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Hydrodynamics and Shocks
Satyam Ladva

Lecture 2
Note: Lecture 1 was the introduction lecture and outlined the course aims
...
1 Concepts and Simplications
1
...
1 Fluids, Liquids and Gases
•

The dierences between liquids and gases is mainly focused on the compressibility i
...


Gases are more

compressible than liquids since gas density is lower than liquid density
...


•

In general, 1 atmospheric pressure

Note: The Aluminium

P vs
...


diagram (above) does not have any phase transition information i
...
The boundaries

are innitely small and there is no middle region
...


very high outward pressure forces
...


1

1 Fluid Dynamics

•

2

As a result of this small density change, only height scales with Pressure
where

h

∴ P = ρgh+Psurf ace

so

ρ ∼ constant

is the sea depth, atmospheric altitude etc
...
This equation is known as Pascal's

Principle which is an isotropic pressure equation
...
1
...
)

•

The ow line is dened as the path of an individual particle/uid element

dA

and these particles follow the

same ow lines if the net ow is constant with time i
...
steady ow
...


•

The separation of the ow lines is

•

Stream lines are curves formed by the tangent, in the direction of the ow vector at any one time
...


•

dm
dt = ρvA where the continuity equation (conservation of mass)
1
implies a constant mass ux
...
which means that v ∝
A
...


•

The closer the ow lines, the faster the uid
...


Lecture 3
1
...
3 Navier Stokes (NS) Equation
•

A set of non-linear equations which determine the equation of motion of a uid element wrt external forces
...

ˆ

1 Fluid Dynamics

• P = F/A
•

3

and

F = ma

where

A = ∆y∆z

for the above uid element case
...
In addition, the
Fluid element is deemed incompressible and independent of temperature
...

dt

This

equation just governs the acceleration in 1D
...
h v where h represents

the viscosity tensor,

v represents the tensor gradient and h v is the deviatoric (Forces which aren't isotropic)

=

=

=

stress tensor
...


•

Combining the 4 forces above gives rise to the NS equation:

ρ dv = − P − ρg −
...

dt
=

This equation denes

the motion of the element in the frame of reference moving with the uid
...
i
...
Changes in time and space
...
This yields the relationship
dt = v
...
The v
...


•

NS at a xed point reduces to

ρ ∂v + ρ(v
...
h v
=

be rewritten using vector identity relations to give the NS equation:

× v) × v

ρ(
•

where

×v

where the 2nd term on the LHS can

ρ ∂v = − P − ρg −
∂t


...
e
...


The new NS equation shows that Acceleration = Pressure gradient + Gravity + Viscosity + Inertia + vorticity
...


 ρ(v
...
v
...

 −ρg - Force due to gravity
...
h v - Shear stress force term wrt the viscosity tensor h
...
e
...


Lecture 4
1
...
4 Drag and Ram Pressure
•

The drag acting on an object depends on the deviation of the streamlines which ow around the object i
...

The more streamlined an object, the lower the drag
...


ρ ∂v =
∂t

1 Fluid Dynamics

•

4

To calculate the drag

Fdrag = − 1 ρv 2 ACd
2

where the area of the object

A⊥

direction of ow and

Cd

is the

drag co-ecient which can be determined experimentally or computationally, and it is dependent on object
shape, wind speed etc
...
There is a sphere of

1
2
2 ρv term is referred to as Ram Pressure
...
Only when there is ecient hotspot heating Photspot ≥ ρv does the
2
hot fuel (hotspot) inside a sphere of dense imploding fuel
...


1
...
5 Incompressible Flow
•

The form of the NS equation used in 1
...
3 assumed

ρ =constant
...

∂t
2

This equation is valid for compressible uids, such as gases, also
...


1
P + 2 ρv 2 = const
...
2
...


means that even if there is a change in

1
2
2 ρv , the change in pressure is

small compared to the background pressure
...

v

From

P
ρ which

cs
...


1
...
6 Ideal Flow
•

No vorticity (rotation) or viscosity in ow allows the NS equations to be made simpler
...


Inviscid - There is negligible viscosity and it is determined by the dimensionless parameter, the Reynold's
number

R
...
h v| where you can dene the scalar magnitude of =as η
...

η
ν
ρ
∗ A large R implies that viscosity is small which allows the production of a laminar ow or, if the uid
ow is unstable, a turbulent ow
...




Irrotational - There is negligible voriticity i
...
no rotation
...





Steady - The uid does not accelerate at any point in its ow and as a constant velocity w
...
t time
...


