ME 3560 Fluid Mechanics 1
4.1 The Velocity Field One of the most important parameters that need to be monitored when a fluid is flowing is the velocity. In general the flow parameters are described in terms of the motion of fluid particles rather than individual molecules. Thus, this motion can be described in terms of the velocity and acceleration of the fluid particles. In order to describe the flow parameters it is sought to provide at a given instant in time, a description of any fluid property (such as, p, V, and a) as a function of the fluid's location. This representation of fluid parameters as functions of the spatial coordinates is termed a field representation of the flow. The specific field representation may be different at different times, thus. Thus, to completely specify the velocity, V, in a process, the velocity field must be expressed as: V = V (x, y, z, t). V u( x, y, z, t)ˆ i v( x, y, z, t) ˆj w( x, y, z, t) kˆ 2
V u( x, y, z, y)ˆ i v( x, y, z, y) ˆj w( x, y, z, y) kˆ u, v, and w are the x, y, and z components of the velocity vector. The velocity of a particle is the time rate of change of the position vector for that particle. The position of particle A relative to the coordinate system is given by its position vector, r A, which (if the particle is moving) is a function of time. The velocity is a vector therefore it has both a direction and a magnitude, V= V =(u 2 + v 2 + w 2 ) 1/2 A change in velocity results in an acceleration. This acceleration may be due to a change in speed and/or direction. 3
4.1.1 Eulerian and Lagragian Flow Descriptions There are two general approaches in analyzing fluid mechanics problems: Eulerian method. Measures the flow parameters at fixed locations to then generate the parameters flowfield. Lagrangian method, involves following individual fluid particles as they move about and determining how the fluid properties associated with these particles change as a function of time. 4.1.2 One-, Two-, and Three-Dimensional Flows In general, a fluid flow is a rather complex three-dimensional, timedependent phenomenon. V u( x, y, z, t)ˆ i v( x, y, z, t)ˆj w( x, y, z, t) kˆ 4
However, sometimes it is possible to make simplifying assumptions that allow a much easier understanding of the problem without sacrificing needed accuracy: Assume the flow is two dimensional or one dimensional. Assume the flow is incompressible (even when dealing with gases) One of these simplifications involves approximating a real flow as a simpler one or two dimensional flow. 5
4.1.3 Steady and Unsteady Flows Steady flow. The velocity (or any other parameter: T, p,, etc.) ata given point in space does not vary with time V/t=0. Unsteady flows. Flow field parameters are time dependent V/t0. Among the various types of unsteady flows are nonperiodic flow, periodic flow, and truly random flow. 4.1.4 Streamlines, Streaklines, and Pathlines Streamline is a line that is everywhere tangent to the velocity field. If the flow is steady, nothing at a fixed point (including the velocity direction) changes with time, so the streamlines are fixed lines in space. For unsteady flows the streamlines may change shape with time. x y z u v w dy dx v u 6
Streakline consists of all particles in a flow that have previously passed through a common point. Streaklines can be generated by taking instantaneous photographs of marked particles that all passed through a given location in the flow field at some earlier time. Such a line can be produced by continuously injecting marked fluid (neutrally buoyant smoke in air, or dye in water) at a given location. 7
Pathline is the line traced out by a given particle as it flows from one point to another. The pathline is a Lagrangian concept that can be produced in the laboratory by marking a fluid particle (dying a small fluid element) and taking a time exposure photograph of its motion. Pathlines, Streamlines,andStreaklines are the same for steady flows. For unsteady flows none of these three types of lines need to be the same. 8
4.2 The Acceleration Field A flow can be studied by either (1) following individual particles (Lagrangian description) or (2) remaining fixed in space and observing different particles as they pass by (Eulerian description). In either case, to apply Newton's second law (F =ma) it is necessary to describe the particle acceleration in an appropriate fashion. For the infrequently used Lagrangian method, we describe the fluid acceleration just as is done in solid body dynamics a = a(t) for each particle. For the Eulerian description we describe the acceleration field as a function of position and time without actually following any particular particle, a = a(x, y, z, t). The acceleration of a particle is the time rate of change of its velocity. For unsteady flows the velocity at a given point in space varies with time. Also, a fluid particle may accelerate because its velocity changes as it flows from one point to another in space. 9
10 4.2.1 The Material Derivative z w w y w v x w u t w a z v w y v v x v u t v a z u w y u v x u u t u a z V w y V v x V u t V a z w y v x u t dt D dt DV a () () () () ()
D( )/dt is termed the material derivative or substantial derivative. A shorthand notation for the material derivative operator is D() dt () t ( V )() 4.2.2 Unsteady Effects The material derivative formula contains two types of terms: Terms involving the time derivative ()/t: Local Derivative Terms involving spatial derivatives ()/x, ()/y,and()/z. ()/t represents the effects of the unsteadiness of the flow. V/t is termed the local acceleration.forsteadyflow:()/t=0 Physically, there is no change in flow parameters at a fixed point in space if the flow is steady. There may be a change of those parameters for a fluid particle as it moves about. If a flow is unsteady, its parameter values (V, T,, etc.) at any location may change with time 11
