scholarly journals Nonlinear feedback in a six-dimensional Lorenz Model: impact of an additional heating term

2015 ◽  
Vol 2 (2) ◽  
pp. 475-512
Author(s):  
B.-W. Shen
Keyword(s):  
Rayleigh Number ◽  
Numerical Solutions ◽  
Critical Rayleigh Number ◽  
Critical Value ◽  
Small Scale ◽  
Solution Stability ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Additional Heating ◽  
The Impact

Abstract. In this study, a six-dimensional Lorenz model (6DLM) is derived, based on a recent study using a five-dimensional (5-D) Lorenz model (LM), in order to examine the impact of an additional mode and its accompanying heating term on solution stability. The new mode added to improve the representation of the steamfunction is referred to as a secondary streamfunction mode, while the two additional modes, that appear in both the 6DLM and 5DLM but not in the original LM, are referred to as secondary temperature modes. Two energy conservation relationships of the 6DLM are first derived in the dissipationless limit. The impact of three additional modes on solution stability is examined by comparing numerical solutions and ensemble Lyapunov exponents of the 6DLM and 5DLM as well as the original LM. For the onset of chaos, the critical value of the normalized Rayleigh number (rc) is determined to be 41.1. The critical value is larger than that in the 3DLM (rc ~ 24.74), but slightly smaller than the one in the 5DLM (rc ~ 42.9). A stability analysis and numerical experiments obtained using generalized LMs, with or without simplifications, suggest the following: (1) negative nonlinear feedback in association with the secondary temperature modes, as first identified using the 5DLM, plays a dominant role in providing feedback for improving the solution's stability of the 6DLM, (2) the additional heating term in association with the secondary streamfunction mode may destabilize the solution, and (3) overall feedback due to the secondary streamfunction mode is much smaller than the feedback due to the secondary temperature modes; therefore, the critical Rayleigh number of the 6DLM is comparable to that of the 5DLM. The 5DLM and 6DLM collectively suggest different roles for small-scale processes (i.e., stabilization vs. destabilization), consistent with the following statement by Lorenz (1972): If the flap of a butterfly's wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado. The implications of this and previous work, as well as future work, are also discussed.

scholarly journals Nonlinear feedback in a six-dimensional Lorenz model: impact of an additional heating term

2015 ◽  
Vol 22 (6) ◽  
pp. 749-764 ◽  
Author(s):  
B.-W. Shen
Keyword(s):  
Rayleigh Number ◽  
Numerical Solutions ◽  
Critical Rayleigh Number ◽  
Critical Value ◽  
Small Scale ◽  
Solution Stability ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Additional Heating ◽  
The Impact

Abstract. In this study, a six-dimensional Lorenz model (6DLM) is derived, based on a recent study using a five-dimensional (5-D) Lorenz model (LM), in order to examine the impact of an additional mode and its accompanying heating term on solution stability. The new mode added to improve the representation of the streamfunction is referred to as a secondary streamfunction mode, while the two additional modes, which appear in both the 6DLM and 5DLM but not in the original LM, are referred to as secondary temperature modes. Two energy conservation relationships of the 6DLM are first derived in the dissipationless limit. The impact of three additional modes on solution stability is examined by comparing numerical solutions and ensemble Lyapunov exponents of the 6DLM and 5DLM as well as the original LM. For the onset of chaos, the critical value of the normalized Rayleigh number (rc) is determined to be 41.1. The critical value is larger than that in the 3DLM (rc ~ 24.74), but slightly smaller than the one in the 5DLM (rc ~ 42.9). A stability analysis and numerical experiments obtained using generalized LMs, with or without simplifications, suggest the following: (1) negative nonlinear feedback in association with the secondary temperature modes, as first identified using the 5DLM, plays a dominant role in providing feedback for improving the solution's stability of the 6DLM, (2) the additional heating term in association with the secondary streamfunction mode may destabilize the solution, and (3) overall feedback due to the secondary streamfunction mode is much smaller than the feedback due to the secondary temperature modes; therefore, the critical Rayleigh number of the 6DLM is comparable to that of the 5DLM. The 5DLM and 6DLM collectively suggest different roles for small-scale processes (i.e., stabilization vs. destabilization), consistent with the following statement by Lorenz (1972): "If the flap of a butterfly's wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado." The implications of this and previous work, as well as future work, are also discussed.

