Numerical Study on the Behavior of Air Layers Used for Drag Reduction

Transcription

1 28 th Symposium on Naval Hydrodynamics Pasadena, California, September 2010 Numerical Study on the Behavior of Air Layers Used for Drag Reduction Jin-Keun Choi and Georges L. Chahine (DYNAFLOW, INC., U.S.A.) ABSTRACT This contribution addresses achieving higher speed and higher propulsion efficiency by reducing drag of a hull. Placing an air layer beneath a hull to reduce the frictional resistance is a simple and attractive approach. However, one of the major issues of a practical air layer system is the stability of the air layer free surface with regards to the ship motion and the upstream conditions. We address this issue of unsteady air layer behavior, using a boundary element method approach applied to the conditions of recent experiments with a large flat plate conducted in the Large Cavitation Channel. The numerical studies indicate that inclusion of small upstream unsteadiness is necessary to recover the experimentally observed air layer behavior and the critical air flow rates correctly. The numerical model is also applied to the unsteady excitation experiments conducted with imposed motion of a flapping gate. INTRODUCTION Reducing hull drag is of great interest in naval and marine applications. Resistance of a ship is conventionally decomposed into three components; the wave-making resistance, the frictional drag, and the viscous form drag. In the past, much of the research was focused on the reduction of the wave-making resistance and form drag through modification of hull forms (Wigley, , Eggers et al., 1967, Inui, 1980, Celebi, 2000). The reduction of frictional drag has been researched more recently in a number of ways including injection of polymers, surfactants, fibers, micro-bubbles, vorticity manipulation with riblets, large eddy break-up devices, ventilated cavities, and active turbulence control (Sellin & Moses, 1989, Gyr, 1990, Bushnell & Hefner, 1990, Madavan et al., 1984, Sanders, 2006). All these methods of reducing friction drag have been demonstrated, but have posed efficiency and practicality issues for general applications. Creating an air layer (also known as air cavity, ventilated cavity, or air plenum) has been proposed as a method to reduce frictional resistance of ship hulls (Kostilainen & Salmi, 1972, Latorre, 1997, Butuzov et al., 1999, Matveev, 1999). In addition to the reduced resistance, hull designs with air layers may result in better underwater sound insulation, shock mitigation, and sea keeping performance (Choi & Annasami, 2006). In the air layer approach, the frictional drag is reduced by covering parts of the hull surface with an air layer, which, in effect, results in a decrease of the wetted area of the ship hull. This idea was not new and was patented in the nineteenth century (Latorre, 1997). However, a stable formation of such a thin cavity suffers from interference with the ship motion, the changes in buoyancy forces, and other instabilities at gas-liquid interfaces (Kostilainen & Salmi, 1972). Recent large Reynolds number experiments with a flat plate (Lay et al., 2008) and 15 m long prototype ship trials in Europe (Trauthwein, 2010) have shown that the idea is promising, provided that the issues of stability of the air layer free surface and its sensitivity to the upstream flow conditions are addressed properly. In addition, the effects of the ship motion and maneuvers on the air layer, the mechanics of closure at the trailing edge of the air layer, air leakage, and the amounts of air to inject require better understanding for good design and control of the air layer dynamics. In order to understand the air layer behavior better, we have conducted numerical studies of air layer behavior using our Boundary Element Method code, 3DYNAFS_BEM. DYNAFLOW has been developing software to model drag reduction on vessels using air plenums and air tunnels (Choi et al., 2004, 2005, Choi & Chahine, 2004, Choi & Annasami, 2006, Choi, 2007). These studies concentrated on the modeling of the air-water interfaces, their interactions with the vessel, plenum walls, and the surrounding ocean free surface. In these studies, we addressed the startup stability of the ship from rest and the air layer behavior in inclined hulls. For ships having air tunnels, 1

