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International Journal of Rotating Machinery 2005:1, 45–52c© 2005
Hindawi Publishing Corporation
Analysis of Secondary Flows in Centrifugal Impellers
Klaus BrunMechanical and Fluids Engineering Department,
Southwest Research Institute, 6220 Culebra Road,San Antonio, TX
78238-5166, USAEmail: [email protected]
Rainer KurzSolar Turbines Incorporated, 9330 Sky Park Court, San
Diego, CA 92123-5398, USAEmail: kurz rainer [email protected]
Received 12 October 2003
Secondary flows are undesirable in centrifugal compressors as
they are a direct cause for flow (head) losses, create
nonuniformmeridional flow profiles, potentially induce flow
separation/stall, and contribute to impeller flow slip; that is,
secondary flowsnegatively affect the compressor performance. A
model based on the vorticity equation for a rotating system was
developed todetermine the streamwise vorticity from the normal and
binormal vorticity components (which are known from the
meridionalflow profile). Using the streamwise vorticity results and
the small shear-large disturbance flow method, the onset,
direction, andmagnitude of circulatory secondary flows in a
shrouded centrifugal impeller can be predicted. This model is also
used to estimatehead losses due to secondary flows in a centrifugal
flow impeller. The described method can be employed early in the
designprocess to develop impeller flow shapes that intrinsically
reduce secondary flows rather than using disruptive elements such
assplitter vanes to accomplish this task.
Keywords and phrases: pumps, compressors, secondary flows,
vorticity, circulating flows.
1. INTRODUCTION
Strong circulatory secondary flows (vortex flows) are ob-served
in mixed-flow impellers such as axial/centrifugalpumps, turbines,
and compressors. These vortex flows areundesirable as they are
responsible for head losses, flow non-uniformity, and slip. To
reduce secondary flows and slip, tur-bomachinery designers often
employ flow guiding/disruptiveelements such as splitter vanes and
other hardware modifi-cations (which themselves negatively affect
the efficiency ofthe machine) rather than focusing on the actual
causes andintrinsic physical mechanism that generate vortex
secondaryflows.
Most modern commercially available three dimensional(3D)
computational fluid dynamics codes can predict circu-latory
secondary flows in rotating machinery with a reason-able accuracy.
However, the aim of this work is not to de-velop another
“black-box” flow prediction tool (such as a 3DNavier-Stokes flow
solver), but rather to derive a simplifiedmodel from the governing
equations to study the underlying
This is an open access article distributed under the Creative
CommonsAttribution License, which permits unrestricted use,
distribution, andreproduction in any medium, provided the original
work is properly cited.
physics of the secondary vortex flow phenomena. Resultsfrom this
paper will provide the designer with a more fun-damental
understanding of how circulatory secondary flowsbehave and are
affected by operational and geometric param-eters of the
turbomachine.
To limit the topic somewhat, this paper will focus only
onprediction of vortex secondary flows in shrouded centrifu-gal
compressors using the streamwise vorticity equation. (Inunshrouded
centrifugal compressors the viscous influence ofthe rotating
wall/shroud and tip leakage effects complicatethe secondary flows
such that a simple model based on thevorticity equation is not
directly applicable.) As the influenceof the density gradient terms
within the streamwise vorticityequation will be demonstrated to be
negligible, results fromthis paper are certainly also applicable to
centrifugal pumps.
Thus, a model to determine the rotational direction andmagnitude
of the passage circulatory secondary flows was de-rived based on
the streamwise vorticity equation. The modelapplies known
meridional velocities to the streamwise vor-ticity equations to
determine normal, binormal, and stream-wise vorticity, circulatory
secondary velocities, and associ-ated head losses. The influence of
nondimensional operat-ing parameters (Reynolds number, Rossby
number) on vor-tex secondary flows is also analyzed.
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46 International Journal of Rotating Machinery
2. BACKGROUND AND REVIEW OFRELEVANT LITERATURE
During the past 50 years a number of researchers have stud-ied
the flow fields in mixed-flow turbomachinery employingboth
analytical and experimental methods. For the sake ofbrevity only
analysis relevant to the secondary flow field andjet/wake effects
(as it relates directly to secondary flows) ispresented herein.
Eckardt [1] used the laser-2-focus (L2F) method to mea-sure
velocities in radially bladed and backward swept im-pellers with
diffusers. Suction side separation and wake flowwas observed. More
uniform flow was observed in the back-swept impeller.
Fister et al. [2] used L2F to measure flow in simulatedbends of
multistage radial-flow impellers. Results were com-pared to
predictions from a 3D, turbulent, viscous Navier-Stokes code.
Reasonable agreement was found between ex-perimental and
computational results; however, the Navier-Stokes code failed to
predict existing flow separation re-gions.
Krain [3] used L2F to study the effect of vaned and vane-less
diffusers on impeller flow in a centrifugal compressor.Rapid
boundary layer growth and wake flow was shown atthe midpassage on
the blade suction sides. Similarly, Hayamiet al. [4] employed an
L2F to measure velocities in the in-ducer of a centrifugal
compressor. Also, static pressure mea-surements were taken at the
shroud. Small separation regionswere shown near the inducer inlet,
while the flow along theblades was mostly stable, except near the
shroud.
Hamkins and Flack [5], Flack et al. [6], and Miner etal. [7]
used a two-directional Doppler laser velocimeter tomeasure the flow
of a shrouded and unshrouded centrifu-gal impeller with logarithmic
spiral volutes. Measurementswere taken in the impeller and the
volute. Nonuniform asym-metric flow was shown at impeller
off-design conditions, butjet/wake flow was not observed.
Fagan and Fleeter [8] measured the flows in a centrifu-gal
compressor impeller using a laser velocimeter and a shaftencoder.
Significant flow changes as compressor stall ap-proached were
identified. Hathaway et al. [9] measured ve-locities in a large
low-speed centrifugal fan using a laservelocimeter. Results were
favorably compared to five-holeprobe and other data. McFarland and
Tiederman [10] used atwo-directional laser velocimeter to measure
flows in an ax-ial turbine stator cascade. Unsteady flow due to the
turbineupstream wakes were seen to affect the flow field
throughoutthe stator.
Strong secondary flows were observed in the mixed-flowpump of a
torque converter by Gruver et al. [11] and Brunet al. [12].
Comparable secondary flows in other turboma-chinery geometries were
studied by a number of researchers.For example, Moore [13, 14] used
a hot wire probe to de-termine the secondary flow field in a
rotating radial-flowpassage and to compare the results to
predictions from apotential flow code. Hawthorne [15], Kelleher et
al. [16],and Sanz and Flack [17] studied secondary flows in
station-ary circular and rectangular bends. Ellis [18]
experimentally
and analytically studied the induced vorticity in a
centrifugalcompressor. Krain [19], Moore and Moore [20],
Eckardt[1], Howard and Lennemann [21], and Brun et al. [12]
ex-perimentally determined the secondary flows in
mixed-flowcentrifugal impellers. Finally, Johnson and Moore [22]
de-termined secondary flow mixing losses in a centrifugal im-peller
from pressure probes installed in the rotating im-peller.
A number of analytical models have been derived for
thedevelopment of streamwise vorticity and, thus, vortex sec-ondary
flows in turbomachines. Wu et al. [23], Smith [24],Hill [25],
Horlock and Lakshminarayana [26], and Laksh-minarayana and Horlock
[27] derived equations for the de-velopment of streamwise vorticity
in stationary and rotat-ing systems. Analytical solutions of these
equations were pre-sented by Hawthorne [15] for stationary bends
and rotatingradial-flow passages. Later, Johnson [28] solved these
equa-tions to predict secondary flows in a rotating bend.
3. VORTEX FLOW THEORY
Secondary flows are always caused by an imbalance betweena
static pressure field and the kinetic energy in the flow.An example
is the well-documented horseshoe vortex, wherethe incoming boundary
layer flow meets a stagnation linewhich causes a motion of the
fluid along the wall, and sub-sequently the formation of a vortex.
The important obser-vation herein is that the strength of the
vortex is mostly de-termined by the starting conditions and the
further develop-ment of the vortex is determined by the
conservation of itsangular momentum. In a rotating system the
analogy is thatthe vortex flows are principally generated by the
meridionalflow field while the centrifugal and Coriolis forces only
actto change the vortex vector direction (tilting of the
vortexplane).
