Random Walk Particle Tracking to Model Dispersion in Steady Laminar and Turbulent Pipe Flow

Abstract

To accurately model a two-dimensional solute transport in drinking water pipes and determine the effective dispersion coefficients for one-dimensional water quality models of water distribution systems, a random walk particle tracking approach was developed to analyze the advection and dispersion processes in circular pipes. The approach considers a solute particle’s two-dimensional random movement caused by molecular or turbulent diffusion and associated velocity profile, and can simulate any mixing time and accurately model the longitudinal distribution of the solute concentration. For long mixing times, the simulation results agreed with a previous analytically derived solution. For turbulent flow conditions, simulations showed that the longitudinal dispersion of the solute is very sensitive to the utilized cross-sectional velocity profiles. This approach is easy to implement programmatically and unconditionally stable. It can predict the mixing characteristics of a pipe under various initial and boundary conditions.

Introduction

To better protect public health, it is critical to understand the transport mechanism of water quality components, such as disinfectants and intentionally or unintentionally introduced contaminants, within the drinking water pipes. Although the flow in many parts of drinking water distribution systems (DWDS) corresponds to turbulent flow regimes—where dispersion processes are not expected to significantly affect water qualities—molecular/turbulent diffusion should not be ignored in the water quality analysis because both laminar and critical flow conditions can exist in the peripheral parts of the DWDS (Buchberger et al. 2003; Tzatchkov et al. 2002). Additionally, more attention has been focused on the water quality problems within premise plumbing systems, which operate on a much smaller scale than the traditional DWDS (Abokifa and Biswas 2017; Burkhardt et al. 2020). From a public health point of view, a quantitative microbial risk assessment (QMRA) based on drinking water quality data demands microscopic analysis of a contaminant’s pathway from the point of entry to the point of consumption (Besner et al. 2011). There is a need for a more detailed modeling capability to study the advection, dispersion, and reaction processes of contaminant transport within drinking water piping systems.

Usually, a one-dimensional (1D) advection-dispersion equation (ADE) is solved to simulate the longitudinal spread of a solute in a water pipe. An effective longitudinal dispersion coefficient is introduced to account for the combined effect of the diffusion and the shear dispersion due to the nonuniformity of the velocity profile. Basha and Malaeb (2007) highlighted the importance of including the dispersion effect in the 1D transport equation. Abokifa et al. (2016) showed that after the dispersion process was considered, water quality simulation results at network dead ends got closer to field measurements. Tzatchkov et al. (2002) found that, in pipes with low flow velocities, the time series of the measured concentration was more closely represented by the ADE than the advection-only EPANET model (Rossman 2000). All these studies emphasized the importance of having appropriate values for the effective dispersion coefficient to accurately model water quality within DWDS.

The one-dimensional ADE with effective longitudinal dispersion coefficient is described as

$$\frac{\partial C}{\partial t}+U\frac{\partial C}{\partial z}=D_l\frac{{\partial }^{2}C}{\partial z^2}$$ (1)

where z = the longitudinal distance along the pipe; C = the cross-sectionally averaged concentration; U = the cross-sectionally averaged velocity; t = time; and D_l = the effective dispersion coefficient in the longitudinal direction.

Early studies of the effective longitudinal dispersion within a circular pipe (Taylor 1953, 1954b) showed that, after an initialization period, the response of the solute’s concentration profile to an upstream impulse input can be approximated as a Gaussian distribution and the 1D ADE can be applied to model the cross-sectionally averaged concentration within a pipe. Instead of being a physical constant, the effective longitudinal dispersion coefficient is mathematically derived based on the pipe diameter and the flow conditions. It is a measure of the rate at which the introduced solute spreads out in the flowing water. The expressions for the effective longitudinal dispersion coefficient, D_l, within a fully developed laminar flow and a highly turbulent flow, are:

$$\text{Laminar}{D}_l=\frac{a^2{U}^2}{48D_m}$$ (2)

$$\text{Turbulent}{D}_l=10.1a{u}_{\mathrm{*}}$$ (3)

where a = pipe radius; D_m molecular diffusion coefficient; and u_* shear velocity.

The primary difficulty in applying Taylor’s formulas of the effective longitudinal dispersion coefficient to solving the 1D ADE is that in laminar flow the molecular diffusion is the only radial mixing mechanism, and it takes a long time to achieve complete cross-sectional mixing of the solutes. The initial mixing period can last for days for a typical DWDS pipe under laminar conditions (Lee 2004). Eq. (2) can be validly applied to the 1D ADE only after the initial mixing period. The mixing time is usually expressed in a dimensionless form:

$$\tau =\frac{D_{m}t}{a^2}$$ (4)

Taylor (1954a) showed that the condition under which the longitudinal dispersion of a solute can be approximated with the 1D ADE is $\tau \gg 0.25$. Gill and Sankarasubramanian (1970) estimated that $\tau$ needs to be larger than 0.5 to achieve good cross-sectional mixing.

