ASSESSMENT OF STATISTICAL EDDY-VISCOSITY TURBULENCE …
Transcript of ASSESSMENT OF STATISTICAL EDDY-VISCOSITY TURBULENCE …
Paper ID: ETC2019-047 Proceedings of 12th European Conference on Turbomachinery Fluid dynamics & Thermodynamics ETC13, April 8-12, 2019; Lausanne, Switzerland
OPEN ACCESS Downloaded from www.euroturbo.eu
1 Copyright © by the Authors
ASSESSMENT OF STATISTICAL EDDY-VISCOSITY
TURBULENCE MODELS FOR UNSTEADY FLOW AT PART AND
OVERLOAD OPERATION OF CENTRIFUGAL PUMPS
N. Casimir1 - Z. Xiangyuan2– G. Ludwig3– R. Skoda1
1Chair of Hydraulic Fluid Machinery, Ruhr University Bochum, Germany
2 School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, P. R. China
3Chair of Fluid Systems, Technical University Darmstadt, Germany
ABSTRACT
The accuracy of linear eddy-viscosity URANS turbulence modelling is assessed for off-
design operation of a single stage radial centrifugal pump. The open source software
OpenFOAM is utilized. Hot wire probes have been utilized for measurement of spatially and
temporally resolved validation data at the impeller discharge of the air-operated pump. The
ensemble-averaged mean flow angle is in qualitatively good agreement to data for the entire
operation range, while minor deviations occur close to the volute tongue due to impeller-volute
interaction. Turbulence statistics from the simulation are compared to ensemble-averaged
velocity RMS from measurements. RMS distribution is also qualitatively well reproduced for
near-design and overload operation. A more pronounced RMS mis-prediction occurs in part
load in the region of high impeller-volute tongue interaction and reveals limitations of
turbulence modelling in highly unsteady flow regions of centrifugal pumps.
KEYWORDS
URANS, OPENFOAM, PART LOAD, PUMP, TURBULENCE MEASUREMENTS
NOMENCLATURE
Latin characters
b Volute inlet width [m] Tp Periodic time scale [s] c Velocity [m/s] TT Integral turbulent time scale [s]
d Diameter [m] t Blade spacing [m]
n Rotational speed [1/s] TI Turbulent Intensity [-]
G1 Coarse mesh [-]
u Impeller circumferential
velocity [m/s]
G2
Fine mesh [-] x Circumferential distance [m] k
Turbulent kin. Energy [m2/s2] between trailing edge and ns Specific speed [-] measurement point p Pressure [Pa] y Axial direction [m] Q Flow rate [m3/s] Z Number of blades [-] q Flow rate related to [-] nominal Flow rate
2
Greek characters
α Flow-angle [°] TKE Turbulent Kinetic ε Volute angle [°] Energy ν Kinematic viscosity [m2/s] u Circumferential
Ψ Pressure coefficient [-] Abbreviations
ρ Density [kg/m3] CFD Computational Fluid Dynamics
ω Turbulent frequency [1/s] Exp Experimental Data
Subscripts GGI Generalized Grid Interface
0 Nominal value Hub Hub side
1 Impeller inlet LES Large Eddy Simulation
2 Impeller outlet MP Measurement Point
3 Measurement plane Num Numerical data
abs Absolute PISO Pressure Implicit with
exp Experimental Separation of Operators
m Meridional SIMPLE Semi Implicit Method for
num Numerical Pressure Linked Equations
out Outlet property Shr Shroud side
ref Reference position SST Shear-Stress-Transport
RMS Root Mean Square URANS Unsteady Reynolds Averaged
t Total Navier-Stokes
INTRODUCTION
Increasing demands on centrifugal pumps, e. g. by legislature, require higher efficiencies even at
off-design operation, i.e. part load and overload, which are characterized by a highly unsteady and
turbulent flow field due to flow separation and impeller-stator interaction. CFD methods are
increasingly integrated in the design and optimization process of pumps. Statistical (i.e. URANS)
eddy-viscosity turbulence models are widely used for pump flow simulations. In this class of
turbulence models, a significant simplification is introduced by an a-priori time-average of the
turbulent fluctuations. Thus, the resulting Reynolds-stress tensor has lost any spectral information of
the turbulence field. Although statistical models may yield a good prediction of pump characteristics
at close-to-design operation, they may increasingly fail towards off-design pump operation. There are
several studies that show such limitations of statistical models and benefits of scale-resolving models
(LES), e.g. for the prediction of in-homogenous flow distribution through impeller blade channels at
part load (Byskov et al., 2002, 2003), head characteristics instability by Kato et al. (2003), part load
instabilities and flow separation (Tang et al., 2007; Tokay et al., 2006; Zhang, 2010; Wang et al.,
2013; Posa et al., 2011, 2015; Si et al. 2014) or tip vortices by Shen et al. (2013). However, the
computational effort of LES is tremendous because at least 80 % of the spectral energy must be
resolved (Pope, 2000; Fröhlich et al., 2008). Since Reynolds numbers typically encountered in
centrifugal pumps are high, and thus the spectrum of turbulent eddies spreads over several spatial and
temporal orders of magnitude, a soundly resolved LES demands an extensively high number of
computational cells and time steps. It is thus attractive to exploit the moderate computational effort
of statistical models after their careful validation on spatially and temporally resolved flow field data
and thorough evaluation of their particular limitations. Spence and Amaral-Teixeira (2008, 2009) and
Barrio et al. (2010) presented URANS simulations and a validation on measured integral
characteristics i.e. head, power and efficiency. However, a comparison with measured spatially and
temporally resolved velocity has not been presented yet for centrifugal volute casing pumps, to the
knowledge of the authors, and is thus the aim of the present study.
