Paul Clavin

Finite-time singularity associated with the deflagration-to-detonation transition on the tip of an elongated flame-front in a tube

Keywords: Deflagration-to-detonation transition, Laminar flow, Finite-time singularity

The critical condition for the deflagration to detonation transition (DDT) on the tip of elongated flames in tubes is revisited. Outstanding experiments and numerics, performed in 2010, have shown that a train of successive shock waves are produced by the self-accelerating flame-front.

Just prior to the DDT, a pre-conditioned state is thus generated in a slice of unreacted gas adjacent to the flame front with a temperature too low for self-ignition, leaving unexplained the abrupt transition. Using a simplified one-dimensional model, the objective of the theoretical analysis of this short note is to identify the pre-conditioned state just prior to the spontaneous formation of a flow singularity on the flame front. This finite-time singularity responsible for the DDT is produced by the least further increase in propagation velocity beyond a critical value. The attention is focused on the unsteady compression waves emitted in the unreacted gas by the self-accelerating flame front. The unsteady effects were overlooked by the previous theoretical analyses.

Introduction

Various mechanisms of the deflagration-to-detonation transition (DDT) in gaseous mixtures are mentioned in the modern literature. The attention will be limited here to a tentative explanation of the DDT on the tip of a self-accelerating curved flame propagating in a radius tube larger than the laminar flame thickness but small enough for the flow to be laminar. The real problem is multidimensional. However the DDT mechanism can be enlightened in plane geometry with a flame treated as a discontinuity pushed from behind by a flow of burned gas perpendicular to the flame front [START_REF] Clavin | Formation of the preheated zone ahead of a propagating flame and the mechanisms underlying the deflagration-to-detonation transition[END_REF] [START_REF] Clavin | Asymptotic solution of two fundamental problems in gaseous detonations[END_REF]. In the real process, this burned gas flow, called back flow in the following, is generated by the lateral wings of the elongated flame and is thus proportional to the elongation of the curved front. The attention is focused on a small rate of elongation, smaller than the inverse of the transit time across the laminar flame speed. In one centimeter radius tubes, this occurs typically some time after the formation of the tulip-shaped flame. Extending the pioneering analysis of [START_REF] Deshaies | Flame-speed sensitivity to temperature changes and the Deflagration-to-Detonation Transition[END_REF] [START_REF] Deshaies | Flame-speed sensitivity to temperature changes and the Deflagration-to-Detonation Transition[END_REF] (DJ), strong arguments have been recently provided in [START_REF] Clavin | Formation of the preheated zone ahead of a propagating flame and the mechanisms underlying the deflagration-to-detonation transition[END_REF] and [START_REF] Clavin | Asymptotic solution of two fundamental problems in gaseous detonations[END_REF] indicating that the abrupt transition (1µs) observed in the outstanding 2010 experiments and numerics [START_REF] Kuznetsov | Experimental study of the preheated zone formation and deflagration to detonation transition[END_REF] [START_REF] Liberman | Deflagrationto-detonation transition in high reactive combustible mixtures[END_REF] could correspond to a critical condition similar to the one identified long ago by DJ for the self-similar solutions. The DJ analysis describes a thermal feed back which can be summarized as follows; the laminar flame speed is an increasing function of the temperature which is controlled by the lead shock, the strength of which increases with the flame speed. Considering a plane turbulent flame in the wrinkled regime with a propagation velocity larger than the laminar flame speed by a wrinkling factor σ > 1, DJ showed that, because of a nonlinear effect associated with the high thermal sensitivity of the laminar flame speed, the self-similar solutions present a turning point for a critical value of σ about 10, beyond which no solution exists. Such a turning point exists also for the laminar flame at the tip of the curved front for a critical elongation [START_REF] Clavin | Formation of the preheated zone ahead of a propagating flame and the mechanisms underlying the deflagration-to-detonation transition[END_REF]. Moreover a finite-time singularity of the flow was shown to develop on the flame when the increasing elongation reaches the critical value [START_REF] Clavin | Formation of the preheated zone ahead of a propagating flame and the mechanisms underlying the deflagration-to-detonation transition[END_REF] [START_REF] Clavin | Asymptotic solution of two fundamental problems in gaseous detonations[END_REF]. Unfortunately, these analyses are based on self-similar solutions that are quasi-static in the sense that the flow is homogeneous and steady between the flame and the lead shock, each of them propagating with a constant velocity, subsonic and supersonic respectively. A key element of the sudden transition is overlooked by the selfsimilar solutions, namely the unsteadiness and non-homogeneity of the compression waves that are generated by a self-accelerating flame front. The purpose of this short note is precisely to take into account the strong unsteady effects of the compression waves emitted by the self-accelerating flame front. The objective is to find out a generic expression for the critical propagation velocity of the tip of the curved flame front at which the DDT is produced quasi-spontaneously by the least further increase on flame elongation.

As mentioned by Liberman and co-workers [START_REF] Kuznetsov | Experimental study of the preheated zone formation and deflagration to detonation transition[END_REF] [5], a train of successive shockwaves is issued from the self-accelerating flame and generates a critical pre-conditioned state for the DDT. The attention is focused here on the theoretical identification of this pre-conditioned state just prior to the transition. Focusing the attention on the unsteady process in the one-dimensional flow of unreacted gas, a finite-time singularity of the solution of the Euler's equations is shown to occur systematically on the self-accelerating flame front. This divergence of the flow gradient on the flame front is expected to be the basic explanation of the DDT observed in experiments and numerics [START_REF] Kuznetsov | Experimental study of the preheated zone formation and deflagration to detonation transition[END_REF] [5].

One-dimensional model

The model is the same as in [START_REF] Clavin | Formation of the preheated zone ahead of a propagating flame and the mechanisms underlying the deflagration-to-detonation transition[END_REF] [START_REF] Clavin | Asymptotic solution of two fundamental problems in gaseous detonations[END_REF]. The flame acts as a semi-transparent piston with a velocity U P (t) proportional to the laminar flame velocity U b (T u ) relative to the burned gas for the same temperature of unburned gas T u (t) as on the tip of the elongated flame. Assuming an internal structure of the flame in quasi-steady state, the piston velocity in the laboratory frame is, according to the mass conservation, U P = u b + U b = u u + U L where u b is the back flow of burned gas, u u is the flow of unburned gas and U L is the laminar flame speed relative to the unreacted gas. The lateral skirt of the flame front being quasi-parallel to the lateral wall of the tube, a radial flow of burned gas is emitted with a velocity U b from the lateral wall. Following [START_REF] Clanet | On the tulip flame phenomenon[END_REF], the mass conservation in the turning flow of burned gas leads to a longitudinal back flow u b impinging on the flame tip from behind, taking the form u b ≈ [2L(t)/R]U b (T u ). The elongation of the curved front 2L(t)/R > 1 increasing like the length of the flame skirt L(t) (R denotes the radius of the tube), the piston velocity U P takes the form

u b = [2L(t)/R]U b (T u ) ⇒ U P (t) = S (t)U b T u (t) , S (t) ≡ 1 + 2L(t)/R ≈ 2L(t)/R, ( 1
)
where S is the elongation parameter. The flow velocity of the fresh mixture just ahead of the flame is u u = U P -(T u /T b )U b where T u is the local temperature of the unburned gas and T b = T u +q/c p the flame temperature, q and c p being respectively the chemical energy and the specific heat per unit mass, U L = (T u /T b )U b . If the elongation parameter is sufficiently larger than unity, the unreacted gas flow on the piston is approximately equal to the piston velocity and the piston acts like an impermeable piston,

S (t) 1 : u u ≈ U P , U P (t) = S (t)U b T u (t) , S (t) ≈ 2L(t)/R. ( 2
)
This approximation simplifies the algebra and can easily be removed without modifying qualitatively the results. An impermeable piston is the simplest one-dimensional model for pointing out a fundamental mechanism of DDT on the tip of elongated flames.

A second ingredient is that the laminar flame speed is an increasing function of the temperature, characterized by a parameter of thermal sensitivity

1 b > 1, b ≡ T u U b (T u ) dU b dT u > 1, (3)
with, typically b 2 for energetic mixtures [START_REF] Clavin | Formation of the preheated zone ahead of a propagating flame and the mechanisms underlying the deflagration-to-detonation transition[END_REF]. For simplicity, we will use the exponential form associated with an Arrhenius law in the limit of large activation energy,

U b T u (t) U b T u (0) = exp b T u (t) T u (0) -1 . (4)
The analysis can be performed with a temperature dependence different from (4) to the price of a more complex algebra. The essential ingredient is a function U b (T u ) increasing strongly enough with the temperature. The detailed temperature-dependence is not important and the exponential form ( 4) is convenient for an analytically description.

