Solution of Self-similar Equations of the k-ε Model in the Shear Turbulent Mixing Problem and Its Numerical Simulation

The paper presents the k-ε model equations of turbulence with a single set of constants chosen by the authors, which is appropriate to simulate a wide range of turbulent flows. The model validation has been performed for a number of flows and its main results are given in the paper. The turbulent mixing of flow with shear in the tangential velocity component is discussed in details. An analytical solution to the system of ordinary differential equations of the k-ε model of turbulent mixing has been found for the self-similar regime of flow. The model coefficients were chosen using simulation results for some simplest turbulent flows. The solution can be used for the verification of codes. The numerical simulation of the problem has been performed by the 2D code EGAK using this model. A good agreement of the numerical simulation results with the self-similar solution, 3D DNS results and known experimental data has been achieved. This allows stating that the k-ε model constants chosen by the authors are acceptable for the considered flow.

Spatial structure of turbulent mixing of an anticyclonic mesoscale eddy in the northern South China Sea

Upper turbulent mixing in the interior and surrounding areas of an anticyclonic eddy in the northern South China Sea(SCS)was estimated from underwater glider data(May 2015)in the present study,using the Gregg-HenyeyPolzin parameterization and the Thorpe-scale method.The observations revealed a clear asymmetrical spatial pattern of turbulent mixing in the anticyclonic eddy area.Enhanced diffusivity(in the order of 10–3 m2/s)was found at the posterior edge of the anticyclonic mesoscale eddy;on the anterior side,diffusivity was one order of magnitude lower on average.This asymmetrical pattern was highly correlated with the eddy kinetic energy.Higher shear variance on the posterior side,which is conducive to the triggering of shear instability,may be the main mechanism for the elevated diffusivity.In addition,the generation and growth of sub-mesoscale motions that are fed by mesoscale eddies on their posterior side may also promote the occurrence of strong mixing in the studied region.The results of this study help improve our knowledge regarding turbulent mixing in the northern SCS.

Observations of upper layer turbulent mixing in the southern South China Sea

A turbulent microstructure experiment was undertaken at a low latitude of 10°N in the South China Sea in late August 2012. The characteristics of the eddy diffusivity above 650 m were analyzed, which is one order of magnitude larger than that in the open ocean at that low latitude. Enhanced eddy diffusivities by strong shears and sharp changes in topography were observed. The strongest eddy diffusivity occurred in the mixed layer, and it reached O（10^-2 m^2/s）. Strong stratification in the thermocline inhibited the penetration of surface eddy diffusivities through the thermocline, where the mixing was weakest. Below the thermocline, where the background eddy diffusivity was approximately O（10^-6 m^2/s）, the eddy diffusivity increased with depth, and its largest value was O（10^-3 m^2/s）.

Spatial and temporal variability of turbulent mixing in the near-field of the Changjiang River

Based on field hydrological,microstructural,and shipboard Acoustic Doppler Current Profiler data,we quantified the spatial and temporal variability of turbulent mixing in the near-field Changjiang(Yangtze)River plume.The surface dissipation rate(ε)changed by three orders of magnitude from near-field(10^-4 W/kg)to far-field(10^-7 W/kg)plumes,indicating a decrease with distance from the river mouth.Below the river plume,εchanged with depth to 10^-8 W/kg,and increased to 10^-6 W/kg at the layer where the Taiwan Warm Current(TWC)intruded.Thus,εin the near-field plume showed three layers:surface layer in the river plume,middle layer,and lower TWC layer.In the river plume,the strongestεand turbulent diffusivity(Kz)were greater than 10^-4 W/kg and 10^-2 m^2/s,respectively,during strong ebb tides.A three-orders-of-magnitude change inεand Kz was observed in the tidal cycle.The depth of the halocline changed with tidal cycles,and stratification(N 2)varied by one order of magnitude.Stratification in the TWC layer followed the distribution of the halocline,which is opposite to the dissipation structure.Tidal currents led to intrusion and turbulent mixing in the TWC layer.During ebb tides,εand Kz were as strong as those measured in the river plume,but did not last as long.The structure of the velocity shear was similar to the dissipation rate in both the river plume and TWC layer,whereas the velocity shear in the TWC layer did not match the stratification structure.In the high dissipation rate area,the gradient Richardson number was smaller than the critical value(Ri g<1/4).The Ri g structure was consistent with shear and dissipation distributions,indicating that turbulent mixing in the near-field plume was controlled by a combination of shear induced by the discharged river flow and tidal current.

