Mathematical concepts and their physical foundation in the nonstandard analysis theory of turbulence

Main mathematical concepts and their physical foundation in the nonstandard analysis theory of turbulence are presented and discussed. The underlying fact is that there does not exist the absolute zero fluid-volume. Therefore, the physical object corresponding to the absolute point is just the uniform fluid-particle. The fluid-particle, in general, corresponds to the monad. The uniform fluid-particle corresponds to the uniform monad, while the nonuniform fluid-particle to the nonuniform monad. There are two kinds of the differentiations, one is based on the absolute point, and the other based on the monad. The former is adopted in the Navier-Stokes equations, and the latter in the fundamental equations presented in this paper for the nonstandard analysis theory of turbulence. The continuity of fluid is elucidated by virtue of the concepts of the fluid-particle and fluid-particle at a lower level. Furthermore, the characters of the continuity in two cases, i.e. in the standard and nonstandard analyses, are presented in this paper. And the difference in discretization between the Navier-Stokes equations and the fundamental equations given herein is also pointed out.

ON THE CLOSURE PROBLEM OF TURBULENCE MODEL THEORY

It is a wrong viewpoint that the turbulence closure problem is due to thenon-linearity, of N-S equation, because if we omit the non-linear terms in N-Sequation,many, physical quantities can not be obtained other than the mean-values. Inthis paper, we proof that the closure problem of turbulence be induced by lack ofstatistical disiribution in present turbulence theory. And the restriction of turbulencemodel theory and shortcoming of direct numerical simulation of N-S to solve theturbulence have been pointed out.

A Closure for Isotropic Turbulence Based on Extended Scale Similarity Theory in Physical Space

The closure of a turbulence field is a longstanding fundamental problem, while most closure models are introduced in spectral space. Inspired by Chou＇s quasi-normal closure method in spectral space, we propose an analytical closure model for isotropic turbulence based on the extended scale similarity theory of the velocity structure function in physical space. The assumptions and certain approximations are justified with direct numerical simulation. The asymptotic scaling properties are reproduced by this new closure method, in comparison to the classical Batchelor model.

“Equilibrium”and“non-equilibrium”turbulence

We review the concept of ‘‘equilibrium＇＇ in turbulence. It generally means a property of the energy spectrum, it can also be understood in terms of a scalar property, the Taylor–Kolmogorov formula relating the dissipation rate to the total energy and integral length scale. The implications of equilibrium and strong departure from equilibrium for turbulence modeling are stressed.

SOME ISSUES OF TURBULENCE STATISTICS

The issue of dropping the random force f(i) and the arbitrariness of choosing the basic variable in the variational approach to turbulence closure problem, pointed out recently by the Russian scientists Bazdenkov and Kukharkin, are discussed. According to the mean-square estimation method, the random force f(i) should be dropped in the error expression of the LFP (Langevin-Fokker-Planck) model. However, f(i) is not neglected, its effect has been taken into account by the variational approach. In order to optimize the perturbation solution of the Liouville equation, the LFP model requires that the basic variable is as near to Gaussian as possible. Hence, the velocity, instead of the vorticity, should be chosen as the basic variable in the three-dimensional turbulence. Although the LFP model and the zero-order Gaussian term of PDF (probability density function) imply whiteness assumption (zero correlation time of f(i)), the higher-order non-Gaussian terms of PDF correspond to the nonwhiteness of turbulence dynamics, the variational approach does calculate the nonwhiteness effect properly.