Gill and Sankarasubramanian (1970) derived an exact expression for the time-dependent dispersion coefficient in steady laminar flow through a pipe. Lee (2004) provided a simplified form of their expression. Neglecting molecular diffusion in the longitudinal direction, the time-dependent dispersion coefficient for laminar flow at mixing time t is

$$D_l\left(t\right)=\frac{a^2{U}^2}{48D_m}\left[1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-16\frac{D_{m}t}{a^2}\right)\right]$$ (5)

Eq. (5) applies to any instant in time. For the solute spreading process in a pipe, it is important to quantify the longitudinal distribution at the end of the pipe. For such application, the averaged dispersion coefficient between time 0 and t is likely a more reasonable variable to characterize the transport. Lee (2004) also provided the averaged dispersion coefficient formula as

$$\overline{D_l\left(t\right)}=\frac{a^2{U}^2}{48D_m}\left[1-\left[\frac{1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-16\frac{D_{m}t}{a^2}\right)}{16\frac{D_{m}t}{a^2}}\right]\right]$$ (6)

Eqs. (5) and (6) are derived from the analytical calculation of the longitudinal variance of the solute distribution (Crank 1956). Both the instantaneous and the averaged time-dependent dispersion coefficient can be estimated by the method of moments (Fischer et al. 1979)

$$D_l(t)=\frac{1}{2}\frac{{\sigma }_z^2(t)-{\sigma }_z^2(t-\text{Δ}t)}{\text{Δ}t}$$ (7)

$$\overline{D_l\left(t\right)}=\frac{1}{2}\frac{{\sigma }_z^2\left(t\right)}{t}$$ (8)

where ${\sigma }_z^2\left(t\right)=$ variance of the longitudinal distribution as a function of the mixing time, t; and $\mathrm{\Delta }t=$ the temporal increment for the calculation of the instantaneous change rate of the variance.

The initial period prior to complete cross-sectional mixing is relatively short under turbulent flow conditions. The limitation of applying Taylor’s effective longitudinal dispersion coefficient in turbulent flow is that Taylor derived the formula assuming fully developed turbulence with a Reynolds number (Re) larger than 20,000. As summarized in Hart et al. (2016), it has been well reported in the literature that for turbulent flows with Re < 20,000, the effective dispersion coefficients obtained experimentally are larger than what Eq. (3) predicts. Hart et al. (2016) compared the experimentally derived values reported in the literature with what Taylor’s formula predicts, conducted tracer experiments, and provided an empirical equation to calculate the effective dispersion coefficient for Reynolds numbers between 3,000 and 50,000.

Due to the limitation of the 1D ADE and the difficulty of determining the effective dispersion coefficients, computational fluid dynamics (CFD) approaches have been applied to solve the two-dimensional (2D) ADE for circular pipes.

Romero-Gomez and Choi (2011) used a finite–volume-based CFD solver (ANSYS FLUENT) to simulate the cross-sectional distribution of the solute concentrations under laminar flow, and calculated the averaged concentrations based on the CFD results. They proposed a direction-dependent formula for the effective dispersion coefficient for laminar flow, with the dimensionless time, $\tau$, being less than 0.01. The two parameters in the proposed formula were optimized using Newton’s method to minimize the difference between the simulated 1D ADE and the cross-sectionally averaged 2D CFD results.

Ozdemir et al. (2021) assumed a constant and isotropic diffusion coefficient for a given Reynolds number and used the velocity profiles reported in the literature to solve the 2D ADE with an alternate direction implicit method. The constant diffusion coefficient was calibrated based on a comparison of the steady state experimental data with the numerical simulation results. Two empirical equations of the diffusion coefficient as a function of the Reynolds number were derived: one for chlorine and the other for fluoride. The impact of the mixing time on the dispersion coefficient in laminar flow condition was not considered in this study.

In the present study, a 2D stochastic approach called random walk particle tracking (RWPT) was developed to model the movement of an instantaneous pulse of solute particles under both laminar and turbulent conditions. This method is applied to numerically calculate the effective dispersion coefficients under various flow conditions and evaluate the appropriateness a of 1D ADE equation in water quality modeling of a pipe network system. RWPT approaches have been used extensively for the simulation of passive solutes within fluids (Thompson and Wilson 2012; Hunter et al. 1993). However, a RWPT approach has rarely been applied to the solute transport analysis within drinking water pipes. Houseworth (1984) is one of the few reported works, but it was limited to laminar flow conditions with a homogeneous radial diffusion coefficient. As in the simplified approach proposed by Hathhorn (1997) for contaminant transport analysis, the spatial distribution of the diffusion coefficients (typical in turbulent flow conditions) was not considered.