1 Fluid Dynamics

•

5

Many of the principles e
...


Bernoulli, can still be used in a non-ideal uid ow however they just require

modications to the derivation for ideal ow
...
1
...
Close to the ball's surface, viscosity and friction are important
i
...
boundary layers
...
e
...

Can use potential ow method to calculate ow velocity, in this region
...
v = 0

and, using continuity equation,

×v = 0

where

v=− ψ

where

ψ

since

ρ = const
...
Substitution of this into

the dot product gives the Laplacian
...


•

The ball in the uid ow is analagous to a polarisable ball placed in a steady electric eld

ˆ
E = E0 z
...
A charge build up at either end results in the ball becoming a dipole whereby the

potential outside is the sum of the potential due to applied uniform eld

•

V = −E0 z
...

•

i
...
The ow is undisturbed far from the ball
...
vr = ∂ψ |θ → 0
...


Drag

= 0,

r = a, v

which means nal velocity of ball

=

initial velocity
...


This implies that the eective surface area of the ball is 0 which means there should be no drag
...


Drawing Flow Net Diagrams
•

Assume that the equipotentials are

⊥

to the sphere's surface but are equispaced and moving vertically

away from the sphere, due to a uniform velocity acting away from the spherical body
...


=0

and irrotational

(

× v)
...
2 Applications of Bernoulli's Principle
1
...
1 NS for a steady, inviscid, irrotational ow
•

The form of this type of NS equation is:
gravitational potential




•

Steady:
Invisid:

g=

φ

ρ ∂v = − P − ρ φ −
∂t


...
h v = 0
=

Irrotational:

ρ(

× v) × v = 0

(P + ρφ + 1 ρv 2 ) = 0
2
whereby the equation in the brackets is a conserved quantity (Bernoulli's principle)
...
e
...
, then Bernoulli's obtained P ∝
v 2
...
gives NS of the form:

principle explains lift, spin, waves and instability
...
[(

× v) × v] = 0
...
(P + ρφ + 1 ρv 2 ) = 0
2

which means

the quantity within the brackets is a constant along the velocity ow line
...
2
...
The ow
velocity can be measured by using the Pressure change equation to determine the height of the water in the
verticle tubes attached
...
2
...
P2

and

Assume, for this example, the uid is water which is initially at rest
...
The ow is horizontal,

•

φ =const
...

P1 > P2 , the water begins to

and the ow is incompressible
...
Note: assume P1 − P2 = 104 and that the area of the constriction A2 = 100 A1
...


1
...
3 The Magnus Eect
•

A spinning object will generate a rotation in the uid owing around it
...
The air near the ball's surface moves with a rotational
velocity, which adds to the uid velocity or decreases it, depending on direction of rotation
...
The potential ow

created is a superposition between an unspinning ball and a vortex
...

2

between the dierent sides of the ball
...


E
...
David Beckham's freekick - displacement sideways was 2
...
The slower the
object, the further the sideways turn
...


1 Fluid Dynamics

8

Lecture 7
1
...
4 Linear/Airy wave theory
•

Ocean surface waves gain from the wind (strength of wind eects intensity of waves) but propagate due to
gravity (rising and falling motion)
...
Airy theory describes this behaviour
...


A = A(ω, z)
...
e
...


•

By transforming into a frame moving with the phase velocity, surface of the wave is now static and water is
moving for a shallow water wave (d

•

λ)
...
Using Bernoulli's equation P +
1
2
2
=
2 ρv + ρgz√ const
...
This is the ow velocity in a frame where the surface is static whereas, in stationary frame,

from the continuity equation, ow velocity has to be higher

this is the phase velocity
...


•

The general solution of the linear wave theory assumes inviscid, irrotationaly and incommpressible, but
unsteady, ow
...


1 Fluid Dynamics


•

9

∂ζ
∂ψ
∂t = vz = ∂z |z=0
...


Sea Surface:

v and ψ also
...

∂t

Unsteady ow results in a time derivative of

(P +

Bernoulli's equation

•

Is

Pair ≈ PH2O

•

By assuming

1
2
2 ρv

+ ρgz +

ρ ∂v =
∂t

ρ ∂ψ
∂t

near boundary, amplitude small and surface tension neglected:

which will give a modied

gζ = − ∂ψ |z=0
...

kz
For deep water ψ → 0 as z → ∞, solution is of the form f (z) = e
...
e
...
Can determine
∂t = ∂z and gζ = − ∂t and using ψ ∝ e e
ψ

is a plane wave of the form

phase and group velocity using normal procedures
...



...


obtain an oscillatory wave with a group and phase velocity
...
2
...
This rear moves faster and eventually catches up with the
front of the wave
...


•

The shear force between this interaction

∂vx
∂z introduces a vorticity in the wave
...


1
...
6 Tsunamis
•

Tsunami's are caused by two key processes:
1
...
This process is only dangerous when there is shoaling (shallowing) at the shore
...


2
...
The
process is locally very devastating but the eect doesn't propagate over large distances
...
3 Compressible ow
1
...
1 Cavitation

•

By considering Bernoulli's principle for a large instantaneous change in ow velocity (as in the construction
above)

2
2
∆P = 1 ρ(v1 − v2 )
2

for

v1

v2
...


•

Impurities, surfaces etc
...


If these bubbles are forced between a constriction (between 2 atmospheric pressure

regions) the high extremal pressure will result in the bubble imploding
...


The same process occurs if the water undergoes rapid deviation when

impacting a solid surface or a bullet is red in the water etc
...
3
...


•

At this point in time, the velocity of the water driving the implosion is high and its inertia

Pbackground

Pvapour

implosion will begin to occur
...


Rf inal
Rinitial are limited by symmetry, viscosity etc
...


•

The convergence

•

For an adiabatic gas:

ρ∝

1
r3 ,

5

T ∝ ρ 3 −1 ∝

1
r 2 therefore

P ∝

1
r5
...


Lecture 11
1
...
4
...


This is the resistance of the liquid to shear ow

which is dened as shear stress
...


1 Fluid Dynamics

11

1
...
2 Boundary layer
•

As in the above diagram, the faster moving molecules further from the surface pull the slower moving molecules
forward whereas the slower moving ones pull the faster moving ones back
...


•

Viscosity in gases arises from molecular diusion (mainly) which is responsible for transporting momentum
between layers of ow
...
e
...


•

The shear stress, as function of the vertical direction y:

s = η ∂vx
...
Can use N2 law to compute the deceleration:
∂t
(velocity diuses from high speed region to slow speed region)
...

∂y
2

∂s
∆y ∂y = η ∂ v2x ∆y
∂y

where the mass per

η ∂ 2 vX
ρ ∂y 2 which is a 1D diusion equation

As in the above diagram, slowness propagates up from the solid surface, which implies that there is no
stationary solution
...


•

Boundary layer thickness is :

δ=

2ηt
ρ where

η = |h|
=

- This is only valid for a frame moving with the uid

elements
...

ρ

δ(x) =

2ηx
ρvx

≈x

η
R where the Reynold's number

ρ
R = |v|x η = |v| x
ν

where

This means that a more viscous uid is thicker with reduced shear
...
4
...


•

For a ow entering a narrow pipe, the viscosity implies that the ow on the walls is 0 because the velocity
shear propagates from the walls into the ow
...
e
...

A cylindrical uid element, length

∆z

and radius

r,

inside a larger cylinder of radius

a

can be solved to

determine the velocity prole
...
When the assumed constant pressure gradient driving the ow is

balanced by the viscous drag, a constant velocity prole is reached
...


vz = 0
...
i
...
The inlet length can be considered parabolic
...
5 Vorticity
1
...
1 Vorticity
•

Viscosity within the boundary layer causes drag between layers with dierent velocity
...
e
...

∂vy
z
∂z (ˆ) exists as a result of shear
...
Initially, if vorticity is
0 then it remains 0 but in this situation, we are forcing a boundary condition at vx = 0 - which is a source of
By taking

∂vx
∂
∂y of the viscous deceleration: ∂t

=

vorticity that diuses into the ow
...
Outside,
the boundary layer vorticity tends to 0 and the potential ow approximation is used
...


1
...
2 Types of Vortex
•

A vortex is a circulating uid with closed streamlines
...
e
...
There is an outer boundary to prevent

•

circulation:
Free vortex
...
dS = 2πrvθ (r)

direction:

vθ ∝ r resulting
vθ → ∞ when r → ∞
...


1
× v = 1 ∂(rvθ ) z = 0 despite
r and therefore:
r ∂r ˆ
being nite and constant
...
e
...
As in baths and hurricanes, this structure lives long
...
This means that there is a
vθ → 0
...


•

The Rankine model of a vortex is problematic because

∂vθ
∂r

→∞

at

r = a
...


1
...
3 Vortex shedding, vortex streets
•

By changing the Reynold's number when a ow meets a cylindrical object (assume R is only velocity dependent) the vorticity ow pattern changes
...


