Ch5-Introduction to Differential Analysis of Fluid Motionerac.ntut.edu.tw/ezfiles/39/1039/img/832/Ch5... · · 2010-03-09Chapter 5: Introduction to Differential Analysis of Fluid

Chapter 5: Introduction to Differential Analysis of Fluid Motion

Fluid Mechanics Y.C. Shih Spring 2009

Chapter 5 Introduction to Differential Analysis of Fluid Motion

5-1 Conservation of Mass5-2 Stream Function for Two-Dimensional 5-3 Incompressible Flow5-4 Motion of a Fluid Particle (Kinematics)5-5 Momentum Equation5-6 Computational Fluid Dynamics (CFD)5-7 Exact Solutions of the Navier-Stokes

Equation

Chapter 5: Introduction to DifferentialAnalysis of Fluid Motion


5-1 Conservation of Mass (1)

Basic Law for a System:

5-1




Rectangular Coordinate System:

5-2





“Continuity Equation” Ignore terms higher than order dx

5-3





“Del” Operator

5-4





Incompressible Fluid:

Steady Flow:

5-5




Cylindrical Coordinate System:

5-6





“Del” Operator

5-7





Incompressible Fluid:

Steady Flow:

5-8



Alternative Method:Divergence Theorem:Divergence theorem allows us to transform a volume integral of the divergence of a vector into an area integral over the surface that defines the volume.


5-9



Rewrite conservation of momentum

Using divergence theorem, replace area integral with volume integral and collect terms

Integral holds for ANY CV, therefore:


5-10



Alternative form:


5-11



5-2 Stream Function for Two-DimensionalIncompressible Flow (1)Two-Dimensional Flow:

Stream Function ψ

5-12

This is true for any smoothfunction ψ(x,y)



Why do this?Single variable ψ replaces (u,v). Once ψ is known, (u,v) can be computed.Physical significance

Curves of constant ψ are streamlines of the flowDifference in ψ between streamlines is equal to volume flow rate between streamlines

5-2 Stream Function for Two-DimensionalIncompressible Flow (2)

5-13



Physical Significance: Recall that along astreamline

∴ Change in ψ along streamline is zero


5-14



Physical Significance:

Difference in ψ between streamlines is equal to volume flow rate between streamlines


5-15




Cylindrical Coordinates:

Stream Function ψ(r,θ)

5-16



Lagrangian Description:

5-3 Motion of a Fluid Particle (Kinematics) (1)

Lagrangian description of fluid flow tracks the position and velocity of individual particles.Based upon Newton's laws of motion. Difficult to use for practical flow analysis.

Fluids are composed of billions of molecules.Interaction between molecules hard to describe/model.

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

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

5-17



Eulerian Description:

( ), , ,V V x y z t=r r

Eulerian description of fluid flow: a flow domain or control volume is defined by 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).

( ) ( ) ( ), , , , , , , , ,V u x y z t i v x y z t j w x y z t k= + +rr r r

( ), , ,a a x y z t=r r

( ) ( ) ( ), , , , , , , , ,x y za a x y z t i a x y z t j a x y z t k= + +rr rr


5-18




Fluid Translation: Acceleration of aFluid Particle in a Velocity FieldFluid RotationFluid Deformation

Angular DeformationLinear Deformation

5-19



Acceleration Field

particle particle particleF m a=r r

Consider a fluid particle and Newton's second law,

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, chain rule must be used.

particleparticle

dVa

dt=

rr

( ) ( ) ( )( ), ,particle particle particle particleV V x t y t z t=r r

particle particle particleparticle

dx dy dzV dt V V Vat dt x dt y dt z dt

∂ ∂ ∂ ∂= + + +

∂ ∂ ∂ ∂

r r r rr


5-20



, ,particle particle particledx dy dzu v w

dt dt dt= = =

particleV V V Va u v wt x y z

∂ ∂ ∂ ∂= + + +

∂ ∂ ∂ ∂

r r r rr

Since

In vector form, the acceleration can be written as


5-21




Fluid Translation: Acceleration of aFluid Particle in a Velocity Field:

5-22



The total derivative operator d/dt is call the material derivative and is often given special notation, D/Dt.Advective acceleration is nonlinear: 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 substantialderivative.