2 the model included a simplified compressible air flow model inside the tunnel open to the upstream ocean. The model also included air mass balance between supply and leakage, air blowing fans, non-uniform pressure distribution on the air plenum due to air blowing, etc. We also conducted a preliminary study on the interaction of an underwater explosion bubble and the air plenum (Choi & Annasami, 2006, Matveev, 2006). One of the conclusions from these studies is that the air layer behavior is very closely related to that of the free surface around the ship and there is a very strong interaction between the two free surfaces. The objective of the present research is to develop a computational fluid dynamics tool which can assist our understanding of air layer dynamics and can enable designs of future practical hulls with air layers. In this work, we extended 3DYNAFS_BEM and its numerical modeling to the geometry and conditions of large plate experiments conducted at the U.S. Navy William B. Morgan Large Cavitation Channel (LCC). We also compared the predicted air layer behaviors with the observations in the LCC experiments. APPROACH Our approach aims at concentrating attention on the large scale behavior of the air layer, ignoring for the moment fine turbulent mechanisms occurring at the air layer water interface, and model the air layer as a well defined deformable air-liquid interface, which separates the injected air from the bulk water. This is similar to what is conventionally used to study ocean free surface waves and sheet cavities among others. Such a model enables tracking of the interface, its stability, and its extent starting from the air jet cavity issued from the injection port. The position of the layer free surface is an unknown that is sought as a part of the solution. The model accounts for air influx into the layer and air loss into the bulk water from the air layer closure area. While the influx is related to the air injection conditions, the layer closure needs a modeling to address the air mass equilibrium, at which point air loss should balance the air input. We have used such an equilibrium model in our previous work for an air plenum in the bottom part of a ship (Choi et al., 2004, 2005, Choi & Chahine, 2004, Choi, 2007, Choi & Annasami, 2006) or in the air tunnels (Choi, 2007), where flat vertical walls and beach type angled plates were considered in the closure region of the plenum. We further extended the concept in the present work. NUMERICAL METHODS The basic numerical scheme of this work is based on the 3DYNAFS_BEM, a three-dimensional non-linear boundary element method (BEM) code. This code has been utilized for simulations of complex free surface dynamics such as bubbles (Chahine et al., 1997, Chahine & Perdue, 1989), breaking waves (Goumilevski et al., 2000), motion of floating bodies (Cheng et al., 2001, Kalumuck et al., 1999), cavitation (Chahine & Hsiao, 2000, Hsiao & Chahine, 2004, Choi & Chahine, 2007), and fish swimming (Cheng & Chahine, 2001). The accuracy of the code in these applications has been proven through comparisons with experiments for these various applications. A brief summary of the BEM method used in this study is provided here. Except for very thin boundary layers near the solid boundaries where viscous effects are important, the general features of the hydrodynamic problem can be obtained by assuming the flow is incompressible and inviscid. The flow is then described with a velocity potential, φ, which satisfies in the liquid domain, D: 2 φ = 0, in D. (1) The Bernoulli equation provides the pressure with ρ being the liquid density and g the acceleration of gravity: 1 2 φ p + ρ φ + ρ + ρgz = c() t, (2) 2 t where c is a time constant. The boundary value problem for the potential is solved by applying the Green identity to domain D: φ G Ω ( P) φ( P) = G φ ds n n, (3) S where Ω ( P) is the solid angle under which the point P sees the fluid domain D. G = 1/ MP is the Green s function, where M belongs to the boundary surface S and P belongs to the fluid domain D. By knowing either φ or φ / n on the boundary S, the other quantity on S can be obtained from (3). To solve this equation numerically, all surfaces are discretized into panels on which linear distributions of φ or φ / n are assumed. The corresponding surface integrals are then represented as a summation over all panels. Writing this at each discrete node yields a system of linear algebraic equations, and their solution provides the values of φ or φ / n at all nodes. An example of discretized hull with air tunnel is shown in Figure 1. The boundary surface S of the fluid domain may include the wetted hull surface, the air layer surface, the ocean free surface, and any nearby boundary or sea bottom surface. The boundary 2

