the off-design analysis of flow in axial...
TRANSCRIPT
C.P. No. 1234
MINISTRY OF DEFENCE (PROCUREMENT EXECUTIVE)
AERONAUTICAL RESEARCH COUNCIL
CURRENT PAPERS
The Off-Design Analysis
of Flow in Axial Compressors
H. Daneshyar and M. R. A. Shaalan,
Department of Engineering,
University of Cambridge
LONDON: HER MAJESWS STATIONERY OFFICE
1972
Price 8Op net
CP No. 123l+*
March 1971
The Off-Design Analysis of Flow m Axxl Compressors
9. Daneshyar and M. R. A. Shaalan Department of Engineering,
University of Cambridge
SUIllMARY
The existence and uniqueness of the solutions obtained from the stre&nlme curvature method of calculating flow through turbomachines' are examined for several operazmg points of Rolls-Royce compressor. It 1s shown that under certain conditions the truncation errors m the numerical solution can become large and hence give rise to the violation of the uniqueness conditions. The computer programme may then give wrong answers to the physical problem. The condztzons for existence and unxqueness may be vlolated when the merl-llonal velocities are small (e.g., near stall) or when there are regions of choked flow. Flow for an operating point m the stall region is computed by suitable modlficatlons to minimize the truncation errors and hence to obtain a unique solutxon. This is oompared with the results of the actuator disc theory and experxment reported i2 Ref. 8. Also the effect of variation of losses on the calculation IS examined together 171th the effect of a correction term due to a dissipative body force, which should be included in the momentum equation, when losses are introduced.
* Replaces A.a.C.32 727
-2-
Notation
a c C
P f F
Fr
G
h
H i
I lt
m . %
M n
P 0 r r m
x s
T
Urn
w
a
P
a
Y
6
velocity of sound absolute velocity specific heat at constant pressure
equation (11)
equation (12)
radial component cf the dissipative body force
equation (11)
enthalpy equation (II) incidence
relative stagnation enthslpy blockage coefficient
meridional direction
specified mass flew rate
Machnumber number of estimates for the streamline pattern
pI-eSSW.3 entropy function radius radius of curvature a? meridicnd projection
of a streamline gas constant
entropy temperature blade speed at the mean radius
velocity relative to the rotor
blade angle
flow angle ted 3 t ) cm
9o" -B
ratio of specific heats ratio of the bcdy force term to the other terms
on the right hand side of equation (1)
-J-
6 r streamline shift
P relaxation factor
P density
0 rotational speed of the rotor Ei total pressure loss coefficient
Subscripts
h hub
i upstream reference station m meridional direction R relative t tip x axul direction
0 clrcuderential direction 0 stagnation condition - also value at radius
1 inlet of the blade row 2 outlet of the blade row
52
Introduction/
-4-
Introduction
During recent years the numerical methods of streamline curvature' ad matrix throughflOV?, both using digitsl computers, have been widely used for the calculation of throughflow in turbomachinery. The two methods are based on the same mathematical models but ddfer in their numerical techniques. The general procedure, common to both methods, is to start with an initial approximation of the streamline positions and then by using equations of motion, energy and continuity calculate more accurate positions. It is found that this process overcorrects the initial errors and that the change in such p-eters as streamline position or radius of curvature must be damped by a factor less than 1 to obtain convergence.
In general the above procedures are ideaLsed and cannot alwa s be realised in practice. Choking and negative velocities (flow reversals 7 may occur in the process a? completing the cslculat~on. If these conditions occur for an intermetite streamline pattern and are not present in the flnel solution, then the basic techniques may be modified to deal mth them. However, if the foal solution involves choked or reversed flows (reversed flows occur mainly when a compressor is operated in the surge region), then the techmques can not be used without substantial modifications. In ths case the only way the system of equations can be satisfied is to allow for negative nerdional velocities to occur, a condition not allowed for m either of the methods at present.
Mathematically when choking or zero mentional velocities occur, the conditions for the existence and unzqueness of the solution of the governing system of differential-integral equations is violated. Moreover, since these equations are solved numerically, it is the behaviour of the finite ddference equations (replacing the original differential-integral equations in the computer programme) which is of interest. Numerical errors may give rise to violation of the existence an& uniqueness conditions and if solutions exist the programme msy give wrong answers for the physxal problem. For the throughflow cdcu- lations this conditzon may occur at small values of the merldlonal velocity 1.e. near stall when large losses ape present.
One object of the present investigation was to study the flow in this region in order to determine the 1imd.s of applxabdi.ty of the throughflow methods. It was therefore essential to investigate the conditions for the existence and uniqueness of the finite difference equations used in the programme. Since flow in this region is highly rrreversible, It was necessary to include high losses in the calculation. It has been shown by Horlock3 that when losses are included in the computation a disslpatlve body force should be included III the momentum equation. This gives rise to a 'body force term' in the differ- entlal equation for the meridxonal throughflow velocity. This term is usually neglected if the losses are small. Another obJect of the present work was to study the effect of inclusion of this term on the accuracy of the streamline curvature calculation when losses are hxgh.
Analysis
The flow field is calculated by using the laws of conservation of mass and energy and Newton's second law of motion.
-5-
Momentum equation a
For amsymmetric flow - = 0 , the momentum equations ae
and the equation of state may be used to give the radial variation of the merdxond velocity' as
acp 2L+2&p~
s3n$ ac -- 2
00s 6 cose2 5 +-+ -
ar C am r 2 m m
(Jay
0 ar
a + g-
COP 3 cot* p + - +
2 0 cot/?
%l 1 C,' ar r
= 2sins5 -1 aza
t --+ 0 ar
where Q is a function of the entropy
- s/c Qze
and 3 = (90 -j?) with ,Q the flow angle downstream of the blade row, measured along a streamline
Behind a stationary blade row the angle p represents the absolute flow angle with the angular velocity w = 0. For a rotating blade row (O finite) p is the flow angle relative to the blade.