 R > 1, boundary layer width decreases resulting in ow shear at the surface and vorticity increasing
...
You get trapped

vortices in a small region close to the cylinder back
...
This shedding occurs on one side then the
other, resulting in asymmetric ow - Structure created is the Karman vortex street
...
e
...


 R > 104

results in the vortices breaking up further resulting in turbulence which dominates rather than

viscosity
...


This results in the object oscillating and if this oscillation frequency equals the resonant

frequency of the object, it can be destructive
...
5
...


The ow is from high

pressure (below the wing) to low pressure (above the wing) resulting in the rotation at the wings
...


2 Aerodynamics and the principles of ight

•

14

In addition, the vortex decay time limits airplane takeo frequency on runaways
...


•

Ekranoplan - The proximity to the surface increases lift by increasing the pressure under the wing thereby
decreasing drag by extinguishing wingtip vortices due to contact with the surface
...
5
...


•

The NS equation (neglecting gravity and viscosity) gives:



In a frame moving with the uid element,

v





v

ρ ∂v = − P − 1 ρ v 2 − ρ(
∂t
2

× v) × v

changes but, at a xed point i
...
lab frame,

∂v
∂t

=0

because

is constant
...


By applying Bernoulli's principle to the remaining terms:

1
(P + 2 ρv 2 ) = 0
...
This means that the pressure gradient force creates a centripetal

The Pressure gradient force then is:
mensional analysis,

P ∝ ρr−3 ∝

force in the rotating uid element
...
This
means that a second term is required to balance the pressure gradient force, in addition to the centrifugal
force
...


This

implies a cyclone is not a perfect Rankine vortex
...


 K
...
T otal (per unit of height)=

´R

1
2
2 ρ(v|r=a ) 2πrdr where R is a outer radius and a is the inner radius
...

a
a
a

 This results in K
...
=
 R prevents the energy from becoming innite however, the majority of the energy is close to a so choice
of




R

is arbitrary
...

A small atmospheric pressure change results in massive wind speed changes
...
1 Forces acting on an aircraft
2
...
1 The Lift
•

When an aircraft is moving at a steady speed it is stated to be in equilibrium and, just like any moving object,
this is due to the forces acting on it being balanced
...
e
...




To solve any equation use the relationship:

Fweight = mg = v dmair = vρAv
dt

where the latter is the drag

formula
...

2
 Combine the two and solve for velocity
...


•

Setting an airplane wing at a specic angle

θ

where

θ = θAT

generates the maximum lift
...


•

An issue with the above model is that the airow also acts above the airplane wing which results in a
displacement of the wing downwards
...


2 Aerodynamics and the principles of ight

•

16

The potential ow solution for aerofoil - the ow above has a shape as usual but the ow below takes a shorter
path - undisturbed
...

a consequence of Bernoulli's principle which requires:

1
2
2 ρ(vupper

Plower − Pupper =

This implies that lift is

2
− vlower ) −→continuity

equation
...


•

Incorrect explanation 1: Equal transit time assumption - A ow which is broken by an object does not have to
reconnect after passing that object
...
g
...

The idea is that distance changes but time to ow remains constant
...


•

An angled plate can get both kinds of lift which means that the shape of the aerofoil is a compromise between
minimising drag and maximising lift
...


2
...
2 Types of aircraft drag
•

Assume that the drag forces on an object move wrt air ow
...


Low speed aerofoils with a large angle of attack have a

component of drag associated with lift - lift induced drag
...




Vortex drag - Flow decelerates in order to ow around vortexes which increases form drag
...




•

Viscous drag - A result of strong velocity shear within the boundary layer
...


For low speed ows, a large angle of attack will increase the lift but it will result in lift induced drag and
vortex drag (it dominates)
...
Boundary layer shear increases resulting
in increased viscous and skin friction drag
...

Can reduce drag by ying at higher altitudes (low air pressure) but the lift generated is reduced
...
1
...


(a)

A hemispherical disc reduces this drag as a result of moving the ow line until they are

to the hemispherical

surface
...


•

cd ≈ 0
...


(b)

Modifying the hemisphere design by making it into an ice cone shape will reduce vortex drag
...
Shear, vorticity and vortex sizes
are reduces resulting in

•

cd ≈ 0
...


(c)

Further delays/prevention of boundary layer separation can reduce drag further
...
1
...
This results in shear and vortex
formation
...


•

For very large

θAT

(>

15◦ )

the vortex region is extended along the wing in addition to having the ow

separation point being earlier on the wing
...


