Liquid Impact on a Permeable Solid Body

Transcription

1 Liquid Impact on a Permeable Solid Bod Yuri A. Semenov a, Guo Xiong Wu a a Department of Mechanical Engineering, Universit College London, Torrington Place, London WCE 7JE, UK Abstract The free surface flow and the hdrodnamic loads generated b impact between a liquid wedge and a permeable solid bod are investigated. The stud is carried out within the framework of self-similar solution, which is realistic for this kind of configuration and over the short period of impact. We stud the effect of liquid penetration through the porous/perforated solid surface on the pressure distribution and flow pattern. An integral hodograph method is emploed to convert the differential equation in the fluid domain into integral equations along the aes of a parameter plane, from which the problem corresponding to the impermeable solid surface is a special case. The sstem of integral equations are solved numericall using the method of successive approimations. The results are presented for streamline patterns, pressure distribution along the solid surface of permeable wedges. Ke words: Liquid/solid impact; Permeable bod ; Self-similar flow; Integral hodograph method.. Introduction Liquid/structure impact is a widel observed natural phenomenon. Eamples include wave impacts on marine structures and coastline, slamming of ships, landing of aircrafts on a water surface, motion of planing crafts and droplet impact on aircrafts. High speed liquid impacts can generate ver high loading addresses: semenov@a-teleport.com (Yuri A. Semenov), g.wu@ucl.ac.uk (Guo Xiong Wu) Preprint submitted to Elsevier June 8, 5

2 on a solid bod that ma cause structural damage or failure. Perforated and porous solid structures have been proposed as a wa to reduce hdrodnamic loading during wave impacts []. A perforated bod allows liquid penetration though its surface, which decreases the hdrodnamiressure. Eamples of perforated structures include wave breakers and stabilizer of the Roseau tower which consists of an open-ended square bo attached to the frame of the tower subjected to wave impacts. Another eample of a perforated structure is a tubular frame, usuall used as a protection for subsea installations on the seabed. Arras of renewable energ devises or their elements in some cases also can be considered as a perforated structure for its interaction with incoming waves. Review of practical applications of perforated structures in offshore and coastal engineering as well as methods predicting hdrodnamic forces was presented b Molin []. In order to understand better the fluid structure interaction of the processes mentioned above and provide an effective wa to predict the wave loadings, it is necessar to account for permeabilit of the structure in the wave impact processes. Impacts of interest usuall lasts for a ver short period of time, during which the pressure and fluid velocit var rapidl both with time and in space. In general, the process is full transient and the temporal and spatial variables are full independent. One wa to solve this kind of problem is based on the Wanger s theor together with the technique of matched asmptotic epansions, especiall in the contet of marine applications [, 3, 4]. However, in man cases, especiall at initial stage and in some local areas, the flow ma be treated as self similar. Moreover, at this stage the hdrodnamic force ma reach its maima value that is important to estimate safet and reliabilit at the design stage. In the studies mentioned above, the bodies in the impact are usuall considered as rigid and impermeable. However there are publications, notabl those b Molin and Korobkin [5], Iafrati and Korobkin [6], and Iafrati et al. [7], in which the effect of permeabilit on the hdrodnamic force during wave impacts is investigated. In this stud we will obtain analtical/numerical solution for self similar flow to stud an effect of permeabilit of a solid structure on h-

3 drodnamic loads which are important for assessing safet and reliabilit of the structure. The Integral hodograph method [8, 9] which is an etension of the classical hodograph method is applied. The method enables the original partial differential equation with nonlinear boundar conditions to be converted into a set of integral equations along the straight lines, which are then solved numericall. It has been successfull used in variet of water impact problems [8,, ]. However the application of the method to the present problem has some new major challenges. On the impermeable solid surface, the normal velocit is unknown a priori and depends on the pressure which has to be determined from the solution. The boundar condition is given in terms of a relationship between the pressure and the normal component of the velocit trough the bod surface which is satisfied under iteration procedure. Various case studies related to the permeabilit of the solid bod are presented. Both the solid bod and the wave front are considered in the form of a wedge of various angles. It includes the cases of impact of the wave crest on to the perforated wall as well as water entr of the perforated wedge into half - space of the liquid. The terms porous and perforated solid surfaces are used to distinguish the difference in the boundar condition for penetration of the liquid through the bod surface. In the first one a linear relationship between the pressure and normal velocit through non deforming bod boundar is emploed, that corresponds to a porous bod when the diameter of holes are much smaller that its length and the holes are finel distributed in the bod [6]. In the second case, a perforated bod is considered, for which the quadratic relationship between the pressure and the normal velocit trough the non deforming bod surface is used [5]. These two laws of the liquid penetration are related to the applications in coastal and offshore engineering [, 5, ] where a massive of wave breakers or a cover subsea installations, forms the porous/perforated structure, used to reduce the fluid impact on a structure. To outline briefl, the present stud generalizes the previous integral form of solution [8, ] for liquid impacting onto impermeable rigid bodies to permeable ones. Section describes the mathematical formulation and the solution proce- 3

