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.

A multifractal model for linking Lagrangian and Eulerian velocity structure functions

A multifractal model is developed to connect the Lagrangian multifractal dimensions with their Eulerian counterparts. We propose that the characteristic time scale of a Lagrangian quantity should be the Lagrangian time scale, and it should not be the Eulerian time scale which was widely used in previous studies on Lagrangian statistics. Using the present model, we can obtain the scaling exponents of Lagrangian velocity structure functions from the existing data or models of scaling exponents of Eulerian velocity structure functions. This model is validated by comparing its prediction with the results of experiments, direct numerical simulations, and the previous theoretical models. The comparison shows that the proposed model can better predict the scaling exponents of Lagrangian velocity structure functions, especially for orders larger than 6.

Tide-induced Lagrangian residual velocity and dynamic analysis based on field observations in the inner Xiangshan Bay,China

In the Xiangshan Bay at the east coast of China,coastal marine pollution is conspicuous and severe in recent years.As transport of the pollutants is closely related to the coastal circulation,there is a great practical significance to investigate the circulation in this area.In this work,the surface pattern and vertical profiles of Lagrangian residual velocity(LRV)were studied based on field observation data from the inner Xiangshan Bay.By tracking GPS-GPRS drifters’trajectories,the surface LRV pattern is going out in the central deep trough and flowing inwards near the shoreside.Combined with data from two mooring stations,vertical profiles of LRV is flowing out at surface and flowing in at the bottom,consistent with the gravitational circulation induced by baroclinic effects at the estuary.However,according to the diagnostic analysis,the main mechanism driving the residual current is barotropic rather than baroclinic.The LRV equation is controlled by the tidally-averaged barotropic pressure gradient force,tidal body force and tidally-averaged turbulent stress,while the tidallyaveraged baroclinic pressure gradient force is one order of magnitude less than other forces.Additionally,the tidally mean eddy viscosity coefficient which is used in the expression of tidally-averaged turbulent stress might be not adequate and requires further studies.

A Lagrangian mean theory on coastal sea circulation with inter-tidal transports I.Fundamentals

A brief review is made on the theory of the Lagrangian residual circulation and inter-tidal transports in a convectively weakly nonlinear system. In the review the emphasis is put on the systematical development of the theory and its weakness of convectively weakly nonlinear approximation. The fundamentals of a Lagrangian tidally-averaged theory on circulation with inter-tidal transport processes have been proposed for a general nonlinear coastal/estuarine system. The Lagrangian residual velocity is strictly de- fined, and it has been verified to be able to embody the velocity field of circulation. A new concept of the concentration for inter- tidal transport processes is presented. The concentration describing the inter-tidal transport processes should be a ＂Lagrangian inter-tidal concentration＂ defined and named, but not the Eulcrian tidally-averaged concentration used traditionally. The circulation described here contains a set of infinite temporal-spatial fields of velocity/concentration, each of which corresponds to a specific value of tidal phases varying continuously over one tidal cycle. When the convectively weakly nonlinear condition（ with a smaller order of eddy diffusion and sources） is approximately satisfied, a set of infinite temporal-spatial fields of velocity/concentration can be reduced to a single one.. the mass transport velocity/the Eulerian tidally averaged concentration as exhibited traditionally.

A Lagrangian mean theory on coastal sea circulation with inter-tidal transports Ⅱ. Numerical Experiments

The results of the new concept of coastal sea circulation are demonstrated by numerical simulations for the first time.The numerical experiments in three types of rectangular model seas illustrate the dependence of circulation on tidal phases due to the convectively nonlinear effect which is estimated by a newly defined drift dispersion index.Then,the present theory is applied in the Bohai Sea of China.At the Bohai Straits and the Huanghe River mouth area the circulation direction even reverses owing to different initial tidal phases which shows that the theory copes with nonlinearity well.The calculated M2 tide-induced residual circulation shows that a clockwise gyre exists in the center of an anticlockwise gyre in the central Bohai Sea due to the topographic features.In the Bohai Gulf the tide induced circulation shows a 3D structure with outflow at the surface and the inflow at the bottom which can partly explains the spread of the Huanghe River fresh water out of the Bohai Gulf and the inflow of the sediment from the Huanghe River.

On residual velocities in sigma coordinates in narrow tidal channels

In shallow coastal regions where water surface fluctuations are non-negligible compared to the mean water depth,the use of sigma coordinates allows the calculation of residual velocity around the mean water surface level.Theoretical analysis and generic numerical experiments were conducted to understand the physical meaning of the residual velocities at sigma layers in breadth-averaged tidal channels.For shallow water waves,the sigma layers coincide with the water wave surfaces within the water column such that the Stokes velocity and its vertical and horizontal components can be expressed in discrete forms using the sigma velocity.The residual velocity at a sigma layer is the sum of the Eulerian velocity and the vertical component of the Stokes velocity at the mean depth of the sigma layer and,therefore,can be referred to as a semi-Lagrangian residual velocity.Because the vertical component of the Stokes velocity is one order of magnitude smaller than the horizontal component,the sigma residual velocity approximates the Eulerian residual velocity.The residual transport velocity at a sigma layer is the sum of the sigma residual velocity and the horizontal component of the Stokes velocity and approximates the Lagrangian residual velocity in magnitude and direction,but the two residual velocities are not conceptually the same.