Introduction to Astrophysical Gas Dynamics

1
Introduction to Astrophysical Gas Dynamics
Part 6

• Bram Achterberg
• a.achterberg_at_astro.uu.nl
• http//www.astro.uu.nl/achterb/aigdppt

2
Rotation

• Two aspects of rotation in fluid dynamics
• Vorticity swirling motions within a flow as
• a dynamical entity long-lived structures
• due to Kelvins circulation theorem
• Large-scale rotation
• - rotating frame-of-reference
• - Coriolis - and centrifugal forces

3
Applications
Meteorology Cyclones Tornados

• Astrophysics
• Jupiters Great Red Spot
• Accretion Disks

4
(No Transcript)
5
Vortex Shedding
Flow direction
Obstacle
Fluctuating lift force
6
(No Transcript)
7
Jupiters Great Red Spot
8
Smoke ring from volcanic vent on Mnt. Etna
9
Definition vorticity
Vorticity field is the rotation (curl) of the
velocity field
Vorticity field is divergence-free closed field
lines
10
Vorticity in component form
11
Equation of motion for vorticity
Step 1 take curl of equation of motion
12
Equation of motion for vorticity
Step 1 take curl of equation of motion
Step 2 use vector identity
13
Step 3 some more manipulation
14
Step 3 some more manipulation
Equation of motion for vorticity
15
Equation of motion for vorticity
Yet another vector identity
16
Equation of motion for vorticity
Yet another vector identity
Another form of the vorticity equation
17
Vorticity equation
Mass conservation
18
Vorticity equation
Mass conservation
Final best form of the equation
19
Interpretation of the vorticity equation
Vortex stretching
20
Interpretation of the vorticity equation
Vortex stretching
Vorticity generation
21
Interpretation of the vorticity equation
Vortex stretching
Vorticity generation
Ideal gas law
Condition for vorticity generation
22
Velocity at each point equals fluid velocity
Definition of tangent vector
Equation of motion of tangent vector
23
Vortex Stretching
Equation of motion for curve carried by flow
Vorticity equation without generation term
Conclusion vortex lines are carried by the flow
24
Definition vortex line
Vortex lines are the field lines of the vorticity
field
25
Definition vortex line
Vortex lines are the field lines of the vorticity
field
Vortex lines are carried by the flow
26
Vortex tubes and the circulation theorem
Definition of circulation integral
dr is carried by the flow!
Use of Stokes theorem!
27
Circulation number of vortex lines piercing
surface O vortex flux
28
Change of circulation
Deformation of surface
Change of vorticity
29
Deformation of surface-element carried passively
by the flow
Definition of surface-element
30
Deformation of surface-element carried passively
by the flow
Definition of surface-element
Change of the two vector-elements carried by flow
31
Deformation of surface-element carried passively
by the flow
Definition of surface-element
Change of the two vector-elements carried by flow
Use of chain rule
32
Smart choice
Use determinant form of cross-product
33
Smart choice
Use determinant form of cross-product
Write out determinants
34
And now for some horrific algebra
Add and subtract the same term!
35
Equation for surface change
Divergence effect of isotropic
compression/expansion
36
Equation for surface change
Divergence effect of isotropic
compression/expansion
New animal velocity gradient tensor effect of
surface warping
37
Change of circulation
Deformation of surface
38
Change of circulation
Deformation of surface
39
Use vorticity equation
40
Use vorticity equation
Kelvins theorem
41
Important consequence for barotropic fluid with
?P ??
Circulation is conserved!
Stretching the tube increases Vorticity!
42
Alternative derivation
43
(No Transcript)
44
Stokes Theorem
45
Change of circulation (1)
Vorticity equation of motion (2)
12 together yield Kelvins Circulation Theorem
46
(Fluid)dynamics in a rotating frameand
curvilinear coordinates
Introduction a overview over linear
vector-algebra
Vector in terms of its components
Orthonormal base vectors
Vector components as a scalar product
47
Change of a vector field(difference vector)
Change of the components
Change of the base vectors
Components of the difference vector
48
Change of base-vectors
(i1, 2, 3)
Example cylindrical polar coordinates
49
Cylindrical Polars
50
Distances
In three dimensions
51
Gradient of a function
52
Surfaces and volumes
53
Derivative of a vector
Definition of V-grad
j-th component
Change of components
Change of unit vectors
