Improvement of Prandtl mixing length theory and application in modeling of turbulent flow in circular tubes

In order to correctly predict tube cross section time-smoothed velocity distribution, friction factor and mass transfer behavior, two models for turbulent flow in circular tubes based on classical Prandtl mixing length theory and a modified mixing length were established. The results show that the modified mixing length includes the introduction of a damping function for the viscous sublayer and the second-order derivative to approximate eddy velocity. The calculated dimensionless time-smoothed velocity from the model based on Prandtl mixing length is much better than the result from the concept of eddy viscosity. The calculated eddy viscosity from the model based on modified mixing length is much better than the result from the model based on the classical Prandtl mixing length theory. And the friction factor calculated from the model based on the modified mixing length agrees well with the reported empirical relationships.

Similarity solutions of Prandtl mixing length modelled two dimensional turbulent boundary layer equations

The exact similarity solutions of two dimensional laminar boundary layer were obtained by Blasius in 1908,however,for two dimensional turbulent boundary layers,no Blasius type similarity solutions(special exact solutions)have ever been found.In the light of Blasius’pioneer works,we extend Blasius similarity transformation to the two dimensional turbulent boundary layers,and for a special case of flow modelled by Prandtl mixing-length,we successfully transform the two dimensional turbulent boundary layers partial differential equations into a single ordinary differential equation.The ordinary differential equation is numerically solved and some useful quantities are produced.For numerical calculations,a complete Maple code is provided.

Velocity transformation for compressible wall-bounded turbulence——An approach through the mixing length hypothesis

In an endeavor to establish a connection between the mean velocity profile in compressible wall-bounded turbulence and its incompressible analogue,a refined version of the Trettel and Larsson's(TL)transformation is systematically derived and rigorously assessed across diverse flow scenarios.Incorporating the recently proposed intrinsic compressibility effects and modeling the multi-layer structure of mixing lengths,the proposed transformation demonstrates exceptional performance in collapsing 57canonical flow cases,including cooled channel and pipe flows,channel flows with pseudo heat sources,as well as adiabatic and diabatic boundary layer flows.Furthermore,the transformation seamlessly extends to low Reynolds number cooled channel and pipe flows,achieving a level of accuracy unparalleled by other transformations in the current state-of-the-art.

NUMERICAL STUDY OF WAVE EFFECTS ON SURFACE WIND STRESS AND SURFACE MIXING LENGTH BY THREE-DIMENSIONAL CIRCULATION MODELING

The effects of waves on Surface Drag Coefficient （SDC） and surface mixing length were analyzed and discussed by carrying out three-dimensional current modeling for the Bohai Sea in the present work. A threedimensional coupled hydrodynamical-ecological model for regional and shelf seas （COHERENS） incorporating the influences of wave-current interactions was coupled with the third-generation wave model swan taking into account time-varying currents. The effects of waves on currents were included in the SDC, surface mixing length and bottom drag coefficient. Firstly, the formulations in Donelan were incorporated into the COHERENS to account for wave-dependent SDC. In order to compare simulation results for the wave-dependent SDC, the simulation for wind-dependent SDC was also carried out. Second, Wave-Induced Surface Mixing Length （described as WISML sometimes in this paper） was incorporated into the COHERENS. Four numerical experiments were conducted to discuss the effects of two kinds of wave processes. Generally, the values of time series of current velocity and water surface elevation given by the simulation with all of the three wave processes have a good agreement with observed data. The existence of WISML changes obviously current vertical profiles and the existence of the wave dependent SDC modifies the current field of both top and bottom layers with the wind-dependent SDC.

Numerical and experimental investigations of liquid mixing in two-stage micro-impinging stream reactors

A two-stage micro-impinging stream reactor(TS-MISR) that combined a first pre-mixing stage with a second micro-impinging stream reacting stage for continuous multi-component reacting systems has been built from commercial T-junctions and steel micro-capillaries. Both of operating parameters and reactor configurations,such as jet Reynolds number(Rej), volumetric flow ratio(R), the first-stage junction angle(φ), the connecting capillary length(Lc) and connecting capillary diameter(dc), had significant effects on the micromixing efficiency of the reactor. Such effects were investigated for both of the two stage structures, respectively, by experimental and CFD methods and were optimized for the best micromixing performance. Intensified micromixing among at least three reacting components can be achieved in a continuous mode by using TS-MISR; therefore, it is expected that the TS-MISR will produce products of higher quality with more uniform and stable element distribution.

Weighted Integral Means of Mixed Areas and Lengths Under Holomorphic Mappings

This note addresses monotonic growths and logarithmic convexities of the weighted （（1-t2）αdt2, -∞〈α〈∞, 0〈t〈1） integral means Aα,β（ f ,·） and Lα,β（ f ,·） of the mixed area （πr2）-βA（ f ,r） and the mixed length （2πr）-βL（ f ,r） （0≤β≤1 and 0〈r〈1） of f （rD） and？f （rD） under a holomorphic map f from the unit disk D into the finite complex plane C.

