Basics: Fluid Mechanicswp.kntu.ac.ir/mojra/FM1-chap3.pdf• Kinematics: how the fluid flows? • Lagrangian description:we follow a mass of fixedidentity. Lagrangian & Eulerian Descriptions

Objectives

• Fluid kinematics deals with describing the motion of fluids withoutnecessarily considering the forces and moments that cause the motion.

• In this chapter, we introduce several kinematic concepts related toflowing fluids.

• Reynolds transport theorem (RTT)

material derivative

Lagrangian description offluid flow

Eulerian description offluid flow

a C.V. of the flow field

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Objectives

• We then discuss various ways to visualize flow fields:streamlines, streaklines, pathlines

• and we describe three ways to plot flow data:Profile plots, vector plots, and contour plots

• The concepts of vorticity, rotationality, and irrotationality in fluid flowsare also discussed.

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Kinematic Concepts of Flow Field

• Kinematics: how the fluid flows?

• Lagrangian description: we follow a mass of fixed identity.

Lagrangian & Eulerian Descriptions

Difficult!

From a microscopic point of view, a fluid iscomposed of billions of molecules that are

continuously banging into one another

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• A more common method of describing fluid flow is the Euleriandescription of fluid motion.

• We do not need to keep track of the position and velocity of a mass offluid particles of fixed identity. Instead, we define field variables,functions of space and time, within the control volume.

a finite volume called a flow domain or control volume is defined,through which fluid flows in and out.

We define variables at specific time & at specific location.

: = ( , , , ): = ( , , , ) : = ( , , , )

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• Collectively, these (and other) field variables define the flow field:

, , , = , , , + , , , + , , ,5/66


In the Eulerian description we don’t really care what happens toindividual fluid particles;

rather we are concerned with the pressure, velocity, acceleration,etc., of whichever fluid particle happens to be at the location of

interest at the time of interest.

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While there are many occasions in which the Lagrangian description isuseful, the Eulerian description is often more convenient for fluidmechanics applications.

Furthermore, experimental measurements are generally more suited tothe Eulerian description.

In a wind tunnel, for example, velocity or pressure probes are usuallyplaced at a fixed location in the flow, measuring velocity/pressure field.

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However, whereas the equations of motion in the Lagrangiandescription following individual fluid particles are well known (e.g.,Newton’s second law), the equations of motion of fluid flow are not soreadily apparent in the Eulerian description and must be carefullyderived.

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Velocity vectors for the velocity field.

The stagnation point is indicated bycircle.

Solid black curves represent theapproximate shapes of somestreamlines, based on the calculatedvelocity vectors.

The shaded region represents aportion of the flow field that canapproximate flow into an inlet.

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• The equations of motion for fluid flow (such as Newton’s second law) arewritten for an object of fixed identity, taken here as a fluid particle ormaterial particle.

• Material position vector:

Acceleration Field

Lagrangian Descriptions( , , )However, some mathematical manipulation is then necessary to convert

the equations of motion into forms applicable to the Eulerian description.2 : = : =

= ( , , , )

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= = = ( , , , )= + + + ( , , , ) = = + + +

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• apple = + + += + + += + + +

Local acceleration(steady flow=0)

Convective (advective)acceleration

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Flow Visualization

• “whole picture” rather than merely a list of numbers and quantitativedata.

Flow Visualization

Spinning baseball

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Flow Visualization

• A streamline is a curve that is everywhere tangent to the instantaneouslocal velocity vector.

• Streamlines are useful as indicators of the instantaneous direction offluid motion throughout the flow field.

• regions of recirculating flow and separation of a fluid off of a solid wallare easily identified by the streamline pattern.

Streamlines and Streamtubes

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Flow Visualization

• Consider an infinitesimal arc length: = + +along a streamline;

must be parallel to the local velocity vector: = + + bydefinition of the streamline.

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Flow Visualization

• In 2D:

: = = = =− : =

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Flow Visualization

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Flow Visualization

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Flow Visualization

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Flow Visualization

• A streamtube consists of a bundle of streamlines

• Since streamlines are everywhere parallel to the local velocity, fluidcannot cross a streamline by definition. By extension, fluid within astreamtube must remain there and cannot cross the boundary of thestreamtube.