54
Example fluid acceleration in circular polar
coordinates
55
Summary so far
Definition vorticity
Kelvins circulation theorem Vortices in ideal
fluids are long-lived
56
Rotating coordinates
57
Rotating coordinates
Equation of motion for unit vectors
58
Rate-of-change of a vector
Rate-of-change of unit vector
Rate-of-change of an arbitrary vector
59
Interpretation
Rate-of-change in Inertial Frame
Rate-of-change in Rotating Frame
60
Interpretation
Rate-of-change in Inertial Frame
Rate-of-change in Rotating Frame
61
Applications
Basic relation for any vector
Apply to velocity
62
Basic relation for any vector
Apply to acceleration
Put in relation between velocities
63
Basic relation for any vector
Apply to acceleration
64
Summary
Acceleration as seen by a rotating observer
Coriolis force term
Centrifugal force term
Euler force
65
Illustration Coriolis Force
Ball moves with constant velocity in the
inertial (laboratory) frame NO FORCE!!
66
Illustration Coriolis Force
67
(No Transcript)
68
(No Transcript)
69
(No Transcript)
70
(No Transcript)
71
(No Transcript)
72
(No Transcript)
73
(No Transcript)
74
(No Transcript)
75
(No Transcript)
76
(No Transcript)
77
(No Transcript)
78
Fluid equation in a rotating frame
Step1 use relation between acceleration in
inertial and rotating frames (? constant)
79
Fluid equation in a rotating frame
Re-order terms
80
Fluid equation in a rotating frame
Use definition of comoving time-derivative
81
Effective gravity for rotationalong z-axis
true gravity centrifugal force
82
Application cyclones
- ?P
L
Vh
83
Approximate balance between the Coriolis force
and the pressure force
Gradient in horizontal plane!
Component of ? in vertical direction
84
Approximate balance between the Coriolis force
and the pressure force
Take vector product with vertical unit vector
85
Approximate balance between the Coriolis force
and the pressure force
Use property of double vector product
86
Geostrophic Flow(Coriolis term dominates!)
Flow is to lowest order- along isobars!
87
Equation for vorticity
Vorticity in rotating frame
Vorticity equation
Influence of Coriolis force!
88
Definition absolute vorticity
Relative vorticity
Planetary vorticity
Alternative form vorticity equation for ?
constant
89
Vorticity equation
Thermal wind equation
90
(No Transcript)
91
Ideal gas law
Incompressible flow
Thermal wind equation
92
Ideal gas law
Incompressible flow
Thermal wind equation
Density gradient mostly vertical due to gravity
Atmospheric scale-height
Temperature gradient Equator-to-pole!
93
(No Transcript)
94
Example of Thermal WindGlobal Eastward
Circulation!
95
More extreme example Jupiters cloud bands
Great Red Spot
96
Shallow water theory
97
Equations of motion in horizontal plane
Shallow water assumption
98
Vertical direction hydrostatic equilibrium
Weight/unit area of overlying layer
Atmospheric pressure
99
Barometric formula
Horizontal pressure gradients
100
Barometric formula
Horizontal pressure gradients
substitute into eqn. of motion
Variations in depth drive the motions in the
horizontal plane!
101
Height variations and mass conservation
Volumehorizontal area x depth
Surface change law two-dimensional volume-chang
e law!
2D divergence
102
Constant-density flow
Surface-change law
103
Equation for layer depth
104
Summary the shallow water equations
105
Application I water waves
Water waves are surface waves, leading to
varying depth
Assume that unperturbed flow is at rest Small
velocity perturbations
106
Linarized equations
107
Standard approach seek plane-wave solutions
Result set of three linear algebraic equations
108
Solution if determinant 3x3 matrix vanishes
Determinant
109
Solution if determinant 3x3 matrix vanishes
Determinant
Dispersion relation
110
Solution if determinant 3x3 matrix vanishes
Determinant
Dispersion relation
Wave frequency
111
Physical interpretation
Compare
1. Sound waves in rotating cylinder
2. Shallow-water waves
Pressure at bottom unperturbed layer
112
Shallow-water vorticity
Velocity in horizontal plane
Vorticity associated with horizontal motions
113
Shallow-water approximation
Exact (3D) velocity
Constant-density flow
114
z-component of vorticity equation
115
z-component of vorticity equation
116
Conservation of the potential vorticity
117
Finally the Great Red Spot
118
(No Transcript)
119
Merging of like-signed vortices
(Dye visualization, TU Delft)
120
A
C
A
A
C
C
C
A
121
(No Transcript)