PREDICTION OF CAVITY VOLUMES ON ROTATING BLADESAND SCALE EFFECTS Текст научной статьи по специальности «Физика»

ТЕОРИЯ КОРАБЛЯ И СТРОИТЕЛЬНАЯ МЕХАНИКА

DOI: 10.24937/2542-2324-2021-3-397-13-24 UDC 532.528+629.5.025.1

E. Amromin

Mechmath LLC, Federal Way, USA

PREDICTION OF CAVITY VOLUMES ON ROTATING BLADES AND SCALE EFFECTS

Object and purpose of research. Pressure pulsations induced by cavitating blades substantially contribute to flow-induced loads and amplify structural vibration. These pulsations depend on oscillation of the volume of cavities over blades. Prediction of them usually involves model tests and there are three kinds of scale effects influencing the cavity volumes. The first one is associated with the non-uniform inflows. The second one is associated with the combined influence of the blade boundary layer and surface tension on the cavity surface. The third one is associated with the cavity buoyancy. Materials and methods. Because of complexity of blade flows, a qualitative analysis of similar unsteady non-uniform flows around 3D hydrofoils is useful. This paper presents such an analysis for a hydrofoil with the sections copied from a marine propeller blade. The inflows correspond to the wakes of a ship and of her model. Computations carried out using an analysis of viscous-inviscid interaction.

Main results. The qualitative explanation of observed trends and scale effects is obtained due to this analysis. In particular, the role of pressure side cavitation in full scale conditions is pointed out.

Conclusion. The difference of model and ship wakes results in the substantial difference in blade section angles of attack at the same blade loading. Therefore, in model tests the suction side cavitation is more extensive, whereas the pressure side cavitation may not appear, though it exists on full-scale ship propeller blade. This substantial scale effect has been usually out of previous considerations.

Keywords: cavitation, blades, wake, boundary layer, surface tension, scale effects. The author declares no conflicts of interest.

NAVAL ARCHITECTURE

DOI: 10.24937/2542-2324-2021-3-397-13-24 УДК 532.528+629.5.025.1

Э.Л. Амромин

Mechmath LLC, Federal Way, USA

ПРОГНОЗИРОВАНИЕ ОБЪЕМА КАВЕРН НА ВРАЩАЮЩИХСЯ ЛОПАСТЯХ И МАСШТАБНЫЕ ЭФФЕКТЫ

Объект и цель научной работы. Пульсации давления, вызываемые кавитирующими лопастями, вносят существенный вклад в нагрузки, вызванные потоком, и усиливают вибрацию конструкции. Эти пульсации зависят от колебаний объема каверн на лопасти. Их прогнозирование обычно включает модельные испытания, и существуют три вида масштабных эффектов, влияющих на объемы каверн. Первый связан с неравномерным притоком. Второй связан с совместным воздействием пограничного слоя лопатки и поверхностного натяжения на поверхность каверны. Третий связан с плавучестью каверны.

For citations: Amromin E. Prediction of cavity volumes on rotating blades and scale effects. Transactions of the Krylov State Research Centre. 2021; 3(397): 13-24.

Для цитирования: Амромин Э.Л. Прогнозирование объема каверн на вращающихся лопастях и масштабные эффекты. Труды Крыловского государственного научного центра. 2021; 3(397): 13-24.

Материалы и методы. Из-за сложности обтекания лопастей полезен качественный анализ подобного нестационарного неоднородного обтекания трехмерных подводных крыльев. Представлен такой анализ для подводного крыла с сечениями, скопированными c лопасти морского винта. Набегающие потоки соответствуют следам за кораблем и его моделью. Расчеты проводились с использованием анализа вязко-невязкого взаимодействия.

Основные результаты. Получено качественное объяснение наблюдаемых тенденций и масштабных эффектов благодаря этому анализу. В частности, подчеркивается роль кавитации на нагнетающей стороне лопасти в полномасштабных условиях.

Заключение. Различие модели и следа корабля приводит к существенному различию углов атаки сечения лопасти при одинаковой нагрузке лопасти. Следовательно, при модельных испытаниях кавитация на стороне всасывания более обширна, тогда как кавитация на стороне нагнетания может не появиться, хотя она существует на полномасштабной лопасти гребного винта корабля. Этот эффект существенен и не рассматривался ранее. Ключевые слова: кавитация, лопасти, след, пограничный слой, поверхностное натяжение, масштабные эффекты.

Автор заявляет об отсутствии возможных конфликтов интересов.

Introduction

Введение

The very frequent and disturbing phenomenon occurring on ship propeller blades during their rotation in the ship wake consists of periodical appearances and disappearances of cavities. This phenomenon occurs mainly due to periodical variations of the angles of attack a of the blade sections in the non-uniform inflow in the wake (like variations shown in Fig. 1). The variations of cavitation number due to blade rotation are shown in Fig. 2.

Though such cavities usually cover only some leading part of the blade (and can sometimes merge with pulsating cavities in tip vortices, as shown in [1], f.e.), the corresponding periodical variations of

a, degrees

l\ 4\ / " »\ t\

/ * " -W *

Y /• = 0.9 yt /

\ sr = 0.4 // /V

0 60 120 180 240 G, degrees

Fig. 1. Typical variation of the blade section angle of attack; solid curves relates to the model test/MT wake, dashed curve relate to the full-scale/FS ship wake

Рис. 1. Типичная вариация угла атаки сечения лопасти: модельное (сплошная кривая) и натурное (пунктир) поле скоростей

the cavity volume substantially contribute to pulsation of pressure and, as a result, to the flow-induced loads on the ship hulls and their vibration, as well as to the propeller-induced noise. Unlike to the loads transmitted to hulls by shafting, these loads cannot be mitigated by damping devices, A similar impact of non-uniform inflow leads to erosion of Kaplan turbine blades [2].