Observed characteristics of flow,water mass,and turbulent mixing in the Preparis Channel

Preparis Channel is the very important exchange path of energy and materials between the northern Bay of Bengal and Andaman Sea(AS).A set of hydrographic measurements,a microstructure profiler,and a deep mooring were used to determine the characteristics of water masses,turbulent mixing,and flows in the Preparis Channel.The unprecedented short-term mooring data reveal that a deep current in the deep narrow passage(below 400 m)of the Preparis Channel flows toward the Bay of Bengal(BoB)with a mean along-stream velocity of 25.26 cm/s at depth of 540 m;above the deep current,there are a relatively weak current flows toward the AS with a mean along-stream velocity of 15.46 cm/s between 500 m and 520 m,and another weak current flows toward the BoB between 430 m and 500 m.Thus,a sandwiched vertical structure of deep currents(below 400 m)is present in the Preparis Channel.The volume transport below 400 m is 0.06 Sv(1 Sv=106 m^(3)/s)from the AS to the BoB.In the upper layer(shallower than 300 m),the sea water of the AS is relatively warmer and fresher than that in the BoB,indicating a strong exchange through the channel.Microstructure profiler observations reveal that the turbulent diffusivity in the upper layer of the Preparis Channel reaches O(10−4 m^(2)/s),one order larger than that in the interior of the BoB and over the continental slope of the northern AS.We speculate that energetic high-mode internal tides in the Preparis Channel contribute to elevated turbulent mixing.In addition,a local“hotspot”of turbidity is identified at the deep mooring site,at depth of about 100 m,which corresponds to the location of elevated turbulent mixing in the Preparis Channel.

Distribution of Vertical Turbulent Mixing Parameter Caused by Internal Tidal Waves and Solitary Waves in the South Yellow Sea

Many observations show that in the Yellow Sea internal tidal waves (ITWs) possess the remarkable characteristics of internal Kelvin wave, and in the South Yellow Sea (SYS) the nonlinear evolution of internal tidal waves is one of the mechanisms producing internal solitary waves (ISWs), which is different from the generation mechanism in the case where the semidiurnal tidal current flows over topographic drops. In this paper, the model of internal Kelvin wave with continuous stratification is given, and an elementary numerical study of nonlinear evolution of ITWs is made for the SYS, using the generalized KdV model (GKdV model for short) for a continuous stratified ocean, in which the different effects of background barotropic ebb and flood currents are considered. Moreover, the parameterization of vertical turbulent mixing caused by ITWs and ISWs in the SYS is studied, using a parameterization scheme which was applied to numerical experiments on the breaking of ISWs by Vlasenko and Hutter in 2002. It is found that the vertical turbulent mixing caused by internal waves is very strong within the upper layer with depth less than about 30m, and the vertical turbulent mixing caused by ISWs is stronger than that by ITWs.

The influence of turbulent mixing on the subsurface chlorophyll maximum layer in the northern South China Sea

We present observations from deployments of turbulent microstructure instrument and CTD package in the northern South China Sea from April to May 2010.From them we determined the turbulent mixing(dissipation rateεand diapycnal diffusivityκ),nutrients(phosphate,nitrate,and nitrite),nutrient fluxes,and chlorophyll a in two transects(A and B).Transect A was located in the region where turbulent mixing in the upper 100 m was weak(κ~10-6-10-4 m^(2)/s).Transect B was located in the region where the turbulent mixing in the upper 100 m was strong(κ~10-5-10-3 m^(2)/s)due to the influence of internal waves originating from the Luzon Strait and water intrusion from the Western Pacific.In both transects,there was a thin subsurface chlorophyll maximum layer(SCML)(>0.25 mg/m^(3))nested in the upper 100 m.The observations indicate that the effects of turbulent mixing on the distributions of nutrients and chlorophyll a were different in the two transects.In the transect A with weak turbulent mixing,nutrient fluxes induced by turbulent mixing transported nutrients to the SCML but not to the upper water.Nutrients were sufficient to support a local SCML phytoplankton population and the SCML remained compact.In the transect B with strong turbulent mixing,nutrient fluxes induced by turbulent mixing transported nutrients not only to the SCML but also to the upper water,which scatters the nutrients in the water column and diffuses the SCML.

