Fluid-Structure Interaction Response of Planar

29th Symposium on Naval Hydrodynamics Gothenburg, Sweden, 26-31 August 2012

Fluid-Structure Interaction Response of Planar Surface Effect Ship Seals M.R. Kramer & Y.L. Young (University of Michigan, Ann Arbor, MI, USA) g Kβi Lc Li Lw Lh Lout p∗ Patm Pc P¯c Psi Qleak T U u (xh , yh ) D L M α βi βio ρi µi νi σ

ABSTRACT Surface Effect Ships (SESs) are high speed vessels that are partially supported by a pressurized air cushion, which is typically contained beneath the hull by two rigid sidehulls and flexible bow and stern seals. The seals, being highly deformable, can experience large drag forces, particularly at low Froude numbers, and can provide a significant contribution to the total resistance. In order to ensure adequate powering, the drag of the seals must be computed over the entire speed range. Since the seals are highly deformable, and are responsible for maintaining the air cushion, methods for total drag prediction of SESs must account for the FluidStructure Interaction (FSI) response of the seals in order to consider the highly coupled nature of the air cushion, seals, sidehulls, and free surface flow. The objectives of this work are to (1) develop a simple, effective model to simulate the FSI response of 2-D planar SES seals, (2) investigate the influence of the angle of attack on the hydrodynamic response of planar seals both with and without consideration for the air cushion via one- and two-plate simulations, and (3) investigate the effects of bow seal stiffness and cushion pressure on the equilibrium geometry and resulting hydrodynamic response for a 2-D representation of an SES in calm water and fixed in sinkage and trim. The proposed FSI methodology is shown to be effective and computationally inexpensive for predicting the steady-state equilibrium geometry for simplified, flatplate SES bow seals. The bow seal stiffness is found to have a large impact on the seal equilibrium geometry, which in turn greatly affects the seal forces and moment, the free surface profiles, and the air leakage flow rate under the pressure cushion for a fixed sinkage and trim.

1 1.1

INTRODUCTION Background

A Surface Effect Ship (SES) is a type of high-speed vessel that is partially supported by a captured air cushion, which is typically contained between two rigid sidehulls and flexible bow and stern seals (see Fig. 1). SESs can offer several benefits compared to traditional displacement hulls, including higher attainable speeds, better seakeeping performance, and higher transport efficiency (Butler, 1985). However, the design and analysis of SESs introduce several additional challenges compared to displacement vessels due to the interactions of the free surface with the air cushion, sidehulls, and bow

SYMBOLS CQ Fr F rp

Gravitational acceleration Torsional spring constant of spring i Cushion length Immersed length Wetted length Hinged length Length to upstream pressure outlet Local fluid dynamic pressure Atmospheric pressure Pressure of air cushion (gage) Nondimensional cushion pressure, = Pc /ρw gT Internal pressure of seal i (gage) Air leakage flow rate Draft of bow seal Uniform inflow speed Local fluid velocity Hinge point Drag force Lift force Moment about hinge point Fluid volume fraction Angle of Attack (AOA) of plate i Undeformed AOA of plate i Density of fluid i Dynamic viscosity of fluid i Kinematic viscosity of fluid i, = µi /ρi Standard deviation

2 1/2 Leakage coefficient, = Qleak (ρ √a /2Pc Lc ) Vessel Froude number, = U/ √ gLc Plate Froude number, = U/ gLh

1

complicated than for conventional displacement vessels. One difficulty lies in the scaling of forces from model to full scale (see, e.g., Doctors and McKesson, 2006; Kramer et al., 2010; McKesson and Doctors, 2011; Wilson et al., 1979). For displacement hulls, the resistance is typically expressed as the sum of frictional and wavemaking components, which are assumed to scale with the Reynolds number and Froude number, respectively. In addition to the wave resistance and sidehull skin friction, the resistance of an SES includes contributions from air drag (which can be significant at higher speeds), momentum drag (from intake to the cushion fans), transom drag (from hydrostatic pressure difference between bow and stern), spray drag, and seal drag. The scaling relations for these additional components are typically obtained using empirically derived methods based on model test results. Since seal drag is a very important component, and the seal response directly affects the vessel motion, it warrants specialized study in order to characterize the major scaling relations. For accurate prediction of the performance of an SES, a simulation tool must include the capability to (1) model the flow past the sidehulls with consideration for the free surface and viscous effects, (2) model the air cushion, which interacts with the free surface beneath the vessel, and (3) model the flow past the seals, the shape of which is determined by material deformation under the internal seal pressure, cushion pressure, fluid pressure, and shear stress. Several methods have been applied for items (1) and (2), including both potential flow methods and fully nonlinear Computational Fluid Dynamics (CFD) methods. A brief review of past modeling efforts will now be given. The first efforts associated with wave resistance prediction for SESs, and Air Cushion Vehicles (ACVs) in general, can be traced back to Newman and Poole (1962) and Barratt (1965). In these early works, the air cushion is modeled using linearized potential flow theory by applying a modified dynamic boundary condition (i.e. fixed pressure) at the free surface to account for the higher pressure in the air cushion. It should be noted that these early studies were primarily aimed towards ACVs and the contribution of the seals was neglected. The theory was found to accurately predict the hump speed(s), however the magnitudes of the resistance humps at low Froude numbers were largely overestimated. A means of resolving this overprediction was developed by Doctors and Sharma (1972), where a smoothing function was applied at the edge of the pressure cushion to better model the drop-off from cushion pressure to atmospheric pressure. Using this approximation, the predicted wave resistance was found to agree more closely with measured experimental results. Several studies have shown potential methods to compare well with experi-

Fig. 1: Diagram of an SES showing air cushion, flexible seals, and sidehulls (from Butler (1985))

