- Email: [email protected]
Analysis of interaction between ship bottom air cavity and boundary layer
Applied Ocean Research 59 (2016) 451–458
Contents lists available at ScienceDirect
Applied Ocean Research journal homepage: www.elsevier.com/locate/apor
Analysis of interaction between ship bottom air cavity and boundary layer E.L. Amromin ∗ Mechmath LLC, Federal Way, WA 98003, USA
a r t i c l e
i n f o
Article history: Received 23 August 2015 Received in revised form 4 February 2016 Accepted 14 March 2016 Keywords: Ship air bottom cavity Boundary layer Air demand Multi-phase compressible flow
a b s t r a c t The successful designs of hulls for ships employing drag reduction by air bottom cavitation have been based on solutions of inverse problems of the theory of ideal incompressible fluid. However, prediction of the drag reduction ratio, the air demand by ventilated cavities and the cavity impact on the hull–propeller interaction is impossible in the framework of this theory because all mentioned characteristics depend on interaction of air cavities with the ship boundary layers. Because the known CFD tools are not fitted to ventilated cavitation at low Froude numbers, an analysis of this interaction requires a novel flow model. This model includes the incompressible air flow in the ventilated cavity, the compressible flow of a water–air mixture in the boundary layer on cavities and downstream of them and the curl-free incompressible outer water flow. The provided 2D computations employing this model allows for explanations of the earlier observed effects and for prediction of the air demand by ventilated cavities. The computed velocity profiles downstream of cavities are in the accordance with the available experimental data. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Air bottom cavitation has been proven as the effective ship drag reduction technology since 1960s. The drag reduction rates up to 20% were reported by Butuzov et al. [1] for river ships and barges. Later the power saving rate close to 25% were predicted by Amromin et al. [2] and Gorbachev et al. [3] on the basis of towing tank tests for models of various sea ships with air bottom cavities. Sverchkov [4] and Amromin and Gorbachev [5] provided more detail on these successes. The interest to this technology has recurred during the last decade in various countries and the experimental results were reported also by Foeth [6], Kopriva et al. [7], Lay et al. [8], Mäkiharju et al. [9], Zverkhovskyi et al. [10]. This technology became successful because the special design of ship bottoms allowed for simultaneous elimination of the wall friction under the air cavity and suppression of the cavity tail pulsations. The proven methods of the successful ship hull design or the hull retrofits to this technology are based on solving inverse problems of ideal fluid theory similar to the linear problem considered by Butuzov [11] a half century ago. However, the effects related to interaction of the ventilated cavities with the ship boundary layers were left behind solutions of these problems, though some of these effects are significant and very important practically.
∗ Tel.: ++1 952 402 9642. E-mail address: [email protected] http://dx.doi.org/10.1016/j.apor.2016.03.009 0141-1187/© 2016 Elsevier Ltd. All rights reserved.
First, the drag reduction and power saving rates cannot be predicted in the framework of ideal fluid theory. This theory can satisfactory predict the shape of ventilated cavities (like done by Choi and Chahine [12]), but these rates are not directly proportional to the ratio of the hull surface area covered by the cavity to the total hull wetted surface area. In particular, these rates were lower in the experiment [10], but they were higher in the experiment [2]. Second, the friction reduction by the cavity and an air escape from the cavity affects the thicknesses of ship boundary layers and wakes, as well as the velocity profiles across them. The profile changes influence the propeller inflow field and should be taken into account during design of ship propellers. Third, there are substantial scale effects on air demand by partial ventilated cavities. The power saving rate evidently depends on this demand. As was reported in [1] for very small Fr and later confirmed in [2] for much higher Fr, there is a saturation of drag reduction by ventilated cavitation with an increase of air supply and an excessive air supply gives even negative results. However, as seen in Fig. 1 combining experimental data of Arndt et al. [13], Amromin et al. [2] and Mäkiharju et al. [9], the common trend of the air demand as a function of Fr for various tests does not exist even for Fr < 1 (the shapes of models the tested in [2,9,13] are shown in Figs. 2–4). Moreover, as reported in [13], the ventilated cavities under small models can be maintained at small Reynolds numbers even without any air supply. Further, as one may find in the paper of Kawakami and Arndt [14], i.e., it looks impossible to derive a dependency of Q on
452
E.L. Amromin / Applied Ocean Research 59 (2016) 451–458
Nomenclature B ship (model) beam CD drag coefficient dP/d sound speed c=
Fr = U∞ / gL Froude number g gravity acceleration cavity thickness H L ship (model) length Lc cavity length M Mach number normal to S* N Pc pressure in cavity P∞ ambient pressure volumetric air demand by cavity Q q 104 Q/LBU∞ Re = LU∞ / Reynolds number r(ˇ) = ˇε + 1 − ˇ mixture normalized (dimensionless) density combination of all boundaries of inviscid flow S* Sc the part of S* under the cavity U = |grad˚|/U∞ velocity of inviscid flow U∞ ship (model) speed u velocity in the boundary layer u0 velocity on the cavity surface air flow velocity components ua , va v* friction velocity X2 abscissa of the cavity end friction velocity v* y lateral coordinate in boundary layer zC vertical coordinate of the cavity surface normalized by L void fraction in the boundary layer ˇ ı thickness of water boundary layer
ı
ı∗
=
(1 − ru/U)dy displacement thickness 0
ı ı∗∗ =
ru/U(1 − u/U)dy momentum thickness 0 H
ı∗a =
(1 − ua /u0 )dz the airflow displacement thicknesses 0
the ratio of air density to water density ε water density water kinematical viscosity 2 cavitation number = 2(P∞ − Pc )/U∞ ˚ velocity potential
Fig. 1. Air demand to maintain cavities under a 12.9 m 2D model in a water tunnel (squares, from [9]), a 0.5 m 2D model in another water tunnel (triangles, from [13]) and a 4.6 m ship model towed in a tank at Fr-depended pitch (rhombs, from [2]).
Fig. 2. Computed sections of the water tunnel model with cavities tested in [9] at various Fr.
cavitation number from experimental data even for the same body. So, the known experimental data on interaction of ventilated cavities with boundary layers are insufficient for their empirical generalization and there is the practical necessity of a numerical analysis of the interaction of ventilated cavities with boundary layers over the already designed hull. Kinzel et al. [15] numerically manifested a high impact of turbulent mixing on air demand by ventilated supercavitating flows at moderate Re and Fr 1 and the necessity to include a boundary layer impact in the analysis of ventilated cavitation became clear. One may think that such analysis is a simple task because the numerous computational studies on natural cavitation in viscous fluids have been already described in the literature. Indeed the flow models employed in these studies are not suitable for the air ventilated cavitation. First of all, they did not give a satisfactory description of air escape through the cavity surface. One group of these computational studies has been carried out using various versions of viscous–inviscid interaction methods; this group neglected by the gas flows within the cavities and its escape from them (as in Brewer and Kinnas [16] or in Amromin [17]). Another group of computational studies has been carried out using for the whole flow the model of a bubble cloud. Such model was suggested by Kubota et al. [18] for the cavity closure zone and there is no cavity surface in this model, though one can clearly see such surface in photos of ventilated cavities (as by Kawakami and Arndt [14]) and no bubbly clouds inside ventilated cavities are seen there. Further, all computational studies are based on semi-empirical approaches
Fig. 3. Sections of the ship model towed [2] with computed cavity at Fr = 0.48 (at the model measured pitch).
E.L. Amromin / Applied Ocean Research 59 (2016) 451–458
453
Fig. 4. Ventilated cavity in a bottom niche of a 0.5 m 2D model in the water tunnel [13].
Fig. 5. Ship models with bottom recesses (niches) for cavities tested in Krylov Ship Research Institute (left) and David Taylor Model Basin (right).
(like any study of turbulent flows) and therefore, the corresponding solvers are unavoidably fitted to the certain kinds of flow. Meanwhile, though ventilated cavitation has been broadly used as the experimental tool for modeling natural cavitation (since Reichardt [19]), there is the measurable differences in the characteristics of ventilated and natural cavities on the same body at the same Fr and cavitation number. One can see this difference in [7]. Moreover, such solvers were validated at high values of Fr and the Fr effects were mainly reasonably omitted in them (one can find in [14] that |∂LC /∂| ≈ (160 ÷ 300) |∂LC /∂Fr| at 9.1 < Fr < 34.0), but such omission is not acceptable for small Fr, where the Fr influence may be comparable or even higher than the influence (one can find in Amromin [20], that |∂LC /∂| ≈ |∂LC /∂Fr| at 0.5 < Fr < 0.55). So, a numerical analysis of the interaction of air ventilated cavities with boundary layers requires a special flow model. Such a special model can be validated even in its 2D version because the flows near the ship hull bottom are quite close to 2D flows due to installation of keels restricting ventilated cavities (as seen in Fig. 5) and the following analysis is limited by steady 2D flows, like the analysis carried out by Shiri et al. [21].
