scholarly journals Thermodynamics/Dynamics Coupling in Weakly Compressible Turbulent Stratified Fluids

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Rémi Tailleux
Keyword(s):  
Equation Of State ◽  
Nonlinear Equation ◽  
Potential Energy ◽  
Mechanical Energy ◽  
Fluid Flows ◽  
Boussinesq Approximation ◽  
Gravitational Potential Energy ◽  
Transient Flows ◽  
Diffusive Effects ◽  
Boussinesq Fluid

In traditional and geophysical fluid dynamics, it is common to describe stratified turbulent fluid flows with low Mach number and small relative density variations by means of the incompressible Boussinesq approximation. Although such an approximation is often interpreted as decoupling the thermodynamics from the dynamics, this paper reviews recent results and derive new ones that show that the reality is actually more subtle and complex when diabatic effects and a nonlinear equation of state are retained. Such an analysis reveals indeed: (1) that the compressible work of expansion/contraction remains of comparable importance as the mechanical energy conversions in contrast to what is usually assumed; (2) in a Boussinesq fluid, compressible effects occur in the guise of changes in gravitational potential energy due to density changes. This makes it possible to construct a fully consistent description of the thermodynamics of incompressible fluids for an arbitrary nonlinear equation of state; (3) rigorous methods based on using the available potential energy and potential enthalpy budgets can be used to quantify the work of expansion/contraction in steady and transient flows, which reveals that is predominantly controlled by molecular diffusive effects, and act as a significant sink of kinetic energy.

scholarly journals Understanding mixing efficiency in the oceans: do the nonlinearities of the equation of state for seawater matter?

Ocean Science ◽  
2009 ◽  
Vol 5 (3) ◽  
pp. 271-283 ◽  
Author(s):  
R. Tailleux
Keyword(s):  
Equation Of State ◽  
Potential Energy ◽  
Energy Conversion ◽  
Internal Energy ◽  
Molecular Diffusion ◽  
Rate Of Change ◽  
Gravitational Potential Energy ◽  
Mixing Efficiency ◽  
Buoyancy Frequency ◽  
Boussinesq Fluid

Abstract. There exist two central measures of turbulent mixing in turbulent stratified fluids that are both caused by molecular diffusion: 1) the dissipation rate D(APE) of available potential energy APE; 2) the turbulent rate of change Wr, turbulent of background gravitational potential energy GPEr. So far, these two quantities have often been regarded as the same energy conversion, namely the irreversible conversion of APE into GPEr, owing to the well known exact equality D(APE)=Wr, turbulent for a Boussinesq fluid with a linear equation of state. Recently, however, Tailleux (2009) pointed out that the above equality no longer holds for a thermally-stratified compressible, with the ratio ξ=Wr, turbulent/D(APE) being generally lower than unity and sometimes even negative for water or seawater, and argued that D(APE) and Wr, turbulent actually represent two distinct types of energy conversion, respectively the dissipation of APE into one particular subcomponent of internal energy called the "dead" internal energy IE0, and the conversion between GPEr and a different subcomponent of internal energy called "exergy" IEexergy. In this paper, the behaviour of the ratio ξ is examined for different stratifications having all the same buoyancy frequency N vertical profile, but different vertical profiles of the parameter Υ=α P/(ρCp), where α is the thermal expansion coefficient, P the hydrostatic pressure, ρ the density, and Cp the specific heat capacity at constant pressure, the equation of state being that for seawater for different particular constant values of salinity. It is found that ξ and Wr, turbulent depend critically on the sign and magnitude of dΥ/dz, in contrast with D(APE), which appears largely unaffected by the latter. These results have important consequences for how the mixing efficiency should be defined and measured in practice, which are discussed.

scholarly journals Understanding mixing efficiency in the oceans: do the nonlinearities of the equation of state for seawater matter?