& stern seals. The resistance of an SES is typically characterized by the presence of several large “humps,” including a primary hump at a Froude number of F r ≈ 0.6, caused by interference of the wave patterns produced by the sidehulls and air cushion, and a secondary hump at F r ≈ 0.45, which is primarily driven by the flow past the flexible bow and stern seals (Heber, 1977; Yin and Bliault, 2000). The magnitude of the secondary drag hump is dependent on the effective stiffness of the seals, which is primarily driven by relative differences between the internal seal pressure, cushion pressure, and dynamic fluid pressure. Most of the first SES development efforts focused on the very high speed regime (F r & 1.5) with intended top speeds of 80+ knots (see, e.g., Mantle, 1973). Due to the very high levels of installed power required to reach these higher speeds, surpassing the secondary drag hump was of relatively minor concern. Consequently, the low-speed performance of the seals was not of large importance. More recently, SESs have been designed to perform in the low to moderate speed regimes, with top speeds in the range of 40 to 50 knots. The power requirements at these speeds can be on the same order of magnitude as the power required to surpass the primary and secondary drag humps (Kramer and Wilson, 2009; Kramer et al., 2010; Van Dyck, 1972). Installing adequate powering is thus very important, as is the requirement to minimize weight. Therefore, in order to ensure an accurate estimate for the required power, methods must be developed for predicting the resistance throughout the entire speed range. Since seal effects have been found to be very important at lower speeds, accurate methods for predicting the behavior of the seals at these speeds must be developed. Due to the dynamics of the air cushion and flexible seals, the computation of resistance for SESs is more

2

mental results at higher speeds, but not at lower speeds, due to the significant contribution of the seals to the total resistance (Doctors, 2009; Doctors and McKesson, 2006; Wilson et al., 1979). Nonlinear CFD methods have also been used for calculation of calm-water resistance. Maki et al. (2009) used a single-phase level set method to model an SES in calm water, where the air cushion was considered by directly applying a fixed pressure boundary condition to the free surface with similar smoothing parameters to Doctors and Sharma (1972). Since the air phase was not considered, seals were not necessary to maintain the cushion pressure and were not simulated. The results were compared to the linear potential method of Doctors and Sharma (1972) and it was found that the CFD results tended to underpredict the resistance, while the linearized potential method agreed well with experimental measurements. It should be noted that the successful agreement of the linearized potential method with experimental results is in part due to the successful tailoring of hydrodynamic form factors and friction coefficients based on years of past experience. All of the aforementioned methods represented the air cushion by directly applying a constant pressure boundary condition on the free surface. In reality, the air cushion is generated by a set of fans and the pressure within the cushioned space may vary temporally and spatially due to a ride control system, localized leakage, acoustic wave generation and propagation, and interactions with water waves. To more realistically model the air cushion, Donnelly (2010) and Donnelly and Neu (2011) used a Volume of Fluid (VOF) method, which tracks the interface between the air phase and water phase, to simulate the flow past the U.S. Navy (USN) T-Craft. A momentum source model was used to represent the inflow from the fans. Both inviscid and viscous fluid models were applied and the resistance was computed in both calm water and in regular waves. The seals were assumed to be rigid, and the height of the seals, along with the fan source strength, were adjusted to avoid excessive seal immersion, which would corrupt both the predicted wave profile and drag computations. The resistance predictions agreed well with the experimental results of Bishop et al. (2009), however the method requires the cushion pressure and seal height to be known a priori, which is not possible unless experiments are conducted beforehand for that particular SES system at the operating conditions of interest. A similar model was used in Young et al. (2011) to investigated the interactions of the air cushion and waterjet propulsion system by performing simulations of a full-scale SES with a complete waterjet propulsion system. An in-depth discussion about air cushion modeling can be found in Doctors (1993), Milewski et al. (2008),

and Connell et al. (2011), where various simplified models are proposed for dynamic simulation of ACVs and SESs. Several of the proposed models were applied in Bhushan et al. (2011), where an SES was simulated for both resistance and seakeeping predictions. In addition to varying the cushion model, a selection of simplified seal models were investigated with varying levels of success. The computational results were found to compare fairly well with experimental results for the resistance, heave, and pitch motions of the T-Craft SES model presented in Bishop et al. (2009). In their model, the cushion pressure is prescribed, and the bow and stern seals were modeled as rigid bodies and allowed to rotate about a body-fixed hinge point. The seals were subject to both external fluid pressure and internal cushion pressure, and a simple equation of motion was formulated to find the equilibrium seal position. SES bow seals are typically constructed of flexible rubber membrane materials in various configurations, including semi-rigid planing, lobed, bag and finger, and finger-only designs (Heber, 1977). Analysis of the structural behavior of SES seals differs from more traditional structures in their stiffness depends on the relative difference between the internal pressure (which yields a restoring force), cushion pressure, and external fluid pressure and shear stress. Due to the large flexibility of the seals, a method for calculating the Fluid-Structure Interaction (FSI) response of the seals is necessary in order to predict the deformed shape for different operating conditions. Due to the complexities inherent in predicting the deformable behavior of the seals, most past prediction methods have used empirical results from model- and full-scale experiments. While experimental results are very useful in understanding the seal behavior, full parametric experimental studies are expensive, difficult to perform, and pose significant scaling challenges. Numerical FSI models have been developed for prediction of SES seal resistance by representing the seal as a Timoshenko beam (Doctors, 1977) or by assuming a fixed seal geometry (Doctors, 2009; Doctors and McKesson, 2006) and directly coupling the linear structural equations of motion with a linear potential flow model. While these models have been shown to improve the resistance predictions for certain cases, they have not compared well at lower speeds. More recently, Yang et al. (2011) presented coupled Smoothed Particle Hydrodynamics (SPH)-Finite Element Method (FEM) simulations for the flow past ACV skirts (and SES seals). SPH methods show promise for predicting deformations under extreme loadings, such as wave impact, however they are not as accurate in predicting resistance due to the difficulty of refining the near-wall regions in a Lagrangian particle framework.

