# The Zoo of Modes of Convection in Liquids Vibrated along the Direction of the Temperature Gradient

^{*}

## Abstract

**:**

_{ω}= 10

^{4}to Ra

_{ω}= 10

^{6}. It is shown that a kaleidoscope of possible variants exist whose nature and variety calls for the simultaneous analysis of their temporal and spatial behavior, thermofluid-dynamic (TFD) distortions, and the Nusselt number, in synergy with existing theories on the effect of periodic accelerations on fluid systems.

## 1. Introduction

## 2. Mathematical Model

#### 2.1. The Geometry

#### 2.2. Balace Equations and Boundary Conditions

^{2}/α, α/L, ρα

^{2}/L

^{2},and ΔT, respectively (where α is the fluid thermal diffusivity and ρ is the fluid density), the balance equations for mass, momentum, and energy can be cast in compact form as:

^{2}in place of the standard gravity g).

#### 2.3. Embedded Symmetries

- (ss): The symmetric–symmetric mode. This mode is characterized by an even number of rolls along the two coordinate axes. It reduces to a configuration with the central symmetry if the same number (m) of rolls affects both the x and y directions, i.e., m
_{x}= m_{y}(whereas a columnar arrangement is obtained if m_{y}> m_{x}). - (sa): The symmetric–antisymmetric mode. This mode displays symmetry only with respect to the y-axis; accordingly, the flow typically features an odd number of rolls along y and an even number of rolls along the other axis.
- (as): The antisymmetric–symmetric mode. This mode displays symmetry only with respect to the x-axis; accordingly, the flow typically features an odd number of rolls along x and an even number of rolls along the other axis.
- (aa): The antisymmetric–antisymmetric mode. No symmetry is retained in this case, as the number of rolls is odd along both axes (a single column being obtained for m
_{y}> m_{x}= 1).

## 3. Numerical Method

## 4. Validation

^{4}, (b) Ω = 500, $R{a}_{\omega}$ = 10

^{5}, and (c) Ω = 200, $R{a}_{\omega}$ = 10

^{5}(the velocity signals for these cases are readily available to the reader in the original study, making the comparison with the current results straightforward). As the reader will realize by inspecting Figure 2, the present results are in excellent agreement with the original signals reported by Hirata et al. [31], both in shape and periodicity.

## 5. Grid Refinement Study

## 6. Results

^{6}.

#### 6.1. Regime Classification

^{4}≤ $R{a}_{\omega}$ ≤ 10

^{5}, the constraint on the upper value being essentially an outcome of the limited computational resources available at that time). This figure is instructive also for another reason. It shows the well-known stabilization of thermovibrational flow when the frequency of vibrations is increased (Simonenko and Zen’kovskaja [50]; Simonenko [51]; Gershuni and Zhukhovitskii [52]; Gershuni et al. [53]; Gershuni and Zhukhovitskii [54]); we will come back to this fundamental concept later.

^{3}, with the addition of a very high value of Ω = 10

^{4}. As explained before, a fluid with Pr = 15 has been considered in place of Pr = 7. Another distinguishing mark of the present analysis is the extension to $R{a}_{\omega}$ = 10

^{6}.

^{5}< $R{a}_{\omega}$ < 10

^{6}). These new findings have implicitly led to the need to introduce a distinction within the synchronous regime whereby a flow may be synchronous and periodic or synchronous and non-periodic (SY-P or SY-NP). Relevant examples of such new solutions are shown in Figure 5a,b, respectively. In particular, Figure 5a relates to the circumstances where the velocity signal is identical over each period for all periods (therefore, the signal can be considered synchronous and periodic). By contrast, although the signal shown in Figure 5b is still synchronous in time with the applied forcing (i.e., the vibrations), it also exhibits turbulent busts every period; moreover, the signal is not periodic, as the bursts display a more or less erratic evolution in time.

