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)