3

1.2

Objective

Patm , ρa , νa

g

The objectives of the current work are to: U , ρw , νw

3. Investigate the influence of effective stiffness on the equilibrium angle and resultant resistance of a single planar seal supported by a torsional spring

5. Investigate the influence of the pressure cushion and stern seal on the bow seal performance PROBLEM DESCRIPTION

g

Patm , ρa , νa

Generalized 2-D Representation of an SES β1

Due to the complexity of the problem of predicting the performance of a complete SES, several modeling simplifications are made in the current study to focus on the governing physics. The first simplification is to consider a 2-D slice through the vessel centerline, ignoring the impact of the sidehulls (see Fig. 2). The SES is depicted with a single-lobed bow and stern seal, which itself is a significant simplification from more realistic bag and finger seal designs, to introduce the nomenclature. The model is assumed to be fixed in sinkage and trim. Each seal is pressurized above atmospheric pressure (Patm ) via a fan system, where Ps1 , Ps2 , and Pc are the nominal gage pressure of the bow seal, stern seal, and air cushion, respectively. The vessel is subject to steady, uniform flow at a speed U . The fluids are assumed to be incompressible with densities ρw & ρa and kinematic viscosities νw & νa for water and air, respectively. The speed is √ non-dimensionalized as the Froude number, F r = U/ gLc , where Lc is the cushion length and g is gravitational acceleration. 2.2

Lc

degree of freedom, represented by its angle of attack, denoted βi . The spring stiffness of seal i, Kβi , is related to the internal seal pressure, i.e. Kβi = fi (Psi ). The functions fi may be determined via experimental or numerical models. It is assumed that the behavior of the rotational system is characteristically representative of the seal stiffness, and hence is appropriate for a simplified study.

4. Investigate the effects of bow seal stiffness and cushion pressure on the equilibrium geometry and resulting hydrodynamic response for a 2-D SES in calm water and fixed in sinkage and trim.

2.1

Ps 2

Fig. 2: Simplified 2-D representation of an SES using two inflated single-lobed seals, which are stiffened by internal pressure. The superstructure is ignored. Although the geometry is highly generalized, the figure summarizes the important variables for the problem and serves as a useful presentation of the nomenclature.

2. Investigate the influence of geometry, as represented by the angle of attack, on the hydrodynamic response of a single, planar seal

2

Pc

Ps 1

1. Develop a simple, effective method for simulating the FSI response of deformable seals for a Twodimensional (2-D) representation of an SES

Kβ1

Pc

β2

Kβ2

U , ρw , νw Lc Fig. 3: Simplified 2-D representation of an SES using two flat plates supported by torsional springs to represent the seals

For the current study, the bow seal is modeled as deformable, while the stern seal is either ignored or fixed. A method is first presented for calculating the FSI response of a single plate without a pressure cushion, and this model is then extended to include the effects of the pressure cushion and stern seal. The nomenclature for a flat-plate bow seal supported by a torsional spring is shown in Fig. 4. The plate geometry is defined using a fixed length from the trailing edge to the hinge point, denoted as Lh . The system has one degree of freedom, which is the rotation angle, or angle of attack of the plate, denoted β1 . The√localized plate Froude number is defined as F rp = U/ gLh . For the current study, a coordinate system (x, y) is placed where the plate intersects the calm-water free surface at an angle of attack β1 = 10◦ . The hinge point is assumed to be fixed at a point (xh , yh ). The spring force is assumed to be linear in β1 with a spring constant Kβ1 . The immersed length Li is geometrically related to the angle of attack and the height of the hinge through Eq. (1). The hinge location was chosen such that Li = 0.5 m

Flat-plate Representation of SES Seals

In general, the seals are deformable surfaces and the overall stiffness of each seal is dependent on its internal pressure. A further simplification of the problem can be made by representing each of the seal surfaces by a planar surface supported by a torsional spring above the free surface, as shown in Fig. 3. Each seal has a single 4

Kβ1

y

The volume fraction satisfies the advection equation:

β1

x

U , ρw , νw Lw

s

g

∂α + ∇ · (αu) + ∇ · (α(1 − α)ur ) = 0 ∂t

Li

where u = u(x, t) is the local fluid velocity. The third term in Eq. (4) corresponds to a free surface compression term, responsible for sharpening the interface between the two fluid phases. The flow velocity and pressure are solved by satisfying the Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations, which may be written as:

Lh Fig. 4: Definition of nomenclature for a planing flat-plate bow seal

at β1 = 10◦ . Li = Lh −

yh sin β1

(1) ∇·u=0

The wetted length Lw is dependent on the flow solution and is defined as the length from the trailing edge to the stagnation point. The fixed dimensions, material properties, and operating parameters used in this study are tabulated in Tab. 1.

3 3.1

Value 1.0 4.77 −0.4924 0.0868 10 − 10000 998.2 1.225 1.0048 × 10−6 1.4604 × 10−5 9.81

where p∗ ≡ p − ρg · x is the dynamic pressure, g is the gravitational acceleration vector, and µeff = µeff (x, t) is the generalized dynamic viscosity of the fluid mixture, which is dependent on the shear stress model that is used (e.g. inviscid, laminar, turbulent). For the current study, laminar flow is assumed, so µeff = µ(x, t), as in Eq. (3).