#### 6.2. Map Extension

^{4}), the predominant flow regime apparent for high vibrational Rayleigh numbers ($R{a}_{\omega}$ > 5 × 10

^{5}) is the aforementioned SY-NP case denoted by . The red sinusoidal signal included in the insets represents the forcing applied to the system; it is instrumental in making evident that the turbulent bursts for the SY-NP mode occur synchronously with the forcing but do not display the same behavior for each period. These solutions are less ordered than those found for the lower vibrational Rayleigh numbers whereby they appear in less organized clusters. The appearance of the SY-P regime is more sporadic. However, this mode of convection also exists at extremely high Rayleigh numbers ($R{a}_{\omega}$ = 10

^{6}), which indicates that an increase in $R{a}_{\omega}$ does not systematically lead to a more chaotic system (in some regions of parameters (Ω = 10

^{3}), the flow reverts from an NS-NP back to an SY-P state).

#### 6.3. Themofluid-Dynamic Disturbances

_{diff}represents the temperature field that would be established in the absence of convection (in other words, a purely diffusive temperature profile, which using the reference system indicated in Figure 1 would simply read T

_{diff}= y).

_{o}≤ t ≤ t

_{o}+ τ (where τ=2π/Ω)

^{5}for which the TFD

_{averaged}tends to 0).

^{4}, i.e., when the Gershuni regime is approached (Savino and Lappa [55]). As originally argued by Birikh at al. [56], indeed, in the limit as the frequency tends to infinite, if temperature distortions with respect to the purely diffusive case are present, the major role of the mean vibration force is that of forcing isotherms to turn and become perpendicular to the vibration direction. To elucidate further the significance of this observation, one should keep in mind that in other words, this simply means that an intrinsic property of thermovibrational convection induced by vibrations parallel to the imposed temperature difference is to tend naturally to a quiescent thermally diffusive state as Ω is increased (which provides the sought physical justification for the ST states reported in the existence map).

#### 6.4. Evaluation of the Nusselt Number

_{max}= max (Nu) for t

_{o}≤ t ≤ t

_{o}+ τ.

^{5}and the SY-NP and NS-NP regimes are considered. As witnessed by Figure 10, a remarkable decrease in Nu

_{max}occurs for Ω = 1000 (Nu

_{max}ideally tending to 1 in the limit as Ω→∞). However, as still evident in this figure, a peak is located Ω = 100. For all values of $R{a}_{\omega}$ at low frequencies (Ω < 20), Nu

_{max}remains constant; then, it grows for intermediate frequencies (50 < Ω < 100) and finally decreases for high values of Ω (Ω > 1000).

_{max}tends to become higher for all values of Ω (Figure 11).

#### 6.5. Streamlines and Patterning Behaviors

^{5}< $R{a}_{\omega}$ ≤ 10

^{6}, as these circumstances were not covered in the earlier study by Hirata et al. [31].

^{6}, Ω = 10

^{4}, i.e., a condition for which the flow is almost negligible (“stable state”). As shown by Figure 12, it manifests itself as an (ss) convective mode characterized by two rolls along each coordinate axis. This extremely weak flow starts as a four-roll configuration. From there, small rolls nucleate at the corners of the cavity and, as time passes, they tend to merge with their respective neighbors until the original quadrupolar arrangement is recovered. This nucleation occurs twice in the space of a period rapidly regaining the four-roll configuration. The periodicity of this evolutionary scenario is consistent with the velocity signal (which is sinusoidal and synchronous with the forcing period).

^{5}. As illustrated in Figure 13, this regime presents periodically identical instantaneous velocity fields and streamlines. In this case, the quadrupolar (four-roll) configuration is interrupted at each period by the genesis of two small rolls in the center of the lower part of the cavity, which are eventually flattened, hence allowing the flow to return to the original pattern.

^{5}, Ω = 100 is taken as a representative example (Figure 15). As a fleeting glimpse into this figure would immediately confirm, the fluid becomes almost quiescent over a fixed sub-interval of each period.

