_ch4-fluid.pdf

Chapter 4: Fluid Kinematics

4-1 Lagrangian g g and Eulerian Descriptions p 4-2 Fundamentals of Flow Visualization 4-3 Kinematic Description p 4-4 Reynolds Transport Theorem (RTT)

Fluid Mechanics Y.C. Shih February 2011

Chapter 4: Fluid Kinematics

4-1 Lagrangian and Eulerian Descriptions (1) Lagrangian description of fluid flow tracks the position i i andd velocity l i off individual i di id l particles. i l Based upon Newton's laws of motion. Diffi l to use for Difficult f practical i l flow fl analysis. l i Fluids are composed of billions of molecules. Interaction between molecules hard to describe/model. describe/model

However, useful for specialized applications Sprays particles, particles bubble dynamics, dynamics rarefied gases. gases Sprays, Coupled Eulerian-Lagrangian methods.

Named after Italian mathematician Joseph p Louis Lagrange (1736-1813). 4-1

Fluid Mechanics Y.C. Shih February 2011

Chapter 4: Fluid Kinematics

4-1 Lagrangian and Eulerian Descriptions (2) Eulerian description of fluid flow: a flow domain or control volume is defined by b which hich fluid fl id flows flo s in and out. o t We define field variables which are functions of space and time. Pressure field, P=P(x,y,z,t) r r Velocity field, V = V ( x, y, z, t ) r r r r V = u ( x, y , z , t ) i + v ( x, y , z , t ) j + w ( x, y , z , t ) k r r a = a ( x, y , z , t ) r r r r a = a x ( x , y , z , t ) i + a y ( x, y , z , t ) j + a z ( x, y , z , t ) k

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). (1707 1783)

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Chapter 4: Fluid Kinematics

4-1 Lagrangian and Eulerian Descriptions (3) Acceleration Field Consider a fluid particle and Newton Newton'ss second law law, r r Fparticle = m particle a particle The acceleration of the particler is the time derivative of the dV particlel particle'ss velocity particle velocity. r a particle = dt However particle However, r velocity r at a point is the same as the fluid V particle = V ( x particle ( t ) , y particle ( t ) , z particle ( t ) ) velocity, To take the time of, chainr rule must ber used. r derivative r ∂V dt ∂V dx particle ∂V dy particle ∂V dz particle r a particle = + + + ∂t dt ∂x dt ∂y dt ∂z dt 4-3

Fluid Mechanics Y.C. Shih February 2011

Chapter 4: Fluid Kinematics

4-1 Lagrangian and Eulerian Descriptions (4) Since

dx particle dt

= u,

r a pparticle

dy particle dt

= v,

dz particle dt

=w

r r r r ∂V ∂V ∂V ∂V = +u +v +w ∂t ∂x ∂y ∂z

In vector form, the acceleration can be written as

r r r r r dV ∂V r a ( x, y , z , t ) = = + V ∇ V dt ∂t

(

)

First term is called the local acceleration and is nonzero only for unsteady flows. Second term is called the advective acceleration and accounts for the effect of the fluid particle moving to a new location in the flow, where the velocity is different. 4-4

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Chapter 4: Fluid Kinematics

4-1 Lagrangian and Eulerian Descriptions (5) The total derivative operator d/dt is call the material derivative p notation,, D/Dt. and is often ggiven special

r r r r r r DV dV ∂V = = + V ∇ V Dt dt ∂t

(

)

Advective acceleration is nonlinear: source of many phenomenon h andd primary i challenge h ll in i solving l i fluid fl id flow fl problems. Provides ``transformation'' transformation between Lagrangian and Eulerian frames. Other names for the material derivative include: total,, p particle,, Lagrangian, Eulerian, and substantial derivative. Steady:

v ∂V =0 ∂t

Fluid Mechanics Y.C. Shih February 2011

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Chapter 4: Fluid Kinematics

4-2 Fundamentals of Flow Visualization (1) Flow visualization is the visual examination of flowfield features. Important for both physical y experiments and numerical (CFD) solutions. Numerous methods Streamlines and streamtubes Pathlines Streaklines Timelines Refractive techniques Surface flow techniques 4-6

Fluid Mechanics Y.C. Shih February 2011

Chapter 4: Fluid Kinematics

4-2 Fundamentals of Flow Visualization (2) A Streamline is a curve that is everywhere tangent to the instantaneous local velocity vector. C Consider id an arc llength th

r r r r dr = dxi + dyj + dzk

r dr must be parallel to the local velocity vector

r r r r V = ui + vj + wk

Geometric arguments results in the equation for a streamline

dr dx dy dz = = = V u v w

Fluid Mechanics Y.C. Shih February 2011

Chapter 4: Fluid Kinematics

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4-2 Fundamentals of Flow Visualization (3) NASCAR surface pressure contours and streamlines li

