The analysis of vibration effect on non-isothermal fluid in closed cavity is important for planning technological experiments in space. Control and optimization of these processes critically depend on the understanding of liquid response to the vibrations. With this aim the theoretical investigation for infinite plane and cylindrical fluid layers are performed. We investigated simple case of the fluid response-thermal vibrational convection in a cylindrical fluid layer with rigid conducting boundaries. It is found that steady modes of thermal vibrational convection are subjected to various bifurcations. Bifurcations cause sharp changes in heat transfer. The Lorenz model is generalized (GLM) and used to conduct the analysis of bifurcations caused by the changing of the cavity shape and vibrational Rayleigh number. The shape of steady-state surface in 3D space of the streamfunction of mean flow, vibrational Rayleigh number and the cavity curvature is found. The numerical 2D solution is performed for plane and cylindrical fluid layers. The results of the analysis based on the GLM model are compared with the data obtained by direct numerical simulation. The bifurcation curve with extremum is found. Thus, bifurcations of complex shape could be observed. Comparisons with space experiments and its discussions are presented.
Deep Dive into Bifurcations of Free Thermal Vibrational Convection in Cylindrical Fluid Layer in Micro-Gravity: Numerical and Analytical Research.
The analysis of vibration effect on non-isothermal fluid in closed cavity is important for planning technological experiments in space. Control and optimization of these processes critically depend on the understanding of liquid response to the vibrations. With this aim the theoretical investigation for infinite plane and cylindrical fluid layers are performed. We investigated simple case of the fluid response-thermal vibrational convection in a cylindrical fluid layer with rigid conducting boundaries. It is found that steady modes of thermal vibrational convection are subjected to various bifurcations. Bifurcations cause sharp changes in heat transfer. The Lorenz model is generalized (GLM) and used to conduct the analysis of bifurcations caused by the changing of the cavity shape and vibrational Rayleigh number. The shape of steady-state surface in 3D space of the streamfunction of mean flow, vibrational Rayleigh number and the cavity curvature is found. The numerical 2D solution is p
Let us consider closed cavity filled in with fluid. Assume that temperature distribution on the cavity boundaries sets steady vertical temperature gradient inside the cavity with no fluid displacement. Following Lorenz [4], the behavior of fluid can be described by the system of ordinary differential equations: where Ra and Pr -Rayleigh and Prandtl criteria, b -positively defined geometrical parameter, Ψ -stream-function, ϑ 1 and ϑ 2 -the characteristics of the deviation of temperature field from equilibrium. Lorenz applied this model for the study of fluid instability with high values of Ra. The chaotic behavior in the system (1) is widely known as strange attractor of Lorenz. We use (1) for the analysis of the bifurcations arising in numerical experiments applying full system of partial differential equations to free thermal and vibrational convection.
Lorenz’s model (1.1) can be modified to take into consideration the variations in direction of heating [2]:
In addition to the numbers of Rayleigh and Prandtl, the cavity inclination α (0 2 )
In spite of simplicity, this model correctly reflects the bifurcations of steady states of non-uniformly heated fluid in closed cavity for arbitrary heating directions.
System (1.2) has steady solutions with stream-functions satisfying the following equation:
The thermal components of these solutions is connected to stream-function Ψ by the following relation:
For α π = (heating from the top) equation ( 1.3) has single solution 0 Ψ = corresponding to the fluid at rest. The linear analysis shows the stability of the solution for any 0 Ra > . For high values of Rayleigh number small instabilities attenuate harmonically. Such a behavior is similar to well-known results of analysis of full equations of free thermal convection. For The stream-function Ψ uniquely defines the steady solutions. Because of that, the threedimensional phase space of the system (1.2) can be considered as one-dimensional for steady case. Equation (1.3) describes the steady-state surface in space of ( , , )
. This space represents the product of phase space Ψ and space of parameters ( , ) Ra α . Figure 1 illustrates the shape of the steady-state surface ( , )
- for b=8/3. (heating strait from the bottom). The proof based on the bifurcations theory [5] is contained in [2].
Numerous computational experiments on system (1.2) were performed. In particular, experiments on structural stability of the system -the system response on adding to and removing out the members of the first equation. Thus, removing the component 2 sin ϑ α
does not cause the qualitative change of the steady-state surface.
Let us introduce the parameter γ and call it the cavity curvature. Assume the low and high limits for γ : -1 and +1 correspondingly. Assume no cavity curvature for 0 γ = , and new system coincides with the Lorenz’s model. The system for analysis of the cavity curvature can be written as
The steady-state conditions of system (2.1) is derived from the equation:
The shape of the steady-state surface of the system is represented at Figure 2.
Pr
given condition is satisfied. The shape of the steady-state surface for system (2.3) is shown at Figure 5. frequency ω are performed along the direction given by the unit vector k r .Governing equations for thermal vibrational convection in form of generalized Gelmgolts equations are [3]: For cylindrical fluid layer with r 2 =2r 1 the flow mode with counter-clock dominated vortex circulation is observed when G ν < 3.6*10 3 (curve 3). The stream function has extremum in the vortex center. There is one more stable flow mode (curve 4) with clockwise circulation of the main vortex. Curves 3 and for are approaching curves 1 and 2 with increasing of the layer curvature γ = (r 2 -r 1 )/(r 2 +r 1 ).
The solution analysis, and in particular, analysis of Ψ m (γ) shows that the steady-state surface Ψ m (G ν , γ) has the singularity for G ν =2.1*10 3 , γ = 0. Steady state surface have a lot of another type of singularities. This conclusion is fallowed from our current calculations presented below.
Problem (3.1)-(3.7) has been solved analytically for approach of small values of vibrational Grachof number v G . In this approach the influence of mean flow on temperature field in annulus was neglected. Stream function of mean flow is given by formula:
Formula (3.8) shows that along angular coordinates flows has four cells, like on Figure 7. Number of cells along radius depends from dimensionless value of inner cylinder a . One can obtain from (3.8) -for 0.27 a > (thin annulus) in annulus must be two floors of eddies. This analytical results looks like paradox. On the next figures 7 -13 presented some results of solving problem for Pr=1 by using finite-difference method on net with 20 nodes along radius and 90 -angular. Figure 7 has two floor of eddies in agreement with formula (3.8)
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