## 7. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Nomenclature | |

b | Vibration amplitude |

Nu | Nusselt number |

p | Pressure |

Pr | Prandtl number |

Ra | Rayleigh number |

s | Displacement |

T | Temperature |

t | Time |

u | Velocity component along x |

V | Velocity |

v | Velocity component along y |

x | Horizontal coordinate |

y | Vertical coordinate |

Greek Symbols | |

α | Thermal diffusivity |

β_{T} | Thermal expansion coefficient |

ν | Kinematic viscosity |

ρ | Fluid density |

ω | Dimensional angular frequency |

Ω | Non-dimensional angular frequency |

ΔT | Temperature difference |

τ | Non-dimensional period of vibrations |

δ | Thickness |

ζ | Kolmogorov length scale |

Subscripts | |

BL | Boundary layer |

Cold | Cold |

diff | Diffusive |

Hot | Hot |

max | Maximum |

Superscripts | |

Lab | Laboratory |

## References

- Hadley, G. Concerning the cause of the general trade winds. Philos. Trans. R. Soc. Lond.
**1735**, 29, 58–62. [Google Scholar] - Mizushima, J. Onset of thermal convection in a finite two–dimensional box. J. Phys. Soc. Jpn.
**1995**, 64, 2420–2432. [Google Scholar] [CrossRef] - Mizushima, J.; Adachi, T. Sequential Transitions of the Thermal Convection in a Square Cavity. J. Phys. Soc. Jpn.
**1997**, 66, 79–90. [Google Scholar] [CrossRef] - Goldhirsch, I.; Pelz, R.B.; Orszag, S.A. Numerical simulation of thermal convection in a two-dimensional finite box. J. Fluid Mech.
**1989**, 199, 1–28. [Google Scholar] [CrossRef] - Villermaux. Memory-induced low frequency oscillations in closed convection boxes. Phys. Rev. Lett.
**1995**, 75, 4618–4621. [Google Scholar] [CrossRef] - Kadanoff, L.P. Turbulent Heat Flow: Structures and Scaling. Physics Today
**2001**, 54, 34–39. [Google Scholar] [CrossRef] - Lappa, M. Some considerations about the symmetry and evolution of chaotic Rayleigh–Bénard convection: The flywheel mechanism and the “wind” of turbulence. C. R. Mécanique
**2011**, 339, 563–572. [Google Scholar] [CrossRef] - Monti, R.; Langbein, D.; Favier, J.J. Influence of residual accelerations on fluid physics and material science experiments. In Fluid and material Science in Space: A European Perspective; Walter, H.U., Ed.; Springer: Berlin, Germany, 1987; Chapter XVIII; pp. 637–680. [Google Scholar]
- Alexander, J.I.D. Low gravity experiment sensitivity to residual acceleration: A review. Microgravity Sci. Technol.
**1990**, 3, 52–68. [Google Scholar] - Alexander, J.I.D.; Ouazzani, J.; Rosenberger, F. Analysis of the low gravity tolerance of Bridgman-Stockbarger crystal growth, II. Transient and periodic accelerations. J. Cryst. Growth
**1991**, 113, 21–38. [Google Scholar] [CrossRef] - Alexander, J.I.D.; Garandet, J.-P.; Favier, J.J.; Lizee, A. g-jitter effects on segregation during directional solidification of tin-bismuth in the MEPHISTO furnace facility. J. Cryst. Growth
**1997**, 178, 657–661. [Google Scholar] - Feonychev, A.I.; Dolgikh, G.A. Influence of vibration on heat and mass transfer in microgravity conditions. Microgravity Q.
**1994**, 4, 233–240. [Google Scholar] - Gershuni, G.Z.; Lyubimov, D.V. Thermal Vibrational Convection; Wiley: Chichester, UK, 1998. [Google Scholar]
- Monti, R.; Savino, R.; Lappa, M. Microgravity sensitivity of typical fluid physics experiment. In Proceedings of the 17th Microgravity Measurements Group Meeting, Cleveland, OH, USA, 24–26 March 1998; Published in the Meeting Proceedings in NASA CP-1998-208414. pp. 1–15. [Google Scholar]
- Naumann, R.J. An analytical model for transport from quasi-steady and periodic accelerations on spacecraft. Int. J. Heat Mass Transf.
**2000**, 43, 2917–2930. [Google Scholar] [CrossRef] - Mialdun, A.; Ryzhkov, I.I.; Melnikov, D.E.; Shevtsova, V. Experimental Evidence of Thermal Vibrational Convection in a Nonuniformly Heated Fluid in a Reduced Gravity Environment. Phys. Rev. Lett.
**2008**, 101, 084501. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Melnikov, D.; Ryzhkov, I.; Mialdun, A.; Shevtsova, V. Thermovibrational Convection in Microgravity: Preparation of a Parabolic Flight Experiment. Microgravity Sci. Technol.
**2008**, 20, 29–39. [Google Scholar] [CrossRef] - Lyubimova, T.P.; Perminov, A.V.; Kazimardanov, M.G. Stability of quasi-equilibrium states and supercritical regimes of thermal vibrational convection of a Williamson fluid in zero gravity conditions. Int. J. Heat Mass Transf.
**2019**, 129, 406–414. [Google Scholar] [CrossRef] - Bouarab, S.; Mokhtari, F.; Kaddeche, S.; Henry, D.; Botton, V.; Medelfef1, A. Theoretical and numerical study on high frequency vibrational convection: Influence of the vibration direction on the flow structure. Phys. Fluids
**2019**, 31, 043605. [Google Scholar] [CrossRef] [Green Version] - Shevtsova, V.; Lyubimova, T.; Saghir, Z.; Melnikov, D.; Gaponenko, Y.; Sechenyh, V.; Legros, J.C.; Mialdun, A. IVIDIL: On-board g-jitters and diffusion controlled phenomena. J. Phys. Conf. Ser.
**2011**, 327, 012031. [Google Scholar] [CrossRef] - Shevtsova, V.; Mialdun, A.; Melnikov, D.; Ryzhkov, I.; Gaponenko, Y.; Saghir, Z.; Lyubimova, T.; Legros, J.C. The IVIDIL experiment onboard the ISS: Thermodiffusion in the presence of controlled vibrations. C. R. Mécaniq
**2011**, 339, 310–317. [Google Scholar] [CrossRef] - Maryshev, B.; Lyubimova, T.; Lyubimov, D. Two-dimensional thermal convection in porous enclosure subjected to the horizontal seepage and gravity modulation. Phys. Fluids
**2013**, 25, 084105. [Google Scholar] [CrossRef] - Vorobev, A.; Lyubimova, T. Vibrational convection in a heterogeneous binary mixture. Part I. Time-averaged equations. J. Fluid Mech.
**2019**, 870, 543–562. [Google Scholar] [CrossRef] [Green Version] - Lappa, M. Control of convection patterning and intensity in shallow cavities by harmonic vibrations. Microgravity Sci. Technol.
**2016**, 28, 29–39. [Google Scholar] [CrossRef] [Green Version] - Lappa, M. The patterning behavior and accumulation of spherical particles in a vibrated non-isothermal liquid. Phys. Fluids
**2014**, 26, 093301. [Google Scholar] [CrossRef] [Green Version] - Lappa, M. Numerical study into the morphology and formation mechanisms of threedimensional particle structures in vibrated cylindrical cavities with various heating conditions. Phys. Rev. Fluids
**2016**, 1, 25. [Google Scholar] [CrossRef] [Green Version] - Lappa, M. On the multiplicity and symmetry of particle attractors in confined non-isothermal fluids subjected to inclined