Arpita Mondal and R. Gayen

Department of Mathematics, Indian Institute of Technology, Kharagpur, 721302, India

Omdehghiasi Hamed, Mojtahedi Alireza*and Lotfollahi-Yaghin Mohammad Ali

Department of Water Resources Engineering, Faculty of Civil Engineering, University of Tabriz, Tabriz, Iran

Wave Interaction with Dual Circular Porous Plates

Arpita Mondal and R. Gayen*

Department of Mathematics, Indian Institute of Technology, Kharagpur, 721302, India

In this paper we have investigated the reflection and the transmission of a system of two symmetric circular-arc-shaped thin porous plates submerged in deep water within the context of linear theory. The hypersingular integral equation technique has been used to analyze the problem mathematically. The integral equations are formulated by applying Green’s integral theorem to the fundamental potential function and the scattered potential function into a suitable fluid region, and then using the boundary condition on the porous plate surface. These are solved approximately using an expansion-cum-collocation method where the behaviour of the potential functions at the tips of the plates have been used. This method ultimately produces a very good numerical approximation for the reflection and the transmission coefficients and hydrodynamic force components. The numerical results are depicted graphically against the wave number for a variety of layouts of the arc. Some results are compared with known results for similar configurations of dual rigid plate systems available in the literature with good agreement.

water wave scattering; circular-arc-shaped plates; hypersingular integral equation; Green’s integral theorem; reflection coefficient; energy identity; hydrodynamic force

1 Introduction1

Interaction of water waves with the obstacles of various geometrical shapes and sizes has been widely studied in the modern literature, due to the huge number of applications in the modeling of breakwaters that are constructed mainly to protect sheltered areas such as harbours, marinas,etc. from the impact of rough seas. An account of the detailed literature in the field of wave scattering problems involving barriers can be found in Mandal and Chakrabarti (2000). Ursell (1947) obtained the first rigorous mathematical analysis for an obstacle in the form of a partially immersed thin vertical plate, or a submerged thin vertical barrier extending infinitely downwards, by using an integral equation formulation. Since then a large number of research papers have been published by various researchers in this fascinating area of applied mathematics. Evans (1970) used the complex variable technique to investigate the wave scattering problem of a vertical plate completely submerged in deep water. Applying a reduction procedure and an integral equation formulation, Porter (1972) solved the problem of wave transmission through a small gap of arbitrary width.

However if the plate is curved, it is not possible to obtain an explicit analytical solution for the related water wave scattering problems. For such problems, some approximation techniques are used to obtain numerical values for the things of physical interest, namely the reflection and the transmission coefficients. Parsons and Martin (1994) investigated water wave scattering by a submerged thin curved plate, convex upwards and symmetric about a vertical line passing through the centre of the arc, and a surface piercing inclined plate in deep water, using the hypersingular integral equation technique. Subsequently, McIver and Urka (1995) used multipole potentials and a matching procedure to obtain numerical results for the reflection coefficient for a circular arc shaped plate submerged in deep water. Their motivation was to compare the reflective properties between a circular arc shaped plate and a submerged full circle, for assessing the acceptability of using circular plates in the construction of a water wave lens that might be helpful in focusing waves prior to extracting energy from them. Kanoria and Mandal (2002) investigated the problem of wave scattering by a submerged thin circular-arc-shaped plate, not necessarily symmetric about the vertical through its centre, submerged in infinitely deep water using a hypersingular integral equation technique.

Water wave scattering problems involving double or multiple barriers are fairly common in the modern literature. Seminal investigations involving the problems of scattering of water waves by two thin and impermeable vertical barriers of different configurations were performed by Levine and Rodemich (1958), Jarvis (1971), Newman (1974), Daset al. (1997), Neelamani and Vedagiri (2002), Deet al. (2009; 2010) applying various mathematical techniques. In the field of porous structures, the scattering of water waves by double or multiple porous structures are somewhat rarely available in the literature. Research articles involving double permeable structures can be found in Twu and Lin (1991); Losadaet al. (1992); Isaacsonet al. (1999); Koraimet al. (2011). The response of double curved barriers towards water wave interaction can only be found in Mandal and Gayen (2002). They used the hypersingular integral equation technique as described in Parsons and Martin(1994) for investigating the scattering of water waves by two symmetric circular-arc-shaped rigid plates submerged in deep water. This use of hypersingular integral equations is most acceptable in the sense that it can be adapted to study water-wave-interaction problems involving obstacles in the form of thin curved plates having any geometrical shape. The only limitation of this methodology is that it might not be directly extended to the case of a plate with finite width. However, in that case the problem can be formulated in terms of the discontinuity in the displacement across a two-dimensional cross section of the plate. The rectangular cross section may further be transformed into a circular disc by some conformal mapping. This will produce a two-dimensional hypersingular integral equation of order three which can be solved using the Fourier-Gegenbauer expansion method.

Here we have extended the work of Mandal and Gayen (2002) in the case of porous structures, namely double circular-arc-shaped permeable plates, placed symmetrically about they-axis in deep water. Earlier Lu and He (1989) analyzed the phenomenon of the reflection and transmission of water waves by a thin curved permeable barrier and showed that a well planned curved porous plate was very effective in trapping waves within a large frequency range.

This paper deals with a water wave scattering problem involving two symmetric circular arc shaped thin porous plates submerged in deep water. Submerged plates are used to protect the shore-ward area of the breakwater from the hazards of rough seas by diminishing the effects of incoming waves. Such structures are effective as they allow the free exchange of water mass through them so that the water pollution in the sheltered area is minimized. The submerged structures are also capable of absorbing some wave energy, they break up waves and thus control shore erosion.

