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Diffusion of a solute in turbulent flows through a circular pipe or tunnel is an important aspect of environmental safety. In this study, the diffusion coefficient of turbulent flow in circular pipe has been simulated by the Discrete Tracer Point Method (DTPM). The DTPM is a Lagrangian numerical method by a number of imaginary point displacement which satisfy turbulent mixing by velocity fluctuations, Reynolds stress, average velocity profile and a turbulent stochastic model. Numerical simulation results of points’ distribution by DTPM have been compared with the analytical solution for turbulent plug-flow. For the case of turbulent circular pipe flow, the appropriate DTPM calculation time step has been investigated using a constantβ, which represents the ratio between average mixing lengths over diameter of circular pipe. The evaluated values of diffusion coefficient by DTPM have been found to be in good agreement with Taylor’s analytical equation for turbulent circular pipe flow by givingβ=0.04 to 0.045. Further, history matching of experimental tracer gas measurement through turbulent smooth-straight pipe flow has been presented and the results showed good agreement.

The diffusion of gas and other particulate matter in pipe or channel flows is important aspect to meet the safety requirements. It controls the longitudinal spreading and the residence time of gases or other particulate matter throughout the pipe. Diffusion occurred in turbulent flow in circular airway has been investigated for a century. Several researches were done by conducting experimental works or numerical approaches. When a pulsed substance or solute is injected into a straight pipe flow, it is advected and diffused to a relative reference moving with certain average velocity. Diffusion in the turbulent pipe flow is mainly characterized by axial velocity profile and velocity fluctuation in flow direction, because the radial gradient of solute concentration is much less than that of flow direction and also radial diffusion is limited by its pipe wall. Furthermore, turbulent mixing motions at different radial positions enhance the diffusion degree in flow direction.

Taylor [1,2] and Aris [^{4}) where viscous sublayer and transition layer are negligibly thin. Taylor [^{2}/s), against pipe diameter and turbulent shear velocity given by following equations:

where, τ (Pa) and u* (m/s) are shear stress and friction velocity in an arbitrary sub layer of the flow, ρ (kg/m^{3}) is fluid density, d (m) is pipe diameter, f (-) is DarcyWeisbach friction factor and U_{m} (m/s) is cross sectional average velocity.

There are numbers of studies especially for atmospheric pollutant dispersion using random walk as basic method. Most of researches addressed the ideal homogenous turbulence. The pioneer was Taylor [

Pulsed injection measurement method by using NaCl into water stream in smooth glass pipe was conducted by Sittel, Threadgill and Schnelle [

Measurements of diffusion of gas-phase in smooth pipe flows were carried out by Keyes [^{4}, due to the different sets of velocity profile. Thus, the velocity profile used in calculations has a sensitive effect on evaluated values of diffusion coefficient.

prediction compared to [

The advantages of proposed DTPM are that the calculations of concentration gradient in space or time domain, which are commonly employed in numerical simulations, are not required. It may reduce the computational time. It is also free of grid requirement and the visualization of points’ distribution is simple.

The scheme of present numerical simulations has been developed by moving points with regards to velocity profile and turbulent intensities in a turbulent circular pipe flow, depends on radial position of the point in circular cross section.

velocities in r and φ directions can be zero.

Turbulent flow is characterized by its stochastic properties. The velocity at a given specific position fluctuates around its mean value. The velocity fluctuation intensity is known as root mean square (r.m.s) values which vary as function of r. The time of averaged flow velocities and the turbulent intensities at a certain position in cylindrical coordinate system (x, r, φ) are defined as (U, 0, 0) and (, ,) respectively. The moving of points may be treated easily on Cartesian coordinate system (x, y, z), therefore, in present calculations, turbulent intensities in (x, r, φ) directions are transformed into (x, y, z) directions of Cartesian coordinate. Assumed instantaneous turbulent fluctuations are (, ,) in (x, y, z) directions, these can be obtained by transforming velocity fluctuations (, ,) as follows (see

Suppose the position of m^{th} point dosed into the flow at origin is denoted with superscript showing the elapsed time, t = 0, its moving distances (Δx, Δy, Δz) during time step Δt are given by;

Its displacement is expressed at t + Δt and t by;

Laufer [

If dimensionless value of longitudinal average velocity is defined as:

Dimensionless distance from the wall is given by:

where υ (m^{2}/s) is kinematic viscosity. According to Kenyon [^{+} and y^{+} have been presented by Nikuradse with equations for three regions, that are viscous sub layer (y^{+} ≤ 5), buffer layer (5 < y^{+} ≤ 30) and turbulent zone (y^{+} > 30).

