Lecture 4 FluidKinematics

Slide 1: 

Kinematics means the study of motion without considering the forces and moments that cause the motion. Kinematics involves position, velocity, and acceleration, not force. Fluid kinematics is the study of how fluids flow and how to describe fluid motion without considering the forces and moments that cause the motion. Fluid kinematics describes how a fluid particle translates, distorts, and rotates, and how to visualize flow fields . Introduction

Slide 2: 

Topics Scalar and Vector Fields, Flow Field Descriptions of fluid flow. Material Derivative or Substantial Derivative Fundamentals of Flow visualization. Plots of fluid flow data. Fundamental kinematic properties of fluid motion and deformation. Reynolds Transport Theorem

Field Representation: 

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

Scalar and Vector Fields : 

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 .

Descriptions of Fluid Flow: 

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

Lagrangian Vs Eulerian: 

Lagrangian Vs Eulerian X A X B X C V B V A V C P(x, y, z) (x, y, z ) V (x, y, z ) In the lagrangian description, one must keep track of the position and velocity of individual 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 Lagrangian description of fluid flow tracks the position and velocity of individual particles . (eg. Brilliard ball on a pooltable.) 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 (1736-1813).

Eulerian Description: 

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 for formulation of initial boundary-value problems (PDE's). Named after Swiss mathematician Leonhard Euler (1707-1783).

Slide 9: 

Eulerian and Lagrangian descriptions of temperature of a flowing fluid.

Slide 10: 

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

Slide 11: 

The use of numerous temperature-measuring devices fixed at various locations would provide the temperature field, T = T (x, y, z, t). The temperature of a particle as a function of time would not be known unless the location of the particle were known as a function of time. The use of many such measuring devices moving with various fluid particles would provide the temperature of these fluid particles as a function of time. The temperature would not be known as 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

Acceleration Field: 

Acceleration Field Consider Newton's second law applied to a fluid particle , The acceleration of the particle is the time derivative of the particle's velocity. However, particle velocity at a point is the same as the fluid velocity, To take the time derivative of velocity, chain rule must be used.

Slide 13: 

Since In vector form, the acceleration can be written as First term is called the local acceleration and is nonzero only for unsteady flows. Second term is called the advective or convective acceleration and accounts for the effect of the fluid particle moving to a new location in the flow, where the velocity is different.

Material Derivative: 

Material Derivative The total derivative operator d/dt is given a special name material derivative D/Dt which is formed by following a fluid particle as it moves through the flow field . t t + dt t + 2 dt t +3 dt

Slide 15: 

Material Derivative

Slide 16: 

the above equation is valid for any particle, we can drop the reference to particle A and obtain the acceleration field from the velocity field as

Slide 17: 

The above result is often written in shorthand notation as is termed the material derivative or substantial derivative An often-used shorthand notation for the material derivative operator is the rate of change of temperature as

Remarks about Material Derivative: 

Remarks about Material Derivative The total derivative operator d/dt is called the material derivative and is often given special notation, D/Dt. Advective acceleration is nonlinear. It is the source of many phenomenon and primary challenge in solving fluid flow problems. Provides transformation between Lagrangian and Eulerian frames. Other names for the material derivative include: total, particle, Lagrangian, Eulerian, and substantial derivative.

Slide 19: 

Consider an example which will help to reinforce the physical meaning of the substantial derivative Imagine that you are hiking in the mountains, and you are about to enter a cave. The temperature inside the cave is cooler than outside. Thus, as you walk through the mouth of the cave, you feel a temperature decrease--this is analogous to the convective derivative . However, imagine that, at the same time, a friend throws a snowball at you such that the snowball hits you just at the same instant you pass through the mouth of the cave. You will feel an additional, but momentary, temperature drop when the snowball hits you--this is analogous to the local derivative. The net temperature drop you feel as you walk through the mouth of the cave is therefore a combination of both the act of moving into the cave, where it is cooler, and being struck by the snowball at the same instant--this net temperature drop is analogous to the substantial derivative.

Slide 20: 

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

Streamlines: 

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

Slide 22: 

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 given streamtube must remain the same.

Following points about streamtube are worth noting: 

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: 

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: 

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).

Slide 26: 

Stream Line Path Line This is an imaginary curve in a flow field for a fixed instant of time, tangent to which gives the instantaneous velocity at that point . This refers to a path followed by a fluid particle over a period of time. 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: 

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: 

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: 

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.

Refractive Flow Visualization Techniques: 

Refractive Flow Visualization Techniques Based on the refractive property of light waves in fluids with different index of refraction, one can visualize the flow field: shadowgraph technique and schlieren technique .

Kinematic Description: 

Kinematic Description In fluid mechanics, an element may undergo four fundamental types of motion. Translation Rotation Linear strain Shear strain Because fluids are in constant motion, motion and deformation are described in terms of rates velocity: rate of translation angular velocity: rate of rotation linear strain : rate of linear strain shear strain : rate of shear strain

Rate of Translation : 

Rate of Translation To be useful, these rates must be expressed in terms of velocity and derivatives of velocity The rate of translation vector is described as the velocity vector. In Cartesian coordinates :

Rate of Rotation: 

Rate of Rotation Rate of rotation or angular velocity at a point in the xy plane is equal to the time derivative of the average rotation angle. The rate of rotation vector in Cartesian coordinates:

Linear Strain Rate: 

Linear Strain Rate Linear Strain Rate is defined as the rate of increase in length per unit length. Linear strain rate in Cartesian coordinates Volumetric strain rate in Cartesian coordinates Since the volume of a fluid element is constant for an incompressible flow, the volumetric strain rate must be zero.