Here the curved plates are mounted so that the centers of the circular arcs are placed along a horizontal line joining the centers of two circles. The circular arcs are symmetric about they-axis. Utilizing the geometrical symmetry of the plates, the velocity potential for the fluid motion when a train of regular, small amplitude surface water waves are striking the plates, is divided into two parts, namely the symmetric and the anti-symmetric potential functions. Then appropriate use of Green’s integral theorem to the suitable functions in the fluid region, followed by utilization of the boundary condition on the porous plate surfaces produces two hypersingular integral equations of the second kind involving discontinuities of the symmetric, and the anti-symmetric potential functions across one of the two porous plates. These hypersingular integral equations are then solved numerically using an expansion-cum-collocation method, where the unknown discontinuities of the potential functions across the plates are approximated using two finite series involving Chebyshev polynomials of the second kind. The zeros of the Chebyshev polynomials are used as the collocation points. The numerical estimates for the reflection and transmission coefficients and the hydrodynamic forces are then computed using the solutions of the hypersingular integral equations. The numerical results for the reflection and transmission coefficients for a set of values of the depth parameters, arc lengths of the plates, and separation length between the centers of the circles whose arcs indicate the positions of the plates, are plotted graphically against the wave number in a number of figures. Some results are compared with the results of Mandal and Gayen (2002) for two circular arc shaped rigid plates submerged in deep water by taking the zero value of the porosity parameter. A very good agreement has been found in each case. Some new results are also provided here showing the effect of porosity in the reflection and transmission of the waves by a system of two curved porous plates. A significant result is achieved by taking two very closely spaced semi-circular arc shaped plates. Such a configuration serves as the cross section of a horizontal circular cylinder. It is observed that in this case there is almost no reflection. Ursell (1950) and Mandal and Gayen (2002) showed that a rigid horizontal circular cylinder offers no hindrance to incoming waves. Comparison of our results with those in Mandal and Gayen (2002) shows that a horizontal porous circular cylinder is more transparent to the incoming waves than the rigid one.

2 Mathematical formulation

We choose a two-dimensional Cartesian co-ordinate system with they-axis directed vertically downwards passing through the midpoint of the line joining the centers of the circular arcs. Thexz-plane denotes the position of the undisturbed free surface. The fluid occupies the region 0＜y＜∞,-∞＜x＜∞. Two circular arc shaped thin porous platesΓi（i=1,2） are situated inside the water as given in the Fig. 1. The vertical section of each of the plates are in the form of an arc of a circle of radiusbwith their centers at（）,a db±+. The plates are considered to be infinitely long in thez-direction and we take a vertical cross-section in thexy-plane. Thus the motion is taken to be two-dimensional in thexy-plane.

Fig. 1 Geometry of the dual porous plates

The upper and the lower end points of each circular arc make anglesαandβrespectively with the upwardvertical. The parametersa,b,αandβwill always satisfy the identity

Considering linear theory, incompressible and inviscid fluid, and irrotational motion, a train of surface water waves coming from the direction of positive infinity can be described by the potential function Re｛φ0（x,y）e-iσt′｝, wheret′ is the time,σis the frequency and

and the condition on the porous plate surface as given by

whereγis the porosity;f*is the resistance force coefficient;Sis the inertial force coefficient andd1is the thickness of the porous medium. The velocity potentialφalso satisfies the tip condition given by

whereris the distance of any fluid particle from either of the submerged tips ofiΓ,i=1, 2, the bottom condition as given by

φ（x,y） has asymptotic behavior

whereRandTdenote the reflection and transmission coefficients respectively, and are to be obtained.

3 Method of solution

Due to the geometrical symmetry aboutx=0, the velocity potentialφ（x,y） can be divided into two parts namely the symmetric and the anti-symmetric partsφs（x,y） andφa（x,y） so that

where

Therefore we can confine our analysis to the regionx≥0 only. Thenφs,a（x,y） will satisfy the Eqs. (3)-(4), (6)-(7) together with

and

Let the behaviour ofφs,a（x,y） for largexbe expressed by

where the factorsRs,aare the unknown constants related toRandTby the relation

Now, we reduce the boundary-value problem for the velocity potential to a boundary integral equation over1Γ. To do this, we incorporate an appropriate fundamental solution with an application of Green’s integral theorem. We use the fundamental solution

Now we employ the Green’s theorem to the functions

and

in the fluid region bounded externally by the lines

and internally by a small circle of radiusεand centered at（ξ,η） and the contour enclosing the plateΓ1. We finally makeX,Y→∞ andε→0 and shrink the contour enclosing the plateΓ1into both sides ofΓ1and obtain the integral representation ofφs,a（ξ,η） as

whereq≡(x,y) is a point onΓ,Fs,a（q） are the

1have square-root singularities there.

Now we finally need to apply the boundary condition on the porous plate surface rewritten as

wherep≡（ξ,η）∈Γ1. For that purpose we take the normal derivative on the both sides of the Eq. (18) at a pointponΓ1and using the boundary condition (19) we get

that

We now can interchange the order of the integration and the normal differentiation which is legitimate (cf. Martin and Rizzo (1989)) provided that the integral is then to be treated as the Hadamard-finite part integral. By applying this procedure we get

The integral Eq. (22) is a hypersingular integral equation of the second kind, to be solved subject to the boundary conditions (21). The （）·∫ implies that the integral is to be considered as a two-sided finite part integral of order two.

where

and an appropriate parameterization of the curved plate is taken asx=a+bsinθt,y=d+b（1-cosθt）,-1≤t≤1, whereq≡（x,y）.

The pointp≡（ξ ,η） onΓ1has the same parameterisation, but withtreplaced byτ.Using this parameterization we find that

where

with

The right hand side of the Eq. (22) can be expressed as

where

Now the Eq. (22) can finally be rewritten as

where

andfs,a（t） stands forFs,a（q）.IfG=0, the Eq. (29) reduces to a first kind hypersingular integral equation ( cf. Mandal and Gayen (2002)) as follows

The hypersingular integral Eq. (29) are to be solved subject to the end condition thatfs,a=0.

Now, using the method of contour integration the integrals in (26) can be expressed as

whereUn（t） is the Chebyshev polynomial of the second kind and the unknown factors,an’s are to be determined. The square-root factor in (31) ensures thatfs,a（t） have the right behavior at each tip of the plates, where the potential difference vanishes. Substituting the expansion (31) in the place of allfs,a（t） in (29) we finally get

where

Next, we collocate atτ=τj,j=0,1,...,N, whereτj’s are chosen as

these being the zeros ofTn+1（τ）, the Chebyshev polynomial of the first kind. Golberg (1983; 1985) has shown that the Eq. (34) is a good choice since it provides a uniform convergent method. The rate of convergence of his method depends on the smoothness of the kernels,aMin the Eq. (29).

1) Reflection and transmission coefficients.

From the Eq. (14) it is evident that to determine|R|and|T|we first have to compute the unknown quantitiesRs,a. We first makeξ→∞ in the representation ofφs,a（ξ ,η） as given in Eq. (18) and then compare it with (13) (with (x,y) replaced by（）,ξ η). Also we need to make use of the asymptotic results

Finally we find that

Thus, once we get the numerical estimates ofRs,aby solving the integrals in Eq. (36),RandTcan be computed using the Eq. (14). We have depicted the results forRandTin a number of figures in section 4.

2) Hydrodynamic forces on the plates.

The hydrodynamic forces acting on each curved porous plate can be determined as follows:

The fluid pressure （）,,p x y t′ at a point (x,y) is connected to the velocity potential （）,,x y tΦ′ by the expression

and

The numerical estimates for these quantities can be evaluated using the solutions of the hypersingular integral Eq. (29).

4 Numerical results

In this section we will discuss the effect of different parameters i.e. arc-length, porosity, depthetc. on the reflection and the transmission coefficients and the hydrodynamic forces acting on the porous plates. We have made different physical quantities dimensionless with respect to the radiusb. While taking the values of ,|R||T|it has always been verified that the computed values of these two quantities are always satisfied by the following energy identities

and

where

This will give a partial check on the correctness of the numerical results obtained here.

Table 1 Reflection and the transmission coefficients

Fig. 3 stands for a comparison in the reflective and transmissive properties between the systems of two closely placed semi-circular arc-shaped plates: one consists of two permeable plates and the other consists of two impermeable plates. Such a configuration serves almost as the cross section of a horizontal circular cylinder. It is observed that for both the cases (G=0 or 1) the amount of reflection is very low. This is consistent with the phenomenon of zero reflection by a horizontal circular cylinder. This result was established long ago by Ursell (1950) for a rigid horizontal cylinder and was verified by Mandal and Gayen (2002) by taking two semi-circular rigid plates whose vertical diameters are very close to each other. The solid line represents the data of Mandal and Gayen (2002), and the dashed line in the curve of reflection coefficient stands for the results obtained by the present method. It can be seen that a horizontal porous cylinder also offers no hindrance tothe propagation of water waves. The reflection as well as the transmission for the porous cylinder is even less than that for a rigid one.

Fig. 2 Effect of separation on|R| for a system of two half-circular porous plates

Fig. 3 Reflection and transmission coefficients for two half-circular impermeable and permeable plates

Fig. 4 Reflection and transmission coefficientsfor different arc-lengths of two curved porous plates

Fig. 5 Reflection coefficientfor different depths of two permeable and impermeable curved plates

Fig. 6 Dissipated energy for two curved porous plates, for variousG

Fig. 7 Reflection and transmission coefficients for two almost full circles

Fig. 8 Reflection and transmission coefficient for smallβ-α

Fig. 9 Amplitude of the hydrodynamic forces acting onΓ1

Fig. 10 Amplitude of the hydrodynamic forces acting on

5 Conclusions

The problem of water wave scattering by two symmetric circular-arc-shaped porous plates submerged in deep water has been discussed with linear theory. The plates are situated symmetrically about vertical lines passing through the midpoint of the line joining the centers of circles whose arcs are assumed to be the positions of the plates. Exploiting the geometrical symmetry of the plates, the velocity potential has been divided into its symmetric and anti-symmetric parts. Appropriate use of Green’s integral theorem to the suitable functions in the fluid region, followed by the utilization of the boundary condition on the porous plate surface, yielded two hypersingular integral equations of the second kind. These are solved using an expansion-collocation method. Using the numerical solutions of these hypersingular integral equations, the reflection and the transmission coefficients and the hydrodynamic forces on the plates have been determined and depicted graphically in a number of figures. The numerical results for the reflection coefficient for two circular-arc-shaped rigid barriers have been obtained as a special case and matched with the corresponding published paper of Mandal and Gayen (2002). A good agreement has been achieved. After analyzing the numerical results for the reflection coefficient obtained here, it may be concluded that the incoming waves experience less reflection due to the presence of the dual plates system consisting of two curved porous plates compared to that consisting of two similar rigid plates. Thus a dual circular plate system provides an effective model for porous barriers, as the system effectively reduces the height of the reflected wave, which is useful in reducing the occurrence of wave resonance inside harbours. It is also established that, like a rigid horizontal circular cylinder, a porous horizontal cylinder is also transparent to the incoming waves.

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The 12th International Conference on Hydrodynamics (ICHD 2016)

September 18-23, 2016

Delft, The Netherlands

Hydrodynamics has always been an important and fundamental subject for many disciplines involving the science of forces acting on or exerted by fluids, and engineering applications including ship and marine engineering, ocean and coastal engineering, mechanical and industrial engineering, environmental engineering, hydraulic engineering, petroleum engineering, biological & biomedical engineering, and so on.

While many engineering questions have been answered, there are still many more that need to be addressed through field trial, verification and fundamental research and development (R&D). The International Conference on Hydrodynamics (ICHD) is the forum for participants from around the world to review, discuss and present the latest developments in the broad discipline of hydrodynamics and fluid mechanics.

The first International Conference on Hydrodynamics (ICHD) was initiated in 1994 in Wuxi, China. Since then, 11 more ICHD conferences were held in Hong Kong, Seoul, Yokohama, Tainan, Perth, Ischia, Nantes, Shanghai, St Petersburg and Singapore. Evidently the ICHD conference has become an important event among academics, researchers, engineers and operators, working in the fields closely related to the science and technology of hydrodynamics.

