streamlines across the span of a multistage fan, compressor or turbine (Smith, 1966). ..... sitions, Rocket Performance, Incident and Re ected Shocks,.
AXISYMMETRIC AERODYNAMIC NUMERICAL ANALYSIS OF A TURBOFAN ENGINE Mark E. M. Stewart NYMA Inc. Brook Park, Ohio
ABSTRACT
; ;
An equation and numerical solution procedure for a steady-state aerodynamic analysis of a turbofan engine in the meridional plane is presented. An axisymmetric average of the three-dimensional uid ow equations within the engine is presented. The equations simultaneously represent the major engine components and include terms for blade forces, loss, combustor heat addition, blockage, bleeds and convective mixing in the mixer. These general equations are specialized with modeled terms and advanced to a steady-state with a time-marching scheme on a multiblock grid. The procedure is veri ed with comparisons to one-dimensional analytic solutions and a compressor stage. How this procedure can capture component interactions is demonstrated with a solution for General Electric's Energy Ecient engine, which is representative of modern turbofan engines.
?
Subscripts b c I p, s
1
1, 2
Superscripts
?0
NOMENCLATURE b - blockage E; e - total, internal energy f - force; fuel H; h - total, static enthalpy I, I - rotational Inertia; rothalpy
l
N, N P; p q R,r s T, T U; W
w
Z
~ 00
- absolute, relative, meridional air angles - cell or arbitrary region within domain - eciency - circumferential coordinate - Angular Velocity - blade - combustor - ideal - pressure, suction side of blade - loss - ambient or far eld - leading, trailing edge of blade - circumferential average - perturbation from circumferential average - density weighted average - perturbation from density average
INTRODUCTION
The introduction of aircraft jet engines brought revolutionary changes in the speed and economics of air transportation. However, the current emphasis in their design is evolutionary improvements which reduce the time, cost and risk of engine development. One strategy for meeting this goal is to exploit computer and algorithmic improvements and use numerical simulations to develop and test engines and their components while decreasing the need for experimental tests. With few exceptions, the trend in jet engine numerical analysis has been to apply advanced numerical techniques
- streamwise arc length - wheel speed (rpm); Number of blades - Total, static pressure - Heat addition per unit volume - Gas constant; Radial coordinate - Entropy - Temperature; Turning distribution - Absolute, Relative frame velocities - Vector of dependent variables - Compressibility factor 1
nent simulations must be included because of the problem's complex physics and the modeling expertise of these codes. However, instead of patching these codes together at boundaries with boundary conditions, a single set of equations is solved throughout the domain and some equation terms are interpolated from the component simulations. This approach is used since it simulates a complete engine with uniform modeling delity and uniform computational requirements. Further the accuracy and stability properties of interpolating onto a grid are better understood than the accuracy and stability properties of boundary conditions. To consider the system of components, an axisymmetric equation is derived in Section 2 which represents the engine's major aerodynamic components. The equation includes terms which represent blade forces, loss, blockage, combustion heat addition, bleed and cooling ows. One way these terms can be implemented in a numerical simulation is detailed in Section 3. The following sections explain the numerical method, grid scheme and verify the procedure. Finally, results are shown for General Electric's Energy Ef cient engine (EEE).
to isolated components to understand the details of their operation. Meanwhile, analysis and testing of these components as a system is limited to performance maps and one-dimensional analysis in preliminary design and experimental testing of prototypes. The goal of the current work is to allow these component calculations to be coupled to simulate a full turbofan engine and capture the axisymmetric interactions. In current preliminary design practice, the complete engine is simulated with performance maps and one-dimensional analysis. Both methods are dependent on databases of experimental results to quantify component performance at a range of operating conditions. However, they do allow trends in engine performance to be examined and design decisions to be made. As design progresses, component simulations and experiments are done which calibrate the component performance maps and one-dimensional analysis. Traditionally, the workhorse of turbomachinery simulation has been the steamline curvature method which calculates multiple streamlines across the span of a multistage fan, compressor or turbine (Smith, 1966). Development of this method has emphasized improved component performance prediction with improvements such as coupled quasi-three-dimensional blade calculations (Jennions and Stow, 1985) and spanwise mixing (Adkins and Smith, 1982). The method cannot predict shocks and endwall losses, and geometrical generality has limited its extension to full engines. The use of multi-dimensional numerical techniques and complex geometry grid generation to solve the Euler or Navier-Stokes equations in complex geometries has re ned component prediction further, by allowing transonic, viscous, heat transfer and chemistry e ects to be studied. For blades, quasi-three-dimensional (Chima, 1986) and threedimensional Navier-Stokes solutions (Hah, 1984) have been found which allow transonic behavior to be predicted. Methods for simulating multistage transonic turbomachines have also been developed (Adamczyk, 1985). Numerical techniques have been used to simulate other components such as nacelles (Crook et al., 1982) and nozzles (Povinelli et al., 1984). However, to extend these methods to an entire engine is currently beyond the computational power of existing computers. Although component simulations may predict performance at unique component design points, engine performance depends on components working together and their interactions can be important. The outlet conditions of a transonic blade row will dramatically change shock position. The outlet conditions of a compressor will change mass ow, pressure rise and eciency, and the interactions between a supersonic inlet and the compressor determine the inlet's stability. Performance maps and one-dimensional analysis attempt to capture these interactions, however component simulations have become very sophisticated and an axisymmetric, full engine analysis can use these improved predictions to better represent component interactions. In doing a full engine calculation, it is believed that compo-
AXISYMMETRIC ENGINE EQUATIONS
The axisymmetric engine equations are derived from the three-dimensional uid ow equations in a rotating cylindrical coordinate system, their boundary conditions and a torque equation for each engine shaft. The uid ow equations for the conservation of mass, momentum and rothalpy expressed in the blade frame ( = 0 for stators ) are
I w d V ol = ? ( F; G; H ) n dA @? Z ? (k + kc + km + k ) d V ol ? 0 1 0 Wx 1 BB Wx CC BB Wx2 + p CC w = B@ Wr CA F = B@ Wr Wx CA W W Wx
@ Z @t ?
E R 0 Wr 1 B WxWr C G = BB@ Wr2 + p CCA W Wr Wr I
W I
x 0 W 1 B WxW C H = BB@ W2r W CCA W + p W I 1 0 0 _f 0 C BB 0 B 0 k = BB ? 1r (Wp + (W + r)2) CCC kc = BB@ 0 @ 0 (W + 2 r) A r
r
?q_
0
0 ?m1() 1 0 0 B ?m2(Wx ) C B ?f km = BB@ ?m3 (Wr ) CCA k = BB@ ?f ?f ?m4 (W )
x
r
?m5 (I )
2
(1)
0
1 CC CA
1 CC CA
0 1 0 _ f 1 0 B 0 C f + r)2) CCC kc = BBB 00 CCC k = BBB ? 1r (p + (W @ 0 A @ Wer (W f + 2 r) A ?qe_ 0 0 m1() 1 0 0 1 fx) CC BB m2(W BB ?fe CC B f C km = B m3 (Wr ) C k = BB ?fe CC @ m 4 ( W @ ?fe A f ) A 0 m5 (Ee) 0 0 1 0 0 1 0 _bc 1 @b B ?p @x C fx C B ?fb C B _bcW fr CC kB = BBB ?p @r@b CCC kf = BB@ ?fb CCA kbc = BB@ _bcW f A @ 0 A ?fb _bc W 0 _bc Ie 0 p = Z ( p; Te ) Rgas Te(e): (5)
p = Z ( p; T ) Rgas T (e) and they model compressible rotational ow and allow entropy and vorticity production across shocks. The equations include terms for the addition of fuel mass, _f , and heat, q_, in the combustor and mixing, km , within the mixer. However, they di er from the Euler or Navier-Stokes equations since they include terms, f , which increase entropy within blade rows or the combustor (Horlock, 1971). The equations' boundary conditions include the force of blades on the ow, and bleed and coolant ow. A torque equation for each shaft, j , balances the torques due to blade forces, power extracted, Pj , and coolant acceleration, and it is
_ j Ij =
I
( 0; pr; 0 ) n dA +
r
x r
Pj + m_ r2: (2) bc j
x r
j
Jennions and Stow (1985) have derived an axisymmetric form of the uid ow equations within blade rows using the circumferential and circumferential-mass averages (x; r) =
Z
1 s ? p p (x; r; ) d
(3:1)
e(x; r) = :
(3:2)
s
The torque equation for each spool, j , is
P
_ j Ij = j + m_ bc j r2 +
j
X Spool j Rotors
These operators average between the suction and pressure sides of a blade passage or around an annular duct when outside a blade row. Perturbations from these averages are denoted (x; r; ) = (x; r) + 0 (x; r; ) = e(x; r) +
00
(x; r; ):
Z ?
w b d V ol = ?