Analysis of the critical dynamics

Preliminary considerations

Starting from a small elongation of the finger flame, a train of successive shock waves is created in the unburned mixture at finite distance from the piston (flame front). Just before the abrupt transition observed in [START_REF] Kuznetsov | Experimental study of the preheated zone formation and deflagration to detonation transition[END_REF] [5], the rate of increase of U P is not larger than the rate of increase of the elongation 1/t evol which is typically smaller than the inverse of the transit time of a fluid particle across the flame. Moreover, the gradient of the unreacted flow is small near the piston in the fresh gas between the flame front and the last shock wave formed ahead of the flame. Therefore the flow velocity near the piston is nearly equal to the piston velocity and the temperature is quasiuniform. However, this temperature depends on the past history of the piston acceleration. When a train of successive shock waves is produced, the temperature on the piston is not the same as in a self-similar solution characterized by a single lead shock propagating in the unburned gas at rest. This is a consequence of the nonlinearity of the Rankine-Hugoniot relations. What can be said about the critical state adjacent to the flame just prior to the transition ? This is the first question addressed in this note, the second one concerns the sharp flow gradient occurring suddenly on the flame front.

Formulation

Starting at time t = 0 after the last formation of a shock at finite distance from the flame, the flow in a slice of the fresh mixture adjacent to the piston is quasi-uniform nearby the flame front and equal to the piston velocity u(x, t = 0) = U P (0) with a gas temperature also quasiuniform, T ≈ T P (0). The subscript u has been removed for saving the notation. This situation is similar locally to a self-similar solution but the thermodynamic state of the unburned gas adjacent to the flame depends now on the past history of the piston acceleration. Compression waves are then launched from the self-accelerating piston. Before the formation of a new shock on the leading edge of the compression wave, the dissipative processes (heat conduction and viscosity) are negligible so that the compression waves are quasi-isentropic. More precisely, the fluctuation of entropy generated by the shock are assumed small without noticeable effect on the state near the piston. This is the case before the leading edge of the compression wave reaches the last formed shock. According to the classical theory of simple compression waves in a perfect gas (Riemannn 1860), the temperature of the gas is related to the unsteady flow by a quadratic relation,

T T (0) = 1 + 1 2 (γ -1) u -U P (0) a(0) 2 . ( 5
)
This relation can be linearized as long as the flow velocity varies by an amount smaller than the speed of sound 0 u -U P (0 a(0)

1 : T T (0) -1 ≈ (γ -1) u -U P (0) a(0) . (6)
Using ( 6) on the piston, a relation between the temperature of the gas on the piston and the piston velocity is obtained, on the piston: u = U P and T T (0)

-1 = (γ -1) U P (t) -U P (0) a(0) . ( 7
)
According to ( 2) and ( 4), the chemical kinetics provides us with an additional relation between T

and U P which involves the thermal sensitivity of the laminar flame speed b, defined in (3),

U P -U P (0) U P (0) = S S (0) exp b T T (0) -1 -1. (8)
A key point is that the variation of the laminar flame speed U b (T ) with the temperature in ( 4) is stronger than the temperature dependence (5) of the isentropic flow in a compression wave.

From an asymptotic point of view, the association of a strongly nonlinear relation [START_REF] Kagan | On the transition of Deflagration to Detonation in narrow tubes[END_REF] with a linear approximation ( 7) is consistent for a large thermal sensitivity b 1 and a small propagation Mach number of the laminar flame U b /a 1, in the distinguished limit lim˜b →∞ b(γ -1)U b (0)/a(0) = O(1).

Critical elongation and pre-conditioned state

Introducing the notation

Θ P ≡ b [T P /T (0) -1] = O(1)
for the temperature T P on the piston and the parameter B ≡ b (γ -1)U b (0)/a(0) < 1 depending on both thermal sensitivity and initial state, equations ( 7)-( 8) lead to a nonlinear relation between the gas temperature on the piston Θ P and the elongation parameter S ,

Θ P (t) = BS (0) S (t) S (0) exp Θ P (t) -1 with S (0) S max (0) where S max (0) ≡ 1 B > 1, (9)
the last expression is discussed few lines below.

A first outcome of ( 9) is that the temperature on the piston depends on the time only through the elongation S (t). The parameter B ≡ b (γ -1)U b (0)/a(0) = O(1) of a unity order in the limit b → ∞ is smaller than unity (B < 1) because the Mach number of the laminar flame speed U b (0)/a(0) of a very energetic mixture is typically ten times smaller than the inverse of the parameter b (γ -1) controlling the thermal sensitivity, (γ -1) bU b /a ≈ 10 -1 . The nonlinear relation linking the temperature and the elongation can be obtained numerically using an expression of the laminar flame speed more general than (4). For example, this was done in [START_REF] Clavin | Formation of the preheated zone ahead of a propagating flame and the mechanisms underlying the deflagration-to-detonation transition[END_REF] for the self-similar problem.

The simplicity of the nonlinear equation ( 9) allows a simple analytical description leading to an accurate order of magnitude of the solution.

A second outcome of ( 9) is the existence of an upper bound of the initial elongation S (0)

S max (0) = 1/B. For an initial elongation S (0) S max (0), the flame temperature and the flame velocity increases with the time up to a critical value S * , as it should be on a physical branch of solutions. The solution of [START_REF] Wu | Yetter Flame acceleration and the transition to detonation of stoichiometric ethylee/oxygen in microscale tubes[END_REF] shows effectively the existence of a critical elongation S * larger than its initial value but smaller than S max (0), S (0) S * S max (0),

S * S max (0) = exp S (0) S max (0) -1 1 where S max (0) = 1 b (γ -1)U b (0)/a(0) (10) 0 < S (0) S max (0) 1 ⇒ S (0) S * S max (0), 1 e < S * S max (0) 1 (11)
and

Θ * P ≡ b T * P T (0) -1 = 1 - S (0) S max (0) 0, (12)
T * P is the critical gas temperature on the piston, T * P = T P (S * ). The critical elongation S * in (10) corresponds to a turning point on the C-shaped curve (9) "temperature versus elongation" Θ P (S ), reminiscent of the critical wrinkling factor in the pioneering analysis [START_REF] Deshaies | Flame-speed sensitivity to temperature changes and the Deflagration-to-Detonation Transition[END_REF] above which self-similar solutions no longer exist. Here, according to (9), isentropic compression waves can no longer be launched from the self-accelerating flame if the elongation is larger than its critical value (S > S * ) since the boundary condition on the piston can no longer be satisfied. Before discussing in more detail the C-shaped curve T P (S ) in [START_REF] Wu | Yetter Flame acceleration and the transition to detonation of stoichiometric ethylee/oxygen in microscale tubes[END_REF], it is worth mentioning that a singularity of the flow will be shown in § 3.4 to be formed on the flame front when S reaches S * , enlightening the underlying dynamical nature of the critical condition near the turning point S = S * .

A single parameter characterizing the unburned gas at rest ahead of the lead shock was involved in the C-shaped curve of the self-similar solutions, see [START_REF] Deshaies | Flame-speed sensitivity to temperature changes and the Deflagration-to-Detonation Transition[END_REF] and (19) in [START_REF] Clavin | Formation of the preheated zone ahead of a propagating flame and the mechanisms underlying the deflagration-to-detonation transition[END_REF], so that the critical condition was the same for any initial point on the C-shaped curve. In contrast to the self-similar solutions, the C-shaped curve T P (S ) in ( 9) and also the critical elongation S * in (10) both depend on the initial condition S (0) pointing out the limited interest of the self-similar solutions. However, for an initial elongation S (0) closer and closer to its maximum value S max (0), the critical elongation S * approaches quickly S max (0) from below, as shown by an expansion of S (0) around S max (0) in [START_REF] Wu | Reaction propagation modes in millimeter-scale tubes for ethylene/oxygen mixtures[END_REF]

