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1 UBBWT5TW" ' I-.' '.. ' 1 " ' '! " A/ O 2 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL NOTE 2750 MATRIX AND RELAXATION SOLUTIONS THAT DETERMINE SUBSONIC THROUGH FLOW IN AN AXIAL-FLOW GAS TURBINE By Chung-Hua Wu Lewis Flight Prpulsin Labratry Cleveland, Ohi NACA i Washingtn July 1952 Reprduced Frm Best Available Cpy is'* sfck' % (V* Möö-0%**JLJL3ö

2 P^_, i t, ' _«..,i ; ^ :! =- **. NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL NOTE 2750 MATRIX AND RELAXATION SOLUTIONS THAT DETERMINE SUBSONIC THROUGH FLOW IN AN AXIAL-FLOW GAS TURBINE N By Chung -Hua Wu SUMMARY A methd recently develped fr determining the steady flw f a nnviscus cmpressible fluid alng a relative stream surface between adjacent blades in a turbmachine was applied t investigate the subsnic thrugh flw in a single-stage axial-flw gas turbine. A freevrtex type f variatin in tangential velcity was prescribed alng the stream surface. Cylindrical bunding walls were specified in rder t avid radial flw at the walls. The flw variatins n the stream surface fr incmpressible and cmpressible flws were btained by using the relaxatin methd with hand cmputatin and the matrix methd n bth an IBM Card Prgrammed Electrnic Calculatr and a UNIVAC. In all slutins cnsidered in this investigatin, cnvergence was btained withut difficulty. A cmparisn between the relaxatin and the matrix methds shwed that mre accurate results were btained with the matrix methd in a shrter interval f time. The results f. these accurate calculatins prvide a basis fr evaluatin f simpler, mre apprximate methds fr cmputing subsnic thrugh flw in turbines. Cnsiderable radial flw was btained fr bth incmpressible and cmpressible flws because, f the radial twist f the stream surface^ required by the prescribed velcity diagram and the cmpressibility f the gasj the radial twist f the stream surface and the cmpressibility f the gas had equally imprtant effects, and the nnlinear nature f the equatins defining the flw was quite evident. The shape f the stream surface was fund t be sensitive t the axial psitin f the radial element f the stream surface in the statr. When the radial element f the stream surface in the. rtr was near the midaxial psitin in the rtr, a large negative gradient f axial-velcity was bserved in all cases ahead f the rtr.

3 üi NACA TN 2750 INTRODUCTION In the design f gas turbines having relatively lng blades in the radial directin, a first apprximatin f the radial variatin f the state f the gas is cmmnly btained by assuming the gas t be nnviscus and t flw n cylindrical surfaces. Axial symmetry is assumed in such ne-dimensinal slutins fr the radial variatin in the gas state upstream and dwnstream f a blade rw and they are usually referred t as the "simple-radial-equilibrium" r "simplified-radial-equilibrium" slutins (see references 1 and 2 fr examples). Even turbines designed frm a free-vrtex velcity diagram, hwever, have cnsiderable radial displacement f gas particles acrss the blade rw due t the cmpressibility f the gas; this displacement, f curse, vilates the assumptin f flw n cylindrical surfaces. The radial displacement f gas particles acrss the blade rw and the curvature in the streamline caused by the radial mtin have a significant effect n the radial variatin in the gas state (see reference 2). * N ^ CO Several thrugh-flw methds have been develped t cnsider bth, radial and axial variatins in flw cnditins (see, fr example, references 2 and 3). In these methds an infinite number f blades and a nnviscus fluid are assumed. The interpretatin f such a slutin, its mdificatin fr any finite number f thick blades, and it's extensin t a cmplete three-dimensinal slutin fr nnviscus fluids are given in reference 4. In actual turbines, the mtin f the gas is further cmplicated by the secndary flw caused by the bundary layers alng the hub and casing walls (fr example, reference 5). Nevertheless, it is believed that the detailed flw analysis based n an assumptin f a nnviscus gas will permit a clearer understanding f the individual cntributins t the cmplete flw and help t explain the develpment f the viscus bundary layer alng the walls. The theretical methd f reference 4 fr btaining a cmplete three-dimensinal slutin is based n an apprpriate cmbinatin f a number f mathematically tw-dimensinal flws n tw kinds f relative stream surface. The first kind f relative stream surface extends frm the suctin surface f ne blade t the pressure surface f the adjacent blade and deviates, in general, frm a surface f revlutin abut the axis f the turbmachine; in reference 4 this surface is called S lv The secnd kind f relative stream surface extends frm hub t casing and rughly apprximates the shape f the mean camber surface f the blades; in reference 4 this surface is called S 2. The equatins defining the flw n these tw kinds f stream surface are, hwever, similar and the methds f successive apprximatin used in their slutin are essentially the same. The mechanics f btaining a numerical slutin f either f these tw-dimensinal prblems by the methds f reference 4 has nt previusly been investigated. If cnvergence f ne f these

4 r NACA TN 2750 tw-dimensinal slutins can "be btained withut difficulty, the ther ^ slutin shuld then cnverge as well. The methd f reference 4 will :* then very likely be practical fr analyzing three-dimensinal flws fcmpressible nnviscus fluids in turbmachines having finite numbers f thick blades and arbitrary hub and casing shapes. The primary bjective f this investigatin made at the MCA Lewis f labratry is t study in detail the technique f slving fr the sub- N snic flw alng a relative stream surface f the secnd kind (S2) by -H using the methds f references 4 and 6. At the same time, f curse, the effects f cmpressibility and radial twist f the stream surface are btainedj the examinatin f these effects therefre cnstitutes h the secndary purpse f this investigatin. If highly accurate slutins can be btained in this way, these slutins will prvide a basis fr evaluatin f ther simpler but mre apprximate slutins fr sub-, snic thrugh flw in turbines. " Fr these calculatins, a single-stage axial-flw turbine was selected as the machine type t be analyzed. The annulus walls were ;.g' chsen t be cylindrical and t have a hub-tip radius rati f 0.6. The radial distributin f the tangential cmpnent f velcity was f the free-vrtex type..the turbine-wrk parameter -AH/U+ 2 was 0.96 ' where -AH represents the stagnatin-enthalpy drp and U t, the bladetip speed. The tangential cmpnent f velcity at the rtr exit was assigned equal t zer. Fr a cmpressible fluid, the assigned cnditins resulted in an abslute Mach number f 0.99 at the rt and exit f the statr and a relative Mach number f abut 0.70 at the rt and entrance f the rtr. The effects f radial blade frce were varied by changing (l) the axial distributin f aerdynamic lading and (2) the axial lcatin f the radial element f the stream surfaces in I'.' '..' the statr. The fllwing three methds were emplyed in making the calculatins r (l) Relaxatin methd with the use f a hand-perated desk calculatr.. ' (2) Matrix methd n an IBM Card Prgrammed Electrnic Calculatr (3) Matrix methd n a UNIVAC The matrix factrs btained in the matrix slutins can be used fr similar calculatins fr cmpressrs and turbines which have a cnstant hub-tip radius rati f 0.6.

5 4 NACA TN 2750 SYMBOIS The fllwing symbls are used in this reprt. (Bld symbls indicate vectrs n the stream surface; nnbld symbls indicate the cmpnents f these vectrs.) a velcity f sund B variable defined by equatin (2c) "B 1 ' differentiatin cefficients in equatin (26) used t 4 <J multiply functin value at grid pint j t give m tü derivative at grid pint i using plynmial f furth degree [C],[EJ,LFJ, square matrices.". '. [G},CL],[U3 JL differentiatin with respect t time fllwing mtin f gas particle F vectr having dimensins f frce per unit mass f gas; defined "by equatin (6) H ttal, r stagnatin, enthalpy per unit mass f gas, h+vijh*^ h enthalpy per unit mass f gas i,j,k grid pints M Mach number U nnhmgeneus term f principal equatin as given in equatin (5) n unit vectr nrmal t stream surface p static pressure f gas q any dependent variable E gas cnstant r radial distance, r*r^

6 " ; ' ; ''', " J» % NACA TN 2750 s T t U entrpy per unit mass f gas, str temperature f gas time velcity f "blade at radius r Jf V abslute gas velcity, V*Ut -* «* w relative gas velcity, W*Ut z axial distance, z*r t {a},{ß},{*} clumn matrices 1. r rati f specific heats, r> 8 grid spacing in r- and z-directins, respecti z ^rt> ^z r t... P mass density, p p^,. 2,4 general variables used in density table «P angular psitin t 2 stream functin, i(r*p T iu t r t CD angular speed f blade * Subscr ipts: e 'h i exit hub inlet v% j grid pint 0 refers t psitin where stream surface has i ment r where F r = 0 r radial cmpnent T ttal, r stagnatin, state

