scholarly journals The effect of sudden source buoyancy flux increases on turbulent plumes

2009 ◽  
Vol 635 ◽  
pp. 137-169 ◽  
Author(s):  
M. M. SCASE ◽  
A. J. ASPDEN ◽  
C. P. CAULFIELD
Keyword(s):  
Vortex Ring ◽  
Scaling Laws ◽  
Scaling Law ◽  
Buoyancy Flux ◽  
Time Dependent ◽  
Strength Increase ◽  
Source Strength ◽  
Ambient Fluid ◽  
Rise Height ◽  
Pure Plume

Building upon the recent experimentally verified modelling of turbulent plumes which are subject to decreases in their source strength (Scase et al., J. Fluid Mech., vol. 563, 2006b, p. 443), we consider the complementary case where the plume's source strength is increased. We consider the effect of increasing the source strength of an established plume and we also compare time-dependent plume model predictions for the behaviour of a starting plume to those of Turner (J. Fluid Mech., vol. 13, 1962, p. 356).Unlike the decreasing source strength problems considered previously, the relevant solution to the time-dependent plume equations is not a simple similarity solution. However, scaling laws are demonstrated which are shown to be applicable across a large number of orders of magnitude of source strength increase. It is shown that an established plume that is subjected to an increase in its source strength supports a self-similar ‘pulse’ structure propagating upwards. For a point source plume, in pure plume balance, subjected to an increase in the source buoyancy flux F0, the rise height of this pulse in terms of time t scales as t3/4 while the vertical extent of the pulse scales as t1/4. The volume of the pulse is shown to scale as t9/4. For plumes in pure plume balance that emanate from a distributed source it is shown that the same scaling laws apply far from the source, demonstrating an analogous convergence to pure plume balance as that which is well known in steady plumes. These scaling law predictions are compared to implicit large eddy simulations of the buoyancy increase problem and are shown to be in good agreement.We also compare the predictions of the time-dependent model to a starting plume in the limit where the source buoyancy flux is discontinuously increased from zero. The conventional model for a starting plume is well approximated by a rising turbulent, entraining, buoyant vortex ring which is fed from below by a ‘steady’ plume. However, the time-dependent plume equations have been defined for top-hat profiles assuming only horizontal entrainment. Therefore, this system cannot model either the internal dynamics of the starting plume's head or the extra entrainment of ambient fluid into the head due to the turbulent boundary of the vortex ring-like cap. We show that the lack of entrainment of ambient fluid through the head of the starting plume means that the time-dependent plume equations overestimate the rise height of a starting plume with time. However, by modifying the entrainment coefficient appropriately, we see that realistic predictions consistent with experiment can be attained.

Buoyancy-driven mean flow in a long channel with a hydraulically constrained exit condition

1999 ◽  
Vol 398 ◽  
pp. 155-180 ◽  
Author(s):  
TH. GRIMM ◽  
T. MAXWORTHY
Keyword(s):  
Scaling Laws ◽  
Scaling Law ◽  
Buoyancy Flux ◽  
Experimental Results ◽  
Density Difference ◽  
Mean Flow ◽  
Natural Systems ◽  
Lateral Contraction ◽  
The Persian Gulf ◽  
Long Channel

Convection plays a major role in a variety of natural hydrodynamic systems. Those in which convection drives exchange flows through a lateral contraction and/or over a sill form a special class with typical examples being the Red and Mediterranean Seas, the Persian Gulf, and the fjords that indent many coastlines. The present work focuses on the spatial distribution and scaling of the density difference between the inflowing and outflowing fluid layers. Using a long water-filled channel, fitted with buoyancy sources at its upper surface, experiments were conducted to investigate the influence of the geometry of the strait and the channel as well as the magnitude of the buoyancy flux. Two different scaling laws, one by Phillips (1966), and one by Maxworthy (1994, 1997) were compared with the experimental results. It has been shown that a scaling law for which g′ = kB02/3x/h4/3 best describes the distribution of the observed density difference along the channel, where B0 is the buoyancy flux, x the distance from the closed end of the channel, h its height at the open end (sill) and k a constant that depends on the details of the channel geometry and flow conditions. This result holds for the experimental results and appears to be valid for a number of natural systems as well.

