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A Lagrangian stochastic model for particle trajectories in non-Gaussian turbulent flows
FLUIDDYNAMICS RESEARCH
ELSEVIER
Fluid Dynamics Research 19 (1997) 277-288
A Lagrangian stochastic model for particle trajectories in non-Gaussian turbulent flows A.M. Reynolds Silsoe Research Institute, Wrest Park, Silsoe, Bedford MK45 4HS, UK
Received 22 April 1996; revised 11 September 1996; accepted ! October 1996
Abstract
A Lagrangian stochastic model, a generalization of the Langevin equation to inhomogeneous flows, is developed for the simulation of particle trajectories within inhomogeneous turbulent flows. For homogeneous flows the model yields a Gaussian distribution of velocity and is exactly consistent with Thomson's well-mixed condition. For inhomogeneous flows: the model takes explicit account of the non-Gaussian distribution of velocity; Thomson's well-mixed condition is satisfied approximately to second order; whilst third- and higher-order moments of the velocity distribution are determined by the modelled dynamics. The model is shown to predict mean particle concentrations within a threedimensional inhomogeneous turbulent flow (a ventilating air flow) in accord with experimental findings. These predictions are contrasted with those obtained using Thomson's model which exactly satisfies the well-mixed condition for inhomogeneous Gaussian turbulence. The results support the view that third- and higher-order moments of velocity are of secondary importance in determining particle dispersion in highly inhomogeneous turbulent flows when compared to the effects of strong mean-streamline straining and large gradients in Reynolds-stress.
1. Introduction Much progress has been made (see e.g. Sawford, 1988; Pope, 1987; Thomson, 1987) in understanding how to correctly formulate Lagrangian stochastic models (also known as random flight models) of particle trajectories in inhomogeneous high Reynolds number turbulent flows. These models are particularly suited to the prediction of passive scalar dispersion in inhomogeneous turbulent flows, where other techniques (e.g. eddy-diffusivity) are inappropriate or invalid. In principle, they can take account of inhomogeneities, unsteadiness and even non-Gaussian velocity distributions. Thomson showed that random flight models which satisfy the so-called 'well-mixed' condition (i.e. the model gives the correct steady-state distribution of particles in phase-space) are essentially 'correct' in the sense that (i) the Eulerian equations derived from the model are compatible with the true Eulerian equations, (ii) their forward and reverse time formulations of dispersion are 0169-5983/97/$17.00 (~5 1997 The Japan Society of Fluid Mechanics Incorporated and Elsevier Science B.V. All rights reserved. PII S0 1 6 9 - 5 9 8 3 ( 9 6 ) 0 0 0 5 0 - 0
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consistent and (iii) they give the correct small-time behaviour of the velocity distribution of particles from a point source. In practice, however, a number of difficulties arise when formulating random flight models to satisfy the well-mixed condition. First, for three-dimensional flows, random flight models which exactly satisfy the well-mixed condition have only been formulated for inhomogeneous Gaussian turbulence [-Thomson (1987), Borgas (see Sawford and Guest (1988)]. In this case the specified probability distribution function is jointly normal, even though the first two moments of the distribution, the mean velocity and the Reynolds stresses, vary with position. However, such a specification is physically incorrect: it does not occur in the turbulent flow governed by the Navier-Stokes equation. This is because in inhomogeneous flows the third-order moments of velocity, (UiU~Uk), are non-zero and hence turbulence cannot be Gaussian. Constructing random flight models which exactly satisfy the well-mixed condition for other than Gaussian turbulence remains a formidable challenge. To date, random flight models which exactly satisfy the well-mixed condition for non-Gaussian turbulence have only been constructed for particle motion in onedimension (see e.g. Thomson, 1987; Luhar and Britter, 1989). Second, no account is taken of the dependency of the joint probability distribution of velocity on dynamical processes within the flow. That is, there is no dynamical content in such an approach. In this paper, a random flight model for particle trajectories in three-dimensional inhomogeneous turbulent flows is presented which overcomes many of the difficulties associated with random flight models constructed to satisfy the well-mixed condition for a specified (i.e. Gaussian) velocity distribution. For homogeneous flows the model yields a Gaussian velocity distribution and is exactly consistent with Thomson's well-mixed condition. For inhomogeneous flows: the model takes explicit account of the non-Gaussian distribution of velocity; Thomson's well-mixed condition is satisfied approximately to second-order; whilst third- and higher-order moments of the velocity distribution are determined by the modelled dynamics. The random flight model is shown to be successful in predicting mean particle concentrations within a complex inhomogeneous flow within a mechanically ventilated building. Comparisons are made with predictions obtained using Thomson's model which exactly satisfies the well-mixed condition for inhomogeneous Gaussian turbulence. An assessment is made of the importance of non-Gaussian velocity statistics in determining particle dispersion.
2. Model formulation
The random flight model adopted was first proposed by Pope (1983) and later developed by Haworth and Pope (1986, 1987). It is a natural extension of the Langevin equation to inhomogeneous flows. The generalized Langevin equation takes the form dxi = Ui dr,
dUi -
t~2 ( U i )
O ( p ) dt + v - - d t c~xi 0xj 0xj
+ Gij(Uj - ( U j ) ) d t + (Co(e))1/2 dW/.
(1)
Here U(x, t) is the fluid particle velocity, p(x, t) is the fluid pressure divided by the fluid density, v is the kinematic viscosity, Co is the Lagrangian structure constant, e is the average rate of dissipation
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of turbulent kinetic energy divided by the fluid density and the dWi are increments of a vectorvalued Wiener process with independent components. The increments d W/ are Gaussian with mean zero and covariance dt; increments dW,., dWs occurring at different times, or at the same time with i :Aj are independent. The Eulerian quantities of the right-hand side of Eq. (1) are evaluated at the fluid particle location, x(t). In the current application of the random flight model, which is used in conjunction with a previously determined flow field, the angular brackets denote specified time-averaged quantities. This differs from previous applications (Pope, 1983; Haworth and Pope, 1986, 1987) of the model in such the time-averaged quantities were calculated by the ensemble averaging quantities predicted by the model. The first two terms on the right-hand side of Eq. (1) ensure consistency with the time-averaged Navier-Stokes equation,
(DU,)
.
. Dt
.
.
O(p) c3x~
+ v
02(U,) 0x s ~3xs
(2)
The form of last term (the diffusion term) on the right-hand side of Eq. (1) ensures that the Lagrangian structure function is consistent with inertial range scaling. However, because the drift term (involving the tensor Gij.) is linear in Us, not all aspects of the small-time behaviour of particles can be correct (Thomson, 1987; van Dop et al., 1986). In particular, in unsteady or inhomogeneous flows the small-time behaviour of the velocity variance and higher moments will be incorrect. The tensor G~s, which is a modelled function of local mean velocity gradients, O(U~)/c~x~, Reynolds-stresses, -(u~u~), and mean dissipation rate (e), is determined by the requirement that the consistency conditions (i.e. equivalence of calculated and specified moments of velocity) be satisfied. The procedure for achieving this, which closely follows that described by Pope (1987, 1994a), is given below. Corresponding to the generalized Langevin equation, Eq. (1), there is a modelled evolution equation (the Fokker-Planck equation) for the one-point Eulerian probability distribution function, g(v; x, t), of velocity u(x, t), which takes the form -0t+(vp) -
gOXp
6~Xp
#Xq
~l)p
_
63Vp
Gpq
6~Xq
gVq q- ½ C O( 8 ) ~l)p G~l)p'
(3) where r is the sample-space velocity. It is clear that there is no choice for Gq that can yield an arbitrary evolution of the one-point Eulerian joint velocity distribution. Thus, the condition of complete consistency cannot be satisfied. However, it is possible to choose Gis so the generalized Langevin equation yields approximately the specified mean velocity and Reynolds stresses, so that Thomson's well-mixed condition is approximately satisfied to second order. The evolution equation for the Reynolds stresses is obtained from Eq. (3) by multiplying by vivs and integrating over all r
~(uius)o O(uius)c ~(upuiu~)c ~t + (up) aXp + ~Xp + ( U p U j ) c~Xp - - + ( u p u i ) c -~Xp = Gjp(blpUi) c ~- Gip(UpUj) c -~- Co(8)(~ij ,
where the subscript c denotes calculated quantities.
