The Spectrum of Steady State Turbulent Convection

1971 ◽  
Vol 26 (7) ◽  
pp. 1140-1146
Author(s):  
F. Winterberg
Keyword(s):  
Integral Equation ◽  
Steady State ◽  
Energy Spectrum ◽  
Buoyancy Force ◽  
Radiative Heat ◽  
Order Differential Equation ◽  
Stellar Atmospheres ◽  
Turbulent Convection ◽  
First Order ◽  
Model Equations

Abstract Based on Heisenberg's statistical theory of turbulence, a model for steady state turbulent convection is herein proposed, and on the basis of this model, equations for the energy spectrum for steady state turbulent convection are derived. The spectrum is obtained from the solution of a nonlinear integral equation. After the integral equation is brought into a universally valid nondimensional form, it is transformed into a nonlinear first order differential equation to be solved numerically, with the Rayleigh number appearing as the only parameter. The energy spectrum has a substantial deviation from the Kolmogoroff law, as a result of the buoyancy force acting on the rising and falling eddies. The presented theory may be applicable to convection in planetary and stellar atmospheres wherein the radiative heat transport is small.

Lambertian Enclosures — A First Step towards Fast Room Acoustics Simulation

2003 ◽  
Vol 10 (1) ◽  
pp. 33-54 ◽  
Author(s):  
D. Alarcão ◽  
J. L. Bento Coelho
Keyword(s):  
Integral Equation ◽  
Markov Chain ◽  
Steady State ◽  
Statistical Method ◽  
Room Acoustics ◽  
Homogeneous Markov Chain ◽  
Acoustical Parameters ◽  
First Order ◽  
Time Discretisation ◽  
Reflecting Boundaries

A statistical method for calculation of the acoustical parameters of lambertian enclosures (that is, enclosures with diffusely reflecting boundaries) is presented. The theory considers the distribution of sound particles over the boundaries of the enclosures. The method includes the familiar Kuttruff Integral Equation. A homogeneous Markov Chain of first order is obtained through the time discretisation of the equations. Applications of this method are demonstrated for the case of a long enclosure and for the case of a real-shaped room. Decay calculations as well as steady-state sound distributions are obtained. The results show that the method is reliable, flexible, and that computation times are low.

A Fast and Reliable Dynamic Tyre-Road Contact Model for ABS Braking Manoeuvre Simulations

2006 ◽  
Author(s):  
Francesco Braghin ◽  
Federico Cheli ◽  
Emiliano Giangiulio ◽  
Federico Mancosu ◽  
Daniele Arosio
Keyword(s):  
Differential Equation ◽  
Steady State ◽  
Order Differential Equation ◽  
Transient Behavior ◽  
Slip Angle ◽  
Relaxation Length ◽  
Steady State Condition ◽  
Tyre Model ◽  
First Order ◽  
Significant Parameter

Due to the dimensions of the tyre-road contact area, transients in a tyre last approximately 0.1s. Thus, in the case of abrupt maneuvers such as ABS braking, the use of a steady-state tyre model to predict the vehicle’s behavior would lead to significant errors. Available dynamic tyre models, such as Pacejka’s MF-Tyre model, are based on steady-state formulations and the transient behavior of the tyre is included by introducing a first order differential equation of relevant quantities such as the slip angle and the slippage. In these differential equations the most significant parameter used to describe the transient behavior is the so-called relaxation length, i.e. the distance traveled by the tyre to settle to a new steady–state condition once perturbated. Usually this parameter is assumed to be constant.

A Dynamic Tire Model for ABS Maneuver Simulations

2010 ◽  
Vol 38 (2) ◽  
pp. 137-154 ◽  
Author(s):  
Francesco Braghin ◽  
Edoardo Sabbioni
Keyword(s):  
Steady State ◽  
Order Differential Equation ◽  
Transient Behavior ◽  
Slip Angle ◽  
Tire Model ◽  
Relaxation Length ◽  
Steady State Condition ◽  
First Order ◽  
Significant Parameter ◽  
Tire Models

Abstract Due to the dimensions of the tire-road contact area, transients in a tire last approximately 0.1 s. Thus, in the case of abrupt maneuvers such as ABS braking, the use of a steady-state tire model to predict the vehicle’s behavior would lead to significant errors. Available dynamic tire models, such as Pacejka’s MF-Tire model, are based on steady-state formulations and the transient behavior of the tire is included by introducing a first order differential equation of relevant quantities such as the slip angle and the slippage. In these differential equations the most significant parameter used to describe the transient behavior is the so-called relaxation length, i.e., the distance traveled by the tire to settle to a new steady-state condition once perturbated. Usually this parameter is assumed to be constant.

