Low frequency acoustic waves from explosive sources in the atmosphere

a a

Low frequency acoustic waves from explosive

sources in the atmosphere

C. Millet1, C. Roblin1,2, J.-C. Robinet2 and X. Gloerfelt21 : CEA, Paris, France

2 : SINUMEF, Paris, France

59th Annual Meeting of the APS Division of Fluid Dynamics Nov. 19-21, 2006 1

a a

E. Blanc, CEA, Private Communication

Misty Picture Event (White Sands Missile Range, NM), May 14 1987

4.7 kt of ANFO at ground level (= 3.8 kt TNT)


Data from CEA / Sandia Nat. Lab. / Los Alamos Nat. Lab.a a


Data from CEA / Sandia Nat. Lab. / Los Alamos Nat. Lab.a a

18 19 20 21 22 23−1000

0

1000

2000

Kinney law

Admin Park 7.26 km


1 Dispersion-Relation-Preserving methodsa a

◦ High order finite difference DRP (Dispersion-Relation-Preserving) schemes

→ optimized spatial and time discretization

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

k∆x

k∆x

' ikf̂

ikf̂ '

1

∆x

8

<

:

NX

j=−M

ajeijk∆x

9

=

;

f̂

E =

Z π2

ε

˛

˛

˛

˛

˛

˛

iα −

NX

j=−M

ajeijα

˛

˛

˛

˛

˛

˛

d(ln α)

fx(x) '

1

∆x

NX

j=−M

ajf(x + j∆x)

(5,5)

(0,6)

(2,4)

For a given (M, N), the coefficients aj are chosen to minimize the error E over

the range of waves with wavelength longer than four ∆x (or α = k∆x < π/2)

→ radiation and outflow boundary conditions

→ perfect reflexion at the ground ⇒ py(y = 0) = 0

→ fully non linear Euler equations


1 Dispersion-Relation-Preserving methodsa a

◦ Normal mode solution

→ numerical integration of the Rayleigh (or wave) equation

−0.01 0 0.01 0.02 0.03 0.04−1

−0.5

0

0.5

1

1.5

2

2.5x 10−3

0

50

100

150

200

250

300

-1 -0.5 0 0.5 10

50

100

150

200

250

300

-0.4 -0.2 0 0.2 0.4

1

2

1

2

20 Hz

40 Hz φ1 φ2

Residue theorem

k-plane

p(r, z) ∼

X

n

φn(z)H(1)0 (knr)kn

Dk(kn)

D(kn; ω) = 0 : dispersion relation

Im(kn) < 0 : amplified wave

Im(kn) > 0 : damped wave

→ complex boundary conditions→ linearized Euler equations

◦ Ray tracing

→ the dispersion relation is given by the eikonal equation

→ high frequency approximation of the wave equation


Temperature and wind profiles at ground zeroa a

−60 −40 −20 0 200

0.5

1

1.5

2

2.5 x 105

−20 0 20 400

0.5

1

1.5

2

2.5 x 105

−20 0 200

0.5

1

1.5

2

2.5 x 104

0 500 10000

0.5

1

1.5

2

2.5 x 105

radiosonde

rocketsonde

ECMWF

MSISE 90HWM 93

damping zonev

u

: empirical global model of horizontal winds (HWM 93)

: radiosonde upper air observations at Stallion

: European Centre for Medium-range Weather Forecasts

u (m/s) v (m/s) T (K)

Z (m)


2 The atmospheric waveguidea a

0 200 400 600 800 10000

100

200

300

400

500

600

700

800

River Side

Silver City

Alpine

White River

Roosevelt

Lake Havasu

Las Vegas

Time−Distance/340 (s)

Dist

ance

(km

) 0 200 400 600 800 10000

100

200

300

400

500

600

700

800

River Side

Silver City

Alpine

White River

Roosevelt

Lake Havasu

Las Vegas

Time−Distance/340 (s)Di

stan

ce (k

m)

dis

tance

(km

)

Roosevelt (416 km West)

: travel-time by the ray tracing

time - distance/340 (sec)

distance (km)

alt

itude

(km

)



0 200 400 600 800 10000

100

200

300

400

500

600

700

800

River Side

Silver City

Alpine

White River

Roosevelt

Lake Havasu

Las Vegas


Dist

ance

(km

) 0 200 400 600 800 10000

100

200

300

400

500

600

700

800

River Side

Silver City

Alpine

White River

Roosevelt

Lake Havasu

Las Vegas


stan

ce (k

m)

Stratospheric phase IS

Thermospheric phase IT

IS

IT

dis

tance

(km

)




distance (km)

alt

itude

(km

)



0 200 400 600 800 10000

100

200

300

400

500

600

700

800

River Side

Silver City

Alpine

White River

Roosevelt

Lake Havasu

Las Vegas


Dist

ance

(km

) 0 200 400 600 800 10000

100

200

300

400

500

600

700

800

River Side

Silver City

Alpine

White River

Roosevelt

Lake Havasu

Las Vegas


stan

ce (k

m)

Stratospheric phase IS

Thermospheric phase IT

cusp singularity

IS

IT

dis

tance

(km

)




distance (km)

alt

itude

(km

)


3 Three-dimensional waveguidesa a

0.1 Hz

1.0 Hz1.0 Hz

Normal mode solution

◦ tropospheric phase

limitations of ray tracing

f = 1.0 Hz

◦ finite amplitude

f → ∞

◦ 3D waveguide

◦ caustics

◦ shadow zones

O. Gainville, CEA, Private Communication



0.1 Hz

1.0 Hz1.0 Hz


f → ∞

◦ 3D waveguide

◦ caustics

◦ shadow zones




0.1 Hz

0.1 Hz1.0 Hz


f → ∞

◦ 3D waveguide

◦ caustics

◦ shadow zones



4 Acoustic and instability wavesa a

0 1 2 3 4x 10−4

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.8 0.805 0.81 0.815 0.82 0.825 0.83 0.835 0.84 0.845 0.85−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

X: 0.8281Y: −0.07798

Re

Im

0.1 Hz

1.0 Hz

x 1E−04

4800 sec

5600 sec

400 sec

300 sec

acoustics

Kelvin Helmholtz

∼ u

∼ c + u

k2 - plane

multiscale spectral collocation technique

Normal modes by the


Conclusionsa a

◦ The Misty Picture experiment was studied using a DRP finite

difference scheme to solve the low frequency component of non linear

Euler equations over large distances. Most of phases were identified

using 2D calculations.

◦ Realistic temperature and winds profiles are required to compute

waveforms that are in agreement with measurements.

◦ The method is capable to capture features of Kelvin-Helmholtz insta-

bility waves and acoustics phases. Very good agreements were obtained

with the normal mode solution in the linear regime. Three-dimensional

tropospheric phases were identified.


Low frequency acoustic waves from explosive sources in the atmosphere

Documents

Transcript of Low frequency acoustic waves from explosive sources in the atmosphere