Nonlinear Feedback in a Five-Dimensional Lorenz Model

2014 ◽  
Vol 71 (5) ◽  
pp. 1701-1723 ◽  
Author(s):  
Bo-Wen Shen
Keyword(s):  
Numerical Solutions ◽  
Three Dimensional ◽  
Solution Stability ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Dissipative Term ◽  
Higher Dimensional ◽  
The Impact ◽  
Underlying Processes

Abstract In this study, based on the number of modes, the original three-dimensional Lorenz model (3DLM) is generalized with two additional modes [five-dimensional Lorenz model (5DLM)] to examine their role in the predictability of the numerical solutions and to understand the underlying processes that increase the solution stability. As a result of the simplicity of the 5DLM with respect to existing generalized Lorenz models (LMs), the author is able to obtain the analytical solutions of its critical points and identify the role of the major nonlinear term in the solution’s stability, which have previously not been documented in the literature. The nonlinear Jacobian terms of the governing equations are analyzed to highlight the importance of selecting new modes for extending the nonlinear feedback loop of the 3DLM and thus effectively increasing the degree of nonlinearity (i.e., the nonlinear mode–mode interactions) in the 5DLM. It is then shown that numerical solutions in the 5DLM require a larger normalized Rayleigh number r for the onset of chaos and are more predictable than those in the 3DLM when r is between 25 and 40 and the Prandtl number σ is 10. The improved predictability is attributable to the negative nonlinear feedback enabled by the new modes. The role of the (negative) nonlinear feedback is further verified using a revised 3DLM with a parameterized nonlinear eddy dissipative term. The finding of the increased stability in the 5DLM and revised 3DLM with respect to the 3DLM is confirmed with the linear stability analysis and the analysis of the Lyapunov exponents using different values of r and σ. To further understand the impact of an additional heating term, results from the 5DLM and a higher-dimensional LM [e.g., the six-dimensional LM (6DLM)] are analyzed and compared.

Numerical simulation of two-dimensional compressible convection

1975 ◽  
Vol 70 (4) ◽  
pp. 689-703 ◽  
Author(s):  
Eric Graham
Keyword(s):  
Rayleigh Number ◽  
Numerical Solutions ◽  
Fluid Velocity ◽  
Critical Rayleigh Number ◽  
Finite Amplitude ◽  
Two Dimensions ◽  
Onset Of Convection ◽  
Local Sound ◽  
Constant Viscosity ◽  
Convection Problem

A procedure for obtaining numerical solutions to the equations describing thermal convection in a compressible fluid is outlined. The method is applied to the case of a perfect gas with constant viscosity and thermal conductivity. The fluid is considered to be confined in a rectangular region by fixed slippery boundaries and motions are restricted to two dimensions. The upper and lower boundaries are maintained at fixed temperatures and the side boundaries are thermally insulating. The resulting convection problem can be characterized by six dimension-less parameters. The onset of convection has been studied both by obtaining solutions to the nonlinear equations in the neighbourhood of the critical Rayleigh number Rc and by solving the linear stability problem. Solutions have been obtained for values of the Rayleigh number up to 100Rc and for pressure variations of a factor of 300 within the fluid. In some cases the fluid velocity is comparable to the local sound speed. The Nusselt number increases with decreasing Prandtl number for moderate values of the depth parameter. Steady finite amplitude solutions have been found in all the cases considered. As the horizontal dimension A of the rectangle is increased, the length of time needed to reach a steady state also increases. For large values of A the solution consists of a number of rolls. Even for small values of A, no solutions have been found where one roll is vertically above another.

Flow and Decay Characteristics of Submerged Yield-Pseudoplastic Jets

2014 ◽  
Author(s):  
Khaled J. Hammad
Keyword(s):  
Flow Structure ◽  
Numerical Solutions ◽  
Shear Thinning ◽  
Critical Value ◽  
Momentum Conservation ◽  
Recirculation Region ◽  
Centerline Velocity ◽  
General Increase ◽  
Flow Recirculation ◽  
The Impact

The flow structure and decay characteristics of submerged yield-pseudoplastic jets are investigated. Numerical solutions to the governing mass and momentum conservation equations, along with the Herschel-Bulkley rheological model, are obtained using a finite-difference scheme. A large recirculation region exists for Newtonian and pseudoplastic non-Newtonian jets. However, the extent and strength of the recirculation region substantially diminish with the yield number and, to a much lesser extent, when the shear-thinning index is decreased from 1 to 0.6. Increasing the yield number beyond a critical value eliminates flow recirculation. The centerline velocity and momentum decay of yield-pseudoplastic jets, in general, increase with the yield number. The impact of shear-thinning on the flow structure and decay characteristics of the jet is more pronounced at low yield numbers.