scholarly journals Structure evolution of suspensions under time-dependent electric or magnetic field

2021 ◽  
Author(s):  
Konstantinos Manikas ◽  
Markus Hütter ◽  
Patrick D. Anderson
Keyword(s):  
Magnetic Field ◽  
Field Strength ◽  
Brownian Dynamics ◽  
Thermal Fluctuations ◽  
Time Dependent ◽  
Structure Evolution ◽  
Strength Increase ◽  
Large Rotation ◽  
Dynamics Simulations ◽  
Brownian Dynamics Simulations

AbstractThe effect of time-dependent external fields on the structures formed by particles with induced dipoles dispersed in a viscous fluid is investigated by means of Brownian Dynamics simulations. The physical effects accounted for are thermal fluctuations, dipole-dipole and excluded volume interactions. The emerging structures are characterised in terms of particle clusters (orientation, size, anisotropy and percolation) and network structure. The strength of the external field is increased in one direction and then kept constant for a certain amount of time, with the structure formation being influenced by the slope of the field-strength increase. This effect can be partially rationalized by inhomogeneous time re-scaling with respect to the field strength, however, the presence of thermal fluctuations makes the scaling at low field strength inappropriate. After the re-scaling, one can observe that the lower the slope of the field increase, the more network-like and the thicker the structure is. In the second part of the study the field is also rotated instantaneously by a certain angle, and the effect of this transition on the structure is studied. For small rotation angles ($$\theta \le 20^{{\circ }}$$ θ ≤ 20 ∘ ) the clusters rotate but stay largely intact, while for large rotation angles ($$\theta \ge 80^{{\circ }}$$ θ ≥ 80 ∘ ) the structure disintegrates and then reforms, due to the nature of the interactions (parallel dipoles with perpendicular inter-particle vector repel each other). For intermediate angles ($$20<\theta <80^{{\circ }}$$ 20 < θ < 80 ∘ ), it seems that, during rotation, the structure is altered towards a more network-like state, as a result of cluster fusion (larger clusters). The details provided in this paper concern an electric field, however, all results can be projected into the case of a magnetic field and paramagnetic particles.

scholarly journals Asymmetric Unimodal Maps with Non-universal Period-Doubling Scaling Laws

2020 ◽  
Vol 379 (1) ◽  
pp. 103-143
Author(s):  
Oleg Kozlovski ◽  
Sebastian van Strien
Keyword(s):  
Scaling Laws ◽  
Scaling Law ◽  
Period Doubling ◽  
Cantor Sets ◽  
Unimodal Maps ◽  
Renormalization Theory

Abstract We consider a family of strongly-asymmetric unimodal maps $$\{f_t\}_{t\in [0,1]}$$ { f t } t ∈ [ 0 , 1 ] of the form $$f_t=t\cdot f$$ f t = t · f where $$f:[0,1]\rightarrow [0,1]$$ f : [ 0 , 1 ] → [ 0 , 1 ] is unimodal, $$f(0)=f(1)=0$$ f ( 0 ) = f ( 1 ) = 0 , $$f(c)=1$$ f ( c ) = 1 is of the form and $$\begin{aligned} f(x)=\left\{ \begin{array}{ll} 1-K_-|x-c|+o(|x-c|)&{} \text{ for } x<c, \\ 1-K_+|x-c|^\beta + o(|x-c|^\beta ) &{} \text{ for } x>c, \end{array}\right. \end{aligned}$$ f ( x ) = 1 - K - | x - c | + o ( | x - c | ) for x < c , 1 - K + | x - c | β + o ( | x - c | β ) for x > c , where we assume that $$\beta >1$$ β > 1 . We show that such a family contains a Feigenbaum–Coullet–Tresser $$2^\infty $$ 2 ∞ map, and develop a renormalization theory for these maps. The scalings of the renormalization intervals of the $$2^\infty $$ 2 ∞ map turn out to be super-exponential and non-universal (i.e. to depend on the map) and the scaling-law is different for odd and even steps of the renormalization. The conjugacy between the attracting Cantor sets of two such maps is smooth if and only if some invariant is satisfied. We also show that the Feigenbaum–Coullet–Tresser map does not have wandering intervals, but surprisingly we were only able to prove this using our rather detailed scaling results.

