Advanced Compressor Loss Correlations , Part I : Theoretical Aspects

Reliable efficiency calculation of high-subsonic and transonic compressor stages requires a 
detailed and accurate prediction of the flow field within these stages. Despite the tremendous 
progress in turbomachinery computational fluid mechanics, the compressor designer still 
uses different loss correlations to estimate the total pressure losses and thus the efficiency of 
the compressor stage. A new loss model is presented in this article. Special attention is paid 
to the shock and profile losses, since they contribute significantly to the total pressure loss 
balance, specifically for transonic compressor stages. A new shock loss model is presented 
that calculates the shock position and the shock total pressure losses. The available experimental 
data were used to establish new loss correlations that account for experimental 
findings.


INTRODUCTION
The development in the field of turbomachinery com- putational fluid dynamics (CFD) has reached an ad- vanced level that allows a detailed calculation of the complex three-dimensional viscous flow through a compressor stage using Navier-Stokes codes.Recent case studies presented at the 1994 IGTI-Conference displayed the capability of different CFD methods to calculate various flow quantities in detail.However, the efficiency and loss calculations revealed a significant discrepancy between the experiment and numer- ical calculation.To predict the compressor stage effi- ciency accurately, the compressor designer often uses loss correlations that reflect different loss mecha- nisms within the compressor stage flow field.In the early fifties, Lieblein and his co-workers [1953, 1954,  1956, 1957, 1959] conducted fundamental research in compressor cascade and stage aerodynamics.Their research work, NASA-Report SP-36 [1965], is a guideline for compressor designers.Miller and Hartmann [1958], Miller et al. [1961], and Schwenk et al.  [1957] initiated their fundamental research on tran- sonic compressors, where they primarily investigated the shock losses.Gostelow et al. [1968], Gostelow  [1971], Seylor and Smith 1967], Seylor and Goste- *Corresponding author.Tel.: (409)  845-0819.low [1967], Gostelow and Krabacher [1967], Krabacher and Gostelow [1967a,b] focused their experi- mental research on single stage high Mach number compressor stages.Their comprehensive experimen- tal research includes the performance evaluation of several rotors.Monsarrat et al. [1969] performed sim- ilar investigations on single stage high Mach number compressor stages.Koch and Smith [1976] presented a method for calculating the design point efficiency potential of a multi-stage compressor.Schobeiri  [1987] investigated the individual loss mechanisms that occur in an advanced compressor stage.He de- veloped a new shock loss model, introduced a modi- fied diffusion factor, and re-evaluated the significant experimental data published by NASA (see previous references).Recent investigations by K6nig et al.  [1994a,b] investigate the loss and deviation angles for transonic bladings.This paper focuses on three issues: (1) A new mod- ified diffusion factor that describes the blade loading for the rectilinear and annular cascades, as well as for the entire compressor stage.This new diffusion fac- tor, which includes the compressibility effects, allows the loss parameters to be systematically correlated with the diffusion factor.(2) A new shock loss model is presented that overcomes the weaknesses of the existing loss models described by Levine [1957], Balzer 1970] and Swan 1961 ]. (3) The existing pub- lished data are re-evaluated and detailed correlations are presented.that causes wake defect, mixing, and thus additional entropy increase.From an experimental point of view, these two losses are not separable, since the total pressure measurements occur at a certain dis- tance downstream of the trailing edge plane and in- herently include the wake total pressure defect.The combination of these two losses is frequently called a profile loss.(2) Shock losses are encountered in com- pressor stages with high transonic to supersonic inlet flow conditions.Based on an angle incidence and the shock position, these losses may generate consider- able entropy increase that results in a significant de- terioration of the stage efficiency.The shock losses are approximately of the same order of magnitude as the profile losses.(3) Secondary losses due to the end wall boundary layer development and blade-wall clearances.(4) Secondary flow losses are also present for compressor blades with shrouds.A comprehen- sive treatment of losses is found in Schobeiri [1987].