(4)
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With the initial condition (u~u~)~ = (uiuj), the consistency condition is satisfied to second order if (?(u~u;)¢/& = (?(u~uj)/& is satisfied everywhere. Thus, from Eq. (4), a necessary and sufficient condition for the consistency condition to be satisfied to second order is
Gjp(u~u)i + Gi~(u~u)j = Qi~,
(5)
where
Q~J-
a~
+(Up)
+ (upu;)
ox~ +
Ox----~+ (UpUi)
axp Ox~
Co(e)(~ij.
(6)
Therefore, the consistency condition is satisfied to second order if the t e n s o r Gij satisfies Eq. (6). However, this condition removes only six of the nine degrees of freedom in G~j (Pope, 1987). In principle, this non-uniqueness could be exploited to refine the model by improving the degree of consistency at higher orders. This is not attempted here; instead the simplest choice for G~j, namely Gij =-½Qik(UkUj)-1 is adopted, where --(UiUk) -1 is the inverse of the Reynolds-stress tensor
-(u~uk). A corresponding non-uniqueness arises when random flight models are constructed to satisfy the well-mixed condition for a specified Eulerian probability distribution for velocity. There the non-uniqueness is non-trivial as Sawford and Guest (1988) showed that two different random flight models (due to Thomson and Borgas), both satisfying the well-mixed condition for inhomogeneous Gaussian turbulence, give slightly different predictions for particle dispersion. For Gaussian turbulence, the Thomson model is favoured simply because it is the simplest random flight model satisfying the well-mixed condition. For homogeneous turbulence, the generalized Langevin equation, with an appropriate choice for G~i, is completely consistent. That is, it exactly satisfies the well-mixed condition for Gaussian turbulence. This is because a Gaussian distribution is uniquely determined by its first- and second-order moments, and because for homogeneous turbulence the generalized Langevin equation yields a Gaussian distribution (Pope, 1994b). For inhomogeneous turbulence, it is not clear how the tensor Gij can be chosen so that the consistency condition, Eq. (5), is exactly satisfied. This is because Q~ is dependent upon the calculated third-order moments of velocity, (upu~uj)c, whose evolution, determined from Eq. (3), is dependent itself upon Gij and fourth-order velocity correlations (uqupu~uj)c, which in turn are dependent upon fifth-order velocity correlations. Indeed, except for the case of Gaussian turbulence, exactly satisfying the consistency condition to second order requires exactly satisfying the consistency condition to all orders. However, a tensor, Gzj, which approximately satisfies the consistency condition, Eq. (5), can be found by replacing calculated with specified moments of velocity, at some order, thereby truncating the hierarchy of equations governing the evolution of moments of velocity. In this paper two such truncations, discussed below, are considered. The simplest truncation is obtained by approximating the term in Eq. (5) involving the calculated gradients in the third-order moments of velocity, c~(upu~u~)¢/OXp by a term involving specified gradients in the third-order moments of velocity, O(upuiu~)s/~x r. These specified
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quantities are taken to be prescribed by the generalized gradient approximation (Daly and Harlow, 1970),
Oupu,uj)s Oxp -
Cs
(!
(7)
This approximation is appropriate because both calculated and specified quantities are expected to be physically reasonable. Calculated third-order moments in velocity are expected to be physically reasonable because the Lagrangian stochastic model, Eq. (1), from which they are derived, is consistent with the Navier-Stokes equations. Indeed, Pope (1994a) showed that, for appropriate choices of the tensor G~; (corresponding to realistic predictions for the flow field), the calculated third-order moments are identical to those proposed on physical grounds by Launder et al. (1975). The generalized gradient approximation is expected to be physically reasonable approximation because, second-order moment closures the Reynolds-differential models which utilize it, have been successful in predicting the statistical properties of many inhomogeneous turbulent flows. An improved approximation for the tensor G~j is obtained by calculating third-order moments of velocity, ~blpUilAj)c, from an approximate form of their governing evolution equation. The evolution equation for third-order moments of velocity, (upuiuj)s, obtained from Eq. (3) by multiplying by VpViV; and integrating over all v is 0
0
0
+ ,(u uj)c + (8)
An approximate solution to this equation, for the calculated third-order moments of velocity,
(u,u~uj)¢, is obtained by approximating the terms involving calculated fourth-order velocity correlations, (U, UpU~Uj)~, by terms involving specified quantities, (U, UpUiUj)s. Here the specified quantities are taken to be prescribed by the Millionshchikov approximation,
(u, upuiu;)~ = (u~up)~(uiuj)~ + (u, ui)~(UpUj)~ + (u, uj)~(UpUi)~,
(9)
which has been widely adopted in statistical mechanics. In Section 5 an assessment is made of the dependency of predicted mean particle concentrations on the level of truncation of the hierarchy of evolution equations for moments of the velocity distribution.
3. Experimental facility Mean particle concentrations were measured in a full-scale section of the building, designed to study air flow patterns under controlled climatic conditions. The facility is a cross-section of
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a typical U K intensive livestock building of one bay, 3 m wide. The dimensions and geometry of the building are shown in Fig. 1. The section is surrounded by an outer temperature-controlled shell allowing precise ambient conditions to be established. The temperature of the inlet air and the air surrounding the building section were set at 20 °C. The room is ventilated by a jet ventilation system with air entering horizontally in each direction from an adjustable baffled ceiling inlet. The jet m o m e n t u m number was taken to be 0.0076 and the inlet Reynolds number 5 x 104. Air is exhausted through two fans located in the side walls of the building. The total ventilation rate was 1.22 m 3 s- 1, which is equivalent to 44 air changes per hour. Dust was generated using a TSI fluidized-bed aerosol generator. The dust material was Arizona Road dust, with density p = 2650 k g m -3 and with 99% of particles having diameters between 0 and 2.5 x ]0 -6 m. Mean concentrations of particles with diameters between 1 and 2 × 10 -6 m were measured on a vertical plane through the building using a R I O N optical particle counter. The
Air outlet
Y x Air
inlet 3.34 m
2.60 m
Y
Jr"
4-
4-
-4-~ ' " ~ - . ~
2.O5m
+
+
+
+
+
+
+
1.5Om
+
+
+
+
+
+
+
++
o,g5 rn
4-
4-
+
4-
4-
+
+
+
%-
0.4o m
4-SI+
4-
+
4-
+
4-
+
+ $2
0.14. m
4- 1140.00 m
-I"F ,.3.75 m
-{" 5.00 m
-It-F 1.25 m
4-}2.50 m
Z
y
L
Air
outlet
1.85 m
©
0,96 m
~-I / 5.88 m
0,0 m
0.0 rn
J 2.22 m
3.09 m
X
Fig. 1. Dimensions and geometry of the building section. Location of the sample points (+). The y - z plane containing the sampling points lies at x = 1.545 m. In the coordinate shown, the locations, S1 and $2, of the dust generator are (0.5, 0.2, 0.5 m) and (0.5, 0.2, 5.38 m). The locations of S1 and $2 are indicated by solid circles (O).
