Mean Entrainment Velocity Research Articles

A comparison of entrainment velocity and displacement speed statistics in different regimes of turbulent premixed combustion

The statistics of entrainment velocity, defined as the displacement speed of an enstrophy isosurface, which can be taken to be the interface between turbulent/non-turbulent regions, have been analysed using a Direct Numerical Simulation database of statistically planar H2-air flames with a range of different Karlovitz numbers. It has been found that the component of the entrainment velocity arising from molecular dissipation plays a leading order role for all values of Karlovitz number, whereas the relative importance of the baroclinic torque and dilatation rate weakens with increasing Karlovitz number. By contrast, the relative contribution of the entrainment velocity component arising from vortex-stretching strengthens with increasing Karlovitz number Ka. The mean entrainment velocity remains positive for the case representing the corrugated flamelets regime (i.e. Ka<1), whereas it assumes negative values in the cases with large values of Karlovitz number (i.e. Ka≫1). The magnitude of the ratio of the mean values of entrainment velocity to the mean values of flame displacement speed conditional upon non-dimensional temperature within the flame front remains of the order of unity irrespective of Karlovitz number. However, the probability density functions of entrainment velocity exhibit considerably higher probabilities of finding large magnitudes than in the case of flame displacement speed. The alignments between the normal vector on the enstrophy isosurface and local principal strain rates have been found to be qualitatively similar to the corresponding alignments between flame normal and local principal strain rates, and the same holds true for the distributions of curvature shape factor of reaction progress variable and enstrophy isosurfaces. These findings indicate that the isosurface topologies and the alignments of normal vectors with local principal strain rates for enstrophy and reaction progress variable exhibit qualitatively similar behaviours. Consequently, turbulence and combustion modelling strategies cannot be considered in isolation in premixed turbulent flames.

Estimates of entrainment in closed cellular marine stratocumulus clouds from the MAGIC field campaign

Entrainment of warm, dry air from above the boundary layer into the cloud layer has a significant impact on stratocumulus clouds in the marine boundary layer. During the MAGIC field campaign, the Atmospheric Radiation Measurement (ARM) mobile facility was deployed aboard a container ship that made regular transects between Los Angeles, California and Honolulu, Hawaii. Observations made during MAGIC transects were collocated with observations from the Geostationary Operational Environmental Satellite (GOES‐15) and European Centre for Medium‐range Weather Forecasting (ECMWF) reanalysis model. From these data, hourly estimates of entrainment velocities in closed cellular stratocumulus cloud conditions were calculated from the mixed‐layer mass budget equation, modified to accommodate observations sampled from a moving platform. The technique is demonstrated using observations collected during Leg 15A (46 h) and then extended to 178 h of data. The average entrainment velocity was 7.83 ± 5.23 mm/s, and the average large‐scale vertical air motion at cloud top (obtained from reanalysis) was −2.56 ± 3.31 mm/s. The vertical air motion at cloud top was positive (upward) during 36 h (∼20%) with a mean of 2.68 mm/s. Entrainment velocity is highly variable and on average the MAGIC observations show no dependence of entrainment velocity on longitude or any pronounced diurnal cycle. When binned by inversion strength, the mean entrainment velocity and mean large‐scale vertical air motion mirrored each other, with both exhibiting substantial variability. Collectively, our results suggest a mean entrainment‐velocity behaviour associated with the background state, with large changes in entrainment velocity forced by strong variability in internal boundary‐layer properties like turbulence, radiation, and inversion strength. This cautions against using climatological mean estimates of entrainment velocities or neglecting instances with upward large‐scale vertical air motion.

Characterization of wind-shear effects on entrainment in a convective boundary layer

Direct numerical simulations are used to characterize wind-shear effects on entrainment in a barotropic convective boundary layer (CBL) that grows into a linearly stratified atmosphere. We consider weakly to strongly unstable conditions $-z_{enc}/L_{Ob}\gtrsim 4$, where $z_{enc}$ is the encroachment CBL depth and $L_{Ob}$ is the Obukhov length. Dimensional analysis allows us to characterize such a sheared CBL by a normalized CBL depth, a Froude number and a Reynolds number. The first two non-dimensional quantities embed the dependence of the system on time, on the surface buoyancy flux, and on the buoyancy stratification and wind velocity in the free atmosphere. We show that the dependence of entrainment-zone properties on these two non-dimensional quantities can be expressed in terms of just one independent variable, the ratio between a shear scale $(\unicode[STIX]{x0394}z_{i})_{s}\equiv \sqrt{1/3}\unicode[STIX]{x0394}u/N_{0}$ and a convective scale $(\unicode[STIX]{x0394}z_{i})_{c}\equiv 0.25z_{enc}$, where $\unicode[STIX]{x0394}u$ is the velocity increment across the entrainment zone, and $N_{0}$ is the buoyancy frequency of the free atmosphere. Here $(\unicode[STIX]{x0394}z_{i})_{s}$ and $(\unicode[STIX]{x0394}z_{i})_{c}$ represent the entrainment-zone thickness in the limits of weak convective instability (strong wind) and strong convective instability (weak wind), respectively. We derive scaling laws for the CBL depth, the entrainment-zone thickness, the mean entrainment velocity and the entrainment-flux ratio as functions of $(\unicode[STIX]{x0394}z_{i})_{s}/(\unicode[STIX]{x0394}z_{i})_{c}$. These scaling laws can also be expressed as functions of only a Richardson number $(N_{0}z_{enc}/\unicode[STIX]{x0394}u)^{2}$, but not in terms of only the stability parameter $-z_{enc}/L_{Ob}$.

Wind Shear Effects on Radiatively and Evaporatively Driven Stratocumulus Tops

Abstract Direct numerical simulations resolving meter and submeter scales in the cloud-top region of stratocumulus are used to investigate the interactions between a mean vertical wind shear and in-cloud turbulence driven by evaporative and radiative cooling. There are three major results. First, a critical velocity jump exists, above which shear significantly broadens the entrainment interfacial layer (EIL), enhances cloud-top cooling, and increases the mean entrainment velocity; shear effects are negligible when the velocity jump is below . Second, a depletion velocity jump exists, above which shear-enhanced mixing reduces cloud-top radiative cooling, thereby weakening the large convective motions; shear effects remain localized within the EIL when the velocity jump is below . The critical velocity jump and depletion velocity jump are provided as a function of in-cloud and free-tropospheric conditions, and one finds and for typical subtropical conditions. Third, the individual contributions to the mean entrainment velocity from mixing, radiative cooling, and evaporative cooling strongly depend on the choice of the reference height where the entrainment velocity is calculated. This result implies that the individual contributions to the mean entrainment velocity should be estimated at a comparable height while deriving entrainment-rate parameterizations. A strong shear alters substantially the magnitude and the height where these individual contributions reach their maxima, which further demonstrates the importance of shear on the dynamics of stratocumulus clouds.

Entrainment at multi-scales across the turbulent/non-turbulent interface in an axisymmetric jet

We consider the scaling of the mass flux and entrainment velocity across the turbulent/non-turbulent interface (TNTI) in the far field of an axisymmetric jet at high Reynolds number. Time-resolved, simultaneous multi-scale particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF) are used to identify and track the TNTI, and directly measure the local entrainment velocity along it. Application of box-counting and spatial-filtering methods, with filter sizes $\unicode[STIX]{x1D6E5}$ spanning over two decades in length, show that the mean length of the TNTI exhibits a power-law behaviour with a fractal dimension $D\approx 0.31{-}0.33$. More importantly, we invoke a multi-scale methodology to confirm that the mean mass flux, which is equal to the product of the entrainment velocity and the surface area, remains constant across the range of filter sizes. The results, within experimental uncertainty, also show that the entrainment velocity along the TNTI exhibits a power-law behaviour with $\unicode[STIX]{x1D6E5}$, such that the entrainment velocity increases with increasing $\unicode[STIX]{x1D6E5}$. In fact, the mean entrainment velocity scales at a rate that balances the scaling of the TNTI length such that the mass flux remains independent of the coarse-grain filter size, as first suggested by Meneveau &amp; Sreenivasan (Phys. Rev. A, vol. 41, no. 4, 1990, pp. 2246–2248). Hence, at the smallest scales the entrainment velocity is small but is balanced by the presence of a very large surface area, whilst at the largest scales the entrainment velocity is large but is balanced by a smaller (smoother) surface area.