7 NACA TN 2750 t u z tip tangential, r circumferential, cmpnent axial cmpnent Superscripts: i.j.k grid pints <u * dimensinless value N EQUATIONS GOVERNING FLOW ON A RELATIVE STREAM SURFACE BETWEEN TWO ADJACENT BLADES The present reprt is cncerned with the inverse slutin (design prblem) fr the steady cmpressible flw, n a relative stream surface abut midway between tw adjacent blades (see fig. l). The shape f this stream,surface is nt knwn in advance but the variatin in tangential velcity f the gas and the psitin f the radial element f the surface are prescribed in the design. The shape f the stream surface 9 as well as the state f the gas flwing alng the surface is described by the tw independent variables r and z. '. ' In reference 4, the fllwing cntinuity equatin fr steady flw n the stream surface is btained: a(rbpwr) 9(rBpW z ) ' ' ' Yl) ar az In equatin (l) the bld partial derivative sign dentes the rate f change f the dependent variable n the stream surface with respect t the independent variable and is related t the rdinary partial derivative with respect t the crdinates r, cp, and z as fllws: 9 = a _Ü l ä. ' ' (2a) 3r dr riß r cp ia = a _ ^i a r 2 b) 3z z r^ r cp The angular variatin f the variable is thus implicitly included althugh its value can be calculated nly after the shape f the stream

8 NACA TN 2750 i surface is btained in the slutin. The variable B in equatin (l) is related t the angular variatin f the gas velcities and the shape f the stream surface by B P/nr ÖW r iöw ü nz ÖW Z \ N k This variable B can als be interpreted as a variable angular thickness f a thin stream sheet whse mean surface is the stream surface shwn n figure 1 (reference 4). Because. B alng the mean stream surface is very clsely related t the rati f circumferential channel width t pitch (reference 7), it can be apprximately estimated in the design by cnsidering desirable axial and radial variatins in blade thickness required t prvide, fr example, adequate strength and a cling passage f sufficient size; with these variatins taken int accunt, B can then'be used in the design calculatin f the meanstream surface. Fr the limiting case f an infinite number f blades f zer thickness, B becmes a cnstant and is taken equal t 1. ^ Equatin (l) is the necessary and sufficient cnditin that a stream functin ifr exists with f =rb P W z (3) i = -rbpw (4) 9z With the tangential velcity f the gas specified;, the subsnic flw f the gas is btained, by the slutin f if in the fllwing principal equatin, which is btained frm the equatin f mtin in the radial directin with the use f relatins (3) and (4): where ö'.ia + s5i[ ai in (5) K },2 ± r ar + CM. 9z ai72 c Q T C N = 1 \~*(ßP) *± + 3(Bp) 3i] _ (rbp) 2 ~ V u 3 ( V u r ) _ 3H + T 8s + p 1 Bp _ >r ar z azj aty/är L r ar ar 3r r J

9 8 NACA TN 2750 In equatin (5), F r is the radial cmpnent f a vectr F, which is defined as fllws: F = 1 /b m ÖS \ 1 dp n u rp ^> (6) Fr the limiting case f an infinite number f blades, F becmes the blade frce. Fr the evaluatin f F r, which represents the influence f the radial twist f the stream surface and f the circumferential pressure gradient n the gas flw, the ther cmpnents f F are first cmputed by the equatin f mtin in the tangential and axial directins: F u r = rbp 2i 3z 8(V) 3r at 9r 8(V. iz J (7) F z - - r az 1 3 z +» /ar \_Br 3r J i at f^i i?i + &. JL r^iifii M + 9 ( B P) i]\ ^2äT ar 2' r ar + az2 " BpLar ar az azjf (8) The cmpnent F r is then btained by the use f the fllwing equa- tin, which is derived frm the integrability cnditin, which insures that the stream surface t be btained is a cntinuus integral surface : (9) The radial derivative f entrpy s in equatin (5) is determined frm its radial distributin at the inlet and the fllwing cnditin f the cnstancy f s alng a streamline n the stream surface fr reversible adiabatic flw:. 5 s. = Dt (10)

10 2S NACA TN 2750 The radial derivative f H in equatin (5) is determined frm the inlet variatin and the fllwing equatin: DH D(V u r) Dt = ü) Dt (11) c- Equatin (ll) is btained (reference 4) by the use f the equatin f mtin n the stream surface and the fllwing equatin expressing the rthgnal relatin between the resultant relative velcity and the surface nrmal n r its parallel vectr F: w r*r + Vu + *Jz = (12) Because the three equatins f mtin have already been emplyed in the slutin, there is nly ne mre independent relatin in equatins (ll) and (12). In the fllwing, equatin (ll) is used and is cnsidered t represent the rthgnal relatin (12). The variatin f density included in N in equatin (5) is determined frm the i(r-derivatives by use f the fllwing relatin between density, enthalpy, and entrpy, between the lcal static cnditin and the ttal cnditin at the inlet: f>t,] H i (vf 2H-JT 2 (IT +(g)' 2H i (rbp)2 1 r-i Sm J -S e Tfl (13) PRESCRIBED DESIGN CONDITIONS In the present study f the thrugh flw in a gas.turbine, the effects f sme f the design variables are cnsidered. Cylindrical bunding walls are specified in rder t avid radial flw at the walls. The meridinal sectin f the turbine is shwn in figure 2. A hub-tip radius rati f 0.6 and a blade aspect rati f 2.67 (which is based n axial chrd and crrespnds t the blade-rw aspect rati f 2 used in reference 2) are chsen in rder t cmpare sme f the results with thse previusly btained in\reference 2 by an apprximate methd.

11 10 NACA TN 2750 Flw cnditins were cmputed fr six sets f assigned cnditins. These sets f cnditins are designated cases A, B, C, D, E, and F and are summarized in the fllwing table:. Case Fluid Lading distri- - butin Axial lcatin in statr f radial element f stream surface * z CM O A B C D E F Incmpressible Incmpressible Cmpressible Cmpressible Cmpressible Cmpressible Unifrm Nnunifrm Nnunifrm Nnuriifrm Nnunifrm Nnunifrm (F r neglected) The prescribed variatin f tangential velcity, r the angular mmentum per unit mass f gas V*r*, is such that at a cnstant-z plane, V^r* n the stream surface is cnstant with respect t r *; and at all T r* * fixed values f r*, the variatin f n the stream surface with respect t z* is as shwn in figure 5. Tw kinds f variatin with respect t z* * are cnsidered. The dashed line shws a linear variatin,- and the slid line shws a cmpsite variatin in which a cnstant rate is maintained fr the first half f the blade chrd and a rate linearly decreasing t zer is used fr the secnd half. These tw variatins are called unifrm and -nn-unifrm lading, respectively. In bth cases the ttal change f V u r divided by U t r t (r -AH/U t ) acrss the blade rw is 0.96, which is used in reference 2. The expressins fr the dimensinless specific angular mmentum are as fllws: (a) Unifrm lading ' Statr: 0<z*< 0.15, * 3(V*r*) 0.96 az* 0 15 V u r = 0 n z* (14) (15)

12 NACA TN Rtr: 0.20 < z* < 0.35, I «I (t>) Hnunifrm lading *«**')_ 0.96 az* 0.15 V u r = 0 0 Statr: 0<z*< 0.075, (0.35 * ) (16) (17) 8 (V* r *) z * * * V r = z u < z*<0.15, (18) (19)»frfr*) 8 0^96/. z*\ 9z* \ " 0.15/ (20) yy X \0.15j J (21) Rtr: 0.20 < z* < 0.275, #) _ az" (22) v r* = (z* - 0.2) u v ' (23) < z*< 0.35,»Pfr*) / z*-q. 2 ) az* \ X 0.15 / K^>