5-23



There is a direct analogy between the transformation from Lagrangian to Eulerian descriptions (for differential analysis using infinitesimally small fluid elements) and the transformation from systems to control volumes (for integral analysis using large, finite flow fields).


5-24



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 streamtubesPathlinesStreaklinesTimelinesRefractive techniquesSurface flow techniques


5-25



dr dxi dyj dzk= + +rr rr

V ui vj wk= + +rr r r

drr

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

dr dx dy dzV u v w

= = =

Streamlines


5-26



NASCAR surface pressure contours and streamlines

Airplane surface pressure contours, volume streamlines, and surface streamlines


Streamlines

5-27



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

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.

start

t

startt

x x Vdt= + ∫rr r

Pathline


5-28



A Streakline is the locus of fluid particles that have passed sequentially through a prescribed point in the flow.Easy to generate in experiments: dye in a water flow, or smoke in an airflow.

Streakline:


5-29



For steady flow, streamlines, pathlines, and streaklines are identical. For unsteady flow, they can be very different.

Streamlines are an instantaneous picture of the flow fieldPathlines 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.


5-30



In fluid mechanics, an element may undergo four fundamental types of motion. a) Translationb) Rotationc) Linear straind) Shear strain

Because fluids are in constant motion, motion and deformation is best described in terms of rates a) velocity: rate of translationb) angular velocity: rate of rotationc) linear strain rate: rate of linear

straind) shear strain rate: rate of shear

strain


5-31




Fluid Translation: Acceleration of aFluid Particle in a Velocity Field:

5-32




Fluid Translation: Acceleration of aFluid Particle in a Velocity Field (Cylindrical):

5-33




Fluid Rotation:

5-34



The vorticity vector is defined as the curl of the velocity vectorVorticity 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.

Vζ = ∇ ×r r r

Vorticity and Rotationality:

2ζ ω=r r

w v u w v ui j ky z z x x y

ζ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞= − + − + −⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠

rr r r

( )1 z r z rr z

ruuu u u ue e er z z r r

θθθζ

θ θ⎛ ⎞∂∂∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞= − + − + −⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠

r r r r


5-35



Vorticity and Rotationality:


5-36



Comparison of Two Circular Flows:

( ) ( )2

0,

1 1 0 2

r

rz z z

u u r

rru u e e er r r r

θ

θ

ω

ωζ ω

θ

= =

⎛ ⎞∂⎛ ⎞∂ ∂ ⎜ ⎟= − = − =⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠

r r r r

Special case: consider two flows with circular streamlines

( ) ( )

0,

1 1 0 0

r

rz z z

Ku ur

ru Ku e e er r r r

θ

θζθ

= =

⎛ ⎞ ⎛ ⎞∂ ∂∂= − = − =⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠⎝ ⎠

r r r r


5-37




Fluid Deformation:Angular Deformation

5-38



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:

1 1 1, ,2 2 2xy zx yz

u v w u v wy x x z z y

ε ε ε⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞= + = + = +⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠


5-39



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

1 12 2

1 12 2

1 12 2

xx xy xz

ij yx yy yz

zx zy zz

u u v u wx y x z x

v u v v wx y y z y

w u w v wx z y z z

ε ε εε ε ε ε

ε ε ε

⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂⎛ ⎞+ +⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠⎜ ⎟⎛ ⎞ ⎜ ⎟⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂⎜ ⎟ ⎜ ⎟= = + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ∂ ∂ ∂ ∂ ∂⎜ ⎟⎝ ⎠ ⎝ ⎠⎜ ⎟⎝ ⎠ ⎜ ⎟⎛ ⎞∂ ∂ ∂ ∂ ∂⎛ ⎞⎜ ⎟+ +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠


5-40




Fluid Deformation:Linear Deformation

5-41



Linear Strain Rate is defined as the rate of increase in length per unit length.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.

