Large Eddy Simulation of High Atwood Number Rayleigh-Taylor Mixing

E3S Web of Conferences 128, 08001 (2019) https://doi.org/10.1051/e3sconf/201912808001 ICCHMT 2019

Large eddy simulation of high atwood number rayleigh-taylor mixing

1, Ilyas Yilmaz ⇤ 1Department of Mechanical Engineering, Faculty of Engineering and Natural Sciences, Istanbul Bilgi University, 34060, Eyup, Istanbul, Turkey

Abstract. Large eddy simulation of Rayleigh-Taylor instability at high Atwood numbers is performed using recently developed, kinetic energy-conserving, non-dissipative, fully-implicit, ﬁnite volume algorithm. The algorithm does not rely on the Boussinesq assumption. It also allows density and viscosity to vary. No interface capturing mechanism is requried. Because of its advanced features, unlike the pure incompressible ones, it does not su↵er from the loss of physical accuracy at high Atwood numbers. Many diagnostics including local mole fractions, bubble and spike growth rates, mixing eﬃciencies, Taylor micro-scales, Reynolds stresses and their anisotropies are computed to analyze the high Atwood number e↵ects. The density ratio dependence for the ratio of spike to bubble heights is also studied. Results show that higher Atwood numbers are characterized by increasing ratio of spike to bubble growth rates, higher speeds of bubble and especially spike fronts, faster development in instability, similarity in late time mixing values, and mixing asymmetry.

1 Introduction his paper was discussed by Burton[5]. Additionally, only the overall characteristics of growth and mixing were stud- Rayleigh-Taylor Instability (RTI) occurs when a heavy ied. Dimonte and Schneider[6] studied RTI up to A = 0.96 fluid of density ⇢h on top is supported against the gravity, using Linear Electric Motor. They provided scaling laws g, by a light fluid of density ⇢l on bottom [1–3]. When it between spike to bubble penetration ratio and heavy to is subject to perturbations, the fluids which are initially in light density ratio, and observed greater asymmetry at high hydrostatic equilibrium start to interpenetrate each other Atwood numbers. They also found that the bubble sizes due to the baroclinic vorticity generated by the opposite increase while the spikes become narrower. Large Eddy density and pressure gradients. It can be observed in flows Simulation (LES) of RTI up to A = 0.96 were also pre- such as type Ia supernovae, Inertial Confinement Fusion sented by Burton[5]. It was one of the most comprehensive (ICF), cavitation bubbles, oceanic and atmospheric cur- works on this subject and provided many diagnostics and rents and many other natural and engineering flows. In measures. A series of high-resolution simulations with an terms of flow modeling, RTI serves as very challenging aim to provide benchmark solutions for use in engineering test case for algorithms, as it includes many aspects such models were also performed by Youngs[7]. Recently, an- as body force treatment, turbulent mixing, di↵usion and other numerical study was reported by Shmony et. al.[8] interface capturing. RTI is characterized by the Atwood ↵ ⇢ ⇢ to examine the density ratio e ects on the mixing process number, A, which is given as A = h− l . Penetration of the ⇢h+⇢l and the e↵ective Atwood number. However, the literature light fluid into the heavy one as bubbles and penetration on this area is limited and not complete, and it is worth to of the heavy fluid into the light one as spikes can be mod- enhance it by adding more results obtained from advanced 2 eled in terms of the penetration lengths as hb = ↵bAgt and simulation methods. h = ↵ Agt2. ↵ and ↵ are the growth rates for bubble s − s b s The aim of this study is twofold. Firstly, to show that and spike, respectively. The mixing zone and its rate of the present LES algorithm is able to capture the evolution 2 change in time are then found as, h = hb hs = ↵Agt and characteristics of the RTI at moderate Atwood num- ˙ − and h = 2↵Agt, where ↵ = ↵b + ↵s is the total growth rate bers in a physically more correct way and then to investi- factor. gate the e↵ects of increasing Atwood numbers (i.e., higher In some flows in engineering and astrophysics, such density ratios) on the development of the RTI and con- as ICF and supernovae collapse, the Atwood number can tribute to the literature. reach very high values. Although RTI at low to interme- diate Atwood numbers have been investigated extensively, experimental and numerical studies on high and very high 2 Numerical Methodology Atwood number three-dimensional multi-mode RTI are Fully compressible, non-dimensional, Favre-weighted rarely available in the literature. It was first reported by Navier-Stokes equations with buoyancy term are solved Youngs[4]. Suitability of the numerical method used in using a recently proposed, non-dissipative, fully-implicit, ⇤e-mail: [email protected] kinetic energy conserving LES algorithm[9]. It is a

© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). E3S Web of Conferences 128, 08001 (2019) https://doi.org/10.1051/e3sconf/201912808001 ICCHMT 2019

second-order, pressure-correction type algorithm based on with a resolution of 64 192 64. H is set to 2⇡ . The ⇥ ⇥ an iterative predictor-corrector approach to update the flow interface is located at y=0. variables[10]. A pressure relaxation procedure was also The stream-wise and the span-wise boundary condi- added to remove the possible oscillations and speed-up tions are periodic. The no-slip wall boundary condition the solution[11]. The well-known Wall Adapting Local is applied to the velocity at top and bottom. The homo- Eddy (WALE) viscosity model was employed to model the geneous Neumann boundary condition for the thermody- subgrid-scale viscosity[12]. namic variables are used at walls. Usually, the interface is perturbed to trigger the instability. However, this may require an additional grid 3 Solver smoothing, if the amplitude of the perturbation is smaller An in-house, fully-implicit, fully-parallel, structured, than the grid resolution. In addition, a small initial pertur- single-block, finite-volume LES solver, written using bations for velocity may be necessary for consistency at PETSc[13] with Fortran syntax, is used. It was pre- the inter-facial region[16]. viously validated for various transitional and turbulent Jun et. al.[17] showed that directly perturbing the ve- flows. See for example [9, 11, 14, 15]. Incomplete LU locity field is superior to perturbing the interface position, preconditioned-GMRES with no fill-in, ILU(0), solves the since no smoothing is required and resolution-independent linear systems stemming from the implicit discretization. perturbations are obtained which yield almost identical Domain decomposition and load balancing are determined results. Following the latter way, perturbation is added at the beginning by PETSc. See [9] for other details. to the vertical velocity component as v0(y) = A0 Rand (1 + cos(2⇡y/Ly)) with the amplitude A0 is 0.005 and the random number, Rand, between -1 and +1. The pertur- 4 Problem setup bation spectrum is a white noise almost down to the grid scale, meaning that at early times high-k modes are dom- A series of runs are performed in order to properly ana- inant. The perturbations are also small amplitude, zone- lyze the Atwood number (i.e., density ratio) e↵ects. In to-zone perturbations and decrease toward to boundaries. each run, the flow is initialized with a specified Atwood The random number generator implemented in PETSc, number, ranging from 0.5 to 0.9, which corresponds to the which generates purely real random number and creates density ratios from 3 to 19 respectively. There are five At- normally (Gaussian) distributed values within the given wood numbers studied in total. range, is used to generate random values. The fluids are initially at rest with constant downward The non-dimensional time step is decreased from 1 gravitational acceleration and in hydrostatic equilibrium 3 4 5 5 ⇥ 10− (5 10− ⌧) to 5 10− (3.5 10− ⌧) as the Atwood p = p ⇢gy. p is the reference pressure. The refer- ⇥ ⇥ ⇥ 0 − 0 number increases. The simulations are carried out up to a ence density ⇢ is the light fluid density. The height of the 0 very late non-linear regime where spikes reach the domain domain is taken as the reference length l . The free-fall 0 boundaries. velocity is used as the reference