Unit m m m m N − m/deg kg/m3 kg/m3 m2 /s m2 /s m/s2

3.2 3.2.1

Governing Fluid Equations

To calculate the hydrodynamic solution, the OpenFOAM CFD library is used. The fluid is assumed to be incompressible and laminar, and the flow is modeled as a single multiphase fluid using the Volume of Fluid (VOF) method, where the air and water phases are considered by introducing the volume fraction α = α(x, t), where α = 1 corresponds to water and α = 0 corresponds to air. The fluid mixture properties are calculated as a volume-weighted average of the two phases, i.e. (2)

µ(x, t) = αµw + (1 − α) µa

(3)

Mesh Topology Computational Domain

To minimize potential sources of error resulting from grid non-orthogonality, a structured mesh was chosen. For the problem of a single planing flat plate, the air phase was found to have a negligible effect on the flow solution in the water phase (which is the region of interest), particularly for cases where the free surface reaches a steady state. Therefore, the computational domain was constructed with the goal of minimizing the number of extraneous cells in the air phase. A convergence study will be shown in Section 4, however it is useful to show the converged mesh at this point. The converged mesh for the single plate case is shown in Fig. 5. The mesh was refined in three key locations: near the stagnation point (to capture the large pressure gradient), along the plate surface (to capture the large velocity gradient and shear stress), and near the free surface (to resolve the free surface profile). The converged mesh for the single-plate cases had approximately 125,000 cells with 325 faces along the plate surface. The dimensions of the two-plate case were chosen to be as similar as possible to the geometry presented in Wiggins et al. (2011) to facilitate future collaboration and comparison of experimental and numerical results. The converged mesh for the two-plate cases is shown in Fig. 6. For these cases, the aft plate was created

NUMERICAL FORMULATION

ρ(x, t) = αρw + (1 − α) ρa

(5)

∂ (ρu) + ∇ · (ρuu) = −∇p∗ − g · x∇ρ ∂t

+ ∇ · µeff ∇u + ∇uT (6)

Tab. 1: Fixed geometric and fluid properties Item Lh Lc xh yh Kβ1 ρw ρa νw νa g

(4)

where ρ is the mixture density, µ is the mixture dynamic viscosity, x is the spatial location vector, and t is the time variable. The kinematic viscosity is related to the dynamic viscosity through the relation ν = µ/ρ. 5

(a) Near-field view

(a) Near-field view

(b) Far-field view

(b) Far-field view

Fig. 5: Near- and far-field views of the fine mesh for single-plate simulations with β1 = 10◦

Fig. 6: Near- and far-field views of the fine mesh for two-plate simulations with β1 = 10◦ , β2 = 30◦

by duplicating the mesh near the bow plate, given certain restrictions governed by the geometry. The mesh consisted of approximately 635,000 cells with 330 faces along the bow plate and 225 faces along the stern plate. 3.2.2

3.2.3

The upstream height of the domain is governed by the total length of the plate, which was adjusted in order to prevent unsteady overturning of the jet root. The jet root region for β1 = 10◦ is shown in Fig. 7 for illustration. In potential formulations for the planing flat plate problem, the jet root (the portion of the flow that remains upstream of the stagnation point) is assumed to have a negligible effect on the flow once its shape has been established. Solving the problem using a CFD method introduces several challenges with regards to the jet root, since gravitational effects will cause the jet root to fall and re-impact the upstream free surface, introducing a large level of unsteadiness to the solution. Since the simulations are 2-D, it is likely that this unsteadiness is largely overestimated, compared to 3-D cases. In order to prevent this unphysical behavior, an atmospheric pressure outlet is placed upstream of the stagnation point in order to prevent the jet root from corrupting the steady free surface profile. The pressure outlet location was chosen to be far enough upstream to have negligible impact on the plate forces, but far enough downstream to prevent the jet root from falling onto the upstream free surface.

Boundary Conditions

The boundary conditions for the simulations are listed in Tab. 2. The inlet boundary conditions are consistent with a multiphase velocity inlet and the outlet is consistent with a pressure outlet. For all seal and SES surfaces, a no-slip wall boundary condition is used. In order to model the pressure condition for the two-plate cases, a fixed-value pressure boundary condition is applied on a portion of the SES wetdeck. A flux-corrected uniform velocity boundary condition is used to ensure that the flow from the “fan” is uniform in both pressure and velocity. Further details may be found in OpenFOAM (2011). Tab. 2: Boundary conditions for CFD simulations Boundary Inlet Water Inlet Air Outlet Bottom Top Plates Upstream Outlet Cushion Fan

Variable p∗ ∗ ∂p =0 ∂n ∂p∗ =0 ∂n p∗ = 0 symmetry symmetry ∂p∗ u=0 =0 ∂n ∂u =0 p∗ = 0 ∂n u = Uf p∗ = Pc u u = Uˆı u = Uˆı ∂u =0 ∂n

Upstream Pressure Outlet

α α=1 α=0 ∂α =0 ∂n

3.3

Structural Model

The flat-plate bow seal has a single rotational degree of freedom, β1 , and the equilibrium equation of motion for steady-state response can be expressed as:

∂α ∂n ∂α ∂n

=0 =0 α=0

Kβ1 (β1o − β1 ) = M(β1 )

6

(7)

The modified structural equilibrium equation can be written as: ˜ 1) Kβ1 (β1o − β1 ) = M(β

˜ 1 ) is obtained by using piecewise cubic inwhere M(β terpolation of the CFD computations obtained at fixed values of β1 . Equation (9) is nonlinear and is solved numerically using a Newton-Raphson method. It should be noted that, although the current paper considers deformation of the bow seal only, it may be extended to include a deformable flat-plate stern seal as well.

Fig. 7: Close-up view of jet root region for β1 = 10◦ , F rp = 1.0. The fluid is flowing from left-to-right. A pressure outlet is placed upstream of the stagnation point to prevent the jet root from impacting the free surface, which would corrupt the steady-state solution.

