K A Ivanova

S L Gavrilyuk
email: sergey.gavrilyuk@univ-amu.fr

Structure of the hydraulic jump in convergent radial flows

Keywords: turbulent hydraulic jump, convergent radial flow, radial symmetry breaking, nonconservative hyperbolic equations

We are interested in modelling of multi-dimensional turbulent hydraulic jumps in convergent radial flow. To describe the formation of intensive eddies near the free surface (rollers) at the front of the hydraulic jump, a new model of shear shallow water flows is used. The governing equations form a non-conservative hyperbolic system with dissipation source terms. The structure of equations is reminiscent of generic Reynolds-averaged Euler equations for barotropic compressible turbulent flows.

Two types of dissipative terms are studied. The first one corresponds to Chézy-like dissipation rate, and the second one to a standard energy dissipation rate commonly used in compressible turbulence. Both of them guarantee the positive definitness of the Reynolds stress tensor. The equations are rewritten in polar coordinates and numerically solved by using an original splitting procedure. Numerical results for both types of dissipation are presented and qualitatively compared with the experimental works. The results show both experimentally observed phenomena (cusp formation at the front of the hydraulic jump) as well as new flow patterns (the shape of the hydraulic jump becomes a rotating square).

Introduction

Much work has been done in the study of the hydraulic jumps in a divergent radial flow [START_REF] Andersen | Separation vortices and pattern formation[END_REF], [START_REF] Bush | The influence of surface tension on the circular hydraulic jump[END_REF], [START_REF] Eyo | Two dimensional laminar flow of a liquid with circular hydraulic jump[END_REF], [START_REF] Kasimov | A stationary circular hydraulic jump, the limits of its existence and its gasdynamic analogue[END_REF], [START_REF] Labousse | The hydraulic bump: The surface signature of a plunging jet[END_REF], [START_REF] Liu | The hydraulic jump in circular jet impingement and in other thin liquid films[END_REF], [START_REF] Martens | Model for polygonal hydraulic jumps[END_REF], [START_REF] Ray | Standing and travelling waves in the shallow-water circular hydraulic jump[END_REF], [START_REF] Rojas | A progressive correction to the circular hydraulic jump scaling[END_REF], [START_REF] Teymourtash | Experimental and numerical investigation of circular hydraulic jump[END_REF]. Such a flow can easily be created by a vertical impact of a jet of water on a solid flat surface. After the impact, the flow spreads radially outwards and at some distance of the point of impingement an abrupt depth transition occurs (hydraulic jump appears). Despite the radial symmetry of the initial flow and corresponding boundary conditions, the hydraulic jump is not necessary circular: its shape can have even polygonal structure with sharp corners (see [START_REF] Bush | An experimental investigation of the stability of the circular hydraulic jump[END_REF][START_REF] Ellegaard | Creating corners in kitchen sinks[END_REF][START_REF] Ellegaard | Cover illustration: Polygonal hydraulic jumps[END_REF]).

An interesting experimental study was also recently performed by [START_REF] Foglizzo | The explosion mechanism of corecollapse supernovae: progress in supernova theory and experiments[END_REF][START_REF] Foglizzo | Shallow water analogue of the standing accretion shock instability: Experimental demonstration and a two-dimensional model[END_REF] for convergent radial flows. A radial symmetry breaking was also observed: under some flow conditions a rotating singular point (a cusp) appears at the front of the hydraulic jump.

A reliable mathematical model which is capable to capture this complex phenomenon of symmetry breaking is still lacking. The aim of this article is to model such a phenomenon by using the model of shear shallow water flows [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 shallowwater flows[END_REF][START_REF] Teshukov | Gas dynamic analogy for vortex free-boundary flows[END_REF]. Shear effects are necessary to describe surface rollers of the hydraulic jump. In 1D-case, such a model complemented by friction terms was used for the study of travelling waves down inclined plane (roll waves) and turbulent hydraulic jumps. A strong physical adequacy of the model with the experimental observations was found [START_REF] Ivanova | Formation and coarsening of roll-waves in shear shallow water flows down an inclined rectangular channel[END_REF][START_REF] Richard | Elaboration d'un modèle d'écoulements turbulents en faible profondeur: application au ressaut hydraulique et aux trains de rouleaux[END_REF][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 shallowwater flows[END_REF]. The classical Saint-Venant equations were not able to describe these phenomena, because they do not take into account the vorticity effects. The new model was able to form rollers in the hydraulic jumps through the Rankine-Hugoniot relations for the corresponding hyperbolic and conservative equations. In particular, the model was able to describe the shape of the 1D hydraulic jump as well as oscillations of the jump toe appearing when the Froude number is larger than approximately 1.5. A natural step would be to apply this model for the description of multi-dimensional turbulent hydraulic jumps, and, in particular, to see if the model of shear shallow water flows is able to describe the breaking of radial symmetry in hydraulic jumps observed experimentally.

The governing equations are obtained by depth averaging of multi-dimensional Euler equations [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 shallowwater flows[END_REF][START_REF] Teshukov | Gas dynamic analogy for vortex free-boundary flows[END_REF]. The hypothesis of smallness of the horizontal vorticity (the hypothesis of weakly sheared flows) allows us to keep the second order depth averaged correlations in the governing equations but neglect the third order correlations, and thus to close the governing system in dissipationless limit.

The corresponding multi-dimensional model of shear shallow water flows is a hyperbolic system of equations which is reminiscent of a generic Reynolds-averaged Euler for barotropic turbulent flows. The model has three families of characteristics corresponding to the propagation of surface waves, shear waves and average flow. The main difficulty is non-conservativity of governing equations: for six unknowns (the fluid depth, two components of the depth averaged horizontal velocity, and three independent components of the symmetric Reynolds stress tensor) one has only five conservation laws : conservation of mass, momentum, energy and mathematical "entropy" . The last one determines the evolution of the determinant of the Reynolds stress tensor. The non-conservative nature of the multi-dimensional equations represents an enormous difficulty from the mathematical and numerical point of view. The definition and computation of discontinuous solutions for non-conservative hyperbolic equations is a challenging problem. Recently, a numerical method (a splitting procedure) was developed for solving this non-conservative system [START_REF] Gavrilyuk | Multi-dimensional shear shallow water flows: problems and solutions[END_REF]. Cartesian coordinates were used for the description of multi-dimensional roll waves. In the present work we extend this technique to polar coordinates.

Another difficulty is to introduce the dissipative terms into the model. The introduction of dissipation terms is always an empirical procedure. In particular, in [START_REF] Gavrilyuk | Multi-dimensional shear shallow water flows: problems and solutions[END_REF] we proposed the dissipation terms which ensure the positive definiteness of the stress tensor P. Also, in 1D limit we recovered the dissipation law developed in [START_REF] Richard | The classical hydraulic jump in a model of shear shallowwater flows[END_REF]. In particular, this dissipation law allowed us to describe the "fingering" phenomenon (i.e. the formation of transverse waves on the front of roll waves). We will test here another type of dissipation used earlier in [START_REF] Gavrilyuk | Spilling breakers in shallow water: applications to Favre waves and to the shoaling and the breaking of the solitary wave[END_REF] to study the surface wave propagation in shear flows.

Different dissipation laws give us different flow patterns. The first type of dissipation give us hydraulic jumps with formation of rotating cusp at the front of the hydraulic jump as in experiments of [START_REF] Foglizzo | Shallow water analogue of the standing accretion shock instability: Experimental demonstration and a two-dimensional model[END_REF]. The second type of dissipation gives us an unusual structure, which represent a smooth "square". The square has no sharp corners and rotates with constant velocity. This is different from sharp corner structures of hydraulic jumps obtained in divergent radial flows ( [START_REF] Ellegaard | Creating corners in kitchen sinks[END_REF][START_REF] Ellegaard | Cover illustration: Polygonal hydraulic jumps[END_REF]).

The structure of the article is organized as follows. In Section 2 the "shear shallow water" equations are presented. Two types of dissipative terms are proposed in Section 3. Experimental works are briefly discussed in Section 4. Governing equations of sher flows rewritten in polar coordinates and the corresponding initial and boundary conditions presented in Sections 5, 6. The numerical results are discussed in Section 7. In Appendix A, the numerical scheme for the non-conservative equations is written in polar coordinates.