3 condition on the solid surface such as the hull surface enforces equality of the normal velocity of the liquid and the surface: φ = [ V + ( ϖ r)] n, (4) n b b where V and b ϖ are the linear and angular velocities b of the solid surface. Figure 1: Grid of a complex hull with tunnel air plenums connected to the upstream ocean free surface. View from underwater. Colors represent z-coordinate. solid surface. The free surface is advanced using the liquid velocities, and frequent regridding is used to prevent extreme stretching or movement of panels. MODELING OF AIR LAYER The modeling of the air layer free surface is very general. Depending on the modeling needs, the junction between the air layer surface and the solid boundary can be set by either of the following two conditions: The internal walls of the air plenum are discretized, and the air layer free surface is tracked along these internal walls. An advantage of this approach is that the nonlinear motion of the air layer free surface moving up and down along the internal walls can be simulated directly. An air layer trailing edge modeled with this approach is shown in Figure 2. The air layer interface is attached to fixed solid nodes at the edges of the plenum. This condition is appropriate for the leading edge of the air layer where the liquid flow detaches from a sharp corner of the solid geometry. Each free surface such as the ocean surface and the injected air layer surface can be described by a general equation, F( x, t) = z ζ ( x, y, t) = 0. On this surface, there are two conditions. The kinematic condition expresses that a fluid particle in its motion remains at the free surface, df( x, t)/ dt = 0. (5) The second dynamic condition requires the balance of pressures at the interface: pg φ ρgz+ ρ + ρ φ γc = 0, (6) t 2 where γ is the surface tension parameter, C is the local surface curvature, and p is the gas pressure in g the gas side of free surface (atmospheric pressure in case of the ocean free surface, air layer gas pressure in the case of an injected air layer). The air layer interface is treated as a free surface in its full rights. The airwater interface is allowed to move freely according to the local flow. The numerical scheme proceeds in time. At each time step, the matrix equation resulting from the discretization of Green s equation is solved, and the normal velocity at the free surface and the potential on the solid surfaces are obtained. The Bernoulli equation is then used to calculate dφ / dt and update the potential at the free surface and the pressures on the Free surface Figure 2: Side view (bottom left) and bottom view (bottom right) of the air cavity closure. With the air cavity boundary model, the cavity boundary can move along the sloped beach of the hull near the cavity closure (top). Three types of air plenum pressure conditions were considered so far: (a) constant pressure, (b) conservation of air mass in the plenum, and (c) air jet pressure imposed on the interface, for instance, a pressure distribution from a blowing fan. The second condition is the most useful in this study to determine 3

4 the proper air plenum pressure needed to obtain a stable cavity. The conservation of air mass is based on the polytropic pressure-volume compression law, p k V o g = pg,0 V, (7) where p and g,0 V o are the initial pressure and volume of the air in the plenum and p and g V are the current values. This condition is modified as shown later to account for continuous supply and leakage of air. MODELING OF LCC PLATE EXPERIMENTS We extended the studies presented above to the case of an injected air layer used for drag reduction and used the experimental work in the LCC to validate and improve the code. Details of the experimental setup can be found in Lay et al. (2008), and a brief overview of the setup is described here. The LCC experimental setup with the large plate is shown in Figure 3. The nominal test section dimensions are 13 m (length) by 3.05 m x 3.05 m (width and height). In the absence of the plate, test-section flow speeds can be set from 0.5 to 18.3 m/s. The free-stream turbulence level in the LCC is below 0.5 %. The flat plate is 12.9 m long, and the air layer is generated by injecting air at the vertical face of a 178 mm (7 in) step beneath the flat plate. Near the end of the plate on the bottom side, there is a sloped beach and a beach flat on which the trailing edge of the air layer closes and air leakage occurs. Figure 3: Side view of the plate installed in the LCC test section. Taken from Lay et al. (2008). The above experimental setup is modeled using 3DYNAFS_BEM. A schematic sketch of the modeling is shown in Figure 4. The plate, tunnel top, and bottom are modeled as solid walls. The leading edge of the air layer free surface is attached to the step corner, and the location of the trailing edge of the air layer is predicted by the code. The air layer free surface moves freely following the liquid flow and air pressure dynamics. The simulation starts with an arbitrary very short air layer attached to the step and most of the plate bottom nodes are flagged as solid nodes. As the injected air fills the cavity, the pressure inside the cavity rises and the air layer grows. If the air layer trailing edge node moves too far away vertically from the plate solid surface or horizontally too close to the next downstream solid node, then the next downstream solid node is turned into a free surface node. Both vertical and horizontal distance criteria in this study were 5% of the local panel size. If the air layer reaches the end of the beach flat, air is assumed to leak at an air flow rate Q loss equal to the horizontal component of the local flow velocity vector at the last free node V multiplied by horizontal projection AL, TE height of the last panel h where the cavity closure AL, TE occurs (Figure 5). Q = V h. (8) loss AL, TE AL, TE Also, if the air layer free surface touches the solid plate (Figure 6), the air pocket downstream of such a location is considered lost with a volume loss V. The air layer trailing edge is redefined to that loss location, and all nodes downstream of the location are moved to the plate surface with those nodes turned in to solid nodes. The volume loss V loss is considered in the air mass balance equation (9). The pressure in the lost air pocket is assumed to be equal to the pressure in the air plenum at the moment of detachment. To take the air supply and air leakage into account, (7) is modified to the following air mass balance equation: p V g = p V o o + pinqindt pgqlossdt p V, (9) g loss where, p is the air injection pressure, in Q is the air in injection flow rates, and V loss is the occasional volume loss due to air pocket detachment. The inflow velocity is specified at the upstream boundary. We experimented with uniform inflow conditions and inflow with sinusoidal oscillations. The sinusoidal inflow velocity U in is defined as follows in this study: z Uin = U + uin cos π sin ( 2π f t), (10) H where U is the mean inflow velocity, u in is the amplitude of fluctuations, z is the vertical coordinate ( z = 0 at the air layer at the step corner), H is the 4