Airplane surface pressure contours, volume streamlines,, and surface streamlines

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Chapter 4: Fluid Kinematics

4-2 Fundamentals of Flow Visualization (4) A Pathline is the actual path traveled by an individual fluid particle over some time period. Same as the fluid particle's position vector material p

(x

particle

( t ) , y particle ( t ) , z particle ( t ) )

Particle location at time t: dx ⎞ = u ( x, y , z , t ) ⎟ dt ⎠ particle dy ⎞ = υ ( x, y , z , t ) ⎟ dt ⎠ particle dz ⎞ = w( x, y, z , t ) ⎟ dt ⎠ particle

r r x = xstart +

t

∫

r Vdt

t start

Particle Image Velocimetry (PIV) is a modern experimental technique to measure velocity field over a plane in the flow field. 4-9

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Chapter 4: Fluid Kinematics

4-2 Fundamentals of Flow Visualization (5)

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Chapter 4: Fluid Kinematics

4-2 Fundamentals of Flow Visualization (6) A Streakline is the locus of fluid particles that have passed p sequentially through a pprescribed ppoint in the flow. Easy to generate in experiments: dye in a water flow, flow or smoke in an airflow. 4-11

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Chapter 4: Fluid Kinematics

4-2 Fundamentals of Flow Visualization (7)

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Chapter 4: Fluid Kinematics

4-2 Fundamentals of Flow Visualization (8) Comparison For steady flow, streamlines, pathlines, and streaklines are identical identical. For unsteady flow, they can be very different. Streamlines St li are an instantaneous i t t picture i t off the th flow fl field fi ld Pathlines and Streaklines are flow patterns that have a time them history associated with them. Streakline: instantaneous snapshot of a time-integrated flow pattern. pattern Pathline: time-exposed flow path of an individual particle. 4-13

Fluid Mechanics Y.C. Shih February 2011

Chapter 4: Fluid Kinematics

4-2 Fundamentals of Flow Visualization (9) EXAMPLE2.1 Streamlines and Pathlines in TwoDimensional Flow A velocity field is given by ; the units of velocity are m/s; x and y are given in meters; . (a) Obtain an equation for the streamlines in the xy plane. (b) Plot the Streamlines passing through the point (c) Determine the velocity of particle at the point (2, 8). ((d)) If the pparticle passing p g through g the ppoint ( is marked at time t = 0, determine the location of the particle at time t = 6 s . (e) What is the velocity of this particle at time t = 6 s ? (f) Show that the equation of the particle path (the pathline) is the same as the equation of the Streamline. 4-14

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Chapter 4: Fluid Kinematics

4-2 Fundamentals of Flow Visualization (10)

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4-3 Kinematic Description (1) In fluid mechanics, an element may undergo four fundamental types of motion. a) Translation b) Rotation c) Linear strain d) Shear strain Because fluids are in constant motion, motion and deformation is best described in terms of rates a) velocity: rate of translation b) angular velocity: rate of rotation c) linear strain rate: rate of linear strain d) shear strain rate: rate of shear strain Fluid Mechanics Y.C. Shih February 2011

Chapter 4: Fluid Kinematics

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4-3 Kinematic Description (2) 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:

r r r r V = ui + vj + wk

Rate of rotation at a point is defined as the average rotation rate i i i ll perpendicular di l lines li h intersect i h point. i off two initially that at that The rate of rotation vector in Cartesian coordinates: 1 ⎛ ∂w ∂v ⎞ r 1 ⎛ ∂u ∂w ⎞ r 1 ⎛ ∂v ∂u ⎞ r − ⎟i + ⎜ − ω= ⎜ ⎟ j + ⎜ − ⎟k 2 ⎝ ∂y ∂z ⎠ 2 ⎝ ∂z ∂x ⎠ 2 ⎝ ∂x ∂y ⎠ r

Fluid Mechanics Y.C. Shih February 2011

Chapter 4: Fluid Kinematics

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4-3 Kinematic Description (3) Linear Strain Rate is defined as the rate of increase in length per unit length. In Cartesian coordinates

∂u ∂v ∂w ε xx = , ε yy = , ε zz = ∂x ∂y ∂z

Volumetric strain rate in Cartesian coordinates

1 DV ∂u ∂v ∂w = ε xx + ε yy + ε zz = + + ∂x ∂y ∂z V Dt Since the volume of a fluid element is constant for an incompressible flow, the volumetric strain rate must be zero.

