The temporal evolution of the energy flux across scales in homogeneous turbulence

A temporal study of energy transfer across length scales is performed in 3D numerical simulations of homogeneous shear flow and isotropic turbulence. The average time taken by perturbations in the energy flux to travel between scales is measured and shown to be additive. Our data suggests that the propagation of disturbances in the energy flux is independent of the forcing and that it defines a ‘velocity’ that determines the energy flux itself. These results support that the cascade is, on average, a scale-local process where energy is continuously transmitted from one scale to the next in order of decreasing size.

Viscous coupling of shear-free turbulence across nearly flat fluid interfaces

The interactions between shear-free turbulence in two regions (denoted as + and − on either side of a nearly flat horizontal interface are shown here to be controlled by several mechanisms, which depend on the magnitudes of the ratios of the densities, ρ+/ρ−, and kinematic viscosities of the fluids, μ+/μ−, and the root mean square (r.m.s.) velocities of the turbulence, u0+/u0−, above and below the interface. This study focuses on gas–liquid interfaces so that ρ+/ρ− ≪ 1 and also on where turbulence is generated either above or below the interface so that u0+/u0− is either very large or very small. It is assumed that vertical buoyancy forces across the interface are much larger than internal forces so that the interface is nearly flat, and coupling between turbulence on either side of the interface is determined by viscous stresses. A formal linearized rapid-distortion analysis with viscous effects is developed by extending the previous study by Hunt & Graham (J. Fluid Mech., vol. 84, 1978, pp. 209–235) of shear-free turbulence near rigid plane boundaries. The physical processes accounted for in our model include both the blocking effect of the interface on normal components of the turbulence and the viscous coupling of the horizontal field across thin interfacial viscous boundary layers. The horizontal divergence in the perturbation velocity field in the viscous layer drives weak inviscid irrotational velocity fluctuations outside the viscous boundary layers in a mechanism analogous to Ekman pumping. The analysis shows the following. (i) The blocking effects are similar to those near rigid boundaries on each side of the interface, but through the action of the thin viscous layers above and below the interface, the horizontal and vertical velocity components differ from those near a rigid surface and are correlated or anti-correlated respectively. (ii) Because of the growth of the viscous layers on either side of the interface, the ratio uI/u0, where uI is the r.m.s. of the interfacial velocity fluctuations and u0 the r.m.s. of the homogeneous turbulence far from the interface, does not vary with time. If the turbulence is driven in the lower layer with ρ+/ρ− ≪ 1 and u0+/u0− ≪ 1, then uI/u0− ~ 1 when Re (=u0−L−/ν−) ≫ 1 and R = (ρ−/ρ+)(v−/v+)1/2 ≫ 1. If the turbulence is driven in the upper layer with ρ+/ρ− ≪ 1 and u0+/u0− ≫ 1, then uI/u0+ ~ 1/(1 + R). (iii) Nonlinear effects become significant over periods greater than Lagrangian time scales. When turbulence is generated in the lower layer, and the Reynolds number is high enough, motions in the upper viscous layer are turbulent. The horizontal vorticity tends to decrease, and the vertical vorticity of the eddies dominates their asymptotic structure. When turbulence is generated in the upper layer, and the Reynolds number is less than about 106–107, the fluctuations in the viscous layer do not become turbulent. Nonlinear processes at the interface increase the ratio uI/u0+ for sheared or shear-free turbulence in the gas above its linear value of uI/u0+ ~ 1/(1 + R) to (ρ+/ρ−)1/2 ~ 1/30 for air–water interfaces. This estimate agrees with the direct numerical simulation results from Lombardi, De Angelis & Bannerjee (Phys. Fluids, vol. 8, no. 6, 1996, pp. 1643–1665). Because the linear viscous–inertial coupling mechanism is still significant, the eddy motions on either side of the interface have a similar horizontal structure, although their vertical structure differs.

Inhomogeneous two-dimensional turbulence in the atmosphere

Faced with the strongly nonlinear and apparently random behaviour of the energy-containing scales in the atmosphere, geophysical fluid dynamicists have attempted to understand the synoptic-scale atmospheric flow within the context of two-dimensional homogeneous turbulence theory (e.g. FJØRTOFT [1]; LEITH [2]). However atmospheric observations (BOER and SHEPHERD [3] and Fig.1) show that the synoptic-scale transient flow evolves in the presence of a planetary-scale, quasi-stationary background flow which is approximately zonal (east-west). Classical homogeneous 2-D turbulence theory is therefore not strictly applicable to the transient flow. One is led instead to study 2-D turbulence in the presence of a large-scale (barotropically stable) zonal jet inhomogeneity.

