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Chapter4 - 57:020 Fluid Mechanics Professor Fred Stern Fall...

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57:020 Fluid Mechanics Chapter 4 Professor Fred Stern Fall 2008 1 Chapter 4: Fluids Kinematics 4.1 Velocity and Description Methods Primary dependent variable is fluid velocity vector V = V ( r ); where r is the position vector If V is known then pressure and forces can be determined using techniques to be discussed in subsequent chapters. Consideration of the velocity field alone is referred to as flow field kinematics in distinction from flow field dynamics (force considerations). Fluid mechanics and especially flow kinematics is a geometric subject and if one has a good understanding of the flow geometry then one knows a great deal about the solution to a fluid mechanics problem. Consider a simple flow situation, such as an airfoil in a wind tunnel: k ˆ z j ˆ y i ˆ x r ˆ ˆ ˆ ( , ) V r t ui vj wk x r U = constant

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57:020 Fluid Mechanics Chapter 4 Professor Fred Stern Fall 2008 2 Velocity: Lagrangian and Eulerian Viewpoints There are two approaches to analyzing the velocity field: Lagrangian and Eulerian Lagrangian: keep track of individual fluids particles (i.e., solve F = Ma for each particle) Say particle p is at position r 1 (t 1 ) and at position r 2 (t 2 ) then, Of course the motion of one particle is insufficient to describe the flow field, so the motion of all particles must be considered simultaneously which would be a very difficult task. Also, spatial gradients are not given directly. Thus, the Lagrangian approach is only used in special circumstances. Eulerian: focus attention on a fixed point in space In general, where, , ,
57:020 Fluid Mechanics Chapter 4 Professor Fred Stern Fall 2008 3 This approach is by far the most useful since we are usually interested in the flow field in some region and not the history of individual particles. However, must transform F = Ma from system to CV (recall Reynolds Transport Theorem (RTT) & CV analysis from thermodynamics) V can be expressed in any coordinate system; e.g., polar or spherical coordinates. Recall that such coordinates are called orthogonal curvilinear coordinates. The coordinate system is selected such that it is convenient for describing the problem at hand (boundary geometry or streamlines). Undoubtedly, the most convenient coordinate system is streamline coordinates: ) t , s ( e ˆ ) t , s ( v ) t , s ( V s s However, usually V not known a priori and even if known streamlines maybe difficult to generate/determine. Ex. Flow around a car e ˆ v e ˆ v V r r j ˆ cos i ˆ sin e ˆ j ˆ sin i ˆ cos r e ˆ sin r y cos r x

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57:020 Fluid Mechanics Chapter 4 Professor Fred Stern Fall 2008 4 4.2 Acceleration Field and Material Derivative The acceleration of a fluid particle is the rate of change of its velocity.
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Chapter4 - 57:020 Fluid Mechanics Professor Fred Stern Fall...

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