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Prediction of cavitation inception in slots
International Journal of Multiphase Flow 119 (2019) 217–222
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International Journal of Multiphase Flow journal homepage: www.elsevier.com/locate/ijmulflow
Prediction of cavitation inception in slots E.L. Amromin Mechmath LLC, Federal Way, WA 98003, United States
a r t i c l e
i n f o
Article history: Received 11 February 2019 Revised 26 May 2019 Accepted 24 July 2019 Available online 29 July 2019 Keywords: Cavitation inception Surface irregularities Numerical prediction
a b s t r a c t Cavitation inception in slots on the flat walls is analyzed in the steady flow approach using a viscousinviscid interaction method. It was assumed that, in the accordance with known observations, cavities appear as spherical bubbles in the cores of vortices near the slot corners. Computed cavitation inception numbers are compared with published experimental results of other authors within wide ranges of inflow parameters. The comparison shows a satisfactory agreement of computations with the measured data. © 2019 Published by Elsevier Ltd.
1. Introduction
2. Definition of cavitation inception number
Cavitation inception has been the topic of intensive experimental studies for many decades. One of the best reviews of the early studies was prepared by Lindgren and Johnson (1966) more than a half century ago. The diversity of experimental data in this review did not allow for determination of certain dependencies. Also, the substantial scale effects were seen there. No computational tool for cavitation inception existed at that time to support a theoretical analysis. Computational tools capable to analyze initial forms of cavitation were developed several decades later. The successful predictions of cavitation inception numbers σ I for bodies/hydrofoils of smooth surfaces made with the employment of a viscous-inviscid interaction method (as by Amromin et al., 1986) and the methods for a fully turbulent mixture (as by Coutier-Delgosha et al., 2007) were reported. For surface irregularities in the form of lugs/obstacles mounted on a wall, the successful prediction of σ I was reported by Amromin (2016), but the flow model used in that study was irrelevant to surfaces with slots or with very small gaps between irregularity elements. This study is aimed on computation of cavitation inception number for slots. The detailed observations of slot flows by Liu and Katz (2013) are used to elaborate the flow scheme. The employed computational method can be considered as a modification of the method described in Amromin (2016). Its validation is carried out with the experimental data published by Arndt et al. (1979) and by Liu and Katz (2008).
First of all, it is necessary to precisely define the cavitation inception. A student would say that cavitation inception is the appearance of a region filled by gas (usually, by the vapor) within a liquid and that this region was developed either from imperfections on the body surface, as discussed in the monograph of Knapp et al. (1970), or from a nucleus existing in the inflow, as noted in compendium edited by Dern et al. (2015). A student also would recall that inception can be characterized by a jump of the vapor volume fraction in a small region, but because of the difficulty to observe randomly appearing cavities determination of cavitation inception number σ i has been frequently substituted by determination of desinence of the already existing cavities due to the gradual decrease of the cavitation number σ . In such a way a set of observations of steady flows has been used to characterize a rather unsteady phenomenon. An engineer would first recall a jump in the noise radiation as the most important effect assisting cavitation inception. The old practice of visual determination of cavitation inception is not in the controversy with the prediction of such a jump because the radiated energy is proportional to the cavity volume variations. However, the observed σ i and cavitation desinence number σ d substantially depends on the vision of an observer. Generally, there is no established lower threshold of this jump. The more modern techniques as holography (used by Katz (1984), f. i.) allow for the more accurate optical determination of appearing cavities than the direct visualization has allowed for. However, this does not help to define the difference between the smallest cavity and the biggest nucleus. There is no established minimal cavity size even in the papers of Meyer et al. (1992) or of Waniewski and Brennen (1999), where registration of cavitation events was
E-mail address: [email protected] https://doi.org/10.1016/j.ijmultiphaseflow.2019.07.017 0301-9322/© 2019 Published by Elsevier Ltd.
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E.L. Amromin / International Journal of Multiphase Flow 119 (2019) 217–222
described. The measured distributions of bubbles by their size in the books by Blake (1986) and by Dern et al. (2015) or in the review by Arndt (1981) manifested the increase of the number of nuclei with the decrease of their size. However, how to distinguish the nuclei from the smallest cavities? The absence of the unique definition of cavitation inception number urges us adjust the employed definition to the conditions of the experiments selected as validation tests. In particular for the experiment of Liu and Katz (2008), there are photos of incipient cavitation allowing for the assumption that the radius RC of an incipient cavitating bubble is between 0.1 mm and 0.15 mm. For the observation of Arndt et al. (1979), it could be greater. As stated by Liu and Katz (2008), “observations on occurrence of cavitation reveal that the first site of cavitation inception is located on top of the downstream corner of the cavity, regardless of the free stream speed and dissolved air content in the water”. Taking into account surface tension, one calculates cavitation number 2 for a bubble as σ = 2(P∞ − Pc )/ρU∞ 2 σ = −CP − 4χ /ρU∞ RC
(1)
The maximum value of σ can be considered as σ I . Here PC is the pressure inside the cavity, P∞ is the inflow pressure, U∞ is the free stream speed and χ is the surface tension coefficient. The dimensionless pressure coefficient CP must be calculated with taking into account the vortex contribution because cavitation in separation zones appears in the vortex cores. If the core radius R exceeds RC , the pressure in the cavity must be calculated as
PC = P (R ) + ρ
R
RC
uˆ2 2χ dr + , r RC
(2)
The pressure P(R) is the difference of some unperturbed pressure at the vortex location and [ /(2π R)]2 . For turbulent flows, as shown by Amromin (2007, 2016) with the use of an exact solution of Reynolds equation in a region with insignificant variation of turbulent stress component u v , the circumferential velocity in the core is
uˆ (r ) = W (1 − ln |r/R| )r/R,
(3) √
√
where W = Г/(2π R) and R = ν / −π < u v >. As illustrated by Fig. 1, this core is in much better agreement with the known experimental data than the broadly used Rankine core is. As shown in Fig. 2, the difference in velocity distributions in the vortex core results in the significant difference of pressure in vortex cores. As shown by Agarwal (2010), employment of Eq. (3) allows for the much better prediction of cavitation inception num-
Fig. 1. Theoretical (from Amromin, 2007, solid curve) and experimental (from Arndt et al., 1991 and Castro et al., 1997), symbols) circumferential velocities in vortex cores.
Fig. 2. Comparison of the pressure drop within laminar and turbulent cavitationfree vortex cores (solid lines) and within a cavity of the radius r in turbulent core (dashed line); dP = P (r ) − P (R ).
ber in vortices. However, calculation of σ I with employment of Eqs. (1)–(3) requires determination of Cp, Г and u v . Further, in the majority of experimental studies the vapor pressure PV is employed instead of the actual cavity pressure PC in calculation of σ . As described by Wide and Acosta (1966) longtime ago, the difference PC −PV significantly increases with the increase of inflow air content. The impact of this difference on values of σ is evident, but experimental determination of pressure in appearing cavities is too difficult. So, a correction to σ (like one derived by Amromin (2016), f. i.) should be used to take into account this impact must be used during comparison of the computed results with the experimental data. 3. Scheme of separated flow around a slot The separated flow around a slot may seem similar to the flow behind a backward facing step. Indeed these flows are similar for the ratios L/H > 7, where L and H are length and depth of the slot, because for L/H > 7 reattachment of separated boundary layer to the slot bottom can occur. Otherwise, two separated flows are dissimilar and reattachment take place just downstream of the slot, as one can understand with flow scheme in Fig. 3. In the experiments of Liu and Katz (2008) cavities appear in vortices just upstream of the slot trailing edge (as shown in their Fig. 3) and this fact is reflected in the employed scheme. For L/H ∼ 1, the bottom vicinity has a little influence on other parts of flow (and these parts were excluded from the measurements of Liu and Katz). Therefore, this flow scheme and the employed viscous-inviscid interaction computational method has derivations from the scheme and method employed by Amromin (2016, 2018) for prediction of cavitation inception in others separated flows. On the other hand, in spite of the general unsteadiness
Fig. 3. Scheme of flow around a slot; doted curve is section of displacement body, dashed curve limits separation zone.
E.L. Amromin / International Journal of Multiphase Flow 119 (2019) 217–222
of cavitation inception phenomenon, the corresponding mathematical problem is also considered as the time-independent problem. In such problems, as shown by Amromin and Arndt (2019), the hypothesis on the cavity content has a little impact on their solutions. As usual for viscous-inviscid interaction methods, the entire flow is divided into parts. In this problem these parts include turbulent boundary layer, inviscid flow over it, a separation zone under it and a stationary vortex located in the slot near its trailing edge. The inviscid flow influences the boundary layer by pressure distribution, but the pressure coefficient Cp is computed at the surface S of body displacement. The pressure within the vortex core also depends on this coefficient. On the other hands, the displacement thickness δ ∗ predetermining shape of S depends on the boundary layer growth. Over the slot this growth depends on the water flux through the frontier between separation zone and boundary layer (on the velocity v normal to this frontier).
Here h is the distance between initial and corrected shapes of S. One should substitute Eq. (9) into Eq. (8) and integrate it with the initial condition h(0) = 0. The Eq. (9) was derived using inversion of the Cauchy singular integral appearing in the original form of quasi-linearized Eq. (6) written with the use of a potential of sources and sinks of the assumable small intensity q(x). However, as one can find in Zabreyko et al. (1975), this inversion exists only with the additional condition
X3
[C + C1 f1 (x ) + C2 f2 (x ) − U0 (x )]dx
0
= 0
(4)
with the condition
∂ =0 ∂ N S
(5)
C2
δ∗
d f2 − a f2 dx
Here U = C + C1 f1 (x ) + C2 f2 (x ) is defined with the undetermined coefficients C, C1 , C2 and with the use of functions f1 = 2 min{1; (x/X2 ) (3 − 2x/X2 )}, f2 = [max{0; 3ξ 2 − 2ξ 3 )}], the x axis is directed along the wall and counted from the slot leading edge, ξ = (x − X2 )/(X3 − X2 ), X2 = 0.6 L, X3 = 1.1 L. This approximation of U reflects the typical behavior of pressure in separation zones. The problem with boundary conditions (5)–(7) for linear Eq. (4) is nonlinear even for the known values of these coefficients because two conditions must be satisfied on the same part of the boundary S. Therefore, the shape of S must be fitted in iterations. These iterations include two steps. The first step consists of determination of U0 = grad( ) after solving Eqs. (4), (5), (7) out of the considered known approach to S. This step can be solved using various boundary element methods. The second part consists of determination of the necessary deformation of S by solving quasi-linearized Eqs. (5) and (6) that can be written here as
d (hUh ) q = , dx 2
q (x ) =
2 x(X3 − x )
π
(8) 0
X3
[C + C1 f1 (z ) + C2 f2 (z ) − U0 (z )]dz
(x − z ) z(X3 − z )
.
(9)
δ
(11)
[1 − u¯ ]dy, δ is the boundary layer thickness,
dU dδ ∗ ∗ (δ ∗ +2δ ∗ ∗ ) + U 2 + v(U − u0 ) = 0. dx dx
Here δ ∗∗ =
(7)
− bC = bC1
y is counted across this layer, the velocity profile u¯ = u0 /U + (1 − u0 /U )(3η2 − 2η3 ) is used in the separated boundary layer, η = y/δ , u0 is the tangent velocity component at y = 0, C1 = 0.01 and b = const. So, C and C2 can be found from Eqs. (10)–(11), but the values of δ ∗ (L) must be found by integration of boundary layer equations. There are two parameters in the velocity profile in the separated layer. These parameters are the layer thickness δ (x) and u0 (x). Therefore, two equations are necessary for its description. The first one is the momentum integral equation
|grad ( )||Sˆ = U
grad ( )|x→±∞ → 0.
(10)
0
U
for tangent velocity on separation zone upper boundary. There is also condition of perturbation absence in the infinity
= 0.
Here a = const.,δ ∗ =
for the normal velocity component on the inviscid flow boundary and the condition
(6)
x(X3 − x )
The condition of boundary layer reattachment at x = L is similar to the widely used criterion of boundary layer separation (see Castillo et al. (2004), f. i,) because both of them are conditions at the point of contact between attached and separated boundary layers. Taking into account the approximation used for U, one can write this condition as
4. Computational method Generally speaking, prediction of cavitation inception requires a multi-zone multi-scale analysis, as manifested by Amromin (2018) or by Hsiao et al. (2017). The diverse parts of the flow should be analyzed using diverse physical assumptions and equations. The inviscid flow is assumed to be curl free and velocity potential can be defined there. The boundary-value problem for this potential includes the Laplace equation
219
δ
(12)
u¯ [1 − u¯ ]dy and the function v(x) is ordered with re-
0
gards to the experimental data of Liu and Katz (2013). This equation is a modification of the well-known von Karman equation. The third term in its left-hand side is usually zero because usually the ordinate y is counted from the rigid walls, where v(0) = 0. However, for this separated layer integration starts from a line inside of fluid. The second one is the differential momentum equation considered along this line
d u0 (U − u0 ) dU +U =A . dx U δ∗ dx 2
u0
(13)
The initial conditions for Eqs. (12), (13) are u0 (0) = 0 and
δ ∗ (−0) = δ ∗ (+0), where δ ∗ (−0) is the known value, and the coefficient A must be fitted to meet the condition u0 (L) = 0. As shown by Amromin (2018), the employed three-term approximation of U allows for a good agreement with measured pressure distributions in separation zones behind steps. Comparison of the computed result with the pressure distribution measured by Liu and Katz (2013) in Fig. 4 shows such an agreement for slots as well. As illustrated by Fig. 5, the employed description of separated boundary layer is in the satisfactory agreement with the experimental data of Liu and Katz (2008, 2013). The maximum values u0 /U∞ was about 0.26. This value is also in the range of experimental data. Finally, it is necessary to compute σ using Eq. (2) for the selected value of RC . As noted above, computation of pressure for this equation requires know the vortex intensity . Besides there is the
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Fig. 4. Computed (line) and measured (squares) pressure coefficients along the slot at y = 0 at inflow speed 5 m/s.
Fig 6. Comparison of computed and observed (by Liu and Katz) cavitation inception number. Lines 1 and 2 provides maximum and minimum of observed σ I for the basic level of dissolved oxygen, triangles show average σ I for fivefold higher oxygen level. Line 3 shows computed σ I for Rc = 0.1 mm, line 4 –for 0.15 mm. Line 5 shows σ I + dσ fitted to the basic oxygen level using line 4, line 6 shows σ I + dσ recalculated from line 4 for fivefold its level.
Fig. 5. Computed (line) and measured (squares) velocity along the line ∂ u¯ /∂ y = 0 at inflow speed 5 m/s.
R-dependent integral in the right-hand side of Eq. (2). Moreover, R also depends on . For determination of vortex intensity the approximation
= Duδ3
(14)
with δ 3 (x) = δ ∗ (X2 )−δ ∗ (x) and a constant D was used. Its righthand side is calculated at x = 0.95 L. The difference between Eq. (14) and the approximations used by Amromin (2016) for surface irregularities mounted on walls is associated with the different boundary layer effects. The non-uniformity of velocity across boundary layer has a little impact on the flows within slots. Rather variation of its thickness along the slot is important. The value D = 1.0 was fitted as basic for this coefficient. 5. Computational results and consideration of uncertainties First of all, it would be desirable to compare the obtained computational results with computational results of computations of other authors. Computational studies of cavitation inception compose a very small fraction of numerous papers on numerical analysis of cavitation. Possibly, only Shams and Apte (2010) published a LES computational result for slot cavitation inception. They calculated σ I for conditions of Liu and Katz (2008) experiment using the unexplained assumption σ I = –CP −CPRMS . Their prediction gave σ I = 0.93 at U∞ = 5 m/s, but their computed averaged (timeindependent) pressure coefficient and its RMS substantially overestimates observed both |CP | and CPRMS (in particular, giving threefold higher CPRMS than was measured) and neglected by surface tension. These results deserve some comments. Generally speaking, cavitation inception is an unsteady phenomenon. In the particular case of cavitation in the slot, cavitation appears via capturing randomly drifting nuclei by about stationary vortices (vortex shed-
ding behind surface-mounted irregularities is not similar to the vortex behavior in the slot). However, as noted by Harsha (1977), a single computation of unsteady turbulent flow is not informative enough because some randomly perturbed initial conditions must be used on the boundaries of the region of computation and some averaging must be applied to a set of numerical solutions to obtain results comparable to experiments. On the other hand, as noted by L. I. Sedov, the averaging procedure is the most important factor in computations of turbulent flows. Therefore, RANS and LES methods can give expectably different results and may have substantial disagreements with experimental data for some flows. Further the comparisons with experimental data were considered. Let us recall that the inflow characteristics in these experiments were time-independent. Therefore, cavitation inception in this case is rather quasi-steady phenomenon under a random impact and it was allowable to determine the flow characteristics in the framework of steady theory. The results of the above-described computations are plotted in Fig. 6 together with the data of Liu and Katz (2008). The computations were carried out for two values of Rc that look consistent with their photos of appearing cavities. Let us discuss the computational uncertainties. Purely computational uncertainties inherent to the above-described viscous-inviscid interaction method were already evaluated by Amromin (2002). Another uncertainty is associated with definition of Rc corresponding to incipient cavities. Uncertainties (discrepancies) of the corresponding measurements for the basic (low) inflow air content α are also shown in Fig. 6 (this discrepancy can be comprehended as the result of random distribution of nuclei in the inflow). The experimental data were presented by these authors versus cavitation number calculated using vapor pressure. Therefore, a correction to such a cavitation number should be employed for comparisons with the experimental data and instead of computed σ I the sum σ I + dσ should be employed in the plot. Here
dσ =
2(PC − PV ) α ∼ 2 2 ρU∞ U∞
(15)
The proportionality of dσ to α and its inverse proportional2 has been noted by many researchers starting at least ity to U∞ from Hammit (1980). Recalculation of the computed dependency
E.L. Amromin / International Journal of Multiphase Flow 119 (2019) 217–222
221
core, the weak point of described computational method is in definitions of the coefficients D and u v . The integral method employed for boundary layers does not allow for determination of these coefficients. Meanwhile, viscous-inviscid interaction methods allow also for computation of boundary layers with more advanced differential approaches, like it was manifested since Kwon and Pletcher (1986). A satisfactory computation of u v was manifested by Mokhtarpoor et al. (2016), f. i., in a quite similar separated flow (within a recess on the wall). Also, it may be easier to √ √ compute directly R and find then using the formula R = ν / −π < u v >. 6. Conclusions Fig. 7. Cavitation inception number versus the parameter with ε = 0.041 and λ = 0.51. ; rhombs – values measured by Arndt et al. (1979); lines – computed dependencies for various Rc .