Equations (8) to (10) have been confirmed to agree well with equations proposed by Reichardt [

A stochastic approach was applied to determine points’ diffusion process in the turbulent flow. In present numerical model, it is supposed that the point movements were based on turbulent eddy motion, which satisfies Gaussian probability density function (hereinafter GPDF) with a standard deviation equal to the turbulent intensities or the r.m.s value of velocity fluctuations in each direction (see Rouse [

As described previously, Laufer’s measurement results of turbulent intensities were presented in normalized values over shear velocity, u^{*}, which is function of cross sectional average velocity, U_{m}, and friction factor, f. The relationship between f and Re was presented with empirical equation by Colebrook [

In turbulent shear flow, fluid particles are translated from slower region to faster one and Reynolds stress is generated. It shows a time-averaged correlation between longitudinal and radial velocity fluctuations in the shear flows. In present study, effects of Reynolds stress correlations have been investigated by giving relationship between x axis velocity fluctuation, , with velocity fluctuation in r direction, , expressed as;

In the simulations, the value of including its sign was firstly given as a random number following GPDF described previously in preceding chapter, then absolute values of || were generated using similar method, but the signs of u’ were decided to satisfy Equation (12).

In fact, point’s displacement near wall and its wall interaction has not been well understood to simulate breaking sub-layer. The boundary condition for Lagrangian random walk still needs a calculation model. Several boundary models have been proposed in previous studies. Those methods depend on the physical and numerical factors considered in the simulation [

In present DTPM simulations, the reflection boundary condition at the airway wall is satisfied by a repositioning numerical treatment if points are out of the flow region. Thus, by this boundary condition the zero-flux condition at wall can be satisfied. A reflection boundary scheme is modeled as shown in

Another boundary condition namely rearrangement model has been investigated. In this model, random number is continuously generated until the point is repositioned within the flow regime. Szymczak and Ladd [

Other zero-flux boundary conditions also have been proposed. Drazer and Koplik [

Hulin, Koplik and Hinch [

Taylor [1,2] proposed that concentration distribution of solute after certain time, t, is symmetrically distributed follows the partial differential equation. He proposed concentration gradient at x and r direction which move to x direction with constant average cross sectional velocity, U_{m};

Several studies (see Wen and Fan [

The solution of Equation (13) can be obtained by assuming that molecular diffusion is neglected and concentration gradient in radial direction is negligible. The variable, E shows effective diffusion coefficient in axial direction. Since the center of dispersed solute is assumed to be at x = U_{m}t after elapsed time, t, the solution of Equation (13) can be given with an equation similar with Gaussian distribution:

where G (m^{3}) is total volume of released solute at x = 0, and A (m^{2}) is cross sectional area of pipe.

In the DTPM simulation, concentration of diffused points, C, can be calculated by;

where Δm is number of point counted in the numerical control volume which located at a certain downstream position, given by ΔV = ΔXπd^{2}/4 and ΔX = U_{m}Δt. Suppose the total number of points released from the origin is M_{D}, The normalized concentration of points is expressed as C_{T}A/M_{D}.

The DTPM simulations on turbulent flow of a straightsmooth airway have been carried out. _{m} = 1.5 m/s, d = 0.5 m). It can be observed that the asymmetry of points distribution is gradually diminished as the travelled distance increases.

Figures 8 and 9 show the results of simulation for flow with d = 2 m, U_{m} = 4 m/s (f = 0.01315) and d = 4 m, U_{m} = 5 m/s (f = 0.0112) respectively after t = 100 s. It can be inferred that different calculation time step, Δt, gives different evaluated value of effective diffusion coefficient. To consider this effect, the evaluated value of effective diffusion coefficient is plotted against dimensionless value, β (-) defined as:

where is r.m.s value of streamwise velocity fluctuation in the center of pipe. It may be possible to

regard β as ratio of the average mixing length to pipe diameter.

Figures 10(a)-(d) shows evaluated values of effective diffusion coefficient of different time step for Re = 5 × 10^{5}, 7.5 × 10^{5}, 9.4 × 10^{5} and 1.25 × 10^{6}. The values of E calculated by Equation (1) are also presented as comparisons. From the results, it can be seen that the evaluated values of E from present DTPM for β = 0.04 ~ 0.045 intersect with those Taylor analytical solution given by Equation (1).

The results of evaluated E of DTPM showing relatively non-linear and inversely proportional correlation to the value of β, before gradually attain constant value as higher value of Δt is applied. Further simulations were done by setting β = 0.040, 0.043 and 0.045 at different flow conditions. As shown in

Tracer gas diffusion experiment was conducted at a laboratory scale by Widodo [

For the straight airway, he used a pipe with smooth lining 0.025 m diameter, 30 m length and placed horizontally. Tracer gas was released by breaking balloons filled with methane (CH_{4}) and measured arriving concentration at the end of pipe. Methane concentration was measured by an original infrared adsorption gas detector

using 3.4 mm He-Ne laser with an infrared (IR) sensor, air pump, air mass flow meter and amplifier as shown in

Beforehand, the conversion of voltage reading by data logger to gas concentration was calibrated based on calibration data with the standard methane-air mixture. The validity of concentration reading from IR sensor was crosschecked using gas chromatograph and showed good linear fit.

The measurement was conducted for Reynolds number, Re = 6085 (U_{m} = 3.87 m/s, d = 2R = 0.025 m). Measurements for higher Re were also done; however, the data were inadequately acquired due to insufficient sampling rate to compensate higher flow velocity.

For the DTPM simulation, calculation parameters were decided based on the experimental properties such as pipe’s length, average cross sectional velocity and pipe’s diameter. Value of friction factor, f, was calculated using Equation (11) and calculation time step was defined using Equation (16) as Dt = 0.0092 s. To verify the effect of initial points’ distribution, three different conditions were considered; 1) point source (x_{0} = 0, r_{0} = 0); 2) uniformly distributed at x_{0} = 0 (0 ≤ r_{0} < R); 3) uniformly distributed at x_{0} ± 2d.

In the real case, the utilization of tracer gas measurement is not merely straight pipe, but also tunnels network [19,32-35]. This kind of network is assembled in the form of interconnecting tunnels which allow airflow to be separated or rejoined at junction. The mechanism of points tracking as proposed in this study can be developed to consider network flow by combining with a scheme to treat flow separation. The developed scheme is supposed to be able to simulate point’s distribution at arbitrary position in the network and allow easy dispersion evaluation of gas or other particulates spreading.

In this study, effective diffusion coefficients of turbulent flow in circular pipe or channel have been evaluated by the Discrete Tracer Point Method (DTPM), a Lagrangian numerical simulation method. The present study is summarized as follows:

1) DTPM simulation procedures have been presented to

simulate the displacement of points released into straightsmooth circular airway flow by giving turbulent average velocity, intensity of velocity fluctuation and Reynolds stress;

2) A simple procedure to represent diffusion of points in the airway flow has been employed by generating random number which satisfies Gaussian probability functions with value of turbulent intensities in each direction as standard deviations;

3) Appropriate calculation time step was proposed by matching with Taylor’s analytical equation by considering the ratio between the average mixing lengths to airway’s diameter, β @ 0.043;

4) The result of matching curve of DTPM with experimental result by considering different initial points distribution has shown that the present DTPM can be used to simulate turbulent advection-diffusion in smoothstraight pipe representing the ideal model of straight pipe or channel;

5) Present DTPM simulation has promising possibility to simulate the network flow by combining with a scheme to treat flow separation at junction.

Authors would like to thank Japan Ministry of Education, Culture, Sports, Science and Technology and Kyushu University Global COE program for the financial support.

A = Cross-sectional area of pipe (m^{2})

C = Gas concentration (−)

C_{T} = Number of points located within numerical control volume (−)

d = Diameter of pipe (m)

E = Longitudinal diffusion coefficient (m^{2}/s)

f = Darcy-Weisbach friction factor (−)

M_{D} = Total number of released points (−)

r = Radial position of a point (m)

R = Radius of pipe (m)

Re = Reynolds number (−)

t = Elapsed time (s)

U = Time-averaged flow velocity in x direction u^{*} = Friction velocity (m/s)

U^{+} = Dimensionless value of flow velocity (−)

y^{+} = Dimensionless distance from the wall (−)

Um = Cross-sectional average velocity (m/s)

= Stream-wise velocity fluctuation in the pipe center (m/s)

(, ,) = Turbulent intensities in cylindrical coordinate (r, φ, x) (m/s)

(, ,) = r.m.s value of turbulent intensities in cylindrical coordinate (r, φ, x) (m/s)

(Δx, Δy, Δz) = Displacement vector of a point (m)

β = ratio between average mixing in stream-wise direction to the diameter of pipe

Γ = Released volume of tracer gas (m^{3})

Δt = Numerical time step (s)

ΔV = Numerical control volume (m^{3})

λ = Time adjustment factor for interruption boundary condition (−)

ρ = Density of fluid (kg/m^{3})

τ = Shear stress (Pa)

υ = Kinematic viscosity of fluid (m^{2}/s)