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Comparison of Flow Characteristics Around Refractive and Right-angled Groins in Barotropic and Baroclinic Conditions

Omdehghiasi Hamed, Mojtahedi Alireza*and Lotfollahi-Yaghin Mohammad Ali

Department of Water Resources Engineering, Faculty of Civil Engineering, University of Tabriz, Tabriz, Iran

DOI: 10.1007/s11804-015-1330-x

Abstract:Groins are employed to prevent nearshore areas from erosion and to control the direction of flow. However, the groin structure and its associated flow characteristics are the main causes of local erosion. In this study, we investigate the flow patterns around refractive and right-angle groins. In particular, we analytically compare the flow characteristics around a refractive groin and study the degree of accuracy that can be achieved by using a right-angle groin of various projected lengths. To compare the flow characteristics, we replaced the right-angle groin with an approximation of a refractive groin. This replacement had the least effect on the maximum velocity of flow in the channel. Moreover, we investigated the distribution of the density variables of temperature and salinity, and their effects on the flow characteristics around the right-angle groin. A comparison of the flow analysis results in baroclinic and barotropic conditions reveals that the flow characteristic values are very similar for both the refractive and right-angle groins. The geometry of the groin, i.e., right-angle or refractive, has little effect on the maximum speed to relative average speed. Apart from the angular separation, the arm length of the groin in downstream refractive groins has less effect on other flow characteristics than do upstream refractive groins. We also correlated a number of non-dimensional variables with respect to various flow characteristics and groin geometry. These comparisons indicate that the correlation between the thalweg height and width of the channel and groin arm’s length to projection length have been approximated using linear and nonlinear formulas regardless of inner velocity in the subcritical flow.

Keywords:refractive groin; right-angle groin; recirculation area; turbulence modeling; flow characteristic; baroclinic conditions; barotropic conditions

Article ID:1671-9433(2015)04-0376-13

1 Introduction1

Groins are dynamic structures that spread from the beach to the sea and are built in various shapes. Their main functions are to trap sediment deposits, promote surf zone activities, and return transported sediments to the sea. Experiments have shown that when waves are under poor equilibrium conditions, groins disperse the energy of the moving water and drive it onto the adjacent sandy beach. This phenomenon causes sediment accumulation. In stormy conditions, in which waves generally radiate in a direction perpendicular to the shore, groins do not function effectively and the beach is washed out. Because of the undesirable side effects that may arise locally, it is doubtful that the groin will ever be employed as the sole coastal protection structure. The performance of groins improves when they are combined with other coastal protection strategies. By detailed coastal investigations, especially during storm periods, a better understanding can be gained of the seasonal changes and their interaction with groins and other coastal structures such as jetties, breakwaters, and seawalls. In particular, we can determine the role they play in scouring at the base of coastal structures and the geotechnical parameters of perpendicular shore sediment transport to the shore. Using local and appropriate geotechnical laboratory testing during stable conditions, we can determine the rate of sea-level fluctuations and evaluate short and long-term changes in the coastal profile, the effects of storm waves, and the design of coastal structures.

Breakwaters are wave energy barriers designed to protect landforms or the water areas behind them from direct wave assault. However, because of the higher cost of these offshore vs. onshore structures (e.g., seawalls), breakwaters have been mainly used for harbor protection and navigational purposes.

Offshore breakwaters can lead to sand accretion in the lee of the breakwater in the form of a sandbar (also called a tombolo), which grows from the shore toward the structure, as well as the associated down-drift beach erosion. Shore-connected breakwaters protect harbors from wave action and have the advantage of a shore arm to facilitate the construction and maintenance of the structure. In recent years, shore-parallel breakwaters, built in short detached groupings, have provided adequate protection from large storms without adversely affecting the longshore transport (Coastal Engineering Research Center, 1984). However, groins are generally constructed to protect the shore from longshore currents.

One of the major environmental uses of groins is related to biological functions. Groins protect organisms from natural hazards such as floods. Material exchange processes in the vicinity of groins are crucial for ecological habitats since the developed flow structure and resultant mixing may be advantageous for fish, plankton species, and other marineorganisms. Since nutrients can accumulate in the recirculation area, this region is also ideal for juvenile fish (Hentschel and Anlauf, 2002; Zsugyelet al., 2014). As shown in the red area in Fig. 1, a groin protects organisms by calming the flow during flooding.

Fig. 1 Physiological function of a groin (Sea Angler, 2011)

There are a number of different groin structure types, and the flow and sediment patterns are different around each type. Studies have shown that groins angled downstream exacerbate erosion around groin abutment and create deep hole erosion (Uijttewaal, 2005).

Downstream groins enhance sedimentation in the groin field when compared with right-angle groins. Suspended sediment is deposited (especially in the upstream area) because of the creation of vortices. Downstream groins are not well suited for protecting riverbanks from erosion. The penetration of secondary currents into the groin field has often caused destruction and is also hazardous to the stability of the structure itself. Right-angle groins are designed to extend the protected area. However, the downstream groin is used to protect flanks from erosion by promoting sedimentation and the formation of a thick sedimentary layer in the spaces between the groins.

In addition to the length and direction of groins, designing them to have specific shapes can also be effective in enhancing their function (Uijttewaal, 2005). In right-angle groins, a wider abutment is used to reduce erosion and prevent the spread of localized erosion holes. There are three distinct modes of right-angle groin: a flat abutment (usually circular), a T-shaped groin, and an L-shaped groin.

By identifying various factors such as the flow pattern around a structure, the processes of sedimentation and transport can be recognized. Flow patterns around structures are described based on their representative features, including factors such as the hydraulic and sediment conditions, current type, geometry of the environment, regional hydrology, shape and materials, and the structural characteristics and structure type (Kanget al., 2011; 2012). In three-dimensional analyses, additional factors are considered, such as vertical shear stress combined with horizontal stress, vertical vortices, and turbulences. The influence of these factors should be addressed in two-dimensional (2D) studies (Blanckaertet al., 2010). Vertical shear stresses are illustrated in Fig. 2.

Fig. 2 Vertical vortices in a groin group (McCoyet al., 2007)

Some studies have examined three-dimensional (3D) simulations of turbulent flow around a right-angle groin in a channel, using rectangular grids to mesh the channel. They then compared these results with those from experiments. Researchers have also studied shear stress on the shore bottom and sedimentation processes around groins with respect to the conduction of materials (Ouillon and Dartus, 1997).

In some studies, shear stresses of the shore bed were measured experimentally around the groin in a perpendicular channel using various Froude numbers. A significant increase in shear stress has been observed near the tip of the upstream area, close to the groin (Rajaratnam and Nwachukwu, 1983).

A 3D numerical model was also developed to determine the flow pattern and its features behind a groin. The vortex center position varies in the perpendicular direction and it moves from the end of groin to the vicinity of the channel wall and near the water surface (Penget al., 1997).