5 tunnel height, and f is the frequency of the fluctuations. The LCC experiments included tests with gate and flap. In some tests, the flap was at a fixed angle; while in some other tests sinusoidal flap motion was applied. A schematic diagram for such experiments is shown in Figure 6. The flap motion and the free surface downstream of the flap are also modeled. As explained above, when the air layer free surface touches the plate, the portion of air downstream of the point is considered lost forming an air pocket. In the future, we will model the air pocket and small breakup bubbles from it using DYNAFLOW s 3DYNAFS_DSM. solution of the potential, and this condition also enforces velocity vectors at the outlet parallel to the tunnel walls. In the figure, an initial cavity that spans a few panels downstream of the injection step is also shown. Gate & Flap Air Supply Q in Free Surface Beach Flow Air Leakage V loss DBM Bubbles Air Supply Q in Beach Figure 6: Modeling of the plate experiments with gate and flap. Scales are non-uniformly stretched for illustration. U in Flow Beach Flat Air Leakage Q loss Figure 4: Sketch of the LCC experiment shown in Figure 3 modeled in 3DYNAFS. Scales are non-uniformly stretched for illustration. Inlet dφ/dn = 0 Outlet φ = 0 Plate T.E. Basic flow U = 10 Air Layer Beach Flat h ALTE Initial cavity V ALTE Figure 5: Modeling of air leakage at the trailing edge of the air layer. NUMERICAL RESULTS WITH UNIFORM INFLOWS A 3D grid for simulations of the tests without gate is shown in Figure 7. Only half of the flow domain is modeled and symmetry of the tunnel center plane is used. The boundary element method was setup for the perturbed potential after subtracting the background basic uniform flow corresponding to the tunnel inflow velocity U in =10 m/s. The outlet boundary condition φ = 0 is needed to enforce unique Figure 7: Modeled flow domain and boundary conditions for simulation of LCC experiments without the gate. Figure 8 shows a side view of the solution after the air layer has evolved fully for the case of an air injection rate Q in = 0.7 m 3 /s. The trailing edge of the air layer fluctuated over the beach flat toward the end of simulation. Figure 9 shows the time history of the air layer evolution. In the beginning, the air layer length and volume grew, then there was a sudden loss because the middle of the air layer touched the plate above it at about t = 0.9 s. Then, the air layer grew again until it reached the beach flat. The pressure in the air layer was continuous across the air pocket detachment (t = 0.9 s) as expected, and pressure and volume behaved out of phase. 5

6 Figure 8: Air layer shape obtained for U in = 10 m/s, Q in = 0.7 m 3 /s. Vertical axis is stretched by a factor of 2. Figure 10: Effect of air pressure on the air layer shape for a fixed air layer detach and reattachment points. U in = 4.0 m/s. Vertical scale is stretched. Figure 9: Time evolution of the air layer volume, the air layer pressure, and the air layer length. U in = 10 m/s, Q in = 0.7 m 3 /s. To study the effect of air layer pressure, the leading edge and the trailing edge of the air layer were fixed to the step corner and the end of the beach flat. Figure 10 shows the effect of air pressure, which was varied from 99,300 Pa to 103,300 Pa while the ambient pressure at the level of air layer leading edge was kept at 101,300 Pa with the inflow velocity of 4.0 m/s. In this test, the air pressure was enforced and no air injection and leakage were modeled. As expected the higher air pressure resulted in a fuller air layer shape. With a similar computational setup, the effect of tunnel wall is demonstrated as shown in Figure 11. In this case, both the ambient and air layer pressure were set to 101,300 Pa, while two tunnel wall locations corresponding to 3 m and 100 m tunnel heights were simulated. For a 3 m tunnel height, the bottom wall was at z = 1.5 m, and the top wall at z = 1.5 m. It is clear that the tunnel wall at the proximity of 1.5 m has a strong effect on the shape of the air layer. Figure 11: Effect of the tunnel wall proximity on air layer shape. U in = 4.0 m/s. Vertical scale is stretched. The effects of air injection rates were studied with the air layer trailing edge set free to move and with leaking as described earlier (9). The inflow velocity was 4 m/s, and the ambient and air injection pressure were both set at 101,300 Pa. The comparison of air layer behavior for three air injection flow rates, Q in = 3.6, 0.6, and 0.1 m 3 /s are shown in Figure 12. These air injection rates in this paper are the 2D volume rates for a unit length (1 m) span of the plate for later comparison with the experimental results. It is clear from Figure 12 that the larger the air flow rate is, the more energetic the air layer behavior is. The 3.6 m 3 /s case showed large fluctuations at the air layer closure, while for the 0.1 m 3 /s injection the air layer grows and reaches smoothly to a stable final shape. The corresponding air layer volumes are compared in Figure 13. The volume evolved steadily and reached the final value for the 0.1 m 3 /s injection case, while the 0.6 m 3 /s case showed the largest fluctuations in volume. 6