vibrations. Int. J. Multiphase Flow
**2017**, 93, 71–83. [Google Scholar] [CrossRef] [Green Version] - Lappa, M. On the formation and morphology of coherent particulate structures in non-isothermal enclosures subjected to rotating g-jitters. Phys. Fluids
**2019**, 31, 11. [Google Scholar] [CrossRef] [Green Version] - Lappa, M.; Burel, T. Symmetry Breaking Phenomena in Thermovibrationally Driven Particle Accumulation Structures. Phys. Fluids
**2020**, 32, 23. [Google Scholar] [CrossRef] - Lappa, M. Thermal Convection: Patterns, Evolution and Stability; John Wiley & Sons, Ltd.: Chichester, UK, 2009. [Google Scholar]
- Hirata, K.; Sasaki, T.; Tanigawa, H. Vibrational effects on convection in a square cavity at zero gravity. J. Fluid Mech.
**2001**, 445, 327–344. [Google Scholar] [CrossRef] - Harlow, F.H.; Welch, J.E. Numerical calculation of time-dependent viscous incompressible flow with free surface. Phys. Fluids
**1965**, 8, 2182–2189. [Google Scholar] [CrossRef] - Chorin, A.J. Numerical solutions of the Navier-Stokes equations. Math. Comput.
**1968**, 22, 745–762. [Google Scholar] [CrossRef] - Temam, R. Sur l’approximation de la solution des èquations de Navier-Stokes par la mèthode des pas fractionnaires (I). Arch. Rat. Mech. Anal.
**1969**, 33, 377–385. [Google Scholar] [CrossRef] - Gresho, P.M.; Sani, R.T. On pressure boundary conditions for the incompressible Navier-Stokes equations. Int. J. Numer. Meth. Fluids
**1987**, 7, 1111–1145. [Google Scholar] [CrossRef] - Gresho, P.M. Incompressible fluid dynamics: Some fundamental formulation issues. Ann. Rev. Fluid Mech.
**1991**, 23, 413–453. [Google Scholar] [CrossRef] - Guermond, J.-L.; Quartapelle, L. On stability and convergence of projection methods based on pressure Poisson equation. Int. J. Numer. Meth. Fluids
**1998**, 26, 1039–1053. [Google Scholar] [CrossRef] [Green Version] - Guermond, J.-L.; Minev, P.; Shen, J. An Overview of Projection Methods for Incompressible Flows. Comput. Methods. Comput. Methods Appl. Mech. Eng.
**2006**, 195, 6011–6045. [Google Scholar] [CrossRef] [Green Version] - Ladyzhenskaya, O.A. The Mathematical Theory of Viscous Incompressible Flow, 2nd ed.; Gordon and Breach: New York, NY, USA; London, UK,, 1969. [Google Scholar]
- Lappa, M. Strategies for parallelizing the three-dimensional Navier-Stokes equations on the Cray T3E. In Science and Supercomputing at CINECA; Voli, M., Ed.; CINECA: Bologna, Italy, 1997; Volume 11, pp. 326–340. ISBN 10:88-86037-03-1. [Google Scholar]
- Lappa, M.; Boaro, A. Rayleigh-Bénard convection in viscoelastic liquid bridges. J. Fluid Mech.
**2020**, 904. [Google Scholar] [CrossRef] - Issa, R.I. Solution of the implicitly discretized fluid flow equations by operator-splitting. J. Comp. Phys.
**1986**, 62, 40–65. [Google Scholar] [CrossRef] - Rhie, C.M.; Chow, W.L. Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA J.
**1983**, 21, 1525–1532. [Google Scholar] [CrossRef] - Lappa, M.; Inam, S. Thermogravitational and hybrid convection in an obstructed compact cavity. Int. J. Thermal Sci.
