Download - CE 204 FLUID MECHANICS - Okan University · Chapter 4Kinematics of Fluids Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation CE 204 FLUID MECHANICS Onur AKAY Assistant

Chapter 4 Kinematics of Fluids

Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation

CE 204

FLUID MECHANICS

Onur AKAY

Assistant Professor

Onur Akay, Ph.D. CE 204 Fluid Mechanics 1

Assistant Professor

Okan University

Department of Civil Engineering

Akfırat Campus

34959 Tuzla-Istanbul/TURKEY

Phone: +90-216-677-1630 ext.1974

Fax: +90-216-677-1486

E-mail: [email protected]



Kinematics of Fluids:

The relationship between pressure and flow velocity is important in many engineering

applications.

- Blood pressure (flow of blood through veins and arteries.

- Weather forecasts (pressure readings of atmospheric flow patterns)


- Stirring a cup of coffee (pressure variations enhance mixing)

- Design of tall structures (pressure forces from the wind)

- Aircraft design (lift and drag)

- Flow systems (heating and air conditioning)

The eye of a

hurricane.

The space

shuttle during

liftoff.



Streamlines and Flow Patterns:

- Flow pattern shows the flow direction.

- Streamlines are drawn tangent to local velocity vectors.


The flow pattern provided by the streamlines

represent an instantaneous visualization of the

flow field.MOVIES



Uniform Flow: The velocity does not change along a fluid path.

s: the distance traveled by a fluid particle along a path.

t: time




Nonuniform Flow: The velocity changes along a fluid path.

s: the distance traveled by a fluid particle along a path.

t: time




Steady Flow: The velocity at a given point on a fluid path does not change with time.

Unsteady Flow: The velocity at a given point on a fluid path changes with time.


Streamlines give no indication of the steadiness or unsteadiness of the flow!!!

Steady or unsteady?



Laminar Flow: Well-ordered state of flow in which adjacent flow layers move smoothly with

respect to each other..


MOVIETurbulent Flow: Unsteady flow characterized by intense cross-stream mixing.

Instantaneous

velocity profile



Pathlines and Streaklines:




Pathlines and Streaklines:

Examples:

http://www.tecplot.com/showcase/contours/article.aspx?issue=35&article=2

http://en.wikipedia.org/wiki/Streamlines,_streaklines,_and_pathlines


MOVIE



One-Dimensional and Multi-Dimensional Flows:

The dimensionality is characterized by the number of spatial dimensions needed to

describe the velocity field.

One-dimensional


Two-dimensional

Three-dimensional



Acceleration:

Acceleration is the rate of change of the particle’s velocity with time.

Velocity


Velocity

Acceleration

an : normal component of acceleration

(centripetal)

at : tangential component of acceleration



Acceleration:


convective

local

centripetal



Acceleration:

convective

local

centripetal

In steady flow the local acceleration is zero.


In steady flow the local acceleration is zero.

In uniform flow the convective acceleration is zero.



Lagrangian and Eulerian Approach:

Objective: To determine the pressure and velocities at arbitrary locations in a flow field.




Euler’s Equation:

Hydrostatic equations were derived by equating the

sum of forces on a fluid element equal to zero.

Leonhard Euler

Born: 15 April 1707, Basel, Switzerland

Died: 18 September 1783, St. Petersburg, Russia


Same ideas are applied to a moving fluid, but now we are executing Newton’s second law.



Euler’s Equation:




Euler’s Equation:


For an incompressible flow, γ is constant

Euler’s Equation:



Euler’s Equation:

In a static body of fluid, Euler’s equation reduces to:

(Hydrostatic Equation)


REMEMBER



Pressure Distribution in Rotating Flows: Euler’s Equation

Apply Euler’s equation outward from the center of rotation.

Since the flow is steady, the partial derivative has been

replaced by an ordinary derivative:


Uniform or nonuniform?



Bernoulli Equation:

Bernoulli equation is developed by applying Euler’s

equation along a pathline:

Daniel Bernoulli

Born: 8 February 1700, Groningen, Netherlands

Died: 8 March 1782, Basel, Switzerland


For steady flow:

piezometric kinetic pressure



Bernoulli Equation:
