1、cm /I,LTECHNICAL NOTE 3583CHARTS OF BOUNDARY -LAYER MASS FLOW AND MOMENTUMFOR INLET PERFORMANCE ANALYSISMACH NUMBER RANGE, 0.2 TO 5.0By Paul C. Simon and Kenneth L. KowalskiLewis Flight Propulsion LaboratoryCleveland, OhioWashingtonNovember 1955t.,I =!. F“,.t Lbs JLh LL b:m! . -n.,_+ -,.-. . . . . .
2、-. - ,- - -. . . . -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TECH LIBRARY KAFB, NMNATIONAL ADVISORY COMMITTEEFORTECHNICALNOTE 3583AERONAUTICSCHARTS OF BOUNRARY- MASS FLOW AND MOMENTUMFOR INLET PERFORMANCEANALYSISMACH NUMBER RANGE, 0.2 TO 5.0By
3、 Paul C. Sinmn and Kenneth L. KowalskiSUMMARYMass flow and momentum for various fractions of a turbulentbound-ary layer are presented in chart form for a range of velocity profilesand for Mach nunbers from 0.2 to 5.0. These charts are intended toassist the analyst and designer of boundary,-layerremo
4、val systems. Ap-plication of the charts to inlets of arbitrary shape and to the deter-mination of the pressure recovery of rectangubr, normal-shock inletsimmersed in boundary layer is described.INTRODUCTIONIn the design of auxiliary air inlets or boundary-ler renmvalsystems, an estimate of the enter
5、ingboundary-layermass flow and momsn-tummust be made. The mass flow must be knuwn in order to correctsize the inlet, while the inlet nnnentum is frequently required to es-timate the potential pressure recovery, or the thrust ndnus drag of theinstallation,or both. To aid the designer or analyst confr
6、ontedtithsuch problems, charts of mass flow, total monadmm, and momentum ratiofor various fractions of a two-dimensional,turbulentboundary layer arepresented. The Mach nwiber range covered is from 0.2 to 5.0 and thepowers of the nonitkensionalvelocity profiles considered are 1/5, 1/7,1/9, and 1/11.
7、The charts presented -e adaptableto inlets eitherattached or detached to the boundary-layer generating surface.An approximatemethod for determiningthe boundary-layerflowparameters for inlets of arbitrary shape is included in an appendix.In addition, the average normal-shockpressure recovery of inlet
8、s ofrectangular cross section is determinedfrom the mass-momentummethodof reference 1 and a range of values for a l/7-powerprofile is charted. . . - 2Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 NACA TN 3583DESCRIPTIONOF CHARTSTwo-dimensional,bo
9、undary-layer flow parameters can be evaluatedanalytically once the velocity profile parameter N and the boundpresented in figures 1 to 6. They canbe applied directly to rectangular mshaped inlets. Application of the charts to arbitrary shaped inlets isdemonstratedin appendixB.All boundsry-layer rati
10、os are establishedby evaluating the denomi-nators at local-stream conditions at the edge of the boundary layer.The parameters may be easily referenced to conditions at infinity if itis desirable.Boundsxy-i+yer Mass-FlowRatioThe boundary-layermass-flow ratio m/m7 is-defined as the ratioof the mass fl
11、ow in a givenflow in the local stream ofsketch,t 7fraction of the oundary layer to the massequal area. As seen from the followingBoundary-layer profile, V/V, = (y/5).1.0 *Boundary- y/6 blayergenerating“surface=r */ / / / / / / / / / / / /o 1.0WZ . . - Provided by IHSNot for ResaleNo reproduction or
12、networking permitted without license from IHS-,-,-NACA TN 3583m/mz can be written asIf the boundary-layervelocity profile, describedbyv ol/h= gz3m(2)is substitutedinto equation (1), and it is assumed that the static pres-sure and total temperatureremain constant throughout the boundary layer,the int
13、egral form of the mass-flow ratio becomes,=.-:& ,3)Upon inteation of equation (3)for inlet height ratios r /5 = O (at-%tached inlets) and r2/5 1.0 for incrementsof r2/5 m%were determinedfrom equation(4)plus local-streammass flow etween y/6 = 1.0 and r2/5.values of m/, presented in figure l(b)(N= 7),
14、 are cross-plottedin figure 2 as a function of r2/b. The use of these curves improvesthe accuracy of problems involvedwith fractional inlet height ratios byavoiding the necessity of interpolation.Boundary-LayerTotsl-MomentumRatioThe boundary-layer total-momentumratio of the total mommtum in a givent
15、he total momentum in the local streamwritten asratio / is&fraction o theof equal area.From a development slar to the mass-flow, it can being constant static pressure across the boundary layerdefined as theboundary layer toThis ratio can be(5)shown that assum-yields the fol-lowing general equation of