A.M. Reynolds/Fluid Dynamics Research 19 (1997) 277-288
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Source S 1
i Source $2
Experimental
Thomson (Gaussian) Model
Non-Goussian Model (Second-order closure)
Fig. 2. Measured and predicted particle concentrations within the building section for sources S1 and $2. Particle concentrations have been normalized by the maximum particle concentrations. Predictions are shown for Co = 5 and Cs = 0.2. Calculated gradients in third-order moments of velocity were approximated by specified quantities, prescribed by the generalized gradient approximation.
measured mean particle concentrations corresponding to two locations, S1 and $2 (see Fig. 1), of the dust generator are shown in Fig. 2. A more complete description of the experimental facility and the measurements of mean particle concentrations can be found in Boon and Andersen (1995).
4. Numerical implementation In predicting particle dispersion it is assumed that the presence of the particle did not affect the statistical properties of the air flow. The statistical properties of the air flow could then be predicted independently of the particles. The statistical properties of the air flow,~, ~and
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Eqs. (5), (6), (8) and (9) were solved numerically, using an iterative process, for the third-order moments of velocity, (upuiuj), and the tensor, Gij. The iterative process was taken to be converged when the whole field residuals, corresponding to each component of the third-order moments velocity, (upuiu~), were all less than 0.1. The size of the time-step, At, used in the numerical integration of the random flight model, Eq. (1), should be chosen to be smaller than the Lagrangian time-scale and be sufficiently smaller so that inhomogeneities in the flow field are resolved. This cannot be done appropriately. Instead, the sensitivity of predicted mean particle concentrations to the size of the time-step, At, was investigated in numerical experiments. All predicted mean particle concentrations presented in this paper are insensitive to a reduction in the size of the time-step, At, that was adopted. Turbulent fluctuations in velocity, u, were updated at the end of each time-step, At. During each time-step, At, the numerical integration of the random flight model, Eq. (1), was performed using a fourth-order Runge-Kutta method. Particles are treated as passive scalars until they are transported toward a surface within one 'stopping distance' of a surface, characterized by the particles relaxation time and the wallward velocity (Friedlander and Johnstone, 1957). At this point, particles are assumed to deviate from the local turbulent fluid motion, arriving at the surface through 'free-flight' by virtue of their inertia. Free-flight deposition is widely believed (see eg. Brooke et al., 1994 and references within) to be the main contributor to particle deposition. The other mechanisms of particle deposition include Brown diffusion and turbulent diffusion. In this paper, free-flight deposition is taken to be the only mechanism of particle deposition and, once deposited, particles are assumed not to be re-entrained. Nodes in the computational grid, used in predicting the statistical properties of the air flow, formed 50 parallel x y planes, each containing 45 x 48 cells. Piecewise linear interpolation was used to interpolate, in the z-direction, the predicted statistical properties of the air flow from the nodes of the computational grid, surrounding the particle, to the particle location. Nodes in the x-y planes did not form an orthogonal system. The Rinka and Cline method (1984) was used to interpolate, in the x- and y-directions, the predicted values of the statistical properties of the air flow from the nodes of the computation grid, surrounding the particle, to the particle location. Ensemble mean particle concentration fields were calculated from the simulated independent trajectories of 25 000 particles. All predicted mean particle concentrations fields presented in this paper are statistically stationary; that is independent of increasing the number of particles within the ensemble. Multiple contributions to the ensemble averages due to particles revisiting points in the flow domain were removed. This enables direct comparisons between predicted mean particle concentrations and measured particle concentrations, obtained by extracting particles at given points within the flow domain, to be made. In the experiments, a single-particle detector was used. The detector being moved to different locations. Consequently, it is appropriate, in the simulations, to prevent particles from making contributions to the predicted mean particle concentration field at more than one location of the detector.
5. Predicted particle dispersion Before the random flight model can be implemented, appropriate values for the Lagrangian structure constant, Co, and the skewness parameter, C~, must be determined.
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Sawford (1991) showed for that random flight models which assume that the collective evolution of (x, u) is Markovian, e.g. the non-Gaussian random flight model, Eq. (1), and Thomson's (Gaussian) model, the supposedly universal Lagrangian structure constant, Co, is not universal but rather depends upon Reynolds number. He subsequently attributed the variability in the 'best' estimates for Co obtained using such models to variations in the Reynolds number across the various dispersion experiments. The dependency of predicted mean particle concentrations on the value of Co has been investigated in numerical studies. The 'best' correspondence between measured and predicted particle concentrations was obtained with Co in the range 5 ~< Co ~< 10. This range of values is in accord with experimental (Hanna, 1991; Sawford and Guest, 1988), numerical (Reynolds, 1995) and theoretical estimates (Sawford, 1991). Predicted mean particle concentrations obtained using Co = 5 and 10 and the generalized gradient approximation with Cs = 0.2 for gradients in the third-order moments of velocity, ~uruiuj)/Oxp, are shown in Figs. 2 and 3. Shown for comparison, in Figs. 2 and 3, are the measured mean particle concentrations and predictions obtained using Thomson's (Gaussian) model (Reynolds, 1996). Intriguingly, the dependency of particle dispersion on the value of Co predicted by the non-Gaussian random flight model, Eq. (1), is seen to be somewhat less than that predicted by Thomson's (Gaussian) model. The dependence of predicted mean particle concentrations on the value of the skewness parameter, Cs, has also been investigated in numerical studies. The 'best' correspondence between measured and predicted mean particle concentrations was obtained with Cs in the range 0.2 ~< Cs ~< 0.3. This is in accord with the range of values for Cs, 0.15 ~< C, ~< 0.25, which have been adopted in second-moment closure models of the Reynolds-stress equations (see e.g. Launder, 1992).
Source S 1
J Source $2 ~°~
Experimental
Thomson (Oaussiun) Model
~
°3
Non-Gaussion Model
(Second-order closure) Fig. 3. Measured and predicted particle concentrations within the building section for sources S I and $2. Particle concentrations have been normalized by the maximum particle concentrations. Predictions are shown for Co = 10 and Cs = 0.2. Calculated gradients in third-order moments of velocity were approximated by specified quantities, prescribed by the generalized gradient approximation.
286
A.M. Reynolds/Fluid Dynamics Research 19 (1997) 277-288 n
Source S 1 / f ' - - ' ~ - ~ . ° ~
Source $2
Non-Oaussian Model (Third-order closure) Fig. 4. Measured and predicted particle concentrations within the building section for sources SI and $2. Particle concentrations have been normalized by the maxiumum particle concentrations. Predictions are shown for Co = 5. Third-order moments of velocity were calculated from an approximation form of the governing evolution equation in which terms involving calculated fourth-order moments of velocity, {ur upuiu~)c, were approximated by terms involving specified fourth-order moments of velocity, {urupuiuj)~, prescribed by the Millionshchikov approximation.