The quantity I is defined as
and r Sirp$b a+
sin$ ac l+Meg + + tJn$ -
P ar -2 = -
rm cos $h s-s (2a)
cm am 1 -MB m
-6-
in a duct region,
in a bladed region, where the tangential component of the blade force is finite. Note that by a duct region it is meant that rCe is constant along a streamline and by a bladed regicn it is meant that NO = Cm tanfi where @ is a specified blade angle, giving tangential component of the blade force.
Equation (I) is derived by introducing changes in entropy into the equations of motion for inviscid adiabatic flow. As pointed cut by Horloc d the manipulation of the loss problem in this manner 1s inconsistent with the assumption of inviscid adiabatic flow unless dissipative body forces are intro- duced in the momentum equations. Horlock obtains an expression for the radial component of the dissipative body force as
c Tp aa Fr = -p -ssirp~sin$
0 am
Introducing this component of the dissipative force into the radial component of the equation of motion, Horlock arrives at a modified equatxon for Cm,
2F with the correction term 2 added to the right hand side of equation (1).
P
One object of the present investigation was to find the effect of this correction term on the accuracy of the calculation when the losses are large.
Continuity equation
Therefore The mass flew rate must be equal to the specified value mt.
r. t . “t = 2xr p cm cos $ ar
rh
This integral equation gives the constant of integration of equation (I).
Additional/
-7-
Additional equatxns
% ' These equations relate the density p, the tangential velocity
and the Mach numbers M and m 4l to cm.
The enthalpy rise across a blade row can be found from the steady flow energy equation and from the equation relating change of angular momentum to the moment of tangential forces (including viscous forces) about the axis of rotation. This leads to
Ah0 = (r w c,lp- (r w Ce) I --- (4)
where
% = cm cot a --* (5)
Note that this equation does not depend on the dxsipatlve body force and for incompressible flow remains the same with or without this force. Assuming a perfect gas,
Thus starting with bow-n conditions at a given axial statlon (station 1) and for specified values of Cm at the next axial station (statlon 2),
values of To at statlon 2 can be obtained from equations (4), (5) and (6) (applied along each streamline between station (I) and (2)).
From the definltlon of stagnation temperature
T = To - - 2c
P so that
1 y-1
I- - t
ce* + c,' 4 a = a 0
a,’ ,I s-s (7)
2
and the Mach numbers Mm and s can be evaluated. In order to obtain
the density at station (2), first the value of the entropy function Qp 1s obtained from the specified losses. It is assumed that the following total pressure loss coeffxient IS available:
ij = 'OIR - '02R
'01X - '1
where OIR and 02R denote respectively the inlet and outlet stagnation conditions relatxve to the blade row. Ideally the loss data should account for losses due to skin frxtlon on blade surfaces, tip clearance, secOndSy
flow/
-8-
flow and shocks. If the radrral shift of the streamlines is small, then G can be estimated from two-dimensional cascade data.
If the effect of the radial movement of streamlines is Ignored, then TO,,, = TOZB and the change of entropy along a streamline is given by
'OIR - '1 =
'OIR
The change in the entropy funotlon (Q) can be found from the definition
QP Sp - Si = -Cp log - t ) Qi
Hence
QP 'OIR - '1 -= Ql 'OIR )I
and in terms of absolute and relative total temperatures and absolute Mach number at inlet to the blade row, we have
as -= I -0 Qi
[ (
y-1 - I Y
'i~,i
'-- (9) Y -1
l+- MiP 2
and the density can be found from
Numerical solution
The flow through a turbomachine of given geometry can be analysed by using equations (I) and (3) and the auxiliary equations (2) and (4) to (10). Equation (1) must be solved by the method of successive approrA.mations since it is necessary to assume a set of streamlines that satisfy both equations (1) and (3). These two equations form a system of non-linear simultaneous bfferentlal-mtegral equations and can not be solved explxitly. The numerical solution outlined by Novak' is used here. Some details of the present solution will now be given sx~ce the actual numerical technique used
for/
-V-
for integration of equation (1) has to be decided and the method used for streamlines can differ. calculating slope and curvature of the meridional
Also the conditions for the existence and unzqueness oi' the numerical solu- tion have to be investigated.
Equation (1) (mcluding Horlock's correctzon term) is of the form
8C m= 4
- GbJ,)
ar L cm *a’ (11)
where
H(r,C,) = 2 sinDp sin$ ac cos $
--A!+-+ cm am I- m
and a
+& cot’ 3 cot 3
- cot’3 + + 20 - ar r 5u 1
7 G(r,C,) = 2 sic* ,t?
w*ra 1 aQ CTaQ ---+ 2 -sin$
2 Q ar Q am J
Equation (11) IS solved with an lnltlal condition /
Cmbo) = cm 0
The conditionskfor the existence and uniqueness of solutions of equation (II) are well known . Briefly if f 1s continuous m some regxon of (r,C,) plane, containing the point (ro,Cm ) then equation (11) has at least
0 af one solution. If moreover the partial derlvatlve - exxts and is
ac m continuous III that region then the equation has precisely one solution.
The behaviour of f depends on the manner in whch the numerwal so1ut1on 1s set up. The ?xsent solution was based on the "off design alternative" i.e., the equation 1s solved with the following input data.
(i JZnthslpy distribution at inlet to the blade row. (= Losses for each streardllne (specified ii)).
(ill) Flud leaving angles relative to the blade row for each streamline.
Losses and leaving angles are derived from cascade data and the enthalpy dis- tributlon 1s specified at the first axial station and subsequently found from equation (4).
The/
- 10 -
The complete method of solution is s unnarieed below.