•

The stall speed is determined for very large

•

Stall also occurs for higher speeds - Airplane diving followed by a rapid climbing motion
...


θAT

if a lift is required to be generated
...
2 Transonic Flight
2
...
1 Transonic ight
•

Wave drag occurs for aircrafts that approach supersonic speeds
...
8 (approaching

cs )

create air ows with increasing velocity at the top of the wing

which may result in parts of the ow becoming supersonic
...
e
...


•

Shock front forms

⊥

top of the wing
...


2
...
2 Wave drag
•

Shock deviated air ow increases drag dramatically
...


•

•

Shocks increasing pressure on the control surfaces causes stalling in old aircrafts, upon diving
...
For
the sound barrier moto
...
Rocket powered vehicles can generally achieve this
...
3 Supersonic ight
•

An object is said to be in supersonic ight if the relative velocity of the air and wing is greater than the speed
of sound

•

cs
...


2 Aerodynamics and the principles of ight

19

2
...
1 Mach's construction
•

Consider an airplane with a siren i
...
continuous sound source emitting from the tip of the aircraft
...


•

As the speed of the airplane (at a large scale, can model this airplane as a point particle) increases the
uniformity of the sound wave is disturbed, set o-centre
...


•

Flow lines close together indicate compression i
...
an increase in pressure
...
Greater than Mach 1, the point moves faster than
the spherical waves
...


i
...
This surface corresponds to the shock front position
...


2 Aerodynamics and the principles of ight

20

2
...
2 Shock formation
•

For normal shocks, the air ow is

⊥

the shock front and this allows discontinuities in

ρ, T, p

and

v

to be

determined by the Rankine-Hugoniot conditions
...
(=1
...


Air which passes thrugh the shocks compress and heat up resulting in an energy trasfer
...


2
...
3 Oblique shocks
•

The ow enters the shock at an oblique angle as a result of the conical shock front emanating from the nose
of the aircraft
...


The shock angle
velocity can be

dependence of the discontinuity equations and, as a result, the post shock

The angle of the inclination of the leading nose of the aircraft

tanθ = 2cotβ
•

is determined by the equation:

2
M1 sin2 β − 1

...

2
γM1 sin2 β − γ−1
2

1+

(2)

If the tip of the aircraft is blunt, a bow shock is produced
...

Earth
...
g
...
The oblique shock wave becomes detached

Mach number
...


2
...
4 Supersonic aircraft design
•

Conventional aircraft designs suer from ow separation and loss of life with supersonic ight speeds, in
addition to increased drag
...


This prevents the

oblique shocks create at the nose from reaching the wings
...


•

The swept-back wing, delta wings, maximise the area inside a mach cone whilst maintaining the small angle
...


•

The leading edge is sharp (θ is small) in order to keep the shock attached to the surface and prevent bow
shock induced drag
...
3
...


•

They achieve this through a use of a combustion chamber which increases the pressure applied to the uid
...

equation
...

Beyond this point, the ow becomes supersonic which results in the gas expanding creating a density reduction
...
e
...


•

The pressure gradient along the axis accelerates the ow to higher velocities
...


2
...
6 Diamond exhaust patterns
•

If

Pe = Patm then the radial
(Patm < Pe )
...


•

The diamond pattern is created by reecting shocks - a feature common for high altitude engine designs
operating at high

Patm

on take o
...
4 Hypersonic ight
•

There is so dened point of transition from supersonic to hypersonic ight
...
g
...
It
is usually beyond Mach 5
...
e
...
For an aircraft attempting re-entry, Mach number is about 20-25
...


θ

with the shock attached, the nose cannot withstand the heat loading and deforms
...


2 Aerodynamics and the principles of ight

23

2
...
1 Detached shocks
•

The nose is shaped as such in order to follow the curved bow shock
...


•

A buer, composed stagnated air or ablated material, decelerates the supersonic ow
...
e
...
Apollo command modules used epoxy
resin lled quartz bres and phenolic micro balloons
...


2
...


2
...
1 Wind tunnels
•

Full scale wind tunnels provide the best results but they are impractical due to high cost, lack of space and
inability to place real objects inside
...


i
...


ow shape must be made the

same
...


2
...
2 Dimensionless parameters and Buckingham π theory
•

The ability of
where

•

cs =

P
P
ρ

to cause a change in the momentum

ρ ∂v
∂t

depends on the ratio:

2
1
2 ρv

P

=

v2

(

2P
ρ

)

≈

v2
c2
s

≈ M2


...


Magnitude of inertia to viscous force gives the Reynolds number

R=

1
|2ρ

v2 |


...