Predicted mean particle concentrations obtained using third-order moments of velocity, calculated from the approximate evolution equation, Eqs. (8) and (9), are shown in Fig. 4. A comparison of Figs. 2 and 4 shows that predicted mean particle concentrations obtained using the two truncation schemes are not to be significantly different. This suggests that the violation of the thermodynamic constraint resulting the approximate closure schemes does not seriously invalidate predictions for mean particle concentrations. And that, when predicting mean particle concentrations, at least for the highly inhomogeneous flow under consideration here, it is sufficient to approximate calculated gradients in the third-order moments of velocity, O(upu~uj)/~?xp, by specified quantities prescribed by the generalized gradient approximation. Thereby truncating the hierarchy of equations governing the evolution of moments of velocity at second order. Thomson's (Gaussian) model is seen to have remarkable success in predicting mean particle concentrations. The shape of contours of constant particle concentration and the location of the maximum particle concentration for source S1 are in accord with experimental findings. Less well predicted, however, are gradients in mean particle concentration and the location of the maximum particle concentration for source $2. The success of the Thomson model has been attributed (Reynolds, 1996) to the dominance of mean-streamline streaming and gradients in Reynolds stresses in determining particle dispersion within the highly inhomogeneous flow. The non-Gaussian model is found to be leading to some improvements in predictions for several aspects of particle dispersion. In particular, the positions of maximum particle concentrations, for both source S1 and $2, are seen to be better predicted. This is especially the case for Co = 5. Furthermore, the proportion of particles from source S1 which are exhausted out of the building through the fan on the left-hand side of the building section is better predicted by the non-Gaussian
(uruiu j),
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model. The experimental estimate for this quantity is 44%. The predicted value of this quantity obtained using Thomson's (Gaussian) model is between 15% (Co = 5) and 11% (Co = 10) whilst that obtained using the non-Gaussian model is between 27% (Co = 5) and 25% (Co = 10). However, both Thomson's (Gaussian) model and non-Gaussian model significantly underpredict this quantity. Both models also significantly overpredict gradients in mean particle cncentrations. The inability of the random flight models to predict correctly these features of particle dispersion can be attributed, at least in part, to the neglect of unsteadiness of the air flow. The air flow was predicted by assuming stationarity. However, experimental observations with a sonic anemometer suggest that there are slow (period about 28 s) large-scale fluctuations in air velocity within the building section. This unsteadiness of the air flow is expected to enhance the proportion of particles which can cross the buildings plane of symmetry to be eventually exhausted out of the building through the left-hand fan. This is because the predicted residence time for a particle from source S1 (113 s for Thomson's model) is much longer than the period of oscillation of the air flow. When stationarity is assumed then only turbulent fluctuations in the air flow can cause a particle to cross the buildings plane of symmetry. The inclusion of unsteadiness, by enhancing particle dispersion, is also expected to reduce the predicted gradients in mean particle concentrations.
6. Discussion For quasi-homogeneous flows, e.g. in convective boundary-layers, third-order moments of velocity have an important effect upon dispersion and the inclusion of these moments in particle trajectory models have found to lead to improved predictions for mean particle concentrations (Thomson, 1987; Luhar and Britter, 1988). In contrast, the remarkable success of Thomson's model in predicting particle dispersion within strongly inhomogeneous turbulent flows (see e.g. Flesch and Wilson, 1991; Reynolds, 1995, 1996) suggests that third- and indeed higher-order moments of velocity are of secondary importance in determining particle dispersion when compared with the effects of strong mean-streamline straining and strong gradients in Reynolds stress. In this paper, further support for the view has been provided by numerical studies of particle dispersion in a strongly inhomogeneous turbulent flow using a random flight model which takes explicit account of the non-Gaussian distribution of velocity arising in such flows. Predicted mean particle concentrations obtained using this model were shown to be only marginally better than those obtained using Thomson's model. However, the results tentatively suggest that this random flight model does provide a better description of particle dispersion in complex inhomogeneous flows than that given by Thomson's model.
Acknowledgements The author would like to thank M. Andersen and C.R. Boon for making available their unpublished findings and B.B. Harral for making available his predictions for properties of the air flow within the building section. I am indebted to an anonymous referee for clarifying the conditions under which the thermodynamic constraint is satisfied.
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References Boon, C.R., M. Andersen, B.B. Harral and A.M. Reynolds (1994) Dynamics of particulate pollutants in an experimental livestock building, in: Proc. ASAE Meeting, Dec. 1994, Paper No. 94-4586. Brooke, J.W., T.J. Hanratty and J.B. McLaughlin (1994) Free-flight mixing and deposition of aerosols, Phys. Fluids 6, 3404-3415. Daly, B.J. and F.J. Harlow (1970) Transport equations in turbulence, Phys. Fluids 13, 2634-2649. van Dop, H., F.T.M. Nieuwstadt and J.C.R. Hunt (1985) Random walk models for particle displacements in inhomogeneous unsteady turbulent flows, Phys. Fluids 26, 1639 1653. Flesch, T.K. and J.D. Wilson (1992) A two-dimensional trajectory-simulation model for non-Gaussian, inhomogeneous turbulence within plant-canopies, Bound. Layer Meteorol. 61,349 374. Friedlander, S.K. and H.F. Johnstone, Deposition of suspended particles from turbulent gas streams, Ind. Eng. Chem. 49, 1151-1156. Hanna, S.R. (1991) Lagrangian and Eulerian time-scale relations in the day-time boundary layer, J. Appl. Metorol. 20, 242-249. Harral, B.B. and C.R. Boon (1996). Comparison of predicted and measured air flow patterns within a mechanically ventilated livestock building, J. Agr. Eng. Res., in press. Haworth, D.C. and S.B. Pope (1986) A generalized Langevin model for turbulent flows, Phys. Fluids 29, 387-405. Haworth, D.C. and S.B. Pope (1987) A pdf modelling study of self-similar turbulent free shear flow, Phys. Fluids 30, 1026-44. Launder, B.E., G.J. Reece and W. Rodi (1975) Progress in the development of a Reynolds-stress turbulence closure, J. Fluid Mech. 68, 537. Launder, B.E. (1992) Turbulence Modelling for CFD Applications (UMIST, Manchester). Luhar, A.K. and R.E. Britter (1989) A random walk model for dispersion in inhomogeneous turbulence in a convective boundary layer, Atmos. Environ. 23, 1911-1924. Pope, S.B. (1983) A Lagrangian two-time probability density function equation for inhomogeneous turbulent flows, Phys. Fluids 26, 3448 50. Pope, S.B. (1987) Consistency conditions for random-walk models of turbulent dispersion, Phys. Fluids 30, 2374-2379. Pope, S.B. (1994a) On the relationship between stochastic Lagrangian models of turbulence and second moment closures, Phys. Fluids 6, 973 985. Pope, S.B. (1994b) Lagrangian pdf methods for turbulent flows, Ann. Rev. Fluid Mech. 26, 23 63. Reynolds, A.M. (1995) Modelling particle dispersion within a ventilated airspace, Int. J. Multiphase Flows (submitted) preprint. Reynolds, A.M. (1996) On the application of Thomson's model to the prediction of particle dispersion within a ventilated airspace, J. Wind Eng. Ind. Aerodyn. (submitted) preprint. Rinka, R.L. and A.K. Cline (1984) A triangle-based C 1 interpolation method, Rocky-Mountain J. Maths 14, 223-237. Sawford, B.L. and F.M. Guest (1988) Uniqueness and universality of Lagrangian stochastic models for turbulent dispersion, Proc. 8th Syrup. Turbulent Diffusion AMS, San Diego, CA, pp. 96-99. Sawford, B.L. (199 l) Reynolds number effects in Lagrangian stochastic models of particle trajectories in turbulent flows, Phys. Fluids A 3, 1577-1586. Thomson, D.J. (1987) Criteria for the selection Of stochastic models of particle trajectories in turbulent flows, J. Fluid Mech. 180, 529-556. Thomson, D.J. and M.R. Montgomery (1994) Reflection boundary conditions for random walk models of dispersion in non-Gaussian turbulence, Atmos. Environ. 28, 1981-1987.
