flow field specification
DESCRIPTION
FLOW FIELD SPECIFICATION. Eulerian and Lagrangian descriptions:. Eulerian. Lagrangian. Velocity field and trajectories. Streamlines. Parallel (tangent) to the velocity vector → boundaries are streamlines. Represent velocity trajectories. slope of streamlines. - PowerPoint PPT PresentationTRANSCRIPT
FLOW FIELD SPECIFICATIONEulerian and Lagrangian descriptions:
txVV ,
Eulerian
txVV ,0
Lagrangian
Velocity field and trajectories
Streamlines Parallel (tangent) to the velocity vector → boundaries are streamlines. Represent velocity trajectories.
uv
dxdy
slope of streamlines
www.me.bme.hu/~karolyi/results/advect/advection.html
What can we say about these streamlines?
Pathlines Trajectory of an identified water parcel – represent one particle at different times.
http://projects.ict.usc.edu/animation/fluidsim.htm
0
0
0
0
t
t
dtvyy
dtuxx
(e.g. progressive vector diagrams)
Streaklines Pathlines that move more than a single point through the flow. Represent multiple particles at one time.
https://visualization.hpc.mil/wiki/Streaklines
Effective way to identify flow discontinuities through large separation of the points in the Streakline.
Streamlines = Pathlines = Streaklines if the flow is steady, i.e., 0t
Stream functionIn two dimensional, non-divergent flow we can define a stream function Ψ:
xv
yu
,
http://www.atmos.albany.edu/student/gareth/diagnostics.html
x
y A
B
What is the mass flux ρQ across the line AB?
If AB II to y, thenQ = u dy
But in general AB can be defined by (Δx, Δy); and for convenience the normal vector can be written as: (dy,-dx), so that
Qvdxudydxdyvu ),(),(
Qvdxudydxdyvu ),(),(
ddxx
dyy
Q
If A and B are on the same streamline, then Q = 0, so Ψ must be a constant along a streamline.
udx
vdy
uv
dxdy
;Another way to look at it:slope of streamlines
y
dx
x
dy
xv
yu
,
0
ddyy
dxx
Ψ must be a constant along a streamline.
Differentiation following a fluid parcel
dtdz
zQ
dtdy
yQ
dtdx
xQ
tQ
dtdQ
Adopting the Eulerian description, the rate of change of properties following a parcel:
wzQv
yQu
xQ
tQ
dtdQ
DtDQQu
tQ
dtdQ
localadvective
(chain rule)