B Collapseofaliquidcolumn:numericalsimulationandexperimentalvalidation

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Comput. Mech. (2007) 39: 453–476 DOI 10.1007/s00466-006-0043-z

O R I G I NA L PA P E R

Marcela A. Cruchaga · Diego J. Celentano Tayfun E. Tezduyar

Collapse of a liquid column: numerical simulation and experimental validation

Received: 19 December 2005 / Accepted: 11 January 2006 / Published online: 11 February 2006

Abstract This paper is focused on the numerical and exper- imental analyses of the collapse of a liquid column. The measurements of the interface position in a set of experiments carried out with shampoo and water for two different initial column aspect ratios are presented together with the corresponding numerical predictions. The experimental procedure was found to provide acceptable recurrence in the observation of the interface evolution. Basic models describing some of the relevant physical aspects, e.g. wall friction and turbulence, are included in the simulations. Numerical experiments are conducted to evaluate the influence of the parameters involved in the modeling by comparing the results with the data from the measurements. The numerical predictions reasonably describe the physical trends.

Keywords Moving interfaces· Two-fluid flows ·

Computational fluid mechanics· Experimental validation 1 Introduction

Modeling two-liquid interfaces or free-surface flows is an active research field due to the wide range of their engineering applications. This motivates the development of formulations capable of accurately representing this kind of transient problems. Various numerical techniques were proposed, with moving or fixed domain discretizations, to overcome the chal- lenges related to updating the interface intrinsically coupled with the fluid dynamics equations (e.g. see [1–34], and references therein). Work was carried out also in the development of physical models to represent real problems at laboratory scales and, in particular, to assess the numerical performance M.A. Cruchaga (

B

⁾· D.J. Celentano

Departamento de Ingeniería Mecánica, Universidad de Santiago de Chile, Av. Bdo. O’Higgins 3363, Santiago, Chile E-mail: mcruchag@lauca.usach.cl T.E. Tezduyar

Team for Advanced Flow Simulation and Modeling (T*AFSM), Mechanical Engineering, Rice University - MS 321,

Houston, TX 77005, USA

of the proposed numerical techniques by comparing their predictions with the measurements obtained from experiments.

Focusing on the description of the collapse of a water column, experiments were presented in [11, 35]. This problem was adopted by several researchers as a benchmark test to validate the numerical performance of the proposed formulations in solving two-liquid interfaces or free-surface flows [11, 12, 15, 21, 23, 24, 26, 28, 29, 32].

In this paper, in addition to numerical results, we report results from experiments for the collapse of liquid columns.

The experiments are carried out using two different fluids:

shampoo and water, and two different column aspect ratios (height to width ratio of the initial liquid column). The pur- pose of these experiments is to provide data for the interface evolution with very different fluid properties. The experimental data is used for evaluating the performance of the numerical formulations we propose for solving this class of problems.

We use finite element techniques with fixed meshes. The interface is “captured” by solving an advection equation governing the time-evolution of an interface function that marks the interface location [16–33]. A version of this approach presented in [32] can capture the interface while maintain- ing the global mass conservation and accurately representing the sudden changes in the materials properties. That formulation is called the edge-tracked interface locator technique (ETILT), which was introduced in [19]. More recent versions of the ETILT were introduced in [30, 31, 33]. Although some new techniques related to the ETILT are presented in this work, the simulations are mainly devoted to evaluation of the numerical performance of our core techniques in describing some of the physical aspects such as wall friction and turbulence. The influence of the parameters involved in modeling is also analyzed.

The governing equations are presented in Sect. 2. The ETILT and the new aspects included in its current version are summarized in Sect. 3. The details of the experimental procedure are presented in Sect. 4. Experimental and numerical results are presented and discussed in Sect. 5. Concluding remarks are presented in Sect. 6.

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454 M.A. Cruchaga et al.

Fig. 1 Collapse of a liquid column: experimental setup and illustrative picture

2 Governing equations

The Navier–Stokes equations of unsteady incompressible flows are written as follow:

ρ∂u

∂t + ρu∇ · u + ∇ p − ∇ · (2με) = ρf in × Y, (1)

∇ · u = 0 in × Y, (2)

whereρ, u, p, μ, ε and f are the density, velocity, pressure, dynamic viscosity, strain-rate tensor, and the specific body force. In these equations, denotes an open-bounded domain with a smooth boundary, and Y is the time interval of inter- est. This system of equations is completed with a set of initial and boundary conditions:

u= u0 in (3)

u= g in g× Y (4)

σ · n = h in ^h× Y (5)

where u0 is the initial value of the velocity field, g repre- sents the velocity imposed on the part of the boundary^g, and h is the traction vector imposed over^h (^g∪ ^h =  and^g∩ ^h = ∅), typically taken as traction-free condition:

h= 0.

