1、NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL NOTE 3400 ANALYSIS OF ERRORS INTRODUCED BY SEVERAL METHODS OF WEIGHTING NONUNIFORM DUCT FLOWS By DeMarquis D. Wyatt Lewis Flight Propulsion Laboratory Cleveland, Ohio Washington March 1955 Provided by IHSNot for ResaleNo reproduction or networkin
2、g permitted without license from IHS-,-,-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECRNICAL NOTE 3400 ANALYSIS OF ERRORS INITODUCED BY SEVERAL METHODS OF WEIGHTING NONUNIFORM DUCT FLOWS By DeMarquis D. Wyatt SUMMARY Various weighting methods are applied to typical nonuniform duct flow profiles to
3、 determine average flow properties. a range of subsonic duct Mach numbers, but is confined to flows having uniform static pressure and total temperature. The analysis covers An averaging method is developed which yields uniform properties that reproduce the mass and momentum of the nonuniform flow.
4、In con- trast, it is shown that the use of conventional weighting methods may result in large errors in these properties. These errors are shown to have varying significance depending on the applications to which the data are applied. It is also shown that nonuniform flows through variable-area duct
5、 passages will cause changes in average flow properties that are not as- sociated with the real thermodynamic flow path. INTRODUCTION In most calculations involving duct air-flow properties, it is not convenient to consider local flow variations within the duct. the properties of the flow are treate
6、d as though they were uniformly distributed, and one-dLmensiona1 equations are applied to this uniform flow. of interest, the equivalent uniform flow must be determined by some method of averaging the properties of the real flow. Therefore, Inasmuch as the real flow seldom approaches uniformity at p
7、lanes This report presents the results of an analytical study made to de- termine the accuracy with which several commonly used averaging or weighting methods reproduce the real flow properties. of inherent errors is illustrated for several common applications of duct flow data. dimensional relation
8、s to the uniform flow are briefly examined. The significance Errors introduced through the application of one- Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 NACA TN 3400 The study considers several typical velocity gradients but is con- fined to
9、subsonic compressible flows with uniform static pressures and stagnation temperatures. (Since the present analysis was completed, it has been found that a more generalized, qualitative analysis of the same problem is contained in ref. 1.) ANALYSIS A uniform flow representing the flow properties of a
10、 nonuniform duct flow should satisfy the total energy, mass, and momentum of the real flow. For the special case considered herein in which the flow is assumed to arise from a uniform temperature source and to flow adia- batically to the measuring station, the total energy of the real flow can be re
11、produced by the assumption of constant total temperature in the uniform flow at the source value. The determination of a uniform flow that will simultaneously satisfy the mass flow and the momentum in the real flow is more difficult. Mass-Momentum Method For the special case in which the static pres
12、sure and total tem- perature are constant across the duct, the mass flow is given by the equation n 1 where M is the axial component of the local duct Mach number. (All symbols are defined in appendix A.) In order for the mass flow in the representation to equal this in- tegrated mass flow, the unif
13、orm flow must satisfy the relation 1 where pe and M, are the effective static pressure and Mach number, respectively. The integrated momentum of the real flow can be expressed by UI cn K) M Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3400
14、 3 Thus, the effective static pressure and Mach number must also satisfy the relation By combining equations (1) to (4), the expression for the effective I Mach number required to satisfy the total energy, mass flow, and momen- tum of the real flow becomes cp 1 + yMg (5) where m and CP are integrate
15、d values determined from equations (1) and (3). Although for this analysis the static pressure is assumed to be constant across the real duct flow, this measured value of pressure cannot be used in conjunction with the effective Mach number determined from equation (5) to satisfy the real flow prope
16、rties. effective static pressure must be determined from either the momentum or the mass-flow equations as Instead, a new n 1 This effective static pressure is never identical to the measured pres- sure if velocity gradients are present in the real flow. To complete the definition of the equivalent