A single stage volute casing radial centrifugal pump (𝑛𝑆 = 26 𝑚𝑖𝑛−1) is investigated including
off-design operation. The pump is operated with air in order to use hotwire anemometry for a high
temporal and spatial resolution of the velocity field near the impeller discharge and thus enables a
comparison with turbulence statistics from the simulation. Experimental data origins from
3
measurements at the Chair of Fluid Systems, Technical University Darmstadt, see Hergt et al. (2004)
and Meschkat and Stoffel (2002) and is reviewed and re-edited within the present study to yield
ensemble-averaged mean flow angle and turbulence field (velocity RMS) at the impeller discharge
for comparison with CFD results. The statistical, linear eddy-viscosity SST turbulence model by
Menter (1994) is employed due to its wide use for pump flow simulation.
EXPERIMENTAL SETUP
Details of the measurements were presented by Hergt et al. (2004) and Meschkat and Stoffel
(2002) and briefly summarized here. A centrifugal pump with a specific speed of 𝑛𝑆 = 26 𝑚𝑖𝑛−1 and a trapezoidally shaped volute casing is operated in an open test rig with air. A special volute casing
construction allows circumferential and axial probe traversals across a cylindrical plane 6 𝑚𝑚
downstream of the impeller trailing edge, see Figure 1 a and b, with a spatial resolution of 2 𝑚𝑚 in
axial and 1.8° in circumferential direction. Since the maximum Mach number is evaluated to be below
0.3, it is assumed that compressibility effects on the velocity field are of minor relevance. The volute
casing is designed for about 80 % nominal impeller flow rate because a nearly constant
circumferential distribution of the static volute wall pressure has been achieved for that flow rate, see
Hergt et al., 2004. Thus, 𝑞 = 𝑄 𝑄0 = 80 %⁄ is considered to be the design operation of the pump. Pump
impeller data are summarized in Table 1, and more details can be found in Hergt et al. (2004) and
Meschkat and Stoffel (2002).
Table 1: Pump impeller data.
Parameter Notation Value Unit
Specific speed (water and air) 𝑛𝑆 = 𝑛[𝑚𝑖𝑛−1] ∙√𝑄0[
𝑚3
𝑠]
(𝐻)3/4 26 𝑚𝑖𝑛−1
Specific speed (non- 𝑛𝑆 = 𝑛[𝑠−1] ∙
√𝑄0[𝑚3
𝑠]
(𝐻𝑔)3/4 0.08 -
dimensional)
Number of blades 𝑍 7 -
Rotational speed (air) 𝑛 3000 𝑚𝑖𝑛−1
Nominal flow rate (air) 𝑄0 879 𝑚3 ℎ⁄
Suction flange diameter 𝑑1 230 𝑚𝑚
Impeller diameter 𝑑2 405 𝑚𝑚
Measurement plane diameter 𝑑3 417 𝑚𝑚
Impeller discharge width 𝑏 34 𝑚𝑚
The spatially and temporally resolved velocity field in the measurement plane was recorded for
operation points from 𝑞 = 20 % to 𝑞 = 130 %, while only the range of 𝑞 = 40 % to 𝑞 = 120 % is
considered in the present study. The transient flow angle was evaluated by single wire measurements
with a sampling frequency of 42 𝑘𝐻𝑧 and ensemble-averaged in dependence on the angular impeller
position. Velocity RMS was determined by cross wire probe measurements with a sampling rate of
37 𝑘𝐻𝑧 and also ensemble-averaged. By the cross wire probe arrangement, it was ensured that the
velocity fluctuations in all three spatial directions are detected to reflect the three dimensional
turbulence field. The uncertainties of both, single and cross wire probe measurements, are 5 % of the
measured value, according to Meschkat (2004), which are suitably low for the assessment of
qualitative velocity and turbulence distributions. The flow-angle 𝛼 was evaluated with Equation 1
and ranges from −180° to 180°.
𝛼 =
|𝑐𝑚|
𝑐𝑚∙ 𝑎𝑐𝑜𝑠 (
𝑐𝑢
𝑐𝑎𝑏𝑠) (1)
4
Negative angles indicate a meridional backflow into the impeller. Values between 0° and 90° indicate that 𝑐𝑢 is oriented towards impeller rotation direction, and values between 90° and 180° in
the opposite direction.
Mean flow angle and RMS velocities are presented as functions of the axial position 𝑦 𝑏⁄ , see
Figure 1 a, and the circumferential coordinate 𝑥 𝑡⁄ , which corresponds to the time range when one
blade spacing passes the measurement position, see Figure 1 c. Results are presented for selected
circumferential measurement positions 𝜀 = 0° and 120° in the volute casing, see Figure 1 b.
The total pressure cross-area distribution was recorded 2,5 ∙ 𝐷 upstream of the impeller inlet
(position 𝑟𝑒𝑓) as well as at the volute discharge by transient single-hole cylindrical pneumatic probes
and was area-averaged for the evaluation of the pressure coefficient according to Equation 2.
Ψ𝑡 =𝑝𝑡,𝑜𝑢𝑡 − 𝑝𝑡,𝑟𝑒𝑓
0,5 ∙ �̅� ∙ 𝑢22 (2)
An averaged density �̅� is evaluated between suction pipe and discharge and assumed to be
constant within the pump. According to Meschkat (2004) uncertainties of pressure coefficient
measurements are at 1.4 % at 𝑞 = 40% partial load and increasing to 4% at 𝑞 = 120% overload,
illustrated in Figure 3c as error bars.
Figure 1. a) Meridional contour and radial probe position; b) Circumferential probe positions;
c) Illustration of coordinate 𝒙, which corresponds to the time range when one blade spacing
passes the measurement position.
