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Lecture 5

# ENB221 Fluid Mechanics Week 5.docx

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Course Code
ENB205
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Negareh Ghasemi

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ENB221 Lecture Fluid Mechanics Fluid Kinematics The study of motion in fluids, two distinct ways to describe fluid motion:  Langrangian – Sitting in a boat (in flow)  Eulerian – Sitting on a bank (observing flow) System A system is defined as a fixed, identifiable quantity of mass; the system boundaries separate the system from the surroundings. The boundaries of the system may be fixed, or movable but no mass crosses them. Also known as a Closed system.  Heat and work may cross the boundaries of the system  The quantity of matter within the system boundaries remains fixed Control Volume A control volume is an arbitrary volume In space through which fluid flows in and out. Control volume may also be known as an open system or a flow domain. The geometric boundary of the control volume is called the control surface. The control surface may be real or imaginary; it may be at rest or in motion Flow Visualisation Streamlines A streamline is a curve that is everywhere tanget to the instantaneous local velocity vector. Useful as indicators of the instantaneous direction of fluid motion throughout the flow field. Cannot be directly observed experimentally – except in steady flow fields. Streamtubes A series of streamlines drawn through every point on the perimeter of a small cross-sectional area. As streamlines are everywhere parallel to the local velocity, fluid cannot cross a streamline by definition. Fluid within the streamtube must remain there and cannot cross the boundary of a streamtube. Just like streamlines, streamtubes are an instantaneous quantity. The mass flow rate, m, at any instant passing through any cross-sectional slice of a stream tube must be equal. The diameter of the streamtube must decrease as the velocity increases to conserve mass. Pathlines A pathline is the actual path, or trajectory, travelled by an individual moving fluid particle over some time period. To determine a pathline, we might idenify a fluid particle at a given instant of time, for example, by the use of dye and then take a long expsoure photograph of its subsequent motion, the line is the pathline. 1 Week 5 Tuesday, 27 August 2013 ENB221 Lecture Fluid Mechanics Streaklines A streakline is the locus (position) of fluid particles that have passed sequentially through a prescribed point in the flow. Shown by introducing a steady stream of dye or tracer fluid from a single point in the fluid. NOTE: Whilst streamlines, pathlines and streaklines sound the same, they are not  A streamline represents an instantaneous flow pattern at a given time.  Streaklines and Pathlines are flow patterns that have some time history attached to them.  A Streakline is an instantaneous snapshot of a time integrated flow pattern.  A Pathlineis a time exposed flow path of an individual particle over some period of time However, if the flow is steady they are identical Lagrangian Description Follows the path of individual objects and uses Newton’s laws to describe motion. Can accurately predict where they go, and predict how much momentum and kinetic energy are exchanged from one object to another. Involves keeping track of the position and velocity vector of each object with respect to time. Also referred to as system analysis and it follows a mass of fixed identity. Limitations  Difficult because of large numbers of particles o from a microscopic point of view a fluid is comprised of billions of molecules that are continuously banging into each other.  Difficult to identify and define fluid particles  Fluid parcles continuely deform as they move  Easier to think of a fluid as a continuum (macroscopic point of view) o Makes intereactions not as easy to define as those from more distinct objects (e.g. billiard balls) Application  Tracking of passive scalars in a flow o Fluid contaminants  Rarefied gas dynamics calculations concerning reentry of a spaceship into the Earth’s atmosphere.  The development of flow measurement systems based on particle imaging. Eulerian Description Flow happens through a control volume (or a flow domain), through which fluid flows in and out. We don’t track the mass of fluid particles with fixed identity, rather we define field variables, functions of space and time, within the control volume. Field Variables General unsteady three dimensional fluid flow in Cartesian coordinates ( ) ( ) ( ) ( ) ( ) ( ) In the Eulerian description we define field variables at any location in the control volume and at any instant in time. It also does not matter what happens to individual fluid particles, we are only concerned with what happens to the fluid particles that are at the point of interest at the time of interest. 2 Week 5 Tuesday, 27 August 2013 ENB221 Lecture Fluid Mechanics Field Flow Collectively the field variables define the field flow. There variables are expanded in cartesian coordinates: ( ) ( ) ( ) The same treatment applies to the other field variables. From a practical perspective In most experiements it is more convenient to measure what is happening at a particular point with respect to time. This is using the Eulerian method of analysis, unfortunately the analysis with this method can be more challenging (although much more practical because you don’t have to consider the fluid as particles but rather as a continuum). In the Lagrangian method the equations of motion are well established and simple to use, care must be taken with the Eulerian method in describing the motion. Steady Uniform Flow Conditions do not change with position or time, the velocity and cross-sectional area of the stream of fluid are the same at each cross-section. Flow of a liquid through a pipe of uniform bore running completely full at constant velcoity. Steady Non-Uniform Flow Conditions change from point-to-point but not with time, the velocity and cross sectional area of the stream may vary from cross-section to cross- section. For each cross section they will not vary with time Flow of a liquid at a constant rate through a tapering pipe running completely full Unsteady Uniform Flow At a given instant of time the velocity at every point is the same, but the velocity will change with time. Accelerating flow of a liquid from a pipe of uniform bore running full. Unsteady Non-Uniform flow The cross-sectional area and velocity vary from point-to-point and also change with time. A wave travelling along a channel. Frames of Reference Whether fluid motion is steady or unsteady depends on the location of the observer; e.g. a wave in a ch
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