KINEMATICS OF FLUID MOTION

Transcription

1 KINEMATICS OF FLUID MOTION The Velocity Field The representation of properties of flid parameters as fnction of the spatial coordinates is termed a field representation of the flo. One of the most important flid ariables is the elocity field. V = ( x, y, z, t)ˆ i ( x, y, z, t) ˆj ( x, y, z, t) kˆ Where, and are the x, y and z components of the elocity ector. The speed of flid is ; Figre V = V = ( 2 ) 1

2 Example 1 The elocity field of a flo is gien by V = ( 5z 3)ˆ i ( x 4) ˆj (4y) kˆ (m/s) Determine the flid speed at the origin (x=y=z=0) And on the x-axis (y=z=0) Example 2 The elocity field of a flo is gien by 20y x V iˆ 20 ˆj x y x y = (m/s) Determine the flid speed at points along the x-axis and along the y-axis. What is the angle beteen the elocity ector and the x-axis at point (x,y)=(5,0), (5,5) and (0,5) Example 3 The components of a elocity ector field are gien by =xy, =xy 3 16 and =0. Determine the location of any stagnation points (V=0) in the flo field. 2

3 Elerian and Lagrangian Flo Descriptions There are to general approaches in analyzing flid mechanics problems. The first method is called the Elerian method, the second method is called the Lagragian method. From Elerian method e obtain information abot the flo in terms of hat happens at fixed points in space as the flid flos past those points. Lagragian method inoles folloing indiidal flid particles as they moe abot and determining ho the flid properties associated ith these particles change as a fnction of time. That is, the flid particles are tagged or identified, and their properties determined as they moe. Figre 2 3

4 Steady and Unsteady Flo We hae assmed steady flo is the elocity at a gien point in space does not ary ith time. V t In reality, almost all flos are nsteady in some sense. The nsteady flos are sally more difficlt to analyze and to inestigate experimentally than are steady flos. Laminar flo smooth flo = 0 Trblent flo irreglar flo 4

5 Streamlines Streamline is a line that is eeryhere tangent to the elocity field. Streamlines are obtained analytically by integrating the eqations defining lines tangent to the elocity field as illstrated in the Figre 3. Figre 3 dy = dx 5

6 Streaklines and pathlines A streakline consists of all particles in a flo that hae preiosly passed throgh a common point. Streaklines are more of a laboratory tool than an analytical tool. They can be obtained by taking instantaneos photographs of marked particles that all passed throgh a gien location in the flo field at some earlier time. A pathline is the line traced ot by a gien particle as it flos from one point to another. The pathline is a Lagrangian concept that can be prodced in the paboratory by marking a flid particle and taking a time exposre photograph of its motion. Figre 4 6

7 The Acceleration Field For the infreqently sed Lagrangian method, e described the flid acceleration jst as is done in solid body dynamics, a = a(t) for each particle. For the Elerian description e describe the acceleration field as a fnction of position and time ithot actally folloing any particlar particle. This is analogos to describing the flo in terms of the elocity field, V = V ( x, y, z, t) The acceleration of a particle is the time rate of change of its elocity. 7

8 The Material Deriatie Figre 5 Consider a flid particle moing along its pathline as shon in Figre 5. In general, the particle s elocity, denoted V A for particle A, is a fnction of its location and the time. That is ; VA = VA ( ra, t) = VA[ x A ( t), y A ( t), z A ( t), t] 8

9 Then, the acceleration field from the elocity field for any particle obtained as ; z V y V x V t V a = This is a reslt hose scalar components can be ritten as ; (x-axis) z y x t a x = (y-axis) z y x t a y = (z-axis) z y x t a z = 9

10 Conectie Effects The portion of the material deriatie represented by the spatial deriaties is termed the conectie deriatie. It represents the fact that a flo property associated ith a flid particle may ary becase of the motion of the particle from one point in space here the parameter has one ale to another point in space here its ale is different. That portion of the acceleration is termed the conectie acceleration. For example, the temperatre of a ater particle changes as it flos throgh a ater heater. Figre 6 10

11 Example 4 The x and y components of a elocity field are gien by =x 2 y and =-xy 2. Determine the eqation for the streamlines of this flo. Example 5 A elocity field is gien by =cx 2 and =cy 2, here c is a constant. Determine the x and y components of the acceleration. At hat point (points) in the flo is the acceleration zero. Example 6 Determine the acceleration field for a three-dimensional flo ith elocity components, =-x, =4x 2 y 2 and =x-y. 11

12 Example 7 Figre 7 A hydralic jmp is a rather sdden change in depth of a liqid layer as it flos in an open channel as shon in Figre 7. If V 1 =0.4m/s, V 2 =0.1m/s and l = 0.07m, estimate the aerage deceleration of the liqid as it flos across the hydralic jmp. Ho many G s deceleration does this represent? Example 8 V 0 =40m/s and V = 3 2 V 0 sinθ Figre 8. Determine the streamise and normal components of acceleration at point A if the radis of the sphere is a=0.20m. 12

13 Streamline Coordinates Figre 9 In many flo sitations, it is conenient to se a coordinate system defined in terms of the streamlines of the flo. The flos can be described either in (x, y) Cartesian coordinate or (r, θ) polar coordinate system. 13

14 Control Volme and System Representations Figre 10 A control olme, on the other hand, is a olme in space (a geometric entity, independent of mass) throgh hich flid may flo. A system is a specific, identifiable qantity of matter. It may consist of a relatiely large amont of mass, or it may be an infinitesimal size. The system may interact ith its srrondings by arios means. It may continally change size and shape, bt it alays contains the same mass. 14