Numerical Models in Fluid-Structure Interaction
Chapter 3
FEM on nonlinear free-surface flow
K.J. Bai1 & J.W. Kim2 1Seoul National University, Korea 2American Bureau of Shipping, USA
3.1 Introduction
Recently there has been a growing need for the treatment of the nonlinear water- wave problems. In treating the nonlinearity, there are basically two different approaches: One is based on the long-wave approximations or a shallow-water theory that gives an approximate formulation including the effect of the nonlinearity. The other is to treat the Laplace equation defined in the fluid domain with the exact nonlinear free-surface boundary conditions and the exact body-boundary condition if a body is present. The former is to treat a simplified nonlinear formulation defined only in a horizontal free-surface plane, which is one dimension less than the original fluid domain. The free-surface flow problems have been of interest to many naval architects and ocean and coastal engineers for a long time. The generation and evolution of the water waves and their interaction with the man-made structures are the main concerns in the problem. The theoretical investigations on the topics have been usually made in the scope of the potential theory by assuming the fluid is inviscid and incompressible. The most distinctive feature of this problem is the
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WIT Transactions on State of the Art in Science and Engineering, Vol 18, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) doi:10.2495/978-1-85312-837-0/03 Numerical Models in Fluid-Structure Interaction presence of the free surface. This is one of the typical free-boundary problems where the part of the boundary should be obtained as a part of the solution. Due to this complexity the theoretical investigation on the problem has been restricted until recently. In the past, the linearized problem has been mainly treated. Sometimes, the nonlinear effect is considered by treating the second-order problem obtained in a systematic perturbation expansion. Another approach, where the leading nonlinearity is included, is to use approximate theoretical models such as the well known KdV and Boussinesq equations based on the long-wave approximation or shallow-water theory. Recently, the application of the Green–Naghdi equation is also proved to be an efficient method for treating nonlinear free-surface flow problems (Ertekin et al., 1986). In this chapter we will present the finite-element method applied to the nonlinear water-wave problems, i.e., in ship motions, wave resistance, lifting problems, and initial-value problems. The present numerical scheme can be used for the validation of the existing approximate theories, i.e., the KdV and the Boussinesq equations or Green–Naghdi equation as well as for a better prediction for more realistic physical models. In Section 3.2, we will describe the finite-element method applied to a two- dimensional hydrofoil problem. In this numerical scheme we introduced a buffer zone where the forced damping is introduced to reduce the wave elevation in matching the nonlinear numerical solution with the far-field linear solution. However, in three dimensions the nonlinear wave amplitude will reduce as it propagates to infinity. Thus the introduction of the fictitious damping term is not essential. In Section 3.3, we treat an initial-value problem for a physical model of an axi-symmetric container filled with water freely falling into a flat solid surface. In this problem we included the surface tension. In the numerical procedures, the finite-element subdivisions are made such that the free-surface elevation can be described as the multi-valued functions, since the surface tension as well as the gravity is important in the flow. In Section 3.4, the generation of solitons in a shallow-water towing-tank near the critical speed as well as a numerical towing- tank simulation for arbitrary tank conditions is discussed. In Section 3.5 a preliminary result of a sloshing problem by the same finite-element method used in Section 3.4 is discussed. In Section 3.6 a nonlinear diffraction problem is discussed. In Section 3.7, a brief concluding remark is given.
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3.2 Nonlinear steady waves due to a two-dimensional hydrofoil
An application is described of the localized finite-element method to a steady nonlinear free-surface flow past a submerged two-dimensional hydrofoil at an arbitrary angle of attack. The earlier investigations with the linear free-surface boundary condition in Bai (1978) have shown some disagreement between the computed results and the experimental measurements for the cases of shallow submergence. The computational method of solution is the localized finite- element method based on the classical Hamilton’s principle. In the present study, a notable step is introduced in the matching procedure between the fully nonlinear and the linear sub-domains. Additional numerical results of wave resistance, lift force, and circulation strength are presented. Details can be found in Bai and Han (1994). It should be noted that in the earlier treatment of the linear hydrofoil problems, the velocity-potential formulation was used. However, in the following treatment of the nonlinear problem, a stream-function formulation is used since it has a distinct advantage in the numerical treatment.
