Paper ID: ETC2019-047 Proceedings of 12th European Conference on Turbomachinery Fluid dynamics & Thermodynamics ETC13, April 8-12, 2019; Lausanne, Switzerland

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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.

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