Three-Dimensional Tidal Model and Its Application to Numerical Simulation of Water Quality in Coastal Waters

The turbulence mechanism plays an important part in the mixing process and momentum transfer of turbulence. A three-dimensional Prandtl mixing length tidal model has been developed to simulate tidal flows and water quality. The eddy viscosities and diffusivities are computed from the Prandtl mixing length model. In order to model the water quality of an estuary or coastal area many interdependent processes need to be simulated. These may be conveniently separated into three main groups: transport and mixing processes, biochemical interaction of water quality variables and the utilization and re-cycling of nutrients by living matter. The model simulates full oxygen and nutrient balance, primary productivity and the transport, reaction mechanism and fate of pollutants over tidal time-scales. The model is applied to numerical simulation of tidal flows and water quality in Dalian Bay. The model has been calibrated against a limited data set of historical water quality observations and in general demonstrates excellent agreement with all available data.

Analytical solution of velocity distribution for flow through submerged large deflection flexible vegetation

An analytical solution for predicting the vertical distribution of streamwise mean velocity in an open channel flow with submerged flexible vegetation is proposed when large bending occurs. The flow regime is separated into two horizontal layers： a vegetation layer and a free water layer. In the vegetation layer, a mechanical analysis for the flexible vegetation is conducted, and an approximately linear relationship between the drag force of bending vegetation and the streamwise mean flow velocity is observed in the case of large deflection, which differes significantly from the case of rigid upright vegetation. Based on the theoretical analysis, a linear streamwise drag force-mean flow velocity expression in the momentum equation is derived, and an analytical solution is obtained. For the free water layer, a new expression is presented, replacing the traditional logarithmic velocity distribution, to obtain a zero velocity gradient at the water surface. Finally, the analytical predictions are compared with published experimental data, and the good agreement demonstrates that this model is effective for the open channel flow through the large deflection flexible vegetation.

Vertical Distribution of Momentum Exchange Coefficient and Sediment Concentration in Estuarine and Coastal Waters

Vertical structures of momentum exchange coefficient and sediment concentration are the keys in the research on es- tuarine and coastal suspended sediment transport. Based on the parabolic mixing length distribution pattern, the distribu- tion pattern of vertical momentum exchange coefficient which is suitable for estuarine and coastal waters is constructed. A compari^n with steady flow and measured momentum exchange coefficient during one tidal cycle in the Menai Strait of England shows that the result of this pattern is closer to the measured values than those of commonly used Rouse＇ s and van Rijn＇s patterns, and the pattern is also suitable for estuarine and coastal waters. Successively, based on the precon- dition that momentum exchange coefficient is equivalent to sediment turbulent diffusion coefficient, and combining with the Diffusion Theory, we obtain the exponential vertical distribution pattern of sediment concentration, which is also suitable for estuafine and coastal waters, Thereby, using measured vertical stratified sediment concentration data of the South and the North Passages of the Yangtze Estuary and Zhoushan archipelago waters for fitting calculation, and comparing the results with those from Rouse＇s （1938） and Zhang et al. ＇s （1989） formulas, the results show that the exponential vertical distribution pattern of sediment concentration obtained in the present study not only overcomes the defect of Rouse＇s formula that the surface sediment concentration will be 0, but also has a holistic higher precision along the vertical lines than those of Rouse＇s and Zhang et al. ＇s formulas.

Distribution of Time-Averaged Suspended Sediment Concentration Due to Waves

According to the mechanism of sediment suspension under waves, namely, the main reason of sediment suspension changes from the turbulent mixing in the bottom boundary layer to the periodic motion of the water particle near the free water surface, a three-layer model of sediment concentration distribution due to waves is presented along the whole water depth based on the concept of the finite mixing length. 1he determination of the parameters in the model is discussed and an empirical formula is suggested. Comparisons between the calculated results and the measurements indicate that the resuits of the model agree well with the data from both the large and small scale flume experiments.

Study on a Boundary-layer Numerical Model with Inclusion of Heterogeneous Multi-layer Vegetation

On the basis of improving the algorithm of the mixing length in and above forest canopies, a PBL numerical model including the multi-layer, heterogeneous vegetation is developed. Simulations indicate thatdifferent treatments of mixing length can make a great difference in the wind field especially for dense forest.and results fi.om the improved mixing length scheme are in better agreement with observations than thosefrom the original scheme. It may be expected that the improved mixing length scheme can lead to more rational turbulent transfer than the original one. From the sensitivity experiments, we obtain the characteristics of both wind and temperature profiles in and above plant canopies, e.g., during the daytime, a stablethermal stratitication exists near the surface in the canopies, but a neutral or slightly unstable condition appears above plant canopies. while at night the reverse situations occur; the increase of the temperature of thedense-forest case is less than that of the sparse-forest case, the windspeed is reduced within the canopy layer and the large wind shear occurs near the treetop, etc.