7 These air layer behaviors are compared with the experimental observation of Lay et al. (2008) in Figure 14. The figure shows a map of critical minimum air flow rates needed to maintain an air layer in the experiments. In the experiments at 4.0 m/s inflow, the air layer could be maintained around 0.03 m 3 /s air injection flow rate with oscillatory behavior while around 0.02 m 3 /s air injection a stable air layer resulted. Between the experiments and the predictions, the trend is that larger air flow rates result in more unstable air layer behavior. From Figure 12, this can be explained by the fact that a larger air injection introduces a thicker and wavy air layer trailing edge which generates larger air leakages, while a smaller air injection rate arrives at a more stable thin leakage state. However, the predicted border line between the stable and unstable behavior was much higher than the experimental observation. That is, the numerical solution is more stable than in practice. This suggests that tunnel upstream fluctuations should be modeled. q crit (m 3 /s) Q air =0.6, 3.6: Oscillatory (numerical) Q air =0.1: Stable (numerical) Oscillatory in exp. Stable in exp. Figure 14: Comparison of predicted (red symbols) and experimentally observed air layer behaviors (black symbols). Experimental data taken from Lay et al. (2008). Only black squares correspond to the setup with a step studied here. Figure 12: The effect of air injection rates on air layer geometries at different times. From top to bottom, the air injection rates are 3.6, 0.6, and 0.1 m 3 /s. Note that the geometry is stretched 5 times vertically to show the air layer better. Figure 13: Comparison of the time variations of the air layer volumes for the three cases in Figure 12. NUMERICAL RESULTS WITH FLUCTUATING INFLOWS The tunnel upstream turbulence is modeled with the fluctuations in the inlet velocity using (10). Figure 15 shows the evolution of air layer volume for the case of an air injection rate Q in = m 3 /s, mean inflow velocity U = 4.0 m/s, amplitude of inflow fluctuation u in = 0.1 m/s, and a frequency of fluctuation of f = 1 Hz. The excitation frequency 1 Hz is noticeable as small fluctuations in the cavity volume curve. After the air layer builds up fully on the beach flat at about t = 25 s, the leakage starts to occur periodically at a frequency of about 0.33 Hz. This suggests that the air layer has its own dynamics with a certain natural frequency, which can be excited by upstream fluctuations. Three upstream excitation frequencies were tested to observe their effect on the air layer behavior. Figure 16 shows the comparison of air layer behaviors with upstream excitation frequencies of 0.1, 1.0, and 5.0 Hz. The other parameters were kept the same at Q in = 0.05 m 3 /s, U = 4.0 m/s, and u in = 0.1 m/s. The 5 Hz excitation resulted in larger amplitude fluctuations of the air layer compared to lower frequency excitations. This is further evident in the air layer volume curves shown in Figure 17. The volume losses due to leakage are compared in Figure 18. After the air layer behavior has developed fully reaching somewhat regular intervals of leakage, the period becomes quite regular. It can be observed that the 1 Hz excitation case results in the most number of leaks, while other excitation 7

8 frequency results in skips of some of the leaks with the predicted leakage period around 1.5 s. This shows that the air layer under this condition has a preferred oscillation at period 1.5 s, independent of upstream excitation frequency. The effect of upstream excitation amplitude is also studied. Figure 19 shows the air layer volumes for two different upstream excitation amplitude u in = 0.1 and 0.4 m/s. The other parameters were kept the same at Q in = 0.05 m 3 /s, U = 4.0 m/s, and f = 1 Hz. As expected, a larger amplitude excitation results in larger air layer volume oscillations. Figure 20 compares the corresponding leakage volume. It can be observed that the leakage frequency of the two cases is similar, but larger amplitude excitation results in larger leaks. Through the above numerical simulations, we found that the prediction of air layer stability is not too sensitive to the selection of the frequency and amplitude of upstream fluctuations. We selected u in = 0.1 m/s at f = 1 Hz in our simulations shown next. Figure 17: Effect of inflow fluctuation frequency on air layer volume. Q in = 0.05 m 3 /s, U = 4.0 m/s, u in = 0.1 m/s. Figure 18: Effect of inflow fluctuation frequency on air layer leakage. Q in = 0.05 m 3 /s, U = 4.0 m/s, u in = 0.1 m/s. Figure 15: Total and leaked air layer volume. Q in = m 3 /s, U = 4.0 m/s, u in = 0.1 m/s, f = 1 Hz. q=0.015 m 2 /s U inlet = 4.0m/s f=1.0hz U'=0.4m/s U'=0.1m/s Vol 0.5 Figure 16: Effect of upstream excitation frequency (0.1, 1.0, and 5.0 Hz from top to bottom) on air layer geometries at different times. Q in = 0.05 m 3 /s, U = 4.0 m/s, u in = 0.1 m/s Time Figure 19: Effect of upstream excitation amplitude on air layer volume. u in = 0.1 vs. 0.4 m/s. Q in = m 3 /s, U = 4.0 m/s, and f = 1 Hz. 8