0 < S max (0) -S (0) S max (0) 1 ⇒ S max (0) -S * S max (0) ≈ S max (0) -S (0) S max (0) 2 1. ( 13
)
This limiting case S * ≈ S max (0) for S (0) ≈ S max (0) leads to a universal critical Mach number b /a * u , the greater the critical elongation L * /R. The result (14) differs from the self-similar solution recalled in § 3.2 of [START_REF] Clavin | Formation of the preheated zone ahead of a propagating flame and the mechanisms underlying the deflagration-to-detonation transition[END_REF] by the fact that the critical condition is related to the state of the fresh gas adjacent to the flame front and not to the state of the initial mixture at rest in front of the lead shock. The thermodynamic state of this pre-conditioned state is not fully determined by the first equation ( 14) relating the elongation to the flame temperature through [START_REF] Kuznetsov | Experimental study of the preheated zone formation and deflagration to detonation transition[END_REF]. The critical state depends on the past history of the self-accelerating flame. A self-similar solution is relevant to describe the DDT only if the flame speed is quasi-constant from the initial condition so that no secondary shocks are formed. This is not the case in [START_REF] Kuznetsov | Experimental study of the preheated zone formation and deflagration to detonation transition[END_REF] [START_REF] Liberman | Deflagrationto-detonation transition in high reactive combustible mixtures[END_REF] showing the formation of a train of successive shock waves near the flame front before the DDT on the flame tip. Most of the time this occurs in tubes during a phase of reacceleration after a transitory stage of deceleration associated with the formation of a tulip-shaped flame. Numerical simulations have also observed transition in the fold of a tulip flame [7] [8]. This multidimensional phenomenon is not considered in this short note in which the attention is focused on the DDT occurring before or after the formation of a tulip-shaped flame. In any case, no tulip flame is formed in very narrow tubes in which the DDT was nevertheless observed [START_REF] Wu | Yetter Flame acceleration and the transition to detonation of stoichiometric ethylee/oxygen in microscale tubes[END_REF] [10].

U * P /
If the initial elongation S (0) is not close enough to its upper bound S max (0), more precisely if the time delay for reaching the critical condition is longer than the time delay for a shock wave to be created on the leading edge of the isentropic rarefaction wave, the condition on the piston will not correspond well to the critical condition of spontaneous transition. Consequently, the pre-conditioned state of the slice of unburned mixture adjacent to the flame front just prior to the transition [START_REF] Kuznetsov | Experimental study of the preheated zone formation and deflagration to detonation transition[END_REF] [START_REF] Liberman | Deflagrationto-detonation transition in high reactive combustible mixtures[END_REF], should correspond roughly to (14). In any case, according to (14), the Mach number U * P /a * u cannot be larger than 1/[(γ -1) b]. This confirms a conclusion obtained previously with the self-similar solution: the DDT is produced for a flame velocity much smaller than the one which would be required for self-igniting the reactive gaseous mixture ahead of the flame front. According to (14), the critical flow velocity corresponds to a Mach number of the shock propagating in the gas at rest smaller than 2.5, which is effectively too small for self-ignition of the compressed reactive gaseous mixture, in full agreement with the experiments.

The solution of the unsteady flow presented in the following subsection enlightens the DDT mechanism by exhibiting the singular character of the underlying dynamics of the flow near the turning point of the C-shaped curve [START_REF] Clavin | Asymptotic solution of two fundamental problems in gaseous detonations[END_REF].

Finite-time singularity of the solution of the Euler's equation

If S (t) is a prescribed function increasing with the time without bound, a singularity of the solution of the Euler's equations appears spontaneously on the flame front at the critical time t * , S (t * ) = S * . This is why isentropic compression waves can no longer be launched from the flame front for t > t * (S (t) > S * ). Such a scenario was mentioned in [START_REF] Clavin | Formation of the preheated zone ahead of a propagating flame and the mechanisms underlying the deflagration-to-detonation transition[END_REF] within the context of self-similar solutions, hoping that the inconsistency of the static nature of the solutions could be overcome in some way, see the foot note 3 p. 9 in [START_REF] Clavin | Formation of the preheated zone ahead of a propagating flame and the mechanisms underlying the deflagration-to-detonation transition[END_REF]. The proper way is to solve the fully unsteady problem by considering the compression waves, instead of the self-similar solutions.

The finite-time singularity of the isentropic flow generated by a self-accelerating piston reaching a finite velocity in a finite time with a divergence of the acceleration is predicted by the Cshaped curve [START_REF] Wu | Reaction propagation modes in millimeter-scale tubes for ethylene/oxygen mixtures[END_REF] and takes a universal form near the critical condition [START_REF] Clavin | Asymptotic solution of two fundamental problems in gaseous detonations[END_REF], regardless of the context. The analysis is briefly outlined below for making the present short note self-contained.

Additional comments are also included. Only the main steps are recalled; the reader is referred to [START_REF] Clavin | Asymptotic solution of two fundamental problems in gaseous detonations[END_REF] for the technical details. The critical dynamics is solved by the Riemann's method, taking full advantage of the quadratic relation S (U P ) ∝ (1 -U P /U * P )2 near the critical point (maximum of the function S (U P ) in ( 8)-( 9)). Following Riemann, the equation for the flow velocity of a simple compression wave satisfying the downstream boundary condition lim x→∞ u = U P (0) reads

∂ [u -U P (0)] ∂t + γ + 1 2 [u -U P (0)] + [a(0) + U P (0)] ∂ [u -U P (0)] ∂x = 0. ( 15
)
The solution u(x, t) can be written in the form 2 ,

x = γ + 1 2 [u -U P (0)] + [a(0) + U P (0)] t + f (u) (16)
where the function f (u) is determined by the boundary condition on the piston, x = X P (t) = t 0 U P (t)dt : u = U P (t). Thanks to the quadratic form S (U P ) ∝ (1 -U P /U * P ) 2 , a generic law for the piston velocity is easily obtained [ 17) is the same for any value of the critical velocity U * P and any growth rate of the elongation, as small as it could be. Coming back to the flow field, the boundary condition on the piston x = X P (t) = t 0 U P (t)dt : u = U P (t) then shows that the function f (u) is a cubic polynomial, see (67)-(68) in [START_REF] Clavin | Asymptotic solution of two fundamental problems in gaseous detonations[END_REF], so that, according to (16), the problem reduces to solve a third-degree polynomial in (1u/U * P ), see ( 74)-(75) in [START_REF] Clavin | Asymptotic solution of two fundamental problems in gaseous detonations[END_REF]. Close to the critical velocity and for (1u/U * P ) a * u /U * P , the cubic term is negligible and the problem is reduced to solve a second-order algebraic equation. Moreover this equation further simplifies for (t *t)/t * (a * u /U * P ) 2 yielding the following analytical expression of the flow field u(x, t)

U * P = 1 - (t * -t) t * evol + [x -X P (t)] a(0)t * evol . ( 18
)
The condition for the validity of ( 18) is less drastic than in [START_REF] Clavin | Asymptotic solution of two fundamental problems in gaseous detonations[END_REF]. More particularly, (18) is not limited to a small Mach number of the piston U * P /a * relative to the gas flow x wd (t) -X P (t) = a(0)t, as it is checked by using (17). As already mentioned, close to the critical point, the initial sound speed a(0) can be replaced by a * u . According to (14) and (18), the gradient of the flow

t * evol ∂u(x, t) ∂x = - U * P a * u 1/2 (t * -t) t * evol + [x-X P (t)] a * u t * evol where U * P a * u = 1 (γ -1) b ( 19
)
presents a finite-time singularity on the piston at the critical time t * which, according to (17), corresponds to a divergence of the acceleration of the piston dU P /dt ∝ 1/ (t *t)/t * evol when the piston velocity reaches the critical value U * P . The flow (18) is indeed solution of the linear version of (15) so that the wave-breaking mechanism which is responsible for the formation of shock waves, is not involved in the genesis of the finite-time singularity of the flow on the flame front.

The singularity (19) comes from the quadratic form S (U P ) ∝ (1 -U P /U * P ) 2 near the turning point S = S * on the curve U P (S ).

Could this finite-time singularity of the gradient of the flow on the flame front (19) be responsible for the formation of a shock wave inside the flame structure leading to a quasi-instantaneous DDT? This is likely because the temperature inside the flame structure is higher than in the unreacted gas ahead of the flame where the temperature is not large enough for self-igniting the unreacted gas in the compressed gas behind the shock waves emitted by the self-accelerating flame front. This problem requires to investigate the unsteady flame structure. In that respect, the ZFK flame model with a back flow u b ≈ [2L(t)/R]U b (T u ) applied at the reaction sheet is worth investigating by solving analytically (or numerically) a free boundary problem yielding the instantaneous velocity of the reaction sheet U P (t) in the form of an eigenvalue. A finite-time singularity of the flow gradient, associated with the singularity of the acceleration dU P (t)/dt at a critical value U * P = U P (t * ), is expected to occur on the reaction sheet. The analysis is beyond the scope of the present note. The achievement of such a difficult analysis is the price to pay for a full understanding of a fundamental problem, namely the sudden transition of a subsonic reaction-diffusion wave (the flame) into a supersonic wave (the detonation).