2D shear shallow water flows

The system describing multi-dimensional shear shallow water flows over a flat bottom without friction effects can be written in the form ( [START_REF] Teshukov | Gas dynamic analogy for vortex free-boundary flows[END_REF][START_REF] Gavrilyuk | Geometric evolution of the Reynolds stress tensor[END_REF][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 shallowwater flows[END_REF]):

                       ∂h ∂t + div (hU) = 0, ∂ (hU) ∂t + div ( hU ⊗ U + gh 2 2 I + hP ) = 0, DP Dt + ∂U ∂x P + P ( ∂U ∂x ) T = 0.
(1

)
The system is obtained by depth averaging the Euler equations and using the hypothesis of smallness of the horizontal vorticity (weakly sheared flows). Here t is the time, x = (x, y) T are the Cartesian coordinates, h is the fluid depth, g is the gravity, U = (U, V ) T is the depth averaged horizontal velocity, D/Dt means the material derivative with respect to the mean motion :

D Dt = ∂ ∂t + U T ∇, (2)
P = P T is the symmetric stress tensor which measures the distortion of the instantaneous horizontal velocity profile Ũ(t, x, y, z) depending of the vertical coordinate z. The definitions of U(t, x) and P are as follows:

U(t, x) = 1 h ∫ h 0 Ũ(t, x, y, z)dz, P = 1 h ∫ h 0 ( Ũ -U ) ⊗ ( Ũ -U ) dz.
The tensor P is positive definite. The positive definiteness of P is a consequence of the Cauchy-Schwarz inequality. The sign ⊗ means the tensor product, I is the identity tensor. Equations (1) admit the energy conservation law :

∂ ∂t { h ( 1 2 |U| 2 + e i + e T )} + div { hU ( 1 2 |U| 2 + e i + e T ) + ( gh 2 2 I + hP ) U } = 0, (3)
with e i = 1 2 (gh), e T = 1 2 tr(P) , and an additional conservation law:

∂hΨ ∂t + div (hUΨ) = 0, Ψ = Det (P) h 2 . ( 4
)
The variable Ψ will be referred to as "entropy" (mathematical) because this quantity is transported along the mean flow in the same way as the true entropy for the Euler equations of compressible fluids. Also, we will see that this quantity will increase across the shocks in analogy with the conventional entropy. The 1D system also admits an analogue of Ψ having a clear physical meaning. This quantity was associated with the enstrophy (squared vorticity). For example, for hydraulic jumps the enstrophy increase corresponds to the vorticity creation : a roller appears at the forward slope of the hydraulic jump [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 shallowwater flows[END_REF][START_REF] Ivanova | Formation and coarsening of roll-waves in shear shallow water flows down an inclined rectangular channel[END_REF]. The system (1) is composed of the equations of mass balance, horizontal momentum and evolution equation for the stress tensor. The governing equations are hyperbolic, if P is positive definite, but not in conservative form (for proof, see [START_REF] Gavrilyuk | Multi-dimensional shear shallow water flows: problems and solutions[END_REF]). For discontinuous solutions, the Rankine-Hugoniot relations come only from the mass, momentum and energy equations. For six unknowns (the fluid depth, two components of the depth averaged horizontal velocity, and three independent components of the symmetric Reynolds stress tensor) one has only five conservation laws (conservation of mass, momentum, energy and mathematical "entropy"). However, for 1D flows the system of Rankine-Hugoniot relations is closed.

Dissipative terms compatible with the positive definiteness of the Reynolds stress tensor

Now we add dissipation terms into the model. Even if this process is always empirical, some constraints should be respected. For example, the dissipation law should always be compatible with the energy decrease and guarantee the positive definiteness of the Reynolds stress tensor. Two types of dissipation terms will be tested below.

Let us add the dissipation terms in the following form :

                   h t + div(hU) = 0, (hU) t + div ( hU ⊗ U + gh 2 2 I + hP ) = -C f |U|U, DP Dt + ∂U ∂x P + P ( ∂U ∂x ) T = D, D = D T .
(

) 5
In particular, the equations for P imply:

h 2 D Dt ( det(P) h 2 ) = tr(P)tr(D) -tr(PD). ( 6
)
The friction force in the momentum equation is a classical expression, where C f is the Chézy coefficient. The dissipation tensor D should be somehow defined. The equations (5) should satisfy the energy conservation law :

∂ ∂t ( h ( 1 2 |U| 2 + e i + e T )) + div ( hU ( 1 2 |U| 2 + e i + e T ) + ( gh 2 2 I + hP ) U ) = -C f |U| 3 -Q, ( 7
)
where the dissipative source term Q should be positive. The positivity of Q is the analogue of the second law of thermodynamics. The compatibility of ( 5) and [START_REF] Foglizzo | The explosion mechanism of corecollapse supernovae: progress in supernova theory and experiments[END_REF] implies the expression for Q in terms of D:

tr (D) = - 2 h Q. ( 8
)
By analogy with the Stokes hypotheses for the Navier-Stokes equations, we assume that the dissipation tensor D is an isotropic tensor function of P. Then, for two-dimensional case, D is linear in P :

D = aP + bI,
where a and b are functions of invariants of P. Consider the simplest case where b = 0. This choice allows us to obtain a natural reduction to the Saint-Venant equations in the limit P = 0.

First type of dissipation terms

The first type of dissipation is :

D = aP, a = -2α h |U| 3 , ( 9
)
where a = τ -1 , τ is the relaxation time, α has the dimension s 2 m -2 . The multiplier -2|U| 3 /h is for convenience only. In particular, this choice implies the equation for Ψ in the form :

h 2 D Dt ( det (P) h 2 ) = - 4α h |U| 3 det (P) . ( 10
)
Equations ( 8) and ( 9) imply the following relation between Q and α :

Q = αtr (P) |U| 3 . ( 11
)
We will introduce now a new dimensionless parameter κ = αtr (P). To recover the 1D case studied priviously ( [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 shallowwater flows[END_REF]), we will finally choose :

κ = max ( 0, C r tr(P) h 2 -φ tr(P) h 2 ) ≥ 0. ( 12
)
Here φ and C r are the model constants : φ is associated with the enstrophy of small vortexes in the vicinity of the bottom, and C r is the coefficient associated with the roller dissipation. We will search the values of these parameters to have at least a qualitative agreements with experiments.

As it follows from ( 10) and ( 12), "entropy" Ψ is decreasing on continuous solutions, but always stays positive. This means that the dissipation law also guarantees the positive definiteness of P.

Second type of dissipation terms

Another type of dissipation tensor D can be considered in this form:

D = aP, a = -2 h κtr(P) 1 2 . ( 13
)
As a consequence of ( 11), one has:

Q = κtr(P) 3 2 > 0. ( 14
)
This type of dissipation is Galilean invariant and was used, in particular, to study the solitary wave breaking ( [START_REF] Gavrilyuk | Spilling breakers in shallow water: applications to Favre waves and to the shoaling and the breaking of the solitary wave[END_REF]). Moreover, ( 14) corresponds to the classical energy dissipation rate with the dissipation rate scale h ( [START_REF] Townsend | The structure of turbulent shear flow[END_REF]). Such a choice also guarantees that P is positive definite for any time, if initially it was positive definite.

Experiments on hydraulic jumps in a radial flow

The circular hydraulic jumps are commonly observed in a kitchen sink when a vertically falling tap water jet strikes a horizontal plate and then spreads radially outwards ([1], [START_REF] Bush | The influence of surface tension on the circular hydraulic jump[END_REF], [START_REF] Bush | An experimental investigation of the stability of the circular hydraulic jump[END_REF], [START_REF] Eyo | Two dimensional laminar flow of a liquid with circular hydraulic jump[END_REF], [START_REF] Labousse | The hydraulic bump: The surface signature of a plunging jet[END_REF], [START_REF] Liu | The hydraulic jump in circular jet impingement and in other thin liquid films[END_REF], [START_REF] Martens | Model for polygonal hydraulic jumps[END_REF], [START_REF] Ray | Standing and travelling waves in the shallow-water circular hydraulic jump[END_REF]). Using a more viscous liquid, [START_REF] Bush | An experimental investigation of the stability of the circular hydraulic jump[END_REF][START_REF] Ellegaard | Creating corners in kitchen sinks[END_REF][START_REF] Ellegaard | Cover illustration: Polygonal hydraulic jumps[END_REF] observed breaking of radial symmetry of the hydraulic jump front. More exactly, [START_REF] Ellegaard | Creating corners in kitchen sinks[END_REF][START_REF] Ellegaard | Cover illustration: Polygonal hydraulic jumps[END_REF] obtained polygonal shapes of hydraulic jumps in divergent radial flow with sharp corners: "the sharp corners of the polygons carry a large radial flux, while the sides generate resistance to the stream". [START_REF] Bush | An experimental investigation of the stability of the circular hydraulic jump[END_REF] also noted that: "some polygonal ... forms were subject to weak time-dependent fluctuations, typically characterized by a net rotational motion of the entire jump structure, or the propagation of wave-like disturbances towards a single point on the jump".

Our numerical study will concern the hydraulic jump in a convergent and not divergent radial flow. Such experiments were recently performed by [START_REF] Foglizzo | The explosion mechanism of corecollapse supernovae: progress in supernova theory and experiments[END_REF][START_REF] Foglizzo | Shallow water analogue of the standing accretion shock instability: Experimental demonstration and a two-dimensional model[END_REF]. In these experiments, the fluid is radially injected inwards and strikes a hollow cylinder placed in the center, through which the water is evacuated. As a consequence, a hydraulic jump is formed separating convergent supercritical flow from deeper subcritical one. The hydraulic jump also exhibits the appearance of radial asymmetries: after formation of the hydraulic jump, it starts to oscillate randomly, then the amplitude of oscillations grows and, finally, the flow becomes very asymmetric and develops a singularity of the jump front that starts to rotate with a constant angular velocity.