• R determines the importance of viscosity in the boundary layer
...


•

An exact similitude requires that the magnitudes are the same for:





•

the shape
the initial conditions of the ow
the boundary conditions
all dimensionless parameters

If an equation, in terms of n variables e
...


ρ, v

(NS equation) etc
...
g
...
then the equation can be rewritten in terms of n-k dimensionless variables the mach number
...
The signicance of the viscosity to the thermal
conduction is given by the Prandtl number
...


For an ideal gas, the equation of state is given by:
with the reservoir/surroundings: PV=constant
...


Compressing a sample, without heating, is governed by the Van der waals equation of state:

N kT

where

P+

N 2a
V2

(V − N b) =

a is a measure of the attraction between molecules and b is the average volume of molecules
...


•

If one keeps compressing the uid, the pressure will eventually reach a critical value which will cause a
transition of state i
...
A gas to liquid transition, or a liquid to solid transition etc
...

The use of lower temperatures allows the compression of a uid directly into a solid (or liquid) without the
need for a transition stage (As shown by the second gure)
...




K =

The greater the compressive force acting on the material, the greater the atomic repulsion becomes
...
This

model is useful for planetary formation
...
This EOS relates the pressure
change inside a solid as a function of changing internal energy, at constant volume
...


An EOS is based on multiple factors including: Electronic structure, Thermal motion, Phase transitions (inc
...


3 Shocks in Solids

25

3 Shocks in Solids
3
...
1
...
e
...


•

For the example of a piston, the launching of a piston onto a sample generates a pressure wave where, at the
time of instantaneous impact i
...


•

t = 0
...
2 Rankine-Hugoniot conditions
3
...
1 Hugoniots

•

The physical variables before a shock are:
where

E

P0 , V0 , E0 , up0 , ρ0 whilst after a shock the variables are P1 , V1 , E1 , up , ρ
1
V is the specic volume ∝ ρ
...


(us − up )∆t
...


ρ0 us A∆tup
...


Lecture 19
•

P Aup ∆t
...

p

Energy is dened as the work done on the object thus the energy of the shock is
in K
...
and internal thermal energy (heats up particles inside object)
...
E
...

p
p

•

From the K
...
relation, 3 generalised expressions are formed:




• us

Mass:

ρ0 (up0 − us ) = ρ(up − us )

Momentum:
Energy:

and

up

P − P0 = ρ0 (us − up0 )(up − up0 )

1
1
1
E − E0 = 2 (P + P0 )( ρ0 − ρ )

are the main variables measured since they can be determined from only a single experimental setup

whereby the two are linearly related to each other
...
The locus of points produced gives the Hugoniot
...
i
...
There is a start and end but the middle of the path is undened
since the shock is discontinuous
...


•

Can measure density by using ash x-ray radiography
...


•

Shock time gates, Intereferometry and magnetised wire gauge systems can measure the velocities
...
2
...


Substitution of the relationship into the Pressure relationship:
at

t = 0
...






The particle velocity inside the target:
Replace the
Replace

up(imp) )
...


Pimp = ρ0(imp) us(imp) (v − up(imp) )
...
i
...
The

This gives an exponentially decaying relationship between

Hugoniot relationship
...
e
...


3
...
3 Solving Problems using RH conditions
•

Substitute the particle velocity condtion into the continuity equation and substitute this equation into the
Pressure relationship to obtain:

•

P =

a2 (V0 −V )
(V0 −b(V0 −V ))2


...
V gives an adiabat (decreasing exponential as a lack of heat ow)
...


•

For small pressure shockwaves, the Hugoniot lies above this adiabat - there is a third order dierence in entropy,
density and pressure
...


3 Shocks in Solids

28

Lecture 20
•

The 3 types of Hugoniots covered are:

P

vs
...
up and

us

vs
...


However, the shocks leaving the solids are also interesting
...
3 Reections, Rarefactions and Spall
•

When a shock is responsible for remaking a surface, the shock inducing particles continue to move within
the solid, resulting in the build up of tension
...
This compression direction opposes the rarefaction
wave
...
In general, the shock wave disperses resulting in a spreading of energy
...


•

On a space time graph, where

θ

is dened as the forming angle - The angle of the shock spreading - the eect

looks like the following:

•

After a time t, the shock travels through both the impactor and target after the impact however, the spreading
of this shock is not a universal constant and depends on various factors including material type, defects,
elasticity of the object etc
...