16、 the boundary-layer total-momentumratio X/t,in closed form, for rl/6 = O, r2/6 1.0, and odd values of N:+= (1+ &)-l ( )+C2 1+=12 c -#-2- N N-2-”- 1(6)where.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3853and=The values of ,and N = d5, 7,
17、9, &nd ,was determinedfrom equation (6) for T = 1.4are presented in figure 3 as a function of M7for incrementsof r2/5 of 0.1 to 2.0. These same values are cross- Uplotted in figure 4 as a function of r2/5, but for N = 7 only.An application of the total-momentumratio to the estimation ofboundary-laye
18、r inlet total-pressurerecovery is discussed in appendix C.Boundary-LayerMomentum RatioThe boundary-layermoqentum ratio Q/Q2 is defined as the ratio ofthe momentum in a given fraction of the boundary layer to the momentumin the local stream of equal area. This ratio is written asor as a function of t
19、otal momentum and local stream Mach nunber(7)(7a)The momentum ratios for y = 1.4 and N= 5, 7, 9, and 11 srepresentedin figure 52.0. CrossIn theinlet totalas a function of MZ for increm&9 of r /5 frmO.1 toplats of the same value-sfor N = f7 are g ven in figure 6.Total Momentumfor Drag .evaluation of
20、inlet drag, it is common procedure to utilizenxmentum which incorporatesthe term -P=A. Thus, equation._.-.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6(5) couverts toNACA TN 3583or in terms of local-streamMach ntier(8)(8a)Total momentum for drag
21、can be written as a function of the momentumratio =ws m)If the conditions are such that pl = p the total momentum ratiofor drw /2 reduces simply to the momntum ratio (p/qZ.Effect of Velocity Profile ParameterVariations of boundary-layer mass-flow, total-momentum,and momentumratios with changes in th
22、e shape of the velocity profile are presentedin figure 7. There is a pronounced increase in all three boundary-layerflow parameters at all Mach nunibers,when the profile parameter N isincreased from 5 to Il. These plots can be used to determine the errorinvolved in selectingvalues of boundary-layerp
23、arameters from the limitednumiberof charts presented rather thsm the actual N values involved inan analysis. For-example, if in aof 6 at a of 3.0 and a r2/5to 5 percent. .problem N is taken to be 7 insteadof 1.0, the error in m/% mounts. Provided by IHSNot for ResaleNo reproduction or networking per
24、mitted without license from IHS-,-,-.NACA TN 3583 7Summaxy of EquationsThe application of inlets to boundary-layer control problems maylead to inlet esign heights either equal, greater, or less than theboundsry-layerthickness and either attached or detached from the gen-erating surface. The equation
25、s in table I, in conjunctionwith theworking charts,wilJ serve to evaluate the flow parameters desired.Lewie Flight Propulsion LaboratoryNationalAdvisory Committeefor AeronauticsCleveland, Ohio, Septenber 16, 1955. Provided by IHSNot for ResaleNo reproduction or networking permitted without license f
26、rom IHS-,-,-NACA TN 3583APPENDIXA.,flow area, sqinlet height,Mach numberSYMBOLSft(r2 - rl), ftmass flow, pVA, slugs/seeratio of mass flow in a given fraction of boundary layer tomass flow in local stream of equal area (see eq. (l)velocity profile paramtertotal pressure, lb/sq ftstatic pressure, lb/s
27、q ft(seeeq. (2)normal Ustance from boundary-layer generating surface to inletlip, fttotal temperature, %velocity, ft/secnormal distance from boundary-layer generating surface, ftratio of specific heats, 1.4boundary-layer thickness,mass density, slugs/cu fttotal momentum, mV + PA=ftrpM2A+ PA, lbratio
28、 of total momentum in a given fraction of boundary layerto local-streamtotal momentum of equal srea (see eq. (5)total momentum for drag, mV+ (p - p=)A . y_pM2A+ (p - p=)Aratio of total momentum for drag in a given fraction oflayermomentum, mV = M2A, lbratio of momentum in a given fraction of boundar
29、y layerlocal-streammomentumof equal area (see eq. (7)boundaryto,.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3583Subscripts:z local stream conditions outside the boundary layerm free-stream conditions (at iufinity)1 lower lip of inlet or
30、inlet entrance station in appendix C2 upper lip of inlet.- . . .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-10 NACA TN 3583APPENDIXB4“APPLICATION OF CHARTS TO ARBITRARY-SHAPEDINLETS. The mass-flow and momentum ratios presented in figures 1 to 6 a
31、reapplicableto boundary-layer inlets of rectangular cross section.However, these integratedvalues may be used to approximatethe criticalmaas flow and nmmentum of other shaped inlets.Consider an inlet of arbitrary shape (see sketch)with a total areaA/A &ivided into small vertical elements of width-w
