Stability of stationary solutions of the forced Navier-Stokes equations on the two-torus

We study the linear and nonlinear stability of stationary solutions of the forced two-dimensional Navier-Stokes equations on the domain [0,2π]x[0,2π/α], where α ϵ(0,1], with doubly periodic boundary conditions. For the linear problem we employ the classical energy{enstrophy argument to derive some fundamental properties of unstable eigenmodes. From this it is shown that forces of pure χ2-modes having wavelengths greater than 2π do not give rise to linear instability of the corresponding primary stationary solutions. For the nonlinear problem, we prove the equivalence of nonlinear stability with respect to the energy and enstrophy norms. This equivalence is then applied to derive optimal conditions for nonlinear stability, including both the high-and low-Reynolds-number limits.

On the energetics of stratified turbulent mixing, irreversible thermodynamics, Boussinesq models and the ocean heat engine controversy

In this paper, the available potential energy (APE) framework of Winters et al. (J. Fluid Mech., vol. 289, 1995, p. 115) is extended to the fully compressible Navier– Stokes equations, with the aims of clarifying (i) the nature of the energy conversions taking place in turbulent thermally stratified fluids; and (ii) the role of surface buoyancy fluxes in the Munk & Wunsch (Deep-Sea Res., vol. 45, 1998, p. 1977) constraint on the mechanical energy sources of stirring required to maintain diapycnal mixing in the oceans. The new framework reveals that the observed turbulent rate of increase in the background gravitational potential energy GPEr , commonly thought to occur at the expense of the diffusively dissipated APE, actually occurs at the expense of internal energy, as in the laminar case. The APE dissipated by molecular diffusion, on the other hand, is found to be converted into internal energy (IE), similar to the viscously dissipated kinetic energy KE. Turbulent stirring, therefore, does not introduce a new APE/GPEr mechanical-to-mechanical energy conversion, but simply enhances the existing IE/GPEr conversion rate, in addition to enhancing the viscous dissipation and the entropy production rates. This, in turn, implies that molecular diffusion contributes to the dissipation of the available mechanical energy ME =APE +KE, along with viscous dissipation. This result has important implications for the interpretation of the concepts of mixing efficiency γmixing and flux Richardson number Rf , for which new physically based definitions are proposed and contrasted with previous definitions. The new framework allows for a more rigorous and general re-derivation from the first principles of Munk & Wunsch (1998, hereafter MW98)’s constraint, also valid for a non-Boussinesq ocean: G(KE) ≈ 1 − ξ Rf ξ Rf Wr, forcing = 1 + (1 − ξ )γmixing ξ γmixing Wr, forcing , where G(KE) is the work rate done by the mechanical forcing, Wr, forcing is the rate of loss of GPEr due to high-latitude cooling and ξ is a nonlinearity parameter such that ξ =1 for a linear equation of state (as considered by MW98), but ξ <1 otherwise. The most important result is that G(APE), the work rate done by the surface buoyancy fluxes, must be numerically as large as Wr, forcing and, therefore, as important as the mechanical forcing in stirring and driving the oceans. As a consequence, the overall mixing efficiency of the oceans is likely to be larger than the value γmixing =0.2 presently used, thereby possibly eliminating the apparent shortfall in mechanical stirring energy that results from using γmixing =0.2 in the above formula.

Joint modelling of obstacle induced and mesoscale changes: current limits and challenges

Obstacles considerably influence boundary layer processes. Their influences have been included in mesoscale models (MeM) for a long time. Methods used to parameterise obstacle effects in a MeM are summarised in this paper using results of the mesoscale model METRAS as examples. Besides the parameterisation of obstacle influences it is also possible to use a joint modelling approach to describe obstacle induced and mesoscale changes. Three different methods may be used for joint modelling approaches: The first method is a time-slice approach, where steady basic state profiles are used in an obstacle resolving microscale model (MiM, example model MITRAS) and diurnal cycles are derived by joining steady-state MITRAS results. The second joint modelling approach is one-way nesting, where the MeM results are used to initialise the MiM and to drive the boundary values of the MiM dependent on time. The third joint modelling approach is to apply multi-scale models or two-way nesting approaches, which include feedbacks from the MiM to the MeM. The advantages and disadvantages of the different approaches and remaining problems with joint Reynolds-averaged Navier–Stokes modelling approaches are summarised in the paper.