I
@?
( F; G ) n b dA ?
ZZ
? 2 Rotor
s
r p0
p
dr dx
Some perturbation terms which arise from averaging are detailed in Appendix 1. The circumferential average analytically changes boundary conditions in the three-dimensional equations into terms in (4). Physically, the blade force terms, efb, arise from the difference in pressure between the blades' suction and pressure sides, and averaging produces the terms
(3:3)
Z Z @ @ 1 p0 ( @x ; @r ; ? r ) dV ol ? P ? ? (7) where P are perturbation terms. Although efb and blade ZZ
Applying this average to (1) and (2), manipulating with Liebnitz rule yields the equations,
@ Z @t ?
N ( Rotor )
(6)
(4)
efbb dV ol = N 2
s
p
force terms are used synonymously they refer only to the isentropic blade forces and do not include the losses incurred through blade rows. The thickness of blades also reduces the cross-sectional area of the annulus by the ratio
(k b + kc + km b + k b + kB + kf b + kbc b ) d V ol
0 1 fx C B W fr CC w = BB@ W f A W Ee 1 0 W 0 W fx fr 1 BB W fx2 + p C BB W fxW fr C 2 +pC C B f fr W fx C F = BB W G = W B@ fr f CCA A @ W fx C f W W Wr fx fr IeW IeW
bb = (s ?2 p ) N
(8)
and introduces the blockage term b. Similarly, bleed and coolant ow through blade surfaces and endwalls introduce the additional terms kbc . The equations admit shocks as solutions and shocks can exist in the rotating blade frame. is only non-zero in rotor blade rows hence the equations change reference frame across the domain. 3
TERM EVALUATION AND IMPLEMENTATION
The component simulations (Sullivan, 1983), (Holloway et al., 1985), (Timko, 1993), (Cherry, 1993) give design point values of these coecients on a set of streamlines. These coecients and blade leading edge solution data give total pressure losses which are related to ef by
In the current work, it is believed that a tractable way of solving these equations for a full engine is to use the modeling expertise of component codes to evaluate some terms. In an analogous way, component analysis and experiments are used to determine component performance maps which are coupled to predict engine performance. Although different methods and component solutions could be used, one approach to evaluating the blade forces, fb , loss, ef , and blockage terms, b, from streamline curvature component solutions is presented. Modeling of the combustor heat addition, bleed, mixer mixing terms and the equation of state is also presented. If isentropic blade forces, fb , were evaluated and directly included in (4), there could be a force component parallel to the ow in o -design conditions. Since this loss-like component would unphysically change entropy, these force terms must be constructed without this component. Consequently, turning distributions are used within blade rows to iteratively nd the isentropic blade forces. The turning disfx; W fr ; W f )= jW f j, which for stators tribution is T = ( W and rotors is ; tan ) T = p((11+; tan (9:1) tan2 + tan2 ) (9:2) T = p(( 11;+tantan ;2 ?+tantan 2) ) : The air angles, , and , are interpolated from values given in streamline curvature solutions at each component's design point (Sullivan, 1983), (Holloway et al., 1985), (Timko, 1993), (Cherry, 1993). At each stage, n, of the numerical scheme (15) and at each blade row cell, ?, a force update, 4fbn, is calculated ( gure 1) n fn T ) T ? W f n ] (10) 4 fbn = 4v = 1 [ ( W
4t
ZP
@p = dh = ?T ds = ef dl: P2 ; 4s = 0 I2
The integral is known from the thermodynamic properties of the gas (Gordon and McBride, 1976). The equations (4) will admit shocks and the accompanying entropy increase. While the axisymmetric average analytically gives the blade blockage, bb , as (8), viscous boundary layers on blade surfaces and endwalls further reduce the annulus crosssectional area relative to a simulation which does not model boundary layers. These viscous e ects signi cantly in uence the velocities and air angles within compressors and bv theoretically satis es
I
@?