Molecular Dynamics Study of Cohesionless Granular Materials: Size Segregation by Shaking

1993 ◽  
Vol 07 (09n10) ◽  
pp. 1865-1872 ◽  
Author(s):  
Toshiya OHTSUKI ◽  
Yoshikazu TAKEMOTO ◽  
Tatsuo HATA ◽  
Shigeki KAWAI ◽  
Akihisa HAYASHI
Keyword(s):  
Molecular Dynamics ◽  
Steady State ◽  
Specific Gravity ◽  
Pressure Gradient ◽  
Granular Materials ◽  
Buoyancy Force ◽  
Temporal Evolution ◽  
Particle Radius ◽  
Size Segregation ◽  
Increasing Function

The Molecular Dynamics technique is used to investigate size segregation by shaking in cohesionless granular materials. Temporal evolution of the height h of the tagged particle with different size and mass is measured for various values of the particle radius and specific gravity. It becomes evident that h approaches the steady state value h∞ independent of initial positions. There exists a threshold of the specific gravity of the particle. Below the threshold, h∞ is an increasing function of the particle size, whereas above it, h∞ decreases with increasing the particle radius. The relaxation time τ towards the steady state is calculated and its dependence on the particle radius and specific gravity is clarified. The pressure gradient of pure systems is also measured and turned out to be almost constant. This suggests that the buoyancy force due to the pressure gradient is not responsible to h∞.

scholarly journals Non-Linear First-Order Differential Boundary Problems with Multipoint and Integral Conditions

2021 ◽  
Vol 5 (1) ◽  
pp. 15
Author(s):  
Misir J. Mardanov ◽  
Yagub A. Sharifov ◽  
Yusif S. Gasimov ◽  
Carlo Cattani
Keyword(s):  
Integral Equation ◽  
Integral Boundary Conditions ◽  
Contraction Mapping Principle ◽  
Boundary Problems ◽  
Integral Boundary ◽  
Multipoint Problem ◽  
First Order ◽  
Banach Contraction ◽  
First Time ◽  
Banach Contraction Mapping Principle

This paper considers boundary value problem (BVP) for nonlinear first-order differential problems with multipoint and integral boundary conditions. A suitable Green function was constructed for the first time in order to reduce this problem into a corresponding integral equation. So that by using the Banach contraction mapping principle (BCMP) and Schaefer’s fixed point theorem (SFPT) on the integral equation, we can show that the solution of the multipoint problem exists and it is unique.

scholarly journals Bubble wall velocity at strong coupling

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Francesco Bigazzi ◽  
Alessio Caddeo ◽  
Tommaso Canneti ◽  
Aldo L. Cotrone
Keyword(s):  
Phase Transitions ◽  
Steady State ◽  
Friction Force ◽  
Bubble Velocity ◽  
Strongly Coupled ◽  
First Order ◽  
Black Hole Background ◽  
Different Dimensions ◽  
Holographic Description ◽  
First Order Phase

Abstract Using the holographic correspondence as a tool, we determine the steady-state velocity of expanding vacuum bubbles nucleated within chiral finite temperature first-order phase transitions occurring in strongly coupled large N QCD-like models. We provide general formulae for the friction force exerted by the plasma on the bubbles and for the steady-state velocity. In the top-down holographic description, the phase transitions are related to changes in the embedding of $$ Dq\hbox{-} \overline{D}q $$ Dq ‐ D ¯ q flavor branes probing the black hole background sourced by a stack of N Dp-branes. We first consider the Witten-Sakai-Sugimoto $$ D4\hbox{-} D8\hbox{-} \overline{D}8 $$ D 4 ‐ D 8 ‐ D ¯ 8 setup, compute the friction force and deduce the equilibrium velocity. Then we extend our analysis to more general setups and to different dimensions. Finally, we briefly compare our results, obtained within a fully non-perturbative framework, to other estimates of the bubble velocity in the literature.

scholarly journals An exact expression of positive periodic solution for a first-order singular equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yun Xin ◽  
Xiaoxiao Cui ◽  
Jie Liu
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Lower Bounds ◽  
Topological Degree ◽  
Order Differential Equation ◽  
Positive Periodic Solution ◽  
Exact Expression ◽  
Upper And Lower Bounds ◽  
Singular Equation ◽  
First Order

Abstract The main purpose of this paper is to obtain an exact expression of the positive periodic solution for a first-order differential equation with attractive and repulsive singularities. Moreover, we prove the existence of at least one positive periodic solution for this equation with an indefinite singularity by applications of topological degree theorem, and give the upper and lower bounds of the positive periodic solution.

Asymptotics for third-order nonlinear differential equations: Non-oscillatory and oscillatory cases

2021 ◽  
pp. 1-19
Author(s):  
Calogero Vetro ◽  
Dariusz Wardowski
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Oscillatory Behavior ◽  
Third Order ◽  
First Order ◽  
Third Order Differential Equation ◽  
Comparison Technique ◽  
First Order Differential Equations

We discuss a third-order differential equation, involving a general form of nonlinearity. We obtain results describing how suitable coefficient functions determine the asymptotic and (non-)oscillatory behavior of solutions. We use comparison technique with first-order differential equations together with the Kusano–Naito’s and Philos’ approaches.

scholarly journals Kinetics of Biofilm Detachment

1992 ◽  
Vol 26 (9-11) ◽  
pp. 1995-1998 ◽  
Author(s):  
B. M. Peyton ◽  
W. G. Characklis
Keyword(s):  
Steady State ◽  
Utilization Rate ◽  
Carbon Utilization ◽  
Biofilm Thickness ◽  
First Order ◽  
Detachment Rate ◽  
Biofilm Detachment ◽  
Biofilm Modeling ◽  
Kinetics Of ◽  
Sensitive Variable

In predictive biofilm modeling, the detachment rate coefficient may be the most sensitive variable affecting both the predicted rate and the extent of biofilm accumulation. At steady state the detachment rate must be equal to the net growth rate in the biofilm. In systems where organic carbon is growth-limiting, the substrate carbon utilization rate determines the net biomass production rate and, therefore, the steady state biomass detachment rate. Detachment rates, first order with biofilm thickness, fit the experimental data well, but are not predictive since the coefficients must be determined experimentally.

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