We want to understand if model ( 1) is able to reproduce, at least qualitatively, the last experiments. Also, it would be interesting to study the influence of different types of dissipation on the solution structure and to understand if new patterns can appear.

Governing equations in polar coordinates

To model the hydraulic jumps in radial flow, we rewrite the system (1) in polar coordinates (see Appendix A). In dissipationless case over a flat bottom the system (1) becomes:

∂(hr) ∂t + ∂(rhU r ) ∂r + ∂(hU θ ) ∂θ = 0, (15)
∂(rhU r ) ∂t + ∂ ∂r { r ( hU 2 r + gh 2 2 + hP rr )} + ∂ ∂θ (hU r U θ + hP rθ ) = h(U 2 θ + P θθ ) + gh 2 2 , ( 16
)
∂ (rhU θ ) ∂t + ∂ ∂r {rh (U r U θ + P rθ )} + ∂ ∂θ ( hU 2 θ + gh 2 2 + hP θθ ) = -h (U r U θ + P rθ ) , ( 17
)
DP rr Dt -4 U θ r P rθ + 2 ( ∂U r ∂r P rr + P rθ r ∂U r ∂θ ) = 0, ( 18
)
DP rθ Dt + U θ r (P rr -2P θθ ) + P θθ r ∂U r ∂θ + P rr ∂U θ ∂r + P rθ r { ∂U θ ∂θ + ∂(rU r ) ∂r } = 0, ( 19
)
DP θθ Dt + 2 U θ r P rθ + 2 { ∂U θ ∂r P rθ + P θθ r ( ∂U θ ∂θ + U r )} = 0, ( 20
)
∂ ∂t { hr ( 1 2 |U| 2 + E )} + ∂ ∂r { r [ hU r ( 1 2 |U| 2 + E ) + gh 2 2 U r + h(P rr U r + P rθ U θ ) ]} + + ∂ ∂θ { hU θ ( 1 2 |U| 2 + E ) + gh 2 2 U θ + h(P rθ U r + P θθ U θ ) } = 0, (21)
where the material derivative is defined as:

D Dt = ∂ ∂t + U r ∂ ∂r + U θ r ∂ ∂θ .
The system ( 15)-( 21) admits the "entropy" conservation law:

D Dt ( det(P) h 2 ) = 0. (22)
We have now to add into the model the dissipation terms and bottom topography. The bottom topography consists of an inclined plane of a mild slope superposed with a bump modeling the hollow cylinder (see Figure 1). Analytically, the expression of the bottom is given by:

b(r) =      A [(r-R --L1) 2 -L 2 1 ] 2 L 4 1 , if 0 < r -R -< 2L 1 , (r -R --2L 1 ) tan β, if 2L 1 + R -< r < R + . ( 23
)
Here β is a small inclination angle, A is the bump amplitude, 2L 1 is the radial bump width, R - is the internal boundary radius, R + is the external boundary radius. The formula ( 23) is used to model the "physical" cylinder of hight A which is placed at r = R -+ L 1 .

Initial and boundary conditions

Now, we need to impose boundary conditions on the cylinder. The number of boundary conditions is equal to the number of characteristics entering the flow domain. For stationary flow a natural boundary condition to impose on the top of the cylinder is the criticality condition (Froude number is equal to one). When the flow is nonstationary and transcritical, the problem of boundary conditions is an open question even for the classical Saint-Venant equations (cf. [START_REF] Shiue | Boundary value problems for the shallow water equations with topography[END_REF]). In particular, if the flow is subcrictical, the perturbations can propagate inside the domain and change the solution. We need thus a remedy to impose "good" boundary conditions. For this, we will compute the solution in a larger domain by introducing a bump, where the downslope flow accelerates and becomes supercritical. If the flow is supercritical at both boundaries r = R -and r = R + , the boundary conditions are simple to impose. In particular, at r = R -there is no characteristics entering the flow domain, so we can use Neumann boundary conditions. At r = R + we impose all unknowns. Such an approach allowing us to control the fluid flow at the downstream crest was also used by [START_REF] Winters | The turbulent transition of a supercritical downslope flow: sensitivity to downstream conditions[END_REF]. We numerically solved the equations ( 21) where, additionally, the dissipation and bottom topography are added (see Appendix A) in the computational domain:

h 0 [m] β [rad] C r C f φ [s -2 ] q 0 [m 3 /s] R + [m] R -[m] A [m] L 1 [m] 0.
R -≤ r ≤ R + , 0 ≤ θ ≤ 2π.
The initial conditions are :

h(r, θ, t = 0) =    1.5h 0 [m]; if r -R -≤ 0.3(R + -R -), h 0 [m];
else U r (r, θ, t = 0) = -q 0 /(rh), U θ (r, θ, t = 0) = 0, P rr = φh 2 , P rθ = 0, P θθ = ϵφh 2 .

Here ϵ = 10 -8 , h 0 is a given flow depth at r = R + , q 0 > 0 is a given flow discharge. The other parameters are shown in Table 1. Obviously, we respect the inequality 2L 1 < 0.3(R + -R -). We have to underline that the initial conditions are inimportant, because for large time the solution is definied only by the boundary conditions. To avoid the degeneracy of determinant of P and ensure the hyperbolicity of the governing equations, we take a small positive value of P θθ . This initial radial flow is perturbed by boundary conditions at r = R + in the following form:

h = h 0 , U r = -q 0 (1 + 0.01 sin(nθ))/(R + h 0 )), U θ = 0, P rr = P θθ = 0.5φh 2 0 , P rθ = 0. ( 24
)
We used here isotropic expression of P. Also, for this case one has κ = 0. We also added a small perturbation of the radial velocity in θ-direction with n being any natural number. We took n = 16 that corresponds to the number of injecting pumps used, in particular, in experiments by [START_REF] Foglizzo | Shallow water analogue of the standing accretion shock instability: Experimental demonstration and a two-dimensional model[END_REF]. The periodic boundary condition is used in θ-direction. We control that at the inner boundary r = R -in radial direction the corresponding generalized Froude number is larger than one to guarantee the condition of supercritical flow :

F g = U r √ gh + 3P rr > 1.
The numerical scheme is based on the splitting technique developed in [START_REF] Gavrilyuk | Multi-dimensional shear shallow water flows: problems and solutions[END_REF]. The first-order Godunov scheme with HLLC Riemann solver was used (for details, see Appendix A). The code is parallelized using the Message Passing Interface (MPI). The first order Godunov method with HLLC Riemann solver was used with 300 × 300 mesh cells. In both cases, the hydraulic jump was stationary. The radius of hydraulic jump for the quadratic friction is a little bit larger than that for the linear friction.

Numerical results

Classical Saint-Venant equations

The numerical resolution of the classical shallow water equations (Saint-Venant equations) with quadratic and linear friction give us only stationary hydraulic jumps without any oscillations or formation of rotating patterns (see Figure 2). The radius of hydraulic jump for the quadratic friction is a little bit larger of that corresponding to the linear friction. The same type of discharge perturbations was used as in [START_REF] Teshukov | Gas dynamic analogy for vortex free-boundary flows[END_REF] with the same bottom topography [START_REF] Shiue | Boundary value problems for the shallow water equations with topography[END_REF].

Shear shallow water model with first type of dissipation

We present now the numerical results for the first type of dissipation given by ( 9). The scenario of the hydraulic jump formation in convergent radial flow consists of the following three different stages. First, almost radially symmetric hydraulic jump is formed at the time instant approximately 25 s (see Figure 3). One can see only small amplitude nonstationary transverse perturbations on the front of the hydraulic jump. This is clearly visible on the Figure 3 showing the distribution of Ψ = det(P) h 2 . Then, the hydraulic jump is destabilized into a pattern exhibiting sloching type oscillations at the time instant about 50 s. The oscillation period is approximately 4 seconds. In Figure 4 the schlieren images are shown at the time instants 50 s and 52 s (after the half-period of oscillations). Remarkably, in a certain domain of parameters, as the amplitude of the oscillations grows, a new free surface pattern appears. More exactly, a singular point (cusp) at the free surface of the hydraulic jump appears which starts to rotate. In Figure 5 the cusp rotating in positive direction is denoted by S. The sense of rotation can easly be understood from the Figure 5. Indeed, ahead of the cusp the front is smooth, and behind the cusp the front is rough. The sense of rotation can be changed by a small initial perturbation. If the initial condition for the tangential velocity are taken in the form : U θ = ±µr (µ is a small positive parameter), the sense

φ = 2 s -2 φ = 5 s -2 φ = 10 s -2 φ = 20 s -2 C r = 1 + + ± - C r = 0.5 + + ± - C r = 0.3 + + ± - Table 2:
In this table, "+" means that for corresponding values of φ and C r , the rotating cusp is formed, "-" means that the cusp is absent."±" means the limit behaviour.

of rotation will be determined by the sign of U θ . The radius of the hydraulic jump increases with time. One can also see the emergence of double jump structures appearing at time instant 150 s, where the free surface changes abruptly (Figure 5). Double jump structures were also observed in the case of divergent circular hydraulic jump in [START_REF] Bush | An experimental investigation of the stability of the circular hydraulic jump[END_REF].