Discussion of the results

The results ( 14) and ( 18)-(19) predict the characteristics of the pre-conditioned state of the slab of fresh mixture adjacent to the self-accelerating flame, observed just prior to DDT in a laminar regime [START_REF] Kuznetsov | Experimental study of the preheated zone formation and deflagration to detonation transition[END_REF]

Declaration of Competing Interset

The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

  a * u of the piston velocity (i.e. the critical velocity of the tip of the elongated flame) depending only on the reactive mixture through the thermal sensitivity b of the laminar flame speed, P /a * u ≈ 1.25 for b ≈ 2, in good agreement with the experiments [4] [5]. In the last relation in (14) yielding the critical value of the piston velocity U * P , the temperature of the unburned gas just ahead of the flame occurs only through the speed of sound which increases weakly with the temperature like √ T . Therefore, according to (2) and (14), the upper bound of the piston velocity (velocity of the flame tip) U * P = 2(L * /R)U * b = a * u /[(γ -1) b] varies as the inverse of the thermal sensitivity b with a coefficient of proportionality which does not change much. The smaller the Mach number U *

u 1 which

 1 , according to (14), would require a too large value of b 1, larger than in very energetic mixtures. The flow field (18) satisfies the boundary condition (17) on the piston, as it should be. A weak discontinuity is exhibited at the leading edge x = x wd (t) where the initial flow, namely the initial velocity of the piston, is recovered, u(x, t)| x=x wd (t) = U P (0). The point x = x wd (t) propagates effectively at the speed of sound

  [START_REF] Liberman | Deflagrationto-detonation transition in high reactive combustible mixtures[END_REF]. The finite-time singularity of the solution of the Euler's equation in the laminar flow of unreacted gas ahead of the flame front could be the key of the DDT. The solution of the unsteady flame structure remains to be carried out in order to confirm this statement. The analysis is in process.The result (14) explains why the DDT in the laminar regime can be observed only with strongly energetic mixtures, never with gaseous mixtures in air, as discussed now. The critical length in the first equation (14) is reached only if the Mach number of the laminar flame velocity U b /a u is large enough to make the critical elongation L/R accessible. The elongation of curved flames in tubes is due to hydrodynamical mechanisms. It is typically not larger than 10 in centimeter radius tubes. According to (14) with γ = 1.4 and b ≈ 2, a critical elongation L * /R about 10 requires U b /a u not smaller than 0.06. Such a fast flame occurs in stoichiometric mixtures of hydrogen or acetylene in pure oxygen whose laminar flame speed U L (relative to the fresh mixture gas) and gas expansion parameter T b /T u are respectively close to 10 m/s and 10. In ordinary gaseous combustible mixtures in air the Mach umber U b /a u is between ten to a hundred times smaller. On the opposite case of highly energetic mixtures, the critical ratio L * /R can be small enough to concern the cellular structure of self-accelerating flames propagating freely in open space under the Rayleigh-Taylor instability. This could be the case in astrophysics for flames sustained by nuclear reactions. Another interesting result in (14) is that the critical Mach number U * P /a * u (measured in the laboratory frame) of the flame pushed from behind by the back flow depends only on the thermal sensitivity of the laminar flame speed (3).

  * evol is the gtowth rate of the elongation at the critical time t * . The law (

	where 1/t
				1]
	0	(t * -t) t * evol	1 :	U * p -U P (t) U * P	=	t * -t t * evol	,	U * p -U P (0) U * P	=	t * evol t *	(17)

The thermal sensitivity of the laminar flame velocity of energetic mixtures is mainly due to the pre-factor of the Arrhenius law, as explained in[START_REF] Clavin | Formation of the preheated zone ahead of a propagating flame and the mechanisms underlying the deflagration-to-detonation transition[END_REF]

Equation (15) is called in the modern literature "Burger's equation" even though it was mentioned a century before; the form (16) was anticipated by Poisson (1808), see p.118 of the textbook[START_REF] Courant | Supersonic flow and shock waves[END_REF]. Its derivation from the Euler's equations can be found p.

of[START_REF] Landau | Fluid Mechanics[END_REF]

Acknowledgments

Enlightening discussions on the finite-time singularities with Yves Pomeau, Christophe Josserand and Sergio Rica during a 2021 stay at "la fondation des Treilles" of Tourtour (France) are greatly acknowledged. The work in this note is partially supported by the French agency, Agence Nationale de la Recherche, under the project 18-CE05-0030 'ReDDT' (Revisiting Deflagration to Detonation Transition in the context of carbon-free energy production). Lively interactions with PhD students Hassan Tofaili at CORIA in Rouen and Raúl Hernández Sánchez at IRPHE in Marseilles, both involved in the numerical aspect of this project, are acknowledged.


I A Milne

O Kimmoun

J M R Graham

B Molin

An experimental and numerical study of the resonant flow between a hull and a wall

Keywords: Gap resonance, vortex shedding, hydrodynamic damping

The wave-induced resonant flow in a narrow gap between a stationary hull and a vertical wall is studied experimentally and numerically. Vortex shedding from the sharp bilge edge of the hull gives rise to a quadratically-damped free surface response in the gap, where the damping coefficient is approximately independent of wave steepness and frequency. Particle image velocimetry (PIV) and direct numerical simulations (DNS) were used to characterise the shedding dynamics and explore the influence of discretisation in the measurements and computations. Secondary separation was identified as a particular feature which occurred at the hull bilge in these gap flows. This can result in the generation of a system with multiple vortical regions and asymmetries between the inflow and outflow. The shedding dynamics were found to exhibit a high degree of invariance to the amplitude in the gap and the spanwise position of the barge. The new measurements and the evaluation of numerical models of varying fidelity can assist in informing offshore operations such as the side by side offloading from Floating Liquid Natural Gas (FLNG) facilities.

Introduction

The resonant flow in the narrow gap between a carrier and a second vessel or a solid terminal often governs operability [START_REF] Zhao | Current practice and research directions in hydrodynamics for FLNG-side-by-side offloading[END_REF]. Near resonance, the potential flow damping is small and viscous effects can provide a significant contribution to the total hydrodynamic damping. Compared to the inviscid response, the viscous effects are typically significantly more challenging to determine, such that the free surface amplitudes at resonance are difficult to predict.

Given the practical importance, numerous studies have been carried out to investigate resonances in narrow gaps between vessels and terminals, as well as in moonpools. These have been approached theoretically, numerically and experimentally (see e.g. [START_REF] Faltinsen | On damping of two-dimensional piston-mode sloshing in a rectangular moonpool under forced heave motions[END_REF]; [START_REF] Zhao | Gap resonance and higher harmonics driven by focused transient wave groups[END_REF]; [START_REF] Molin | On natural modes in moonpools and gaps in finite depth[END_REF] and references therein). Such studies have demonstrated how the addition of artificial damping to potential flow solutions, or vortex methods, or viscous solvers with high spatial and temporal resolution are generally required for reliable predictions of the resonant amplitudes. However, the underlying complex flow dynamics around the bilge of the ship and within the gap (or moonpool) which underpins the response remains comparatively less understood.

Email address for correspondence: ian.milne@uwa.edu.au Provided that the edge or bilge is sufficiently sharp and the Reynolds number is not small, flow separation occurs and vortex shedding can be expected to have a dominant contribution to the damping over other viscous effects such as skin friction [START_REF] Kristiansen | Application of a vortex tracking method to the piston-like behaviour in a semi-entrained vertical gap[END_REF][START_REF] Feng | Numerical investigation of viscous effects on the gap resonance between side-by-side barges[END_REF]. In such cases, the flow dynamics are generally considered to be nearly independent of Reynolds number, with the flow separating at the edge and viscosity only affecting the decay of the shed vortices. However, compared to the flow around an isolated edge, such as the case considered by [START_REF] Graham | The forces on sharp-edged cylinders in oscillatory flow at low Keulegan-Carpenter numbers[END_REF], for instance, a narrow gap presents additional complexities and gives rise to asymmetries between the oscillatory half-cycles. In particular, the confinement of the flow may be expected to influence the displacement of the vortices which can remain close to the hull and affect the separation, but which has seldom been studied.