3.2.1 Mathematical formulation We consider here a steady uniform flow past a fixed two-dimensional hydrofoil submerged in a fluid. The coordinate system is right-handed and rectangular as shown in Fig. 3.1. The y-axis is directed opposite to the force of gravity, and the x-axis coincides with the undisturbed free surface. The unit normal vector nr is always directed outward from the fluid domain. We neglect surface tension and assume that the fluid is inviscid, incompressible, and that the motions are irrotational. Here we assume the water depth, H, is constant. The steady two-dimensional flow is described by a total stream function
Ψ(x, y) = Uy +ψ (x, y) , (3.1) where U is the incoming uniform flow velocity in the upstream and ψ (x, y) is the perturbation stream function. The total stream function Ψ of the incoming uniform flow is set to 0 on the undisturbed free surface and −UH on the bottom. The perturbation stream function ψ must satisfy
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y
S F 0 x
S U 0
H c S TE S R1 R2 D
S B
Figure 3.1: Sketch of boundary configurations.
∇2ψ (x, y) = 0 (3.2) in the fluid domain D. The kinematic and dynamic boundary conditions on the free surface SF are as follows:
ψ = −Uη or Uη +ψ = 0 x x 1 on S , y =η(x) , (3.3a,b) Uψ + gη + ()∇ψ ⋅∇ψ = 0 F y 2 where y =η(x) and g are the free-surface elevation and the gravitational acceleration, respectively. The boundary condition on the hydrofoil surface S0 can be
Ψ(x, y) = C on S0 , (3.4) where the constant C will be determined as a part of the solution. To determine the constant C, we require at the trailing edge denoted by TE an additional condition, i.e., the Kutta condition, which states that the pressure on the upper and lower surfaces is continuous. Therefore, the tangential velocities on the upper and lower surfaces have the same magnitude at the trailing edge and bounded, that is,
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∂ ∂ ()Uy +ψ + = (Uy +ψ − ) at TE, (3.5) ∂n ∂n where ∂ / ∂n is the normal derivative and ψ + and ψ − are the perturbation stream function ψ on the upper and lower surfaces, respectively. The boundary condition on the bottom, SB , i.e., y = −H is
ψ = 0 on SB . (3.6)
y
x x x 1 S 2 3 F 0 x
S R1 S U 0 S H J J R2 c TE 2 3 J 1 D 1.1 D N DB D1.2
SB
Figure 3.2: Subdomains in nonlinear calculation.
As the radiation condition, we require that no disturbances exist far upstream, that is,
∇ψ → 0 as x → −∞ , (3.7) and that the perturbed flow velocity is bounded in the far downstream, i.e.,
∇ψ is bounded as x → +∞ . (3.8)
We hereby assume that the solution of eqns (3.2~3.8) can be determined uniquely.
3.2.2 Variational principle A mathematical formulation of the nonlinear free-surface flow phenomena of an incompressible ideal fluid can be described by variational principles based on the
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WIT Transactions on State of the Art in Science and Engineering, Vol 18, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Numerical Models in Fluid-Structure Interaction classical Hamilton’s principle as given in Kim and Bai (1990). Therefore, the formulation given in the previous section can be replaced by an equivalent variational functional formulation as follows. Now the nonlinear problem defined by eqns (3.2–3.8) is replaced by an equivalent Hamilton’s principle for the Lagrangian. The Lagrangian L is defined as L = T – V + W, (3.9) where T, V, and W are the kinetic energy, the potential energy, and the work done by external pressure, respectively. Then the Lagrangian L for the above nonlinear boundary-value problem can be defined as 1 g L[ψ ,η] = ∫∫ ∇ψ ⋅ ∇ψ dxdy − ∫ η 2dx , (3.10) 2 D 2 SF with the kinematic constraints eqns (3.3a), (3.4–3.6). SF denotes the projection of the free surface on the y=0 plane. Equation (3.10) can also be written in the following form by using the kinematic constraint of eqn (3.3a),
1 −ψ (x,η) U g L[ψ ,η] = ∇ψ ⋅∇ψ dxdy − ψ 2 (x,η)dx . (3.11) 2 2 ∫∫−H 2U ∫ SF
In steady nonlinear free-surface problems, the numerical treatment of the radiation condition (3.7) and (3.8) is one of the most difficult obstacles. In linear problems, the radiation condition has been effectively treated by the localized finite-element method, where the numerical solutions in the computational domain were matched to the complete set of the analytic solutions known in truncated sub-domains (Bai and Yeung 1974, Bai 1977, 1978). As an extension of the application of this method to nonlinear problems, a modified variational form is developed in which the degree of the free-surface nonlinearity is artificially reduced to that of the linear problem, by introducing a nonlinear-to- linear transition buffer subdomain between the fully nonlinear computational subdomain and the truncated linear infinite subdomain. This modified variational form was successfully applied to a two-dimensional wave-resistance problem for a nonlifting body in the stream-function formulation in Bai et al (1990).