9 0.008 q=0.015 m 2 /s U inlet =4.0m/s f=1.0hz U'=0.1m/s U'=0.4m/s stable and unstable regions reported by Lay et al. (2008). Figure 21 shows the comparison, where the agreement between the predictions and experiments is excellent. Vol_loss q crit (m 3 /s) Time Figure 20: Effect of upstream excitation amplitude on leakage volume. u in = 0.1 vs. 0.4 m/s. Q in = m 3 /s, U = 4.0 m/s, and f = 1 Hz. Oscillatory Stable Collapsing In the LCC experiments by Lay et al. (2008), they observed that the air layer collapsed from its fully developed state to the injector over a time period of 65~70 s after the air injection was stopped. Through numerical modeling of the same condition, we found that such a collapse along a long flat part of the plate was predicted much slower than the experimental observation if we used the upstream fluctuations as the only mechanism to cause the air leakage. Although the numerical method is good at predicting relatively large leakage due to air layer free surface dynamics near the beach flat, but it needs improvements in order to capture the small leakage at the trailing edge of a relatively stable air layer ending on the beach and its upstream. In order to take this stable leakage into account, we introduced a minimum air layer thickness, h. Physically, this minimum height corresponds to min the height of the air layer trailing edge where the turbulence and other small scale instability continuously break the layer in to small shedding bubbles. With this minimum thickness concept, the air leakage flux introduced earlier in (8) is now modified to: Q = V max h, h. (11) ( ) loss AL, TE AL, TE min That is, air is leaking at the air layer trailing edge through a gap of h min or higher. In this study, we used h min = m, which was inferred from the measured critical air flux of m 3 /s at 5 m/s flow velocity. Using the above discussed upstream excitation and the minimum leakage computed using (11), we conducted numerical simulations corresponding to conditions around the border line of Figure 21: Comparison of the measured and the predicted critical air flow rates. Measurements (black symbols) are from Lay et al. (2008). Only black squares correspond to the setup with a step studied here. Predictions (red and blue symbols) with u in = 0.1 m/s at f = 1 Hz. NUMERICAL RESULTS WITH THE GATE Numerical simulations with the gate and flap require the modeling of the free surface above the plate (Figure 6). When the flap is at an angle, the plate has a lift, and this requires modeling of the wake behind the plate. Before simulating the more complex air layer cases, we tested the code with the simpler case of a hydrofoil translating below a free surface studied by Duncan (1983). Figure 22 shows the foil which has a m chord length NACA0012 submerged m below the free surface with the water depth of m. The computational domain extended ±10 m in the horizontal direction. The translating velocity was U in = 0.80 m/s and the angle of attack was α=5º. Figure 23 compares the predicted free surface shape with the measurement of Duncan (1983). The two free surface shapes agree well m m Figure 22: Modeling of NACA0012 hydrofoil translating below the free surface. 9