, ,xx yy zzu v wx y z

ε ε ε∂ ∂ ∂= = =

∂ ∂ ∂

1xx yy zz

DV u v wV Dt x y z

ε ε ε ∂ ∂ ∂= + + = + +

∂ ∂ ∂


5-42



5-5 Momentum Equation (1)

Newton’s Second Law:

5-43




Forces Acting on a Fluid Particle:

5-44




Differential Momentum Equation:

5-45

or



Stress Tensor:

σ ij =σ xx σ xy σ xz

σ yx σ yy σ yz

σ zx σ zy σ zz

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=−p 0 00 −p 00 0 −p

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ +

τ xx τ xy τ xz

τ yx τ yy τ yz

τ zx τ zy τ zz

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟⎟⎟

Viscous (Deviatoric) Stress Tensor


5-46



Unfortunately, this equation is not very useful10 unknowns

Stress tensor, σij : 6 independent componentsDensity ρVelocity, V : 3 independent components

4 equations (continuity + momentum)6 more equations required to close problem!


5-47



First step is to separate σij into pressure and viscous stresses

Situation not yet improved6 unknowns in σij ⇒ 6 unknowns in τij + 1 in P, which means that we’ve added 1!

σ ij =σ xx σ xy σ xz

σ yx σ yy σ yz

σ zx σ zy σ zz

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=−p 0 00 −p 00 0 −p

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ +

τ xx τ xy τ xz

τ yx τ yy τ yz

τ zx τ zy τ zz

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟⎟⎟

Viscous (Deviatoric) Stress Tensor


5-48



Reduction in the number of variables is achieved by relating shear stress to strain-rate tensor.For Newtonian fluid with constant properties

Newtonian closure is analogousto Hooke’s Law for elastic solids

Newtonian fluid includes most commonfluids: air, other gases, water, gasoline


5-49



Substituting Newtonian closure into stress tensor gives

Using the definition of εij


5-50



Substituting σij into Cauchy’s equation gives the Navier-Stokes equations

This results in a closed system of equations!4 equations (continuity and momentum equations)4 unknowns (U, V, W, p)

Incompressible NSEwritten in vector form


5-51



In addition to vector form, incompressible N-S equation can be written in several other forms

Cartesian coordinatesCylindrical coordinatesTensor notation

Navier-Stokes Equation:


5-52




Newtonian Fluid: Navier-Stokes Equations:

5-53



Tensor and Vector notation offer a more compact form of the equations.

Continuity

Conservation of MomentumTensor notation Vector notation

Vector notationTensor notation

Repeated indices are summed over j (x1 = x, x2 = y, x3 = z, U1 = U, U2 = V, U3 = W)


5-54




Special Case: Euler’s Equation

5-55



Alternative method:

Body Force

Surface Force

σij = stress tensor

Using the divergence theorem to convert area integrals


5-56



*Recognizing that this holds for any CV, the integral may be dropped

This is Cauchy’s EquationCan also be derived using infinitesimal CV and Newton’s 2nd Law


5-57



Alternate form of the Cauchy Equation can be derived by introducing

Inserting these into Cauchy Equation and rearranging gives


5-58



5-6 Computational Fluid Dynamics (1)

Some Applications:

5-59



5-6 Computational Fluid Dynamics (2)

Discretization:

5-60



5-7 Exact Solutions of the Navier Stokes Equation (1)

RecallChap 4: Control volume (CV) versions of the laws of conservation of mass and energyChap 5: CV version of the conservation of momentum

CV, or integral, forms of equations are useful for determining overall effectsHowever, we cannot obtain detailed knowledge about the flow field inside the CV ⇒ motivation for differential analysis