Observations of turbulence within the surf and swash zone of a field-scale sandy laboratory beach

Current coastal-evolution models generally lack the ability to accurately predict bed level change in shallow (<~2 m) water, which is, at least partly, due to the preclusion of the effect of surface-induced turbulence on sand suspension and transport. As a first step to remedy this situation, we investigated the vertical structure of turbulence in the surf and swash zone using measurements collected under random shoaling and plunging waves on a steep (initially 1:15) field-scale sandy laboratory beach. Seaward of the swash zone, turbulence was measured with a vertical array of three Acoustic Doppler Velocimeters (ADVs), while in the swash zone two vertically spaced acoustic doppler velocimeter profilers (Vectrino profilers) were applied. The vertical turbulence structure evolves from bottom-dominated to approximately vertically uniform with an increase in the fraction of breaking waves to ~ 50%. In the swash zone, the turbulence is predominantly bottom-induced during the backwash and shows a homogeneous turbulence profile during uprush. We further find that the instantaneous turbulence kinetic energy is phase-coupled with the short-wave orbital motion under the plunging breakers, with higher levels shortly after the reversal from offshore to onshore motion (i.e. wavefront).

On the relation between the conditional moment closure and unsteady flamelets

We consider the relation between the conditional moment closure (CMC) and the unsteady flamelet model (FM). The CMC equations were originally constructed as global equations, while FM was derived asymptotically for a thin reaction zone. The recent tendency is to use FM-type equations as global equations. We investigate the possible consequences and suggest a new version of FM: coordinate-invariant FM (CIFM). Unlike FM, CIFM complies with conditional properties of the exact transport equations which are used effectively in CMC. We analyse the assumptions needed to obtain another global version of FM: representative interactive flamelets (RIF), from original FM and demonstrate that, in homogeneous turbulence, one of these assumptions is equivalent to the main CMC hypothesis.

The modeling of turbulent reactive flows based on multiple mapping conditioning

A new modeling approach-multiple mapping conditioning (MMC)-is introduced to treat mixing and reaction in turbulent flows. The model combines the advantages of the probability density function and the conditional moment closure methods and is based on a certain generalization of the mapping closure concept. An equivalent stochastic formulation of the MMC model is given. The validity of the closuring hypothesis of the model is demonstrated by a comparison with direct numerical simulation results for the three-stream mixing problem. (C) 2003 American Institute of Physics.

A spectral view of nonlinear fluxes and stationary-transient interaction in the atmosphere

Nonlinear spectral transfers of kinetic energy and enstrophy, and stationary-transient interaction, are studied using global FGGE data for January 1979. It is found that the spectral transfers arise primarily from a combination, in roughly equal measure, of pure transient and mixed stationary-transient interactions. The pure transient interactions are associated with a transient eddy field which is approximately locally homogeneous and isotropic, and they appear to be consistently understood within the context of two-dimensional homogeneous turbulence. Theory based on spatial wale separation concepts suggests that the mixed interactions may be understood physically, to a first approximation, as a process of shear-induced spectral transfer of transient enstrophy along lines of constant zonal wavenumber. This essentially conservative enstrophy transfer generally involves highly nonlocal stationary-transient energy conversions. The observational analysis demonstrates that the shear-induced transient enstrophy transfer is mainly associated with intermediate-scale (zonal wavenumber m > 3) transients and is primarily to smaller (meridional) scales, so that the transient flow acts as a source of stationary energy. In quantitative terms, this transient-eddy rectification corresponds to a forcing timescale in the stationary energy budget which is of the same order of magnitude as most estimates of the damping timescale in simple stationary-wave models (5 to 15 days). Moreover, the nonlinear interactions involved are highly nonlocal and cover a wide range of transient scales of motion.

