_3-Cavitation-Cavitation Prodiction in Hydraulic Machinery

Cavitation Prediction in Hydraulic Machinery

Author Firm / Institution City, Country Lecturer (x)

Moritz Frobenius, Rudolf Schilling Institute of Fluid Mechanics,

Munich University of Technology

Institute of Fluid Mechanics,

Munich University of Technology

Garching, Germany

Garching, Germany x

The paper presented consists of two parts. The first one deals with the basic investigations of

the prediction of the time-averaged leading edge cavitation and the head drop behaviour of a centrifugal pump having a specific speed of n q=26 1/min.

Résumé

Cet article est constitué de deux parties. La première partie traite de la recherche de base sur

l’écoulement avec cavitation en domaine dépendant du temps et autour d'un profil incliné en deux dimensions. Le but de cette recherche est de développer un modèle dynamique des bulles. La deuxième partie traite des applications de ce modèle sur des pompes centrifuge

pour prédire la moyenne temporelle des effets de cavitation sur le bord d'attaque et la diminution du hauteur du refoulement. La pompe aura une vitesse précise de n q=26 1/min. Nomenclature

Symbols:

f frequency Str = f·l / u Strouhal number

H head T temperature k turbulent kinetic energy Tu turbulent intensity

l length t time

n speed of rotation u velocity

n cell number of bubbles per computing

cell u in inflow

velocity

n0number of bubbles per m3 fluid V v cell volume

occupied by vapor

NPSH Net Positive Suction Head V fl cell

volume

occupied by fluid

NPSH3%Three percent head drop V cell total cell volume NPSH IC incipient head drop x space coordinate

p static pressure y+ dimensional wall

distance

p B static pressure at the bubble

αvapor void fraction boundary

p0static pressure at the suction

βangle of attack nozzle

p s1, p s2static pressure at the inlet, outlet εdissipation rate

p∞static pressure at the ambient cellμmolecular viscosity pressure μt turbulent viscosity p v vapor

Q Flow rate ρdensity

radius σ=(p-p v)/(1/2·ρ·u2) cavitation number R bubble

Re = ρ ·u·l / μReynolds number τviscious stress

tensor Subscripts:

cor corrected v vapor

l liquid ∞reference value Opt at design point of pump in inflow

Introduction

In the design and use of hydraulic turbo-machines cavitation and the effects related to cavitation are playing an important role. Of special interest are the mechanisms of cavitation yielding to loss of efficiency, erosive damage and noise generation of the exposed surfaces. To avoid or to reduce the effects of cavitation by design and operation, there is a persistent need of improving the detailed understanding of the physical phenomena and their modelization for numerical calculations.

Thus, a cavitation model has been integrated in the CFD-Code NS3D developed at the Institute of Fluid Mechanics, Munich University of Technology. The CFD-Code solves the Reynolds-averaged Navier-Stokes equations coupled with a single fluid cavitation model. For modeling the cavitation the bubble dynamic approach developed by SAUER (Ref 1)is applied, which is based on a modified Volume of Fluid (VoF)-Method and considers the cavitating structures as a homogeneous liquid-vapor mixture, with a no-slip-condition between the two phases.

Cavitation model

(),

0)(=??+??i i u x t

r ρρ (1) .)()()(i

ij i i j i i x p x u u x u t ??+??

?=??+??τρρr r r (2)

The constitutive relations for the density and dynamic viscosity of the mixture are:

()().

1,1l v l v μαμαμραραρ??+?=??+?=

(3) The subscripts l and v stand for the properties of pure liquid and pure vapour, which are supposed to be constant.

Additionally, a transport equation for the vapour fraction α is required. The vapour void fraction α can be written as

,13

403

340334R

n R n V V R n V V l v cell cell v ππα?+?=+?== (4) where V cell is the volume of the computational cell, V v and V l are the volumes occupied by vapor and liquid, n cell is the number of bubbles in the computational cell, whereas n 0 is the number of bubbles or nuclei per cubic meter. That means n 0 is a constant parameter depending on the considered liquid.

The bubble growth is described by the Rayleigh-relation:

l

B B B p T p p T p p T p dt dR R

ρ∞∞∞???==)(32)()(& (5) Here, p B (T) is the pressure in the liquid at the bubble boundary which is assumed equal to the vapour pressure p v , which depends on the temperature. p ∞ is the ambient cell pressure.