velocity u0 = gl0. This leads to the reference time scale ⌧ = l0 . The Eckert Ag p 5 Results and discussion number can be approximated as Ec = (γq1)M2. The refer- − r p0 As the primary flow diagnostic, the local mole fraction ence Mach number, Mr, is also defined as, Mr = u0/ γ ⇢0 (LMF) field based on heavy fluid is computed as and is set to 0.1. p0 is computed using Mr. The referenceq dynamic viscosity µ here can be obtained using the fol- ⇢(x, y, z, t) ⇢l 0 χ(x, y, z, t) = − (1) lowing grid-dependent relation, µ = ⇢ ! Ag∆3, where ⇢ ⇢ 0 0 h − l ! is the coefficient taken as 0.21 for A = 0.5[18], and is p Then, it is averaged over the homogeneous directions to increased up to 3.6 for A = 0.9. These choices lead to the ⇢ u l give the mixing zone growth as scaling(working) Reynolds number, Re = 0 0 0 , ranging µ0 from 1.7 104 to 8 103. The Froude number, Fr, which is 1 ⇥ ⇥ χ(x, y, z, t) xz = χ(x, y, z, t) (2) the ratio of the reference velocity to the free-fall velocity h i N N x z N is equal to unity. The initial non-dimensional temperature Xt field, T , is found from the non-dimensional equation of Using the traditional measures based on LMF which are state. Then, the non-dimensional dynamic viscosity, µ, can χ(x, y, z, t) 1 ✏ and χ(x, y, z, t) ✏ where ✏ is h ixz  − h ixz ≥ be computed using µ = T n. The value of n is set to 0.71 traditionally set to 0.01, the bubble and spike penetration in the simulations. It is also possible to obtain the non- lengths are obtained, respectively. Now, it is easy to com- dimensional fields for T and µ at initial by dividing their pute the total extent of mixing zone, its rate of change in reference values. However it is not preferred. Since the ap- time, the corresponding growth rates and the total growth proach followed here allows not only the density but also rate from the equations given in Introduction. The mixing the temperature and the dynamic viscosity to vary during efficieny ✓ which quantifies the amount of mixing can also the simulation, and it is expected to give physically more be derived from LMF via accurate results, especially at high Atwood numbers. + 1 χ(1 χ) dy The domain size is H 3H H in the stream-wise, h − ixz ⇥ ⇥ ✓ = +−1 (3) the normal-wise and the span-wise directions respectively, R1 χ xz (1 χ) xzdy −1 h i h − i R 2 E3S Web of Conferences 128, 08001 (2019) https://doi.org/10.1051/e3sconf/201912808001 ICCHMT 2019 second-order, pressure-correction type algorithm based on with a resolution of 64 192 64. H is set to 2⇡ . The ⇥ ⇥ an iterative predictor-corrector approach to update the flow interface is located at y=0. 0.1 variables[10]. A pressure relaxation procedure was also The stream-wise and the span-wise boundary condi- added to remove the possible oscillations and speed-up tions are periodic. The no-slip wall boundary condition b α the solution[11]. The well-known Wall Adapting Local is applied to the velocity at top and bottom. The homo- 0.05 Eddy (WALE) viscosity model was employed to model the geneous Neumann boundary condition for the thermody- subgrid-scale viscosity[12]. namic variables are used at walls. A=0.5 A=0.6 Usually, the interface is perturbed to trigger the in- 0 stability. However, this may require an additional grid A=0.7 A=0.8 3 Solver smoothing, if the amplitude of the perturbation is smaller A=0.9 An in-house, fully-implicit, fully-parallel, structured, than the grid resolution. In addition, a small initial pertur- -0.05