ELSEVIER
Fluid Dynamics Research 19 (1997) 277-288
A Lagrangian stochastic model for particle trajectories in non-Gaussian turbulent flows A.M. Reynolds Silsoe Research Institute, Wrest Park, Silsoe, Bedford MK45 4HS, UK
Received 22 April 1996; revised 11 September 1996; accepted ! October 1996
Abstract
A Lagrangian stochastic model, a generalization of the Langevin equation to inhomogeneous flows, is developed for the simulation of particle trajectories within inhomogeneous turbulent flows. For homogeneous flows the model yields a Gaussian distribution of velocity and is exactly consistent with Thomson's well-mixed condition. For inhomogeneous flows: the model takes explicit account of the non-Gaussian distribution of velocity; Thomson's well-mixed condition is satisfied approximately to second order; whilst third- and higher-order moments of the velocity distribution are determined by the modelled dynamics. The model is shown to predict mean particle concentrations within a threedimensional inhomogeneous turbulent flow (a ventilating air flow) in accord with experimental findings. These predictions are contrasted with those obtained using Thomson's model which exactly satisfies the well-mixed condition for inhomogeneous Gaussian turbulence. The results support the view that third- and higher-order moments of velocity are of secondary importance in determining particle dispersion in highly inhomogeneous turbulent flows when compared to the effects of strong mean-streamline straining and large gradients in Reynolds-stress.
1. Introduction Much progress has been made (see e.g. Sawford, 1988; Pope, 1987; Thomson, 1987) in understanding how to correctly formulate Lagrangian stochastic models (also known as random flight models) of particle trajectories in inhomogeneous high Reynolds number turbulent flows. These models are particularly suited to the prediction of passive scalar dispersion in inhomogeneous turbulent flows, where other techniques (e.g. eddy-diffusivity) are inappropriate or invalid. In principle, they can take account of inhomogeneities, unsteadiness and even non-Gaussian velocity distributions. Thomson showed that random flight models which satisfy the so-called 'well-mixed' condition (i.e. the model gives the correct steady-state distribution of particles in phase-space) are essentially 'correct' in the sense that (i) the Eulerian equations derived from the model are compatible with the true Eulerian equations, (ii) their forward and reverse time formulations of dispersion are 0169-5983/97/$17.00 (~5 1997 The Japan Society of Fluid Mechanics Incorporated and Elsevier Science B.V. All rights reserved. PII S0 1 6 9 - 5 9 8 3 ( 9 6 ) 0 0 0 5 0 - 0
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consistent and (iii) they give the correct small-time behaviour of the velocity distribution of particles from a point source. In practice, however, a number of difficulties arise when formulating random flight models to satisfy the well-mixed condition. First, for three-dimensional flows, random flight models which exactly satisfy the well-mixed condition have only been formulated for inhomogeneous Gaussian turbulence [-Thomson (1987), Borgas (see Sawford and Guest (1988)]. In this case the specified probability distribution function is jointly normal, even though the first two moments of the distribution, the mean velocity and the Reynolds stresses, vary with position. However, such a specification is physically incorrect: it does not occur in the turbulent flow governed by the Navier-Stokes equation. This is because in inhomogeneous flows the third-order moments of velocity, (UiU~Uk), are non-zero and hence turbulence cannot be Gaussian. Constructing random flight models which exactly satisfy the well-mixed condition for other than Gaussian turbulence remains a formidable challenge. To date, random flight models which exactly satisfy the well-mixed condition for non-Gaussian turbulence have only been constructed for particle motion in onedimension (see e.g. Thomson, 1987; Luhar and Britter, 1989). Second, no account is taken of the dependency of the joint probability distribution of velocity on dynamical processes within the flow. That is, there is no dynamical content in such an approach. In this paper, a random flight model for particle trajectories in three-dimensional inhomogeneous turbulent flows is presented which overcomes many of the difficulties associated with random flight models constructed to satisfy the well-mixed condition for a specified (i.e. Gaussian) velocity distribution. For homogeneous flows the model yields a Gaussian velocity distribution and is exactly consistent with Thomson's well-mixed condition. For inhomogeneous flows: the model takes explicit account of the non-Gaussian distribution of velocity; Thomson's well-mixed condition is satisfied approximately to second-order; whilst third- and higher-order moments of the velocity distribution are determined by the modelled dynamics. The random flight model is shown to be successful in predicting mean particle concentrations within a complex inhomogeneous flow within a mechanically ventilated building. Comparisons are made with predictions obtained using Thomson's model which exactly satisfies the well-mixed condition for inhomogeneous Gaussian turbulence. An assessment is made of the importance of non-Gaussian velocity statistics in determining particle dispersion.
2. Model formulation
The random flight model adopted was first proposed by Pope (1983) and later developed by Haworth and Pope (1986, 1987). It is a natural extension of the Langevin equation to inhomogeneous flows. The generalized Langevin equation takes the form dxi = Ui dr,
dUi -
t~2 ( U i )
O ( p ) dt + v - - d t c~xi 0xj 0xj
+ Gij(Uj - ( U j ) ) d t + (Co(e))1/2 dW/.
(1)
Here U(x, t) is the fluid particle velocity, p(x, t) is the fluid pressure divided by the fluid density, v is the kinematic viscosity, Co is the Lagrangian structure constant, e is the average rate of dissipation
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of turbulent kinetic energy divided by the fluid density and the dWi are increments of a vectorvalued Wiener process with independent components. The increments d W/ are Gaussian with mean zero and covariance dt; increments dW,., dWs occurring at different times, or at the same time with i :Aj are independent. The Eulerian quantities of the right-hand side of Eq. (1) are evaluated at the fluid particle location, x(t). In the current application of the random flight model, which is used in conjunction with a previously determined flow field, the angular brackets denote specified time-averaged quantities. This differs from previous applications (Pope, 1983; Haworth and Pope, 1986, 1987) of the model in such the time-averaged quantities were calculated by the ensemble averaging quantities predicted by the model. The first two terms on the right-hand side of Eq. (1) ensure consistency with the time-averaged Navier-Stokes equation,
(DU,)
.
. Dt
.
.
O(p) c3x~
+ v
02(U,) 0x s ~3xs
(2)
The form of last term (the diffusion term) on the right-hand side of Eq. (1) ensures that the Lagrangian structure function is consistent with inertial range scaling. However, because the drift term (involving the tensor Gij.) is linear in Us, not all aspects of the small-time behaviour of particles can be correct (Thomson, 1987; van Dop et al., 1986). In particular, in unsteady or inhomogeneous flows the small-time behaviour of the velocity variance and higher moments will be incorrect. The tensor G~s, which is a modelled function of local mean velocity gradients, O(U~)/c~x~, Reynolds-stresses, -(u~u~), and mean dissipation rate (e), is determined by the requirement that the consistency conditions (i.e. equivalence of calculated and specified moments of velocity) be satisfied. The procedure for achieving this, which closely follows that described by Pope (1987, 1994a), is given below. Corresponding to the generalized Langevin equation, Eq. (1), there is a modelled evolution equation (the Fokker-Planck equation) for the one-point Eulerian probability distribution function, g(v; x, t), of velocity u(x, t), which takes the form -0t+(vp) -
gOXp
6~Xp
#Xq
~l)p
_
63Vp
Gpq
6~Xq
gVq q- ½ C O( 8 ) ~l)p G~l)p'
(3) where r is the sample-space velocity. It is clear that there is no choice for Gq that can yield an arbitrary evolution of the one-point Eulerian joint velocity distribution. Thus, the condition of complete consistency cannot be satisfied. However, it is possible to choose Gis so the generalized Langevin equation yields approximately the specified mean velocity and Reynolds stresses, so that Thomson's well-mixed condition is approximately satisfied to second order. The evolution equation for the Reynolds stresses is obtained from Eq. (3) by multiplying by vivs and integrating over all r
~(uius)o O(uius)c ~(upuiu~)c ~t + (up) aXp + ~Xp + ( U p U j ) c~Xp - - + ( u p u i ) c -~Xp = Gjp(blpUi) c ~- Gip(UpUj) c -~- Co(8)(~ij ,
where the subscript c denotes calculated quantities.