Turbulent flows have also been simulated numerically and experimentally around groins. Using a third-order interpolation method in a 2D domain, the authors measured the current speed and the intensity of the vortices by varying the location of the groins (Giriet al., 2004). The effects of the geometry on the flow field around the groin, including the ratio of length to width in the separation zone, are represented by the numbers and shapes of the vortex. When this ratio is close to one, a vortex is created. If the ratio is higher, there is sufficient space for the creation of two stable vortices (Uijttewaalet al., 2001).

The finite difference method (FDM) is typically applied on a structured mesh for solving with differential equations. In addition, in the finite volume method (FVM), integral equations are solved to handle discontinuities. Therefore, FVM is mostly used to integrate equations for natural complex geometries (Sleighet al., 1998). The simulation of large vortices around groins has been correctly predicted in numerous studies. A small-scale meshing model was developed by combining the Poisson equation used in the analysis of secondary flow near a groin and the FVM for discretizing Navier-Stokes equations (Tanget al., 2006).

The flow around complex geometries, such as hydrofoils,has also been considered by a number of studies. Because of the difficult geometry of these structures, the authors suggest that an appropriate mesh be applied to achieve greater accuracy (Celicet al., 2014 and Lotfollahi Yaghimet al., 2011).

In this study, we investigated the flow characteristics in a channel four meters in length and two meters in width using an image processing technique. In a previous study, groin flow characteristics had been determined for various input velocities, arm lengths, and angles of refraction (Kanget al., 2011). We used Kanget al.’s paper to validate our study results with respect to the refractive groin.

This study presents the flow characteristics around a refractive groin in a perpendicular channel with the assumption of the influence of a right-angle, rather than refractive, groin. Since the flow characteristics around refractive groins have been previously considered experimentally, we investigated the numerical flow results for right-angle groins experimentally and with less error.

Density is a function of salinity, temperature, and pressure, and changes in each of these parameters has some effect on the flow characteristics examined in this study. The size and number of vortices significantly influence erosion; therefore, vortices that arise from flow should be investigated carefully. In this study, to improve groin functions, we consider the advantages and disadvantages of the use of refractive and right-angle groins. In particular, we also consider the effect of replacing right-angle groins with refractive groins on flow characteristics. In addition, we reduce channel length to reduce the analysis time with no significant resulting error.

2 Materials and methods

2.1 Overview

The flow area around the groin can be divided into the main flow area and the recirculation area that forms on the downstream side of the groin. The main area where flow is interrupted by the groin is the susceptible area. Thus, as the thalweg height changes, flow width decreases and flow velocity increases. Thalweg height refers to the line at which there is maximum flow velocity. Diverse flows in the recirculation area diminish the flow velocity, as compared with that in the existing river bed. Thus, it can protect the embankment from erosion, provide ecological space for various types of underwater species, and serve as a point of refuge in times of flood.

The refractive groin is a structure with right-angle and arm-length sections (Fig. 3). In fact, this structure can be considered to be a combination of a right-angle and a downstream or upstream groin (Fig. 4).

Flow change around the groin mainly depends on the length ratio (L/B), in whichLandBare the length and width of the channel, respectively, and the penetration rate of the groin (p). Groins are classified as either permeable or impermeable, with respect to their construction materials. An impermeable groin is one through which there is no fluid transmission. The penetration rate is defined as the ratio of the reflected energy to the incident energy of the groin. In this research, we have assumed the groin to be impermeable (Yeoet al., 2005).

Fig. 4 Refractive groin (Kanget al., 2011)

Fig. 3 Types of groin

As mentioned above, one significant parameter of flow characteristics is the geometry of the groin structure. By changing the angle of the arm and its length, different flow patterns and characteristics may result. In this paper, we selected a 40 meters long and 2 meters wide channel, and the groin is installed at a distance of fifteen meters from the channel entrance.

As also mentioned, we have decreased the channel length to 11.5 meters. In addition, according to the current specifications, we applied as boundary conditions the output hydrography or surface elevation resulting from the analysis of the main channel along the inner and outer borders of the reduced channel length.

We modeled cases with angles of 45 and 135 degrees and various groin lengths. The results were then compared with those of the reduced channel model with arm lengths equal to the length of the right-angle body of the groin. Moreover, we also analyzed the flow around the projected length of the refractive groin and compared the results with the experimental results to evaluate the effectiveness of the groin arm. We also note that, while density is mostly a function of pressure, salinity, and temperature, we first assumed density to be barotropic or independent of these factors; then, we studied the dense current (baroclinic) distribution and its effect on the flow characteristics in the presence of a right-angle groin (based on the hydrostatic pressure assumption).

2.2 Principles and methods

2.2.1 Physical model

Regarding the types and shapes of groins, it seemednecessary to use different kinds of meshes around it. We actually used a rectangular mesh for the right-angle groins, but in the refractive groins we applied a combination of rectangular and triangular meshes. For the areas between the different mesh types, it was necessary to specify a transition zone. We used rectangular meshes in the near-channel wall region and triangular meshes in the near-groin arm region. Only unstructured triangular meshes were used in the model. This meshing of the model had significant effects on calculation accuracy. Lastly, the flow characteristics should be independent of the mesh dimensions, as the flow parameters must not vary by the fineness of the mesh.

2.2.2 Governing equations

After meshing the computing environment, we next solved the governing equations. Equilibrium equations are used for the model analysis, including those for shallow water. This model is based on a solution for 3D incompressible Reynolds-averaged Navier-Stokes equations, based on the assumptions of Boussinesq approximation and hydrostatic pressure.

Continuity and momentum equations are used to describe the model. The shallow water equations are shown in Eqs. (1) to (3), as follows:

whereηis the surface elevation;his the total water depth (h=η+d) (Fig. 5);u,v, andware the velocity components in thex,y, andzdirection, respectively;fis the Coriolis parameter;Sijis the matrix of the radiation stress tensors;vtis the vertical turbulent (or eddy) viscosity;FuandFvare horizontal stress terms that are described using a gradient stress relation;Pais the atmospheric pressure;ρ0is the reference density of water.Sis the magnitude of the discharge due to point sources, andusandvsare the velocity at which the water is discharged into the ambient water in various directions. The horizontal stress terms are shown in Eqs. (4) and (5), as follows:

whereAis the horizontal turbulent viscosity. If the density is baroclinic—a function of salinity and temperature—then other types of equations (transport equations) must be added to the previous equation and these should be solved simultaneously.