NUMERICAL SET-UP
Numerical method
URANS simulations are performed with the open source software Foam-Extend version 4.01. The
pimpleDyMFoam solver is utilized, i.e. a combined PISO (Issa, 1986) and SIMPLE (Patankar and
Spalding, 1972) algorithm for incompressible unsteady flows, combined with moving mesh
capabilities. The statistical eddy-viscosity SST turbulence model by Menter (1994) with automatic
wall functions is employed due to its wide use for pump flow simulation.
The computational domain contains the impeller, volute casing, side chambers and the suction
and pressure pipe, see Figure 2 d. The suction pipe has been elongated to a length of 5 ∙ 𝐷 to avoid an
impact of boundary conditions on the part load vortex. A block structured hexahedral grid as shown
in Figure 2 a, b and c with approximately 3 million nodes (named G1) is generated with the software
1 https://foam-extend.fsb.hr/
d2
d3
d1
Measurement
plane
Hub
b
Front shroud
Rear shroud
Volute
Casing
a) Pump meridional view b) Volute axial view
120 d1 d3
c) Tongue region axial view
𝑦 Measurement
plane
.
.
MP: ε=0
MP: ε=0
Measurement
position:ε=0°y
Impeller
rotation
5
ANSYS ICEM 182. For a grid study, the grid G1 is successively refined to approximately 24 million
nodes (named G2) by bisection of node distances in each direction. For G1, average and maximum
𝑦+ values equal about 10 and 60, and for G2 about 5 and 30.
A Dirichlet inlet boundary condition is set for velocity according to flow rates from experiment,
together with a Neumann (zero-gradient) condition for static pressure. At the volute discharge, a
Neumann boundary conditions is set for velocity (zero gradient) and a Dirichlet condition for static
pressure. Temporal and spatial second order discretization methods are used according to table 2.
Time step is set according to an impeller rotation of 1° on the coarse and 0.5° on the fine grid to maintain a constant Courant number during the grid study.
For the evaluation of convergence of the non-linear iterative SIMPLE-PISO algorithm,
preliminary investigations have shown that a drop of the non-linear non-dimensional residual sum
norm of each equation below a value of 10-5 is a suitable convergence criterion. It is noteworthy that
we evaluate the residual of the pressure equation in the first PISO loop and not in the last one, which
is a more stringent convergence evaluation. For assessment of the temporal convergence, an
ensemble-average of velocity magnitude in the measurement plane in the same way as in the
experiment is performed. Temporal convergence is defined when the standard deviation of ensemble-
averaged velocity is below 1 % which is achieved after less than one revolution at design point 𝑞 =80 %, about two revolutions in overload 𝑞 = 120 % and about four revolutions in part load 𝑞 = 40 %.
Figure 2. a) Computational grid near volute tongue, b) Conformal interface meshing between
rotor and stator domain, c) Computational grid at the blade leading edge; d) Computational
domain.
2 https://www.ansys.com
Volute
Impeller and side
chambers
Outlet
Suction pipe Inlet
c) Computational domain
a) Polyhedral interface meshing (schematic) b) Blade leading edge meshing
Polyhedral Interface
Conformal Interface
Homogeneous interface grid
Rotor
Stator
Volu
te
Impel
ler
and
side
cham
ber
s
Outl
et
Suct
ion
pip
eIn
let
c) C
om
puta
tional
dom
ain
a) P
oly
hed
ralin
terf
ace
mes
hin
g(s
chem
atic
)b)
Bla
de
lead
ing
edge
mes
hin
g
Poly
hed
ral
Inte
rfac
e
Confo
rmal
Inte
rfac
e
Hom
ogen
eous
inte
rfac
egri
d
Roto
r
Sta
tor
Trailing edge
a) Volute tongue meshing (G1) b) Polyhedral interface meshing (G1)
c) Blade leading edge meshing d) Computational domain
Fine
G2
Coarse
G1
6
Table 2: Overview of the numerical setup
Parameter Description
Fluid Incompressible air, 𝜐 = 1.54 ∙ 10−5 𝑚2 𝑠⁄
Simulation type Transient, 1° time step (G1), 0.5° time step (G2)
Time discretization Second-order backward Euler
Spatial discretization of convection Momentum equations: Warming and Beam (1976)
Turbulence equation 𝑘 and 𝜔: Van Leer (1979)
Solver convergence criteria Non-linear non-dimensional residual < 1 ∙ 10−5
Temporal convergence criteria Deviation of ensemble averaged results and RMS < 1%
Rotor-stator interface
The rotating domain includes the suction pipe, impeller and side chambers, and the stationary
domain includes the volute-casing and discharge pipe, see Figure 2 d. The governing equations are
solved in the absolute frame of reference so that a moving grid is applied for the rotating domain.
Transient coupling of rotor and stator domain is achieved by a GGI algorithm by Beaudoin and Jasak
(2008) that allows a non-conformal grid, i.e. non-matching grid nodes on both, rotor and stator side
of the interface. The GGI algorithm is based on corrected area weighted flux interpolations.
Projections of faces along the normal-vector of neighboring faces are calculated to determine the
weighting factors. For planar interfaces, this algorithm yields perfectly conservative flux
transmission. However, the rotor-stator interface in radial pumps forms a cylindrically shaped surface.
A non-conformal interface - i.e. that in circumferential direction grid nodes on rotor and stator side
of the interface do not match - leads to radial gaps and overlapping mesh cells. The projections are
distorted, and an error in the flux interpolation results in non-conservative flux transmissions which
cannot be resolved by interpolation weighting factors of the GGI. As a consequence, transient pump
head curves show significant high-frequency oscillations, see Figure 3 a. These oscillations are caused
by the locally non-conservative flux transmission which was revealed by a FFT analysis comparing
the frequency spectra of mass flux differences at the interface and pump head oscillations. In
preliminary simulations, a merge of rotor and stator interface by insertion of polyhedral cells
(“stitching”) within each time step avoids gaps and overlapping cells, ensures a conservative flux-
transmission and thus effectively suppresses the oscillations. However, the necessary changes of mesh
topology within each time step results in a high computational effort. Thus, a different approach is
proposed for the present study.
