Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSystematic comparison between the generalized Lorenz equations and DNS in the two-dimensional Rayleigh-Benard convection
77
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The classic Lorenz equations were originally derived from the two-dimensional Rayleigh-Benard convection system considering an idealised case with the lowest order of harmonics. Although the low-order Lorenz equations have traditionally served as a minimal model for chaotic and intermittent atmospheric motions, even the dynamics of the two-dimensional Rayleigh-Benard convection system is not fully represented by the Lorenz equations, and such differences have yet to be clearly identified in a systematic manner. In this paper, the convection problem is revisited through an investigation of various dynamical behaviors exhibited by a two-dimensional direct numerical simulation (DNS) and the generalized expansion of the Lorenz equations (GELE) derived by considering additional higher-order harmonics in the spectral expansions of periodic solutions. Notably, the GELE allows us to understand how nonlinear interactions among high-order modes alter the dynamical features of the Lorenz equations including fixed points, chaotic attractors, and periodic solutions. It is verified that numerical solutions of the DNS can be recovered from the solutions of GELE when we consider the system with sufficiently high-order harmonics. At the lowest order, the classic Lorenz equations are recovered from GELE. Unlike in the Lorenz equations, we observe limit tori, which are the multi-dimensional analogue of limit cycles, in the solutions of the DNS and GELE at high orders. Initial condition dependency in the DNS and Lorenz equations is also discussed.
rate research
Read More
The dynamics of inertial particles in Rayleigh-B{e}nard convection, where both particles and fluid exhibit thermal expansion, is studied using direct numerical simulations (DNS). We consider the effect of particles with a thermal expansion coefficient larger than that of the fluid, causing particles to become lighter than the fluid near the hot bottom plate and heavier than the fluid near the cold top plate. Because of the opposite directions of the net Archimedes force on particles and fluid, particles deposited at the plate now experience a relative force towards the bulk. The characteristic time for this motion towards the bulk to happen, quantified as the time particles spend inside the thermal boundary layers (BLs) at the plates, is shown to depend on the thermal response time, $tau_T$, and the thermal expansion coefficient of particles relative to that of the fluid, $K = alpha_p / alpha_f$. In particular, the residence time is constant for small thermal response times, $tau_T lesssim 1$, and increasing with $tau_T$ for larger thermal response times, $tau_T gtrsim 1$. Also, the thermal BL residence time is increasing with decreasing $K$. A one-dimensional (1D) model is developed, where particles experience thermal inertia and their motion is purely dependent on the buoyancy force. Although the values do not match one-to-one, this highly simplified 1D model does predict a regime of a constant thermal BL residence time for smaller thermal response times and a regime of increasing residence time with $tau_T$ for larger response times, thus explaining the trends in the DNS data well.
We analyse the nonlinear dynamics of the large scale flow in Rayleigh-Benard convection in a two-dimensional, rectangular geometry of aspect ratio $Gamma$. We impose periodic and free-slip boundary conditions in the streamwise and spanwise directions, respectively. As Rayleigh number Ra increases, a large scale zonal flow dominates the dynamics of a moderate Prandtl number fluid. At high Ra, in the turbulent regime, transitions are seen in the probability density function (PDF) of the largest scale mode. For $Gamma = 2$, the PDF first transitions from a Gaussian to a trimodal behaviour, signifying the emergence of reversals of the zonal flow where the flow fluctuates between three distinct turbulent states: two states in which the zonal flow travels in opposite directions and one state with no zonal mean flow. Further increase in Ra leads to a transition from a trimodal to a unimodal PDF which demonstrates the disappearance of the zonal flow reversals. On the other hand, for $Gamma = 1$ the zonal flow reversals are characterised by a bimodal PDF of the largest scale mode, where the flow fluctuates only between two distinct turbulent states with zonal flow travelling in opposite directions.
We study the stability of steady convection rolls in 2D Rayleigh--Benard convection with free-slip boundaries and horizontal periodicity over twelve orders of magnitude in the Prandtl number $(10^{-6} leq Pr leq 10^6)$ and five orders of magnitude in the Rayleigh number $(8pi^4 < Ra leq 3 times 10^7)$. The analysis is facilitated by partitioning our modal expansion into so-called even and odd modes. With aspect ratio $Gamma = 2$, we observe that zonal modes (with horizontal wavenumber equal to zero) can emerge only once the steady convection roll state consisting of even modes only becomes unstable to odd perturbations. We determine the stability boundary in the $(Pr,Ra)$-plane and observe remarkably intricate features corresponding to qualitative changes in the solution, as well as three regions where the steady convection rolls lose and subsequently regain stability as the Rayleigh number is increased. We study the asymptotic limit $Pr to 0$ and find that the steady convection rolls become unstable almost instantaneously, eventually leading to non-linear relaxation osculations and bursts, which we can explain with a weakly non-linear analysis. In the complementary large-$Pr$ limit, we observe that the stability boundary reaches an asymptotic value $Ra = 2.54 times 10^7$ and that the zonal modes at the instability switch off abruptly at a large, but finite, Prandtl number.
We perform a bifurcation analysis of the steady state solutions of Rayleigh--Benard convection with no-slip boundary conditions in two dimensions using a numerical method called deflated continuation. By combining this method with an initialisation strategy based on the eigenmodes of the conducting state, we are able to discover multiple solutions to this non-linear problem, including disconnected branches of the bifurcation diagram, without the need of any prior knowledge of the dynamics. One of the disconnected branches we find contains a s-shape bifurcation with hysteresis, which is the origin of the flow pattern that may be related to the dynamics of flow reversals in the turbulent regime. Linear stability analysis is also performed to analyse the steady and unsteady regimes of the solutions in the parameter space and to characterise the type of instabilities.
Two different types of perturbations of the Lorenz 63 dynamical system for Rayleigh-Benard convection by multiplicative noise -- called stochastic advection by Lie transport (SALT) noise and fluctuation-dissipation (FD) noise -- are found to produce qualitatively different effects, possibly because the total phase-space volume contraction rates are different. In the process of making this comparison between effects of SALT and FD noise on the Lorenz 63 system, a stochastic version of a robust deterministic numerical algorithm for obtaining the individual numerical Lyapunov exponents was developed. With this stochastic version of the algorithm, the value of the sum of the Lyapunov exponents for the FD noise was found to differ significantly from the value of the deterministic Lorenz 63 system, whereas the SALT noise preserves the Lorenz 63 value with high accuracy. The Lagrangian averaged version of the SALT equations (LA SALT) is found to yield a closed deterministic subsystem for the expected solutions which is found to be isomorphic to the original Lorenz 63 dynamical system. The solutions of the closed chaotic subsystem, in turn, drive a linear stochastic system for the fluctuations of the LA SALT solutions around their expected values.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
