Verification of the LOGOS Software Package for Tsunami Simulations

geosciences

Article Veriﬁcation of the LOGOS Software Package for Tsunami Simulations

Elena Tyatyushkina 1,2, Andrey Kozelkov 1,2,3, Andrey Kurkin 2,*, Eﬁm Pelinovsky 2,4 , Vadim Kurulin 1, Kseniya Plygunova 1 and Dmitry Utkin 1

1 Russian Federal Nuclear Center—All-Russian Research Institute of Experimental Physics, Nizhny Novgorod Region, 607188 Sarov, Russia; [email protected] (E.T.); [email protected] (A.K.); [email protected] (V.K.); [email protected] (K.P.); [email protected] (D.U.) 2 Department of Applied Mathematics, Nizhny Novgorod State Technical University n.a. R.E. Alekseev, 603155 Nizhny Novgorod, Russia; [email protected] 3 Sarov Institute of Physics and Technology, National Research Nuclear University MEPhI, Nizhny Novgorod Region, 607188 Sarov, Russia 4 Institute of Applied Physics, Russian Academy of Sciences, 603155 Nizhny Novgorod, Russia * Correspondence: [email protected]

Received: 22 August 2020; Accepted: 22 September 2020; Published: 26 September 2020

Abstract: Verification results for the LOGOS software package as applied to numerical simulations of tsunami waves are reported. The module of the LOGOS software package that is used for tsunami simulations is based on the numerical solution of three-dimensional Navier–Stokes equations. The verification included two steps. The first step involved the verification of LOGOS free-surface flow simulations on the test cases of a collapsing water column and gravity water sloshing in a tank and the known test cases of wave generation by objects falling into water or lifted out of it. The verification of LOGOS specifically for tsunami simulations was performed using a reference set of international benchmarks including the propagation and run-up of a single wave onto a flat slope and a vertical wall, the sliding of a wedge-shaped body down a slope, flow around an island and wave run-up over an obstacle. The results of the verification simulations demonstrate that LOGOS provides sufficient accuracy in numerical simulations of tsunami waves, namely, their generation, propagation and run-up.

Keywords: Navier–Stokes equations; VOF method; numerical simulation; free surface; tsunami waves; LOGOS software package

1. Introduction The adequate description of the generation, propagation and run-up of tsunami waves is one of the most critical issues in the problem of marine natural disasters [1–8]. The immediate cause of most tsunami waves is a change in the bottom conﬁguration as a result of an earthquake leading to large slumps, gaps, etc. Other causes of tsunami waves include landslides, volcanic eruptions and meteorological sources, rockslides into water and falls of celestial bodies [9–11]. Present-day methods of tsunami studies are most often based on the shallow-water theory and its disperse extensions. An overview of the existing analytical solutions within this theory is provided in [9,12]. The numerical implementation of the non-linear shallow-water equations [13,14] enabled simulations of tsunamis in numerous historical events and delivered reasonable assessments [8,15]. In all these cases, two-dimensional partial diﬀerential equations were solved. Despite the progress achieved, calculations of tsunami characteristics are rather challenging, because tsunami waves tend to be severely non-linear and highly dispersive. For seismic and volcanic

Geosciences 2020, 10, 385; doi:10.3390/geosciences10100385 www.mdpi.com/journal/geosciences Geosciences 2020, 10, 385 2 of 28 tsunamis, these processes are particularly important in the run-up phase, while for tsunamis generated by falling celestial bodies, they also need to be taken into account near the source of tsunami. Three-dimensional Navier–Stokes equations [16], which are now at an early stage of their use for tsunami simulations [7,10,17–19], represent the most exhaustive set of equations that capture tsunami details in all tsunami phases, from outbreak through to run-up. Before conducting a numerical experiment, one should identify the range of validity, where a computational physics model is able to simulate a physical phenomenon under its natural conditions. The validity of a computational physics model designed to describe one process or another can be established by measuring the error and uncertainty it introduces relative to a real experiment. The quantification of the uncertainty and error of a computational physics model can be performed by testing (validation). Validation involves the testing of model outputs by their comparison with experimental data to establish whether the numerical simulation results comply with real physics. The basic validation and verification principles for Computational Fluid Dynamics (CFD) methods are described in [20,21]. There is an international set of tsunami simulation benchmarks called NTHMP (National Tsunami Hazard Mitigation Program) [22]. These benchmarks include the propagation and run-up of a single wave in a constant-depth tank, propagation and run-up of a single wave in a varied-depth tank, sliding of a wedge-shape body down a slope, and flow around an island. This set of benchmarks offers data to test whether the computational physics model can correctly describe the process of tsunami propagation and run-up. It contains wave gage data, based on which one can quantify the error in mathematical simulation relative to an experiment. Benchmarks from this set and an additional test case of a wave running up over an obstacle [23] were used to verify our software package LOGOS. Simulations of cosmogenic tsunamis by LOGOS were verified on internationally recognized test cases, such as waves generated by a body falling into water [24] or waves induced by lifting a body out of water [25]. This paper presents verification results for LOGOS simulations of free-surface problems in general and tsunami waves in particular [26–28]. LOGOS is a three-dimensional multi-physics code for convective heat and mass transfer and aerodynamic and hydrodynamic simulations on parallel computers that is already used for tsunami simulations [7,10,18,19]. To assess how accurate LOGOS simulations are, below, we present LOGOS verification results for free-surface problems, including the collapse of a liquid column, gravity water sloshing in a tank, the known problems of wave generation by bodies falling into water or lifted out of water, and tsunami simulations using the NTHMP benchmark problems [22] with an additional test case of wave run-up over an obstacle. We show that the results calculated by the LOGOS software package closely match both analytical and experimental data and results obtained by other authors.

2. Governing Equations and Computational Method Let us consider a system of air and water as a set of two interfaced incompressible media. We use the one-velocity approximation, in which water and air have the same continuity and momentum equations solved for the resulting medium whose properties are linearly dependent on the volume fraction [29]. This approach is quite common and provides good results in calculations of free-surface problems [30,31], including those of tsunami waves [7,10,17]. The dynamics of this system are described by incompressible Navier–Stokes equations, including a continuity equation, a momentum equation, and an equation for volume fractions of the phases [16,19,29]:   →u = 0,  ! ∇·  P P P  ∂ (k) (k) → = (k) (k) → → + (k) (k) 2→ + →  ∂t ρ α u ρ α u u µ α u p gρ, (1)  k − k ·∇ k ∇ − ∇   ∂ α(k) + α(k)→u = 0. ∂t ∇· Geosciences 2020, 10, 385 3 of 28

Here, →u is the three-dimensional velocity vector, ρ(k) is the density of phase k, α(k) is its volume P fraction ( α(k) = 1), p is the pressure, µ(k) is the molecular viscosity of phase k, and →g is the acceleration k due to gravity. This system is solved without using the Reynolds averaging and subsequent closure of turbulence moment equations, i.e., turbulence is resolved by Direct Numerical Simulations (DNS). This allows resolving turbulent structures as small as the mesh size. Note that resolving only large eddies relaxes the fairly strict requirements for the scheme viscosity and dissipation of the numerical schemes to be used for the modeling of turbulent flows [27]. Equation (1) is discretized by finite volumes on an arbitrary structured mesh and solved numerically by a fully implicit method [21,32,33] based on the known SIMPLE algorithm. Free-surface flow modeling involves certain modifications of the SIMPLE algorithm, which are described in detail in [19,32–34]. To describe the finite-volume discretization procedure, let us consider a transfer equation for a scalar quantity. The equation has the form

∂ρφ ∂ ∂ + ρφuj = τij (2) ∂t ∂xj ∂xj

Here, ϕ is a scalar quantity and τij is a stress tensor. The ﬁnite-volume method provides for a transition from diﬀerentials to algebraic expressions withGeosciences respect 2020 to, 10 mesh, x FOR cells. PEER ConsiderREVIEW two adjacent cells of an unstructured mesh as shown in Figure4 of 129.

Figure 1. Two neighbor cells of a computational mesh. Figure 1. Two neighbor cells of a computational mesh. The points i and m are the centers of the adjacent cells; k is a set of all faces of the i-th cell comprised The discretization of the diffusion term in Equation (2) is performed using the over-relaxed of a set of its interior faces k and a set of its exterior faces ks. The vector from the center of the i-th cell approach [35], which servesint for non-orthogonal mesh discretization. As a result of the discretization, to the center of the m-th cell along the face k is denoted by d = r rm. The vector from the center the diffusion term takes the form int i,kint i − of the i-th cell to the center of face ks is denoted by di,ks = rks ri, where ri is the radius vector. − α The time derivative in Equation()()μ u S (2)* − is approximatedμ u S* + ()μ by∇ theu second-orderS + scheme  m m k  i i k  k k k,i k k k I n+1 n n 1 ∂φ∂ 3φ 4φ + φ − (5)   uj   i 2  i  ∂ui   + μ  dVi   + − μ  k V iδ  (3) i ∂t nj ≈Sk 2∆t i nj ijSk ,   ∂x   3   ∂x   k Vi  i k  k   k k  where the superscript n denotes the time step number, the subscript* = Sk,iSikdenotes,i α the= i-th− cell,* and Vi is the where dk,j is the distance from the cell center to the face k; Sk ; Sk,i Sk,i Sk . volume of the i-th cell. dk,jSk,j TheThis spaceset of discretization equations must of the be derivativessupplemented in Equation by boundary (1) includes conditions. their integrationFor an incompressible over the cell volumeflow, if Vthei and boundary transition of using the thecomputational Gauss–Ostrogradsky domain formulaconsists to of surface solid integrals,walls, all which the arevelocity then representedcomponents asat athe sum boundary of volume must ﬂuxes be taken across as the equa faces:l to their corresponding velocity components of the solid surface; i.e., the fluidZ can neither slipZ along the fluid/solid interface nor move normally to it. ∂ X X The velocity and shearρujφ stresses= ( ρatu iφthe)dV fluid/fluidi = ρφ uinterfacejdSj mustρkφk ubej,k Scontinuous.j.k = ρkφ AirkFk at the upper(4) ∂xj ∇ ≈ k k boundary has zero static Vpressure.i The discreSj tized equations above incorporate the following physical boundary conditions:

  ∂p − = = 0 Inlet—incoming-flow boundary: u uinl , ∂ ; n  ∂u − Pressure—given-pressure boundary: pp= , = 0 ; press ∂n  ∂p − Wall—rigid boundary: u = 0 , = 0 ; ∂n  ∂p ∂u − Symmetry—symmetry plane: = 0 , = 0 . ∂n ∂n Thus, in its discretized form, the equation of continuity is written as

 α ∂ρ n+1 − n + n−1 α   = −  ξ ξ 3pi 4pi pi + ξ ρ −ρ  ui,kSi,k Vi  ξ,kui,kSi,k ξ ui,kSi,k    ρξ ∂p 2τ ρξ   k ξ   k k 

∂ρξ which, considering the equality ρξ = p and the boundary conditions, takes the final form ,i ∂p i

Geosciences 2020, 10, 385 4 of 28

where Sj,k is the surface area of the face k and Fk is the volume flux across the face k of the i-th cell. The value of the quantity ϕk is determined by the numerical scheme being used. The discretization of the diffusion term in Equation (2) is performed using the over-relaxed approach [35], which serves for non-orthogonal mesh discretization. As a result of the discretization, the diffusion term takes the form P P P u S u S + u Sα + µm m k∗ µi i k∗ µk k k,i k − k k h∇ i ∂u (5) +P j + 2 P ∂uk µi ∂x nj Sk 3 µi ∂x nj δijSk , k i k k k k

Sk,iSk,i α where dk,j is the distance from the cell center to the face k; S∗ = ; S = Sk,i S∗. k dk,jSk,j k,i − k This set of equations must be supplemented by boundary conditions. For an incompressible flow, if the boundary of the computational domain consists of solid walls, all the velocity components at the boundary must be taken as equal to their corresponding velocity components of the solid surface; i.e., the fluid can neither slip along the fluid/solid interface nor move normally to it. The velocity and shear stresses at the fluid/fluid interface must be continuous. Air at the upper boundary has zero static pressure. The discretized equations above incorporate the following physical boundary conditions:

∂p → = → = – Inlet—incoming-ﬂow boundary: u u inl, ∂n 0; = ∂→u = – Pressure—given-pressure boundary: p ppress, ∂n 0; ∂p → = = – Wall—rigid boundary: u 0, ∂n 0; ∂p = ∂→u = – Symmetry—symmetry plane: ∂n 0, ∂n 0. Thus, in its discretized form, the equation of continuity is written as    3pn+1 4pn + pn 1 X Xαξ ∂ρξ i i i − αξ X X  u S =  − V +  ρ u S ρ u S  i,k i,k −  ρ ∂p 2τ i ρ  ξ,k i,k i,k − ξ i,k i,k k ξ ξ ξ k k