Experimentalists have pursued various means to investigate gap and moonpool resonance. The problem has often been simplified by generating idealised two-dimensional flow conditions in a wave tank. [START_REF] Kristiansen | Application of a vortex tracking method to the piston-like behaviour in a semi-entrained vertical gap[END_REF] and [START_REF] Ravinthrakumar | A two-dimensional numerical and experimental study of piston and sloshing resonance in moonpools with recess[END_REF], for instance, used the forced heave response of two rigidly connected hull sections separated by a small gap to study the moonpool problem. The terminal and side-byside problems have been investigated using fixed hulls positioned either close to a wall [START_REF] Kristiansen | Studies on resonant water motion between a ship and a fixed terminal in shallow water[END_REF] or separated by a small gap [START_REF] Tan | A viscous damping model for piston mode resonance[END_REF], subjected to surface waves. Fewer studies have been performed in wave basins, which although potentially enable realistic hull forms or three-dimensional resonances to be investigated (see e.g. [START_REF] Peri | An experimental study of the wave excitation in the gap between two closely spaced bodies, with implications for LNG offloading[END_REF]; [START_REF] Zhao | Gap resonance and higher harmonics driven by focused transient wave groups[END_REF]), are less amendable to flow visualisation such as particle image velocimetry (PIV) to study the vortex shedding dynamics.

Experimental studies have also served to provide empirical corrections to inviscid models to enable computationally efficient prediction of the free surface responses in narrow gaps or moonpools, which may facilitate parametric studies. As demonstrated by [START_REF] Kimmoun | Wave drift force on a rectangular barge by a vertical wall[END_REF] for a rectangular barge next to a vertical wall, not accounting for flow separation at the bilges results in unrealistically large predictions of the free surface amplitudes when considering the linear diffraction/radiation problems alone. The effect of vortex shedding has typically been treated indirectly within inviscid models by incorporating a fictitious lid in the gap, tuned to provide appropriate (typically quadratic) damping (e.g. [START_REF] Molin | Experimental and numerical study of the gap resonances in-between two rectangular barges[END_REF]). Deploying such models in practice however, requires understanding the universality of the additional viscous correction terms since the data for their determination are generally limited.

The application of viscous or vortex methods is necessary to resolve the underlying flow dynamics and vorticity fields around the bilge and within the gap. Solvers of the unsteady Navier-Stokes equations for three-dimensional multiphase flow have been implemented by [START_REF] Feng | Numerical investigation of viscous effects on the gap resonance between side-by-side barges[END_REF] and [START_REF] Wang | Resolving wave and laminar boundary layer scales for gap resonance problems[END_REF] for the pure diffraction response for two identical stationary barges in close proximity. However, significant supercomputing resources are required to fully resolve flow separation which in many cases approximates to two-dimensional flow over extensive lengths of the hull. As such, two-dimensional Navier-Stokes computations are more practical to study the problem and determine the flow field over the entire wave tank as demonstrated by [START_REF] Moradi | Effect of inlet configuration on wave resonance in the narrow gap of two fixed bodies in close proximity[END_REF] and [START_REF] Lu | Two-dimensional numerical study of gap resonance coupling with motions of floating body moored close to a bottom-mounted wall[END_REF], for instance. This also avoids possible issues related to the coupling of viscous solvers to capture flow separation with inviscid solutions as pursued by [START_REF] Kristiansen | Gap resonance analyzed by a new domaindecomposition method combining potential and viscous flow DRAFT[END_REF] and [START_REF] Ravinthrakumar | A two-dimensional numerical and experimental study of piston and sloshing resonance in moonpools with recess[END_REF]. Notably, [START_REF] Lu | Two-dimensional numerical study of gap resonance coupling with motions of floating body moored close to a bottom-mounted wall[END_REF] considered a barge heaving and swaying next to a vertical wall and presented vorticity contours to support the conclusion that vortex shedding around the square bilge dominated the damping of the gap response.

Vortex methods are particularly well suited to predicting flows with well-defined separation points such as sharp-edged bilges. [START_REF] Graham | The forces on sharp-edged cylinders in oscillatory flow at low Keulegan-Carpenter numbers[END_REF] applied an inviscid, discrete vortex technique to show that the drag coefficient due to the vortex force for oscillatory flow around an infinite right-angled edge at low Keulegan-Carpenter number (K C ) was independent of K C . For relatively small gap oscillations where the vortices remain close to the bilge, this would suggest that the damping coefficient is independent of the geometry. A vortex model combined with an inviscid boundary element solution was also used by [START_REF] Kristiansen | Application of a vortex tracking method to the piston-like behaviour in a semi-entrained vertical gap[END_REF] to study the moonpool problem. While vortex models omit diffusion and boundary layer separation, their superior computational efficiency compared to CFD is particularly useful for conducting parametric studies and estimating the vortex contribution to the damping while still encapsulating the rich physics.

To this end, the objectives here are to devise an experiment and appropriate numerical models to study not only the free surface response but also the flow fields around the bilge and within the gap in resonant conditions. Specifically, the intention is to provide new insights on the fundamental flow separation and vortex shedding dynamics and, in turn, the capabilities of PIV and numerical models to investigate the problem and determine the vortex contribution to the hydrodynamic damping. The response associated with regular waves incident to a two-dimensional section of a fixed barge located next to a vertical wall in a wave tank is considered. The setup is regarded as a suitable analogue to common gap resonance scenarios in practice, such as the side by side offloading from Floating Liquid Natural Gas (FLNG) facilities to LNG carriers in the open sea, and appropriate for evaluating the efficacy of PIV and numerical models of varying fidelity.

Experiment

Facility and set-up

The experiment was conducted in the wave tank at École Centrale Marseille and the set-up is depicted in figure 1. The wave tank is 0.65 m wide and the vertical wall was positioned at a distance of L 12.02 m from the wavemaker, with the water depth set at h 0.52 m. The hull had a beam of b 0.6 m, draft of d 0.12 m and spanned the width of the tank. The sides of the hull were vertical and the bilges were right-angled (i.e. square edges). For the purposes of this study the hull motion was fully restrained and only the pure diffraction response was considered. The vertical wall extended to the bottom of the tank and the gap between the wall and the barge was w 0.073 m (i.e. w~b 0.12).

The barge model was constructed from 250 mm thick Poly(methyl methacrylate) (PMMA) to provide the required rigidity. The vertical wall was a 25 mm thick Polyvinyl Chloride (PVC) plate. It was affixed to the walls and bottom of the tank as to ensure that it did not deform under the significant hydrodynamic pressures generated during the experiment. All dimensions and angles were confirmed using a mechanics rule, with tolerances of less than 0.5 mm.

The flat-type wavemaker was used to generate regular surface waves of constant wave steepness. A linear ramp function was applied to the wave amplitude for the first three wave periods in order to reduce the effect of initial transients. All measurements were acquired having ensured that the water in the tank was well mixed and the uniformity of the temperature was confirmed regularly using probes. The water surface elevation was measured at several locations upstream of the barge as well as in the gap using resistance wave gauges. With this set-up, based on a typical wave period of T 1.1 s, the reflected wavetrain would be re-encountered at the hull after around t 35 s, providing a sufficient uncompromised test duration.

Measurement of the flow field

The flow field around the bilge closest to the wall (i.e. within the gap and below) was observed using a PIV system which comprised two 50 mJ lasers pulsed at 100 Hz. The acquisition frequency was set to 200 Hz by adjusting the interval between the two laser pulses. A Phantom V641 camera with a resolution of 2560 1600 pixels was synchronized with the laser pulses. The water was seeded with silver coated hollow glass spheres of around 10 to 14 µm in diameter.

The vorticity (ω ∂v~∂x ∂u~∂y) was computed using an adaptive PIV algorithm based on that described by [START_REF] Thomas | A robust motion estimation algorithm for PIV[END_REF]. The same algorithm was also applied to compute the vorticity from the CFD velocity data for consistency. The 10-point scheme has the advantage of reducing the effect of fluctuations. A consequence of employing this scheme, however, is that the first two points on the hull wall are subject to a relatively large error. For the smallest discretisation used in the PIV, this affected a region of up to 0.3 mm from the hull wall. The computation of the flow field from the PIV images was also affected by the reflection from the laser sheet onto the hull. As such, it was not possible to reliably compute the flow field in the region of this intense light. This affected a region around 1 mm from the hull, depending on the angle of incidence of the light.

Several fields of view which corresponded to different discretisations (pixel sizes) were tested, as summarised in table 1. For each case, the field was centred on the bilge. Except for the smallest field of view, these images also provided an independent measurement of the free surface in the gap to verify the wave probe measurements. While the majority of the measurements were obtained near the centre of the hull at approximately z 0, where z is in the spanwise direction, data were also obtained at z 0.225 m which allowed for an assessment of spatial variability.