s single-block, finite-volume LES solver, written using bations for velocity may be necessary for consistency at α PETSc[13] with Fortran syntax, is used. It was pre- the inter-facial region[16]. Jun et. al.[17] showed that directly perturbing the ve- -0.1 viously validated for various transitional and turbulent 1 2 3 4 5 flows. See for example [9, 11, 14, 15]. Incomplete LU locity field is superior to perturbing the interface position, t/τ preconditioned-GMRES with no fill-in, ILU(0), solves the since no smoothing is required and resolution-independent linear systems stemming from the implicit discretization. perturbations are obtained which yield almost identical results. Following the latter way, perturbation is added Figure 3. Evolution of growth rates in time for bubbles and Domain decomposition and load balancing are determined Evolution of local mole fraction iso-surfaces in time to the vertical velocity component as v (y) = A Rand Figure 1. spikes at di↵erent Atwood numbers at the beginning by PETSc. See [9] for other details. 0 0 (from top to bottom, t/⌧ = 1, 3, 5) at di↵erent Atwood numbers + ⇡y/ (1 cos(2 Ly)) with the amplitude A0 is 0.005 and the (from left to right, A=0.5,0.7,0.9), respectively. (χ = 0.5) random number, Rand, between -1 and +1. The pertur- 4 Problem setup bation spectrum is a white noise almost down to the grid is mainly dominated by di↵usive forces and a non-linear scale, meaning that at early times high-k modes are dom- A series of runs are performed in order to properly ana- phase characterized by strong interactions lead to transi- inant. The perturbations are also small amplitude, zone- lyze the Atwood number (i.e., density ratio) e↵ects. In 0.5 tion to turbulent. The spikes are the most a↵ected by the to-zone perturbations and decrease toward to boundaries. each run, the flow is initialized with a specified Atwood higher Atwood numbers (i.e., larger density ratios). For The random number generator implemented in PETSc, 0.4

number, ranging from 0.5 to 0.9, which corresponds to the b all the cases, the spikes penetrate faster than the bubbles. which generates purely real random number and creates 0.3 density ratios from 3 to 19 respectively. There are five At- The largest spike velocity is achieved at A = 0.9. h /h is normally (Gaussian) distributed values within the given s b wood numbers studied in total. 0.2 predicted as 1.24 at A = 0.5. This value is well-consistent range, is used to generate random values. A=0.5 The fluids are initially at rest with constant downward 0.1 A=0.6 with the previous numerical and experimental data that de- The non-dimensional time step is decreased from 1 gravitational acceleration and in hydrostatic equilibrium 3 4 5 5 ⇥ 0 A=0.7 fine a range of 1.2-1.3 [5, 16, 18]. The ratio increases 10− (5 10− ⌧) to 5 10− (3.5 10− ⌧) as the Atwood p = p ⇢gy. p is the reference pressure. The refer- ⇥ ⇥ ⇥ A=0.8 to 1.72 at A = 0.9. Burton[5] gives 1.71. The scaling 0 − 0 number increases. The simulations are carried out up to a -0.1 ence density ⇢0 is the light fluid density. The height of the A=0.9 between hs/hb and ⇢h/⇢l is also of interest. The power- very late non-linear regime where spikes reach the domain -0.2 0.18 domain is taken as the reference length l0. The free-fall law scaling relation found here is hs/hb = 1.02(⇢h/⇢l)

boundaries. h

s -0.3 velocity is used as the reference velocity u0 = gl0. This h which is in agreement with the two other results provided l0 -0.4 by Dimonte and Schneider[6] and Burton[5] as hs/hb = leads to the reference time scale ⌧ = Ag . Thep Eckert 0.33 0.18 5 Results and discussion -0.5 1.01(⇢h/⇢l) and as hs/hb = 0.96(⇢h/⇢l) respectively. 2 0 1 2 3 4 5 6 number can be approximated as Ec = (γq1)Mr . The refer- − t/τ p0 As the primary flow diagnostic, the local mole fraction ence Mach number, Mr, is also defined as, Mr = u0/ γ ⇢0 (LMF) field based on heavy fluid is computed as The corresponding growth rates are plotted in Fig.3. and is set to 0.1. p is computed using Mr. The reference For A = 0.5, ↵ is found as 0.027 at late times. The re- 0 q Figure 2. Evolution of bubble and spike penetration lengths in b dynamic viscosity µ here can be obtained using the fol- ⇢(x, y, z, t) ⇢l ported values are between 0.02 and 0.03 [5, 16, 18]. The 0 χ(x, y, z, t) = − (1) time at di↵erent Atwood numbers lowing grid-dependent relation, µ = ⇢ ! Ag∆3, where ⇢ ⇢ late time total growth rate (not shown here) is also ob- 0 0 h − l ! is the coefficient taken as 0.21 for A = 0.5[18], and is tained as 0.067 which falls into the range of 0.01-0.07 p Then, it is averaged over the homogeneous directions to increased up to 3.6 for A = 0.9. These choices lead to the [5, 16]. Unlike the many other numerical methods suf- ⇢ u l give the mixing zone growth as scaling(working) Reynolds number, Re = 0 0 0 , ranging which produces a value between 0 (immiscible) and 1 fering from excessive dissipation, the present algorithm µ0 from 1.7 104 to 8 103. The Froude number, Fr, which is 1 (complete mixing). takes the advantage of non-dissipativeness and produces a ⇥ ⇥ χ(x, y, z, t) xz = χ(x, y, z, t) (2) the ratio of the reference velocity to the free-fall velocity h i N N Fig.1 shows the evolution of RTI in terms of LMF iso- higher value close to the experimental data (i.e., the upper x z N is equal to unity. The initial non-dimensional temperature Xt surfaces. Three of the five Atwood numbers are selected limit). No significant e↵ect of the higher Atwood num- field, T , is found from the non-dimensional equation of Using the traditional measures based on LMF which are to represent. The development of RTI is clearly observed. bers on the bubble growth rate is observed. However, the state. Then, the non-dimensional dynamic viscosity, µ, can χ(x, y, z, t) 1 ✏ and χ(x, y, z, t) ✏ where ✏ is Initially small structures merge and evolve into large ones. spike growth rates increase slightly. A late time steady- h ixz  − h ixz ≥ be computed using µ = T n. The value of n is set to 0.71 traditionally set to 0.01, the bubble and spike penetration Higher Atwood numbers characterized by higher bubble state value is reached. in the simulations. It is also possible to obtain the non- lengths are obtained, respectively. Now, it is easy to com- and spike velocities. As a result, they reach boundaries In Fig.4, it is seen that mixing decreases rapidly during dimensional fields for T and µ at initial by dividing their pute the total extent of mixing zone, its rate of change in earlier. Additionally, the morphologies of the structures the di↵usive phase. Then, it starts to rise, beginning from reference values. However it is not preferred. Since the ap- time, the corresponding growth rates and the total growth change, as Atwood number increases. While the bub- the non-linear phase, and reaches a nearly steady value at proach followed here allows not only the density but also rate from the equations given in Introduction. The mixing bles are getting large in diameter, the spikes are becom- very late times. For A = 0.5 case, ✓ is around 0.85 con- the temperature and the dynamic viscosity to vary during efficieny ✓ which quantifies the amount of mixing can also ing narrower. This also agrees well with Dimonte and sistently with the values in the literature [5, 16, 19]. This the simulation, and it is expected to give physically more be derived from LMF via Schneider[6] and Burton[5]. result indicates that molecular mixing is well-captured by accurate results, especially at high Atwood numbers. + How the bubble and spike penetration lengths vary in the algorithm even on coarser grids compared to highly 1 χ(1 χ) xzdy The domain size is H 3H H in the stream-wise, h − i time is shown in Fig.2. It suggests that there two phases of resolved simulations. Fig.4 also suggests that ✓min moves ⇥ ⇥ ✓ = +−1 (3) the normal-wise and the span-wise directions respectively, R1 χ xz (1 χ) xzdy the development of the RTI: an initial linear phase which back to earlier times as A increases, and is getting smaller −1 h i h − i R 3 E3S Web of Conferences 128, 08001 (2019) https://doi.org/10.1051/e3sconf/201912808001 ICCHMT 2019