= ∂ρξ which, considering the equality ρξ,i ∂p pi and the boundary conditions, takes the ﬁnal form   n+1 n n 1 n+1 P P αξ ∂ρξ 3p 4p +p − αξ P ∂ρ =  i − i i +  n+1 + ui,kSi,k  ρ p 2τ Vi ρ  p pm Fkint −  ξ ∂ ξ  ∂ m k ξ kint  n+1 n+1 n+1 (6) P ∂ρ ∂ρ P ∂ρ P  + pn+1 F pn+1 F pn+1 F  ∂p press kpress ∂p i kint ∂p press kpresst . kpress press − i k − press kpress 

The convective term on the left side of momentum Equation (1), considering the boundary conditions, is expressed as

N ! N ! N ! P P P P P n+1 P = + + ukFk ρξ,kαξ,k ui Fk ρξ,kαξ,k ui Fk ρξ,iαξ,i k ξ=1 − k ξ=1 k+ int ξ=1 ! int ! ! P PN P PN P PN + un+1 F ρ α + un+1F ρ α + un+1F ρ α m,k k− ξ,m ξ,m inl kinl ξ,kinl ξ,kinl i kpress ξ,i ξ,i k int− int ξ=1 k ξ=1 k ξ=1 int− inl press N ! N ! N ! n+1 P P n+1 P P n+1 P P + u Fk ρξ,iαξ,i u Fk ρξ,mαξ,m u Fkinl ρξ,kinl αξ,kinl i + int i int− i − k ξ=1 − k ξ=1 − kinl ξ=1 − int int− N ! N ! N ! n+1 P P P n+1 P P n+1 P u Fkpress ρξ,iαξ,i = u Fk ρξ,mαξ,m + u Fk ρξ,k αξ,k i m,k− int− inl inl inl inl − kpress ξ=1 k int ξ=1 k ξ=1 − int− ! inl ! P PN P PN un+1 F ρ α un+1 F ρ α . i k− ξ,m ξ,m i kinl ξ,kinl ξ,kinl − k int ξ=1 − k ξ=1 int− inl Geosciences 2020, 10, 385 5 of 28

The diﬀusion term Equation (5) in Equation (1) takes the form

∂u P P + P α + P j + µmumS∗ µiuiS∗ µi u kS µi x nj S k − k h∇ i k,i ∂ i k k k k k ! k ! 2 P ∂uk P n+1 P n+1 P n+1 + µi nj δijS = u + µ + S∗+ u µ + S∗+ + u µk S∗ 3 ∂xk k k m,k k k i k k m,k int− k− k k+ int int int −k+ int int k int− int − int int int− (7) P P P P un+1µ S + un+1µ S un+1µ S + µ u Sα + i k− k∗ k kinl k∗ i kinl k∗ i kint k ,i −k int int− k inl inl − k inl k h∇ i int int− inl inl int ∂u +P j + 2 P ∂uk µi ∂x nj Sk 3 µi ∂x nj δijSk . k i k k k k

The ﬁnal form of the discretized momentum equation is the following:

N n+1 n n 1 N ! N ! P 3u 4u +u − P P P P ρ α i − i i V + un+1 F ρ α + un+1F ρ α ξ,i ξ,i 2τ i m,k k− ξ,m ξ,m inl kinl ξ,kinl ξ,kinl ξ=1 k int− int ξ=1 k ξ=1 − int− inl N ! N ! (8) + P P + P P j P un 1 F ρ α un 1 F ρ α = p + D + g n S ρ . i k− ξ,m ξ,m i kinl ξ,kinl ξ,kinl i i,k i k k − k int ξ=1 − k ξ=1 − ∇ k int− inl

Here, the coeﬃcient D denotes expression Equation (7). The volume fraction transfer Equation (3), after discretization, takes the form

n+1 n + n 1 3αξ,i 4αξ,i αξ,−i P P P − V + αn+1 Fn+1 + αn+1F αn+1 Fn+1 2τ i m,k k inl kinl i k k int− int− k − k int− − int− inl int− ! (9) αn ρn+1 ρn n+1 P = n ξ,i ξ − ξ + n+1P n+1 α Fkinl α G n τ Vi u ρ Si,k , i i ρξ i k − kinl − − k where the symbol G denotes the right side of expression Equation (3). The approximation we use reduces the scalar quantity transfer Equation (2) to a system of linear algebraic equations formulated with respect to computational cells: X + = Aiφi Am,kint φm,kint Ri (10) kint

where Ai is the diagonal matrix coefficient, kint are the interior cell faces, Am,kint are the off-diagonal matrix coefficients, and Ri is right-side vector component. The discretized equivalents of Equation (1) reduced to the system of linear algebraic Equation (10) can be solved by the SIMPLE algorithm [19,32–34], one of the most common algorithms in computational fluid dynamics. At the predictor step, in order to obtain the first approximation of the sought velocity field, a discrete equivalent of the momentum equation with respect to velocity is constructed without expanding the pressure gradient term. For the i-th cell, the equation takes the form ! X j ∂p A u + A u = R V i i m,kint m,kint i i (11) − ∂xi i kint

where ui is the velocity component in the i-th cell and um,kint is the velocity component in the near-boundary cell m at the face kint. Geosciences 2020, 10, 385 6 of 28

The matrix coeﬃcients and the right-side members of the system of linear algebraic Equation (11) have the form

N N ! N ! j 3Vi P j+1 j+1 P P P P A = ρ α Fk ρξ,mαξ,m Fk ρξ,k αξ,k + i 2τ ξ,i ξ,i int− inl inl inl ξ=1 − k− ξ=1 − kinl ξ=1 P int P P + + + + µk S∗+ µk S∗ µkinl S∗ , + int k int− k− kinl k int k int kinl int int−

N j X j+1 A = F ρ α µ S k k− ξ,m ξ,m k k∗ int− int − int− int− ξ=1

j j+1 A = µ S∗+ k+ k+ k int − int int

j j 1 ! N − N N j 4ui Vi P j+1 j+1 ui Vi P j+1 j+1 P n+1 P j+1 j+1 P R = ρ α ρ α + µk u S∗ u F ρξ,k αξ,k + i 2τ ξ,i ξ,i 2τj+1 ξ,i ξ,i inl kinl kinl kinl kinl inl inl ξ=1 − ξ=1 kinl − kinl ξ=1 u P n α P ∂ j 2 P ∂uk + µk u S + µk nj S + µi nj δijS + giρiVi. int kint k int ∂xi k 3 ∂xk k k k h∇ i int k kint int k int int For the corrector step, an expression for velocity in the i-th cell must be derived from Equation (11):

  ! ! 1  j X  1 ∂p V ∂p u = R A u  V = H i (12) i  i m,kint m,kint  i i Ai  −  − Ai ∂xi i − Ai ∂xi i kint   1  j P  where Hi = R Am k um k . Ai  i , int , int  − kint Interpolating velocity Equation (12) from the cells to the exterior faces using the geometric weighting factor λk: ! ! ! Vi,k ∂p λVi ∂p Vm ∂p Hi,k = λHi + (1 λ)Hm (1 λ) (13) − Ak ∂xi k − − Ai ∂xi i − − Am ∂xi m and inserting Equation (13) into continuity Equation (1) gives ! ! ! ! ∂ λkVi ∂p Vm ∂p X αξ dρξ λkHi + (1 λk)Hm (1 λk) = (14) ∂xi − − Ai ∂xi − − Am ∂xi − ρ dt i m ξ ξ

In practice, the weight λk can be determined in diﬀerent ways—for example, basing it on the cell geometric center-to-face distances or the values of the diagonal coeﬃcients Ai and Am, or letting λk = 0.5. Equation (14) must also be discretized with respect to pressure; then, by analogy with Equation (10), it can be re-written for the i-th cell: X j n+1 n+1 j B p + Bm,k p = M (15) i i int m,kint i kint

j j B B M where i is the diagonal equation coefficient, m,kint are the off-diagonal coefficients, and i is a right-side vector component. Geosciences 2020, 10, 385 7 of 28

The members in the system of linear algebraic Equation (15) are as follows:

S Sk j P λVi (1 λ)Vm kint P Vi presst αξ ∂ρξ αξ ∂ρξ P B = + − Vi + Fk+ i − Ai Am dim − Ai dgran − τρξ ∂p ρξ ∂p kint | | kpress | | k αξ ∂ρξ P αξ ∂ρξ P + F + F , ρξ ∂p k− ρξ ∂p kpress ik int ik int− press " # λV (1 λ)Vm Sk αξ ∂ρξ B = i + − int F m,kint k− A Am d − ρ ∂p int i | im| ξ m j P V ppress PD E P D E P M = i S + H S + H S + F s i Ai d kpress kint kint kpress kpress inlet αξ −k gran k · k · k − − press | |  int press  inlet j P αξ ∂ρξ pi αξ P ∂ρξ n+1   V + p Fpress.  ρξ ∂p τ i ρξ ∂p press  − ξ  k press  int− Solving Equation (15) allows us to find the pressure field, which is used to calculate the mass flux fields with the formula ! ! Vi,k ∂p Fk = Hi,k Si,k (16) − Ak ∂xi k Then, the predictor-corrector procedure is repeated with the updated velocity, pressure and mass flux fields until the desired accuracy is achieved. Next, the volume fraction transfer Equation (9) is solved, which also needs to be formulated as a system of linear algebraic equations for the i-th cell:

j X j j A αξ,i + A αξ,m,k = R i m,kint int i kint

Here, the matrix coefficients and the right-side members of the system of linear algebraic equations are equal to j 3Vi X X A = + Fk + Fk i 2τ int− out k k int− out j A = Fk m,kint int− n  n+1 n n 1  X α 3ρ 4ρ + ρ − X j j 2Vi j 1 Vi ξ,i  ξ ξ ξ n+1 n+1  R = α α − α F  − V + u ρ S  i ξ,i τ − ξ,i 2τ − ξ,inl kinl − ρn  2τ i i k i,k kinl ξ k where kout = k kpress denotes the outgoing-flow boundary condition. This algorithm can be extended inlet ∪ by an original method by taking into account the gravity term for free-surface flow simulations [36]. Thus, using this algorithm, Equation (1) can be solved numerically. This algorithm has been implemented in LOGOS and adapted for massively parallel computations on multiprocessor computers [19,26]. Simulations of moving bodies employ an approach based on superimposed multi-domain (Chimera) meshes [37]. In this case, the main computational domain is mapped onto a background mesh, while a separate mesh surrounding the solid body is moving atop (Figure2). The resulting geometric models are combined into a single mesh representing the initial problem. Interaction between the background mesh and the mesh atop is implemented using an interpolation template intended to ensure correct interaction between the topologically disjoint regions. The new interfaces generated as a result of constructing the template and tagging the cells of the combined mesh enable communication between the coupled regions and produce the required initial data for the interpolation template [36]. One of the advantages of this approach is that it allows using Geosciences 2020, 10, x FOR PEER REVIEW 8 of 29

Here, the matrix coefficients and the right-side members of the system of linear algebraic equations are equal to

j 3Vi A = + F − + F i  k  kout 2τ − int kint kout

j A = F − m,kint kint

αn  ρn+1 − ρn + ρn−1  j j 2Vi j−1 Vi ξ,i 3 ξ 4 ξ ξ n+1 n+1 R = αξ − αξ − αξ F −  V + u ρ S  i ,i ,i ,inl kinl n i i k i,k τ 2τ  ρξ  2τ   kinl  k 

where kout = kinlet ∪ kpress denotes the outgoing-flow boundary condition. This algorithm can be extended by an original method by taking into account the gravity term for free-surface flow simulations [36]. Thus, using this algorithm, Equation (1) can be solved numerically. This algorithm has been implemented in LOGOS and adapted for massively parallel computations on multiprocessor computers [19,26]. Simulations of moving bodies employ an approach based on superimposed multi-domain (Chimera) meshes [37]. In this case, the main computational domain is mapped onto a background Geosciencesmesh,2020 while, 10, 385a separate mesh surrounding the solid body is moving atop (Figure 2). The resulting8 of 28 geometric models are combined into a single mesh representing the initial problem. Interaction ratherbetween large computationalthe background domains. mesh and The the mesh mesh can atop be is coarse implemented in the regions using faran frominterpolation the solid template body andintended ﬁne in its to vicinity. ensure correct Boundary interaction tracking be istween performed the topologically automatically. disjoint regions.