The rotor and stator domain are meshed with a narrow radial gap at each side of the cylindrical
interface region, and the gap is filled on rotor and stator side with hexahedral interface cells with a
circumferentially homogenous node distribution, named “Homogeneous interface grid” in Figure 2
b. Their circumferential distribution matches the circumferential rotor movement within each time
step, yielding matching interface nodes and thus a conservative interface at each time step, referred
to as conformal interface. A non-matching interface, however, remains between homogeneous
interface grid and the remaining domain of rotor and stator, referred to a polyhedral interface in Figure
2 b. Therefore, the gaps and overlaps at the polyhedral interface on rotor and stator side are stitched
by the StitchMesh-tool of Foam-Extend during pre-processing and yield polyhedral cells and a closed
and thus fully conservative domain. Since this procedure is done only once during pre-processing, no
significant increase of computational effort is present. Of course, a fixed time step modulated to the
interface grid spacing needs to be chosen for the simulation runs.
As a result, transient head curves do not show any high-frequency oscillations, see Figure 3 b. It
is noteworthy to mention, that the pressure coefficient in Figure 3 b is shown for the last revolution
of the simulation, while the time average in Figure 3 c has been obtained over the last four revolutions
to get a stable time-converged average value.
7
Figure 3. Transient pressure coefficient without (a) and with (b) conformal rotor-stator
interface at 40% partial load; c) Pump head characteristics in terms of time-averaged total
pressure coefficient.
Evaluation of turbulence statistics
In URANS simulations, the turbulent momentum transfer is described by a time average of the
second moment velocity fluctuation correlations, yielding the Reynolds stress tensor. In the present
simulations, also the resolved, i.e. time averaged velocity �̂�(𝑡) shows a strongly transient behavior
which is most pronounced in part load. For the evaluation of the RMS velocity in the measurement,
the velocity fluctuations cannot be separated into turbulent and non-turbulent fluctuations. Thus, for
a sound comparison with measured RMS velocity, the velocity fluctuations which are resolved in the
simulation need to be considered in addition to the normal Reynolds stresses in terms of the turbulent
kinetic energy which is illustrated in what follows.
URANS methods separate the velocity magnitude into a Reynolds-averaged part �̂�(𝑡) and its
deviation 𝑐′(𝑡): 𝑐(𝑡) = �̂�(𝑡) + 𝑐′(𝑡) with �̂� =
1
∆𝑇∫ 𝑐𝑑𝑇
𝑇2
𝑇1 (3)
The averaging time scale ∆𝑇 must be longer than the integral turbulent time scale 𝑇𝑇 for a valid
URANS, a conditions that is verified at the end of the result section of the present paper. The RMS
of the ensemble-averaged velocity field is calculated by Equation (4):
𝑅𝑀𝑆(𝑐) = √1
𝑁∑(𝑐𝑖 − 𝑐̅)2𝑁
𝑖=1
= √1
𝑁∑(𝑐2 − 2𝑐𝑐̅ + 𝑐̅2)𝑖
𝑁
𝑖=1
(4)
Inserting the Reynolds separation of Equation (3) into 𝑐2 − 2𝑐𝑐̅ + 𝑐̅2 in Equation (4) and
Reynolds-averaging yields:
�̂�2̂ + 2�̂�𝑐 ′̂ + 𝑐′𝑐′̂ − 2�̂�𝑐̅̂ − 2𝑐′𝑐̅̂ + 𝑐̅2̂ = �̂�2 + 𝑐′𝑐′̂ − 2�̂�𝑐̅ + 𝑐̅2 = (�̂� − 𝑐̅)2 + 𝑐′𝑐′̂ (5)
The correlation term 𝑐′𝑐′̂ is described in terms of the turbulent kinetic energy 𝑘:
2𝑘 = (𝑐𝑚′𝑐𝑚
′̂ + 𝑐𝑢′𝑐𝑢
′̂ + 𝑐𝑎𝑥′𝑐𝑎𝑥
′̂ ) = (𝑐′𝑐′̂ ) (6)
The Reynolds-averaged turbulence intensity (𝑇𝐼) is calculated inserting Equation (5) and
Equation (6) into (4) and dividing by the impeller circumferential trailing edge velocity 𝑢2:
1.1
1.2
1.00 π/2 π 3/2π 2π
a) ANSYS-CFX: Q/Q0=40%
Pre
ssure
Co
eff.
Ψ(t
) [-
]
1.1
1.2
1.00 π/2 π 3/2π 2π
b) Foam-Extend: Q/Q0=40%P
ress
ure
Co
eff.
Ψ(t
) [-
]
1.1
1.2
1.00 π/2 π 3/2π 2π
c) FE Conformale: Q/Q0=40%
Pre
ssure
Co
eff.
Ψ(t
) [-
]Rotation [rad] Rotation [rad] Rotation [rad]
b) Ψ𝑡, with conformal Interface
1.1
1.2
1.00 π/2 π 3/2π 2π
a) ANSYS-CFX: Q/Q0=40%
Pre
ssure
Co
eff.
Ψ(t
) [-
]
1.1
1.2
1.00 π/2 π 3/2π 2π
b) Foam-Extend: Q/Q0=40%P
ress
ure
Co
eff.
Ψ(t
) [-
]
1.1
1.2
1.00 π/2 π 3/2π 2π
c) FE Conformale: Q/Q0=40%
Pre
ssure
Co
eff.
