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Show Physical interpretations can be given to many of the terms in equation (2). The first term, 1/2Uor2, describes the streamlines or mass flow of the original, undisturbed vortex in solid body rotation. The constants D and E can be interpreted as the wave strength, the maximum displacement of the vortex from an initial state of uniform axial velocity and solid body rotation. The symbol k can be directly related to the swirl number and, for simple pipe flow in a tube of radius a, it can be shown as k = 2n/U = 4S/a. In F1. gure 2 , the form these 0w ave-like motions take for various values of the wave strength D are shown. An important point to note is that the governing equations are linear and thus no assumption concerning small amplitudes of D, the wave strength, has been made. o small O/Uo V //////////////////////// 1 r~ I L . tube ax.s Fig. 2 - Inviscid cylindrical flows (from equation 2) The equation of motion, equation (1), can also be solved for tubes of slowly expanding or contracting cross sections when the initial flow satisfies the initial conditions of solid body rotation and uniform axial velocity. By assuming any diameter change is gradual, i.e., quasi-cylindrical, the stream function can be assumed to be a function of r alone, which allows equation (1) to be written as [4]: d2F/dr2 + 1 dF + (k2 __1 _)F r dr 2 o (3) r Some examples of cylindrical transitions for which this equation can be applied are shown in Figure 3. Equation (3) can be solved to give the U(r) and W(r) velocity profiles at a new flow diameter: U = l 1! = U + MkJO(kr) + NkY O (kr) (4) r dr 0 wher-e--' , again, JO' J 1 , YO and Y1 denote Bessel 91 _~-Ib a J_~ _ ~ I11} a l ~ ~- f f~ , :~: r fbI a2~b2 ~ Fig. 3 - Examples of cylindrical trans1tions 2n 'II W = U r = Qr + MkJ1 (kr) + NkY1 (kr) (5) o functions. The two constants, M and N, can be determined by using the known values of 'II at the inner and outer radius of the upstream and downstream flow. This implies M = (6) and N is of the same form with J 1 and Y 1 interchanged and k = 2n/U = ~ o a (7 ) The boundary conditions necessary for determining M and N when the inner and outer flow position is not known are, in general, quite complex and the interested reader should refer to [3]. Figure 4 shows the U( r) and W( r) profiles predicted by equations (4) and (5) in an expanding quarl. The expanding swirling flow equations do not directly yield information about the flow in the IRZ, although the size and shape of the IRZ is obtained. The inviscid equations have several important implications which can directly improve our understanding of swirling flows. The solution to the equation of motion for expanding swirling flow, equations (4) and (5), implies that the flow properties will be determined by the initial swirl number, ka1 , and the amount by which the flow expands, (ka1)(b1/a1) = kb 1• The geometry of the burner (for example, the blockage ratio of the burner a2/al) should also be important. The solut10n to the equation of motion, equation (2), when the outer flow boundary is cylindrical, also has several interesting properties [4]. Depending on the value of the swirl number, ka/4, equation (2) is either real or |