| OCR Text |
Show H U U; 1,0 OS Q7~------~--------~ 0,5 -'---- -4- -I--J ::!!.... Uo o +-------~--~----~ o \0 rio 2,0 1,0 ~ Uo QS 10 .:!1... Uo QS lD rio 1,4 Fig. 4 - U(r) and W(r) velocities in an expanding quar1 before and after the first stagnation point (inviscid predictions) imaginary, representing either a potential or wave-like solution. For a flow having a distribution of energy and angular momentum along streamlines corresponding to solid body rotation, the transition from potential to wave-like flow occurs at a swirl number of 0.96 (ka = 3.83) [4]. If the swirl number is less than 0.96, the fluid cannot support upstream propagation of inertial waves, while if S>O.96, inertial waves can propagate upstream into the flow [4]. These two flows states have been labeled supercritica1 and subcritical, respectively, and subcritica1 flows have the special property that they can be drastically altered by downstream conditions through upstream inertial wave propagation. ISOTHERMAL MODEL FOR EXPANDING SWIRLING FLOWS - An isothermal model has been developed which is capable of quantitative prediction of the • U(r) and W(r) velocities in the expanding flow, • IRZ strength, size and shape, • mixing energy dissipated, as a function of the • swirl profile and level, and • the burner and furnace geometry . A full description of the model can be found in reference [3]. The swirling flow model is, in many ways, analogous to other analytic procedures used in fluid mechanics which connect a basically inviscid upstream flow with an inviscid downstream flow of a lower total energy level through a small region where viscosity dominates. Some well-known examples of this technique are shock waves in compressible flows and a hydraulic jump in channel flow. Some liberties have been taken in the case of the inviscid swirling flow model but, as will be shown later, the experimental results have verified the overall concept. The model is based on combining the two solutions to the equation of motion. The model uses the expanding flow, equation (4), to obtain the velocities in the expanding flow and to obtain the magnitude of the disturbance to a new swirling flow with a lower total energy level governed by the cylindrical equation of motion, equation (2). The new downstream flow has the distribution of angular momentum and energy along streamlines corresponding to a uniform axial velocity and solid body rotation. Since the new flow diameter is larger than the initial diameter, the new uniform axial velocity is lower than Uo. The cylindrical flow is initially displaced in an inviscid manner from solid body rotation in the manner illustrated in Figure 2. The constant, D, in equation (2) is determined by using equation (4) to determine the position of the IRZ boundary after expansion. The model assumes both that the region of intense energy loss is confined to a small region after the flow stops expanding and that, during the energy loss, the flow remains cylindrical. The model separates the flow field into 4 regions which are illustrated 92 in Figure 5: • Region 1 the initial swirling flow; • Region 1-2 the inviscid expanding flow; • Region 2-3 the highly turbulent region of energy loss; • Region 3-4 the downstream inviscid, cylindrical flow. o r--- ...... -- 20 Su~rcr.t.co l Subcnt.col EI "101 ttlrU$t for I A~"". 1 flow H x_ Fig. 5 - Physical model for the flow transition The inviscid equations of motion, when combined through the model proposed here, can be used to quantify the importance of several swirl and/or geometrical parameters on the following near isothermal field flow properties: • the swirling flows which are likely to form an IRZ; • the velocities in the expanding swirling flow, when quar1 confined; • the strength of the IRZ |