13 12 NACA TN 2750 V*r* = X 6.96 u ^ (25) Fr a given axial distributin f tangential velcity, a change in the axial lcatin f the radial element f the stream surface will alter the radial and axial distributins' f bth axial and radial velcities and thereby change the shape f the stream surface. In the rtr, the axial lcatin f the radial element f the relative stream N surface is 45 percent f the axial blade chrd frm the leading edge O (z*= ) fr case A and 41.7 percent (z* = ) fr cases B, D, E, and F. With this prescribed axial variatin in V u r, the radial deriva- tive f V u r cntained in the equatins given in the preceding sectin drps ut. Als, fr the present investigatin, the inlet flwis cnsidered t be unifrm in entrpy and ttal enthalpy. Then, fr adiabatic frictinless flw, the radial and axial derivatives f s / vanish. The radial derivative f H als vanishes fr the specified inlet cnditin and V u r. Fr the-present investigatin, B is taken t be a cnstant (a value f l.is used in the calculatin). In a sense, the slutins thus btained d nt depend n any particular blade cnfiguratins. But the slutin is crrect nly fr thse bladings whse gemetrical cnfiguratin is such that the angular thickness f the mean stream sheet is essentially cnstant. It als gives the limiting slutin fr an infinite number f infinitely thin blades. T interpret the results btained as this limiting slutin, the prescribed variatin f V u r crrespnds exactly t the free-vrtex type with ö(v u r)/ör equal t zer. Fr the general interpretatin f the results btained as the slutin fr the flw alng a mean stream surface (subject t the assumed cnstant value f B), the prescribed cnditin that a(v u r)/3r be equal t zer des nt give ö(v u r)/ör equal t zer. The inlet flw f the turbine example given in reference 2 is used in the present investigatin and is. as fllws: V i V? = TT = '. t M. = l

14 NACA TN Fr this Mach number, pf = = Prj^i H i Ut2 H* 7 x = ö = METHOD OF SOLUTION The principal equatin t be slved is equatin (5). This partial differential equatin is first replaced by a number f finite-difference equatins representing the differential equatin at a number f grid pints cvering the dmain. Because f the nnlinear nature f the prblem, these equatins are slved by the general methd f successive apprximatins. In each cycle f calculatin, the nnhmgeneus term W is evaluated by emplying any apprpriate apprximate slutin at the start and by using equatins (7) t (ll) and equatin (13) tgether with the flw variatin btained in the preceding cycle; the results are then taken as given values in the slutin f t in the fllwing cycle. The slutin f t frm the finite-difference frm f the principal equatin is btained by the relaxatin methd (reference 8) in the mdified frm as given in reference 6 and by the matrix methd discussed in reference 6. The accuracy f the slutin depends n the accuracy f the finite-difference representatin f the partial differential equatin, the size f the residual left in the slutin f i r, and the number f cycles cmpleted fr cnvergence. Chice f Grid System and Degree f Plynmial Representatin In these calculatins, a single grid size was used fr bth the relaxatin and the matrix calculatins. The results f reference 2 were useful in selectin f the grid size. Frm the results btained in reference 2, the stream functin i r is expected t increase smthly with respect t radius at cnstant-z planes and t vary apprximately as a sine curve with respect t z at cnstant r-values. With the necessity f cvering a large dmain in rder t satisfy the bundary cnditins given far upstream f the statr and far dwnstream f the rtr, and with such a smth variatin f \ r ver the dmain, the use f a furth-degree plynmial representatin rather than f the usual secnd-degree ne is suggested s that the number f grid pints may be reduced. Frm experience gained while btaining relaxatin slutins, a final grid size f 0.05 r t in the radial directin and

15 14 NACA TN rt in the axial directin in the meridinal plane are used (fig. 4). These grid pints n the meridinal plane are fr reference, r recrding, purpses nly. The variables invlved in the slutin t are thse n the stream surface (fig. 4). The use f this grid size gives seven radial and axial statins n the stream surface between the hub and the casing and acrss each blade rw. With the present variatin f f in the radial directin, ' ' sufficiently accurate results are expected fr radial derivatives. The ^ accuracy f the axial derivatives is analyzed fr a simple sinusidal ^ variatin in the axial directin with a perid f 20 5 ^ Z, and the furthdegree plynmial representatin is fund t give first- and secndrder derivatives accurate t within 0.02 percent. Fr simplicity (in rder t give a unifrm frmula in the z-directin), the same grid size in the z-directin is used fr the entire dmain. It is fund in the relaxatin slutin that the radial distributin f i r had n axial variatin in the first five significant figures after eight f these z-statins either upstream f the statr r dwnstream f the rtr. The matrix factrs fr matrix slutins therefre cver a range f z* varying frm -0.5 t 0.85, which includes 10 statins each way upstream f the statr and dwnstream f the rtr. The rder f matrices is thus 7X55 = 385, which is als the ttal number f interir grid pints. At the first and the last z-statins and a few statins nearby, sufficiently accurate z-derivatives can be btained by the use f a three-pint differentiatin frmula. Fr simplicity in setting up the matrix factrizatin, hwever, the same central-pint furth-degree differentiatin frmulas are used fr the entire dmain. The use f these frmulas means that the same bundary values f \ r are used fr the tw statins utside the first (z* = - 0.5) and the last (z* = 0.85) z-statins., Finite-Difference Frm f Principal Equatin With the grid sizes f 0.05 and chsen fr & r and 6 Z, respectively, the differentiatin cefficients fr the first- and the secnd-rder derivatives are cmputed. If these cefficients at grid pint i are dented by B^ and -^B^, respectively, the finitedifference frm f the principal equatin at any grid pint i becmes

16 NACA TN f i \ A. * 4 A -W T * ' A 1, J = 0 \ 4 J r* 4 <V 4 * V 1, = 1 X (26) k=0 4 k where \frj and \ r k dente the values f if n the stream surface crrespnding t "the grid pints alng cnstant-z and cnstant-r lines n the meridinal plane, respectively (fig. 4). Calculatin f Bundary Values f i f The value f ijr alng the hub is chsen t "be zer. The value f a dimensinless \ r alng the casing is chsen as fllws: At statin i-i, with the use f equatin (3), t w t *t J r h (el t i PT,i U t r t 2 PT,i u t" r t 2 r t 2 " r h 2 B ipi ^z,± s- = = I B i p*<,± C 1 " 1 "* 2 ) 1*1,1 U t r t Fr B i equal t 1 and the chsen values f p? and V* *, X Z X * if* = which is a cnstant alng the intersecting curve f the stream surface and the casing. At the inlet statin i-i, which is ne 5 z -distance upstream f the first statin (z* = -0.5), and the exit statin e-e, which is ne ö z -distance dwnstream f the last statin (z* = 0.85) (see fig. 2), the radial distributin f f* is cmputed as fllws: r -r *i ^r) = *e ^ = h ' (27) r t " r h

17 16 NACA TN 2750 This simple variatin results frm the fact that with n tangential and radial velcities f the gas and with unifrm values in H and s, the» axial velcity and density are als unifrm at bth these statins. ( This radial variatin in f is maintained cnstant frm cycle t cycle and the prblem is therefre treated as a bundary-value prblem f the first kind. If the nnhmgeneus term U is everywhere equal t zer, the differential equatin (5) wuld be satisfied by the f-functin as given by equatin (27), which means that the gas flws n cylindrical sur- g faces. The present prblem is then essentially t determine the change g f t frm this simple distributin due t a certain distributin f K in the dmain resulting frm the nnzer values f density derivatives and F r - Calculatin f Nnhmgeneus Term Calculatin f density by the use f general table. - The definitin öf N in the principal equatin (5) shws N t cnsist f tw.* terms fr the present investigatin. One is cnnected with the cmpressibility f the gas, and the ther, with the tangential pressure ^ gradient f the gas and the radial twist f the stream surface. The first term in K vanishes fr incmpressible flw, and its evaluatin fr cmpressible flw is as fllws: ' ' '. In each cycle f cmputatin starting with a given variatin f t, its derivatives with respect t r and z are first btained by numerical differentiatin. These derivatives are squared, added, and used in the fllwing frmula t btain a functin $.' -, ) *te.7j(-"-^w)-fe-;s *2 \ r-i (28) The last tw factrs in the right side f the preceding equatin are btained frm the given inlet cnditin and equatin (ll). Frm the value f $, a value f E is read frm table l(b) given in reference 4 (with r = 4 / 3 )> and p* is btained by * = * _:* 2 (29) *