One can also notice that the cusp appears only in some region of parameters φ and C r (see Table 2). The existence of such a cusp allows us to easily determine the period of rotation of pattern. In particular, in our case the period of rotation was about 6 seconds. It does not depend too much on the values of φ and C r . In the experiments of [START_REF] Foglizzo | Shallow water analogue of the standing accretion shock instability: Experimental demonstration and a two-dimensional model[END_REF] the period was was about 3 s, but the geometry was different : the radius R + was 0.33 m. The main reason why we did not use the same geometry was that we could not guarantee the supercritical flow at the outlet (at r = R -) which was necessary to assure correct boundary conditions for the governing equations.

The results obtained are qualitatively similar to those observed experimentally. It means that our mathematical model of shear shallow water flows and the corresponding numerical method are capable to simulate new flow patterns appearing in radial fluid flows. The transverse structures (formation of the "triangles" behind the moving cusp) that are clearly visible in Figure 5, is a physical phenomenon and not a numerical instability. They could also be seen in the experiments of [START_REF] Foglizzo | Shallow water analogue of the standing accretion shock instability: Experimental demonstration and a two-dimensional model[END_REF].

When φ vanishes, we do not obtain any transverse structures: the circular hydraulic jump is only formed. Recall, that φ is associated to the enstrophy of small vortexes at the vicinity of the bottom [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 shallowwater flows[END_REF].

In Table 2 we studied how parameter φ and dissipation coefficient C r influence the flow behavior. Numerical tests showed that the more sensible parameter is φ rather than C r . For φ larger than 10 s -2 the cusp disappears, but the singular behaviour is still visible (see Figure 6).

Numerical "cusp" solution was also obtained in [START_REF] Foglizzo | Shallow water analogue of the standing accretion shock instability: Experimental demonstration and a two-dimensional model[END_REF] by using the Saint-Venant equations with linear friction, but for a different bottom topography. They used the criticality condition at the top of the cylinder what corresponds to the hypothesis of the steady flow assumption.

A surprising fact also mentioned in the experiments is that the sense of rotation of the cusp and the tangential fluid velocity near the hole is opposite. This fact is also described by our model.

Shear shallow water model with second type of dissipation

Using the second type of dissipation terms [START_REF] Ivanova | Formation and coarsening of roll-waves in shear shallow water flows down an inclined rectangular channel[END_REF], one can see new unusual structures: the formation of rotating square type hydraulic jump shape (see Figure 7). The shape doesn't contain sharp corners. In our case, the "corners" are rather smooth and the shape between "corners" is convex, while it was concave in experiments of [START_REF] Ellegaard | Creating corners in kitchen sinks[END_REF]. This is the main difference between convergent and divergent radial flows. A detailed comparison of structures of divergent and convergent hydraulic jumps in shear flows will be a future work.

Conclusion

Numerical modeling of multi-dimensional turbulent hydraulic jumps formed in convergent radial flow is performed and qualitatively compared to the experimental observations. Two types of "well posed" dissipation terms are proposed and studied. For the first type of dissipation, in some region of parameters φ and C r , the formation of a rotating cusp (angular point) on the hydraulic jump front was found. For the second type of dissipation, a smooth rotating square structure of hydraulic jump was found.

The numerical validations demonstrated the capability of the model and numerical method to reproduce the multi -dimensional hydraulic jump dynamics.

Futher development of this multi-dimensional model would be to add the dispersive effects to describe the multi-D wave propagation and breaking as was performed in 1D case in [START_REF] Gavrilyuk | Spilling breakers in shallow water: applications to Favre waves and to the shoaling and the breaking of the solitary wave[END_REF][START_REF] Gavrilyuk | Interaction of a subsurface bubble layer with long internal waves[END_REF].

Appendix A

Governing equations in polar coordinates

We will rewrite now the system of governing equations (1) in polar coordinates. The relation between Cartesian and polar coordinates are:

x = r cos θ, y = r sin θ, r = √ x 2 + y 2 , θ = arctan ( y x ) .
We introduce the vectors: e r = (cos θ, sin θ) T , e θ = (-sin θ, cos θ) T .

We have:

∂e r ∂r = 0, ∂e θ ∂r = 0, ∂e θ ∂θ = -e r , ∂e r ∂θ = e θ . ( 25
)
One can decompose: Using [START_REF] Teymourtash | Experimental and numerical investigation of circular hydraulic jump[END_REF], [START_REF] Townsend | The structure of turbulent shear flow[END_REF]

U = U r e r + U θ e θ , ∂U ∂x = ∂U ∂r ⊗ e r + 1 r ∂U ∂θ ⊗ e θ . ( 26
)
One can finally obtain the system (1) in polar coordinates:

                                                                 ∂(
( ∂U θ ∂θ + U r )} = 0, ∂ ∂t { hr ( 1 2 |U| 2 + E )} + ∂ ∂r { r [ hU r ( 1 2 |U| 2 + E ) + gh 2 2 U r + h(P rr U r + P rθ U θ ) ]} + + ∂ ∂θ { hU θ ( 1 2 |U| 2 + E ) + gh 2 2 U θ + h(P rθ U r + P θθ U θ ) } = 0. ( 34

) Numerical splitting technique in polar coordinates

The method follow the splitting technique proposed in [START_REF] Gavrilyuk | Multi-dimensional shear shallow water flows: problems and solutions[END_REF] in Cartesian coordinates. We adapt this approach to polar coordinates. For this, we solve first the model in r-direction, then in θdirection. For each direction, the two subsystems, proposed below, are solved separately. Each subsystem has the following properties:

• is hyperbolic • admits the energy conservation law.

Subsystem 1 in r-direction

The first subsystem in r-direction is:

                                       ∂(rh) ∂t + ∂(rhUr) ∂r = 0, (a)
+ ∂(rhUrP θθ ) ∂r = -2h(P rθ U θ + P θθ U r ), (f ) (35)
The subsystem (35) admits the energy conservation law

∂ ∂t { rh ( 1 2 |U| 2 + E )} + ∂ ∂r { r [ hU r ( 1 2 |U| 2 + E ) + gh 2 2 U r + hP rr U r ]} = 0, (36)
where E = (gh + P rr + P θθ )/2. The system is hyperbolic with the eigenvalues given by :

λ 1,2,3,4 = U r , λ 5,6 = U r ± √ gh + 3P rr . ( 37
)
The equation (35.d) is not conservative. The jump relation for this equation is not well defined. The value of P rr will be corrected using the energy equation (36). Numerically, we solve first the system (35) where the equation (35.d) is replaced by

∂(rhP rr ) ∂t + ∂(rhU r P rr ) ∂r = 0. ( 38
)
We update then rhP rr by using (36).

Godunov type method and correction of the non-conservative term rhP rr

The subsystem 1 can be rewritten in the following form:

∂W ∂t + ∂(rF) ∂r = S r , ( 39
)
where the vector of "conservative" variables W, the vector of fluxes F(W) and the geometric source term vector S r (W) here are W = ( rh, rhU r , rhU θ , rhP rr , rP rθ , rhP θθ , rh

( E + 1 2 |U| 2 )) T , F(W) = ( hU r , hU 2 r + p, hU r U θ , hU r P rr , U r P rθ , hU r P θθ , hU r ( 1 2 |U| + E ) + pU r ) T , ( 40
) S r = (0, h ( U 2 θ + P θθ ) + gh 2 2 , -h (U r U θ + P rθ ) , 4hP rθ U θ , U θ (2P θθ -P rr ) -P rθ U r , -2h (P rθ U θ + P θθ U r ) , 0) with p = gh 2 /2 + hP rr , E = (gh + P rr + P θθ )/2. ( 41
)
Let us consider now a fixed grid of size ∆r = r i+1/2 -r i-1/2 , the time increment is defined as ∆t = t n+1 -t n that must respect the Courant-Friederichs-Lewy (CFL) condition. The discrete values of the vector-function W at (r i , t n ) will be denoted by W n i . Integrating the conservation laws (39) on [r i-1/2 , r i+1/2 ]×[t n , t n+1 ], one obtains the conservative finite volume Godunov scheme on a fixed grid

W n+1 = W n i - ∆t ∆r ( r i+1/2 (F * ,n i+1/2 -S r,n i ) -r i-1/2 (F * ,n i-1/2 -S r,n i ) ) , ( 42
)
where F * ,n i+1/2 and F * ,n i-1/2 are the numerical fluxes. They are constant across interfaces between cells during the time step. For computing the fluxes

F * ,n i+1/2 = F * ,n i+1/2 (W n i , W n i+1 ) and F * ,n i-1/2 = F * ,n i-1/2 (W n i-1 , W n i ),
we use approximate solution of the Riemann problem between cells i, i + 1 and i -1, i respectively.