ON THE MECHANISM OF TURBULENT COHERENT STRUCTURE (I)──THE PYSICAL MODEL OF THE COHERENT STRUCTUREFOR THE SMOOTH BOUNDARY LAYER

In this paper. based on the discussion of properties of the turbulent coherentstructure for smooth boumdary layer. a physical model has been established. The widthof slowly-moving streak line and bursting time of coherent structure expressed bymomentum thickness Reynolds mumber of boundary layer layer have been obtained,whichare in agreement with The experimental data￣[ 3. 4. 5.7 ]

A Theoretical Solution for the Reynolds Stress: Validations in Canonical Flow Geometries

A theoretical approach is developed for solving for the Reynolds stress in turbulent flows, and is validated for canonical flow geometries (flow over a flat plate, rectangular channel flow, and free turbulent jet). The theory is based on the turbulence momentum equation cast in a coordinate frame moving with the mean flow. The formulation leads to an ordinary differential equation for the Reynolds stress, which can either be integrated to provide parameterization in terms of turbulence parameters or can be solved numerically for closure in simple geometries. Results thus far indicate that the good agreement between the current theoretical and experimental/DNS (direct numerical simulation) data is not a fortuitous coincidence, and in the least it works quite well in sensible ways in canonical flow geometries. A closed-form solution for the Reynolds stress is found in terms of the root variables, such as the mean velocity, velocity gradient, turbulence kinetic energy and a viscous term. The form of the solution also provides radically new insight on how the Reynolds stress is generated and distributed.

Decentralized PID neural network control for a quadrotor helicopter subjected to wind disturbance

A decentralized PID neural network(PIDNN) control scheme was proposed to a quadrotor helicopter subjected to wind disturbance. First, the dynamic model that considered the effect of wind disturbance was established via Newton-Euler formalism.For quadrotor helicopter flying at low altitude in actual situation, it was more susceptible to be influenced by the turbulent wind field.Therefore, the turbulent wind field was generated according to Dryden model and taken into consideration as the disturbance source of quadrotor helicopter. Then, a nested loop control approach was proposed for the stabilization and navigation problems of the quadrotor subjected to wind disturbance. A decentralized PIDNN controller was designed for the inner loop to stabilize the attitude angle. A conventional PID controller was used for the outer loop in order to generate the reference path to inner loop. Moreover, the connective weights of the PIDNN were trained on-line by error back-propagation method. Furthermore, the initial connective weights were identified according to the principle of PID control theory and the appropriate learning rate was selected by discrete Lyapunov theory in order to ensure the stability. Finally, the simulation results demonstrate that the controller can effectively resist external wind disturbances, and presents good stability, maneuverability and robustness.

SKEWNESS FACTOR OF TURBULENT VELOCITY DERIVATIVE

Using nonequilibrium statistical mechanics closure method, it is shown that the skewness factor of the velocity derivative of isotropic turbulence ap- proaches a constant -0.515 when the Reynolds number is very high, which is in agree- ment with the DNS (direct numerical simulation) result of Vincent and Meneguzzi (1991).

ON THE SOLUTIONS OF NAVIER-STOKES EQUATIONS AND THE THEORY OF HOMOGENEOUS ISOTROPIC TURBULENCE

The present paper is a further development of our previous work in solving the wholeproblem of the homogeneous isotropic turbulence from the nitial period to the final period ofdecay. An expansion method is developed to obtain the axinlly symmetrical solution of theNavier-Stokes equations of motion in the form of an infinite set of nonlinear partial differen-tial equations of the second order. For the present we solve the zeroth order approximation.By using the method of Fourier transform, we get a nonlinear nitegro-differential equationfor the amplitude function in the wave number space.It is also the dynamical equation forthe energy spectrum. By choosing a suitable initial condition, we solve this equation numerically. The energyspectrum function and the energy transfer spectrum function thus calculated satisfy the spec-trum form of the karman-Howarth equation exactly. We Lave computed the energy spectrumfunction, the energy transfer function the decay of turbulent energy, the integral scale, Taylormicroscale, the double and triple velocity correlations on the whole range from the initialperiod to the final period of decay. As a whole all these calculated statistical physicalquantities agree with experiments very wall except a few cases with small discrepancies at largeseparations.