Fig. 5 Total water depth definition

The transport equations are shown in Eqs. (6) and (7) as follows:

whereTandSare local temperature and salinity,Dvis the vertical turbulent diffusion coefficient,His the source term due to the heat exchange with the atmosphere,TsandSsare the temperature and salinity of the source, andFTandFSare the horizontal diffusion terms of temperature and salinity, respectively. The horizontal diffusion terms are calculated using Eqs. (8) and (9), as follows:

The spatial discretization of the primitive equations is performed using a cell-centered FVM. The spatial domain is discretized by subdividing the continuum into non-overlapping cells. In the horizontal plane, an unstructured grid is used; however, in the vertical domain in the 3D model, a structured mesh can be used. In the 2D model, the elements can be triangles or quadrilateral elements. In the 3D model, the elements can be prisms or bricks whose horizontal faces are triangles and quadrilateral elements, respectively (MIKE BY DHI, 2011).

Initial and boundary conditions must be defined in the transport equation, in addition to the conditions of the momentum equations for hydrodynamic analysis. We used an FVM to solve discrete equations by applyingunstructured meshing to cover the channel domain. Moreover, we used a high resolution or total variation diminishing (TVD) method to solve these equations (Darwish and Moukalled, 2003; Jawaher and Kamath, 2000).

For the shallow water equations in Cartesian coordinates, we defined the Courant-Friedrich-Levy (CFL) number in order to control model instability and prevent error distribution. As such, if an error is entered in the numerical analysis, the error in the next steps should not increase. Since we have solved two types of equations—hydrodynamic and transport—two CFL numbers should be defined. Therefore, we define two types of stability numbers for the hydraulic (CFLHD) and transport (CFLAD) equations. These CFL numbers for the hydrodynamic and transport equations are calculated using Eqs. (10) and (11), as follows:

wherehis the total water depth,uandvare the velocity components in thexandydirections, respectively,gis the gravitational acceleration, Δxand Δyare the characteristic length scale of the elements in thexandydirections, respectively, and Δtis the time step interval. The characteristic length scales, Δxand Δy, are approximated by the minimum edge length for each element, and the water depth and the velocity components are evaluated at the element center (DHI, 2011).

2.2.3 Turbulence modeling

Turbulence is the irregular movement of fluids, including gases and liquids, as the fluid emerges from a solid surface or a neighbor in the fluid flow. Turbulent flows have large Reynolds numbers. A high Reynolds number is a relative concept. For low viscosity liquids, such as water and air, a high Reynolds number is defined as anything greater than tiny twists. Most methods for this type of analysis involve linearization equations of motion. Although this approach has achieved some success, the nonlinearity of Navier-Stokes equations precludes analytical descriptions of significantly disturbed phenomena. In real or viscous fluids, instabilities occur in the interaction between the inertia and viscosity terms of the Navier-Stokes equation. These interactions are very complex, rotational, three-dimensional, and time dependent (Wilcox, 1994). Smagorinsky proposed the expression of sub-grid scale transport by the relationship of effective eddy viscosity to a characteristic length scale. The sub grid scale eddy viscosity is defined in Eq. (12), as follows:

whereCSis Smagorinsky constant (the Smagorinsky constant is selected between 0.25 to 1), andLis the characteristic length and strain rate is shown in Eq. (13) (Smagorinsky, 1963):

2.2.4 Analysis process

The mesh generation for the computational domain is one of the most significant steps required in engineering analysis. The mesh generation must be independent of the computation, which implies that the computational mesh should be the greatest mesh size, and that by making it finer, the results do not change significantly. Next, the sensitivity analysis phase is considered. At this stage, the model-sensitive parameters are identified. After determining the sensitive parameters, the calibration phase begins. At this stage, at every step, while keeping constant all other parameters, sensitive parameters are calibrated by comparing the results with those from the experimental work. After calibrating all sensitive parameters, the model is then validated by the experiment results. At this point, one analysis has been conducted. The comparison of analysis results of this phase is then conducted for the rest of the experimental results not used in the calibration phase. If the comparison of the analysis results with the experimental data reveal significant faults, the sequence of choosing the calibration parameter must be modified.

Groin structures include downstream, upstream, and right-angle types. For cases with inner velocities (Uin) of 0.25 m/s and 4.0 m/s, there were five cases wherein the arm length (La) sizes were directed against the body of the groin (La/L)—0.2, 0.4, 0.6, 0.8, and 1—which were specified as cases 1 to 5, respectively. A total of 30 cases were investigated. The right-angle body length of the groin was considered to be 0.3 meters in all cases (Fig. 6), and the groin width was assumed to be 4 cm.

With the current regime, which was subcritical and turbulent, we applied the velocity and surface elevation as the inner and outer boundary conditions, respectively. Because we used a reduced channel length, we derived the results of the overall channel length analysis, where the velocity at the entrance and the surface elevation profile produced by the real channel length were applied as the boundary of the reduced channel length. We applied a zero velocity condition at the land boundaries (channel walls). The symbols used in each analysis for the shape, type, and physical parameters of the groins are listed in Table 1. URG, DRG, and RG refer to upstream, downstream refractive and right-angle groins, respectively. URGLD and DRGLD refer to a right-angle groin in dense flow, in which the physical parameters and boundaries are the same as in the RG cases. These results were compared with the upstream and downstream refractive groins. Fig. 6 shows the hydraulic channel, the location of the groin in the channel, the channel length, width, and groin geometry dimensions, and the channel water depth.

Fig. 6 Scheme and dimensions of the groin (Kanget al., 2011)

Table 1 Model characteristics

3 Results

3.1 Velocity profile qualities

The flow at the channel entrance was fully developed. Due to the non-slipping condition of the velocity at the channel wall, it shows zero velocity. Fig. 7 shows the velocity profiles at various distances and at various locations. After encountering the groin, since the flow rate was constant and there was negative velocity, the maximum velocity increased. Before the flow encountered the groin, we observed its influence on the thalweg line and the location of the maximum rates of the currents and vortices.