Using the energy conservative law, the non-conservative term rhP rr is updated by the following way:

(rhP rr ) = 2(rhE) - g(rh) 2 r -(rhP θθ ). (43)
The "mathematical" entropy Ψ corresponding to the vorticity formation is increasing through the shock at this step (see for proof [START_REF] Gavrilyuk | Multi-dimensional shear shallow water flows: problems and solutions[END_REF]).

Subsystem 2 in r-direction

The subsystem 2 in r-direction is:

                                     ∂(rh) ∂t = 0, (a)
It admits the energy conservation law :

∂ ∂t { rh ( |U| 2 2 + E )} + ∂ ∂r (rhU θ P rθ ) = 0. ( 45
)
The system is hyperbolic with the eigenvalues given by :

λ 1,2,3,4 = 0, λ 5,6 = ± √ P rr . ( 46
)
in equation (44.f). In the following this equation will be replaced by ∂(rhP θθ ) ∂t = 0 at the first step, and then the energy conservation law will be used for update the value of P θθ .

In particular, the estimation of P rθ at time instant (t + ∆t) is given by: (rP rθ ) (r, t + ∆t) = (rP rθ ) (r, t) + (rP rr )(r, t)

( U * θ, i-1/2 (t) -U * θ, i+1/2 (t) ) ∆t ∆r . ( 47
)
Here U * θ it is speed at the contact discontinuity. The non-conservative equation for P rθ we discretize in the following way: P rθ (r, t + ∆t) = P rθ (r, t) + P rr (r, t)

( U * θ, i-1/2 (t) -U * θ, i+1/2 (t) ) ∆t ∆r . ( 48
)
Here U * θ is the speed of the contact discontinuity. Equation for (rhP θθ ) is also non-conservative. This equation is solved numerically first with zero flux, and then the energy conservation law was used to update the value of rhP θθ :

(rhP θθ ) = 2(rhE) -g(rh) 2 r -(rhP rr ).

(49)

The "mathematical" entropy Ψ is conserved at this step.

We will give now a shorter explication of splitting in θ-direction, because it is completely analogous to that in r-direction.

Subsystem 1 in θ -direction

The first subsystem in θ -direction is :

                                      
The subsystem (50) admits the energy conservation law :

∂ ∂t { rh ( 1 2 |U| 2 + E )} + ∂ ∂θ { hU θ ( 1 2 |U| 2 + E ) + gh 2 2 U θ + hP θθ U θ } = 0. ( 51
)
The system is hyperbolic with the eigenvalues given by :

λ 1,2,3,4 = U θ r , λ 5,6 = U θ ± √ gh + 3P θθ r . ( 52
)
The "mathematical" entropy Ψ is increasing through the shock at this step.

Subsystem 2 in θ -direction

The subsystem 2 in θ -direction is : The system is hyperbolic with the eigenvalues given by : The subsystem (53) admits the energy conservation law ∂ ∂t ( rh

                                    
( 1 2 |U| 2 + E )) + ∂ ∂θ (hP rθ U r ) = 0. (55)
The mathematical "entropy" Ψ is conserved at this step.

Integration of source terms

To add the dissipation (source terms), we integrate the ordinary differential equation:

dW dt = S(W), (56)
with unknown vector W = ( rh, rhU r , rhU θ , rhP rr , rP rθ , rhP θθ , rh

( E + 1 2 |U| 2
)) T .

The vector of source terms is :

S(W) = ( 0, -grh ∂b ∂r -C f r √ U 2 r + U 2 θ U r , -rC f √ U 2 r + U 2 θ U θ , rhD rr , rD rθ , rhD θθ , -grh ∂b ∂r U r -rC f ( √ U 2 r + U 2 θ ) 3 -rQ ) T . ( 57
)
The initial condition W| t=0 is obtained from the previous hyperbolic step after solving the previous four subsystems. The first order Euler method is used for such an integration.

Figure 1 :

 1 Figure 1: The bottom topography used in numerical simulations.

Figure 2 :

 2 Figure 2: Schlieren images of the fluid depth for Saint-Venant equations are shown with quadratic friction (F = -C f |U|U, C f = 0.0036, β = 0.05[rad]) on the left, and with linear friction (F = -ν h U, β = 0.05[rad] with kinematic viscosity ν = 10 -6 [m 2 /s 2 ]) on the right. The white boundaries inside the domain correspond to strong gradients of h. Central white part corresponds to the hole. The "schlieren" function was used in standard form: exp(-||∇h||/||∇h|| max ), where || • || means the Euclidean norm. The same "schlieren" function was used for all computations.The first order Godunov method with HLLC Riemann solver was used with 300 × 300 mesh cells. In both cases, the hydraulic jump was stationary. The radius of hydraulic jump for the quadratic friction is a little bit larger than that for the linear friction.

Figure 3 :

 3 Figure 3: The schlieren images of h, U θ and Ψ = det(P) h 2 shown. The first order Godunov method with HLLC Riemann solver is used with 500 × 500 mesh cells, C r = 1, φ = 2 s -2 . Central white part corresponds to the hole. White line in the internal area corresponds to front of the hydraulic jump.

Figure 4 :

 4 Figure 4: The schlieren images of h, U θ and Ψ = det(P) h 2 are shown. The first order Godunov method with HLLC Riemann solver is used with 500 × 500 mesh cells, C r = 1, φ = 2 s -2 . The hydraulic jump is destabilized into a pattern which corresponds to the sloching type oscillations.

Figure 5 :

 5 Figure 5: The schlieren images of h, U θ and Ψ = det(P) h 2 are shown. S corresponds to a cusp (singular point). The first order Godunov method with HLLC Riemann solver is used with 500 × 500 mesh cells, C r = 1, φ = 2 s -2 . A cusp rotating in positive direction is formed. Transverse triangular structures appearing at the front follow this singular point.

Figure 6 :Figure 7 :

 67 Figure 6: Schlieren images of the hydraulic jump corresponding to φ = 2 s -2 (on the top left), φ = 5 s -2 (on the bottom left), φ = 10 s -2 (on the top right) and φ = 20 s -2 (on the bottom right) are shown with 500 × 500 mesh cells at time instant 290 s. The value of C r was always taken as 0.5. The cusp is formed only for φ = 2 s -2 and φ = 5 s -2 .In the case φ = 10 s -2 and φ = 20 s -2 the cusp degenerates into a moving singular point through which we have always the transition from "smooth" to "rough" region. The corresponding singular point (denoted by S) rotates in negative sense for all cases, excepting the case φ = 20 s -2 where the sense of rotation is positive. The sense of rotation can easly be understood because ahead of the singular point the front is smooth, while behind the singular point the front is rough.

  θθ ) ∂t + ∂(hU θ P θθ ) ∂θ + 2hP θθ ∂U θ ∂θ = 0. (f )

λ 1 ,

 1 2,3,4 = 0, λ 5,6 = ± √ P θθ r . (54)

Table 1 :

 1 Parameters for numerical tests.

	003	0.07	0.3 -1.0 0.0036 2 -20	1.2 • 10 -3 1.0	0.08	0.005 0.006

  , we obtain: P = P rr e r ⊗ e r + P rθ e r ⊗ e + P θr e θ ⊗ e r + P θθ e θ ⊗ e θ , P rθ = P θr .

	∂U ∂x	=	∂U r ∂r	e r ⊗ e r +	∂U θ ∂r	e θ ⊗ e r +	(	1 r	∂U r ∂θ	-	U θ r	)	e r ⊗ e θ +	1 r	( U r +	∂U θ ∂θ	)	e θ ⊗ e θ .	(27)
	(27) implies :
											divU =	1 r	∂U θ ∂θ	+	1	∂ ∂r	(rU r ) ,	(28)
	Also, one has:
																					(29)
	Then,
											div(hP) = (div(P))h + P∇h,	(30)
	where			div(P) =	(	∂P rr ∂r	+	1 r	∂P rθ ∂θ	+	1 r	(P rr -P θθ ),	∂P rθ ∂r	+	1 r	∂P θθ ∂θ	+	2 r	P rθ	)	.	(31)
	Also,
											DP Dt	=	∂P ∂t		+ U r	∂P ∂r	+	U θ r	∂P ∂θ	.	(32)
	Using (25), one obtains:
			∂P ∂r =	 	∂Prr ∂r ∂P rθ ∂r		∂P rθ ∂r ∂P θθ ∂r	  , ∂P ∂θ =	 	∂P rθ ∂θ + P rr -P θθ ∂Prr ∂θ -2P rθ	∂θ + 2P rθ ∂P θθ ∂P rθ ∂θ + P rr -P θθ	  .	(

Acknowledgment

The authors thank B. Nkonga, J. Massoni, S. Hank and N. Favrie for helpful discussions. This work was partially supported by l'Agence Nationale de la Recherche, France (grant numbers ANR-11-LABEX-0092, and ANR-11-IDEX-0001-02).