Profile Losses
Lieblein and Roudebusch [1956] derived the expres- sion for profile loss coefficient as a function of cas- cade geometry, flow angles, and the boundary layer parameters shown in Eq. (1): ' \sin2J -00" H12 C

TOTAL PRESSURE LOSSES IN AN ADVANCED COMPRESSOR
The total pressure losses encountered in an advanced compressor stage are: (la) The blade primary losses generated by the wall shear stress, which is propor- tional to the local velocity deformation.The primary losses are restricted to the middle of the blade surface area and thus not affected by the secondary vortices generated by the secondary flow on the blade hub and tip.(lb) The trailing edge mixing losses are due to the thickness of the trailing edge and the boundary layer thicknesses on the suction and pressure surfaces with the form factor H32 f(H12).Among the boundary layer parameters in Eq. ( 1), the momentum thickness 0 is of primary importance.It gives a direct relationship between the separation point and the freestream velocity gradient (or pressure gradient) shown in the following yon Khrmtn integral equation for incompressible flow: with % as the wall shear stress and HI2 the form parameter.For a highly loaded compressor blade, the velocity distribution may separate.Consequently, the wall shear stress vanishes and Eq. ( 2) may reduce to: dO 0 dV dx -(2 + HI2 --X (3) Equation ( 3).shows a direct relation between the blade velocity gradient and the momentum thickness.Appropriate differences can approximate the above differentials.As an appropriate measure for the veloc- ity gradient, Lieblein et al. [3] introduced the equiv- alent diffusion factor: with Vm, as the maximum velocity on the suction surface.This velocity ratio, which changes by changing the flow deflection given by the velocity triangle, properly reflects the blade loading situation.However, it requires the knowledge of the maximum ve- locity at different flow deflections, which is not al- ways given.From a compressor designer point of view, relating the blade loading to the actual velocity triangle is most appropriate.

New Diffusion Factor
We introduce the dimensionless parameter 1 V/V1, x/b and approximate the dimensionless velocity distribution q by: N q an n_ a0 + a + a22 + anon 0 (5) The Taylor expansion in the near of max results in" dq d2"q A 2 + (6) n nm.x + ae + with A m,x" Neglecting the higher order terms, Eq. ( 6) can reduce to The velocity slope is found from Eq. ( 5): () -a| +2a2ma (8) max Incorporating Eq. ( 8) into Eq.( 7) results in: qq Tlmax -}-CI C 2 (9) with the constants: C| a| + 2a2ma and C 2 -Clmax.The tangential component of the force act- ing on the blade (see Fig. 1) is calculated by integrat- ing the pressure distribution along the blade suction and pressure surfaces using the Bernoulli equation: b T= T s Tp =-p (Vs V2) dx (10)   0 Since only the contribution of the suction surface is considered for estimating the diffusion factor, the sec- ond term in the integrand may be set equal to zero.
Incorporating Eq. (9) into Eq.(10) results in: The force component T can also be calculated using the momentum equation in the tangential direction: T-spV(Vtl Vt2 (12) with s and p as the blade spacing and the flow den- sity, V,, and V as the axial and tangential velocity components.Inserting the velocity components de-  Equating Eq. ( 11) and ( 13) results in the following with the coefficients D as: blade C.4, for the optimum flow condition denoted by (*), the constants in Eq. ( 15) are experimentally de- termined by Lieblein [1959]" The solution of Eq. ( 14) after neglecting the higher order terms in D 3 yields: Equation ( 14) is generally valid for any arbitrary inlet flow angle including the off-design incidence angles.
However, the compressor designer prefers to relate the off-design Tlmax to the design point condition.Lie- blein [1975] introduced an empirical correlation for VmaxlV for positive angles of attack as a function of the incidence angle Equation ( 15) exhibits a special case of Eq. ( 14) and gives an explicit relationship between the maximum velocity Vmax and the cascade circulation function G sinZf31 / cr (cot[31 cot[32) in the bracket.Us- ing the NACA-65(A10) series and the circular arc where a 0.0117 for the NACA 65(A10) and a 0.0070 for the C.4 circular arc blades.Eq. ( 17) accu- rately estimates the maximum velocity ratio for a positive incidence.However, Eq. ( 17) cannot be used for negative incidence angles [3 [3" because of the rational exponent of the argument [3 [3".Intro-  ducing the angle [3 [3" + into Eq.( 14) eliminates this deficiency and after some rearranging we obtain: N Neglecting terms with n > 2, Eq. ( 18) becomes Re-evaluating the experimental results by Lieblein  [1957] leads to a 0.746 and a 2 0.65.Equation ( 19) enables the calculation of the velocity ratio and thus the diffusion factor for any off-stall operation range.Figure ( 2) compares the results of Eq. ( 19) with the experiments, where the velocity ratio is plotted against the circulation function G with the inci- dence angle [ [3" as a parameter.Compared G (-)   FIGURE 2 Velocity ratio as a function of G with the incidence angle as parameter.with Lieblein's correlation Eq. ( 17), the new correla- tion Eq. ( 19) yields more accurate results.This state- ment is also true for Fig.
(3), where the maximum velocity ratio is plotted versus the incidence angle with the lift coefficient Co as parameter.