for Rc = 0.15 mm with the use of Eq. (15) and a proportionality coefficient put the recalculated dependency practically within the strip of observed cavitation inception numbers. Its recalculation for much higher inflow air content (for the fivefold higher oxygen content) gives a satisfactory approximation of the related experimental data as well. No information on discrepancy of measured σ I was noted for the experimental data Arndt et al. (1979) for slots at various L = H. These data were results of observations with an unknown observer vision capability. This urged to carry out computations for various Rc . As seen in Fig. 7, the computations for Rc = 0.25 mm are in a good agreement with the experimental data. The comparisons in this plot are provided versus the aggregate variable = (H/δ )ε (U∞ δ /ν )λ introduced by Arndt et al. (1979). Their data manifested a very weak dependency on H/δ for slots (ε = 0.041), unlikely to their data for the wall-mounted surface irregularities (approximated using 0.36 < ε <0.74). The correction (15) was not applied to the computed dependencies for these slots because of very low total inflow air content in the experiments. The above-described computations were carried out using D = 1.0 and u v = 0.005. The sensibility of computational results to variation of these coefficients is illustrated by the computations presented in Fig. 8. Inaccuracy in values of u v are less influential. With the proven general feasibility of the employed multizone scheme of cavitating flow and description of vortex viscous
Fig. 8. Impact of coefficient variation on the computed σ I for a slot with L = 12.7 mm, Rc = 0.5 mm and δ ∗ (0) = 0.1 L. Solid curve corresponds to D = 1.0 and u v = 0.005, dashed curve - to D = 1.0 and u v = 0.0025, dashed-dotted curve – to D = 0.75 and u v = 0.005.
Cavitation inception in slots on the flat walls was analyzed in the steady flow approach using the viscous-inviscid interaction method. This method is a modification of the method earlier employed by the author for prediction of cavitation inception behind surface irregularities mounted on the walls. The modification was caused, first, by a relatively weak influence of the wall boundary layer thickness on the flows in slots and, second, by the about stationary position of incipient cavities; in the accordance with the known observations, these cavities appear as bubbles in the cores of quasi-stationary vortices near the slot trailing corners. Computed cavitation inception numbers were compared with the published experimental results of other authors within the wide ranges of inflow parameters. Impact of the size of smallest observed cavities on predicted cavitation inception numbers was manifested. The effect of inflow air content on experimental data and its influence on comparison of numerical results with the observations was discussed and illustrated by computational examples. The sensibility of computational results to the employed approximation coefficients was shown. The comparison shows a satisfactory agreement of computations with the measured data. Though the feasibility of suggested computational scheme with the emphasis on the turbulent vortex core and on the surface tension influence was proven, the path of the following enhancement of applied numerical methods was also discussed.
References Agrawal, A., 2010. Analysis of vortex cores in turbulent flows. J. Turbul. 11, 1–9. Amromin, E.L., 2002. Scale effect of cavitation inception on a 2D eppler hydrofoil. ASME J. Fluids Eng. 124, 186–193. Amromin, E.L., 2007. Analysis of vortex core in steady turbulent flow. Phys. Fluids 19, 118108. Amromin, E.L., 2016. Analysis of cavitation inception and desinence behind surface irregularities. Phys. Fluids 28, 075106. Amromin, E.L., 2018. Prediction of cavitation inception within regions of flow separation. ASME J. Fluids Eng. 140, 011103. Amromin, E.L., Alexandrov, K.A., Levkovsky, Y.L., 1986. Determination of cavitation inception condition for bodies with separation and reattachment of its boundary layer. J. App. Mech. Tech. Phys. 27, 184–189. Amromin, E.L., Arndt, R.E.A., 2019. Analysis of influence of cavity content on flow pulsations. Intern. J. Multiph. Flows 110, 108–117. Arndt, R.E.A., 1981. Cavitation in fluid machinery and hydraulic structures. Ann. Rev. Fluid Mech. 13, 273–328. Arndt, R.E.A., Arakeri, V.H., Higuchi, H., 1991. Some observations of tip-vortex cavitation. J. Fluid Mech. 229, 269–289. Arndt, R.E.A., Holl, J.W., Bohn, J.C., Bechtel, W.T., 1979. Influence of surface irregularities on cavitation performance. J. Ship Res. 23, 150–170. Blake, W.K., 1986. Mechanics of Flow-Induced Sound and Vibration. Academic Press, New York. Castillo, L., Wang, X., George, W.K., 2004. Separation criterion for turbulent boundary layers via similarity analysis. ASME J. Fluids Eng. 126, 297–304. Castro, E., Crespo, A., Manuel, F., Fruman, D.H.J., 1997. Equilibrium of ventilated cavities in tip vortices. ASME J. Fluids Eng. 119, 759–767. Coutier-Delgosha, O., Deniset, F., Astolfi, J.A., Leroux, J.-.B., 2007. Numerical prediction of cavitating flow on a two-dimensional symmetrical hydrofoil and comparison to experiments. ASME J. Fluids Eng. 129, 279–292.
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Dern, J.-C., Quenez, J.-M., Willson, P., 2015. Compendium on Naval Hydrodynamics. ENSTA, Paris, France. Hamitt, F.G., 1980. Cavitation and Multiphase Flow Phenomena. McGraw-Hill Int. Book Comp. Harsha, P.T., 1977. Kinetic energy methods. In: Handbook of Turbulence, 1. Plenum, New York, pp. 187–236. Hsiao, C.-.T., Ma, J., Chahine, G.L., 2017. Multiscale tow-phase flow modeling of sheet and cloud cavitation. Int. J. Multiph. Flows 90, 102–117. Katz, J., 1984. Cavitation phenomena within regions of flow separation. J. Fluid Mech. 140, 397–436. Knapp, R.T., Daily, J.W., Hammitt, F.G., 1970. Cavitation. McGraw-Hill. Kwon, O.K., Pletcher, R., 1986. A viscous-inviscid interaction procedure. ASME J. Fluids Eng. 108, 64–75. Lindgren, H., Johnson, C.A., 1966. Cavitation inception on head form ITTC: comparative experiments. 14th ITTC Conference. Liu, X., Katz, J., 2008. Cavitation phenomena occurring due to interaction of shear layer vortices with the trailing corner of a two-dimensional open cavity. Phys. Fluids 20, 041702.
Liu, X., Katz, J., 2013. Vortex-corner interactions in a cavity shear layer elucidated by time resolved measurements of the pressure field. J. Fluid Mech. 728, 417–457. Meyer, R.S., Billet, M.L., Holl, J.W., 1992. Freestream nuclei and traveling-bubble cavitation. ASME J. Fluids Eng. 116, 672–679. Mokhtarpoor, R., Heinz, S., Stoellinger, M., 2016. Dynamic unified RANS-LES simulations of high Rseynolds number separated flows. Phys. Fluids 28, 095101. Shams, E., Apte, S., 2010. Prediction of small-scale cavitation in a high speed flow over an open cavity using large-eddy simulation. ASME J. Fluids Eng. 132, 111301. Wade, R.B., Acosta, A.J., 1966. Experimental observation of the flow past a plano– convex hydrofoil. ASME J. Basic Eng. 88, 273–283. Waniewski, T.A., Brennen, C.E., 1999. The evolution of cavitation events with speed and scale of the flow. ASME paper FEDSM99-7916. Zabreyko, P.P., Koshelev, A.I., Krasnosel’skii, M.A., Mikhlin, S.G., Rakovshik, L.S., Stet’senko, V.Ya, 1975. Integral Equations – a Reference Text. Noordhoff Intern. Publishing, Leyden.
Contents lists available at ScienceDirect
International Journal of Multiphase Flow journal homepage: www.elsevier.com/locate/ijmulflow
Prediction of cavitation inception in slots E.L. Amromin Mechmath LLC, Federal Way, WA 98003, United States
a r t i c l e
i n f o
Article history: Received 11 February 2019 Revised 26 May 2019 Accepted 24 July 2019 Available online 29 July 2019 Keywords: Cavitation inception Surface irregularities Numerical prediction
a b s t r a c t Cavitation inception in slots on the flat walls is analyzed in the steady flow approach using a viscousinviscid interaction method. It was assumed that, in the accordance with known observations, cavities appear as spherical bubbles in the cores of vortices near the slot corners. Computed cavitation inception numbers are compared with published experimental results of other authors within wide ranges of inflow parameters. The comparison shows a satisfactory agreement of computations with the measured data. © 2019 Published by Elsevier Ltd.
1. Introduction
2. Definition of cavitation inception number
Cavitation inception has been the topic of intensive experimental studies for many decades. One of the best reviews of the early studies was prepared by Lindgren and Johnson (1966) more than a half century ago. The diversity of experimental data in this review did not allow for determination of certain dependencies. Also, the substantial scale effects were seen there. No computational tool for cavitation inception existed at that time to support a theoretical analysis. Computational tools capable to analyze initial forms of cavitation were developed several decades later. The successful predictions of cavitation inception numbers σ I for bodies/hydrofoils of smooth surfaces made with the employment of a viscous-inviscid interaction method (as by Amromin et al., 1986) and the methods for a fully turbulent mixture (as by Coutier-Delgosha et al., 2007) were reported. For surface irregularities in the form of lugs/obstacles mounted on a wall, the successful prediction of σ I was reported by Amromin (2016), but the flow model used in that study was irrelevant to surfaces with slots or with very small gaps between irregularity elements. This study is aimed on computation of cavitation inception number for slots. The detailed observations of slot flows by Liu and Katz (2013) are used to elaborate the flow scheme. The employed computational method can be considered as a modification of the method described in Amromin (2016). Its validation is carried out with the experimental data published by Arndt et al. (1979) and by Liu and Katz (2008).
First of all, it is necessary to precisely define the cavitation inception. A student would say that cavitation inception is the appearance of a region filled by gas (usually, by the vapor) within a liquid and that this region was developed either from imperfections on the body surface, as discussed in the monograph of Knapp et al. (1970), or from a nucleus existing in the inflow, as noted in compendium edited by Dern et al. (2015). A student also would recall that inception can be characterized by a jump of the vapor volume fraction in a small region, but because of the difficulty to observe randomly appearing cavities determination of cavitation inception number σ i has been frequently substituted by determination of desinence of the already existing cavities due to the gradual decrease of the cavitation number σ . In such a way a set of observations of steady flows has been used to characterize a rather unsteady phenomenon. An engineer would first recall a jump in the noise radiation as the most important effect assisting cavitation inception. The old practice of visual determination of cavitation inception is not in the controversy with the prediction of such a jump because the radiated energy is proportional to the cavity volume variations. However, the observed σ i and cavitation desinence number σ d substantially depends on the vision of an observer. Generally, there is no established lower threshold of this jump. The more modern techniques as holography (used by Katz (1984), f. i.) allow for the more accurate optical determination of appearing cavities than the direct visualization has allowed for. However, this does not help to define the difference between the smallest cavity and the biggest nucleus. There is no established minimal cavity size even in the papers of Meyer et al. (1992) or of Waniewski and Brennen (1999), where registration of cavitation events was
E-mail address: [email protected] https://doi.org/10.1016/j.ijmultiphaseflow.2019.07.017 0301-9322/© 2019 Published by Elsevier Ltd.
