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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: onur.akay@okan.edu.tr
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
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)
Onur Akay, Ph.D. CE 204 Fluid Mechanics 2
- 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.
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
Streamlines and Flow Patterns:
- Flow pattern shows the flow direction.
- Streamlines are drawn tangent to local velocity vectors.
Onur Akay, Ph.D. CE 204 Fluid Mechanics 3
The flow pattern provided by the streamlines
represent an instantaneous visualization of the
flow field.MOVIES
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
Uniform Flow: The velocity does not change along a fluid path.
s: the distance traveled by a fluid particle along a path.
t: time
Onur Akay, Ph.D. CE 204 Fluid Mechanics 4
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
Nonuniform Flow: The velocity changes along a fluid path.
s: the distance traveled by a fluid particle along a path.
t: time
Onur Akay, Ph.D. CE 204 Fluid Mechanics 5
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
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.
Onur Akay, Ph.D. CE 204 Fluid Mechanics 6
Streamlines give no indication of the steadiness or unsteadiness of the flow!!!
Steady or unsteady?
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
Laminar Flow: Well-ordered state of flow in which adjacent flow layers move smoothly with
respect to each other..
Onur Akay, Ph.D. CE 204 Fluid Mechanics 7
MOVIETurbulent Flow: Unsteady flow characterized by intense cross-stream mixing.
Instantaneous
velocity profile
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
Pathlines and Streaklines:
Onur Akay, Ph.D. CE 204 Fluid Mechanics 8
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
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
Onur Akay, Ph.D. CE 204 Fluid Mechanics 9
MOVIE
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
One-Dimensional and Multi-Dimensional Flows:
The dimensionality is characterized by the number of spatial dimensions needed to
describe the velocity field.
One-dimensional
Onur Akay, Ph.D. CE 204 Fluid Mechanics 10
Two-dimensional
Three-dimensional
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
Acceleration:
Acceleration is the rate of change of the particle’s velocity with time.
Velocity
Onur Akay, Ph.D. CE 204 Fluid Mechanics 11
Velocity
Acceleration
an : normal component of acceleration
(centripetal)
at : tangential component of acceleration
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
Acceleration:
Onur Akay, Ph.D. CE 204 Fluid Mechanics 12
convective
local
centripetal
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
Acceleration:
convective
local
centripetal
In steady flow the local acceleration is zero.
Onur Akay, Ph.D. CE 204 Fluid Mechanics 13
In steady flow the local acceleration is zero.
In uniform flow the convective acceleration is zero.
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
Onur Akay, Ph.D. CE 204 Fluid Mechanics 14
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
Lagrangian and Eulerian Approach:
Objective: To determine the pressure and velocities at arbitrary locations in a flow field.
Onur Akay, Ph.D. CE 204 Fluid Mechanics 15
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
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
Onur Akay, Ph.D. CE 204 Fluid Mechanics 16
Same ideas are applied to a moving fluid, but now we are executing Newton’s second law.
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
Euler’s Equation:
Onur Akay, Ph.D. CE 204 Fluid Mechanics 17
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
Euler’s Equation:
Onur Akay, Ph.D. CE 204 Fluid Mechanics 18
For an incompressible flow, γ is constant
Euler’s Equation:
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
Euler’s Equation:
In a static body of fluid, Euler’s equation reduces to:
(Hydrostatic Equation)
Onur Akay, Ph.D. CE 204 Fluid Mechanics 19
REMEMBER
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
Onur Akay, Ph.D. CE 204 Fluid Mechanics 20
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
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:
Onur Akay, Ph.D. CE 204 Fluid Mechanics 21
Uniform or nonuniform?
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
Onur Akay, Ph.D. CE 204 Fluid Mechanics 22
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
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
Onur Akay, Ph.D. CE 204 Fluid Mechanics 23
For steady flow:
piezometric kinetic pressure
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
Bernoulli Equation:
Onur Akay, Ph.D. CE 204 Fluid Mechanics 24
Chapter 4 Kinematics of Fluids
Descriptions of Fluid Flow, Euler’s Equation, Bernoulli Equation
Onur Akay, Ph.D. CE 204 Fluid Mechanics 25
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