(I) Some initial guess is made of the streamline pattern in the meridional pMe such that a certain percentage of total mass flow will pass through each stream tube. This enables the slope and curvature to be com- puted at each meridional station.
Initially the flow is assumed to be one-dimensional and the annulus height is divided into a suitable number of annuli of equal areas (to pass equal mass flows).
The most critical part of the streamline-curvature calculation is the estimation of the curvature of the streamlines. A study was made of the curve-fitting methods used for this purpose. In Ref. 5 a reliable technique was proposed and compared with other methods. In brief the proposed technique is based on the spline curve-fitting method and is referred to as the "double spline fit*. Instead of fitting one spline to the data points and Wfer- entiating it once for slope and once more for curvature, a second spline is fitted to the slopes obtained from the first spline and this is differsntiated for curvatures. This "double spline fit" was found to give satisfactory results when the ratio of the wave length to the point spacing a? the primary harmonics constituting the curve were greater than 5. However when the calou- lation stations are between blades this ratio is smaller than 5 and even the "double spline fit" may not be suffxciently accurate. willrinson6 shows that finite difference techniques using p01yn0mi.d.s to find the ourvature give better results for the curvature than the wave length to point spacing is large (>
s line fits, provided the ratio of 10 P but these methods are also in-
accurate for the present computations.
(2) Some Initial guess is made of the value of the meridional velocity C m at radius r. , for the axial station at which the calculation is being
0 done.
(3) From this assumptxon of Cm the local speed of sound is computed (equations (5) and (7)).
(4) The term 2 3 , needed to find cn am
G(r,C,) is found from
either equation (28) or (2b).
008 $4 cosec' j (5) m.5 h-m - ,
52
2
2 al cot /Y needed to find H(r,Cm) are obtained from the specified gas
c m leaving an&es 6, slope and curvature of streamlines, and the estimated Cm.
- 11 -
aa aIQ w (6) the quantzties - , - , - are obtained from results of
am ar ar the previous iteration. (Initially these are set equal to zero). All the derivatives in the present programme are obtained by fitting spllnes to the functions and dzfferentxting the splines.
(7) Steps (1) to (6) enable the wantlties to be evaluated.
G(rA,) and HbJ,)
(8) The differential equatxon (11) is integrated by the Runga-Kutta method and the value of Cm at the next radxl station is found. ThiS
procedure would fail If the conbtiono for the existence and unxqueness are violated for any radial step. rows) l.f cm f 0 and Mmfi
For a duct region (stations between blade then the functions f and
af - are both continuous provided the specified angles 3 and loss dzs- ac m
trlbution Q do not give rxe to singi;laritles. conditions become and M noting that
cm # 0 Elf" In a bladed region the
These results are obtained by
(i) If Cm = 0 then f(r,C,) is not continuous because GbJ,) 1s divided by Cm. Also the errors XI the numerIca integration procedure become large if Cm IS small.
(Ii) lf c m 1s finite then only the behaviour of HhC,) (equation (11)) needs to be investigated since G(Gm) 33 normally
continuous. The only term 111 G(r,C,) which can most likely become singu-
sin$ ac 1arLS - 2 and the singularity oxurs when the local meridional
Cm am Mach numbers becorws unity.
Therefore provided numerical errors are small, for finite meridional velocities (Cm), and Mach numbers f?, a unique solution exists for Cm
at each radial locatIon. However for trszxonic flows the cowktions for existence and unlquencss are vlolated and the numerxal solution outlined here may fail.
(9) If a unique meri&onal velocity profile (Cm versus r) ezsts
then the density prOfile can be obtained by using equstlons (4) to (10).
(10) The mass flow rate across the an~ui~us is round from equation (3) taking into account the effect of annulus wall bomdary layers 111 the form of a blockage factor wkach may vary through the nachme. If the computed mass flow rate does not agree wzth the specrfxd mass flow rate rmthul a suitable
accuracy/
- 12 -
accuracy range then C m is adJUSted and the steps (I) to (10) are 0
repeated. If the process by means of wLch C m 1s adJUSted is convergent 0
then a value of Cm can be found to give a mass flow rate which agrees 0
with the specified value witbin any accuracy range.
The reqmed value of C m should satisfy the following 0
equation
F(C, ) = ;nt - ;n (Cm ) = 0 0 0
. where m t is the true mass flow rate and
A(cm ) = i
Y‘t 27tr cm p cos $ dr
0
rh
If FCC, ) is well behaved, l.e., It changes monotonically as shown in 0
Fig. I, then the method of False Position' can be used to find the next approxinttion to C In'
0
F (Cm01
// 0 a C,'$ b
C m0
required root
In this method, the next approxlmatlon to the root of equation (12) is given by
afb) - bf'(a) 0 =
f'(b) - f(a)
where o is the point where the chord AB intersects the axis. The procedure can be repeated usmg values at B and C to obtain c' and so on.
- 13 -
Now provided the conditions for the ex~stenoe and uniqueness of
ac solution for C m are satisfied the vslue of -! at sny radial position
ar is unique and the family of meridional velocity profiles (C m 5 r) obtained
(at a given sxial station) for different values of Cm cannot intersect. 0
Considering this fact Marsh7 has shown that the mass flow rate, obtained from the basic differentd-integrsl equations (1) and (3) increases monotonically with cm. This w3J.l also be true for the finite difference equations used
0
for the numerical solution provided numerxal errors are small so that the behaviour of the two systems is the same. The numerxal errors will be smsll if c everywhere along the radius remains sufficiently large, for the Runga- &ttamprocedure to remain accurate. A difficulty arises when for a value of C
mO’
the meridional velocity at a radius becomes nearly eero. The programme
is arranged to discontinue the calculation of the mendionsl velocity profile with this Cm and calculate a new profile with Cm increased by a small
0 0
increment. This procedure overcomes the difficulty provided the final results do not include reverse flows.