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The entire fluid domain D is subdivided into four subdomains as shown in
Fig. 3.2. We denote the two linear infinite subdomains by DL1 in the upstream and by DL2 in the downstream, respectively. The solutions in these linear subdomains can be represented by the known complete set of analytic solutions in the same way as in linear problems. The nonlinear subdomain and the nonlinear-to-linear transition buffer subdomain are denoted by DN and DB, respectively. The juncture boundary between the upstream linear subdomain and the nonlinear subdomain is denoted by J1 . The juncture boundary between the downstream linear subdomain and the transition buffer subdomain is denoted by
J3 . The two intersection points, i.e., the x-coordinates of J1 and J3, are denoted by x = x1 and x = x3 , respectively. The juncture boundary between the nonlinear subdomain and the transition buffer subdomain is denoted by J2 with its x - coordinate x = x2 . In the nonlinear-to-linear transition buffer subdomain DB , a modification has been made on the Lagrangian and the free-surface boundary conditions in the following way. By introducing a locally linearizing parameter, ε(x) , in the transition buffer subdomain DB , the Lagrangian L given in eqn (3.11) can be rewritten as
1 x3 −ε (x)ψ (x,η) U g L[ψ ,η]= ∇ψ ⋅∇ψ dxdy − ψ 2 (x,η)dx , (3.12) 2 2 ∫x1 ∫−H 2U ∫SF with the constraints of eqns (3.4–3.6). In the fully nonlinear subdomain, DN , ε(x) equals 1 and in the two linear subdomains, DL1 and DL2 , ε(x) equals 0, whereas the value of ε(x) is gradually reduced from 1 to 0 in the nonlinear-to- linear transition buffer subdomain in the downstream. There is no need to introduce such a nonlinear-to-linear transition buffer region in the upstream, since the upstream free-surface elevation decays very rapidly. It should be noted that the generations of upstream solutions are absent in the range of the Froude numbers treated here. One can find the application of the present method to this problem in Bai et al. (1989) and Choi et al. (1990). The new Lagrangian defined in eqn (3.12) is equivalent to the original nonlinear formulation given in eqns (3.2), and (3.4–3.8). However, the following modified free-surface boundary conditions, which replace eqns (3.3a) and (3.3b), are obtained by the new Lagrangian,
Uη +ψ =0 on y =ε(x)η(x) (3.13a)
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ε(x) 2 Uψ + gη + ∇ψ =0 on y =ε(x)η(x) , (3.13b) y 2 where 0 ≤ ε(x) ≤1 . It should be noted that the nonlinear problem reduces to a linearized free- surface problem locally in the neighborhoods of the two juncture boundaries, i.e.,
J1 and J3 , where the two complete sets of analytic solutions known in DL1 and DL2 are valid. As shown in Kim and Bai (1990), it is easy to show that the Lagrangian formulation given in eqn (3.12) is identical to the bilinear functional formulation in linear problems, and the free-surface conditions given in eqns (3.13a) and (3.13b) can be combined to obtain the linearized free-surface condition. Accordingly, the matching procedure is the same as in the localized finite-element method of Bai (1975, 1977, 1978). However, in the nonlinear subdomain DN , the modified Lagrangian defined in eqn (3.12) recovers the original Lagrangian given in eqn (3.11). For finite-element approximations, the Lagrangian L in eqn (3.12) with the constraints of eqns (3.4–3.6) goes into an operational form in the following way. The m-dimensional subspace of the admissible function space is chosen and let Ni (i = 1,2,....,m) be the basis for the m -dimensional subspace. Then the solution is assumed to be
m ψ (x, y) = ∑ψ i Ni (x, y) , (3.14) i=1
where ψ i are coefficients to be determined in eqn (3.14) on the free surface,
mF ψ (x) = ψ M (x) ; M (x) = N (x, y) (k =1,....,m ) , (3.15) ∑ k k k ik y=η F k =1
where mF is the number of nodes on the free surface, ik is the node number of the basis function Ni that correspond to the k -th node on the free surface. By substituting eqns (3.14) and (3.15), the Lagrangian L in eqn (3.12) finally reduces to
* 1 g L = ψ i Kijψ j − ψ kTklψ l , (3.16) 2 2U 2
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where Kij = ∇Ni ⋅∇N jdxdy ; Tkl = M K M l ds . ∫∫D ∫S F
From Hamilton’s principle, eqn (3.16) must satisfy the following equations,
∂L* = 0 ∂ψ i . (3.17) ∂L* = 0 ∂ψ k
The final nonlinear matrix equation in eqn (3.17) for the computational domains, i.e., DN and DB , can be solved by an iterative procedure of Newton's method. In the procedure of imposing the appropriate upstream and downstream radiation conditions, asymmetric sets of test and trial functions are taken from and the bilinear functional. The detail procedures in the treatments of the radiation condition and the Kutta condition can be found in Bai and Han (1994). After the stream function has been obtained, the pressure can be computed by Bernoulli’s equation ρ P = −ρUψ − (ψ 2 +ψ 2 ) , (3.18) y 2 x y where the hydrostatic pressure has been omitted. Then the nondimensional pressure coefficient CP is defined as
2 P ∇ψ CP = =1− , (3.19) 1 2ρU 2 U 2
and CP is unity at the stagnation point.