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Flow Visualization

• both streamlines and streamtubes are instantaneous quantities, definedat a particular instant in time according to the velocity field at thatinstant.

• In an unsteady flow, the streamline pattern may change significantlywith time.

• Nevertheless, at any instant in time, the mass flow rate passing throughany cross-sectional slice of a given streamtube must remain the same.

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Flow Visualization

• A pathline is the actual path traveled by an individual fluid particle oversome time period.

Pathlines

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Flow Visualization

• Pathline: fluid particle’s material position

• A modern experimental technique called particle image velocimetry(PIV) utilizes particle pathlines to measure the velocity field over anentire plane in a flow

• tiny tracer particles are suspended in the fluid• two flashes of light (usually from a laser)• two bright spots on the film or photosensor for each moving particle.

Then, both the magnitude and direction of the velocity vector at eachparticle location can be inferred, assuming that the tracer particles aresmall enough that they move with the fluid.

( , , )

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Flow Visualization

• Pathlines can also be calculated numerically for a known velocity field:

: = +

If the velocity field is steady, individual fluid particles will follow streamlines.Thus, for steady flow, pathlines are identical to streamlines.

If the velocity field is steady, individual fluid particles will follow streamlines.Thus, for steady flow, pathlines are identical to streamlines.

= ; = ; =

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Flow Visualization

• A streakline is the locus of fluid particles that have passed sequentiallythrough a prescribed point in the flow.

• in a wind or water tunnel, the smoke or dye is injected continuously, notas individual particles, and the resulting flow pattern is by definition astreakline.

Streaklines

if the flow is steady, streamlines, pathlines, and streaklinesare identical

if the flow is steady, streamlines, pathlines, and streaklinesare identical

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Flow Visualization

• The main difference is that a streamline represents an instantaneousflow pattern at a given instant in time, while a streakline and a pathlineare flow patterns that have some age and thus a time history associatedwith them.

• To find the streakline, use the integrated result for the pathline retainingtime as a parameter. Now, find the integration constant which causes thepathline to pass through ( , , ) for a sequence of times < . Theneliminate .

: = +

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Flow Visualization

• Example: an idealized velocity distribution is given by:

• calculate and plot: 1) the streamlines 2) the pathlines 3) the streaklineswhich pass through ( , , ) at t=0.

= 1 + ; = 1 + 2 ; = 0

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Flow Visualization

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Flow Visualization

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Flow Visualization

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Flow Visualization

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Plots of Fluid Flow Data

• profile plots, vector plots, and contour plots

• A profile plot indicates how the value of a scalar property varies alongsome desired direction in the flow field.

• For scalar variable (pressure, temperature, density)

• most common app.: velocity profile plot

• since velocity is a vector quantity, we usually plot either the magnitude ofvelocity or one of the components of the velocity vector as a function ofdistance in some desired direction.

Plots Of Fluid Flow Data

Profile Plot

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• Finally, it is common to add arrows to velocity profile plots to make themmore visually appealing, although no additional information is providedby the arrows.

• If more than one component of velocity is plotted by the arrow, thedirection of the local velocity vector is indicated and the velocity profileplot becomes a velocity vector plot.

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• A vector plot is an array of arrows indicating the magnitude anddirection of a vector property at an instant in time.

Vector Plots

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• A contour plot shows curves of constant values of a scalar property (ormagnitude of a vector property) at an instant in time.

• The maps consist of a series of closed curves, each indicating a constantelevation or altitude.

• Contours can be filled in with either colors or shades of gray; this iscalled a filled contour plot.

Contour Plots

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Fluid Deformation

• In fluid mechanics, as in solid mechanics, an element may undergo fourfundamental types of motion or deformation, as illustrated in twodimensions (a) translation, (b) rotation, (c) linear strain (d) shear strain.