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E.L. Amromin / International Journal of Multiphase Flow 119 (2019) 217–222
described. The measured distributions of bubbles by their size in the books by Blake (1986) and by Dern et al. (2015) or in the review by Arndt (1981) manifested the increase of the number of nuclei with the decrease of their size. However, how to distinguish the nuclei from the smallest cavities? The absence of the unique definition of cavitation inception number urges us adjust the employed definition to the conditions of the experiments selected as validation tests. In particular for the experiment of Liu and Katz (2008), there are photos of incipient cavitation allowing for the assumption that the radius RC of an incipient cavitating bubble is between 0.1 mm and 0.15 mm. For the observation of Arndt et al. (1979), it could be greater. As stated by Liu and Katz (2008), “observations on occurrence of cavitation reveal that the first site of cavitation inception is located on top of the downstream corner of the cavity, regardless of the free stream speed and dissolved air content in the water”. Taking into account surface tension, one calculates cavitation number 2 for a bubble as σ = 2(P∞ − Pc )/ρU∞ 2 σ = −CP − 4χ /ρU∞ RC
(1)
The maximum value of σ can be considered as σ I . Here PC is the pressure inside the cavity, P∞ is the inflow pressure, U∞ is the free stream speed and χ is the surface tension coefficient. The dimensionless pressure coefficient CP must be calculated with taking into account the vortex contribution because cavitation in separation zones appears in the vortex cores. If the core radius R exceeds RC , the pressure in the cavity must be calculated as
PC = P (R ) + ρ
R
RC
uˆ2 2χ dr + , r RC
(2)
The pressure P(R) is the difference of some unperturbed pressure at the vortex location and [ /(2π R)]2 . For turbulent flows, as shown by Amromin (2007, 2016) with the use of an exact solution of Reynolds equation in a region with insignificant variation of turbulent stress component u v , the circumferential velocity in the core is
uˆ (r ) = W (1 − ln |r/R| )r/R,
(3) √
√
where W = Г/(2π R) and R = ν / −π < u v >. As illustrated by Fig. 1, this core is in much better agreement with the known experimental data than the broadly used Rankine core is. As shown in Fig. 2, the difference in velocity distributions in the vortex core results in the significant difference of pressure in vortex cores. As shown by Agarwal (2010), employment of Eq. (3) allows for the much better prediction of cavitation inception num-
Fig. 1. Theoretical (from Amromin, 2007, solid curve) and experimental (from Arndt et al., 1991 and Castro et al., 1997), symbols) circumferential velocities in vortex cores.
Fig. 2. Comparison of the pressure drop within laminar and turbulent cavitationfree vortex cores (solid lines) and within a cavity of the radius r in turbulent core (dashed line); dP = P (r ) − P (R ).
ber in vortices. However, calculation of σ I with employment of Eqs. (1)–(3) requires determination of Cp, Г and u v . Further, in the majority of experimental studies the vapor pressure PV is employed instead of the actual cavity pressure PC in calculation of σ . As described by Wide and Acosta (1966) longtime ago, the difference PC −PV significantly increases with the increase of inflow air content. The impact of this difference on values of σ is evident, but experimental determination of pressure in appearing cavities is too difficult. So, a correction to σ (like one derived by Amromin (2016), f. i.) should be used to take into account this impact must be used during comparison of the computed results with the experimental data. 3. Scheme of separated flow around a slot The separated flow around a slot may seem similar to the flow behind a backward facing step. Indeed these flows are similar for the ratios L/H > 7, where L and H are length and depth of the slot, because for L/H > 7 reattachment of separated boundary layer to the slot bottom can occur. Otherwise, two separated flows are dissimilar and reattachment take place just downstream of the slot, as one can understand with flow scheme in Fig. 3. In the experiments of Liu and Katz (2008) cavities appear in vortices just upstream of the slot trailing edge (as shown in their Fig. 3) and this fact is reflected in the employed scheme. For L/H ∼ 1, the bottom vicinity has a little influence on other parts of flow (and these parts were excluded from the measurements of Liu and Katz). Therefore, this flow scheme and the employed viscous-inviscid interaction computational method has derivations from the scheme and method employed by Amromin (2016, 2018) for prediction of cavitation inception in others separated flows. On the other hand, in spite of the general unsteadiness
Fig. 3. Scheme of flow around a slot; doted curve is section of displacement body, dashed curve limits separation zone.
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of cavitation inception phenomenon, the corresponding mathematical problem is also considered as the time-independent problem. In such problems, as shown by Amromin and Arndt (2019), the hypothesis on the cavity content has a little impact on their solutions. As usual for viscous-inviscid interaction methods, the entire flow is divided into parts. In this problem these parts include turbulent boundary layer, inviscid flow over it, a separation zone under it and a stationary vortex located in the slot near its trailing edge. The inviscid flow influences the boundary layer by pressure distribution, but the pressure coefficient Cp is computed at the surface S of body displacement. The pressure within the vortex core also depends on this coefficient. On the other hands, the displacement thickness δ ∗ predetermining shape of S depends on the boundary layer growth. Over the slot this growth depends on the water flux through the frontier between separation zone and boundary layer (on the velocity v normal to this frontier).