Wind Shear and Buoyancy Reversal at the Top of Stratocumulus

Abstract A numerical experiment is designed to study the interaction at the stratocumulus top between a mean vertical shear and the buoyancy reversal due to evaporative cooling, without radiative cooling. Direct numerical simulation is used to eliminate the uncertainty introduced by turbulence models. It is found that the enhancement by shear-induced mixing of the turbulence caused by buoyancy reversal can render buoyancy reversal comparable to other forcing mechanisms. However, it is also found that (i) the velocity jump across the capping inversion Δu needs to be relatively large and values of about 1 m s−1 that are typically associated with the convective motions inside the boundary layer are generally too small and (ii) there is no indication of cloud-top entrainment instability. To obtain these results, parameterizations of the mean entrainment velocity and the relevant time scales are derived from the study of the cloud-top vertical structure. Two overlapping layers can be identified: a background shear layer with a thickness (⅓)(Δu)2/Δb, where Δb is the buoyancy increment across the capping inversion and a turbulence layer dominated by free convection inside the cloud and by shear production inside the relatively thin overlap region. As turbulence intensifies, the turbulence layer encroaches into the background shear layer and defines thereby the entrainment velocity. Particularized to the first research flight of the Second Dynamics and Chemistry of the Marine Stratocumulus (DYCOMS II) field campaign, the analysis predicts an entrainment velocity of about 3 mm s−1 after 5–10 min—a velocity comparable to the measurements and thus indicative of the relevance of mean shear in that case.

Turbulence dynamics near a turbulent/non-turbulent interface

AbstractThe 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 ($\varepsilon $), turbulence length scales, and pressure statistics are derived, showing an excellent agreement with results from direct numerical simulations (DNS). Interestingly, the normalized inviscid flow statistics at the T/NT interface do not depend on the form of the assumed TKE spectrum. Outside the turbulent region, where the flow is irrotational (except inside a thin viscous boundary layer),$\varepsilon $decays as${z}^{\ensuremath{-} 6} $, where$z$is the distance from the T/NT interface. The mean pressure distribution is calculated using RDT, and exhibits a decrease towards the turbulence region due to the associated velocity fluctuations, consistent with the generation of a mean entrainment velocity. The vorticity variance and$\varepsilon $display large maxima at the T/NT interface due to the inviscid discontinuities of the tangential velocity variances existing there, and these maxima are quantitatively related to the thickness$\delta $of the viscous boundary layer (VBL). For an equilibrium VBL, the RDT analysis suggests that$\delta \ensuremath{\sim} \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.

IGen 0.1: the automated generation of a parameterisation of entrainment in marine stratocumulus

Abstract. In a previous paper we described a new technique for automatically generating parameterisations using a program called iGen. iGen generates parameterisations by analysing the source code of a~high resolution model that resolves the physics to be parameterised. In order to demonstrate that this technique scales up to deal with models of realistic complexity we have used iGen to generate a parameterisation of entrainment in marine stratocumulus. We describe how iGen was used to analyse the source code of an eddy resolving model (ERM) and generate a parameterisation of entrainment velocity in marine stratocumulus in terms of the large-scale state of the boundary layer. The parameterisation was tested against results from the DYCOMS-II intercomparison of ERM models and iGen's parameterisation of mean entrainment velocity was found to be 5.27 × 10−3 ± 0.62 × 10−3 m s−1 compared to 5.2 × 10−3 ± 0.8 × 10−3 m s−1 for the DYCOMS-II ensemble of large eddy simulation (LES) models.

The evaporatively driven cloud-top mixing layer

Direct numerical simulations of the turbulent temporally evolving cloud-top mixing layer are used to investigate the role of evaporative cooling by isobaric mixing locally at the stratocumulus top. It is shown that the system develops a horizontal layered structure whose evolution is determined by molecular transport. A relatively thin inversion with a constant thickness h = κ/we is formed on top and travels upwards at a mean velocity we ≃ 0.1(κ |bs|χc2)1/3, where κ is the mixture-fraction diffusivity, bs &lt; 0 is the buoyancy anomaly at saturation conditions χs and χc is the cross-over mixture fraction defining the interval of buoyancy reversing mixtures. A turbulent convection layer develops below and continuously broadens into the cloud (the lower saturated fluid). This turbulent layer approaches a self-preserving state that is characterized by the convection scales constructed from a constant reference buoyancy flux Bs = |bs|we/χs. Right underneath the inversion base, a transition or buffer zone is defined based on a strong local conversion of vertical to horizontal motion that leads to a cellular pattern and sheet-like plumes, as observed in cloud measurements and reported in other free-convection problems. The fluctuating saturation surface (instantaneous cloud top) is contained inside this intermediate region. Results show that the inversion is not broken due to the turbulent convection generated by the evaporative cooling, and the upward mean entrainment velocity we is negligibly small compared to the convection velocity scale w* of the turbulent layer and the corresponding growth rate into the cloud.