3.2.3 Numerical results Computations are made for the flow around a hydrofoil with chord length c in water of finite depth H. The depth submergence h is measured from the undisturbed free surface, y = 0, to the trailing edge as shown in Fig. 3.3. Throughout the computations, we choose the 12% thick symmetric Joukowski hydrofoil at 5-deg angle of attack, which was previously investigated in the
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Froude number FUc = / gcHchc , /, /, where U is the incoming uniform flow velocity. In presenting the pressure distributions on the hydrofoil and the wave profiles on the free surface, the coordinates x (shown in Fig. 3.3) and the free-surface elevation are nondimensionalized by the chord length c such as x / c, / c , and / c , respectively. y
0 x
c
U ⌧ H TE
Figure 3.3: Schematic diagram of a hydrofoil.
The computations are made for the case when a hydrofoil is submerged near to the free surface to find the effect of the nonlinearity in the free-surface condition. The pressure distributions on the hydrofoil are computed and compared with experimental results by Parkin et al. (1956) as well as with the linear results of Bai (1978). The wave profiles on the free surface are also computed and compared with the linear results. The results of the wave profiles are shown not only in the fully nonlinear subdomain but also in the nonlinear-to- linear transition buffer subdomain in order to show the entire computed numerical results by the present numerical scheme. However, the computed profiles in the neighborhood of the nonlinear-to-linear transition buffer subdomain should be discarded since we introduced an artificial linearization for a smooth matching in the region.
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The computations are made for several values of Froude numbers and the depths of submergence. Figure 3.4 shows the result for a Froude number of 0.989 with a depth of submergence, h c = 0.2. The pressure distributions on the hydrofoil are shown in Fig. 3.4(a); there is remarkable agreement with the experimental results compared with the linear result. In Fig. 3.4(b) the nonlinear and the linear wave profiles are also compared. According to Parkin et al., the wave profile at this relatively high Froude number has a smooth shape. In the numerical computations, the convergence of the iteration process is fairly good.
Figure 3.4(a): Pressure distribution ( Fc = 0.989, H / c = 6,h / c = 0.2 ).
3.3 Axi-symmetric transient problem
In this section the time-dependent motion of a fluid in a vertical circular cylindrical container with a free surface subject to an impact is discussed. The free surface abruptly rises to a very high level just after this impact. Therefore it seems to be necessary to apply the nonlinear free-surface condition for the problem. This problem is originally investigated by Milgram (1969), who performed a series of experiments and linear analysis for a vertical impact due to
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Figure 3.4(b): Wave profiles on free surface
( FHchcc ====−0.989, / 6, / 0.2,buffer 15 19 ).
In the present computations we introduced an automatic mesh generation at some appropriate time to trace the free surface that changes very much. We also introduced splines along which the nodes at the free surface moves up and down. By this method we can represent a multi-valued surface elevation caused by the surface-tension force.
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3.3.1 Physical model and assumptions
v = 0 . v = - g v = - 2gh v = 0
h
t = t t = 0 t > 0 0
Figure 3.5: Perfect plastic collision of the container.