The first zone consists of the inviscid incompressible flow of water outside of the hull boundary layer and wake (this zone is under the dashed curve in Fig. 6). The second zone consists of the cavity filed by air (its boundary is plotted as a dotted curve in Fig. 6); an air escape from the cavity is included in the model. The third zone consists of the ship boundary layer on the cavity surface and on the hull wetted surface; the wake behind the hull is the continuation of this zone. The boundaries between these zones are initially unknown and must be determined in the course of computations. Because the outer inviscid flow in the first zone is considered as curl-free, the velocity potential can be employed there. Besides of Laplace equation ˚ = 0,
(1)
the boundary-value problem for ˚ includes the condition on the velocity projection on the normal to the inviscid flow boundary
∂˚/∂N
S∗
= 0,
(2)
the pressure constancy condition on the cavity surface written with taking into account the buoyancy force
2. Model of cavitating flow and employed numerical techniques Cavitating flows are quite complex and include zones with the qualitatively different features. In the suggested model of ventilated by air bottom cavitation the whole flow is divided into three zones (a sketch of a part of the flow with three zones is given in Fig. 6).
Fig. 6. Sketch of ventilated bottom cavitation.
U 2 + 2Fr −2 (z∞ − zC )
Sc
= 1 + ,
(3)
and the condition of flow uniformity in the infinity |grad˚||x|→∞ → {U∞ , 0}.
(4)
One part of the boundary S* covers the cavity surface, another part covers the hull wetted surface. The last part includes a region just downstream of the cavity, where the flow parameters may experience a jump (like pressure experiences a jump on the Riabouchinsky solid closing cavities in the Riabouchinsky scheme described by Knapp et al. [22] and in many other monographs). The above problem for the linear Eq. (1) becomes nonlinear due to the necessity to satisfy two conditions on the same boundary Sc. Because of this necessity, the boundary shape cannot be arbitrary, but it should be found in iterations. There are several known methods to solve such problems, from the BEM-based method described by Amromin [23] to the method described by Faltisen and Semenov [24]. The employed method is based on the subsequent quasilinearization of Eqs. (2) and (3). An illustration of its convergence
454
E.L. Amromin / Applied Ocean Research 59 (2016) 451–458
Fig. 7. An example of distribution of computational error along the cavity.
is shown in Fig. 7, where modulus of the difference between the left-hand side and the right-hand side of Eq. (3) is defined as the computational error. One will see there that by iteration number 90 this modulus is below 0.0005 at any point of Sc. Regarding to the experimental validation of this method, it is necessary to point out that the computed cavity shape with the thinnest cavity tail in Fig. 2 corresponds to exactly the same Fr = 0.71 that corresponded to the most stable cavity among cavities observed in [9]. Another example of validation by comparisons of computed and observed shapes of ventilated cavities determined with by solving Eqs. (1)–(4) were provided in [13]. So, computations in the first zone are based on the already proven flow model and computational technique. In the second zone, there is the need in a novel and quite simple flow model. Let us neglect by compressibility of the air within cavities because the air velocity components are small there. Experimental data [2,3] make it possible to estimate that these components are below 1 m/s. It is possible to model the flow in this zone as a turbulent air boundary layer between the bottom and the flow of the water–air mixture. Therefore one can write the air mass conservation equation integrated across the cavity as u0
dı∗a dx
− (H − ı∗a )
du0 dQ − = 0. dx dx
(5)
Here va = dQ/dx, the selected air velocity profile across the ˆ ˛ + B ˆ , ˛ = 1/7, ˆ = z/H, A = u0 /(1 − ˛/ ), cavity is ua = A B = u0 /(1 − /˛), = ˛(Q/Hu0 − 1) because on the cavity surface the tangential velocity of air coincides with the tangential velocity of water. The air entrainment by water on the cavity surface is implicitly taken into account here by matching the air and water velocities at the surface of their contact. The third zone indeed includes three sub-zones. The first one is located upstream the cavity, and the hull boundary layer is considered as the air-free layer there. The second on is under the cavity. The third one is downstream from it. From the cavity detachment point water entrains air from the cavity and the flow in this zone becomes a two-phase flow. At the cavity tail, there is usually some time-average air flux from the cavity even with the perfect ship bottom design because in the reality the design condition are usually perturbed, and therefore, the total air flow experiences a jump at the boundary layer reattachment to the hull downstream of the cavity. For such complex zone, an integral computational method is employed because of the simplicity reason. Thus, along the cavity surface, the boundary layer void fraction varies in the accordance with the equation d(ıˇ) dQ + = −ˇϑ. dx u0 dx
Fig. 8. Mixture sound speed (m/s) versus air void fraction at various ambient pressure values.
Here W = 1 − u0 /U, M = U/c. Derivation of Eq. (7) follows to the steps described by Cebeci and Bradshow [25], but here the second term in the left-hand side reflects flow through the layer inner boundary. Unlikely to the air cavity, the boundary layer on the cavity and downstream of it must be considered as the compressible layer because of the low sound speeds in air–water mixtures. As follows from Fig. 8, the local Mach numbers M =
2 (1 − ˇ − ˇε) + M 2 ˇ/ε]/(1 − εˇ) can be significant in (1 − ˇ)[Mw a the ship boundary layers; here Mw and Ma are water Mach number and air Mach number. The sound speed is calculated for Fig. 8 at 20 ◦ C, but one can found the similar data for the broad range of temperatures and comparison with experimental data in the paper of Kieffer [26]. Additionally to Eqs. (6) and (7), the water differential momentum equation directly derived from the 2D Reynolds equation on the cavity surface is considered
u0
du0 ∂u u
∂u v
− . =− dx ∂N ∂x
The right-hand side of Eq. (8) can be approximated by the sum (UW)2 /ı + bgdzC /dx because its second term should be proportional to U2 . An analogy of the boundary layer over the cavity with a body wake is used there and according to the approximation of the experimental data of Bourgoyne et al. [28] for turbulent wakes shown in Fig. 9, one can assume that u v ∼U 2 Re−1/2 . Therefore = C Re−0.5 , where C = const. The experimental data [9] for Fr < 0.75 were used here to find C = 3.0 and b = −0.05. Because the equations describing the air flow in the cavity and the flow of an air–water mixture in the boundary layer on the cavity are strongly coupled, they must be solved together. The unknown functions in Eqs. (5)–(8) include Q and three boundary layer parameters {ı, u0 , ˇ}. As usual for the integral methods, the velocity profiles across the layer must be selected. For this turbulent boundary layers on the cavity the velocity profile is selected as
(6)
Here ϑ = const is assumed. The momentum equation integrated across this layer is written with taking into account the air escape dı∗∗ εW dQ (M 2 − 2)ı∗∗ − ı∗ + ı dU + = . U dx U dx dx
(7)
(8)
Fig. 9. Approximation of experimental data for turbulent wakes.
E.L. Amromin / Applied Ocean Research 59 (2016) 451–458
455
u = u0 (x) + [U(x) − u0 (x)]FP (x, ), where FP = 32 − 23 , = y/ı. Generally, a profile of ˇ can be also introduced. The void fraction profiles in a two-phase flow with microbubbles were considered by Kunz et al. [29], but flows with air layers studied by Elbing et al. [30] look more similar to the cavity boundary layer. This similarity allows for the assumption that the void fraction in the cavity boundary layer sharply drops to a constant level in the main part of the layer. The momentum integral equation for the boundary layer on the hull wetted surface has the form
dı∗∗ v∗ =r U dx
2
−
(2 − M 2 )ı∗∗ + ı∗ dU . U dx
(9)
Eq. (9) is very similar to Eq. (7). The employed velocity profile is u(x, y) = F(x, y) + CU FP (x, ), where F = 2.5v∗ (ln |yv∗ /| + B), B(x) depends on the wall roughness, CU = U(x) − F(x, ı). Upstream of the cavity M is negligibly small. The mass conservation law for this boundary layer is written with an empirical coefficient K* d[U(ı − ı∗ )] = rUK ∗ , dx
(10)
where the expression for K* as a function of (ı − ı*)/ı** recommended by Cebeci and Bradshaw [25] for the single phase flows is used. Also, the air escape from boundary layer is described by the equation d(ıˇ) = −ˇϑ. dx
(11)
obtained from Eq. (6) with taking into account that Q = 0 out of cavity. In the hull boundary layer downstream of cavities the unknown boundary layer parameters are {v*, ı, ˇ}. It is necessary to determine their initial values at the cavity end x = X2 . The conditions applied for their determination include the continuity condition for the displacement body ordinate ı∗ (X2 + 0) = ı∗ (X2 − 0) + H(X2 − 0)
(12)
a generalization of the experimental data of Kline et al. [31] for separated boundary layers ı(X2 + 0)/ı∗ (X2 + 0) = 1.42 + r
(13)
Fig. 10. Computed (lines) and measured [34] (symbols) distributions of the friction ratio Cf /Cf0 in air–water mixture and pure water along the same flat plate. Distance x is counted from the plate leading edge. The first numbers in the legend indicate the flow speed (m/s). The second numbers indicate the value of air volumetric concentration.