3.2 Assessment of velocity contours

A derivation of the flow pattern was required to identify areas susceptible to erosion and the deposition of sediment. Based on the flow characteristics, we were able to identify the areas susceptible to erosion, and can illustrate these areas qualitatively. Notice that the order of the terms DRG (URG)xvydenotes a downstream or upstream groin for whichyis the input speed in centimeters per second andxdenotes the groin arm length, for modes 1 to 5 (Table 1).

The analysis for the right-angle groin was carried out with input velocities of 0.25 m/s and 0.4 m/s for the five projection lengths of the refractive groin—a total of 10 cases. The main purpose here is to compare the basic feature results with the analysis results, which was performed for a refractive groin. In other words, by placing a refractive groin with a right-angle groin of a projected length, we could check the accuracy of our results.

Since the experimental work by Kanget al. (2011) was carried out around a refractive groin, rather than a right-angle groin, the flow characteristics produced from this analysis was based on the experimental work. We performed this analysis to evaluate which flow characteristics showed less discrepancy with experiments for a right-angle groin. It is not surprising that some of the flow characteristics of the right-angle groin showed great discrepancy with those from around the refractive groin by Kanget al. (2011). Therefore, if some of the flow characteristics show acceptable results that are more similar to those of the experiments, the idea to replace the right-angle groin with a refractive groin of the same projected length would be reasonable, given those characteristics.

Two vortices were created after the flow passed the groin, and the intensity of the secondary vortex in the right-angle groin was generally weakened. The intensity of the secondary vortex in the refractive groin plays a significantrole in the deposition pattern. Moreover, with the presence of the groin arm, the maximum speed occurred at a significant distance from the groin tip. This is resulted in a smaller boundary layer near the groin on the opposite wall. The length of the secondary vortex tended to increase with increasing arm length of the refractive groin. Figs. 8 to 13 show the velocity contours and streamlines for the downstream and upstream refractive groins and the right-angle groin.

It was obvious that three vortex centers were created due to the refractive groin, and that the URG had greater intensity than the DRG groin.

Fig. 7 Sketch of velocity profile qualities for the URG groin

Fig. 8 Velocity contour lines of DRG05v40

Fig. 9 Stream line of DRG05v40

Fig. 10 Velocity contour lines of URG05v40

Fig. 11 Stream line of URG05v40

Fig. 12 Velocity contour lines of RG01v25

Fig. 13 Stream line of RG01v25

Some significant points may be drawn from Figs. 8 to 13. Due to groin conduction, three vortex centers were created. These vortices play a major role in the associated erosion and sedimentation processes. Groin types and shapes have a significant impact on the size and number of vortices. In right-angle groins, by reducing the groin length, smaller vortices tend to disappear and afterwards, only one vortex center remains. In refractive groins, the type of the downstream or upstream refractive groin has a major role in vortex generation. In the downstream refractive groin, the vortex that is created prior to encountering the groin has a more critical value than the secondary vortex that appears after the groin. In the upstream refractive groin, the vortex created after the flow encounters the groin is the more critical. Therefore, the numbers and size of the vortices are important indicators of the sediment transport processes.

3.3 Barotropic flow characteristics

In this section, we compare the flow characteristics, including the length and width of the recirculation area, the maximum velocity, the thalweg height, and the separation angle of the groin tip with those in the experiments. In Figs. 14(a) to 14(e), the horizontal axis is the interval error and the vertical axis is the error of the frequency percentage of the desired state. Frequency error refers to the comparison of the analysis results with the experimental results in all cases, and the error range is the discrepancy interval. For example, in Fig. 14(a), 15.79% of the total cases (20 refractive groin analyses) for DRG groins show less than 10% discrepancy in separation length characteristic. DRGL and URGL results show characteristics similar to right-angle groins and were compared with experimental results for downstream and upstream groins, respectively.

Fig. 14 Comparison of flow characteristics with those from experimental work (Kanget al., 2011) on barotropic flow

Thus, the geometry of groins, in terms of being right-angled or refractive, has little effect on the maximum speed to the relative average speed. Apart from the angular separation, the arm length of the groin in the DRG groin had less effect than that in the URG groin.

3.4 Baroclinic flow characteristics

3.4.1 Density dispersion

In this section, density is assumed to be a function of salinity and temperature, and that these variations have changed over time and space. The flow is still incompressible and pressure is assumed to be hydrostatic.

The execution model is the temperature-salinity (TS) model in which eddy viscosity and shear stress change due to density variations. The TS model sets up additional transport equations for temperature and salinity. Additionally, the calculated temperature and salinity are fed back to the hydrodynamic equations via buoyancy forcing induced by density gradients. We used the horizontal dispersion formulated using a scaled eddy viscosity formulation, in which the dispersion coefficient is calculated as the eddy viscosity in the flow equation solutions multiplied by a scaling factor. We determined the density, salinity, and temperature distributions associated with the right-angle groin. The boundary conditions were considered to be 25 PSU for salinity and 25 °C for the inlet temperature. We found that the factors of salinity, temperature, and density do not result in large changes at the back side of right-angle groins (Fig. 15), but over the groin area, two vortex centers formed. These vortex centers had minimum density, salinity, and temperature values. We numbered the vortex centers according to their proximity to the groin sequentially as 1 and 2. The values of the groin’s vortex centers are indicated as min1 and min2. Moreover, the values of the vortex centers that refer to either salinity, temperature, or density are indicated by min1(2)s(t or d)1(2). The second 1 or 2 modes indicate the input velocity of either 0.25 m/s or 0.4 m/s, respectively. Figs. 16(a) to 16(c) show the distance of the minimum vortex centers of salinity, temperature, and density from the groin. Fig. 17 shows the values of these parameters.

According to Fig. 15, the best groin performance is obtained when two vortices flow as close to each other as possible.

From Figs. 16(a) to 16(c), we see that by increasing the inner velocity, the vortex centers of salinity, temperature, and density that are close to the groin (No. 1) have travelled a shorter distance from the groin and the further vortex centers (No. 2) are farther away from the groin.