4

CONVERGENCE AND VALIDATION STUDY

A convergence study was performed for the singleplate simulations at β1 = 10◦ and F rp = 1.0 in order to ensure that the mesh was adequately refined in order to accurately predict the forces and pressure distribution along the plate. Both inviscid and laminar simulations were performed. A laminar flow model was chosen to facilitate calculation of the stagnation point, which determines the wetted length. Four meshes of varying levels of refinement were generated. The results for the pressure distributions and free surface profile were compared with the linearized potential flow method of Doctors (1974). These comparisons are shown in Fig. 8. The results were found to compare fairly well in both the pressure profile, wetted length, and free surface profile. Since the potential method predicts a pressure singularity at the stagnation point, the stagnation pressure based on the Bernoulli equation, which should be 1.0, was used as a measure of refinement as well. In addition to convergence of the pressure and free surface profiles, it was important to examine the convergence of the forces, moment, and wetted length. Since the simulations are performed in the time domain, the steady-state forces and moments were obtained by computing a time-average after the initial transient oscillations have subsided. The results for the convergence study for forces, moment, and wetted length are shown in Tab. 3. The drag, lift, and moment coefficients are non-dimensionalized as follows:

where M(β1 ) is the fluid moment taken about the hinge point, defined as positive in the counterclockwise (+z) direction, and β1o is the undeformed reference angle, which is assumed to be equal to 15◦ . The moment is calculated about the hinge point (see Fig. 4) by integrating the pressure distribution along the plate: Z Lout M(β1 ) = p(β1 , s)(Lh − s) ds (8) 0

where s is defined as the distance from the trailing edge (see Fig. 4) and Lout is the distance from the trailing edge to the upstream pressure outlet along the plate, which identifies the end of the computational domain. It should be noted that the shear stress does not contribute to the moment, but it does contribute to the lift and drag forces. 3.4

(9)

FSI Coupling Algorithm/Methodology

Many challenges arise when performing FSI simulations involving light/slender structures in incompressible, viscous flow, including numerical instabilities caused by the virtual added mass effect (see, e.g., Belanger and Paidoussis, 1995; Causin et al., 2005; Forster et al., 2007; Young et al., 2012), and computational cost, which can be significant, particularly if a deforming mesh method is used. The computational cost makes these methods impractical for design or parametric studies. As an alternative, a coupled method is proposed in this paper that represents the fluid moment via a metafunction. The steady moment is calculated by interpolating the results from a series of static simulations performed at angles from β1 = 5◦ to 15◦ in increments of 2.5◦ . Using this method, the effects of seal stiffness can be studied in the post-processing stage, which can significantly improve the computational cost of the parametric study and avoid numerical instability issues due to FSI.

D 1/2ρw U 2 Lh L Cl = 1/2ρw U 2 Lh M Cm = 1/2ρw U 2 L2h Cd =

(10) (11) (12)

where D, L, and M are the drag, lift, and moment, respectively. The Fine mesh was found to be of sufficient quality for the remainder of the simulations. 7

Tab. 3: Convergence of force coefficients, moment coefficient, and non-dimensional wetted length for various levels of mesh resolution for a single plate with β1 = 10◦ , F rp = 1.0

Mesh Coarse Medium Fine Finest




Cd , Cl , Cm

FOAM - coarse FOAM - medium FOAM - fine FOAM - finest Doctors (1974)

2

U

0.6

Cm 0.0280 0.0293 0.0312 0.0308

Lw /Lh 0.948 0.965 0.971 0.982

Cd Cl Cm

0.30 0.25 0.20 0.15 0.10 0.05 0.00 4

6

8

4

6

8

10

12

14

16

10

12

14

16

1.3 1.2

Lw /Lh

w

Cl 0.1673 0.1680 0.1685 0.1684

0.35

0.8

atm)/0.5

Cd 0.0299 0.0301 0.0302 0.0303

Faces on Plate 195 260 325 390

0.40

1.0

0.4

1.1 1.0 0.9 0.8 0.7

(

p

p

# Cells 45k 80k 125k 180k

0.6

0.2

1

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Fig. 9: Force coefficients, moment coefficient, and wetted length for single-plate simulations at F rp = 1.0 as a function of angle of attack, β1 . The symbols represent discrete simulation results while the curves represent a piecewise cubic interpolation of the results. Error bars are drawn at ±σ to display the level of unsteadiness.

s/Lh

(a) Pressure coefficient

FOAM - coarse FOAM - medium FOAM - fine FOAM - finest Doctors (1974)

5

4

3

y/T

2

1

0

1

2 1.0

0.5

0.0

0.5




1.0

x/

1.5

2.0

5 5.1

2.5

[deg]

RESULTS AND DISCUSSION Effects of Angle of Attack for Single Plate

To better understand the influence of geometry on the hydrodynamic response of a single, planar seal, the angle of attack was varied at a plate Froude number F rp = 1.0. The hydrodynamic response was calculated for five fixed angles of attack from β1 = 5◦ to 15◦ in increments of 2.5◦ . The objective of this section is to investigate the single-plate problem to identify the most important physics to ensure that these phenomena are captured in more detailed models. The quasi-steady force and moment coefficients are shown in Fig. 9 (nondimensionalized using Eqs. (10) through (12)). The discrete simulations are shown as symbols, and the curves represent a piecewise cubic interpolation, which is used for the FSI analysis. Error bars are shown at ± one standard deviation (σ) in order to show the level of unsteadiness in the solution. The solution was found to become unsteady when a breaking wave was observed at β1 ≈ 15◦ . The free surface profiles for all five cases are shown

3.0

(b) Free surface profile

Fig. 8: Convergence of pressure distribution on the plate, and the free surface profiles, for various levels of mesh resolution for a single plate with β1 = 10◦ , F rp = 1.0. Here, λ = 2πU 2 /g is the wavelength of the resulting longitudinal waves and T = Li sin β1 is the draft at the plate trailing edge.