Ψ(t
) [-
]Rotation [rad] Rotation [rad] Rotation [rad]
Impeller position [-] Impeller position [-]
Ψ𝑡[-
]
Ψ𝑡[-
]Q/Q0 [-]
c) Head characteristicsa) Ψ𝑡, w/o conformal Interface
Ψ𝑡[-
]
Ψ𝑡 [
-]
Ψ𝑡 [
-]
Ψ𝑡 [
-]
𝑞 [-]Impeller position[-]Impeller position[-]
a) Ψ𝑡, w/o conformal Interface b) Ψ𝑡, with conformal Interface c) Head characteristics
Sim-G2Sim-G1
Exp
1.1
1.2
1.00 π/2 π 3/2π 2π
a) ANSYS-CFX: Q/Q0=40%
Pre
ssure
Co
eff.
Ψ(t
) [-
]
1.1
1.2
1.00 π/2 π 3/2π 2π
b) Foam-Extend: Q/Q0=40%P
ress
ure
Co
eff.
Ψ(t
) [-
]
1.1
1.2
1.00 π/2 π 3/2π 2π
c) FE Conformale: Q/Q0=40%
Pre
ssure
Co
eff.
Ψ(t
) [-
]Rotation [rad] Rotation [rad] Rotation [rad]
b) Ψ𝑡, with conformal Interface
1.1
1.2
1.00 π/2 π 3/2π 2π
a) ANSYS-CFX: Q/Q0=40%
Pre
ssure
Co
eff.
Ψ(t
) [-
]
1.1
1.2
1.00 π/2 π 3/2π 2π
b) Foam-Extend: Q/Q0=40%P
ress
ure
Co
eff.
Ψ(t
) [-
]
1.1
1.2
1.00 π/2 π 3/2π 2π
c) FE Conformale: Q/Q0=40%
Pre
ssure
Co
eff.
Ψ(t
) [-
]Rotation [rad] Rotation [rad] Rotation [rad]
Impeller position [-] Impeller position [-]
Ψ𝑡[-
]
Ψ𝑡[-
]Q/Q0 [-]
c) Head characteristicsa) Ψ𝑡, w/o conformal Interface
Ψ𝑡[-
]
8
𝑇𝐼𝑆𝑖𝑚 =
√𝑅𝑀𝑆2(�̂�) + 2�̅�
𝑢2 (7)
Ensemble averaged turbulent kinetic energy and the variance (RMS²) of the unsteady Reynolds-averaged
velocity are used for 𝑇𝐼𝑆𝑖𝑚 according to Equation (7), which can immediately be compared with
measured 𝑇𝐼𝐸𝑥𝑝 in Equation (8):
𝑇𝐼𝐸𝑥𝑝 =
𝑅𝑀𝑆(𝑐𝑎𝑏𝑠)
𝑢2 (8)
RESULTS
Grid study
The grid influence is assessed by the pump head characteristics in terms of the time-averaged total
pressure coefficient (Figure 3 c) and the spatial-temporal distribution of mean flow angle 𝜶 (Figure 4
a) as well as 𝑻𝑰 (Figure 4 b). The head shows slight differences between G1 and G2 towards overload,
see Figure 3 c. Since the grid dependence of head is lower than the difference to measurement data,
sufficient grid independence of total pressure coefficient is assumed.
Since 𝛼 and 𝑇𝐼 are time dependent, their ensemble-averaged distribution is presented. Ensemble-
averaged results vs. the axial coordinate 𝑦 and direction of blade spacing 𝑥 are discussed in Figure 4. Note that this kind of illustration should not be confused with a snapshot of the flow field in the
measurement plane in direction of blade spacing 𝑡. Rather, the 𝑥 𝑡⁄ axis illustrates the time range when
one blade spacing passes the measurement position 𝜀. Grid dependence is exemplarily presented for
𝑞 = 40 %, since at the other operation points the deviations between both grids are lower than at part
load. Two exemplary measurement locations, 𝜀 = 0° and 120° are discussed. The flow angle 𝛼 is
essentially grid independent, see Figure 4 a. A more pronounced grid dependence is present for 𝑇𝐼, see Figure 4 b. 𝑇𝐼 increases with higher grid resolution. Since these differences are again lower than
the difference to measurement (as will be shown further below) and the qualitative distribution is
essentially unaffected, also for 𝑇𝐼 a sufficient grid independence is assumed. In what follows, G2
results are presented.
Figure 4. Grid dependency of mean flow angle 𝜶 (a) and turbulence intensity 𝑻𝑰 (b) for part
load 𝒒 = 𝟒𝟎 %.
Spatially and temporally resolved flow angle and turbulence intensity
In Figure 5, the flow angle 𝛼 is illustrated for 𝑞 = 40 %, 80 % and 120 % for two circumferential
measurement location, a close-to volute tongue location 𝜀 = 0° and far away from the tongue 𝜀 =120°. Note that for 𝜀 = 0°, flow angle results at 𝑞 = 100 % are presented to illustrate overload
operation instead of 120 % due to defective measurement data for 𝑞 = 120 % near the volute tongue.
It was approved, that turbulence intensity measurements were not affected by the defective
𝜀 = 0° 𝑞 = 40 %
𝑦/𝑏
𝑥/𝑡 [-]
𝑦/𝑏
𝑥/𝑡 [-] 𝑥/𝑡 [-] 𝑥/𝑡 [-]
𝜀 = 120° 𝑞 = 40 % 𝜀 = 0° 𝑞 = 40 % 𝜀 = 120° 𝑞 = 40 %
a) Flow angle 𝛼 [°] b) Turbulence Intensity 𝑇𝐼 [°]
-10 20 0 0.18
9
measurement data and thus shown in Figure 6 c for the highest overload operation, 𝑞 = 120 %. Close
to the tongue at overload (Figure 5 c), a high temporal change of the flow angle is present due to
deflection of the flow from circumferential direction to the volute discharge pipe direction, when the
blade trailing edge passes the tongue. Note that this flow deflection is also verified by flow
visualizations in a relative frame, which are not shown here. Due to that strong deflection, the 𝛼
scaling was adjusted in Figure 5 c for clarity of the image. Summarizing, the qualitative agreement
of simulation to measurement data is very good for 𝛼, irrespective of the operation point and the
circumferential position 𝜀.
