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CE 150 Fluid Mechanics

CE 150 Fluid Mechanics. G.A. Kallio Dept. of Mechanical Engineering, Mechatronic Engineering & Manufacturing Technology California State University, Chico. Fluid Kinematics. Reading: Munson, et al., Chapter 4. Introduction.

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CE 150 Fluid Mechanics

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  1. CE 150Fluid Mechanics G.A. Kallio Dept. of Mechanical Engineering, Mechatronic Engineering & Manufacturing Technology California State University, Chico CE 150

  2. Fluid Kinematics Reading: Munson, et al., Chapter 4 CE 150

  3. Introduction • In this chapter we consider fluid kinematics, which addresses the behavior of fluids while they are flowing without concern of the actual forces necessary to produce the motion • Specifically, we will address • fluid velocity • fluid acceleration • flow pattern description and visualization CE 150

  4. Fluid Models • Continuum model: fluids are a collection of fluid particles that interact with each other and surroundings; each particle contains a sufficient number of molecules such that fluid properties (e.g., velocity) can be defined. • Molecular model: the motions of individual fluid molecules are accounted for; not a practical model unless fluid density is very small or flow over very small objects are considered. CE 150

  5. Flow Descriptions • Lagrangian description: properties of individual fluid particles are defined as a function of time as they move through the fluid; the overall fluid motion is found by solving the EOMs for all fluid particles. • Eulerian description: properties are defined at fixed points in space as the fluid flows past these points; this is the most common description and yields the field representation of fluid flow. CE 150

  6. The Velocity Field • Consider an array of sensors that can simultaneously measure the magnitude and direction of fluid velocity at many fixed points within the flow as a function of time; in the limit of measuring velocity at all points within the flow, we would have sufficient information to define the velocity vector field: CE 150

  7. The Velocity Field • u, v, and w are the x, y, and z components of the velocity vector • The magnitude of the velocity, or speed, is denoted by V as • Velocity field may be one- (u), two- (u,v)or three- (u,v,w) dimensional • Steady vs. unsteady flows: CE 150

  8. Visualization of Fluid Flow • Three basic types of lines used to illustrate fluid flow patterns: • Streamline: a line that is everywhere tangent to the local velocity vector at a given instant. • Pathline: a line that represents the actual path traversed by a single fluid particle. • Streakline: a line that represents the locus of fluid particles at a given instant that have earlier passed through a prescribed point. CE 150

  9. Streamlines • Streamlines are useful in fluid flow analysis, but are difficult to observe experimentally for unsteady flows • For 2-D flows, the streamline equation can be determined by integrating the slope equation: • The resulting equation is normally written in terms of the stream function:  (x,y) = constant CE 150

  10. Pathlines & Streaklines • The pathline is a Lagrangian concept that can be visualized in the laboratory by “marking” a fluid particle and taking a time exposure photograph of its trajectory • The streakline can be visualized in the laboratory by continuously marking all fluid particles passing through a fixed point and taking an instantaneous photograph • Streamlines, pathlines, and streak-lines are identical for steady flows CE 150

  11. Acceleration Field • Acceleration is the time rate of change of velocity: • Using the Eulerian description, we note that the total derivative of each velocity component will consist of four terms, e.g., CE 150

  12. Acceleration Field • Collecting derivative terms from all velocity components, • The operator is termed the material, or substantial, derivative; it represents the rate at which a variable (V in this case) changes with time for a given fluid particle moving through the flow field CE 150

  13. Acceleration Field • The term is called the local acceleration; it represents the unsteadiness of the fluid velocity and is zero for steady flows. • The terms are called convective accelerations; they represent the fact that the velocity of the fluid particle may vary due to the motion of the particle from one point in space to another; it can occur for both steady and unsteady flows. CE 150

  14. The Control Volume • A control volume is a volume in space through which fluid may flow; in some cases, the volume may move or deform • The control volume has a boundary which separates it from the surroun-dings and defines a control surface • In the study of fluid dynamics, the control volume approach is used to analyze fluid flow and fluid machinery • The control volume approach is consistent with the Eulerian description CE 150

  15. The Reynolds Transport Theorem • The basic laws governing the motion of a fluid (e.g., conservation of mass, momentum, and energy) are usually written in terms of a fixed quantity of mass, or system* • Because a control volume does not always have constant mass, the basic laws must be rephrased • The Reynolds Transport Theorem is a tool that allows one to shift from a system viewpoint (fixed mass) to a control volume viewpoint * In thermodynamics, a system is defined more generally as a fixed mass or control volume CE 150

  16. The Reynolds Transport Theorem • Let B = any fluid parameter, such as mass, velocity, temperature, momentum, etc. • Let b = B/m, a fluid parameter per unit mass • The mass m may be that contained in a system or a control volume CE 150

  17. The Reynolds Transport Theorem • Example 4.7 (B = m, b = 1) CE 150

  18. The Reynolds Transport Theorem CE 150

  19. The Reynolds Transport Theorem • Reynolds Transport Theorem (RTT) for fixed control volume with one inlet, one exit and uniform properties: • LHS term is Lagrangian • RHS terms are Eulerian CE 150

  20. The Reynolds Transport Theorem • A general control volume may have multiple inlets and outlets, three-dimensional flow, and nonuniform properties; the general form of the RTT is: • for a control volume moving at constant velocity Vcv, replace V by V-Vcv CE 150

  21. Physical Interpretation • The RTT allows one to translate the time rate of change of some parameter B of the system in terms of the time rate of change of B of the control volume and the net flow rate of B across the control surface • A material derivative is used because the translation consists of an unsteady term ( )/t and convective effects associated with the flow of the system across the control surface CE 150

  22. Steady Flow • For steady flow, • For B = m (mass), the LHS is zero since the mass of a system is constant • For B = V (velocity), the LHS is nonzero in general • For B = T (temperature), the LHS is also nonzero in general CE 150

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