Transcript CVE 240 – Fluid Mechanics
Fluid Mechanics
CHAPTER 4
EULER’S EQUATION
Dr . Ercan Kahya
Engineering Fluid Mechanics 8/E
by Crowe, Elger, and Roberson Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Review of Definitions
•
Steady flow:
•
Unsteady flow: velocity is constant with respect to time velocity changes with respect to time
•
Uniform flow: velocity is constant with respect to position
•
Non-uniform flow: velocity changes with respect to position
•
Local acceleration:
– change of flow velocity with respect to
time
– occurs when flow is unsteady •
Convective acceleration:
– change of flow velocity with respect to
position
– occurs when flow is non‐uniform
EULER’S EQUATION
• To predict
pressure variation in moving fluid
• Euler’s Equation is an extension of the hydrostatic equation for accelerations other than gravitational • RESULTED FROM APPLYING NEWTON SECOND LAW TO A FLUID ELEMENT IN
THE FLOW OF INCOMPRESSIBLE, INVISCID FLUID
Assume that the viscous forces are zero
EULER’S EQUATION
F l
ma l
ΔP Δl
γsinα
ρa
l Taking the limit of the two terms at left side at a given time as Δl → 0
l
(
p
γz) ρa l
ACCELERATION IS IN THE DIRECTION OF DECREASING PIEZOMETRIC PRESSURE!!! When “a = 0” → Euler equation reduces to hydrostatic equation!
In the x direction, for example:
x
(
p
γz) ρa x (
p
γz) 2 (
p
γz) 1
x
ρa x “2” and “1” refer to the location with respect to the direction l (When l = x direction, then “2” is the right-most point. When l = z direction, “2” is the highest point.)
EULER’S EQUATION
An example of Euler Equation is to the uniform acceleration of in a tank: Open tank is accelerated to the right at a rate a x For this to occur ; a net force must act on the liquid in the x-direction To accomplish this ; the liquid redistributes itself in the tank (A’B’CD) – The rise in fluid causes a greater hydrostatic force on the left than the right side → this is consistent with the requirement of “ F = ma ” – Along the bottom of tank, pressure variation is hydrostatic in the vertical direction
EULER’S EQUATION
l
(
p
γz)
ρa
l •
The component of acceleration in the l direction : a
x cosα
Apply the above equation along A’B’
d dl
( γz)
ρa x Cos
Apply the above equation along DC tan -
a x g dz dl
-
a x Cos
g
sin
Example 4.3
: Euler’s equation
A • The truck carrying gasoline (
γ = 6.60 kN/m3) and is slowing down
at a rate of 3.05 m/s 2 .
• 1) What is the pressure at point A?
• 2) Where is the greatest pressure & at what value in that point?
l
(
p
γz) ρa l
Solution:
• Apply Euler’s equation along the top of the tank; so z is constant • Assume that deceleration is constant • Pressure does not change with time
d dl
(
p
γz)
ρa
l Along the top the tank
dp dl
ρa l
p
ρa l
l
C
Euler’s equation in vertical direction: ( Note that a z =0 )
d dz
(
p
γz) ρa z (
p bottom
γz
bottom
) (
p top
γz
top
) Pressure variation is hydrostatic in the vertical direction
Centripetal (Radial) Acceleration
a r
V t
2
r
2
r
• For a liquid rotating as a rigid body:
V = ω r
• a r = centripetal (radial) acceleration, m/s 2 • V t = tangential velocity, m/s • r = radius of rotation, m • ω = angular velocity, rad/s
Pressure Distribution in Rotating Flow
• A common type of rotating flow is the flow in which the fluid rotates as a rigid body. • Applying Euler Equation in the direction normal to streamlines and outward from the center of rotation (
OR
INTEGRATING EULER EQUATION IN THE RADIAL DIRECTION FOR A ROTATING FLOW ) results in
p
z
2
2
r g
2
C Pressure variation in rotating flow
Note that this is not the Bernoulli equation • When flow is rotating, fluid level will rise away from the direction of net acceleration
Example 4.4:
Find the elevation difference between point 1 and 2
p
1
z
1 2 2
r
1 2
g
p
2
z
2 2 2
r
2 2
g
p 1 = p 2 = 0 and r 1 = 0 , r 2 = 0.25m then →
z
1
z
2 2 2
r
2 2
g
z 2 – z 1 = 0.051m & Note that the surface profile is parabolic
Pressure Distribution in Rotating Flow
p
1
z
1 2 2
r
1 2
g
p
2
z
2 2 2
r
2 2
g Another independent equation;
The sum of water heights in left and right arms should remain unchanged
p = pressure, Pa γ = specific weight, N/m3 z = elevation, m ω = rotational rate, radians/second r = distance from the axis of rotation
Bernoulli Equation
Integrating Euler’s equation along a streamline in a steady flow of an
incompressible, inviscid
fluid yields
t
he Bernoulli equation:
V
2 2
g
P
z
C
z: Position p/ γ : Pressure head V 2 /2g: Velocity head C: Integral constant
Application of Bernoulli Equation Bernoulli Equation: – Piezometric pressure :
p + γz
– Kinetic pressure :
ρV 2 /2
For the steady flow of incompressible fluid inviscid fluid the sum of these is constant along a streamline
Application of Bernoulli Equation: Stagnation Tube
V 1 2 2 g p 1 V 2 2 2 g p 2
V
1 2 2 (
P
2
P
1 )
P
1
d
P 2 (
l
d
)
V
1 2 2 ( (
l
d
)
d
)
V
1 2
gl
Stagnation Tube
p 1 / p 2 / 1 2 h=V 2 /2g V 1 2 2 g p 1 V 2 2 2 g p 2 V 2 =0 & z 1 = z 2 V 1 2 g p 2 p 1 2 g h
Application of Bernoulli Equation: Pitot Tube
Bernoulli equation btw static pressure pt 1 and stagnation pt 2; V 1 2 2 g p 1 z 1 V 2 2 2 g p 2 z 2 H V 2 = 0 then
Pitot tube equation
; h 1 1 2 1
V
1 2
g
p
2
p
1 2
p
Stagnation point p 1 h 1 1 h h 1 h p 2 p 2 p 1 h 1 1 h s 1 s 1
VENTURI METER
The Venturi meter device measures the flow rate or velocity of a fluid through a pipe. The equation is based on the Bernoulli equation, conservation of energy, and the continuity equation.
Solve for flow rate Solve for pressure differential
Class Exercises:
(Problem 4.42)
Class Exercises:
(Problem 4.59)