Figure 5. Flow angle for different operation points and measurement positions .
In Figure 6, 𝑇𝐼 is presented in the same way as the flow angle in Figure 5, in dependence of 𝑦 𝑏⁄
and 𝑥 𝑡⁄ . The 𝑇𝐼 maximum location is attributed to the trailing edge wake and therefore essentially
the same for any operation point 𝑞 and circumferential measurement position 𝜀. Far away from the
tongue at 𝜀 = 120°, the location (in terms of 𝑦 𝑏⁄ ) and the instant (in terms of 𝑥 𝑡⁄ ) of 𝑇𝐼 maxima is
well captured by the simulation, albeit the 𝑇𝐼 maximum magnitude is overestimated for any operation
point. Essentially the same observation can be made close to the tongue, for 𝜀 = 0° and design and
overload operation, 𝑞 = 80 % and 120 %. While for these operation points, a clear local 𝑇𝐼 maximum
is discernible, for 𝜀 = 0° and 𝑞 = 40 %, a rather broad region of increased 𝑇𝐼 is discernible in the
measurement as well as simulation data, see Figure 6 a. This broad 𝑇𝐼 region and the absence of a
distinct local maximum is attributed to highly transient flow field at part load in combination with
blade-tongue interaction. For 𝜀 = 0° and 𝑞 = 40 %, the magnitude of 𝑇𝐼 is significantly overestimated
by the simulation.
𝜀 = 0° 𝑞 = 40 % 𝜀 = 0° 𝑞 = 80 % 𝜀 = 0° 𝑞 = 100 %
𝜀 = 120° 𝑞 = 120 %𝜀 = 120° 𝑞 = 80 %𝜀 = 120° 𝑞 = 40 %
a) Flow angle part load b) Flow angle design point c) Flow angle overload
𝑦/𝑏
𝑥/𝑡 [-]
𝑦/𝑏
𝑦/𝑏
𝑦/𝑏
𝑥/𝑡 [-] 𝑥/𝑡 [-]
𝑥/𝑡 [-] 𝑥/𝑡 [-] 𝑥/𝑡 [-]
𝛼 [ ]
𝛼 [ ]
10
Figure 6. Turbulence intensity for different operation points and measurement positions .
In order to reveal the origin of the 𝑇𝐼 overestimation by the simulation, two components of 𝑇𝐼𝑆𝑖𝑚
in terms of the first and second term in the square root of Equation (7) are illustrated separately in
Figure 7, exemplarily for 𝑞 = 40 %. The components of 𝑇𝐼𝑆𝑖𝑚 are referred to 𝑇𝐼𝑅𝑀𝑆 = 𝑅𝑀𝑆(�̂�) 𝑢2⁄ and
𝑇𝐼𝑇𝐾𝐸 = √2𝑘 𝑢2⁄ . As can be seen by comparing Figure 7 b and c, the maximum of 𝑇𝐼𝑅𝑀𝑆 corresponds
roughly to the measured 𝑇𝐼𝐸𝑥𝑝. Even by 𝑇𝐼𝑇𝐾𝐸 alone, see Figure 7 a, the measured 𝑇𝐼𝐸𝑥𝑝 is significantly
overestimated. It can be thus assumed that the mismatch of simulation to measurement results is
attributed to the turbulence model whose performance is immediately reflected in the too high 𝑘
distribution. The turbulence model shortcomings are most pronounced for highly transient flow as it
occurs in part load close to the tongue. This conclusion is supported by an illustration of the 𝑇𝐼𝑅𝑀𝑆
and 𝑇𝐼𝑇𝐾𝐸 distribution over the entire range of 𝜀 which is discussed next.
Figure 7. Ensemble-averaged contributions to 𝑻𝑰 at the volute tongue 𝜺 = 𝟎° for 𝒒 = 𝟒𝟎 %.
𝑇𝐼, averaged over 𝑦 𝑏⁄ and time is presented in Figure 8 for the entire range of circumferential
positions 𝜀, exemplarily for 𝑞 = 40 % and 120 %. Note that 𝑇𝐼𝑆𝑖𝑚 in Figure 8 is evaluated according
to Equation 7, and that 𝑇𝐼𝑅𝑀𝑆 and 𝑇𝐼𝑇𝐾𝐸 cannot simply be added to obtain 𝑇𝐼𝑆𝑖𝑚, because the sum of
𝑅𝑀𝑆2 and 2𝑘 is within the square root of Equation 7. Thus, the relation between 𝑇𝐼𝑅𝑀𝑆, 𝑇𝐼𝑇𝐾𝐸 and
𝜀 = 0° 𝑞 = 40 % 𝜀 = 0° 𝑞 = 80 % 𝜀 = 0° 𝑞 = 120 %
𝜀 = 120° 𝑞 = 120 %𝜀 = 120° 𝑞 = 80 %𝜀 = 120° 𝑞 = 40 %
a) 𝑇𝐼 at part load b) 𝑇𝐼 at design point c) 𝑇𝐼 at overload𝑦/𝑏
𝑥/𝑡 [-]
𝑦/𝑏
𝑦/𝑏
𝑦/𝑏
𝑥/𝑡 [-] 𝑥/𝑡 [-]
𝑥/𝑡 [-] 𝑥/𝑡 [-] 𝑥/𝑡 [-]
𝑇𝐼 [-]
𝑇𝐼 [-]
𝜀 = 0° 𝑞 = 40 % 𝜀 = 0° 𝑞 = 40 % 𝜀 = 0° 𝑞 = 40 %
a) 𝑇𝐼𝑇𝐾𝐸 b) 𝑇𝐼𝑅𝑀𝑆 c) 𝑇𝐼𝐸𝑥𝑝
𝑦/𝑏
𝑥/𝑡 [-] 𝑥/𝑡 [-] 𝑥/𝑡 [-]
𝑇𝐼 [-]
11
𝑇𝐼𝑆𝑖𝑚 in Figure 8 is rather qualitative. For 𝑞 = 40 % (Figure 8 a), both components contribute
significantly to 𝑇𝐼𝑆𝑖𝑚, and even 𝑇𝐼𝑇𝐾𝐸 alone leads to an overestimation of measured 𝑇𝐼𝐸𝑥𝑝. This
overestimation is more pronounced close to the tongue, towards rotation direction, i.e. in the range
𝜀 = 0°…180°. For overload (Figure 8 b), the contribution of 𝑇𝐼𝑅𝑀𝑆 is minor, and only a rather slight
overestimation and thus a better prediction of the measured 𝑇𝐼𝐸𝑥𝑝 by the simulation result 𝑇𝐼𝑆𝑖𝑚 is
achieved.