The experiments with marine propeller blades have shown significant differences of pressure harmonics in the water tunnels and the full scale conditions [3]. The blade passage effects are modeled in most experimental facilities by geometric similarity of the propeller and ship hull with the correct propeller loading. The inflow velocities at the propeller disc are strongly retarded in its top sector and, as pointed out in [4], the local similarity of the propeller model test inflow to the full scale inflow cannot be obtained in such a wake peak sector, where cavi-tation is expected. On the other hand, after tuning the blade advance ratio to obtain the wake peak similarity, cavities tend to appear on the blade pressure side and the similarity of cavitation will be lost for another sector of this disk. Thus, there is no way to simulate the local blade load in the wake simultaneously with the average propeller load. As a result, the pressure harmonics are usually higher in the model tests [5]. Therefore, a theoretical/numerical analysis of cavitation-caused pressure pulsations is very desirable.

At the distances greater than the blade diameter, the pressure pulsations p can be estimated with simulation of the oscillating cavity of a volume V by the source of a time-dependent intensity Q. Therefore one can write

p ~ d 2V / dt2. (1)

So, the accuracy of prediction of pressure pulsations generated by cavitating hydrofoils or blades directly depends on the accuracy of prediction of their cavity volumes.

Contemporary computations of blade cavitation are mainly based on the ideal fluid theory (as noted in the review [6]), but this theory usually overestimates cavity volumes and periods of their existence in the wake flows (as in [5], [7] in particular). On the other hand, the known attempts to employ RANS solvers to determine cavitation-induced pressure pulsations were not successful enough. In particular, employment of ANSYS [8] and of FLUENT [9] resulted in the substantial underestimation of the second harmonics. So, as noted in [10], "It is concluded that predictions of pressure fluctuations from a potential flow BEM code ... give, so far, the most reliable results." Nevertheless, seeking an estimation of scale effects, it would be questionable to limit consideration by a potential flow analysis.

Because propeller cavitation is the very complex phenomenon, a comprehensive comparison of its multiple effects is difficult. Therefore, the presented analysis is carried out simply for a system of hydrofoils, but their shapes are typical for marine propeller blade sections and the inflows are typical for these sections. A similar approach with twisted hydrofoils was used in [11], [12].

Flow scheme and employed numerical technique

Схема течения и использованный численный метод

As became clear after the experiments [13], sheet cavi-tation is indeed a special type of viscous separation depending on both Reynolds number Re and cavitation number c. The following study [14] has shown also the surface tension influence on the cavity location and size for bodies of smooth shapes; this influence can be described with Weber number We. The further studies of sheet cavitation [15, 16] with the combined account of {c, Re, We} basically consisted of inserting a cavity within a viscous separation zone generated by the cavity itself. The same model of sheet cavitation is employed here.

The presented computations are quasi two-dimensional. This means that the three-dimensional nature of the real flow around rotating blades is implicitly included, first, in the characteristics of the unsteady incoming flows, second, in the period-averaged radial lift distributions [17] (shown in Fig. 3), and third, in the

0.75

0.5

0.25

2 r/D = 0.4, FS

2 r/D = 0.9, FS

90

180

240 0, degrees

Fig. 2. Variation of section cavitation number with the blade rotation

Рис. 2. Вариация числа кавитации сечения при повороте лопасти

definition of the section cavitation numbers c by the formula

с =

2P¥o + rgD cos 8 - 2PC p(nD)2 [j2 + (nr )2 ]

r cos 8

êJ2 + (nr)2] Fr2 [l + (nr / J)2 ]'

(2)

Here D is the propeller diameter, r = 2r/D, r is the blade section radius, 0 is the time-dependent blade azimuth, J is the blade advance ratio, p is the water density. Froude number Fr is calculated using D, the ambient pressure Pw0 is calculated at the submergence of the rotation axis, Pc is the pressure within the cavity. So, though the entire three-dimensional cavity depends on the single cavitation number cn defined in Eq. (2), its two-

■■■—jCi &tJ= 0.95

4,

N..

Fig. 3. Geometrical blade section characteristics and their average lift coefficients at two advance ratios J Рис. 3. Геометрические характеристики сечения лопасти и их средние коэффициенты подъемной силы на двух поступях J

Fig. 4. Scheme of two-dimensional cavity; flow goes from the left

Рис. 4. Схема двухмерной каверны: поток набегает слева

dimensional sections are analyzed in the two-dimensional flows around the blade sections at cavitation numbers c varying with 0 and r. The scheme of such two-dimensional cavitating flow is shown in Fig. 4.

For the selected r, constant Reynolds number Re, Weber number We, and time-depending c, the time-dependent locations of the cavity edges (X1(t), X2(t)} over the section and the cavity thickness H(x, t) must be found. Here this problem is solved using a viscous-inviscid interaction method [15, 16]. In such methods the entire flow is divided into viscous and inviscid parts with their interaction based on two effects: First, the pressure gradient influences the thicknesses of boundary layers and wakes, but, second, these thicknesses influence the pressure itself.