Tides and Turbulent Mixing in the North of Taiwan Island

Microstructure and hydrological profiles were collected along two cross-shelf sections from the deep slope to the shallow water in the north of Taiwan Island in the summer of 2006.While the tidal currents on the shelf were dominated by the barotropic tide with the current ellipse stretched across the shelf,significant internal tides were observed on the slope.The depth-mean turbulent kinetic energy(TKE)dissipation rate on the shelf was 10^-6W kg^-1,corresponding to a diapycnal diffusivity of 10^-2 m^2s^-1.The depth-mean TKE dissipation rate on the slope was 1×10^-7 Wkg^-1,with diapycnal diffusivity of 3×10^-4m^2s^-1.The shear instability associated with internal tides largely contributed to the TKE dissipation rate on the slope from the surface to 150 m,while the enhanced turbulence on the shelf was dominated by tidal or residual current dissipations caused by friction in the thick bottom boundary layer(BBL).In the BBL,the Ekman currents associated with the northeastward Taiwan Warm Current were identified,showing a near-bottom velocity spiral,which agreed well with the analytical bottom Ekman solution.

Eulerian-Lagranigan simulation of aerosol evolution in turbulent mixing layer

The formation and evolution of aerosol in turbulent flows are ubiquitous in both industrial processes and nature. The intricate interaction of turbulent mixing and aerosol evolution in a canonical turbulent mixing layer was investigated by a direct numerical simulation （DNS） in a recent study （Zhou, K., Attili, A., Alshaarawi, A., and Bisetti, F. Simulation of aerosol nucleation and growth in a turbulent mixing layer. Physics of Fluids, 26, 065106 （2014））. In this work, Monte Carlo （MC） simulation of aerosol evolution is carried out along Lagrangian trajectories obtained in the previous simulation, in order to quantify the error of the moment method used in the previous simulation. Moreover, the particle size distribution （PSD）, not available in the previous works, is also investigated. Along a fluid parcel moving through the turbulent flow, temperature and vapor concentration exhibit complex fluctuations, triggering complicate aerosol processes and rendering complex PSD. However, the mean PSD is found to be bi-modal in most of the mixing layer except that a tri-modal distribution is found in the turbulent transition region. The simulated PSDs agree with the experiment observations available in the literature. A different explanation on the formation of such PSDs is provided.

REVIEW OF TURBULENT MIXING MODELS

Fluid mixing is an important phenomenon in many physical applications from supernova explosions to genetic structure formations. In this paper, we overview some theoretical and empirical dynamic mix models, which have been developed over the recent decades, in particular, the ensemble-average micro physical mix model, the multifluid interpenetration mix model, the phenomenological and hybrid turbulent mix models, the buoyancy drag mix model, the single fluid turbulence mix model, and the large eddy simulation mix model. The similarities, distinctions, and connections between these models and their applications are discussed.

The cruise observation of turbulent mixing in the upwelling region east of Hainan Island in the summer of 2012

The turbulent mixing in the upwelling region east of Hainan Island in the South China Sea is analyzed based on in situ microstructure observations made in July 2012. During the observation, strong upwelling appears in the coastal waters, which are 3℃ cooler than the offshore waters and have a salinity 1.0 greater than that of the offshore waters. The magnitude of the dissipation rate of turbulent kinetic energy ε in the upwelling region is O（10–9 W/kg）, which is comparable to the general oceanic dissipation. The inferred eddy diffusivity Kρ is O（10–6 m2/s）, which is one order of magnitude lower than that in the open ocean. The values are elevated to Kρ≈O（10–4 m2/s） near the boundaries. Weak mixing in the upwelling region is consistent with weak instability as a result of moderate shears versus strong stratifications by the joint influence of surface heating and upwelling of cold water.The validity of two fine-scale structure mixing parameterization models are tested by comparison with the observed dissipation rates. The results indicate that the model developed by Mac Kinnon and Gregg in 2003 provides relatively better estimates with magnitudes close to the observations. Mixing parameterization models need to be further improved in the coastal upwelling region.