In the present simulations, the condition at a friction wall is given by the following simple model [21]:

h= −ρu |u|

C²_f in(h)w× Y, (6)

where Cf is a friction parameter and(h)w is a subset of

h, representing the walls where the friction conditions are applicable.

A simple model to compute the energy dissipated by the turbulent effects could be obtained by modifyingμ as [32]:

μ = min(μ + l_mix² ρ√

2ε : ε; μmax), (7)

where lmix is a characteristic mixing length (in the present work, lmix = C^thUGNwith Ct being a modeling parameter and hUGNa characteristic element length [20]) andμmaxis a cut-off value.

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Collapse of a liquid column 455

Fig. 2 Collapse of a liquid column for aspect ratio Ar= 1. Experimental results for the time evolution of the interface position: a dimensionless horizontal position (x/L) at y = 0.0 m (bottom wall) and b dimensionless vertical position (y/L) at x = 0.0 m (left wall). Comparison with the results obtained by Martin and Moyce (M&M) [35]

The interface between the two fluids (Fluids 1 and 2) represents a strong discontinuity in the fluid properties and the gradients of the velocity and pressure. Nevertheless, these variables are interpolated as continuous functions across the interface. Other types of discontinuities at the interface, e.g.

surface tension, are not included in the present model. The interface motion is governed by an advection equation:

∂ϕ

∂t + u · ∇ϕ = 0 in × Y, (8)

whereϕ is a function marking the location of the interface.

The weak form of the Navier–Stokes equations is obtained in the context of a finite element formulation using a Gen- eralized Streamline Operator technique [34]. The stabilized nature of the technique allows the use of equal-order inter- polation functions for velocity and pressure. In the context of two-fluid flow analysis, all the matrices and vectors de- rived from the discrete form are computed including the discontinuity in fluid properties [32]. Therefore, the algorithm used for updating the interface is coupled with the finite element solution of the Navier–Stokes equations. The time integration is performed using a standard backward Euler scheme.

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Fig. 3 Collapse of a liquid column for aspect ratio Ar= 2. Experimental results for the time evolution of the interface position: a dimensionless horizontal position (x/L) at y = 0.0 m (bottom wall) and b dimensionless vertical position (y/L) at x = 0.0 m (left wall). Comparison with the results obtained by Martin and Moyce (M&M) [35] and Koshizuka and Oka (K&O) [11]

3 Interface update

In the present work, the version of the ETILT presented in [32] is used for updating the interface. At each time step, Eqs. (1) and (2) are computed in the entire domain using the density and viscosity distributions expressed as:

ρ^h = ϕ^heρ1+ (1 − ϕ^he)ρ2, (9.1)

μ^h= ϕ^heμ1+ (1 − ϕ^he)μ2, (9.2) whereϕ^heis the edge-based representation ofϕ.

To determine u^h_n₊₁, p^h_n₊₁andϕn^he+1at a time level n+ 1 from u_n^h andϕn^he at time level n, the velocity and pressure are computed from Eqs. (1) and (2). To computeϕn^he+1, given ϕ^hen , first a nodal representationϕ^his computed. This is done by using a constrained least-squares projection as given in [30, 33]:

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Fig. 4 Collapse of a shampoo column with aspect ratio Ar = 1. Interface positions at different instants: experimental (left) and numerical results computed without (Simulation 1, middle) and with (Simulation 2, right) wall friction effects

ψ^h

ϕn^h− ϕn^he

d

+

n_ie

k=1

ψ^h(x^k)λPEN

ϕn^h(x^k) − 0.5

= 0. (10)

Hereψ^h is the test function, nieis the number of the inter- face edges (i.e., the edges crossed by the interface), xkis the

coordinate of the interface location along the k^th interface edge andλPEN is a penalty parameter. This penalty parameter ensures that the projection topologically preserves the interface position along edges. In four-noded quadrilateral elements, this condition is activated also along the element diagonals crossed by the interface. After this projection, we update the interface by using a discretization based on the streamline upwind/Petrov–Galerkin (SUPG) formulation [1]

of the advection equation (8) governingϕ:

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Fig. 5 Collapse of a shampoo column with aspect ratio Ar = 1: a dimensionless horizontal position (x/L) at y = 0.0 m (bottom wall), b dimensionless vertical position (y/L) at x = 0.0 m (left wall), c dimensionless vertical position (y/L) at x = 0.27 m (middle of the box) and d dimensionless vertical position (y/L) x = 0.42 m (right wall). Numerical results computed without (Simulation 1) and with (Simulation 2) wall friction effects

ψ^h

∂ϕ^h

∂t + u^h.∇ϕ^h

d

+

n_el

e=1

^e

τSUPGu^h.∇ψ^h ∂ϕ^h

∂t + u^h.∇ϕ^h

d

+

nel

e=1

^e

∇ψ^hνDCID∇ϕ^hd = 0 (11)

Here n_elis the number of elements,τSUPGis the SUPG stabilization parameter [20], andνDCIDis the discontinuity-capturing interface dissipation (DCID) proposed as:

νDCID = Cs

2 h²_UGN√

2ε : ε∇ϕ^hh_UGN

ϕref , (12)

where Csis a discontinuity-capturing parameter andϕrefis a reference value set to 1. We note that the justification behind the expression given by Eq. (12) is combining some of the features we see in Eq. (7) and the discontinuity-capturing

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Fig. 5 (Contd.)

directional dissipation (DCDD) given in [20].ϕn^h+1is computed from Eq. (11) by using a Crank–Nicholson time integration scheme.

Fromϕn^h+1we obtainϕn^he+1by a combination of a least- squares projection:

ψn^he+1

P

(ϕn^he+1)P− ϕn^h+1

d = 0 (13)

and corrections to enforce volume conservation for chunks of Fluids 1 and 2 [32]. The subscript P in Eq. (13) is used for representing the intermediate values following the projection, but prior to the corrections for the volume conservation.

In this technique,(ϕn^he+1)^P = H

ϕn^h+1− 0.5

, where H is the Heaviside function. The volume conservation condition is given by the following equation:

ϕn^he+1− ϕn^he

d = Q, (14)

where Q is the mass inflow/outflow in the time interval [n, n + 1]. An iterative procedure is employed to satisfy Eq.

(14). The mass balance ratio is defined as Rm =

⎡

⎣

(ϕn^he+1− ϕn^he)d

⎤

⎦

Q for Q= 0, (15)

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Fig. 6 Collapse of a shampoo column with aspect ratio Ar= 2. Interface positions at different instants: experimental (left) and numerical results computed without (Simulation 1, middle) and with (Simulation 2, right) wall friction effects

Rm =

⎛

⎝

ϕn^he+1d

⎞

⎠

ϕn^hed

for Q= 0. (16)

The residual Rm needs to be equal to 1.0 for volume conservation. To achieve this, ϕn^h+1 is corrected iteratively as follows:

ϕ^hn+1,i+1= ϕn^h+

ϕn^h+1,i− ϕn^h

/ (R^m)^k, (17)

where k= sign

ϕ_n^h_+1,i− ϕn^h

and i is the iteration counter.

The iterations continue until the volume conservation condi-

tion is reached, i.e.|R^m− 1.0| < ε^R, whereε^Ris an acceptable tolerance. When the convergence is reached,ϕ^hn+1gen- erates a volume-conserving value forϕn^he+1. This value serves as the starting point for the computations in marching from n+ 1 to n + 2.

4 Experimental procedure

The collapse of a water column was extensively studied in open channels [35] and in closed containers [11]. In the

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Fig. 6 (Contd.)

present work, a simple experimental setup was built to compare the experimental data with the numerical results to be evaluated in Sect. 5. The apparatus used is shown in Fig. 1 and consists of a glass box with two parts delimited by a gate with a mechanical release system. The left part of the box is initially filled with a liquid: shampoo or coloured water.

In both cases two different liquid aspects ratios were tested:

Ar = 1 and A^r = 2 (A^r = H/L, with H and L being the initial height and width of the column respectively, with L = 0.114 m).

Different experiments were carried out and recorded to a film. To measure the interface position, a ruler with tics 0.02 m apart is drawn surrounding the box and over the frame of the gate. A red marker that jointly moves with the gate helps to observe its position during the gate opening. The measurements reported in the present work are average values from the measurements registered for a minimum of five experiments. The maximum experimental errors in space and time are estimated as ±0.005 m and ±0.025 s, respectively.

