Numerical Simulation of Two-Phase Flow for Wave Propagation andBreaking near Submerged and Vertical Breakwaters

R. Bakhtyar1,2, C.E. Kees1, M.W. Farthing1, and C.T. Miller21U.S. Army Engineer Research and Development Center; 2University of North Carolina at Chapel Hill

IntroductionSubmerged breakwaters avoid the generation of major reflected waves that affectthe beach. They are very useful for erosion control and beach protection by de-creasing the wave height[7]. In addition, standing waves can play a significant rolein hydrodynamics near the vertical breakwater. However, an accurate understandingof the wave propagation/breaking over a submerged breakwater, and wave overtop-ping in front of a vertical breakwater has yet to be achieved.

Model EquationsThe Navier-Stokes (NS) equations are used to model Newtonian and incompress-ible turbulent flow for the time interval [0,Ts] and a physical domain Ω, which arespecified as [5]:

The general model formulation is specified by an overall conservation of mass equa-tion given as

∂ρ

∂t+∇· (ρv) = 0, Ω ∈ IR2, t ∈ [0, Ts] (1)

and the general NS equation for the case of a constant dynamic viscosity given by

∂(ρv)

∂t+∇· (ρv ⊗ v) +∇p− 2µ∇·d +

2

3µ∇ [(I:d) I]− ρg = 0,

Ω ∈ IR2, t ∈ [0, Ts] , (2)

and the rate of strain tensor is defined as

d =1

2

[∇v + (∇v)T

](3)

where ρ is the density, t is time, v is the fluid velocity, Ts is the extent of the temporaldomain, p is the fluid pressure, µ is the dynamic viscosity, I is the identity tensor,and g is the gravitational acceleration vector.

For the case in which the material derivative of the density vanishes, the so-calledincompressible case, the conservation of mass equation reduces to

∇·v = 0, Ω ∈ IR2 (4)

and the incompressible NS equation can be written as

ρ∂v

∂t+ ρv·∇v +∇p− µ∇2v − ρg = 0,

Ω ∈ IR2, t ∈ [0, Ts] . (5)

Turbulence modeling The inner surf and swash zones are complex regions ofturbulent flow [8]. Since turbulence has a profound influence on the flow field andmomentum transfer, it must be modeled [1]. Because of the complexity of turbulentflows, direct numerical simulation (DNS) to resolve all length scales of the flow fieldis typically not feasible for the surf and swash zones.

A popular alternative to DNS is modeling the conservation of momentum usingthe Reynolds Averaged NS (RANS) equation, which yields an expression for the av-erage velocity but depends upon a modeling approximation of the dyadic product offluctuation velocities that arise in the Reynolds stress tensor. The RANS approachthus produces an estimate of the average velocity, while approximating the Reynoldsstress effects on this velocity. A key point is that RANS models only yield averagedbehavior.

A middle ground between DNS and RANS are approaches that attempt to re-solve the velocity directly, similar to DNS, but only over a set of dominant energy-containing length scales. The shorter length scale effects are modeled using aclosure scheme for the stress contributions due to velocity fluctuations, similar toa RANS approach. Large eddy simulation (LES) is an approach that resolves arange of length scales known to contain the majority of the kinetic energy in the sys-tem, while approximating the unresolved scales by modeling the stress contributionsusing a filtered velocity field. The LES model thus provides a means to simulate im-portant, highly energetic, aspects of the velocity field, while still greatly reducing thecomputational burden associated with a DNS approach.

In the LES model, the velocity is written in filtered form as

v = v + v′ = 〈v〉 + v′ , (6)

where an implicit spatial filter resulting from the numerical approximation is used,and v′ is the fluctuation velocity. Applying the spatial filter to Eqn (4) yields

∇·v = 0 , (7)

and applying the spatial filter to Eqn (5) and simplifying yields the filtered incom-pressible NS equation given by

∂v

∂t+∇·vv +

1

ρ∇p− ν∇2v − g = 0 , (8)

where ν is kinematic viscosity.

Define the sub-grid scale stress tensor as

τ SGS = vv − v v , (9)

and combine with Eqn (8) to give the filtered NS equation of the form

∂v

∂t+ v·∇v +∇·τ SGS +

1

ρ∇p− ν∇2v − g = 0 . (10)

The Smagorinsky model was used to approximate the sub-grid-scale stresstensor as

τ SGS = −2νTd , (11)

where eddy viscosity is approximated as

νT = (Csh)2(2d:d

)1/2, (12)

the filtered rate of strain tensor is

d =1

2

[∇v + (∇v)T

], (13)

h is the grid spacing in the numerical approximation, and a dynamic approximationof the Smagorinsky constant is computed as [9]

CS =

0 Reh ≤ 1

0.027× 10−3.23Re−0.92

Reh > 1(14)

and the grid Reynolds number is

Reh =h2(2d:d

)1/2ν

. (15)

Free surface tracking There are several existing techniques to capture the air-water interfaces. A hybrid level set/Volume Of Fluid (LS-VOF) method denotes oneof the approaches for tracking air-water interface that has been used effectively tosolve problems in various water resources fields [3, 10, 12]. In the VOF method (Fig.2a), the filled fraction of computational cells is used for defining the interface [4]. Inthe LS technique (Fig. 2b) the interface is defined implicitly as the zero level set ofscalar function [11] and, consequently, no mesh adjustment is needed to define theair-water interface (i.e., level sets with zero, positive and negative values representair-water interface, water and air phases). Accurate and suitable approximations offree surface are obtained, by coupling the LS and VOF methods [6]. A hybrid LS-VOF method is an Eulerian free surface technique with high-order estimations, andis very compatible with the NS type equations. A precise free surface tracking with asatisfactory computational cost can be obtained using a hybrid LS-VOF technique.This hybrid method reduces the mass conservation errors that exist in the classicallevel set methods.