18 3S NACA TN ? «r After p* is evaluated, its derivatives are cmputed and cmbined with the -^-derivatives t frm the first term in N. w g Calculatin f F r by equatin (9) and apprximate frmulas. - The secnd term in N invlves the use f p and 8t/9r based n the new values f i(r in each cycle and the evaluatin f F r in the cycle. In rder t evaluate F r, equatins (7) t (9) are used. The cmputatin f F u r by equatin (7) is relatively simple (the secnd term drps ut fr the present prblem). Fr the cmputatin f F z f equatin (8), the last factr f the last term a 2 f _ i at e>2jf _ i / ap at ap at g r 2 'r3r 2 " p I ar ar az QZ is available in the slutin by the relaxatin methd because it is invlved in the cmputatin f the residual. In the matrix slutin, hwever, this factr is nt available. In such a case, the last term is replaced by the fllwing expressin thrugh the use f equatin (l) **&*? at/ar r in which the value f F r f the previus cycle is used. After F z is cmputed, it is divided by F u r and differentiated with respect t z. With z 0 chsen between tw grid pints, the frmula given in reference 9 is used fr the integratin t determine F r. The fllwing apprximate expressin fr F r is btained frm equatins (7) t (9) by assuming at/ar cnstant and neglecting small terms: F - 2 8(V u r) r r3 az pz z (V) dz (30) When equatins (14) t (25) are substituted int the preceding equatin, the fllwing expressins fr dimensinless F r are btained:

19 18 NACA TN 2750 (a) Unifrm lading - 2 /0.96\ 2 z " z Statr: F r = ^3~iö35/ 2, (31) * 2 /0.96\2 Rtr: F* = - -5^ /0.96y '\0.15J 0.35 (z*-0.2) - z* (32) (b.) Nnunifrm lading Statr: 0 <z*< 0.075, * 2_^ F r ~ r * fj_ * 2 -z* 2 (33) < z* < 0.15, F* 2 * / z*\ *\T4 h 0.96/ z* is^'.isyjj 0.32 (z*-0.075) + i^ (z* ) - 1 z* (34) Rtr: 0.2 < z* <0.275,, *r " ", (z*-z*) - i (z*-0.2) 2 -(zj-0.2) 2 (35) < z* <0.35,

20 NACA TN F r * = --L- 8 0^9. /l- 2 *- - 2 \(.96( Z *)- r» \ 0.15 )] O.OTS^^-O.Z) 2, 3 0Ö5 + 1 ' 28 ( Z ~ Z S) IliU [C Z *- - 2 > 2 - (0.075)2 ( Z *-0.2) l (. 3X0.15 JJ V ' Slutin f Principal Equatin by Relaxatin Methd Cmputatin f the residual. - After the nnhmgeneus term Iris btained at each grid pint, it is subtracted frm the left side f equatin (26). The difference btained is the residual at the grid pint i. Reductin f residuals. - The cefficients used t relax residuals (relaxatin pattern) are btained accrding t the five-pint firstand secnd-rder differentiatin cefficients in the left side f equatin (26). These cefficients fr a change in the f-value at the grid pint are given in table I. These and similar cefficients fr ther values f change in ^ listed n cards were fund t be cnvenient in calculatin. As a cmbinatin f checking and time saving, it is fund cnvenient t relax a given set f residuals by using nly the central three r five majr cefficients at first and, when the residual is small enugh, t cmpute fr the ttal changes in f made at each grid pint the crrect resultant residual everywhere, and then t relax the residuals further with n.1'1 cefficients. The technique f verrelaxing r underrelaxing and line-relaxing is very helpful. In mst cases, the relaxatin f 385 pints (mst relaxatin is dne in the blade regin) is cmpleted within 16 hurs by hand cmputatin. The greater prtin f wrk fr the present prblem is btaining, the residuals, which takes abut 40 and 60 hurs, by hand cmputatin, fr the incmpressible and the cmpressible slutins, respectively. In sme f the later relaxatin slutins btained herein, this wrk f cmputatin f residual is dne n the IBM 604 Calculating Punch and an IBM Card Prgrammed Electrnic Calculatr (hereinafter called CPEC), except the cmputatin f F r, which is mre difficult t set up. In this way, the calculatin f residuals fr the cmpressible flw prblem takes abut 8 hurs f machine time and a few hurs f hand cmputatin fr F r.

21 20 NACA TN 2750 Slutin f Principal Equatin by Matrix Methd The 385 simultaneus algebraic equatins (26) fr the 385 interir grid pints can be written in a cmpact matrix frm by denting the cmbined cefficients f i r«5 's and t k 's in equatin (26) at the pint i by C4 and denting the sum f W 1 and the prduct f the knwn bun- dary \ r-values and their crrespnding cefficients by a 1 : C 1 t J = a. 1 OT KI {*}= } (37) 3 ^ ^ The matrix [c] is shwn in figure 5 in terms f submatrices [E], [F], and \ß~}. Because there are seven interir grid pints in the radial directin, these submatrices are f the rder seven, and because f the cylindrical walls and the use f unifrm spacing and the same differentiatin frmula, they repeat regularly alng the diagnal f [c]. All the ther submatrices are zer. The elements f PO, [X],[G3 are given in figure 6 in terms f the grid spacings &? and f. In [pj and [G] there are nnzer elements nly n the diagnal. With the use f 5? and 5* equal t 0.05 and 0.025, respectively, in rder t get the elements f the matrices in shrt rund numbers, the fllwing multiplicatins are made: ' Rw f rci Value Multiplif r* catin factr 1, 8, 15, , 9, 16,, , 10, 17, , 11, 18, , 12, 19, ,6, 13, 20, , 14, 21, The resultant submatrices are shwn in figure 7. The same rw multi-, pliers are als applied t the elements f Ia\ befre slving fr -T\A. In the slutin f equatin (37), because f the afrementined special nature f \C], it is best t factr [c] int tw triangular matrices which als have nt mre than 15 nnzer elements in a rw running alngside the diagnal (references 6, and 10 t 13). Thus

22 NACA TN [C] = [L] [u] (38a) where [u] has elements which are unity alng the diagnal, and equatin (37) becmes [L][u]{>} = {a} (38b)» The slutin f {i r} is then btained by frward and backward substi- J: tutin nrcesses prcesses as fllws: fllws* Let *l Then {ß}'-[*]{*}' ' ( 39 ) [L] {ß} = {a} (40) Slve }ß\ frm equatin (40) and then \ A frm equatin (39). This matrix slutin fr the present investigatin was made n an IBM CPEC and n a UNIVAC. Nine digits are used n the CPEC and eleven digits n the UNIVÄC. At n place in the dmain d the results btained fr {\ r} differ by mre than 5 in the sixth digit. After the slutin f {t} is btained in each cycle, it is substituted int equatin (37) fr an ver-all check. The residual fund at any interir grid pint ib less than 1 in the eighth digit. Thus, the residual at every interir grid pint as fund in each cycle f calculatin is reduced t practically zer with reference t the accuracy f the grid size chsen. A cmparisn between the relaxatin and the matrix methds shwed that mre accurate results were btained with the matrix methd in a smaller amunt f ttal man- and machinehurs. RESULTS AND DISCUSSION Incmpressible Slutins Fr adiabatic flw f a nnviscus gas with unifrm s and H at the inlet, the deviatin f gas flw frm that n cylindrical surfaces in the incmpressible case is due entirely t the term cntaining F r in equatin (5). In rder t study this effect, slutins are btained fr the tw types f lading as given by equatins (14) t (21). The ne with unifrm lading is designated case A, and the ne with nnunifrm lading, case B.

23 22 NACA TN 2750 Cnvergence f slutin. - Similar appraches are used in btaining the slutin fr bth cases. The calculatin fr case A is started with an assumptin f straight cylindrical flw and successively crrected until the slutin cnverges. The calculatin fr case B is started frm an apprximate slutin f case A. The initial calculatins in g CO bth cases are made by the relaxatin methd with 8 r = 0.1 and 5* = The values 5^ = 0.05 and 8^ =0.025 are used in later calculatins. In each cycle, the t-values are difficult t imprve beynd the fifth decimal place. The final relaxatin slutin is further imprved by the matrix methd. The \ r-values btained in each matrix slutin are resubstituted int equatin (26) and the residuals calculated at any pint are fund t be less than 1 in the eighth decimal place. In the furth matrix slutin - all values are sufficiently cnverged fr the grid size and differentiatin frmula chsen. As an indicatin f cnvergence f the slutin, the successive values f t at the mean radius r* = 0.8 in the matrix slutins f case B are shwn in table II. The change in if* in the last cycle is less than 3 in the fifth significant figure r less than percent. The variatins f i r*, at*/ 9 ' 1 "*; 3t*/ 9z *> *ur*, F*, F, and N* at tw pints z* = 0.10 and 0.25 at the same radius btained in the matrix slutins as well as the last three relaxatin slutins are shwn in table III. All results given in the fllwing paragraphs are based n the last matrix slutin and are the values n the mean stream surface as indicated in figure 4. Variatin f F cmpnents and shape f streamline. - The calculatin is first started by using F r determined by the apprximate fr- mula (30). It was later refined by using equatins (7) t (9). The variatins f F* r * and F* with z* at several"radii are shwn in figures 8(a) and 8(b), respectively. The variatin f F*r* is very similar t that f s(v*r*)/9z* shwn in figure 3, being mdified nly by the variatin in V*. The final values f F at the same radii are shwn in fig- ure 8(c). The starting values determined by the apprximate frmulas (31) t (36) are als shwn in the same figure and are seen t give reasnably gd apprximate values. The magnitude f F r is seen t be f the rder f a quarter f F u (figs. 8(a) and 8(c)). This large value f F r is due t a cmbinatin f the large deflectin f the gas in passing the turbine blading and the large radial twist f the stream surface f this type f velcity diagram and influences the flw distributins significantly. CO