scholarly journals Hierarchical scale dependence associated with the extension of the nonlinear feedback loop in a seven-dimensional Lorenz model

2016 ◽  
Author(s):  
Bo-Wen Shen
Keyword(s):  
Stability Analysis ◽  
Prandtl Number ◽  
Analytical Solutions ◽  
Feedback Loop ◽  
Critical Value ◽  
Scale Dependence ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Onset Of Chaos ◽  
Hierarchical Scale

Abstract. In this study, we construct a seven-dimensional Lorenz Model (7DLM) to discuss the impact of an extended nonlinear feedback loop on solutions' stability and illustrate the hierarchical scale dependence of chaotic solutions. Compared to the 5DLM, the 7DLM includes two additional high wavenumber modes that are selected based on an analysis of the nonlinear temperature advection term, a Jacobian term (J(ψ, θ)), where, ψ and θ represent the streamfunction and temperature perturbations, respectively. Fourier modes that represent temperature in the 7DLM can be categorized into three major scales as the primary (the largest scale), secondary, and tertiary (the smallest scale) modes. Further extension of the nonlinear feedback loop within the 7DLM can provide negative nonlinear feedback to stabilize solutions, thus leading to a much larger critical value for the Rayleigh parameter (rc ∼ 116.9) for the onset of chaos, as compared to an rc of 42.9 for the 5DLM as well as an rc of 24.74 for the 3DLM. The rc is determined by an analysis of ensemble Lyapunov exponents (eLEs) with Prandtl number (σ) of 10. To examine the dependence of rc on the value of the Prandtl number, a linear stability analysis is performed near the nontrivial critical point using a wide range of Rayleigh parameter (40 ≤ r ≤ 195) and Prandtl number (5 ≤ σ ≤ 25). Then an eLE analysis is conducted using selected values of the Prandtl number. The linear stability analysis is done by solving for the analytical solutions of the critical points, by linearizing the 7DLM with respect to the analytical solutions, and by calculating the eigenvalues of the linearized system. Within the range of (5 ≤ σ ≤ 25), the 7DLM requires a larger rc for the onset of chaos than the 5DLM. In addition to the negative nonlinear feedback illustrated and emulated by the quasi-equilibrium state solutions for high wavenumber modes, the 7DLM reveals the hierarchical scale dependence of chaotic solutions. For chaotic solutions with r = 120, the Pearson correlation coefficients (PCCs) between the primary and secondary modes (i.e., Z and Z1) and between the secondary and tertiary modes (i.e., Z1 and Z2) are 0.988 and 0.998, respectively. Here, Z, Z1, and Z2 represent the time-varying amplitudes of the primary, secondary, and tertiary modes, respectively. High PCCs indicate a strong linear relationship among the modes at various scales and a hierarchy of scale dependence. Future work will be undertaken to examine how higher dimensional LMs may produce a larger critical value for the Rayleigh parameter for the onset of chaos and reveal stronger hierarchical scale dependence.

Non-linear Rayleigh-Benard Magnetoconvection in Temperature-sensitive Newtonian Liquids with Variable Heat Source

2021 ◽  
Vol 88 (1-2) ◽  
pp. 08
Author(s):  
A. S. Aruna ◽  
V. Ramachandramurthy ◽  
N. Kavitha
Keyword(s):  
Rayleigh Number ◽  
Heat Source ◽  
Variable Viscosity ◽  
Critical Rayleigh Number ◽  
Lorenz Model ◽  
Onset Of Convection ◽  
Variable Heat ◽  
Non Linear ◽  
Source Sink ◽  
Rayleigh Benard

The present paper aims at weak non-linear stability analysis followed by linear analysis of nite-amplitude Rayleigh-Benard magneto convection problem in an electrically conducting Newtonian liquid with heat source/sink. It is shown that the internal Rayleigh number, ther- morheological parameter, and the Chandrasekhar number in uence the onset of convection. The generalized Lorenz model derived for the prob- lem is essentially the classical Lorenz model but with some coecient depending on the variable heat source (sink), viscosity, and the applied magnetic eld. The result of the parameters' in uence on the critical Rayleigh number explains their in uence on the Nusselt number. It is found that an increasing strength of the magnetic eld is to stabilize the system and diminishes heat transport whereas the heat source and variable viscosity in-tandem to work system unstable and enhances heat transfer.