4t
P I 2 ? P 2 > 0: P 1 ? p1
f i n bb dA = bv i W
I
@?
f v n bb dA (13) v W
where the subscripts i and v indicate inviscid and viscous solutions. The combined e ective blockage factor is b = bv bb . bb is known from blade geometry, and bv values are taken directly from component simulations (Sullivan, 1983) (Holloway et al., 1985). The aerodynamic e ects of the combustor are represented in (4) by the rate of addition of fuel mass, _f , heat, qe_ and an entropy increase due to losses. Although the exhaust gas composition is important for emissions, it has a negligible e ect on the equation of state (5). The fuel mass ow is iteratively updated from the current solution's core mass
ow and by imposing the fuel's stoichiometric fuel-to-air ratio and the combustor's design equivalence ratio, '. The heat addition is determined from this fuel mass ow, fuel @q , and combustion eciency, c heat of combustion, @
and added to the force fbn+1 = fbn + 4fbn. Any deviation in velocity direction from the turning distribution e ects a f , is restorative force. At convergence, the uid velocity, W tangent to the turning distribution, force changes 4fbn are f) = 0 zero, and the net force e ects no loss since (4 fbn W by construction. This iterative evaluation procedure can be shown to be numerically stable and demonstrates stable behavior during starting. In transonic blade rows, the equations admit shocks which turn the ow and this turning distribution must be accounted for. While the isentropic blade force, fb , is perpendicular to the ow, the loss force, ef , which gives an entropy increase, is parallel to and opposed to the ow. In this implementation, this vector is scaled so that across the blade chord or combustor length a given total pressure loss coecient is matched. For a streamline in a stator or rotor blade row (blade frame) the total pressure loss coecients are
P1 ? P2 > 0 P 1 ? p1
(12)
@q : qe_ = _ f (?) @ c
(14)
Burrus (1984) gives values of the equivalence ratio, combustion eciency, and typical pressure losses, and Odgers (1986) gives fuel properties. These values are assumed to be constant. A further term in the equations is kbc which accounts for bleed and cooling ow in the engine. Fluid is bled or removed from some engine parts only to be reintroduced to cool others. The corresponding bleed and cooling ows de ne a cooling line in which mass and rothalpy are conserved. Further, the bleeds are assumed to be isentropic, and design mass ows are imposed. The mixer in an engine mixes the bypass and core ows thereby increasing thrust and eciency while decreasing noise. Without a mixer, a velocity discontinuity can exist between the core and bypass ows which results in a
(11:1; 11:2) 4
practice, the numerical solution is initialized to a uniform
Kelvin-Helmholtz instability. In engine simulations without a mixer, this instability has been observed and the discontinuity rolls up and oscillates in time. Mixing is due to convection and not turbulent mixing alone. Consequently, the mixing term, km , is approximated with the rst-order dissipation term, D(w ), and applied at the mixer's location. Within a jet engine gas temperatures and pressures can reach 1500o C and 30 atm where the gas is perfect to within 1.5% but not calorically perfect. Consequently, it is assumed in (5) that Z (p; T ) = 1, and thermodynamic data (Gordon and McBride, 1976) is used to represent the calorically imperfect gas with a polynomial approximation to Te(e) for dry air.
M1 = 0:3 ow and advanced by applying the numerical
solution procedure (15) at a sequence of operating points speci ed by the input parameters ( N1 ; N2 ; c ; M1 ; T1 ). First-order dissipation, D(w ) is necessary during starting for numerical stability. A conservative starting schedule involves 150 iterations of (15) at each of 20 operating points.
GEOMETRY AND GRIDS
The axisymmetric equations (4,5) and their boundary conditions are discretized on a multiblock grid (Stewart, 1992a,b). In this method, the major internal ducts of the engine, the inlet, nozzle, and external eld are represented with non-overlapping, boundary conforming blocks which meet at common interfaces. Coordinate lines pass through these interfaces without loss of continuity, and in solving the equations, data is passed between blocks so that interfaces are transparent to the numerical method and solution. Coordinate lines coincide with the leading and trailing edges of blades to distinguish the di erent blade reference frames.