(11) When the overall continuity is satisfied a new streamline pattern is derived as follows:
The local mass flow function zmi? p Cm cos # is plotted against
r and the area under the curve is divided into radial increments such that each stream tube will contain the percentage of mass flow chosen in 1. To do this a spline is fitted to the points of 2nrI? p Cm cos # versus r (Fig. 2a) -
mass-flow function 2nt Kp Cm cos 4
rh radius
(al
‘t
FIG.2
I 2nrKPC,cos +dr
/
‘h radius 7 6r
Pt
lb)
- IL -
The spline is then integrated and the values of the integral are plotted versus radius as shown in Fig. (2b). The ordinate of the curve at rh is then divided into the number of stream tubes specafied c.n the first approximation. The new positions of the streamlines are then obtained by a suitable method of inverse interpolation.
Experience has shown that the computation process outlined above overcorrects the initlal errors and the change in the streamline positions,
aIQ and the parameters whxh depend on the streamline positions (such as - ,
ar w - etc.), must be damped by a factor less than 1-O to obtain convergence. am Recommended damping factors are based almost entirely on experience and may range from O-5 to O-025. Usually for any new case the damping factor is found by trial and error so that a number of calculations with different values must be done. Even when convergence has been obtained there is the possibility that a slightly different damping factor would have given the answer in fewer interactions. For a simple twc-dimensional flow Wilkinson6 derives explicit expressions for the optimum damping factor as a function of grid aspect ratio, Mach number and the differentiation method used for fintig slope and curvature of the meridional streamlmes. A similar analysis to give the damping factor for the present problem (flow in axial compressors) would be very complex and is therefore not attempted. In the present com- puter programme the relaxation is applied to
(i) Streamline shrft
where 6r = streamline shift
r n-i = previous streamline position
r = n new streamline positlon (unweighted)
r = new streamline posltion (weighted) gz = relaxation factor
aIQ aQ aQ (ii) quantities -,- and - whxh depend on the streamline
ar ar am
location. For example:
a IQ - = (1 -pa) ar
Again suffuses (n-l) and n denotc any two consecutave approximations. tithe presentprograme P'I = 0.2 and ps = 015.
Results/
- 15 -
Results and Discussion
The objects of the investigation were:-
(4 effect of lnclusicn of losses on the streamline curvature calculation
(b) limits of applicability of the through-flow calculations (Matru through-flow - streamline curvature) for axial ccillpresscrs
(0) effect Of Borlcck's correction term on the accuracy of streamline curvature cslculaticns
Test data for a Rolls Royce axial flow compressor was available and this machine was chosen for the present investlgaticn. Axial velocities had been measured for this machine and compared with the actuator disc theory (Ref. 8): a comparison between the latter and the streamline curvature was therefore also possible.
Data was provided for several possible compressor builds but the sXl'aIlgement chosen for the present calculations consisted of three rcws cf blades; a set of stationary inlet guide vanes, a rotating row and a stationary rcw. The hub-tip ratlo is constant (O-4) through the compressor and the ratio of blade height to blade axial spacing is 2.1. The tip diameter is I4 in. and the txp speed is of the order of 370 ft/sec or 283 ft/sec depending on the gear ratm of the drive gear box.
In the through-flow calculations it is usual to relate the losses and angles tcr -;he last (n-l) iteration using a plot of loss and angle against i, the blade row IrlcIdence, entering the curves at in-, = p,-, - a, where B is the entry gas angle and a the blade angle. A relaxation factor may be applied to the losses as above.
However in the present calculation the main emphasis was on the stability of the flow calculation md on the effect of the dissipative force. It was therefore decided to accept the angle and loss distributions (except8 loss distributicrs II) determined from Horlock's actuator disc calculations , and not to change these during the successive iterations described here. Essentially therefore we have investigated the flow through three rows of blading which prcvide prescribed unchanging distributmns of loss and cutlet angle.
Fig. (3) Illustrates the position of the calculating planes, the gas leaving angles and the loss distributmns used in the calculations. The program was run for three flow coefficients (cxl& = O-67, O-585, 0.417)
and the results willbe discussed in turn.
(iI/
- 16 -
CX.
(i) 1 = 0.67 (Figs. 4, 5) u m
This flow coefficient is well. within the compressor (away from the stdl region).
operat Figs. (4) and (5 compare axial "k:
range of the
velocity profiles downstream of rotor and stator, from experiment, actuator disc theory and streamline curvature cslculation with and without losses. The agreement between various calculations and experiment is reasonable
(note the large scale for $ 1. i
The losses are small for this flow coefficient and little difference is made in the velocity profiles calculated from the streamline-curvature method with and without losses. Two loss distributions have been tried: the distribution with higher losses near the tip of the rotor (Fig. 3) produces a dip in the axial velocity profile (Fig. 4), as might be expected. Since the velocity distribution must satisfy the continuity equation, the sxial velocity is increased near the hub (the flow can be considered incom- pressible, maximum Mach No. < 0.15). The axial velocity profile downstream of the stator is dependent on the axial velocity profile downstream of the rotor and the stator loss distribution. The comparatively hi&er losses near the root of the stator (Fig. 3) push the flow towards the tip of the stator: thus the dip in the axial velocity profile downstream of the rotor is not apparent downstream of the stator.
For the streamline curvature calculation no difficulties were experienced with convergence, ne igible shift being noted in the streamline patterns in less than 20 cycles t? each cycle using a new streamline pattern), with or without losses. Also the effect of Horlock's correction term was negligible (see Appendix I).