If a circular container filled with water or some other fluid falls down on the ground as in Fig. 3.5, the fluid begins to move abruptly due to the impact and accordingly the free surface starts its motion. For a mathematical model of this physical problem we introduce the following assumptions. The fluid is assumed to be inviscid and incompressible and the motion of the fluid is assumed to be irrotational. A perfect inelastic collision between the container and the ground is also assumed, i.e., the container is stationary after the collision. If the container begins to fall at the height h with its initial velocity zero, its speed just before the collision will be 2gh , where g is the gravitational acceleration. We assume that the shape of a free surface before the collision is in equilibrium with the surface-tension force for simplicity, since the domain is axisymmetric, it is convenient to use the cylindrical coordinate system. The body-fixed coordinate system Orz is defined in Fig. 3.6. The z-axis points upward opposite to the gravity force at the center of the container and the r-axis is attached to the bottom of it. The radius of the container is R, and the contact angle between the solid wall and the free surface is θ0 , which is assumed to be constant. Because the solution will be independent of θ , the numerical calculation may be performed on the half-plane ( r > 0 ) of the vertical cross section D of the container. The unit normal vector nr points outward on the boundary and τr denotes the unit tangential vector.
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z
n S F 0
S W D
0 R r Figure 3.6: Definition sketch.
3.3.2 Mathematical model The velocity field of an inviscid and incompressible fluid can be described by a scalar field if the motion of the fluid is irrotational. The scalar function, which is the velocity potential function φ in this case, must satisfy the Laplace equation at all points in the domain D ,
∇2φ = 0 in domain D . (3.20)
Because the normal velocities on the solid wall, SW , vanish, we can obtain the following condition, ∂φ = 0 on solid wall, S . (3.21) ∂n W The kinematic condition of the free surface is as follows,
∂φ = V on free surface S , (3.22) ∂n n F
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where Vn is the normal velocity of the free surface. The dynamic condition on the free surface can be obtained from the Bernoulli equation.
∂φ 1 1 1 1 + ∇φ ⋅∇φ + ()g + v&(t) z = Pa −σ + on free surface, SF . (3.23) ∂t 2 ρ R1 R2
Here, Pa is an atmospheric pressure on the surface, ρ is the density of the fluid and σ is the surface-tension coefficient. R1 and R2 are principle radii of curvatures of the free surface. If we suppose that the fluid is in equilibrium just before the impact, the velocity potential at the free surface may arbitrarily be set to zero. The velocity potential at the free surface just after the impact can be obtained by integrating eqn (3.23) over the infinitesimal time interval during the impact, i.e., 0+ φ = − v(t)dt z = − 2gh z on S , (3.24) t=0+ ∫ & F 0− because the physical quantities, other than the vertical velocity of the container, are continuous at t=0.
3.3.3 Method of solution The initial- and boundary-value problem given in eqns (3.20) through (3.24) is a free-boundary problem. For a given time t , the shape of the fluid domain as well as the velocity potential given in the domain should be obtained. In this problem the free-surface elevation and the velocity potential on it are the canonical variable for the time evolution (Mi1es, 1977). The surface elevation and the velocity potential on it are evolved in time by the free-surface conditions, eqns (3.22) and (3.23). Then the remaining boundary conditions and the governing equation constitute a boundary-value problem from which the velocity potential in the fluid domain can be obtained. For a numerical computation, as a result, it is important to specify how to represent the free surface and to trace the velocity potential on it. In this section two methods representing the free surface are presented. One is for a single-valued free surface and the other is for a multi- valued free surface. Since the first is a special case of the second, we present the general case of the mu1ti-valued free surface. For a given time tt= , the r 0 position of the free surface can be represented by a curve (,rz )= X0 (), s where s is a parameter such as nodal number. For an interval of time, say (t0 ,t1) , we
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WIT Transactions on State of the Art in Science and Engineering, Vol 18, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Numerical Models in Fluid-Structure Interaction specify the direction of the surface evolution as er(s) . Then the position vector of the free surface at time t is given as r r (r, z) = X (s,t) = X 0 (s) + h(s,t)e(s) , t0 < t < t1 , (3.25) so that the velocity of the free surface can be represented by a scalar function h(s,t) . We also define the velocity potential on the free surface as