NUMERICAL METHODS
The equations (4) are discretized, and with the blade force, loss, blockage and heating terms described above, they are advanced to a steady-state in the manner of Jameson et al. (1981) using a multistage scheme with stabilizing arti cial dissipation D(w ) :
w(1) =wi + 14t(Q(wi) + D(wi)) w(2) =wi + 24t(Q(w(1)) + D(w(1))) w(3) =wi + 34t(Q(w(2)) + D(w(1))) w(4) =wi + 44t(Q(w(3)) + D(w(1))) wi+1 =wi + 54t(Q(w(4)) + D(w(1))):
RESULTS
This mathematical model and numerical procedure have been veri ed and demonstrated in three ways. First, high hub-to-tip radius ratio duct simulations with each term of (4) have been compared with one-dimensional solutions. Also, convergence has been veri ed and grid resolution studies have been done to ascertain the order of accuracy of the scheme. Second, a solution from General Electric's compressor o -design code, Perch, for the rst stage of the EEE compressor is used for comparison. Finally, to demonstrate this procedure for a full turbofan engine, a simulation of the full EEE engine is presented. In high hub-to-tip radius ratio annular ducts radial variations are small and solutions approach one-dimensional solutions which can be found analytically. To verify fundamental aspects of the scheme, a test case was devised with an annular duct with rhub =rtip = 0:99, length 0:02rhub and at least four grid resolutions up to 256x32. Separate tests were done for each term of (4). In each test solution parameters such as mass ow and temperature were compared with one-dimensional solutions and used in grid resolution studies to verify the order of accuracy. In each case these parameters approached expected values with second-order accuracy. Also solution convergence was checked and found to be at least six orders of magnitude. A second test was devised to quantify errors in the term evaluation and solution procedures. A solution for the rst 1.5 stages of the EEE compressor at 76.6 % design corrected speed was obtained from General Electric's compressor o design code, Perch (Brown, 1994). Perch's o -design models have been calibrated against experimental data taken at this condition. Turning distributions (9.1), total pressure losses (11.1,11.2), and blockage (13) were derived from this Perch solution in the manner described above. Predicted variables such as mass ow, spanwise pressure and meridional velocity
(15)
Q(w) is the RHS of (4) for a region ? and i are coecients. The time step, 4t, is chosen to be the local maximum which
sacri ces time accuracy for convergence speed. At the domain boundaries, conditions representing solid walls, free stream conditions or the centerline axis are imposed. The torque equation (6) for each spool can also be evaluated to determine changes in j . Two forms of dissipation have been used in these calculations. One is a robust but dissipative rst-order seconddi erence, and the other is the preferred and less dissipative third-order fourth di erence which switches to rst-order dissipation near shocks. When Q(w) is dominated by convection, as in the Euler equations, this scheme has been shown to be stabilized by the dissipation, D(w). However, when the blade forces, fb, of (7) dominate Q(w) the blade forces (10) must be designed to maintain stability. A time scaling of the increments 4fb (10) can be shown to be numerically stable when used with either form of dissipation. For the EEE engine, this system of equations (4,5,6) and boundary conditions cannot be impulsively started from the uniform initial condition corresponding to the desired operating point. High Mach numbers will choke an unstarted compressor and rapidly increasing combustor heat addition can reverse ow in the compressor. One e ective starting procedure mimics the real engine starting procedure by beginning at low subsonic speeds, gradually increasing wheel speeds and combustor heating followed by increasing the free stream Mach number to the design condition. In 5
of the two models. The cycle deck and solution agree on half the values of mass ow, total pressure and temperature to within 2%. They agree on engine net thrust to within 3%. Mass ow results at various stations agree to within 3%, with two exceptions. Similarly, total temperature results agree to within 3% with three exceptions. The biggest discrepancies between the two solutions occur in total pressure within the compressor and turbines, and are as great as 20 %. To resolve these discrepancies, General Electric's compressor o -design code PERCH will be used to examine the compressor in greater detail.