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Fig. 7 Collapse of a shampoo column with aspect ratio Ar = 2: a dimensionless horizontal position (x/L) at y = 0.0 m (bottom wall), b dimensionless vertical position (y/L) at x = 0.0 m (left wall), c dimensionless vertical position (y/L) at x = 0.27 m (middle of the box) and d dimensionless vertical position (y/L) x = 0.42 m (right wall). Numerical results computed without (Simulation 1) and with (Simulation 2) wall friction effects. Simulation 3 was carried out with the wall friction effects and a finite gate opening speed

To evaluate the performance of the experimental procedure, the observed evolution of the interface at the bottom and left walls of the box are compared in Figs. 2 and 3 with those reported in [11] and [35], up to the instant that the liquid impinges on the right wall. The dimensionless hori- zontal position (x/L) at the bottom wall versus dimension- less time (t[A^rg/L]^1/2, with t being the real time and g the gravity modulus) is plotted in Figs. 2a and 3a for the two

aspect ratios and both liquids. Figures 2b and 3b show for the same cases the evolution of the dimensionless vertical position (y/L) at the left wall (time scaled as in Figs. 2a and 3a). Figures 2 and 3 show that the measurements are in reasonably good agreement with those reported in the literature (in particular for water with Ar = 2). Note that the interface positions are delayed in time for the shampoo due to its higher viscosity.

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Fig. 7 (Contd.)

5 Comparison of numerical and experimental results In [32] we discussed some performance aspects of the ETILT:

sensitivity to mesh and time-step sizes, performance for struc- tured and unstructured meshes, quadrilateral and triangu- lar finite element discretizations, different sets of properties for the two liquids, and laminar and turbulent flow simulations. Also, in that a work the collapse of a water column was analyzed with a low value of the initial column width (L = 0.05715 m). The computed results, up to the instant of the water impingement on the right wall, were

compared with those obtained using a Lagrangian technique and other numerical techniques and measurements reported in the literature.

In the present paper, the numerical simulations are focused on studying the long-term transient behavior for the collapse of a liquid column described in Sect. 4. In particular, the effects of the parameters used in modeling some of the physical aspects (wall friction and turbulence) as proposed in Sect. 3 are evaluated by comparing them with the experimental results. The influence of the DCID is also analyzed. In the simulations, the liquid column is initially at rest and confined

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Fig. 8 Collapse of a shampoo column with aspect ratio Ar= 2. Interface positions at different instants: effect of the gate opening

Fig. 9 Collapse of a shampoo column with aspect ratio Ar= 2. Interface positions at different instants: 3D analysis

between the left wall and the gate. The pressure is set to zero at the top of the rectangular computational domain.

5.1 Collapse of a shampoo column

The fluid properties are:ρ1=1, 042 kg/m³andμ1=8 kg/m/s for the shampoo (measured at a laboratory using a standard procedure), andρ2= 1 kg/m³andμ2= 0.001 kg/m/s for the air (lower values for the air viscosity do not play a significant role in the numerical predictions, as observed in [32]).

As the shampoo-glass contact effects are observed in experiments, simulations with slip (Simulation 1) and fric-

tion (Simulation 2) conditions at the walls were carried out to assess the influence of this effect on the interface response.

In Simulation 2, we set Cf = 190 [see Eq. (6)] based on the numerical tests described below. A mesh composed of 60× 45 four-noded isoparametric elements is used with a time-step size of 0.001 s.

Column of aspect ratio Ar = 1 Figure 4 shows the experimental interface positions at various instants together with the corresponding predictions computed via Simulations 1 and 2. A detailed description of the interface evolution is shown in Fig. 5.

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Fig. 10 Collapse of a water column with aspect ratio Ar= 1. Interface positions at different instants: experimental (left) and numerical results computed without (Simulation 1, middle) and with (Simulation 2, right) discontinuity-capturing dissipation

The interface evolution at the bottom of the box during the spread of the liquid under the gravity is plotted in Fig. 5a up to the instant when the shampoo interface reaches the right wall (t ≈ 0.4 s). The motion of the horizontal position of the interface is delayed due to the wall friction effects.