24 KACA TW ^ Because f this F r -di-stributin, the nnhmgeneus term W in equatin (5) r (26) is psitive upstream f the radial element in the statr and dwnstream f the radial element in the rtr and is negative between them. The term N als increases in magnitude tward the hub. This distributin f W requires a general increase in \ r at any pint inside the blade rw frm its inlet value at the same radius and results in a streamline shape as shwn in figure 9. In bth cases A and B, the gas flws radially inward in the statr and utward in the rtr. The difference between the tw cases is rather small. Variatin f radial and axial velcities. - The variatin f radial velcity with respect t z at several radii is shwn in figure 10(a). In bth cases A and B, this velcity has a minimum and a maximum abut in the middle f the statr and the rtr, respectively. It is practically zer abut 1 axial chrd upstream and dwnstream. The radial lcatin f the largest radial velcity ccurs arund r* = The variatin f axial velcity at five radii is shwn in figure 10(b). Its deviatin frm the inlet value is rather large alng the casing and the hub, especially in the space between the tw blade rws. * Cnditin in plane nrmal t turbine axis and between statr and rtr. - In a plane nrmal t the turbine axis and between the statr and the rtr, the axial velcity is seen t decrease with an increase in the radius (fig. 10(b))'. Fr radially unifrm H and s, the equatin f mtin in the radial directin is 3z ~ 3r F r Because Fr is zer in the space between the statr and the rtr (fig. 8(c)), the radial velcity must decrease with an increase in z in rder t satisfy this equatin. This negative slpe in V r with respect t z is clearly shwn in case B by the values f radial velcity btained at the regular grid pints at z* = 0.15, 0.175, and 0.20 (fig. 10(a)). Fr case A, where V r is nearly equal t zer arund z*= 0.15, this negative slpe is apparent after the values f V r at z* = and are cmputed by using the three- pint differentiatin frmula and the values f i f at the regular grid pints. It may be nted that this scillatry variatin f velcities in the space between the statr and the rtr is entirely due t the specified variatin f V u r in that space. Because V u r is specified t

25 24 NACA TN 2750 be cnstant in that space (fig. 4), F u asd. F r n the stream surface are equal t zer in that space accrding t equatins (7) and (9). These values are apprached when the number f blades f the statr and rtr appraches infinity. In the actual case f a stream surface between tw adjacent blades a finite distance apart, the streamlines turn fr a shrt distance bth upstream and dwnstream f the statr and the rtr (see, fr example, reference 7), and cnsequently, bth F r and the velcities wuld vary smthly in the space between the statr and the rtr. _, Cnturs f cnstant velcity. - A cntur plt f cnstant values f abslute velcity in the statr and relative velcity in the rtr fr case B is given in figure 11. The maximum velcities leaving the statr and entering the rtr are, respectively, 1.77 and 1.25 times the,rtr tip speed. Cmpressible Slutin Neglecting F r.' A cmpressible slutin fr an inlet Mach number f is first btained by neglecting F r in the principal equatin (case C). This slutin is fund in rder t see the effect f the cmpressibility term in equatin (5) alne and t see the errr invlved in neglecting ' Fr fr cmpressible flw. Nnunifrm lading is used. The deviatin f flw in this slutin frm cylindrical flw is due t the density term in N, which is mainly determined by the prduct (ä a ) Th 3-8 generally psitive value f N inside the statr and the rtr requires a general decrease in f at any pint inside the statr and the rtr frm its inlet value at the same radius, thereby resulting in a streamline shape as shwn in figure 12. In this slutin, the starting values fr the relaxatin slutin are btained by using the values fr the gas state at z* = btained in reference 2 and by assuming that the streamline shape varies as a simple sine curve in the meridinal plane. The slutins cnverged quickly. The successive values f if*, 8\ r*/3r*, Bi r*/8z*, p*, and N* at r* = 0.8 btained in the matrix and relaxatin slutins are given in tables IV and V. The change' in \ r* in the last cycle is less than 2 in the fifth significant figure r abut percent. ', The variatins f V*, V*, and p* at several radii btained in the final matrix slutin are shwn fr case C in figures 13(a), 13(b), and 14, respectively (the curves in thse figures fr cases D and E will be discussed in the fllwing sectin). Because f the decrease in density acrss the turbine, the axial velcity rises t a higher value dwnstream f the rtr (fig. 13(b)).

26 4S NACA TN Cmpressible Slutins Including F r * Slutins are next btained fr cmpressible flw by cnsidering simultaneusly the effects f cmpressibility and f F r - As in case C, nnunifrm lading is used. Three axial psitins f the radial element f the stream surface in the statr (z* = , 0, and 0.15 fr t t cases D, E, and F, respectively} see fig«15) are cnsidered in rder t investigate the pssibility f minimizing the effect f radial flw by the chice f this psitin. (Cnceivably, the stream surface within the statr may have n radial element, its inclinatin with respect t a radial line being arbitrarily chsen t prduce a certain desired effect.) Because f the very large value f F r and the accmpanying large radial flw, the slutin fr case F was stpped at an early stage, and nly the value f Fr btained in the early relaxatin slutin and that btained by the apprximate frmula (30) are given. The ther tw cases were carried further by the relaxatin methd and checked by tw t three matrix cycles. Althugh the slutins are ht s far cnverged as thse in cases A, B, and C, they are accurate enugh fr rdinary purpses. The successive values f ifr and ther pertinent variables at a number f typical pints fr case D are given in tables VI and VII in a manner similar t that fr cases B and C. Nnlinearity f cmpressible slutin. - In the calculatin f case D, the values f the later relaxatin slutins f case C were used as starting values. The results sn made clear, hwever, that quicker cnvergence wuld have been btained if the calculatin had been started frm cylindrical flw. Because f the nnlinear nature f the principal equatin, the cmplete slutin fr cmpressible flw fr case D cannt be estimated frm the slutins btained in cases B and C. Fr the tw cntributins in the nnhmgeneus term N, the first term due t cmpressibility remains abut the same as that in case C,' but the secnd term, cntaining F r and p, is greatly changed because f the change in density. The distributin f the resultant values f N (and cnsequently the ^-distributin) is therefre quite different frm that in case B r C r that btained by directly cmbining the N-values in the tw cases. Variatin f F cmpnents. - The variatins f F u r, F z, and F r are pltted as befre in figures 16(a), 16(b), and 16(c), respectively. The variatins f Fur and F z are similar t thse in case B, which has the same type f lading. A cmparisn f the Fr-distributin in the statr fr the three values f z Q chsen shws that alining the stream surface at the statr exit (z* = 0.15) causes very large negative values f F r in

27 26 NACA TN 2750 «the statr, which will prduce large amunts f radial flw. Fr z* = , F r is negative fr the first prtin and psitive fr the ^ secnd prtin. Fr z$«=-.0, F r is psitive everywhere.in the statr.' The apprximate frmula (30)' fr F r still cmpared very well with the final values. Shape f streamline. - The meridinal prjectin f the streamlines -btained fr cases D and E is shwn in figure 17. In case D, the gas flws radially utward and then inward in the statr, whereas in case EN the gas flws radially inward and then utward in the statr. The $ streamline in case D deviates frm a cylindrical surface mre than that in case E at the leading edge f the statr but less at the leading edge f the rtr. Variatin f velcities and density. - The variatins f V r and V z in cases D and E at several radii are shwn in figure 13. The radial velcities in these cases are f abut the same rder f magnitude as the velcity in case C but have mre cmplicated shapes (fig. 13(a)). Figure 13(b) shws that there are significant differences between the values V z f the three cases C, D, and E, especially at the casing and the hub. These differences are due almst entirely t the different shapes in the streamlines because the differences in density amng the * three cases are very small (fig. 14). Cntur plts f static pressure and Mach number. - Because f the lw value f the hub-tip radius rati, there is an ver-all staticpressure rise acrss the rtr fr values f r belw 0.7 (fig- 18). An even greater static-pressure rise accmpanied by a subsequent reexpansin t the exit pressure ccurs within the rtr-blade rw; this increase in pressure rise is a result f the flw-area increase assciated with the assumptin that B is equal t 1. ' t The cnturs f cnstant Mach number (abslute Mach number in statr and relative Mach number,in rtr) f cases D and E are shwn in - figure 19. The maximum Mach numbers at the statr exit are and fr cases D and E, respectively. If' the effects f radial flw are cmpletely ignred (simplified radial equilibrium), the maximum Mach number at the statr exit is nly :_. Eadial variatin f axial velcity ahead f rtr. - In the simplified-radial-equilibrium calculatin, the axial velcity des nt vary radially fr free-vrtex-type blading. The present slutins shw, hwever, that there is cnsiderable radial variatin in axial^ velcity fr bth incmpressible and cmpressible flws. An errr in # the axial velcity f the gas entering a blade.rw prduces an errr in Mach number and angle f attack, and thereby the range f efficient