The inviscid part of flow must be determined by solving the nonlinear problem for the velocity potential

A® = 0; (3)

дФ

dN

U 2 + дФ

дН ;

"дГ;

dt

1 - CP( x);

U|x= {cosa; sina}-

(5)

(6)

These equations are written in the dimensionless form. Here U = grad(®), the blade chord C is used as

the length unit and nD^J2 + (nr)2 is used as the ve-

locity unit, S is the inviscid flow boundary around a hydrofoil/blade section and the surface S* is a part of S, CP' is a given function. Additionally to Eqs. (3-6), the total vorticity conservation law is included in the problem, with the vortex sheet shedding from the section/hydrofoil trailing edge in the direction of incoming flow. Unlikely to cavitation in ideal fluid (illustrated by Fig. 5, where segments are used as Riabouchinsky solids), there is a distance Ç between S and the surface of the hydrofoil or of the cavity. On a cavitation-free surface Ç = 5*, where ô* is the boundary layer displacement thickness.

A condition for determination of the lift coefficient CL is necessary to complete the problem formulation. The classical Kutta-Joukovskii condition overestimates CL. Nevertheless, within ranges of angles of attack inherent to blade sections, the linear dependency of CL on a takes place. This allows for the approximation of the total intensity of the vortices distributed over the section itself as

(4) y = -0,5C'(a - a0).

(7)

The factor C' = (1 + 0.877)(1 - e12-5^-0^-0186) is used here in the according to the approximation [18] of hydrofoil lift in viscous fluid, T is the section dimen-sionless thickness. Employment of Eq. (7) assumes that the Kutta-Joukovskii condition is disregarded, but this allows for the much better pressure prediction on the main part of the contour, whereas an unlimited rise of U occurs only in a small vicinity of the trailing edge.

x/C

Рис. 5. Частичные каверны на крыльевом профиле ЫАСА16009 в гидролотке при а = 5°: расчет для условий идеальной жидкости; цифры показывают соответствующие значения числа кавитации а

Fig. 5. Partial cavities on hydrofoil NACA16009 in a water tunnel at a = 5° computed in ideal fluid; numbers indicate the corresponding values of cavitation number a

S

S

The parameter ao for the blade sections must be initially found by computation of the steady flow at the period-averaged angle of attack with the average lift. Three-dimensional effects of the flow are implicitly included in determination of ao.

The right-hand side part of Eq. (5) in viscous fluid is different from used in ideal fluid theory (this difference is clarified by Fig. 6); this part is defined here as

[c\(x - x*)2 +V1 + g J for X0 < x < x*;

1 + g for x* < x < X2;

[c2^2(3-2|) + VT+GJ for X2 < x < X3.

(8)

dU dx

dU dx

(10)

It is more important to fix just the pressure (velocity) derivative because the function U itself cannot get large variations at the small distance X.

The third of these conditions predetermines the boundary layer reattachment to the cavity

H + 5* - ç|x- =

Cp / 1 1 1

, Ideal cavity 1 1 / Cp-

Within B] Го x/C

Here ^ = (x - X2)/(X4 - X2) and locations of the abscissas used in Eq. (8) are explained in the flow scheme in Fig. 4. In the above-formulated ideal fluid problem with the given coefficients C1, C2 and abscissas x*, X0, X, X2, X3, X4, X2 can be shifted to obtain c prescribed by Eq. (2), but seven other abscissas and all coefficients require additional conditions.

The first of these conditions is the condition of boundary layer separation upstream of the cavity. Because the cavity head is submerged in the boundary layer, this head induces velocities like a surface irregularity of a curvature 1/r*. For estimation of the induced velocity, such irregularity can be replaced by a source. Such a source induces the normal velocity proportional to 5*X-2r*(1 + cosP), where X = X1 - X0 and p depends on blade material wettability (for metals p ^ n). So, the separation criterion for the laminar boundary layer upstream of cavity is

cc + 5* U = [! + cos( p)]2 (9)

5* Re U dx (CC X / 5*)3

According to theory of thin laminar boundary layer, CX = 1.1, whereas the value of CC is the result of tuning to empirical data [13] on cavitation of axisym-metric bodies.

The second of these conditions is the condition of the pressure gradient continuity at x = X0

Fig. 6. Comparison of pressure around a cavity at the same a in flow with a boundary layer (dashed-dotted line) and in ideal fluid (solid line)

Рис. 6. Сравнение давления вокруг каверны при одном и том же числе кавитации a в потоке с погранслоем (штрих-пунктир) и в идеальной жидкости (сплошная кривая)

The cavity thickness H in Eq. (11) is obtained by integration of the equation d2H/dx2[1+(dH/dx)2]-3/2 = = We(c + Cp*)/2 with the Cauchy conditions H = 0, dH/dx = tgP at x = X.

The fourth and fifth of these conditions relate to the boundary layer reattachment downstream of the cavity and can be written as

5*( X3) = Z( X3);

дСр *

5

ds

■ = a .

(12) (13)

The coefficient a = 0.015 is selected here. Also, two semi-empirical formulas are used:

X3 = 0.6 X4 + 0.4 X 2;

x = 1.5Xj - 0.5X0.

(14)

(15)

Computation of 5 is carried out with the use of integral methods for boundary layers. For the boundary layer over the cavities, the dimensionless velocity profile is written using the variable n = (y-H)/5:

U(n) = и / U + (1 - и / U)n2 (3 - 2n).