Here h is the distance between initial and corrected shapes of S. One should substitute Eq. (9) into Eq. (8) and integrate it with the initial condition h(0) = 0. The Eq. (9) was derived using inversion of the Cauchy singular integral appearing in the original form of quasi-linearized Eq. (6) written with the use of a potential of sources and sinks of the assumable small intensity q(x). However, as one can find in Zabreyko et al. (1975), this inversion exists only with the additional condition
X3
[C + C1 f1 (x ) + C2 f2 (x ) − U0 (x )]dx
0
= 0
(4)
with the condition
∂ =0 ∂ N S
(5)
C2
δ∗
d f2 − a f2 dx
Here U = C + C1 f1 (x ) + C2 f2 (x ) is defined with the undetermined coefficients C, C1 , C2 and with the use of functions f1 = 2 min{1; (x/X2 ) (3 − 2x/X2 )}, f2 = [max{0; 3ξ 2 − 2ξ 3 )}], the x axis is directed along the wall and counted from the slot leading edge, ξ = (x − X2 )/(X3 − X2 ), X2 = 0.6 L, X3 = 1.1 L. This approximation of U reflects the typical behavior of pressure in separation zones. The problem with boundary conditions (5)–(7) for linear Eq. (4) is nonlinear even for the known values of these coefficients because two conditions must be satisfied on the same part of the boundary S. Therefore, the shape of S must be fitted in iterations. These iterations include two steps. The first step consists of determination of U0 = grad( ) after solving Eqs. (4), (5), (7) out of the considered known approach to S. This step can be solved using various boundary element methods. The second part consists of determination of the necessary deformation of S by solving quasi-linearized Eqs. (5) and (6) that can be written here as
d (hUh ) q = , dx 2
q (x ) =
2 x(X3 − x )
π
(8) 0
X3
[C + C1 f1 (z ) + C2 f2 (z ) − U0 (z )]dz
(x − z ) z(X3 − z )
.
(9)
δ
(11)
[1 − u¯ ]dy, δ is the boundary layer thickness,
dU dδ ∗ ∗ (δ ∗ +2δ ∗ ∗ ) + U 2 + v(U − u0 ) = 0. dx dx
Here δ ∗∗ =
(7)
− bC = bC1
y is counted across this layer, the velocity profile u¯ = u0 /U + (1 − u0 /U )(3η2 − 2η3 ) is used in the separated boundary layer, η = y/δ , u0 is the tangent velocity component at y = 0, C1 = 0.01 and b = const. So, C and C2 can be found from Eqs. (10)–(11), but the values of δ ∗ (L) must be found by integration of boundary layer equations. There are two parameters in the velocity profile in the separated layer. These parameters are the layer thickness δ (x) and u0 (x). Therefore, two equations are necessary for its description. The first one is the momentum integral equation
|grad ( )||Sˆ = U
grad ( )|x→±∞ → 0.
(10)
0
U
for tangent velocity on separation zone upper boundary. There is also condition of perturbation absence in the infinity
= 0.
Here a = const.,δ ∗ =
for the normal velocity component on the inviscid flow boundary and the condition
(6)
x(X3 − x )
The condition of boundary layer reattachment at x = L is similar to the widely used criterion of boundary layer separation (see Castillo et al. (2004), f. i,) because both of them are conditions at the point of contact between attached and separated boundary layers. Taking into account the approximation used for U, one can write this condition as
4. Computational method Generally speaking, prediction of cavitation inception requires a multi-zone multi-scale analysis, as manifested by Amromin (2018) or by Hsiao et al. (2017). The diverse parts of the flow should be analyzed using diverse physical assumptions and equations. The inviscid flow is assumed to be curl free and velocity potential can be defined there. The boundary-value problem for this potential includes the Laplace equation
219
δ
(12)
u¯ [1 − u¯ ]dy and the function v(x) is ordered with re-
0
gards to the experimental data of Liu and Katz (2013). This equation is a modification of the well-known von Karman equation. The third term in its left-hand side is usually zero because usually the ordinate y is counted from the rigid walls, where v(0) = 0. However, for this separated layer integration starts from a line inside of fluid. The second one is the differential momentum equation considered along this line
d u0 (U − u0 ) dU +U =A . dx U δ∗ dx 2
u0
(13)
The initial conditions for Eqs. (12), (13) are u0 (0) = 0 and
δ ∗ (−0) = δ ∗ (+0), where δ ∗ (−0) is the known value, and the coefficient A must be fitted to meet the condition u0 (L) = 0. As shown by Amromin (2018), the employed three-term approximation of U allows for a good agreement with measured pressure distributions in separation zones behind steps. Comparison of the computed result with the pressure distribution measured by Liu and Katz (2013) in Fig. 4 shows such an agreement for slots as well. As illustrated by Fig. 5, the employed description of separated boundary layer is in the satisfactory agreement with the experimental data of Liu and Katz (2008, 2013). The maximum values u0 /U∞ was about 0.26. This value is also in the range of experimental data. Finally, it is necessary to compute σ using Eq. (2) for the selected value of RC . As noted above, computation of pressure for this equation requires know the vortex intensity . Besides there is the
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Fig. 4. Computed (line) and measured (squares) pressure coefficients along the slot at y = 0 at inflow speed 5 m/s.
Fig 6. Comparison of computed and observed (by Liu and Katz) cavitation inception number. Lines 1 and 2 provides maximum and minimum of observed σ I for the basic level of dissolved oxygen, triangles show average σ I for fivefold higher oxygen level. Line 3 shows computed σ I for Rc = 0.1 mm, line 4 –for 0.15 mm. Line 5 shows σ I + dσ fitted to the basic oxygen level using line 4, line 6 shows σ I + dσ recalculated from line 4 for fivefold its level.
Fig. 5. Computed (line) and measured (squares) velocity along the line ∂ u¯ /∂ y = 0 at inflow speed 5 m/s.