The RWPT approach discussed herein is applicable to spatially variable diffusion coefficients and compatible with a variety of initial and boundary conditions. Solute particles are tracked, and the longitudinal distribution of the solute is calculated based on the locations of the particles. The method bridges the gaps between 1D advection-dispersion modeling and 2D CFD approaches without the heavier computational burden of the traditional CFD softwares The effective dispersion coefficient in 1D modeling can be numerically estimated with RWPT simulation results. The proposed RWPT method is a Lagrangian approach and therefore is unconditionally stable without significant numerical dispersion. It is easy to implement numerically and can replace expensive lab experiments and CFD simulations to obtain the effective dispersion coefficients under both laminar and turbulent flow conditions.

Methodology

Stochastic Differential Equation

Stochastic differential equations (SDEs) are used to describe stochastic processes and contain terms involving random variables. To model the diffusion process with the SDEs, the diffusion equation is usually written in the form of a Fokker-Planck equation, which provides the time evolution of the probability density function (PDF) of the particles that represent the component released into the fluid system. Details of the Fokker-Planck equation and its connection with SDEs can be found in Gardiner (2004).

Ignoring the non-diagonal elements of the diffusion tensor, the three-dimensional advection-diffusion equation can be written as

$$\frac{\partial c}{\partial t}=-\frac{\partial (uc)}{\partial z}-\frac{\partial (vc)}{\partial x}-\frac{\partial (wc)}{\partial y}+\frac{\partial }{\partial z}\left(D_z\frac{\partial c}{\partial z}\right)+\frac{\partial }{\partial x}\left(D_x\frac{\partial c}{\partial x}\right)+\frac{\partial }{\partial y}\left(D_y\frac{\partial c}{\partial y}\right)$$ (9)

where x, y, and z = Cartesian coordinates; u, v, and w = flow velocities in lateral (x), vertical (y), and longitudinal (z) directions, respectively; and c = concentration.

In pipe flow cases, it is assumed that advection is only in the longitudinal direction (z), and that the longitudinal diffusion can be ignored (Taylor 1953, 1954)

$$\frac{\partial c}{\partial t}=-\frac{\partial \left(uc\right)}{\partial z}+\frac{\partial }{\partial x}\left(D_x\frac{\partial c}{\partial x}\right)+\frac{\partial }{\partial y}\left(D_y\frac{\partial c}{\partial y}\right)$$ (10)

Assuming that diffusion is isotropic cross-sectionally and the advective flow velocity does not change in the longitudinal direction, Eq. (10) can be rewritten as

$$\frac{\partial c}{\partial t}=-u\frac{\partial \left(c\right)}{\partial z}+\frac{\partial D}{\partial x}\frac{\partial c}{\partial x}+\frac{\partial D}{\partial y}\frac{\partial c}{\partial y}+D\frac{{\partial }^{2}c}{\partial x^2}+D\frac{{\partial }^{2}c}{\partial y^2}$$ (11)

The resulting equivalent SDEs for particle position change are (Gardiner 2004):

$$x_{t+\text{Δ}t}=x_t+\frac{\partial D}{\partial x}\text{Δ}t+\sqrt{2D\text{Δ}t}{\xi }_1(t)$$ (12)

$$y_{t+\mathrm{\Delta }t}=y_t+\frac{\partial D}{\partial y}\mathrm{\Delta }t+\sqrt{2D\mathrm{\Delta }t}{\xi }_2\left(t\right)$$ (13)

$$z_{t+\mathrm{\Delta }t}=z_t+u\mathrm{\Delta }t$$ (14)

where $\xi =$ the standard Wiener process increment.

For a circular pipe, it is assumed that the diffusivity is a function of the r. Because

$$\frac{\partial r}{\partial x}=\frac{x}{r}$$ (15)

$$\frac{\partial r}{\partial y}=\frac{y}{r}$$ (16)

where r = radial distance to the center of the pipe.

SDEs corresponding to Eq. (11) are written as

$$x_{t+\text{Δ}t}=x_t+\frac{x}{r}\frac{\partial D}{\partial r}\text{Δ}t+\sqrt{2D\text{Δ}t}{\xi }_1(t)$$ (17)

$$y_{t+\mathrm{\Delta }t}=y_t+\frac{y}{r}\frac{\partial D}{\partial r}\mathrm{\Delta }t+\sqrt{2D\mathrm{\Delta }t}{\xi }_2\left(t\right)$$ (18)

$$z_{t+\mathrm{\Delta }t}=z_t+u\mathrm{\Delta }t$$ (19)

Eqs. (17) and (18) imply that, with the existence of inhomogeneous diffusivity, the solute particles tend to move in the direction of increasing diffusivity based on the gradient of the diffusion coefficient, $(\partial D/\partial r)$, even in the absence of a mean advective flow in the radial direction (Monin and Yaglom 1971). The additional radial movements are deterministic rather than stochastic, and the failure to incorporate a term for them leads to particle accumulation in the low diffusivity regions (Hunter et al. 1993).

Calculation of Turbulent Diffusion Coefficient

Instead of using complex CFD softwares, velocity profiles as functions of Reynolds number are used in the present study to calculate the turbulent diffusivity.