et al. [34] in Fig. 10 is one of comparisons shown in [27]. Here it can be considered as a validation example of the described method for two-phase boundary layers. 3. Prediction of air demand to maintain the cavity The problem of prediction of the air demand by cavities in steady flow conditions is the first problem considered with the abovedescribed computational model. The obtained numerical solutions are compared in Fig. 11 with the experimental data. The data [9] on air demand necessary to maintain cavities in steady flows under a huge 2D hydrofoil in the water tunnel are quite unique. Other water tunnel experiments are not suitable for such validation because their trend is substantially affected by laminar-turbulent transition (as provided in [13]). The comparison of the described computations with experimental data [2] for a ship model in a towing tank is also shown in Fig. 11. The common point of all sets of data on Q used in Fig. 11 is that the air demand minima were achieved at the design values of Fr corresponding to the drag minima. The agreement of computations with experimental data in Fig. 11 looks satisfactory.
and the mass conservation law written with taking into account the air escaped the cavity
4. Cavities under flat bottoms
U(ı − ı∗ )
The majority of known studies on drag reduction by ventilation relates to flows under flat bottoms. So, it is important to consider such flows in this analysis. In particular for cavities behind backward-facing steps under flat bottoms, the computed dependencies for cavity sizes should be quite similar to predicted in the framework of the linear ideal fluid theory [11]. So, one may
X2 +0
= εQ + U(ı − ı∗ )
X2 −0
(14)
Generally, the described model belongs to the viscous–inviscid interaction approach and the boundary S* should be displaced on a distance ı* from the body surface, etc. (as done in [17]). However, because the cavity detachment for the bottom ventilated cavities are usually fixed, it is accurate enough to solve the whole problem with the assumption of the weak viscous–inviscid interaction around the cavity. This actually means that the inverse influence of the boundary layer thickness on the pressure can be neglected during determination of the cavity shape. On the other hand, there is a substantial impact of ı* on the pressure at the transaction of the boundary layer into the wake behind the hull. This zone is a zone of strong viscous–inviscid interaction. This wake can be computed with the integral method described in [27]. Let us point out that the role of the described multi-zone model of cavitating flows is much more important than the role of the numerical methods assigned to each zone. Indeed, viscous–inviscid interaction approach has been used with integration of differential equations for boundary layers, too (as by Briley and Mcdonald [32], Cebeci et al. [33] and in others papers later). The comparison of the computed characteristics of such multiphase boundary layer with its measured characteristics by Sanders
Fig. 11. Comparison of computations (lines) and experimental data (symbols) air demands by cavities. Data A are taken from [2] data M – from [9].
456
E.L. Amromin / Applied Ocean Research 59 (2016) 451–458
Fig. 12. Computed cavitation numbers versus ventilated cavity lengths behind a backward step under a flat bottom, L/H = 100. The right edges of curves correspond to cavities of the maximum Fr-dependent length.
consider the presented computations for flat bottoms as a qualitative validation for the above-described combined method. Indeed, the computations results obtained by solving the nonlinear problem (1)–(4) and shown in Fig. 12 exhibit a substantial dependency of Lc on Fr (though it is not absolutely proportional to Fr2 , as the linear theory predicts) and the possibility to maintain a cavity of the same length at very different cavitation numbers. As seen in Fig. 12, cavitation numbers are negative there. So, an excessive air pressure is necessary to maintain such cavities. However, according to computations, a very moderate increase of the 2 ÷ 0.02U 2 ) is sufficient for a 25% cavity pressure (dPC ≈ 0.01U∞ ∞ increase of the cavity length in the considered Fr range. Further, the flow in the boundary layer downstream of the cavity is significantly different from the cavitation-free situation. As illustrated by Fig. 13 containing comparisons with the pure water boundary layer characteristics over the flat plate, the behavior of the displacement thickness downstream of the cavity is dissimilar to its behavior in the cavitation-free flow. This is important because a decrease of ı* should lead to a reduction of the wake thickness and, as a result, of the form resistance. 5. Influence of pressure gradient on flow behind cavities and scale effect The flows under ships under hulls of ships and their models are much more complex. In particular, as shown in [2], a significant pitch and heavy can take place for ships with ventilated cavities or their models at Fr > 0.45 and this leads to a complex combination of drag reduction effects. However, because the following computations relate to the smaller Fr values, only negligibly small pitch values will be considered below. Further, the real ships always have zones of substantial pressure gradients at their sterns and, as clear from Eq. (9), the flow compressibility effects is coupled with the
Fig. 13. Variation of thickness displacement along the flat bottom downstream of cavity for Fr = 0.355, Re = 3.5 × 107 . Here q = 30 corresponds to the minimum air demand to maintain the cavity.
Fig. 14. Buttock of a ship (upper plot) and pressure coefficient over her stern (lower plot).
pressure gradient effects. The computed void fraction in some parts of the boundary layer may go at least up to 0.1 for full scale ships. Returning to Fig. 8, one can estimate that the local sound speed may be below 40 m/s and the Mach number can be quite high there. On the other hand, for the towing tank models of the length from 5 m to 10 m at Fr < 1, U ∼ L1/2 and the local Mach number is very small everywhere. The buttock employed in the following computations and shown in Fig. 14 has a cavity seal/locker behind the cavity (like the buttocks shown in Figs. 2 and 3). For the computational examples shown in these figures, X2 = 0.73L, and one can see in Fig. 14 that the cavity slightly affects the pressure gradient downstream from the cavity locker (at x/L > 0.78). The pressure gradients in 2D flows are higher than they would be over a 3D ship stern with the same buttock. Therefore the real pressure gradient influence on 3D multi-phase flows downstream of cavities will rather qualitatively correspond to the dependencies computed for a ship buttock and plotted in Figs. 15–17. These computations were carried out for a ship model of the length L = 10 m (Reynolds number Re = 4 × 107 ). The corresponding fullscale ship length was selected as L = 160 m (Re = 2.56 × 109 at the same Fr = 0.4). It is seen in Fig. 15 that the wall friction significantly decreases in the zone of the adverse pressure gradient (0.85 ≤ x/L ≤ 0.95) in the cavitation-free pure water flow, but there is no similar decrease downstream of the cavities.
Fig. 15. Wall friction downstream of cavities with the same q at Re = 4 × 107 (solid lines) and Re = 2.56 × 109 (dashed lines) and without cavity at Re = 2.56 × 109 (squares).
E.L. Amromin / Applied Ocean Research 59 (2016) 451–458
Fig. 16. Displacement thickness (in the right) downstream of cavities with the same q at Re = 4 × 107 (solid lines) and Re = 2.56 × 109 (dashed lines) and without cavity at Re = 2.56 × 109 (triangles).
457
Fig. 18. Normalized velocity profiles at x = 0.95L for flows behind cavities. Dashed line relates to q = 13 and Re = 2.56 × 109 , solid line–to q = 13 and Re = 4 × 107 , rhombs–to q = 40 and Re = 4 × 107 .
As shown in Figs. 15 and 16, both the wall friction and the displacement thickness downstream of the cavities are lower than they were on the cavitation-free stern. The decrease of the thickness displacement (and of the wake thickness) behind the cavity is very significant. Because the form resistance is roughly propor-
L
tional to
L
(∂[U( )ı∗ ( )]/∂ )(d /x − )dx in 2D flows and
Nx U(x) 0
0
this resistance can be similarly estimated in 3D flows, a measurable reduction of the form resistance for ships with ventilated cavity is expectable. This reduction is the substantial advantage of drag reduction by air cavities in comparison with microbubbles drag reduction, were the total drag reduction is smaller that friction drag reduction under the affected bottom area (as reported by Sato et al. [35]). Also, the changes of the wall friction and the boundary layer thicknesses downstream of ventilated cavities evidently lead to changes of the marine propeller inflow. The location of the propeller disk is selected as x ≈ 0.95 for the buttock shown in Fig. 14. An illustration of the cavitation effect and scale effect on the velocity profiles at x = 0.95L is provided in the left part of Fig. 17. The right part of this figure provides experimental data on the velocity profiles (inflow field) in a propeller disk extracted from the recent paper by Gorbachev et al. [36]. There is not enough data on the model hull in [36] for an attempt of a direct comparison. So, one can find only in two parts of this figure only the same qualitative effect (similar velocity differences between flows with and without cavitation at the fixed points). The Fig. 17 is supplemented by the illustration of two abovementioned effects on the product of velocity and density in Fig. 18. As one can conclude from Figs. 17 and 18, the inflow of a marine propeller is significantly affected by the air bottom cavitation.
Fig. 19. Air supply impact on velocity profiles at x = 0.95L.