Generation of inertia-gravity waves in the rotating thermal annulus by a localised boundary layer instability

Waves with periods shorter than the inertial period exist in the atmosphere (as inertia-gravity waves) and in the oceans (as Poincaré and internal gravity waves). Such waves owe their origin to various mechanisms, but of particular interest are those arising either from local secondary instabilities or spontaneous emission due to loss of balance. These phenomena have been studied in the laboratory, both in the mechanically-forced and the thermally-forced rotating annulus. Their generation mechanisms, especially in the latter system, have not yet been fully understood, however. Here we examine short period waves in a numerical model of the rotating thermal annulus, and show how the results are consistent with those from earlier laboratory experiments. We then show how these waves are consistent with being inertia-gravity waves generated by a localised instability within the thermal boundary layer, the location of which is determined by regions of strong shear and downwelling at certain points within a large-scale baroclinic wave flow. The resulting instability launches small-scale inertia-gravity waves into the geostrophic interior of the flow. Their behaviour is captured in fully nonlinear numerical simulations in a finite-difference, 3D Boussinesq Navier-Stokes model. Such a mechanism has many similarities with those responsible for launching small- and meso-scale inertia-gravity waves in the atmosphere from fronts and local convection.

An ‘ideal’ form of decaying two-dimensional turbulence

In decaying two-dimensional Navier-Stokes turbulence, Batchelor's similarity hypothesis fails due to the existence of coherent vortices. However, it is shown that decaying two-dimensional turbulence governed by the Harney-Hasegawa-Mima (CHM) equation ∂/∂t (V^2 φ-λ^2 φ)+J(φ,∇^2 φ)=D where D is a damping, is described well by Batchelor's similarity hypothesis for wave numbers k ≪ λ (the so-called AM regime). It is argued that CHM turbulence in the AM regime is a more `ideal' form of two-dimensional turbulence than is Navier-Stokes turbulence itself.

Irreversible compressible work and APE dissipation in turbulent stratified fluid

Although it plays a key role in the theory of stratified turbulence, the concept of available potential energy (APE) dissipation has remained until now a rather mysterious quantity, owing to the lack of rigorous result about its irreversible character or energy conversion type. Here, we show by using rigorous energetics considerations rooted in the analysis of the Navier-Stokes for a fully compressible fluid with a nonlinear equation of state that the APE dissipation is an irreversible energy conversion that dissipates kinetic energy into internal energy, exactly as viscous dissipation. These results are established by showing that APE dissipation contributes to the irreversible production of entropy, and by showing that it is a part of the work of expansion/contraction. Our results provide a new interpretation of the entropy budget, that leads to a new exact definition of turbulent effective diffusivity, which generalizes the Osborn-Cox model, as well as a rigorous decomposition of the work of expansion/contraction into reversible and irreversible components. In the context of turbulent mixing associated with parallel shear flow instability, our results suggests that there is no irreversible transfer of horizontal momentum into vertical momentum, as seems to be required when compressible effects are neglected, with potential consequences for the parameterisations of momentum dissipation in the coarse-grained Navier-Stokes equations.

The integrated WRF/urban modelling system: development, evaluation, and applications to urban environmental problems

To bridge the gaps between traditional mesoscale modelling and microscale modelling, the National Center for Atmospheric Research, in collaboration with other agencies and research groups, has developed an integrated urban modelling system coupled to the weather research and forecasting (WRF) model as a community tool to address urban environmental issues. The core of this WRF/urban modelling system consists of the following: (1) three methods with different degrees of freedom to parameterize urban surface processes, ranging from a simple bulk parameterization to a sophisticated multi-layer urban canopy model with an indoor–outdoor exchange sub-model that directly interacts with the atmospheric boundary layer, (2) coupling to fine-scale computational fluid dynamic Reynolds-averaged Navier–Stokes and Large-Eddy simulation models for transport and dispersion (T&D) applications, (3) procedures to incorporate high-resolution urban land use, building morphology, and anthropogenic heating data using the National Urban Database and Access Portal Tool (NUDAPT), and (4) an urbanized high-resolution land data assimilation system. This paper provides an overview of this modelling system; addresses the daunting challenges of initializing the coupled WRF/urban model and of specifying the potentially vast number of parameters required to execute the WRF/urban model; explores the model sensitivity to these urban parameters; and evaluates the ability of WRF/urban to capture urban heat islands, complex boundary-layer structures aloft, and urban plume T&D for several major metropolitan regions. Recent applications of this modelling system illustrate its promising utility, as a regional climate-modelling tool, to investigate impacts of future urbanization on regional meteorological conditions and on air quality under future climate change scenarios. Copyright © 2010 Royal Meteorological Society

How to parametrize urban-canopy drag to reproduce wind-direction effects within the canopy

The mean wind direction within an urban canopy changes with height when the incoming flow is not orthogonal to obstacle faces. This wind-turning effect is induced by complex processes and its modelling in urban-canopy (UC) parametrizations is difficult. Here we focus on the analysis of the spatially-averaged flow properties over an aligned array of cubes and their variation with incoming wind direction. For this purpose, Reynolds-averaged Navier–Stokes simulations previously compared, for a reduced number of incident wind directions, against direct numerical simulation results are used. The drag formulation of a UCparametrization ismodified and different drag coefficients are tested in order to reproduce the wind-turning effect within the canopy for oblique wind directions. The simulations carried out for a UC parametrization in one-dimensional mode indicate that a height-dependent drag coefficient is needed to capture this effect.

