Eulerian Description

• A finite volume called a flow domain or control volume is defined through which fluid flows in and out.• There is no need to keep track of the position and velocity of a fluid particle of fixed identity.• Field variables such as pressure, velocity and acceleration fields are defined.

• Pressure field:

Velocity field:

Acceleration field:

( , , , )p p x y z t

( , , , )V V x y z t

( , , , )a a x y z t

Difference Between Lagrangian and Eulerian Description

• Imagine a person standing beside a river measuring its properties. In the Lagrangian approach, he throws in a probe that moves downstream with the water. In the Eulerian approach, he anchors the probe at a fixed location in the water.• Experimental measurements are more suitable to the Eulerian Description.• Equations of motion of fluid flow in Lagrangian description are well defined (Newtons’ second law), but needs to be carefully derived for the Eulerian description

Example 1 A Steady 2-dimensional Velocity Field

A steady, incompressible, two dimensional velocity field is

given by

where the x- and y-coordinates are in meters and themagnitude of velocity is in m/s. A stagnation point isdefined as a point in the flow field where the velocity is zero. (a) Determine if there are any stagnation points in this flow field and, if so, where? (b) Sketch velocity vectors at several locations in the domain between x = -2 m to 2 m and y = 0 m to 5 m; qualitatively

•describe the flow field.

( , ) (0.5 0.8 ) (1.5 0.8 )V u v x i y j

Acceleration Field( , , , )particle particle particle particle

particle

dV x y z tdV dVa

dt dt dt

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particle particle particle

particle particle particle

dx dy dzV dt V V V

t dt x dt y dt z dt

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partial derivative poerato d total derivative poerator

, ,particle particle particledx dy dzu v w

dt dt dt

( , , , ) ( . )dV V

a x y z t V Vdt t

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where gradient or del operator��������������

( , , , )particledV V V V V

a x y z t u v wdt t x y z

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Transforming from the Lagrangian to the Eulerian frams of reference ,

Acceleration Field

gradient or del operator=( , , )=i j kx y z x y z

��������������������������������������������������������

in Cartesian soordinate,

x

u u u ua u v w

t x y z

y

v v v va u v w

t x y z

z

w w w wa u v w

t x y z

/ local acceration;( . ) advective accerationV t V ������������������������������������������

For steady flow, local acceleration is zero, but advective acceleration is nonzero even for steady flows.

Example 2

• Nadeen is washing her car, using a nozzle. The nozzle is

3.90 in (0.325 ft) long, with an inlet diameter of 0.420 in

(0.0350 ft) and an outlet diameter of 0.182 in. The volume

flow rate through the garden hose (and through the nozzle)

is V =0.841 gal/min (0.00187 ft3/s), and the flow is steady.

Estimate the magnitude of the acceleration of a fluid

particle moving down the centerline of the nozzle

Material Derivative

Material derivative: ( . )D d

VDt dt t

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Material acceleration:

Material derivative of pressure:

( , , , ) ( . )dV V

a x y z t V Vdt t

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( . )DP dP P

V PDt dt t

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Material derivative is also called total, particle, Lagrangian, Eulerian and substantial derivative

Material Acceleration of a Steady VelocityField

Consider the steady incompressible two-dimensional

velocity field. (a) Calculate the material acceleration at

the point (x=2 m, y= 3m).

( , ) (0.5 0.8 ) (1.5 0.8 )V u v x i y j

Flow visualization

Spinning baseball at speed

77 ft/s, rotated at 630 ppm.

dr dx dy dz

V u v w

Streamlines: A streamline is a curve that is everywheretangent to the instantaneous local velocity vector. They are useful indicators of the instantaneous direction of fluid motion.

along a streamline( )dy u

dx v

Stremline in the xy - plane

Example 3

Consider the steady incompressible two-dimensional

velocity field. Plot several streamlines in the right half of

the flow (x>0)

( , ) (0.5 0.8 ) (1.5 0.8 )V u v x i y j

Note: Velocity vectors are superimposed on the streamlines. The speed can not be determined directly from the streamlines alone.

Streamtube• A streamtube consists of a bundle of streamlines. Since streamlines are everywhere parallel to the local velocity, fluid cannot cross a streamline.• Fluid within a streamtube must remain there and cannot cross the boundary of the streamtube.• Both streamlines and streamtubes are instantaneous quantities, defined at a particular instant in time according to the velocity field at that instant.

• In an unsteady flow, the streamline pattern may change significantly with time, but the mass flow rate passing through any cross-sectional slice of a given streamtube must remain the same.• For example, in a converging portion of an incompressible flow field, the diameter of the streamtube must decrease as the velocity increases. Likewise, the streamtube diameter increases in diverging portions of the incompressible flow

Pathlines

• A pathline is the actual path traveled by an individual fluid particle over some time period.• A pathline is a Lagrangian concept in that we simply follow the path of an individual particle.• For steady flow pathlines are identical to streamlines

Streaklines• A streakline is the locus of fluid particles that have passed sequentially through a prescribed point in the flow.• Streaklines are the most common flow pattern generated in a physical experiment. If a small tube is inserted into a flow and introduce a continuous stream of tracer fluid (dye in a water flow or smoke in an airflow), the observed pattern is a streakline.

• Streaklines are often confused with streamlines or pathlines.

• While the three flow patterns are identical in steady flow, they can be quite different in unsteady flow.

• The main difference is that a streamline represents an instantaneous flow pattern at a given instant in time, while a streakline and a pathline are flow patterns that have some age and thus a time history associated with them.

• A streakline is an instantaneous snapshot of a time integrated flow pattern.

• A pathline, on the other hand, is the time-exposed flow path of an individual particle over some time period.