The time histories for the vertical position at the left wall, middle of the box and right wall are shown in Figs. 5b, c and d. From these figures, it is apparent that, in the experiment, the

shampoo motion practically stops once the interface impinges on the right wall. Although similar predictions of the interface vertical position at the left wall are obtained for Simula- tions 1 and 2, some differences with the experimental results persist. As can be seen, the maximum interface vertical position at the middle of the box is captured reasonably well in the simulations, but the results exhibit a time delay. With the friction wall model, the maximum value of the interface vertical position at the right wall is reduced. Nevertheless,

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Fig. 10 (Contd.)

Fig. 5 shows that the overall trends for the interface behavior are captured reasonably well.

Column of aspect ratio Ar = 2 Figure 6 shows the experimental interface positions at different instants and the numerical results obtained with Simulations 1 and 2.

To evaluate the effect of the gate opening, an additional computation is carried out with the gate not opening instan- taneously but with a finite speed (Simulation 3). The gate opening speed was extracted from the experiments. In this simulation, an average gate opening speed of 0.86 m/s is used,

while the friction conditions are used on both the walls and the gate.

Evolutions of the horizontal position of the interface at the bottom of the box and the vertical position at the left wall, middle of the box and right wall are shown in Fig. 7 together with the experimental results.

The simulations show that the interface, as it was also observed in the experiments, still moves after the first impingement on the right wall (t ≈ 0.3 s). The reflected wave realistically represents the real liquid motion. The results computed with Simulation 2 satisfactorily capture the

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maximum and minimum interface vertical positions and the time of the first impingement on the right wall.

The gate opening effect included in Simulation 3 slightly changes the results obtained in Simulation 2. With the gate opening effect, more advanced interface positions at early instants of the analysis are obtained at the bottom of the box.

This is due to the “orifice effect” induced at the beginning of the gate opening, which increases the velocity at the bottom of the box. This situation quickly changes and the liquid

attaches to the right wall later than it does in Simulation 2. The interface positions at different instants obtained with Simulation 3 are shown in Fig. 8.

In the present study, numerical analyses with different values of the friction coefficient Cf were carried out for the shampoo columns with aspect ratios 1 and 2 (results not shown). The value Cf = 190 was selected on the basis that it leads to minimum differences between the numerical predictions of the interface position and the corresponding

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Fig. 11 Collapse of a water column with aspect ratio Ar = 1: a dimensionless horizontal position (x/L) at y = 0.0 m (bottom wall), b dimen- sionless vertical position (y/L) at x = 0.0 m (left wall), c dimensionless vertical position (y/L) at x = 0.27 m (middle of the box) and d dimensionless vertical position (y/L) at x = 0.42 m (right wall). Numerical results computed without (Simulation 1) and with (Simulation 2) discontinuity-capturing dissipation

measurements. Such differences can not be fully eliminated with just a friction model, because other aspects, such as surface tension (not included in the model), should also play a role.

In the computations reported above, the shampoo was considered a Newtonian fluid. A more realistic representation of the material behavior for the shampoo can be based on choosing a viscosity that decreases with the increasing shear rate. To evaluate this effect, an additional computation was carried out using a simple constitutive

model based on the power law of the shear rate. The preliminary results (not shown) did not show any quantitative changes.

We also carried out a 3D computation based on the conditions of Simulation 1. These results are included to briefly assess the performance of the proposed technique in a 3D case. The interface position is shown in Fig. 9, and we see that the interface evolution is close to what we obtained in the 2D analysis. This trend agrees with the fact that 3D effects were not observed in the experiments.

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5.2 Collapse of a water column

In this case the gate is assumed to be suddenly removed at time t = 0 s. Slip conditions are assumed at the solid surfaces. The fluid properties are:ρ1 = 1, 000 kg/m³and μ1

= 0.001 kg/m/s for the water, and ρ2 = 1 kg/m³ and μ2

= 0.001 kg/m/s for the air.

Two simulations are carried out: Simulations 1 and 2.

Both simulations include a simple turbulence model (defined by Eq. (7) and with Ct = 3.57). Simulation 2 also has the DCID (defined by Eq. (12) and with Cs = 10). The value

of the coefficient Cs was selected from numerical tests by trying to minimize the differences between the numerical predictions and the experimental measurements for both aspect ratios.

The mesh is composed of 100× 75 four-noded isoparametric elements. The time-step size is 0.001 s.

Column of aspect ratio Ar= 1 The experimental and numerical interface positions at various instants are shown in Fig. 10.