scholarly journals Scaling and Similarity of the Anisotropic Coherent Eddies in Near-Surface Atmospheric Turbulence

2018 ◽  
Vol 75 (3) ◽  
pp. 943-964 ◽  
Author(s):  
Khaled Ghannam ◽  
Gabriel G. Katul ◽  
Elie Bou-Zeid ◽  
Tobias Gerken ◽  
Marcelo Chamecki
Keyword(s):  
Scaling Laws ◽  
Velocity Structure ◽  
Scaling Law ◽  
Longitudinal Velocity ◽  
Structure Functions ◽  
Length Scale ◽  
Near Surface ◽  
Vegetation Canopies ◽  
Velocity Spectra ◽  
Attached Eddies

Abstract The low-wavenumber regime of the spectrum of turbulence commensurate with Townsend’s “attached” eddies is investigated here for the near-neutral atmospheric surface layer (ASL) and the roughness sublayer (RSL) above vegetation canopies. The central thesis corroborates the significance of the imbalance between local production and dissipation of turbulence kinetic energy (TKE) and canopy shear in challenging the classical distance-from-the-wall scaling of canonical turbulent boundary layers. Using five experimental datasets (two vegetation canopy RSL flows, two ASL flows, and one open-channel experiment), this paper explores (i) the existence of a low-wavenumber k−1 scaling law in the (wind) velocity spectra or, equivalently, a logarithmic scaling ln(r) in the velocity structure functions; (ii) phenomenological aspects of these anisotropic scales as a departure from homogeneous and isotropic scales; and (iii) the collapse of experimental data when plotted with different similarity coordinates. The results show that the extent of the k−1 and/or ln(r) scaling for the longitudinal velocity is shorter in the RSL above canopies than in the ASL because of smaller scale separation in the former. Conversely, these scaling laws are absent in the vertical velocity spectra except at large distances from the wall. The analysis reveals that the statistics of the velocity differences Δu and Δw approach a Gaussian-like behavior at large scales and that these eddies are responsible for momentum/energy production corroborated by large positive (negative) excursions in Δu accompanied by negative (positive) ones in Δw. A length scale based on TKE dissipation collapses the velocity structure functions at different heights better than the inertial length scale.

Clocking the buildup dynamics of light-induced states through attosecond transient absorption spectrum

2021 ◽  
Author(s):  
Xiaoxia Wu ◽  
Shaofeng Zhang ◽  
Difa Ye
Keyword(s):  
Floquet Theory ◽  
Transient Absorption ◽  
Scaling Law ◽  
Time Lag ◽  
Time Dependent ◽  
Transient Absorption Spectroscopy ◽  
Transient Absorption Spectrum ◽  
Wide Range ◽  
Maximal Shift ◽  
Buildup Time

Abstract The buildup processes of the light-induced states (LISs) in attosecond transient absorption spectroscopy are studied by solving the time-dependent Schrödinger equation and compared with the quasistatic Floquet theory, revealing a time lag of the maximal shift and strongest absorbance of the LIS with respect to the zero delay that is referred to as the buildup time. We analytically derive a scaling law for the buildup time that confirms the numerical results over a wide range of detunings. Our theory verifies the commonly accepted scenario of nearly instantaneous response of matter to light if the pump field is blue-detuned, but some differences are found in the near-resonant and red-detuning cases. Implications of the buildup time in petahertz optoelectronics are discussed.

