Competition of overstability and stabilizing effects in viscoelastic thermovibrational flow

Attention is paid to a specific form of thermal convection which encompasses viscoelastic and thermovibrational effects in a single problem or framework. The main objective is understanding the relationship between the phenomenon of overstability and periodic forcing through numerical solution of the governing equations in their complete, time-dependent, and nonlinear form. Fluid motion is found for values of the control parameter one order of magnitude smaller than the threshold to be exceeded in the equivalent Newtonian case. When the disturbances saturate their amplitude, patterns emerge that are reminiscent of the superlattice structures typical of complex order. In the present case, such peculiar modes of convection are driven by the coexistence of two distinct spatial scales, each displaying a different temporal dependence, driven by the interplay of the time-varying (stabilizing or destabilizing) acceleration induced by vibrations and the ability of the fluid to store and release elastic energy.

Figure 1

Sketch of the geometry and scheme of the problem.

Figure 2

Time evolution of the Nusselt number for two different densities of the mesh, $200\times 35\times 200$ (black line) and $300\times 35\times 300$ (blue line), and the qualitative evolution of the acceleration (red line), from 3D simulations.

Figure 3

3D isosurfaces of the axial velocity for the case with ${\text{Pr}}_g=7, \mathrm{\Omega }=26.5, {\text{Ra}}_{\omega }=1170, \xi =0.1, \vartheta =0.38, t=14.92$, and (a) mesh $200\times 35\times 200$ and (b) mesh $300\times 35\times 300$.

Figure 4

Time evolution of the axial velocity signal. The probe is located in the center of the layer. The numerical simulation is with $\mathcal{R}=15, {\text{Pr}}_g=7, \mathrm{\Omega }=26.5, \xi =0.1$, and (a) Newtonian fluid and ${\text{Ra}}_{\omega }=19750$, (b) $\vartheta =0.06$ and ${\text{Ra}}_{\omega }=3500$, (c) $\vartheta =0.24$ and ${\text{Ra}}_{\omega }=1475$, and (d) $\vartheta =0.38$ and ${\text{Ra}}_{\omega }=1170$. The black line represents the signal and the red the evolution of ${\mathbf{a}}_{\omega }$.

Figure 5

Angular frequency of the axial velocity signal. The probe is located in the center of the layer. The numerical simulation is with $\mathcal{R}=15, {\text{Pr}}_g=7, \mathrm{\Omega }=26.5, \xi =0.1$, and (a) Newtonian fluid and ${\text{Ra}}_{\omega }=19750$, (b) $\vartheta =0.06$ and ${\text{Ra}}_{\omega }=3500$, (c) $\vartheta =0.24$ and ${\text{Ra}}_{\omega }=1475$, and (d) $\vartheta =0.38$ and ${\text{Ra}}_{\omega }=1170$.

Figure 6

Time evolution of $\text{Nu}\left(t\right)$. The numerical simulation is with $\mathcal{R}=15, {\text{Pr}}_g=7, \mathrm{\Omega }=26.5, \xi =0.1$, and (a) Newtonian fluid, ${\text{Ra}}_{\omega }=19750$, and $\overline{\mathrm{N}\mathrm{u}}=1.16$; (b) $\vartheta =0.06, {\text{Ra}}_{\omega }=3500$, and $\overline{\mathrm{N}\mathrm{u}}=1.22$; (c) $\vartheta =0.24, {\text{Ra}}_{\omega }=1475$, and $\overline{\mathrm{N}\mathrm{u}}=1.24$; and (d) $\vartheta =0.38, {\text{Ra}}_{\omega }=1170$, and $\overline{\mathrm{N}\mathrm{u}}=1.21$. The black line represents $\text{Nu}\left(t\right)$ and the red the evolution of ${\mathbf{a}}_{\omega }$.

Figure 7

Thermovibrational convection in a layer of Newtonian fluid delimited by differentially heated solid walls (snapshots of the isosurfaces of the $y$ component of velocity evenly distributed over the first half period $T_{\mathrm{\Omega }}$), with $\text{Pr}=7, \mathrm{\Omega }=26.5$, and ${\text{Ra}}_{\omega }=19750$.