The cut-off value of the viscosity in Eq. (7) is set as μmax = 1.5 kg/m/s. In the experiments, it is observed that

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Fig. 12 Collapse of a water column with aspect ratio Ar = 2. Interface positions at different instants: experimental (left) and numerical results computed without (Simulation 1, middle) and with (Simulation 2, right) discontinuity-capturing dissipation

separated flows are developed close to the interface at specific instants (e.g., during the liquid impingement on the walls).

The separated fluid quickly merges with the bulk liquid and no bubbles are seen. However, bubbles are obtained in Sim- ulation 1 for t ≥ 0.6 s. Although they are caused due to the folding of the interface, they have no physical meaning beyond that. Note that the formulation does not include a bubble model. When the DCID is activated (Simulation 2), the formation of bubbles in the bulk liquid are inhibited and the interface is represented as a continuous front.

Figure 11 shows, for the experiments and Simulations 1 and 2, the evolutions of the interface horizontal position at the bottom of the box and the interface vertical position at the left wall, middle of the box and right wall. The results are in reasonably good agreement with the experimental data.

The computed vertical position of the interface at the left wall practically coincides with the experimental results up to t = 0.9 s. Additionally, the magnitude of its second maxi- mum is well described. The numerical behavior of the interface vertical position at the middle of the box also shows good

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agreement between the numerical and experimental results up to t = 0.6 s. After that, we see some discrepancies in the maximum values and a delay in the occurrence of the second maximum. The computed vertical interface position at the right wall captures reasonably well the instant of the first impingement (t = 0.25 s) and first maximum (t = 0.45 s).

The maximum value, however, is lower than that observed experimentally. The second maximum is described reasonably well in magnitude but has a delayed time.

Column of aspect ratio Ar=2 In this case μmax=3.0 kg/m/s.

Figure 12 shows the interface positions at various instants.

The time history of the interface horizontal position along the bottom of the box and the interface vertical position at selected locations are shown in Fig. 13.

The numerical results obtained in this case show trends that are similar to those observed in the previous analysis. In particular, the numerical evolutions of the interface positions at the bottom of the box and along the left wall

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satisfactorily match the corresponding measurements up to t = 1.5 s. Moreover, the time of the first impingement on the right wall is captured well. The amplitude of the reflected wave is predicted reasonably well but has a delayed time.

6 Conclusions

The performance of the ETILT was assessed in simulation of the collapse of liquid columns. A set of experiments have

been carried out to obtain the evolution of the interface position. Two different fluids, shampoo and water, and two different column aspect ratios, 1 and 2, were studied.

Simple concepts were included in the model to describe the wall friction and turbulence effects in the analyses of the shampoo and water, respectively. Additionally, the influence of some proposed stabilization parameters was assessed.

The wall friction effect is particularly apparent in the collapse of the shampoo column due to its high viscosity.

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No separated flow was observed in the experiments. This aspect was also seen in the simulations where neither turbulence model nor discontinuity-capturing dissipation was used. The effect of the gate opening was found not to change the overall behavior of the interface. In addition, the 3D analysis provides results similar to those computed under the 2D assumption.

In the case of the collapse of the water column, since it is practically an inviscid fluid, the friction wall effect does

not play much role in the analyses. No special model was included to represent the bubbles. Hence, once the bubbles are generated, the interface becomes unrealistic. The discontinuity-capturing dissipation included in the formulation helps to model the overall interface behavior reasonably well and inhibits the bubble formation in the bulk liquid. The turbulence model controls the height of the liquid columns on the walls and the wave evolution. A high mesh refinement is required to better represent the turbulence effects, and

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Fig. 13 Collapse of a water column with aspect ratio Ar = 2: a dimensionless horizontal position (x/L) at y = 0.0 m (bottom wall), b dimen- sionless vertical position (y/L) at x = 0.0 m (left wall), c dimensionless vertical position (y/L) at x = 0.27 m (middle of the box) and d dimensionless vertical position (y/L) at x = 0.42 m (right wall). Numerical results computed without (Simulation 1) and with (Simulation 2) discontinuity-capturing dissipation

reasonable results are obtained with the coefficients used in the turbulence model.

Overall, the numerical results compare satisfactorily with the measurements.

Acknowledgements The support provided by the Chilean Council of Research and Technology CONICYT (FONDECYT Projects No 1020029 and 7020029) and the Department of Technological and Scien- tific Research at the University of Santiago de Chile (DICYT-USACH) is gratefully acknowledged.

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