().1413

34020dt dR R

n R n dt d παα+?= (6) The transport equation for the void fraction is extended by a source term on the right hand side:

().)(i i

j i u x dt d x u t r

??+=??+??α

ααα (7)

Because of the bubble growth, the velocity field is no longer divergence-free. As proposed by

SAUER (Ref 1) or SPALDING (Ref 13), the continuity equation is used in its non-conservative form:

().11dt d dt d x u t u x v l i i

i i αρρρρρρρρ?≈?=???

???????+???=??r r (8) With this expression and eqn. 6, the transport equation for the void fraction can be written as

().1)(3

3

4

33400R dt d

R

n n x u t j i ππαα?+=??+??

(9)

The source terms for the vapour fraction equation and for the pressure correction equation have the same form. The continuity equation can be used in its incompressible form, only an additional source term has to be considered. A more detailed description of the

implementation of the cavitation model can be found in FROBENIUS ET AL. (Ref 3).

Numerical procedure

The cavitation model has been integrated in the CFD-Code NS3D, developed at the Institute for Fluid Mechanics, Munich University of Technology. The basic code NS3D is based on a co-located, cell-centred and block-structured finite volume method using the SIMPLE algorithm, see Ref 1. For the interpolation of the mass fluxes at the cell faces, RHIE &

CHOW’s method is used, see Ref 4. The discretization of the convective terms is realized by the second order MINMOD-scheme of HARTEN (Ref 5). The code is able to calculate general multi-block topologies with matching and non-matching interfaces in 2D and 3D domains. The parallelization of the code is realized by means of MPI-libraries (Message Passing Interface), and the sets of linear equations are solved using the Strongly Implicit Procedure of STONE (Ref 6). For the simulation of cavitating flows a linear k-ε turbulence model, see Ref 7, assuming a density variable, but incompressible flow field together with the realizability condition for the normal stresses suggested by MOORE & MOORE (Ref 14). For unsteady simulations a three-level implicit time-discretization-scheme is used, so the time step is not limited. A detailed description of the code can be found in SKODA (Ref 9).

2D Unsteady cavitating flow

To validate the developed code with experimental data, the 2D flow around a hydrofoil with a semi-circular leading edge and the 3D flow around a hydrofoil with swept leading edge are computed. The simulations were carried out for an angle of attack β = 5° and different values of the cavitation number between σ = 2.0 and σ =2.7 to vary the characteristics of the

cavitation occurring on the suction side of the hydrofoils. Inflow velocities of u in = 13 m /s and u in = 16 m /s corresponding to Reynolds-numbers of Re = 1.3·106 and Re = 1.6·106 respectively were assumed. The experimental investigations performed with this profile are described in Ref 10 and Ref 11.

The flow is assumed isothermal and fluid properties are supposed to be constant at a given temperature for the entire flow domain. For the simulations presented here, cold water at a constant temperature T = 296 K with 108 nuclei per m 3 water having a minimal nuclei radius of 30 microns is assumed to match the experimental conditions. The vapor density was set to a constant value of ρv = 0.1 kg /m 3 and the vapor pressure to p v = 2809 Pa corresponding to the water temperature T= 296 K . The incoming fluid has a turbulence intensity of Tu = 2%. At the inlet of the computational domain the flow rate is prescribed, at the outlet a “Dirichlet”-condition for the static pressure is assumed.

For the simulation of the cavitating flow around the hydrofoil a 4-block structured grid was used, see Figure 1. The inner O-grid around the hydrofoil is connected to its neighbour-block by a non-matching block-interface, which allows refining the inner block without changing the size of the other blocks. A grid-independent solution is achieved with 237 x 60 cells for the inner block. Figure 1 Multi-block grid topology for the simulation of the CLE-hydrofoil.

An example of the simulated unsteady behaviour of the 2D cavitating flow around the hydrofoil with an angle of attack of 5° is shown in Figure 2. Here successive shapes of the cavitation zone obtained during a complete cavitation cycle are shown. The flow is from left to right. The plotted velocity vectors show the development of the re-entrant-jet and the

shedding of the cavitation cloud.

Figure 2 Predicted vapor void fraction and velocity vectors during one period of the unsteady cavitation loop. CLE-foil with Re = 1.3 · 106, σ=2.0.