(ii) 2 = o-585 (Figs. 6, 7) m
Again comparatively higher losses near the tip of the rotor (Fig. 3) decrease the axial velocity there: axial velocity near the hub is then increased to satisfy the continuity equation. The rotor losses are small and therefore cause little modification in the axisl velocity profile down- stream of the rotor. The effect of comparatively high losses near the tip of the stator (Fig. 3) can be seen on Fig. 7 where the axlsl velocities near the tip are appreciably reduced and mass flow near the hub is increased. Again no diffxulties were experienced with the convergence of the streamline curvature calculation and the effect of' Horlcck's correction term was negligible.
(iii)/
- 17 -
CX. (iii) 1 P 04l3 (Figs. 9-17)
U m
This operating point was within the stall region of the compressor characteristics. It was chosen to study the accuracy of the streamline curvature calculation when the meridional through-flow velocities may become small for a given streamline pattern. The trunoatlon error in the Bunga Kutta procedure, used to integrate equation (II), is a function of f,
ar af --ma-. The actual expression is too complicated to be of much ar acm practical use, but for small Cm, these values may become large and hence the error may become large. Referring to Fag. 8, it can be seen that the numerical solution cannot be used for calculating Cm below the value C where the numerical errors become unacceptable. %
Cm
/
‘h
t- \
(al
C YrnE
0
FIG.8
approximate
F - ‘t
TO Overcome ths difflcdty the programme can be motifled in either of two ways. If for the gzven streamline pattern the velocity profile to give the correct mass flow rate 1s above line AB, then the programme can be arranged to discontinue the computation if Cm falls below AB and mm-ease C untd this condition is satisfied. m However, if this
0
procedure gives too large a vslue for mass flow rate then the velocity profile must oross AB. In this ease values of Cm below AE? are
replaced by a small positive value so that the resulting merlbonal velocity profile, to satisfy the nass flow, would appear as II~ Fig. (8b). This is of course inaccurate III regions CD and EF where the true velocities may be even negative but the nume~cal method can not be used to predict the true velocities z.n thx region,
Three/
- 18 -
Three calculations were performed for this flow coefficient, one without losses and two with different rotor loss distributions, to study the effect of variation of losses (when the losses are high).
LOSS distribution 1 (Figs. 9-11)
The programme was first run without incorporating the above modifications to mintise truncation errors for small through-flow velocities. This computation was stopped for the 7th streamline pattern because multiple vslues of cm were found to give the specified mass flow rate, downstream
0 of the stator, where some through-flow velocities were small (Figs. II and lla). Therefore, due to growth of numerical errors, the uniqueness conditions for the solution had been violated. Figs. Ila and Ilb illustrate that the com- puted mass flow rate is not monotonically increasing with Cm and numerical
0
errors have considerably modified the behaviour predxted from consideration of the basic differential-mtegral equations (1) and (3).
The programme was modified by stopping the computation of C m by
the Runga-Kutta procedure, when Cm was small -s?_<O*O5 and CX. ) 1
replacing the remaining values of Cm by a small value t
C ~=O.Ol . CX. 1 1
It can be seen from Figs. lla and lib that the behaviour is considerably improved but there are still small oscillations in the graphs. These are attrzbuted to errors in the numerical integration of equations (1) and (3). Apart from the s&&t inaccuracies due to the oscillations, the uniqueness condition is now satufied and the final results (20th streamline pattern) are compared with the results of the actuator disc theory and experimeot reported in Ref. 8. In general the agreenent between theory ,%ld experiment seems reasonable. Since this operatlng point is in the stdll region of the compressor the losses are not accurately known, and quantitative agreement between theory and experiment is not expected. This is true even when the effect of snnulus wall boundary layers are taken Into account.
Finally the effect of Horlock's correction term on the accuracy of the calculation was found to be negligible (the difference between results with and without this term was negligible). Conditions for which this term nay be neglected are examined in Appendix 1.
Loss distribution II (Figs. 12-14)
Again the programme was first run without the necessary modzfica- txon to reduce tnrncatlon errors in the RuEga-Kutta procedure. Multzple Clues of Cm mere found downstream of stator, for the 6th streamline
0
patter,l, giving the specified nass flow rate (Fig. lb). The uniqueness con- ditxon was therefore violated due to the growth of the truncation errors. The modrrfled program (described above) is seen (Figs. ILa, 14b) to give unique values for Cm , and the final results obtained from this program=
0
are/
- 19 -
are given in Figs. 12-14. Loss distribution II gives higher losses near the tip of the rotor as compared with loss distribution I, but gives the same loss distribution downstream of the stator. Comparison of Figs. 10 and 13 shows that the axial velocity near the tip of the rotor is reduced for loss distribution II as compared with the values for loss distribution I. The same trend is obtained downstream of the stator (compare Figs. 11 snd 14).
Finally Figs. IQ and l&c demonstrate that Horlock's correction term does not appreciably affect the accuracy of this computation.
No losses (Figs. 15-17)
Results of the tests for uniqueness are given in Figs. 17a to 17d. Again the modified programrae gives a unique solution. However the oscills- tions in the graphs of is versus Cm are not completely eliminated when
0
the programme is modified. These oscillations indicate the existence of appreciable numerical errors in the region where they are large. The curves
are well behaved in the region near the specified mass flow rate so that for the present operating point a unique solution can be obtained with reasonable sccuracy. The final results (20th streamline pattern) can be compared with the results for flow with losses to study the effect of losses on the oslou- lation. Axial velocity profiles downstream of the guide vanes (Figs. 9, 12, 15) are very similar. For both cases the guide vanes were assumed to intro- duce sero losses and the changes are due to the change in the slope and curvature of streamlines. The axial velocity profile downstream of tbs rotor for flow without losses (Fig. 16) is more uniform as compared with flow with losses (Figs. 10, 13) but the differences are not large. However flow iown- stream of the stator is changed considerably due to losses. Theaxial velocity downstream of the stator with no losses (Fig. 17) is small near the hub; the velocity for floe with losses (Figs. 11 and 14) is small near the tip. Losses can therefore modify the velocity profiles to a large extent and hence have large effects on the convergence and uniqueness of the solution.
Conclusxns/
- 20 -
Conolusions
The conbtions for uniqueness of the solutions obtained from the streamline-curvature method of calculating flow through turbomachines may be violated when the meridional through-flow velocity becomes small (e.g. near stall), or when there are regions of choked flow. For a Rolls Royce compressor, flow in the stall region is computed by suitable modifica- tions of the method to minimise the truncation errors and hence to obtain a unxque solution. There is agreement between this solution and the results of the actuator disc theory and experiment reported in Ref. 8.
The effect of the losses on the computation is significant when losses are large, but for the axial machine investigated here the effect of the correction term, arising from the inclusion of a dissipative force in the momentum equation, is negligible.
AcIz-iowledgements
The authors are indebted to Professor J. H. Horlock and Dr. H. Marsh for many helpful discussions.
Aupendix I/
- 21 -
A.PFENllXl
Effect of the dissipative body force on the accuracy of the streamline curvature calculation
The dissipative body force gives rise to the ter!n
2?? 2 = -
2C T ilQ - sins p sin 9
P Q am
on the right hand sde of equation (1). Ifthistem with the other terns on the right hand side
is negligible compared
i.e. 2 sin*j5 , au Q2P - - + - (i Z)) ) then the body force will 0 ar 2
not effect the accuracy of the calculation. The ratio of the terms is
cp w - sin$o Q am
6 = aIQ 4rp ag -+- - ar 2 c ) ar
aQ For the present calculations the term - is only finite downstream of the
am rotor and downstream of the stator; the magnitude of 6 can be readily estimated at either station by noting that for the Rolls Royce compressor
CPT = ho >> u?rp
and Q= I
Therefore, downstream of the stator
aQ - sin $2
6 e am aQ 1 ah -+- 0 ar ho ar
The/
- 22 -
The derzvatlves can be obtained from Figs. Ai, A.2 and A3 and values of 6, calculated from these denvatxves, are given in Table 1.
Table I Table I
It can be seen that 6 is negligible. However if there are regions where the denominator can become small and if $ can become large (e.g. in centrifugal machines) then the effect of the dissipative body force can become appreciable.
References/
- 23 -
References
E?s
1
2
3
6
7
a
Author(s~
R. A. Novak
H. Marsh
J. H. Horlock
3. xreysz1g
M. R. A. Shaalan and 2. ilaneshyar
II. H. Rllkinson
H. Marsh
J. H. Horlock
Title, etc.
Streamline curvature computing procedures for fluid-flaw problems. Trans. A.S.M.E. Journal of basx engmeering for pcwer, A. 89, 473490. 1967.
A digital computer programme for the through-flow fluid mechsnxs in an arbitrary turbomachine, using a matrzx method. ARC R.& M. No.3509. 1966.
On entropy production in adiabatic flow in turbomachines. &S.K.E., 7!-FE-J. 1971.
Advanced engineering mathematics. 41ely. 1966.
)Eethods of calculatnxg slope and curvature of streamlines III fluid flow Problems. Cambridge 'Jnlversity khgineerlng Laboratories. Report UC. TR2. 1969.
Stability, convergence, and accuracy of lxo-dimens~onal stresmlme curvature methods using quasi-orthogonals. I. Mech. E. Thermo. and Fluid Mech. Convention, Paper 35. 1570.
Frlvste communlcatwns.
3xperimental and theoretical investigation of the flow of air through two single-stage conpressors. L-C R.& I\!. ?Io.3031. 1955.
BMG
I2 3 45678 9 IO II 12 13 14 I5 cs- &
Q J I G ;R S I I I
-10 -5 0 5 x IInches) 15 20 25
0.6
llpo h9W:
02
9
30
20
IO VI ; 0
ET -u
20
0
Computing statrons -I 06 /---
I - / L
‘lz,c,z
04
0.2
9 I 06 AadiusltVradius
1 4 IO I 06 .
Radius/tit kdius IO
Slator
Total Dressure loss-coefflclents
I I
4 06 00 I Radlusltipradlus Rotor (relative1
I
6Ot
0 L I 04 04 08 IO
Radius/tip radius Stator
FIG.3 Downstream gas angles
I*2 I o No losses
o Loss distribution I
x Loss distribution II
& Actuator disc
v Experiment
K~UIUS I tip raaius
FIG.4 Axial velocity profiles downstream of the rotor
CXI - ~0.67 Urn
I* 2 -
o No losses
o Loss distribution I
x Loss distribution II
A Actuator dlcs
1@15- v Experiment
FIG. 5 Axial velocity proflles downstream of the stator
Cxi - = 0.67 Urn
o No losses x With losses
0 *90 0.4
I 0.5
I I I 0.6 0.7 04
Radius / tip radius
I I 09 I.0
&la1 velocity profiles downstream of rotor
cxl_ = 0.585 Urn
1.1 -
l*OS-
I.0 -
X’Z o-95- UY I
0 .- .$ o-90-
L
% r
5 pl O-85-
b
I X
o No losses x With losses
0.70 0.4
I
05 I I I
0.6 0.7 08 Radius / tip radius
I
0.9 I.0
FIG 7 Axial velocity proflles downstream of stator
CXI - = Urn
0.585