Energy consistent discontinuous Galerkin methods for the Navier–Stokes–Korteweg system

We design consistent discontinuous Galerkin finite element schemes for the approximation of the Euler-Korteweg and the Navier-Stokes-Korteweg systems. We show that the scheme for the Euler-Korteweg system is energy and mass conservative and that the scheme for the Navier-Stokes-Korteweg system is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to viscous effects, that is, there is no numerical dissipation. In this sense the methods are consistent with the energy dissipation of the continuous PDE systems. - See more at: http://www.ams.org/journals/mcom/2014-83-289/S0025-5718-2014-02792-0/home.html#sthash.rwTIhNWi.dpuf

Wind-tunnel modelling of dispersion from a scalar area source in urban-like roughness

A wind-tunnel study was conducted to investigate ventilation of scalars from urban-like geometries at neighbourhood scale by exploring two different geometries a uniform height roughness and a non-uniform height roughness, both with an equal plan and frontal density of λ p = λ f = 25%. In both configurations a sub-unit of the idealized urban surface was coated with a thin layer of naphthalene to represent area sources. The naphthalene sublimation method was used to measure directly total area-averaged transport of scalars out of the complex geometries. At the same time, naphthalene vapour concentrations controlled by the turbulent fluxes were detected using a fast Flame Ionisation Detection (FID) technique. This paper describes the novel use of a naphthalene coated surface as an area source in dispersion studies. Particular emphasis was also given to testing whether the concentration measurements were independent of Reynolds number. For low wind speeds, transfer from the naphthalene surface is determined by a combination of forced and natural convection. Compared with a propane point source release, a 25% higher free stream velocity was needed for the naphthalene area source to yield Reynolds-number-independent concentration fields. Ventilation transfer coefficients w T /U derived from the naphthalene sublimation method showed that, whilst there was enhanced vertical momentum exchange due to obstacle height variability, advection was reduced and dispersion from the source area was not enhanced. Thus, the height variability of a canopy is an important parameter when generalising urban dispersion. Fine resolution concentration measurements in the canopy showed the effect of height variability on dispersion at street scale. Rapid vertical transport in the wake of individual high-rise obstacles was found to generate elevated point-like sources. A Gaussian plume model was used to analyse differences in the downstream plumes. Intensified lateral and vertical plume spread and plume dilution with height was found for the non-uniform height roughness

A combined BIE-FE method for the Stokes equations

A numerical algorithm for the biharmonic equation in domains with piecewise smooth boundaries is presented. It is intended for problems describing the Stokes flow in the situations where one has corners or cusps formed by parts of the domain boundary and, due to the nature of the boundary conditions on these parts of the boundary, these regions have a global effect on the shape of the whole domain and hence have to be resolved with sufficient accuracy. The algorithm combines the boundary integral equation method for the main part of the flow domain and the finite-element method which is used to resolve the corner/cusp regions. Two parts of the solution are matched along a numerical ‘internal interface’ or, as a variant, two interfaces, and they are determined simultaneously by inverting a combined matrix in the course of iterations. The algorithm is illustrated by considering the flow configuration of ‘curtain coating’, a flow where a sheet of liquid impinges onto a moving solid substrate, which is particularly sensitive to what happens in the corner region formed, physically, by the free surface and the solid boundary. The ‘moving contact line problem’ is addressed in the framework of an earlier developed interface formation model which treats the dynamic contact angle as part of the solution, as opposed to it being a prescribed function of the contact line speed, as in the so-called ‘slip models’. Keywords: Dynamic contact angle; finite elements; free surface flows; hybrid numerical technique; Stokes equations.

Mean flow and turbulence statistics over groups of urban-like cubical obstacles

Direct numerical simulations of turbulent flow over regular arrays of urban-like, cubical obstacles are reported. Results are analysed in terms of a formal spatial averaging procedure to enable interpretation of the flow within the arrays as a canopy flow, and of the flow above as a rough wall boundary layer. Spatial averages of the mean velocity, turbulent stresses and pressure drag are computed. The statistics compare very well with data from wind-tunnel experiments. Within the arrays the time-averaged flow structure gives rise to significant 'dispersive stress' whereas above the Reynolds stress dominates. The mean flow structure and turbulence statistics depend significantly on the layout of the cubes. Unsteady effects are important, especially in the lower canopy layer where turbulent fluctuations dominate over the mean flow.