Pathlines
Pathlines behind two
square cylinder
obstructions, Re=1000,
SLFCD package demo
Trajectories of moving fluid
particles
dx i
=ui (x i ,t )
dt
To find a pathline going
through x0,y0,z0, integrate
these equations with initial
conditions
x=x0,y=y0,z=
Streamlines
Lines whose
tangents are
parallel to
velocity field:
Vorobieff et al., Phys. Fluids 11, 2167 (1999)
dy v dz w
= ,
= ,
dx u dx u
dz w
=
dy v
or.
dx dy dz
= = =ds
u
v w
Increment along the
streamline
Critique of streamlines
Illustrate flow topology (critical points, etc.)
Reference-frame dependent
Time-dependent
Can be produced in experiment with steady flow
(/t=0)
Can be misleading (in transient flow, they do not
show where stuff ends up)
Streaklines
Tracer (dye, smoke)
continuously injected into
flow at a given point
Easy experimental
implementation, tricky
analytical treatment
H. Babinsky, Cambridge
University
Analytical construction of a streakline
Start with obtaining pathline equations
For a moment t, find the pathline produced by a
particle that was at the injection point x0,y0,z0
at t=t
For a pathline at t=t0, repeat this process for all
tt0
x i=x i (x 0 ,
Equations of streamline in Einstein notation
d xi
=u i (x , t)
ds
Only valid for a specific t!
Integrate to get equation of streamline passing
through x0,y0,z0 at given t parameters
x i=x i ( x 0, y 0, z 0, t , s )
variable
2. Flow kinematics
2.1. Flow lines
Streamlines
Pathlines
Parallel to velocity field
trail left by a moving fluid particle
Streaklines
Produced by stationary tracer injection into moving
flow
Critique of pathlines
Show where a fluid volume will end up
Can be traced experimentally by following a
blob of tracer inserted into the flow
Construction requires knowledge of the time
history of the flow
In stationary flow, will be identical to streamli
Streakline critique
Easily implemented experimentally
Can be misleading
Same as streamline and streakline in stationary
flow
Not easy to construct analytically