440 JOURNAL OF APPLIED . SEPTEMBER, 1953

it is very important when the two-dimensional analog}' is con- to applying the relaxation technique automatically and instan- sidered. For this application a note of warning should be added taneously at every node on the bearing surface. The question for future users of the hydraulic analogy, as pointed out pre- arises as to the accuracy of the solutions obtained using this viously by the writer.3 It must be remembered that the group electrical method. The author answers this question for the velocity of water-wave propagation will not be constant unless case of a journal bearing of infinite length. However, it is desira- the water is V-i in. deep. The group velocity is the actual rate ble to know what results this method would give when applied of propagation of the energy of the wave and is the only analogy to bearings of finite length. to the speed of sound in a gas. The group velocity is less than In order to obtain such results and to determine what diffi- the so-called wave velocity which has no analogy to the speed of culties, if any, would be encountered in applying this resistance- sound unless the water is y4 in. deep so the wave velocity be- network method to the more difficult finite-bearing problems, comes independent of wave length and equal to the group the apparatus, shown in Fig. 1 of this discussion, was built. velocity. Therefore we must restrict ourselves to 1/, in. water With this device it is possible to obtain the distributions depths and weak hydraulic jumps in order to obtain equations for bearings having an L/B ratio equal to 1 (L = length per-

that are comparable to those corresponding to a two-dimensional pendicular to direction of motion, B = length in direction of Downloaded from http://asmedigitalcollection.asme.org/appliedmechanics/article-pdf/20/3/440/6748130/440_2.pdf by guest on 02 October 2021 perfect gas flow with a specific ratio of 2. motion). As precision were not available, plain '/j-watt carbon-type resistors were used throughout the apparatus. The AUTHOR'S CLOSURE commercial tolerance of the network resistors (located in Fig. 1 The author agrees with Professor Laitone's comments on approximately on the center of the board between the two banks the principles of the hydraulic analogy to compressible flow. The of switches and rheostats) was given as ±5 per cent. However, main virtue of this analogy is the fact that the behavior of the measurement of the individual resistors showed that they did not two analogous systems is governed by a similar set of nonlinear deviate more than 2 per cent from each other. This deviation equations of motion, the nonlinearity resulting in the transition was not considered serious for Liebman5 has shown that because from an elliptic to a hyperbolic type. The physical implica- of the averaging properties of resistance networks it is possible tion of this property is the creation of discontinuities appearing to obtain solutions ranging in accuracy from one part in 103 to in the form of bores in the hydraulic system and of shock waves one part in IO4 (depending upon the geometry and scale of the in the compressible flow. The electrical field does not exhibit network) when precision resistors accurate to ±1 per cent were this property and consequently the electrical system cannot be used. used for this purpose. On the other hand, the details of the proc- In the tests conducted it was thought desirable to obtain values esses such as the velocity profiles, viscosities, and so on, are not of the pressure at ten different points along the bearing surface in the same, and in particular, the energy conditions are quite dif- the direction of motion and at ten different points across the ferent. Hence the analogy can only yield information on the bearing surface, perpendicular to the direction of motion. How- "behavior or nature of the solution," as Professor Laitone rightly ever, advantage was taken of the symmetry of the pressure dis- pointed out, and any attempts to utilize it as a basis for an analog tributions which necessitated the determination of the computer, similarly as it is usually done with electrical analogies, at only five different points across the bearing surface and thus are deemed to be futile. reduced the size of the required network by a factor of 2. Hence the network, rectangular in shape, was made of 10 X 5 = 50 resistors and consisted of 11 X 6 = 66 nodes. An average An Electrical Method for Determin- value of Ri (i.e., the resistors which lie along the center line of ing Journal-Bearing Characteristics1 the bearing) was 1020 ohms; Ri (all other resistors in network) averaged 515 ohms. These values were made very low in com- A. A. RAIMONDI.2 By suitable transformations, the author parison to the resistances used for regulating the input currents has succeeded for the case of journal bearings in reducing Reyn- la, for the reason set forth by the author. The bank of switches olds equation to the form3 and rheostats shown at the left in Fig. 1 were used to regulate the input currents I0, while the bank of switches and rheostats 111 at the right were used to set the leak resistances rL, to their proper !Hf f values. A precision-type potentiometer and a sensitive gal- vanometer were used for accurately setting the input currents, The partial derivatives in Equation [1 ] can be approximated by leak resistors, and for measuring the node . finite-difference equations with the result that it becomes Early in the tests it was discovered that when a 135- B- in B\2 battery was used for supplying the input currents it was not ( — — ) ("3 + «4 — 2«o) — possible to check the node- readings any closer than 4 + ff(?o) = 0 [2] per cent in some cases, depending upon the amount of time the battery was delivering current. A more satisfactory stable which can now be solved for u by applying Southwell's4 "relaxa- voltage supply was obtained by using the circuit shown in Fig. tion" method. However, the author has substituted for the 2, herewith. With this arrangement it was possible to repeat manual relaxation process an electrical method which is analogous node-voltage readings to better than 0.5 per cent accuracy. 3 "A Study of Transonic Gas Dynamics by the Hydraulic Analogy," The first pressure distribution obtained with this apparatus by E. V. Laifone, Journal of the Aeronautical Sciences, vol. 19, April, was that of a 120-deg journal bearing with Type I boundary 1952, pp. 265-272. condition having an e = 0.2, dA = 67.4 deg, and L/B = 1. In 1 By D. S. Carter, published in the March, 1952, issue of the this problem it was required to simulate four negative leak JOURNAL OF APPLIED MECHANICS, Trans. ASME, vol. 74, pp. 114- 118. resistors near the entering edge of the bearing along a line of 2 Research Engineer, Mechanics Department, Westinghouse Elec- nodes perpendicular to the direction of motion. This condition tric Corporation, East Pittsburgh, Pa. Jun. ASME. was handled with little difficulty by the method given by the 3 The nomenclature used in this discussion is the same as that used in the paper. 6 "Solution of Partial Differential Equations With a Resistance 4 "Relaxation Methods in Engineering Science," by R. V. South- Network Analogue," by G. Liebman, British Journal of Applied well, Clarendon Press, Oxford, England, 1940. Physics, vol. 1, 1950, pp. 92-103.