Romain Boissier

Vital Hevia

Harman Max Bruins

Klemens Budde

Arnaldo Figueiredo

Enrique Lledó-García

Jonathon Olsburgh

Heinz Regele

Rhana Claire Fraser Taylor

Hassan Zakri

Enrique Lledó García

Rhana Hassan Zakri

Cathy Yuhong Yuan

Dr Alberto Breda

The Risk of Tumour Recurrence in Patients Undergoing Renal Transplantation for End-stage Renal Disease after Previous Treatment for a Urological Cancer: A Systematic Review

Keywords: Renal transplantation, Prostate cancer, Renal cancer, Urothelial carcinoma, Testicular cancer, Systematic review

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.

Introduction

Renal transplantation is the gold standard renal replacement therapy in end-stage renal disease (ESRD) owing to its superior survival and quality of life compared with other replacement therapies [START_REF] Laupacis | A study of the quality of life and cost-utility of renal transplantation[END_REF][START_REF] Wolfe | Comparison of mortality in all patients on dialysis, patients on dialysis awaiting transplantation, and recipients of a first cadaveric transplant[END_REF].

The standard procedure for transplantation candidates includes systematic screening for the presence of any active/latent cancer or a history of cancer [START_REF] Kälble | EAU guidelines on renal transplantation[END_REF]. In transplantation candidates with a previous history of urological cancer, it can be challenging to decide if patients are suitable for transplantation and if so how long the waiting period prior to transplantation should be. So far, the clinical decision has been mainly based on the Cincinnati Registry, which essentially considers the type of tumour and the time between its treatment and kidney transplantation [START_REF] Penn | Evaluation of transplant candidates with pre-existing malignancies[END_REF]. The waiting period varies from less than 2 years to at least 5 years according to the Registry. However, the Cincinnati Registry has several deficiencies: the treatment and the staging of the disease are not defined; the epidemiology of tumours, the diagnostic and therapeutic procedures/tests have changed and the prognostic tools have improved.

The objective of this systematic review was to appraise all available evidence on the risk of cancer recurrence in ESRD patients who underwent transplantation after having been successfully treated for a urological cancer. The primary objectives were to determine in patients with chronic kidney disease (CKD) 4/5 and a history of urological cancer, the risk of tumour recurrence after transplantation compared with renal replacement therapy (peritoneal and hemodialysis). The secondary objectives were to report on the overall and cancer-specific survival of transplanted and non-transplanted patients with a history of malignancy and to determine for each urological cancer the minimum tumour-free waiting period prior to transplantation.

Evidence acquisition

Data sources and searches

The systematic review was performed according to the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) [START_REF] Moher | Preferred reporting items for systematic reviews and meta-analyses: the PRISMA statement[END_REF]. The systematic review protocol was

Study selection

There was no restriction on types of study design. All randomized control studies (RCTs), non-randomized comparative studies (NRCS) and non-comparative studies (single-arm cohort studies, case reports) and meta-analyses published in the English language were included.

Types of participants

The study population was adults (≥ 18 years) with CKD 4/5 and previous urological cancer under renal replacement therapy or who subsequently undergo renal transplantation.

Data collection and data extraction

Following de-duplication of abstracts, two reviewers (R.B. and V.H.) screened all abstracts and full-text articles independently. Disagreement was resolved by a third party (M.B.).

References cited in all full-text articles were also assessed for additional relevant articles

Risk of bias assessment

Two reviewers (R.B. and V.H.) independently assessed the "risk of bias (RoB)" of each included study any discrepancies were revolved by a third reviewer (M.B.). The Cochrane Risk of Bias (RoB) tool was used for RoB assessment. For non-randomized studies the tool was modified to include an additional domain to assess the risk of confounding bias. A list of five important potential confounders was developed a priori with clinical content experts (European Association of Urology Renal Transplantation Guidelines Panel) [START_REF] Higgins | Cochrane handbook for systematic reviews of interventions version 5.1. 0[END_REF][START_REF] Reeves | 13 Including non-randomized studies[END_REF]. The confounders included were: type of urological cancer, baseline risk of tumour recurrence, means of diagnosis of tumour recurrence, duration of free period prior to transplantation and tumour-free status prior to transplantation. For each study, it was assessed whether each confounder was considered and whether, if necessary, the confounder was controlled in analysis. The RoB was considered to be high if the confounder had not been considered, was imbalanced between patients or not corrected for during analysis. This approach is detailed in the study protocol (ref PROSPERO).

Data synthesis

Methodological and clinical heterogeneity of the included studies meant that meta-analysis was inappropriate, therefore, a narrative synthesis of the data was performed (https://www.york.ac.uk/crd/guidance/). The primary outcome was cancer recurrence at <1-year, 1-5-year and >5-year time points and is summarized in descriptive tables.

Secondary outcomes were cancer-specific and the overall survival at <1-year, 1-5-year and >5-year time points. Possible reasons for heterogeneity were explored using the available information such as differences in the population studied, the treatment given, or the way in which the outcomes were assessed. Intended formal subgroup analysis was not possible due to the inclusion of NRCS. Therefore, any subgroup differences were discussed narratively to explore potential effect size differences. A planned sensitivity analysis to assess the robustness of our review results, by repeating the analysis only including studies with an overall medium to low risk of bias, was not possible.

Evidence synthesis

Quality of the studies

A total of 1,891 studies were identified by the literature search and two reviewers screened all study abstracts independently. Of these, 53 studies were selected for full-text screening and 20 studies were eligible for inclusion. After an additional search in the references of these 20 studies and the elimination of duplicates with the selection of the most up-to-date publication for each series, 32 studies (4 retrospective comparative studies, 21 retrospective noncomparative studies and 7 case reports) were eligible for inclusion (Figure 1). The quality of studies was assessed as described above. Risk of bias is summarized in Figure 2. Overall there was a low risk of bias across all studies. Study design was retrospective for all studies.

Characteristics of studies, population and interventions (Table 1)

In total 2,519 patients were included, suffering from: renal cell carcinoma (RCC) (n=1,810, 72%), prostate cancer (PC) (n=213, 8%), upper urinary tract carcinoma/bladder cancer (UUTUC/BC) (n=451, 18%) and testicular cancer (TC) (n=45, 2%). The time on dialysis before cancer treatment ranged from 1 month to 14 years. Among these patients with a history of urological cancer, 1,733 (69%) remained on dialysis, while 786 (31%) had a transplantation with a waiting period ranging from 0 to > 5 years.

Oncological outcomes

Renal Cell Carcinoma

Seventeen included studies evaluated patients suffering from RCC (Table 2). Mean age at diagnosis ranged from 37 to 57 years. The primary treatment was radical nephrectomy. Mean follow-up after nephrectomy was 3.6 years [ranging from 1.2-10.0 years]. The mean waiting period before transplantation ranged from 0 to 10 years. The recurrence rates for transplanted vs. dialyzed patients at < 1 year, 1-5 years and > 5 years were 0-8% vs. 0%, 0-27% vs. 0-9% and 0-41% vs. 0-48% respectively. The 5-year cancer specific survival rates for transplantation vs. dialyzed patients were 79-100% vs. 77-100%, respectively. Overall 5-year survival rates for transplantation vs. dialyzed patients were 80-100% vs. 76-100%, respectively.

Prostate cancer

Eight included studies evaluated patients suffering from PC (Table 3). Mean age at diagnosis ranged from 57 to 69 years. Treatments were prostatectomy alone (61%), prostatectomy + adjuvant radiotherapy (13%), External Beam Radiotherapy (EBRT) (10%), brachytherapy (1%) and androgen deprivation therapy (7%). The type of treatment was not specified in 8%.

Average follow-up after treatment ranged from 0.5 to 4 years. The mean interval between cancer treatment and transplantation ranged from 3 months to 4 years. Recurrence rates for transplantation patients at < 1 year and > 5 years were 0-9%, and 4-20%, respectively. The 1-5-year cancer specific survival rates for transplantation patients ranged from 96% to 100%.

Overall 1-5-year survival rates for transplantation patients ranged from 62% to 100%. No study reported results with a median follow-up beyond 5 years.

Urothelial Carcinoma : Bladder cancer and Upper Urinary Tract Urothelial

Carcinoma

Nine studies evaluated patients suffering from UC (Table 4). Mean age on diagnosis ranged from 37 to 66 years. The treatments performed for UUTUC ranged from unilateral or bilateral nephroureterectomy to complete exenteration of the urinary tract. Bladder tumours were treated with endoscopic resection or cystectomy. Mean follow-up after treatment was 1.6 to 9 years. The interval between cancer treatment and transplantation was specified in one case report: 6 years.