Iso-surface mass flow density and its implications for turbulent mixing and combustion

A new result is derived for the mass flow rate per unit volume through a scalar iso-surface – called here the ‘iso-surface mass flow density’. The relationship of the surface mass flow density to the local entrainment rate per unit volume in scalar mixing and to the local reaction rate in turbulent premixed combustion is considered. In inhomogeneous flows, integration of the surface mass flow density across the layer in the direction of the mean scalar inhomogeneity yields the mean entrainment velocity in scalar mixing and the turbulent burning velocity in premixed combustion. For non-premixed turbulent reacting flow, this new result is shown to be consistent with the classical result of Bilger (Combust. Sci. Technol. vol. 13, 1976, p. 155) for fast one-step irreversible chemical reactions. Direct numerical simulation data for conserved scalar mixing, isothermal reaction front propagation and turbulent premixed flames are analysed. It is found that the entrainment velocity in the conserved scalar mixing case is sensitive to a threshold value. This suggests that the entrainment velocity is not a well-defined concept in temporally developing mixing layers and that scaling laws for the viscous superlayer warrant further investigation. In the isothermal reaction fronts problem, the characteristics of iso-surface propagation in a low Damköhler number regime are investigated. In premixed flames, the effects of non-stationarity on the turbulent burning velocity are addressed. The difference from the existing methods for determining turbulent burning velocity, and the implications of the present results for flames with multi-dimensional complex geometry are discussed. It is also shown that the surface mass flow density is related to the turbulent scalar flux in statistically stationary one-dimensional premixed flames. Variations of the local propagation characteristics due to departure from an unstretched laminar flame structure are shown to decrease the tendency to counter-gradient transport in turbulent premixed flames.

Entrainment results from the Flatland boundary layer experiments

A primary objective of the 1995 and 1996 Flatland boundary layer experiments, known as Flatland95 and Flatland96, was to measure and characterize entrainment at the top of the convective boundary layer. The experiments took place in the area near the Flatland Atmospheric Observatory near Champaign‐Urbana, Illinois, in August‐September 1995 and June‐August 1996. The site is interesting because it is extraordinarily flat, has uniform land use, and is situated in a prime agricultural area. Measurements in the entrainment zone are difficult to make due to the time and space scales involved. We will present entrainment estimates derived from budget calculations with data from UHF wind profiling radars and from radiosondes. The results demonstrate that the remote sensing instruments produce results comparable to radiosondes and have significant advantages for boundary layer studies. Surface flux measurements are also used in the calculations. Direct heating by shortwave radiation absorbed by aerosols in the boundary layer is found to be an important component of the boundary layer heat budgets. The entrainment virtual temperature flux and the ratio of entrainment to surface flux found from the budget calculations are somewhat larger than expected. Advection of warm air, which is not accounted for in the budget calculations, is probably a factor in some periods but may not be significant in the full data set. For the full data set, the mean entrainment velocity found from the heat budget is 0.03±0.01 m s−1, slightly less than the mean rate of change of the boundary layer height. The mean entrainment ratio AR is 0.51±0.12 and the median is 0.43, comparable to results from some other studies in comparable conditions.

Lateral entrainment in baroclinic currents

The strong mesoscale velocity fluctuations (''eddies'') observed in the deep Gulf Stream region maybe dynamically necessary to incorporate the Recirculation Gyres which account for the downstream increase in transport. The eddy-current interaction leading to entrainment is studied in a two layer quasi-geostrophic model with piecewise uniform potential vorticity. It is shown that the volume entrained by a bottom eddy of strength Gamma*(2) initially located near the edge of a bottom current depends mainly on the lateral shear st of this current, rather than on the much stronger shear of the upper layer. It is suggested that the ''mean entrainment velocity'' into an isopycnal layer of nearly uniform vertical thickness is proportional to (Gamma*(2s2)), where Gamma*(2) (cm(2)/sec) is proportional to the integrated potential vorticity anomaly of the eddy.

Seasonal cycle of sea surface temperature and mixed layer heat budget in the tropical Pacific Ocean

Climatological data derived from the Comprehensive Ocean‐Atmosphere Data Set (COADS) combined with the climatological temperature fields from XBT observations in the tropical Pacific Ocean are used to assess the importance of various oceanic and atmospheric processes in the seasonal variation of the mixed layer temperature. The analyses indicate that in most areas of the extra‐equatorial oceans, where the mixed layer is deep and oceanic mixing is weak, the seasonal change in SSTs can be attributed to the net local heating received by the mixed layer. However, in other parts of the equatorial oceans, oceanic processes are found to play an important role in determining the seasonal fluctuations of the SST. In particular, in the eastern equatorial Pacific Ocean where the mixed layer is shallow, the thermal‐inertia of the upper ocean is reduced significantly by the intense vertical mixing so that the response of the SST is nearly in phase with the surface heating. Using the Kraus‐Turner type of entrainment parameterization, the analyses further demonstrate that the vertical heat flux induced by the mean entrainment velocity contributes significantly to the mixed layer heat budget.