5-61



Example: incompressible Navier-Stokes equations

We will learn:Physical meaning of each termHow to deriveHow to solve


5-62



For example, how to solve?Step Analytical Fluid Dynamics

Verify and plot results6

Apply I.C.’s and B.C.’s to solve for constants of integration5

Integrate equations4

Simplify PDE’s3

List all assumptions, approximations, simplifications, boundary conditions

2

Setup Problem and geometry, identify all dimensions and parameters1


5-63



Navier-Stokes equations

This results in a closed system of equations!4 equations (continuity and momentum equations)4 unknowns (U, V, W, p)

Incompressible NSEwritten in vector form


5-64



Continuity

X-momentum

Y-momentum

Z-momentum


5-65



Solutions can also be classified by type or geometry1. Couette shear flows2. Steady duct/pipe flows3. Unsteady duct/pipe flows4. Flows with moving

boundaries5. Similarity solutions6. Asymptotic suction flows7. Wind-driven Ekman flows

There are about 80 known exact solutions to the NSE (Navier-Stokes Equation)The can be classified as:

Linear solutions where the convective

term is zeroNonlinear solutions where convective term is not zero


5-66



No-slip boundary condition:

For a fluid in contact with a solid wall, the velocity of the fluid must equal that of the wall


5-67



Interface boundary condition:When two fluids meet at an interface, the velocity and shear stress must be the same on both sides

If surface tension effects are negligible and the surface is nearly flat


5-68



Interface boundary condition:Degenerate case of the interface BC occurs at the free surface of a liquid.Same conditions hold

Since μair << μwater,

As with general interfaces, if surface tension effects are negligible and the surface is nearly flat Pwater = Pair


5-69



Example exact solution: Fully Developed Couette Flow

For the given geometry and BC’s, calculate the velocity and pressure fields, and estimate the shear force per unit area acting on the bottom plateStep 1: Geometry, dimensions, and properties


5-70



Step 2: Assumptions and BC’sAssumptions1. Plates are infinite in x and z2. Flow is steady, ∂/∂t = 03. Parallel flow, V=04. Incompressible, Newtonian, laminar, constant properties5. No pressure gradient6. 2D, W=0, ∂/∂z = 07. Gravity acts in the -z direction,

Boundary conditions1. Bottom plate (y=0) : u=0, v=0, w=02. Top plate (y=h) : u=V, v=0, w=0


5-71



Step 3: Simplify3 6

Note: these numbers referto the assumptions on the previous slide

This means the flow is “fully developed”or not changing in the direction of flow

Continuity

X-momentum2 Cont. 3 6 5 7 Cont. 6


5-72



Step 3: Simplify, cont.Y-momentum

2,3 3 3 3,6 7 3 33

Z-momentum

2,6 6 6 6 7 6 66


5-73



Step 4: Integrate

Z-momentum

X-momentum

integrate integrate

integrate


5-74



Step 5: Apply BC’sy=0, u=0=C1(0) + C2 ⇒ C2 = 0y=h, u=V=C1h ⇒ C1 = V/hThis gives

For pressure, no explicit BC, therefore C3 can remain an arbitrary constant (recall only ∇P appears in NSE).

Let p = p0 at z = 0 (C3 renamed p0)

1. Hydrostatic pressure2. Pressure acts independently of flow


5-75



Step 6: Verify solution by back-substituting into differential equations

Given the solution (u,v,w)=(Vy/h, 0, 0)

Continuity is satisfied0 + 0 + 0 = 0

X-momentum is satisfied


5-76



Finally, calculate shear force on bottom plate

Shear force per unit area acting on the wall

Note that τw is equal and opposite to the shear stress acting on the fluid τyx(Newton’s third law).


5-77

Documents

Ch5-Introduction to Differential Analysis of Fluid Motionerac.ntut.edu.tw/ezfiles/39/1039/img/832/Ch5... · · 2010-03-09Chapter 5: Introduction to Differential Analysis of Fluid