4.2.3 Convective Effects The portion of the material derivative represented by the spatial derivatives is termed the convective derivative. The convective derivative represents the fact that a flow property associated with a fluid particle may vary because of the motion of the particle from one point in space to another point in space. This contribution to the time rate of change of the parameter for the particle can occur whether the flow is steady or unsteady. It is due to the convection, or motion, of the particle through space in which there is a gradient [( )=()/xî+()/x ĵ + ()/z k] The portion of a due to the term (V )V is the convective acceleration 12
4.3 Control Volume and System Representations A system is a collection of matter of fixed identity (always the same atoms or fluid particles), which may move, flow, and interact with its surroundings. A system is a specific, identifiable quantity of matter. It may consist of a relatively large amount of mass or it may be an infinitesimal size. A system may interact with its surroundings by various means (by the transfer of heat or the exertion of a pressure force, for example). A system may continually change size and shape, but it always contains the same mass. 13
A control volume, is a volume in space (a geometric entity, independent of mass) through which fluid may flow. In fluid mechanics, it is difficult to identify and keep track of a specific quantity of matter. In several cases, the main interest is in determining the forces put on a device rather than in the information obtained by following a given portion of the air (a system) as it flows along. For these situations it is more adequate to use the control volume approach. Identify a specific volume in space (a volume associated with the device of interest) and analyze the fluid flow within, through, or around that volume. 14
In general, the control volume can be a moving volume, although for most situations we will use only fixed, nondeformable control volumes. The matter within a control volume may change with time as the fluid flows through it. The amount of mass within the volume may change with time. The control volume itself is a specific geometric entity, independent of the flowing fluid. 15
All of the laws governing the motion of a fluid are stated in their basic form in terms of a system approach. For example, the mass of a system remains constant, or the time rate of change of momentum of a system is equal to the sum of all the forces acting on the system. 16
4.4 The Reynolds Transport Theorem The Reynolds transport theorem provides a mathematical way to relate the properties of a flow between a control volume and a system. Extensive Property, is a property whose value is directly proportional to the amount of the mass being considered. -Mass -Energy -Momentum Intensive Property, is a property whose value is independent of the amount of mass. -Density -Temperature -Velocity -Pressure 17
B = m b=1 B = mv b=v B = E b=e An extensive property B is related to its corresponding intensive property b by B = mb In general, for a system, an extensive property B can be determined as: B b dv sys The rate of change of an extensive property in a system is expressed as db dt sys sys d b dv sys dt 18
In a similar way, the rate of change of an extensive property in a control volume is expressed as d b dv db cv cv dt dt 4.4.1 Derivation of the Reynolds Transport Theorem Consider two instants in time: t and t+t. Assume that at t the control volume and the system coincide, thus B sys ( t) B ( t) cv 19
Then, at a time t+t B sys ( t t) B ( t t) B ( t t) B ( t t) cv I II The change in the amount of B in the system in the time interval δt divided by this time interval is given by Bsys Bsys ( t t) Bsys ( t) t t Bsys Bcv ( t t) BI ( t t) BII ( t t) t t B sys ( t) 20
Since at the initial time t we have B sys (t)=b cv (t), B t sys B cv ( t t) Bcv ( t) BI ( t t) BII ( t t) t t t For δt 0, the left-hand side this equation is DB sys /Dt. DB sys /Dt represents the time rate of change of property B associated with a system (a given portion of fluid) as it moves along. For δt 0,thefirsttermontheright-handsideofthisequationisthe time rate of change of the amount of B within the control volume lim t0 B cv ( t t) t B cv ( t) B t cv cv bdv t 21
The term B II ( t t) ( 2 b2 )( VII ) 2b2 A2V 2t Represents the amount of the extensive parameter B flowing out of the control volume, across the control surface. Thus, the rate at which this property flows from the control volume is: B out lim t0 B II ( t t) t A V 2 2 2 b 2 22
The term B I ( t t) ( 1 b1 )( VI ) 1b1 AV 1 1t Represents the amount of the extensive parameter B flowing into the control volume, across the control surface Thus, the rate at which this property flows from the control volume is: B in lim t0 B I ( t t) t AV 1 1 1 b 1 23
By combining the previous equations, a relation between the time rate of change of B for the system and that for the control volume is given by DB Dt DB Dt sys sys B t B t cv cv B out B in 2 A2V 2b2 1AV 1 1b1 A generalization of the previous equation is given by the Reynolds Transport Theorem DB Dt sys t cv b dv cs bv nda ˆ 24
Selection of a Control Volume CV is typically fixed and non deforming. CV includes all relevant inlets and outlets where information is available or required. If possible CS should be perpendicular to the velocity vector to simplify V Contributions to the surface integrals must be simple and relevant. In CS other than inlets and outlets, select solid boundaries where V =0. Sign of V nˆ nˆ n n V in V nˆ in V in V V out nˆ out V out 25