Numerical Study of Transient High Rayleigh Number Convection in an Attic-Shaped Porous Layer

1983 ◽  
Vol 105 (3) ◽  
pp. 476-484 ◽  
Author(s):  
D. Poulikakos ◽  
A. Bejan
Keyword(s):  
Rayleigh Number ◽  
Numerical Simulations ◽  
Porous Layer ◽  
Numerical Solutions ◽  
Numerical Study ◽  
Critical Rayleigh Number ◽  
Scale Analysis ◽  
Geometric Ratio ◽  
Rayleigh Numbers ◽  
Transient And Steady State

Scaling arguments and numerical analysis are used to document the transient and steady-state regimes of natural convection in a triangular porous layer cooled from above (along the sloping wall). The numerical simulations are conducted in the high Rayleigh number domain, Ra = 100, 1000, where Ra is the Darcy-modified Rayleigh number based on height, H. The scale analysis predicts the existence of distinct thermal boundary layers if Ra1/2 (H/L) > 1, where H/L is the height/length geometric ratio of the attic-shaped porous layer. The numerical simulations confirm the scaling results, as well as the prediction that the flow consists primarily of an elongated horizontal counterflow driven by the cold wall. In addition, the numerical solutions show the presence of a Be´nard-type instability at high enough Rayleigh numbers. For example, if H/L = 0.2, the instability is present when Ra > 620; this critical Rayleigh number is found to increase as H/L increases.

scholarly journals Hierarchical scale dependence associated with the extension of the nonlinear feedback loop in a seven-dimensional Lorenz model

2016 ◽  
Vol 23 (4) ◽  
pp. 189-203 ◽  
Author(s):  
Bo-Wen Shen
Keyword(s):  
Stability Analysis ◽  
Prandtl Number ◽  
Analytical Solutions ◽  
Feedback Loop ◽  
Critical Value ◽  
Scale Dependence ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Onset Of Chaos ◽  
Hierarchical Scale

Abstract. In this study, we construct a seven-dimensional Lorenz model (7DLM) to discuss the impact of an extended nonlinear feedback loop on solutions' stability and illustrate the hierarchical scale dependence of chaotic solutions. Compared to the 5DLM, the 7DLM includes two additional high wavenumber modes that are selected based on an analysis of the nonlinear temperature advection term, a Jacobian term (J(ψ, θ)), where, ψ and θ represent the streamfunction and temperature perturbations, respectively. Fourier modes that represent temperature in the 7DLM can be categorized into three major scales as the primary (the largest scale), secondary, and tertiary (the smallest scale) modes. Further extension of the nonlinear feedback loop within the 7DLM can provide negative nonlinear feedback to stabilize solutions, thus leading to a much larger critical value for the Rayleigh parameter (rc  ∼  116.9) for the onset of chaos, as compared to an rc of 42.9 for the 5DLM as well as an rc of 24.74 for the 3DLM. The rc is determined by an analysis of ensemble Lyapunov exponents (eLEs) with a Prandtl number (σ) of 10. To examine the dependence of rc on the value of the Prandtl number, a linear stability analysis is performed near the nontrivial critical point using a wide range of the Rayleigh parameter (40  ≤  r  ≤  195) and the Prandtl number (5  ≤  σ  ≤  25). Then an eLE analysis is conducted using selected values of the Prandtl number. The linear stability analysis is done by solving for the analytical solutions of the critical points, by linearizing the 7DLM with respect to the analytical solutions, and by calculating the eigenvalues of the linearized system. Within the range of (5  ≤  σ  ≤  25), the 7DLM requires a larger rc for the onset of chaos than the 5DLM. In addition to the negative nonlinear feedback illustrated and emulated by the quasi-equilibrium state solutions for high wavenumber modes, the 7DLM reveals the hierarchical scale dependence of chaotic solutions. For chaotic solutions with r  =  120, the Pearson correlation coefficients (PCCs) between the primary and secondary modes (i.e., Z and Z1) and between the secondary and tertiary modes (i.e., Z1 and Z2) are 0.988 and 0.998, respectively. Here, Z, Z1, and Z2 represent the time-varying amplitudes of the primary, secondary, and tertiary modes, respectively. High PCCs indicate a strong linear relationship among the modes at various scales and a hierarchy of scale dependence. Future work will be undertaken to examine how higher-dimensional LMs may produce a larger critical value for the Rayleigh parameter for the onset of chaos and reveal stronger hierarchical scale dependence.