The meridional flow in centrifugal/mixed flow pumpsand
compressors is usually highly nonuniform, dominatedby significant
jet/wake flow with separation regions block-ing up to half of the
passage through-flow areas. At highReynolds numbers, the peak
meridional velocities are lo-cated at the blade hub-pressure sides
due to potential floweffects; at low Reynolds numbers, the viscous
jet/wake flowcauses the flow to separate at the hub and, thus, the
peakvelocities are seen at the blade tip-pressure side.
Thesethrough-flow profiles can be accurately predicted using
sim-ple jet/wake flow models (for low Reynolds number, low
spe-cific speed impellers), empirical models (based on the
sig-nificant amount of experimental data available in the pub-lic
domain), or even Euler flow solvers (for high Reynoldsnumber
impellers). Once the meridional flow profile is de-termined, the
normal and binormal vorticity can be numer-ically evaluated and the
results are directly applied to therotating system vorticity
equations to calculate the stream-wise vorticity. Using simple
potential flow solvers and theviscous dissipation function, the
passage circulatory (vortex)secondary flow and associated head
losses can be estimated,respectively.
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Analysis of Secondary Flows in Centrifugal Impellers 47
4. SECONDARY FLOW MODEL
Streamwise vorticity and, hence, rotating secondary flowswill
develop whenever a moving fluid with a gradient of thereduced
stagnation pressure (Prs = P + 1/2ρ(V 2 − ω2R2))turns around a bend
or is rotated about a fixed axis (John-son [28]). A gradient in the
reduced stagnation pressure, Prs,might result from a nonuniform
velocity profile or a reduc-tion of Prs due to boundary layer
viscous dissipation. In cen-trifugal impellers the meridional flow
field is highly nonuni-form because of the jet/wake flow phenomena,
the nonuni-form flow field is both turned around a bend with a
radiusof curvature r, and is rotated around the shaft at an
angu-lar speed of ω. Hence, high values of streamwise vorticityand
strong associated circulatory secondary flows are antici-pated.
Equations for the generation of streamwise vorticity, and,thus,
circulatory secondary flows, in an intrinsic rotating co-ordinate
system were first derived by Hawthorne [15], Smith[29], Smith [24],
and Ellis [18]. These equations were thengeneralized to include
viscous terms and compressibility ef-fects by Howard [30] and
Lakshminarayana and Horlock[27]. The equations as derived by
Lakshminarayana and Hor-lock were employed for the analysis
herein.
Since a detailed derivation of these equations is availablein
the literature, only the main steps for the inviscid,
incom-pressible streamwise vorticity generation equations are
de-scribed here. The steady-state, incompressible
Navier-Stokesequations for a rotating system are given in vector
form by
(V · ∇)V = −∇(P
ρ
)+ υ∇2V + 2ω ×V + ω × (ω × r),
(1)
where V is the velocity vector, P is the total pressure, υ isthe
kinematic viscosity, and ω is the angular velocity of thesystem.
One should note that the fourth term in (1) rep-resents the
Coriolis force and the fifth term represents thecentrifugal force.
Also, note that the equations are being de-rived with the
assumption of incompressible flow; the subjectanalysis found that
for the generation of streamwise vorticityin centrifugal
compressors, the fluid compressibility effectswere found to be
negligible (see more detailed explanationbelow).
Introducing the vector identity,
(V · ∇)V = 12∇(V ·V)−V × (∇×V), (2)
defining the vorticity vector, ξ, as
ξ = ∇×V , (3)
and taking the curl of (1) with the knowledge that the curl ofa
gradient of a scalar always equals zero (∇×∇Φ = 0), oneobtains
∇× (V × ξ) = υ∇2ξ + 2∇× (ω ×V). (4)
Exits
bn
Tip
Pressure Suction
Hub
ω
nb s Inlet
s-Streamwisen-Normalb-Binormal
Figure 1: Intrinsic coordinate system in centrifugal
impeller.