In a circular pipe, the shear stress as a function of the radial position is expressed as (Lin et al. 1953)

$${\tau }_r=-\rho \nu \frac{\partial u}{\partial r}-\rho {\nu }_t\frac{\partial u}{\partial r}$$ (20)

where τ_r = shear stress at radial position r; ρ = liquid density; ν = molecular kinematic viscosity; and ν_t turbulent kinematic viscosity.

Because the shear stress ${\tau }_r=(r/a){\tau }_w$, where ${\tau }_w$ is the shear stress at the pipe wall and ${\tau }_w=\rho u_{\mathrm{*}}^2$, the following relationships can be developed:

$${\nu }_t=\frac{{\tau }_r}{-\rho \frac{\partial u}{\partial r}}-\nu =\frac{{\tau }_{w}r}{-a\rho \frac{\partial u}{\partial r}}-\nu =\frac{\rho u_{\mathrm{*}}^{2}r}{-a\rho \frac{\partial u}{\partial r}}-\nu =\frac{u_{\mathrm{*}}^{2}r}{-a\frac{\partial u}{\partial r}}-\nu$$ (21)

Taylor (1954b) assumed that the Reynold’s analogy was true and the turbulent diffusion coefficient, D_t, which is also called turbulent diffusivity or eddy diffusivity, equals the turbulent kinematic viscosity, ${\nu }_t$. In a study with Re > 20,000, Taylor (1954b) neglected the viscous shear stress to develop the following relationships for D_t:

$$D_t={\nu }_t=\frac{{\tau }_r}{-\rho \frac{\partial u}{\partial r}}=\frac{u_{\mathrm{*}}^{2}r}{-a\frac{\partial u}{\partial r}}$$ (22)

In a smooth pipe, the ratio of mean velocity to shear velocity, $U/u_{\mathrm{*}}$, depends only on the Reynolds number (Taylor 1954b). In the present study, the pipe is assumed to be smooth, and the shear velocity values are found according to the flow mean velocity and the Reynolds number. The key step to identify the turbulent diffusivity is to obtain the velocity profile and compute the gradient of the velocity as a function of the radial position.

Taylor (1954b) assumed a universal velocity distribution in a pipe under turbulent flow with Re > 20,000. The velocity at radial position r in a pipe of radius a is given by

$$\frac{u_0-u\left(\eta \right)}{u_{\mathrm{*}}}=f\left(\eta \right)$$ (23)

where $u_0=$ the maximum flow velocity at the pipe centerline; and $\eta =r/a$. The function $f\left(\eta \right)$ applies to any straight pipe with a circular cross-section, and Taylor (1954b) provided the continuous log law of $f\left(\eta \right)$ for $0.9<\eta <1.0$, and 14 discrete values for $\eta \le 0.9$. Taking Eqs. (22) and (23) together results in

$$D_t={\nu }_t=\frac{u_{\mathrm{*}}^{2}r}{-a\frac{\partial u}{\partial r}}=\frac{a\eta u_{\mathrm{*}}}{\frac{\partial f\left(\eta \right)}{\partial \eta }}$$ (24)

For highly turbulent flow in which the viscous sublayer can be neglected, Guo and Julien (2003) proposed a modified log-wake law for smooth pipes. The velocity profile and turbulent diffusivity distributions are, respectively, described by the following equations:

$$f(\eta )=\frac{u_0-u(\eta )}{u_{\text{*}}}=-\kappa \left(\text{ln}(1-\eta )+\frac{1-{(1-\eta )}^3}{3}\right)+2{\text{cos}}^2\left(\frac{\pi (1-\eta )}{2}\right)$$ (25)

$$\frac{v_t}{a{u}_{\mathrm{*}}}=\frac{\eta }{\frac{\partial f\left(\eta \right)}{\partial \eta }}={\left[\kappa \left(\frac{1}{1-\eta }+2-\eta \right)+\frac{\pi \mathrm{s}\mathrm{i}\mathrm{n}\left(\pi \right(1-\eta \left)\right)}{\eta }\right]}^{-1}$$ (26)

where κ = the von Karman constant and is taken as 0.40 in this study to be consistent with the log law used in Taylor (1954b) for 0.9 < η<1.0.

The comparisons of $f\left(\eta \right)$ and $v_t/a{u}_{\mathrm{*}}$ from the velocity profiles proposed by Taylor (1954b) and Guo and Julien (2003) are shown in Fig. 1. Although the velocity profiles are close to each other, the turbulent diffusivity profiles have some significant differences, especially in the regions away from the pipe wall. The reason is that turbulent diffusivity is a function of the radial gradient of the velocity rather than of the velocity itself. We used both velocity profiles in the simulation studies and these differences are further explored in the context of RWPT simulation results.

Fig. 1.

Profiles of the flow velocity and turbulent diffusivity for turbulent flow with Re > 20,000.