ON THE MECHANISM OF TURBULENT COHERENT STRUCTURE（II）──A PHYSICAL MODEL OF COHERENT STRUCTURE FOR THE ROUGH BOUNDARY LAYER

In this paper, the differences of turbulent coherent structure beween the smooth and rough boundary lavers are analysed.Based on the discussing the transient properties from the smooth wall tothe rough wall,the physicalmodel of coherent structure for the rough boundary layer are established.The width of slwly-moving turbulent spot and the bursting time are obtained,which are in agreement with experimental results.

Stochastic Particle Acceleration in Blazar Jets

The bulk kinetic energy of jets can be dissipated via generating tur- bulen plasma waves. We examine stochastic particle acceleration in blazar jets to explain the emissions of all blazars. We show that acceleration of electrons by plasma turbulence wavs with a spectrum W(k) ~ k^(-4/3) produces a nonthermal population of relativistic electrons whose peak frequency of synchrotron emission can fit the observational trends in the spectral energy distribution of all blazars. The plasma nonlinear processes responsible for the formation of turbulent spectrum are investigated. Increases in the ioteraction time of turbulent waves can produce a fiatter spectrum leading to efficient particle acceleration.

Global expressions for high-order structure functions in Burgers turbulence

Since the famous work by Kolmogorov on incompressible turbulence,the structure-function theory has been a key foundation of modern turbulence study.Due to the simplicity of Burgers turbulence,structure functions are calculated to arbitrary orders,which provides numerous implications for other compressible turbulent systems.We present the derivation of exact forcing-scale resolving expressions for high-order structure functions of the burgers turbulence.Compared with the previous theories,where the structure functions are calculated in the inertial range based on the statistics of shocks,our expressions link high-order structure functions in different orders without extra information on the flow structure and are valid beyond the inertial range,therefore they are easily checked by numerical simulations.

A STUDY OF 1.5-ORDER CLOSURE TURBULENCE MODEL

Based on turbulence theory,a 1.5-order closure turbulence model is established.The model incorporating with the ground surface energy budget equation is constructed by means of a vertical one-dimensional(1-D)40-level grid-mesh.The numerical results reveal the 24-h evolution of the clear planetary boundary layer comparing with the Wangara boundary layer data of days 33—34.The model also takes into account some physical processes of radiative transfer and baroclinicity,revealing some important characteristics observed in the boundary layer,especially for the evolution of the mixed layer and low-level jet.The calculated results are in good agreement with the observational data. On the other hand,we also run the high-resolution model of the planetary boundary layer in the Mesoscale Model Ver- sion 4(MM4)with the same physical processes and initial conditions.The results show that the high-resolution model can not reveal those important characteristics as the 1.5-order closure model did.In general,it is shown that the 1.5-or- der closure turbulence model based on turbulence theory is better in rationality and reality.

Influence of advection on the characteristics of turbulence over uneven surface in the oasis and the Gobi Desert

Utilizing experimental data of the atmospheric surface layer in the Gobi Oasis of Jinta in a comparative study, we demonstrate that under the condition of unstable stratification, the normalization variances of temperature in the oasis and Gobi Desert meetφs （z∧）= φθ（Z/∧） =αθ（-Z/ ∧）-1/3 while normalization variances of both humidity and CO2 in the oasis meet φ（Z/∧）= αs （1 - βs z /∧）-1/3 ; the normalization variance of temperature in the oasis is large due to disturbance by advection, whereas variance of CO2 in the Gobi Desert has certain degree of deviation relative to Monin-Obukhov （M-O） scaling, and humidity variance completely deviates from variance M-O scaling. The above result indicates that under the condition of advection, hu-midity variance meets the relation δ2 sm=D2 δ2SA ＋ δ2SB and it is determined by relative magnitude of scalar variance of ad- midity variance meets the relation δsm = D2δsA ＋ δsB vection transport. Our study reveals that, if the scalar variance of humidity or CO2 transported by advection is much larger than local scalar variance, observation value of scalar variance will deviate from M-O scaling; when scalar variance of advection transport is close to or less than local scalar variance, the observation value of scalar variance approximately meets M-O scal- ing.