Furthermore, introducing the vector identities,
∇× (V × ξ)=V(∇ · ξ)−ξ(∇ ·V)−(V · ∇)ξ+(ξ · ∇)V ,∇ · ξ = ∇ ·∇×V =
0,
(5)
and knowing that for an incompressible flow from the conti-nuity
equation,
∇ ·V = 0, (6)one obtains
(V · ∇)ξ = (ξ · ∇)V − υ∇2ξ − 2∇× (ω ×V), (7)
which is the vorticity transport equation for an incompress-ible
flow in a rotating system (Greenspan [31]).
Unit vectors are now defined along the streamline,
s(streamwise), in the inward radius of curvature direction,
n(normal), and along the b (binormal) direction, so that s, n,and b
form a right-handed set of unit vectors. In a centrifu-gal
compressor these directions approximately correspond
tos-through-flow direction, n-tip-to-hub side direction, and
b-suction-to-pressure side direction (see Figure 1).
Taking the dot product of (7) (and neglecting the vis-cous term)
with the streamwise unit normal vector, s, andusing the dot product
relations given by Bjørgum [32] for anintrinsic coordinate system,
one obtains (Lakshminarayanaand Horlock [27])
|V |∂ξs∂s− |V |ξn
r= ξs ∂|V |
∂s+ |V |ξn
r+ 2ω · ∇V , (8)
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48 International Journal of Rotating Machinery
where r is the radius of curvature and s is along the
stream-wise direction. This expression can be simplified to
obtainthe fundamental generation of streamwise vorticity
equationfor a rotating system (Lakshminarayana and Horlock
[27]),
∂
∂s
(ξs|V |
)= 2 ξn|V |r − 2
s · (ω× ξ)|V |2 , (9)
where ξ is the total vorticity vector, ξn is the normal
vorticitycomponent, ξs is the streamwise vorticity component, and
ris the radius of curvature. The first term in (9) is a
streamlinecurvature term and the second is a Coriolis force term.
Foran impeller with a fixed axis of rotation (such as a
centrifugalcompressor) this equation can be further reduced to
∂
∂s
(ξs|V |
)= 2 ξn|V |r − 2
ωξb sin κ|V |2 , (10)
where κ is the meridional flow angle relative to the axis
ofrotation and ξb is the binormal component of vorticity.
The normal and binormal components of vorticity (ξn,ξb) in (10)
can be directly determined from
ξn = − 1A
∫∫(∇×U) · dA = − 1
A
∫∫ (∂v
∂z− ∂w∂y
)dA, (11)
ξb = − 1A
∫∫(∇×V) · dA = − 1
A
∫∫ (∂u
∂z− ∂w∂x
)dA, (12)
where A is the through-flow area. For the purpose of
thisanalysis the term ∂v/∂z in (11) and the term ∂u/∂z in (12)can
be neglected as they are small compared to the merid-ional
gradients (∂w/∂y and ∂w/∂x). Thus,
ξn = 1A
∫∫∂w
∂ydx dy,
ξb = 1A
∫∫∂w
∂xdx dy.
(13)
Values for ξn and ξb can be calculated from (13) if the
merid-ional flow profile is known by discretizing the equationsand
numerical integration. Namely, the normal and binor-mal components
of the vorticity can be determined for aknown meridional flow
profile (from jet/wake models, 2Dflow through-flow models, and/or
experimental data).
Equation (10) shows that there are primarily two forceterms that
contribute to the generation of streamwise vortic-ity in a
centrifugal impeller. The first term shows that stream-wise
vorticity is generated whenever a flow that is nonuni-form in the
pressure-to-suction direction, and thus, has anormal vorticity
component, ξn, follows a curved bend witha radius of curvature, r.
The second term shows streamwisevorticity generation whenever flow
that is nonuniform inthe hub-to-tip direction (ξb) is rotated
around a centerline(shaft). The sine-term indicates that streamwise
vorticity isonly generated by the second term when a radial-flow
com-ponent (κ �= 0◦) exists.