Figure 2. Computational mesh: (a) mesh of the exterior region; (b) mesh around the body; (c) resulting Figure 2. Computational mesh: (a) mesh of the exterior region; (b) mesh around the body; (c) superimposed mesh. resulting superimposed mesh. 3. Verification Results for the LOGOS Software Package The new interfaces generated as a result of constructing the template and tagging the cells of the combinedThe quantitative mesh enable assessment communication of simulation between results the must coupled be performed regions and based produce on the the error required in calculatedinitial data and for experimental the interpolation data at template given points. [36]. One Changes of the inadvantages the sea level of this caused approach by tsunami is that it propagationallows using are rather detected large by computational wave gages—instruments domains. The that mesh perform can be measurements coarse in the regions and continuous far from the computerizedsolid body and recording fine in ofits variationsvicinity. Bounda in thery sea tracking level at is a performed given accuracy. automatically. The instrumental error of wave gages can range from several millimeters to several centimeters [38]. Therefore, in wave simulations,3. Verification it is reasonable Results for to the introduce LOGOS a meanSoftware error Package and calculate it as a sum of the relative errors at each point in space or time divided by the number of such points. In some tests, we compare a The quantitative assessment of simulation results must be performed based on the error in wave profile relative to the experimental data, while in other cases, we evaluate the maximum value of calculated and experimental data at given points. Changes in the sea level caused by tsunami the waveform. propagation are detected by wave gages—instruments that perform measurements and continuous Before we proceed to the verification of the tsunami simulations, let us present some results of LOGOS verification against free-surface test problems, including the collapse of a liquid column, gravity water sloshing in a tank, the fall of a sphere into a liquid, the fall of a parallelepiped into a liquid and wave flows induced by lifting a rectangular beam out of water. The verification results allow us to assess the accuracy of the predicted changes in the sea level in confined space.

3.1. Collapse of a Liquid Column This problem considers a column of liquid (water) collapsing onto a tank bottom. The experimental results for the collapsing water column are reported in [30]. The parameters of the experimental setup and initial location of the water/air interface are shown in Figure3. The problem was solved using a block-structured mesh model, a mesh of uniform hexahedral cells, consisting of 8400 cells: 120 cells in the horizontal direction and 70 cells in the vertical direction; thus, ∆x = 0.033a and ∆y = 0.0286a. No-slip boundary conditions were applied to the left, right and bottom walls; the front and the back walls were slip boundaries; the top boundary had a static pressure held ﬁxed at 0 MPa. The problem was run in single mode on an Intel Core i5 series CPU; the total computational time was approximately 10 min. Figure4a shows the results of the numerical modeling. The solid line represents the air /water interface, and the black dashes are the velocity vectors. The results are plotted for diﬀerent time points: 0.2, 0.6 and 1.0 s. The results are presented in dimensional form as presented by Ubbink [30]. Geosciences 2020, 10, x FOR PEER REVIEW 9 of 29 computerized recording of variations in the sea level at a given accuracy. The instrumental error of wave gages can range from several millimeters to several centimeters [38]. Therefore, in wave simulations, it is reasonable to introduce a mean error and calculate it as a sum of the relative errors at each point in space or time divided by the number of such points. In some tests, we compare a wave profile relative to the experimental data, while in other cases, we evaluate the maximum value of the waveform. Before we proceed to the verification of the tsunami simulations, let us present some results of

GeosciencesLOGOS 2020verification, 10, 385 against free-surface test problems, including the collapse of a liquid column,9 of 28 gravity water sloshing in a tank, the fall of a sphere into a liquid, the fall of a parallelepiped into a liquid and wave flows induced by lifting a rectangular beam out of water. The verification results Figureallow 4usb showsto assess the the numerical accuracy modeling of the predicted results changes obtained in in the [ 30 sea]. Figurelevel in4c confined shows the space. experimental photographs [30]. By t = 0.6 s, the wave has reached the right boundary of the model (experimental) domain,3.1. Collapse and of its a hydraulicLiquid Column impact on the wall generates a wave. The time t = 0.6 s corresponds to the most active phase of collapse with the highest wave. The calculated results demonstrate an air pocket This problem considers a column of liquid (water) collapsing onto a tank bottom. The forming at the left boundary of the domain; its shape is similar in both computations. It can also be experimental results for the collapsing water column are reported in [30]. The parameters of the identiﬁed in the experimental photograph. At time 1.0 s, the ﬂuid hits the left wall. experimental setup and initial location of the water/air interface are shown in Figure 3.

Geosciences 2020, 10, x FOR PEER REVIEW 10 of 29 calculated position of the leading edge relative to the experimental data does not exceed 10%. One can see that the numericalFigure 3.resultsSchematic agree representation well with ofthe the experiment experimental and setup. the numerical solution Figure 3. Schematic representation of the experimental setup. reported in [30] both qualitatively and quantitatively. The problem was solved using a block-structured mesh model, a mesh of uniform hexahedral cells, consisting of 8400 cells: 120 cells in the horizontal direction and 70 cells in the vertical direction; thus, ∆x = 0.033a and ∆y = 0.0286a. No-slip boundary conditions were applied to the left, right and bottom walls; the front and the back walls were slip boundaries; the top boundary had a static pressure held fixed at 0 MPa. The problem was run in single mode on an Intel Core i5 series CPU; the total computational time was approximately 10 min. Figure 4a shows the results of the numerical modeling. The solid line represents the air/water interface, and the black dashes are the velocity vectors. The results are plotted for different time points: 0.2, 0.6 and 1.0 s. The results are presented in dimensional form as presented by Ubbink [30]. Figure 4b shows the numerical modeling results obtained in [30]. Figure 4c shows the experimental photographs [30]. By t = 0.6 s, the wave has reached the right boundary of the model (experimental) domain, and its hydraulic impact on the wall generates a wave. The time t = 0.6 s corresponds to the most active phase of collapse with the highest wave. The calculated results demonstrate an air pocket forming at the left boundary of the domain; its shape is similar in both computations. It can also Figurebe identified 4. Collapse in the of aexperimental liquid column: photograph (a) results calculated. At time in 1.0 this s, work;the fluid (b) resultshits the calculated left wall. in [30]; Figure 4. Collapse of a liquid column: (a) results calculated in this work; (b) results calculated in [30]; (Thec) experimental results indicate data [30 that]. the calculated data qualitatively agree with the experiment in both (c) experimental data [30]. unsteady characteristics (wave formation and collapse time) and the shape of the free surface. Figure 5 showsThe resultsthe time indicate evolution that of thethe calculatedcolumn height data al qualitativelyong the left wall agree of with the tank, the experiment and Figure in6 is both the unsteadyposition of characteristics the leading (waveedge of formation the liquid and along collapse the tank time) bottom and the as shape a function of the freeof time. surface. The Figure column5 showsheight thealong time the evolution left tank ofwall, the the column position height of the along leading the left edge wall along of the the tank, tank and bottom Figure (measured6 is the positionfrom the of bottom the leading left angle) edge and of thethe liquidtime are along dimens theionless tank bottom quantities. as a functionThe solid of black time. line The represents column heightthe curves along calculated the left tank in wall,this thework, position the dashed of the leadingline shows edge the along results the tank calculated bottom (measuredin [30], and from the themarkers bottom represent left angle) the and experimental the time are data dimensionless [30]. The mean quantities. square deviation The solid blackin the linecalculated represents column the curvesheight calculatedrelative to in the this experimental work, the dashed data linedoes shows not theexceed results 1%. calculated The maximum in [30], anddeviation the markers in the represent the experimental data [30]. The mean square deviation in the calculated column height relative to the experimental data does not exceed 1%. The maximum deviation in the calculated position of the leading edge relative to the experimental data does not exceed 10%. One can see that the numerical results agree well with the experiment and the numerical solution reported in [30] both qualitatively and quantitatively. Figure 5. Height of the liquid column as a function of time (●—experiment; –—LOGOS; --—calculation [30]).

Figure 6. Position of the leading edge of the liquid as a function of time (●—experiment; –—LOGOS; --—calculation [30]).

3.2. Gravity Sloshing of Water in a Tank

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Geosciences 2020, 10, x FOR PEER REVIEW 10 of 29 calculated position of the leading edge relative to the experimental data does not exceed 10%. One calculatedcan see that position the numerical of the leading results edge agree relative well to wi theth experimentalthe experiment data and does the not numerical exceed 10%. solution One canreported see that in [30] the both numerical qualitatively results and agree quantitatively. well with the experiment and the numerical solution reported in [30] both qualitatively and quantitatively.

Figure 4. Collapse of a liquid column: (a) results calculated in this work; (b) results calculated in [30]; Figure(c) experimental 4. Collapse data of a [30]. liquid column: (a) results calculated in this work; (b) results calculated in [30]; Geosciences(c) experimental2020, 10, 385 data [30]. 10 of 28

Figure 5. Height of the liquid column as a function of time (●—experiment; –—LOGOS; Figure 5. Height of the liquid column as a function of time ( —experiment; – —LOGOS; Figure--—calculation 5. Height [30]). of the liquid column as a function of time •(●—experiment; –—LOGOS; - - —LOGOS [30]). --—calculation [30]).

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The problem considers the sloshing of water in a tank under the influence of gravity. The parameters of the numerical scheme and initial interface location (water/air) are shown in Figure 7 [30]. The problem was solved using a block-structured mesh model having a size of 0.1 × 0.065 × 0.01 m and consisting of 33,600 cells, 210 cells in the horizontal direction and 160 cells in the vertical direction, resulting in ∆x = 4.762∙10−4 m and ∆y = 4.0625∙10−4 m. No-slip boundary conditions were applied to the left, right and bottom walls; the front and the back walls were slip boundaries; the top boundary had a static pressure held fixed at 0 MPa. The problem was run in single mode; the total computational time was approximately 15 min. FigureFigure 6.6. PositionPosition ofof thethe leadingleading edgeedge ofof thethe liquidliquid asas aa functionfunction ofof timetime ((●—experiment;—experiment; – –—LOGOS; —LOGOS; The numerical results for the gravity sloshing of water are presented• in Figure 8. The solid line ---—calculation - —LOGOS [30 [30]).]). representsFigure the6. Position water/air of the interface, leading edgegray ofshows the liquid the liasquid a function phase, of and time the (● —experiment;black dashes –—LOGOS;are the velocity 3.2. Gravity--—calculation Sloshing [30]). of Water in a Tank vectors.3.2. Gravity The Sloshing results ofare Water given in afor Tank different points in time; T is the theoretical period of cosine oscillations, which is 0.3739 s in our case [30]. Panels “a” show the results obtained in [30], and 3.2. GravityThe problem Sloshing considers of Water the in sloshinga Tank of water in a tank under the inﬂuence of gravity. The parameters ofpanels the numerical “b”, the results scheme calculated and initial by interface LOGOS. location (water/air) are shown in Figure7[30].

Figure 7. Schematic representation of the experimental setup. Figure 7. Schematic representation of the experimental setup. The problem was solved using a block-structured mesh model having a size of 0.1 0.065 0.01 m × × and consisting of 33,600 cells, 210 cells in the horizontal direction and 160 cells in the vertical direction, resulting in ∆x = 4.762 10 4 m and ∆y = 4.0625 10 4 m. No-slip boundary conditions were applied to · − · − the left, right and bottom walls; the front and the back walls were slip boundaries; the top boundary had a static pressure held ﬁxed at 0 MPa. The problem was run in single mode; the total computational time was approximately 15 min. The numerical results for the gravity sloshing of water are presented in Figure8. The solid line represents the water/air interface, gray shows the liquid phase, and the black dashes are the velocity vectors. The results are given for diﬀerent points in time; T is the theoretical period of cosine

Figure 8. Gravity sloshing: (a) numerical results obtained in [30]; (b) results calculated by LOGOS.

Initially, the interface has a cosine shape. The interface is then released and begins to slosh under the influence of gravity and liquid inertia. At t = T/4, the interface becomes horizontal, and then, at t = T/2, it recovers its cosine shape. The process is repeated in time. Figure 9 shows the time evolution of the sea level along the left wall of the tank.

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oscillations, which is 0.3739 s in our case [30]. Panels “a” show the results obtained in [30], and panels “b”, the results calculatedFigure by 7. LOGOS. Schematic representation of the experimental setup.

FigureFigure 8. 8.GravityGravity sloshing: sloshing: (a)( anumerical) numerical results results obtained obtained in in[30]; [30 (];b) ( bresults) results calculated calculated by by LOGOS. LOGOS. Initially, the interface has a cosine shape. The interface is then released and begins to slosh under Initially, the interface has a cosine shape. The interface is then released and begins to slosh the inﬂuence of gravity and liquid inertia. At t = T/4, the interface becomes horizontal, and then, under the influence of gravity and liquid inertia. At t = T/4, the interface becomes horizontal, and at t = T/2, it recovers its cosine shape. The process is repeated in time. Figure9 shows the time then, at t = T/2, it recovers its cosine shape. The process is repeated in time. Figure 9 shows the time Geosciencesevolution 2020 of,the 10, x sea FOR level PEER along REVIEW the left wall of the tank. 12 of 29 evolution of the sea level along the left wall of the tank.

Figure 9. Oscillations of sea level along the left wall of the tank under the inﬂuence of gravity ( —cosine Figure 9. Oscillations of sea level along the left wall of the tank under the influence of··· gravity (oscillations;∙∙∙—cosine oscillations; – —LOGOS; –—LOGOS; - - —LOGOS --—calculation [30]). [30]). Table1 shows the time errors in the numerical results relative to the theoretical cosine oscillations. Table 1 shows the time errors in the numerical results relative to the theoretical cosine The time error is calculated as oscillations. The time error is calculated as ts tt − 100% t −ttt · s t ⋅100% where ts is the calculated period of oscillations and tt is the theoretical period T [30]. tt where ts is the calculated period of Tableoscillations 1. Error and in numerical tt is the theoretical results. period T [30]. One can see that the numerical results are in close qualitative and quantitative agreement with Relative Time Error, % the theory andTime the numerical solution presented in [30]. Numerical Results [30] LOGOS Numerical Results 2T Table0.00 1. Error in numerical results. 0.70 4T 0.75 0.76 − T Relative Time Error, % 6 Time 0.04 0.03 Numerical Results [30] LOGOS Numerical Results One can see that2 theT numerical results0.00 are in close qualitative and0.70 quantitative agreement with the theory and the numerical4T solution presented−0.75 in [30]. 0.76 6T 0.04 0.03 3.3. Fall of a Sphere into a Liquid 3.3. Fall of a Sphere into a Liquid The problem simulates a fall of a solid sphere into water with low initial velocity. The experiment was conductedThe problem with simulates spheres froma fall di ﬀoferent a solid materials sphere and into reported water in with [39]. low The probleminitial velocity. schematic The is experiment was conducted with spheres from different materials and reported in [39]. The problem schematic is shown in Figure 10. A sphere having a radius r = 1.27 cm and velocity V = 2.17 m/s hits the surface of water. The initial sea level h is 0.2 m. The domain is a cylinder of radius R = 0.25 m and height H = 0.2254 m. Two computational cases were considered: in the first, the sphere was made from polypropylene, and its density was 0.86 of water’s density; in the second, the sphere was made from steel of 7.86 times water’s density.

Figure 10. Schematic representation of the problem.

The problem domain was discretized with a block-structured mesh model refined around the solid body and composed of 800,000 cells (see Figure 11). The cell size near the cylinder is equal to 5% of the cylinder radius. No-slip boundary conditions were applied to the wall inside a cylinder;

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Figure 9. Oscillations of sea level along the left wall of the tank under the influence of gravity (∙∙∙—cosine oscillations; –—LOGOS; --—calculation [30]).

Table 1 shows the time errors in the numerical results relative to the theoretical cosine oscillations. The time error is calculated as − ts tt ⋅100% tt where ts is the calculated period of oscillations and tt is the theoretical period T [30]. One can see that the numerical results are in close qualitative and quantitative agreement with the theory and the numerical solution presented in [30].

Table 1. Error in numerical results.

Relative Time Error, % Time Numerical Results [30] LOGOS Numerical Results 2T 0.00 0.70 4T −0.75 0.76 6T 0.04 0.03

3.3. Fall of a Sphere into a Liquid The problem simulates a fall of a solid sphere into water with low initial velocity. The experiment was conducted with spheres from different materials and reported in [39]. The problem schematic is shown in Figure 10. A sphere having a radius r = 1.27 cm and velocity V = 2.17 m/s hits Geosciencesthe surface2020 of, 10 water., 385 The initial sea level h is 0.2 m. The domain is a cylinder of radius R = 0.25 12m ofand 28 height H = 0.2254 m. shownTwo in Figurecomputational 10. A sphere cases having were a radiusconsidered:r = 1.27 in cm the and velocityfirst, theV =sphere2.17 m /wass hits made the surface from ofpolypropylene, water. The initial and its sea density level h wasis 0.2 0.86 m. of The water’s domain density; is a cylinder in the second, of radius the Rsphere= 0.25 was m andmade height from Hsteel= 0.2254 of 7.86 m. times water’s density.

FigureFigure 10.10. SchematicSchematic representationrepresentation ofof thethe problem.problem. Two computational cases were considered: in the first, the sphere was made from polypropylene, The problem domain was discretized with a block-structured mesh model refined around the and its density was 0.86 of water’s density; in the second, the sphere was made from steel of 7.86 times solid body and composed of 800,000 cells (see Figure 11). The cell size near the cylinder is equal to water’s density. 5% of the cylinder radius. No-slip boundary conditions were applied to the wall inside a cylinder; The problem domain was discretized with a block-structured mesh model refined around the solid body and composed of 800,000 cells (see Figure 11). The cell size near the cylinder is equal to 5% Geosciencesof the cylinder 2020, 10 radius., x FOR PEER No-slip REVIEW boundary conditions were applied to the wall inside a cylinder;13 the of top 29 boundary had a static pressure held fixed at 0 MPa. The problem was run in parallel mode on 120 thestandard top boundary CPUs; the had total a static computational pressure held time fixed was at approximately 0 MPa. The problem 40 min. was run in parallel mode on 120 standard CPUs; the total computational time was approximately 40 min.

(a) (b)

FigureFigure 11. 11. DiscreteDiscrete model: model: ( (aa)) General General view; view; ( (bb)) Mesh Mesh refinement reﬁnement around around the the solid solid body. body.

Figure 1212 illustratesillustrates thethe descent descent dynamics dynamics of of the the steel steel sphere: sphere: (a) experimental(a) experimental photographs photographs [39]; [39];(b) numerical (b) numerical results results obtained obtained for corresponding for corresponding time points. time points. Gray represents Gray represents the liquid the phase, liquid the phase, solid theblack solid line black is the line interface, is the interface, and the sphere and the is shown sphere in is black. shown At int =black.5.9 ms, At thet = 5.9 sphere ms, isthe fully sphere submerged. is fully submerged.At t = 54.9 ms, At at = cavity 54.9 ms, starts a cavity forming starts above forming the sphere. above th Thee sphere. cavity isThe observed cavity is to observed stop growing to stop at growingt = 68.9 ms, at t both= 68.9 numerically ms, both numerically and experimentally. and experimentally. The numerical and the experimental result resultss are in good qualitative agreement. Figure 13 shows the depth of sphere descent in water as a function of time. Results are provided for the two computational cases. Theoretical results are presented in [39].

Figure 12. Descent of the steel sphere: (a) Experimental photographs [39]; (b) Numerical results.

Figure 13 shows the depth of sphere descent in water as a function of time. Results are provided for the two computational cases. Theoretical results are presented in [39].

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the top boundary had a static pressure held fixed at 0 MPa. The problem was run in parallel mode on 120 standard CPUs; the total computational time was approximately 40 min.

(a) (b)

Figure 11. Discrete model: (a) General view; (b) Mesh refinement around the solid body.

Figure 12 illustrates the descent dynamics of the steel sphere: (a) experimental photographs [39]; (b) numerical results obtained for corresponding time points. Gray represents the liquid phase, the solid black line is the interface, and the sphere is shown in black. At t = 5.9 ms, the sphere is fully submerged. At t = 54.9 ms, a cavity starts forming above the sphere. The cavity is observed to stop growing at t = 68.9 ms, both numerically and experimentally. Geosciences 2020, 10, 385 13 of 28 The numerical and the experimental results are in good qualitative agreement.

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Geosciences 2020, 10, x FOR PEER REVIEW 14 of 29 Figure 12. Descent of the steel sphere: (a) Experimental photographs [39]; (b) Numerical results. Figure 12. Descent of the steel sphere: (a) Experimental photographs [39]; (b) Numerical results.

Figure 13 shows the depth of sphere descent in water as a function of time. Results are provided for the two computational cases. Theoretical results are presented in [39].

Figure 13. Depth of sphere’s center-of-mass descent in water as a function of time (▪—polypropylene—theory; ——polypropylene—LOGOS; ●—steel—theory; ‒ ‒— steel—LOGOS).

3.4. Fall of a Parallelepiped into a Liquid The verification of the algorithm described above as applied to the description of waves traveling across a free surface can be performed by the numerical modeling of an experiment described in [24]. In the experiment, a rectangular parallelepiped of height H1 and length l falls freely fromFigure height 13.H intoDepth waterof along sphere’s the end center-of-mass wall of a tank. descent The intank water is rectangular, as a function with ofa horizontal time Figure 13. Depth of sphere’s center-of-mass descent in water as a function of time bottom,(—polypropylene—theory; filled with water to h < — H —polypropylene—LOGOS;1. In the real experiment, the—steel—theory; tank was 4.3 –m – long — steel—LOGOS). and 0.2 m wide. In (▪—polypropylene—theory; ——polypropylene—LOGOS; ●—steel—theory; ‒ ‒— steel—LOGOS). the unperturbed condition, water is quiescent. The problem• geometry is shown in Figure 14. 3.4. FallA number of a Parallelepiped of problem into configurations a Liquid were considered with different body dimensions, heights 3.4. Fall of a Parallelepiped into a Liquid of fall,The and veriﬁcation water depths of the in algorithm the tank. describedThese parameters above as have applied the tofollowing the description dimensionless of waves form traveling [24]: The verification of the algorithm described above as applied to the description of waves across a free surface can be performed0 by0 the numerical0 modeling0 of an experiment0 described in [24]. traveling across a free surfaceH = H/h,can beH 1performed = H1/h, l =by l/h, the ρ =numerical (ρ1 − ρ)/ρ, xmodeling = x/h, of an experiment In the experiment, a rectangular parallelepiped of height H1 and length l falls freely from height H described in [24]. In the experiment, a rectangular parallelepiped of height H1 and length l falls freely into water along the end wall of a tank. The tank is rectangular, with a horizontal bottom, ﬁlled0 with0 where ρ1 and ρ are the density of the solid body and the liquid, respectively. The values of h, H , H1 fromwater height to h < H into. In thewater real along experiment, the end the wall tank of wasa tank. 4.3 mThe long tank and is 0.2rectangular, m wide. In with the unperturbeda horizontal , l0, ρ0 and x0 vary1 depending on the configuration. bottom,condition, filled water with is water quiescent. to h < The H1. problemIn the real geometry experiment, is shown the tank in Figure was 4.3 14 m. long and 0.2 m wide. In the unperturbed condition, water is quiescent. The problem geometry is shown in Figure 14. A number of problem configurations were considered with different body dimensions, heights of fall, and water depths in the tank. These parameters have the following dimensionless form [24]:

0 0 0 0 0 H = H/h, H1 = H1/h, l = l/h, ρ = (ρ1 − ρ)/ρ, x = x/h,

0 0 where ρ1 and ρ are the density of the solid body and the liquid, respectively. The values of h, H , H1 , l0, ρ0 and x0 vary depending on the configuration.

Figure 14. Problem geometry. Figure 14. Problem geometry.