The vertical distribution characteristics of integral turbulence statistics in the atmospheric boundary layer over an urban area in Beijing

Turbulence data(2008–2012) from a 325 m meteorological tower in Beijing, which consisted of three layers(47,140, and 280 m), was used to analyze the vertical distribution characteristics of turbulent transfer over Beijing city according to similarity theory. The conclusions were as follows.(1) Normalized standard deviations of wind speeds/ui * were plotted as a function only of a local stability parameter. The values under near-neutral conditions were 2.15, 1.61, and 1.19 at 47 m, 2.39, 1.75,and 1.21 at 140 m, and 2.51, 1.77, and 1.30 at 280 m, showing a clear increase with height. The normalized standard deviation of wind components fitted the 1/3 law under unstable stratification conditions and decreased with height under both stable and unstable conditions.(2) The normalized standard deviation of temperature fitted the.1/3 law in the free convection limit, but was quite scattered with different characteristics under near-neutral conditions. The normalized standard deviations of humidity and the CO2 concentration fitted the.1/3 law under unstable conditions, and remained constant under near-neutral and stable stratification. The normalized standard deviation of scalars, i.e., temperature, humidity, and CO2 concentration, all increased with height.(3) Compared with momentum, and the water vapor and CO2 concentrations, the turbulence correlation coefficient for heat was smaller under near-neutral conditions, but larger under both stable and unstable conditions. A dissimilarity between heat, and the water vapor and CO2 concentrations was observed in urban areas. The relative correlation coefficients between heat and each of momentum, humidity, and CO2 concentration(|rwT/ruw|, |rwT/rwc| and |rwT/ruq|) in the lower layers were always larger than in higher layers, except for the relative correlation coefficient between heat and humidity in an unstable stratification. Therefore, the ratio between heat and each of momentum, humidity, and CO2 concentration decreased with height.

Analysis of diurnal boundary layer development in boreal forests: measurements and simulations

Aims Combining field data analysis and modeling,this study investigates factors influencing the diurnal boundary layer(BL)development in boreal forest.Methods Field data analysis used both air sounding and surface flux measurements collected during the Boreal Ecosystem–Atmosphere Study field campaigns in central Canada.Model study applied a non-local transilient turbulence theory(TTT)to simulate the impact of the heterogeneous boundary conditions together with initial conditions on the BL development at the Candle Lake and Thompson release sites over boreal forests.Boundary conditions were characterized by the integrated surface flux measurements from different forest stands.The lake effect was included in constructing the surface fluxes at Candle Lake release site.Important Findings Analyses of serial upper air sounding data and tower flux data indicate strong linear impacts of surface sensible heat forcing on the diurnal BL development above boreal forests.The regression slopes on the relationship between the BL development and the surface fluxes reflect the influences of initial boundary conditions to the BL developments.Both the modeled and the measured diurnal BLs show that lakes reduce sensible heat flux,leading to a shallower boundary in Candle Lake than in Thompson.Comparison of the model results and field measurements on the BL profiles indicates that the TTT model has the capability to simulate the BL development above boreal forests for sunny,rainfall or cloudy days.This study demonstrates the importance of lake on surface fluxes and BL development.The modeling effort shows the potential to couple the transilient theory with a land surface process model to study land surface and atmosphere interaction in boreal forest.

THE SOLUTION FOR INCOMPRESSIBLE PLANE TURBULENT JET BY USING THE METHOD OF SUCCESSIVE SUBSTITUTION

Based on P. Y. Chou's theory of turbulence for incompressible viscous fluids, this paper solves the Reynolds' equations of mean motion and the equations of turbulent fluctuation simultaneously using the method of successive substitution. The comparison of the computed results for incompressible plane turbulent jet with the experimental data is reasonably good.