**2020**, 156, 21. [Google Scholar] [CrossRef] - Ouertatani, N.; Cheikh, N.B.; Beya, B.B.; Lili, T. Numerical simulation of two-dimensional Rayleigh-Benard convection in an enclosure. C. R. Mecanique
**2008**, 336, 464–470. [Google Scholar] [CrossRef] - Soong, C.Y.; Tzeng, P.Y.; Chiang, D.C.; Sheu, T.S. Numerical study on mode-transition of natural convection in differentially heated inclined enclosures. Int. J. Heat Mass Transf.
**1995**, 39, 2869–2882. [Google Scholar] [CrossRef] - Russo, G.; Napolitano, L.G. Order of Magnitude Analysis of unsteady Marangoni and Buoyancy free convection. In Proceedings of the 35th Congress of the International Astronautical Federation, Lausanne, Switzerland, 7–13 October 1984. [Google Scholar]
- Shishkina, O.; Stevens, R.J.A.M. Grossmann, S.; Lohse, D. Boundary layer structure in turbulent thermal convection and its consequences for the required numerical resolution. New J. Phys.
**2010**, 12, 17. [Google Scholar] [CrossRef] - De, A.K.; Eswaran, V.; Mishra, P.K. Scalings of heat transport and energy spectra of turbulent Rayleigh-Bénard convection in a large-aspect-ratio box. Int. J. Heat Fluid Flow
**2017**, 67, 111–124. [Google Scholar] [CrossRef] [Green Version] - Simonenko, I.B.; Zen’kovskaja, S.M. On the effect of highfrequency vibrations on the origin of convection. Izv. Akad. Nauk SSSR. Ser. Meh. Zhidk. Gaza
**1966**, 5, 51–55. [Google Scholar] - Simonenko, I.B. A justification of the averaging method for a problem of convection in a field rapidly oscillating forces and other parabolic equations. Mat. Sb.
**1972**, 129, 245–263. [Google Scholar] [CrossRef] - Gershuni, G.Z.; Zhukhovitskii, E.M. Free thermal convection in a vibrational field under conditions of weightlessness. Sov. Phys. Dokl.
**1979**, 24, 894–896. [Google Scholar] - Gershuni, G.Z.; Zhukhovitskii, E.M.; Yurkov Yu, S. Vibrational thermal convection in a rectangular cavity. Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza
**1982**, 4, 94–99. [Google Scholar] [CrossRef] - Gershuni, G.Z.; Zhukhovitskii, E.M. Vibrational thermal convection in zero gravity. Fluid Mech. Sov. Res.
**1986**, 15, 63–84. [Google Scholar] - Savino, R.; Lappa, M. Assessment of the thermovibrational theory: Application to g-jitter on the Space-station. J. Spacecraft Rockets
**2003**, 40, 201–210. [Google Scholar] [CrossRef] - Birikh, R.V.; Briskman, V.A.; Chernatynski, V.I. Roux, B. Control of thermocapillary convection in a liquid bridge by high frequency vibrations. Microgravity Q.
**1993**, 3, 23–28. [Google Scholar] - Hof, B.; Lucas, G.J.; Mullin, T. Flow state multiplicity in convection. Phys. Fluids
**1999**, 11, 2815–2817. [Google Scholar] [CrossRef] - Leong, S.S. Numerical study of Rayleigh-Bénard convection in a cylinder. Numer. Heat Transf. Part A
**2002**, 41, 673–683. [Google Scholar] [CrossRef] - Lappa, M. On the Nature of Fluid-dynamics. In Understanding the Nature of Science; Lindholm, P., Ed.; Science, Evolution and Creationism, BISAC: SCI034000; Nova Science Publishers Inc.: Hauppauge, NY, USA, 2019; Chapter 1; pp. 1–64. ISBN 978-1-53616-016-1. Available online: https://novapublishers.com/shop/understanding-the-nature-of-science/ (accessed on 10 October 2020).
- Gershuni, G.Z.; Zhukhovitskii, E.M. Convective instability of a fluid in a vibration field under conditions of weightlessness. Fluid Dyn.
**1981**, 16, 498–504. [Google Scholar] [CrossRef]