For turbulent flows with Re < 20,000, literature data of pipe velocity profiles were summarized in Tichacek et al. (1957). The authors emphasized that the calculated longitudinal dispersion coefficient D_l is very sensitive to the velocity profile used—an approximately 3% difference in the velocity profiles can lead to a 50% difference in the estimated D_l. As an illustration, the velocity profile data for Re = 3,040, 4,000, 6,000, and 10,000 are used in the present study for the RWPT simulation and the estimation of the D_l with the method of moments. The velocity profiles are shown in Fig. 2. Because only velocity data is available, the gradient of the velocity needs to be calculated by the difference scheme at discrete radial positions.

Fig. 2.

Velocity profile for turbulent flow with Re < 20,000 (Tichacek et al. 1957).

RWPT Implementation

It is assumed that the solute particles enter the pipe at time zero. The spatial distribution of the particles’ initial position is determined by the given initial and boundary conditions. For each time step, the locations of the particles are first updated based on Eqs. (17) and (18). Under laminar flow conditions, the molecular diffusion coefficient does not depend on the radial location. But, in turbulent flow, both the local diffusivity (D_t) and the gradient of the diffusivity $\left(\partial D_t/\partial r\right)$ are calculated based on the velocity profile in the radial direction. With the diffusivity value, the polar method (Marsaglia and Bray 1964) was used to sample the 2D random movement of a particle in the cross section. (The details of using the polar method to generate the 2D movement are described in Appendix S2.) Due to the existence of pipe wall, a particle’s position is not changed if the random movement in radial direction would move it out of the pipe. After the location of a particle in the cross section is updated in the cross section, it is moved longitudinally according to Eq. (19).

The longitudinal position of the particles at a given time are used to calculate the longitudinal distribution of the particles or the longitudinal distribution of the concentration if the particles are associated with the mass. A pipe is divided into multiple bins with the same length and the particles in each bin are counted to calculate the particle concentration in the unit of particle number per meter (particle number divided by the bin length).

Simulation Study and Results

The spatial distribution of the particles is calculated as the number of particles per unit length (meter) in the longitudinal direction. In water quality analysis applications, this unit can be easily converted into mg/L given the mass represented by the particles and the radius of the pipe

$$C_m=\frac{MC}{\pi a^{2}N}$$ (27)

where M = total mass injected (g); C = concentration in number of particles per meter; C_m = cross sectionally averaged concentration in (mg/L); N = total number of particles; and a = pipe radius (m). The shape of the distribution curve is not affected by the unit used. Assuming 10 grams of contaminant is injected at time zero and at the upstream end of a pipe, the longitudinal distribution curves of the contaminant concentration are included in Appendix S1.

When possible, the longitudinal distributions of the particles are compared with the asymptotic solutions provided by Taylor (1953, 1954b) to verify the application of RWPT in pipe flows. Effective 1D dispersion coefficients calculated based on the particle distribution are compared with those reported in the literature.

Laminar Flow

For laminar flow, the molecular diffusion is assumed to be the only radial mixing mechanism and a constant diffusion coefficient is assumed. The parabolic velocity profile for laminar flow is well known as

$$u_r=u_0\left(1-\frac{r^2}{a^2}\right)$$ (28)

where u_r = flow velocity at radial position r; and u₀ = maximum flow velocity at pipe centerline; and u₀ = 2U.

In the simulation study, the pipe radius was set to be 0.02 m and the molecular diffusion coefficient to be $1.25\times {10}^{-9}{\mathrm{m}}^2/\mathrm{s}$. The mean flow velocity was set to be 0.01 m/s and Eq. (28) was used as the velocity profile. For water with a kinematic viscosity of $1.10\times {10}^{-6}{\mathrm{m}}^2/\mathrm{s}$, the corresponding Reynolds number is 363.6. The simulation time step was selected to be 1 s and a total number of 50,000 particles were released uniformly in the cross-section at t = 0 and z = 0. Sensitivity analysis was done to find the number of particles and the time step length, and determined that further increasing the particle number and/or reducing the time step would not change the results significantly (see Table S1).

Simulations were run for dimensionless times τ = 0.1, 0.25, 0.5, and 0.75. The calculated D_l based on Eq. (2) was $D_l=\left(a^2{U}^2/48D_m\right)=\left({0.02}^2{0.01}^2/48\times 1.25\times {10}^{-9}\right)=0.6667{\mathrm{m}}^2/\mathrm{s}$. With the effective longitudinal dispersion coefficient, the 1D ADE with an impulse input has the following analytical solution for concentration in the unit of number of particles per meter:

$$C(z,t)=\frac{M}{\sqrt{4\pi D_{l}t}}\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{-(z-Ut{)}^2}{4D_{l}t}\right]$$ (29)

where M = initial input. In the present simulation study, M = 50,000, which is the number of particles released at t = 0 and z = 0.

