Argiris I Delis
email: adelis@science.tuc.gr

Hervé Guillard
email: herve.guillard@inria.fr

Yih-Chin Tai
email: yctai@mail.ncku.edu.tw.

Numerical simulations of hydraulic jumps with the Shear Shallow Water model

Keywords: Shear Shallow Water model, Shallow Water equations, Turbulent Hydraulic jumps, Free surface flows, Finite Volumes, Well-balancing

, is considered in this work. The model equations are able to describe the oscillatory nature of turbulent hydraulic jumps and as such correct the deficiency of the classical non-linear shallow water equations in describing such phenomena. The model equations, originally developed for horizontal flow or flows occurring over small constant slopes, are straightforwardly extended here for modeling flows over non-constant slopes and numerically solved by a second-order well-balanced finite volume scheme. Further, a new set of exact solutions to the extended model equations is derived and several numerical tests are performed to validate the numerical scheme and its ability to predict the oscillatory nature of hydraulic jumps under different flow conditions.

Introduction

Hydraulic jumps are commonly observed in laboratory experiments, river flows or coastal areas. The classical description of this phenomenon uses the shallow water hyperbolic system of equations (SWE) and assimilates hydraulic jumps to a discontinuity of zero length governed by jump relations between the supercritical upstream state and the subcritical downstream one as:

[h(u -D)] = 0; hu(u -D) + gh 2 /2 = 0 (1.1)
where here [•] stands for the jump (•) R -(•) L between the upstream state (•) L and the downstream one (•) R , h is the water height, u the flow velocity, g the gravity acceleration and D the jump velocity. This classical description of hydraulic jumps has well-known shortcomings. As observed in many laboratory experiments, see e.g [START_REF] Chachereau | Free-surface fluctuations and turbulence in hydraulic jumps[END_REF], hydraulic jumps have a non-zero width and their shapes are different according to the upstream Froude number. In particular, for large Froude numbers (typically larger than 1.7) the jump becomes turbulent; the water depth rapidly increases, large scale turbulent eddies, forming a roller, appear beneath the free surface and intense vorticity generation is observed. Oscillations with well-defined frequencies of the toe of the hydraulic jump have been also reported [START_REF] Mok | Relation of surface roller eddy formation and surface fluctuation in hydraulic jumps[END_REF][START_REF] Chanson | Convective transport of air bubbles in strong hydraulic jumps[END_REF].

Building on an earlier work devoted to roll waves in [START_REF] Richard | A new model of roll waves: comparison with brock's experiments[END_REF], and more recently in [START_REF] Ivanova | Formation and coarsening of roll-waves in shear shallow water flows down an inclined rectangular channel[END_REF], Richard and Gavrilyuk have proposed in [START_REF] Richard | The classical hydraulic jump in a model of shear shallow-water flows[END_REF] a new model for turbulent hydraulic jumps, called the Shear Shallow A.I Delis, H. Guillard, et al.

Water Equations (SSWE), that attempt to correct most of the deficiencies of the classical shallow water description of these phenomena. In particular, their model is able to compute the jump's toe oscillations with a frequency that compare reasonably well with experiments. This SSWE model was obtained by taking into account the fluctuation of the horizontal velocity along the vertical direction. This model is also interesting since it contains only two adjustable parameters with a well-defined physical interpretations. These two parameters, namely the wall enstrophy φ and the energy dissipation coefficient C r have been determined in Richard and Gavrilyuk's model from the experiments by Hager and Bremen [START_REF] Hager | Classical hydraulic jump: sequent depths[END_REF].

The model of [START_REF] Richard | The classical hydraulic jump in a model of shear shallow-water flows[END_REF] has been developed for horizontal flows or flows occurring on a small constant slope. The main purpose of this work is to investigate, also numerically, a straightforward extension of this model for flows on non-constant slopes. In particular, no attempts have been made to adjust the parameters of the model to this new situation hence, we compute these values as proposed in [START_REF] Richard | The classical hydraulic jump in a model of shear shallow-water flows[END_REF]. The remainder of this paper is the following: In Section 2, we recall some salient features of the model proposed in [START_REF] Richard | The classical hydraulic jump in a model of shear shallow-water flows[END_REF] and describe its extension for flows over a non-constant topography. Then, in Section 3, we describe the numerical method used to approximate the solutions of the model. Actually, the model of [START_REF] Richard | The classical hydraulic jump in a model of shear shallow-water flows[END_REF] is a conservative hyperbolic system with a stiff source term whose approximation is not straightforward. The design of a second-order (in space and time) finite volume numerical scheme used for its extension on non-constant topography necessitates to integrate some modifications related to the preservation of equilibrium (lake at rest) solutions. In the last section, we present some numerical tests and corresponding results. Some new analytic solutions of the SSWE over non-constant topography are derived, by modifying some well-known benchmark cases used in shallow water and open channel flow studies. This consists another novel aspect of this work. Finally, a numerical comparison with experimental data from a forced hydraulic jump is also presented.

The shear shallow water model

For a detailed derivation of this model, we refer to [START_REF] Richard | A new model of roll waves: comparison with brock's experiments[END_REF] and [START_REF] Richard | The classical hydraulic jump in a model of shear shallow-water flows[END_REF] and here briefly summarize the rationale leading to this model. The shallow water model uses the conservation of mass and momentum and thus does not consider explicitly the physical principle of energy conservation. In contrast, the SSWE deals explicitly with the conservation of energy that is split into three components: kinetic, potential and turbulent energy, the latter being associated with the fluctuation of the horizontal velocity along the vertical direction. For hydraulic jumps, these fluctuations are themselves associated with the presence of small scale vortices near the bottom associated to the boundary layer and to the existence of a large scale roller close to the water surface. Using physical arguments, the turbulent energy is expressed in term of a variable called enstrophy with two components : a small scale enstrophy φ s associated with the near bottom boundary layer and a large scale enstrophy Ψ associated with the roller flow. Incorporating in this description, a space varying bottom topography, b ≡ b(x) leads to the following system written in one-dimensional conservative form [START_REF] Richard | The classical hydraulic jump in a model of shear shallow-water flows[END_REF] :

∂ t U + ∂ x F(U) = S b + S f , (2.1)
where the vectors of conserved variables U, fluxes F and source terms S b and S f are defined as

U =   h q E   , F =     q q 2 h + P q(E + P )/h     , S b =   0 -gh∂ x b 0   , S f =       0 -C f |q|q h 2 -C f + C r Φ -φ s Φ |q| 3 h 3       (2.2)
Numerical simulations of hydraulic jumps with the Shear Shallow Water model with h being the water depth, q = hu the unit discharge, u the average velocity, Φ the total enstrophy and is E the total energy defined as

E = h u 2 2 + g h 2 2 + ghb + Φ h 3 2 .
In this relation, the first two terms represent, respectively, the kinetic and potential energies while the last one stands for a turbulent energy defined thanks to the total enstrophy Φ. For more details on the physical interpretation of the model we refer to [START_REF] Richard | The classical hydraulic jump in a model of shear shallow-water flows[END_REF]. Further, P is the pressure given as

P = gh 2 2 + Φh 3 .
By defining an "internal energy", e, by

e = gh 2 + gb + Φ h 2 2 ,
the following equation of state can be verified

P = 2he - gh 2 2 -2ghb = 2E -hu 2 - gh 2 2 -2ghb.
The two source terms in (2.1) S b and S f stand, respectively, for the topography and the friction terms. In the latter, C f is the friction coefficient, related to the Darcy-Weisbach coefficient f by C f = f /8 and C r is a drag coefficient corresponding to the dissipation of the entrophy in the turbulent roller generated in a hydraulic jump. Further, the total enstrophy is decomposed as

Φ = Ψ + φ s
where φ s is the small scale enstrophy describing the intensity of the vortices in the boundary layer near the bed, and Ψ is the large scale (roller) enstrophy. Moreover, it can be seen that system (2.1) implies the following equation for the enstrophy

∂ t Φ + u∂ x Φ = - 2 h 3 C r Φ -φ s Φ |u| 3 . (2.3)
System (2.1) admits the following quasi-linear form, setting W = (h, u, Φ) T ,

∂ t W + A∂ x W = S b + S f where A =   u h 0 g + 3hΦ u h 2 0 0 u   , S b =   0 -g∂ x b 0   , S f =      0 -C f |u|u h - 2 h 3 C r Φ -φ s Φ |u| 3     
.

The characteristic polynomial of A is then given as

χ(A) = (u -λ) (u -λ) 2 -(gh + 3Φh 2 )
with roots λ 1 = u -α, λ 2 = u, λ 3 = u + α where α = gh + 3Φh 2 , verifying the hyperbolic nature of system (2.1).

A.I Delis, H. Guillard, et al.

An important property of (2.1), related to the source terms, is that admits non-trivial steady-states. Following from (2.1), these may be given as

∂ x q = 0 q∂ x u + gh∂ x (h + b) + ∂ x (Φh 3 ) = -C f |q|q h 2 ∂ x (u(E + P )) = - 2 h 3 C r Φ -φ s Φ |q| 3 h 3
From the above equilibria, some classes of steady-state solutions can be derived which can help to asses the performance of a numerical scheme and will be considered later on in Section 4.3.

An important elementary solution is the so-called flow at rest that is easily obtained assuming u = q = 0 and h + b = η(x, 0) = η 0 (constant). For the SSWE we see that from the momentum equation a steady equilibrium is obtained in this case if the enstrophy verifies also

∂ x (Φh 3 ) = 0.
This relation is verified if Φ = 0 that is consistent with the physical requirement that no large or small scale enstrophy is created in the stagnant water case. Therefore,W 0 = (η 0 -b, 0, 0) T defines a class of exact steady solution of the model.