(16)

For determination of two unknown parameters u(x) and 5(x) in Eq. (16), two equations must be employed. The first equation is the Karman equation in the form

U — (5* + 25**) + U2 d5

dU dx

dx

- + v(U - и) = 0.

(17)

(11)

Here v is the water velocity component normal to the cavity surface. The water flux through this surface is equal to its supply with the reentrant jet in the cavity-

1.6

1.4

1.2

0.8

Xx = 0.0225 С X2 = 0.05 С

0.02225 C\

0.1 С

\а2с

X

0.022

0

0.2

0.4 0.6

Hydrofoil chord С, m

Fig. 7. Dependencies of {X1, X2} on {C, a} for the EN-hydrofoil at a = 4.2° and Re = 1.5-106 Рис. 7. Зависимости {X1, X2} от {C, a} для крыльевого профиля EN при a = 4,2° и Re = 1,5-106

X2/C 0.6 0.4

0.2

ч V

1 ^ ^ -—

0.7

0.9

1.1

Fig. 8. Cavity lengths on EN-hydrofoil at a = 4.2°; dashed curves are results of ideal fluid theory with Brillouin-Villat condition, solid curves - of the presented theory for Re = 1.5-106, vertical segments show discrepancy of measured lengths Рис. 8. Длины каверн на профиле EN при a = 4,2°. Результаты по теории идеальной жидкости с условием Бриллуэна-Вилла (пунктир) и по представленной теории для Re = 1,5106 (сплошная кривая): на вертикальных сегментах показано расхождение измеренных длин

у/С

0.05

-0.05

Measu Ideal fluid \ - rements B IV;:/

у- A ^ В—в ✓ ___ir— \ar

0

0.2

х/С

Fig. 9. Computed in ideal fluid and measured cavity thickness over a propeller blade section Рис. 9. Толщина каверны на сечении лопасти гребного винта: расчет для идеальной жидкости и результаты измерений

tail, but the water motion in the cavity is not analyzed. Friction on the cavity surface is omitted in Eq. (17) as negligibly small. The second equation is the momentum differential equation for water on the cavity surface as on a zero friction line. This equation is used in the Clauser form

udu=0.015-U-u- - UdU.

dx U 5 dx

(18)

Thus, for the given X1 and X2 at the fixed Re, the corresponding pairs (c, We} can be found. Researchers, however, usually recorded C instead of We. Because We = p(vRe)2/(xC) = const/C, the pair (c, We} can be replaced by the pair (c, C}. An example of coupled dependencies (Xb X2} on (c, C} is presented in the left part of Fig. 7 for EN hydrofoil described in [19]. One can see there that the cavity leading edge depends on both c and C, but a tenfold increase of the chord at c = 1.1 leads to 0.0005C displacement of Xi only. Such a trend exists because of high pressure gradients at the hydrofoil leading edges.

So, as shown in Fig. 8 and earlier noted [20], the effect of cavity detachment location on the cavity length and volume can be significant (ideal fluid flow can be considered here as the extreme situation, as an asymptotic for C ^ <»). It occurs because even for such small variations of X1, the corresponding slope of in-viscid flow streamline significantly varies.

The similar tendency exists for propeller blades, as one can find in the right part of Fig. 9 plotted with the data [5], where the authors addressed the difference between ideal fluid computations and measurements to some experimental errors. Indeed, this difference is caused mainly by the difference in the location of cavity leading edge in computations and in the reality.

Blade shape and cavitating flows around its sections

Форма лопасти и кавитационное обтекание ее сечений

There is no attempt to carry out computation for an actual marine propeller. The blade section characteristics shown in Fig. 3 were selected for the presented computations. Such sections are typical for marine propeller blades, as well as the selected NACA profiles for them. The wake characteristics presented in Fig. 1 were used. The presented computations correspond to the advance ratio J = 0.95. The full scale diameter D = 4 m, the model test D = 0.4 m, though smaller blades are also usable in model tests (f.e., with D = 0.2 m). The

model test inflow speed was selected as 10m/s, the full scale speed at this J is 10.4 m/s.

The examples of related velocity distributions over a blade section are presented in Fig. 10. As easy to see, there is already a clear scale effect. The substantial difference in the velocity distributions for model test and full scale conditions has two sources. The weaker of them is the effect of Re described by Eq. (7), but the most powerful is in the difference of the section angles of attack seen in Fig. 1 (up to 1.5 degrees). This difference occurs due to the necessity to obtain in the model test the blade normalized load corresponding to its value in full scale conditions. So, this effect is coupled with the necessity to have the same load in the substantially different incoming flows.

Velocity peaks and cavitation on the blade pressure side appears in full scale conditions, possibly, because, as often in the practice, the blade design has been made with taking into account just the model test results. Theoretically (and certainly in the water tunnel model tests), it is possible to vary cavitation number computing cavitating flows for the same U. Such computations are illustrated here in Fig. 11 by the dependencies of cavitation detachment point abscissa on cavitation number. On can see in Fig. 11 that for the section suction side, there is a small difference between the full-scale results (marked as C = 1.25 m) and the results of ideal fluid theory. For the section pressure side, the difference is much greater, whereas computations for the model test conditions (C = 0.15 m) give no cavita-tion in the considered range of cavitation number due to the surface tension (Weber number) effect. As already briefly noted [21], this effect leads also to the difference in the cavity length and volume (the cavity section areas).