10 Figure 24 shows the case of U in = 2.0 m/s inflow over a solid plate with a fixed wake and the flap at 0º. This computation started with a flat free surface. Then a wavy free surface emerged with the waviness propagating only downstream over the plate. Downstream of the plate, the wave propagated both upstream and downstream for a while until the free surface settled to its final shape. Figure 23: Comparison of free surface shape above the NACA0012 hydrofoil. Chord length m, α=5º, and translating velocity U in = 0.80 m/s. Experiments by Duncan (1983). When we deal with free surface flows over a shallow bottom, we need to pay attention to the depth Froude number defined as: F d U =, (12) gd where U is the flow velocity, g the gravity acceleration, and d the water depth. If F d < 1 the flow is subcritical; if F d > 1 the flow is supercritical. The critical velocity U crit is the velocity corresponding to F d = 1: Ucrit = gd. (13) For the air layer experiments conducted at the LCC, there are two water depths and thus two Froude numbers: one corresponding to the flow over the plate and the other to the flow over the tunnel bottom. With the given dimension of the experiments with the flap fixed at 0º (horizontal), the critical velocity over the plate is 2.6 m/s and that over the tunnel bottom is 4.8 m/s. Thus, the following three scenarios are possible: Subcritical flow everywhere: For example, an inflow of 0.5 m/s (0.65 m/s after the flap) results in subcritical flow everywhere. Supercritical flow over the plate and subcritical flow over the bottom: For example, an inflow of 2.0 m/s (2.6 m/s after the flap) produces this flow regime. Supercritical flow everywhere: For example, an inflow of 5.0 m/s. 3DYNAFS_BEM correctly predicted the transient behaviors of the above three flow regimes. Figure 24: Time overlapped free surface shapes over a solid plate with a fixed wake. U in = 2.0 m/s, flap at 0º. NUMERICAL RESULTS WITH FLAP MOTIONS The flap motion is simulated by imposing the necessary angular motion in time on the corresponding nodes. The rest of the numerical scheme is the same to above. We found that code was very robust and a simulation of relatively large amplitude motions can be done as shown in Figure 25. In this case, oscillation between 0º ~ 30º at 1 Hz was imposed, and the inflow velocity was U in = 5.0 m/s. The computation started with the steady state solution established with the flap at 0º, and the flap started its motion by swinging down. Then, the free surface waves developed as the flap oscillated between 0º and 30º, and the mean water level dropped after a while. The free surface waves generated on the top free surface are seen to interact strongly with the air layer free surface and the plate wake sheet. Figure 26 compares air layer volumes for four different air injection rates Q in = ~ 0.2 m 3 /s with the flap motion 0º ~ 30º at 1 Hz, and U in = 5 m/s. The flap was at 0º position during the first 22 s with a large air injection of Q in = 0.2 m 3 /s to establish the steady state air layer, and then the flap motion was activated. There was a large loss of air immediately after the initiation of the flap motion, and then the air layer reestablished at different rates. Large air injections quickly recovered, while small injections took a longer 10

11 time in reestablishing the full volume. In the case of the smallest air injection Q in = m 3 /s, the air layer slowly collapsed because leakage was larger than supply on average. The oscillation of air layer volume after t = 22 s became at the frequency of 1 Hz because the excitation by the flap was 1 Hz. the plate than along the plate top. By trial and error, we found that an upstream velocity of 3.8 m/s resulted in a flow velocity fluctuating around 7.4 m/s as shown in Figure 27. The nose has a stagnation point, and the flow velocity between the air layer and the tunnel bottom is quite uniform and almost steady. The velocity above the plate and below the top free surface oscillates due to the flap motion, but the two are almost the same at a given time. The flap motion for Run 656 is shown in Figure 28. Table 1: Conditions of the three experimental cases simulated. Figure 25: Free surface and wake geometries at different times for a 30 degree motion of the flap. U in = 5 m/s, Q in = 0.1 m 3 /s. Run # Flap motion (º) Period (s) U at cavity (m/s) Q (m 2 /s) Air layer ~ Lost ~ Lost ~ Stable Q air = 0.2 m 3 /s No flap motion Figure 26: Predicted air layer volume at various air injection rates. Flap motion 0º ~ 30º at 1 Hz, U in = 5 m/s, Q in = ~ 0.2 m 3 /s Figure 27: Time overlap plot of the predicted velocity for upstream inflow velocity of 3.8 m/s. This upstream velocity produced mean velocity at the air layer fluctuating around 7.4 m/s. Geometry to scale Very recently, Ceccio s group analyzed their tests results with flap motions. We selected three representative cases from their results (Makiharju, 2010), and conducted numerical simulations under those experimental conditions. The three cases are summarized in Table 1. One minor difficulty concerning the cases is that the upstream velocity is unknown from the experiments. The upstream velocity predicted from flow rates and tunnel cross sectional area does not produce correct flow velocity near the air layer. Due to the flow angle generated by the gate and flap, higher flow velocities exist along the bottom of 5º Figure 28: Flap motion 20º ~ 25º simulated for Run

