N u m e r i c a l s t a b i l i t y a n d accuracy considerations for the solution of Lagrangian stochastic particle dispersion model equations

Brian N. Bailey USDA-ARS, Corvallis, OR

bailebri@oregonstate.edu

Abstract Based on the classical Langevin equation for molecular diffusion, Lagrangian stochastic dispersion models are a popular method for describing the turbulent diffusion of particles in the atmospheric boundary layer. Pioneering work in the 1980’s led to the formulation of a consistent framework for the formulation of Langevin-based Lagrangian stochastic dispersion models. Although such models are theoretically consistent in that they agree with their corresponding macroscopic Eulerian equation, they have been commonly observed to break down when applied to flows that contain substantial inhomogeneity or anisotropy. Modeled particle plumes have been reported to violate the second law of thermodynamics (i.e., entropy increases), and contain “rogue” particles whose velocities approach infinity. As a result, some type of ad hoc intervention is frequently

needed to eliminate unphysical behavior, which can also lead to questionable results. The present work identifies the source of such unphysical model behavior, and presents a straightforward remedy that is consistent with theory. It was found that “rogue” trajectories are associated with numerical stability, while numerical accuracy dictates the degree to which the second law of thermodynamics or “well-mixed condition” is satisfied. Simple verification tests are suggested that future workers can implement to ensure computed particle plumes not only satisfy the second law of thermodynamics, but more fundamentally that they agree with Eulerian velocity statistics specified as inputs.

22nd Symposium on Boundary Layers and Turbulence ◉ Salt Lake City, UT ◉ June 20-24, 2016

Background

dw = − C0ε2σ 2 wdt +

121+ w

2

σ 2

⎛⎝⎜

⎞⎠⎟∂σ 2

∂zdt +

t

t+Δt

∫t

t+Δt

∫w(t )

w(t+Δt )

∫ C0ε( )1/2 dWt

t+Δt

∫ (1)

“Rogue” Trajectories

Lagrangian Stochastic Dispersion Modeling

Adding in the “drift correction” term results in a model that theoretically satisfies the well-mixed condition (second law of thermodynamics).

The source of rogue trajectories

Unconditionally stable schemesRe-write using the total derivative

Isotropic turbulence model

Generalization to anisotropic turbulence (Thomson’s “simplest” model)

Eulerian profiles

References

Web

Forward Euler scheme

Article: Bailey, B.N. (2016) Numerical considerations for Lagrangian stochastic dispersion models: Eliminating rogue trajectories, and the importance of numerical accuracy. Boundary-Layer Meteorol. In press.

Recommendations

Acknowledgements

Langevin velocity equation - First consider 1D steady isotropic Gaussian turbulence

energy dissipation “drift” correction turbulent diffusion

“Toy” Problem

wn+1 = wn − C0ε2σ 2

⎛⎝⎜

⎞⎠⎟n

wnΔt + 121+

wn( )2σ 2( )n

⎛

⎝⎜⎜

⎞

⎠⎟⎟

∂σ 2

∂z⎛⎝⎜

⎞⎠⎟

n

Δt + C0εn( )1/2 ΔW (3)

Figure 2. Particle concentration with (a) no drift correction term, (b) with drift correction term. Note that Δt = 0.1.

Rogue trajectories arise from an unstable numerical integration scheme.

This can be remedied by:1. Reducing the timestep2. Using a stable numerical integration scheme

For real flows, a minuscule timestep often cannot be afforded!

w ∂σ 2

∂z→ dσ 2

dtdw = − C0ε

2σ 2 wdt +12

∂σ 2

∂z+ wσ 2

dσ 2

dt⎛⎝⎜

⎞⎠⎟dt +

t

t+Δt

∫t

t+Δt

∫w(t )

w(t+Δt )

∫ C0ε( )1/2 dWt

t+Δt

∫ (4)

now linear in w

Effect of timestep

Δt < 4σ 2 C0ε −w∂σ 2

∂z⎛⎝⎜

⎞⎠⎟

−1

• Homogeneous model is unconditionally stable

• Inhomogeneous model is only stable when:

Rij must be realizable! (positive semidefinite)

Rkk > 0R11R22 + R11R33 + R22R33 − R12

2 − R132 − R23

2 > 0

det Rij( ) > 0

• Unconditionally stable • Accuracy still depends on Δt

Nomenclature:

Figure 3. Effect of integration t i m e s t e p o n p a r t i c l e concentration profiles for the explicit forward Euler scheme: (a) Δt = 0.1, (b) Δt = 0.05, (c) Δt = 0.01.