(4)
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With the initial condition (u~u~)~ = (uiuj), the consistency condition is satisfied to second order if (?(u~u;)¢/& = (?(u~uj)/& is satisfied everywhere. Thus, from Eq. (4), a necessary and sufficient condition for the consistency condition to be satisfied to second order is
Gjp(u~u)i + Gi~(u~u)j = Qi~,
(5)
where
Q~J-
a~
+(Up)
+ (upu;)
ox~ +
Ox----~+ (UpUi)
axp Ox~
Co(e)(~ij.
(6)
Therefore, the consistency condition is satisfied to second order if the t e n s o r Gij satisfies Eq. (6). However, this condition removes only six of the nine degrees of freedom in G~j (Pope, 1987). In principle, this non-uniqueness could be exploited to refine the model by improving the degree of consistency at higher orders. This is not attempted here; instead the simplest choice for G~j, namely Gij =-½Qik(UkUj)-1 is adopted, where --(UiUk) -1 is the inverse of the Reynolds-stress tensor
-(u~uk). A corresponding non-uniqueness arises when random flight models are constructed to satisfy the well-mixed condition for a specified Eulerian probability distribution for velocity. There the non-uniqueness is non-trivial as Sawford and Guest (1988) showed that two different random flight models (due to Thomson and Borgas), both satisfying the well-mixed condition for inhomogeneous Gaussian turbulence, give slightly different predictions for particle dispersion. For Gaussian turbulence, the Thomson model is favoured simply because it is the simplest random flight model satisfying the well-mixed condition. For homogeneous turbulence, the generalized Langevin equation, with an appropriate choice for G~i, is completely consistent. That is, it exactly satisfies the well-mixed condition for Gaussian turbulence. This is because a Gaussian distribution is uniquely determined by its first- and second-order moments, and because for homogeneous turbulence the generalized Langevin equation yields a Gaussian distribution (Pope, 1994b). For inhomogeneous turbulence, it is not clear how the tensor Gij can be chosen so that the consistency condition, Eq. (5), is exactly satisfied. This is because Q~ is dependent upon the calculated third-order moments of velocity, (upu~uj)c, whose evolution, determined from Eq. (3), is dependent itself upon Gij and fourth-order velocity correlations (uqupu~uj)c, which in turn are dependent upon fifth-order velocity correlations. Indeed, except for the case of Gaussian turbulence, exactly satisfying the consistency condition to second order requires exactly satisfying the consistency condition to all orders. However, a tensor, Gzj, which approximately satisfies the consistency condition, Eq. (5), can be found by replacing calculated with specified moments of velocity, at some order, thereby truncating the hierarchy of equations governing the evolution of moments of velocity. In this paper two such truncations, discussed below, are considered. The simplest truncation is obtained by approximating the term in Eq. (5) involving the calculated gradients in the third-order moments of velocity, c~(upu~u~)¢/OXp by a term involving specified gradients in the third-order moments of velocity, O(upuiu~)s/~x r. These specified
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quantities are taken to be prescribed by the generalized gradient approximation (Daly and Harlow, 1970),
Oupu,uj)s Oxp -
Cs
(!
(7)
This approximation is appropriate because both calculated and specified quantities are expected to be physically reasonable. Calculated third-order moments in velocity are expected to be physically reasonable because the Lagrangian stochastic model, Eq. (1), from which they are derived, is consistent with the Navier-Stokes equations. Indeed, Pope (1994a) showed that, for appropriate choices of the tensor G~; (corresponding to realistic predictions for the flow field), the calculated third-order moments are identical to those proposed on physical grounds by Launder et al. (1975). The generalized gradient approximation is expected to be physically reasonable approximation because, second-order moment closures the Reynolds-differential models which utilize it, have been successful in predicting the statistical properties of many inhomogeneous turbulent flows. An improved approximation for the tensor G~j is obtained by calculating third-order moments of velocity, ~blpUilAj)c, from an approximate form of their governing evolution equation. The evolution equation for third-order moments of velocity, (upuiuj)s, obtained from Eq. (3) by multiplying by VpViV; and integrating over all v is 0
0
0
+ ,(u uj)c + (8)
An approximate solution to this equation, for the calculated third-order moments of velocity,
(u,u~uj)¢, is obtained by approximating the terms involving calculated fourth-order velocity correlations, (U, UpU~Uj)~, by terms involving specified quantities, (U, UpUiUj)s. Here the specified quantities are taken to be prescribed by the Millionshchikov approximation,
(u, upuiu;)~ = (u~up)~(uiuj)~ + (u, ui)~(UpUj)~ + (u, uj)~(UpUi)~,
(9)
which has been widely adopted in statistical mechanics. In Section 5 an assessment is made of the dependency of predicted mean particle concentrations on the level of truncation of the hierarchy of evolution equations for moments of the velocity distribution.
3. Experimental facility Mean particle concentrations were measured in a full-scale section of the building, designed to study air flow patterns under controlled climatic conditions. The facility is a cross-section of
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a typical U K intensive livestock building of one bay, 3 m wide. The dimensions and geometry of the building are shown in Fig. 1. The section is surrounded by an outer temperature-controlled shell allowing precise ambient conditions to be established. The temperature of the inlet air and the air surrounding the building section were set at 20 °C. The room is ventilated by a jet ventilation system with air entering horizontally in each direction from an adjustable baffled ceiling inlet. The jet m o m e n t u m number was taken to be 0.0076 and the inlet Reynolds number 5 x 104. Air is exhausted through two fans located in the side walls of the building. The total ventilation rate was 1.22 m 3 s- 1, which is equivalent to 44 air changes per hour. Dust was generated using a TSI fluidized-bed aerosol generator. The dust material was Arizona Road dust, with density p = 2650 k g m -3 and with 99% of particles having diameters between 0 and 2.5 x ]0 -6 m. Mean concentrations of particles with diameters between 1 and 2 × 10 -6 m were measured on a vertical plane through the building using a R I O N optical particle counter. The
Air outlet
Y x Air
inlet 3.34 m
2.60 m
Y
Jr"
4-
4-
-4-~ ' " ~ - . ~
2.O5m
+
+
+
+
+
+
+
1.5Om
+
+
+
+
+
+
+
++
o,g5 rn
4-
4-
+
4-
4-
+
+
+
%-
0.4o m
4-SI+
4-
+
4-
+
4-
+
+ $2
0.14. m
4- 1140.00 m
-I"F ,.3.75 m
-{" 5.00 m
-It-F 1.25 m
4-}2.50 m
Z
y
L
Air
outlet
1.85 m
©
0,96 m
~-I / 5.88 m
0,0 m
0.0 rn
J 2.22 m
3.09 m
X
Fig. 1. Dimensions and geometry of the building section. Location of the sample points (+). The y - z plane containing the sampling points lies at x = 1.545 m. In the coordinate shown, the locations, S1 and $2, of the dust generator are (0.5, 0.2, 0.5 m) and (0.5, 0.2, 5.38 m). The locations of S1 and $2 are indicated by solid circles (O).
A.M. Reynolds/Fluid Dynamics Research 19 (1997) 277-288
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Source S 1
i Source $2
Experimental
Thomson (Gaussian) Model
Non-Goussian Model (Second-order closure)
Fig. 2. Measured and predicted particle concentrations within the building section for sources S1 and $2. Particle concentrations have been normalized by the maximum particle concentrations. Predictions are shown for Co = 5 and Cs = 0.2. Calculated gradients in third-order moments of velocity were approximated by specified quantities, prescribed by the generalized gradient approximation.
measured mean particle concentrations corresponding to two locations, S1 and $2 (see Fig. 1), of the dust generator are shown in Fig. 2. A more complete description of the experimental facility and the measurements of mean particle concentrations can be found in Boon and Andersen (1995).
4. Numerical implementation In predicting particle dispersion it is assumed that the presence of the particle did not affect the statistical properties of the air flow. The statistical properties of the air flow could then be predicted independently of the particles. The statistical properties of the air flow,
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Eqs. (5), (6), (8) and (9) were solved numerically, using an iterative process, for the third-order moments of velocity, (upuiuj), and the tensor, Gij. The iterative process was taken to be converged when the whole field residuals, corresponding to each component of the third-order moments velocity, (upuiu~), were all less than 0.1. The size of the time-step, At, used in the numerical integration of the random flight model, Eq. (1), should be chosen to be smaller than the Lagrangian time-scale and be sufficiently smaller so that inhomogeneities in the flow field are resolved. This cannot be done appropriately. Instead, the sensitivity of predicted mean particle concentrations to the size of the time-step, At, was investigated in numerical experiments. All predicted mean particle concentrations presented in this paper are insensitive to a reduction in the size of the time-step, At, that was adopted. Turbulent fluctuations in velocity, u, were updated at the end of each time-step, At. During each time-step, At, the numerical integration of the random flight model, Eq. (1), was performed using a fourth-order Runge-Kutta method. Particles are treated as passive scalars until they are transported toward a surface within one 'stopping distance' of a surface, characterized by the particles relaxation time and the wallward velocity (Friedlander and Johnstone, 1957). At this point, particles are assumed to deviate from the local turbulent fluid motion, arriving at the surface through 'free-flight' by virtue of their inertia. Free-flight deposition is widely believed (see eg. Brooke et al., 1994 and references within) to be the main contributor to particle deposition. The other mechanisms of particle deposition include Brown diffusion and turbulent diffusion. In this paper, free-flight deposition is taken to be the only mechanism of particle deposition and, once deposited, particles are assumed not to be re-entrained. Nodes in the computational grid, used in predicting the statistical properties of the air flow, formed 50 parallel x y planes, each containing 45 x 48 cells. Piecewise linear interpolation was used to interpolate, in the z-direction, the predicted statistical properties of the air flow from the nodes of the computational grid, surrounding the particle, to the particle location. Nodes in the x-y planes did not form an orthogonal system. The Rinka and Cline method (1984) was used to interpolate, in the x- and y-directions, the predicted values of the statistical properties of the air flow from the nodes of the computation grid, surrounding the particle, to the particle location. Ensemble mean particle concentration fields were calculated from the simulated independent trajectories of 25 000 particles. All predicted mean particle concentrations fields presented in this paper are statistically stationary; that is independent of increasing the number of particles within the ensemble. Multiple contributions to the ensemble averages due to particles revisiting points in the flow domain were removed. This enables direct comparisons between predicted mean particle concentrations and measured particle concentrations, obtained by extracting particles at given points within the flow domain, to be made. In the experiments, a single-particle detector was used. The detector being moved to different locations. Consequently, it is appropriate, in the simulations, to prevent particles from making contributions to the predicted mean particle concentration field at more than one location of the detector.
5. Predicted particle dispersion Before the random flight model can be implemented, appropriate values for the Lagrangian structure constant, Co, and the skewness parameter, C~, must be determined.
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Sawford (1991) showed for that random flight models which assume that the collective evolution of (x, u) is Markovian, e.g. the non-Gaussian random flight model, Eq. (1), and Thomson's (Gaussian) model, the supposedly universal Lagrangian structure constant, Co, is not universal but rather depends upon Reynolds number. He subsequently attributed the variability in the 'best' estimates for Co obtained using such models to variations in the Reynolds number across the various dispersion experiments. The dependency of predicted mean particle concentrations on the value of Co has been investigated in numerical studies. The 'best' correspondence between measured and predicted particle concentrations was obtained with Co in the range 5 ~< Co ~< 10. This range of values is in accord with experimental (Hanna, 1991; Sawford and Guest, 1988), numerical (Reynolds, 1995) and theoretical estimates (Sawford, 1991). Predicted mean particle concentrations obtained using Co = 5 and 10 and the generalized gradient approximation with Cs = 0.2 for gradients in the third-order moments of velocity, ~uruiuj)/Oxp, are shown in Figs. 2 and 3. Shown for comparison, in Figs. 2 and 3, are the measured mean particle concentrations and predictions obtained using Thomson's (Gaussian) model (Reynolds, 1996). Intriguingly, the dependency of particle dispersion on the value of Co predicted by the non-Gaussian random flight model, Eq. (1), is seen to be somewhat less than that predicted by Thomson's (Gaussian) model. The dependence of predicted mean particle concentrations on the value of the skewness parameter, Cs, has also been investigated in numerical studies. The 'best' correspondence between measured and predicted mean particle concentrations was obtained with Cs in the range 0.2 ~< Cs ~< 0.3. This is in accord with the range of values for Cs, 0.15 ~< C, ~< 0.25, which have been adopted in second-moment closure models of the Reynolds-stress equations (see e.g. Launder, 1992).
Source S 1
J Source $2 ~°~
Experimental
Thomson (Oaussiun) Model
~
°3
Non-Gaussion Model
(Second-order closure) Fig. 3. Measured and predicted particle concentrations within the building section for sources S I and $2. Particle concentrations have been normalized by the maximum particle concentrations. Predictions are shown for Co = 10 and Cs = 0.2. Calculated gradients in third-order moments of velocity were approximated by specified quantities, prescribed by the generalized gradient approximation.
286
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Source S 1 / f ' - - ' ~ - ~ . ° ~
Source $2
Non-Oaussian Model (Third-order closure) Fig. 4. Measured and predicted particle concentrations within the building section for sources SI and $2. Particle concentrations have been normalized by the maxiumum particle concentrations. Predictions are shown for Co = 5. Third-order moments of velocity were calculated from an approximation form of the governing evolution equation in which terms involving calculated fourth-order moments of velocity, {ur upuiu~)c, were approximated by terms involving specified fourth-order moments of velocity, {urupuiuj)~, prescribed by the Millionshchikov approximation.
Predicted mean particle concentrations obtained using third-order moments of velocity, calculated from the approximate evolution equation, Eqs. (8) and (9), are shown in Fig. 4. A comparison of Figs. 2 and 4 shows that predicted mean particle concentrations obtained using the two truncation schemes are not to be significantly different. This suggests that the violation of the thermodynamic constraint resulting the approximate closure schemes does not seriously invalidate predictions for mean particle concentrations. And that, when predicting mean particle concentrations, at least for the highly inhomogeneous flow under consideration here, it is sufficient to approximate calculated gradients in the third-order moments of velocity, O(upu~uj)/~?xp, by specified quantities prescribed by the generalized gradient approximation. Thereby truncating the hierarchy of equations governing the evolution of moments of velocity at second order. Thomson's (Gaussian) model is seen to have remarkable success in predicting mean particle concentrations. The shape of contours of constant particle concentration and the location of the maximum particle concentration for source S1 are in accord with experimental findings. Less well predicted, however, are gradients in mean particle concentration and the location of the maximum particle concentration for source $2. The success of the Thomson model has been attributed (Reynolds, 1996) to the dominance of mean-streamline streaming and gradients in Reynolds stresses in determining particle dispersion within the highly inhomogeneous flow. The non-Gaussian model is found to be leading to some improvements in predictions for several aspects of particle dispersion. In particular, the positions of maximum particle concentrations, for both source S1 and $2, are seen to be better predicted. This is especially the case for Co = 5. Furthermore, the proportion of particles from source S1 which are exhausted out of the building through the fan on the left-hand side of the building section is better predicted by the non-Gaussian
(uruiu j),
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model. The experimental estimate for this quantity is 44%. The predicted value of this quantity obtained using Thomson's (Gaussian) model is between 15% (Co = 5) and 11% (Co = 10) whilst that obtained using the non-Gaussian model is between 27% (Co = 5) and 25% (Co = 10). However, both Thomson's (Gaussian) model and non-Gaussian model significantly underpredict this quantity. Both models also significantly overpredict gradients in mean particle cncentrations. The inability of the random flight models to predict correctly these features of particle dispersion can be attributed, at least in part, to the neglect of unsteadiness of the air flow. The air flow was predicted by assuming stationarity. However, experimental observations with a sonic anemometer suggest that there are slow (period about 28 s) large-scale fluctuations in air velocity within the building section. This unsteadiness of the air flow is expected to enhance the proportion of particles which can cross the buildings plane of symmetry to be eventually exhausted out of the building through the left-hand fan. This is because the predicted residence time for a particle from source S1 (113 s for Thomson's model) is much longer than the period of oscillation of the air flow. When stationarity is assumed then only turbulent fluctuations in the air flow can cause a particle to cross the buildings plane of symmetry. The inclusion of unsteadiness, by enhancing particle dispersion, is also expected to reduce the predicted gradients in mean particle concentrations.
6. Discussion For quasi-homogeneous flows, e.g. in convective boundary-layers, third-order moments of velocity have an important effect upon dispersion and the inclusion of these moments in particle trajectory models have found to lead to improved predictions for mean particle concentrations (Thomson, 1987; Luhar and Britter, 1988). In contrast, the remarkable success of Thomson's model in predicting particle dispersion within strongly inhomogeneous turbulent flows (see e.g. Flesch and Wilson, 1991; Reynolds, 1995, 1996) suggests that third- and indeed higher-order moments of velocity are of secondary importance in determining particle dispersion when compared with the effects of strong mean-streamline straining and strong gradients in Reynolds stress. In this paper, further support for the view has been provided by numerical studies of particle dispersion in a strongly inhomogeneous turbulent flow using a random flight model which takes explicit account of the non-Gaussian distribution of velocity arising in such flows. Predicted mean particle concentrations obtained using this model were shown to be only marginally better than those obtained using Thomson's model. However, the results tentatively suggest that this random flight model does provide a better description of particle dispersion in complex inhomogeneous flows than that given by Thomson's model.
Acknowledgements The author would like to thank M. Andersen and C.R. Boon for making available their unpublished findings and B.B. Harral for making available his predictions for properties of the air flow within the building section. I am indebted to an anonymous referee for clarifying the conditions under which the thermodynamic constraint is satisfied.
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References Boon, C.R., M. Andersen, B.B. Harral and A.M. Reynolds (1994) Dynamics of particulate pollutants in an experimental livestock building, in: Proc. ASAE Meeting, Dec. 1994, Paper No. 94-4586. Brooke, J.W., T.J. Hanratty and J.B. McLaughlin (1994) Free-flight mixing and deposition of aerosols, Phys. Fluids 6, 3404-3415. Daly, B.J. and F.J. Harlow (1970) Transport equations in turbulence, Phys. Fluids 13, 2634-2649. van Dop, H., F.T.M. Nieuwstadt and J.C.R. Hunt (1985) Random walk models for particle displacements in inhomogeneous unsteady turbulent flows, Phys. Fluids 26, 1639 1653. Flesch, T.K. and J.D. Wilson (1992) A two-dimensional trajectory-simulation model for non-Gaussian, inhomogeneous turbulence within plant-canopies, Bound. Layer Meteorol. 61,349 374. Friedlander, S.K. and H.F. Johnstone, Deposition of suspended particles from turbulent gas streams, Ind. Eng. Chem. 49, 1151-1156. Hanna, S.R. (1991) Lagrangian and Eulerian time-scale relations in the day-time boundary layer, J. Appl. Metorol. 20, 242-249. Harral, B.B. and C.R. Boon (1996). Comparison of predicted and measured air flow patterns within a mechanically ventilated livestock building, J. Agr. Eng. Res., in press. Haworth, D.C. and S.B. Pope (1986) A generalized Langevin model for turbulent flows, Phys. Fluids 29, 387-405. Haworth, D.C. and S.B. Pope (1987) A pdf modelling study of self-similar turbulent free shear flow, Phys. Fluids 30, 1026-44. Launder, B.E., G.J. Reece and W. Rodi (1975) Progress in the development of a Reynolds-stress turbulence closure, J. Fluid Mech. 68, 537. Launder, B.E. (1992) Turbulence Modelling for CFD Applications (UMIST, Manchester). Luhar, A.K. and R.E. Britter (1989) A random walk model for dispersion in inhomogeneous turbulence in a convective boundary layer, Atmos. Environ. 23, 1911-1924. Pope, S.B. (1983) A Lagrangian two-time probability density function equation for inhomogeneous turbulent flows, Phys. Fluids 26, 3448 50. Pope, S.B. (1987) Consistency conditions for random-walk models of turbulent dispersion, Phys. Fluids 30, 2374-2379. Pope, S.B. (1994a) On the relationship between stochastic Lagrangian models of turbulence and second moment closures, Phys. Fluids 6, 973 985. Pope, S.B. (1994b) Lagrangian pdf methods for turbulent flows, Ann. Rev. Fluid Mech. 26, 23 63. Reynolds, A.M. (1995) Modelling particle dispersion within a ventilated airspace, Int. J. Multiphase Flows (submitted) preprint. Reynolds, A.M. (1996) On the application of Thomson's model to the prediction of particle dispersion within a ventilated airspace, J. Wind Eng. Ind. Aerodyn. (submitted) preprint. Rinka, R.L. and A.K. Cline (1984) A triangle-based C 1 interpolation method, Rocky-Mountain J. Maths 14, 223-237. Sawford, B.L. and F.M. Guest (1988) Uniqueness and universality of Lagrangian stochastic models for turbulent dispersion, Proc. 8th Syrup. Turbulent Diffusion AMS, San Diego, CA, pp. 96-99. Sawford, B.L. (199 l) Reynolds number effects in Lagrangian stochastic models of particle trajectories in turbulent flows, Phys. Fluids A 3, 1577-1586. Thomson, D.J. (1987) Criteria for the selection Of stochastic models of particle trajectories in turbulent flows, J. Fluid Mech. 180, 529-556. Thomson, D.J. and M.R. Montgomery (1994) Reflection boundary conditions for random walk models of dispersion in non-Gaussian turbulence, Atmos. Environ. 28, 1981-1987.