Download Talk - University of Rochester Physics and Astronomy

Jonathan Carroll-Nellenback
Center for Integrated Research Computing
University of Rochester
Turbulence Workshop
th
August 4 2015
Introduction to Turbulence in the context of gaseous flows
Supersonic Turbulence
Outflow driven turbulence
Turbulent support of Molecular Clouds
Modeling Molecular Clouds with AstroBEAR
The Navier-Stokes equation defines the motion of fluids

 

v

  v  v  p    2 v
t
Convective Acceleration
(Non-linear)
Viscosity
(Friction/Dissipation)
Pressure
(Resistance to Compression)
The relative size of various terms in the equation can be used to characterize the fluid motion.
The lower the viscosity, the larger the Reynolds number and the more turbulent the flow becomes.
rvL
Re =
=
m
-1
1 -3
0
1 -1
-1 1
10
kg
m
10
m
s
10
(
)(
)( m )
-4
1 -1 -1
10
kg
m s )
(
=10 2
Transition to turbulence occurs around Re=103
rvL
Re =
=
m
-1
1 -3
0
1 -1
1 1
10
kg
m
10
m
s
10
(
)(
)( m )
-4
1 -1 -1
10
kg
m s )
(
=10 4
Astrophysical flows generally have very large Reynolds numbers, 104 - 106
The Navier-Stokes equation defines the motion of fluids

 

v

  v  v  p    2 v
t
Convective Acceleration
(Non-linear)
Viscosity
(Friction/Dissipation)
Pressure
(Resistance to Compression)
 
v  v v 2
 2  M 2 where M is the mach number.
p
cs
The relative size of various terms in the equation can be used to characterize the fluid motion.
The lower the viscosity, the larger the Reynolds number and the more turbulent the flow becomes.
The higher the mach number, the less effective pressure forces are at resisting compression resulting in
strong shocks and large jumps in density and temperature.
Mach number = .5
Incompressible
Compressible
Mach number = 10
Astrophysical flows often have large Mach numbers.
For incompresible flows, the conservation of mass implies that the velocity field is solenoidal.
This permits a simplified evolution equation for the vorticity
Taking the curl of the Navier Stokes equation (and dividing by the density) gives the vorticity equation:
Viscosity
(Friction/Dissipation)
Advection of vorticity
Vortex stretching
As you stretch a vortex, you must have compression in the plane of the vortex so the vortex shrinks in
diameter.
As the diameter of the vortex shrinks, the vorticity must increase to conserve angular momentum.
“Big whorls have little whorls,
Note in 2D, there is no vortex stretching!
https://www.youtube.com/watch?v=Ax688IYmxf8
which feed on their velocity, And
little whorls have lesser whorls,
and so on to viscosity.”
- Lewis Richardson -
We can then compute the energy as a function of
scale.
Note that
1
Energy
Turbulence involves motions on a range of scales.
One way to characterize turbulence is by looking at
the velocity in Fourier space.
Energy Spectrum
3
2
Size of initial eddies
Large Scales --------------------------- Small Scales
k
In 3D, vortex stretching leads to a cascade in
energy from large scales to small scales
Energy
Energy Spectrum
Driving Scale
1
2
3
Large Scales --------------------------- Small Scales
Simulation of mach 6 turbulence computed on a 40963 grid
http://ppmac.ucsd.edu/Isothermal/2048Ma6/
Between the driving scale and the dissipation scale, lies the inertial scale where the cascade becomes scale free and the
spectrum obeys a power law. For incompressible turbulence, the spectral index is -5/3 (Kolmogorov 1941)
Energy Cascade Rate h º
Eddy turn-over time

Energy µ v 2 µ E ( k ) dk

v
v
v
t µ Þhµ µ
ℓ
t
ℓ
v µ ℓ1/3 µ k -1/3
2
E (k) µ
v (k)
dk
Energy
Time
µ
v2
t
Inertial Range
3
Dissipation Scale
Driving Scale
2
µk
-5/3
Small k -----------------------------------
Large k
Between the driving scale and the dissipation scale lies the
inertial scale where the cascade becomes scale free and the
spectrum obeys a power law.
For incompressible turbulence, the spectral index is
experimentally -5/3
However for compressible turbulence, the velocity spectrum
steepens to a spectral index of -2.
Inertial Range
Dissipation Scale
Driving Scale
Small k -----------------------------------
Large k
Astrophysical turbulence is often highly supersonic and compressible.
Studies of the velocity spectra for supersonic turbulence indicate a steeper spectral index of -2, however the meaningfulness
of this is complicated by the fact that a network of overlapping shocks is expected to give the same spectral index.
Energy cascade arguments have focused on the spectra of ρ1/3v which seem to follow the Kolmogorov scaling (at least for
solenoidal forcing).
Federrath 2013 demonstrated that the spectral index for ρ1/3v depends on the degree of compressive motions, which
depends on the nature of the driving (solenoidal vs. compressive).
For purely compressive driving, the spectral index for ρ1/3v is closer to -2.1 consistent with Galtier & Banerjee’s model that
predicts a spectral index for for ρ1/3v of -19/9 for highly compressive turbulence.
Stars do not usually form in isolation
Dense regions (like NGC 1333) are the primary sight of star
formation.
As these new stars form, some of the gravitational energy is
released back into the surrounding material in the form of
powerful outflows, jets, and winds.
How long will outflows expand unimpeded?  t 
 m 
Take outflows of momentum P  
 t 
m
in an environme nt with density  0  3 
 
L
1 
occuring at a rate per volume S  3 .
 t 
P
ρ0
What will the velocity spectra look like?
03/ 7
  3/ 7 4 / 7
P S
How much mass will outflows have swept up? M m
0 4 / 7 P3/ 7
M
S 3/ 7
How big will outflows grow unimpeded? L 
L 
P 1/ 7
 01/ 7S 1/ 7
 
How fast will outflows be travellin g? V  
t 
P 4 / 7S 3 / 7 L
V 

4/7
0

Matzner 2007
Velocity Spectra
  5/3
 k
 3

Outflow Scale L
 2
Energy is present at large scales , but why does this energy not cascade as in isotropically
forced case?
Turbulence energy deposited at outflow scale should cascade in eddy turnover time:
Eddy turnover time is the same as the outflow time scale which is the length of time outflows
will grow unimpeded. It is also the length of time a fixed point in space will experience before
being overrun by the next outflow
Eddies are swept up before they have time to break up.
Outflows suppress cascade of energy from larger scales!
What about outflows/feedback within an isotropically driven turbulent cascade?
Cascade
Both
Outflows
Above the outflow scale the energy spectra is dominated by the
cascade.
Below the outflow scale the energy spectra is dominated by
outflow driving.
The density spectra is much flatter when outflows are present.
Outflows disrupt large scale density structures.
D = 40 pc
V = 8.25 km/s
Res =
3
2048
It is thought that molecular clouds may
form as the result of colliding streams of
gas in the ISM.
This can lead to rapid cooling and collapse
and explain the apparent simultaneity of
cloud formation and star formation
This also provides an explanation for the
low star formation efficiency of molecular
clouds as a whole. Clouds need not be
gravitationally bound.
Smooth
Clumpy
Modelling astrophysical flows often requires capturing
physics simultaenously over a wide range of scales. This
requires an Adaptive Mesh Refinement (AMR)
AstroBEAR is a parallel AMR code that scales well out to
10’s of 1000’s of cores.
Questions?