17、uniform flow, an ef- fective total pressure can be determined from the expression Y The flow quantities defined by this method of averaging would be those obtained by mixing the measured profile to a uniform flow in a constant-area duct without wall friction. Mixing losses are inherently contained i
18、n the average flow quantities. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4 NACA TN 3400 Conventional Weighting Methods The weighting or averaging methods commonly used to obtain uniform flow representations of nonuniform duct flows require eith
19、er less com- plicated data-collection methods or less tedious calculation techniques than does the exact weighting procedure. herent errors in the representation of one or more of the basic proper- ties of the real flow. The required assumptions and applicable equa- t.i,ocs for three of the more com
20、monly utilized methods fol$ow. Such methods result in in- Mass-derived method. - When the mass flow in the duct is known from some independent measurement, the measured static pressure at a station can be used in conjunction with the geometrical flow area A to define a uniform duct Mach number Mc eq
21、uation that satisfies the mass flow by the From this average Mach number and the measured static pressure, an average total pressure Pc can be calculated as Y - PC The momentum calculated average Mach number becomes (9) from the measured static pressure and the It is evident that the mass flow and t
22、otal energy of the real duct flow are inherently satisfied by the mass-derived method of determining an average flow. fy the momentum of the real flow. There is no attempt in this method, however, to satis- Mass-flow-weighting method. - A pitot-static survey of the flow at the desired duct station i
23、s frequently employed to determine an average uniform flow. If it is assumed that the measured nonuniform flow can be brought to rest without mixing losses, the resultant pressure can be determined from the equation Provided by IHSNot for ResaleNo reproduction or networking permitted without license
24、 from IHS-,-,-NACA TN 3400 5 For the special case in which the static pressure and total temperature are constant across the duct, the compressible form of equation (lla) becomes A w w a cn The mass flow and momentum of the uniform flow having a total pres- sure defined by equations (11) are not uni
25、que values. depend upon the nature of additional assumptions about the properties of the uniform flow. Their magnitudes The measured static pressure at the duct station is often assumed to be the static pressure of the average flow. With this assumption, a uniform duct Mach number can be defined by
26、the relation 1 The momentum for this uniform flow is given by equation (10). The calculated mass flow becomes 1 The mass flow determined from equation integrated mss flow which was used to sure in equation (lla). (13) will not correspond to the determine the average total pres- This anomaly between
27、the integrated and calculated mass flows can be avoided by defining an average static pressure with the average total pressure from equations (ll), will satisfy the integrated mass flow. mass flow under these conditions is given by pc which, when used The average Mach number required to satisfy the
28、Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6 NACA TN 3400 and the resultant static pressure becomes pC l- P, = - (1 + 9 The momentum calculated from equation (10) with either the measured or calculated values of static pressure and corresponding
29、 Mach number will not equal the integrated momentum. (15) can be determined which would yield a static pressure and Mach num- ber for the uniform flow that would satisfy the real flow momentum. These flow properties would not satisfy the mass flow, however, and are not conventionally employed. Equat
30、ions similar to (14) and Area-weighting method. - When Pitot-static flow surveys are em- ployed, the complications of the calculation procedure can be reduced by using an area-weighted average total pressure determined fromthe e quat ion n 7 In cn M M The remaining properties of the uniform flow are
31、 calculated by the equations used with the mass-flow-weighting method. As in the former method, several solutions for these properties are possible. the static pressure is assumed equal to the measured value. If inde- pendent mass measurements are available, a static pressure may be cal- culated to
32、satisfy the mass flow. tegrated momentum will not be satisfied with either assumption. (For the incompressible case, a uniform flow defined by the total pressure from eq. (16) and the measured static pressure will duplicate the real flow momentum. ) Generally, With compressible duct flow, the in- NU
33、MERICAL CALCULATIONS The uniform flow properties of three arbitrary duct profiles were calculated by the mass-momentum weighting procedure and by the conven- tional weighting methods discussed in ANALYSIS. For simplicity, the ducts were assumed square with symmetrical two-dimensional profiles. The p