As well known, the cavity volume in ideal fluid substantially depends of the cavity closure scheme. Even for the quite similar schemes, the volume variation with the scheme variation may be up to 25 % (as in the examples with Riabouchinsky solid and Efros-Gilbarg reentrant jet closure schemes provided in [22]).

The scheme selected for the illustrations in Figs. 12 and 13 is a closed scheme with the single-parametric distribution of U along fictitious body; the length of this body is selected as 10 % of the cavity length. One can see in these figures that the difference between ideal fluid and full-scale results may be very moderate for the suction side, but it remains significant for the pressure side.

The combination of two-dimensional results for cavity section areas (two-dimensional cavity volumes)

Fig. 10. Examples of distribution of inviscid flow velocities around the blade section r = 0.8 in its two azimuth positions at J = 0.95. Solid lines correspond to model test conditions, dashed lines - to full scale conditions Рис. 10. Примеры распределения скоростей невязкой жидкости вокруг сечения лопасти r = 0,8 в нулевом угловом положении при J = 0,95. Сплошными кривыми показаны результаты модельных испытаний, пунктиром -натурные значения

Хх!С

0.012

0.006

С =1.25 m, PS

C= 0.15 m, SS C= 1.25 m, SS

B-W, PS

B-W, SS

0.4

0.6

0.8

Fig. 11. Dependencies of cavity detachment locations on cavitation number for a section suction side and its pressure side. B-W marks the location corresponding to the Brillouin-Villat condition Рис. 11. Зависимость места отрыва каверны от числа кавитации для всасывающей и нагнетающей сторон сечения. Буквами B-W показано место, соответствующее условию Бриллуэна-Вилла

L/C 0.6 0.4 0.2

0

\ Pressure \ / % side % IF i

y \ FS « Suction \4 / >ide

\ \ V ч % 4

s

0.3

0.4

0.5

0.6

Fig. 12. Comparison of ideal fluid and full scale cavity lengths for r = 0.9 at J = 0.95

Рис. 12. Сравнение длин каверны для r = 0,9 при J = 0,95: идеальная жидкость и результаты измерений

Normalized V

0.3 0.4 0.5 0.6 a

Fig. 13. Comparison of ideal fluid and full scale normalized cavity section areas for r = 0.9 at J = 0.95; solid line corresponds to solid line in Fig. 12, dashed line - to dashed line in Fig. 12, etc.

Рис. 13. Сравнение нормализованных площадей сечения каверны для r = 0,9 при J = 0,95: кривые (сплошная, пунктирная и т.п.) - см. рис. 12

at the varying values of azimuth 9 presented in Figs. 14 and 15 are in an evident correlation with the dependencies of Fig. 1. The wake structure in the computed example is more complex than in the majority of known studies operating with the wakes of the single-screw hulls (as in [7], [9]) and a small second peak of a at 9 ~ 270° was not typical for these studies.

Substantially greater dimensionless cavities at the model test conditions have been mentioned by many researchers. There is the possibility to reduce model test cavities and the first pressure oscillation harmonic down to the full scale level by simultaneous reduction of the blade propeller thrust coefficient and increase of

its advance ratio [23]. However, such modification of the test procedure leaded to an increase of the higher harmonics that would significantly exceeded their full scale level then.

Comparison of variation of cavity volume at model test and full scale conditions

Сравнение вариаций объема каверны в модельных и натурных условиях

The volumes of the entire cavities were calculated by integration of the above-presented two-dimensional results for their sections. There was no direct account of tip vortex cavitation in these computations, but some implicit account was included with the assumption of the constant cavity section area for 0.9 < r < 1.0. The comparison of evolutions of full scale and model test cavities during a blade revolution is given in Fig. 16.

Prediction of pressure harmonics on the basis of cavity volume computations is a separate numerical procedure because, as is clear from Eq. (1), such a prediction is associated with calculation of second derivatives of numerical solutions of the nonlinear problems. Also, the blade elasticity can affect harmonics. Thus, for a qualitative analysis of scale effects, one can consider only harmonics the volumes themselves, presented in Fig. 17.

The difference of full scale and model test harmonics, of course, depends on the hull shape and on the difference between the value of actual advance ratio J and its design value. In particular, the ratios of the first

Fig. 14. Dimensionless cavity section areas versus blade azimuth for model test conditions

Рис. 14. Безразмерные площади сечения каверны в зависимости от углового положения лопасти: модельные испытания

Fig. 15. Dimensionless cavity section areas versus blade azimuth for full-scale conditions

Рис. 15. Безразмерные площади сечения каверны в зависимости от углового положения лопасти: натурные условия

and the second harmonics depend of the actual wake. Also, the rigid wall of the ship hull amplifying volume oscillation is closer to the blade just in its top position (within wake peak). Nevertheless, the general tendencies, including substantially greater first and second model test harmonics, are quite similar. Seeking a possibility to match model test harmonics with full scale harmonics with employment of some coefficient/factor, one may compare normalized harmonics, as done here in Fig. 18.

Also, the contribution of the pressure side cavities is separately estimated in that plot. In the considered flow, the normalized first and second harmonics are close for the model test and full scale conditions, but there is no similarity of higher harmonics. So, there is no a coefficient/ factor that could uniformly reduce model test harmonics down to their full-scale level.