According to Figs. 17(a) to 17(c), the inner velocity value of 0.4 m/s and the minimum salinity, temperature, and density values are obviously unchanged by the increased length of the groin, as indicated by the further centers of the vortex. With respect to the inner velocity value of 0.25 m/s, by increasing the groin length, the minimum values of the variables at the center of the groin are reduced. However, forthe inner velocity value of 0.4 m/s, the salinity and density at the vortex centers closer to the groin have the lowest values in case 4, with a length of 0.47 m.

Fig. 15 Density distribution of RGd04v25

Fig. 16 Distance of vortex centers from right-angle groin

Fig. 17 Minimum values at the vortices centers

3.4.2 Flow characteristics of dense flow

As mentioned above, when the density is baroclinic and varies in time and space, the hydrodynamic model should be updated with observations of the dispersion feedback, but the extent of its influence on the flow characteristics is unclear. In this study, we examined the effects of dense flow on the flow characteristics in right-angle groins and compared their barotropic conditions. In fact, we have conducted and compared two types of analyses with the experimental case studies of the refractive groin. Note that the order of the URG(DRG)L and URG(DRG)LD refers to the results of the right-angle groin, as compared with the URG(DRG) results in barotropic and baroclinic conditions for the refractive groin, respectively. In Fig. 18, we compare the flow characteristics around the right-angle groin for both baroclinic and barotropic flow conditions with the experimental refractive groin results produce by Kanget al. (2011) of the associated flow characteristics. Because of the geometry differences, errors are expected, but there is actually only low error in one of the flow characteristic results which illustrate this paper’s purpose.

Fig. 18 Comparison of flow characteristics with experimental results for right-angle groins (Kanget al. 2011) with baroclinic flow

In Fig. 18(c), the maximum velocity per mean velocity in the right-angle groin shows good agreement with Kanget al.’s experimental work for the refractive groin. Also, the frequency error is almost identical in baroclinic and barotropic conditions. In the DRG groin, the parameters of separation height and the flow separation angle in dense flow are better predicted. In the URG groin, in addition to the above parameters, separation length was better predicted than in barotropic flow conditions. Also, predictions of the maximum velocity parameters were almost the same in the two types of analyses. Due to the large error in the thalweg height parameter prediction, the replacement of refractive groins with a right-angle groins does not show good accuracy for this parameter.

4 Discussion

As flow characteristics were acquired, we plotted dimensionless charts, as per the recommendation of the reference case study. Each of these diagrams, including the schematic comparisons with the experimental results, have been estimated with a linear equation (Figs. 19 to 22). In some cases, the correlations are not good approximations of the linear equation, but the relationship between the thalweg height and the aspect ratio of the groin arm length to the projection of groin length is relatively significant.

Thus, by increasing or decreasing the ratio of the arm length to the projection of the groin, regardless of the input rate variation, the thalweg height increases and decreases,respectively. Therefore, we used Eq. (14) to correlate and estimate this parameter. As shown in Fig. 23, Eq. (14) was achieved:

whereTCL,B,LaandLare the thalweg height, channel width, groin arm length, and projected length of groin, respectively.

Fig. 19 Variation of separation angle to arm length rate

Fig. 20 Variation of separation height rate to arm length rate

Fig. 21 Variation of separation length rate to arm length rate

Fig. 22 Variation of thalweg height rate to arm length rate

Fig. 23 Nonlinear variation of thalweg height rate to arm length rate

5 Conclusions

In this study, we investigated the flow pattern around refractive and right-angle groins with respect to barotropic and baroclinic flows and then compared the flow characteristics for different scenarios. Also, with respect to baroclinic flow, we studied the dispersion of salinity, temperature and density around right-angle groins with different input velocities. In the case of the barotropic flow, by introducing some dimensionless parameters and then comparing the flow characteristics and geometries of the channel and groin, we found a significant relationship in one of the cases. The results from this paper can be generalized to coastal areas under special conditions outlined in the Shore Protection Manual (Coastal Engineering Research Center, 1984). Finally, based on the analysis results, we make the following conclusions:

1) The dimensions of the recirculation area tend to be larger for increased refractive-groin arm length.

2) Groin geometry, with respect to right-angle or refractive types, had few effects on the maximum velocity to mean velocity ratio among the flow characteristics.

3) Thalweg height increased with increasing arm length at different input velocities. A maximum increase in the input velocity of 0.4 m/s and 0.25 m/s, yielded 9% and 8% increases in the DRG groin and 8% and 8% increases in the URG groin, respectively.

4) In a baroclinic current, by increasing the inner velocity, the vortex centers of salinity, temperature, and density close to the groin had a shorter distance from the groin and a greater distance from the further vortex centers.

5) By increasing the right-angle groin length, with a 0.4 m/s input velocity, the minimum values of salinity, temperature, and density at the vortex center that were far away from the groin were appreciably unchanged, but the minimum of this value in the vortex center and the input speed of 0.25 m/s was reduced by increasing the groin length.

The analysis in baroclinic conditions compared to barotropic conditions showed that the some of the flow characteristic parameter values in right-angle groins were similar to those in refractive groins.

The relationship between the parameters of the thalwegheight and the aspect ratio of the arm length to projected length were diagnosed as being fairly significant. We found a nonlinear relationship between these parameters. Regardless of the input velocity, by increasing the ratio of the arm length to the projected length of the refractive groin, thalweg height increased.

Nomenclatures

URG Upstream refractive groin

DRG Downstream refractive groin

RG Right-angle groin

URGL Right-angle groin in comparison with URG

DRGL Right-angle groin in comparison with DRG

URGLD Right-angle groin in comparison with URG in baroclinic currents

DRGLD Right-angle groin in comparison with DRG in baroclinic currents

RGD Right-angle groin in baroclinic currents

LGroin right-angle length

LaGroin arm length

L′ Groin projected length

dWater depth

θArm angle of refractive groin

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10.1007/s11804-015-1325-7

1671-9433(2015)04-0366-10

Received date: 2015-02-10.

Accepted date: 2015-05-15.

Foundation item: Partially Supported by the Department of Science and Technology Through a Research Grant to RG (No. SR/FTP/MS-020/2010). *Corresponding author Email: rupanwita@maths.iitkgp.ernet.in

? Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2015

Received date: 2015-02-13.

Accepted date: 2015-09-01.

*Corresponding author Email: mojtahedi@tabrizu.ac.ir

? Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2015