Figure 8

Thermovibrational convection in a layer of Oldroyd-B fluid delimited by differentially heated solid walls (snapshots of the isosurfaces of the vertical component of velocity evenly distributed over the interval ${\mathcal{I}}_1$), with ${\text{Pr}}_g=7, \mathrm{\Omega }=26.5, {\text{Ra}}_{\omega }=3500, \xi =0.1$, and $\vartheta =0.06$.

Figure 9

Thermovibrational convection in a layer of Oldroyd-B fluid delimited by differentially heated solid walls (snapshots of the isosurfaces of the vertical component of velocity evenly distributed over the interval ${\mathcal{I}}_2$), with ${\text{Pr}}_g=7, \mathrm{\Omega }=26.5, {\text{Ra}}_{\omega }=3500, \xi =0.1$, and $\vartheta =0.06$.

Figure 10

Thermovibrational convection in a layer of Oldroyd-B fluid delimited by differentially heated solid walls (snapshots of the isosurfaces of the vertical component of velocity evenly distributed over the interval ${\mathcal{I}}_3$), with ${\text{Pr}}_g=7, \mathrm{\Omega }=26.5, {\text{Ra}}_{\omega }=3500, \xi =0.1$, and $\vartheta =0.06$.

Figure 11

Thermovibrational convection in a layer of Oldroyd-B fluid delimited by differentially heated solid walls (snapshots of the isosurfaces of the vertical component of velocity evenly distributed over the interval ${\mathcal{I}}_1$), with ${\text{Pr}}_g=7, \mathrm{\Omega }=26.5, {\text{Ra}}_{\omega }=1475, \xi =0.1$, and $\vartheta =0.24$.

Figure 12

Thermovibrational convection in a layer of Oldroyd-B fluid delimited by differentially heated solid walls (snapshots of the isosurfaces of the vertical component of velocity evenly distributed over the interval ${\mathcal{I}}_2$), with ${\text{Pr}}_g=7, \mathrm{\Omega }=26.5, {\text{Ra}}_{\omega }=1475, \xi =0.1$, and $\vartheta =0.24$.

Figure 13

Thermovibrational convection in a layer of Oldroyd-B fluid delimited by differentially heated solid walls (snapshots of the isosurfaces of the vertical component of velocity evenly distributed over the interval ${\mathcal{I}}_1$), with ${\text{Pr}}_g=7, \mathrm{\Omega }=26.5, {\text{Ra}}_{\omega }=1170, \xi =0.1$, and $\vartheta =0.38$.

Figure 14

Thermovibrational convection in a layer of Oldroyd-B fluid delimited by differentially heated solid walls (snapshots of the isosurfaces of the vertical component of velocity evenly distributed over the interval ${\mathcal{I}}_2$), with ${\text{Pr}}_g=7, \mathrm{\Omega }=26.5, {\text{Ra}}_{\omega }=1170, \xi =0.1$, and $\vartheta =0.38$.

Figure 15

Color field of axial velocity ($y$ component of $\mathbf{u}$) as a function of the time and the $x$ coordinate in the center of the cavity, with $0<x<15, y=0.5, z=7.5$, and (a) $\vartheta =0.06$, (b) $\vartheta =0.24$, and (c) $\vartheta =0.38$.

Figure 16

Color field of $\mathrm{tr}\left(\tilde{\mathit{\tau }}\right)$ as a function of the time and the $x$ coordinate in the center of the cavity, with $0<x<15, y=0.5, z=7.5$, and (a) $\vartheta =0.06$, (b) $\vartheta =0.24$, and (c) $\vartheta =0.38$.

Figure 17

Distribution of $\mathrm{tr}\left(\tilde{\mathit{\tau }}\right)$ on the $xz$ plane at $y=1$ (cold plate), (a) $\vartheta =0.06(t=t_0)$, (b) $\vartheta =0.24(t=t_0)$ and (c) $\vartheta =0.38(t=t_0)$, (d) $\vartheta =0.06(t=t_0+T_{\mathrm{\Omega }/2})$, (e) $\vartheta =0.24(t=t_0+T_{\mathrm{\Omega }/2})$ and (f) $\vartheta =0.38(t=t_0+T_{\mathrm{\Omega }/2})$.