0 0 0 In the first configuration, the following values were used: h = 8 cm, H = 3.75, H1 = 2.26, l = 0 0 0 0 1.15, and ρ = 0.215. For the second computational configuration, h = 8 cm, H = 2.90, H1 = 2.26, l = 0.575, and ρ0 = 0.215. A wave gage, which detects changes in the sea level, was placed at x0 = 16. 0 0 0 0 In the third configuration, h = 4 cm, H = 4.75, H1 = 4.5, l = 1.15, and ρ = 0.215, and the wave gage was placed at x0 = 31.5. To model the fall of a parallelepiped into water, the 4.3 × 0.4 × 0.5 m domain was discretized with a block-structuredFigure 14. Problem mesh geometry. of about 20,000 cells. The mesh was refined in the regions of the fall and the wave gage. No-slip boundary conditions were applied to the left, right

0 0 0 In the first configuration, the following values were used: h = 8 cm, H = 3.75, H1 = 2.26, l = 0 0 0 0 1.15, and ρ = 0.215. For the second computational configuration, h = 8 cm, H = 2.90, H1 = 2.26, l = 0.575, and ρ0 = 0.215. A wave gage, which detects changes in the sea level, was placed at x0 = 16. 0 0 0 0 In the third configuration, h = 4 cm, H = 4.75, H1 = 4.5, l = 1.15, and ρ = 0.215, and the wave gage was placed at x0 = 31.5. To model the fall of a parallelepiped into water, the 4.3 × 0.4 × 0.5 m domain was discretized with a block-structured mesh of about 20,000 cells. The mesh was refined in the regions of the fall and the wave gage. No-slip boundary conditions were applied to the left, right

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A number of problem conﬁgurations were considered with diﬀerent body dimensions, heights of fall, and water depths in the tank. These parameters have the following dimensionless form [24]:

H0 = H/h, H0 = H /h, l0 = l/h, ρ0 = (ρ ρ)/ρ, x0 = x/h, 1 1 1 − 0 0 0 where ρ1 and ρ are the density of the solid body and the liquid, respectively. The values of h, H , H1, l , ρ0 and x0 vary depending on the configuration. 0 0 0 In the first configuration, the following values were used: h = 8 cm, H = 3.75, H1 = 2.26, l = 1.15, 0 0 0 0 and ρ = 0.215. For the second computational configuration, h = 8 cm, H = 2.90, H1 = 2.26, l = 0.575, and ρ0 =Geosciences0.215. 2020 A wave, 10, x FOR gage, PEER which REVIEW detects changes in the sea level, was placed at x0 =1516. of 29 0 0 0 0 In theand thirdbottom configuration, walls; the top boundaryh = 4 cm, hadH zero= 4.75, staticH pressure.1 = 4.5, lThe= 1.15,experiment and ρ was= 0.215,run in parallel and the wave gage wasmode placed on 32 atCPUs;x0 = the31.5. total To computational model the time fall ofwas a approximately parallelepiped 15 min. into water, the 4.3 0.4 0.5 m × × domain wasFigure discretized 15 shows with the a block-structuredwaves produced by mesh a falling of about parallelepiped 20,000 cells. in the The first mesh problem was refined in the regionsconfiguration. of the The fall black and line the is wave the outline gage. of the No-slip parallelepiped. boundary Gray conditions represents werethe liquid applied phase. to At the left, tGeosciences = 0.25 s, 2020we, observe10, x FOR PEERan early REVIEW phase in the development of a cavity and a splash. At t = 0.315 s, ofthe 29 right and bottom walls; the top boundary had zero static pressure. The experiment was run in parallel splash continues to develop. By 0.4 s, the tank bottom becomes clear of water, and the rate of splash and bottom walls; the top boundary had zero static pressure. The experiment was run in parallel mode onformation 32 CPUs; slows the down. total computationalThe late phase intime the evolutio was approximatelyn of the cavity (its 15 collapse) min. and the splash is mode on 32 CPUs; the total computational time was approximately 15 min. Figureshown 15 at shows t = 0.75 the s. Note waves that produced the photography by a falling time points parallelepiped are not specified in the in first [24], problemso our numerical configuration. Figure 15 shows the waves produced by a falling parallelepiped in the first problem The blackresults line were is the obtained outline for ofthe the time parallelepiped. points chosen at our Gray discretion. represents the liquid phase. At t = 0.25 s, configuration. The black line is the outline of the parallelepiped. Gray represents the liquid phase. At we observet = 0.25 anearly s, we phaseobserve in an the early development phase in the of development a cavity and of aa splash.cavity and At at =splash.0.3 s, At the t splash= 0.3 s, the continues to develop.splash By continues 0.4 s, the to tankdevelop. bottom By 0.4 becomes s, the tank clear botto ofm becomes water, and clear the of water, rate of and splash the rate formation of splash slows down. Theformation late phase slows indown. the evolutionThe late phase of the in the cavity evolutio (itsn collapse) of the cavity and (its the collapse) splash and is shown the splash at t is= 0.75 s. Note thatshown the photography at t = 0.75 s. Note time that points the photography are not specified time points in [ 24are], not so specified our numerical in [24], so results our numerical were obtained for the timeresults points were chosenobtained atfor our the discretion.time points chosen at our discretion.

Figure 15. Waves induced by a falling parallelepiped in the first configuration: (a) Numerical results for the time points 0.25, 0.3, 0.4 and 0.75 s; (b) Experimental data.

Figures 16 and 17 are the amplitudes of water sloshing in the tank as a function of time for the second and the third configurations, respectively. The dots in the plot represent the experimental Figuredata 15.measuredWaves by induced a fixed bywave a falling gage [24]. parallelepiped The solid black in the line first shows configuration: the amplitude (a )of Numerical water sloshing results Figure 15. Waves induced by a falling parallelepiped in the first configuration: (a) Numerical results forin the the time tank points obtained 0.25, in 0.3, the 0.4calculations. and 0.75 s; (b) Experimental data. Comparisonfor the time points of the 0.25, gage 0.3, readings0.4 and 0.75 demonstrates s; (b) Experimental close data.agreement for the first incoming wave. FiguresFor the 16 second and 17configuration, are the amplitudes the difference of betw watereen sloshing the calculated in the and tank the experimental as a function amplitudes of time for the Figures 16 and 17 are the amplitudes of water sloshing in the tank as a function of time for the second andof the the first third wave configurations, is 4%, and for the respectively. third configur Theation, dots 5%. inThe the calculated plot represent amplitude the of experimental the second data second and the third configurations, respectively. The dots in the plot represent the experimental wave (“negative”) also closely matches the experimental data, but a difference is observed in measureddata by measured a fixed wave by a fixed gage wave [24]. gage The [24]. solid The black solid lineblack shows line shows the the amplitude amplitude of of water water sloshingsloshing in the subsequent surface oscillations of millimeter amplitude. tank obtainedin the tank in the obtained calculations. in the calculations. Comparison of the gage readings demonstrates close agreement for the first incoming wave. For the second configuration, the difference between the calculated and the experimental amplitudes of the first wave is 4%, and for the third configuration, 5%. The calculated amplitude of the second wave (“negative”) also closely matches the experimental data, but a difference is observed in subsequent surface oscillations of millimeter amplitude.

Figure 16. h H0 FigureAmplitude 16. Amplitude of water of water sloshing sloshing in in the the tank tank inin the second second configuration conﬁguration (h = (8 cm,= 8 H cm,0 = 2.90) = 2.90) (- —calculation; —experiment). (-—calculation;• ●—experiment).

Figure 16. Amplitude of water sloshing in the tank in the second configuration (h = 8 cm, H0 = 2.90) (-—calculation; ●—experiment).

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Figure 17. Amplitude of water sloshing in the tank in the second configuration (h = 4 cm, H0 = 4.75) (-—calculation; ●—experiment).

3.5. WaveGeosciences Flows 2020 Induced, 10, x FOR by PEER Lifting REVIEW a Be am Partially Immersed into Shallow Water 16 of 29 This problem considers the lifting of a solid body from the surface of a liquid and associated wave flows [25]. The problem geometry (Figure 18) is a horizontal-bottom rectangular prismatic channel with a beam partially immersed into shallow water. The modeling was performed without −6 Figurefriction 17. andAmplitude the surface of tension water sloshing of water, in whose the tank viscosity in the was second taken configuration as equal to 10 (h .= 4 cm, H0 = 4.75) FigureThe 17. rectangular Amplitude channelof water is sloshing filled with in thean incompressibletank in the second liquid, configuration into which a(h beam = 4 cm, is partiallyH0 = 4.75) (- —calculation; —experiment). (-—calculation;immersed. The ●•beam—experiment). has length 2L, where L = 1 m, and the same width as the channel. The channel Comparisondepth is h0 = 20 of cm, the and gage the readings depth under demonstrates the beam is closeh1 = 5 agreementcm. Both thefor liquid the and first the incoming beam are wave. initially at rest. The beam is lifted out of water at a constant velocity of 7.5 cm/s. For3.5. theWave second Flows configuration, Induced by Lifting the di a ffBeerenceam Partially between Immersed the calculated into Shallow and the Water experimental amplitudes of The beam motion is simulated using an approach based on superimposed multi-domain the firstThis(Chimera) wave problem is meshes. 4%, considers and According for the the third liftingto this configuration, oftechnique, a solid bodythe 5%. mesh Thefrom calculatedis thedivided surface amplitudeinto of a abackground liquid of the and second mesh, associated wave (“negative”) also closely matches the experimental data, but a difference is observed in subsequent wave coveringflows [25]. all the The computation problem geometrydomain, and (Figure an over 18)set mesh,is a horizontal-bottom representing the domain rectangular of a moving prismatic surfacechannelbody oscillations with [37]. a beamThe of problempartially millimeter domain immersed amplitude. was intodiscretize shallowd with water. an Theunstructured modeling computational was performed mesh without Figure 17. Amplitude of water sloshing in the tank in the second configuration (h = 4 cm, H0 = 4.75) frictionconsisting and the of surface truncated tension polyhedron of water,s (Figure whose 19). viscosityHighlighted was in Figutakenre as19 isequal the superimposed to 10−6. mesh. 3.5. Wave Flows(-—calculation; Induced by ●—experiment). Lifting a Beam Partially Immersed into Shallow Water TheThe rectangularmesh had ~30,000 channel cells, is and filled its basicwith sizean incompressiblewas 0.02 m. Next liquid,to the liquid into whichsurface, a an beam additional is partially refined block was constructed with a mesh size of ∆y = 0.005 m. The experiment was carried in immersed.This problem3.5. The Wave beam Flows considers has Induced lengththe by Lifting lifting 2L, awhere Be ofam a Partially solid L = 1 body m,Immersed and from theinto the Shallowsame surface widthWater of aas liquid the channel. and associated The channel wave parallel mode on 16 CPUs; the total computational time was approximately 20–30 min. flowsdepth [25is ].h0 The= 20This problem cm, problem andgeometry considersthe depth the (Figure underlifting of 18the a )solid isbeam a body horizontal-bottom is fromh1 = the5 cm.surface Both of rectangular athe liquid liquid and associated prismaticand the beam channel are waveFigure flows 20 shows [25]. The the problempictures geometryof wave propagation (Figure 18) is as a ahorizontal-bottom result of lifting the rectangular beam at differentprismatic time withinitially a beam at rest. partially The beam immersed is lifted into out shallow of water water. at a Theconstant modeling velocity was of performed 7.5 cm/s. without friction and points.channel with a beam partially immersed into shallow water. The modeling was performed without 6 the surfaceThe frictionbeam tension andmotion ofthe water, surface is simulated whosetension of viscosity water, using whose wasan viscosity takenapproach aswas equal takenbased toas equal 10on− superimposed.to 10−6. multi-domain (Chimera) meshes.The rectangular According channel to is thisfilled technique,with an incompressible the mesh liquid, is divided into which into a beam a backgroundis partially mesh, covering allimmersed. the computation The beam has domain,length 2L, whereand anL = over1 m, andset themesh, same representingwidth as the channel. the domain The channel of a moving depth is h0 = 20 cm, and the depth under the beam is h1 = 5 cm. Both the liquid and the beam are body [37]. The problem domain was discretized with an unstructured computational mesh initially at rest. The beam is lifted out of water at a constant velocity of 7.5 cm/s. consisting of truncatedThe beam motionpolyhedron is simulateds (Figure using 19). an Highlightedapproach based in onFigu superimposedre 19 is the multi-domain superimposed mesh. The mesh(Chimera) had ~30,000 meshes. cells, According and its to basic this technique,size was the0.02 mesh m. isNext divided to the into liquid a background surface, mesh, an additional refined blockcovering was all constructedthe computation with domain, a mesh and an size over ofset ∆mesh,y = 0.005representing m. The the experimentdomain of a moving was carried in body [37]. The problem domainFigure Figurewas 18.discretize 18. ProblemProblemd with geometry. geometry. an unstructured computational mesh parallel mode on 16 CPUs; the total computational time was approximately 20–30 min. consisting of truncated polyhedrons (Figure 19). Highlighted in Figure 19 is the superimposed mesh. TheFigure rectangularThe 20 mesh shows had channelthe~30,000 pictures cells, is filledand of waveits with basic propagation ansize incompressiblewas 0.02 m.as Nexta result to liquid, the of liquid lifting into surface, whichthe beam an aadditional beam at different is partially time immersed.points. refined The beamblock was has constructed length 2L ,with where a meshL = size1 m, of and∆y = the0.005 same m. The width experiment as the was channel. carried Thein channel parallel mode on 16 CPUs; the total computational time was approximately 20–30 min. depth is h0 = 20 cm, and the depth under the beam is h1 = 5 cm. Both the liquid and the beam are Figure 20 shows the pictures of wave propagation as a result of lifting the beam at different time initially atpoints. rest. The beam is lifted out of water at a constant velocity of 7.5 cm/s. The beam motion is simulated using an approach based on superimposed multi-domain (Chimera) meshes. According to this technique, the mesh is divided into a background mesh, covering all

the computation domain, and an overset mesh, representing the domain of a moving body [37]. The problem domain was discretized withFigure an unstructured 19. Mesh fragment. computational mesh consisting of truncated polyhedrons (Figure 19). Highlighted in Figure 19 is the superimposed mesh. The mesh had

~30,000 cells, and its basic size was 0.02Figure m. 18. Next Problem to the geometry. liquid surface, an additional reﬁned block was constructed with a mesh size of ∆yFigure= 0.005 18. Problem m. The geometry. experiment was carried in parallel mode on 16 CPUs; the total computational time was approximately 20–30 min.