Cascade, Stage
Lieblein's correlations were further extended in previous section for the maximum velocity ratio and their experimental verifications were based exclu- sively on the two-dimensional incompressible cas- cade flow situation.Significant effects such as com- pressibility and three-dimensionality were not consid- ered.Furthermore, changes of axial velocity component and streamline curvature, which are al- ways present in a modern compressor, were ignored.Finally, the effect of rotationa!motion on circulation was disregarded.This section includes the above FIGURE 3 Velocity ratio as a function of incidence angle with the lift coefficient as parameter.mentioned effects in the maximum velocity ratio by employing a generalized circulation concept that leads to a modified diffusion factor.Starting from the Kutta-Joukowsky's lift equation (lift force/unit span) with p, V as the freestream density, velocity, and F as the circulation: , Using the definition in Figs. ( 4) and (5) for linear and annular cascades, the circulation is expressed as: where V1 and V2 represent the tangential velocity components at the inlet and exit and F23 -F41.s.For a stator cascade with conical streamline (Figs.4b, 5b), different spacings at the inlet and exit are present that relate to each other by the radius of the streamline curvature, s AOr 1, s2 AOr.Using the following velocity ratios: and defining the specific circulation function /, we obtain the following relations for the linear cascade /c, stator /s and rotor /R: Wal G1 rl Val rl cot2 (24)   1 (1 1)2) cot[31 Equation ( 24) exhibits a generalized relation for the specific circulation function.As seen, to calculate cir- culation the absolute velocity components Vtl and are utilized and refer to the absolute circulation rather than the relative one.Special cases such as the linear cascade, stator with cylindrical streamlines, and sta- tor with axisymmetric streamlines can follow imme- diately by setting: case 1: u 0, + oo, ta= l, v l(linearcascade, cylindrical stator) case 2 u 0, b 0% la 4= 1, v 4= l(axisymmetric stator) case 3 u 4 0, b 4= 0% a 4 1, v l(cylindrical rotor) case 4 u 4= 0, qb 4= % a 4= 1, v 4= l(axisymmetric rotor) Using the most general case (Eq.24), we obtain the circulation function for the rotor as: Correspondingly, we obtain the off-design maximum velocity ratio by using Eq. ( 19)" In Eqs. ( 26) and ( 27), the individual quantities de- noted by the subscripts C, S, and R pertain to cas- cade, stator and rotor, respectively.The Lieblein's equivalent diffusion factor is then: Compressibility Effect To consider the effect of compressibility on the max- imum velocity ratio and thus on the diffusion factor, we modify the specific circulation function for the simplest case, namely a linear cascade, by introduc- ing the inlet density pl:

Ac
Po (VI1 Vt2) YCc 191 VVal s Pl al (29)   The second subscript c refers to compressible flow.The freestream density p can be expressed in terms of the density at the inlet and a finite increase p pl + kp.Outside the boundary layer we assume a po- tential flow that is not influenced by small perturbations.With this assumption, the Euler equation com- bined with the speed of sound may be applied: VdV C 2 d9 (30) P with C as the speed of sound.For small changes, the flow quantities can be related to the quantities at the inlet: V=V +AV; C-C +AC p= p + Ap; d0-Ap We introduce the above relations into Eq.( 30) and approximate the differentials by differences and neglect the higher order terms.After some rearrang- ing we obtain the density changes by: Pl 11 (31) Introducing Eq. ( 31) into the relation poo 91 + zip results in: (24), the specific circulation of compressible and in- compressible flows are related to each other by the density ratio and thus the Mach number.Considering the simplest case, namely the linear cascade de- scribed by Eq. ( 34), because of the compression pro- cess with 2 < 1, the bracket representing the com- pressibility effect is always greater than unity.With Eq. ( 34) and the condition for case 1 to case 4, the circulation function for cascade, stator, and rotor with compressible flow are: The subscripts C, S and R refer to the cases discussed above.Using Eq. ( 35) for optimum conditions, the velocity ratio for compressible flow is obtained from: 1-M (32) P \sin[32J \sin2 Implementing Eq. (32) into Eq.( 29) obtains the spe- cific circulation functions for linear cascade and sta- tors with cylindrical streamlines: The expression in the above bracket reflects the Mach number effect on the specific circulation function.Using the same principle, the generalized circulation function for the rotor is obtained by: This equation allows calculation of the specific circu- lation function for cases 1 to 4. As seen from Eqs.
(33) and ( 34) and comparing them with Eqs. ( 23) and Introducing Eq. ( 36) into the relationship for the equivalent diffusion factor for the rotor as a generalized case, reads: )+ a2 (1-)2 + bl The angles used in the above equations correspond to those defined in Figs.
(1) and (6).Equations ( 36) and (37) show the direct relationship between the maxi- mum velocity ratio and the specific circulation func- tion.Equation (37) inherently includes the compress- ibility effect and the actual and optimum flow angles.An alternative diffusion factor, which was first pro- posed by Smith [1995], is: r2V2 F1Vtl The crucial part of this equation is the expression in the bracket, which is identical to the specific circula- tion function in Eqs. ( 34) and (37).Introducing the compressibility effect results in a modified version of Eq. ( 39): The diffusion factors previously discussed are used to establish correlations for the individual losses as well as the total losses.

