VVR 120 Fluid Mechanics
6. Basic definitions, basic equations I (4.24.4)
•
Stationary and non-stationary flow, streamline,
streamtube
• One-, two-, and three-dimensional flow
• Laminar and turbulent flow
• Reynolds´ number
• System and control volume
• Continuity equation
Exercises: C1, C2, C4, and C7
VVR 120 Fluid Mechanics
CLASSIFICATION OF FLOWS
Flow characterized by two parameters – time and distance.
Division of flows with respect to time:
• Steady flow – time independent
• Unsteady flow – time dependent
• Quasi-steady flow – slow changes with time
Division of flows with respect to distance:
• Uniform flow – constant section area along flow path
• Non-uniform flow – variable section area
VVR 120 Fluid Mechanics
Examples of flow types:
Steady uniform flow:
flowrate (Q) and section area (A)
are constant
Steady = time independent
Uniform = constant section
Steady non-uniform flow:
Q = constant, A = A(x).
VVR 120 Fluid Mechanics
Unsteady uniform flow:
Q = Q(t), A = constant
Unsteady non-uniform flow:
Q = Q(t), A = A(x).
Steady = time independent
Uniform = constant section
VVR 120 Fluid Mechanics
VISUALIZATION OF FLOW PATTERNS
Streamline:
a curve that is drawn in such a way that it is tangential
to the velocity vector at any point along the curve. A
curve that is representing the direction of flow at a
given time. No flow across a stream line.
VVR 120 Fluid Mechanics
Streamtube:
A set of streamlines arranged to form an imaginary
tube. Ex: The internal surface of a pipeline
VVR 120 Fluid Mechanics
Potential flow: Flow that can be represented by streamlines.
Streakline = path made by injected colour in a flow field
VVR 120 Fluid Mechanics
Example streamline and streakline
A flowfield is periodic in such a way that the streamline pattern is
repeated at fixed intervals. During the first second the fluid is
moving upwards to the right at a 45o angle and during the next
second the fluid is moving downwards to the right at a 45o angle etc
according to Fig. a). The flow velocity is constant = 10 m/s. After 2.5
s the particle track for a particle that is released at point A at time
zero is shown in Fig. b). If colour is injected continuously at point A
from time 0 how will the resulting streakline look like after 2.5 s?
A•
Fig. a) Streamlines
Fig. b) Particle track
VVR 120 Fluid Mechanics
TWO WAYS OF DESCRIBING FLUID MOTION
• Lagrangian view: the path, density, velocity and other
characteristics of each fluid particle in a flow is traced.
• Eulerian view: study the flow characteristics (velocity,
pressure, density, etc.) and their variation with time at
fixed points in space.
VVR 120 Fluid Mechanics
LAMINAR AND TURBULENT FLOW
Laminar flow
• Flow along parallel paths
• Shear stress proportional to
velocity gradient (τ = μ⋅du/dy)
• Disturbances in the flow are
rapidly damped by viscous
action
Turbulent flow
• Fluid particles moves in a
random manner and not in
layers
• Length scales >> molecular
scales in laminar flow
• Rapid continuous mixing
• Inertia forces and viscous
forces of importance
VVR 120 Fluid Mechanics
Reynolds experiment
Q and P variables
•
•
•
Small velocities ⇒ line of
dye intact, movement in
parallel layers ⇒ laminar
flow
High velocities ⇒ rapid
diffusion of dye, mixing ⇒
turbulent flow
Critical velocity ⇒ line of
dye begin to break-up,
transition between laminar
and turbulent flow
VVR 120 Fluid Mechanics
Reynolds´ number
• Reynolds generalized his results by introduction of a
dimensionless number (Reynolds number):
ρVD
Re =
=
ν
μ
VD
ν= μ/ρ, V=Q/A
ν = kinematic viscosity
μ = dynamic viscosity
D = diameter (for pipes)
VVR 120 Fluid Mechanics
Reynolds numbers for pipe flow
• Laminar flow
Re < 2000
• Transitional flow Re = 2000 to 4000
• Turbulent flow
Re > 4000
Two thresholds:
Upper critical velocity – transition of laminar flow to
turbulent flow
Lower critical velocity – transition of turbulent flow to
laminar flow
VVR 120 Fluid Mechanics
The critical Reynolds number, Rc, defining the division
between laminar and turbulent flow, is very dependent on
the geometry for the flow.
• Parallel walls: Rc ≅ 1000 (using mean velocity V and
spacing D)
• Wide open channel: Rc ≅ 500 (using mean velocity V
and depth D)
• Flow about sphere: Rc ≅ 1 (using approach velocity V
and sphere diameter D)
VVR 120 Fluid Mechanics
C1 When 0.0019 m3/s of water flow in a 76
mm pipeline at 20°C, is the flow laminar or
turbulent?
VVR 120 Fluid Mechanics
VVR 120 Fluid Mechanics
C2 What is the maximum speed at which a
spherical sand grain of diameter 0.254 mm
may move through water (20°C) and the flow
regime be laminar?
VVR 120 Fluid Mechanics
FLUID SYSTEM AND CONTROL VOLUME
• Fluid system: Specified mass of fluid within a closed
surface
• Control volume: Fix region in space that can’t be moved
or change shape. Its surface is called control surface.
VVR 120 Fluid Mechanics
CONTINUITY EQUATION
• Steady flow
ρ1⋅V1⋅A1 = ρ2⋅V2⋅A2
m1
m2
(m1 = m2)
Control volume
• Incompressible flow
V1⋅A1 = V2⋅A2 or Q1 = Q2
(Q = V⋅A)
V: Average velocity at a section (m/s)
A: Cross-section area (m2)
Q: Flow rate (m3/s)
Fluid system
volume
VVR 120 Fluid Mechanics
Continuity equation applied to changing
pipe diameter
V1⋅A1 = V2⋅A2 or Q1 = Q2
(Q = V⋅A)
Q = constant, A =(x)
Q2
Q1
Q1=V1 ⋅ A1
Q2=V2 ⋅ A2
Control volume
VVR 120 Fluid Mechanics
•
Flow in a pipe junction
•
Channel flow (unsteady)
d(Vol)/dt = Q1– Q2
(Q12 = 0)
Q1 + Q2 = Q3
or
V1⋅A1 + V2⋅A2 = V3⋅A3
Vol: Volume of water in
channel between section 1 and
2
VVR 120 Fluid Mechanics
C4 Water flows in a pipeline composed of
75 mm and 150 mm pipe. Calculate the
mean velocity in the 75 mm pipe when that
in the 150 mm pipe is 2.5 m/s. What is its
ratio to the mean velocity in the 150 mm
pipe?
VVR 120 Fluid Mechanics
C7 Using the Y and the control volume in the fig. find
the mixture flowrate and density if freshwater (ρ1 =
1000 kg/m3) enters section 1 at 50 l/s, while
saltwater (ρ2 = 1030 kg/m3) enters section 2 at 25
l/s.