The scale effect on the velocity profiles in the boundary layer downstream of ventilated cavities is even smaller than the cavitation effect on these profiles. So, the marine propellers should be rather redesigned in the course of a retrofit of the existing ships to drag reduction by air bottom cavitation. It is also interesting that, as shown in Fig. 19, the intensity of air supply significantly affects the velocity profile at a fixed pair {Re, Fr}. So, model tests with the excessive air supply may be useful for the ship propeller design. Some illustration on compressibility impact on flows downstream of the cavity could be also useful. Distribution of thickness displacement was selected for such illustration because of the possibility to show this impact for the whole stern. Computations for full-scale conditions with the actual Mach number and computations with M = 0 in Eqs. (7) and (9) are compared in Fig. 20. It is clearly seen in Fig. 20 that the compressibility impact increases with the air supply rate. Unlike to situation with microbubble drag reduction [27] with much higher air demand, it may be acceptable to neglect the flow compressibility at design conditions with quite low air escapes from cavities, but such simplification can be unacceptable at design off.
Fig. 17. In the left: computed velocity profiles at x = 0.95L. Dashed line relates to cavitation-free flow at Re = 4 × 107 , solid line–to q = 13 and Re = 2.56 × 109 , triangles – to q = 13 and Re = 4 × 107 ; q = 13 corresponds to the minimum air demand by the cavity. In the right: Measured [36] velocity profile in the model propeller disk at two azimuth values: symbols jointed by lines related to flow without cavity, isolated symbols – to flow with cavity; r – distance from the disk center, D – its diameter.
458
E.L. Amromin / Applied Ocean Research 59 (2016) 451–458
Fig. 20. Illustration of compressibility impact on thickness displacement.
6. Conclusions The provided analysis of air cavitation under ship bottoms is carried out with a novel flow model and a combination of the earlier developed computational techniques. Employment of this model allowed for several important quantitative and qualitative results related to interaction of ventilated cavities with ship boundary layers. In particular, the minimum air demand necessary for the cavity maintenance in steady flows can be predicted satisfactory with this model. The reported computations also confirm that this minimum air for a cavity without a locker/seal is much greater than such minimum for the design with the cavity locker/seal. The reported computations of the wall friction and thickness displacement over the sterns make it possible to better comprehend the synergy of drag reduction effects for ships with bottom ventilated cavitation and, in particular, reduction of their form resistance. The effect of ventilated cavities on propeller inflows (on the velocity profiles at the propeller disks) was estimated. It was shown that cavities mitigate scale effects on these profiles. The feasibility of the described novel flow model was proven for 2D flows with the quite simple description of boundary layers (with employment of integral methods). This model can be generalized for 3D flows with the employment of the more modern CFD techniques. References [1] Butuzov AA, Gorbachev YN, Ivanov AN, Kaluzhny VG, Pavlenko AN. Ship drag reduction by artificial gas cavities. Sudostroenie 1990;n#11:3–6 [in Russian]. [2] Amromin EL, Metcalf B, Karafiath G. Synergy of resistance reduction effects for a ship with bottom air cavity. ASME J Fluids Eng 2011;133:021302 [7 pages]. [3] Gorbachev YN, Buyanov AS, Sverchkov AV. Ships with air cavity: real method to enhance energy efficiency and ecological safety. Sea Fleet 2015;n#1:26–35 [in Russian]. [4] Sverchkov AV. Application of air cavities on high-speed ships in Russia. In: International conf. on ship drag reduction. 2010. [5] Amromin EL, Gorbachev YN. Technologies of ship resistance reduction. In: Compendium on naval hydrodynamics. Paris, France: ENSTA; 2015 [Chapter 4]. [6] Foeth EJ. Decreasing of frictional resistance by air lubrication. In: 20th int. Hiswa symposium on Yacht design and Yacht construction. 2008.
[7] Kopriva J, Amromin EL, Arndt REA. Improvement of hydrofoil performance by partial ventilated cavitation in steady flow and periodic gusts. ASME J Fluids Eng 2008;130:031301 [7 pages]. [8] Lay KA, Yakushiji R, Makiharju S, Perlin M, Ceccio SL. Partial cavity drag reduction at high Reynolds numbers. J Ship Res 2010;54:109–19. [9] Mäkiharju SA, Elbing BR, Wiggins A, Schinasi S, Vanden-Broeck J-M, Perlin M, et al. On the scaling of air entrainment from a ventilated partial cavity. J Fluid Mech 2013;732:47–76. [10] Zverkhovskyi O, Terwisga T, van Guning M, Westerwell J, Delfos R. Experimental study on drag reduction by air cavities on a ship model. In: 30th sym. naval hydrodynamics. 2014. [11] Butusov AA. Extreme parameters of vented cavity on the top surface of horizontal wall. Fluids Dyn 1966;1:167–70. [12] Choi J-K, Chahine GL. Numerical study on the behavior of air layers used for drag reduction. In: 28th sym. naval hydrodynamics. 2010. [13] Arndt REA, Hambleton WT, Kawakami E, Amromin EL. Creation and maintenance of cavities under horizontal surfaces in steady and gust flows. ASME J Fluids Eng 2009;131:111301 [10 pages]. [14] Kawakami E, Arndt REA. Investigation of the behavior of ventilated supercavities. ASME J Fluids Eng 2011;133:091305 [11 pages]. [15] Kinzel MP, Lindau JW, Peltier J, Zajaczkowski F, Arndt REA, Wosnik M, et al. Computational investigation of air entrainment, hysteresis and loading for large-scale, buoyant cavities. In: 9th int. conference on numerical ship hydrodynamics. 2007. [16] Brewer WH, Kinnas SI. Experiment and viscous flow analysis on a partially cavitating hydrofoil. J Ship Res 1997;41:161–71. [17] Amromin EL. Scale effect of cavitation inception on a 2D Eppler hydrofoil. ASME J Fluids Eng 2002;124:186–93. [18] Kubota A, Kato H, Yamagushi H. A new modeling of cavitating flows: a numerical study of unsteady cavitation of a hydrofoil section. J Fluid Mech 1992;240:59–96. [19] Reichardt H. The law of cavitation bubbles at axially symmetric bodies in a flow. Reports and translations, vol. 766. UK: Ministry of Aircraft Production; 1946. [20] Amromin EL. Design of bodies with drag reduction by partial cavitation as an inverse ill-posed problem for velocity potential. In: 9th int. conference on numerical ship hydrodynamics. 2007. [21] Shiri A, Leer-Andersen M, Bensow RE, Norrby J. Hydrodynamics of a displacement air cavity ship. In: 29th sym. naval hydrodynamics. 2012. [22] Knapp RT, Daily JW, Hammit FG. Cavitation. NY: McCraw-Hill; 1970. [23] Amromin EL. Ships with ventilated cavitation in seaways and active flow control. Appl Ocean Res 2015;50:163–72. [24] Faltisen OM, Semenov YA. The effect of gravity and cavitation on a hydrofoil near the free surface. J Fluid Mech 2008;597:371–94. [25] Cebeci T, Bradshaw P. Physical and computational aspects of convective head transfer. NY: Springer-Verlag; 1984. [26] Kieffer SW. Sound speed in liquid–gas mixtures, water–air and water–steam. J Geophys Res 1977;82:2895–904. [27] Amromin EL. Microbubble drag reduction downstream of ventilated partial cavity. ASME J Fluids Eng 2010;132:051302 [5 pages]. [28] Bourgoyne DA, Ceccio SL, Dowling DR. Vortex shedding from a hydrofoil at high Reynolds number. J Fluid Mech 2005;531:293–324. [29] Kunz RF, Maxey MK, Tryggvason G, Fontaine AA, Ceccio SL, Gibeling HL, et al. Validation of two-fluid Eulerian CFD modeling for microbubble drag reduction across a wide range of Reynolds numbers. ASME J Fluids Eng 2006;129:66–79. [30] Elbing BR, Mäkiharju SA, Wiggins A, Perlin M, Dowling DR, Ceccio SL. On the scaling of air layer drag reduction. J Fluid Mech 2013;717:484–513. [31] Kline SJ, Bardina JG, Strawn RC. Correlation of the detachment of twodimensional boundary layers. AIAA J 1983;21:68–73. [32] Briley WR, Mcdonald H. Numerical prediction of incompressible separation bubbles. J Fluid Mech 1976;69:631–56. [33] Cebeci T, Clark RW, Chang KC, Halsey ND, Lee K. Airfoils with separation and the resulting wakes. J Fluid Mech 1986;163:323–47. [34] Sanders WC, Winkel ES, Dowling DR, Perlin M, Ceccio SL. Bubble friction drag reduction in a high Reynolds number flat plate turbulent boundary layer. J Fluid Mech 2006;552:353–80. [35] Sato T, Nakata T, Takeshita M, Tsuchiya Y, Miyata H. Experimental study on friction reduction of a model ship by air lubrication. J Soc Naval Archit Jpn 1997;182:121–8. [36] Gorbachev YN, Sverchkov AV, Galushina MV. Propulsion of displacement ships with the single bottom cavities. Sudostroenie 2015;n#1:17–23 [in Russian].
