3 Fluid and Flow Properties
3.3 Modeling and Visualizing Flow
Basic courses in dynamics introduce the kinematics and kinetics of moving bodies.
Kinematics and
kinetics Kinematic describes the movement of matter and kinetic connects the motion with its causes, i.e. the forces acting on the objects. In particular, you studied Newton’s second
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3.3 Modeling and Visualizing Flow 33
law stating that the inertia force𝑚 𝑎is equal to the resultant external force acting on the body.
𝑚 𝑎 = 𝐹external = ∑
𝑖
𝐹𝑖
Following individual bodies and describing their motions is the Lagrangian formulation Lagrangian
mechanics
of mechanics. It is named after Joseph-Louis Lagrange (* – †), an Italian mathematician who succeeded Leonard Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin. You can find details in your textbook on dynamics or, for starters, read the article on Langrangian mechanics on Wikipedia (http://en.wikipedia.org/wiki/Lagrangian_mechanics).
The Lagrangian description states physical quantities (velocity, momentum, etc.) as properties of a piece of matter with mass𝑚. This is necessary to formulate Newton’s second law of motion. However, the Lagrangian approach does not lend itself well to describe the motion of fluids because we would have to track an infinite number of fluid particles to entirely describe a flow.
It is more practical to study the development of physical properties over time at various Eulerian
mechanics
positions in the fluid domain. Jean-Baptiste le Rond d’Alembert, a French mathemati- cian, physicist, philosopher, and music theorist, introduced this type of mathematical modeling of continua. However, we now call it the Eulerian description. It is named after Swiss mathematician and physicist Leonard Euler (pronounced ‘Oiler’, * –
†), who used this method to formulate the basic equations of motions for an invis- cid fluid. We will study these later in Chapters and . In the Eulerian formulation, physical properties of the flow are described as functions of space𝑥𝑇 = (𝑥, 𝑦, 𝑧)and time𝑡. For example, the pressure function is𝑝=𝑝(𝑥, 𝑦, 𝑧, 𝑡). This is also known as field theory. Chapter reviews the substantial derivative, which converts flow properties from Eulerian into Lagrangian coordinates.
The description of a flow field in Eulerian coordinates has two advantages:
(i) For many flow patterns, a coordinate system exists in which the Eulerian descrip- tion becomes independent of time. For example, describing the flow around a body moving forward with constant speed in a coordinate system which is fixed to the body (moves with the same speed), makes the flow field steady. When you are a passenger on a ship moving steadily in calm water, the wave pattern you observe behind the vessel will not change over time.
(ii) The Eulerian description concurs with our instruments measuring flow properties.
We do not have pressure sensors which can be attached to a fluid particle and track the changes in velocity along the particle’s path. However, we stick Pitot-static tubes into the fluid to measure velocity over time at selected locations. Pitot-static tubes are named after French hydraulic engineer Henri Pitot (*–†). They actually measure dynamic pressure which is proportional to the squared flow velocity.
There are two old but very instructive videos which explain the differences between Lagrangian and Eulerian description of fluid flows. Part : http://www.youtube.com/watch?v=XgL-dnUZc and Part :
http://www.youtube.com/watch?v=BRjBptzhnA
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34 3 Fluid and Flow Properties
Since density and temperature are considered constants in ship hydrodynamics, the
Flow properties
objective is to find the spatial and time dependent distribution of velocity and pressure.
velocity vector 𝑣 =
⎛⎜
⎜⎝
𝑢(𝑥, 𝑦, 𝑧, 𝑡) 𝑣(𝑥, 𝑦, 𝑧, 𝑡) 𝑤(𝑥, 𝑦, 𝑧, 𝑡)
⎞⎟
⎟⎠
(.)
pressure 𝑝 = 𝑝(𝑥, 𝑦, 𝑧, 𝑡) (.)
Knowledge of these flow properties enables us to compute forces acting on a vessel.
Three basic principles of continuum mechanics are available to formulate a mathemati-
Basic principles
cal model of the flow:
• conservation of mass,
• conservation of momentum, and
• conservation of energy.
We will concentrate on the first two of these principles. Conservation of mass leads to the continuity equation. Conservation of momentum is equivalent to Newton’s second law and results in the Navier-Stokes equations. For applications in ship hydrodynamics, the equation for the conservation of energy is often not directly used because the remaining two are sufficient to describe flows with constant temperature, viscosity, and density. However, conservation of energy is often employed to model turbulence.
Flow patterns can be visualized in different ways. If you follow an object and record
Pathline
its path, you see its pathline. Just observe the flight of a bird or watch a leaf floating down a stream. Of course, observing the pathline of a fluid particle in a flow is almost impossible, because we cannot easily identify individual particles. For visualization purposes, however, small, highly reflective particles may be used.
Flow can also be visualized by inserting a constant stream of colored fluid or smoke
Streaklines
into the flow. The smoke enters the flow at a fixed point. The visible lines of smoke emanating from that point are called streaklines. For instance, the smoke rising from a candle which has just been extinguished forms a streakline.
Most widely known are streamlines. These are lines which are tangent to the velocity
Streamlines
vector at all points they include. Since the flow is tangent to the velocity vector, fluid does not cross streamlines. Therefore, streamlines divide the flow into regions. Streamlines may be computed from the Eulerian formulation. Visualizing streamlines, however, is not quite as simple. You may seed a flow with reflective particles and take a photo with a suitable long exposure time. Particles will appear as short dashes on the picture which may be interpreted as velocity vectors. Drawing curves tangent to the vectors reconstructs the streamlines. In an unsteady flow, streamlines will constantly change.
For time independent flows (steady flows), pathlines, streaklines, and streamlines are identical, which makes their visualization a lot easier.
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