28 NACA TN peratin is reduced. The radial variatin f axial velcity ahead f. the rtr (at z* = 0.175) is pltted fr cases A t E in figure 20 in * which the variatin btained in reference 2 is als pltted fr cmparisn. In all cases except simplified radial equilibrium, the axial velcity decreases with an increase in radius, and the rate f decrease. is greatest at the hub. The effect f mving the radial element f the stream surface in the statr frm the statr entrance (case E) t the midaxial psitin in the statör (case D) was t decrease the radial M gradient in axial velcity at the rtr entrance. The apprximate slu tin f reference 2 is very clse t the slutin f case D. Fr incmpressible flw, this variatin is due t the shapes f the streamlines (fig. 9), as influenced by the F r -distributins (fig. 8(c)), whereas fr the cmpressible flw this variatin is due mainly t the increasingly larger drp in density acrss the statr tward the hub. SUMMARY OF RESULTS A methd recently develped fr determining the steady flw f a nnviscus cmpressible fluid alng a relative stream surface between adjacent blades in a turbmachine was applied t investigate the subsnic thrugh flw in a single-stage axial-flw gas turbine. A freevrtex type f variatin in tangential velcity was prescribed alng the stream surface. Cylindrical bunding walls were specified in rder t avid radial flw at the walls. The flw variatins n the stream surface fr incmpressible and cmpressible flws were btained by using the relaxatin methd with hand cmputatin and the matrix methd n bth an IBM Card Prgrammed Electrnic Calculatr and a UNIVAC. In all slutins cnsidered in the present investigatin, cnvergence was btained withut difficulty. A cmparisn between the relaxatin and the matrix methds shwed that mre accurate results were btained with the matrix methd in a shrter interval f time. The results f these highly accurate thrugh-flw calculatins frm a basis fr evaluatin f simpler but less accurate methds. Fr incmpressible flw, cnsiderable radial flw was btained because f the circumferential pressure gradient f the gas and the radial twist f the stream surface. The gas flwed radially inward in the statr and radially utward in the rtr. This radial flw resulted in a large negative radial gradient f axial velcity in the space between the statr- and rtr-blade rws. Fr cmpressible flw, the cmpressibility f the gas and the radial twist f the stream surface had equally imprtant effects n

29 0 NACA TN 2750 d the flw distributin and the nnlinear behavir f the principal equa-.-: tin defining the flw was quite evident. The shape,, r twist, f the stream surface and thus the amunt f radial flw were sensitive t the axial lcatin f the radial element f the stream surface in the statr. The largest radial flw ccurred when this radial element was lcated at the exit f the statr. The effect f mving the radial element f the -. stream surface in the statr frm the statr entrance t the midaxial psitin in the statr was t increase the deviatin frm cylindrical flw ahead f the statr and t reduce the deviatin ahead f the rtr; ^ the radial gradient in axial velcity at the rtr entrance was decreased. When the radial.element f the stream surface in the rtr N was near the midaxial, psitin in the rtr, a large negative gradient f axial velcity was bserved in all cases ahead f the rtr. Lewis Flight Prpulsin Labratry Natinal Advisry Cmmittee fr Aernautics Cleveland, Ohi, April 18, 1952 ' REFERENCES 1. Eckert and Krbacher: The Flw Thrugh Axial Turbine Stages f Large Radial Length. NACA TM 1118, ~., 2. Wu, Chung-Hua, and Wlfenstein, Lincln: Applicatin f Radial- Equilibrium Cnditin t Axial-Flw Cmpressr and Turbine Design. NACA Rep. 955, (Supersedes MCA TN 1795.) 3. Wu, Chung-Hua: A General Thrugh-Flw Thery f Fluid Flw with Subsnic r Supersnic Velcity in Turbmachines f Arbitrary Hub, and Casing Shapes. NACA TN 2502, :.. ;. \ 4. Wu, Chung-Hua: A General Thery f Three-Dimensinal Flw in Subsnic and Supersnic Turbmachines f Axial-, Radial-, and Mixed- Flw Types. NACA TN 2604, Carter, A. D.S. : Three-dimensinal-flw Theries fr Axial Cmpressrs and Turbines. iwar Emergency Issue N. 41, pub. by Inst. Mech. Eng. (Lndn). (Reprinted in U. S. by A.S.M.E., April 1949, pp )..-'.. 6. Wu, Chung-Hua: Frmulas and Tables f Cefficients fr Numerical Differentiatin with Functin Values Given at Unequally Spaced Pints and Applicatin t Slutin f Partial Differential Equa- - * tins. NACA TN 2214, 1950.

30 NACA TN Wu, Chung-Hua, and Brwn, Curtis A.: Methd f Analysis fr Cmpressible Flw Past Arbitrary Turbmachine Blades n General Surface f Revlutin. NACA TN 2407, Suthwell, R. V.: Relaxatin Methd in Theretical Physics. Clarendn Press (Oxfrd), Lwan, Arnld N., and Salzer, Herbert E.: Table f Cefficients fr Numerical Integratin Withut Differences. Jur. Math, and Physics vl. XXIV, n. 1,.Feb. 1945, pp Crut, Presctt D.: A Shrt Methd fr Evaluating Determinants and Slving Systems f Linear Equatins with Real r Cmplex Cefficients. AIEE Trans. (Suppl.), vl. 60, 1941, pp vn Neumann, Jhn, and Gldstine, H. H.: Numerical Inverting f Matrices f High Order. Bull. Am. Math. Sc, vl. 53, n. 11, Nv. 1947, pp Turing, A. M.: Rund-Off Errrs in Matrix Prcesses. Mech. and Appl. Math., vl. 1, 1948, pp Quart. Jur. 13. Fx, L.', Huskey, H. D., and Wilkinsn, J. H.: Ntes n the Slutin f Algebraic Linear Simultaneus Equatins. Quart. Jur. Mech. and Appl. Math., vl. 1, 1948, pp ,

31 NACA TN 2750 TABLE I - RELAXATION COEFFICIENTS r 1 Fr change f f at grid pint (r 1,z 1 ),.. r t change residual at (r >z) by r z z 1-25 z - 1 -», i z z z + 25 z ' '

32 NACA' TN S3 3 TABLE II - SUCCESSIVE VALUES OF i r IN MATRIX SOLUTIONS AT r* = 0.8 FOR CASE B * z Starting value Matrix slutins First Secnd Third Furth CFEC UNIVAC it ' ,