Matching conditional moments in PDF modelling of nonpremixed combustion

The rate of generation of fluctuations with respect to the scalar values conditioned on the mixture fraction, which significantly affects turbulent nonpremixed combustion processes, is examined. Simulation of the rate in a major mixing model is investigated and the derived equations can assist in selecting the model parameters so that the level of conditional fluctuations is better reproduced by the models. A more general formulation of the multiple mapping conditioning (MMC) model that distinguishes the reference and conditioning variables is suggested. This formulation can be viewed as a methodology of enforcing certain desired conditional properties onto conventional mixing models. Examples of constructing consistent MMC models with dissipation and velocity conditioning as well as of combining MMC with large eddy simulations (LES) are also provided. (c) 2005 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Stochastic generation of homogeneous isotropic turbulence with well-defined-spectra

A precise and simple computational model to generate well-behaved two-dimensional turbulent flows is presented. The whole approach rests on the use of stochastic differential equations and is general enough to reproduce a variety of energy spectra and spatiotemporal correlation functions. Analytical expressions for both the continuous and the discrete versions, together with simulation algorithms, are derived. Results for two relevant spectra, covering distinct ranges of wave numbers, are given.

Stochastic generation of homogeneous isotropic turbulence with well-defined-spectra

A precise and simple computational model to generate well-behaved two-dimensional turbulent flows is presented. The whole approach rests on the use of stochastic differential equations and is general enough to reproduce a variety of energy spectra and spatiotemporal correlation functions. Analytical expressions for both the continuous and the discrete versions, together with simulation algorithms, are derived. Results for two relevant spectra, covering distinct ranges of wave numbers, are given.

Coherent structures in statistically-stationary homogeneous shear turbulence

La región cerca de la pared de flujos turbulentos de pared ya está bien conocido debido a su bajo número de Reynolds local y la separación escala estrecha. La región lejos de la pared (capa externa) no es tan interesante tampoco, ya que las estadísticas allí se escalan bien por las unidades exteriores. La región intermedia (capa logarítmica), sin embargo, ha estado recibiendo cada vez más atención debido a su propiedad auto-similares. Además, de acuerdo a Flores et al. (2007) y Flores & Jiménez (2010), la capa logarítmica es más o menos independiente de otras capas, lo que implica que podría ser inspeccionado mediante el aislamiento de otras dos capas, lo que reduciría significativamente los costes computacionales para la simulación de flujos turbulentos de pared. Algunos intentos se trataron después por Mizuno & Jiménez (2013), quien simulan la capa logarítmica sin la región cerca de la pared con estadísticas obtenidas de acuerdo razonablemente bien con los de las simulaciones completas. Lo que más, la capa logarítmica podría ser imitado por otra turbulencia sencillo de cizallamiento de motor. Por ejemplo, Pumir (1996) encontró que la turbulencia de cizallamiento homogéneo estadísticamente estacionario (SS-HST) también irrumpe, de una manera muy similar al proceso de auto-sostenible en flujos turbulentos de pared. Según los consideraciones arriba, esta tesis trata de desvelar en qué medida es la capa logarítmica de canales similares a la turbulencia de cizalla más sencillo, SS-HST, mediante la comparación de ambos cinemática y la dinámica de las estructuras coherentes en los dos flujos. Resultados sobre el canal se muestran mediante Lozano-Durán et al. (2012) y Lozano-Durán & Jiménez (2014b). La hoja de ruta de esta tarea se divide en tres etapas. En primer lugar, SS-HST es investigada por medio de un código nuevo de simulación numérica directa, espectral en las dos direcciones horizontales y compacto-diferencias finitas en la dirección de la cizalla. Sin utiliza remallado para imponer la condición de borde cortante periódica. La influencia de la geometría de la caja computacional se explora. Ya que el HST no tiene ninguna longitud característica externa y tiende a llenar el dominio computacional, las simulaciopnes a largo plazo del HST son ’mínimos’ en el sentido de que contiene sólo unas pocas estructuras media a gran escala. Se ha encontrado que el límite principal es el ancho de la caja de la envergadura, Lz, que establece las escalas de longitud y velocidad de la turbulencia, y que las otras dos dimensiones de la caja debe ser suficientemente grande (Lx > 2LZ, Ly > Lz) para evitar que otras direcciones estando limitado también. También se encontró que las cajas de gran longitud, Lx > 2Ly, par con el paso del tiempo la condición de borde cortante periódica, y desarrollar fuertes ráfagas linealizadas no físicos. Dentro de estos límites, el flujo muestra similitudes y diferencias interesantes con otros flujos de cizalla, y, en particular, con la capa logarítmica de flujos turbulentos de pared. Ellos son exploradas con cierto detalle. Incluyen un proceso autosostenido de rayas a gran escala y con una explosión cuasi-periódica. La escala de tiempo de ruptura es de aproximadamente universales, ~20S~l(S es la velocidad de cizallamiento media), y la disponibilidad de dos sistemas de ruptura diferentes permite el crecimiento de las ráfagas a estar relacionado con algo de confianza a la cizalladura de turbulencia inicialmente isotrópico. Se concluye que la SS-HST, llevado a cabo dentro de los parámetros de cílculo apropiados, es un sistema muy prometedor para estudiar la turbulencia de cizallamiento en general. En segundo lugar, las mismas estructuras coherentes como en los canales estudiados por Lozano-Durán et al. (2012), es decir, grupos de vórticidad (fuerte disipación) y Qs (fuerte tensión de Reynolds tangencial, -uv) tridimensionales, se estudia mediante simulación numérica directa de SS-HST con relaciones de aspecto de cuadro aceptables y número de Reynolds hasta Rex ~ 250 (basado en Taylor-microescala). Se discute la influencia de la intermitencia de umbral independiente del tiempo. Estas estructuras tienen alargamientos similares en la dirección sentido de la corriente a las familias separadas en los canales hasta que son de tamaño comparable a la caja. Sus dimensiones fractales, longitudes interior y exterior como una función del volumen concuerdan bien con sus homólogos de canales. El estudio sobre sus organizaciones espaciales encontró que Qs del mismo tipo están alineados aproximadamente en la dirección del vector de velocidad en el cuadrante al que pertenecen, mientras Qs de diferentes tipos están restringidos por el hecho de que no debe haber ningún choque de velocidad, lo que hace Q2s (eyecciones, u < 0,v > 0) y Q4s (sweeps, u > 0,v < 0) emparejado en la dirección de la envergadura. Esto se verifica mediante la inspección de estructuras de velocidad, otros cuadrantes como la uw y vw en SS-HST y las familias separadas en el canal. La alineación sentido de la corriente de Qs ligada a la pared con el mismo tipo en los canales se debe a la modulación de la pared. El campo de flujo medio condicionado a pares Q2-Q4 encontró que los grupos de vórticidad están en el medio de los dos, pero prefieren los dos cizalla capas alojamiento en la parte superior e inferior de Q2s y Q4s respectivamente, lo que hace que la vorticidad envergadura dentro de las grupos de vórticidad hace no cancele. La pared amplifica la diferencia entre los tamaños de baja- y alta-velocidad rayas asociados con parejas de Q2-Q4 se adjuntan como los pares alcanzan cerca de la pared, el cual es verificado por la correlación de la velocidad del sentido de la corriente condicionado a Q2s adjuntos y Q4s con diferentes alturas. Grupos de vórticidad en SS-HST asociados con Q2s o Q4s también están flanqueadas por un contador de rotación de los vórtices sentido de la corriente en la dirección de la envergadura como en el canal. La larga ’despertar’ cónica se origina a partir de los altos grupos de vórticidad ligada a la pared han encontrado los del Álamo et al. (2006) y Flores et al. (2007), que desaparece en SS-HST, sólo es cierto para altos grupos de vórticidad ligada a la pared asociados con Q2s pero no para aquellos asociados con Q4s, cuyo campo de flujo promedio es en realidad muy similar a la de SS-HST. En tercer lugar, las evoluciones temporales de Qs y grupos de vórticidad se estudian mediante el uso de la método inventado por Lozano-Durán & Jiménez (2014b). Las estructuras se clasifican en las ramas, que se organizan más en los gráficos. Ambas resoluciones espaciales y temporales se eligen para ser capaz de capturar el longitud y el tiempo de Kolmogorov puntual más probable en el momento más extrema. Debido al efecto caja mínima, sólo hay un gráfico principal consiste en casi todas las ramas, con su volumen y el número de estructuras instantáneo seguien la energía cinética y enstrofía intermitente. La vida de las ramas, lo que tiene más sentido para las ramas primarias, pierde su significado en el SS-HST debido a las aportaciones de ramas primarias al total de Reynolds estrés o enstrofía son casi insignificantes. Esto también es cierto en la capa exterior de los canales. En cambio, la vida de los gráficos en los canales se compara con el tiempo de ruptura en SS-HST. Grupos de vórticidad están asociados con casi el mismo cuadrante en términos de sus velocidades medias durante su tiempo de vida, especialmente para los relacionados con las eyecciones y sweeps. Al igual que en los canales, las eyecciones de SS-HST se mueven hacia arriba con una velocidad promedio vertical uT (velocidad de fricción) mientras que lo contrario es cierto para los barridos. Grupos de vórticidad, por otra parte, son casi inmóvil en la dirección vertical. En la dirección de sentido de la corriente, que están advección por la velocidad media local y por lo tanto deforman por la diferencia de velocidad media. Sweeps y eyecciones se mueven más rápido y más lento que la velocidad media, respectivamente, tanto por 1.5uT. Grupos de vórticidad se mueven con la misma velocidad que la velocidad media. Se verifica que las estructuras incoherentes cerca de la pared se debe a la pared en vez de pequeño tamaño. Los resultados sugieren fuertemente que las estructuras coherentes en canales no son especialmente asociado con la pared, o incluso con un perfil de cizalladura dado. ABSTRACT Since the wall-bounded turbulence was first recognized more than one century ago, its near wall region (buffer layer) has been studied extensively and becomes relatively well understood due to the low local Reynolds number and narrow scale separation. The region just above the buffer layer, i.e., the logarithmic layer, is receiving increasingly more attention nowadays due to its self-similar property. Flores et al. (20076) and Flores & Jim´enez (2010) show that the statistics of logarithmic layer is kind of independent of other layers, implying that it might be possible to study it separately, which would reduce significantly the computational costs for simulations of the logarithmic layer. Some attempts were tried later by Mizuno & Jimenez (2013), who simulated the logarithmic layer without the buffer layer with obtained statistics agree reasonably well with those of full simulations. Besides, the logarithmic layer might be mimicked by other simpler sheardriven turbulence. For example, Pumir (1996) found that the statistically-stationary homogeneous shear turbulence (SS-HST) also bursts, in a manner strikingly similar to the self-sustaining process in wall-bounded turbulence. Based on these considerations, this thesis tries to reveal to what extent is the logarithmic layer of channels similar to the simplest shear-driven turbulence, SS-HST, by comparing both kinematics and dynamics of coherent structures in the two flows. Results about the channel are shown by Lozano-Dur´an et al. (2012) and Lozano-Dur´an & Jim´enez (20146). The roadmap of this task is divided into three stages. First, SS-HST is investigated by means of a new direct numerical simulation code, spectral in the two horizontal directions and compact-finite-differences in the direction of the shear. No remeshing is used to impose the shear-periodic boundary condition. The influence of the geometry of the computational box is explored. Since HST has no characteristic outer length scale and tends to fill the computational domain, longterm simulations of HST are ‘minimal’ in the sense of containing on average only a few large-scale structures. It is found that the main limit is the spanwise box width, Lz, which sets the length and velocity scales of the turbulence, and that the two other box dimensions should be sufficiently large (Lx > 2LZ, Ly > Lz) to prevent other directions to be constrained as well. It is also found that very long boxes, Lx > 2Ly, couple with the passing period of the shear-periodic boundary condition, and develop strong unphysical linearized bursts. Within those limits, the flow shows interesting similarities and differences with other shear flows, and in particular with the logarithmic layer of wallbounded turbulence. They are explored in some detail. They include a self-sustaining process for large-scale streaks and quasi-periodic bursting. The bursting time scale is approximately universal, ~ 20S~l (S is the mean shear rate), and the availability of two different bursting systems allows the growth of the bursts to be related with some confidence to the shearing of initially isotropic turbulence. It is concluded that SS-HST, conducted within the proper computational parameters, is a very promising system to study shear turbulence in general. Second, the same coherent structures as in channels studied by Lozano-Dur´an et al. (2012), namely three-dimensional vortex clusters (strong dissipation) and Qs (strong tangential Reynolds stress, -uv), are studied by direct numerical simulation of SS-HST with acceptable box aspect ratios and Reynolds number up to Rex ~ 250 (based on Taylor-microscale). The influence of the intermittency to time-independent threshold is discussed. These structures have similar elongations in the streamwise direction to detached families in channels until they are of comparable size to the box. Their fractal dimensions, inner and outer lengths as a function of volume agree well with their counterparts in channels. The study about their spatial organizations found that Qs of the same type are aligned roughly in the direction of the velocity vector in the quadrant they belong to, while Qs of different types are restricted by the fact that there should be no velocity clash, which makes Q2s (ejections, u < 0, v > 0) and Q4s (sweeps, u > 0, v < 0) paired in the spanwise direction. This is verified by inspecting velocity structures, other quadrants such as u-w and v-w in SS-HST and also detached families in the channel. The streamwise alignment of attached Qs with the same type in channels is due to the modulation of the wall. The average flow field conditioned to Q2-Q4 pairs found that vortex clusters are in the middle of the pair, but prefer to the two shear layers lodging at the top and bottom of Q2s and Q4s respectively, which makes the spanwise vorticity inside vortex clusters does not cancel. The wall amplifies the difference between the sizes of low- and high-speed streaks associated with attached Q2-Q4 pairs as the pairs reach closer to the wall, which is verified by the correlation of streamwise velocity conditioned to attached Q2s and Q4s with different heights. Vortex clusters in SS-HST associated with Q2s or Q4s are also flanked by a counter rotating streamwise vortices in the spanwise direction as in the channel. The long conical ‘wake’ originates from tall attached vortex clusters found by del A´ lamo et al. (2006) and Flores et al. (2007b), which disappears in SS-HST, is only true for tall attached vortices associated with Q2s but not for those associated with Q4s, whose averaged flow field is actually quite similar to that in SS-HST. Third, the temporal evolutions of Qs and vortex clusters are studied by using the method invented by Lozano-Dur´an & Jim´enez (2014b). Structures are sorted into branches, which are further organized into graphs. Both spatial and temporal resolutions are chosen to be able to capture the most probable pointwise Kolmogorov length and time at the most extreme moment. Due to the minimal box effect, there is only one main graph consist by almost all the branches, with its instantaneous volume and number of structures follow the intermittent kinetic energy and enstrophy. The lifetime of branches, which makes more sense for primary branches, loses its meaning in SS-HST because the contributions of primary branches to total Reynolds stress or enstrophy are almost negligible. This is also true in the outer layer of channels. Instead, the lifetime of graphs in channels are compared with the bursting time in SS-HST. Vortex clusters are associated with almost the same quadrant in terms of their mean velocities during their life time, especially for those related with ejections and sweeps. As in channels, ejections in SS-HST move upwards with an average vertical velocity uτ (friction velocity) while the opposite is true for sweeps. Vortex clusters, on the other hand, are almost still in the vertical direction. In the streamwise direction, they are advected by the local mean velocity and thus deformed by the mean velocity difference. Sweeps and ejections move faster and slower than the mean velocity respectively, both by 1.5uτ . Vortex clusters move with the same speed as the mean velocity. It is verified that the incoherent structures near the wall is due to the wall instead of small size. The results suggest that coherent structures in channels are not particularly associated with the wall, or even with a given shear profile.