4 a) O z=+i v np A A n s b) B i = + i v d C B B O D A C Figure : Sketch of the problem for impact between a liquid wedge (dotted line at the time of impact) and an permeable solid wedge: (a) similarit plane and (b) parameter plane. dure based on the integral hodograph method which reduces the problem to a sstem of integral equations. The numerical results, their analsis and phsic implications based on the flow patterns, pressure distribution and free surface, as well as the flu into the permeable bod are discussed in Section 3.. Formulation of the problem and the solution procedure We consider the impact problem between a liquid and solid permeable wedge of half-angle α and α A, respectivel. The liquid is assumed to be ideal and incompressible, the flow to be irrotational, and the incoming velocit is constant. The gravit and surface tension effects are ignored. The flow is self similar and will be studied in the frame of reference with its origin attached to the solid wedge. A sketch of the problem is shown in figure. The Cartesian coordinate sstem with origin at point A is defined with ais along the line of the flow smmetr. The liquid wedge has uniform velocit V along the -ais, which is marked as v in the similarit plane in figure a. 4

5 The fluid particle at point A when t = coincides with point O. After impact, a jet with a tip of angle µ moves to point O along the permeable wedge side. The smbol v n in the figure is the normal velocit due to bod surface permeabilit, and it tends to zero at the contact point O, where the pressure tends to ambient pressure. For a constant impact velocit of the liquid wedge, the time-dependent problem in the phsical comple plane Z = X + iy can be written in the stationar similarit plane z = + i in terms of the self-similar variables = X/(V t), = Y/(V t), where t is time. As a result the inflow velocit at infinit v = at points B and C in figure a. The comple velocit potential W (Z, t) for the self-similar flow can be written as W (Z, t) = V tw(z) = V t[ϕ(, ) + iψ(, )] () The problem is to determine the function w(z) which conformall maps the similarit plane z onto the comple potential region w. We choose the first quadrant of the plane in figure b as the parameter region to derive epressions for the non dimensional comple velocit, dw/dz, and for the derivative of the comple potential,dw/dζ, both as functions of the variable ζ. Once these functions are found, the velocit field and the relation between the parameter region and the phsical flow region can be determined as follows: v iv = dw (ζ), z(ζ) = z() + dz ζ dw dζ /dw dz dζ, () where v and v are the - and -components of the velocit nondimensionalized b V. Conformal mapping allows us to fi three arbitrar points in the parameter region, which are chosen O, B (a point at infinit) and A as shown in figure b. In this plane, the positive part of the imaginar ais ( < η <, ξ = ) corresponds to the free surface OB. The interval ( < ξ <, η = ) of the real ais corresponds to the wetted part of the wedge, and the rest of the positive real ais ( < ξ <, η = ) corresponds to the smmetr line AC. The point ζ = in figure b is the image of the point A from the similarit plane ζ. 5

6 .. Epressions for the comple velocit and the derivative of the comple potential, dw/dζ. When the bod surface is an impermeable rigid, the normal component of the velocit relative to the bod surface equals zero, and therefore the velocit on the bod surface is directed along its tangential direction. In the present problem we consider a permeable bod surface, where the normal component of the velocit is nonzero and the direction of the velocit on the bod surface becomes unknown. It presents an additional challenge to the solution procedure. The boundar-value problem for the comple velocit function is formulated in the parameter plane. At this stage we introduce function β(ξ) = arg (dw/dz) along the interface OA, i.e. on the interval < ξ < of the real ais of the parameter plane, and function v(η) which is the velocit modulus along the free surface OB, or along the positive part of the imaginar ais of the ζ - plane. With these notations, we have β(ξ), < ξ <, η =, χ(ξ) = arg(dw/dz) = π/, < ξ <, η =. (3) v(η) = dw dz, < η <, ξ =. (4) When we approach point A along the interface OA, the velocit direction β(ξ) = tan (v /v ) ξ ε, ε, tends to the value π/, or coincides with the direction of the -ais, since v (ξ) ξ ε due to the flow smmetr, and v (ξ) ξ ε > for an small permeabilit of the interface. For the impermeable interface there is a jump in the function β(ξ) at point ξ =, since the velocit direction coincides with the direction of the interface, or β(ξ) = tan (v /v ) ξ ε π/ α, ε. The formulation of the problem corresponds to the permeable bod surface, and the case of the impermeable bod surface is a particular one for which the liquid flowrate through the bod tends to zero. Thus, for both cases the function χ(ξ) in Eq.(3) is continuous function at point A and along the whole real ais. 6