Contents lists available at ScienceDirect
Applied Ocean Research journal homepage: www.elsevier.com/locate/apor
Analysis of interaction between ship bottom air cavity and boundary layer E.L. Amromin ∗ Mechmath LLC, Federal Way, WA 98003, USA
a r t i c l e
i n f o
Article history: Received 23 August 2015 Received in revised form 4 February 2016 Accepted 14 March 2016 Keywords: Ship air bottom cavity Boundary layer Air demand Multi-phase compressible flow
a b s t r a c t The successful designs of hulls for ships employing drag reduction by air bottom cavitation have been based on solutions of inverse problems of the theory of ideal incompressible fluid. However, prediction of the drag reduction ratio, the air demand by ventilated cavities and the cavity impact on the hull–propeller interaction is impossible in the framework of this theory because all mentioned characteristics depend on interaction of air cavities with the ship boundary layers. Because the known CFD tools are not fitted to ventilated cavitation at low Froude numbers, an analysis of this interaction requires a novel flow model. This model includes the incompressible air flow in the ventilated cavity, the compressible flow of a water–air mixture in the boundary layer on cavities and downstream of them and the curl-free incompressible outer water flow. The provided 2D computations employing this model allows for explanations of the earlier observed effects and for prediction of the air demand by ventilated cavities. The computed velocity profiles downstream of cavities are in the accordance with the available experimental data. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Air bottom cavitation has been proven as the effective ship drag reduction technology since 1960s. The drag reduction rates up to 20% were reported by Butuzov et al. [1] for river ships and barges. Later the power saving rate close to 25% were predicted by Amromin et al. [2] and Gorbachev et al. [3] on the basis of towing tank tests for models of various sea ships with air bottom cavities. Sverchkov [4] and Amromin and Gorbachev [5] provided more detail on these successes. The interest to this technology has recurred during the last decade in various countries and the experimental results were reported also by Foeth [6], Kopriva et al. [7], Lay et al. [8], Mäkiharju et al. [9], Zverkhovskyi et al. [10]. This technology became successful because the special design of ship bottoms allowed for simultaneous elimination of the wall friction under the air cavity and suppression of the cavity tail pulsations. The proven methods of the successful ship hull design or the hull retrofits to this technology are based on solving inverse problems of ideal fluid theory similar to the linear problem considered by Butuzov [11] a half century ago. However, the effects related to interaction of the ventilated cavities with the ship boundary layers were left behind solutions of these problems, though some of these effects are significant and very important practically.
∗ Tel.: ++1 952 402 9642. E-mail address: [email protected] http://dx.doi.org/10.1016/j.apor.2016.03.009 0141-1187/© 2016 Elsevier Ltd. All rights reserved.
First, the drag reduction and power saving rates cannot be predicted in the framework of ideal fluid theory. This theory can satisfactory predict the shape of ventilated cavities (like done by Choi and Chahine [12]), but these rates are not directly proportional to the ratio of the hull surface area covered by the cavity to the total hull wetted surface area. In particular, these rates were lower in the experiment [10], but they were higher in the experiment [2]. Second, the friction reduction by the cavity and an air escape from the cavity affects the thicknesses of ship boundary layers and wakes, as well as the velocity profiles across them. The profile changes influence the propeller inflow field and should be taken into account during design of ship propellers. Third, there are substantial scale effects on air demand by partial ventilated cavities. The power saving rate evidently depends on this demand. As was reported in [1] for very small Fr and later confirmed in [2] for much higher Fr, there is a saturation of drag reduction by ventilated cavitation with an increase of air supply and an excessive air supply gives even negative results. However, as seen in Fig. 1 combining experimental data of Arndt et al. [13], Amromin et al. [2] and Mäkiharju et al. [9], the common trend of the air demand as a function of Fr for various tests does not exist even for Fr < 1 (the shapes of models the tested in [2,9,13] are shown in Figs. 2–4). Moreover, as reported in [13], the ventilated cavities under small models can be maintained at small Reynolds numbers even without any air supply. Further, as one may find in the paper of Kawakami and Arndt [14], i.e., it looks impossible to derive a dependency of Q on
452
E.L. Amromin / Applied Ocean Research 59 (2016) 451–458
Nomenclature B ship (model) beam CD drag coefficient dP/d sound speed c=
Fr = U∞ / gL Froude number g gravity acceleration cavity thickness H L ship (model) length Lc cavity length M Mach number normal to S* N Pc pressure in cavity P∞ ambient pressure volumetric air demand by cavity Q q 104 Q/LBU∞ Re = LU∞ / Reynolds number r(ˇ) = ˇε + 1 − ˇ mixture normalized (dimensionless) density combination of all boundaries of inviscid flow S* Sc the part of S* under the cavity U = |grad˚|/U∞ velocity of inviscid flow U∞ ship (model) speed u velocity in the boundary layer u0 velocity on the cavity surface air flow velocity components ua , va v* friction velocity X2 abscissa of the cavity end friction velocity v* y lateral coordinate in boundary layer zC vertical coordinate of the cavity surface normalized by L void fraction in the boundary layer ˇ ı thickness of water boundary layer
ı
ı∗
=
(1 − ru/U)dy displacement thickness 0
ı ı∗∗ =
ru/U(1 − u/U)dy momentum thickness 0 H
ı∗a =
(1 − ua /u0 )dz the airflow displacement thicknesses 0
the ratio of air density to water density ε water density water kinematical viscosity 2 cavitation number = 2(P∞ − Pc )/U∞ ˚ velocity potential
Fig. 1. Air demand to maintain cavities under a 12.9 m 2D model in a water tunnel (squares, from [9]), a 0.5 m 2D model in another water tunnel (triangles, from [13]) and a 4.6 m ship model towed in a tank at Fr-depended pitch (rhombs, from [2]).
Fig. 2. Computed sections of the water tunnel model with cavities tested in [9] at various Fr.
cavitation number from experimental data even for the same body. So, the known experimental data on interaction of ventilated cavities with boundary layers are insufficient for their empirical generalization and there is the practical necessity of a numerical analysis of the interaction of ventilated cavities with boundary layers over the already designed hull. Kinzel et al. [15] numerically manifested a high impact of turbulent mixing on air demand by ventilated supercavitating flows at moderate Re and Fr 1 and the necessity to include a boundary layer impact in the analysis of ventilated cavitation became clear. One may think that such analysis is a simple task because the numerous computational studies on natural cavitation in viscous fluids have been already described in the literature. Indeed the flow models employed in these studies are not suitable for the air ventilated cavitation. First of all, they did not give a satisfactory description of air escape through the cavity surface. One group of these computational studies has been carried out using various versions of viscous–inviscid interaction methods; this group neglected by the gas flows within the cavities and its escape from them (as in Brewer and Kinnas [16] or in Amromin [17]). Another group of computational studies has been carried out using for the whole flow the model of a bubble cloud. Such model was suggested by Kubota et al. [18] for the cavity closure zone and there is no cavity surface in this model, though one can clearly see such surface in photos of ventilated cavities (as by Kawakami and Arndt [14]) and no bubbly clouds inside ventilated cavities are seen there. Further, all computational studies are based on semi-empirical approaches
Fig. 3. Sections of the ship model towed [2] with computed cavity at Fr = 0.48 (at the model measured pitch).
E.L. Amromin / Applied Ocean Research 59 (2016) 451–458
453
Fig. 4. Ventilated cavity in a bottom niche of a 0.5 m 2D model in the water tunnel [13].
Fig. 5. Ship models with bottom recesses (niches) for cavities tested in Krylov Ship Research Institute (left) and David Taylor Model Basin (right).
(like any study of turbulent flows) and therefore, the corresponding solvers are unavoidably fitted to the certain kinds of flow. Meanwhile, though ventilated cavitation has been broadly used as the experimental tool for modeling natural cavitation (since Reichardt [19]), there is the measurable differences in the characteristics of ventilated and natural cavities on the same body at the same Fr and cavitation number. One can see this difference in [7]. Moreover, such solvers were validated at high values of Fr and the Fr effects were mainly reasonably omitted in them (one can find in [14] that |∂LC /∂| ≈ (160 ÷ 300) |∂LC /∂Fr| at 9.1 < Fr < 34.0), but such omission is not acceptable for small Fr, where the Fr influence may be comparable or even higher than the influence (one can find in Amromin [20], that |∂LC /∂| ≈ |∂LC /∂Fr| at 0.5 < Fr < 0.55). So, a numerical analysis of the interaction of air ventilated cavities with boundary layers requires a special flow model. Such a special model can be validated even in its 2D version because the flows near the ship hull bottom are quite close to 2D flows due to installation of keels restricting ventilated cavities (as seen in Fig. 5) and the following analysis is limited by steady 2D flows, like the analysis carried out by Shiri et al. [21].