φ (s,t) = φ(X (s,t),t). (3.26)
Then the evolution equations of the two canonical variable h(s,t) and φ (s,t) are given as 1 ∂φ h = (3.27.a) t e ∂n
∂φ P σ 1 1 1 a = − + + + G − gz , (3.27.b) ∂t ρ ρ R1 R2 2 where r r r r en = e ⋅n , eτ = e ⋅τ (3.28.a)
2 ∂φ ∂φ 2 G = ()hten + 2hteτ − . (3.28.b) ∂τ ∂τ
In the case of the multi-valued free surface, the direction vector er(s) is taken as the unit normal vector at t = t0 . For a single-valued tee surface er(s) is taken simply as the unit upward vector. The velocity potential in the fluid domain is determined from a boundary -value problem,
∇2φ = 0 in domain D (3.29.a)
φ = φ on free surface SF (3.29.b)
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∂φ = 0 on solid wall S . (3.29.c) ∂n W
3.3.3.1 Weak formulation The boundary value problem given in eqns (3.29a,b,c) can be equivalently formulated by the following variational problem. Find a function φ(r, z) ∈ H such that
∫∫∇ψ ⋅ ∇φ r drdz = 0 (3.30.a) D for all test function ψ (,rz )∈ H, with constraints
φ = φ and ψ = 0 on SF . (3.30.b) Here, H is the tiral and test function space whose elements have square integrable functions and their derivatives in D. The normal derivative of the velocity potential on the free surface can be calculated by the right-hand side of eqn (3.31) by substituting the weak solution for eqn (3.30),
∂φ ψ r dl = ∇ψ ⋅∇φ r drdz (3.31) ∫∫∫∂n SDF for all ψ (r, z)∈ H . It should be noted that the present boundary-value problem might also be replaced by an equivalent form by a well-known classical quadratic variational functional that is very similar to the present weak formulation.
3.3.3.2 Discretization The foregoing variational problem is approximated by the finite-element method. The fluid domain is approximated by a polygon with a finite number of vortices. On the free surface, the direction vectors esr() are defined piece-wise as shown in Fig. 3.7. Then the domain is discretized by triangular meshes. For a single-valued free surface, the meshes can be simply generated as described in Bai et al. (1989). In the case of a multi-valued free surface, an automatic mesh-generation scheme is developed here. This scheme will be described in the next section.
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ni- 1 e i n i ei- 1 i- 1 e i+ 1 i
B Xi XB i- 1 B Xi+1
0 Figure 3.7: Discretization of the free surface.
To solve the evolution equations, (3.27.a) and (3.27.b), it is required to evaluate ∂φ from the given values of the velocity potential on the free surface. The ∂n normal derivative of the velocity potential can be obtained from the following procedures. For a given fluid domain at a fixed time, t , the approximate solution may be expressed as a linear combination of the finite-element basis functions, i.e.,
N N φ(r, z) = ∑φ j N j (r, z) , (3.32) j=1 where N N is the number of nodes of the finite elements and {N j (r, z), j = 1,L, N N } is a piece-wise bilinear basis function whose nodal values are unity at node j and zero at the other nodes. If we make the numbering of the nodes such that the first N F nodes are on the free surface, the test functions are given as
ψ i (r, z) = Ni (r, z), i = N F +1,K, N N . (3.33)
Substituting eqns (3.32) and (3.33) into the variational problem, eqns (3.30.a,b), we obtain the following algebraic equations.
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φ = φ , i =1,K, N F (3.34.a)
N N ∑ Kijφ j = 0 , i = N F +1, N N , (3.34.b) j=1 where
Kij = ∫∫∇Ni ⋅ ∇N j r drdz , D and {}φ , i =1,K, N F is the nodal values of φ , obtained from the previous time step. Then we approximate the normal derivative of the velocity potential on the free surface as N ∂φ F = ∑ϕ j N j (r, z) on S F . (3.35) ∂n j=1 Then the variational equation, (3.31), can be written as
NF ∑ M ijϕ j , i = 1,K, N F ; M ij = ∫ Ni N J r drdz . (3.36) j=1 SF The two principal radii of curvature, which are required to evaluate the surface-tension terms given in eqn (3.8.b), are approximated as follows. The first curvature 1/ R1 at the i-th node of the discrete free surface can be approximated as 1 2 = sign × (τˆi −τˆi−1) , (3.37) R1 li−1 + li where li is the length of the i-th element at the discrete free surface and τˆi is the unit tangential vector of the i-th free surface element. The sign convention of the curvature is defined in Fig. 3.8.
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WIT Transactions on State of the Art in Science and Engineering, Vol 18, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Numerical Models in Fluid-Structure Interaction
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