2009 ◽  
Vol 6 (1) ◽  
pp. 371-387
Author(s):  
R. Tailleux
Keyword(s):  
Equation Of State ◽  
Potential Energy ◽  
Linear Equation ◽  
Internal Energy ◽  
Molecular Diffusion ◽  
Rate Of Change ◽  
Gravitational Potential Energy ◽  
Mixing Efficiency ◽  
Nonlinear Character ◽  
Boussinesq Fluid

Abstract. There exist two central measures of turbulent diffusive mixing in turbulent stratified fluids, which are both caused by molecular diffusion: 1) the dissipation rate D(APE) of available potential energy APE; 2) the rate of change Wr,mixing of background gravitational potential energy GPEr. So far, these two quantities have often been regarded as representing the same kind of energy conversion, i.e., the irreversible conversion of APE into GPEr, owing to the well known result that D(APE)≈Wr,mixing in a Boussinesq fluid with a linear equation of state. Here, this idea is challenged by showing that while D(APE) remains largely unaffected by a nonlinear equation of state, Wr,mixing is in contrast strongly affected by the latter. This result is rationalized by using the recent results of Tailleux (2008), which argues that D(APE) represents the dissipation of APE into one particular subcomponent of internal energy called the "dead" internal energy IE0, whereas Wr,mixing represents the conversion between a different subcomponent of internal energy – called the "exergy" IEexergy – and GPEr. It follows that the concept of mixing efficiency, which represents the fraction of the stirring mechanical energy ultimately dissipated by molecular diffusion is related to D(APE), not Wr,mixing, which ensures that it should be largely unaffected by the nonlinear character of the equation of state, and therefore correctly described in the context of a Boussinesq fluid with a linear equation of state. The variations of GPEr, on the other hand, are sensitive to the linear or nonlinear character of the equation of state.

scholarly journals Available potential energy density for a multicomponent Boussinesq fluid with arbitrary nonlinear equation of state

2013 ◽  
Vol 735 ◽  
pp. 499-518 ◽  
Author(s):  
Rémi Tailleux
Keyword(s):  
Equation Of State ◽  
Kinetic Energy ◽  
Nonlinear Equation ◽  
Potential Energy ◽  
Reference State ◽  
Constant Density ◽  
Production Term ◽  
Reference Position ◽  
Available Potential Energy ◽  
Boussinesq Fluid

AbstractIn this paper, the concept of available potential energy (APE) density is extended to a multicomponent Boussinesq fluid with a nonlinear equation of state. As shown by previous studies, the APE density is naturally interpreted as the work against buoyancy forces that a parcel needs to perform to move from a notional reference position at which its buoyancy vanishes to its actual position; because buoyancy can be defined relative to an arbitrary reference state, so can APE density. The concept of APE density is therefore best viewed as defining a class of locally defined energy quantities, each tied to a different reference state, rather than as a single energy variable. An important result, for which a new proof is given, is that the volume-integrated APE density always exceeds Lorenz’s globally defined APE, except when the reference state coincides with Lorenz’s adiabatically re-arranged reference state of minimum potential energy. A parcel reference position is systematically defined as a level of neutral buoyancy (LNB): depending on the nature of the fluid and on how the reference state is defined, a parcel may have one, none, or multiple LNB within the fluid. Multiple LNB are only possible for a multicomponent fluid whose density depends on pressure. When no LNB exists within the fluid, a parcel reference position is assigned at the minimum or maximum geopotential height. The class of APE densities thus defined admits local and global balance equations, which all exhibit a conversion to kinetic energy, a production term by boundary buoyancy fluxes, and a dissipation term by internal diffusive effects. Different reference states alter the partition between APE production and dissipation, but neither affects the net conversion between kinetic energy and APE, nor the difference between APE production and dissipation. We argue that the possibility of constructing APE-like budgets based on reference states other than Lorenz’s reference state is more important than has been previously assumed, and we illustrate the feasibility of doing so in the context of an idealized and realistic oceanic example, using as reference states one with constant density and another one defined as the horizontal-mean density field; in the latter case, the resulting APE density is found to be a reasonable approximation of the APE density constructed from Lorenz’s reference state, while being computationally cheaper.

scholarly journals Energy Conversion, Mixing Energy, and Neutral Surfaces with a Nonlinear Equation of State