Direct numerical simulation of statistically stationary and homogeneous shear turbulence and its relation to other shear flows

Statistically stationary and homogeneous shear turbulence (SS-HST) is investigated by means of a new direct numerical simulation code, spectral in the two horizontal directions and compact-finite-differences in the direction of the shear. No remeshing is used to impose the shear-periodic boundary condition. The influence of the geometry of the computational box is explored. Since HST has no characteristic outer length scale and tends to fill the computational domain, long-term simulations of HST are “minimal” in the sense of containing on average only a few large-scale structures. It is found that the main limit is the spanwise box width, Lz, which sets the length and velocity scales of the turbulence, and that the two other box dimensions should be sufficiently large (Lx ≳ 2Lz, Ly ≳ Lz) to prevent other directions to be constrained as well. It is also found that very long boxes, Lx ≳ 2Ly, couple with the passing period of the shear-periodic boundary condition, and develop strong unphysical linearized bursts. Within those limits, the flow shows interesting similarities and differences with other shear flows, and in particular with the logarithmic layer of wall-bounded turbulence. They are explored in some detail. They include a self-sustaining process for large-scale streaks and quasi-periodic bursting. The bursting time scale is approximately universal, ∼20S−1, and the availability of two different bursting systems allows the growth of the bursts to be related with some confidence to the shearing of initially isotropic turbulence. It is concluded that SS-HST, conducted within the proper computational parameters, is a very promising system to study shear turbulence in general.

