Hydrology BC ENV 3025
Principles of Fluid Dynamics
- the dynamics of fluids are the foundation of the understanding of
water
movement in streams and in the subsurface
- we need to understand this in order to figure out how to measure
river
discharge, for example
- the basic principles also apply to the flow of air, lava,
glaciers, and
the Earth's mantle
Definitions and Properties
- we usually classify matter as either solid, liquid, or gas, based
on
macroscopic
properties
- a gas takes on the shape and volume of a container,
- a liquid takes the shape of the portion of the
container that
it
fills but retains a fixed volume
- a solid has its own defined shape as well as volume
- liquids and gases are called fluids
- shear stress is a tangential force per unit area acting on a
surface
- the property of a fluid that describes the resistance to motion
under
an
applied shear stress is termed viscosity
- experiment to determine the viscosity of water (Fig3.1)
- F/A = μ Uplate/d
(dimension/unit
of
viscosity?)
- viscosity as a function of temperature (FigA2.1)
- unit of viscosity: Pa s, dimension?
- for some fluids, density changes
Forces on fluids
- the basic forces that make fluids move are gravity, pressure
differences,
and surface stresses
- pressure is a normal stress
Fluid statics
- hydrostatic equation (Fig3.3)
- p = F/A = m*g/A = V*ρ*g/A
= ρ*g*d (d: depth)
Bernoulli equation
- let us look at the movement of a fluid in
a
pipe
(Fig3.5)
- the statement of the conservation of
energy
for a
frictionless fluid along a flowline is the following:
- p*V + m*g*z + 1/2*mu2 =
constant,
means:
- work due to pressure + potential energy +
kinetic
energy
- by dividing this equation by m = v*ρ*g
we get the Bernoulli equation:
- u2/(2*g) + z +p/(ρ*g)
= constant, means:
- velocity head + elevation head + pressure
head =
total head
- this head can be measured by looking at
the
level
to which the water rises in a vertical tube stuck into the pipe
- assumptions we are making:
- no friction (viscosity = 0)
- incompressible fluid
- homogeneous fluid
- flow steady with time
- example: tank with steady flow of water (Fig3.6)
=> u2 = (2*g*d)1/2
- perform the actual experiment
- example: expansion joint in a hose (Fig3.7)
- apply continuity equation Q = u*A =
const,
then Bernoulli
equation
- decrease of mean velocity results in
increase of
pressure
- experiment: blow on sheet of paper
Friction
- in reality, if we do an experiment, we do
see
a loss
in head along a pipe due to friction (Fig3.8)
- instead of having a uniform value in the
pipe, there
is a velocity profile (Fig3.9)
- u2/(2*g) + z +p/(ρ*g)
= constant needs to be modified to: u2/(2*g) + z
+p/(ρ*g)
+hL= constant
- the head loss can be described as: hL
= f*L*u2/(D*2*g), f being the friction factor
- the friction factor has been measured
under a
range
of circumstances (Fig3.10
for smooth pipes)
- The key parameter to describe a flow
situation is
the Reynolds number: R = ρ*U*D/μ
- at R>2000 something happens, the flow
becomes turbulent
- Reynolds experiment, laminar
("layered") flow,
turbulent ("restless") flow (Fig3.11)
- is the flow through the NYC
water tunnels laminar or turbulent?
- is flow in a straw laminar or turbulent?
- let us calculate R for an example:
- U=2 m/s
- D = 0.03 m
- ρ= 103
kg/m3
- μ =1.139
10-3
Pa s
- how can we make the flow laminar?
- reducing the velocity;
- reducing the diameter;
- reducing the density of the fluid; or
- increasing the viscosity. reduce
- where in nature is turbulent flow
important?