Figure 8: 𝑻𝑰 and its contributions in the simulation 𝑻𝑰𝑹𝑴𝑺 and 𝑻𝑰𝑻𝑲𝑬, averaged over 𝒚 𝒃⁄ and
𝒙 𝒕⁄ vs. 𝜺 for a) part load and b) overload.
The assumption of Reynolds-averaging is only valid if the integral turbulent time step 𝑇𝑇 and the
periodic time step 𝑇𝑃 are clearly separated, i.e. 𝑇𝑇 𝑇𝑃⁄ ≪ 1. 𝑇𝑇 is estimated from the numerical results
by the dissipation rate, 𝑇𝑇 = (0.09 ∙ 𝜔)−1, and 𝑇𝑃 is assumed to correspond to the blading-tongue
interaction time scale, 𝑇𝑃 = (𝑛 ∙ 𝑍)−1. The ratio 𝑇𝑇 𝑇𝑃⁄ is presented in Figure 9. Interestingly, the
criterion 𝑇𝑇 𝑇𝑃⁄ < 1 is fulfilled close to the tongue (𝜀 = 0°), where 𝑇𝐼 prediction is worst, and rather
failed for 𝜀 = 120°. It can be concluded that the URANS approach, i.e. the Reynolds-separation and
averaging, per se may not be the only origin of the turbulence model failure at 𝜀 = 0° but rather the
particular choice of the SST turbulence model as a representative of linear eddy-viscosity models that
is the common choice for pump flow simulations.
Figure 9: Ratio of integral turbulent time scale 𝑻𝑻 to periodic time scale 𝑻𝑷
CONCLUSIONS
It was shown that significant deviations to data occur regarding the turbulence field in terms of
𝑇𝐼, most pronounced at part load and in the vicinity of the tongue, where a highly unsteady and
recirculation flow field is present. This mismatch is mainly attributed to an overprediction of the
turbulent kinetic energy. Since the integral turbulent time scale is estimated to be in a suitable range
for URANS modeling, it may be the particular properties of the SST turbulence model as e.g. a linear
eddy-viscosity ansatz that promote its failure in the highly complex part load flow field. Albeit the
a) 𝑇𝐼 contributions at 𝑞 = 40 % b) 𝑇𝐼 contributions at 𝑞 = 120 %
[-][-]
𝜀 [ ] 𝜀 [ ]
a) 𝑇𝐼 contributions at 𝑞 = 40 % b) 𝑇𝐼 contributions at 𝑞 = 120 %
[-][-]
𝜀 [ ] 𝜀 [ ]
𝜀 = 0° 𝑞 = 40 % 𝜀 = 120° 𝑞 = 40 %𝜀 = 0° 𝑞 = 120 % 𝜀 = 120° 𝑞 = 120 %
𝑦/𝑏
𝑥/𝑡 [-] 𝑥/𝑡 [-] 𝑥/𝑡 [-] 𝑥/𝑡 [-]
𝑇𝑇/𝑇𝑃 [-]
0 1
12
local turbulence field is significantly mis-predicted in the vicinity of the volute tongue at part load, in
the far-from-tongue regions as well as for nominal and overload, a sound qualitative prediction of 𝑇𝐼
is obtained. What is more, the head and even local flow field in terms of flow angle are predicted
quite well by the present choice of the turbulence model for any operating point.
In further studies, more-advanced URANS approaches as non-linear eddy-viscosity models
should be evaluated by the present highly-resolved measurement data. The choice of scale-resolving
models (LES) seems particularly promising since the separation of resolved (𝑇𝐼𝑅𝑀𝑆) and averaged
(𝑇𝐼𝑇𝐾𝐸) velocity fluctuations is of much less significance as for statistical modelling. For that purpose,
the detailed measurement data by Meschkat and Stoffel (2002) and Hergt et al. (2004) will be further
exploited e.g. by the local velocity field also in the suction region of the impeller and by spatially and
temporally resolved pressure measurement data in the wake of the impeller.
ACKNOWLEDGEMENTS
The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-
centre.eu) for providing computing time through the John von Neumann Institute for Computing
(NIC) on the GCS Supercomputers JURECA and JUWELS at Jülich Supercomputing Centre (JSC).
The authors also gratefully acknowledge the financial support by the “Forschungskuratorium
Maschinenbau e.V.”. This project is funded by the Federal Ministry for Economic Affairs (BMWi).