Critical dynamics and renormalization group (RG)

2021 ◽  
pp. 875-898
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Renormalization Group ◽  
Time Evolution ◽  
Langevin Equation ◽  
Scaling Laws ◽  
Transport Coefficients ◽  
Time Dependent ◽  
Critical Systems ◽  
Static Properties ◽  
Critical Domain ◽  
Dynamical Scaling

Time evolution, near a phase transition in the critical domain of critical systems not far from equilibrium, using a Langevin-type evolution is studied. Typical quantities of interest are relaxation rates towards equilibrium, time-dependent correlation functions and transport coefficients. The main motivation for such a study is that, in systems in which the dynamics is local (on short time-scales, a modification of a dynamic variable has an influence only locally in space) when the correlation length becomes large, a large time-scale emerges, which characterizes the rate of time evolution. This phenomenon called critical slowing down leads to universal behaviour and scaling laws for time-dependent quantities. In contrast with the situation in static critical phenomena, there is no clean and systematic derivation of the dynamical equations governing the time evolution in the critical domain, because often the time evolution is influenced by conservation laws involving the order parameter, or other variables like energy, momentum, angular momentum, currents and so on. Indeed, the equilibrium distribution does not determine the driving force in the Langevin equation, but only the dissipative couplings are generated by the derivative of the equilibrium Hamiltonian, and directly related to the static properties. The purely dissipative Langevin equation specifically discussed, corresponding to static models like the f4 field theory and two-dimensional models. Renormalization group (RG) equations are derived, and dynamical scaling relations established.

scholarly journals Multiphase plumes in a stratified ambient

2019 ◽  
Vol 869 ◽  
pp. 292-312 ◽  
Author(s):  
Nicola Mingotti ◽  
Andrew W. Woods
Keyword(s):  
Oil And Gas ◽  
Fluid Phase ◽  
Convective Transport ◽  
Maximum Depth ◽  
Buoyancy Flux ◽  
Return Flow ◽  
Ambient Fluid ◽  
Plume Height ◽  
Cylindrical Column ◽  
Fall Speed

We report on experiments of turbulent particle-laden plumes descending through a stratified environment. We show that provided the characteristic plume speed $(B_{0}N)^{1/4}$ exceeds the particle fall speed, where the plume buoyancy flux is $B_{0}$ and the Brunt–Väisälä frequency is $N$, then the plume is arrested by the stratification and initially intrudes at the neutral height associated with a single-phase plume of the same buoyancy flux. If the original fluid phase in the plume has density equal to that of the ambient fluid at the source, then as the particles sediment from the intruding fluid, the fluid finds itself buoyant and rises, ultimately intruding at a height of about $0.58\pm 0.03$ of the original plume height, consistent with new predictions we present based on classical plume theory. We generalise this result, and show that if the buoyancy flux at the source is composed of a fraction $F_{s}$ associated with the buoyancy of the source fluid, and a fraction $1-F_{s}$ from the particles, then following the sedimentation of the particles, the plume fluid intrudes at a height $(0.58+0.22F_{s}\pm 0.03)H_{t}$, where $H_{t}$ is the maximum plume height. This is key for predictions of the environmental impact of any material dissolved in the plume water which may originate from the particle load. We also show that the particles sediment at their fall speed through the fluid below the maximum depth of the plume as a cylindrical column whose area scales as the ratio of the particle flux at the source to the fall speed and concentration of particles in the plume at the maximum depth of the plume before it is arrested by the stratification. We demonstrate that there is negligible vertical transport of fluid in this cylindrical column, but a series of layers of high and low particle concentration develop in the column with a vertical spacing which is given by the ratio of the buoyancy of the particle load and the background buoyancy gradient. Small fluid intrusions develop at the side of the column associated with these layers, as dense parcels of particle-laden fluid convect downwards and then outward once the particles have sedimented from the fluid, with a lateral return flow drawing in ambient fluid. As a result, the pattern of particle-rich and particle-poor layers in the column gradually migrates upwards owing to the convective transport of particles between the particle-rich layers superposed on the background sedimentation. We consider the implications of the results for mixing by bubble plumes, for submarine blowouts of oil and gas and for the fate of plumes of waste particles discharged at the ocean surface during deep-sea mining.