2011 ◽  
Vol 41 (1) ◽  
pp. 28-41 ◽  
Author(s):  
Jonas Nycander
Keyword(s):  
Equation Of State ◽  
Nonlinear Equation ◽  
Potential Energy ◽  
Gravitational Potential ◽  
Stationary Flow ◽  
Gravitational Potential Energy ◽  
Surrounding Water ◽  
Mixing Energy ◽  
Nonlinear Character ◽  
Water Parcel

Abstract A local neutral plane is defined so that a water parcel that is displaced adiabatically a small distance along the plane continues to have the same density as the surrounding water. Since such a displacement does not change the density field or the gravitational potential energy, it is generally assumed that it does not produce a restoring buoyancy force. However, it is here shown that because of the nonlinear character of the equation of state (in particular the thermobaric effect) such a neutral displacement is accompanied by a conversion between internal energy E and gravitational potential energy U, and an equal conversion between U and kinetic energy K. While there is thus no net change of U, K does change. This implies that a force is in fact required for the displacement. It is further shown that displacements that are orthogonal to a vector P do not induce conversion between U and K, and therefore do not require a force. Analogously to neutral surfaces, which are defined to be approximately orthogonal to the dianeutral vector N, one may define “P surfaces” to be approximately orthogonal to P. These P surfaces are intermediate between neutral surfaces and surfaces of constant σ0 (potential density reference to the surface). If the equation of state is linear, there exists a well-known expression for the mixing energy in terms of the diapycnal flow. This expression is here generalized for a general nonlinear equation of state. The generalized expression involves the velocity component along P. Since P is not orthogonal to neutral surfaces, this means that stationary flow along neutral surfaces in general requires mixing energy.

scholarly journals Gravitational Potential Energy Balance for the Thermal Circulation in a Model Ocean

2006 ◽  
Vol 36 (7) ◽  
pp. 1420-1429 ◽  
Author(s):  
Rui Xin Huang ◽  
Xingze Jin
Keyword(s):  
Energy Balance ◽  
Equation Of State ◽  
Kinetic Energy ◽  
Potential Energy ◽  
Gravitational Potential ◽  
Mechanical Energy ◽  
Vertical Mixing ◽  
Gravitational Potential Energy ◽  
Thermal Circulation ◽  
Model Ocean

Abstract The gravitational potential energy balance of the thermal circulation in a simple rectangular model basin is diagnosed from numerical experiments based on a mass-conserving oceanic general circulation model. The vertical mixing coefficient is assumed to be a given constant. The model ocean is heated/cooled from the upper surface or bottom, and the equation of state is linear or nonlinear. Although the circulation patterns obtained from these cases look rather similar, the energetics of the circulation may be very different. For cases of differential heating from the bottom with a nonlinear equation of state, the circulation is driven by mechanical energy generated by heating from the bottom. On the other hand, circulation for three other cases is driven by external mechanical energy, which is implicitly provided by tidal dissipation and wind stress. The major balance of gravitational energy in this model ocean is between the source of energy due to vertical mixing and the conversion from kinetic energy at low latitudes and the sink of energy due to convection adjustment and conversion to kinetic energy at high latitudes.

scholarly journals Estimating Lorenz’s Reference State in an Ocean with a Nonlinear Equation of State for Seawater

2015 ◽  
Vol 45 (5) ◽  
pp. 1242-1257 ◽  
Author(s):  
Juan A. Saenz ◽  
Rémi Tailleux ◽  
Edward D. Butler ◽  
Graham O. Hughes ◽  
Kevin I. C. Oliver
Keyword(s):  
Equation Of State ◽  
Potential Energy ◽  
Frequency Distribution ◽  
Ocean Circulation ◽  
Density Profile ◽  
Mechanical Energy ◽  
Boussinesq Approximation ◽  
Reference State ◽  
Efficient Manner ◽  
Climatological Data