CFD model

The two-dimensional flow field was predicted using a finite-volume solver for the unsteady Navier-Stokes equations for multi-phase flow (i.e. air-water) using direct numerical simulation (DNS) i.e. with no turbulence model included. Consistent with [START_REF] Lu | Two-dimensional numerical study of gap resonance coupling with motions of floating body moored close to a bottom-mounted wall[END_REF], this was justified based on the flow along the side walls of the gap likely being predominantly in the laminar regime. For a typical case with a wave period T 1.1 s, such that Ω 5.71 rad s 1 and with the kinematic viscosity ν 1.15 10 6 m 2 s 1 , the thickness of the Stokes oscillatory boundary layer on both the hull and vertical wall would be expected to be δ 2π » 2ν~Ω 0.6 mm. Taking the gap width as a relevant lengthscale, a Stokes parameter may also be formed as β w 2 ~ν T ¤ 4200. The displacement thickness of the Stokes layer/gap is δ

~w 1 i e iΩt ~2π º
β which has a maximum value of about 0.0025, i.e a very small portion of the gap. The skin friction is similarly proportional to 1~ºβ and is very small compared with the pressure drag force from the vortex shedding. Thus this informs the design of the computational mesh, which should be sufficiently fine to capture the vortex shedding while the resolution of the Stokes layers is likely of secondary importance.

The dimensions of the computational domain were equivalent to the experiment in the x, y-plane. The walls of the hull and exterior domain were set as no-slip boundaries with the upper surface of the domain defined as a pressure-outlet. Regular waves were generated at the inlet with a linear ramp function applied to the wave amplitude for the initial three wave periods, as in the experiment. The computations were performed using a fully structured mesh which was refined in the regions of the hull walls, bilges, the vertical end wall and the free surface, as depicted in figure 2. Solutions were attained for three different meshes. These differed between the sizes of the (square) mesh elements which were uniformly distributed within a region of 100 mm 100 mm centered on the bilge in the gap, as summarised in table 2.

The equations were solved with the widely utilised semi-implicit SIMPLE algorithm [START_REF] Patankar | A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows[END_REF]. The second-order upwind scheme was used for spatial discretisation of momentum and the free surface was tracked using the Volume of Fluid (VoF) technique. The computations used a constant time step not exceeding T ~1000, where T is the wave period. With this, the Courant number was sufficiently low (generally d 1) to ensure simulation accuracy. The free surface time history was extracted at the positions corresponding to the wave gauges in the experiment for comparison with the measurements. The velocity field around the bilge near the wall was sampled at a rate of T ~200 for comparison with the PIV results.

Free surface response

The hydrodynamic response of the free surface in the gap between the hull and wall is first demonstrated, given its central role in driving the vortex shedding around the bilge. Table 2. Total mesh elements in the domain and dimensions of the mesh (dx dy, mm) within a 100 mm 100 mm region centered on the bilge in the gap for the CFD model.

Figure 3(a) shows the gap response amplitude operator (RAO) measured by the wave gauge located at the centre of the gap, as a function of the incident wave period. Here, the RAO A gap ~Ai , where A gap is the wave amplitude in the gap and A i is the incident wave amplitude. The results are for constant wave steepness values of H~λ ¨2A i ~λ 1.3 % and 2.6 %, where λ is the wave length. The resonant peak of the piston mode at a wave period of around T 1.15 s is clearly distinguishable, varying weakly with wave steepness. Figure 3(b) plots the RAOs as function of the wave steepness for a wave period T 1.1 s, thus coinciding close to resonant conditions. The RAO exhibits a decrease with wave steepness at a rate close to H ~λ 1~2 , consistent with a purely quadratically damped response.

The measurements have initially been compared with predictions from a potential flow model with a damping lid in the gap, based on [START_REF] Molin | Experimental and numerical study of the gap resonances in-between two rectangular barges[END_REF] and applied previously by [START_REF] Kimmoun | Wave drift force on a rectangular barge by a vertical wall[END_REF] and [START_REF] Milne | An experimental and numerical study of the vortex shedding dynamics during gap resonance[END_REF]. In the model the inviscid solution is obtained using a matched eigenfunction expansion method and the force applied to the massless lid located at the surface is of the form F 1 2 ρC D L wS ηS η, where η is the free surface elevation in the gap and ρ is the fluid density. Favourable agreement is demonstrated when employing a quadratic damping coefficient of C D L 1.4 and neglecting the Stokes attached flow contribution. This coefficient was determined through iterative tuning to achieve a best fit to the RAO versus H~λ curve and it is emphasised that its value was the same for all wave periods and steepness. The results suggest that the viscous damping coefficient is independent of the oscillatory amplitude in the gap, thus consistent with the K 0 C proportionality expected for an infinite isolated right-angled edge [START_REF] Graham | The forces on sharp-edged cylinders in oscillatory flow at low Keulegan-Carpenter numbers[END_REF]. This is discussed further in Section 6. Consideration is now given to the comparisons with the CFD results which have been obtained for T 1.1 s and varying values of steepness. The predicted RAOs are shown to be consistent with the measurements and those attained using the lid damping model. The results for H~λ 1.6 % corresponding to the three different CFD mesh sizes also serve to show that the computations were satisfactorily converged. As a means of further demonstrating the efficacy of the CFD model, the predicted time histories of the free surfaces at various positions along the tank and in the gap for H~λ 1.3 % are plotted in figure 4. The computed traces are shown to be generally in close alignment with the measurements.

The incident and the reflected amplitudes were subsequently evaluated from both the measured and computed upstream surface elevation data. Here, the technique of [START_REF] Mansard | The measurement of incident and reflected spectra using a least squares method[END_REF] was applied on moving windows two wave periods (2T ) in width with a translation of T ~10. Figure 4 shows the time evolutions of these respective wave amplitudes up to the time in which the wave, having reflected from the hull and then reflected from the wave-maker, re-encountered the first wave gauge. Also plotted are the reflected and dissipation coefficients, which are related through energy considerations by K d » 1 K 2 r . A significant portion (i.e. e 90 %) of the wave energy was dissipated.

The agreement between the measured and computed incident wave amplitudes provides confidence in having replicated the onset flow conditions. However, the reflected wave amplitude is over predicted. It is postulated that turbulence which was not explicitly modelled within the numerical solutions, which most likely affects the flow at scales much smaller than those governed by the oscillatory period, may have been the source of the increased dissipation in the experiment.

Vortex shedding dynamics

Flow around the bilge and in the gap

The flow around the bilge next to the wall and in the gap for near resonant conditions is now examined. Figure 6 shows the measured and computed instantaneous vorticity contours and velocity vectors over a single cycle once quasi-steady resonant conditions have been attained (but prior to encountering reflected waves) for H~λ 1.6 % and T 1.1 s. The results are at intervals of T ~8, labelled consecutively, as the flow enters and is then expelled out of the gap. The PIV data are for a pixel size of 0.116 mm (case II), corresponding to 0.462 mm for the PIV algorithm, with the field of view sufficiently large to visualise the flow up to the free surface. The CFD results are for a mesh size of 1.0 mm (case A).

The plots elucidate the inherent complexities in the flow field and the differences compared to shedding around an isolated edge. As the flow enters the gap (occurring between panels 1 and 2) a thin shear layer emanating from the bilge rolls up to form a concentrated counter-clockwise (CCW) vortex core (of negative vorticity and denoted as 'B'). The CCW vortex pairs with a clockwise (CW) vortex ('A', of positive vorticity) which was shed in the previous half cycle that has entered the gap with the bulk flow.

Owing to the restriction imposed by the narrow gap and the presence of the CW vortex, as the CCW vortex expands it remains near the vertical hull wall (panels 3 to 4). This results in an attached flow away from the edge being induced beneath the primary CCW vortex which is now within a system comprising multiple (negative) vortical regions. The attached flow subsequently separates, generating a secondary CW vortex (denoted as 'C' within panel 5) which initially pairs with core 'B' and convects upward. Without the secondary separation off the surface new positive vorticity would be shed later from the edge when the flow at the edge reverses. As such, here, there is an additional vortex pair ('B C') generated within the system.

The residual positive vorticity which remains attached to the vertical wall is subsequently swept to the edge, forming the next CW vortex ('D', in panel 7) below the hull as the flow reverses. The proximity of this CW vortex to the hull also induces a local attached flow beneath the hull with relatively weak secondary separation. The primary CCW vortex ('B') is ejected from the gap as the free surface is displaced downward. This vortex now pairs with the CW vortex 'D'. The interaction of the vortices, which are of similar but opposite strength, results in a strong local downward velocity below the hull. As the flow reverses again and the free surface moves upward, the CW vortex is reinjected into the gap and the shedding process then repeats. Regions of intense vorticity are present near the hull walls, the thin shear layers and the vortex cores, which are all associated with large flow gradients. Residual vorticity associated with previous shedding cycles is also evident in the flow field, which contributes to the mixing and the decay of the primary vortices.