7 We can confirm that the integral formula [8, 9] for F (ζ) = dw/dz below F (ζ) = v ep ( ) dχ ζ + ξ π dξ ln dξ i ( ) d ln v ζ iη ζ ξ π dη ln dη + iχ, ζ + iη satisfies the given conditions on the real and imaginar aes of the first quadrant in Eqs.(3) and (4), or: χ(ξ) = arg[f (ζ)] ζ=ξ, < ξ <, η =, and v(η) = F (ζ) ζ=iη, < η <, where v = v(η) η, χ = χ(ξ) ξ. Using the known value in the first line of Eq.(3), Eq.(5) becomes dw dz = v ep ( ) dβ ξ ζ dξ ln dξ i ξ + ζ π d ln v dη (5) ( ) iη ζ ln dη iβ. (6) iη + ζ where v = v(η) η= and β = β(ξ) ξ= = π/ α A are the velocit magnitude and direction at point A. The functions β(ξ) and v(η) will be determined later from the free surface kinematic and dnamic boundar conditions. In order to obtain epression for the derivative of the comple potential in the parameter plane, dw/dζ, it is useful to introduce the unit vectors n and τ on the fluid boundar, which are normal and tangent to the surface, respectivel. The former is directed outward from the liquid region, and while moving in the latter direction along the boundar, the spatial coordinate s increases and the liquid region is on the left hand side (see figure a). With this notation, we have dw = (v s + iv n )ds, (7) where v s and v n are the tangential and normal velocit components along the flow boundar, respectivel. Let θ(η) = tan (v n /v s ) and γ(ξ) = tan (v n /v s ) denote the angles between the velocit vector with τ on the flow boundar. The former is defined along the imaginar ais of the parameter and therefore corresponds to the free surface OB in the similarit plane, while the latter is on the real ais and corresponds to the interface and the smmetr line. Eq.(7) allows us to determine the argument of the derivative of the comple potential, 7

8 v n A + B B i O - A - C A + O + v s O A C Figure : (a) Behaviour of the velocit angle to the flow boundar, tan (v n /v s ): the solid lines for the continuous changes while the dashed lines for the step changes. (b) The corresponding variation in the parameter region. ϑ = arg(dw/dζ): ϑ(ζ) = arg ( ) dw = arg dζ ( ) dw +arg ds ( ) ds = dζ γ(ξ), < ξ <, η =, θ(η) + π/, ξ =, < η <. (8) B analzing the behaviour of the velocit angle along the whole flow boundar in figure, we can see the variation of tan (v n /v s ). It is continuous along the free surface OB (defined as θ(η)) and on the permeable surface OA (defined as γ(ξ)) respectivel, as shown b solid lines. The function θ(η) has step changes or discontinuities at points O, B and C, while for the function γ(ξ) the discontinuit occurs at point A. These step changes are shown b dashed lines in figure a. The ma lead to singularities in the epression for the derivative of the comple potential. Now we determine the functions θ(η) and γ(ξ) along the fluid boundar on which the are defined, that is, along the positive parts of the imaginar and real aes of the ζ-plane, respectivel. Between points C and A, < ξ <, function γ(ξ) π, since v n = and v s <. When we move in counter clockwise direction along an infinitesimal semicircle centred at the point ζ = in the parameter plane, the function γ(ξ) = tan, changes from π at A 8