The first zone consists of the inviscid incompressible flow of water outside of the hull boundary layer and wake (this zone is under the dashed curve in Fig. 6). The second zone consists of the cavity filed by air (its boundary is plotted as a dotted curve in Fig. 6); an air escape from the cavity is included in the model. The third zone consists of the ship boundary layer on the cavity surface and on the hull wetted surface; the wake behind the hull is the continuation of this zone. The boundaries between these zones are initially unknown and must be determined in the course of computations. Because the outer inviscid flow in the first zone is considered as curl-free, the velocity potential can be employed there. Besides of Laplace equation ˚ = 0,
(1)
the boundary-value problem for ˚ includes the condition on the velocity projection on the normal to the inviscid flow boundary
∂˚/∂N
S∗
= 0,
(2)
the pressure constancy condition on the cavity surface written with taking into account the buoyancy force
2. Model of cavitating flow and employed numerical techniques Cavitating flows are quite complex and include zones with the qualitatively different features. In the suggested model of ventilated by air bottom cavitation the whole flow is divided into three zones (a sketch of a part of the flow with three zones is given in Fig. 6).
Fig. 6. Sketch of ventilated bottom cavitation.
U 2 + 2Fr −2 (z∞ − zC )
Sc
= 1 + ,
(3)
and the condition of flow uniformity in the infinity |grad˚||x|→∞ → {U∞ , 0}.
(4)
One part of the boundary S* covers the cavity surface, another part covers the hull wetted surface. The last part includes a region just downstream of the cavity, where the flow parameters may experience a jump (like pressure experiences a jump on the Riabouchinsky solid closing cavities in the Riabouchinsky scheme described by Knapp et al. [22] and in many other monographs). The above problem for the linear Eq. (1) becomes nonlinear due to the necessity to satisfy two conditions on the same boundary Sc. Because of this necessity, the boundary shape cannot be arbitrary, but it should be found in iterations. There are several known methods to solve such problems, from the BEM-based method described by Amromin [23] to the method described by Faltisen and Semenov [24]. The employed method is based on the subsequent quasilinearization of Eqs. (2) and (3). An illustration of its convergence
454
E.L. Amromin / Applied Ocean Research 59 (2016) 451–458
Fig. 7. An example of distribution of computational error along the cavity.
is shown in Fig. 7, where modulus of the difference between the left-hand side and the right-hand side of Eq. (3) is defined as the computational error. One will see there that by iteration number 90 this modulus is below 0.0005 at any point of Sc. Regarding to the experimental validation of this method, it is necessary to point out that the computed cavity shape with the thinnest cavity tail in Fig. 2 corresponds to exactly the same Fr = 0.71 that corresponded to the most stable cavity among cavities observed in [9]. Another example of validation by comparisons of computed and observed shapes of ventilated cavities determined with by solving Eqs. (1)–(4) were provided in [13]. So, computations in the first zone are based on the already proven flow model and computational technique. In the second zone, there is the need in a novel and quite simple flow model. Let us neglect by compressibility of the air within cavities because the air velocity components are small there. Experimental data [2,3] make it possible to estimate that these components are below 1 m/s. It is possible to model the flow in this zone as a turbulent air boundary layer between the bottom and the flow of the water–air mixture. Therefore one can write the air mass conservation equation integrated across the cavity as u0
dı∗a dx
− (H − ı∗a )
du0 dQ − = 0. dx dx
(5)
Here va = dQ/dx, the selected air velocity profile across the ˆ ˛ + B ˆ , ˛ = 1/7, ˆ = z/H, A = u0 /(1 − ˛/ ), cavity is ua = A B = u0 /(1 − /˛), = ˛(Q/Hu0 − 1) because on the cavity surface the tangential velocity of air coincides with the tangential velocity of water. The air entrainment by water on the cavity surface is implicitly taken into account here by matching the air and water velocities at the surface of their contact. The third zone indeed includes three sub-zones. The first one is located upstream the cavity, and the hull boundary layer is considered as the air-free layer there. The second on is under the cavity. The third one is downstream from it. From the cavity detachment point water entrains air from the cavity and the flow in this zone becomes a two-phase flow. At the cavity tail, there is usually some time-average air flux from the cavity even with the perfect ship bottom design because in the reality the design condition are usually perturbed, and therefore, the total air flow experiences a jump at the boundary layer reattachment to the hull downstream of the cavity. For such complex zone, an integral computational method is employed because of the simplicity reason. Thus, along the cavity surface, the boundary layer void fraction varies in the accordance with the equation d(ıˇ) dQ + = −ˇϑ. dx u0 dx
Fig. 8. Mixture sound speed (m/s) versus air void fraction at various ambient pressure values.
Here W = 1 − u0 /U, M = U/c. Derivation of Eq. (7) follows to the steps described by Cebeci and Bradshow [25], but here the second term in the left-hand side reflects flow through the layer inner boundary. Unlikely to the air cavity, the boundary layer on the cavity and downstream of it must be considered as the compressible layer because of the low sound speeds in air–water mixtures. As follows from Fig. 8, the local Mach numbers M =
2 (1 − ˇ − ˇε) + M 2 ˇ/ε]/(1 − εˇ) can be significant in (1 − ˇ)[Mw a the ship boundary layers; here Mw and Ma are water Mach number and air Mach number. The sound speed is calculated for Fig. 8 at 20 ◦ C, but one can found the similar data for the broad range of temperatures and comparison with experimental data in the paper of Kieffer [26]. Additionally to Eqs. (6) and (7), the water differential momentum equation directly derived from the 2D Reynolds equation on the cavity surface is considered
u0
du0 ∂u u
∂u v
− . =− dx ∂N ∂x
The right-hand side of Eq. (8) can be approximated by the sum (UW)2 /ı + bgdzC /dx because its second term should be proportional to U2 . An analogy of the boundary layer over the cavity with a body wake is used there and according to the approximation of the experimental data of Bourgoyne et al. [28] for turbulent wakes shown in Fig. 9, one can assume that u v ∼U 2 Re−1/2 . Therefore = C Re−0.5 , where C = const. The experimental data [9] for Fr < 0.75 were used here to find C = 3.0 and b = −0.05. Because the equations describing the air flow in the cavity and the flow of an air–water mixture in the boundary layer on the cavity are strongly coupled, they must be solved together. The unknown functions in Eqs. (5)–(8) include Q and three boundary layer parameters {ı, u0 , ˇ}. As usual for the integral methods, the velocity profiles across the layer must be selected. For this turbulent boundary layers on the cavity the velocity profile is selected as
(6)
Here ϑ = const is assumed. The momentum equation integrated across this layer is written with taking into account the air escape dı∗∗ εW dQ (M 2 − 2)ı∗∗ − ı∗ + ı dU + = . U dx U dx dx
(7)
(8)
Fig. 9. Approximation of experimental data for turbulent wakes.
E.L. Amromin / Applied Ocean Research 59 (2016) 451–458
455
u = u0 (x) + [U(x) − u0 (x)]FP (x, ), where FP = 32 − 23 , = y/ı. Generally, a profile of ˇ can be also introduced. The void fraction profiles in a two-phase flow with microbubbles were considered by Kunz et al. [29], but flows with air layers studied by Elbing et al. [30] look more similar to the cavity boundary layer. This similarity allows for the assumption that the void fraction in the cavity boundary layer sharply drops to a constant level in the main part of the layer. The momentum integral equation for the boundary layer on the hull wetted surface has the form
dı∗∗ v∗ =r U dx
2
−
(2 − M 2 )ı∗∗ + ı∗ dU . U dx
(9)
Eq. (9) is very similar to Eq. (7). The employed velocity profile is u(x, y) = F(x, y) + CU FP (x, ), where F = 2.5v∗ (ln |yv∗ /| + B), B(x) depends on the wall roughness, CU = U(x) − F(x, ı). Upstream of the cavity M is negligibly small. The mass conservation law for this boundary layer is written with an empirical coefficient K* d[U(ı − ı∗ )] = rUK ∗ , dx
(10)
where the expression for K* as a function of (ı − ı*)/ı** recommended by Cebeci and Bradshaw [25] for the single phase flows is used. Also, the air escape from boundary layer is described by the equation d(ıˇ) = −ˇϑ. dx
(11)
obtained from Eq. (6) with taking into account that Q = 0 out of cavity. In the hull boundary layer downstream of cavities the unknown boundary layer parameters are {v*, ı, ˇ}. It is necessary to determine their initial values at the cavity end x = X2 . The conditions applied for their determination include the continuity condition for the displacement body ordinate ı∗ (X2 + 0) = ı∗ (X2 − 0) + H(X2 − 0)
(12)
a generalization of the experimental data of Kline et al. [31] for separated boundary layers ı(X2 + 0)/ı∗ (X2 + 0) = 1.42 + r
(13)
Fig. 10. Computed (lines) and measured [34] (symbols) distributions of the friction ratio Cf /Cf0 in air–water mixture and pure water along the same flat plate. Distance x is counted from the plate leading edge. The first numbers in the legend indicate the flow speed (m/s). The second numbers indicate the value of air volumetric concentration.