12 The predicted air layer behavior for run 656 is shown in Figure 29. The air layer retracted and it could not reattach to the beach flat with its trailing edge remaining near the upstream edge of the beach. The air layer volume is shown in Figure 30. The larger scale fluctuations with a 5 s period are due to the flap motion, while the small scale fluctuation of about 1 s period is due to the natural frequency of the air in the plenum under this flow condition. The condition of Run 656 had twice the amplitude of the flap motion than that of Run 655. As can be seen in Figure 31, the free surface shows larger amplitude motion. However, the air layer behavior looks somewhat similar to that of Run 656. The air layer retracted and stayed near the upstream end of the beach. The corresponding volume variation is shown in Figure 32. The volume fluctuations followed the flap motion of 8.6 s period with large leaks during the air growth phase. Interestingly, the 1 s period fluctuation is weak in this case. Run 661 had an even larger amplitude flap motion, twice the amplitude of Run 656. To accommodate this large motion, the period of the flap motion is also about twice of that of Run 656. The air layer behavior is shown in Figure 33. As expected the free surface moved up and down with much larger amplitude waves. However, the air layer trailing edge always stayed on the beach flat and appeared quite stable. The air layer volume is shown in Figure 34. It changed very smoothly following the imposition of the flap motion period of 16 s. This behavior agrees with the experimental observations that only Run 661 had a stable air layer while the other two runs lost the air layer. These numerical exercises suggest that the excitation frequency is more important than the excitation amplitude because even small amplitude unsteadiness could collapse the air layer if the frequency is close to the natural frequency of the air layer. Figure 30: Predicted air layer volume and volume loss for LCC Run 656. Figure 31: Predicted free surface and air layer behavior for LCC Run 655. Figure 29: Predicted free surface and air layer behavior for LCC Run 656. Figure 32: Predicted air layer volume and volume loss for LCC Run

13 Figure 33: Predicted free surface and air layer behavior for LCC Run 661. other frequencies also resulted in the same period oscillation of the air layer volume. We also studied the effect of excitation amplitude. Larger leakage was observed if a larger amplitude upstream disturbance was applied. However, the amplitude did not affect the air layer leakage frequency. We could conclude that the prediction of air layer stability is not too sensitive to the selection of the frequency and amplitude of upstream fluctuations. The simulated air layer behavior recovered the experimentally observed stability regimes. The 3DYNAFS_BEM code was robust enough to handle large flap motions. The shallow water waves propagating on the tunnel top free surface were found to be the most influential factor in this experimental setup. Both supercritical and subcritical flow regimes were possible in the LCC test setup with gates, and the code predicted the correct wave propagation characteristics in both regimes. The free surface waves generated by the flap motion affected the air layer very strongly. Simulations of three flapping experiments identified the stability characteristics of the three cases correctly. It was also found that the amplitude of the excitation was not as important as the frequency of the excitation. Both experiments and numerical predictions were in agreement that small amplitude high frequency excitations disrupted the air layer while the air layer survived large amplitude low frequency flap motions. Figure 34: Predicted air layer volume and volume loss for LCC Run 661. CONCLUSIONS In this paper, we have presented our efforts to model the air layer behavior as experimented in the Large Cavitation Channel (LCC). We used 3DYNAFS_BEM and modeled all aspects of the experimental setup including the air mass balance in the air layer, flap motion, and the tunnel top free surface. Simulations of earlier experiments without the gate produced solutions more stable than the experimental observations in the absence of imposition of upstream unsteadiness. Upstream unsteadiness is found to be essential to produce realistic stability results agreeing with the experiments. We studied the effect of upstream excitation frequency. For a representative case, 1 Hz excitation resulted in the highest frequency of repetition of air leaks. Excitation frequencies lower and higher than this produced less number of leaks. The air layer studied appeared to have a natural period around 1.5 s because excitations of ACKNOWLEDGEMENT This work was supported by the Office of Naval Research under the contract N C-0516, monitored by Dr. Patrick Purtell. We also appreciate Prof. Steven Ceccio and his group for providing us the experimental data for comparisons. REFERENCES Bushnell, D. M., Hefner, J. N., ed., Viscous drag reduction in boundary layers, Progress in Astronautics and Aeronautics, Vol. 123, AIAA, Washington, D.C., Butuzov, A., Sverchkov, A., Poustoshny, A., Chalov, S., High speed ships. On the cavity: scientific base, design peculiarities and perspectives for the Mediterranean Sea, Proc. Fifth Symposium on High Speed Marine Vehicles, Capri, Italy, March, Celebi, M.S., Computation of transient nonlinear ship waves using an adaptive algorithm, Journal of Fluids and Structures Vol. 14, Chahine, G. L., Hsiao, C.-T., Modeling 3D unsteady sheet cavity using a coupled UnRANS-BEM code, 13

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