The results of the simulations are shown in Fig. 3, together with the longitudinal Gaussian distributions predicted by Eq. (29) for τ = 0.5 and 0.75. Particle concentrations as particle count per meter are displayed with a 10-meter integration interval along the pipe axis. For the τ = 0.1 and 0.25 cases, the longitudinal distribution of particles is hardly Gaussian. As τ increases, the particle distribution increasingly approaches the Gaussian distribution, and at τ = 0.75, the analytical prediction of the 1D AD equation based on the effective dispersion coefficient matches the numerical simulation results very well.

Fig. 3.

Particle axial distribution under laminar flow (U = 0.01 m/s, a = 0.02 m, particle number = 50,000, time step = 1 s).

Because the particle distribution can be analyzed at any mixing time in the RWPT simulation, the instantaneous standard deviation of the particle distribution can be evaluated. Both the instantaneous and the averaged effective dispersion coefficient can be calculated, according to Eqs. (7) and (8), respectively. The results are shown in Fig. 4, in which the instantaneous dispersion coefficient curve converges to the asymptotic value predicted by Eq. (2) (0.6667 m²/s). Compared to the numerical simulation results, the formulas from Lee (2004) provide excellent predictions of both the instantaneous and the averaged effective dispersion coefficients in terms of the variance of the particle distribution. However, it should be noted that, for short mixing times, the shape of the longitudinal distribution cannot be described accurately using the 1D ADE solution.

Fig. 4.

Instantaneous and averaged effective dispersion coefficients (time varying effective dispersion coefficients).

Additionally, simulations were performed using the pipe characteristics of a study in Romero-Gomez and Choi (2011). Flow through a PVC pipe with diameter of 15.6 mm and length of 6.5 m was simulated using the average flow velocity of 0.098 m/s, which corresponds to a Reynolds number of 1,500. The tracer was NaCl, which has a molecular diffusivity of $1.20\times {10}^{-9}{\mathrm{m}}^2/\mathrm{s}$. The focus of these simulations was to develop the concentration-time curve at the end of the pipe. The dimensionless mixing time at the end of the pipe is very short: $\tau =0.00131$. It has previously been stated that Eq. (2) does not apply for such short mixing times, and the 1D ADE is not capable of describing the skewness of the solute distribution along the flow direction.

Like the first simulations, 50,000 particles were simulated to be injected at z = 0 and t = 0 with a cross-sectionally uniform distribution. The simulation time step was set to be 1 s. The particle numbers were counted every 0.2 m to calculate the particle concentration.

In Fig. 5, the RWPT simulation results are displayed, together with the analytical solutions based on the instantaneous and averaged dispersion coefficients. According to Eq. (2), the asymptotic dispersion coefficient is 10.14 m²/s. The instantaneous and averaged dispersion coefficients at $\tau =0.00131$ were calculated using Eqs. (5) and (6), and the results are 0.2100 and 0.1054 m²/s, respectively.

Fig. 5.

Concentration-time curve at the end of a 6.5-m-long pipe (U = 0.098 m/s, I = 0.0078 m, L = 6.5 m, particle number = 50,000, time step = 1 s).

A simulation without incorporating the molecular diffusion movement was also conducted to show that, for very short travel times, molecular diffusion does not significantly affect the transport process. The sharp front of the particle cloud cannot be described by a Gaussian impulse response and the 1D AD model is incapable of accurately describing the concentration-time curve at the end of the short pipe. The Gaussian impulse responses with the effective dispersion coefficients tend to underestimate the concentration at the front end and overestimate the concentration at the tail end. According to Taylor (1953) and Houseworth (1984), the concentration-time relationship without molecular diffusion for pipe under laminar flow conditions is

$$C(z,t)=\left\{\begin{array}{ll}\frac{M}{2Ut},& t>0.5L/U\\ 0,& t\le 0.5L/U\end{array}\right.$$ (30)

where L = pipe length. In Fig. 5, the particle concentration curve calculated by Eq. (30) is compared against the RWPT simulation results and the 1D ADE solutions. The RWPT simulation results with and without considering molecular diffusion are basically the same, with general agreement with Eq. (30), further confirming the validity of RWPT and the negligible impact of the molecular diffusion on solute transport for short mixing times. The peak time of the concentration without molecular diffusion is 0.5L/U (33.2 s) and the concentration is zero before that time. The RWPT simulation without molecular diffusion generated a concentration-time curve with the peak value at t = 34 s and zero values before t = 33 s, while the simulation including molecular diffusion predicts peak concentration at t = 35 s and zero concentration before t = 33 s. The peak concentrations predicted by the RWPT simulations are about 7,500 particles per meter, which is close to the theoretical value without molecular diffusion (Taylor 1953): 50,000/L = 7,692 particles per meter.

It can be inferred here that for extremely short mixing times under laminar flow, an analytical solution without considering diffusion is available to predict the concentration-time curve at the end of the pipe. No effective dispersion coefficient is even needed to achieve accurate water quality modeling for flow in pipes that are too short to experience significant molecular diffusion in the radial direction.