9 to π α A at A +. This is because velocit components v and v >, which gives v s = v cos(α A ) and v n = v sin(α A ). Thus, the jump in the function γ(ξ) at the point A is A = α A. When we move from point A + to point O along the wedge surface, the function γ(ξ) changes continuousl from γ A = π α to γ = pi at point O. When we move in counter clockwise direction along an infinitesimal quarter of the circle centred at the point ζ = in the parameter plane the corresponding line in the phsical plane passes through the tip O of the tip jet. The jump in the function tan (v n /v s ) equals = µ π as it is seen from figure, where µ = θ(η) η =. B taking into account Eq.(8) we can see that the jump in arg(dw/dζ) equals ϑ = + π/ = µ π/. The corresponding change of the argument arg(ζ) equals π/, and so we can epect that function dw/dζ at point O (ζ = ) has a singularit of order dw/dζ ζ ϑ/π. When moving from point O to point B along the imaginar ais of the parameter plane, the function θ(η) changes continuousl from value θ to the value θ B = θ(η) η. We can write function ϑ(ζ) as follows ( ) π, < ξ <, η =, dw ϑ(ζ) = arg = π + dζ A + γ(ξ), < ξ <, η =, θ(η) + ϑ, ξ =, < η <. The problem is then to find the function dw/dζ in the first quadrant of the parameter plane which satisfies the boundar condition (9). This is a homogeneous boundar value problem, or arg(dw/dζ) is given on the entire boundar. It can then be confirmed that the function obtained from the following integral formula [8, 9] G(ζ) = K ep π dϑ dξ ln ( ζ ξ ) dξ + π dϑ dη ln ( ζ + η ) dη + iϑ, (9) () where K is a real factor, ϑ(ζ) = arg[g(ϑ)], < ξ <, η = and < η <, ξ =, ϑ = ϑ(ζ) ζ can meet the condition in Eq.(9). B performing the integration in Eq.() over steps where the value ϑ(ζ) is known, we finall obtain 9

10 the epression for the derivative of the comple potential in the ζ-plane as dw dζ = K ζµ/π ep dγ ( ζ ) α/π π dξ ln ( ξ ζ ) dξ + dθ π dη ln ( ζ + η ) dη. () Integration of Eq.() in the parameter region allows us to obtain the function w(ζ) which conformall maps the parameter region onto the corresponding region in the comple potential plane: ζ ζ µ/π w(ζ) = w A +K ep ( ζ ) α/π π dγ dξ ln ( ξ ζ ) dξ + π dθ dη ln ( ζ + η ) dη dζ, () where w A is the comple potential at point A and can be taken as zero without loss of the generalit. Dividing () b (6), we can obtain the derivative of the mapping function dz = K ζ µ/π ep dγ dζ v ( ζ ) α/π π dξ ln ( ξ ζ ) dξ + dθ π dη ln ( η + ζ ) dη π ( ) dβ ξ ζ dξ ln dξ + i ξ + ζ π d ln v dη ( ) η ζ ln dη + iβ (3) η + ζ The integration of this equation ields the mapping function z = z(ζ) relating the parameter and similarit planes. We notice that in contrast to the impact between the liquid and impermeable solid wedges [], the functions β(ξ) and γ(ξ) characterizing respectivel the direction of the velocit vector with -ais and the angle between the velocit vector and the permeable solid surface now become unknown on AO. The first function appears in the epression for the comple velocit in Eq.(6), and the second appears in the epression for the derivative of the comple potential in Eq.(). In addition, at point A, or ζ =, γ(ξ) is continuous for the impermeable surface but discontinuous for the permeable surface. The latter leads to an additional singularit in the epression for the derivative of the comple potential at point A at, ζ =, as it can be seen in Eq.(), which requires additional attention in the solution procedure.

11 The governing equations (6) and () - (3) enable to determine the pressure coefficient at an point of the flow region. B choosing point A as the reference point in the Bernoulli equation, where w A =, and v A = dw/dz ζ=, and taking the advantage of the self-similarit of the flow, we can determine the pressure coefficient at an point z in the flow region through c p = P P A ρv ( = R w + z dw ) dz dw dz + v A (4) where P is the pressure at the point z and P A is the pressure at the point A. Then, the pressure coefficient based on the ambient pressure, P a, which can be represented b the pressure P O = P a at point O, can be determined as follows (ξ) = P P a ρv = c p(ξ) c p() (5) The governing equations (6) and () - (3) contain the unknown parameter K and the functions γ(ξ), β(ξ), θ(ξ) and v(η), all to be determined from phsical considerations, as well as the dnamic and kinematic boundar conditions on the free surface and the permeable bod surface. At the moment of impact, the tip of the liquid wedge will move into the bod surface due to permeabilit. At the same time it will also spread along the bod surface, which will depart from the bod surface at point D and will eventuall become point O. We notice that when the flow is self similar the velocit at point O is constant. Thus the coordinate position of point O can be decided b its velocit V O = V v relative to point A. This gives Z O = V tz O = V te iβ = V tv e iβ, which further gives z O = V tv e iβ, where v = v(η)) η= and β = β(ξ) ξ=. The same argument was used Semenov, Wu & Oliver [] for the liquid/liquid impact. From this, the distance between points A and O in the similarit plane equals v, which can be used to determine the parameter K: K K dz dζ dξ ζ=ξ = v. (6)