8

0.5

y/Lh

0.0

0.5

1.0

1.5

2.0 2.0




 1

=5

1

=7.5

1

=10

1

=12.5

1

=15

1.5

1.0

0.5

plate will act as a rigid surface and will not deform. In this case, the forces and moments will be large.   yh −1 (13) lim β1eq = sin Kβ1 →0 Lh lim

Kβ1 →∞

0.0

0.5

1.0

1.5

β1eq = β1o

The seal stiffness is shown to have a significant impact on the lift and drag of the seal, which affects the total vessel resistance both directly, via seal drag, and indirectly, by affecting the sinkage and trim. Since the forces are very sensitive to the stiffness, it is important to be able to predict these effects in order to obtain total vessel resistance. For the successful design of SES seals, it is important to minimize the stiffness of the seals, however the stiffness must be large enough to maintain the pressurized air cushion, and to avoid hydroelastic instability issues (e.g. dynamic buckling of the fingers, flutter, resonance, etc) and material failure.

2.0

x/Lh

Fig. 10: Near-plate free surface profiles for five angles of attack, β1 at F rp = 1.0 showing change in wetted length for varying equilibrium angle. It should be noted that each subsequent series of data is plotted 0.4 units below the previous one in order to more clearly show the effects of angle of attack.

in Fig. 10. Here, the free surface in the wake of the plate is clearly shown to be unsteady and breaking for β1 = 15◦ . The angle at which the flow becomes unsteady is dependent on the Froude number and will generally decrease as the Froude number is reduced. Investigation of the effects of Froude number on the response of a single plate is currently underway. Further investigation of Fig. 10 shows that as the angle of attack increases, the water rises further up the plate, causing the wetted length to increase. The ratio of wetted length to hinged length is plotted in Fig. 9 and is shown to increase significantly with β1 . Consequently, the hydrodynamic load coefficients also increase with β1 .

16

1eq

[deg]

14 12 10 8 6 4 10

1

0.7

2

Cd Cl Cm

0.6

Cd , Cl , Cm

10

0.5 0.4 0.3 0.2 0.1

5.2

(14)

Equilibrium Solution for Single Plate with Varying Stiffness

0.0 10

Once the forces and moment are calculated for a range of angles of attack, the effects of seal stiffness on the equilibrium solution and the seal forces may be investigated. For the current study, the seal stiffness is represented by the torsional spring constant Kβ1 . This stiffness is characteristically representative of the internal seal pressure Ps1 for inflated bag seals. ˜ Eq. (9) may be Using the interpolated moment M, solved for a range of spring stiffnesses. The results for the equilibrium angle β1eq as a function of spring stiffness is shown in Fig. 11. The equilibrium angle is shown to vary widely as a function of spring stiffness, and these changes cause large changes in the force and moment coefficients exhibited by the plate. The two limits, which may easily be derived, are shown in Eqs. (13) & (14). At the low-stiffness limit, the trailing edge of the plate will ride on the calm-water free surface. At this limit, the forces and moment on the plate will be nearly zero. At the high-stiffness limit, the

1

10

2

10

3


 10

3

K 1 [N m/deg]

10

10

4

4

Fig. 11: Equilibrium angle β1eq as a function of Kβ1 for a single plate (i.e. no cushion)

5.3

Effects of Angle of Attack and Cushion Pressure for Two-Plate Configuration

The single-plate simulations were extended in order to consider the impact of the pressure cushion and stern seal on the hydrodynamic performance of the bow seal. In order to do so, two-plate simulations were performed for a range of bow seal angles of attack β1 with fixed stern seal √ angle β2 = 30◦ at a high Froude number F r = U/ gLc = 0.8 based on the cushion length. A large Froude number was chosen to yield more steady results than for lower Froude numbers. Future studies will investigate the response at lower Froude numbers. Two cushion pressures were chosen for this study. The cushion pressure is non-dimensionalized to corre-

9

with P¯c = 0.6, F r = 0.8, and β2 = 30◦ . Comparisons of the free surface profile for these three cases are shown in Fig. 13. In contrast with the single-plate case, shown in Fig. 10, the results at 15◦ for the two plate case are steady and non-breaking due to the higher flow speed and existence of the pressure cushion, both of which increase the wavelength (and minimize the steepness) of the longitudinal waves. It should be noted that the bow seal geometry shown is for the β1 = 10◦ case for clarity. As β1 increases, the gap between the free surface and the stern seal increases, and hence the rate of air leakage, CQ , increases, as shown in Fig. 15. Moreover, as shown in Fig. 13, increases in β1 will also lead to increases in wetted length of the bow seal, and consequently the hydrodynamic load coefficients for the bow seal, which is consistent with the single plate results.

(a) Near-field view showing air cushion velocity vectors

0.5


1 =7.5 
1 =10   Two plate -
1 =12.5  Two plate -
1 =15 Two plate -

(b) Far-field view showing longitudinal waves

0.4

Two plate 0.3

Fig. 12: Contours of volume fraction for two plate simulations with β1 = 10◦ , β2 = 30◦ , F r = 0.8, P¯c = 0.6

0.2

0.1

Pc P¯c = ρgT

y/Lc

spond to the lift-to-buoyancy ratio:

0.0

0.1

(15) 0.2

0.3

where T is chosen to be the draft of the bow seal at β1 = 10◦ . The non-dimensional pressures used in this study correspond to values of approximately 0.6 and 0.8, which corresponds to Pc = 500 Pa and 700 Pa, respectively. An example free surface contour is shown in Fig. 12 for β1 = 10◦ . In addition to contours of the free surface, velocity vectors are shown in order to illustrate the air flow in the pressure cushion. The figure shows that a large gap develops under the stern seal due to free surface deformation created by the bow seal and the pressure cushion. The large air gap exists because the model is assumed to be fixed in sinkage and trim, and because the stern seal is assumed to be fixed. In reality, leakage through the air gap under the stern seal will be very transient and lead to an increase in the trim and draft of the vessel, thereby closing the air gap under the stern seal and significantly modifying the hydrodynamic load coefficients and wave patterns. This is consistent with observations in past studies (Heber, 1977), where it was observed that the vessel would tend to trim bow-up with the longitudinal waves at high Froude numbers. Future studies will consider varying F r, Pc , and a 2-D vessel that is free in sinkage and trim to explore these effects. To investigate the impact of the bow seal angle on the hydrodynamic performance, simulations were performed for four values of β1 (7.5◦ , 10◦ , 12.5◦ , & 15◦ )