Conclusion

Заключение

The difference of model and ship wakes results in the substantial difference in blade section angles of attack at the same blade loading. Therefore, in model tests the suction side cavitation is more extensive, whereas the pressure side cavitation may not appear, though it exists on full-scale ship propeller blade. This substantial scale effect has been usually out of previous considerations. The corresponding difference in cavity volume evolutions leads to a difference of spectra of pressure pulsation (and, correspondingly, of excitation spectra for flow-induced ship hull vibration). Thus, because of the wake effect, it is impossible in the same model tests to predict accurately both first and high harmonics.

On the other hand, though the contemporary CFD tools look sufficient for prediction of the full-scale near wake, these tools must be supplemented by validated tools for blade cavitation. As already stated [24], [25], the tools developed on the basis of solvers for fully turbulent flows does not runs well at least for the model test conditions. Nevertheless, it would not be realistic to expect an extensive spread and amelioration of the described viscous-inviscid interaction method with integral boundary layer equations in the near future. It would be more realistic to seek modifications of existing RANS or LES solvers (or their merging with other solvers) to allow for the more accurate analysis of cavitation. Such modifications should take into account laminar flow around blade edges, capillarity and other effects important for cavitation. An intermediate option would be employment of results of the

V!D5

0.02

0.015

0.01

0.005

'* МТ

1 1 ■ 1

1 ** Л \

f ** FS г с-. *

-90

0

90

180

e°

Fig. 16. Dependencies of the dimensionless cavity volumes on blade azimuth

Рис. 16. Зависимости безразмерного объема каверн от углового положения лопасти

0.04

0.03 -

0.02 -

0.01 -

□ мт

■ FS

III........

Гкпгкп

1 2 3 4 5 6 7

9 10 11 12 13 14 15 16 17 18

Fig. 17. Comparison of model test and full scale dimensionless harmonics of V/D3 Рис. 17. Модельные и натурные безразмерные гармоники V/D3

1

0.8 H 0.6 0.4 0.2

□ МТ ■ FS

□ FSw/oPS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Fig. 18. Comparison of normalized harmonics of V/D3; the terms "full scale w/o pressure side" show full-scale results calculated without taking into account cavitation on the pressure side Рис. 18. Нормализованные гармоники V/D3: «натура, без нагнетающей стороны» - натурные результаты, полученные расчетным путем без учета кавитации на нагнетающей стороне

existing solvers for cavitation-free propellers as unperturbed solutions in the perturbation methods broadly used in computation of cavitating flows (in this paper and at least in papers from [26] to [27]).

References

1. G. Kuiper. Cavitation inception on ship propeller models. Wageningen: NSMB, 1981. 209, [16] p. (Publication / Netherlands Ship Model Basin; № 655).

2. O. Puolakka, J. Keto-Tokoi, J. Matusiak. Unsteady load on an oscillating Kaplan turbine runner // Journal of Fluids and Structures. 2013. Vol. 37. P. 22-33. DOI: 10.1016/j.jfluidstructs.2012.12.002.

3. J. Friesch. Correlation Investigation for Higher Order Pressure Fluctuations and Noise for Ship Propellers // Proc. of the 3rd Int. Symposium on Cavitation. Grenoble, 1998. Vol. 1. P. 7-10.

4. H.C.J. van Wijngaarden. Prediction of Propeller-Induced Hull-Pressure Fluctuations: doctoral thesis / H.C.J. Wijngaarden; Delft University of Technology, Department of Maritime Technology. Wageningen: Maritime Research Institute, 2011. 210 p.

5. Computational Analysis of Propeller Sheet Cavitation and Propeller -Ship Interaction / J. Bosschers, G. Vaz, A.R. Starke, E. van Wijngaarden // MARINE CFD 2008: Proc. of the conference. London: Royal Institution of Naval Architects, [2008]. 13 p.

6. Computational Ship Hydrodynamics: Nowadays and Way Forward / F. Stern, J. Yang, Z. Wang [et al.] // International Shipbuilding Progress. 2013. Vol. 60, № 1(Jan.). P. 3-105. DOI: 10.3233/ISP-130090.

7. Theoretical modeling of unsteady cavitation and induced noise / F. Salvatore, C. Testa, S. Ianiello, F. Pereira // Proc. of the 6th Int. Symposium on Cavitation. [S.l.], 2006. 13 p. 1 CD-ROM.

8. Numerical prediction of cavitation and pressure fluctuation around marine propeller / K. Sato, A. Ohshima, H. Egashira, S. Takano // Proc. of the 7th Int. Symposium on Cavitation. [S.l.], 2009. P. CAV2009 - № 141. P. 1-6.

9. K.-J. Paik, H.-J. Park, J. Seo. URANS simulations of cavitation and hull pressure fluctuation for marine propeller with hull interaction // Proc. of 3rd Int. Symposium on Marine Propulsors (SMP'13). Launceston: Australian Maritime College, [2013]. P. 389-396.

10. F. Salvatore, H. Streckwall, T. van. Terwisga. Propeller Cavitation Modelling by CFD - Results from the VIRTUE 2008 Rome Workshop // Proc. of 1st Int. Symposium on Marine Propulsors (SMP'09). Trondheim, 2009. P. 362-371.

11. E.-J. Foeth, C. van Doorne, T. van Terwisga. On the collapse structure of an attached cavity on a three-dimensional hydrofoil // Journal of Fluids Engineering.