12 .. Determination of the functions θ(η) and v(η) on OB from boundar conditions on the free surface. The dnamic boundar conditions on the free surface OB for an arbitrar self-similar flow can be derived in the following form [8], b eploiting the Bernoulli equation and the fact that the acceleration of a liquid particle is orthogonal to the free surface with constant pressure dθ ds = v + s cos θ s sin θ d ln v = d tan θ ds ds [ arg d ln v ds, (7) ( )] dw. (8) dz Multipling both sides of Eqs.(7) and (8) b ds/dη = dz/dζ ζ=iη we obtain the following integro-differential equation for the function θ(η): dθ dη = v + s cos θ d ln v s sin θ dη, (9) where s = s(η) is obtained from integration of the epression dz/dζ ζ=iη along the imaginar ais of the parameter plane. Determining the argument of the comple velocit from Eq.(6) and substituting the result into Eq.(8), the following integral equation for the function d(ln v)/dη is obtained: d ln v tan θ dη + π d ln v dη η η η dη = + η + π dβ ξ dξ ξ dξ. () + η The sstem of equations (9) and () enables us to determine the functions θ(η) and d(ln v)/dη along the imaginar ais of the parameter domain. Then, the velocit magnitude on the free surface can be obtained from d ln v v(η) = v ep dη dη () where v = is the reference velocit at infinit. This gives the velocit at point O, v = v(η) η=..3. Determination of functions γ(ξ) and β(ξ) from the kinematic boundar condition on the permeable solid surface. η In the porous media, the velocit of the fluid flow is proportional to the pressure gradient, based on Darc s law. When the solid becomes ver thin

13 with a pressure jump from one side of the surface to the other side, the velocit normal to the bod is then proportional to the difference of pressures from both sides. In our case, at the initial stage of the impact, which is usuall of main interest, we assume that the pressure on the back side of the solid surface remains to be ambient pressure P a, therefore, the normal component of the velocit through the wetted surface can be written as : V n = α p (P P a ) () where the coefficient α p characterizes the porosit of the thin wall. When α p, V n in Eq.(). This returns to the impermeable boundar condition. On the surface of a perforated bod, the relationship between the normal component of the velocit and the pressure can be written as V n V n = χ (P P a )/ρ, as V n is epected to be positive [, 5] V n = χ (P Pa )/ρ, χ = νκ κ. (3) where ν is a discharge factor, and κ is the ratio between the area of the holes and the total area, ρ is the liquid densit as defined previousl. As κ, the impermeable boundar condition is recovered. In the following, non-dimensional parameters are used based on the definitions of v n = V n /V, α = α p ρv, while χ is alread non-dimensional. Eqs.() and () can be respectivel written as v n = α, (4) v n = χ cp, (5) The tangential component of the velocit on the wedge side OA can be determined with the notation in Eq.(7) as ( ) dw dz v s (ξ) = R = R dz ds ( dw dz ) e iδ A ζ=ξ (6) where δ A = π α is the direction of the vector τ. Then, the function γ(ξ) can be obtained ( ) γ(ξ) = tan v n (ξ) R (e iδ A dw/dzζ=ξ ) 3 (7)