et al. [34] in Fig. 10 is one of comparisons shown in [27]. Here it can be considered as a validation example of the described method for two-phase boundary layers. 3. Prediction of air demand to maintain the cavity The problem of prediction of the air demand by cavities in steady flow conditions is the first problem considered with the abovedescribed computational model. The obtained numerical solutions are compared in Fig. 11 with the experimental data. The data [9] on air demand necessary to maintain cavities in steady flows under a huge 2D hydrofoil in the water tunnel are quite unique. Other water tunnel experiments are not suitable for such validation because their trend is substantially affected by laminar-turbulent transition (as provided in [13]). The comparison of the described computations with experimental data [2] for a ship model in a towing tank is also shown in Fig. 11. The common point of all sets of data on Q used in Fig. 11 is that the air demand minima were achieved at the design values of Fr corresponding to the drag minima. The agreement of computations with experimental data in Fig. 11 looks satisfactory.
and the mass conservation law written with taking into account the air escaped the cavity
4. Cavities under flat bottoms
U(ı − ı∗ )
The majority of known studies on drag reduction by ventilation relates to flows under flat bottoms. So, it is important to consider such flows in this analysis. In particular for cavities behind backward-facing steps under flat bottoms, the computed dependencies for cavity sizes should be quite similar to predicted in the framework of the linear ideal fluid theory [11]. So, one may
X2 +0
= εQ + U(ı − ı∗ )
X2 −0
(14)
Generally, the described model belongs to the viscous–inviscid interaction approach and the boundary S* should be displaced on a distance ı* from the body surface, etc. (as done in [17]). However, because the cavity detachment for the bottom ventilated cavities are usually fixed, it is accurate enough to solve the whole problem with the assumption of the weak viscous–inviscid interaction around the cavity. This actually means that the inverse influence of the boundary layer thickness on the pressure can be neglected during determination of the cavity shape. On the other hand, there is a substantial impact of ı* on the pressure at the transaction of the boundary layer into the wake behind the hull. This zone is a zone of strong viscous–inviscid interaction. This wake can be computed with the integral method described in [27]. Let us point out that the role of the described multi-zone model of cavitating flows is much more important than the role of the numerical methods assigned to each zone. Indeed, viscous–inviscid interaction approach has been used with integration of differential equations for boundary layers, too (as by Briley and Mcdonald [32], Cebeci et al. [33] and in others papers later). The comparison of the computed characteristics of such multiphase boundary layer with its measured characteristics by Sanders
Fig. 11. Comparison of computations (lines) and experimental data (symbols) air demands by cavities. Data A are taken from [2] data M – from [9].
456
E.L. Amromin / Applied Ocean Research 59 (2016) 451–458
Fig. 12. Computed cavitation numbers versus ventilated cavity lengths behind a backward step under a flat bottom, L/H = 100. The right edges of curves correspond to cavities of the maximum Fr-dependent length.
consider the presented computations for flat bottoms as a qualitative validation for the above-described combined method. Indeed, the computations results obtained by solving the nonlinear problem (1)–(4) and shown in Fig. 12 exhibit a substantial dependency of Lc on Fr (though it is not absolutely proportional to Fr2 , as the linear theory predicts) and the possibility to maintain a cavity of the same length at very different cavitation numbers. As seen in Fig. 12, cavitation numbers are negative there. So, an excessive air pressure is necessary to maintain such cavities. However, according to computations, a very moderate increase of the 2 ÷ 0.02U 2 ) is sufficient for a 25% cavity pressure (dPC ≈ 0.01U∞ ∞ increase of the cavity length in the considered Fr range. Further, the flow in the boundary layer downstream of the cavity is significantly different from the cavitation-free situation. As illustrated by Fig. 13 containing comparisons with the pure water boundary layer characteristics over the flat plate, the behavior of the displacement thickness downstream of the cavity is dissimilar to its behavior in the cavitation-free flow. This is important because a decrease of ı* should lead to a reduction of the wake thickness and, as a result, of the form resistance. 5. Influence of pressure gradient on flow behind cavities and scale effect The flows under ships under hulls of ships and their models are much more complex. In particular, as shown in [2], a significant pitch and heavy can take place for ships with ventilated cavities or their models at Fr > 0.45 and this leads to a complex combination of drag reduction effects. However, because the following computations relate to the smaller Fr values, only negligibly small pitch values will be considered below. Further, the real ships always have zones of substantial pressure gradients at their sterns and, as clear from Eq. (9), the flow compressibility effects is coupled with the
Fig. 13. Variation of thickness displacement along the flat bottom downstream of cavity for Fr = 0.355, Re = 3.5 × 107 . Here q = 30 corresponds to the minimum air demand to maintain the cavity.
Fig. 14. Buttock of a ship (upper plot) and pressure coefficient over her stern (lower plot).
pressure gradient effects. The computed void fraction in some parts of the boundary layer may go at least up to 0.1 for full scale ships. Returning to Fig. 8, one can estimate that the local sound speed may be below 40 m/s and the Mach number can be quite high there. On the other hand, for the towing tank models of the length from 5 m to 10 m at Fr < 1, U ∼ L1/2 and the local Mach number is very small everywhere. The buttock employed in the following computations and shown in Fig. 14 has a cavity seal/locker behind the cavity (like the buttocks shown in Figs. 2 and 3). For the computational examples shown in these figures, X2 = 0.73L, and one can see in Fig. 14 that the cavity slightly affects the pressure gradient downstream from the cavity locker (at x/L > 0.78). The pressure gradients in 2D flows are higher than they would be over a 3D ship stern with the same buttock. Therefore the real pressure gradient influence on 3D multi-phase flows downstream of cavities will rather qualitatively correspond to the dependencies computed for a ship buttock and plotted in Figs. 15–17. These computations were carried out for a ship model of the length L = 10 m (Reynolds number Re = 4 × 107 ). The corresponding fullscale ship length was selected as L = 160 m (Re = 2.56 × 109 at the same Fr = 0.4). It is seen in Fig. 15 that the wall friction significantly decreases in the zone of the adverse pressure gradient (0.85 ≤ x/L ≤ 0.95) in the cavitation-free pure water flow, but there is no similar decrease downstream of the cavities.
Fig. 15. Wall friction downstream of cavities with the same q at Re = 4 × 107 (solid lines) and Re = 2.56 × 109 (dashed lines) and without cavity at Re = 2.56 × 109 (squares).
E.L. Amromin / Applied Ocean Research 59 (2016) 451–458
Fig. 16. Displacement thickness (in the right) downstream of cavities with the same q at Re = 4 × 107 (solid lines) and Re = 2.56 × 109 (dashed lines) and without cavity at Re = 2.56 × 109 (triangles).
457
Fig. 18. Normalized velocity profiles at x = 0.95L for flows behind cavities. Dashed line relates to q = 13 and Re = 2.56 × 109 , solid line–to q = 13 and Re = 4 × 107 , rhombs–to q = 40 and Re = 4 × 107 .
As shown in Figs. 15 and 16, both the wall friction and the displacement thickness downstream of the cavities are lower than they were on the cavitation-free stern. The decrease of the thickness displacement (and of the wake thickness) behind the cavity is very significant. Because the form resistance is roughly propor-
L
tional to
L
(∂[U( )ı∗ ( )]/∂ )(d /x − )dx in 2D flows and
Nx U(x) 0
0
this resistance can be similarly estimated in 3D flows, a measurable reduction of the form resistance for ships with ventilated cavity is expectable. This reduction is the substantial advantage of drag reduction by air cavities in comparison with microbubbles drag reduction, were the total drag reduction is smaller that friction drag reduction under the affected bottom area (as reported by Sato et al. [35]). Also, the changes of the wall friction and the boundary layer thicknesses downstream of ventilated cavities evidently lead to changes of the marine propeller inflow. The location of the propeller disk is selected as x ≈ 0.95 for the buttock shown in Fig. 14. An illustration of the cavitation effect and scale effect on the velocity profiles at x = 0.95L is provided in the left part of Fig. 17. The right part of this figure provides experimental data on the velocity profiles (inflow field) in a propeller disk extracted from the recent paper by Gorbachev et al. [36]. There is not enough data on the model hull in [36] for an attempt of a direct comparison. So, one can find only in two parts of this figure only the same qualitative effect (similar velocity differences between flows with and without cavitation at the fixed points). The Fig. 17 is supplemented by the illustration of two abovementioned effects on the product of velocity and density in Fig. 18. As one can conclude from Figs. 17 and 18, the inflow of a marine propeller is significantly affected by the air bottom cavitation.
Fig. 19. Air supply impact on velocity profiles at x = 0.95L.