Turbulent mixing in the upper ocean of the northwestern Weddell Sea, Antarctica

Turbulent mixing in the upper ocean（30-200 m） of the northwestern Weddell Sea is investigated based on profiles of temperature,salinity and microstructure data obtained during February 2014.Vertical thermohaline structures are distinct due to geographic features and sea ice distribution,resulting in that turbulent dissipation rates（ε） and turbulent diffusivity（K） are vertically and spatially non-uniform.On the shelf north of Antarctic Peninsula and Philip Ridge,with a relatively homogeneous vertical structure of temperature and salinity through the entire water column in the upper 200 m,both ε and K show significantly enhanced values in the order of O（10^（-7））-O（10^（-6）） W/kg and O（10^（-3））-O（10^（-2）） m^2/s respectively,about two or three orders of magnitude higher than those in the open ocean.Mixing intensities tend to be mild due to strong stratification in the Powell Basin and South Orkney Plateau,where s decreases with depth from O（10^（-8）） to O（10^（-9）） W/kg,while K changes vertically in an inverse direction relative to s from O（10^（-6）） to O（10^（-5）） m^2/s.In the marginal ice zone,K is vertically stable with the order of10^（-4） m^2/s although both intense dissipation and strong stratification occur at depth of 50-100 m below a cold freshened mixed layer.Though previous studies indentify wind work and tides as the primary energy sources for turbulent mixing in coastal regions,our results indicate weak relationship between K and wind stress or tidal kinetic energy.Instead,intensified mixing occurs with large bottom roughness,demonstrating that only when internal waves generated by wind and tide impinge on steep topography can the energy dissipate to support mixing.In addition,geostrophic current flowing out of the Weddell Sea through the gap west of Philip Passage is another energy source contributing to the local intense mixing.

A climatic dataset of ocean vertical turbulent mixing coefficient based on real energy sources

Using data on wind stress, significant height of combined wind waves and swell, potential temperature, salinity and seawater velocity, as well as objectively-analyzed in situ temperature and salinity, we established a global ocean dataset of calculated wind- and tide-induced vertical turbulent mixing coefficients. We then examined energy conservation of ocean vertical mixing from the point of view of ocean wind energy inputs, gravitational potential energy change due to mixing(with and without artificially limiting themixing coefficient), and K-theory vertical turbulent parameterization schemes regardless of energy inputs. Our research showed that calculating the mixing coefficient with average data and artificial limiting the mixing coefficient can cause a remarkable lack of energy conservation, with energy losses of up to 90% and changes in the energy oscillation period. The data also show that wind can introduce a huge amount of energy into the upper layers of the Southern Ocean, and that tidesdo so in regions around underwater mountains. We argue that it is necessary to take wind and tidal energy inputs into account forlong-term ocean climate numerical simulations. We believe that using this ocean vertical turbulent mixing coefficient climatic dataset is a fast and efficient method to maintain the ocean energy balance in ocean modeling research.

TURBULENT MIXING AND EVOLUTION IN A STABLY STRATIFIED FLOW WITH A TEMPERATURE STEP

Large-Eddy Simulation （LES） is applied to examine the turbulent mixing and evolution in a stably stratified flow with a thermally sharp interface. Turbulent velocity intensities and turbulent kinetic energy are analyzed by considering the mean shear and stratification effects. The evolution of turbulent mixing layer and turbulent structures are mainly investigated. The results show that the streamwise intensities are much larger than the vertical intensities, and vertical fluctuations decay more rapidly at the presence of stratification. The qualitatively computational results suggest that the mixing layer, defined by the mean temperature, inclines to the side with small inlet velocity. The evolution of the half-width of the mixing layer shows two different slopes. The turbulent structure with high vorticity is restricted in the mixing layer especially in strong stratified cases.

Turbulent dissipation and mixing in Prydz Bay

In this paper, we present measurements of velocity, temperature, salinity, and turbulence collected in Prydz Bay, Antarctica, during February, 2005. The dissipation rates of turbulent kinetic energy （e） and diapycnal diffusivities （Ks） were estimated along a section in front of the Amery Ice Shelf. The dissipation rates and diapycnal diffusivities were spatially non-uniform, with higher values found in the western half of the section where E reached 10.7 W/kg and Kz reached 10.2 mVs, about two and three orders of magnitude higher than those in the open ocean, respectively. In the western half of the section both the dissipation rates and diffusivities showed a high-low-high vertical structure. This vertical structure may have been determined by internal waves in the upper layer, where the ice shelf draft acts as a possible energy source, and by bottom-generated internal waves in the lower layer, where both tides and geostrophic currents are possible energy sources. The intense diapycnal mixing revealed in our observations could contribute to the production of Antarctic Bottom Water in Prydz Bay.

Large eddy simulations of a turbulent mixing layer periodically excited with fundamental and third harmonic frequency

A better understanding of the mixing behavior of excited turbulent mixing layers is critical to a number of aerospace applications.Previous studies of excited turbulent mixing layers focused on single frequency excitation or the excitation with fundamental and its second harmonic frequency.There is a lack of detailed studies on applying low and higher frequency excitation.In this study,we have performed large-eddy simulations of periodically excited turbulent mixing layers.The excitation consists of a fundamental frequency and its third harmonic.We have used phase-averaging to identify the vortex structure and strength in the mixing layer,and we have studied the vortex dynamics.Two different vortex paring mechanisms are observed depending on the phase shift between the two excitation frequencies.The influence of these two mechanisms on the mixing of a passive scalar is also studied.It is found that exciting the mixing layer with these low and high frequencies has initially an adverse influence on the mixing process;however,it improves the mixing further downstream of the splitter plate with the excitation using a phase shift ofΔφ=πshowing the best mixing performance.The present works shed lights on the fundamental vortex dynamics,and has great potential for aeronautical,automotive and combustion engineering applications.

Macro and micro issues in turbulent mixing

Numerical prediction of turbulent mixing can be divided into two subproblems: to predict the geometrical extent of a mixing region and to predict the mixing properties on an atomic or molecular scale, within the mixing region. The former goal suffices for some purposes, while important problems of chemical reactions(e.g. flames) and nuclear reactions depend critically on the second goal in addition to the first one. Here we review recent progress in establishing a conceptual reformulation of convergence, and we illustrate these concepts with a review of recent numerical studies addressing turbulence and mixing in the high Reynolds number limit. We review significant progress on the first goal, regarding the mixing region, and initial progress on the second goal, regarding atomic level mixing properties. New results concerning non-uniqueness of the infinite Reynolds number solutions and other consequences of a renormalization group point of view, to be published in detail elsewhere, are summarized here.The notion of stochastic convergence(of probability measures and probability distribution functions) replaces traditional pointwise convergence. The primary benefit of this idea is its increased stability relative to the statistical "noise" which characterizes turbulent flow. Our results also show that this modification of convergence, with sufficient mesh refinement, may not be needed. However, in practice, mesh refinement is seldom sufficient and the stochastic convergence concepts have a role.Related to this circle of ideas is the observation that turbulent mixing, in the limit of high Reynolds number, appears to be non-unique. Not only have multiple solutions been observed(and published) for identical problems, but simple physics based arguments and more refined arguments based on the renormalization group come to the same conclusion.Because of the non-uniqueness inherent in numerical models of high Reynolds number turbulence and mixing, we also include here numerical examples of validation. The algorithm we use here has two essential components. We depend on Front Tracking to allow accurate resolution of flows with sharp interfaces or steep gradients(concentration or thermal), as are common in turbulent mixing problems. The higher order and enhanced algorithms for interface tracking, both those already developed, and those proposed here, allow a high resolution and uniquely accurate description of sample mixing problems. Additionally, we depend on the use of dynamic subgrid scale models to set otherwise missing values for turbulent transport coefficients, a step that breaks the non-uniqueness.