Figure 19. Mesh fragment. Figure 19. Mesh fragment. Figure 19. Mesh fragment.

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Figure 20 shows the pictures of wave propagation as a result of lifting the beam at diﬀerent timeGeosciences points. 2020, 10, x FOR PEER REVIEW 17 of 29

Figure 20. Wave propagation.

Figure 21 shows the wave profiles at different time points. The results calculated by LOGOS are compared with data obtained in [25]. The mean square deviation with respect to the reference numerical data for this test case does not exceed 7%. The diagrams demonstrate that FiguretheFigure results 20.20. WaveWave are in propagation.propagation. close agreement. When the bottom of the beam is fully under water, depression waves are propagating across the region beyond the beam. Then, Figure 21 shows the wave proﬁles at diﬀerent time points. The results calculated by LOGOS whenFigure the beam 21 shows is lifted, the the wave sea profileslevel under at different it increases, time producing points. The a resultsflow directed calculated towards by LOGOS the beam are are compared with data obtained in [25]. The mean square deviation with respect to the reference center.compared Once with the beamdata obtainedis detached in from[25]. water,The mean two divergingsquare deviation waves are with produced. respect to the reference numericalnumerical datadata forfor thisthis testtest casecasedoes doesnot notexceed exceed7%. 7%. The diagrams demonstrate that the results are in close agreement. When the bottom of the beam is fully under water, depression waves are propagating across the region beyond the beam. Then, when the beam is lifted, the sea level under it increases, producing a flow directed towards the beam center. Once the beam is detached from water, two diverging waves are produced.

Figure 21. Wave profiles (—-LOGOS; —calculations [25]). Figure 21. Wave profiles (-—LOGOS; •●—calculations [25]). The diagrams demonstrate that the results are in close agreement. When the bottom of the 3.6. Propagation and Run-Up of a Single Wave over a Flat Slope beam is fully under water, depression waves are propagating across the region beyond the beam. Then,This when problem the beam simulates is lifted, a thesingle sea wave level underof height it increases, H propagating producing in a tank a flow of directedconstant towards depth d the= 1 beamm [22] center. and then Once running the beam up over is detached a slope fromof angle water, β = two 2.88°. diverging The schematic waves arerepresentation produced. of the tank Figure 21. Wave profiles (-—LOGOS; ●—calculations [25]). configuration is shown in Figure 22. This problem was proposed in [22] as a benchmark. 3.6. Propagation and Run-Up of a Single Wave over a Flat Slope 3.6. PropagationThe study [22]and Run-Upconsiders of aseveral Single Waveproblem over configurations a Flat Slope with different H/d ratios. In the first case,This H/d = problem 0.0185, and simulates in the second, a single H/d wave = 0.3. of The height initialH waveformpropagating of the in awave tank is of defined constant by depth This problem simulates a single wave of height H propagating in a tank of constant depth d = 1 d = 1 m [22] and then running up overη a slope= of angle2 ()γβ()=−2.88◦. The schematic representation of the m [22] and then running up over a slope(x ,of0) angleHsech β = 2.88°.x X Thes schematic representation of the(17) tank tank configuration is shown in Figure 22. This problem was proposed in [22] as a benchmark. configuration is shown in Figure 22. This problem was proposed in [22] as a benchmark. The study [22] considers several problem configurations with different H/d ratios. In the first case, H/d = 0.0185, and in the second, H/d = 0.3. The initial waveform of the wave is defined by η = 2 ()γ()− (x,0) Hsech x Xs (17)

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3H g where γ = . The initial velocity of the wave is u(x,0) = η(x,0) , which, in the framework of 4d d the linear shallow-water theory, ensures that the wave is propagating to the right. GeosciencesThe 2020problem, 10, x FOR was PEER simulated REVIEW on a mesh composed of 200,000 cells (Figure 23). The technology18 of 29of mesh model construction for tsunami simulations is described in detail in [19]. The mesh is refined 3H g wherenear the γ =wave run-up. The initial over velocity the slope of theand wave toward is uthe(x,0 )interface= η(x ,to0) , resolvewhich, inthe the details framework of wave of propagation. 4Thed problem was run in parallel mode on 64 standardd CPUs; the total computational theGeosciencestime linear was2020 approximatelyshallow-water, 10, 385 theory,20 min. ensures that the wave is propagating to the right. 17 of 28 The problem was simulated on a mesh composed of 200,000 cells (Figure 23). The technology of meshGeosciences model 2020 ,construction 10, x FOR PEER for REVIEW tsunami simulations is described in detail in [19]. The mesh is refined18 of 29 near the wave run-up over the slope and toward the interface to resolve the details of wave propagation. 3TheH problem was run in parallel mode on 64 standardg CPUs; the total computational where γ = . The initial velocity of the wave is u(x,0) = η(x,0) , which, in the framework of time was approximately4d 20 min. d the linear shallow-water theory, ensures that the wave is propagating to the right. The problem was simulated on a mesh composed of 200,000 cells (Figure 23). The technology of mesh model construction for tsunami simulations is described in detail in [19]. The mesh is refined near the wave run-up over the slopeFigure and 22. towardTank configuration. the interface to resolve the details of wave Figure 22. Tank configuration. propagation. The problem was run in parallel mode on 64 standard CPUs; the total computational time Thewas studyapproximately [22] considers 20 min. several problem configurations with different H/d ratios. In the first case, H/d = 0.0185, and in the second, H/d = 0.3. The initial waveform of the wave is defined by

2 η(x, 0) = Hsec h (γ(x Xs)) (17) − q Figure 22. Tank configuration.q = 3H ( ) = g ( ) where γ 4d . The initial velocity of the wave is u x, 0 d η x, 0 , which, in the framework of the linearFigure shallow-water 23. Mesh fragment theory, (blue ensures represents that the the air wave phase, is red propagating is the water to phase, the right. and black represents Themesh problem cells). was simulated on a mesh composed of 200,000 cells (Figure 23). The technology of mesh model construction for tsunami simulations is described in detail in [19]. The mesh is reﬁnedThe near parameters the wave of run-up both the over water the and slope air and phases toward are given the interface in Table to 2. resolve the details of wave propagation. The problem was run inFigure parallel 22. Tank mode configuration. on 64 standard CPUs; the total computational timeFigure was approximately 23. Mesh fragment 20 min. (blue representsTable the 2. Phaseair phase, properties. red is the water phase, and black represents mesh cells). Phase Molecular Viscosity (kg/(m∙s)) Density (kg/m3) Water 0.001 1000 The parameters of both the water and air phases are given in Table 2. Air 1.85 × 10−5 1.205

Table 2. Phase properties. Figure 23.24 Meshshows fragment the positions (blue represents of the free the airsurface phase, at red dimensionless is the water phase, time and points black t represents= τ d / g for Figuremesh cells). 23. Mesh fragmentPhase (blueMolecular represents Viscosity the air phase, (kg/(m red∙s)) is theDensity water phase, (kg/m and3) black represents the first problem configuration, where τ is an actual physical time. mesh cells). Water 0.001 1000 The parameterslight color shows of both the the mixed water cells and containing air phases areboth given air and in Tablewater.2. This layer is thin. Air 1.85 × 10−5 1.205 The parameterswaveforms of boththe wave the water calculated and air by phases LOGOS are were given compared in Table with2. the analytical data [22]. Figure 25 shows the comparisons for Tablethe first 2. Phase configuration properties. (H/d = 0.0185) at different time points Figure 24 shows the positions of the free surface at dimensionless time points t = τ d / g for t = τ d / g . Phase MolecularTable Viscosity 2. Phase (kgproperties./(m s)) Density (kg/m3) the first problem configuration, where τ is an actual physical· time. PhaseWater Molecular Viscosity 0.001 (kg/(m∙s)) Density 1000 (kg/m3) The light color shows the mixed cells containing5 both air and water. This layer is thin. Air 1.85 10− 1.205 The waveforms Waterof the wave calculated0.001 by× LOGOS were compared 1000 with the analytical data [22]. −5 Figure 25 shows the comparisonsAir for the 1.85 first × 10 configuration (H/d = 0.0185)1.205 at differentp time points Figure 24 shows the positions of the free surface at dimensionless time points t = τ d/g for the t = τ d / g . ﬁrst problemFigure 24 conﬁguration, shows the positions where τ ofis anthe actual free surface physical at time.dimensionless time points t = τ d / g for the first problem configuration, where τ is an actual physical time. The light color shows the mixed cells containing both air and water. This layer is thin. The waveforms of the wave calculated by LOGOS were compared with the analytical data [22]. FigureFigure 25 shows 24. Positions the comparisons of the free surface:for the first(a) τ configuration= 35; (b) τ = 55; (H/d (c) τ= =0.0185) 70 (blue at representsdifferent timeair; red points t = τ representsd / g . water).

Figure 24. Positions of the free surface: (a) τ = 35; (b) τ = 55; (c) τ = 70 (blue represents air; Figure 24. Positions of the free surface: (a) τ = 35; (b) τ = 55; (c) τ = 70 (blue represents air; red red represents water). represents water). The light color shows the mixed cells containing both air and water. This layer is thin. The waveforms of the wave calculated by LOGOS were compared with the analytical data [22].

Figure 25 shows the comparisons for the first configuration (H/d = 0.0185) at different time points p t = τ d/g. Figure 24. Positions of the free surface: (a) τ = 35; (b) τ = 55; (c) τ = 70 (blue represents air; red represents water).

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Figure 25. Waveform comparison for the first configuration H/d = 0.0185 (-—calculation; ♦—analytical solution).

Figure 26 shows positions of the free surface at different time points t = τ d / g for the second

Figureconfiguration.Figure 25. 25.Waveform FigureWaveform comparison27 shows comparison data for co the mparisonsfor first the configuration first for theconfiguration secondH/d = configuration0.0185 H/d (—-calculation;= 0.0185 (H/d =(-—calculation; 0.3) —analyticalat different solution).time♦ points.—analytical solution). The numerical results demonstrate close agreement with the analytical data: The single wave p FiguremovesFigure while 26 26 shows showspreserving positions positions its amplitude ofof thethe freefr untilee surfacesurface it reache at at sdifferent di thefferent slope, time time runs points pointsup onto tt= the=τ τ dslope/ gd/ gforandfor the washes the second second configuration.back. The results Figure calculated 27 shows by data LOGOS comparisons consistently for describe the second both configurationthe propagation (H of/d the= 0.3) wave at and diff erent configuration. Figure 27 shows data comparisons for the second configuration (H/d = 0.3) at different its run-up over the slope. timetime points. points. The numerical results demonstrate close agreement with the analytical data: The single wave moves while preserving its amplitude until it reaches the slope, runs up onto the slope and washes back. The results calculated by LOGOS consistently describe both the propagation of the wave and its run-up over the slope.

Figure 26. Positions of the free surface: (a) τ = 15; (b) τ = 20; (c) τ = 25; (d) τ = 30 (blue represents air; Geosciences 2020, 10, x FOR PEER REVIEW 20 of 29 red representsFigure 26. Positions water). of the free surface: (a) τ = 15; (b) τ = 20; (c) τ = 25; (d) τ = 30 (blue represents air; red represents water).

Figure 26. Positions of the free surface: (a) τ = 15; (b) τ = 20; (c) τ = 25; (d) τ = 30 (blue represents air; red represents water).

FigureFigure 27. Waveform 27. Waveform comparison comparison for the secondfor the conﬁguration second configurationH/d = 0.3 (-H/d —calculation; = 0.3 (-—calculation;—experiment). ♦—experiment).

3.7. Propagation and Run-Up of a Single Wave in a Varied-Depth Basin The problem simulates a single wave propagating in a basin of varied depth [22]. The configuration and the dimensions of the basin are shown in Figure 28. In the reference, three problem configurations are considered with different ratios of wave height H to basin depth d. In the first case, H/d = 0.038 and L = 2.4 m; in the second case, H/d = 0.259 and L = 0.98 m; in the third case, H/d = 0.681 and L = 0.64 m. The spacing between wave gages G1 and G2 is 2.4, 0.98 and 0.64 m in the first, second and third cases, respectively. The initial waveform of the wave is given by Equation (17). The problem was simulated on a computational mesh of 160,000 cells. The mesh was refined towards the interface to provide more accurate simulation. The cell height was chosen such that there were at least 10 cells along the wave height and at least 20 cells along the wavelength. The parameters of both the water and air phases are given in Table 2. The calculated quantitative characteristics of the wave pattern in the basin can be evaluated based on the wave gage data. LOGOS calculations were performed for all the problem configurations. To demonstrate that the problem solution is correct, let us compare the numerical results with the analytical data for the first problem configuration (Figure 29). The results for the two other configurations look similar. Figure 29 shows that the numerical results, both qualitatively and quantitatively, closely match the analytical data. Both the wave incident on the wave gage and the reflected wave are described well. The wave propagates with the same velocity as in the experiment. The maximum deviation in the calculated amplitudes does not exceed 4% for the incoming wave and 13% for the reflected wave at wave gage G6. This error is satisfactory for tsunami simulations.

Figure 28. Tank configuration.

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Figure 27. Waveform comparison for the second configuration H/d = 0.3 (-—calculation; ♦—experiment).

3.7. Propagation and Run-Up of a Single Wave in a Varied-Depth Basin The problem simulates a single wave propagating in a basin of varied depth [22]. The configuration and the dimensions of the basin are shown in Figure 28. In the reference, three problem configurations are considered with different ratios of wave height H to basin depth d. In the first case, H/d = 0.038 and L = 2.4 m; in the second case, H/d = 0.259 and L = 0.98 m; in the third case, H/d = 0.681 and L = 0.64 m. The spacing between wave gages G1 and G2 is 2.4, 0.98 and 0.64 m in the first, second and third cases, respectively. The initial waveform of the wave is given by Equation (17). The problem was simulated on a computational mesh of 160,000 cells. The mesh was refined towards the interface to provide more accurate simulation. The cell height was chosen such that there were at least 10 cells along the wave height and at least 20 cells along the wavelength. The parameters of both the water and air phases are given in Table 2. GeosciencesThe calculated2020, 10, 385 quantitative characteristics of the wave pattern in the basin can be evaluated19 of 28 based on the wave gage data. LOGOS calculations were performed for all the problem configurations. To demonstrate that the problem solution is correct, let us compare the numerical The numerical results demonstrate close agreement with the analytical data: The single wave results with the analytical data for the first problem configuration (Figure 29). The results for the two moves while preserving its amplitude until it reaches the slope, runs up onto the slope and washes other configurations look similar. back. The results calculated by LOGOS consistently describe both the propagation of the wave and its Figure 29 shows that the numerical results, both qualitatively and quantitatively, closely match run-up over the slope. the analytical data. Both the wave incident on the wave gage and the reflected wave are described well.3.7. PropagationThe wave propagates and Run-Up with of a Singlethe same Wave velocity in a Varied-Depth as in the experiment. Basin The maximum deviation in the calculated amplitudes does not exceed 4% for the incoming wave and 13% for the reflected wave at waveThe gage problem G6. This simulates error ais single satisfactory wave propagating for tsunami in simulations. a basin of varied depth [22]. The conﬁguration and the dimensions of the basin are shown in Figure 28.

Figure 28. Tank configuration. Figure 28. Tank configuration. In the reference, three problem configurations are considered with different ratios of wave height

H to basin depth d. In the first case, H/d = 0.038 and L = 2.4 m; in the second case, H/d = 0.259 and L = 0.98 m; in the third case, H/d = 0.681 and L = 0.64 m. The spacing between wave gages G1 and G2 is 2.4, 0.98 and 0.64 m in the first, second and third cases, respectively. The initial waveform of the wave is given by Equation (17). The problem was simulated on a computational mesh of 160,000 cells. The mesh was refined towards the interface to provide more accurate simulation. The cell height was chosen such that there were at least 10 cells along the wave height and at least 20 cells along the wavelength. The parameters of both the water and air phases are given in Table2. The calculated quantitative characteristics of the wave pattern in the basin can be evaluated based on the wave gage data. LOGOS calculations were performed for all the problem configurations. To demonstrate that the problem solution is correct, let us compare the numerical results with the analytical data for the first problem configuration (Figure 29). The results for the two other configurations look similar. Figure 29 shows that the numerical results, both qualitatively and quantitatively, closely match the analytical data. Both the wave incident on the wave gage and the reflected wave are described well. The wave propagates with the same velocity as in the experiment. The maximum deviation in the calculated amplitudes does not exceed 4% for the incoming wave and 13% for the reflected wave at wave gage G6. This error is satisfactory for tsunami simulations.

3.8. Single Wave Flowing around a Conical Island This problem serves for numerically modeling an experiment described in [22,40]. In this experiment, a single wave was produced by a wave maker in a basin of length 25 m and width 30 m filled with water to 0.32 m. The wave flowed around an island of a regular truncated conical shape having a height of 0.625 m, a bottom base diameter of 7.2 m and a top base diameter of 2.2 m. The basin walls were lined with an absorbing material to minimize wave reflection. Changes in the sea level were recorded by means of gages. The problem geometry and locations of the reference gages are shown in Figure 30. Geosciences 2020, 10, 385 20 of 28 Geosciences 2020, 10, x FOR PEER REVIEW 21 of 29

Geosciences 2020, 10, x FOR PEER REVIEW 22 of 29 Figure 29. Wave gage readings for the ﬁrst case (- —calculation; —experiment). Figure 29. Wave gage readings for the first case (-—calculation; ♦—experiment).

3.8. Single Wave Flowing around a Conical Island This problem serves for numerically modeling an experiment described in [22,40]. In this experiment, a single wave was produced by a wave maker in a basin of length 25 m and width 30 m filled with water to 0.32 m. The wave flowed around an island of a regular truncated conical shape having a height of 0.625 m, a bottom base diameter of 7.2 m and a top base diameter of 2.2 m. The basin walls were lined with an absorbing material to minimize wave reflection. Changes in the sea level were recorded by means of gages. The problem geometry and locations of the reference gages are shown in Figure 30.

Figure 30. Problem geometry and gage locations. Figure 30. Problem geometry and gage locations. Two problem configurations with different wave heights were considered. For the first case, H =Two0.016 problem m; for the configurations second case, H with= 0.032 different m. The wave initial heights waveform were of considered. the wave is givenFor the by first Equation case, H (17) = . 0.016 m;The for problem the second was case, solved H = using 0.032 a m. computational The initial waveform mesh refined of the at wave the interfaceis given by and Equation composed (17). of 1.4 millionThe problem cells. The was base solved cell using size is a equal computational to 0.08 m; themesh vertical refined cell at size the nearinterface the interface and composed is equal of to 1.40.005 million m. The cells. basin The walls base cell in the size discrete is equal model to 0.08 were m; the non-reflecting vertical cell boundaries, size near the through interface which is equal waves to 0.005could m. leave The thebasin domain walls withoutin the discrete reflection; model the we bottomre non-reflecting and the island bounda wereries, no-slip through boundaries; which andwaves the could leave the domain without reflection; the bottom and the island were no-slip boundaries; and the top boundary had fixed zero static pressure. The experiment was run in parallel mode on 128 CPUs; the total computational time was about 60 min. Figure 31 shows the calculated results for a wave flowing around an island at individual time points.

Figure 31. Calculated results for a wave flowing around an island at individual time points for H = 0.032 m.

The calculated results show that the wave pattern of the flow around the island corresponds to the classical description of a flow with a detached boundary layer around a bluff-stern body [41].

Geosciences 2020, 10, x FOR PEER REVIEW 22 of 29

Figure 30. Problem geometry and gage locations.

Two problem configurations with different wave heights were considered. For the first case, H = 0.016 m; for the second case, H = 0.032 m. The initial waveform of the wave is given by Equation (17). The problem was solved using a computational mesh refined at the interface and composed of 1.4 million cells. The base cell size is equal to 0.08 m; the vertical cell size near the interface is equal to 0.005 m. The basin walls in the discrete model were non-reflecting boundaries, through which waves Geosciencescould leave2020 ,the10, 385domain without reflection; the bottom and the island were no-slip boundaries;21 ofand 28 the top boundary had fixed zero static pressure. The experiment was run in parallel mode on 128 CPUs; the total computational time was about 60 min. top boundaryFigure 31 had shows ﬁxed the zero calculated static pressure. results for The a experimentwave flowing was around run in an parallel island mode at individual on 128 CPUs; time thepoints. total computational time was about 60 min. Figure 31 shows the calculated results for a wave ﬂowing around an island at individual time points.

Figure 31. Calculated results for a wave flowing around an island at individual time points for H = 0.032 m. Figure 31. Calculated results for a wave flowing around an island at individual time points for H = GeosciencesThe0.032 calculated 2020m. , 10, x FOR results PEER show REVIEW that the wave pattern of the flow around the island corresponds23 of to 29 the classical description of a flow with a detached boundary layer around a bluff-stern body [41]. FigureTheFigure calculated 32 32 shows shows results the the amplitude amplitude show that of ofthe water water wave sloshing sloshing pattern in ofin the thethe basin flowbasin around as as a a function function the island of of time time corresponds for for the the two two to problemtheproblem classical configurations. configurations. description Theof The a dotsflow dots in with thein the plota detached plot represent repr boundaryesent the experimentalthe layerexperimental around data a from databluff-stern [from40]. The [40]. body solid The [41]. black solid lineblack shows line theshows amplitude the amplitude of water of sloshing water sloshing in the basin in the obtained basin obtained in the LOGOS in the LOGOS calculations. calculations.

(a) (b)

FigureFigure 32. 32.Amplitude Amplitude of of water water sloshing sloshing in in the the basin: basin: (a ()aH) H= =0.016 0.016 m; m; (b ()bH) H= =0.032 0.032 m) m) (- (-—calculation; —calculation; —experiment).●—experiment). • Comparison of the gage readings shows their close agreement for the first incident wave both in the waveform and in the amplitude of the wave. However, some differences in further sloshing are observed, which can be related to the mesh quality. The mean square deviation relative to the analytical solution for this problem is about 10%.

3.9. Generation of Waves by a Sliding Wedge-Shaped Body This problem serves for numerically modeling an experiment described in [22,42]. In this experiment, waves were generated by a wedge-shaped body of height 0.455 m, length 0.91 m and width 0.61 m sliding under the influence of gravity down a slope in a tank with water (Figure 33). The problem geometry and gage locations are shown in Figure 33. All the dimensions are given in meters.

(a) (b)

Figure 33. Experimental photographs (a) and problem geometry (b).

Geosciences 2020, 10, x FOR PEER REVIEW 23 of 29

Figure 32 shows the amplitude of water sloshing in the basin as a function of time for the two problem configurations. The dots in the plot represent the experimental data from [40]. The solid black line shows the amplitude of water sloshing in the basin obtained in the LOGOS calculations.

(a) (b)

Figure 32. Amplitude of water sloshing in the basin: (a) H = 0.016 m; (b) H = 0.032 m) (-—calculation; Geosciences●—experiment).2020, 10, 385 22 of 28

Comparison of the gage readings shows their close agreement for the first incident wave both in the waveformComparison and of in the the gage amplitude readings of showsthe wave. their Ho closewever, agreement some differences for the first in further incident sloshing wave both are inobserved, the waveform which andcan inbe the related amplitude to the of mesh the wave. qualit However,y. The mean some square differences deviation in further relative sloshing to the areanalytical observed, solution which for can this be problem related is to about the mesh 10%. quality. The mean square deviation relative to the analytical solution for this problem is about 10%. 3.9. Generation of Waves by a Sliding Wedge-Shaped Body 3.9. Generation of Waves by a Sliding Wedge-Shaped Body This problem serves for numerically modeling an experiment described in [22,42]. In this This problem serves for numerically modeling an experiment described in [22,42]. In this experiment, waves were generated by a wedge-shaped body of height 0.455 m, length 0.91 m and experiment, waves were generated by a wedge-shaped body of height 0.455 m, length 0.91 m and width 0.61 m sliding under the influence of gravity down a slope in a tank with water (Figure 33). width 0.61 m sliding under the influence of gravity down a slope in a tank with water (Figure 33). The problem geometry and gage locations are shown in Figure 33. All the dimensions are given in The problem geometry and gage locations are shown in Figure 33. All the dimensions are given meters. in meters.

(a) (b) Geosciences 2020, 10, x FOR PEER REVIEW 24 of 29 Figure 33. Experimental photographs (a) and problem geometrygeometry ((b).). Gage 1 measures the change in sea level, and Gage 2, the run-up over the slope. Gage 1 measures the change in sea level, and Gage 2, the run-up over the slope. The problem was solved using a composite computational mesh (Figure 34) refined at the The problem was solved using a composite computational mesh (Figure 34) reﬁned at the interface interface and wave propagation area and composed of 2.5 million cells, including a prismatic and wave propagation area and composed of 2.5 million cells, including a prismatic sublayer on sublayer on the slope. The base cell size is equal to 0.02 m. The walls and the bottom of the tank were the slope. The base cell size is equal to 0.02 m. The walls and the bottom of the tank were no-slip no-slip boundaries; the top boundary had fixed zero static pressure. The problem was run in parallel boundaries; the top boundary had ﬁxed zero static pressure. The problem was run in parallel mode on mode on 96 CPUs; the total computational time was 70 min. 96 CPUs; the total computational time was 70 min.