at blade leading and trailing edges were used for comparison and currently match the Perch results to within 2 %. Further improvements are expected. To demonstrate this method for a full turbofan engine, a simulation is presented of the Energy Ecient Engine (EEE) which was designed and tested by General Electric for NASA in the early 1980's. This is the only engine for which detailed design information is publically available. The geometrical description of the EEE engine meridional plane, shown in gure 2, comes from a variety of sources. Flowpath coordinates were provided by GE. A nacelle was designed. Blade positions were determined from component design reports (Sullivan, 1983), (Holloway et al.,1985), (Timko, 1993), (Cherry, 1993). The inner region of a 90,128 cell grid generated from this geometry is shown in gure 3. The EEE simulation is of the engine cruise design point, namely, M1 = 0:8, T1 = ?44:4o C , N1 = 3; 486 rpm, N2 = 12; 405 rpm, and c = 0:994. The information to evaluate T, ef , and b is derived from component streamline curvature solutions at component design points as described above. Since these component design points are not identical to the system or integrated engine design point, each component is o -design in the simulation. For example, the system fan speed, N1 , is 93% of the fan design speed, and the system compressor total pressure ratio is 90% of the compressor design total pressure ratio. General Electric's compressor o -design code PERCH will be used to incorporate improved component o -design data to resolve this de ciency. With third-order dissipation in this simulation, grid maximum and average residuals, @=@t and @ E=@t, converge two to three orders of magnitude. With rst-order dissipation, the solution converges at least ve orders of magnitude as shown in gure 4. On a single processor of a Cray YMP 8E/8128 these simulations run at 9x10?6 s/cell/iteration. Starting the EEE engine and nding a solution typically takes 1-2 hours. Without the startup procedure, engineering solutions can be found in less than 1 CPU hour. The procedure currently includes no convergence acceleration techniques, but multigrid could easily reduce execution time by a factor of 2 to 4. To demonstrate solution results, gure 5 shows the inner domain with ow Mach contours for the cruise design-point mentioned above. Although the torque equation (6) can be evaluated to determine changes in j , if j are xed at the design values, the net torques on the fan and compressor shafts are 1% and 5% of the sum of the absolute torques on each shaft. A cycle deck (Bucy et al., 1983) exists for the EEE engine which gives spanwise averaged mass ow, total temperature and total pressure at nine stations within the engine. This cycle deck and the axisymmetric solution are not identical con gurations of the EEE engine since the cycle deck represents the projected nal engine and the axisymmetric solution incorporates results from early in the design process and involves o -design component operation. A comparison is included despite these questions about the comparability
CONCLUSIONS
This work demonstrates using component solutions to give a complete engine simulation which captures the interactions of aerodynamic components. To this end, an axisymmetric average of the uid ow equations for an engine has been presented. This averaging yields additional terms in the equations which must be evaluated from component solutions. One method of evaluating these terms in a numerical simulation has been presented. The procedure has been veri ed and convergence ensured in two test cases. To demonstrate use of the procedure in a full turbofan engine the Energy Ecient Engine has been simulated. Given the limitations of the component data, the results compare favorably with cycle deck results.
ACKNOWLEDGEMENTS
This work was supported by NASA Lewis Research Center and the Numerical Propulsion System Simulation project under contract # 27186 with Austin Evans and Russell Claus as monitors. Andy Kuchar, David Cherry and Larry Timko have provided information about the EEE engine and Chuck Putt, Bonnie McBride and John Abbott have provided advice on engine modeling. Thanks are due to John Adamczyk and Ian Jennions for very helpful discussions. Special thanks are due to Jim Schmidt, who has generously shared his experience in compressor design and function.