33 1 32 NACA TN »If» g CQ CO in -^ H 10 to H * Til c- r- c H 05 CD in CO CO CO O 00 t t r- # lo IOO t t»- in CD -«^ CO c t CO Ss; en t * t TJI ^ ^jl in in CD CD CD CO CO CO CO CO CO CO * in * ^ r- cn CO O CO -^ in t t t in c c r- c t c r- r- >* 00 t CO CD CD *M > O H 353 cn PH ** in in LO to to H CO CO CO CO CO O» / i* t t *(DO CO c t N rl to 0» H in r- CO to LO CD N> H.t t t t t t jj jj t t t NN N NNN ^ "^p "^ r- r- t- r- r- t»- H H H H H H i i i i ii 1 1 ' I I H H 00 c in r- O O t- CO t t LO * CO ^ CO CO H to CO CO H CO fc 00 t- 00 en c CO c t f- c c *d t CO CO CO CO CD ^ in i LO LO LO ' N t t t t t t in m m in in in iii II i t CO ven r- t- t r- cn r- in O CO t in TJI in in * ^O c en c c in in (M en en» c c c LO LO CO CD f t t t to (M CM CO t t t t t t t V N Ü rt> (D 1 1 II O H t- t» c en ^ 00 CO 00 H 10 to ^ O H f- CD r- CO in c en ** en t ^H NrlN to CO CO CO CO CO CO CO H CO CO Vfn en ) en en cn cn en en en en en en en /Ol<T> ^ ^ ^ ^ ^ *^i "^ ^i ^ ^ H^ ' ««* H en 10 O to -^ LO s-^ c in c CO c cn ^ c r~- c c LO to H H H CO t "^ t c t r- in <# -* * c.en t>- c t cn * cn en en en en en en O» H 00)00 -> CO CO CO CO 00 CO 00 a> en en en en cn en cn n5 zr m in in in in in in in in m in in in in CO CO CO CO CO CO CO CO CO CO CO CO CO CO *N ü «O in in 1 in in in q q i in in in in»fh H H O H H «O, Ö. d. Ö H»H H p -P»? 3 s 3 H 3 u m H O P-) p r-i -p CQ 1 i * H d in X H U P <d s H «3 P s s H > e Ö p a> H s I Ü p (U H! rd O P i Ö H 8 9 H (U -d -d Q) O!> A p

34 5S NACA TN * z TABLE IV - SUCCESSIVE VALUES OF f IN MATRIX SOLUTIONS AT r* = 0.8 FOR CASE C Starting value Matrix s lutins First Secnd Third CREC UNIVAC , i ,

35 34 NACA TN H r- c * CO (M to O ^ O * ^ c t CM CM in * CO H t^- r~ c in in in in s CO O CO CO t c c c c. H a H H EN." e to H *# t 00 H O (M -^ H t-- 00 t- t>- to t- H O O O CO C0 00 CO c r^ c c (M H r-l H H CO 00 CO a CO 00 CO CO 00 c c r~- t c~-» (0\CD O CD rt> g CD CO II t~- O CO CO to to O H O f t- * in in in in c c en, CO * I» 33 ^ ^ rf J=i rf HN CO \< ft O 1 ii i 1 in IM NNN c c in c r- c cq in c 00 to H to N *» in in c in c in in in >[}-< c c CO CO CO CO CO CO CO CO <b\< *# ^ ^ ^ ^4 *<J1 ^fl ^J* ^Jl "tjl * t in in c c c N t «* CO CO N CO t t t t c t- r- yq *<ij 00 CO ^ ^ ^ "<gl ^3> ^5* ^P ^i* CO CO 00 CO CO CO CO CO p IT) lo in in in in in in in in Csl N <N CO CM N N NN N * N O O O 60. -' in in in in m in in in in in * u CO * t CO. d 0 Ö H H -P "E ^ " i a H CO u p s u pa O, * in H <M N O O s

36 NACA TN TABLE VI - SUCCESSIVE VALUES OF \ r IN MATRIX SOLUTIONS AT r* = 0.8 FOR CASE D # Starting ValllG Matrix slutins First Secnd Third CPEC UNIVAC P , '

37 36 NACA TN 2750 H a CO CO CM ^. r r- CD CO CO CM ^ CO CO H CD CO t- CO LO O ^ CO ^ CO * ^ CM CO -<11 CM H CO CO 00 t~- CO CO i) & 00 H O O CM to CO to. D < < i O O CO t. 1 1 CO CO CO CO O O O O O i i CO CD O O. 1 1 CD H t ^ CO LO 00 CO CO O H r- c c CO CO H -^ CM CO - too.rlt^ O CO «# t -*H r- CO CO *fn ' CM <tf * t ^ t O O fr L0 LO LO LO LO LO CM CM CM CM CM CM '»» g ' H cd LO c t t CO 00 CO lo f- LO rl t^ CO CO. O "tf # CO c t- r- H O H CO Nrlrl frrj CD c CO CO c c r^ CO 00 CO 3 * H i-th H <-l r-i rl CO CO CO 00 CO 00 CO a 00 CO 00 CO GO 00 CO r- f- t- r- t -C t *> >1 Q. -p ^ OO) r- H CM CO t t cu CO CM CO CM r- CM LO CO H GO 05 CD LO LO O CO t r- c q - LO CO O CO LQ CO, ^, LO r-\ H H CM CM * 1* O H H H O H H H r-l H H p rlrt> O O O O ä OI«l rt>lrt> gä? ^ H LO CM r- ** H H to CO O CO 00 p- CD t>> r-1 t CO CO CM c LO CO CO.1» * 1* H ^ CO CD CO LO LO LO : LO LO -^ LO ^l ^1 ^Jl c c c c -P CO CO CO CO CO CO CO CO CO CO p, * <0 (0 ^4 ^ T# «*r ^ -^ TT ^H ^ ^ ^ ^ ^ ^ (1) CO U H - ' ^-<tf 00 O ^-vco H CO t m t rw -*W H ^ <# r- c CO LO C- LO 1 H O H H r-l H H H to CM CM CM r-l CO CO CO * CD CO CO CO CO CO CO CD c~-r--c^r^t^-cccli CO 00 CO GO CO CO CO CO ccaoccccc s O-O'O O O O O O O O U CJ CQ CO 6 * N 60 O nt* LO LO LO LO LO LO LO LO LO LO LO LO LO LO CM CM IM CM CM CM CM CM (M CM CM CM CM CM O O O O O O O O O O O O O O O ' l-h O O LO LO LO lo LO O O LO LO LO LO LO H H # fh H H O O H r-i O O t O ' h > a H : % H O CO 1«9- cu * 9. H cö u - -p H -p LO CM O CO. 8 '3 H cd i> +5 I Ö H s- r>5 (U pi ^ -d X > h O -P h cd ft Ö il cd

38 NACA TN Figure 1. Relative stream surface abut midway between adjacent blades.

39 38 NACA TN 2750 i -P 60 H 8 1 O H -d H I.a

40 MCA TN V*r* 10 3(V*r*) 3z*!- -i S~^ r, i 1 1 ' 1 H-l T--+ J ll 111 i II111II1111 i_l_ l.-..::::::_±::_::::::±:::±:::± 8(V*r*) 3Z* Figure 3. - Specified variatin f tangential velcity V*r* with respect t axial distance z*.

41 40 NACA IN 2750 <u 9' H ft d Ö H <d H'!H (U PC u Ö (U <M fh (D u a3 u -p CD d (U (U > Q) d H H <u -p [Q >> CO >d H u -» 00 H P4

42 6S NACA TN E F G F E F G G F E F G G F E F G G F E F \ \ \ \ \ \. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ V N v \ \ \ \ \ G F G E. F G F E F G G F E F G G F E F G. F E Figure 5. - Matrix [c] expressed in terms f submatrices [E], [F], and [G'].

43 42 NACA TN 2750 IM cd U, N \ä H ctt i H X» c r~ IM t~- X in 10 n c +» in N <*l tö + H \$ t i^ ' H in td t * N lo II s CO N N + t IN Cd N cd» N H td X to 1/5 X 0D IVI c O + C\1 1^ «" «# td N t N IN td N H IN ^f cd t m h id Pi t d H w cd H N cd X H 10 rh X» en 05 IV] CO IN a) N H IN cd N H 01 al N : cd N IN t H H Ö in p X H X UJ N LO c- CD IN Ö O td O + + toi N N 03 1 H IN rt *&\ cd N H N t H H td cd Hi i N N t IN t x 1 X N H in» CD Hä N H m i c- n IN Ö + O cd 1 IN tol N it^l + 1 IN >ji cd H in * HI ) 1 t I t H td 1 t 1 H 1 td cd t N N IN t X u, X i-l 5» t 10 lfi N 10 c i U) IN Ö O td O + + tol N l L N H l N _. Ml (fl 1 ^i cd H \& 1 N t H H! <u p <D CO CO (U B 1 C5 I I d. K 14 P & N "Me 1 01 cd to S 10 Iß 0.9 ml cd I t «cv H N + cd N t cd N H CO H

44 NACA TN b b b b b b b* 1 12b b b b b ^ b 2 (b) Submatrix [F]. (c) Submatrix [6]. Figure 6. - Cncluded. Submatrices [E], [F], and [G] expressed in terms f grid spacings 5* and 8*. a = 5*,* b = 6*.

45 44 KACA TN 2750 ^ ,00 CO 8" CO (a) Submatrix [E]. " " Ob) Submatrix [F] V (c) Submatrix [G]. Figure 7. - Submatrices [E], [F], and [G] after rw multiplicatin.