Numerical method

We describe here a relatively simple finite volume (FV) scheme for system (2.1) that is of second-order accuracy in space and time and is well-balanced in the sense of preserving exactly the flow at rest solution thus allowing to have an efficient treatment of the topography source term [START_REF] Bermudez | Upwind methods for hyperbolic conservation laws with source terms[END_REF][START_REF] Vázquez-Cendón | Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry[END_REF]. In practice, schemes preserving this equilibrium (also called the C-propery) give good results even in unsteady cases [START_REF] Delis | A robust high-resolution finite volume scheme for the simulation of long waves over complex domains[END_REF]. In this FV scheme a splitting method is adopted where at the first step we solve the system of equations without the drag term S f and the well-balanced spatial discretization is utilized. At the second step in the splitting method, we take into account the friction term and a system of ordinary equations is to be solved. For this, we will use the fact that due to the special structure of system (2.1) these ODEs can be solved in an exact way.

Advection step and well-balancing

At the first step of the splitting, we consider the following semi-discrete form of the FV scheme for the convection terms and topography source term S b

∆x∂ t U i + F i+1/2 -F i-1/2 = S i+1/2 b + S i-1/2 b (3.1)
where i = 1, . . . , N refers to the center of the computational cell , we need to specify first the form of the numerical fluxes. In this work, we consider the classical Rusanov numerical flux which writes as

I i = [x i-1/2 , x i+1/2 ] = [x i -1 2 ∆x, x i + 1 2 ∆x], F i±1/
F i+1/2 = 1 2 F(U R i+1/2 ) + F(U L i+1/2 ) - C i+1/2 2 U R i+1/2 -U L i+1/2 (3.2)
where (•) L i+1/2 and (•) R i+1/2 denote the left and right (possibly reconstructed) solution and topography values at the i + 1/2 interface and

C i+1/2 = sup U=U R,L i+1/2 sup j=1,2,3 |λ j (U)|
Numerical simulations of hydraulic jumps with the Shear Shallow Water model with λ j being the eigenvalues of the system. For a first-order spatial scheme one has U R i+1/2 = U i+1 and U L i+1/2 = U i . Now, plugging the numerical flux expression (3.2) in the scheme (3.1) and after some simple manipulations, we can write it in the following fluctuation form

∆x∂ t U i + 1 2 F(U R i+1/2 ) -F(U L i+1/2 ) + F(U L i+1/2 ) -F(U i ) - C i+1/2 2 U R i+1/2 -U L i+1/2 + 1 2 F(U R i-1/2 ) -F(U L i-1/2 ) -F(U R i-1/2 ) + F(U i ) + C i-1/2 2 U R i-1/2 -U L i-1/2 = S i+1/2 b + S i-1/2 b .
For the flow at rest equilibrium the above scheme takes the following form, by setting ∂ t U i = 0 and u = q = 0,

1 2    0 g hi+1/2 ∆h i+1/2 0    -    0 g h- i+1/2 ∆h - i+1/2 0    - C i+1/2 2    ∆h i+1/2 0 g bi+1/2 ∆h i+1/2    + 1 2    0 g hi-1/2 ∆h i-1/2 0    +    0 g h+ i-1/2 ∆h + i-1/2 0    + C i-1/2 2    ∆h i-1/2 0 g bi-1/2 ∆h i-1/2    = S i+1/2 b + S i-1/2 b
where the following simple notations have been used

(•) i±1/2 = (•) R i±1/2 + (•) L i±1/2 2 , (•) - i+1/2 = (•) i + (•) L i+1/2 2 , (•) + i+1/2 = (•) i+1 + (•) R i+1/2 2 and ∆(•) i±1/2 = (•) R i±1/2 -(•) L i±1/2 , ∆(•) - i+1/2 = (•) i -(•) L i+1/2 , ∆(•) + i+1/2 = (•) i+1 -(•) R i+1/2 .
We note that for a first-order scheme and following from the above notation the terms g h-i+1/2 ∆h - i+1/2

and g h+ i-1/2 ∆h + i-1/2 vanish. Now, the simplest definition of the discrete source terms, guaranteeing that well-balancedness, is achieved are the following

S i+1/2 b + S i-1/2 b = - 1 2    0 g hi+1/2 ∆b i+1/2 0    +    0 g h- i+1/2 ∆b - i+1/2 0    + C i+1/2 2    ∆b i+1/2 0 g bi+1/2 ∆b i+1/2    - 1 2    0 g hi-1/2 ∆b i-1/2 0    -    0 g h+ i-1/2 ∆b + i-1/2 0    - C i-1/2 2    ∆b i-1/2 0 g bi-1/2 ∆b i-1/2    .
The above discretization exactly preserves the hydrostatic conditions by enforcing at a cell interface that ∆h = -∆b, while the terms g h-i+1/2 ∆b - i+1/2 and g h+ i-1/2 ∆b + i-1/2 , which vanish for a first order scheme, enforce the hydrostatic condition by satisfying ∆h ± = -∆b ± .

Friction and drag terms treatment

Concerning the treatment of the source term S f , at the second splitting step, we use (3.3) as a prediction without the friction and drag terms, i.e.

U i = U n i - ∆t n ∆x F i+1/2 -F i-1/2 + S i+1/2 b + S i-1/2 b (3.3)
where for simplicity the forward Euler time stepping is presented with ∆t n being the current time step and U n i the average value at time t n . At the second splitting step the following system of ODEs has to be solved

dU dt = S f (U) (3.4)
with initial condition U i calculated at the first step. Using the same strategy as in [START_REF] Richard | The classical hydraulic jump in a model of shear shallow-water flows[END_REF], the first two equations of the system can be explicitly solved in each cell I i as

h n+1 i = h i , q n+1 i = q i 1 + ∆t n |q i | C f (h i ) 2 .
(3.5)

The last equation for the energy, following from (2.3), is then

dΨ i dt = - 2C r Ψ i Ψ i + φ s q i h 2 i 3 . (3.6)
Integrating it, Ψ n+1 i can be found in implicit form:

Ψ n+1 i -Ψ i + φ s ln Ψ n+1 i Ψ i = C r C f q i (h i ) 2 2        1 1 + ∆t n |q i | C f (h i ) 2 2 -1        (3.7)
which is then solved by the Newton-Raphson method.

Higher-order spatial and temporal discretizations

As it was mentioned in Section 3.1, a second-order spatial discretization can be obtained by properly define the left and right stated at a cell I i interface. To this end, we utilize the classical MUSCL reconstruction technique, we refer for example in [START_REF] Toro | Shock-Capturing Methods for Free-Surface Shallow Flows[END_REF], which provides the following states at a cell interface for each of the components of U i as.

u L i+1/2 = u i + 0.5s(r i )(u i -u i-1 ), u R i+1/2 = u i+1 -0.5s(r i+1 )(u i+1 -u i ), where r i = u i -u i-1 u i+1 -u i and s(r i ) is a limiter function.
In the present work the classical Van Leer slope limiter has been utilized in the numerical results presented later on in Section 4.

To achieve second-order accuracy in time the optimal, in the sense of the CFL condition, secondorder two-stage Strong Stability Preserving (SSP) Runge-Kutta (RK) method has been implemented. This reads as follows

U (1) i = U n i + ∆t n L(U n i ); U n+1 i = 1 2 U n i + 1 2 U (1) i + 1 2 ∆t n L(U (1) i )
where L(U) is the spatial operator from (3.3) and ∆t n denoted the time-step. It is noted that, the friction and drag term treatment is implemented at the end of each RK step. The stability of the numerical scheme is imposed by the usual CFL condition for explicit schemes and the time-step ∆t n is adaptively computed from it as

∆t n = CFL • ∆x max i {C i+1/2 }
where C i+1/2 is computed as in Section 3.1 for each I i computational cell. First-order schemes are stable if the CFL condition is respected for a CFL number less than one. However, one has to reduce by two (and sometimes by three) the time step to carry out simulations with the second-order scheme to preserve stability. Thus, the CFL value was set to 0.4 in all the computations performed in the next section, unless otherwise stated.

An general algorithmic view of the order that the variables are computed within the numerical scheme is given below.

Algorithm

The splitting method for the SSWE Step 1. Given the initial data at time level n = 0 for h, q and Ψ compute the total energy E.

Step 2. Compute the advective part [START_REF] Delestre | SWASHES: A compilation of shallow water analytic solutions for hydraulic and environmental studies[END_REF] to obtain the prediction values h i , q i and E i .

Step 3. Compute the prediction of the roller enstrophy Ψ i from E i .

Step 4. Solve ( 9) and [START_REF] Goutal | Proceedings of the 2nd workshop on dam-break wave simulation[END_REF]

to get h n+1 i , q n+1 i and Ψ n+1 i to compute E n+1 i .
Remark. We note here that, as it is well-known, using the Rusanov numerical flux in our finite volume approach it is expected to behave in a dissipative way thus, necessitating the use of relative fine meshes. However, this solver has good stability properties without needing to solve the Riemann problem and can be considered as a first approach in solving the extended SSWE. Furthermore, as shown in Section 3.1, it can be extended as to produce a well-balanced scheme. The development of more elaborate well-balanced finite volume schemes for the SSWE is not in the main scope of this presentation but it constitutes an interesting research problem which can benefit from further investigations.

Numerical tests and solutions

In this section, we present numerical tests and results to verify the ability of the proposed numerical scheme to approximate the SSWE model and the phenomena it aims to describe. More precisely, our main interested is for approximating non-stationary hydraulic jumps over constant and non-constant topographies which exhibit an oscillatory behavior in their toes and free surface. Further, and of equal importance, a family of exact solutions to the stationary SSWE is also derived.

Hydraulic jumps with different Froude numbers

These test cases were presented in [START_REF] Richard | The classical hydraulic jump in a model of shear shallow-water flows[END_REF] and are used here as to demonstrate the ability of the numerical scheme to reproduce the expected (non-stationary) behavior for non-stationary hydraulic jumps on a flat bottom. The hydraulic jump features depend on the upstream (of the jump) average Froude number (F r 1 ). As it was demonstrated in [START_REF] Richard | The classical hydraulic jump in a model of shear shallow-water flows[END_REF], the jump toe (with position x 1 (t)) oscillates for all Froude numbers greater than ∼ 1.5. The main frequency of the oscillations decreases as F r 1 increases while the oscillation amplitude increases.

In the cases presented here we consider a channel of length L = 10 m and given the values for φ s , C f and C r , supercritical inflow conditions, at x = 0 m, are imposed by setting h(0, t) = h 0 m, q(0, t) = q 0 m 2 /s and Ψ(0, t) = 0. At x = L subcritical boundary conditions are obtained by imposing a blockage by a sharp-crested weir of height d w with the flow given by the empirical relation [START_REF] Henderson | Open Channel Flow[END_REF]

q(L, t) = f (h) =    0 if h ≤ d w ; 2 3 C d 2g(h -d w ) 3 if h > d w , ( 4.1)
where the discharge coefficient C d is given by

C d = π π + 2 + 0.08 h -d w d w .
To compute the numerical flux F(U) N +1/2 = F(U N ) at the outflow boundary cell N we need to estimate the U N values as a function of U N . From (4.1), the discharge value is set to

q N = f (h N ).
The second condition stems from the fact that Φ N is the Riemann invariant transported by the flow at the boundary x = L which give

Ψ N = Ψ N .
Finally, the water depth h N is computed by the outgoing characteristic as

u N -u N + h N h N gh + 3(φ s + Ψ N )h 2 h dh = 0.
The initial condition for h given in these test cases corresponds to the approximate description of the hydraulic jump by the SWE based on the famous Bélanger formula of sequential depths which results to

h(x, 0) =      h 0 if x ≤ L 10 ; 1 2 h 0 -1 1 + 8F r 2 0 if x > L 10 , (4.2)
where F r 0 = q 0 / gh 3 0 is the inflow Froude number. The rest of the initial conditions are q(x, 0) = q 0 and Φ(x, 0) = φ s . In Figure 4.1 a schematic view of a hydraulic jump, of length L r , forming in a wide rectangular channel is presented. The location of the jump is controlled by the downstream weir. For the numerical results presented in this section, a relative fine grid is used of N = 2000 grid points as to capture the fine features of the oscillatory nature of the hydraulic jumps. Figure 4.2 presents the numerical results for case HJ2 for the water depth, h, and Φ = Ψ + φ s . The numerical solution for water depth h is also compared with the stationary profile obtained, assuming q = uh = cst, by (numerically) solving the system of ODES

L r = h 1 160 tanh F r 1 20 -12h 1 , 2.5 < F
dh dx = -C f q 2 + 2C r Ψq 2 /Φ gh 3 + 3Φh 4 -q 2 ; (4.4) dΨ dx = -2C r Ψq 2 Φh 5 , ( 4.5)
that depends only on the upstream given data while the numerical solution is that of the initial boundary value equations (2.1). System (4.4)-(4.5) can be derived from the steady-state momentum and energy equations of (3) for a horizontal channel (b = 0) since φ s and q = uh are constant. Finally, in Table 4.3 the numerical values obtained for the target values in all three cases are presented along with the space width in which the toe position x 1 oscillates. For the roller length estimation, following [START_REF] Richard | The classical hydraulic jump in a model of shear shallow-water flows[END_REF], a good estimation for its end is considered to be when the large scale (roller) enstrophy Ψ = φ s /2. Comparing Tables 4.2 and 4.3 it can be seen that in all tests presented the expected oscillatory behavior of the hydraulic jump is demonstrated. The values of the water height obtained from the numerical simulation are very close to the ones predicted in Table 4.2 and only the values obtained for the enstrophy Φ are underestimated for cases HJ2 and HJ3. Nevertheless, the enstrophy exhibits the correct behavior with a large and sharp increase at the jump followed by a rapid decrease after the jump and up to the end of the roller. As such its maximum value is of relative importance to the obtained results, see for example [START_REF] Ivanova | Formation and coarsening of roll-waves in shear shallow water flows down an inclined rectangular channel[END_REF], since all other values are in close agreement with the target ones. The oscillatory nature of the jump toe position in terms of frequency and amplitude is in almost perfect agreement with the results presented in [START_REF] Richard | The classical hydraulic jump in a model of shear shallow-water flows[END_REF] for all cases. [START_REF] Macdonald | Steady open channel test problems with analytic solutions[END_REF] Our aim here is to construct some analytic solutions to the steady SSWE over varying topography. Although these solutions are not for oscillatory hydraulic jumps (since their corresponding upstream Froude numbers are less than 1.7) their validity is of importance since they can serve as benchmark solutions to the model equations and numerical schemes. In [START_REF] Macdonald | Analysis and computation of steady open channel flows[END_REF][START_REF] Macdonald | Steady open channel test problems with analytic solutions[END_REF][START_REF] Macdonald | Analytic benchmark solutions for openchannel flows[END_REF], a set of realistic steady open channel flow test cases with analytic solution are given for the Saint-Venant equations. These test problems are often used for code benchmarking and comparison, we refer for example in [START_REF] Delis | TVD schemes for open channel flow[END_REF][START_REF] Delis | Implicit high-resolution methods for modelling one-dimensional open channel flow[END_REF][START_REF] Burguete | Efficient construction of high-resolution TVD conservative schemes for equations with source terms: Application to shallow water flows[END_REF][START_REF] Tseng | Improved treatment of source terms in TVD scheme for shallow water equations[END_REF][START_REF] Kesserwani | Application of a second-order Runge-Kutta discontinuous Galerkin scheme for the shallow water equations with source terms[END_REF][START_REF] Lee | Simple and efficient solution of the shallow water equations with source terms[END_REF][START_REF] Morales-Hernandez | A large time step 1d upwind explicit scheme (CFL>1): Application to shallow water equations[END_REF][START_REF] Delestre | SWASHES: A compilation of shallow water analytic solutions for hydraulic and environmental studies[END_REF][START_REF] Caviedes-Voullième | Benchmarking a multiresolution Discontinuous Galerkin shallow water model: Implications for computational hydraulics[END_REF] among others. Two such cases from [START_REF] Macdonald | Analysis and computation of steady open channel flows[END_REF][START_REF] Macdonald | Steady open channel test problems with analytic solutions[END_REF][START_REF] Macdonald | Analytic benchmark solutions for openchannel flows[END_REF] are utilized and modified here.

Numerical simulations of hydraulic jumps with the Shear Shallow Water model

Modified benchmark test problems of

We first recall how these analytic solutions are constructed for the SWE by looking for steady piece-wise smooth solutions of the SWE characterized by a constant discharge q. Each smooth part of a solution is connected by a hydraulic jump. On each smooth part of the solution, the momentum equation becomes

[1 -F r 2 ]h + C f F r|F r| = -b (4.6)

Numerical simulations of hydraulic jumps with the Shear Shallow Water model

Let x * be a point of discontinuity of the solution, the solution h L on the left side of the discontinuity is arbitrary chosen and (4.6) is used to deduce the left bed slope b : L (x * ). Then the Rankine-Hugoniot relation is used to deduce the water height h * R on the right side of the discontinuity. The form of the right solution h R (x) is then chosen as a function defined by n free parameters. These parameters are computed by solving the algebraic system requiring that h * R = h R (x R * ) and that the bed at x * and its n -1 successive derivatives are continuous. Using relation (4.6) and its derivatives, gives the n -

[1 -F r 2 (h L )]h L + C f F r(h L )|F r(h L )| = -b L . ( 4
1 equations b R (x * ) = b L (x * ), b R (x * ) = b L (x * ), . . . , b (n-1) R (x * ) = b (n-1) L (x * ) that together with h * R = h R (x R * )
are solved for the n free parameters.

Flow with a hydraulic jump on a varying bed

The first example of analytic solutions constructed with this strategy is adapted from the Example 4 of [START_REF] Macdonald | Steady open channel test problems with analytic solutions[END_REF]. This concerns a 1000 m long channel with a q = 2 m 2 /s discharge. The flow is supercritical at inflow with a water depth of h 0 = 0.543853 m, it experiences an hydraulic jump at x * = 500 m and then remains subcritical. The water depth is given by the expression: [START_REF] Macdonald | Steady open channel test problems with analytic solutions[END_REF]. This is due to the fact that the friction law used in [START_REF] Macdonald | Steady open channel test problems with analytic solutions[END_REF] is the Manning one while here the Darcy-Weisbach law with a friction coefficient C f = 0.0053125 has been used. We refer in Appendix A for an explanation and the relation between the two friction formulas.

h(x) =        0.6673794620 -0.1235887893 e -0.
While (4.8) is an exact solution of the nonlinear SWE it is not an exact solution of the shear shallow water model (2.1). Instead of (4.6), steady piece-wise smooth solutions of this model are characterized by the system of equations:

momentum [1 -F r 2 (h) + 3Φh/g]h + h 2 Φ /g + C f F r(h)|F r(h)| = -b (4.9.a) enstrophy h 5 ΦΦ = -2C r |q|q(Φ -φ s ) (4.9.b)
The same strategy as the one used in [START_REF] Macdonald | Steady open channel test problems with analytic solutions[END_REF] can be employed to construct analytic solutions of (2.1) on a varying bed. However, this would result in a bed slope different from the one corresponding to the analytic solution of the shallow water equations. As our objective is also to compare the two models, we have found more interesting to use the same definition of the bed slope (that is thus computed as in [START_REF] Macdonald | Steady open channel test problems with analytic solutions[END_REF] by (4.6)) and to numerically integrate the system of ordinary differential equation (4.9) to get the corresponding solution for the shear shallow water model (2.1). For this model the values of the small scale enstrophy φ s and enstrophy dissipation coefficient C r are constant and have been computed as recommended in [START_REF] Richard | A new model of roll waves: comparison with brock's experiments[END_REF][START_REF] Richard | The classical hydraulic jump in a model of shear shallow-water flows[END_REF]:

φ s := 0.005 • g/h L (x * ), C r := 0.0688 • F r(h L (x * )) 1.337 . (4.10)
In this case the Froude number upstream of the jump is 1.21. Table 4.4 presents the values obtained by the analytic solutions of the SWE and the SSWE upstream and downstream of the jump as well as at the end of the channel for the water depth. [START_REF] Macdonald | Steady open channel test problems with analytic solutions[END_REF] Figures 4.8 and 4.9 compare the exact results for water depth given by the two models on this bed slope as well as the numerical solution for the SSWE obtained with N = 1000 mesh points using a CFL value of 0.2. For the numerical solution of the SSWE the exact solution of the SWE for h is given as initial condition while the discharge was set to q(x, 0) = 2m 2 /s. Setting the relative time variation for water depth as

h L h R Φ L Φ R h(
R(h) = N i h n i -h n-1 i h n i 2
where h n and h n-1 are the water depths at the current and previous time levels, the convergence criterion for a steady state was defined as R(h) < 10 -6 . The convergence history is shown in Figure 4.9 (left panel) where the convergence criterion can be seen to be satisfied around 400s. [START_REF] Macdonald | Steady open channel test problems with analytic solutions[END_REF]: Comparison between the shallow water model (SWE), the shear shallow water model (SSWE) and the numerical solution for SSWE It can be seen that up to the hydraulic jump, the two analytic solutions cannot be distinguished. This is due to the fact that the solution for the enstrophy upwind to the discontinuity location x * is given by : Φ ≡ φ s ; Φ ≡ 0 that is an exact solution of (4.9b). Then (4.9a) differs from the momentum equation of the shallow water model only by the term 3φ s h/g that is never larger than 10 -6 for x < x * (φ s = 0.07538574235). On the discontinuity, the SSW model predicts an increase of the enstrophy and a smaller jump of the water height than in the SWE model. Downwind of the jump, the enstrophy Residual R(h)

Numerical simulations of hydraulic jumps with the Shear Shallow Water model

1.e-8

1.e-7

1.e-6

1.e-5

1.e-4 [START_REF] Macdonald | Steady open channel test problems with analytic solutions[END_REF]: with comparison between the shallow water model (SWE), the shear shallow water model (SSWE) and the numerical solution for SSWE relaxes very rapidly to the value of the small scale enstrophy. This is also confirmed numerically in Figure 4.10 (left panel), although the maximum value of Φ R is slightly underestimated. Nevertheless, the numerical solution of the SSWE for the water depth is almost identical to the analytic solution. The water height slowly increases for the two models, with the value computed by the SSW model being continuously smaller than the value given by the SWE model. Further, the well-balance property of the numerical scheme is verified numerically on the right panel of Figure 4.10 where the computed discharge is presented. Only at the position of the jump a small glitch in the expected steady solution can be observed. This is a common behavior of all well-balanced schemes that satisfy the stagnant water well-balanced property. Convergence studies, verifying the order of accuracy of the numerical scheme using the exact solution from this test case are also given in Appendix B. A.I Delis, H. Guillard, et al.

Flow on constant slope followed by a hydraulic jump

This test case is analogous to the one described in [START_REF] Macdonald | Analysis and computation of steady open channel flows[END_REF] (page 120) as problem 5. The domain is 100 m long and on the left of the discontinuity, located in x * = 50 m, water flows with a constant depth of 0.7 m on a bed of constant slope. Then the bed flattens out smoothly and creates an hydraulic jump. This test case from [START_REF] Macdonald | Analysis and computation of steady open channel flows[END_REF] refers to a prismatic channel and uses the Manning friction law. In order to get results close to the ones in [START_REF] Macdonald | Analysis and computation of steady open channel flows[END_REF] we first have to estimate the discharge. For that we estimate, from the results in [START_REF] Macdonald | Analysis and computation of steady open channel flows[END_REF], the jump of the water height at the discontinuity from the Belanger relation

h R /h L = ( 1 + 8F r 2
L -1)/2 which gives the jump in the water height as a function of the upwind Froude number. From that we deduce the Froude number on the left side of the discontinuity. With h R = 1.306421438, this give a Froude number of 1.635459113 and a discharge of q = 3 m 2 /s. It remains to compute the friction coefficient of the Darcy-Weisbach law. First, from the formula given in [START_REF] Macdonald | Analysis and computation of steady open channel flows[END_REF] (p. 117), we compute that the constant slope on the left is b L = -0.06119282892. Then using relation (4.7) with h ≡ 0 we deduce the friction coefficient C f = 0.02287816294. Then, the water depth is given by the expression:

h(x) =          0.7 for 0 ≤ x < 50 exp (-0.5(x -x * )) 4 i=0 k i x -x * 100 -x * i + r(x) for 50 ≤ x ≤ 100, (4.11)
where r(x) = 1.9 exp(0.0005(x -100)). The k 0 coefficient has the value of h R (x * ) -r(x * ) in order for the water height on the right to be continuous while the other coefficients are given as: k 1 = -9.314064506, k 2 = -73.51227688, k 3 = -225.0402141 and k 4 = 359.7904991. The bed slope profile then results from expression (4.6). This suffices to define the test case for the SWE model. For the SSWE model we need in addition to define the values of the small scale enstrophy and dissipation coefficient of the enstrophy. Using again the relations given in [START_REF] Richard | A new model of roll waves: comparison with brock's experiments[END_REF][START_REF] Richard | The classical hydraulic jump in a model of shear shallow-water flows[END_REF] we obtain: Then, we numerically integrate the system of ordinary differential equations (4.9) to get the corresponding solution for the SSWE model. Table 4.5 compare the exact results for the water depth given by the two models for this bed slope. Again the jump in water height obtained by the SSWE model is smaller than with the plain SWE. However, on the outflow water depth the difference is relatively small with the difference of water height between the two models being smaller than 6cm. Figure 4.11 compares these exact solutions with the numerical solution obtained for the SSWE using N = 1000 grid points. As it can be seen, also in the area close to the jump, the numerical solution of the SSWE model is again in very close agreement with its analytic solution. Again, the enstrophy relaxes very rapidly to its pre-shock value, as shown also in Figure 4.12 (left panel) that displays its numerical solution along with the computed discharge.

h L h R Φ L Φ R h(

Flow over a parabolic bump-transcritical flow with a shock

Here we present a modification of a classical test case, from [START_REF] Goutal | Proceedings of the 2nd workshop on dam-break wave simulation[END_REF], frequently used to asses the performance of well-balanced shock-capturing numerical schemes for approximating steady-state solution for the SWE with topography. If bottom friction is not considered, an analytic solution is available for testing the water level calculations for the SWE. Here, we aim to present the numerical solution obtained by solving the SSWE with all source terms present and asses its behavior. In his test case, the domain length is L = 25 m with a topography given by: b(x) = 0.2 -0.05(x -10) 2 if 8m < x < 12m, 0 else.

Numerical simulations of hydraulic jumps with the Shear Shallow Water model

x(m) Recently, in [START_REF] Cheng | Particle image velocity measurement and mesh-free method modeling study of forced hydraulic jumps[END_REF], an experimental and numerical investigation was presented for the developing stages of forced hydraulic jumps in inclined beds. The main objective of this study was to assess the free surface and velocity field from a forced hydraulic jump by using nonintrusive measuring techniques and a mesh-free numerical method. The test case considered here is for an inclination angle of 5 o in a channel of L = 1.14 m long. A weir of height d w = 0.03 m was placed at the downstream end of the channel. The inflow conditions consist of a discharge q 0 = 5.608 • 10 -3 m 2 /s and h 0 = 0.01 m. The rest of the parameters used for the SSWE have been computed according to the formulas given in Section 4.3. Based on measured average values of F r 1 = 3.96 and h 1 = 0.005858 at the toe of the hydraulic jump in the experimental data, this gives C r = 0.433218407749443, C f = 0.003369082820920 and φ s = 8.373175770887171. Initially, A.I Delis, H. Guillard, et al.

the slope is considered dry and numerically an artificially bed wetting with a wetting parameter of 10 -6 was implemented [START_REF] Toro | Shock-Capturing Methods for Free-Surface Shallow Flows[END_REF].

In the experiments, the development of the forced hydraulic jump started after the fluid contacted the weir and a surge started to develop towards the upstream direction. This backward propagation is essential to the formation of such jumps. This backward propagation eventually stopped in time and it was observed that the toe of the jump continuously oscillated but the entire structure of the jump became gradually approximately consistent. In the experiment the jump is formed around t = 4 s. Although it cannot be excluded that in the initial stages, the flows exhibits some 3D phenomena, the agreement between the experimental results and the 2D numerical simulations reported in [START_REF] Cheng | Particle image velocity measurement and mesh-free method modeling study of forced hydraulic jumps[END_REF] shows that this experiment can serve as a concrete test for the 1D model considered here to test its ability to reproduce some important features of forced hydraulic jumps. Fig. 4.16 presents the comparison between experimental measurements and the numerical solution obtained with the SSWE and the SWE at t ≈ 4 s for the water depth using N = 500 mesh points and a CFL value of 0.25. In Fig. 4.16 approximately the last 30 cm of the channel are shown, since the phenomena, also due to the artificial wetting used, the simulated jump gradually matched the experimental data. Figure 4.17 presents the total enstrophy and velocity profiles as computed by the numerical scheme. The predicted roller length was L r ≈ 0.135 m. Figure 4.18 presents the comparison for the free surface between the numerical and experimental data at later time instances. The shift of the jump toe is evident and the numerical solution gives a relatively accurate representation of the free surface variation. Finally, the oscillatory nature of the computed hydraulic jump is evident from the recording of the toe position x 1 around an average value as presented in Figure 4.19.

x(m)

Conclusions

An extension and numerical solution of a previously developed two-parameter mathematical model describing shear shallow water flows was presented to study its potential of simulating turbulent hydraulic jumps. The mathematical model called the shear shallow water equations (SSWE) consists a conservative hyperbolic system with source terms present with two adjustable parameters, that have a well-defined physical interpretation, the wall enstrophy and energy dissipation coefficient. The SSWE were extended here in a straightforward manner as to incorporate arbitrary non-constant topographies without any adjustments to the model's parameters. To this end, a conservative second-order, in space and time, finite volume scheme was developed to numerically approximate the extended model. The proposed scheme satisfies the well-balance property for quiescent flows over topography. Several numerical test where performed as to access the ability of the numerical model to predict the formation of turbulent hydraulic jumps for upstream (to the jumps) Froude numbers larger than 1.7. The model was capable of predicting the oscillatory nature of such jumps, a phenomenon well-documented in the literature, and several characteristic such as the free surface evolution and jump roller length satisfactorily. Comparisons with experimental data of a forced hydraulic jump were also presented with satisfactory results, given the 1D nature of the model. Finally, an important contribution of this work is the derivation of a set of new analytic steady-state solutions to the SSWE model over variable topography. These exact solutions, although not for oscillatory hydraulic jumps, can serve as benchmark solutions for the validation of any numerical scheme aiming to approximate the model equations.

x B.2, we present convergence results for the full domain solution in terms of the L 1 norm for water depth and water discharge q = hu. As expected, due to the presence of the hydraulic jump the order of the second-order scheme drops to one but the accuracy of the results is much greater compared to that of the first-order scheme.