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Chapter 4: Fluid Kinematics

4-3 Kinematic Description (4) Shear Strain Rate at a ppoint is defined as halff off the rate of decrease of the angle between two initially pperpendicular p lines that intersect at a ppoint. Shear strain rate can be expressed in Cartesian coordinates as:

1 ⎛ ∂u ∂v ⎞ 1 ⎛ ∂w ∂u ⎞ 1 ⎛ ∂v ∂w ⎞ ε xy = ⎜ + ⎟ , ε zx = ⎜ + ⎟ , ε yz = ⎜ + ⎟ 2 ⎝ ∂y ∂x ⎠ 2 ⎝ ∂x ∂z ⎠ 2 ⎝ ∂z ∂y ⎠ 4-19

Fluid Mechanics Y.C. Shih February 2011

Chapter 4: Fluid Kinematics

4-3 Kinematic Description (5) We can combine linear strain rate and shear strain rate into one symmetric second-order tensor called the strain-rate tensor.

⎛ ε xx ⎜ ε ij = ⎜ ε yx ⎜ ε zx ⎝

ε xy ε yy ε zy

⎛ ∂u ⎜ ∂x ⎜ ε xz ⎞ ⎜ 1 ⎛ ∂v ∂u ⎞ ⎟ ε yz ⎟ = ⎜ ⎜ + ⎟ ⎜ 2 ⎝ ∂x ∂y ⎠ ⎟ ε zz ⎠ ⎜ ⎜ 1 ⎛ ∂w + ∂u ⎞ ⎜ 2 ⎝⎜ ∂x ∂z ⎠⎟ ⎝

1 ⎛ ∂u ∂v ⎞ + ⎟ ⎜ 2 ⎝ ∂y ∂x ⎠ ∂v ∂y 1 ⎛ ∂w ∂v ⎞ + ⎟ ⎜ 2 ⎝ ∂y ∂z ⎠

1 ⎛ ∂u ∂w ⎞ ⎞ ⎜ + ⎟⎟ 2 ⎝ ∂z ∂x ⎠ ⎟ 1 ⎛ ∂v ∂w ⎞ ⎟ ⎟ + ⎜ ⎟ 2 ⎝ ∂z ∂y ⎠ ⎟ ⎟ ∂w ⎟ ⎟ ∂z ⎠ 4-20

Fluid Mechanics Y.C. Shih February 2011

Chapter 4: Fluid Kinematics

4-3 Kinematic Description (6) Purpose p of our discussion of fluid element kinematics: Better appreciation of the inherent complexity of fluid dynamics Mathematical sophistication required to fully describe fluid motion

Strain-rate Strain rate tensor is important for numerous reasons. reasons For example, rate Develop relationships between fluid stress and strain rate. Feature extraction and flow visualization in CFD simulations.

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Chapter 4: Fluid Kinematics

4-3 Kinematic Description (7) The vorticity is defined as the curl of the velocity r r vector r vector ζ = ∇ × V Vorticity is to twice the angular velocity of a fluid r equal r particle ζ = 2ω particle. Cartesian coordinates r ⎛ ∂w ∂v ⎞ r ⎛ ∂u ∂w ⎞ r ⎛ ∂v ∂u ⎞ r ζ =⎜ − ⎟i + ⎜ − ⎟ j + ⎜ − ⎟k ⎝ ∂y ∂z ⎠ ⎝ ∂z ∂x ⎠ ⎝ ∂x ∂y ⎠

Cylindrical coordinates

r ⎛ 1 ∂u z ∂uθ ⎞ r ⎛ ∂ur ∂u z ⎞ r ⎛ ∂ ( ruθ ) ∂ur ζ =⎜ − − − ⎟ eθ + ⎜ ⎟ er + ⎜ ∂z ⎠ ∂r ⎠ ∂θ ⎝ ∂z ⎝ r ∂θ ⎝ ∂r

⎞r ⎟ ez ⎠

In regions where z = 0, 0 the flow is called irrotational. irrotational Elsewhere, the flow is called rotational.