Copyright © 1953 by ASME DISCUSSION 441 Downloaded from http://asmedigitalcollection.asme.org/appliedmechanics/article-pdf/20/3/440/6748130/440_2.pdf by guest on 02 October 2021

FIG. 1 RESISTANCE-NETWORK APPARATUS

TO NETWORK NODES.

QC 3/VR IQ5

FIG. 2 VOLTAGE-SUPPLY CIRCUIT author. Needs6 also had obtained the pressure distribution for this same bearing by using the electrolytic-tank method de- veloped by Kingsbury.7 Thus it was possible to compare the pressure distributions obtained by two different experimental methods. Fig. 3 of this discussion shows this comparison, the circles being the test results and the full curves graphical inter- polations from the data of Needs. It is evident that the two methods give pressure values which agree quite closely. The load-carrying capacity of this bearing was obtained using 40 60 80 graphical integration methods and was found to be

HUR*L FIG. 3 COMPARISON OP PRESSURE-DISTRIBUTION METHODS BY IF = 0.316 , (resistance network) THE AUTHOR AND BY NEEDS C2

If this is compared to the load given by Needs for the same bear- Then ing bh d/i dx h, HUR'L h' = — = — — = - - (o - 1) [4a] TF = 0.319 • -, (Needs) of Ox of n C2 it will be seen that the two values agree within 1 per cent. and since The resistance-network method is not restricted to journal bear- ings but is equally applicable to slider bearings and, indeed, H = h'/> = j^a — (a — 1)J [46] bearings having odd film shapes resulting from distortion, mis- alignment, and so on. In order to set up these problems on the then resistance network, it is necessary to introduce the proper equa- tion for the film shape into the general equations developed by -'A .... [4c] the author. For example, for a slider bearing, the film thickness 4 ?i2 L ] at any point, x, along the bearing is expressed by Substituting the foregoing into the author's Equation [5], dividing by fi/io-1/2 and setting q proportional to the new func- h r ho. |'a — -j(a— 1)J [3 ]s tion u where q = Bh2~'/hi. [5] a = hi/lh [3a] the following equation is obtained 6 "Effects of Side Leakage in 120-Degree Centrally Supported Journal Bearings," by S. J. Needs, Trans. ASME, vol. 56, 1934, d2a ( m B V bhi [6] pp. 721-732. d£2 7 "On Problems in the Theory of Film Lubrication, With an Experimental Method of Solution," by A. Kingsbury, Trans. ASME, vol. 53, 1931, pp. 59-75. which although it is identical to the author's Equation [9] is 1 See Fig. 4 for illustration. different in that the functions/(£),