Rayleigh–Bénard convection in the presence of spatial temperature modulations

2011 ◽  
Vol 673 ◽  
pp. 318-348 ◽  
Author(s):  
G. FREUND ◽  
W. PESCH ◽  
W. ZIMMERMANN
Keyword(s):  
Rayleigh Number ◽  
Numerical Solutions ◽  
Critical Rayleigh Number ◽  
Lower Boundary ◽  
Spatial Modulation ◽  
Bénard Convection ◽  
Benard Convection ◽  
Wide Range ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Motivated by recent experiments, we study a rich variation of the familiar Rayleigh–Bénard convection (RBC), where the temperature at the lower boundary varies sinusoidally about a mean value. As usual the Rayleigh number R measures the average temperature gradient, while the additional spatial modulation is characterized by a (small) amplitude δm and a wavevector qm. Our analysis relies on precise numerical solutions of suitably adapted Oberbeck–Boussinesq equations (OBE). In the absence of forcing (δm = 0), convection rolls with wavenumber qc bifurcate only for R above the critical Rayleigh number Rc. In contrast, for δm≠0, convection is unavoidable for any finite R; in the most simple case in the form of ‘forced rolls’ with wavevector qm. According to our first comprehensive stability diagram of these forced rolls in the qm – R plane, they develop instabilities against resonant oblique modes at R ≲ Rc in a wide range of qm/qc. Only for qm in the vicinity of qc, the forced rolls remain stable up to fairly large R > Rc. Direct numerical simulations of the OBE support and extend the findings of the stability analysis. Moreover, we are in line with the experimental results and also with some earlier theoretical results on this problem, based on asymptotic expansions in the limit δm → 0 and R → Rc. It is satisfying that in many cases the numerical results can be directly interpreted in terms of suitably constructed amplitude and generalized Swift–Hohenberg equations.

scholarly journals On the sensitivity of 3-D thermal convection codes to numerical discretization: a model intercomparison

2014 ◽  
Vol 7 (5) ◽  
pp. 2065-2076 ◽  
Author(s):  
P.-A Arrial ◽  
N. Flyer ◽  
G. B. Wright ◽  
L. H. Kellogg
Keyword(s):  
Rayleigh Number ◽  
Numerical Simulations ◽  
Spherical Harmonics ◽  
Thermal Convection ◽  
Mantle Convection ◽  
Numerical Study ◽  
Critical Rayleigh Number ◽  
Initial Condition ◽  
Numerical Discretization ◽  
The Impact

Abstract. Fully 3-D numerical simulations of thermal convection in a spherical shell have become a standard for studying the dynamics of pattern formation and its stability under perturbations to various parameter values. The question arises as to how the discretization of the governing equations affects the outcome and thus any physical interpretation. This work demonstrates the impact of numerical discretization on the observed patterns, the value at which symmetry is broken, and how stability and stationary behavior is dependent upon it. Motivated by numerical simulations of convection in the Earth's mantle, we consider isoviscous Rayleigh–Bénard convection at infinite Prandtl number, where the aspect ratio between the inner and outer shell is 0.55. We show that the subtleties involved in developing mantle convection models are considerably more delicate than has been previously appreciated, due to the rich dynamical behavior of the system. Two codes with different numerical discretization schemes – an established, community-developed, and benchmarked finite-element code (CitcomS) and a novel spectral method that combines Chebyshev polynomials with radial basis functions (RBFs) – are compared. A full numerical study is investigated for the following three cases. The first case is based on the cubic (or octahedral) initial condition (spherical harmonics of degree ℓ = 4). How this pattern varies to perturbations in the initial condition and Rayleigh number is studied. The second case investigates the stability of the dodecahedral (or icosahedral) initial condition (spherical harmonics of degree ℓ = 6). Although both methods first converge to the same pattern, this structure is ultimately unstable and systematically degenerates to cubic or tetrahedral symmetries, depending on the code used. Lastly, a new steady-state pattern is presented as a combination of third- and fourth-order spherical harmonics leading to a five-cell or hexahedral pattern and stable up to 70 times the critical Rayleigh number. This pattern can provide the basis for a new accuracy benchmark for 3-D spherical mantle convection codes.

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