Thus, in a compressor, a pressure-to-suction side non-uniform
flow profile only contributes to the first term of (10)while a
hub-to-tip side nonuniformity only contributes tothe second term of
(10). The meridional flow angle in (10)can be closely approximated
by κ = πs/stotal for a circularcompressor torus. Consequently, (10)
with the normal andbinormal vorticities obtained from the simple 2D
flow mod-els, can be numerically integrated to calculate the
stream-wise vorticity in the compressor and, thus, can be used
toapproximately predict the compressor secondary flow
circu-lation.
One should note that streamwise vorticity can also begenerated
by compressibility effects and viscosity; analyticalterms for these
effects can be found in Lakshminarayana andHorlock [27] but were
found to be negligible in the analysispresented herein. Namely,
compressibility and viscosity havea strong direct influence on the
meridional flow field and,thus, on the normal and binormal
vorticity components, butonly an indirect effect on the streamwise
vorticity. The sec-ondary flow field is “shaped” by gradients in
the reducedstagnation pressure (Johnson [28]) that are principally
de-termined by the meridional flow field, centrifugal forces,
andCoriolis forces. Once the meridional flow field is known,
thesecondary flow field can be determined neglecting
compress-ibility and viscous effects. A vortex will behave mostly
underthe influence of its angular momentum vector. Clearly,
theviscosity of the fluid leads to a small exchange of
momentumbetween the vortex structure and the surrounding flow
field,with the net effect that the radius of the vortex will
increaseand the core vortex strength will decrease as it travels
down-stream in the impeller. However, this influence is
negligiblewhen compared with the inertia of its angular
momentumvector and the effects of centrifugal and Coriolis
forces.
To qualitatively assess the secondary vortex velocity vec-tor
field from the streamwise vorticity, a modified approachto
Hawthorne’s [15] small shear/large disturbance method,as described
by Lakshminarayana and Horlock [33], is used.In this method the
relative displacement of the stream-lines (and the center of
circulation) is determined using thenondimensional meridional
velocity profile (weighted massflow). The small shear/large
disturbance approximation isvalid here because in the centrifugal
compressor flow turning(bending) and not shear (as in an axial
impeller) dominatesthe action on the working fluid (Howard
[30]).
In the small shear/large disturbance method the 2D con-tinuity
equation (∂u/∂x + ∂v/∂y = 0) with the definition ofa stream
function, u = −∂ψ/∂y and v = ∂ψ/∂x, is used toobtain
∇2ψ = ∂2ψ
∂x2+∂2ψ
∂y2= −ξs ·
wi, jwave
· dAi, j , (14)
where wi, j is the local plane through-flow velocity, dAi, j
isthe incremental plane area, and wave is the average
planethrough-flow velocity. Equation (14) must be solved for
thestreamfunction, ψ, with the boundary conditions of ψ = 0on the
walls. A solution for this can be obtained analyticallyin the form
of a Fourier series or (14) can simply be solved by
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Analysis of Secondary Flows in Centrifugal Impellers 49
an iterative numerical approach. Since in a centrifugal
com-pressor the passage planes are not necessarily a perfectly
rect-angular domain, the numerical solution is preferred for
thiscase. A central difference discretization was applied to (14)
toobtain
ψ(new)i, j =14
(ψi, j+1 + ψi, j−1 + ψi+1, j + ψi−1, j
)(old)
− ξs ·wi, jwave
· ∆Ai, j .(15)
This equation was solved for ψ(new) by successive
iterations(updating ψ(old) in each step) marching over the entire
math-ematical domain (plane). The normalized through-flow
ve-locities, wi, j /waverage, are obtained from the predicted
merid-ional flow profiles. Convergence (total residual of ψ of
lessthan 0.01) is typically achieved after approximately 2000
iter-ations (passes) over the domain and a resolution of
100×100grid points is usually adequate. Once a converged solutionis
obtained, the passage secondary vortex velocities can bedetermined
from the numerical partial derivatives of thestreamfunction (u =
−∂ψ/∂y and v = ∂ψ/∂x).
5. HEAD LOSSES DUE TO SECONDARY FLOWS
The total flow head in a compressor or pump increases
astangential kinetic energy is transferred into the fluid by
therotating blades. However, due to fluid friction (viscosity)
thework input to the machine does not equal the isentropicwork out;
that is, the efficiency is always less than 100%.Thus, the viscous
head loss due to secondary flows is an im-portant parameter to
estimate the overall performance of acentrifugal compressor.
Using the streamwise vorticity secondary flow models asdescribed
above, an estimate of this loss can be determinedfrom the laminar
viscous dissipation of the internal flow.(Note that only head
losses due to secondary flows are pre-dicted; other significant
head losses due to turbulence, lam-inar meridional dissipation, and
unsteady viscous dissipa-tion are not evaluated. Typically head
losses due to secondaryflows contribute less than 2% to the total
head loss in a cen-trifugal compressor.)
The laminar viscous dissipation function for incompress-ible
flow (and neglecting all meridional flow terms), Φls, is
Φls = 2µ[(
∂u
∂x
)2+(∂v
∂y
)2]+ µ(∂u
∂y+∂v
∂x
)2. (16)
By integrating the above laminar dissipation, Φlaminar
secondary,across an entire compressor passage, the total head loss
perpassage due to secondary flows, ∆Hpassage, is determined.
Thetotal impeller head loss due to secondary flows, ∆Hsecondary,is
then calculated by multiplying ∆Hpassage by the number ofblade
passages in the compressor, N . Thus
∆Hsecondary = N ·∫ΦlsdV. (17)
Clearly, the secondary flow head loss is directly related tothe
streamwise vorticity function and thus also related to
thenonuniformity of the meridional profile.
6. NONDIMENSIONAL FORCE PARAMETERS
The Rossby number, Ro = V/ωr, is a measure of inertialto
Coriolis force and is commonly employed for turboma-chinery flow
analysis. However, as centrifugal forces have astronger influence
than inertial forces on the secondary flowsin a rotating machine, a
modified Rossby number can be in-troduced:
Rom =FcentrifugalFCoriolis
= ωR2V
. (18)
Namely, Rom is a measure of the relative influence of
thecentrifugal force (in the outward radial direction) versusthe
Coriolis force (in the counter-rotational tangential
direc-tion).
Brun [34] showed that the modified Rossby affects
thepressure-to-suction side meridional (jet/wake) flow profileand
the normal vorticity, ξn. Consequently, the modifiedRossby number
should have a strong effect on the firstterm in the streamwise
vorticity generation term in (10).On the other hand, the binormal
vorticity—the secondterm in (10)—is not affected by the modified
Rossby num-ber directly. However, during actual compressor
operation,the modified Rossby number typically changes with
pumpspeed, ω, and thus indirectly relates the Rossby number tothe
second term of (10).
The Reynolds number, Re = Vr/ν, is an indicatorof inertial
versus viscous forces for a moving fluid. Sincethe Reynolds number
primarily influences the hub-to-tipside meridional flow profile,
the streamwise vorticity shouldmostly affect the second term
(rotational Coriolis force term)of (10) while the influence on the
first term should beweak.
Interestingly, streamwise vorticity theory thus predictsthat
vortex secondary flows are primarily related to the modi-fied
Rossby number (centrifugal/Coriolis force) via first termof (10)
and the Reynolds number (inertial/viscous force) viathe second term
of (10). One should note that the termsof (10) act in the opposite
direction: term one is positiveand acts to generate vortex
secondary flows circulating in theclockwise direction (seen
radial-inward) while term two isnegative and acts in the
counterclockwise direction. That is,a centrifugal compressor can be
designed in which the termsof (10) offset each other and
circulatory secondary flow gen-eration is minimized.
7. PREDICTED RESULTS AND COMPARISON WITHEXPERIMENTAL DATA
Using the above model, parametric studies were performedto
evaluate the relative influence of the nondimensional op-erating
parameters on the centrifugal compressor streamwisevorticity and
subsequent passage vortex secondary flows. Of
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50 International Journal of Rotating Machinery
−25−30−35−40−45−50−55−60−65−70−75
Stre
amw
ise
vort
icit
y(r
ad/s
)
1000 3000 5000 7000 9000 11000
Reynolds number
Figure 2: Streamwise vorticity as a function of Reynolds
number.