0.4

0.5 1.0

0.5

0.0

0.5

1.0

1.5

2.0

x/Lc

Fig. 13: Comparison of free surface profile for varying bow seal angle β1 at fixed β2 = 30◦ , P¯c = 0.6, F r = 0.8. It should be noted that only the geometry for β1 = 10◦ is shown, for clarity, and the axis scales are not equal.

The effects of cushion pressure were investigated by varying the cushion pressure for a fixed angle of attack for β1 = 10◦ at P¯c = 0.6 and 0.8 in addition to P¯c = 0.0, which corresponds to the single-plate simulation at the same flow speed. The free surface profiles for these cases are shown in Fig. 14. The results show that as the cushion pressure increases, the in-cushion free surface is depressed further and the stern seal gap is increased. This depression will generate larger-amplitude waves, which in turn increases the air leakage flow rage, CQ , under the stern seal. The wetted length also increases as the cushion pressure is increased. The amount of air leakage was calculated for these cases by integrating the air velocity over a vertical line

10

0.5

0.4

0.3

¯ One plate - P c

=0.0

¯ Two plate - P c

=0.6

¯ Two plate - P c

=0.8

0.07

¯ =0.6 ¯ =0.8 P

0.2

Pc 0.1

y/Lc

c

0.0

0.06

0.1

CQ

0.2

0.3

0.05

0.4

0.5 1.0

0.5

0.0

0.5

1.0

1.5

0.04

2.0

x/Lc

Fig. 14: Comparison of free surface profile for varying cushion pressure at fixed β1 = 10◦ , β2 = 30◦ , F r = 0.8. It should be noted that the axis scales are not equal.

below the trailing edge of the stern seal: Z Qleak = (1 − α)u · ˆi dy

7

8

9

11

10

1

12

13

14

15

[deg]

Fig. 15: Comparison of air leakage coefficient CQ as a function of β1 for P¯c = 0.6 and P¯c = 0.8 at fixed β2 = 30◦ , F r = 0.8

(16)

where Qleak is the leakage flux below the stern seal and the volume fraction α is used to isolate the air phase. The leakage coefficient is defined as:  1/2 ρa CQ = Qleak (17) 2Pc L2c 0.18

and is shown in Fig. 15. The amount of leakage is shown to increase with increasing β1 and increasing cushion pressure. It should be noted that, although the curves appear to cross, the leakage is higher for all angles of attack for P¯c = 0.8 due to the non-dimensionalization. The amount of leakage is important to quantify for an SES, since the seals must provide an adequate means of maintaining the air cushion. If the leakage is too high, the trim and draft will increase, leading to increased vessel resistance. Moreover, increased leakage will require greater fan power, and hence fuel consumption, to maintain the pressure cushion. The lift, drag, and moment coefficients for varying β1 and P¯c are shown in Fig. 16. Similar to the singleplate case, the forces and moments are found to increase with β1 . Additionally, as the cushion pressure increases, the forces and moments tend to increase as well. The pressure distributions on the bow plate are shown in Fig. 17. The stagnation point is shown to shift towards higher s/Lh (increased wetted length) as the cushion pressure increases, which is consistent with the free surface profiles shown in Fig. 14. Additionally, it can be seen that the pressure at the trailing edge increases due to the Kutta condition, requiring the pressure at the trailing edge to be equal to the cushion pressure.

P¯c =0.0 P¯c =0.6 P¯c =0.8

0.16

Cd

1

0.14 0.12 0.10 0.08 0.06 0.04 7

8

9

10

11

12

13

14

15

7

8

9

10

11

12

13

14

15

7

8

9

10

11

12

13

14

15

0.65 0.60

Cl

1

0.55 0.50 0.45 0.40 0.35 0.30

0.25

Cm

1

0.20

0.15

0.10

0.05

0.00

1

[deg]

Fig. 16: Comparison of bow plate drag, lift, and moment coefficients for varying cushion pressures, and for β1 = 10◦ , β2 = 30◦ , F r = 0.8

11


1.0

¯ =0 0 ¯ =0 6 ¯ =0 8 .

Two plate - Pc

.

[deg]

Two plate - Pc

14

1eq

.

12

10

8

0.6

10

Cd1 , Cl1 , Cm1

0.4

( 0.2

0.0 0.0

0.8

1.2

1.6

2.0

2

10

3

10

4

Cd1 Cl1 Cm1

0.5 0.4 0.3 0.2 0.1

10

1

10

2

10

3

K
1 [Nm/deg]

10

4

Fig. 18: Equilibrium angle β1eq and drag, lift, and moment coefficients on the bow seal as a function of Kβ1 for two-plate simulations for P¯c = 0.6, β2 = 30◦ , F r = 0.8

Fig. 17: Pressure distributions on the bow plate for varying cushion pressures, and for β1 = 10◦ , β2 = 30◦ , F r = 0.8