1 0.6 A=0.5 A=0.6 0.5 0.9 A=0.7 A=0.8 0.4 A=0.9 0.8

22 0.3 Θ A=0.5 b

0.7 A=0.6 A=0.7 0.2 A=0.8 0.6 A=0.9 0.1

0.5 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 t/τ t/τ

Figure 4. The mixing eﬃciency evolution in time at di↵erent Figure 6. Evolution of the vertical component of the anisotropy Atwood numbers tensor b22 in time

0.5 are the measures of the large scale anisotropy. They also A=0.5 also give an insight about the direction of the turbulent 0.4 A=0.6 momentum transport. 0.3 A=0.7 In Fig.5, vertical distribution of the root mean square 0.2 A=0.8 of the vertical component of the Reynolds stress tensor A=0.9 0.1 R22, which is a↵ected mostly, is plotted at the end of the

0 R 0 each simulation. As the Atwood number increases, 22 y/l value increases and the approximate symmetry observed -0.1 at lower Atwood numbers is broken. The peak locations -0.2 are also getting closer to the bottom wall and move into the -0.3 spike-dominated region. Considering the larger spike ve-

-0.4 locities and the penetration lengths at higher Atwood numbers, this result is in consistency with the previous obser- -0.5 0 0.1 0.2 0.3 √ vations presented here. Intensity of the spike interactions R22 with their surroundings is higher than those of the bubbles and this can be considered as the primary mechanism that Figure 5. Vertical distribution of the rms of R22 leads the instability to transition to turbulence and a state of turbulence eventually. Consistently, the largest variations are observed in b22 and smaller. In the non-linear phase, it recovers itself and which is given in Fig.6. As the Atwood number in- reaches a plateau around 0.8. Compared to the interme- creases, the initial fluctuations in di↵usive phase are get- diate Atwood numbers, the recovery period takes longer ting stronger and approach to the upper limit given as 2/3 (i.e., larger time-span). in the literature. However, at later times (i.e., in the non- In the second group of flow diagnostics, the Reynolds linear phase), b22 values reach a steady-state value around 0.3[21] stress tensor Rij, the anisotropy tensor bij, the potential and the kinetic energies are computed using the following How the ratio of the kinetic to the potential energy equations evolves in time is shown in Fig.7. After an initial set- tling period, a rapid increase is observed around t/⌧ = 1.5. ⇢ui00u00j xz R = h i (4) Then, at later times, as observed in many other flow diag- ij ⇢ h ixz nostics, an almost steady-state value around 0.4 is reached. Rij 1 This is in consistent with the range found in previous Di- bij = δij (5) 2ksgs − 3 rect Numerical Simulation (DNS) and LES studies which is 0.4 0.5 [5, 18, 20]. − PE(t) = [⇢(x, y, z, 0) ⇢(x, y, z, t)]gyd3x (6) − ZVol 1 6 Conclusions KE(t) = ⇢(u2 + v2 + w2)d3x (7) 2 ZVol RTI is simulated at high Atwood numbers using an respectively. ksgs is the turbulent kinetic energy. Rij and bij advanced, fully-featured LES algorithm. In addition

4 E3S Web of Conferences 128, 08001 (2019) https://doi.org/10.1051/e3sconf/201912808001 ICCHMT 2019

duce the penetration lengths (i.e., hs/hb ratios) and mix-

1 0.6 1.2 ing, as spikes and bubbles approach the boundaries, as also A=0.5 pointed out in [5]. A=0.6 0.5 1 A=0.5 Since this is an ongoing study, the preliminary results 0.9 A=0.7 A=0.6 presented here will be reﬁned more by performing addi- A=0.8 0.4 0.8 A=0.7 tional simulations especially at high Atwood numbers to A=0.9 0.8 A=0.8 improve the late time statistics and characteristics of RTI. A=0.9 22 0.3 0.6 Θ A=0.5 b