Parts of the measurement data in electronic form were kindly provided by Gunther Treutz, Munsch
Chemie Pumpen GmbH. The authors also would like to thank the anonymous reviewers for their
valuable comments and suggestions to improve the quality of the paper.
REFERENCES Barrio, R., Parrondo, J., & Blanco, E. (2010). Numerical analysis of the unsteady flow in the near-
tongue region in a volute-type centrifugal pump for different operating points. Computers &
Fluids, 39.
Beaudoin, M., & Jasak, H. (2008). Development of a generalized grid interface for turbomachinery
simulations with OpenFOAM. Open source CFD international conference.
Byskov, R., Jacobsen, C., & Pedersen, N. (2003). Flow in a centrifugal pump impeller at design and
off-design conditions – part II: large eddy simulations. J. Fluids Engineering, 125, pp. 73-83.
Byskov, R., Jacobsen, C., Condra, T., & Sorensen, J. (2002). Large eddy simulation for flow
analysis in a centrifugal pump impeller. Advances of LES in Complex Flows, Fluid Mechanics
and its Applications, 65, pp. 217-232.
Fröhlich, J., & von Terzi, D. (2008). Hybrid LES/RANS methods for the simulation of turbulent
flows. Progress in Aerospace Sciences, 44(5), pp. 349-377. doi:10.1016/j.paerosci.2008.05.001
Hergt, P., Meschkat, S., & Stoffel, B. (2004). The flow and head distribution within the volute of a
centrifugal pump in comparison with the characteristics of the impeller without casing. Journal
of Computaional and Applied Mechanics, 5(2), pp. 275-285.
Issa, R. I. (1986). Solution of the implicitly discretised fluid flow equations by operator-splitting.
Journal of Computational Physikcs, 62(1), pp. 40-65.
Kato, C., Mukai, H., & Manabe, A. (2003). Large-eddy simulation of unsteady flow in a mixed-
flow pump. Int. Journal of Rotating Machinery, 9, pp. 345-351.
Menter, F. R. (1994). "Two-equation eddy viscosity turbulence models for engineering
applications". AIAA Journal, 32, pp. 1598-1605.
Meschkat, S., & Stoffel, B. (2002). The local specific head at different circumferential positions in a
volute casing centrifugal pump in comparison to the characteristic curve of the single rotor.
Proceedings of the Hydraulic Machinery and Systems 21st IAHR Symposium. Lausanne.
Meschkat, S. (2004). Experimentelle Untersuchung der Auswirkungen instationärer Rotor-Stator-
Wechselwirkungen auf das Betriebsverhalten einer Spiralgehäusepumpe. Doctoral dissertation,
TU-Darmstadt. Darmstadt, Germany. Retrieved from http://www.tubiblio.ulb.tu-darmstadt.de
13
Patankar, S., & Spalding, D. B. (1972). A calculation procedure for heat, mass and momentum
transfer in three-dimensional parabolic flows. International Journal of Heat and Mass Transfer,
15(10), pp. 1787-1806.
Pope, S. (2000). Turbulent Flows. Cambridge University Press. doi:10.1017/CBO9780511840531
Posa, A., Lippolis, A., & Balaras, E. (2015). Large-eddy simulation of a mixed-flow pump at off-
design conditions. Journal of Fluids Engineering, 137(10).
Posa, A., Lippolis, A., Verzicco, R., & Balaras, E. (2011). Large-eddy simulations in mixed-flow
pumps using an immersed-boundary method. Computers and Fluids, 47, pp. 33-43.
Shen, J., Li, Y., Liu, Z., & Tang, X. (2013). Turbulent flow and pressure fluctuation prediction of
the impeller in an axial-flow pump based on LES. 6th Int. Conf. on Pumps and Fans with
Compressors and Wind Turbines; IOP Conference Series: Material Science and Engineering,
52(032015).
Si, Q., Yuan, J., Wang, W., Zhu, L., & Boris, G. (2014, Februar). Numerical investigation of
pressure fluctuations in centrifugal pump volute based on SAS model and experimental
validation. Advances in Mechanical Engineering.
Spence, R., & Amaral-Teixeira, J. (2008). Investigation into pressure pulsations in a centrifugal
pump using numerical methods supported by industrial tests. Computers & Fluids, 37.
Spence, R., & Teixeira, J. A. (2009). A CFD parametric study of geometrical variations on the
pressure pulsations and performance characteristics of a centrifugal pump. Computers & Fluids,
38, pp. 1243-1257. doi:10.1016/j.compfluid.2008.11.013
Tang, X., Wang, F., & Wu, Y. (2007). An improved large eddy simulation of two-phase flows in a
pump impeller. Acta Mechanica Sinica, 23, pp. 635-643.
Tokay, T., & Constantinescu, S. (2006). Validation of a large eddy simulation model to simulate
flow in pump intakes of realistic geometry. Journal of Hydraulic Engineering, 132(12), pp.
1303-1315.
Van Leer, B. (1979). Towards the ultimate conservative difference scheme V: a second order sequel
to Godunov's method. Journal of Computational Physics, 32, pp. 101-136.
Wang, W., & Wang, Y. (2013). Analysis of inner flow in low specific speed centrifugal pump based
on LES. Journal of Mechanical Science and Technology, 27, pp. 1619-1626.
Warming, R.F., & Beam, M. (1976). Upwind Second-Order Difference Schemes and Applications
in Aerodynamic Flows. AIAA Journal, 14(9), pp. 1241-1249.
Zhang, W. Y. (2010). Numerical simulation of unsteady flow of in centrifugal pump impeller at off-
design condition by hybrid RANS/LES approach. High-Performance Computing and
Applications, Lecture Notes in Computer Science, 5938, pp. 571-578.