I2 Streamline pattern I
I.1
07
06 I I I I I 04 OS 06 0.7 08 09 I 0
Radius/ tip radius
FIG 9 Axial velocity proflles downstream of inlet guide-vanes
cxl =04)3 Loss dlstrlbution 3, Um
l-3 t Actuator disc
"I'= o.g ou
,o
5 0.8
,x G 0
2 0.7
Streamlme pattern I
0.6
0.6 0.7 0.8 Radius / tip radius
FIG.10 Axial velocity profiles downstream of the rotor
Loss distribution I, c$ = 0,413
I.3
I.2
I.1
I.0
o-9
ll.R
0.2/
Actuator disc theory
I Radius /tip radius I 0.4 05 O-6 07 O-8 0.9 IO
FIG. II Axlal velocity profiles downstream of stator
Loss distribution I, e =o 413
I
x WIthout correction for small Cm o With correction for small Cm
Cm0 c
I I I I I I I.0 I.1 I*2 I3 I.4 I5 I.6
FIG Ila Values of Cmo giving the correct mass flow rate
downstream of the stator
Tth streamllne pattern , loss distribution I, E =0.413
o With correction for small Cm x Without correctlon for small Cm
Cmo CXI
I I I I I I
I.0 I .05 I.10 I 15 I.20 I 25 I 30
FIG Ilb Values of Cmo giving the correct mass flow rate
downstream of the stator
6th streamlIne pattern, loss dlstributlon I, cxi= “rn 0 413
0 -61 I I I I I 0.5
I 0.6 0.7 0.8 o-9 I.0 Radius / tip radius
FIG.12 Axial velocity proflles downstream of the gutde vanes
Cxi - = 0,413, Urn
loss distribution II
I.2 t
1 I//////// \ \\\
pattern I
0.41 IO
Radius/tip radius
FIG 13 Axial volocitv Droflles downstream of the rotor
Loss distribution II, ‘:i = 0.413
I.3 Streamline pattern I
I.2
hf
2 . 3 /
0.7
t /
0.4 0.4
I 0.5
I I 0.6 07
Radius / tip radius
FIG.14 Axial velocity proflles downstream of the stator
Loss distribution II, 2 = 0413
I
4 /
/5
9-
7-
6- lY
/
s- I.1
Correct mass flow
fate mt x
I X
I X
r”
/
/’
d x With Horlock’s correction
term A Without Horlock’s
correction term o With correction for small
velocities
I I I 4”p XI
I.30 1.55 I.40 I.45
FIG.14a Values of Cmo giving the correct mass flow rate
downstream of the stator
6th streamline pattern, loss distrrbution II Cxi
> - - 0.413 Urn
x Without correction for small Cm
o With correctlon for small Cm
Cm Cxi
41 I I I I I I I I.0 1.1 I.2 I.3
FIG. l4b Values of Cmo giving the correct mass flow rate
downstream of the stator
7 thstreamline pattern , loss distribution II ,F = 0 413 m
1*3- J
I ,2- /
Xii v L
0*9-
X With Horlock’s correction term
A Without Horlock’s correctlon term
:
0.4 0.4
I 05
I I I 06 0.7 04 Radius / tip radius
FIG 14~ Effect of Horlock’s correctlon term
downstream of the stator
5th streamllne pattern loss dlstrlbutlon II, CXI urn = 9 0.413
0.9
0.8
0.7
StreamlIne pattern I
1 ,o- Streamline pattern 20
0.61 0.4 0.5 O-6 o-7 0.8 09 I.0
Radius / tip radius
FIG.15 Axial velocity profiles downstream of the inlet auide vanes
= 0.413 No losses
0.8
pattern I
o-4’ I I I I , 0.4 0.5 0.6 0.7 08 0.9 I.0
Radius /tip radius
FIG 16 Axial velocity profiles downstream of the rotor
No losses ‘2 = 0 413
Streamhne pattern I A.\ I
I I I
06 0.7 04 Radius / tip radius
I J 0.9 I.0
FIG I7 Axial velocity profiles downstream of the stator
No losses, e = 0.413
1 Correct mass flow rate
6 I.0 I.1 I.2
o With correction for small Cm x Without correction for small Cm
Cm Cxi
I.3
FIGSl7a817b Values of Cmo qlvinq the correct mass flow downstream of the stator and at station II J
7th streamline pattern, no losses, e zo.413
X
Correct mass flow /
rate, x
X
Cm0 Cxr
I I I I I I
0 I.1 I.2 13
o With correction for small Cm
x Without correctlon for small Cm
FIG. l7c Values of Cmo giving the correct mass flow mte
downstream of the stator
Bth streamlne pattern, CXI no losses, - = Urn
0.413
Cm0 - CXI I I I I I 4
IO I 05 I .I0 I.15 I 20 I 25 I.30
o With correctlon for small Cm x Wlthout correctlon for small Cm
FIG. 17d Values of Cmo giving the correct mass flow rate
downstream of the stator
gth streamline pattern, no losses, Um - Ql - 0 413
235 0 I I I I 0.4 0.5 0.6 --ih--- . Radius /
!I; . 0.8 radius
FIG Al Dlstrlbution of stagnation enthalpy downstream of the stator
!Tth streamllne pattern, loss distribution I, 2 = 0.413
I ,
Entropy function Q-e -=/cp
> 0 0 0 0 0 0 0
: z 2 z z s 2 z , w A VI OI -4 Q, a
I I I I I I I
0.999
O-998
0.994
0.993
Downstream of rota!
0.992 I
0.4 05 I I I
0~6 07 0*8 Radius / tip radius
I
0.9
FIGSA Distribution of the entropy function
Downstream of
0
Sth streamline pattern, loss distribution I, z =O. 4 13
I .
- - - - - . - - - - I --__ - ______ I -____ -__ _ _ - - - - - - - - - - - - - - . - - ---.----_--____.--_____________
_ -__ . . . - . . - - - - - - - -
n~t.c.32 727 - Turbo.146 harch, 1971 Daneshyar, H. and Shaalan, 11. R. A.