Anomalous transport driven by ion temperature gradient instability in an anisotropic deuterium-tritium plasma

In this work,the anomalous transport driven by the ion temperature gradient instability is investigated in an anisotropic deuterium-tritium(D-T)plasma.The anisotropic factorα,defined as the ratio of perpendicular temperature to parallel temperature,is introduced to describe the temperature anisotropy in the equilibrium distribution function.The linear dispersion relation in local kinetic limit is derived,and then numerically evaluated to study the dependence of mode frequency on the anisotropic factorαof D and the fraction of T particleεTby choosing three sets of typical parameters,denoted as the cyclone base case,ITER and CFETR cases.Based on the linear results,the mixing length model approximation is adopted to analyze the quasi-linear particle and energy fluxes for D and T.It is found that choosing smallαand largeεTis beneficial for the confinement of particle and energy for D and T.This work may be helpful for the estimation of turbulent transport level in the ITER and CFETR devices.

The role of zonal flows in reactive fluid closures

We will give an overview of results obtained by our reactive fluid model. It is characterised as a fluid model where all moments with sources in the experiment are kept. Furthermore, full account is taken for the highest moments appearing in unexpanded denominators also including full toroidicity. It has been demonstrated that the strength of zonal flows is dramatically larger in reactive fluid closures than in those which involve dissipation. This gives a direct connection between the fluid closure and the level of excitation of turbulence. This is because zonal flows are needed to absorb the inverse cascade in quasi 2D turbulence. This also explains the similarity in structure of the transport coefficients in our model with a reactive closure in the energy equation and models which have a reactive closure because of zero ion temperature such as the Hasegawa-Wakatani model. Our exact reactive closure unifies several well-known features of tokamak experiments such as the L-H transition, internal transport barriers and the nonlinear Dimits upshift of the critical gradient for onset of transport. It also gives transport of the same level as that in nonlinear gyrokinetic codes. Since these include the kinetic resonance this confirms the validity of the thermodynamic properties of our model. Furthermore, we can show that while a strongly nonlinear model is needed in kinetic theory a quasilinear model is sufficient in the fluid description. Thus our quasilinear fluid model will be adequate for treating all relevant problems in bulk transport. This is finally confirmed by the reproduction by the model of the experimental power scaling of the confinement time Te ~ P-2/3. This confirms the validity of our reactive fluid model. This also gives credibility to our ITER simulations including the H-mode barrier. A new result is here, that alpha heating strongly reduces the slope of the H-mode barrier. This should significantly reduce the effects of ELM＇s.

MATHEMATICAL MODEL FOR THE FLOW WITH SUBMERGED AND EMERGED RIGID VEGETATION

The article summarizes previous studies on the flow in open channels with rigid vegetation, and constructs a mathematical model for submerged and emerged rigid vegetation. The model involves the forces balance in the control volume in one-dimensional steady uniform flow. For submerged vegetation, the whole flow is divided into four regions： external region, upper vegetated region, transition region and viscous region. According to the Karrnan similarity theory, the article improves the mixing length expression, and then gives an analytical solution to predict the vertical distribution of stream-wise velocity in the external region. For emerged vegetation, the flow is divided into two region： outer region and viscous region. In the two circumstances, the thicknesses of each region are determined respectively. The comparison between the calculated results and our experimental data and other researchers＇ data proves that the proposed model is effective.

Optimization Study of a Coanda Ejector

The Coanda effect has long been employed in the aerospace applications to improve the performances of various devices. This effect is the ability of a flow to follow a curved contour without separation and has well been utilized in ejectors where a high speed jet of fluid emerges from a nozzle in the ejector body, follows a curved surface and drags the secondary flow into the ejector. In Coanda ejectors, the secondary flow is dragged in the ejector due to the primary flow momentum. The transfer of momentum from the primary flow to the secondary flow takes place through turbulent mixing and viscous effects. The secondary flow is then dragged by turbulent shear force of the ejector while being mixed with the primary flow by the persistence of a large turbulent intensity throughout the ejector. The performance of a Coanda ejector is studied mainly based on how well it drags the secondary flow and the amount of mixing between the two flows at the ejector exit. The aim of the present study is to investigate the influence of various geometric parameters and pressure ratios on the Coanda ejector performance. The effect of various factors, such as, the pressure ratio, primary nozzle and ejector configurations on the system performance has been evaluated based on a performance parameter defined elsewhere. The performance of the Coanda ejector strongly depends on the primary nozzle configuration and the pressure ratio. The mixing layer growth plays a major role in optimizing the performance of the Coanda ejector as it decides the ratio of secondary mass flow rate to primary mass flow rate and the mixing length.