Figure 4. Effect of integration timestep on particle concentration profiles for the implicit backward Euler scheme: (a) Δt = 0.1, (b) Δt = 0.05, (c) Δt = 0.01.Figure 1. Input profiles

for `sinusoidal’ test case . (a ) ve loc i t y variance, (b) viscous dissipation rate, (c) i n i t i a l p a r t i c l e concentration (100,000 particles total).

Bailey, B. N., R. Stoll, E. R. Pardyjak, and W. F. Mahaffee (2014). Effect of canopy architecture on vertical transport of massless particles. Atmos. Env. 95:480-489.

Postma, J. V., E. Yee, and J. D. Wilson (2012). First-order inconsistencies caused by rogue trajectories. Boun.-Lay. Meteorol. 144:431-439.

Postma, J. V. (2015). Timestep buffering to preserve the well-mixed condition in Lagrangian stochastic simulations. Bound.-Lay. Meterol. 156:15-36

Thomson, D. J. (1987). Criteria for the selection of stochastic models of particle trajectories in turbulent flows. J. Fluid Mech. 180:529-556.

Yee, E., and J. D. Wilson (2007). Instability in Lagrangian stochastic trajectory models, and a method for its cure. Bound.-Lay. Meteorol. 122:243-261.

In the end, our goal is not really to satisfy the well-mixed condition. Rather, it is to match the Eulerian profiles specified as inputs! (See Fig. 1, Eq. 2)

Figure 5. Eulerian profiles of particle velocity statistics compared against “exact” values specified as inputs.

• It is preferable to use an implicit scheme in all scenarios I have encountered to date (assuming Gaussian turbulence)

• If the general anisotropic model is used, it is critical that the stress tensor be realizable• See the paper below for a procedure that ensures

realizability• The adequacy of the timestep should be assessed by

comparing computed Eulerian profiles to those specified as inputs (see Fig. 5)

• For skewed Gaussian turbulence, a costly iterative solution is required to formulate an implicit scheme

Supplemental material: Code, test cases, and input data

The author wishes to acknowledge fruitful discussions with Drs. Rob Stoll and Eric Pardyjak in formulating the ideas presented in this work. This research was  supported by U.S. National Science Foundation grants IDR CBET-PDM 113458 and AGS 1255662, and United States Department of Agriculture (USDA) project 5358-22000-039-00D.

To test the models, a simple artificial turbulence field was generated:

σ 2 (z) = 1.1+ sin(z)ε(z) =σ 3(z)

wσ 2

εC0

Our modeling task is to generate an ensemble of particles whose velocity statistics agree with these specified Eulerian statistics and Eq. 2.

S: entropy

R: % particles `rogue’

particle velocityvelocity variance

viscous dissipation rate“universal” constant

However, this often results in unphysical or ‘rogue’ trajectories in which ! (Yee and Wilson 2007; Postma et al. 2012; Bailey et al. 2014; Postma 2015)

Furthermore, we still often don’t actually satisfy the second law of thermodynamics in practice!

w→∞

PE (z) = 2πσ 2 (z)⎡⎣ ⎤⎦−1/2exp − w2

2σ 2 (z)⎡

⎣⎢

⎤

⎦⎥ (2)Implicitly, we assume a

Gaussian velocity p.d.f.:

Δt = 0.1Δt = 0.05

exact

w evaluated at future time

coefficients are “lagged”

Δt = 0.01

uin+1 = ui

n

− C0ε2

Rik−1⎛

⎝⎜⎞⎠⎟n

ukn+1Δt +

Rlj−1

2ΔRilΔt

⎛

⎝⎜⎞

⎠⎟

n

u jn+1Δt + 1

2∂Ril∂xl

⎛⎝⎜

⎞⎠⎟

n

Δt + C0εn( )1/2 ΔWi (6)

wn+1 = wn − C0ε2σ 2

⎛⎝⎜

⎞⎠⎟n

wn+1Δt + 12

∂σ 2

∂z⎛⎝⎜

⎞⎠⎟

n

+ wn+1

σ 2( )nΔσ 2

Δt

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪Δt + C0ε

n( )1/2 ΔW (5)

Rij - stress tensor (Reynolds or subgrid)

My website: people.oregonstate.edu/~bailebri