Fluid Mechanics

FLOWING FLUIDS

Engineering Fluid Mechanics 8/E by Crowe, Elger, and RobersonCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Streamlines & Flow Patterns

Flow Pattern: Construction of streamlines showing the flow direction

Streamlines (light blue): Local velocity vector is tangent to the streamline at every point along the line at a single instant.

Flow through an opening in a tank & over an airfoil section.

Streamline & Pathline

Streamline: line drawn through flow field such that local velocity vector is tangent at every point at that instant

– Tells direction of velocity vector

– Does not directly indicate magnitude of velocity

• Pathline: shows the movement of a particle over time

► In unsteady flow, all can be distinct lines.

► The latter two tells us the history of flow as the former indicates the current flow pattern.

Examples...

Predicted streamline pattern over the Volvo ECC prototype.

Pathlines of floating particles.

TYPES OF FLOW

Uniform: Velocity is constant along a streamline(Streamlines are straight and parallel)

0

s

V

0

s

V

Non-uniform: Velocity changes along a streamline (Streamlines are curved and/or not parallel)

Express velocity V = V(s,t)

Vortex flow

Steady: streamline patterns are not changing over time

Unsteady: velocity at a point on a streamline changes over time

0

t

V

0

t

V

Flow patterns can tell you whether flow is uniform or non-uniform, but not steady vs. unsteady… Why?

Because streamlines are only instantaneous representation of theflow velocity.

TYPES OF FLOW

LAMINAR & TURBULENT FLOW

(a) Experiment to illustrate the type of the flow (b) Typical dye streaks for different cases

(a) (b)

Engineering Fluid Mechanics 8/E by Crowe, Elger, and RobersonCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

LAMINAR & TURBULENT FLOW

DIMENSIONALITY OF FLOW FLIED

→ Characterized by the number of spatial dimensions needed to describe velocity field.

1-D flow:

Axisymmetric uniform flow in a

circular duct

2-D flow:

Uniform flow in a square duct

3-D flow:

Uniform flow in an expanding

square duct

FLOW ACCELERATION (rate of change of velocity with time)

• Consider a fluid particle moving along a pathline...

• There are two components of acceleration:

Tangential to pathline

at : the time-dependent acceleration related to change in speed.

Normal to pathline

an : the centripetal acceleration related to motion along a curved pathline.

Flow Acceleration

Local acceleration – occurs when flow is unsteady (direction or magnitude is changing with respect to time)

Convective acceleration – occurs when flow is nonuniform (acceleration can depend on position in a flow field)

Local acceleration – occurs when flow is unsteady

Centripetal acceleration – occurs when the pathline is curved(normal to the pathline & directed toward the center of rotation)

Example: Convective Acceleration

The nozzle shown below is 0.5 meters long. Find the convective acceleration at x = 0.25 m. The equation describing velocity variation is provided below.

Problem 4.17:

Problem 4.17: (Solution)

Example: