FURTHER STUDIES OF FULLY DEVELOPED FLOW THROUGH DUCTS OF ARBITRARY SECTION BY THE CONTOUR LINE-CONFORMAL MAPPING TECHNIQUE

The paper discusses fully developed flow of viscous fluid through ducts of arbitrary cross-section. The method uses a constant velocity contour line in a typical cross-section of the duch as an independent variable. The amplitude of the oscillatory velocity is then obtained from an ordinary integro-differential equation. Several examples of a practical nature are given, with some that have not yet been discussed in the literature. All details are explained by graphs.


INTRODUCTION
This paper is a further extension to previous work that has been done in the area of oscillatory fluid flow in ducts.In a previous paper [9], a method was proposed for the study of fully developed parallel flow of Newtonian viscous fluid in uniform straight ducsts of very general cross-section.The proposed method is based upon the concept of a family of contour-lines of constant velocity, u(z, y) const., in a typical cross-section of the duct and considering such line as an independent variable.Since the details .,of the method used in this study have been discussed by the first author in an earlier publication [9], only a brief discussion of the method is presented here.
According to the method, the governing equation for the axial velocity component w(z, y, 7-) at any time r is given by claW x/ds + ds iAW (1) d --dz - where (, u, -)= w(, u)', p(z,) P(z)', (3) t/ p The above equation is the integral form of the momentum equation for unsteady flow, ignoring any external forces.Here w represents the frequency of oscillation, and A is a reduced frequency.
The family of contour lines of constant velocity, C,,, are represented by u(z, ) constant (see Fig. 1).If the exact equation for u(z, y) is known, then the governing equation (1) yields the exact solution for the velocity component w.If, however, the exact form of u(z, 1) is not known a priori, then the method ', -'l-_X,-" .-X. u(x,y]=O u (x,y)= constant of conformal transformation can be used to obtain an accurate approximate solution for the velocity distribution.
2. APPLICATIONS 2.1.Solving the unsteady equation for a coaxial elliptical duct As shown in [9], considering the contour function as u(z, y) x2/a y2/b2, the contour integrals in the unsteady equation can be solved easily, so that Eq. becomes #, dz where A1 and A: are arbitrary constants and Io and/to are modified Bessel functions of the first and second kind.The constants AI and A2 are found using the viscous no-slip boundary conditions that occur at duct walls.If the duct is simply connected, then the boundary values of u are u=0 and u=u'=l (9) at the outer boundary and the origin, respectively.This produces the corresponding boundary conditions W(f)lf= 0 and W(f)l]=o 0 (10) and so solving Eq. 8 for A1 and A we get -i dP A (11) , d to(K) and A =0 (12) and thus, Eq.The amplitude velocity, V, is given by IA dz (ber2K + bei2K) u: constant FIGURE 2. Cross-section of a doubly connected ellipse.
(27) bet, be and ker, kei are the real and imaginary parts of Io and Ko respectively.The value, V, thus obtained will give the amplitude distribution of the velocity in the doubly connected elliptical cross-section of a duct.
Since according to the authors' knowledge there is no published results for this problem we will for the sake of comparison and confirmation of our approach, consider the limiting case when two ellipses become two circles.It is interesting to note that if one puts a b and a bx so that the two ellipses reduce to two circles, then the results obtained for this case coincide with the results given in 11 for the case of an annular duct.We further note that in Eq. 7, the only quantity affected by this substitution is the K. Fig. 3 shows the graphs of V/W* for a coaxial elliptical duct, where W" b P -Y-a; I ' and for parameter aspect ratio a/b 2, and for three different values of/3(0.0,0.1,0.5) and various values of the parameter r/= bA.In Fig. 4 the graphs of V/W* are shown with a/b 1, fl(0.0,0.1,0.5) and a set of values of r/.
The results in Fig. 4 agree precisely with those of Tsangaris 11 ].It is further interesting to note that the graph in Fig. 3 corresponding to fl 0 coincides with the graph for an elliptical duct given in [9].

Conformal mapping technique
For more complicated cross-sections an exact equation for the isovelocity contours may not exist, so conformal mapping can be used (see Fig. 5).This assumes a one-to-one correspondance between the U(,) constant contours and the u(x,y) constant contours where z x + iy and Therefore as long as a function exists to map the cross-section onto the unit circle, it is not necessary to know u.The function U is represented as u(, ) = (28) and the contour integrals appearing in the unsteady equation (Eq. 1) are fc v/ds=2fc V/-UedS=4r(1-U)' () is the mapping function for the cross-section.
If we again introduce a variable, F, such that F= l-U, where a depends on the complex variable (.If we make the substitution H'(F) a in Eq. 37, then Eq. 37 will become identical to Eq. 7 where instead of K we have a , and instead of variable f we have F.
We therefore note that Eqn 24 gives the solution for any.duct of'arbitrary cross-section with a value of K; depending on the geometry of the duct boundary.This also applies for the complex mapping technique approach if we replace the parameter K by as.
3. ILLUSTRATIONS 3.1.Simply connected cross-sectional areas (a) Flow through a duct of semi-elliptical cross-section Consider the case of flow through a semi-elliptical-duct (Fig. 6).A complicated conformal mapping is used to get the mapping function of this shape.First we have to map the unit circle onto a semi-circle of radius r using 10] Consider another example of a square duct with rounded corners (Fig. 8).The cross-section for this shape also has a mapping function given by 10]

(( 2_2) (42)
where 2a is the length of the sides of the square.Hence, the quantity a becomes 625 ,ff (43)   and again, by replacing K by a in Eq. 18 we get the solutions for V/W" displayed in Fig. 9 for side length 2a and a set of values r/= a,, where W" -2-g 0.2 V/W connected square with rounded corners for r/= 0, 1,2,3, 4, ( 0.1 and F 0.0,..., 1.0.First consider the case when both inner and outer boundai'ies have polygonal cross-section.Therefore we want to map this shape onto a doubly connected annular region where the outer boundary is the unit circle (see Fig. 10).We do this using the well known Schwartz-Christoffel transformation to approximate the mapping function as it is not known [3].
where a the apothem of the regular polygon, n the number of sides on the polygon, and F, is the mapping coefficient given by r=/ (+)--t.
The numerical values of the mapping coefficients for some regular polygons are given in   and from other results in the literature [3] we know that for a polygon of 4 sides, the mapping function can be approximated by z a [1.0807 0.1081( + 0.045 0.0242( 43 + 0.0174 v 0-012624] (46) This function can be used to map the unit circle onto the outer boundary square.Now we need to map another concentric circle of radius, '7 < 1, onto the inner boundary square.The value of the side length, b, in this square can be denoted by so b aF4 t4)-dt (47) b 0.0126331 0.017'7 xr 0.0242"743 0.045"79 + 0.1081"75 1.0807"7 + 0 (48) must be solved to obtain the concentric inner circle radius, 3'.We can then completely map this cross- section using the mapping function (Eq. 48)to obtain H'(F) after integrating across the contour integrals from 0 to x/1 "-72.The unsteady equation can then be solved, and Fig. 11 shows the graphs of V/W" for side ratios/3 "7/1 < and values oft/= aA, where W* 1 aP I .The results in Fig. l(a) for varying the limiting case,/3 0 are also in excellent agreement with the results for a simply connected four-sided polygon given in [9].

CONCLUSION
We have thus obtained an accurate approximate solution for the analysis of a wide spectrum of fully developed fluid flow problems in simply and doubly-connected ducts of arbitrary cross-sections.The essence of the present approach is to reduce the partial differential equation for the transverse velocity component to an ordinary second order differential equation using the concept of contour lines on a typical cross-section of the duct.Next, the method of conformal transformation is applied to find the velocity distribution in a typical cross-section for a number of duct shapes.The approach presented is quite simple and straightforward.
The examples discussed show a very good agreement between the calculated values and the values existing in the literature, and confirm the usefulness of the method.Furthermore, the present study has been limited to the study of velocity distribution in a duct flow.
Many other problems, for example the dynamic response of MHD (magnetohydrodynamic) flow in conduits when subjected to pressure gradient can be analysed in a like manner, which is to be carried out in subsequent papers in this series.

FIGURE I .
FIGURE I. Isovelocity contour lines.

FFIGURE 9 "
FIGURE 9" Amplitude distribution of the oscillatory flow over the cross-section of a simply FIGURE I. Amplitude sitribution of the oscillatory flow over the cross-section of a doubly-

TABLE I .
Numerical values for mapping coefficients of regular polygonal transformations.