The unsteady cloud shedding only occurs, when the realizability condition is used. By applying the standard k-ε-model without any correction, the simulation leads to a steady solution without bubble cloud shedding. The reason for this is the so-called stagnation point anomaly. With the standard k-ε-model the turbulent kinetic energy in the stagnation point area is highly over-predicted, see Figure 3. The over-estimated turbulent kinetic energy is

transported by the flow along the profile wall, which leads to a very high turbulent kinetic energy and with it a high turbulent viscosity in the area of the re-entrant-jet. This prevents the development of the re-entrant-jet. By the use of the realizability condition, see Figure 4, the

L=107 mm

I II III

IV

Inflow

turbulent kinetic energy is predicted properly and the unsteady behaviour of the cloud

cavitation can occur.

Figure 3 Turbulent kinetic energy near the CLE-profile predicted by the standard k-ε-model. Figure 4 Turbulent kinetic energy near the CLE-profile predicted by the standard k-ε-model using the realizability-condition.

The self-oscillating behaviour shows a characteristic frequency depending on the cavitation number. The experimental investigation of HOFMAN (Ref 10) shows Strouhal-numbers in the range of Str = 0.2 … 0.25, depending on the flow velocity. A comparison of the predicted and measured frequencies of the cloud shedding is shown in Figure 5. Generally, the

predicted frequencies are slightly lower than the measured ones, but show the same tendency:

For increasing cavitation number the frequency of the cloud shedding process also is rising.

Figure 5 Influence of the cavitation number σ on the frequency of the bubble cloud shedding and comparison with experimental data; Re = 1.3 · 106.

A similar profile, with a swept leading edge, was also investigated. Experimental and numerical results of this research can be found in Ref 11 and Ref 12.

Cavitating flow through a centrifugal pump impeller

Unsteady simulations of the cavitating flow are very time-consuming, so for industrial use steady-state-simulations are preferred. For steady-state simulations the source term in the

vapour phase fraction transport equation is modified to account for the different time-scales of the bubble growth and collapse:

()??

?=?+=??+??collapse

bubble for 50/1growth bubble for 50

with ,

1)

(3

3

43

3

400cor cor j i C R dt d R n n C x u t ππαα (10)

Geometry and computational grid

The radial test impeller, see Figure 6, has meridional contours and a 3-dimensional blading which are typical for serial centrifugal pumps. The impeller has a specific speed of n q =261/min . The test impeller is installed in a special pump casing, see Figure 7. The annular casing with 12 outlet openings equally distributed along the circumference was designed to avoid any asymmetry of the outflow conditions for the test impeller. Experiments were performed with a suction side gap width of s = 0.25 mm and s = 0.5 mm .

For the simulation of the cavitating flow through the radial impeller a free impeller calculation was performed, which means, that only one flow channel delivering into the vainless radial diffuser is considered. The shape of the meridional contour, the meridian plot and the conformal mapping of the 3D grid used for the simulation is shown in Figure 8. The grid consists of 112,000 cells; the dimensionless wall distance y+ was set to values between 30 and 300, so that the logarithmic wall law could be applied. To investigate the influence of the suction side gap flow, additionally a multi-block grid including the suction side gap was generated, see also Figure 8.

Figure 6 Test pump RP26. Figure 7 Pump impeller n q =261/min .

Figure 8 Meridional section of the radial impeller (top left); meridional section of the grid without and with considering the suction side gap (top centre and right); conformal mapping of the computational grid.

Head drop curves

For the simulation of the cavitating flow, first a stationary calculation at a high pressure level is performed to ensure non-cavitating conditions in the whole computational domain. Then, the outflow pressure is lowered in small steps. During this process, vapour structures appear in the regions of low static pressure. The more the pressure level is lowered, the more the cavitation zone grows, influencing the impeller head.

Simulations are performed for rotational speeds of n=1750 1/min and n=2000 1/min , the design flow rate being Q = Q opt = 126 m 3/h and Q = Q opt = 143 m 3/h , respectively. The flow rate is set by the inflow boundary condition; the static pressure is defined at the outlet boundary. The head H,

,1

2g

p p H s s ??=

ρ

(11)is computed from the CFD simulation results by averaging the static pressure at the inlet p s1 and the outlet p s2 of the computational domain. The NPSH-value is defined as the total

pressure p 0 of the fluid at the suction nozzle above the vapor pressure p v of the fluid depending on the fluid temperature T, divided by ρ·g :

g

u g p p NPSH in

sat 22

0+

?=ρ (12)

The computed relative head drop H/H 0 for a rotational speed of n=1750 1/min and n=2000 1/min

considering the respective design point Q opt is compared with the experimental results for the small suction side gap width of s = 0.25 mm in Figure 9. In general, the deviation between the predicted and the measured head drop curves has the order of magnitude of the measuring uncertainty. The predicted head drop occurs almost at the same NPSH-value as the measured one.