**Figure 1.**Square cavity with characteristic size L, delimited by solid walls (one at y = 0 cooled, the other at y = 1 heated, perfectly conducting conditions on the remaining sidewalls: T = y for x = 0 and x = 1).

**Figure 2.**Time evolution of velocity at (

**a**) Ω = 200, $R{a}_{\omega}$ = 7 × 10

^{4}, (

**b**) Ω = 500, $R{a}_{\omega}$ = 10

^{5}, and (

**c**) Ω = 200, $R{a}_{\omega}$ = 10

^{5}, respectively. Corresponding to cases (b), (c), and (d) of Figure 4 in Hirata et al. [31].

**Figure 3.**Convergence of the thermofluid-dynamic disturbances as a result of grid refinement for the case Pr = 7, $R{a}_{\omega}$ = 10

^{6}, Ω = 100.

**Figure 4.**Periodicity map for Pr = 7: Syncronous case (SY); 1/2 Subharmonic case (SU); Non-periodic case (NP), Stable case (ST). The shaded area represents the cases where the flow is stationary over a certain time interval. Figure after [31].

**Figure 5.**Cases $R{a}_{\omega}$ = 2 × 10

^{5}, Ω = 50 and $R{a}_{\omega}$ = 7 × 10

^{5}, Ω = 100 respectively (

**a**) shows the case synchronous and periodic(SY-P) and (

**b**) the case synchronous and non-periodic(SY-NP).

**Figure 6.**Response of the velocity field to the imposed periodic acceleration (Pr = 15): Synchronous and periodic case (SY-P); 1/2 Subharmonic case (SU); Synchronous and non-periodic case (SY-NP); Non-periodic and non-synchronous case (NP-NS); Stable case (ST) (quiescent states). The shaded area represents the cases where the flow is stationary over a certain sub-interval of the period of the applied vibrations.

**Figure 8.**Influence of Ω on the global thermofluid-dynamic (TFD) disturbances (the dashed and solid lines indicating instantaneous and time-averaged variants, respectively).

**Figure 9.**Influence of $R{a}_{\omega}$ on the global thermofluid-dynamic (TFD) disturbances (the dashed and solid lines indicating instantaneous and time-averaged variants, respectively).

**Figure 10.**Influence of Ω on the maximum Nusselt number (Nu

_{max}) across the heated wall of the cavity.

**Figure 11.**Influence of $R{a}_{\omega}$ on the maximum Nusselt number (Nu

_{max}) across the heated wall of the cavity.

**Figure 12.**Instantaneous patterning behavior for the case ST, where it is shown that the nucleation of the external rolls occurs at approximately 0.3τ and 0.8τ and that the four-roll configuration is re-established fully when the acceleration tends to zero.

**Figure 13.**Instantaneous streamlines and velocity magnitude over two periods for the case $R{a}_{\omega}$ = 3.5 × 10

^{5}, Ω = 200 (SY-P) accompanied by the velocity signal. The 12 red dots represent the time at which the snapshots are taken (six snapshots for each period).

**Figure 14.**Instantaneous streamlines and velocity magnitude over two periods of forcing (panel (

**a**): first period, panel (

**b**): second period) for the case $R{a}_{\omega}$ = 3.5 × 10

^{5}, Ω = 500 (SU), accompanied by the velocity signal. The 24 red dots represent the time at which the snapshots are taken (six snapshots for each period).

**Figure 15.**Instantaneous streamlines and velocity magnitude over two periods for the case $R{a}_{\omega}$ = 8.5 × 10

^{5}, Ω = 100, accompanied by the velocity signal. The 12 red dots represent the time at which the snapshots are taken (six snapshots for each period).

**Figure 16.**Instantaneous streamlines and velocity magnitude over two periods for the case $R{a}_{\omega}$ = 10

^{6}, Ω = 500 (NS-NP), which are accompanied by the velocity signal. The 12 red dots represent the time at which the snapshots are taken (six snapshots for each period).

Criteria | Theoretical Values | Values Provided by the Numerical Simulation |
---|---|---|

Cell size determined by Kolmogorov length scale | 1.61 × 10^{−2} | 9.80 × 10^{−3} |

Boundary layer thickness (Russo and Napolitano) | 3.16 × 10^{−4} | 2.98 × 10^{−2} |

Number of cells required in boundary layer (Shishkina et al.) | 3 | 3 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Crewdson, G.; Lappa, M.
The Zoo of Modes of Convection in Liquids Vibrated along the Direction of the Temperature Gradient. *Fluids* **2021**, *6*, 30.
https://doi.org/10.3390/fluids6010030

**AMA Style**

Crewdson G, Lappa M.
The Zoo of Modes of Convection in Liquids Vibrated along the Direction of the Temperature Gradient. *Fluids*. 2021; 6(1):30.
https://doi.org/10.3390/fluids6010030

**Chicago/Turabian Style**

Crewdson, Georgie, and Marcello Lappa.
2021. "The Zoo of Modes of Convection in Liquids Vibrated along the Direction of the Temperature Gradient" *Fluids* 6, no. 1: 30.
https://doi.org/10.3390/fluids6010030