14 where the normal component of the velocit, v n, will be detrmined in the following. Taking the argument of Eq.(3), we obtain δ A = β + γ, that gives equation for the function β(ξ) as follows β(ξ) = δ A γ(ξ) (8) The integral equation () with Eq.(9) allows us to obtain the functions θη and v(η), together with the functions γ(ξ) and β(ξ) determined from Eqs.(7) and (8). Once these functions are found, the velocit at point O, v, and the angle of the tip of the splash jet, µ, can also be found. The numerical solution of the sstem of equations, which is strongl nonlinear, is based on the iteration with the relaation coefficient used for the functions βξ and γ(ξ) being /5 of that for the functions θ(η) and v(η). It starts with β(ξ) = β, γ(ξ) = π, v n (ξ) = on the wedge side OA and θ(η) = α, v(η) = on free surface OB, as initial values. The unknown K is obtained from Eq.(6) at each iteration. Eq.(5) gives pressure from which v n is determined from Eqs. (4) or (5), for porous and perforated bodies, respectivel. New functions βξ and γ(ξ) are then determined from Eq, (7) and Eq.(8), respectivel. The functions θ(η) and v(η) are determined from Eq.(9) and Eq.(), respectivel, which give the free surface shape OB. The iteration returns to Eq.(6) and the process is repeated until the convergence has been achieved. 3. Numerical results 3.. Numerical approach The numerical approach emploed in the present stud is based on the method of successive approimations, which is similar to that used b Semenov & Wu [], for solving self-similar impact problems between the impermeable solid and liquid wedges. Let us consider two sets of points distributed along the part of the real ais, < ξ j <, j =... M, and the imaginar ais, < η i < η N, j =..., where η N is sufficientl large. The integrals within each segment in the sstem of equations are evaluated eplicitl, using the linear interpolation for the functions γ(ξ), β(ξ), θ(η) and v(η), on the intervals 4

15 α µ/π ma N=5 N=3 N=6 N=5 N=3 N=6 (I&K) Table : Convergence stud and comparison for water-entr of a porous solid wedge α A = 6 into the half-space of the liquid (α = 9 ). (ξ j, ξ j ) and (η i, η i ), respectivel. The results contain the unknowns coefficients γ j = γ j γ j, β i = β i β i, θ i = θ i θ i and ln(v i /v i ), which are determined from the sstem of equations (9), (), (7), (8). 3.. Impact between liquid and solid porous wedges. Effect of bod porosit on the flow parameters including the slamming pressure peak has been studied using the boundar element method (BEM) b Iafrait & Krobokin [6] for a porous wedge entering the flat water surface. Similar problem of a permeable block sliding along an inclined beach has been studied b Iafrati, Miloh & Korobkin [7], in the contet of violent free surface flow generated b landslide. To evaluate the accurac and mesh-independence of the results, several distributions with different numbers of nodes have been emploed. Table gives results with N = 5, N = 3 and 6 for a porous wedge of half-angle impacting the liquid wedge with α = 9, which is in fact a problem of a solid porous wedge entering a flat free surface. It can be seen from the table that the contact angle µ/π and the maimum pressure coefficient have converged well with the number of nodes. The converged pressure coefficients are virtuall identical to those of Iafrati & Korobkin [6] (I&K) obtained through BEM. 5

16 a) 3 = 6 b) 3 = c) = d) = Figure 3: (a) Streamlines with ψ =. and the free surface shape (solid lines), and the pressure distribution (dashed lines) along the porous surface for α A = 6 and α = 9 : (a) α = ; (b) α =.; (c) α =.3 and (d) α =.5. Further results for streamline patterns, the free surface shape and the pressure distributions for the case in Table are given in figure 3. As discussed b Wu & Sun [3], when a fluid particle moves towards an impermeable bod surface, its path is blocked and has to make a sharp turn to move along the bod surface. This leads to a large acceleration and therefore large pressure gradient near the jet root, as shown in figure 3a. When the bod surface is permeable, the blockage to path of the fluid flow is less solid. The peak pressure near the jet root decreases. This decrease continues as the permeabilit coefficient α increases. This makes the difference between the peak pressure and the ambient pressure on the other side of the bod or insider the jet much smaller. We notice that when α increases, more liquid will move through the bod surface, which can be reflected b v n = α. In fact we can let the stream function ψ = on the smmetr line or the ais. With the same increment ψ =. in figures 3a to 3d, the number of streamlines intersecting the bod surface from the tip of the wedge to the jet root is approimatel equal to the flu rate into the 6

17 a) = = Figure 4: (a) Streamline pattern with ψ =. and the free surface shape (solid lines), and the pressure distribution (dashed lines) along the porous surface for α A = 3 and α = 9 : (a) α = ; (b) α =.3. bod surface. It is evident that this rate increases with increase of α. When more liquid passes through the bod surface, the position of the jet root moves towards the ais. The jet becomes thinner and shorter, and the contact angle µ becomes smaller. When α, Eq.(4) gives to provide the a finite value of v n. This means that the wetted solid surface has become a free surface. The velocit field in the whole flow region becomes undisturbed. The wetted length of the solid wall tends to the distance between tip of the bod and its intersection with the undisturbed liquid wedge. For α A = 3 the results for streamline patterns, the free surface shape and the pressure distributions are shown in figure 4. At α = the pressure on the wedge side is much lower than that for α A = 6, and the pressure peak near the root of the splash jet in figure 3 is absent. This agrees with previous studies of water impact problems [4]. It leads to a smaller normal component of the velocit according to Eq. (4) and, consequentl, smaller flow rate through the porous surface of the wedge. 7