−20−25−30−35−40−45−50−55−60−65−70
Stre
amw
ise
vort
icit
y(r
ad/s
)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Modified Rossby number
Figure 3: Streamwise vorticity as a function of modified
Rossbynumber.
particular interest is the effect on the flow field of
varyingthe nondimensional force parameters—the Reynolds num-ber and
the modified Rossby number—as they represent thechanging operating
conditions an impeller experiences. Forthese studies, normal and
binormal vorticity results fromthe pressure-to-suction and
hub-to-tip jet/wake flow stud-ies as presented by Brun [34] were
used in (10) to deter-mine the streamwise vorticity component. The
subject anal-ysis is based on a 30 cm diameter mixed-flow
centrifugal im-peller geometry, rotating at 1000 rpm with an
incompress-ible, medium viscosity fluid at low Reynolds number
operat-ing conditions.
Figures 2 and 3 show predicted streamwise vorticity asa function
of Reynolds and modified Rossby numbers. Lim-ited experimental data
for a mixed-flow shrouded impelleris shown as a comparison; results
are within the uncertaintybands of the flow measurements.
The streamwise vorticity is seen to decrease with Re andRom.
Consequently, circulatory secondary flow vectors areexpected to
increase their clockwise rotation as Re and Romare increased.
Tip
Pressure Suction
Hub
↓= 3 m/s
Figure 4: Predicted secondary flow field at impeller exit
(Brun[34]).
3.5
3
2.5
2
1.5
Seco
nda
ryfl
owh
ead
loss
(W)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Modified Rossby number
Figure 5: Pump head losses as a function of modified Rossby
num-ber.
Figure 4 shows the predicted secondary velocity fieldfrom (15)
at 100% span. Clearly, a large circulatorysecondary flow vortex,
centered at the suction side of theblade, is seen. The secondary
flow vortex assumes almost theentire passage width. These results
compare favorable to ex-perimental measurements of circulating
secondary flows.
Secondary flow head loss trends as a function of modifiedRossby
number are shown in Figure 5. As a reference, thetotal head loss
for this impeller is approximately 30–40 W.Secondary flow head loss
increases with modified Rossbynumber, which is consistent with the
observation that theclockwise circulatory secondary flows increase
with modifiedRossby number.
8. CONCLUSIONS AND SUMMARY
A model to determine the rotational direction and magni-tude of
the circulatory velocity in shrouded centrifugal im-pellers was
derived based on the streamwise vorticity govern-ing equation. The
model applies known meridional velocitiesto the streamwise
vorticity equations to determine vorticity,
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Analysis of Secondary Flows in Centrifugal Impellers 51
circulatory secondary velocities, and associated head
losses.Streamwise vorticity, and thus circulatory secondary flow,
isseen to be primarily generated by the centrifugal and
Coriolisforces on the meridional flow field. Viscous and
compress-ibility effects must be considered to determine the
merid-ional flow field but can be neglected in the streamwise
vor-ticity equations.
A parametric analysis of the generation of streamwisevorticity
equation showed that
(i) positive vorticity and, thus, counterclockwise sec-ondary
passage flow circulation is generated by theinteraction of the
pressure-to-suction side meridionalflow gradient with the
axial-to-radial turning of theflow in the blade passage,
(ii) negative vorticity and, thus, clockwise secondary pas-sage
flow circulation is generated by the interaction ofthe hub-to-tip
side meridional flow gradient with therotation, ω, of the blade
passage.
Hence, the nondimensional operational force parameters,modified
Rossby number and Reynolds number, directly af-fected the velocity
magnitudes of the vortex secondary flow:
(i) increasing the Reynolds and/or modified Rossby num-ber, Rom,
increases clockwise flow circulation,
(ii) moderating the pressure-to-suction velocity gradient(e.g.,
by backsweeping the blades) increases the coun-terclockwise flow
circulation or moderates the clock-wise flow circulation.
Comparison with experimental data showed that analyticalresults
from the above streamwise vorticity model are withinthe uncertainty
band of flow measurements in a shroudedmixed-flow impeller; namely,
the model can be employedto accurately predict secondary flow
trends. The herein de-scribed method can be employed early in the
design processto optimize impeller flow shapes that intrinsically
reduce sec-ondary flows rather than using flow disruptive elements
suchas splitter to accomplish this task.
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