Five-year recurrence rates for transplantation and dialyzed patients were 4.2-29% vs. 24-57%, respectively. Five-year cancer specific survival rates and overall survival were 86-98% and 43-88%, respectively for patients who remained on dialysis. Cancer specific and overall survival rates were not reported for transplanted patients.

Testicular cancer (Table 5)

Three studies evaluated patients suffering from TC (Table 5)(4,8). The 5-year recurrence rate was 5%. The waiting period before transplantation was not defined. Histology and grade were specified only in 2 case reports: stage I seminoma and teratoma. Cancer-specific and overall survivals of 100% at 1-5 years were reported in the 2 case reports.

Risk of bias and confounding

The risk of bias and confounding was relevant, in particular in the light of the generally low level of evidence studies (Figure 2). The type of urological malignancy and the risk of recurrence were taken into account in most of the studies. The duration of the tumour-free period to transplantation and the confirmation of a tumour-free status prior to transplantation were relevant only for the studies with an intervention arm (transplantation).

Discussion

Principal findings

For RCC, the risk of recurrence was similar between transplantation and dialysis. Stage, grade, histological subtype and solid/cystic component of the tumour were the main prognostic factors for recurrence.

For PC, data were too scarce to reach a conclusion on the impact of transplantation on the risk of recurrence because the majority of the included studies were non-comparative and involved only transplanted patients. Except in the study of Woodle et al which is the last update of the Cincinnati Registry, studies included mainly PC with favorable prognosis: low stage, low grade and low recurrence rates consistent with nomograms (D'Amico, Partin, Kattan, MSKCC) (9).

For UC, the studies mainly included UUTUC in the context of aristolochic acid nephropathy.

In this specific situation, the rate of synchronous bilateral tumour was 10% to 16% and the rate of contralateral recurrence was 31% to 39%. Data on bladder UC and TC were scarce.

For TC, one case report highlighted the possibility of late recurrence (2.3 years) even for a stage I seminoma.

Findings in the context of existing evidence

Organ transplantation is a risk factor of cancer recurrence especially for viral-induced cancers.

However, this risk is modulated by the type of cancer and the type of transplant [START_REF] Viecelli | Cancer-Specific and All-Cause Mortality in Kidney Transplant Recipients With and Without Previous Cancer[END_REF].

In a large cohort of solid organ transplant recipients , Brattströmt et al. reported a 30% increased mortality risk for recipients with a history of cancer. However, this risk was moderately increased for renal transplant recipients (Hazard Ratio [HR] 1.2% 95%: 1.0-1.4, p < 0.05) and mainly concerned recipients of other organs (HR 1.8) [START_REF] Brattström | Overall and cause-specific mortality in transplant recipients with a pretransplantation cancer history[END_REF].

An active cancer is a contraindication for transplantation and a patient with (except active surveillance for prostate cancer) a history of malignancy must be in remission before transplantation. Pre-transplant evaluation includes a systematic search for sub-clinical active or latent tumours. For patients with a history of treated malignancy, the waiting period before transplantation varies from 2 to 5 years after cancer treatment. These recommendations are based on the Israel Penn International Transplant Tumour Registry (4,9).

However, these studies from ten years ago now have several shortcomings and may no longer reflect the epidemiology of patients currently seen in pre-transplantation evaluation. Several oncological data were missing: stage, histological subtype, grade, and type of treatment. For PC, the Penn et al. study essentially included T3 tumours diagnosed in the pre-prostatespecific antigen era. For RCC, the tumours were mainly large and symptomatic. For bladder tumours, the stage, grade, multifocality, type of adjuvant intravesical instillation and the distinction between non-muscle invasive and muscle invasive tumours were not reported (4).

RCC and CKD

In this systematic review, the main histological sub-type was clear cell RCC (53-79%), followed by tubulo-papillary RCC (17-50%). The tumours were mainly of low stage (78-100% of pT1), and/or low grade (51-92% grade I-II) (Table 2).

RCC diagnosed in a context of CKD may have a better prognosis for several reasons. ESRD patients are often monitored more closely than the normal population, favoring early diagnosis with smaller size and lower grade. The main risk factor of RCC is Acquired Cystic Kidney Disease (ACKD), which increases with the duration of dialysis [START_REF] Farivar-Mohseni | Renal cell carcinoma and end stage renal disease[END_REF]. RCC is more common for patients with CKD compared with the general population. The standardized incidence ratio of RCC in dialysis patients is 14 to 17 times higher than that in the general population [START_REF] Ishikawa | Acquired cystic disease of the kidney and renal cell carcinoma-complication of long-term hemodialysis[END_REF]. The systematic review data were consistent with the current knowledge on RCC in CKD patients, reporting an increased prevalence of low-grade tumours (51%-100%)

and the papillary subtype (17%-37%) Table 3) (14-16). The duration of dialysis alters the histological spectrum of tumours: clear cell RCC is the predominant subtype for patients with short dialysis duration, papillary RCC being the predominant type in patients on dialysis for more than 4 years (14).

Prostate cancer and CKD

In this systematic review, the series included were mainly cohorts of patients who were 3).

To date, there is no evidence of worse prognosis for PC in ESRDs. Although the increased prevalence of hypogonadism was suspected to induce PC of worse prognosis, there is no study validating this hypothesis [START_REF] Butler | Cancer incidence among US Medicare ESRD patients receiving hemodialysis[END_REF]. It was not demonstrated that the prognostic performance of pre-therapy (D'Amico, Partin) and post-therapy (Kahn, Kattan) nomograms is altered for CKD patients [START_REF] Mottet | EAU-ESTRO-SIOG Guidelines on Prostate Cancer. Part 1: Screening, Diagnosis, and Local Treatment with Curative Intent[END_REF][START_REF] Breyer | Prostate cancer screening and treatment in the transplant population: current status and recommendations[END_REF].

Urothelial carcinoma and CKD

In this systematic review, the most common tumour site of UC was the upper urinary tract (57% to 100% of the case series). The prevalence of a synchronous bladder tumour was 42-81% and the prevalence of a bilateral tumour was 10-16% [START_REF] Wu | The outcome of patients on dialysis with upper urinary tract transitional cell carcinoma[END_REF][START_REF] Kang | Primary urothelial carcinoma of the upper urinary tract in dialysis patients with 5-year follow-up[END_REF]. Concerning the stage, 72%

[56%-100%] of the patients were diagnosed with a non-muscle invasive tumour. All publications followed the three-stage classification according to the 1973 World Health

Organization classification with a high-grade distribution ranging from 15% to 69% (Table 4).

There were few studies on bladder cancer. Among them, tumour stage (tumour invading muscle or not) and use of adjuvant intravesical therapy were rarely mentioned.

Most studies dealing with UUTUC concerned patients with aristolochic nephropathy and seemed to have a higher risk of recurrence and multifocality than UUTUC induced by tobacco and toxic substances. ( 22)

Testicular cancer

We found no comparative study concerning TC. Testicular cancer had a low risk of recurrence according to Penn et al [START_REF] Penn | Evaluation of transplant candidates with pre-existing malignancies[END_REF]. A single clinical case indicated the possibility of recurrence for stage I seminoma (8).

Implications for practice

Considering the rates of recurrence and the prognostic factors of recurrence reported in this systematic review of the literature, it seems possible to optimize the waiting period between treatment for cancer and transplantation.

Renal Cell Carcinoma

The main prognostic factors of recurrence identified in this systematic review were: stage, tumour size, Fuhrman grade, tumour solid nature, symptomatic tumours on diagnosis, Conventional renal cell carcinoma (CRCC) subtype and lymph node involvement (Table 2).

This systematic review reported a significant risk of recurrence even for low-stage and lowgrade tumours. In Sheashaa et al. and Denton et al, in particular, where ipsilateral nephrectomy was performed systematically during transplantation, there was a risk of recurrence of up to 25% at 5 years, even though these were low-grade and low-stage infraradiological tumours [START_REF] Sheashaa | Impact of accidental discovery of renal cell carcinoma at time of renal transplantation on patient or graft survival[END_REF][START_REF] Denton | Prevalence of renal cell carcinoma in patients with ESRD pre-transplantation: a pathologic analysis[END_REF]. In these studies, contralateral involvement occurred in up to 36% of the cases. Recurrences were essentially recurrent/second contralateral RCC with an early diagnosis. Cancer-specific and overall survival rates were similar between dialyzed and transplanted patients. (Table 2).

The conclusion of this systematic review highlights the requirement for regular annual monitoring of the native kidney in CKD patients and particularly those with a history of RCC.

The reference treatment for a cancer of the native kidneys is radical nephrectomy.