In air-water two-phase flow, the level set governing equations for determiningthe free surface can be written as follows:

∂ψ

∂t+ v·∇ψ = 0 , (16)

‖∇ψ‖ = 1 , (17)

ρ = ρa [1−Hf (ψ)] + ρwHf (ψ) , (18)

µ = µa [1−Hf (ψ)] + µwHf (ψ) (19)

and

Hf (ψ) =

0 ψ < 00.5 ψ = 01 ψ > 0

, (20)

where ψ is a level set function that is used to describe the fluid distribution; ρa andµa are the density and dynamic viscosity of air, respectively; Hf is the Heavisidefunction; and ρw and µw are the density and dynamic viscosity of water, respectively.Kees et al. [5] showed that the LS method, in a complex system like wave breakingprocesses, cannot normally conserve mass and therefore inadequately predicts thefree surface dynamics. In this study, a hybrid LS-VOF method was used in order tofurther preserve mass conservation, which results in the level set equation

∂ψ

∂t+∇· (ψHfv) = 0 . (21)

A detailed formulation of the equations and numerical methods is given by Keeset al. [5].

Prior Experimental ResultsThe numerical model is validated alongside the two data sets [2, 13]. The numericalresults show that a vortex forms when waves spread over a submerged breakwa-ter. The numerical results also indicate that the maximum turbulent kinetic energy,velocities and viscosity occur on the top of breakwater. The results show that forfully standing waves, velocities and wave height are higher in comparison to thecomparable partially standing wave case (for the vertical breakwater). These cal-culations are in overall agreement with earlier observations, while the numericalmodel describes the water and air phase characteristics in greater detail than cur-rent measurements. The model provides a valuable method to advance mechanisticunderstanding of hydrodynamic characteristics near the breakwater in the nearshorearea.

38.0m2.1 32.9 3.0

0.6m

wall

sand bed1:30

0.450.15

h=0.3

6.0wavepaddle

wavescreens

Figure 1: Small experimental tank from [13]

Numerical Results

Figure 2: proteus-mprans numerical results for the experiment from [13]

Figure 3: proteus-mprans numerical reults for [2]

Air/Water Flow Validation Test SetThe use of turbulent two-phase flow models is still relatively imature for applica-tions in coastal and hydraulic structures. In order to verify and validate the proteus-mprans model and place computational modeling on a firmer footing for these appli-cations, we have begun developing a test set spanning dambreak, wave, and flowprocesses in two- and three-dimensions. The test set can be accessed by contact-ing the authors.

Figure 4. The air-water-vv project on github.

References[1] Bakhtyar, R., D. A. Barry, A. Yeganeh-Bakhtiary, and A. Ghaheri, 2009: Numerical simulation of surf-swash zone motions and turbulent flow.

Advances in Water Resources, 32 (2), 250–263.

[2] Chen, J., C. Jiang, S. Hu, and W. Huang, 2010: Numerical study on the characteristics of flow field and wave propagation near submergedbreakwater on slope. Acta Oceanologica Sinica, 29 (1), 88–99.

[3] Farthing, M. W. and C. E. Kees, 2008: Implementation of discontinuous Galerkin methods for the level set equation on unstructured meshes.Tech. Rep. ERDC/CHL CHETN-XIII-2.

[4] Hirt, C. W. and B. D. Nichols, 1981: Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of Computational Physics,39 (1), 201–225.

[5] Kees, C. E., I. Akkerman, M. W. Farthing, and Y. Bazilevs, 2011: A conservative level set method suitable for variable-order approximationsand unstructured meshes. Journal of Computational Physics, 230 (12), 4536–4558.

[6] Kees, C. E. and M. W. Farthing, 2011: Parallel computational methods and simulation for coastal and hydraulic applications using the Proteustoolkit. PyHPC11 Workshop, Supercomputing 11.

[7] Kobayashi, N., L. Meigs, T. Ota, and J. Melby, 2007: Wave transmission over submerged porous breakwaters. ASCE Journal of Waterway,Port, Coastal and Ocean Engineering, 133 (2), 104–116.

[8] Longo, S., M. Petti, and I. J. Losada, 2002: Turbulence in the surf and swash zones: A review. Coastal Engineering, 45, 129–147.

[9] Mattis, S. A., C. N. Dawson, C. E. Kees, and M. W. Farthing, 2012: Numerical modeling of drag for flow through vegetated domains andporous structures. Advances in Water Resources, 39, 44–59.

[10] Osher, S. and R. Fedkiw, 2001: Level set methods: An overview and some recent results. Journal of Computational Physics, 169, 463–502.

[11] Sethian, J. A., 1999: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics,Computer Vision, and Materials Science. Cambridge University Press, New York.

[12] Sethian, J. A., 2001: Evolution, implementation, and application of level set and fast marching methods for advancing fronts. Journal ofComputational Physics, 169, 503–355.

[13] Xie, S., 1981: Scouring pattern in front of vertical breakwaters and their influence on the stability of the foundation of the breakwaters. M.S.thesis, Civil Engineering, TU Delft.