2008. Vol. 130, № 7. P. 071303, p. 1-9. DOI: 10.1115/1.2928345.

12. Numerical simulation of cavity shedding from a three-dimensional twisted hydrofoil and induced pressure fluctuation by large-eddy simulation / X. Luo, X. Peng, H. Xu, M. Nishi, B. Ji // Journal of Fluids Engineering. 2012. Vol. 134, № 4. P. 041202. DOI: 10.1115/1.4006416.

13. V.H. Arakeri. Viscous effects on the position of cavita-tion separation from smooth bodies // Journal of Fluid Mechanics. 1975. Vol. 68, № 4. P. 779-799.

14. E.L. Amromin, A.N. Ivanov. Determination of points for cavity boundary separation from body with consideration of fluid viscosity and capillarity // Doklady Aka-demii Nauk SSSR (Transactions of the USSR Academy of Sciences). 1982. Vol. 262, No. 4. P. 823-826 (in Russian).

15. E.L. Amromin. On calculation of cavitating flows for viscous and capillary fluid // USSR Academy of Sciences. Izvestiya. Mechanics of fluids and gases. 1985. Vol. 20, No. 6. P. 871-877 (in Russian).

16. E.L. Amromin. Scale effect of cavitation inception on a 2D eppler hydrofoil // Journal of Fluids Engineering 2002, Vol. 124, № 1. P. 186-193. DOI: 10.1115/1.1427689.

17. S.D. Jessup, H.C. Wang. Propeller design and evaluation of a high speed patrol boat incorporating iterative analysis with panel method // Proc. of Propeller/Shafting'97 Symposium. Jersey: SNAME, 1997. P. 1101-1125.

18. V.G. Mishkevich. Scale and roughness effects in ship performances in designer's viewpoint // Marine Technology. 1995. Vol. 32, № 2. P. 126-131.

19. H. Yamaguchi, H. Kato. Non-linear theory for partially cavitating hydrofoils // Journal of the Society of Naval Architects of Japan. 1983. Vol. 152, Jan. P. 117-124.

20. C. Pellone, A. Rowe. Effect of separation on partial cavitation // Journal of Fluids Engineering. 1988. Vol. 110, № 2. P.182-189. DOI: 10.1115/1.3243532.

21. E.L. Amromin. Estimations of scale effects on blade cavitation // Journal of Physics: Conference Series. 2015. Vol. 656. P. 012058, p. 1-4. DOI: 10.1088/17426596/656/1/012058.

22. L. Gogish, G. Stepanov. Separating and cavitating flows. Main properties and analytical models. Moscow: Nauka, 1990. 384 p. (in Russian).

23. E. van Wijngaarden, J. Bosschers, T. van Terwisga. On predicting cavitation-induced hull pressure fluctuations -wake scale effects and signal variability // Proc. of IMarEST Noise and Vibration Conf. London: [Institute of Marine Engineering], 2010.

24. E.L. Amromin. Development and validation of CFD models for initial stages of cavitation // Journal of Fluids

Engineering. 2014. Vol. 136, № 8. P. 081303. DOI: 10.1115/1.4026883.

25. The Specialist Committee on Stability in Waves: Final Report and Recommendations to the 25th ITTC // Proc. of 25th ITTC. Fukuoka, 2008. Vol. 2. P. 473-533.

26. A. Ivanov. Cavitating flow around hydrofoils // USSR Academy of Sciences. Izvestiya. Mechanics and engineering. 1960. No. 6. P. 117-120 (in Russian).

27. E.L. Amromin. The origin of patch cavitation // International Journal Multiphase Flows. 2020. Vol. 124. P. 103188. DOI: 10.1016/j.ijmultiphaseflow. 2019.103188.

Список использованной литературы

1. Kuiper G. Cavitation inception on ship propeller models. Wageningen: NSMB, 1981. 209, [16] p. (Publication / Netherlands Ship Model Basin; № 655).

2. Puolakka O., Keto-Tokoi J., Matusiak J. Unsteady load on an oscillating Kaplan turbine runner // Journal of Fluids and Structures. 2013. Vol. 37. P. 22-33. DOI: 10.1016/j.jfuidstructs.2012.12.002.

3. Friesch J. Correlation Investigation for Higher Order Pressure Fluctuations and Noise for Ship Propellers // Proc. of the 3rd Int. Symposium on Cavitation. Grenoble, 1998. Vol. 1. P. 7-10.

4. Wijngaarden H.C.J. van. Prediction of Propeller-Induced Hull-Pressure Fluctuations: doctoral thesis / Wijngaar-den H.C.J. van; Delft University of Technology, Department of Maritime Technology. Wageningen: Maritime Research Institute, 2011. 210 p.

5. Computational Analysis of Propeller Sheet Cavitation and Propeller -Ship Interaction / Bosschers J., Vaz G., Starke A.R., Wijngaarden E. van // MARINE CFD 2008: Proc. of the conference. London: Royal Institution of Naval Architects, [2008]. 13 p.

6. Computational Ship Hydrodynamics: Nowadays and Way Forward / Stern F., Yang J., Wang Z. [et al.] // International Shipbuilding Progress. 2013. Vol. 60, № 1 (Jan.). P. 3-105. DOI: 10.3233/ISP-130090.

7. Theoretical modeling of unsteady cavitation and induced noise / Salvatore F., Testa C., Ianiello S., Pereira F. // Proc. of the 6th Int. Symposium on Cavitation. [S.l.], 2006. 13 p. 1 CD-ROM.