AbstractThe study of the mechanical energy budget of the oceans using the Lorenz available potential energy (APE) theory is based on knowledge of the adiabatically rearranged Lorenz reference state of minimum potential energy. The compressible and nonlinear character of the equation of state for seawater has been thought to cause the reference state to be ill defined, casting doubt on the usefulness of APE theory for investigating ocean energetics under realistic conditions. Using a method based on the volume frequency distribution of parcels as a function of temperature and salinity in the context of the seawater Boussinesq approximation, which is illustrated using climatological data, the authors show that compressibility effects are in fact minor. The reference state can be regarded as a well-defined one-dimensional function of depth, which forms a surface in temperature, salinity, and density space between the surface and the bottom of the ocean. For a very small proportion of water masses, this surface can be multivalued and water parcels can have up to two statically stable levels in the reference density profile, of which the shallowest is energetically more accessible. Classifying parcels from the surface to the bottom gives a different reference density profile than classifying in the opposite direction. However, this difference is negligible. This study shows that the reference state obtained by standard sorting methods is equivalent to, though computationally more expensive than, the volume frequency distribution approach. The approach that is presented can be applied systematically and in a computationally efficient manner to investigate the APE budget of the ocean circulation using models or climatological data.

scholarly journals Walking in simulated reduced gravity: mechanical energy fluctuations and exchange

1999 ◽  
Vol 86 (1) ◽  
pp. 383-390 ◽  
Author(s):  
Timothy M. Griffin ◽  
Neil A. Tolani ◽  
Rodger Kram
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Gravitational Potential ◽  
Inverted Pendulum ◽  
Mechanical Energy ◽  
Center Of Mass ◽  
Metabolic Cost ◽  
Metabolic Energy ◽  
Gravitational Potential Energy ◽  
Reduced Gravity

Walking humans conserve mechanical and, presumably, metabolic energy with an inverted pendulum-like exchange of gravitational potential energy and horizontal kinetic energy. Walking in simulated reduced gravity involves a relatively high metabolic cost, suggesting that the inverted-pendulum mechanism is disrupted because of a mismatch of potential and kinetic energy. We tested this hypothesis by measuring the fluctuations and exchange of mechanical energy of the center of mass at different combinations of velocity and simulated reduced gravity. Subjects walked with smaller fluctuations in horizontal velocity in lower gravity, such that the ratio of horizontal kinetic to gravitational potential energy fluctuations remained constant over a fourfold change in gravity. The amount of exchange, or percent recovery, at 1.00 m/s was not significantly different at 1.00, 0.75, and 0.50 G (average 64.4%), although it decreased to 48% at 0.25 G. As a result, the amount of work performed on the center of mass does not explain the relatively high metabolic cost of walking in simulated reduced gravity.

scholarly journals The Possibility of Zero Limb-Work Gaits in Sprawled and Parasagittal Quadrupeds: Insights from Linkages of the Industrial Revolution

2020 ◽  
Vol 2 (1) ◽  
Author(s):  
J R Usherwood
Keyword(s):  
Potential Energy ◽  
Industrial Revolution ◽  
Mechanical Energy ◽  
Center Of Mass ◽  
Phase Fluctuation ◽  
Vertical Axis ◽  
The Body ◽  
Mechanical Work ◽  
Gravitational Potential Energy ◽  
Pull Out

Synopsis Animal legs are diverse, complex, and perform many roles. One defining requirement of legs is to facilitate terrestrial travel with some degree of economy. This could, theoretically, be achieved without loss of mechanical energy if the body could take a continuous horizontal path supported by vertical forces only—effectively a wheel-like translation, and a condition closely approximated by walking tortoises. If this is a potential strategy for zero mechanical work cost among quadrupeds, how might the structure, posture, and diversity of both sprawled and parasagittal legs be interpreted? In order to approach this question, various linkages described during the industrial revolution are considered. Watt’s linkage provides an analogue for sprawled vertebrates that uses diagonal limb support and shows how vertical-axis joints could enable approximately straight-line horizontal translation while demanding minimal mechanical power. An additional vertical-axis joint per leg results in the wall-mounted pull-out monitor arm and would enable translation with zero mechanical work due to weight support, without tipping or toppling. This is consistent with force profiles observed in tortoises. The Peaucellier linkage demonstrates that parasagittal limbs with lateral-axis joints could also achieve the zero-work strategy. Suitably tuned four-bar linkages indicate this is feasibly approximated for flexed, biologically realistic limbs. Where “walking” gaits typically show out of phase fluctuation in center of mass kinetic and gravitational potential energy, and running, hopping or trotting gaits are characterized by in-phase energy fluctuations, the zero limb-work strategy approximated by tortoises would show zero fluctuations in kinetic or potential energy. This highlights that some gaits, perhaps particularly those of animals with sprawled or crouched limbs, do not fit current kinetic gait definitions; an additional gait paradigm, the “zero limb-work strategy” is proposed.

Mechanics of locomotion in lizards.

1997 ◽  
Vol 200 (16) ◽  
pp. 2177-2188 ◽  
Author(s):  
C T Farley ◽  
T C Ko
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Gravitational Potential ◽  
Inverted Pendulum ◽  
Mechanical Energy ◽  
Center Of Mass ◽  
Gravitational Potential Energy ◽  
Lateral Bending ◽  
Walking Gait ◽  
Bending Force

Lizards bend their trunks laterally with each step of locomotion and, as a result, their locomotion appears to be fundamentally different from mammalian locomotion. The goal of the present study was to determine whether lizards use the same two basic gaits as other legged animals or whether they use a mechanically unique gait due to lateral trunk bending. Force platform and kinematic measurements revealed that two species of lizards, Coleonyx variegatus and Eumeces skiltonianus, used two basic gaits similar to mammalian walking and trotting gaits. In both gaits, the kinetic energy fluctuations due to lateral movements of the center of mass were less than 5% of the total external mechanical energy fluctuations. In the walking gait, both species vaulted over their stance limbs like inverted pendulums. The fluctuations in kinetic energy and gravitational potential energy of the center of mass were approximately 180 degrees out of phase. The lizards conserved as much as 51% of the external mechanical energy required for locomotion by the inverted pendulum mechanism. Both species also used a bouncing gait, similar to mammalian trotting, in which the fluctuations in kinetic energy and gravitational potential energy of the center of mass were nearly exactly in phase. The mass-specific external mechanical work required to travel 1 m (1.5 J kg-1) was similar to that for other legged animals. Thus, in spite of marked lateral bending of the trunk, the mechanics of lizard locomotion is similar to the mechanics of locomotion in other legged animals.

Available potential energy density for Boussinesq fluid flow

2013 ◽  
Vol 714 ◽  
pp. 476-488 ◽  
Author(s):  
Kraig B. Winters ◽  
Roy Barkan
Keyword(s):  
Energy Density ◽  
Potential Energy ◽  
Evolution Equations ◽  
Positive Definite Function ◽  
Boussinesq Approximation ◽  
Exact Expression ◽  
Positive Definite ◽  
Definite Function ◽  
Available Potential Energy ◽  
Boussinesq Fluid

AbstractAn exact expression ${\mathscr{E}}_{a} $ for available potential energy density in Boussinesq fluid flows (Roullet & Klein, J. Fluid Mech., vol. 624, 2009, pp. 45–55; Holliday & McIntyre, J. Fluid Mech., vol. 107, 1981, pp. 221–225) is shown explicitly to integrate to the available potential energy ${E}_{a} $ of Winters et al. (J. Fluid Mech., vol. 289, 1995, pp. 115–128). ${\mathscr{E}}_{a} $ is a positive definite function of position and time consisting of two terms. The first, which is simply the indefinitely signed integrand in the Winters et al. definition of ${E}_{a} $, quantifies the expenditure or release of potential energy in the relocation of individual fluid parcels to their equilibrium height. When integrated over all parcels, this term yields the total available potential energy ${E}_{a} $. The second term describes the energetic consequences of the compensatory displacements necessary under the Boussinesq approximation to conserve vertical volume flux with each parcel relocation. On a pointwise basis, this term adds to the first in such a way that a positive definite contribution to ${E}_{a} $ is guaranteed. Globally, however, the second term vanishes when integrated over all fluid parcels and therefore contributes nothing to ${E}_{a} $. In effect, it filters the components of the first term that cancel upon integration, isolating the positive definite residuals. ${\mathscr{E}}_{a} $ can be used to construct spatial maps of local contributions to ${E}_{a} $ for direct numerical simulations of density stratified flows. Because ${\mathscr{E}}_{a} $ integrates to ${E}_{a} $, these maps are explicitly connected to known, exact, temporal evolution equations for kinetic, available and background potential energies.

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