0.7 A=0.6 KE/ PE A=0.7 0.2 0.4 References A=0.8 0.6 A=0.9 0.1 0.2 [1] Rayleigh, J. W. S., 1883, Investigation of the Char- acter of the Equilibrium of an Incompressible Heavy

0.5 0 0 Fluid of Variable Density, Proc. London Math. Soc., 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 , 170-177. t/τ t/τ t/τ 14 [2] Taylor, G.I., 1950, The Instability of Liquid Surfaces when Accelerated in a Direction Perpendicular to their ﬃ ↵ Figure 4. The mixing e ciency evolution in time at di erent Figure 6. Evolution of the vertical component of the anisotropy Figure 7. Evolution of the energy ratio in time Planes, Proc. Roy. Soc. A, 201, 192-196. Atwood numbers tensor b in time 22 [3] Chandrasekhar, S., 1961, Hydrodynamic and Hydromagnetic Stability, International series of to the unique features of the algorithm such as non- monographs on physics, Clarendon Press. are the measures of the large scale anisotropy. They also 0.5 dissipativeness, kinetic energy conserving, fully-implicit [4] Youngs, D., 1991, Three-dimensional numerical sim- A=0.5 also give an insight about the direction of the turbulent ulation of turbulent mixing by Rayleigh-Taylor insta- 0.4 discretization, no dependence on Boussinesq assumption A=0.6 momentum transport. low-Mach number handling capability and variable prop- bility, Phys. Fluids A, 3, 1312-1320. 0.3 A=0.7 In Fig.5, vertical distribution of the root mean square erty, to the best knowledge of the author, this is the ﬁrst [5] Burton, G. C., 2011, Study of ultrahigh Atwood- 0.2 A=0.8 of the vertical component of the Reynolds stress tensor time the WALE subgrid-scale model is applied to multi- number Rayleigh–Taylor mixing dynamics using the A=0.9 0.1 R22, which is a↵ected mostly, is plotted at the end of the mode RTI at high Atwood numbers. It should also be con- nonlinear large-eddy simulation method, Phys. of Flu-