32、and varying heightsH. It can be seen thatA. L Hkwk=wl+H2w2+ = o Q Hnwnk=l.Therefore, the approximate inlet mass-flow ratio can be written asandifw=w 12 = wk, then.nm=.9 : E(J :kHkk=l(Al)(A2)3r-toProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN
33、 3583 11where values of (m/k are determinedfrom table I and the cues ffigure 1.This development can also be applied to the determinationof themomentum and the total-momentum ratios. . -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-uAPPEND?X cNACA T
34、N 3583APPLICATION TO BOUNDARY-LAYERINLEl?-PRE8SURERECOVERYA mass-momentummethod of averaging a nonuniform flow is outlinedin reference 1. This method may be used to deternihe an approximationto criticalboundary-layer inlet-pressurerecovery. The total-pressurelosses are assumed to be those resulting
35、from a normal shock at the inletentranceplus flow mixing losses in a constant area duct. The resultingvalues of total-pressurerecovery P1/PZ for a range of Mach numbers% andinl et height ratios r2/6 for attached rectangular inlets im-mersed in a l/7-powerboundary-layerprofile sre presented in figure
36、 8.The proceduie used to calculate the recoveries of either attached ordetached inletsby the mass-momentummethod is outlinedas follows:Consider a rectangular inlet operating critically in a boundarylayer of known character. Under the assumption of the conservationofmass flow and total momentum, a on
37、e-dimensionalrepresentationof aunifcrrmflow inside the inlet and behind a normal-shock can be made(station1).Equating the integratedboundary-layermass flow with the inletmass flow and assuming z.= 1 ivesand for the total momentumP,&)(.+Y (:)= P, (w) 5$) (C2)erm +7=+-Yhereres-tsWhen equation C2) is d
38、ividedby the product of equation (Cl) and theor letting .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.NACA TN 3583(tabulatedin ref. 3) equation (C3)becomesThus the value of (F/F*)= determines Mmltiplying both sides of equation (Cl)&yy+l() ZFn2y+l
39、givesor lettingLand (p/P)l. In addition,y+l(Vi-+1)-y+l(Vr+gmr+l2am-+-f+)12r-lMl+ M?(alsotabulated in ref. 3) equation (C5)becomes13(C4)(C5)(C6)Thus, with the aid of figures 1 to 4, gas tables similar to those ofreference 3, and equations (C4) and (C6),thetermined frominlet recovery can be de-(C7). .
40、 . .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-14 NACA TN 3583REFERENCES1. Wyatt, DeMarquis D.: Analysis of Errors Introducedby Several Methods “of Weighting Nonuniform Duct Flows. NACA TN 3400, 1955.2. Tucker, Maurice: Approximate Calculation o
41、f Turbulent Boundary-LayerDevelopment in CompressibleFlow. NACA TN 2337, 1951.3. Keenan, Joseph H., and Kaye, Joseph: Gas Tables - ThermodynanrLcProperties of Air, Products of Combustion and ComponentGsses, and CompressibleFlow Functions. John Wiley & Sons, Inc., 1948. ma71a15Provided by IHSNot for
42、ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3583 15TABLEI. - SUMMARY0 EquATIoris. .- -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-16 NACA TN 35831v .- Local-atrc=m-h UJ 1(a) velociwofle Pter J 5-Fire 1. - Ma
43、as-flow ratio for vaxioua fractiom of bo ImWr.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATIT3583 17.u)ot-N3-$.-cal-&r- h m- 5(b) Velocity profile paramter N, 7.Figure 1. - Continued. MEW-flow ratio for various fractionsof boundary layer. - -
44、. .- ._ . . .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-18 NACA TN 3583.Local-atxesmMach number9. (c) Velocity pr&ilepter N, 9.Figure 1. - Continued. Mass-flow ratio for various fractions of boundary layer.Provided by IHSNot for ResaleNo reprodu
45、ction or networking permitted without license from IHS-,-,-NACA TN 3583 23(c) Velocity profile parameter R, 9.Figure 3. - Continued. l?otal-mmtim ratio for variouE fractlom of IOmdm-y layer. - - _Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-241NAC
46、A TN 3583Local-stream Mach number, Mz(d) PelocityProrileP=Xer I7,11.Fime 3. - Concluded. Total-uomntum ratio for mxiom fractlom of boundary leyer.- -. ._ -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3583.25.L&)%.Inletheightratio,r/8Figure
47、.4.- Totalumnentumratioforvariouslocal-streamMachnumbers. VelocityprofilePera- N, 7. _.-. - . . -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-28 NACA TN 3583ILocal-streamMach number, Ml(c) Velocltyproflle pmameter N, 9.Figure 5. - Contimd. lkmentum ratio for various fractions of lm