As shown in Fig. 6, for a longer pipe (L = 65 m), although the mixing time is still short $(\tau =0.0131)$, the effect of molecular diffusion on solute transport starts to become significant on the particles with residence times more than 1,200 s. But the shape of the impulse response distribution in Fig. 6 is still highly skewed, and the 1D ADE cannot closely approximate the longitudinal distribution of the solute particles.

Fig. 6.

Concentration-time curve at the end of a 65-m-long pipe (U = 0.098 m/s, a = 0.0078 m, L = 65 m, particle number = 50,000, time step = 1 s).

Turbulent Flow

For comparison with one of the studies of Taylor (1954b), a pipe with radius 0.508 m was used in the simulation study for the turbulent flow with Re > 20,000. The average velocity was assumed to be 1.05 m/s and the kinematic viscosity to be $1.10\times {10}^{-6}{\mathrm{m}}^2/\mathrm{s}$. The value of the Re is $9.698\times {10}^5$ and the shear velocity is 0.04018 m/s. Because turbulent diffusion is a significantly faster process than molecular diffusion, a much smaller simulation time step is needed to accurately represent the random movement caused by turbulent diffusion. It was found through trial and error that a time step of 10⁻⁵ s was small enough, and further reduction did not affect the simulation results meaningfully. The total number of particles was again chosen to be 50,000. The impact of time step and particle number is shown in Table S1. Due to the small timestep, the RWPT simulations for turbulent flow conditions were much more computationally expensive than those conducted for laminar flow conditions.

Based on Eq. (3) (Taylor 1954b), the effective dispersion coefficient for this study case was: $D_l=10.1a{u}_{\mathrm{*}}=0.206{\mathrm{m}}^2/\mathrm{s}$. As in the laminar flow studies, the instantaneous dispersion coefficient was calculated according to Eq. (7) and the results are shown in Fig. 7. To demonstrate the impact of the assumed velocity profile, both the universal velocity profile from Taylor (1954b) and the modified log-wake law from Guo and Julien (2003) were implemented in the present study to calculate the turbulent diffusivity. The time series for the instantaneous dispersion coefficient was calculated based on the method of moments, and it asymptotically approaches the value predicted by Eq. (3). An initial period still exists during which the effective dispersion coefficient is much smaller than the asymptotic value, but it is much shorter than the one under laminar flow conditions—about 60 s for this specific case. This mixing time corresponds to a downstream distance of 5.88 m.

Fig. 7.

Effective dispersion coefficient for $\mathrm{R}\mathrm{e}=9.698\times {10}^5$ (U = 1.5 m/s, a = 0.504 m, particle number = 50,000, time step = 10⁻⁵ s).

Fig. 8 presents the RWPT simulation results at time 100 s along the pipe length. The spatial integration interval for the calculation of the particle concentration is 1 meter. The analytical AD solution given U = 1.05 m/s and D_l = 0.206 m²/s is also plotted. The width of particle spread simulated with the modified log-wake law (Guo and Julien 2003) is slightly narrower than when simulated with the universal velocity profile (Taylor 1954b). Both distributions are close to a Gaussian distribution and the 1D analytical solution with the effective dispersion coefficient provided by Eq. (3) generally matches the simulated curve based on the universal velocity profile of Taylor (1954b).

Fig. 8.

Random walk particle tracking results versus prediction using Taylor’s formula for longitudinal particle distribution for $\mathrm{R}\mathrm{e}=$ $9.698\times {10}^5$ (U = 1.5 m/s, a = 0.504 m, particle number = 50,000, time step = 10⁻⁵ s).

Both velocity profiles implemented previously are for highly turbulent flow conditions. For turbulent flow with Re < 20,000, the velocity profile data summarized in Tichacek et al. (1957) were used to calculate the turbulent viscosity of the flow. The pipe radius was chosen to be 0.00476 m (3/8 in.) and the average velocities modeled were 0.3513, 0.4622, 0.6933, and 1.156 m/s, corresponding to Reynolds numbers of 3,040, 4,000, 6,000, and 10,000, respectively. The velocity profiles under these Reynolds numbers are shown in Fig. 2.

Basha and Malaeb (2007) tried to fit the numerical modeling results of Ekambara and Joshi (2003), and reported an empirical equation for the effective dispersion coefficient under turbulent flow:

$$\frac{D_l}{a{u}_{\mathrm{*}}}=\left[10.1+577{\left(\frac{Re}{1000}\right)}^{-2.2}\right]$$ (31)

With experimental data, Hart et al. (2016) also provided a regression-based empirical expression of effective dispersion coefficient for turbulent flow with Re between 3,000 and 50,000. Their expression relates the dispersion coefficient to the pipe diameter and the averaged flow velocity:

$$\frac{D_l}{dU}=1.17\times {10}^{9}R{e}^{-2.5}+0.41$$ (32)

where d = pipe diameter. RWPT simulations with 50,000 particles and 10⁻⁵ s timestep followed by the method of moments found that the effective dispersion coefficients for Re = 3,040, 4,000, 6,000, and 10,000 are 0.00642, 0.00598, 0.00331, and 0.00412 m²/s, respectively. The dimensionless results are shown in Fig. 9 together with the curves described by Eqs. (31) and (32). The RWPT simulations lead to comparable but lower effective dispersion coefficients when compared with the predictions of the two empirical equations, Eqs. (31) and (32). It should be noted that the velocity profile data used here lacked sufficient details and the sensitivities of the RWPT simulation results to the velocity profile data for Re < 20,000 needs further investigation.