This

tension, if greater than the spall strength (strength required to tear the object into smaller pieces), results in
the material being torn from the inside-out and not from the surface forces
...
4 Stress, Strain and Real Solids
•

Real solids are dened as those that are measurable in laboratories
...

1
∴ P = 3 (σx + σy + σz )

•

The pressure is dened as the average stress (averaged in the x, y and z direction)

•

Take a 3D object to undergo uniaxial strain i
...
It is deformed in 1D only therefore, an object in equilibrium
- compressed in the x-direction follows the boundary condition

•

σy = σz

and so

1
P = 3 (σx + 2σy )
...


•

Plastic deformation - The procedure is as above but, once the force is removed, the material will not return
to normal
...





This deformation occurs above the yield strength

Substitution of this expression into the equation for
or can rearrange for



y = σx −σY
P

(Dierence in latitudinal and longitudinal stresses)
...


∆V
In general, Strain=
V
...
5 Isentropic or Ramp loading
•

The input pressure wave can be controlled, in order to prevent shock formation, by using thinner samples or
by controlling the frequency of the pressure waves i
...
Sending them in pulses at a chosen time
...
The current travelling throught the stripline creates a
magnetic eld which attracts/repel the target and impactor
...





The magnetic pressure is dened as

B2
2µ0

−

µ0 I 2
2ω 2
...
e
...


4 The anatomy of Shocks and Shocks in Plasmas

30

Lecture 21
4 The anatomy of Shocks and Shocks in Plasmas
4
...
1
...
Primarily from:




•

Momentum:
Energy:

ρ0 Us = ρ(Us − up )

where

Us

is the shock velocity in the lab frame
...


For shocks in gases and plasmas, the use of a polytropic (thermodynamic process that is reversible and obeys

PV n =

P = (γ − 1)u
γ = n+2
...
e

given by the ratio of specic heats:

γ=

cv
cp
...
V ∝
...


v
v0

4
...
2 Strong and Weak shocks
•

As a shock becomes stronger,
reaching a maximum value of

•

P > P0 thus
(γ+1)
= (γ−1)
...


the ratio of the densities increases (as does the velocity ratio)

ρ
ρ0

P

P0

and Temperature also increase but the jump in velocity and

4 The anatomy of Shocks and Shocks in Plasmas

•

31

5
3
...


Lecture 22
4
...
3 Noh's Problem
•

Consider the case of a steady ow impacting an immobile wall (assume no edge eects)
...


This

technique is used to x the shock in space, so that its structure can be measured experimentally
...
1
...
A z-pinch causes such behaviour through the use of magnetic compression
- directed towards the centre of the cylinder
...

In this case the accumulated mass builds up in front of the wall whilst the pressure, produced by the shocked
material, reaches equilibrium as a result of an external driving force
...
1
...


vpiston by

combining the two equations

thus giving:

 Us =

γ+1
2

P
p, shocked
ρ0

 vpiston =

2
γ+1

P
piston
ρ0

4
...
2
...
An explosion, however, is where there
is a sudden release of energy which forms a point of high pressure which drives a shock into the surrounding
medium
...
g
...


•

Consider a thin shell of dense gas moving supersonically into a stationary background, composed of gas with
density




ρ0
...


Fext = 0, F = 0 =

d
dt (mv)

= m(t) dv +
dt

dm
dt v(t) where the accelerated mass

m(t) = ρ0

The mass accumulation rate determines the deceleration of the blast wave thus, radius

R

´

V (t)dt


...
e
...


Lecture 23
4
...
3
...
What is meant
by this signicant amount?

•

To calculate the relative importance of radiation, the temperature of the shock is required
...


 Pshocked =


Combining the above two equations (with the fact that
of a strong shock (ρ




∼ ρ0 ),

Example: Hydrogen
...


5
3
...


•

The energy obtained by plasma is eventually lost within a time frame, dependent on the radiation power loss

Ploss ,

rate
...


 Ploss = Bn2 T 2 ωm−3
1

where

B = 1
...


5 Fluid Instabilities


•

34

The cooling time:

τcool =

u
Bn2 T

1
2

where

u=

P
γ−1

= 3nkT

and

2
T = 10−9 Us
...
6ns

4
...
2 Radiative transport and radiative precursors
•

If the material ahead of the shock is dense enough, such that it is dened as opaque with respect to the
radiation, it is possible for the radiative energy emitted to be reabsorbed, resulting in the pre-heating of the
unshocked material
...