Figure 34. Computational mesh, side view. Figure 34. Computational mesh, side view. Figure 35 shows the waves generated by the sliding wedge. Figure 35 shows the waves generated by the sliding wedge.

Figure 35. Free surface position.

The numerical modeling confirms that the sea level behind the wedge-shaped body sliding down along a slope is depressed, following which a diverging wave forms. Figure 36 shows the wave amplitude as a function of time for Gages 2 and 4. The plots in Figure 36 demonstrate close quantitative agreement between the numerical and the experimental data. Here, we estimate the waveform relative to the experiment; the mean error, taken

Geosciences 2020, 10, x FOR PEER REVIEW 24 of 29

Gage 1 measures the change in sea level, and Gage 2, the run-up over the slope. The problem was solved using a composite computational mesh (Figure 34) refined at the interface and wave propagation area and composed of 2.5 million cells, including a prismatic sublayer on the slope. The base cell size is equal to 0.02 m. The walls and the bottom of the tank were no-slip boundaries; the top boundary had fixed zero static pressure. The problem was run in parallel mode on 96 CPUs; the total computational time was 70 min.

Figure 34. Computational mesh, side view.

GeosciencesFigure2020 35, 10 shows, 385 the waves generated by the sliding wedge. 23 of 28

Figure 35. Free surface position. Geosciences 2020, 10, x FOR PEER REVIEW 25 of 29 Figure 35. Free surface position. The numerical modeling conﬁrms that the sea level behind the wedge-shaped body sliding down as the mean value of the errors corresponding to the given experimental points, for this problem is alongThe a slope numerical is depressed, modeling following confirms which that the a diverging sea level wavebehind forms. the wedge-shaped Figure 36 shows body the sliding wave about 15%. amplitudedown along as a functionslope is depressed, of time for Gagesfollowing 2 and whic 4. h a diverging wave forms. Figure 36 shows the wave amplitude as a function of time for Gages 2 and 4. The plots in Figure 36 demonstrate close quantitative agreement between the numerical and the experimental data. Here, we estimate the waveform relative to the experiment; the mean error, taken

Figure 36. Wave amplitude (—-calculation; —experiment). Figure 36. Wave amplitude (-—calculation;• ●—experiment). The plots in Figure 36 demonstrate close quantitative agreement between the numerical and the3.10. experimental Wave Propagation data. in Here, a Nonuniform-Botto we estimate them waveformTank with Run-Up relative onto to the an experiment;Obstacle the mean error, takenThis as the problem mean value considers of the a errorswave correspondingtraveling across to a the basin given of length experimental 22 m and points, depth for h0 this = 0.2 problem m. The isbottom about of 15%. the basin starting from x = 10 m has a slope and an obstacle. Experimental studies of such wave propagation are reported in [23]. This problem is a good test to verify the technique described 3.10. Wave Propagation in a Nonuniform-Bottom Tank with Run-Up onto an Obstacle above. ThisFigure problem 37 shows considers the problem a wave geometry. traveling The across wave a is basin formed of length at the left 22 mwall and ofdepth the basinh0 = and0.2 has m. Thea height bottom of H of0 = the 0.07 basin m. starting from x = 10 m has a slope and an obstacle. Experimental studies

(a)

(b)

Figure 37. Schematic representation of the problem (a) and the obstacle (b).

Elevation coordinates of the obstacle are given in Table 3.

Geosciences 2020, 10, x FOR PEER REVIEW 25 of 29 as the mean value of the errors corresponding to the given experimental points, for this problem is about 15%.

Figure 36. Wave amplitude (-—calculation; ●—experiment).

3.10. Wave Propagation in a Nonuniform-Bottom Tank with Run-Up onto an Obstacle Geosciences 2020, 10, 385 24 of 28 This problem considers a wave traveling across a basin of length 22 m and depth h0 = 0.2 m. The bottom of the basin starting from x = 10 m has a slope and an obstacle. Experimental studies of such ofwave such propagation wave propagation are reported are reported in [23]. This in [23 problem]. This problemis a good is test a good to verify test the to verify technique the technique described describedabove. above. FigureFigure 37 37 shows shows the the problem problem geometry. geometry. The The wave wave is is formed formed at at the the left left wall wall of of the the basin basin and and has has a heighta height of ofH 0H=0 =0.07 0.07 m. m.

(a)

Geosciences 2020, 10, x FOR PEER REVIEW 26 of 29

Table 3. Elevation coordinates of the obstacle.

T1 (13.63, 0.008) m T2 (13.67, 0.017) m T3 (13.70,(b )0.025) m T4 (13.73, 0.033) m FigureFigure 37.37. SchematicSchematic representationrepresentation ofof thethe problemproblem ((aa)) andand the the obstacle obstacle ( b(b).). T5 (13.77, 0.042) m T6 (13.80, 0.050) m ElevationElevation coordinatescoordinates ofof thethe obstacleobstacle areare given given in in Table Table3 .3. T7 (13.83, 0.059) m

Table 3. ElevationT8 (13.86, coordinates 0.067) of m the obstacle. T9 (13.97, 0.065) m T1 (13.63, 0.008) m T10 (13.99, 0.055) m T2 (13.67, 0.017) m T11 (14.01, 0.044) m T3 (13.70, 0.025) m T T12 (14.02,(13.73, 0.034) 0.033) m m 4 T5 (13.77, 0.042) m The problem domain wasT6 discretized with a(13.80, three-dimensional 0.050) m unstructured computational T mesh consisting of truncated polyhedrons7 (Figure (13.83,38). 0.059) m T8 (13.86, 0.067) m The mesh had 1.1 million cells. The characteristic cell size was set as 0.1 m. To resolve the T9 (13.97, 0.065) m moving wave in greater detail,T10 an additional mesh(13.99, block 0.055) was constructed m with refinement around y = 0 from −0.1 to 0.2 m, whereT 11the characteristic size(14.01, was 0.044)∆x = 0.02 m m and ∆y = 0.001 m. The problem was run in parallel mode on T6412 CPUs; the total computational(14.02, 0.034) mtime was 20 min. Figure 39 shows the wave profiles calculated by LOGOS in comparison with the experiment at differentThe problemtime points. domain One was can discretized see that withthe acalc three-dimensionalulated and the unstructuredexperimental computational data are in meshclose agreement.consisting of truncated polyhedrons (Figure 38).

Figure 38. Computational mesh.

Figure 39. Calculated vs. experimental wave profiles at different time points: (a) 2.63 s; (b) 2.89 s; (c) 3.01 s; (d) 3.19 s; (e) 3.35 s; (f) 3.71 s (▪—calculation; ●—experiment).

Geosciences 2020, 10, x FOR PEER REVIEW 26 of 29

Table 3. Elevation coordinates of the obstacle.

T1 (13.63, 0.008) m T2 (13.67, 0.017) m T3 (13.70, 0.025) m T4 (13.73, 0.033) m T5 (13.77, 0.042) m T6 (13.80, 0.050) m T7 (13.83, 0.059) m T8 (13.86, 0.067) m T9 (13.97, 0.065) m T10 (13.99, 0.055) m T11 (14.01, 0.044) m T12 (14.02, 0.034) m

The problem domain was discretized with a three-dimensional unstructured computational mesh consisting of truncated polyhedrons (Figure 38). The mesh had 1.1 million cells. The characteristic cell size was set as 0.1 m. To resolve the moving wave in greater detail, an additional mesh block was constructed with refinement around y = 0 from −0.1 to 0.2 m, where the characteristic size was ∆x = 0.02 m and ∆y = 0.001 m. The problem was run in parallel mode on 64 CPUs; the total computational time was 20 min. Figure 39 shows the wave profiles calculated by LOGOS in comparison with the experiment at different time points. One can see that the calculated and the experimental data are in close agreement.

Geosciences 2020, 10, 385 25 of 28

The mesh had 1.1 million cells. The characteristic cell size was set as 0.1 m. To resolve the moving wave in greater detail, an additional mesh block was constructed with refinement around y = 0 from 0.1 to 0.2 m, where the characteristic size was ∆x = 0.02 m and ∆y = 0.001 m. The problem was run in − parallel mode on 64 CPUs; the total computational time was 20 min. Figure 39 shows the wave profiles calculated by LOGOS in comparison with the experiment at different time points. One can see thatFigure the calculated 38. Computational and the mesh. experimental data are in close agreement.

GeosciencesFigureFigure 2020 39.39., 10 Calculated, x FOR PEER vs. vs. REVIEW experimental experimental wave wave profiles profiles at at different different time time points: points: (a)( a2.63) 2.63 s; ( s;b) (2.89b) 2.89 s; 27(c s;) of 29 (c) 3.01 s; (d) 3.19 s; (e) 3.35 s; (f) 3.71 s (—calculation; —experiment). 3.01 s; (d) 3.19 s; (e) 3.35 s; (f) 3.71 s (▪—calculation; ●—experiment).• Figure 40 presents a comparison of the calculated wave profiles with the experimental data and Figure 40 presents a comparison of the calculated wave profiles with the experimental data and calculated velocity fields. calculated velocity fields.

Figure 40. WaveWave profile: proﬁle: ( (aa)—experiment;)—experiment; ( (bb)—calculation;)—calculation; (c (c)—velocity)—velocity field ﬁeld (left—time (left—time 2.63 2.63 s, s, right—time 3.71 s)

The results shown in the figure ﬁgure demonstrate clos closee qualitative agreement between the calculated and the experimental data.

4. Conclusions 4. Conclusions The paper presents verification results for the LOGOS software package as applied to tsunami The paper presents verification results for the LOGOS software package as applied to tsunami simulations. First, LOGOS free-surface flow simulations were verified against the test cases of a simulations. First, LOGOS free-surface flow simulations were verified against the test cases of a collapsing water column and gravity water sloshing in a tank and the known test cases of wave collapsing water column and gravity water sloshing in a tank and the known test cases of wave generation by objects falling into water or lifted out of it. Then, LOGOS verification specifically as generation by objects falling into water or lifted out of it. Then, LOGOS verification specifically as applied to tsunami simulations was performed on international benchmarks, including the propagation and run-up of a single wave onto a flat slope and a vertical wall, the sliding of a wedge-shaped body down a slope, flow around an island and wave run-up onto an obstacle. The results of the verification calculations show that LOGOS enables sufficiently accurate numerical simulations of free-surface flows and delivers acceptable results in tsunami simulations. LOGOS is shown to be good at modeling the formation, propagation and run-up of waves. For all the problems, the error in the calculated results relative to the reference data does not exceed 10%, except for one problem, in which the error was 15%. This level of accuracy is satisfactory for tsunami simulations. Consequently, this technology can be used in numerical simulations of tsunami propagation, including generation, propagation and run-up on a beach.

Funding: The research was supported by grants of the President of the Russian Federation for the state support of research projects by leading scientific schools of the Russian Federation (NSh-2485.2020.5) and by the Russian Foundation for Basic Research (20-05-00162).

Conflicts of Interest: The authors declare no conflict of interest.

Geosciences 2020, 10, 385 26 of 28 applied to tsunami simulations was performed on international benchmarks, including the propagation and run-up of a single wave onto a flat slope and a vertical wall, the sliding of a wedge-shaped body down a slope, flow around an island and wave run-up onto an obstacle. The results of the verification calculations show that LOGOS enables sufficiently accurate numerical simulations of free-surface flows and delivers acceptable results in tsunami simulations. LOGOS is shown to be good at modeling the formation, propagation and run-up of waves. For all the problems, the error in the calculated results relative to the reference data does not exceed 10%, except for one problem, in which the error was 15%. This level of accuracy is satisfactory for tsunami simulations. Consequently, this technology can be used in numerical simulations of tsunami propagation, including generation, propagation and run-up on a beach.

Author Contributions: Conceptualization, A.K. (Andrey Kozelkov), A.K. (Andrey Kurkin) and E.P.; data curation, E.T., V.K., K.P. and D.U.; formal analysis, A.K. (Andrey Kozelkov) and V.K.; investigation, E.T., A.K. (Andrey Kozelkov), A.K. (Andrey Kurkin), E.P., V.K., K.P. and D.U.; methodology, A.K. (Andrey Kozelkov), A.K. (Andrey Kurkin), E.P. and V.K.; software, E.T., V.K., K.P. and D.U.; supervision, A.K. (Andrey Kurkin); validation, E.T., K.P. and D.U.; visualization, E.T., K.P. and D.U.; writing—original draft, A.K. (Andrey Kozelkov), A.K. (Andrey Kurkin) and E.P. All authors have read and agreed to the published version of the manuscript. Funding: The research was supported by grants of the President of the Russian Federation for the state support of research projects by leading scientific schools of the Russian Federation (NSh-2485.2020.5) and by the Russian Foundation for Basic Research (20-05-00162). Conflicts of Interest: The authors declare no conflict of interest.

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