Prostate cancer

In this systematic review, the main prognostic factors for recurrence were: stage, PSA and Gleason score. Radical prostatectomy is the preferred treatment both for staging and curative purposes. Lymph node resection may be best limited to one side of the pelvis to preserve the iliac vessels for the transplantation on the other side [START_REF] Tillou | Prostatectomy for localized prostate cancer to prepare for renal transplantation in end-stage renal disease patients[END_REF]. Transperineal prostatectomy could also be an option to preserve the iliac vessels [START_REF] Yiou | Perineal approach to radical prostatectomy in kidney transplant recipients with localized prostate cancer[END_REF].

The studies reported low recurrence rates consistent with prognostic nomograms. In the Tillou et al. study, the risk of biochemical recurrence at 10 years, calculated using the MSKCC nomogram did not exceed 3% in the worst case and was 1%-2% in all other case. This nomogram has proven its reliability in other studies on large cohorts [START_REF] Ryosaka | Solid-type RCC originating from native kidneys in renal transplant recipients should be monitored cautiously[END_REF]. The nomogram allowed registration of patients on the transplantation waiting list at an earlier stage whilst radical prostatectomy made the decision to put the patient on the transplant list easier. The authors reported no recurrence with a mean follow-up of 3.2 years, suggesting that there could be no waiting period for transplantation in cases of cured low-risk PC [START_REF] Tillou | Prostatectomy for localized prostate cancer to prepare for renal transplantation in end-stage renal disease patients[END_REF].

In

Urothelial carcinoma (Bladder and UUTUC)

The included studies showed that UUTUC has a high-risk of recurrence. The risk of synchronous bilateral involvement is 10%-16% and the risk of contralateral recurrence is 31%-39% (Table 4). For candidates for transplantation with a history of UUTUC, two strategies are justified:

1. Systematic treatment of the contralateral upper urinary tract and/or the bladder by nephroureterectomy and/or even cystectomy;

2. Close monitoring of the bladder and the contralateral upper urinary tract.

Testicular cancer

The level of evidence reported in this study did not allow us to conclude on the risk of testicular tumour recurrence after renal transplantation. The 2 case-reports highlighted the risk of retroperitoneal recurrence after transplantation, even with a long waiting period (Table 5).

A case-by-case discussion is needed to decide on a period of exclusion before transplantation.

Implications for research

Although the findings of this systematic review enabled us to identify the prognostic factors of recurrence and to conclude that immunosuppression did not modify the natural history of urological cancer for selected patients, the literature in this particular area differed according to the type of cancer. With the exception of Tillou et al. (19 patients), we found no comparative study for PC [START_REF] Tillou | Prostatectomy for localized prostate cancer to prepare for renal transplantation in end-stage renal disease patients[END_REF]. There were few studies on bladder cancer, with no data concerning the stage (tumour invading the vesical muscle or not) and the use of adjuvant intravesical therapy.

In practice, it appears to be difficult to evaluate the excess risk of recurrence represented by the initiation of immunosuppressive therapy and thus the possible deleterious impact of transplantation on recurrence and cancer-specific survival. Randomized controlled trials will ethically and logistically be difficult to conduct. However, well-designed prospective cohort studies with homogenous type/stage of cancer and clear predefined oncological outcomes at different time points are needed to strongly support a reduction of the waiting period for low risk RCC and PC, which was a tendency suggested in this systematic review.

Limitations of this study

This report is the first systematic review assessing and appraising all available evidence of the risk of cancer recurrence for CKD patients on dialysis or who have undergone transplantation.

Limitations mainly consist in the low level of the references:

• The included studies are all retrospective and most of them are not comparative. As such, the risk of bias and/or confounding is high in most studies.

• Long-term follow-up is lacking. Several data were not systematically reported: prognostic score or monograms, duration and type of immunosuppressive treatment, type of recurrence (local or systemic).For PC, the oncological results of nontransplanted patients were based on only 6 patients in 2 studies.

• For urothelial tumours, the majority of studies included UUTUC occurring in the particular field of aristolochic acid nephropathy.

Conclusions

Although this systematic review summarized all the available evidence on renal transplantation and history of cancer, it was limited by the level of evidence of the included studies, which mainly consisted of non-comparative retrospective cohort studies of preselected patients. Acknowledging that the comparison is not free from bias, this systematic review indicates that immunosuppression does not seem to alter the natural history of recurrence and mortality for low-risk renal and prostate cancer, and could lead to a shortening of the waiting period in this specific situation

  registered with PROSPERO (http://www.crd.york.ac.uk/PROSPERO/display_record.asp?ID=CRD42016046867). Studies (January 1, 1995, to March 1, 2016) were identified by highly sensitive searches of electronic databases (Medline, Embase, Cochrane library databases). The initial literature search was performed in February 2016 and an updated search performed in March 2017. The search was complimented by additional sources including the reference lists of included studies. The search strategy is described in detail in Appendix 1.

  transplanted after their PC treatment (111 transplanted patients vs. 6 patients who remained on dialysis) with mainly localized PC of excellent prognosis (D'Amico low to intermediate risk or stage ≤II). By contrast, in 2005 the report of Woodle et al. included 12% with stage III PC. In 2001 Chapman et al. did not report the histology and prognosis of the patients included (9) (Table

  their study, Woodle et al. (which was an update of the PC series published by Penn et al. in 1995 and 1997) included a significant number of high-risk extraprostatic tumours. Despite its shortcomings, this study is still unequaled for the study of high-risk prostate cancer before transplantation (11).

  . For high-risk renal and prostate cancer, the historical Cincinnati registry is still an unequalled source of data. Although this registry study suffers from many shortcomings, waiting period should not be modified for high-risk renal and prostate cancers. Urothelial carcinomas are multifocal and highly recurrent tumours. For transplantation candidates, close follow-up or systematic bilateral nephroureterectomy are both possible strategies. For patients with a history of TC a case-by-case discussion is recommended. Full search strategy. 1 exp kidney transplantation/ 2 ((renal or kidney) adj3 (transplant* or allotransplant* or grafting homotransplant* or retransplant*)).tw,kw. 3 1 or 2 4 ((urological or urinary tract or prostate or prostatic or urothel* or ureter* or transitional cell or kidney* or renal or bladder or penis or penile or testicular or testis or testes) adj3 (cancer* or carcin* or malig* or tumour* or tumour* or neoplas*)).tw,kw. 5 exp transitional cell carcinoma/ or exp Carcinoma, Transitional Cell/ 6 exp Ureteral Neoplasms/ or exp ureter tumour/ 7 exp Kidney Neoplasms/ or exp kidney tumour/ 8 exp bladder tumour/ or exp Urinary Bladder Neoplasms/ 9 exp Prostatic Neoplasms/ or exp prostate tumour/ 10 exp penis tumour/ or exp Penile Neoplasms/ 11 exp testis tumour/ or exp Testicular Neoplasms/ 12 or/4-11 13 3 and 12 14 exp chronic kidney disease/ or exp Renal Insufficiency, Chronic/ 15 (chronic and (kidney or renal) and (end stage* or (stage* and ("4/5" or "4" or "5" or "4-5")))).tw,kw. 16 (Chronic and (kidney or renal) and (disease* or disorder* or dysfunction* or insufficien* or failure*)).tw,kw. 17 exp renal replacement therapy/ or exp Renal Dialysis/ 18 ((Renal or kidney) adj3 (dialysis or dialyses or replacement or support)).tw,kw. 19 (hemodialys?s or extracorporeal dialys?s).tw,kw. 20 or/14-19 21 13 and 20 22 ((exp animals/ or exp animal/ or exp nonhuman/ or exp animal experiment/ or animal model/ or animal tissue/ or non human/) not (humans/ or human/)) or ((rats or mice or mouse or cats or dogs or animal* or cell lines) not (human* or men or women)).ti. 23 21 not 22 24 ((child/ or Pediatrics/ or Adolescent/ or Infant/ or adolescence/ or newborn/) not (adult/ or aged/)) or ((baby or babies or child or children or pediatric* or paediatric* or peadiatric* or infant* or infancy or neonat* or newborn* or new born* or kid or kids or adolescen* or preschool or pre-school or toddler*) not (aged or adult* or elder* or senior or men or women)).ti.

	Appendix 1: 25 23 not 24
	26	limit 25 to yr="1995 -Current"
	27	remove duplicates from 26

Acknowledgments: The valuable help of European Association of Urology Guidelines

Office, Arnhem, Netherlands in providing the full text articles and administrative support is gratefully acknowledged. Furthermore, we thank Thomas Lamand Cathy Yuhong Yuan of the EAU Guidelines Office Methods Committee, for providing methodological support

Funding/Support:

This study was supported by the European Association of Urology.

Role of the sponsor:

The European Association of Urology had no role in the design and conduct of the study, collection, management, analysis, and interpretation of the data, preparation, review, or approval of the manuscript nor the decision to submit the manuscript for publication.

Conflict of interest disclosures:

None related to the present work