Chapter 4: Fluid Kinematics

1. PTT 204/3APPLIED FLUID MECHANICSSEM 2 (2012/2013) Chapter 4: Fluid Kinematics

2. Objectives • Understand the role of the material derivative in transforming between Lagrangian and Eulerian descriptions • Distinguish between various types of flow visualizations and methods of plotting the characteristics of a fluid flow • Appreciate the many ways that fluids move and deform • Distinguish between rotational and irrotational regions of flow based on the flow property vorticity • Understand the usefulness of the Reynolds transport theorem

3. 4–1 Lagrangian and EulerianDescriptions Kinematics:The study of motion. Fluid kinematics:The study of how fluids flow and how to describe fluidmotion. There are two distinct ways to describe motion: Lagrangian and Eulerian Lagrangian description:To follow the path of individual objects. This method requires us to track the position and velocity of each individual fluid parcel (fluid particle) and take to be a parcel of fixed identity. With a small number of objects, suchas billiard balls on a pool table,individual objects can be tracked. In the Lagrangian description, onemust keep track of the position andvelocity of individual particles.

4. A more common method isEuleriandescriptionof fluid motion. • In the Eulerian description of fluid flow, a finite volumecalled a flow domainor control volumeis defined, through which fluidflows in and out. • Instead of tracking individual fluid particles, we definefield variables, functions of space and time, within the control volume. • Thefield variable at a particular location at a particular time is the value of thevariable for whichever fluid particle happens to occupy that location at thattime. • For example, the pressure fieldis a scalar field variable. We define the velocity fieldas a vector field variable. Collectively, these (and other) field variables define the flow field. Thevelocity field can be expanded in Cartesian coordinatesas

5. In the Eulerian description we don’t really care what happens to individualfluid particles; rather we are concerned with the pressure, velocity, acceleration,etc., of whichever fluid particle happens to be at the location of interestat the time of interest. • While there are many occasions in which theLagrangian description isuseful, the Eulerian description is often more convenient for fluid mechanicsapplications. In the Eulerian description, onedefines field variables, such as thepressure field and the velocity field, atany location and instant in time.

8. A Steady Two-Dimensional Velocity Field Velocity vectors for the velocity fieldof Example 4–1. The scale is shownby the top arrow, and the solid blackcurves represent the approximateshapes of some streamlines, basedon the calculated velocity vectors.The stagnation point is indicated bythe circle. The shaded regionrepresents a portion of the flow fieldthat can approximate flow into aninlet.

9. Acceleration Field The equations of motion for fluid flow (such as Newton’s second law) are written for a fluid particle, which we also call a material particle. If we were to follow a particular fluid particle as it moves around in the flow, we would be employing the Lagrangian description, and the equations of motion would be directly applicable. For example, we would define the particle’s location in space in terms of a material position vector(xparticle(t), yparticle(t), zparticle(t)). Newton’s second law applied to a fluidparticle; theacceleration vector (grayarrow) is in the same direction as theforce vector (black arrow), but thevelocity vector (red arrow) may actin a different direction. e.g:

10. Velocity field Given: Therefore, HOW? Local acceleration (nonzero for unsteady state, zero for steady state) Advective (convective) acceleration (nonzero for steady and unsteady state

11. The components of the acceleration vector in cartesian coordinates: Local acceleration Advective acceleration Flow of water through the nozzle ofa garden hose illustrates that fluidparticles may accelerate, even in asteady flow. In this example, the exitspeed of the water is much higher thanthe water speed in the hose, implyingthat fluid particles have acceleratedeven though the flow is steady. Figure 4.8

15. Material Derivative The total derivative operator d/dt in this equation is given a special name, thematerial derivative; it is assigned a special notation, D/Dt, in order to emphasizethat it is formed by following a fluid particle as it moves throughthe flow field. Other names for the material derivative includetotal, particle, Lagrangian, Eulerian, and substantial derivative. The material derivative D/Dt isdefined by following a fluid particleas it moves throughout the flow field.In this illustration, the fluid particle isaccelerating to the right as it movesup and to the right.

16. The material derivative D/Dt iscomposed of a local or unsteady partand a convective or advective part.

17. Note: From Example 4.1

18. x

19. Homework • Please read and understand about: • Flow Patterns and Flow Visualization • Streamlines and Streamtubes, Pathlines, • Streaklines, Timelines • Refractive Flow Visualization Techniques • Surface Flow Visualization Techniques • Plots of Fluid Flow Data • Profile Plots, Vector Plots, Contour Plots …………………..to do assignment

20. 4–2 Vorticity and Rotationality Another kinematic property of great importance to the analysis of fluidflows is the vorticity vector, defined mathematically as the curl of the velocityvector Vorticityis equal to twice the angular velocity of a fluid particle The direction of a vector cross productis determined by the right-hand rule. The vorticity vector is equal to twicethe angular velocity vector of arotating fluid particle.