Fig. 9.

Comparison of the calibrated effective dispersion coefficients for turbulent flow with Re < 20,000.

Discussion

An advantage of the RWPT method is that it can be easily configured to account for different types of initial conditions and input scenarios. For example, contaminants can be released uniformly or locally from the pipe wall or enter the pipe at the central part of the cross-section. It is not possible to use 1D AD models to simulate the transport of the contaminants under such complex conditions, because the initial concentrations depend not only on the axial position but also on the radial location and polar angle. In the current RWPT method, the particles can be released from any location within the cross-section as an initial condition of the simulation. To illustrate the impact of the initial distribution of concentration on the transport of a soluble matter within a pipe, RWPT simulations were performed for both the cross-sectionally uniform and the wall uniform initial conditions. The difference in exit concentrations between the two initial conditions is significant (Fig. 10).

Fig. 10.

Impact of initial conditions on exit concentration over time (U = 0.098 m/s, a = 0.0078 m, L = 6.5 m, particle number = 50,000, time step = 1 s).

Accurately modeling dispersion is key to modeling concentration within and leaving a system over time, and Fig. 10 highlights the complexity associated with that modeling problem. The wall uniform case (e.g., modeling dissolution from a pipe wall, or removal of wall-bound material) shows a delayed concentration profile exiting the system due to the lower velocities found near the pipe wall and slow radial mixing process. This can happen in both laminar flow with the parabolic velocity profile, and low-Reynolds-number turbulent flow in which the radial nonuniformity of the advection velocity profile is still significant close to the wall. The case of a cross-sectionally uniform source can also be used to highlight the perfect transition phase between two dissimilar materials in an injection. Based on simulations under this flow condition and without considering reactions, it is expected to take more than 600+ s for the concentration to return to zero concentration, or change to a new state (e.g., introduction of a disinfectant) if we consider this the leading edge rather than the trailing edge of a concentration change. Most real-world cases will likely fall somewhere between the two cases being depicted; however, this highlights the inherent complexity of modeling such problems. In practice, if these differences are not accounted for, erroneous interpretations of results can occur.

Conclusions

A stochastic modeling approach, RWPT, was developed to analyze the advection and dispersion transport of water quality solutes in circular pipes and determine the effective dispersion coefficients in 1D ADE. It considers the spatial variability of the radial diffusivity and therefore can be applied to both laminar and turbulent pipe flows. The RWPT simulation results demonstrated that it is an effective way to model advection and dispersion processes within a circular pipe. The simulation results are comparable to what is reported in the literature.

For laminar flow, both the instantaneous and averaged dispersion coefficients calculated based on the variance of the particle longitudinal positions increase with mixing time and approach the asymptotic value predicted by the formula of Taylor (1953).

The simulation showed that, before the asymptotic value is reached the time-varying variance of the longitudinal distribution can be accurately predicted by the formulas of Lee (2004). But, it should be noted that the 1D advection-dispersion model with effective dispersion coefficient is not capable of describing the shape of the longitudinal distribution when the mixing time is short. The presented studies also found that, for very short pipes and mixing times, a simple analytical solution that considers only advection, i.e., Eq. (30), can accurately predict a pipe’s end concentration under uniform initial conditions. It is not necessary to derive effective coefficient formulas and apply a 1D AD model for such cases.

The simulation results under turbulent flow conditions were found to be sensitive to the chosen radial velocity profile. When the universal velocity profile in Taylor (1954b) was implemented, the asymptotic effective dispersion coefficients calculated with the RWPT simulation were very close to the analytical solution of Taylor (1954b). For turbulent flows with Re < 20,000, RWPT simulations confirmed that the formula of Taylor (1954b) underestimates the effective dispersion coefficient. The calculated effective dispersion coefficients from the RWPT simulation are comparable to the ones predicted by the empirical formulas reported by other researchers (Basha and Malaeb 2007; Hart et al. 2016). Due to the sensitivity of the solute particle distribution to the velocity profile, more detailed velocity profile data under turbulent conditions with Re < 20,000 will be helpful to further improve the accuracy of the RWPT method.

Data Availability Statement

Data associated with this work is available from https://catalog.data.gov/dataset/epa-sciencehub. Please contact the corresponding author for any additional model or data needs.