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Fluid Deformation

حالت حرکت تعریف 4مطالعه سیال حتی از مطالعه جامد پیچیده تر است به این دلیل که اغلب هر •.شده در آن دیده می شود

از آنجاییکه المانهاي سیال دایم در حال حرکت هستند بهتر است که حرکت سیال و یا تغییر شکل آن •.بیان کنیم(rate of motion/deformation)را در غالب نرخ تغییرات

براي اینکه بتوان این پارامترها را در سیال به راحتی اندازه گرفت و بیان کرد بهتر است آنها را بر •.حسب سرعت سیال و مشتقات سرعت سیال بدست آوریم

velocity (rate of translation),angular velocity (rate of rotation),

linear strain rate (rate of linear strain),shear strain rate (rate of shear strain)

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Fluid Deformation

• Rate of translation vector in Cartesian coordinates:

• Rate of rotation (angular velocity) at a point is defined as the averagerotation rate of two initially perpendicular lines that intersect at thatpoint.

= + +

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Fluid Deformation

= +2 = 12 ( − ) = 12 − + 12 − + 12 −

Rate of rotation in Cartesian coordinates:

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Fluid Deformation

• Linear strain rate is defined as the rate of increase in length per unitlength.

• Shear strain rate at a point is defined as half of the rate of decrease of theangle between two initially perpendicular lines that intersect at thepoint.

= ; = ; =Linear Strain Rate in Cartesian coordinates:

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Fluid Deformation

= 12 += 12 += 12 +

Shear Strain Rate in Cartesian coordinates:

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Fluid Deformation

Linear Strain+Shear Strain: Deformation

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Fluid Deformation

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Fluid Deformation

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Fluid Deformation

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Fluid Deformation

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Fluid Deformation

• Rate of rotation vector in Cartesian coordinates:Vorticity and Rotationality

= 12 − + 12 − + 12 −: = × = ( )

= − + − + −51/66

Fluid Deformation

• Two-dimensional flow in Cartesian coordinates:

• Two-dimensional flow in cylindrical coordinates:

= −= ( ) −

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Fluid Deformation

• 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 regionis called rotational.

• Likewise, if the vorticity in a region of the flow is zero (or negligiblysmall), fluid particles there are not rotating; the flow in that region iscalled irrotational.

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Fluid Deformation

Rotation of fluid elements is associated with wakes, boundary layers, flowthrough turbomachinery (fans, turbines, compressors, etc.), and flow with

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Fluid Deformation

• Not all flows with circular streamlines are rotational. To illustrate thispoint, we consider two incompressible, steady, two-dimensional flows,both of which have circular streamlines in the rθ-plane:

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Fluid Deformation

− : = 0 =− : = 0 =

− : = − = 2ω− : = − = 0

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Fluid Deformation

Flow A is rotational. Physically, this means that individual fluidparticles rotate as they revolve around the origin

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Fluid Deformation

By contrast, the vorticity of the line vortex is identically zeroeverywhere (except right at the origin, which is a mathematical

singularity). Flow B is irrotational. Physically, fluid particles do notrotate as they revolve in circles about the origin

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Fluid Deformation

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Fluid Deformation

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Reynolds Transport Theorem

• System: (also called a closed system), defined as a quantity of matter offixed identity.

– The size and shape of a system may change during a process, but no mass crosses itsboundaries.

• Control Volume: (also open system), defined as a region in space chosenfor study.

– A control volume, on the other hand, allows mass to flow in or out across itsboundaries, which are called the control surface.

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• B: extensive property (such as mass, energy, or momentum),• & b=B/m represent the corresponding intensive property.

indicates an increase in the B content62/66


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the time rate of change of the property B of the system is equal to thetime rate of change of B of the control volume plus the net flux of B out

of the control volume by mass crossing the control surface.

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Relationship between MaterialDerivative and RTT

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• The Reynolds transport theorem for finite volumes (integral analysis) isanalogous to the material derivative for infinitesimal volumes(differential analysis). In both cases, we transform from a Lagrangian orsystem viewpoint to an Eulerian or control volume viewpoint.

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Documents

Basics: Fluid Mechanicswp.kntu.ac.ir/mojra/FM1-chap3.pdf• Kinematics: how the fluid flows? • Lagrangian description:we follow a mass of fixedidentity. Lagrangian & Eulerian Descriptions