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Chapter 4: Fluid Kinematics

4-3 Kinematic Description (8)

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Chapter 4: Fluid Kinematics

4-3 Kinematic Description (9) Special case: consider two flows with circular streamlines

ur = 0, uθ = ω r 1 ⎛ ∂ ( ruθ ) ∂ur − ζ = ⎜ ∂θ r ⎝ ∂r r

2 ⎞r ⎞ r 1 ⎛ ∂ (ω r ) r − 0 ⎟ ez = 2ωez ⎟ ez = ⎜⎜ ⎟ ∂r r ⎠ ⎝ ⎠

K r r 1 ⎛ ∂ ( ruθ ) ∂ur ζ = ⎜ − r ⎝ ∂r ∂θ ur = 0, uθ =

⎞ r 1 ⎛ ∂(K ) ⎞r r e = − 0 ⎟ z ⎜ ⎟ e z = 0 ez r ⎝ ∂r ⎠ ⎠ 4-24

Fluid Mechanics Y.C. Shih February 2011

Chapter 4: Fluid Kinematics

4-4 Reynolds Transport Theorem (RTT) (1) A system is a quantity of matter of fixed y No mass can cross a system y identity. boundary. A control volume is a region in space chosen for study. Mass can cross a control surface. f The fundamental conservation laws (conservation of mass, energy, and momentum) apply directly to systems. systems However, in most fluid mechanics problems, control volume analysis is preferred over system analysis (for the same reason that the Eulerian description is usually preferred over the Lagrangian description). Therefore,, we need to transform the conservation laws from a system to a control volume. This is accomplished with the Reynolds transport theorem (RTT). 4-25

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Chapter 4: Fluid Kinematics

4-4 Reynolds Transport Theorem (RTT) (2)

Let B represent any extensive property (such as mass, energy, or momentum), and let b=B/m represent the corresponding di intensive i t i property. t Noting N ti that th t extensive t i properties are additive, the extensive property B of the system at times t and t t can be expressed as

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Chapter 4: Fluid Kinematics

4-4 Reynolds Transport Theorem (RTT) (3)

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Chapter 4: Fluid Kinematics

4-4 Reynolds Transport Theorem (RTT) (4)

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Chapter 4: Fluid Kinematics

4-4 Reynolds Transport Theorem (RTT) (5)

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Chapter 4: Fluid Kinematics

4-4 Reynolds Transport Theorem (RTT) (6) Interpretation of the RTT: Time rate of change of the property B of the system is equal to (Term 1) + (Term 2) Term 1: the time rate of change of B of the control t l volume l Term 2: the net flux of B out of the control volume by mass crossing the control surface

r r ∂ =∫ ρ b ) dV + ∫ ρ bV ndA ( CV ∂t CS dt

dBsys

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Chapter 4: Fluid Kinematics

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4-4 Reynolds Transport Theorem (RTT) (7) ‰ Special Case 1: Steady Flow

Steady flow :

dBsys dt

→ ∧

= ∫ ρb(V ⋅ n)dA CS

‰ Special Case 2: One One-Dimensional Dimensional Flow

One - dimensional flow : dBsys dt

=

d ρb dV + ∑ ρ ebeVe Ae - ∑ ρ i biVi Ai ∫ 1 424 3 in 1 424 3 dt CV out for each exit

for each exit

• • d = m e be - ∑ m i bi ∫ ρb dV + ∑ dt dt CV out in

dBsys

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Chapter 4: Fluid Kinematics

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4-4 Reynolds Transport Theorem (RTT) (8) Material derivative (differential analysis):

r r Db ∂b = + V ∇ b Dt ∂t

(

)

General RTT RTT, nonfixed CV (integral analysis):

r r ∂ ρ b ) dV + ∫ ρ bV ndA =∫ ( CV ∂t CS dt

dBsys

Mass Momentum B, Extensive properties

m

b Intensive properties b,

1

r mV r V

Energy E e

Angular momentum r H r r r ×V

(

)

In Chaps 5 and 6, we will apply RTT to conservation of mass, energy, linear momentum, and angular momentum. 4-32

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Chapter 4: Fluid Kinematics

4-4 Reynolds Transport Theorem (RTT) (9)

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Chapter 4: Fluid Kinematics

4-4 Reynolds Transport Theorem (RTT) (10)

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Chapter 4: Fluid Kinematics

4-4 Reynolds Transport Theorem (RTT) (11)

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Chapter 4: Fluid Kinematics

4-4 Reynolds Transport Theorem (RTT) (12)

There is a direct analogy between the transformation from Lagrangian to Eulerian descriptions (for differential analysis using i infinitesimally i fi i i ll small ll fluid fl id elements) l ) andd the h transformation f i from systems to control volumes (for integral analysis using large, finite flow fields).

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Chapter 4: Fluid Kinematics