tracers Namespace Reference

Functions
	closure_reference (U0, A, D, N=96)
	vel (x, y, U0, A)
	simulate (U0, A, D, Np, dt, T, nper_fpt=(1, 4, 16), seed=0, nsave=400)
	est_msd (ts, X, t1, t2, nb=20)
	est_gk (ts, U, D, Umean, tmax, nb=20)
	est_fpt (tau, L, nb=20)

Variables
int	TWO_PI = 2 * np.pi
	U0
	A
int	D = float(sys.argv[2]) if len(sys.argv) > 2 else 0.1
int	Np = int(sys.argv[3]) if len(sys.argv) > 3 else 20000
int	T = float(sys.argv[4]) if len(sys.argv) > 4 else 600.0
float	dt = 0.01
	Um
int	tD = TWO_PI**2 / D
	t0 = time.time()
	ts
	X
	U
	fpt
	v = X.var(axis=1)
	i = min(len(ts) - 1, int(frac * (len(ts) - 1)))
	j = max(1, i // 2)
	m
	e
	g
	f
	miss

Detailed Description

2-D prototype: dispersion estimators from Brownian tracer trajectories, against an exact
Eulerian reference (Brenner closure solved spectrally).

Flow: periodic cellular flow with mean through-flow on [0, 2pi)^2,
    psi = U0*y + A*sin(x)*sin(y)   ->  u = U0 + A sin x cos y,  v = -A cos x sin y
A > U0 gives closed recirculation cells (an analogue of hold-up regions in a bed).

Reference: B periodic, mean zero,  u.grad B - D lap B = <u> - u   (one per direction)
           D*_aa = D <|e_a + grad B_a|^2>   (no walls -> <grad B> = 0)

Estimators compared on the same trajectories:
  MSD   : late-time slope 1/2 dVar(x)/dt over [t1, t2], error by batch means over particles
  GK    : D + int_0^T <u'(0) u'(tau)> dtau  (velocity autocorrelation only)  -- misses the
          velocity-noise cross-covariance when u_x depends on x
  FPT   : first passage over n periods: D = L^2 Var(tau) / (2 <tau>^3)  (inverse Gaussian)

Function Documentation

closure_reference()

tracers.closure_reference	(		U0,
			A,
			D,
			N = 96
	)

Definition at line 24 of file tracers.py.

vel()

tracers.vel	(		x,
			y,
			U0,
			A
	)

Definition at line 62 of file tracers.py.

Referenced by simulate().

simulate()

tracers.simulate	(		U0,
			A,
			D,
			Np,
			dt,
			T,
			nper_fpt = (1, 4, 16),
			seed = 0,
			nsave = 400
	)

Definition at line 66 of file tracers.py.

References vel().

est_msd()

tracers.est_msd	(		ts,
			X,
			t1,
			t2,
			nb = 20
	)

Definition at line 91 of file tracers.py.

est_gk()

tracers.est_gk	(		ts,
			U,
			D,
			Umean,
			tmax,
			nb = 20
	)

Definition at line 103 of file tracers.py.

est_fpt()

tracers.est_fpt	(		tau,
			L,
			nb = 20
	)

Definition at line 116 of file tracers.py.

References mean().

Variable Documentation

TWO_PI

int tracers.TWO_PI = 2 * np.pi

Definition at line 21 of file tracers.py.

U0

tracers.U0

Definition at line 127 of file tracers.py.

A

tracers.A

Definition at line 127 of file tracers.py.

D

int tracers.D = float(sys.argv[2]) if len(sys.argv) > 2 else 0.1

Definition at line 128 of file tracers.py.

Np

int tracers.Np = int(sys.argv[3]) if len(sys.argv) > 3 else 20000

Definition at line 129 of file tracers.py.

T

int tracers.T = float(sys.argv[4]) if len(sys.argv) > 4 else 600.0

Definition at line 130 of file tracers.py.

dt

float tracers.dt = 0.01

Definition at line 131 of file tracers.py.

Um

tracers.Um

Definition at line 132 of file tracers.py.

tD

int tracers.tD = TWO_PI**2 / D

Definition at line 133 of file tracers.py.

t0

tracers.t0 = time.time()

Definition at line 136 of file tracers.py.

ts

tracers.ts

Definition at line 137 of file tracers.py.

X

tracers.X

Definition at line 137 of file tracers.py.

U

tracers.U

Definition at line 137 of file tracers.py.

fpt

tracers.fpt

Definition at line 137 of file tracers.py.

v

tracers.v = X.var(axis=1)

Definition at line 140 of file tracers.py.

i

tracers.i = min(len(ts) - 1, int(frac * (len(ts) - 1)))

Definition at line 142 of file tracers.py.

j

tracers.j = max(1, i // 2)

Definition at line 143 of file tracers.py.

m

tracers.m

Definition at line 147 of file tracers.py.

e

tracers.e

Definition at line 147 of file tracers.py.

g

tracers.g

Definition at line 150 of file tracers.py.

f

tracers.f

Definition at line 153 of file tracers.py.

miss

tracers.miss

Definition at line 153 of file tracers.py.