scholarly journals Origin of the scaling laws of sediment transport

2017 ◽  
Vol 473 (2197) ◽  
pp. 20160785 ◽  
Author(s):  
Sk Zeeshan Ali ◽  
Subhasish Dey
Keyword(s):  
Sediment Transport ◽  
Froude Number ◽  
Scaling Laws ◽  
Scaling Law ◽  
Bedload Transport ◽  
Inertial Range ◽  
Channel Width ◽  
Densimetric Froude Number ◽  
Relative Roughness ◽  
Contraction Ratio

In this paper, we discover the origin of the scaling laws of sediment transport under turbulent flow over a sediment bed, for the first time, from the perspective of the phenomenological theory of turbulence. The results reveal that for the incipient motion of sediment particles, the densimetric Froude number obeys the ‘(1 +  σ )/4’ scaling law with the relative roughness (ratio of particle diameter to approach flow depth), where σ is the spectral exponent of turbulent energy spectrum. However, for the bedforms, the densimetric Froude number obeys a ‘(1 +  σ )/6’ scaling law with the relative roughness in the enstrophy inertial range and the energy inertial range. For the bedload flux, the bedload transport intensity obeys the ‘3/2’ and ‘(1 +  σ )/4’ scaling laws with the transport stage parameter and the relative roughness, respectively. For the suspended load flux, the non-dimensional suspended sediment concentration obeys the ‘ − Z ’ scaling law with the non-dimensional vertical distance within the wall shear layer, where Z is the Rouse number. For the scour in contracted streams, the non-dimensional scour depth obeys the ‘4/(3 −  σ )’, ‘−4/(3 −  σ )’ and ‘−(1 +  σ )/(3 −  σ )’ scaling laws with the densimetric Froude number, the channel contraction ratio (ratio of contracted channel width to approach channel width) and the relative roughness, respectively.

scholarly journals Scaling laws and warning signs for bifurcations of SPDEs

2018 ◽  
Vol 30 (5) ◽  
pp. 853-868
Author(s):  
CHRISTIAN KUEHN ◽  
FRANCESCO ROMANO
Keyword(s):  
Differential Equations ◽  
Scaling Laws ◽  
Scaling Law ◽  
Warning Signs ◽  
Covariance Operator ◽  
Large Classes ◽  
Practical Applications ◽  
Critical Transitions ◽  
Degenerate Eigenvalues ◽  
Adjoint Operators

Critical transitions (or tipping points) are drastic sudden changes observed in many dynamical systems. Large classes of critical transitions are associated with systems, which drift slowly towards a bifurcation point. In the context of stochastic ordinary differential equations, there are results on growth of variance and autocorrelation before a transition, which can be used as possible warning signs in applications. A similar theory has recently been developed in the simplest setting for stochastic partial differential equations (SPDEs) for self-adjoint operators in the drift term. This setting leads to real discrete spectrum and growth of the covariance operator via a certain scaling law. In this paper, we develop this theory substantially further. We cover the cases of complex eigenvalues, degenerate eigenvalues as well as continuous spectrum. This provides a fairly comprehensive theory for most practical applications of warning signs for SPDE bifurcations.

Scaling Laws From Statistical Data and Dimensional Analysis

2004 ◽  
Vol 72 (5) ◽  
pp. 648-657 ◽  
Author(s):  
Patricio F. Mendez ◽  
Fernando Ordóñez
Keyword(s):  
Dimensional Analysis ◽  
Scaling Laws ◽  
Ad Hoc ◽  
Scaling Law ◽  
Engineering Systems ◽  
User Input ◽  
Backward Elimination ◽  
Dimensionless Groups ◽  
Complex Engineering ◽  
Complex Engineering Systems

Scaling laws provide a simple yet meaningful representation of the dominant factors of complex engineering systems, and thus are well suited to guide engineering design. Current methods to obtain useful models of complex engineering systems are typically ad hoc, tedious, and time consuming. Here, we present an algorithm that obtains a scaling law in the form of a power law from experimental data (including simulated experiments). The proposed algorithm integrates dimensional analysis into the backward elimination procedure of multivariate linear regressions. In addition to the scaling laws, the algorithm returns a set of dimensionless groups ranked by relevance. We apply the algorithm to three examples, in each obtaining the scaling law that describes the system with minimal user input.

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