Equilibrium Solution for Bow Seal for Two-Plate Configuration

project to develop a method for calculating the FSI response of generalized SES seals. The proposed FSI methodology makes use of a metafunction to consider variations in fluid loading via a cubic interpolation. By doing so, the effects of spring/seal stiffness may be considered in post-processing, significantly reducing computational cost. This method must be validated by performing fully-coupled FSI simulations with mesh deformation. This work is currently ongoing. The effects of seal stiffness and cushion pressure on the equilibrium geometry and resultant forces and moments for both single-plate, and two-plate with air cushion representation of an SES were investigated. The results have highlighted the importance the seal stiffness and cushion pressure on the overall performance of the SES, as changes in the seal geometry due to deformation can have a large impact on the vessel sinkage and trim, cushion leakage, and total resistance. Although the current study has assumed the SES to be fixed in sinkage and trim, future work will investigate the impact of the seals on the vessel orientation by allowing the vessel to heave and pitch. The current study has investigated the SES seal problem for simple geometries in order to identify the most important physics. The insights gained have been used as a basis for extending the model to more complex configurations and geometries. In particular, it has been shown that the effects of bow seal flexibility and vessel sinkage and trim are very important for the overall performance, and consequently should not be ignored. It is expected that future studies will focus on investigating the effects of vessel rigid body motions, stern seal flexibility, cushion pressure, and Froude number on the

Once the forces and moments were calculated for the two-plate case, a similar FSI analysis was performed in order to investigate the effect of seal stiffness on the bow seal equilibrium angle and resultant forces. The equilibrium analysis was performed at P¯c = 0.6 with β2 = 30◦ and F r = 0.8 since the data are more complete. Additional equilibrium analyses will be performed for varying P¯c , β2 , and F r, and will be shown at the conference. The results for the P¯c = 0.6 case are shown in Fig. 18. Although the trends are similar, the equilibrium angle was found to be lower for a fixed Kβ1 compared to the single-plate case shown in Fig. 11. The results suggest that the bow seal undergoes a greater deformation, and hence lower β1eq , as P¯c increases due to an increase in moment created by the longer wetted length (see Fig. 16). Consequently, the resulting force and moment coefficients shown in Fig. 18 are, in general, lower than the single-plate case shown in Fig. 11 at the equilibrium angle, β1eq . It should be noted that the changes in forces and moment of the bow seal will lead to different vessel sinkage and trim, which will in turn change the seal equilibrium angle and vessel resistance, and is part of the on-going work. 6

10

0.0 0.4

s/Lh

5.4

1

0.6

p

p

atm)/0.5

w

U

2

0.8

One plate - Pc

CONCLUSIONS

This paper has presented a method for calculating the equilibrium angle of flat, planar SES seals with consideration for FSI effects. Several modeling simplifications have been made in order to systematically study the problem and to identify the relevant physics. The current work represents the preliminary work of a larger

12

seal performance, and the impact of the seals on the total vessel performance. Additionally, the work will be aimed in order to validate with upcoming SES seal experiments (Wiggins et al., 2011).

Doctors, L. J. Theory of compliant planing surfaces. Technical report, Aviation and Surface Effects Department, David W. Taylor Naval Ship Reserach and Development Center, Bethesda, Maryland 20084, June 1977.

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Doctors, L. J. On the use of pressure distributions to model the hydrodynamics of air-cushion vehicles and surface-effect ships. Naval Engineers Journal, 105 (2):69–89, March 1993.

ACKNOWLEDGEMENTS

The authors are grateful for the financial support provided by the Office of Naval Research (ONR) and Ms. Kelly Cooper (program manager) through grant numbers N00014-10-1-0170 and N00014-11-1-0833. Matthew Kramer is supported through the NDSEG fellowship program. REFERENCES

Doctors, L. J. A study of the resistance characteristics of surface-effect-ship seals. In Proceedings of the HighPerformance Marine-Vehicle Symposium, November 2009. Doctors, L. J. and McKesson, C. B. The resistance components of a surface-effect ship. In Proceedings of the 26th Symposium on Naval Hydrodynamics, September 2006.

Barratt, M. J. The wave drag of a hovercraft. Journal of Fluid Mechanics, 22:39–47, 1965. Belanger, F. and Paidoussis, M. P. Time-marchng analysis of fluid-coupled systems with large added mass. AIAA Journal, 33:752–757, 1995.

Doctors, L. J. and Sharma, S. D. Wave resistance of an air-cushion vehicle in steady and accelerated motion. Journal of Ship Research, 16(4):248–260, December 1972.

Bhushan, S., Stern, F., and Doctors, L. J. T-craft calm water resistance and motions, and seakeeping in regular waves. In Proceedings of the 11th Conference on Fast Sea Transportation (FAST ’11), September 2011.

Donnelly, D. J. and Neu, W. L. Numerical simulation of flow about a surface-effect ship. In Proceedings of the 11th Conference on Fast Sea Transportation (FAST ’11), September 2011.

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Donnelly, D. J. Numerical simulation of surface effect ship air cushion and free surface interaction. Master’s thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA, September 2010.

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Forster, C., Wall, W. A., and Ramm, E. Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows. Computational Methods in Applied Mechanical Engineering, 196:1278–1293, 2007.

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Maki, K. J., Doctors, L. J., Broglia, R., McKesson, C., and DiMascio, A. Calm-water resistance prediction of a surface-effect ship. In Proceedings of the 10th International Conference on Fast Sea Transportation (FAST ’09), Athens, Greece, October 2009.

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Mantle, P. J. Development of the USN surface effect ship, SES-100B. Naval Engineers Journal, 85(2):65– 78, October 1973. McKesson, C. and Doctors, L. J. Scaling of resistance from surface-effect ship model tests. In Proceedings of the 11th Conference on Fast Sea Transportation (FAST ’11), September 2011. Milewski, W., Connell, B., Petersen, B., and Kring, D. Initial validation of the ACVSIM model for dynamics of air cushion vehicles. In Proceedings of the 27th Symposium on Naval Hydrodynamics, October 2008. Newman, J. N. and Poole, F. A. P. The wave resistance of a moving pressure distribution in a canal. Schiffstechnik, 9(45):21–26, 1962. OpenFOAM.

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