Fluid Kinematics CEE 311 Chapter 4 Fluid Flow

Fluid Kinematics CEE 311 Chapter 4

Fluid Flow Concepts and Reynolds Transport Theorem • Fluid Kinematics deals with the motion of fluids without considering the forces and moments which create the motion. • Descriptions of: – fluid motion – fluid flow – temporal and spatial classifications • Analysis Approaches – Lagrangian vs. Eulerian • Moving from a system to a control volume – Reynolds Transport Theorem

Field Representation The representation of fluid parameters as functions of the spatial and temporal coordinates is termed a field representation Particle locations in terms of its position vector of the flow

Scalar and Vector Fields • Scalar: Scalar is a quantity which can be expressed by a single number representing its magnitude. Example: mass, density and temperature. – Scalar Field : If at every point in a region, a scalar function has a defined value, the region is called a scalar field. Example: Temperature distribution in a rod. • Vector: Vector is a quantity which is specified by both magnitude and direction. Example: Force, Velocity and Displacement. – Vector Field : If at every point in a region, a vector function has a defined value, the region is called a vector field. Example: velocity field of a flowing fluid.

Velocity Field Magnitude of V: Direction of V: given by q, a, g

Descriptions of Fluid Flow • There are two general approaches in analyzing fluid mechanics problems Lagrangian Description Eulerian Description

Lagrangian Vs Eulerian VA XA XB VB P(x, y, z) VC V (x, y, z ) XC In the lagrangian description, one must keep track of the position and velocity of individual fluid particles In the Eulerian description, one defines field variables, such as the pressure field and the velocity field at any location and instant in time

Lagrangian Description • Lagrangian description of fluid flow tracks the position and velocity of individual particles. (e. g. , Billiard ball on a pool table. ) • Motion is described based upon Newton's laws of motion. • Difficult to use for practical flow analysis. – Fluids are composed of billions of molecules. – Interaction between molecules are hard to describe /model. • However, useful for specialized applications – Sprays, particles, bubble dynamics, rarefied gases. – Coupled Eulerian-Lagrangian methods. • Named after Italian mathematician Joseph Louis Lagrange (17361813).

Eulerian Description • Eulerian description of fluid flow: a flow domain or control volume is defined, through which fluid flows in and out. • We define field variables which are functions of space and time. – Pressure field, P=P(x, y, z, t) – Velocity field, – Acceleration field, These (and other) field variables define the flow field. • Well suited formulation of initial boundary-value problems (PDE's). • Named after Swiss mathematician Leonhard Euler (1707 -1783).

Eulerian and Lagrangian descriptions of temperature of a flowing fluid.

Lagrangian Description Eulerian Description In the Eulerian method one may attach a In the Lagrangian method, one temperature-measuring would attach the temperature- device to the top of the chimney measuring device to a particular (point 0) and record the temperature fluid particle (particle A) and record at that point as a function of time. At that particle’s temperature as it different times there are different moves about. Thus, one would fluid obtain that particle’s temperature particles passing by the stationary device. Thus, one would obtain the temperature, T, for that location (x = xo, y=yo, z= zo) as a function of time. That is, T = T (xo, yo, zo, t) as a function of time, TA = TA(t)

The use of numerous temperature- The use of many such measuring devices fixed at various devices moving with various fluid locations particles would provide the temperature field, T = T (x, y, z, t). temperature of these fluid particles The temperature of a particle as a as function of time would not be known temperature would not be known as unless the location of the particle were known as a function of time. The a function of position unless the location of each particle were known as a function of time. If enough information in Eulerian form is available, Lagrangian information can be derived from the Eulerian data—and vice versa

Flow Visualization Flow visualization is the visual While quantitative study of fluid examination of flow-field features. dynamics Important mathematics, much can be learned for experiments both and physical numerical computational fluid dynamics (CFD) solutions. Numerous methods Streamlines and streamtubes Pathlines Streaklines Timelines Refractive techniques Surface flow techniques requires from flow visualization advanced

Streamlines • A Streamline is a curve that is everywhere tangent to the instantaneous local velocity vector. • Consider an arc length must be parallel to the local velocity vector • Geometric arguments results in the equation for a streamline

Streamtube • A streamtube consists of a bundle of streamlines (Both are instantaneous quantities). • Fluid within a streamtube must remain there and cannot cross the boundary of the streamtube. • In an unsteady flow, the streamline pattern may change significantly with time. • the mass flow rate passing through any cross-sectional slice of a streamtube must remain the same. given

Following points about streamtube are worth noting • Stream tube has finite dimensions • As there is no flow perpendicular to stream lines, there is no flow across the stream surface of the tube • Shape of the stream tube changes from one instant to another because of change in position of streamlines • Examples : - pipes , nozzle, diffuser

Pathlines • A Pathline is the actual path traveled by an individual fluid particle over some time period. • Same as the fluid particle's material position vector • Particle location at time t: • Particle Image Velocimetry (PIV) is a modern experimental technique to measure velocity field over a plane in the flow field.

Pathlines A modern experimental technique called particle image velocimetry (PIV) utilizes (tracer) particle pathlines to measure the velocity field over an entire plane in a flow (Adrian, 1991).

Stream Line Path Line This is an imaginary curve in a flow This refers to a path followed by a field for a fixed instant of time, fluid particle over a period of time. tangent to which gives the instantaneous velocity at that point. Two stream lines can never intersect each other, as the instantaneous velocity vector at any given point is unique. Two path lines can intersect each other or a single path line can form a loop as different particles or even same particle can arrive at the same point at different instants of time.

Streaklines A streak line is the locus of the temporary locations of all particles that have passed though a fixed point in the flow field at any instant of time Easy to generate in experiments: dye in a water flow, or smoke in an airflow.

Features of a Streak Line • While a path line refers to the identity of a fluid particle, a streak line is specified by a fixed point in the flow field. • It is of particular interest in experimental flow visualization. • Example: If dye is injected into a liquid at a fixed point in the flow field, then at a later time t, the dye will indicate the end points of the path lines of particles which have passed through the injection point. • Path taken by smoke coming out of the chimney

Comparisons • For steady flow, streamlines, pathlines, and streaklines are identical. • For unsteady flow, they can be very different. – Streamlines are instantaneous pictures of the flow field – Pathlines and Streaklines are flow patterns that have a time history associated with them. – Streakline: instantaneous snapshot of a time-integrated flow pattern. – Pathline: time-exposed flow path of an individual particle.

Uniform vs. Non-Uniform Flow Using s as the spatial variable along the path (i. e. , along a streamline): Flow is uniform if Examples of uniform flow: Note that the velocity along different streamlines need not be the same! (in these cases it probably isn’t).

Examples of non-uniform flow: a) Converging flow: speed increases along each streamline. b) Vortex flow: Speed is constant along each streamline, but the direction of the velocity vector changes.

Steady vs. Unsteady Flow For steady flow, the velocity at a point or along a streamline does not change with time: Any of the previous examples can be steady or unsteady, depending on whether or not the flow is accelerating:

Turbulent flow in a jet Turbulence is associated with intense mixing and unsteady flow.

Flow around an airfoil: Partly laminar, i. e. , flowing past the object in “layers” (laminae). Turbulence forms mostly downstream from the airfoil. (Flow becomes more turbulent with increased angle of attack. )

Flow inside a pipe: Laminar Turbulent flow is nearly constant across a pipe. Flow in a pipe becomes turbulent either because of high velocity, because of large pipe diameter, or because of low viscosity.