21. If the vorticity at a point in a flow field is nonzero, the fluid particle thathappens to occupy that point in space is rotating; the flow in that region iscalled rotational(ζ ≠ 0 ; rotational). • Likewise, if the vorticity in a region of the flow is zero (ornegligibly small), fluid particles there are not rotating; the flow in that regionis called irrotational. (ζ = 0 ; irrotational). • Physically, fluid particles in a rotational region of flowrotate end over end as they move along in the flow. The difference between rotational andirrotational flow: fluid elements in a rotational region of the flow rotate, butthose in an irrotational region of theflow do not.

22. For a two-dimensional flow in thexy-plane, the vorticity vector alwayspoints in the z-direction or −z-direction. Inthis illustration, the flag-shaped fluidparticle rotates in the counterclockwisedirection as it moves in the xy-plane; its vorticity points in the positivez-direction as shown.

23. For a two-dimensional flow in ther-plane, the vorticity vector alwayspoints in the z (or -z) direction. Inthisillustration, the flag-shaped fluidparticle rotates in the clockwisedirection as it moves in the rθ-plane;its vorticity points in the -z-directionas shown. Note: rotate clockwise = vorticity points in -z direction. counterclockwise = z direction

24. Comparison of Two Circular Flows • Not all flows with circular streamlines are rotational Streamlines and velocity profiles for(a) flow A, solid-body rotation and(b) flow B, a line vortex. Flow A isrotational, but flow B is irrotationaleverywhere except at the origin.

25. A simple analogy can be madebetween flow A and a merry-go-round orroundabout, and flow B and a Ferris wheel. As childrenrevolve around a roundabout, they also rotate at the same angular velocityas that of the ride itself. This is analogous to a rotational flow. In contrast,children on a Ferris wheel always remain oriented in an upright position asthey trace out their circular path. This is analogous to an irrotational flow. A simple analogy: (a) rotational circular flow is analogous to aroundabout,while (b) irrotational circular flow is analogous to a Ferris wheel.

26. 4–3 The Reynolds Transport Theorem Two methods of analyzing thespraying of deodorant from a spraycan: (a)We follow the fluid as itmoves and deforms. This is the systemapproach—no mass crosses theboundary, and the total mass of thesystem remains fixed. (b)We considera fixed interior volume of the can. Thisis the control volume approach—masscrosses the boundary. The relationshipbetween the time rates of change of an extensive property for a system andfor a control volume is expressed by the Reynolds transporttheorem(RTT). The Reynolds transport theorem(RTT) provides a link between thesystem approach and the controlvolume approach.

27. The time rate of change of the property B of the system isequal to the time rate of change of B of the control volume plus the net fluxof B out of the control volume by mass crossing the control surface. This equation applies at any instant in time, where it is assumed that the system and thecontrol volume occupy the same space at that particular instant in time. A moving system (hatched region) anda fixed control volume (shaded region)in a diverging portion of a flow field attimes t and t+t. The upper and lowerbounds are streamlines of the flow.

30. Reynolds transport theorem applied toa control volume moving at constantvelocity. Relative velocity crossing a controlsurface is found by vector additionof the absolute velocity of the fluidand the negative of the local velocityof the control surface.

31. An example control volume in whichthere is one well-defined inlet (1) andtwo well-defined outlets (2 and 3). Insuch cases, the control surface integralin the RTT can be more convenientlywritten in terms of the average valuesof fluid properties crossing each inlet and outlet.

32. Alternate Derivation of the ReynoldsTransport Theorem A more elegant mathematical derivation of the Reynolds transport theorem ispossible through use of the Leibniz theorem The Leibniz theorem takes into account the change of limits a(t) and b(t)with respect to time, as well as the unsteady changes of integrand G(x, t)with time.

35. The equation above is valid for arbitrarily shaped, moving and/or deforming CV at time t. The material volume (system) andcontrol volume occupy the same spaceat time t (the blue shaded area), butmove and deform differently. At alater time they are not coincident.

37. Relationship between Material Derivative and RTT While theReynolds transport theorem deals with finite-size control volumes and thematerial derivative deals with infinitesimal fluid particles, the same fundamentalphysical interpretation applies to both. Just as the material derivative can be applied to any fluid property, scalaror vector, the Reynolds transport theorem can be applied to any scalar orvector property as well. The Reynolds transport theorem forfinite volumes (integral analysis) isanalogous to the material derivativefor infinitesimal volumes (differentialanalysis). In both cases, we transformfrom a Lagrangian or system viewpointto an Eulerian or control volumeviewpoint.