Figure 9 Predicted and measured head drop curves for n=1750 1/min and n=2000 1/min and Q=Q opt .

Figure 10 shows the comparison of the predicted and measured NPSH 3% depending on the relative flow rate and the computed NPSH IC -values for n=1750 1/min and n=2000 1/min . The predicted values for incipient cavitation and for the three percent head drop correlate well with the experiments for the flow rates investigated.

Figure10 Comparison of the predicted and measured NPSH 3%-curves and computed NPSH IC -values for n=1750 1/min (left) and n=2000 1/min (right).

Analysis of the head-drop

A closer look at the head-drop curve for n=2000 1/min and Q= Q opt =143 m 3/h will give some information about the causes of the head-drop, see Figure. The inception of the cavitation occurs at the suction side at NPSH= 3.65 m . The influence of the suction side cavitation on the pump head is very small. At NPSH= 2.57 m cavitation starts to occur at the pressure side for the first time. The distinct head-drop sets in, because the pressure side cavity occurs at the throat of the impeller. At NPSH=2.46 m a head-drop of 3% is reached. The cavitation zones both at the suction- and at the pressure-side of the blade then grow while the pressure level is lowered. When the suction side sheet cavity also reaches the throat and leads to a blockage of

a part of the blade channel, the pump head completely brakes down.

NPSH = 3.65 m H/H 0 = 100 %

NPSH = 2.80 m H/H 0 = 99 %

NPSH = 2.46 m H/H 0 = 97 %

NPSH = 2.36m H/H 0 = 95 %

Figure11 Development of the cavitation zones in the blade channel near the shroud.

Influence of the suction side gap flow

For a huge gap width of s ? 0.5 mm the suction side gap flow can change the development of the cavitation. During the experiments, see Ref 11, it was observed, that for higher gap width of s =0.5 mm cavitation occurs at the pressure side of the blade and also between the suction side gap and the throat.

To investigate the influence of the suction side gap flow on the cavitation, a multi-block grid considering the gap area with a gap width of s = 0.5 mm has been generated, seeFigure 8. The fluid flows from the suction side gap into the suction area of the pump with a spin due to the rotation, see Figure 12. This has a major impact on the incidence angle at the leading edge of the blade and due to this also on the cavitation. A vortex occurs between the suction side gap and the leading edge of the blade. In the centre of the vortex the static pressure is very low. The change of the pressure distribution is also shown in Figure 13. Due to the gap flow, a pressure peak at the pressure side of the blade occurs.

Figure 12 Impact of the suction side gap flow on the incidence angle at the leading edge of the blade. Figure13 Pressure distribution with and without consideration of the suction side gap flow at midspan.

Due to the changed incidence due to the suction side gap flow, cavitation occurs mainly on the pressure side. Because of the vortex, cavitation also developes in the centre of the vortex, which originates past the suction side gap, see Figure 14. Suction side

Pressure side

Figure 14 Cavitations zones predicted by the simulation without and with considering the suction side gap flow. Left: near the suction side; right: near the pressure side of the blade; n=2000 1/min ; Q = Q opt = 143 m 3/h.

Conclusion

A bubble dynamic cavitation model has been integrated in the NS3D-Code. Unsteady and

steady simulations have been performed for cavitating flows around a hydrofoil. Based on

these investigations, the cavitating flow through a centrifugal pump impeller has been simulated in a good agreement with the measurements.

The results of the unsteady simulation show the typical development of a re-entrant-jet and the bubble cloud shedding. The predicted shedding frequencies of the considered hydrofoil agree well with the experimental findings. Also the flow through a radial pump impeller

n q=26 1/min has been simulated. The computed head-drop curves and NPSH3%-values coincide very well with the experiments. A significant influence of the gap flow on the development of the cavitation zone has been found.

References

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