First-order scheme Second-order scheme ∆x L 1 (h) Order L 1 (q) Order L 1 (h) Order L 1 (q) Order 5 30.325 100.06 Water depth (h) and discharge (q) L 1 errors and convergence rates for the first-and second-order schemes for the full domain solution

2

 2 are the numerical fluxes at the cell's interfaces i ± 1/2. To design the topography source terms discretisations at the cell interfaces, S i±1/2 b

Figure 4 . 1 .

 41 Figure 4.1. Schematic view of a hydraulic jump on a flat bed controlled by a downstream weir of height d w

Figure 4 .Figure 4 . 2 .

 442 Figure 4.2. Numerical results at time t = 650s for water depth h and total enstrophy Φ for test case HJ2

Figure 4 . 3 .

 43 Figure 4.3. Oscillations in time of the jump toe position x 1 for test case HJ2

Figure 4 . 4 .

 44 Figure 4.4. Numerical results at time t = 550s for water depth h and total enstrophy Φ for test case HJ3

Figure 4 . 5 .

 45 Figure 4.5. Oscillations in time of the jump toe position x 1 for test case HJ3

Figure 4 . 6 .

 46 Figure 4.6. Numerical results at time t = 350s for water depth h and total enstrophy Φ for test case HJ4

Figure 4 . 7 .

 47 Figure 4.7. Oscillations in time of the jump toe position x 1 for test case HJ4

. 7 )

 7 By deriving relation (4.7), we can also compute the successive left derivatives of the bed in x * : b L (x * ), b L (x * ), . . . , b (n)

1 ( 4 . 8 )

 148 004 x for 0 ≤ x < 500 0.7415327355(1. + a 1 e -0.02 x+10.0 + a 2 e -0.04 x+20.0 for 500 ≤ x ≤ 1000 +a 3 e -0.06 x+30.0 ) + 0.5932261883 e 0.001 x-and equation (4.6) allows to compute the corresponding bed slope. The values of the parameters in (4.8) are a 1 = -0.2935698553, a 2 = 0.4080344466, a 3 = -0.4662074787. Note that the values of the parameters a 1 , a 2 , a 3 are slightly different from the values given in

Figure 4 . 8 .

 48 Figure 4.8. Water depth for the modified Example 4 case of [20]: Comparison between the shallow water model (SWE), the shear shallow water model (SSWE) and the numerical solution for SSWE

Figure 4 . 9 .

 49 Figure 4.9. Residual convergence (left) and zoomed area of water depth for the modified Example 4 case of[START_REF] Macdonald | Steady open channel test problems with analytic solutions[END_REF]: with comparison between the shallow water model (SWE), the shear shallow water model (SSWE) and the numerical solution for SSWE

Figure 4 . 10 .

 410 Figure 4.10. Numerical solution of the SSWE for total enstrophy (left) and discharge (right) for modified Example 4 of[START_REF] Macdonald | Steady open channel test problems with analytic solutions[END_REF]

φ

  s := 0.005 • g/h L (x * ) = 0.07007142855, C r := 0.0688 • F r(h L (x * )) 1.337 = 0.1328081433.(4.12)

Figure 4 . 11 .Figure 4 . 12 .

 411412 Figure 4.11. Water Depth for Example 5 case of [20]: Water depth for the modified Problem 5 of [19]: Comparison between the shallow water model (SWE), the shear shallow water model (SSWE) and the numerical solution the SSWE model

Figure 4 . 14 .

 414 Figure 4.14. Numerical total enstrophy(left) and discharge (right) for the transcritical flow with a hydraulic jump over a parabolic bump

Figure 4 . 15 . 4 . 4 .

 41544 Figure 4.15. Position of the jump toe x 1 (t) for the transcritical flow with a hydraulic jump over a parabolic bump

Figure 4 . 16 .

 416 Figure 4.16. Water depth for the forced hydraulic jump experiment: comparison between numerical solution and experimental data (top) and experimental snapshot (bottom) at t ≈ 4s

Figure 4 . 17 .

 417 Figure 4.17. Total enstrophy (left) and velocity (right) for the forced hydraulic jump experiment

Figure 4 . 18 .Figure 4 . 19 . 2 2 += 1 resulting in h cr = q 2 g 1 / 3 . 5 ) 6 )

 418419221356 Figure 4.18. Water depth for the forced hydraulic jump experiment: comparison between numerical solution and experimental data at t ≈ 5.5 s (top) and t ≈ 7 s (bottom)

Table 4 . 1 .

 41 Test cases of non-stationary hydraulic jumps on a flat bed: parameter values used

	r 1 < 16.	(4.3)

Table 4 . 2 .

 42 Test cases of non-stationary hydraulic jumps on a flat bed: target values

Table 4 . 3 .

 43 x 1 for test case HJ3 Test cases of non-stationary hydraulic jumps on a flat bed: numerical values

	Case	HJ2	HJ3	HJ4
	h 1 (m)	0.0565	0.0118	0.0181
	h (m)	0.09862	0.0258	0.0363
	h 2 (m)	0.1308	0.0877	0.2588
	Ψ	21.4	855	3580
	L r (m)	0.215	0.38	1.46
	x 1 (m) [2.868, 2.91] [0.367, 0.465] [0.85, 1.13]

Table 4 . 4 .

 44 Comparison of exact values between the shallow water model (SWE) and the shear shallow water model (SSWE) for the modified Example 4 of

	1000)

Acknowledgment

This work originates from visits of Yih-Chin Tai to Inria Sophia-Antipolis and of H. Guillard to the National Center for Theoretical Sciences (NCTS) of National Tawain University in the framework of the Inria-MOST Associate team program AMoSS. H. Guillard acknowledges the warm hospitality of NCTS and of the Technical University of Crete during the completion of this work. A.I. Delis wishes to thank the INRIA-Nice Sophia Antipolis University team CASTOR for its hospitality while contracting part of the presented research. The authors would also like to thank Boniface Nkonga, Mario Ricchiuto and Sergey Gavrilyuk for fruitful discussions on the SSWE model and its numerical approximation.

A.I Delis, H. Guillard, et al.

We choose for initial conditions h(x, 0) + b(x) = 0.33 m and q(x, 0) = 0 m 2 /s and the following boundary conditions upstream : q = 0.18m 2 /s, downstream : h = 0.33m .

Due to the steep change in bed elevation, the flow changes from subcritical to supercritical and back to subcritical.

The C f = f /8 = 0.001616907601743 friction coefficient in this case was estimate following [START_REF] Richard | The classical hydraulic jump in a model of shear shallow-water flows[END_REF] by the formula [START_REF] Henderson | Open Channel Flow[END_REF] 1

where Re = 4q/ν is the Reynolds number with ν = 1.309 • 10 -6 being the kinematic viscosity. For the SSWE equations the values for φ s = 0.005 • g/h 1 = 0.607129595246937 and the C r = 0.0688F 1.337 1 = 0.235228180190780. Since the values h 1 and F 1 at the jump toe were not known for the SSWE, these were based on estimates from the numerical results obtained for the SWE solution with friction. The upstream of the jump average Froude number, F r 1 , used to obtain the above values was 2.4 thus, it is expected that the jump toe will oscillate near some average value.

In Figure 4.13 a comparison is presented between the exact SWE solution (with no friction present) and that obtained for the SSWE at t = 200 s using a relatively fine mesh of N = 1000 grid points. For the SSWE the intensity of the jump is lower compared to that of the SWE. The computed total enstrophy, presented in Figure 4.14 (left panel), exhibits the expected behavior i.e. downwind of the jump the enstrophy relaxes very rapidly to the value of the small scale enstrophy. In

Appendix A. Friction formulae relation

We first recall that the Saint-Venant (SV) system results from an integration of the SWE along the cross-section of the channel. In [START_REF] Macdonald | Analysis and computation of steady open channel flows[END_REF][START_REF] Macdonald | Steady open channel test problems with analytic solutions[END_REF][START_REF] Macdonald | Analytic benchmark solutions for openchannel flows[END_REF] the analytic solutions are derived for the Saint-Venant (SV) equations for prismatic channels using the Manning friction formula. Instead of this friction formula, the present work uses the Darcy-Weisbach (DW) formula. Hence, a connection between the SV equations and SWE also in terms of the Darcy-Weisbach (DW) is given here. Considering a we can approximate f from Manning's formula above as

In the SSWE derivation we use in the momentum equation

which results in the obtained form of S f in (2.1).

The analytical solutions for the SWE derived in Section 4 for h(x) and slope b satisfy then the following relation for each friction term used

which corresponds to (4.6).

Appendix B. Mesh convergence study

For the new exact solution from Section 4.2.1 a convergence analysis is performed by first only simulating the smooth part of the solution downstream of the hydraulic jump i.e. for x ∈ (500, 1000] where the flow regime is smooth and purely subcritical (see Figure 9). For the convergence studies we define the L 1 and L ∞ errors for water depth as

were (h ex i ) i=1,...,N and (h n i ) i=1,...,N are, respectively, the exact and the approximate cell mean values at a final time t n = t f . Computations were carried out, for different meshes, until the final time t f = 500s with the first-and second-order versions of the numerical scheme with a CFL=0.1. The reduced CFL value used is justified by the need to ensure that the errors incurred by the temporal integration are negligible. At this time the L 1 and L ∞ errors were several orders of magnitude higher that the corresponding relative variations of the flow variables between consecutive time steps, meaning that, in practice, a steady-state regime has been reached.

The errors for the different meshes and convergence rates are given in Table B.1. The errors computed point out that the second-order convergence is close to optimal for both norms, while the corresponding errors for the first-order version of the scheme are much higher. It is noted that, a similar behavior is obtained for the other flow variables as well.

First-order scheme Second-order scheme ∆x


Hervé Guillard

Jalal Lakhlili

Alexis Loyer

Adrien Loseille

Ahmed Ratnani

Ali Elarif

Tokamesh

Hervé

Tokamesh : A software for mesh generation in Tokamaks

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

-

∇ 1 µr ∇ψ =Jtoro =     

Morse theory

A function f is a Morse function if all its critical points are regular.

Critical points

Let C r be the space of r differentiable scalar field defined on Ω. For r ≥ 2, a point p ∈ Ω is a critical or singular point of f if ∇f = 0. Regular Critical points A critical point is regular (or non-degenerate) if the Hessian of f at p is invertible.

In 2-D if we note µ 1 , µ 2 the two eigenvalues of H f the only possible critical points of a Morse function are therefore :

Maxima index = 2 µ 1 < 0 µ 2 < 0 Saddles index = 1 µ 1 < 0 µ 2 > 0 Minima index = 0 µ 1 > 0 µ 2 > 0
Hervé Guillard EoCoE Final conference, Nicosia, Cyprus 17 -18 September, 2018

Morse theory

Mountainer theorem : Let f be a Morse function defined on Ω, a region defined by a closed iso-contour, then the number of critical points (counting a virtual extremum) verifies the relation : Algorithm for block structured mesh generation G 1 Gluing of patches Given 2 subdomains Ω 1 and Ω 2 described by 2 mappings

C M -C S +
S 1 (s, t) = I 1 i=1 J j=1 P 1 i,j N i (s)N j (t) S 2 (s, t) = I 2 i=1 J j=1 P 2 i,j N i (s)N j (t)
with a common boundary for instance at s = 0 S 1 (0, t) = S 2 (0, t)

G 0 continuity : P 1 0,j = P 2 0,j ∀j G 1 continuity : geometric condition on the control points P 1 1,j , P 1 0,j = P 2 0,j , P 2 1,j have to be aligned.

  EoCoE WP5.1 task: Constructing flux-surface aligned mesh-grids in the poloidal plane Task WP5.1 : Propose unified software for different codes and models gyrokinetic (Gysela) Fluid (Jorek, Tokam3X, SOLPS, SOLEdge, Plato ) different type of meshes block structured -non structured different numerical methods Semi-Lagrangian approaches Finite difference, Finite volume C 0 Finite Element (Lagrange, spectral elements) C 1 Finite Element (Spline or Hermite-Bezier on quadrangles, Powell-Sabin-Clough-Tocher on triangles ) Hervé Guillard EoCoE Final conference, Nicosia, Cyprus 17 -18 September, 2018 Computing the flux surfaces : Grad-Shafranov Equilibrium solver The problem Generate meshes adapted to the magnetic flux surfaces (anisotropy) First part of the problem : Compute the magnetic flux surfaces Magnetic field Ansatz : B = F (ψ)∇φ + ∇ψ × ∇φ Force balance ∇p = J × B and toroidal symmetry yield Non-linear elliptic free boundary problem :

  Figure: Domain decomposition Figure: Associated Reeb graph : Each edge corresponds to a subdomain

G

  1 meshes : example Hervé Guillard EoCoE Final conference, Nicosia, Cyprus 17 -18 September, 2018

Hervé Guillard EoCoE Final conference, Nicosia, Cyprus 17 -18 September, 2018

Data regularization

Tokamesh : A software for mesh generation in Tokamaks Hervé Guillard(Inria Sophia), Jalal Lakhlili (IPP), Alexis Loyer (Inria Sophia), Adrien Loseille(Inria Saclay), Ahmed Ratnani(IPP), Ali Elarif (Inria Sophia)

Tokamesh work flow

Input from equilibrium codes (EFIT, Cedres++) solving Grad-Shafranov equation

Tokamesh work flow

Free boundary Grad-Shafranov Equilibrium

Free boundary Grad-Shafranov Equilibrium solver using Clough-Tocher C 1 finite element method on triangular meshes Regularize the data coming from low-order GS solver Recompute an equilibrium using C 1 high-order method Hervé Guillard EoCoE Final conference, Nicosia, Cyprus 17 -18 September, 2018

Domain decomposition

Segmentation problem : Given a domain Ω : Find n patches Ω n such that :

and there exists a one to one mapping between Ω n and the the unit square : Algorithm for block structured mesh generation

Meshing individual patches

Into each subdomain the level set f -1 = c is unique and closed Choose a finite number of level sets

Approximate each level set by a cubic spline :

Note that P i,j = P i


Mitat Akkoç

Serkan Demirel

Erdinç Öz

Serdar Altın
email: serdar.altin@inonu.edu.tr

Ali Bayri

Vincent Dorcet

Thierry Roisnel

Christian Bruneau

İsmail Özdemir

Sedat Yaşar
email: sedat.yasar@inonu.edu.tr

Cationic versus anionic Pt complex: The performance analysis of a hybridcapacitor, DFT calculation and electrochemical properties

Keywords: Platinum ionic complex, ring-expanded N-heterocyclic carbene, hybrid capacitor, DFT

The synthesis and characterization of cationic versus anionic platinum(II) complex of the type [PtCl3(DMSO)] -[NHC] + (NHC:N-heterocyclic carbene) is reported for the first time. The complex 3 was synthesized from silver(I)-NHC 2, and K2[PtCl4] in DMSO at 80 o C under exclusion of light. The ionic platinum(II) complex was obtained in high yield and fully characterized by 1 H, 13 C, 195 Pt NMR and HRMS spectroscopy, X-ray diffraction and elemental analysis. The structural properties of the complex were investigated by using Gaussian suit and the obtained parameters were crosschecked with the X-ray data. The HOMO-LUMO energies and electrostatic surface mapping of the complex were calculated and the results were discussed with the capacitive properties of the complex. Electrochemical analysis by cyclic voltammetry (CV) of the complex shows reversible redox peaks about platinum center. The complex was used as an electrode material for the capacitor device and result is promising that the highest performance of complex 3 was found as 10.22 F/g.

Introduction

Energy production and storage getting more attention lately from the scientists and technology firms. The production of the energy involves usage of new effective sources such as solar cell and wind energy. In addition to this, we see that the storage techniques are also important especially for portable devices and electric vehicles. In the study of energy technologies, it is seen that the new capacitors which named as hybrid or supercapacitors take an attention at this field. It is known that the storage capacity of the cells is related to the surface area and conducting mechanism of the electrode materials [1]. In addition to this, electrolyte materials are also important for the improving the capacity of the cells. It should be noted that graphene based electrode materials shows high capacity when compared with the other materials. The electronic technology needs different capacitive value of the cells and the intrinsic properties of hybrid capacitors such as high capacity value of 4-250 Fg -1 have gained considerable attention [START_REF] Beguin | Supercapacitors: Materials, systems and applications[END_REF]. The scientists have been trying to find new electrode materials to provide the demands of the electronic sectors. The first study for capacitor device applications of organic complex was performed before to 20th century [3a-c]. The organometallic complexes may be a potential electrode material for production of hybrid capacitors [3d]. Since they may provide higher surface area when compared the other electrode materials. The metal ions in organometallic complex can change the valence state which may provide extra electrochemical properties for the energy storage applications.

Last few decades, distinctive chemistry of N-heterocyclic carbenes (NHCs) were made them one of the most important ligands in organometallic chemistry and catalysis [START_REF]For selected books and recent reviews[END_REF]. NHC's strong -donor and weak -acceptor bond properties bring about more stable metal-ligand bond than many other classes of ligand [5]. The ringexpanded NHCs (re-NHCs), which exhibit different properties, are the types of NHC ligands. Recently, research groups have focused on to explore their properties. The re-NHCs contribute to a significant increase in donor ability and steric hindrance to the metal center than their 5 membered analogues. Thus, these unique properties of ringexpanded-NHCs made them useful ligands in various research areas [6,7]. The pioneer works in this topic were reported by Stahl and Cavell groups. Cavell et all. and other research groups have published their works on synthesis and catalytic application of ring expanded N-heterocyclic carbene complexes [6][7][8][START_REF] Dı´ez-Gonza´lez | N-Heterocyclic Carbenes: From Laboratory Curiosities to Efficient Synthetic Tools[END_REF][10].

Herein, we report the synthesis, characterization, DFT calculation and electro chemical properties of [PtCl3(DMSO)] -[NHC] + type complex. Also, for the first time, the hybrid capacitor property of this type of complex has been investigated.

Experimental Section

General remarks

Unless stated otherwise, all manipulations were performed under an argon atmosphere.

Chemicals were purchased from Sigma Aldrich Co. (Dorset, UK) and used as handled.

Solvents of analytical grade were freshly distilled using an MBraun SPS-800 machine. Mass spectra were recorded by electron spray ionization (ESI) method. The 1 H and 13 C NMR spectra were recorded with a Bruker Avance III 300 or 400 MHz NMR spectrometer with sample solutions prepared in CDCl3. The chemical shifts were reported in δ units downfield from the internal reference (Me4Si) as seen in Figure S1. Coupling constants (J values) are given in hertz. NMR multiplicities are abbreviated as follows: s=singlet, d=doublet, t=triplet, m=multiplet, bs=broad singlet.

Single crystal analysis

(C29H43N2,C2H6Cl3OPtS); M = 799.22. D8 VENTURE Bruker AXS diffractometer [*], Mo-Kα radiation (λ = 0.71073 Å), T = 150 K; monoclinic P 21/c (I.T.#14), a = 10.9339 (11), b = 17.1182(19), c = 17.790(2) Å, β = 93.129(4) °, V = 3324.8(6) Å 3 . Z = 4, d = 1.597 g.cm -3 , μ = 4.551 mm -1 . The crystal structure was solved by direct methods using the SHELXT program [11], and then refined with full-matrix least-squares method based on F 2 (SHELXL-2014) [12]. All non-hydrogen atoms were refined with anisotropic atomic displacement parameters.

H atoms were finally included in their calculated positions. A final refinement on F 2 with 7621 unique intensities and 362 parameters converged at wR(F 2 ) = 0.0802 (R(F) = 0.0306) for 6521 observed reflections with I > 2σ(I). For the ORTEP plot of the crystal structure see supporting info.

Electrochemical activity measurement

CV of the complex 3 was determined by the system of Zive SP1 Potentiostate/galvanostate. The measurement was performed using as Ag/AgCl reference electrode with the three-electrode cell, a platinum rod for the working and counter electrode in the cell, by using acetonitrile and tetrabutylammoniumperchlorate (TBAP) electrolyte. The CV was performed within the range of 0 to 1.5 V with 200-400 mV/s rates.

DFT calculations

DFT optimization study of the produced material was made by the GAUSSIAN 09 W [START_REF] Frisch | Revision D.01[END_REF] software with the B3LYP (Becke, 3-parameter, Lee-Yang-Parr)/LanL2DZ (Los Alamos National Laboratory 2 Double Zeta) basis set. After that, the excited level, the energy level of HOMO and LUMO and electrostatic surface potential (ESP) values were calculated.

Production of the Hybrid-Capacitor

The complex 3 was used as an electrode material for a hybrid capacitor and CR2032 coin cell case was used for the measurements. The cell was designed as electrode/electrolyte/membrane(cellulose)/electrolyte/electrode. The samples used in the electrode fabrication were coated on Ti foil using a doctor blade and dried in vacuum oven at 100 C and then it was punched to circular pieces of 8 mm diameter.

The mixture for film was fabricated using 90% active material, 10% PVDF and enough amount NMP solvent. The purpose of using PVDF was a binder of coated film on Ti substrate. Afterwards, the heat treatment applied at 110 o C to remove the solvent. KOH was selected as electrolyte material since it is the most used electrolyte material in capacitor systems, and, 6 mol KOH/H2O solutions were prepared as electrolyte for this experimental works. The Cyclic voltammetry measurements of complex 3 were performed by using Zive SP1 system. The CV of the capacitor cell in the CR2032 size was measured with two-electrode design and a constant current of ±1 mA between -1 and 1 V with the rate of 200 and 400 mV/s.

Synthesis

Synthesis of 7-IPr.HBr,1

The 7-IPr.HBr was synthesized according to literature [START_REF] Iglesias | [END_REF]. The NMR results obtained are given below. Yield: 2.79 g (56 %). m.p. 259-260 °C. 1 H NMR (300 MHz, CDCl3):δ= 1.17 [d, J=6.8 Hz, 13 C {H} NMR (75 MHz, CDCl3):δ= 24.6 and 125.3,130.9,138.9,144.9 [C6H3((CH(CH3) 2)2 -2,6 ,69.72;H,8.68;N,5.61;Found C,69.80;H,8.72; N, 5.70.

12H, C6H3((CH(CH3)2)2 -2,6], 1.40 [d, J=12.7 Hz, 12H, C6H3((CH(CH3)2)2-2,6], 2.65 [bs, 4H, NCH2CH2 CH2CH2N], 3.24 [sept., J=8 Hz, 4H, C6H3((CH(CH3)2)2 -2,6], 4.70 [bs, 4H, NCH2CH2 CH2CH2N], 7.24 [d, J=8 Hz, 4H, C6H3((CH(CH3)2)2-2,6], 7.29 [s, 1H, NCHN], 7.42 [t, J=8 Hz, 2H, C6H3((CH(CH3)2)2-2,6].
25.0 [C6H3((CH(CH3)2)2-2,6], 24.9[C6H3((CH(CH3)2)2-2,6], 29.0 [NCH2CH2CH2CH2N], 56.2 [NCH2CH2 CH2CH2N],

Synthesis of Silver-NHC complex, 2

The complex 2 was synthesized according to literature [START_REF] Iglesias | [END_REF]. To synthesize complex 2, 2 mmol Ag2O was added to the DCM solution of 1 (1 mmol) under exclusion of light.

After stirring 24h at rt, reaction mixture filtrated upon a short pad of SiO2 and DCM removed under high vacuum. DCM/diethyl ether solution (1:2 v/v) of 2 led to obtain white solid crystals for NMR and other analysis. The obtained NMR results are given below. NMR spectra of complex 2 was measured in DMSO-d6 to determine if complex 2 was formed. As is known, the NCHN proton comes to resonance together with aromatic protons on 1 H NMR in CDCl3. The 1 H NMR in DMSO-d6 shifted NCHN proton to low field at 1 H NMR (see NMR data of complex 3 and compound 1).

Yield: 0.450 g (75 %). 1 H NMR (400 MHz, DMSO-d6):δ= 1.28 and 1.30 [d, J=6.8 ,57.44;H,6.98;N,4.62;Found C,57.32;H,6.89;N,[START_REF]For selected books and recent reviews[END_REF]68.

Synthesis of [PtCl3(DMSO)] -[NHC] + , 3

To synthesize complex 3, 1 mmol K2[PtCl4] was added to the DMSO solution of 2 under exclusion of light. After stirring 2h at 80 o C, reaction mixture was led to cool in room temperature and 60 mL acetone was added in order to precipitate the complex 3. Filtration of solid upon filter paper and washing with acetone (3x10 mL) gave complex as yellow-brown solid. DCM/diethyl ether solution (1:2 v/v) of 3 led to obtain single crystals for X ray diffraction analysis. The synthesis pathway of complex 3 was given in Figure 1. Yield: 0.52 g (65 %). 1 H NMR (400 MHz, DMSO-d6): δ= 1.23 [d, J=6.8

Results and Discussion

Synthesis and characterization

The re-N-heterocyclic carbene precursor 1, has been synthesized according to literature. The obtained NMR data of 1 was compatible with the literature [START_REF] Iglesias | [END_REF]. Upon formation of the re-NHCs precursor, 1 H NMR spectroscopy revealed the appearance of the characteristic formamidinium salt NCHN proton at 7.29 ppm. This 1 H NMR shift of 1 was consistent with reported seven-membered NHC salts [15]. The 13 C NMR spectroscopy revealed the carben carbon of 1 (NCHN) at 157.1 ppm.

Compound 1 was used for synthesis of silver-NHC complex 2 via Ag2O under exclusive of light in dried DCM. The 1 H and 13 C NMR spectra of complex 2 were recorded in CDCl3 and DMSO-d6 in order to clearly understand the transformation of 1 to 2. The 1 H NMR spectra measured in DMSO-d6 showed that characteristic NCHN proton signal totally disappeared.

The NC2N-Ag carbon peak was not detected which is a known situation in the literature [START_REF] Iglesias | [END_REF].

The complex 2 was characterized and used for the synthesis of complex 3.

Initially, the complex we aimed to synthesize was the [Pt(NHC)Cl2DMSO]. Surprisingly, [7-IPr] + [PtCl3(DMSO)] -ionic complex was formed. In the formation of complex 3, we think that the moisture/water that can be present in the reaction medium or in DMSO is effective.

During the transmetalation, re-N-heterocyclic carbene silver complexes can give reversible reaction and sometimes be converted to protonated form of NHC by transferring hydrogen from ambient moisture. According to literature, although there is not any platinum complex similar to the complex 3, there are couple of different type of ionic complexes of ruthenium and palladium [16]. The observation of NCHN proton on 1 H NMR of 3 in DMSO-d6 confirms the formation of [7-IPr] + [PtCl3(DMSO)] -ionic complex. Also, resonance of C2 carbon of re-NHC on ionic Pt(II)-re-NHC complexes in 13 C NMR spectra not changed due to noncoordination to platinum centre (Figure 2). The hydrogens of coordinated DMSO was observed at 2.55 ppm which is shifted to low field at 1 H NMR. The 195 Pt NMR spectra of complex 3 confirm that the compound 3 consist of one platinum fragment. In addition, the HRMS spectra was confirmed the proposed structures for the 3, recorded in methanol. The most intense peaks correspond to molecular fragment ionized by the [PtCl4] 2-[C29H43N2] 2+ in positive-ion mode. In the HRMS spectra of complex 3, [NHC] + , [C29H43N2PtCl4-2H] + and [PtCl4] fragments were observed in the positive and negative MS mode.

Crystallography

According to ORTEP analysis, the molecular structure of the complex 3 which obtained experimentally is given in Figure 3 and it was also theoretically investigated by DFT calculations. The crystallographic results are juxtaposed with experimental data in Table 1 and they are compared with the theoretical results. It should be noted that the calculated and experimentally obtained crystallographic data were compatible with each other. For instance, it can be said that the experimental and computational bond lengths of Pt1-Cl3 and N13 -C14 were 2.3302-2.3806 Å and 1.461-1.4672, respectively. The experimentally obtained and calculated angles of C15 -C14-N13 and O51-S1-Pt1 were 118.6 o -118.907 o and 116.17 o -116.91 o , respectively.

HOMO/LUMO and ESP analysis

The electronic structure of the complex was calculated by the DFT using the optimized structure as seen in the Figure 4a. Structural harmony was seen on the theoretically calculated and experimentally obtained values such as the angles and bond lengths of complex 3. It was used the B3LYP/LanL2DZ basis for energy calculation of the HOMO and the LUMO. The HOMO and LUMO energies were obtained as -5.753 eV and -2.222 eV, respectively. The Eg = (LUMO-HOMO) of complex 3 was found as 3.531 eV.

The HOMO and LUMO states of the complex 3 were given in Figure 4b andc. it should be noted that the decreasing the energy of the LUMO and increasing the energy of the HOMO may be explained by the decrease of the ionization energy of the complex 3.

According to Eg value of the complex 3, it can be said that the electron transfer in complex 3 is irrelevant.

Figure 5 shows the ESP results of the complex that the intense green color shows the positive electrostatic potential region and the red sink color corresponds to the negative electrostatic potential in the structure. According to ESP analysis, it is found that the positively charged area in the structure is lower than that of the negatively charged region in Figure 5. In addition to this, it should be noted that the complex 3 showed nucleophilic behavior which is reactive against electrophilic properties in the environment. The narrow electrophilic region of the complex can be explained by the poor attractive fields for negative charged environment, but the red colored region in ESP figure shows strong electrophilic properties on the region of the chloride ions in the structure.

Redox properties

It is well known that CV measurements are used for the determination of ionization energy, electron affinity and energy gap [17]. The CV measurement of the complex 3 was performed using a 0.05 M acetonitrile solution of n-Bu4NClO4 as the supporting electrolyte including 0.05 mM [7-IPr] + [PtCl3(DMSO)] -and the data was given in the Figure 6. The redox peaks at CV graphs were obtained as Epa = 0.292 V, EPc = 1.211 V, E1/2 = 0.4595 V for the scan rate of 400 mV/s as seen in Table 2. It was predicted that the complexes showed redox peaks which correspond to the Pt(II)-Pt(IV) interconversion. Although it was seen one anodic and one cathodic peaks in CV graph, it should be noted that Pt complex exhibit a doubleelectron redox reaction from Pt(II) to Pt(IV) for the activation/deactivation process [18]. It may be interpreted that the excitations of the electrons in the complex may be observed at almost the same energy and it should gave rise to the peak broadening in the peaks of the CV data. In addition to this, the Pt 2+ ions in the complex structure for square environment should split to 4 different energy levels and the ionization structure of Pt 2+ to Pt 4+ may be at the same energy since the highest occupied energy level of d-ions are paired electrons which shows a diamagnetic properties and the ionization energy of the paired electrons from d-level to vacuum are almost same.

The value of the EPc and Epa was varied by the ascending scan rate of the CV, which can be given with the Randles-Sevcik aproach which means that the peaks at CV are directly related to the square root of the scan rate. The oxidation potential of the structure can be given as below [19]:

𝐸 𝑂𝑥 = 𝐸 𝑜𝑥 𝚤 + 𝐸 𝐴𝑔/𝐴𝑔𝐶𝑙≃ 𝐸 𝑜𝑥 𝚤 + 𝐸 𝑣𝑎𝑐𝑢𝑢𝑚 + 4.4 𝑉 (1)
Where Evacuum is the energy of vacuum level and it may be taken as zero and 𝐸 𝑜𝑥 𝚤 is the onset of oxidation potential. The ionization potential is equal to 𝑒𝐸 𝑂𝑥 which can be calculated from eq.1. It should be noted that eq.1 is a semi-empirical formula and it only gives an information for comparing the CV data. The HOMO and LUMO values were given a previous section and it is not well matched with the eq.1 value since the last two term of the eq.1 should be modified by each system.

Performance Analysis of Capacitors

It should be noted that the theoretical expectation of the CV data of a supercapacitor cell is a rectangle shape and the experimental measurements generally deviates from the expectation. The CV of the capacitor cell used complex 3 as the electrodes using 6 mol KOH/H2O electrolyte solution was shown in Figure 7a. According to Figure 7a, complex 3 has a rectangle CV shapes. 200 and 400 mV/s constant scan rates show us the complex 3 have the supercapacitor property.

Figure 7b shows discharge characteristic data of the capacitors for range of ±1V. It can be seen that the discharge data shows an exponential decay with increasing time as expected shape of a capacitor cell. The time for charging and discharging of the capacitor is directly proportional with the specific capacity of the cell, and it is shown in Figure 7c that the discharging time is shorter than charging time and it is an undesirable result in the structure.

The cell capacity of the cell was explained with the equation below :

𝐶 = 𝐼 • 𝑚 -ΔV/Δt
where m is the mass of the material in the electrodes, I is the applied current which is constant for the measurement, ΔV/Δt is the gradient of the CV curve for charging and discharging in the cell. The capacity of the cells was measured over 1000 cycles, as seen in Figure 7b. The capacity during to charge/discharge process at the different cycles and the calculated capacity fade values of the cells are given in Table 3.

The first charge and discharge capacity values of complex 3 cells have showed 28.15 and 10.22 Fg -1 respectively, and, which the values are almost similar range with the capacity value of a supercapacitors [20]. According to capacity fade values for long term cycle process, the complex 3 has showed too high capacity fade percentage after 1000 cycles with 96.83% and 92.66% for charge and discharge, respectively. Although the fade of capacity is high enough when compared with the material which was used as a capacitor, this is a key study for future investigations. So, we may suggest that the complex materials including metal ions may have a huge potential in capacitive type applications such as supercapacitors and we believe that these types of structures may be used as an electrode material of this type of energy storage applications in the next future after improving their capacitive properties.

Conclusion

The complex 3 was successfully produced and the structures experimental data was compared with the calculated data as given in Table 1. The calculated energy gap between HOMO and LUMO was found as 3.531 eV, which is important for electronic properties.

We used Cr2032 case for the production of a hybrid capacitor and the highest capacity value was found as 10.22 F/g, which has a potential for the capacitor studies of complex structures. It should be noted that the capacitive properties of Pt-complexes for energy technology are a new study area for organometallic industry, and it can be said that organometallic complex materials may be used as a capacitor material and electronic circuit in the next future.

In this study, structural properties are investigated using single crystal XRD and DFT suit. The electrochemical activity of the complex and the capacitive properties were investigated in detail and it is found that the complex has a potential for the energy storage materials in portable devices and electronic circuits.

Appendix A. Supplementary data

CCDC 1845130 contains the supplementary crystallographic data for compound 3. These data can be obtained free of charge via http://www.ccdc.cam.ac.uk/conts/retrieving.html, or from the Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge CB2 1EZ, UK; fax: (+44) 1223-336-033; or e-mail: deposit@ccdc.cam.ac.uk.

  ], 157.1 [NCHN] ppm. LC-MSMS (ESI): m/z 419.35 [M-Br] + (100%) (C29H43N2 requires 419.35); Calcd for C29H43N2Br: C

  Hz, 24H, C6H3((CH(CH3)2)2-2,6], 2.26 [s, 4H, NCH2CH2CH2CH2N], 3.28 [p, J=6.8 Hz, 4H, C6H3((CH(CH3)2)2 -2,6], 4.05 [s, 4H, NCH2CH2CH2CH2N], 7.25-7.36 [m, 6H, C6H3((CH(CH3)2)2-2,6]. 13 C {H} NMR (100 MHz, DMSO-d6):δ= 24.8 and 24.9 [C6H3((CH(CH3)2)2-2,6], 25.2[C6H3((CH(CH3)2)2-2,6], 28.5 [NCH2CH2CH2CH2N], 53.9 [NCH2CH2CH2CH2N], 125.8, 129.1, 145.1 [C6H3((CH(CH3)2)2-2,6], ppm. Calcd for C29H42N2AgBr: C

  13 C {H} NMR (100 MHz, CDCl3):δ= 24.6 and 25.0 [C6H3((CH(CH3)2)2-2,6], 24.9[C6H3((CH(CH3)2)2-2,6], 29.0 [NCH2CH2CH2CH2N], 56.2 [NCH2CH2 CH2CH2N], 125.3, 130.9, 138.9,144.9 [C6H3((CH(CH3)2)2 -2,6], 157.1 [NCHN] ppm. HRMS (ES): m/z 419.3431 [M-Br] + (100%) (C29H43N2 requires 419.3421); m/z 752,8464 [M-2H] + (78.2%); (C29H41N2PtCl4 requires 752,1672).

Figure 2

 2 Figure 2The NMR spectra of compound 1, 2, and 3.

Figure 3 .Figure 4 .Figure 5 .Figure 7 .

 3457 Figure 3. ORTEP drawing of a molecule of complex 3, as present in crystal at

Table 1 .

 1 Analogy of experimental and theoretical computation bond length and angle results of the complex 3

		Bond Leghts (Å)	Angles ( o )	Torsion angles ( o )
		Found Calcd.	Found Calcd.	Found	Calcd.
	Pt1 -S1	2.2093 2.4178 S1-Pt1-Cl1	90.02	88.474 C31-C11-C12-N13 -52.0	-54.89
	Pt1 -Cl1 2.3234 2.4404 S1-Pt1-Cl3	179.03 178.11	N33-C1-N13-C14 173.0	173.22
					6
	Pt1 -Cl3 2.3302 2.3806 Cl1-Pt1-Cl3	89.99	91.359 N33-C1-N13-C12 -13.3	-11.13
	Pt1 -Cl2 2.3328 2.4393 S1-Pt1-Cl2	91.92	89.082 C11-C12-N13-C14 -111.9	-115.45
	S1 -O51 1.471	1.6871 O51-S1-C52	108.2	105.88 C34-C35-C36-C37 1.1	0.871
	S1 -C52	1.773	1.8885 O51-S1-C53	108.12 105.44 C38-C39-C40-C41 -61.8	-67.97
	S1 -C53	1.777	1.8884 O5-S1-Pt1	116.17 116.91 C14-C19-C23-C25 105.6	112.61

Table 2 .

 2 CV properties of the complex 3 on different scan rates

	Scan Rates (mV/s) EPc (eV)	EPa(eV)	E1/2(eV)
	400	1.211	0.292	0.7515
	200	1.127	0.359	0.743

Acknowledgement

Dr. Serdar Altin want to thanks to Inönü University for the financial support with project numbers of G-2016/52 for the measurement of the electrochemical and capacitive properties of complex.