To demonstrate the consistency of the flow dynamics, figure 7 plots the time histories of the vorticity maxima (both positive and negative) within the field of view and its position relative to the bilge. The corresponding free surface elevation in the gap is also shown for reference. The instances of the most intense positive vorticity are associated with the formation of the concentrated CW vortex core which initially remains just below the bilge. The positive vorticity decreases in strength as the CW core is displaced downward, pairs with the CCW vortex and then moves back upward into the gap. The instances of positive vorticity close to the hull and above the bilge correspond to the local attached flow and the formation of the secondary CW vortex. The secondary separation is also identifiable by the smaller secondary spikes in vorticity maxima time history. Further complexities are exhibited in the negative vorticity. In particular, as the flow enters the gap the maxima alternates between the CCW cores within the multiple vorticial region. Similarly, this gives rise to multiple spikes in the vorticity traces.

Sensitivity to discretisation

The ability to detect flow features such as shear layers, concentrated vortex cores and secondary separation is inherently dependent on the respective CFD and PIV discretisations. This is of particular concern when assessing convergence and, moreover, comparing numerical and measured results. The effect of discretisation on the CFD results is demonstrated first, using the H~λ 1.6 %, T 1.1 s case again. In the computational results, the discretisation changes the capability of the model to describe the physics. Figure 8 shows the traces of the vorticity maxima, together with the surface elevation in the gap, computed using smaller CFD mesh sizes around the bilge of dx 0.50 mm and 0.25 mm (i.e. cases B and C). The results are generally consistent with those for the coarser mesh in figure 6. However, the use of a smaller mesh size further elucidates the regions of highly concentrated vorticity close to the hull walls, including those associated with the generation of the secondary vortices.

Figure 9 compares the instantaneous vertical and horizontal velocity maxima computed using the three different mesh discretisations. Importantly, as for the free surface responses in the gap, the computed velocities are shown to be in close alignment. The results therefore indicate that the underlying vortex dynamics within the gap associated with the damping have been resolved and reinforces the particular challenge of demonstrating convergence for a gradient quantity such as the vorticity.

In contrast to the numerical computations, changing the PIV resolution and the field of view modifies the ability to capture the structures while the background physics should be equivalent. The effect of the discretisation on the measured vorticity is demonstrated in figure 10. The larger peaks in the positive vorticity for enhanced PIV resolutions imply greater refinement of the gradients associated with the CW vortex cores below the bilge. The negative vorticity is comparatively less sensitive to the resolution but which is attributed to the thin shear layers and highly concentrated CCW cores within the multiple vortical region which are beyond the limits of the PIV system to resolve.

Figure 11 shows that similar to the numerical computations the velocities measured using the different PIV resolutions agree well. However, since the horizontal velocity maxima can occur further away from the bilge compared to the vorticity maxima there are instances (most evident around the secondary peaks) where those results are affected by the differing fields of view.

Collectively, these insights are particularly valuable when validating numerical computations using measured data. Direct comparisons of the vorticity and vertical velocity maxima between mesh B and PIV case III and between mesh C and PIV case IV , respectively, are made in figure 12. To facilitate the evaluations, here the CFD results are for a field of view equivalent to the PIV. For these cases, the respective discretisations are shown to have enabled near equivalent resolution of the shedding dynamics associated with the positive vorticity. The limitations of the PIV to fully resolve the features associated with the negative vorticity maxima is again highlighted in the comparisons with the finest CFD mesh. As the negative vorticity in the vortex system is particularly sensitive to the secondary separation, it could however be expected that obtaining agreement between the measurements and computations is inherently challenging. The computed and measured velocities, which are less sensitive to both mesh and PIV discretisation, are shown to align closely and provide additional support for having replicated the experiment numerically.

Secondary separation mechanisms

Secondary separation which leads to the generation of multiple vortex systems appears to be an attribute of resonant gap flows. It is similar to the secondary separation which is observed under the primary vortices which form above sharp leading edges of delta wing aircraft at angles of attack (see, e.g. [START_REF] Luckring | The discovery and prediction of vortex flow aerodynamics[END_REF]). As the phenomena have not been reported in oscillatory flows around bilges and other edges, the mechanisms of the secondary vortex separation adjacent to the vertical hull wall are further examined here.

Figure 13 shows the instantaneous vorticity contours and velocity vectors computed in CFD using a mesh sizing of dx 0.50 mm (case B) as the flow enters the gap. The plots ) is instead displaced around the bilge, pairing with the CW vortex 'D' during its formation, before the primary CCW vortex is ejected from the gap. Evidence of the secondary separation and formation of multiple cores is also present in the PIV measurements obtained using 0.071 mm pixels (case though only part of the primary vortex is able to be visualized within the field of view.

As demonstrated by [START_REF] Hummel | On the vortex formation over a slender wing at large incidence[END_REF] for a delta wing, the surface pressure distributions also serve to help inform the separation dynamics. The corresponding instantaneous static pressure distributions along the vertical hull wall computed using CFD are also shown in figure 13. The peaks in the suction associated with the primary and secondary vortices are identifiable. Also evident is the region of the adverse pressure gradient which results in the separation of the attached local flow (induced by the primary vortex and directed towards the edge), forming the secondary CW vortex. The displacement of the primary core initially shifts the location of this adverse pressure gradient upward towards the free surface.

Figure 14 plots the trajectories of the largest vortex structures computed in CFD using dx 0.50 mm (case B) to further illustrate the interactions. Specifically, the primary CCW and CW vortices and the secondary CW vortex have been tracked from their initial formation until they diffuse or are longer visible within the field of view. Their locations at time t 20T and 21T , with respect to the start of the simulation, are also indicated for reference. The interaction between the primary CCW vortex and secondary CW vortices, once the latter has formed, is particularly evident. While the primary CCW vortex is ultimately ejected from the gap, the secondary CW vortex reenters the gap and decays within it.

Self-similarity and spatial invariance

Finally, consideration is given to the variability in the shedding dynamics with the forcing amplitude and along the span of the barge. The spanwise variability, in particular, has seldom been previously reported for gap resonance studies. The measured vorticity maxima for varying wave steepness, based on a pixel size of 0.071 mm (case III) are compared in figure 15(a). The results demonstrate that the underlying vortex dynamics were generally consistent in the range 1.06 d H~λ f 1.62 %, including exhibiting secondary separation. Figure 12(b) plots the vorticity maxima for a relatively large steepness of H~λ 2.6 %. The significant surface amplitudes in the gap necessitate the use of a larger field of view in the PIV (case I), at the expense of a reduction in the resolution. However, the spatial homogeneity is evident in the comparisons of the measurements which were repeated at the two different spanwise locations. This spanwise spatial invariance also

(i) (ii) (iii) (i) (ii) (iv) y (cm) y (cm) y (cm) y (cm) y (cm) y (cm) y (cm) y (cm)
distance from the wall, x (cm) distance from the wall, x (cm) distance from the wall, x (cm) distance from the wall, x (cm) suggests that the process by which the vortex is shed from the bilge is approximately two-dimensional. On this basis, it is postulated that three-dimensionality only becomes once the vortices are relatively small in size and are in the process of decaying. Moreover, this provides justification for the application of two-dimensional numerical models when predicting the response.