R-dependent integral in the right-hand side of Eq. (2). Moreover, R also depends on . For determination of vortex intensity the approximation
= Duδ3
(14)
with δ 3 (x) = δ ∗ (X2 )−δ ∗ (x) and a constant D was used. Its righthand side is calculated at x = 0.95 L. The difference between Eq. (14) and the approximations used by Amromin (2016) for surface irregularities mounted on walls is associated with the different boundary layer effects. The non-uniformity of velocity across boundary layer has a little impact on the flows within slots. Rather variation of its thickness along the slot is important. The value D = 1.0 was fitted as basic for this coefficient. 5. Computational results and consideration of uncertainties First of all, it would be desirable to compare the obtained computational results with computational results of computations of other authors. Computational studies of cavitation inception compose a very small fraction of numerous papers on numerical analysis of cavitation. Possibly, only Shams and Apte (2010) published a LES computational result for slot cavitation inception. They calculated σ I for conditions of Liu and Katz (2008) experiment using the unexplained assumption σ I = –CP −CPRMS . Their prediction gave σ I = 0.93 at U∞ = 5 m/s, but their computed averaged (timeindependent) pressure coefficient and its RMS substantially overestimates observed both |CP | and CPRMS (in particular, giving threefold higher CPRMS than was measured) and neglected by surface tension. These results deserve some comments. Generally speaking, cavitation inception is an unsteady phenomenon. In the particular case of cavitation in the slot, cavitation appears via capturing randomly drifting nuclei by about stationary vortices (vortex shed-
ding behind surface-mounted irregularities is not similar to the vortex behavior in the slot). However, as noted by Harsha (1977), a single computation of unsteady turbulent flow is not informative enough because some randomly perturbed initial conditions must be used on the boundaries of the region of computation and some averaging must be applied to a set of numerical solutions to obtain results comparable to experiments. On the other hand, as noted by L. I. Sedov, the averaging procedure is the most important factor in computations of turbulent flows. Therefore, RANS and LES methods can give expectably different results and may have substantial disagreements with experimental data for some flows. Further the comparisons with experimental data were considered. Let us recall that the inflow characteristics in these experiments were time-independent. Therefore, cavitation inception in this case is rather quasi-steady phenomenon under a random impact and it was allowable to determine the flow characteristics in the framework of steady theory. The results of the above-described computations are plotted in Fig. 6 together with the data of Liu and Katz (2008). The computations were carried out for two values of Rc that look consistent with their photos of appearing cavities. Let us discuss the computational uncertainties. Purely computational uncertainties inherent to the above-described viscous-inviscid interaction method were already evaluated by Amromin (2002). Another uncertainty is associated with definition of Rc corresponding to incipient cavities. Uncertainties (discrepancies) of the corresponding measurements for the basic (low) inflow air content α are also shown in Fig. 6 (this discrepancy can be comprehended as the result of random distribution of nuclei in the inflow). The experimental data were presented by these authors versus cavitation number calculated using vapor pressure. Therefore, a correction to such a cavitation number should be employed for comparisons with the experimental data and instead of computed σ I the sum σ I + dσ should be employed in the plot. Here
dσ =
2(PC − PV ) α ∼ 2 2 ρU∞ U∞
(15)
The proportionality of dσ to α and its inverse proportional2 has been noted by many researchers starting at least ity to U∞ from Hammit (1980). Recalculation of the computed dependency
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core, the weak point of described computational method is in definitions of the coefficients D and u v . The integral method employed for boundary layers does not allow for determination of these coefficients. Meanwhile, viscous-inviscid interaction methods allow also for computation of boundary layers with more advanced differential approaches, like it was manifested since Kwon and Pletcher (1986). A satisfactory computation of u v was manifested by Mokhtarpoor et al. (2016), f. i., in a quite similar separated flow (within a recess on the wall). Also, it may be easier to √ √ compute directly R and find then using the formula R = ν / −π < u v >. 6. Conclusions Fig. 7. Cavitation inception number versus the parameter with ε = 0.041 and λ = 0.51. ; rhombs – values measured by Arndt et al. (1979); lines – computed dependencies for various Rc .
for Rc = 0.15 mm with the use of Eq. (15) and a proportionality coefficient put the recalculated dependency practically within the strip of observed cavitation inception numbers. Its recalculation for much higher inflow air content (for the fivefold higher oxygen content) gives a satisfactory approximation of the related experimental data as well. No information on discrepancy of measured σ I was noted for the experimental data Arndt et al. (1979) for slots at various L = H. These data were results of observations with an unknown observer vision capability. This urged to carry out computations for various Rc . As seen in Fig. 7, the computations for Rc = 0.25 mm are in a good agreement with the experimental data. The comparisons in this plot are provided versus the aggregate variable = (H/δ )ε (U∞ δ /ν )λ introduced by Arndt et al. (1979). Their data manifested a very weak dependency on H/δ for slots (ε = 0.041), unlikely to their data for the wall-mounted surface irregularities (approximated using 0.36 < ε <0.74). The correction (15) was not applied to the computed dependencies for these slots because of very low total inflow air content in the experiments. The above-described computations were carried out using D = 1.0 and u v = 0.005. The sensibility of computational results to variation of these coefficients is illustrated by the computations presented in Fig. 8. Inaccuracy in values of u v are less influential. With the proven general feasibility of the employed multizone scheme of cavitating flow and description of vortex viscous
Fig. 8. Impact of coefficient variation on the computed σ I for a slot with L = 12.7 mm, Rc = 0.5 mm and δ ∗ (0) = 0.1 L. Solid curve corresponds to D = 1.0 and u v = 0.005, dashed curve - to D = 1.0 and u v = 0.0025, dashed-dotted curve – to D = 0.75 and u v = 0.005.
Cavitation inception in slots on the flat walls was analyzed in the steady flow approach using the viscous-inviscid interaction method. This method is a modification of the method earlier employed by the author for prediction of cavitation inception behind surface irregularities mounted on the walls. The modification was caused, first, by a relatively weak influence of the wall boundary layer thickness on the flows in slots and, second, by the about stationary position of incipient cavities; in the accordance with the known observations, these cavities appear as bubbles in the cores of quasi-stationary vortices near the slot trailing corners. Computed cavitation inception numbers were compared with the published experimental results of other authors within the wide ranges of inflow parameters. Impact of the size of smallest observed cavities on predicted cavitation inception numbers was manifested. The effect of inflow air content on experimental data and its influence on comparison of numerical results with the observations was discussed and illustrated by computational examples. The sensibility of computational results to the employed approximation coefficients was shown. The comparison shows a satisfactory agreement of computations with the measured data. Though the feasibility of suggested computational scheme with the emphasis on the turbulent vortex core and on the surface tension influence was proven, the path of the following enhancement of applied numerical methods was also discussed.
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