8. Numerical prediction of cavitation and pressure fluctuation around marine propeller / SatoK., OhshimaA., Egashira H., Takano S. // Proc. of the 7th Int. Symposium on Cavitation. [S.l.], 2009. P. CAV2009 - № 141. P. 1-6.

9. PaikK.-J., ParkH.-J, Seo J. URANS simulations of cavitation and hull pressure fluctuation for marine propeller with hull interaction // Proc. of 3rd Int. Symposium on Marine Propulsors (SMP'13). Launceston: Australian Maritime College, [2013]. P. 389-396.

10. Salvatore F., Streckwall H., Terwisga T. van. Propeller Cavitation Modelling by CFD - Results from the VIRTUE 2008 Rome Workshop // Proc. of 1st Int. Symposium on Marine Propulsors (SMP'09). Trondheim, 2009. P. 362-371.

11. Foeth E.-J, Doorne C. van, Terwisga T. van. On the collapse structure of an attached cavity on a three-dimensional hydrofoil // Journal of Fluids Engineering. 2008. Vol. 130, № 7. P. 071303, p. 1-9. DOI: 10.1115/1.2928345.

12. Numerical simulation of cavity shedding from a three-dimensional twisted hydrofoil and induced pressure fluctuation by large-eddy simulation / Luo X., Peng X., Xu H., Nishi M., Ji B. // Journal of Fluids Engineering. 2012. Vol. 134, № 4. P. 041202. DOI: 10.1115/1.4006416.

13. Arakeri V.H. Viscous effects on the position of cavita-tion separation from smooth bodies // Journal of Fluid Mechanics. 1975. Vol. 68, № 4. P. 779-799.

14. Амромин Э.Л., Иванов А.Н. Определение положения точек отрыва границы каверны от тела с учетом вязкости и капиллярности жидкости // Доклады Академии наук CCCP. 1982. Т. 262, № 4. С. 823-826.

15. Амромин Э.Л. К расчетам кавитационных течений вязкой и капиллярной жидкости // Известия АН CCCP. Механика жидкости и газа. 1985. Т. 20, № 6. С. 871-877.

16. Amromin E.L. Scale effect of cavitation inception on a 2D eppler hydrofoil // Journal of Fluids Engineering 2002, Vol. 124, № 1. P. 186-193. DOI: 10.1115/1.1427689.

17. Jessup S.D., Wang H.C. Propeller design and evaluation of a high speed patrol boat incorporating iterative analysis with panel method // Proc. of Propel-ler/Shafting'97 Symposium. Jersey: SNAME, 1997. P. 1101-1125.

18. Mishkevich V.G. Scale and roughness effects in ship performances in designer's viewpoint // Marine Technology. 1995. Vol. 32, № 2. P. 126-131.

19. Yamaguchi H., Kato H. Non-linear theory for partially cavitating hydrofoils // Journal of the Society of Naval Architects of Japan. 1983. Vol. 152, Jan. P. 117-124.

20. Pellone C., Rowe A. Effect of separation on partial cavi-tation // Journal of Fluids Engineering. 1988. Vol. 110, № 2. P.182-189. DOI: 10.1115/1.3243532.

21. Amromin E.L. Estimations of scale effects on blade cavitation // Journal of Physics: Conference Series. 2015. Vol. 656. P. 012058, p. 1-4. DOI: 10.1088/17426596/656/1/012058.

22. Гогиш Л.В., Степанов Г.Ю. Отрывные и кавита-ционные течения. Основные свойства и расчетные модели. Москва: Наука, 1990. 384 с.

23. Wijngaarden E. van, Bosschers J., Terwisga T. van. On predicting cavitation-induced hull pressure fluctuations -wake scale effects and signal variability // Proc. of IMarEST Noise and Vibration Conf. London: [Institute of Marine Engineering], 2010.

24. Amromin E.L. Development and validation of CFD models for initial stages of cavitation // Journal of Fluids Engineering. 2014. Vol. 136, № 8. P. 081303. DOI: 10.1115/1.4026883.

25. The Specialist Committee on Stability in Waves: Final report and recommendations to the 25th ITTC // Proc. of 25th ITTC. Fukuoka, 2008. Vol. 2. P. 473-533.

26. ИвановА.Н. Кавитационное обтекание профилей крыльев // Известия АН СССР. Механика и машиностроение. 1960. № 6. С. 117-120.

27. Amromin E.L. The origin of patch cavitation // International Journal Multiphase Flows. 2020. Vol. 124. P. 103188. DOI: 10.1016/j.ijmultiphaseflow. 2019.103188.

About the author

Eduard L. Amromin, Dr. Sci. (Eng.), Director, Mechmath LLC. Аddress: 2322 S 284 CT, Federal Way, WA 98003, USA. Tel.: +1 (206) 434-02-44. E-mail: Amromin@aol.com. https://orcid.org/0000-0001-5114-7019.

Сведения об авторе

Амромин Эдуард Львович, д.т.н., директор Mechmath LLC. Адрес: 2322 S 284 CT, Federal Way, WA 98003, USA. Тел.: +1 (206) 434-02-44. E-mail: Amromin@aol.com. https://orcid.org/0000-0001-5114-7019.

Поступила / Received: 27.05.21 Принята в печать / Accepted: 30.07.21 © Eduard L. Amromin. 2021