Insights from an inviscid discrete vortex model

The vortex contribution to the effective damping of the gap response may also be assessed using an inviscid discrete vortex model (DVM). It is applied for the present analysis to an idealised two-dimensional geometry of one end of a semi-infinite rectangular hull separated from an infinite vertical wall by a small gap. Since the flow field is assumed inviscid it may be represented by a velocity potential which Laplace's equation and is invariant under conformal transformation. Transforming the geometry to a halfplane bounded by a wall on one side allows a simple representation of the vortex shedding by a sequence of discrete vortices and their images in the plane boundary. This method follows the procedure of many discrete vortex methods including that for an isolated sharp edge in oscillatory flow [START_REF] Graham | The forces on sharp-edged cylinders in oscillatory flow at low Keulegan-Carpenter numbers[END_REF]. 16 in the form of plots of the discrete vortex distributions and corresponding vorticity contours. The effective force acting on the flow due to the shedding of vortices from the edge is evaluated by computing the additional pressure acting at the free surface caused by the vortex shedding. The additional pressure at the mean free surface due to vortex shedding is given by ∆p ρdφ~dt where φ being invariant under the transformation is the value of the real part of the complex potential W (due to the vortices) in the ζ-plane as ζ 0 (the additional contribution from the vortices to the velocity squared term being negligible at the free surface). Integrating this additional pressure over the free surface in the gap gives the additional force coefficient C F ∆p~1 2 ρV 2 mf s acting at the free surface due to the vortex shedding. A constant coefficient with this Morison definition implies quadratic damping which is equivalent to a conventional (linearly non-dimensionalised) wave-diffraction theory damping coefficient which increases linearly with wave amplitude. Figure 17 10 cycles of flow. This shows that the vortex force is strongly in phase with the velocity which is as predicted for an isolated right-angle edge, the contribution from the inertia force (that part of the force in phase with the free-surface acceleration) being extremely small [START_REF] Graham | The forces on sharp-edged cylinders in oscillatory flow at low Keulegan-Carpenter numbers[END_REF]. Obtaining the component acting in phase with the free surface velocity gives the effective damping coefficient C D acting on the free surface in the gap. In all other far field directions the pressure field due to the vortex shedding decays to zero with distance fast enough to give no contribution.

Results for C D calculated in this way are shown for a range of H~w from 0.2 to 10.0 in figure 17. Also marked in this figure is the empirical value C D 1.4 which was applied to the inviscid diffraction computation to give close agreement with the peak value of the resonant RAO in the gap. C D for the isolated edge as predicted by theory [START_REF] Graham | The forces on sharp-edged cylinders in oscillatory flow at low Keulegan-Carpenter numbers[END_REF] shows that the dependency on Keulegan-Carpenter number should be K 2~32δ~π1 C which implies independence of K C and hence flow amplitude when, as here, the internal edge angle δ π~2. In the present case of an edge close to a vertical wall for small amplitudes of water vertical displacement in the gap C D is approximately independent of changes in H~w. For H~w e 3.0 the effect of the vortex confinement in the gap as H~w increases gradually depresses the values of the drag coefficient below the small amplitude value. The value of the empirical damping coefficient C D 1.4 used in the diffraction computations compares well with the average value of the damping coefficient shown in figure 18. While this inviscid computation can therefore generate useful estimates of vortex effect for separation from a sharp edge it cannot of course predict secondary separation nor the effects of rounding of a bilge in leading to complete or partial suppression of separation.

Conclusion

A new series of wave tank experiments and the application of computational models of varying fidelity have provided new insights into the resonant flow in a narrow gap between a stationary hull and a vertical wall. Vortex shedding from the sharp bilge edge of the hull has been demonstrated to give rise to a quadratically-damped free surface response in the gap at resonance. The shedding dynamics were found to exhibit a high degree of invariance to the amplitude in the gap and spatial uniformity along the span of the barge. Secondary separation was also identified as a particular feature of the separated flow at the hull bilge which led to the generation of a system with multiple vortical regions and asymmetries between the inflow and outflow.

The findings can assist in improving or accelerating the computational methods used to predict resonant gap flows and inform offshore operations such as the side by side offloading from FLNG. The influence of discretisation has been demonstrated to be a particularly important consideration when studying the flow dynamics within the gap and comparing flow quantities between PIV and CFD. While it was shown that the instantaneous velocity maxima were well converged both experimentally and numerically, demonstrating convergence and agreement in the gradient quantities such as vorticity is likely to be more challenging. This is attributed to the concentrated vortex cores, thin shear layers and the complexities associated with the secondary separation. The study has also reinforced the value of more computationally efficient semi-empirical models for accounting for the vortex-induced damping in gap flows, provided their limitations are acknowledged. Using both a damping-lid model and a novel application of a discrete vortex model it has been demonstrated that universal quadratic-damping coefficients, independent of wave steepness and frequency, may be able to be determined. Such computations are well suited to parametric studies and therefore can be useful for facilitating initial operability assessments, though obviously do not directly model secondary separation.

Figure 1 .

 1 Figure 1. Experimental arrangement in the wave tank.

Figure 2 .

 2 Figure 2. CFD mesh; (a) in the region of the hull and wall and (b) in the region of the bilge edge in the gap (bounded by dashed-lines in (a)).

Figure 3 .

 3 Figure 3. Free surface RAOs; (a) as function of wave period for constant wave steepness of H~λ 1.3 % (blue) and 2.6 % (red), with the dashed-line corresponding to the damping-lid model prediction; (b) as a function of wave steepness for T 1.1 s with results for both the lid damping model and CFD shown.

Figure 4 .Figure 5 .

 45 Figure 4. Time histories of the measured (blue) and CFD prediction (red) of the free surface elevation (in cm) at various distances upstream from the vertical wall, for T 1.1 s and H~λ 1.3 %.

  Figure 6. Measured (upper panels, for case II) and computed (lower panels, case A) instantaneous vorticity contours at equivalent intervals of T ~8 (numbered consecutively) during resonance for H~λ 1.6 %, T 1.1 s. The vortex cores are labelled A to E. The extent of the PIV field of view is shown by the dashed lines.

Figure 7 .

 7 Figure 7. Vertical and horizontal positions of the vorticity maxima in the gap measured for case II (panels (a) and (c)) and computed using mesh case A (panels (b) and (d)) for T 1.1 s and H~λ 1.6 %. The colour range corresponding to the intensity of the vorticity is consistent with figure 6. The free surface elevation in the gap and the edges of the hull are shown by the respective thin and thick solid lines. The vorticity intensity is also shown in panels (a) and (c) for reference.

Figure 8 .Figure 9 .

 89 Figure 8. Vorticity maxima computed in CFD using mesh sizes of 0.50 mm (case B) and 0.25 mm (case C), for T 1.1 s and H~λ 1.6 %. The vertical positions of the vorticity maxima for the respective cases are shown in panels (a) and (b) for clarity, with the colour ranges consistent with figure 6. The traces of the free surface elevation in the gap and vorticity intensity are shown in (a) and (c), for reference, where case B is plotted in blue and case C in red.

Figure 10 .Figure 11 .

 1011 Figure 10. Comparison of the measured vorticity maxima using different field of views and pixel sizes for T 1.1 s and H~λ 1.6 %.

Figure 12 .

 12 Figure12. Comparison of the computed and measured vorticity and vertical velocity maxima for T 1.1 s and H~λ 1.6 %. (a,c) PIV with pixels of 0.071 mm (case III) and CFD with a mesh size of dx 0.50 mm (case B); (b,d) PIV with pixels of 0.037 mm (case IV ) and CFD with a mesh size of dx 0.25 mm (case C).

Figure 13 .Figure 14 .

 1314 Figure13. Measured (upper panels, for case III) and computed (lower panels, case B) instantaneous vorticity contours at equivalent intervals of T ~8 (numbered consecutively) during resonance for H~λ 1.6 %, T 1.1 s. The vortex cores are labelled to be consistent with figure6, with the smaller cores also identified in lowercase. The extent of the PIV field of view is shown by the dashed lines. The surfaces pressures (ps (Pa)) along the vertical hull wall computed in CFD are shown in the insets with key features labelled. The peaks at (i) correspond approximately to the primary CCW core. An adverse pressure gradient exists between (i) and the location of the secondary separation (ii). The CW vortex core associated with the secondary separation is at (iii) and the secondary CCW vortex core corresponds to (iv).

Figure 15 .

 15 Figure 15. Measured vorticity maxima T 1.1 s, (a) as a function of wave steepness and (b) as repeated at two different sections of the barge for H~λ 2.6 %.

Figure 16 .

 16 Figure 16. Discrete vortex distributions (a, b) and corresponding vorticity contours (c, d) for t~T 1 and t~T 5.

  plots C F as a time sequence over

Figure 17 .Figure 18 .

 1718 Figure 17. Force coefficient time history computed using DVM.

Table 1 .

 1 Field and view (width height) and dimensions of the pixels corresponding to the camera images and PIV measurements. All values are in mm.

	Case Field of view Camera pixel size PIV pixel size
	I	494 309	0.193	0.772
	II	297 186	0.116	0.464
	III	182 114	0.071	0.284
	IV	96 60	0.037	0.148

Acknowledgment

This research was supported by the ARC Industrial Transformation Research Hub for Offshore Floating Facilities which is funded by the Australian Research Council, Woodside Energy, Shell, Bureau Veritas and Lloyds Register (Grant No. IH140100012).

The authors report no conflict of interest.