18 Although the permeabilit coefficient α is larger than for the case α A = 6 in figure 3, the flowrate trough the permeable surface corresponding to the 4 th streamline, crossing the wedge side in figure 4b, is q = 4 ψ =.4, or about 7% of the total flowrate Q = v tan 3.58 trough the cross-section of the solid wedge and the undisturbed free surface. Here, ψ =. is the increment of the stream function between two neighbour streamlines in figure 4. Thus, although the α is about.6 times larger for the case shown in figure 4b, the relative flowrate trough the permeable wedge side is approimatel the same as for the case α A = 6 in figure 3d. This occurs due to the lower pressure on the permeable wedge side for α A = 6 and α = Impact of the liquid wedge on the perforated bod. Perforated bod is used widel in coastal and offshore engineering [, 5, 6] as a mean to decrease the contact area b allowing the part of the liquid to flow through the bod surface. This also reduces a high-pressure peak and the total force during wave impact. Iafrati, Miloh & Korobkin [7] considered the liquid impact caused a perforated block sliding into the water along a sloping beach. Because the gravit effect is ignored, we can rotate the beach in their case to become the ais and the problem then becomes the same as that shown in figure a. Besides, Iafrati & Korobkin [6] also considered a perforated wedge entering into the calm free surface. In both cases, the permeabilit of the bod surface is characterized b the coefficient χ in Eq. (5). Another model for perforated structures has been proposed b Cooker [5]. He considered a perforated wall as a cascade of impermeable bodies with some gap between them, through which the liquid passes in the form of free streamline jets. Such a model gives the possibilit to determine the coefficient χ. However this is onl for some simple geometries of the wave-breaker structures. Figure 5 show the results for impact of a liquid wedge hitting on a perforated flat wall. Provided are the streamlines, free surface and the pressure distributions along the impermeable wall and a perforated wall with χ =.5. The angle of the liquid wedge has been taken as α =, α = 3 and α = 6. 8

19 . a) -.5 =. b) -.5 = c) = d) = e) = f ) = Figure 5: (a) Streamline pattern, free surface (solid lines), and the pressure distribution (dashed lines) along a flat wall: α =, (a) χ = and (b) χ =.5; α = 3, (c) χ = and (d) χ =.5; α = 6, (e) χ = and (f) χ =.5. As epected, the pressure on the wall will decrease when it is perforated. The reduction of the pressure is more pronounced at the locations where the corresponding pressure on the impermeable wall is higher. This is because at higher pressure more liquid moves into the wall as it can be seen from Eq. (5). The larger pressure is, the bigger pressure reduction occurs. We also notice that the jet root moves towards the centre of the impact for the perforated surface. This effectivel reduces the area affected b the hdrodnamiressure. The tip of the jet also moves towards the centre and the jet itself becomes thinner. In figure 6 the flow configuration, streamlines and pressure distribution are 9

20 a) = 3 b) = c) - = d) - = Figure 6: (a) Streamline patterns, free surfaces (solid lines), and the pressure distribution for impact of liquid and solid wedges with α A = 35 : α =, ψ =.5 (a) χ = and (b) χ =.5; α = 3, ψ =.5 (c) χ = and (d) χ =.3.

21 = = Figure 7: (a) Streamline patterns, free surfaces (solid lines), and the pressure distribution for the liquid and solid wedges α A = 45 and α = 45, increment in the stream function ψ =., and (a) χ = and (b) χ =.. shown for the two deadrise angles f, which is the angle between the solid surface and the undisturbed liquid surface, = 8 α α A = 35 for cases (a) and (b), and f = 5 for (c) and (d). For cases (a) and (c), χ =, the sparse distribution of the streamlines near the ape of the solid wedge indicates the low-velocit, or almost a stagnation region. For f = 35 the low-velocit region also occurs near the ape of the solid wedge. The tip jet, or the jet attached on the bod surface, becomes ver thin at χ =.5. At even larger χ, the jet will be even thinner, which would cause computational difficult. Because of that at f = 5 the maima χ for which the converged results are obtained is.3. Comparing the results in figures 5 and 6 we can see that the smaller deadrise angle causes the larger pressure peak near the root of the tip jet as well as the larger pressure on the entire wedge surface. However, larger permeabilit of the solid wedge leads to larger reduction of the high pressure due to the larger flowrate through the wedge surface for both linear (Eq. (4)) and quadratic (Eq. (5)) relationships between the normal velocit and the pressure.