Shear Strain Rate: 

Shear Strain Rate Shear Strain Rate at a point is defined as half of the rate of decrease of the angle between two initially perpendicular lines that intersect at a point. Shear strain rate can be expressed in Cartesian coordinates as:

Strain - rate Tensor: 

Strain - rate Tensor We can combine linear strain rate and shear strain rate into one symmetric second-order tensor called the strain-rate tensor.

Circulation and Vorticity: 

Circulation : It is defined mathematically as the line integral of the tangential velocity about a closed path (contour) V- velocity in the flow field at the element ds  - angle between V and tangent to the path (in the positive anticlockwise direction along the path) at the point Vorticity ( or ) : It is also defined as circulation per unit of enclosed area . It is a measure of rotation of a fluid particle equal to twice the angular velocity of the fluid particle. Circulation and Vorticity

Vorticity and Rotationality: 

Vorticity and Rotationality The vorticity vector is defined as the curl of the velocity vector Vorticity is equal to twice the angular velocity of a fluid particle. Cartesian coordinates Cylindrical coordinates In regions where z = 0, the flow is called irrotational. Elsewhere, the flow is called rotational.

Vorticity and Rotationality: 

Vorticity and Rotationality

Slide 41: 

Fluid particles within viscous boundary layer near the solid wall are rotational. Fluid particles outside the boundary layer are irrotational. Rotation of fluid elements is associated with wakes, boundary layers, flow through turbomachinery and flow with heat transfer. If a flow originates from the irrotational region it remains irrotational until some non uniform process alters it ( vorticity changes only because of viscosity or non uniform heating or other non uniform phenomena)

Slide 42: 

When torque is applied to the fluid particle it will give rise to rotation; the torque is due to shear stress. The shear stress in turn dependent upon the viscosity, rotational flow occurs where the viscosity effect are predominant. In case were viscosity effects are small it can be assume as irrotational flow

Potential function(): 

Potential function( ) If the curl of a vector is zero, the vector can be expressed as the gradient of a scalar function , called the potential function. In fluid mechanics, vector is the velocity vector, the curl of which is the vorticity vector  and thus we call  the velocity potential. Mathematically  = f(x,y,z,t) ---- unsteady flow  = f(x,y,z) ---- steady flow

Slide 44: 

For an incompressible steady flow the continuity equation is Substituting the value of u,v,w in terms of  in above equation, we obtain the Laplace equation If the velocity potential satisfies the Laplace equation it represents the possible steady, incompressible, irrotational flow. Often an irrotational flow is known as potential flow

Two dimensional Irrotational Regions of flow: 

Two dimensional Irrotational Regions of flow If the flow is 2D, we are able to make use of the stream function as well. The 2D approximation is not limited to flow in the xy plane, nor is it limited to Cartesian coordinates. In fact, we can assume two dimensionality in any region of the flow where only two directions of motion are important and where there is no significant variation in the third direction. The two most common examples are Planar flow ( flow in a plane with negligible variation in the direction normal to the plane) Axisymmetric flow (flow in which there is rotational symmetry about some axis)

The Stream Function: 

The Stream Function Why do this? Single variable  replaces (u,v). Once  is known, (u,v) can be computed. Physical significance Curves of constant  are streamlines of the flow Difference in  between streamlines is equal to volume flow rate between streamlines It can also be defined as the flux or flow rate between two streamlines. The unit of  is m 3 /s (discharge per unit thickness of flow). Existence of  means a possible case of fluid flow

The Stream Function: 

The Stream Function Consider the continuity equation for an incompressible 2D flow Substituting the clever transformation Gives This is true for any smooth function  (x,y)

Slide 48: 

In an irrotational region of flow

Important Remarks about  and : 

Important Remarks about  and  “ The stream function is defined by continuity; the Laplace equations for  results from irrotationality” “The velocity potential is defined by irrotationality; the Laplace equations for  results from continuity”

Streamline & Equipotential Line relationship : 

Streamline & Equipotential Line relationship Curves of constant values of  define streamlines of the flow . Curves of constant values of  define equipotential lines of the flow. In Planar irrotational flow the streamlines and equipotential lines are intersect each other at right angles. Solutions of  and  are called harmonic functions.

The Stream Function Physical Significance: 

The Stream Function Physical Significance Recall along a streamline  Change in  along streamline is zero

The Stream Function Physical Significance: 

The Stream Function Physical Significance Difference in  between streamlines is equal to volume flow rate between streamlines

Cauchy Riemann Equation: 

Cauchy Riemann Equation From the above discussions the following conclusions are arrived: Potential function exist only for irrotational flow Stream function applies to both rotational and irrotational flows In irrotational flow both  &  satisfy the laplace equation as they are interchangeable. CR Equation

Relation between Stream Function and Velocity Potential: 

Relation between Stream Function and Velocity Potential Let two curves  = C &  = C intersect each other at any point . At the point of intersection the slopes are : For  = C : slope = For  = C : slope = It shows that these two sets of curves intersect each other orthogonally at points of intersection.