•

A radiative wave, or pre-cursor, travels ahead of the shock
...
This results in the discontinuities in

•

and

T

of the shock smoothing out
...
For this situation, radiation is coined signicant
if the energy ux

•

P, ρ, u

σT 4 ≥

Kinetic Energy of the ux, dissipated by the shock

2
= 1 ρ0 Us
...


Lecture 24
5 Fluid Instabilities
5
...
1
...
In this situation, the air remains above the water
...
2
...
e
...


This modulation gave a dispersion relation of the form

ω 2 = gk
...
e
...


•

Using the basic laws of Newtonian gravity, since water has a higher density than air, gravitational eects
should cause it to fall into the air (?)

however, with the inclusion of an interface, the formation of a

microscopic ripple on the interface will result in a packet of air forming at a slightly higher position
...


•

Since air is more buoyant than water, it will continue to rise further until it reaches a point where the layer of
uid above it has a lower density/higher buoyancy than it
...
Example: Glass of whisky underneath a water glass experiment with
paper as the interface
...
i
...
Whenever a light uid
tries to accelerate a heavy one, the heavy one is not moving down but the lighter one is trying to move up
(buoyancy)
...


•

Gravitational acceleration is not the only type e
...
A supernova blast wave
...
and becomes much denser than the interstellar
medium
...
Note: The blast wave is not a singular layered
blast wave but a combination of multiple layers, with the denser blast wave in the centre
...


5
...
2 The generalised dispersion relation for uid waves
•

In section 1
...
4, water waves were considered
...
e
...
e
...


and

2

2 ρ2 (v1 −v2
− kx ρ1 (ρ1 +ρ2 )2 )

Without transverse ow:
Atwood number



∂ψ
∂z
...


For the Rayleigh-Taylor instability, the light uid is supporting the heavy uid i
...

√
the positive root,

v
ω = kx ρ1 ρ1 +ρ2 v2 ±
1 +ρ2

are wave numbers with the latter being dependent on

vertical heght
...


1 2
2
 ρ1 ( 1 v1 + gζ + ∂ψ1 ) = ρ2 ( 2 v2 + gζ + ∂ψ2 )
2
∂t
∂t
 With the assumption of a harmonic solution: ζ and ψ are ∝ ei(kx−ωt)
...


ζ∝e

Akh g

(exponential growth with a growth rate

=

√

Akh g )
...
The low density D-T gas becomes the region of greatest ignition (point which gets hottest
quickest) i
...
ignition hotspot
...


Lasers, or X-rays, causes the interior of the

capsule to implode i
...
The onion layer approach
...


The Rayleigh-Taylor instability sets a limit on the maximum velocity and surface roughness (which

provides the perturbation)
...


•

The imploding sphere, which contains the main fuel mass, results in the hotspot compressing thereby increasing
its temperature and density
...
i
...
A light uid (the hotspot) decelerates a heavy uid (main fuel) which implies a
second phase of the Rayleigh-Taylor instability
...
A physical example of such a wave is re burning down a matchstick
...


5 Fluid Instabilities

38

5
...
3 Non-linear growth of Rayleigh-Taylor instability
•

As deduced in section 5
...
2, the amplitude of the instability growth is

√

∝e

Akh gt


...


•

Damping, as a result of viscosity, surface tension or/and magnetic eld tension have greater eects on shorter
wavelengths
...


•

After a large time period, the Rayleigh-Taylor converts horizontal surfaces (those
vertical surfaces (those

•

⊥

to acceleration) into

to acceleration)
...


As a consequence, light uids are freely ascending whilst heavy uids are freely descening
...
At this stage, the amplitude now grows

∝ gt2

where

there is no Rayleigh Taylor on a very small scale
...


•

As the water falls into the air, the interface between the two uids results in very strong vertical shear (which
increases for greater travel distance) resulting in vortex formation due to vorticity eects
...


Lecture 26
5
...


•

Microscopic perturbation of the surface causes dierences in the streamlines on either side of the interface
...

The higher velocity regions (over peaks) correspond to lower pressures as a result of Bernoulli's principle
...


•

The positive feedback eect results in the wave instability to grow
...
e
...

2

This growth rate becomes non-linear, very rapidly, resulting in the overall size of the rippled surface to
dramatically increase in drag, leading to shear, rotation and vortex formation
...


5
...
This shear
drives the Kelvin-Helmholtz instabilty on shorter wavelengths
...
E
...
once the stirring is stopped, the Kelvin-Helmholtz instability produces shorter scale vortices
...
E
...


is larger
...

This implies

5

E(k) ∝ k − 3
...


To model the turbulence eects:
so, for a 3D model,

>

> 104

in each direction

109 − 1010
...




---