SHOCK LOSSES, NEW CORRELATION
Several papers have discussed experimental and the- oretical shock loss investigations.As indicated previ- ously, Miller and Hartmann [1958], Miller et al.  1961 ], and Schwenk et al. 1957] initiated their fun- damental research on transonic compressors where they investigated shock losses.Schwenk et al. [1957]  considered a normal shock in the entrance region of the cascade using a Prandtl-Meyer expansion.Levine  1957], Balzer 1970], and Swan 1961 made efforts to calculate shock losses by estimating the shock po- sition.Their proposed methods, particularly, Levine's  [1957] and Swan's [1961] found their application in compressor design.Similar to Schwenk [1957], the methods by Levine, Balzer, and Swan include the as- sumption of a normal shock.While Levine and Swan considered the acceleration on the suction surface by using the continuity and Prandtl-Meyer expansion, Balzer disregarded the expansion completely and used the continuity requirement.The deficiencies in the existing methods can be summarized as: a) They cannot accurately calculate the shock position, which is a prerequisite for accurately predicting the shock losses, b) The Mach number calculated by the Prandtl-Meyer expansion on the suction surface does not represent the shock Mach number along the chan- nel width.Swan partially corrected this deficiency by building an average Mach number, c) The description of the physical process is not complete: the Prandtl- Meyer expansion combined with the continuity re- quirement is not sufficient to describe the physics.The above deficiencies gave impetus to generate the following new shock loss model.For the develop- ment of this model we assume the passage shock as an oblique shock, whose position changes according to the operating point and may include normal shock as a special case.Furthermore, we assume that the blading has a sufficiently sharp leading edge, where the shocks are attached at least at the design point with no detached bow waves expected.Figure (7) shows the shock situation with the inlet flow angle [3, the metal angle t (camber angle), and the inci- dence angle i.To determine the shock position we use the continuity equation, the Prandtl-Meyer expansion, and the momentum equation.For the control volume in Fig. ( 7), the continuity requirement for a uniform flow is: 91 VlSlsin[3-9, V S, cos h- (41) with h and h, as the height of the stream tube at the inlet and at the shock position.Using the gas dynamics relationships, Eq. ( 41) is written as:  (42) 2 The incidence and expansion angle are coupled by: where v is determined from the Prandtl-Meyer expan- sion law: The momentum equation in tangential (circumferen- tial) direction is given by: As Fig. (7) shows, because of the cascade periodicity, the pressure at point A is identical to the pressure at point C. Furthermore, point B. on the suction surface represents the common end point for both distances AB and CB.This means that the pressure distribu- tions along AB and CB have exactly the same begin- ning and ending values, but may have different dis- tributions between the points AB and CB.Assuming that differences of the pressure integrals along the shock front AB and the blade contour portion CB are approximately equal, their projections in circumfer- ential direction may cancel each other and lead to" unknowns, namely: g, [3 s, 7, and M,.The shock loss is" V, cosl3 Ms + K-1M 2 Finally, we arrive at a geometric closure condition that uses the mean streamline, which is assumed to be identical to the mean camber line of the blade with the radius R.
R [+costcos (t + 0)] -sin 7 (48)   and The shock angle is determined from: With Eqs. ( 43), ( 44), (47), and (51) we have a sys- tem of four equations that easily calculates the four Pb Pa s 1-- where Pb and Pa represent the total pressure before and after the shock.For [3 30 and the incidence angle 0 , the above equation system is used to calculate the shock Mach number, the expansion an- gle 0, the shock position 7, the total pressure ratio, and the shock losses.
Figure ( 8) shows the shock Mach number as a function of spacing ratio S/R with the inlet Mach number M as the parameter.This figure shows that increasing the spacing ratio causes the shock Mach number to continuously increase and approach an as- ymptotic value.These results are similar to those presented by Levine [1957].However, the values are different because of :the simplifications by Levine.Keeping the inlet Mach number constant, the increase of spacing ratio leads to higher expansion angle 0 as shown in Fig. (9).However, increasing the inlet Mach number at a constant spacing ratio S/R leads to a smaller expansion angle.The same tendency can be read from the charts by Levine.For an inlet Mach number M 1.2 and S/R 0.5, the Levine's method gives an expansion angle 0 8 , while the method presented in this paper calculates 0 9.5 .
As indicated previously, this difference is due to the simplified assumptions by Levine.Figure (10) shows the shock angle 7 as a function of spacing ratio S/R.This figure shows the significant effect of the inlet Mach number on the shock position.Finally, Fig. (11) shows the total pressure ratio Pa/Pb as a function of spacing ratio S/R with inlet Mach number as parameter.As shown in Fig. 11, reducing the spacing ratio S/R < 1. causes for inlet Mach numbers M > 1.4   substantial reduction in total pressure ratio.For M1 < 1.4 this tendency is reversed, resulting in higher total pressure ratios.

CONCLUSION
The first part of this paper discussed the basic essen- tial theoretical aspects for generating the loss corre- lations for advanced compressor stages.It focused on two issues: (1) A new modified diffusion factor was introduced that describes the blade loading for the rectilinear and annular cascades as well as for the entire compressor stage.This new diffusion factor, which includes the compressibility effects, allows the loss parameters to be correlated with the diffusion factor in a more systematic manner.(2) A new shock S/R (-) FIGURE 11 Total pressure ratio as a function of the spacing ratio with the inlet Mach number as parameter and the incidence angle 0. Pa, Pb pressure after and before the shock.
loss model was presented that overcomes the weak- nesses of the existing loss models.It accurately cal- culates the expansion angle, the shock position, and the shock losses.The second part of the paper re- evaluated the existing published data and presented detailed correlations.

FIGURE
FIGURESchematic representation of a compressor cascade flow. For FIGURE 5a, b Cross section of two compressor stages with a constant and a variable mean diameter.

FIGURE 7
FIGURE 7 Shock position and angle definition, n, n., n are normal unit vectors at the inlet, shock location and the wall.
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