Persistent mixing bursts in the equatorial Pacific thermocline induced by persistent equatorial waves

A recent study by Liu et al.(2020)suggested that due to the saturation of equatorially trapped planetary waves with different dynamical types,temporal periods,meridional and baroclinic modes,complex layer structures of vertical velocity shear and hence turbulent mixing could frequently occur in the thermocline of the eastern equatorial Pacific.We investigated the occurrence of the interior turbulent mixing as indicated by shear instabilities,above the Equatorial Undercurrent(EUC)core at three equatorial sites along 140°W,170°W,and 165°E,respectively,based mainly on data from the Tropical Atmosphere and Ocean(TAO)mooring array.We found that turbulent mixing bursts persisted in the thermocline of all three sites.Specifically,the interior turbulent mixing layers(ITMLs)could occur in probability of approximately 68%,53%,and 48%at the three sites,respectively.The overall occurrence probability shows obvious and similar biannual variations at 140°W and 170°W,which is higher in boreal from late summer to winter and lower in spring.Vertically,the ITMLs are primarily located above the EUC core and prevail in deeper(shallower)layers from late summer to winter(spring).Most ITMLs(70%)lasted for hours to 3 days,and a few of them(15%)for more than 7 days.The thicknesses of ITMLs were concentrated between 15 and 55 m.At 165°E,the vertical distribution of ITML occurrence probability was different from that at 140°W and 170°W,as it did not show a preference for depths;the durations of ITMLs are short(also from hours to several days)and their thicknesses were between 5 and 25 m.These properties,particularly the high occurrence probability,and short durations demonstrated the persistence of thermocline mixing in the western to eastern equatorial Pacific thermocline and confirmed the generation mechanism by persistent equatorial waves as well.

Analysis of drop deformation dynamics in turbulent flow

Drop breakage and coalescence influence the particle formation in liquid-liquid dispersions. In order to reduce the influencing factors of the whole dispersion process, single drops where coalescence processes can be neglected were analyzed in this work. Drops passing the turbulent vicinity of a single stirrer blade were investi- gated by high-speed imaging. In order to gain a statistically relevant amount of drops passing the area of interest and corresponding breakage events, at least 1600 droplets were considered for each parameter set of this work. A specially developed fully automatic image analysis based on Matlab was used for the evaluation of the resulting high amount of image data. This allowed the elimination of the time-consuming manual analysis and further- more, allowed the objective evaluation of the drops＇ behavior. Different deformation parameters were consid- ered in order to describe the drop deformation dynamics properly. Regarding the ratio of both main particle axes （0axes）, which was therefore approximated through an ellipse, allowed the determination of very small de- viations from the spherical shape. The perimeter of the particle （0peri） was used for the description of highly de- formed shapes. In this work the results of a higher viscosity paraffin oil （ηd =127 mPa. s） and a low viscosity solvent （petroleum, ηd = 1.7 mPa-s） are presented with and without the addition of SDS to the continuous water phase. All results show that the experimentally determined oscillation but also deformation times underlie a wide spreading. Drop deformations significantly increased not only with increasing droplet viscosity, but also with decreasing interfacial tension. Highly deformed particles of one droplet species were more likely to break than more or less spherical particles. As droplet fragmentation results from a variety of different macro-scale de- formed particles, it is not assumed that a critical deformation value must be reached for the fragmentation pro- cess to occur. Especially for highly deformed particles thin particle filaments are assumed to induce the breakage process and, therefore, be responsible for the separation of drops.

Effects of the Reynolds number on a scale-similarity model of Lagrangian velocity correlations in isotropic turbulent flows

A scale-similarity model of a two-point two-time Lagrangian velocity correlation(LVC) was originally developed for the relative dispersion of tracer particles in isotropic turbulent flows(HE, G. W., JIN, G. D., and ZHAO, X. Scale-similarity model for Lagrangian velocity correlations in isotropic and stationary turbulence. Physical Review E, 80, 066313(2009)). The model can be expressed as a two-point Eulerian space correlation and the dispersion velocity V. The dispersion velocity denotes the rate at which one moving particle departs from another fixed particle. This paper numerically validates the robustness of the scale-similarity model at high Taylor micro-scale Reynolds numbers up to 373, which are much higher than the original values(R_λ = 66, 102). The effect of the Reynolds number on the dispersion velocity in the scale-similarity model is carefully investigated. The results show that the scale-similarity model is more accurate at higher Reynolds numbers because the two-point Lagrangian velocity correlations with different initial spatial separations collapse into a universal form compared with a combination of the initial separation and the temporal separation via the dispersion velocity.Moreover, the dispersion velocity V normalized by the Kolmogorov velocity V_η ≡ η/τ_η in which η and τ_η are the Kolmogorov space and time scales, respectively, scales with the Reynolds number R_λ as V/V_η ∝ R_λ^(1.39) obtained from the numerical data.