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
• church glass windows story
  
• 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*U_plate/d
  
• viscosity as a function of temperature (FigA2.1)
• unit of viscosity: Pa s, dimension?
• density ρ of water, unit: kg/m³
  

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 =µ*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:
• m*g*z+ p*V  + 1/2*mu² = constant, means:
• potential energy + work due to pressure +  kinetic energy = constant
  
• by dividing this equation by m*g= v* ρ*g, and some rearranging 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 in relation to a level reference surface
  
• 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, compare predicted and measured u2
  
• 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)  +h_L= constant
• the head loss can be described as: h_L = 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?
1. reducing the velocity;
2. reducing the diameter;
3. reducing the density of the fluid; or
4. increasing the viscosity. reduce
• where in nature is turbulent flow important?