Turbulence dynamics near a turbulent/non-turbulent interface

The characteristics of the boundary layer separating a turbulence region from an irrotational (or non-turbulent) flow region are investigated using rapid distortion theory (RDT). The turbulence region is approximated as homogeneous and isotropic far away from the bounding turbulent/non-turbulent (T/NT) interface, which is assumed to remain approximately flat. Inviscid effects resulting from the continuity of the normal velocity and pressure at the interface, in addition to viscous effects resulting from the continuity of the tangential velocity and shear stress, are taken into account by considering a sudden insertion of the T/NT interface, in the absence of mean shear. Profiles of the velocity variances, turbulent kinetic energy (TKE), viscous dissipation rate (epsilon), turbulence length scales, and pressure statistics are derived, showing an excellent agreement with results from direct numerical simulations (DNS). Interestingly, the normalized inviscid flow statistics at the T/NT interface do not depend on the form of the assumed TKE spectrum. Outside the turbulent region, where the flow is irrotational (except inside a thin viscous boundary layer), epsilon decays as z^{-6}, where z is the distance from the T/NT interface. The mean pressure distribution is calculated using RDT, and exhibits a decrease towards the turbulence region due to the associated velocity fluctuations, consistent with the generation of a mean entrainment velocity. The vorticity variance and epsilon display large maxima at the T/NT interface due to the inviscid discontinuities of the tangential velocity variances existing there, and these maxima are quantitatively related to the thickness delta of the viscous boundary layer (VBL). For an equilibrium VBL, the RDT analysis suggests that delta ~ eta (where eta is the Kolmogorov microscale), which is consistent with the scaling law identified in a very recent DNS study for shear-free T/NT interfaces.

Rossby waves and two-dimensional turbulence in a large-scale zonal jet

The theory of homogeneous barotropic beta-plane turbulence is here extended to include effects arising from spatial inhomogeneity in the form of a zonal shear flow. Attention is restricted to the geophysically important case of zonal flows that are barotropically stable and are of larger scale than the resulting transient eddy field. Because of the presumed scale separation, the disturbance enstrophy is approximately conserved in a fully nonlinear sense, and the (nonlinear) wave-mean-flow interaction may be characterized as a shear-induced spectral transfer of disturbance enstrophy along lines of constant zonal wavenumber k. In this transfer the disturbance energy is generally not conserved. The nonlinear interactions between different disturbance components are turbulent for scales smaller than the inverse of Rhines's cascade-arrest scale κβ[identical with] (β0/2urms)½ and in this regime their leading-order effect may be characterized as a tendency to spread the enstrophy (and energy) along contours of constant total wavenumber κ [identical with] (k2 + l2)½. Insofar as this process of turbulent isotropization involves spectral transfer of disturbance enstrophy across lines of constant zonal wavenumber k, it can be readily distinguished from the shear-induced transfer which proceeds along them. However, an analysis in terms of total wavenumber K alone, which would be justified if the flow were homogeneous, would tend to mask the differences. The foregoing theoretical ideas are tested by performing direct numerical simulation experiments. It is found that the picture of classical beta-plane turbulence is altered, through the effect of the large-scale zonal flow, in the following ways: (i) while the turbulence is still confined to K Kβ, the disturbance field penetrates to the largest scales of motion; (ii) the larger disturbance scales K < Kβ exhibit a tendency to meridional rather than zonal anisotropy, namely towards v2 > u2 rather than vice versa; (iii) the initial spectral transfer rate away from an isotropic intermediate-scale source is significantly enhanced by the shear-induced transfer associated with straining by the zonal flow. This last effect occurs even when the large-scale shear appears weak to the energy-containing eddies, in the sense that dU/dy [double less-than sign] κ for typical eddy length and velocity scales.

Large-scale two-dimensional turbulence in the atmosphere

Global FGGE data are used to investigate several aspects of large-scale turbulence in the atmosphere. The approach follows that for two-dimensional, nondivergent turbulent flows which are homogeneous and isotropic on the sphere. Spectra of kinetic energy, enstrophy and available potential energy are obtained for both the stationary and transient parts of the flow. Nonlinear interaction terms and fluxes of energy and enstrophy through wavenumber space are calculated and compared with the theory. A possible method of parameterizing the interactions with unresolved scales is considered. Two rather different flow regimes are found in wavenumber space. The high-wavenumber regime is dominated by the transient components of the flow and exhibits, at least approximately, several of the conditions characterizing homogeneous and isotropic turbulence. This region of wavenumber space also displays some of the features of an enstrophy-cascading inertial subrange. The low-wavenumber region, on the other hand, is dominated by the stationary component of the flow, exhibits marked anisotropy and, in contrast to the high-wavenumber regime, displays a marked change between January and July.