tank analogy, also worked this same problem and obtained very good agreement with Michell. Thus there are two experimental methods for determining bearing characteristics which will yield solutions in satisfactory («-«]""' ™ agreement with analytical methods. The resistance-network method has the advantage that it is less tedious than the elec- Substituting Equations [3] and [5] of this discussion into trolytic-tank method in that the number of voltage readings Equation [2] of the author, and solving for p yields which must be extracted from the respective apparatus in order to plot a pressure distribution is much less in the former than in v = (a 1} M 181 the latter method. In addition, it may be possible that the ~ J resistance-network method will be applicable for obtaining which is the equation for the pressure at any point on the bearing characteristics of journal bearings operating in the important surface. region of eccentricity ratios greater than 0.8. The electrolytic- Downloaded from http://asmedigitalcollection.asme.org/appliedmechanics/article-pdf/20/3/440/6748130/440_2.pdf by guest on 02 October 2021 Having obtained the functions/(£) and <•/(£) for a slider bearing, tank method is not applicable in this region owing to the dif- the leak resistances and input currents may be calculated ficulty of wetting the model representing the film shape and the through use of Equations [21a, 6] given by the author and readily fact that the minimum bath depth becomes very small. adjusted to their proper values in the apparatus shown in Fig. 1, herewith. The node voltages then may be measured and con- On Asymmetric Flow in an Axial- verted to pressure values through use of the author's Equation [22] and Equation [8] of this discussion. Flow Stage1 The foregoing procedure was followed for a slider bearing having a ratio of entrance to exit film thickness equal to 2, W. D. RANNIE.2 The unsteady flow that arises in the stalled region of operation of turbomachines has been observed and has (i.e., hi/h2 = 2) and L/B = 1. This particular bearing was chosen in order to compare the test data with the data calculated become of interest comparatively recently. Perhaps a descrip- by Michell9 for the same bearing. Pressure values obtained tion of typical behavior of compressor flow in the stalled region is using the method are shown plotted as circles the best way of indicating the difficulties of providing an adequate in Fig. 4, herewith. These values may be compared with those theory. Hot- measurements have been made by one of the writer's students on an axial-flow compressor project supported by the Office of Naval Research at the California Institute of Technologj'. The compressor blading was designed for vortex flow with prerotation in the direction of the wheel speed and a design axial velocity of 45 per cent of the tip speed. The hub ratio was 0.6 and the blade solidity averaged about unity for rotor and stator with nearly constant chord for the rotor. There were three complete stages with 30 blades in the rotor and 32 in the stator. On throttling the flow rate below design value a narrow region of low velocity (i.e., stall) extending over the inner half of the blades was observed to rotate in space in the direction of the rotor with 37 per cent of the rotor speed. There is one such region cir- cumferentially. As the flow rate is decreased this region broadens, then breaks up into two regions eventually 180 deg apart. Further throttling causes a splitting into three regions eventu- ally 120 deg apart. The multiple stalled regions traveled at the same speed as the single region and did not extend beyond mid- radius. All of these patterns were observed within a rather narrow band of flow rates and their appearances and disappear- ances occurred at slightly different flow rates depending on FIG. 4 PRESSURE VALUES OBTAINED BY ELECTRICAL NETWORK whether the throttle was opened or closed. METHOD AND BY CALCULATION At flow rates smaller than the foregoing a single stalled region appeared, but this one involved the entire blade height. It given by the full curves which were obtained by graphical inter- rotated with a speed of 26 per cent of the rotor speed, again in polation from the results of Michell's calculations. In general, the same direction. This pattern appeared to persist down to the agreement is quite good. completely closed throttle although the hot-wire response became The respective load-carrying capacities, obtained by employing very irregular near shutoff. All of the patterns observed oc- graphical integration methods on the curves and circles in Fig. curred throughout the length of the compressor behind the 4 yield the following equations entrance vanes. The stator blades were certainly involved in the pUB*L flow but whether this is the cause or the effect is not yet definitely known. IF = 0.06895 , (Michell) The writer believes that the author's theory is the first impor- tant contribution to an analysis of stalling phenomena. There ulUPL are several obvious limitations to his simplified model; it cannot

1 By W. R. Sears, published in the March, 1953, issue of the JOUR- IF = 0.06871 2 , (resistance network) NAL OF APPLIED MECHANICS, Trans. ASME, vol. 75, pp. 57-62. 9 "The Lubrication of Plane Surfaces," by A. G. M. Michell, 2 Professor of Mechanical Engineering, California Institute of Zeitschriftwhich agre fure withi Mathemalikn 1 per centund. Physik,Kingsbury, vol. 527 usin, 1905g hi, pps electrolytic. 123-137. - Technology, Pasadena, Calif.