Model of free vortexes diffusion

Here the description of free vortexes diffusion in a vortical layer, in which distribution of time-averaged ft>w velocity is known, is offered. It is sup­posed that diffusion occurs due to turbulent pulsations of time-averaged veloc­ity across a layer, i. e. periodic (free) vortexes remain passive. It The initial layer of free vortexes in a vicinity of an airfoil trailing edge (x i =0) is sup­posed a line of contact discontinuity of velocity, in which linear intensity of vortexes may be determined by the formula, following from the Thomson the­orem at absence of difference of static pressure across a layer

1 дГ

n(r,0,yi, t) = 7(0,t) ■ 5(yi) = -8(Vi) (5)

Here y is discontinuity of velocity on a line of contact discontinuity; 5(yi) is delta function; r(t) = ^ Гк ■ e-2nkt is periodic (with period To) velocity

k

circulation on the R blade. The system of coordinates (xi, yi) is used with the beginning in the R airfoil trailing edge and an axis Ox, directed along an axis of a vortical wake behind an airfoil.

The equation of turbulent diffusion in neglect viscosity in a considered case looks like

dUl dUl d f dUl

dt W° dxi dy dy

where w0 is velocity of free vortexes drift, D = const is diffusion constant. The value of w0 is equal to velocity of time-averaged fbw in a vortical wake behind R blade

wo = woo [1 – Ui (xi ,yi)] ,

where w00 is velocity in a fbw core, and u i is additional velocity in a wake.

According to the semiempirical theory of turbulent vortical wakes behind airfoils Ui may be presented in the form

і] = Уі/Фа, ui (0) = 1,

where rather small value x ~ 0,1 is uniquely determined by factor of profile losses. From here the equation (6) becomes

Apparently, the received solution essentially differs from a case x = 0 (w0 = const) that the particles, forming at some moment of time a line, nor­mal to an axis of a vortical wake, at the subsequent moments of time are bent. Free vortexes, diffused to the layers more removed from an axis of a wake, are transferred to the velocity, exceeding velocity of vortexes, located on an axis. As shows the analysis, it in particular results to that for the fixed value y 1 the amplitude of vorticity at increase of x1 aspires to finite (= 0) value, and the axial density of total vorticity across a layer is increased as y/x{.

Prominent feature of the solution (8) is also dependence of distribution across a layer of amplitude function from number of a time harmonic k.

For a determination of connection between coefficients Ak of the series (8) it is possible to take advantage of a condition of total vorticity preservation in a fbw for all time of free vortexes formation. At steady-state fbw (i. e. proceeding indefinitely long) this condition gives a ratio

d

lim 17 [Г (*) + 7(>i, f)] = 0,

xi——o dt xi +0

f f Q (x1,y1) dy1dx1. From here after calculations fol-

o -0

/vrd-x(l + *)

<5>i = [ e ■ fi(rj)drj, Ф2 = / e 4<гf}2f2{n)drj

It is necessary to notice that for x = 0 coefficient Ak unrestrictedly grows at d ^ 0. In this case the layer of diffusion passes in a line of contact dis­continuity, vorticity is described by £-function and coefficients of the series, appropriate to decomposition (8), are determined by the formula

Ak = 27tkqiFk-

Equality (9), obviously, may be used both for a calculated estimation of the decomposition (8) coefficients, if value Г (t) is known, and for definition of the coefficients Гк, if Ak are determined experimentally.