THX OFF-DXSIGN ANALYSIS OF FLW IN AXIAL COh~SSORS
The existence and uniqueness of solutions obtained from the stream line-curvature method of calculating flow through turbomachxxs are examxxd. The conditions for existence and uniqueness may be violated near stall; flow in ths region is computed by sultable notiflcatlons to obtain a urnque solution. Also the effect of a correction term due to a tissslpatlve body force in the momentum equation is emmmed .
A.R.C.32 727 - Turbo.146 March, 1971 Daneshyar, H. and Shaalan, M. R. A.
THE OF!?-DESIGN ANALYSIS OF FLOW IN AXIAL coMPFBssoRS
The existence and uniqueness of solutions obtained from the stream line-curvature method of calculating flow through turbomachrrnes are examined. The conditions for exLstence and. uniqueness may be violated near stall; flow in this region is computed by suitable modifications to obtain a unique solution. Also the effect of a correctIon term due to a dissipative body force in the momentum equation is examined.
A.R.C.32 727 - Turbo.146 March, 1971 Daneshyar, H. and Shaalan, Al. R. A.
THZ OFl?-DESIGN ANALYSIS OF FLO-J IN AXIAL COMFRESSORS
The existence and uniqueness of solutions obtalned from the stream line-curvature method of calculating flow through turbomachlnes are examlned. The conditions for existence and umqueness may be vlolated near stall; flow In this region is computed by suitable motifxations to obtain a unique solution. Also the effect of a correctIon term due to a dissipative body force in the momentum equation is examined.
A.R.C. CP No. 1234 &R.C. CP No. 1234 March, 1971 March, 1971 Daneshyar, H. and Shaalan, M. R. A. Daneshyar, H. and Shaalan, M. R. A.
THE OFF-DESIGN ANALYSIS OF FLOW IN AXIAL COMPRESSORS
THE OFF-DESIGN ANALYSIS OF FLOW IN AXIAL coM.PPzSsoRS
The existence and wuqueness of solutions obtained from the stream line-curvature method of calculatzng flow through turbomachlnes are examined. The condltlons for existence end uniqueness may be violated near stall; flow in this region is computed by suItable modlficatlons to obtain a unique solution. Also the effect of a correctron term due to a disslpatlve body force in the momentum equation 1.s exannned.
The existence and uniqueness of solutions obtained from the stream line-curvature method of calculating flow through turbomachuves are exanned. The condltlons for existence and uniqueness may be vIolated near stall; flow in this region IS computed by sutable modlflcations to obtain a unique solution. Also the effect of a correctwn term due to a disslpatrve body force in the momentum equation 1s examined.
A.R.C. CPNo. 1234 March, 1971 Daneshyar, H. and Shaalan, M. R. A.
THE OFF-DESIGN ANALYSIS OF FLOW IN AXIAL coMPRESsoRS
The existence and uniqueness of solutions obtalned from the steam line-curvature method of calculating flow through turbomachlnes are exanlned. The condltlons for existence and uuqueness may be violated near stall; flow in this region 1s computed by suitable modifications to obtain a unique solution. Also the effect of a correction term due to a dlsslpatlve body force in the momentum equation 1s examined.
A.R.C. CP No. 1234 March, 1971 Daneshyar, H. end Shaalan, M. R. A.
THE OFF-DESIGN ANALYSIS OF FLOW IN AXIAL coMPPJ3soRS
The existence and unlqueness of solutions obtained from the stream line-curvature method of calculating flow through turbomachines are examined. The condltlons for existence and uniqueness may be violated near stall; flow in this region 16 computed by sutable modlflcatlons to obtain a unique solution. Also the effect of a correction term due to a dlssipatlve body force in the momentum equation 1s examined.
A.R.C. C.P. No.1254
Detachable Abstract Cards to replace the incorrect ones originally issued.
A.R.C. CP No. 1234 4.R.C. CP No. 1234 March, 1971 March, 1971 Daneshyar, H. and Shaalan, M. R. A. Daneshyar, H. and Shaalan, M. R. A.
1 THE OFF-DESIGN ANALYSIS OF FLOW IN THE OFF-DESIGN ANALYSIS OF FLOW IN
AXIAL COMPRESSORS AXIAL COMPRESSORS
The existence and uniqueness of solutions obtaned The existence and unlqueness of solutions obtained from the stream line-curvature method of calculating from the stream line-curvature method of calculating
,flow through turbomachlnes are examined. The flow through turbomachlnes are examined. The condltlons for existence and uniqueness may be vlolated condltlons for existence and uniqueness may be vlolated near stall; flow In this region is computed by near stall; flow in ths region 1s computed by suitable modlficatlons to obtain a unique solution. sultable modlflcatlons to obtan a unique solution. Also the effect of a correction term due to a Also the effect of a correction term due to a dissipative body force In the momentum equation 1s dissipative body force In the momentum equation 1s examined. examined.
A.R.C. CP No. 1234 A.R.C. CP No. 1234 March, 1971 March, 1971 Daneshyar, H. and Shaalan, M. R. A. Daneshyar, H. and Shaalan, M. R. A.
THE OFF-DESIGN ANALYSIS OF FLOW IN THE OFF-DESIGN ANALYSIS OF FLOW IN AXIAL COMPR~SORS AXIAL COMPRFSSORS
The existence and unlqueness of solutions obtarned The existence and uniqueness of solutions obtalned from the steam line-curvature method of calculating from the stream line-curvature method of calculating flow through turbomachlnes are examined. The flow through turbomachlnes are examined. The condrtlons for existence and uniqueness may"be violated condltlons for existence and uniqueness may be vlolated near stall; flow In this region 1s computed by near stall; flow in this region is computed by suitable modlflcatrons to obtain a unique solution. sultable modlflcatlons to obtain a unique solution. Also the effect of a correction term due to a Also the effect of a correction term due to a dlsslpative b?dy force In the momentum equation is dlsslpatlve body force in the momentum equation 1s examined. examined.
C.P. No. 1234
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