34、rofiles considered were: (a) A power profile described by 1 M = Kx7 - Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3400 7 (b) A discontinuous, separation profile represented by (c) A linear profile of the form M = K(0.2 t 0.8) Each profile
35、 was evaluated for a range of values of K (correspond- ing to the maximum Mach number at the duct centerline) from 0 to 1.0. Mass-momentum method. - Equation (1) was integrated for each of the The profiles to determine the mass flow actually contained in the duct. integrals for the power and linear
36、profiles were approximated by series expansion. K 1.0) were The resultant expressions for the mass flow (valid for ?!/? = 0.875K + 0.070K3 - 0.00292K5 + 0.00025K7 - . . . (Power profile) PA (204 1 = 0.9K(l + $7 (Separation profile) (20b) = 0.9K + 0.0738K3 - 0.003074K5 + 0.00026K7 - . (Linear profile
37、) (zoc 1 The actual momentum with the assumed profiles was obtained by inte- grating equation (3) with the resultant expressions (Power profile) (zm = 1 + 1.26Kz (Separation profile) (21b 1 = 1 + 1.1387K2 (Linear profile) (21c) - - 1 + 1.08889K2 PA Effective values of duct Mach number, static pressu
38、re, and total pressure were determined from equations (5), (6), and (7), respectively. Mass-derived method. - By using the values of mass flow from equa- tions (ZO), the properties of the uniform flow were determined from equations (8) to (10). Provided by IHSNot for ResaleNo reproduction or network
39、ing permitted without license from IHS-,-,-8 NACA Tm 3400 Mass-flow-weighting method. - The product of the average totalpres- sure and the mass flow was obtained from equation (llb) and modified to the form Equation (22), when integrated, yielded the following expressions : P m$3 = 0.87% + 0.56K3 +
40、0,146 + 0.016K7 + 0.0007K9 (Power profile) P PA (234 4 = 0.9Kk + $) (Separation profile) (2%) = 0.9K+ 0.5904K3 +0.1476K5 +0.01665K7 + 0.00071K9 (Linear profile) (23c 1 The values of integrated mass flow from equations (20) were then used to obtain the average total pressure. Equations (12) to (15) w
41、ere used, as appropriate, to determine the calculated average properties of the flow. from Area-weighting mthod. - The average total pressure was obtained equation (16), which becomes The resultant expressions, after integration, were PC - = 1 +0.5444K2 +0.1114fi +O.O0942K6 +O.O00204K8 - . . . (Pare
42、r profile) P (254 7 = 0.1 + 0.9k + g (Separation profile) = 1+0.5693K2 +0.1177 +0.00988K6 +0.0002 hence, the mass-flow errors indi- cated in figure 5 are included. Similar trends would be observed for the weighting methods in which the integrated mass flow was satisfied. The magnitude of the drag-pa
43、rameter errors would be decreased in the latter case, however. The increasing error in drag parameter with increasing supersonic Mach number does not necessarily imply an increase in the absolute drag-coeff icient error of the same proportion. The total-pressure- recovery term in the denominator of
44、the drag parameter will generally decrease with increasing Mach number. This will compensate in part for the increase in parameter error. For such cases, the anticipated error in drag coefficient may remain relatively constant throughout the super- sonic Mach number range. If, on the other hand, hig
45、hly efficient in- lets are being considered at high Mach numbers, the drag-coefficient error will increase for a given level of flow distortion as compared with the errors resulting at lower Mach numbers. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,
46、-,-IlACA TN 3400 15 Inlet pressure recoveries may be expected to remain at a generally high level throughout the subsonic Mach number range. ipated that the potential error in drag coefficient would therefore in- crease as Mach number is reduced unless there was a concomitant improve- ment in the du
47、ct profile. It would be antic- Variable-area-duct calculations. - In many duct flow applications, uniform-flow properties are calculated at a flow measuring station by one of the weighting methods. used to compute flow properties at other stations in the duct by the assumption of appropriate total-p
48、ressure losses. These resultant prop- erties are affected by the errors previously demonstrated to be associ- ated with the various weighting methods. Additional errors are intro- duced if there are area changes in the duct. 34 D n One-dimensional flow equations are then 34 The nature of the errors
49、introduced in variable-area-duct calcula- tions can be illustrated by the flow shown in figure 15. It has been assumed in this flow that a uniform static pressure exists at each sta- tion and that each filament of the flow expands isentropically between the two stations. Each filament diffuses to a higher static pressure as the flo
