 0.3528— 0.04689«. (a) i, (b) u. Real part : —, Imaginary part : - -.

(a)

(b)

y

y

Figure 4.10: Eigenfunctions for compliant wall (1) of Figure 4.5 atu = 0.125 and R f = 5000 corresponding to position of maximum growth rate. a = 0.2992 - 0.02209«. (a) v, (b) u. Real part : —, Imaginary part : - -.

4 .13.2 W all Eigenfunctions.

Wall eigenfunctions give some insight into the response of the compliant wall to the distur­ mined. The constants of integration C, can be calculated from the zero dispacement condi­ tion at the base of the wall along with the continuity of velocity condition at the wall/flow interface. Given

The wall response for a given set of wall parameters is shown in Figures 4.11 and 4.12 for two different Reynolds numbers, R f = 3000 and 5000, where the greatest growth rate reduction is achieved at R f = 5000. The results explain in part the differences observed in the calculated growth rates for the two cases since at Rb = 5000 both horizontal and vertical wall displacement occurs throughout the whole depth of the layer but at R^ = 3000 the disturbance effects do not penetrate right down to the base. Overall, the displacements are greater in magnitude at Rb = 5000 with in both cases the main horizontal motion being confined to the upper third of the wall and the maximum vertical displacement amplitude occurring at the surface.

A comparison of wall eigenfunctions for different depths is given in Figure 4.13 at a Reynolds number of 5000. Of particular note here is the remarkable similarity in form of the displacements for the two walls with d = 3.6 and d = 5.4. Both vertical and horizontal displacements are almost identical in feature and the corresponding growth rate results for these two cases were also very similar. This provides evidence for the conclusion that disturbances are only able to penetrate to a certain depth and beyond this any increases in wall thickness serve no beneficial purpose. Walls of smaller depth are seen to be affected fully by the boundary layer disturbances with the result of horizontal and vertical displacements throughout the complete wall layer.

bances within the boundary layer and indicate the effect within the wall itself.

Once the eigenvalues have been calculated the wall eigenfunctions can be easily deter­

[i(-</),V(-</),i(0),^(0)]T = R C = [0,0, % i w - ~ v ' w, - v m)T(jjz acO LÜ (4.77) then

0 (a) (b)

Figure 4.11: Wall eigenfunctions for compliant wall (1) of Figure 4.5 at position of maximum growth rate and R f = 3000. (a) Vertical displacement t/, (b) Horizontal displacement Real part : —, Imaginary part : - -.

(a) (b)

Figure 4.12: Wall eigenfunctions for compliant wall (1) of Figure 4.5 at position of maximum growth rate and R f — 5000. (a) Vertical displacement i/, (b) Horizontal displacement f . Real part : —, Imaginary part : - -.

0 -1 - 1 5 - 1 0 - 5 0 A T1 (a) _ - 0 .5 y -i - 1 .5 - 1 5 - 1 0 - 5 0 A *1 (b) - 5

I

Figure 4.13: Wall displacements for walls of different depths. Vertical displace­ ment r/, horizontal displacement £. (a) : d = 1.2, (b) : d = 1.8, (c) d = 3.6, (d) d = 5.. For other wall parameters refer to caption of Figure 4.7. Real part : —, Imaginary part : - -.