APPENDIX 1
Jennions and Stow (1985) detail how the axisymmetric equations (3) are analytically derived from the threedimensional equations (1) and their boundary conditions by using the circumferential averages (2). In this averaging, perturbation terms appear in the momentum and energy equations. In the momentum equations, the perturbation terms, P, are included in fb and take the form
b I B (WgW ; WgW ) ndA (A1:1) x x x r B @? b I B (WgW ; WgW ) ndA Pr = B @ ?Z Z x r r r B W gW d V ol (A1:2) ? Bb ? r
Px =
00
00
00
00
00
00
6
00
00
00
00
P =
b I B (Wg g B @ ?Z Z x W ; Wr W ) ndA b B WgW d V ol: (A1:3) + r B ?r 00
00
00
00
Holloway, P. R., et al., 1985, \Energy Ecient Engine: High Pressure Compressor Detailed Design Report," NASA CR-165558. Horlock, J. H., 1971, \On Entropy Production in Adiabatic Flow in Turbomachines," ASME Journal of Basic Engineering, pp. 587{593. Jameson, A., Schmidt, W., Turkel, E., 1981, \Numerical Solution of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time Stepping Schemes," AIAA Paper 81-1259, pp. 327{356. Jennions, I. K., Stow, P., 1985, \A Quasi-ThreeDimensional Turbomachinery Blade Design System: Part 1 - Through ow Analysis," ASME Journal of Engineering for Gas Turbines and Power, Vol. 107, pp. 301{307. Odgers, J., 1986, Gas Turbine Fuels and Their In uence on Combustion, Abacus Press, Kent, U.K., pp. 48{ 68. Povinelli, L. A., Anderson, B. H., 1984, \Investigation of Mixing in a Turbofan Exhaust Duct, Part II: Computer Code Application and Veri cation," AIAA Journal, Vol. 22, #4, pp. 518{525. Smith, L. H., 1966, \The Radial-Equilibrium Equation of Turbomachinery," ASME Journal of Engineering for Power, pp. 1{12. Stewart, M. E. M., 1992a, \Domain Decomposition Algorithm Applied to Multielement Airfoil Grids," AIAA Journal, Vol. 30, #6 pp 1457{1461. Stewart, M. E. M., 1992b, \A Multiblock Grid Generation Technique Applied to a Jet Engine Con guration," NASA CP-3143. Sullivan, T. J., 1983, \Energy Ecient Engine: Fan Test Hardware Detailed Design Report," NASA CR-165148. Timko, L., 1993, \Energy Ecient Engine: High Pressure Turbine Design Point Information," Private Communication.
00
00
Assuming that streamlines are surfaces of revolution implies constant rothalpy along streamlines, I = 0, and eliminates the perturbation terms of the total energy equation. This assumption contradicts evidence of tip clearance and horseshoe vortices which would mix streamlines, rothalpy and total energy. However, it is consistent with the modeling assumptions in use when the EEE engine was designed. 00
REFERENCES
Adamczyk, J. J., 1985, \Model Equation for Simulating Flows in Multistage Turbomachinery," ASME Paper 85GT-226. Adkins, G. G., Smith L. H., 1982, \Spanwise Mixing in Axial-Flow Turbomachines," ASME Journal of Engineering for Gas Turbines and Power, Vol. 104, pp. 97{110. Brown, R., 1994, \Perch EEE Compressor Solution" Private Communication. Bucy, R. W., et al., 1983, \Energy Ecient Engine: Component Development & Integration, Steady-State Performance Computer Program User's Manual for the IBM 370 and CDC 6600 Computers, GE Reference Computer Program - G0045I," General Electric Report R83AEB553. Burrus, D. L., et al., 1984, \Energy Ecient Engine: Combustor Test Hardware Detailed Design Report," NASA CR168301. Cherry, D., 1993, \Energy Ecient Engine Low Pressure Turbine Design Point Information," Private Communication. Chima, R. V., 1986, \Development of an Explicit Multigrid Algorithm for Quasi-Three-Dimensional Viscous Flows in Turbomachinery," NASA TM-87128. Crook, J.L., et al., 1982, \Isolated Nacelle Performance| Measurement and Simulation," AIAA Paper AIAA-82-0134. Gordon, S., McBride, B. J., 1976, \Computer Program for Calculation of Complex Chemical Equilibrium Compositions, Rocket Performance, Incident and Re ected Shocks, Chapman-Jouguet Detonations," NASA SP-273. Hah, C., 1984, \A Navier-Stokes Analysis of ThreeDimensional Turbulent Flows Inside Turbine Blade Rows at Design and O -Design Conditions," Journal of Engineering for Gas Turbines and Power, Vol. 106, pp. 421{429.
Figure 1: Isentropic blade forces fb are constructed using turning distributions, T.
7