The scale effect on the velocity profiles in the boundary layer downstream of ventilated cavities is even smaller than the cavitation effect on these profiles. So, the marine propellers should be rather redesigned in the course of a retrofit of the existing ships to drag reduction by air bottom cavitation. It is also interesting that, as shown in Fig. 19, the intensity of air supply significantly affects the velocity profile at a fixed pair {Re, Fr}. So, model tests with the excessive air supply may be useful for the ship propeller design. Some illustration on compressibility impact on flows downstream of the cavity could be also useful. Distribution of thickness displacement was selected for such illustration because of the possibility to show this impact for the whole stern. Computations for full-scale conditions with the actual Mach number and computations with M = 0 in Eqs. (7) and (9) are compared in Fig. 20. It is clearly seen in Fig. 20 that the compressibility impact increases with the air supply rate. Unlike to situation with microbubble drag reduction [27] with much higher air demand, it may be acceptable to neglect the flow compressibility at design conditions with quite low air escapes from cavities, but such simplification can be unacceptable at design off.
Fig. 17. In the left: computed velocity profiles at x = 0.95L. Dashed line relates to cavitation-free flow at Re = 4 × 107 , solid line–to q = 13 and Re = 2.56 × 109 , triangles – to q = 13 and Re = 4 × 107 ; q = 13 corresponds to the minimum air demand by the cavity. In the right: Measured [36] velocity profile in the model propeller disk at two azimuth values: symbols jointed by lines related to flow without cavity, isolated symbols – to flow with cavity; r – distance from the disk center, D – its diameter.
458
E.L. Amromin / Applied Ocean Research 59 (2016) 451–458
Fig. 20. Illustration of compressibility impact on thickness displacement.
6. Conclusions The provided analysis of air cavitation under ship bottoms is carried out with a novel flow model and a combination of the earlier developed computational techniques. Employment of this model allowed for several important quantitative and qualitative results related to interaction of ventilated cavities with ship boundary layers. In particular, the minimum air demand necessary for the cavity maintenance in steady flows can be predicted satisfactory with this model. The reported computations also confirm that this minimum air for a cavity without a locker/seal is much greater than such minimum for the design with the cavity locker/seal. The reported computations of the wall friction and thickness displacement over the sterns make it possible to better comprehend the synergy of drag reduction effects for ships with bottom ventilated cavitation and, in particular, reduction of their form resistance. The effect of ventilated cavities on propeller inflows (on the velocity profiles at the propeller disks) was estimated. It was shown that cavities mitigate scale effects on these profiles. The feasibility of the described novel flow model was proven for 2D flows with the quite simple description of boundary layers (with employment of integral methods). This model can be generalized for 3D flows with the employment of the more modern CFD techniques. References [1] Butuzov AA, Gorbachev YN, Ivanov AN, Kaluzhny VG, Pavlenko AN. Ship drag reduction by artificial gas cavities. Sudostroenie 1990;n#11:3–6 [in Russian]. [2] Amromin EL, Metcalf B, Karafiath G. Synergy of resistance reduction effects for a ship with bottom air cavity. ASME J Fluids Eng 2011;133:021302 [7 pages]. [3] Gorbachev YN, Buyanov AS, Sverchkov AV. Ships with air cavity: real method to enhance energy efficiency and ecological safety. Sea Fleet 2015;n#1:26–35 [in Russian]. [4] Sverchkov AV. Application of air cavities on high-speed ships in Russia. In: International conf. on ship drag reduction. 2010. [5] Amromin EL, Gorbachev YN. Technologies of ship resistance reduction. In: Compendium on naval hydrodynamics. Paris, France: ENSTA; 2015 [Chapter 4]. [6] Foeth EJ. Decreasing of frictional resistance by air lubrication. In: 20th int. Hiswa symposium on Yacht design and Yacht construction. 2008.
[7] Kopriva J, Amromin EL, Arndt REA. Improvement of hydrofoil performance by partial ventilated cavitation in steady flow and periodic gusts. ASME J Fluids Eng 2008;130:031301 [7 pages]. [8] Lay KA, Yakushiji R, Makiharju S, Perlin M, Ceccio SL. Partial cavity drag reduction at high Reynolds numbers. J Ship Res 2010;54:109–19. [9] Mäkiharju SA, Elbing BR, Wiggins A, Schinasi S, Vanden-Broeck J-M, Perlin M, et al. On the scaling of air entrainment from a ventilated partial cavity. J Fluid Mech 2013;732:47–76. [10] Zverkhovskyi O, Terwisga T, van Guning M, Westerwell J, Delfos R. Experimental study on drag reduction by air cavities on a ship model. In: 30th sym. naval hydrodynamics. 2014. [11] Butusov AA. Extreme parameters of vented cavity on the top surface of horizontal wall. Fluids Dyn 1966;1:167–70. [12] Choi J-K, Chahine GL. Numerical study on the behavior of air layers used for drag reduction. In: 28th sym. naval hydrodynamics. 2010. [13] Arndt REA, Hambleton WT, Kawakami E, Amromin EL. Creation and maintenance of cavities under horizontal surfaces in steady and gust flows. ASME J Fluids Eng 2009;131:111301 [10 pages]. [14] Kawakami E, Arndt REA. Investigation of the behavior of ventilated supercavities. ASME J Fluids Eng 2011;133:091305 [11 pages]. [15] Kinzel MP, Lindau JW, Peltier J, Zajaczkowski F, Arndt REA, Wosnik M, et al. Computational investigation of air entrainment, hysteresis and loading for large-scale, buoyant cavities. In: 9th int. conference on numerical ship hydrodynamics. 2007. [16] Brewer WH, Kinnas SI. Experiment and viscous flow analysis on a partially cavitating hydrofoil. J Ship Res 1997;41:161–71. [17] Amromin EL. Scale effect of cavitation inception on a 2D Eppler hydrofoil. ASME J Fluids Eng 2002;124:186–93. [18] Kubota A, Kato H, Yamagushi H. A new modeling of cavitating flows: a numerical study of unsteady cavitation of a hydrofoil section. J Fluid Mech 1992;240:59–96. [19] Reichardt H. The law of cavitation bubbles at axially symmetric bodies in a flow. Reports and translations, vol. 766. UK: Ministry of Aircraft Production; 1946. [20] Amromin EL. Design of bodies with drag reduction by partial cavitation as an inverse ill-posed problem for velocity potential. In: 9th int. conference on numerical ship hydrodynamics. 2007. [21] Shiri A, Leer-Andersen M, Bensow RE, Norrby J. Hydrodynamics of a displacement air cavity ship. In: 29th sym. naval hydrodynamics. 2012. [22] Knapp RT, Daily JW, Hammit FG. Cavitation. NY: McCraw-Hill; 1970. [23] Amromin EL. Ships with ventilated cavitation in seaways and active flow control. Appl Ocean Res 2015;50:163–72. [24] Faltisen OM, Semenov YA. The effect of gravity and cavitation on a hydrofoil near the free surface. J Fluid Mech 2008;597:371–94. [25] Cebeci T, Bradshaw P. Physical and computational aspects of convective head transfer. NY: Springer-Verlag; 1984. [26] Kieffer SW. Sound speed in liquid–gas mixtures, water–air and water–steam. J Geophys Res 1977;82:2895–904. [27] Amromin EL. Microbubble drag reduction downstream of ventilated partial cavity. ASME J Fluids Eng 2010;132:051302 [5 pages]. [28] Bourgoyne DA, Ceccio SL, Dowling DR. Vortex shedding from a hydrofoil at high Reynolds number. J Fluid Mech 2005;531:293–324. [29] Kunz RF, Maxey MK, Tryggvason G, Fontaine AA, Ceccio SL, Gibeling HL, et al. Validation of two-fluid Eulerian CFD modeling for microbubble drag reduction across a wide range of Reynolds numbers. ASME J Fluids Eng 2006;129:66–79. [30] Elbing BR, Mäkiharju SA, Wiggins A, Perlin M, Dowling DR, Ceccio SL. On the scaling of air layer drag reduction. J Fluid Mech 2013;717:484–513. [31] Kline SJ, Bardina JG, Strawn RC. Correlation of the detachment of twodimensional boundary layers. AIAA J 1983;21:68–73. [32] Briley WR, Mcdonald H. Numerical prediction of incompressible separation bubbles. J Fluid Mech 1976;69:631–56. [33] Cebeci T, Clark RW, Chang KC, Halsey ND, Lee K. Airfoils with separation and the resulting wakes. J Fluid Mech 1986;163:323–47. [34] Sanders WC, Winkel ES, Dowling DR, Perlin M, Ceccio SL. Bubble friction drag reduction in a high Reynolds number flat plate turbulent boundary layer. J Fluid Mech 2006;552:353–80. [35] Sato T, Nakata T, Takeshita M, Tsuchiya Y, Miyata H. Experimental study on friction reduction of a model ship by air lubrication. J Soc Naval Archit Jpn 1997;182:121–8. [36] Gorbachev YN, Sverchkov AV, Galushina MV. Propulsion of displacement ships with the single bottom cavities. Sudostroenie 2015;n#1:17–23 [in Russian].