22 Results for the angles of the liquid and solid wedges α A = α = 45 are shown in Figure 7. For the impermeable surface in case (a), the pressure decreases almost linearl from the wedge ape to the root of the tip jet, while in case (b) the pressure decreases more mildl and then faster near the root of the tip jet. This is caused b larger pressure reduction near the ape of the wedge due to the larger flowrate into the wedge side there. 4. Conclusions We have developed a method which generalizes previous water-impact studies and accounts for certain essential features associated with a permeabilit of the solid bod. The focus is on some configuration in which the flow can be treated as self similar at earlier stage of the impact. The mathematical approach is based on the integral hodograph method, which enables the original problem in a phsical domain to be reduced to a sstem of integro-differential equations along the straight lines. These equations are solved numericall through the method of successive approimation. The numerical procedure has been verified b convergence stud and comparison with some known results. The presented calculations confirmed the reduction of the hdrodnamic pressure for a porous or perforated wedge entering the flat free surface and revealed the similar effect for a liquid wedge impacting a permeable wedge. This effect magnifies in the case of smaller deadrise angle between the side of the solid and the undisturbed liquid surface. The permeabilit of the bod also affects the tip jet which becomes a much thinner liquid film on the bod surface even at moderate permeabilit condition. 5. Acknowledgements This work is supported b Llods Register Foundation (LRF) through the joint centre involving Universit College London, Shanghai Jiaotong Universit and Harbin Engineering Universit, to which the authors are most grateful. LRF

23 supports the advancement of engineering-related education, and funds research and development that enhances safet of life at sea, on land and in the air. References [] B. Molin, Motion Damping b Slotted Structures. In Hdrodnamics: Computations, Model Tests and Realit, Developments in Marine Technolog. Elsevier, 99. [] S.D. Howison, J.R. Ockendon and S.K. Wilson. Incompressible water entr problems at small deadrise angles. J. Fluid Mech. (99), 5-3. [3] J.M. Oliver Second-order Wagner theor for two-dimensional water-entr problems at small deadrise angles. J. Fluid Mech. 57(7), [4] M. Reinhard, A.A. Korobkin, M. J. Cooker. Water entr of a flat elastic plate at high horizontal speed. J. Fluid Mech. 74(3), [5] B. Molin A.A. Korobkin. Water entr of perforated wedge. In 6th IWWWFB Proc. (Eds. K. Mori and H. Iwashita), Hiroshima, -5 April (), - 4. [6] A. Iafrati, A.A. Korobkin Self-similar solutions for porous/perforated wedge entr problem. In th IWWWFB Proc. Oslo, 9 Ma - June (5). [7] A. Iafrati, T. Miloh, A.A. Korobkin. Oblique water entr of a block sliding along a sloping beach. Proceedings of International Conference on Violent Flows, Fukuoka, Japan (7). [8] Y.A. Semenov, A. Iafrati. On the nonlinear water entr problem of asmmetric wedges. J. Fluid Mech., 547(6), [9] Y.A. Semenov, L.J. Cummings. Free boundar Darc flows with surface tension: analtical and numerical stud. Euro. J. Appl. Math., 7(6),

24 [] Y.A. Semenov, G.X. Wu Asmmetric impact between liquid and solid wedges. Proc. R. Soc. A 469(3). [] Y.A. Semenov, G.X. Wu, J.M. Oliver. Splash Jet Caused b Collision of Two Liquid Wedges. J. of Fluid Mech. 737(3), [] Z. Huang, Y. Li, Y. Liu. Hdraulierformance and wave loadings of perforated/slotted coastal structures: a review. Ocean Engn. 38(), [3] G.X. Wu, S.L. Sun. Similarit solution for oblique water entr of an epanding paraboloid. J. Fluid Mech. 745(4), 398. [4] Z.N. Dobrovol skaa. On Some problems of similarit flow of fluid with a free surface. J. Fluid Mech. 36(969), [5] M. Cooker. A theor for the impact of a wave breaking onto a permeable barrier with jet generation. J. Eng. Math 79(3):-. 4