46 NACA TN Fjr* (a) Fjr*. Figure 8. - Variatin f F in incmpressible slutins.

47 46 NAGA TN 2750 re.35 (b) K- Figure 8. - Cntinued. Variatin f F in incmpressible slutins.

48 NACA TN [ 11 j [ ii 1111 IlkWttHftfttttiittt"' "H"^" 1?^!^ :e se AÜjJttHjfH II fffttr :jffl4jatb totjwjti-be luffiat 6r ttülihhttffftffr : TifltiTtnT -2 : SfflTFFF 'f"'"l' :! ']-T^\Ä 'Ml' ^tj"! 1:i-liOrftj {"- " I -1 '!! r* : 1 I muffi B< CTl'rp, 7Tn_r : ^rtot^^fflffi WtW-H+HKOtOT lll.lllu±tt±: ^tjipfpifp^f l[j l!llltftft Matrix slutin ' j tl^rf zlr ltö : -W^t^lr^^ : F? 0!.40 0 z* (e) F*. Figure 8. - Cncluded. Variatin f F in incmpressible slutins.

49 48 NACA TW DO r

50 7S NACA TN V». (a) Radial velcity V?. Figure Variatin f velcities in incmpressible slutins.

51 50 NACA TN 2750 C\3 O n -" (b) Axial velcity V%. Figure Cncluded. Variatin f velcities in incmpressible slutins.

52 NACA TN >* Statr Rtr r* Figure Cnturs f cnstant abslute velcity in statr and relative velcity in rtr in case B.

53 52 NAÜA TN 2750

54 pfa E! Üifi NACA TN ffmfr p r^ Pi : lip # r US ~ ihi n HH ygt ifei ttgggigh fchq üp frttij* Ell HI ^^ fflisi imm 1 Iff Hi ipeia a Wü ri jiüd : B1! fif^l fe- : J afcl± TWÜM! Ä:Hi mw mi ^W 1BH trh U tjj Pti ij.iiis Hi lip IS Sia iü :^ *? ff pi : ll gn -4J, PMlmmP If «i 1 i * 8! jjtj Stift? in ÄS nf ft«its ir S lil ft flf r^1 s ill "FT! nii : ;'>f^ IHf # ft= pr P 1 ff*^ Pi # H 7 $m, j H' 4 }u r };{= R ÜJi f ^p S' gg IWffljJK ft! Ü? :ffl {:5 ft* rjihhq $ Wf S3 S j8 Si HP ÜÜ jg: ffl s PjH fph, H» ii i' frppwf Hü i#? P Ü i life Hit ttf? tii w pffe m' ^ i : S isftl iifi. 11 Mill mi =TSS St.! rtt^ t^ ig ift; - as I tt t tm^uh 11 ltt fflttt H iii m ff rii-tti BE IN w rf * a 5p Sap s p m ffil 02 P gt tfe ll n fc 2# 1 SB Jffl P tt Jr-p } b fp IPfff ^1 ^B^^Wfflfflffl if. f '1 il llllll^flfjljljlljljj ; *J g±t Ft L ; ftp m w sjiif F ^ 99 nr fe Htß ^Ss &Ü j g pfj$j j^!j jj$] FT? P] : nf f l#j m*i Km if rttiii Ifif ft lb.. ss piiiiiiw * M & b H #.%\#\\\\\\\\\#tf^\\\##i$i * r - ffi m 3f a Ht J P t-(it i E+S fp Itfe isjil Hfi fc. 11 fig Ä ij'ff fe Its!»i Itfel ^^TNAcfcS 04 f H>-tp ffp ms 1 ' (a) Radial velcity V^t. Figure 13. Variatin f velcities in cmpressible slutins.

55 54 NACA" TU' ittitt J^TTi'"'!" l;i^i"''~jj' "I'l: 1 Y : 1" :"lh'--li-4ll i:! l l-ll-- iji, ; ' HjtfeJ J 1 'H rhn. jj ll.fr^itil :!- : j H-- I'li j,.y il -I'U.-i 1 1 "1 u r l --I-I?t1~~!ii "ftl"-' \ 'u\ 'I- Hilf "' Htrj--' :';'"! ' - p T_liL :.. j[ j I'" 'Fi^' ""pm-h^-ir ' : ' : i ili''""'tf^l^l ill' li' "V'^F=^J^rFvT^H ~l : ~l ih^-4' - 1 : " T^F-M 1~ - : : I :'.; [_ j I. ;J.;i -,J r ^_(j^ j j v^,p 4 i ^['ry^ji--' i[ :: LI];]. J ; -^j~~[t l'ih-jlj' I 'ill' rl'"-"-uri''ii FT S '.] ']:\:i\\rf^^jh^\r\viw---\^-\ : iltt^$f--+i!i "l"ü"-#^k,l'fr'=[m'! ; J 1 ^J-4 PW-'^l-l ^jjiiis ^l : sisi-.z^:ie^^^i^;^s^^s^^gffibs^=i= ^mg^^:^:s:ibjiiiissflsij:i^p;^^^^fflpfpffl^ffl=fff millilll 111IH = N^^ (b) Axial velcity Vj.' ' Figure Cncluded. Variatin f velcities in cmpressible slutins.

56 NACA TN p*.8 'Jinil i i IMIIMM^ ; jjllljljill'-'-^-'^lt'"!!li'" ++ [ III j;! II hi lllll'iiiinii 1 IfflitttÜllÜtiiliiil 1 ttittttlltttthtillwhtf : ^^^^^^^^B^B^^^^^^^^^p^#^^Ps! fill Hi 1 '' IrT^^ 11 Trt niiiffif fhimthn^^ - fjfflfff^^ H^ÄÖ 3 ra-'.^iiijij-liii^-!^. 1 :11kl 4--t J^' J- -4 illl 1 llplljjif l!l,l- Ml!' 1 l-j'l Figure Variatin f density in cmpressible slutins.

57 56 NACA TN 2750 Statr Rtr CO Case B and D B,D,E and F z* = 0.15 Figure Axial psitin f radial element f stream surfaces.

58 8S NACA TN rwirlffpff \il ±^^$ffi\v\\\\^\\\\\ 'lllillliiilllttttitnttffliffliititftfl lim,.1 ;.U1L - III :,lilr.ltd -.-, b^ j^i- iijäft 5 wfifflfffiif N S N L : : -. rr-rinii. -'i; \j-~ -t; ^~7, "'].! : : : L: : i i:.'j.4tj - -8 pi l^iym! t" :! ii'-;i:iiil^n3i-];,^^ ll^-;^ilia l:: l'"jüfadtf (a) F^r* fr cases D and E. Figure Variatin f F in cmpressible slutins.

59 58 NACA TN 2750 M h' "4 -> r.-1.1; 11 -il: '' 1 ; i T1 Jbt atr -^t- ptil 4^j^tei" -p F -10 r^ 4f -H:-itr ;TH tzn _j. ji~~ l'-vi WHli: ^iljripi^ r, "CT"tf i -H r Mirii4ii4 i-qji yf -pj-hr* :j~g ~a!b a -j-'-h--h""" '\. \^*^a\' 1 i' 1 ' "ja-u'-j-l-llj... -Tj- { ""! ---"t^. _^juj:: -11 i -VS4J -Ji-*^r^71 ' i 1; I i-j -Tfr : CO -10 (b) F* far cases D and E. Figure Cntinued. Variatin f F in cmpressible slutins.

60 NACA TK Slutin Apprximate frmula (50) Case D (z* = in Statr) E (z* = 0 in Statr) F (ZQ = 0.15 in statr) (c) Cmplete ana apprximate slutins fr F* fr casee D, E, and F. Figure Cncluded. Variatin f F in cmpressible slutins.

61 60 NACA TN 2750 t Ö DO f

62 NACA OT r* (a) Case D. Figure Cnturs f cnstant pressure rati p/p T ^.

63 62 NACA TN 2750 CO s CO 1.00:.60 (b) Case E. Figure Cncluded. Cnturs f cnstant pressure rati P/PT ( I-

64 NACA TN (a) Case D. Figure Cnturs f cnstant Mach number.

65 64 NACA TN Figure Cncluded. (b) Case E. Cnturs f cnstant Mach number.

66 NACA TN Incmpressible Case A Case B ; Simplifiedradialequilibrium slutin Slutin Cmpressible Case C Case 0 Case E Apprximate slutin f reference 2 Simplified-radialequilibrium slutin r zer aspect rati Figure Radial variatin f axial velcity at z* equal t NACA-Langley