0 R 0 each simulation. As the Atwood number increases, 22 sidered that the grid resolution is coarser than many other ids, 23, 045106. y/l value increases and the approximate symmetry observed [6] Dimonte, G. and Schneider, M., 2000, Density ratio -0.1 numerical studies previously reported. at lower Atwood numbers is broken. The peak locations The Atwood numbers studied are ranging from inter- dependence of Rayleigh-Taylor mixing for sustained -0.2 are also getting closer to the bottom wall and move into the mediate to very high values which are 0.5, 0.6, 0.7, 0.8 and impulsive acceleration histories, Physics of Flu- -0.3 spike-dominated region. Considering the larger spike ve- and 0.9. The corresponding density ratios are 3, 4, 5 .7, 9 ids, 12, 304-321. -0.4 locities and the penetration lengths at higher Atwood num- and 19. Such high density ratios may be observed in , for [7] Youngs, D., 2013, The density ratio dependence bers, this result is in consistency with the previous obser- -0.5 example, ICF and astrophysical flows. of self-similar Rayleigh-Taylor mixing, Philosophical 0 0.1 0.2 0.3 √R vations presented here. Intensity of the spike interactions 22 Many flow diagnostics are computed and used to an- transactions. Series A, Mathematical, physical, and with their surroundings is higher than those of the bubbles alyze the complex evolution of the instability. Due to the engineering sciences, 371, 20120173. and this can be considered as the primary mechanism that limited avaliability of space, some of the results are not [8] Shimony A., Malamud G. and Shvarts D., 2017, Den- Figure 5. Vertical distribution of the rms of R22 leads the instability to transition to turbulence and a state presented here. Again, some of them are computed for the sity Ratio and Entrainment E↵ects on Asymptotic of turbulence eventually. first time at high Atwood numbers, such as the Reynolds Rayleigh–Taylor Instability. ASME. J. Fluids Eng., b Consistently, the largest variations are observed in 22 stresses, and these may provide some new insights on the 140, 050906-050906-8. and smaller. In the non-linear phase, it recovers itself and which is given in Fig.6. As the Atwood number in- e↵ects of high Atwood numbers. [9] Yilmaz, I., Saygin, H. and Davidson, L., 2018, Ap- creases, the initial fluctuations in di↵usive phase are get- reaches a plateau around 0.8. Compared to the interme- It is shown that the algorithm is capable of repropduc- plication of a parallel solver to the LES modelling of ting stronger and approach to the upper limit given as 2/3 diate Atwood numbers, the recovery period takes longer ing the complex physics and characteristics of the RTI at turbulent buoyant flows with heat transfer, Progress in in the literature. However, at later times (i.e., in the non- (i.e., larger time-span). A = 0.5. The results are successfully compared with the Computational Fluid Dynamics, an International Jour- linear phase), b values reach a steady-state value around In the second group of flow diagnostics, the Reynolds 22 previous experimental and numerical data where available. nal, 18, 89-107. 0.3[21] stress tensor Rij, the anisotropy tensor bij, the potential They also suggest that the evolution of the RTI has multi- [10] Hou, Y., Mahesh K., 2005, A robust, colocated, How the ratio of the kinetic to the potential energy and the kinetic energies are computed using the following ple stages. Following an early stage di↵usive growth, a implicit algorithm for direct numerical simulation of evolves in time is shown in Fig.7. After an initial set- equations non-linear interactions take place and lead the instability compressible, turbulent flows, Journal of Computa- ⇢ tling period, a rapid increase is observed around t/⌧ = 1.5. ui00u00j xz to a state of turbulence. tional Physics, 205, 205-221. Rij = h i (4) Then, at later times, as observed in many other flow diag- ⇢ It is observed that higher Atwood numbers are charac- [11] Yilmaz, I., Edis, F.O., Saygin, H. and Davidson, L., h ixz nostics, an almost steady-state value around 0.4 is reached. terized by increasing ratio of spike to bubble growth rates, 2014, Parallel implicit DNS of temporally-evolving Rij 1 This is in consistent with the range found in previous Di- b = δ (5) turbulent shear layer instability, Journal of Computa- ij ij rect Numerical Simulation (DNS) and LES studies which higher speeds of bubble and especially spike fronts, faster 2ksgs − 3 tional and Applied Mathematics, , 651-659. is 0.4 0.5 [5, 18, 20]. development in instability, large scale anisotropy, higher 259 − rates of interactions around spikes, similarity in late time [12] Nicoud, F., Ducros, F., 1999, Subgrid-scale stress PE(t) = [⇢(x, y, z, 0) ⇢(x, y, z, t)]gyd3x (6) Vol − values of mixing and some other measures, change in bub- modelling based on the square of the velocity gradi- Z ble and spike morphologies, and mixing asymmetry. A ent tensor, Flow, Turbulence and Combustion,62, 183- 1 6 Conclusions KE(t) = ⇢(u2 + v2 + w2)d3x (7) scaling relation between hs/hb and ⇢h/⇢l is also provided. 200. 2 ZVol RTI is simulated at high Atwood numbers using an Another observation is the e↵ect of solid walls on [13] Balay, S., Gropp, W. D., McInnes, L. C. and Smith, respectively. ksgs is the turbulent kinetic energy. Rij and bij advanced, fully-featured LES algorithm. In addition the development of the instability. This e↵ect may re- B. F., 2009, PETSc Users Manual, Technical Report

5 E3S Web of Conferences 128, 08001 (2019) https://doi.org/10.1051/e3sconf/201912808001 ICCHMT 2019

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