Viscous Flows Introduction

8/10/2019 Viscous Flows Introduction

1/42

Viscous Flow

MECH5001(P01)

Edgar A. MatidaDepartment of Mechanical &

Aerospace Engineering


2/42

Contents

1. Introduction 2. Equations of Motion 3. Boundary Layer Equations 4. Laminar Boundary Layers 5. Boundary Layer Stability and Transition 6. Turbulent Boundary Layers 7. Calculation of Turbulent Boundary Layers 8. Compressible Boundary Layers 9. 3-D Boundary Layers

10. Current Capabilities


3/42

Objectives

Analytical and numerical methods for viscousflow analysis

Report findings (assignments & term project)


4/42

Evaluation

60% - Homework Assignments (not all equallyweighted)

30% - Term Project

10% - Participation


5/42

Recommended Literature

White, F. M., Viscous Fluid Flow, Third Edition,McGraw-Hill


6/42

Accommodations

Students with disabilities requiring academicaccommodations in this course are encouragedto contact the Paul Menton Centre for Students

with Disabilities (500 University Centre) tocomplete the necessary forms. After registeringwith the Centre, make an appointment to meetwith me in order to discuss your needs at least

two weeks before the first in-class test or CUTVmidterm exam. This will allow for sufficient timeto process your request.


7/42

Viscosity

Viscosity is important when

Velocity is very low (creeping flow)

Velocity gradients are changing rapidly (shear flows),for example

Boundary layers

Jets

Wakes Mixing Layers


8/42

Basics Concepts

A) Solid versus fluid For a solid, stress is proportional to strain

For a fluid, stress is proportional to the rate of strain

For a simple parallel laminar shear flow (Couette flow)

Where is the shear stress, du/dy is the rate of shearstrain, and is the viscosity

yd

ud

Newtonian fluid: linear relationship


9/42

Basic Concepts

B) Continuum

Fluid properties (P, , T, ) are treated ascontinuous even though they reflect molecule

properties Pressure is a measure of molecular momentum

Temperature is a measure of molecular kinetic energy

Continuum approach is valid except at very lowdensities


10/42

Basic Concepts

C) Equation of State for Gases For gases, only 2 of P, T, and are independent

They are linked by the equation of state

The perfect gas law will be used in this course

Where M is the molecular weight of the gas

Kkg

J287

Kkmol

kJ314.8where

airR

RM

RR

TRP


11/42

Basic Concepts

D) No-Slip Condition

Surface irregularities and intermolecular forcesprevent relative motion at a solid surface

Due to viscosity, velocity cannot changediscontinuouslythus a thin boundary layerdevelops over which the velocity varies from the

wall value to the freestream value


12/42

Basic Concepts

Temperature dependence


13/42

Role of Viscosity in High Reynolds

Number Flows

The Reynolds number is the ratio between

inertial ( u2) and viscous stresses ( u/ y) givenby

ULRe


14/42

Role of Viscosity in Low Reynolds

Number Flows

Creeping Flow

Viscous forces are important everywhere in the fluid

Low velocities and high viscosities (liquid metal melts)

Very small length scales (micro-organisms, dispersion

of particulate matter, microfluidic devices)

)1(Re O

ULRe


15/42


Number Flows

For most problems on a human scale:

Re = 2 x 104: water flow in a 2.5-cm-diameter bathroomsupply pipe

Re = 2 x 105

: for a baseball thrown by a major league pitcher Re = 1 x 107: car at highway speeds

Re = 5 x 107: modest river (Reynolds based on width)

Re = 2 x 108: commercial jet airplane wing (Reynolds based

on chord length)

Re~1012: For a typical atmospheric low pressure system


16/42


Number Flows

For flows with Reynolds number much largerthan unity, viscous forces will be important onlyin the regions with small length scales or over

very long convective time scales

The length scale ratio (boundary layer thicknessversus height) for a channel flow is

)(Re 2/1OL


17/42


Number Flows

Viscous forces are important everywhere in aturbulent flow, but only at small scales ofmotion

The ratio of the smallest to the largest lengthscales of a turbulent flow is

)(Re 4/3OL


18/42

Viscous Flow Unsteady Simulation

Velocity field using LES (Large EddySimulation, Ilie, 2005)

2-D simulation using 16 processors during aweek (approximately 1.2 million elements)


19/42

Review

Standard vector algebra

Mass, momentum, and energy conservations


20/42

Vector Algebra

A+B=C, A-B=D

Scalar product

Note that the scalar product of two vectors is a scalar

Vector product (cross product)

Where G is perpendicular to the plane of A and B andin a direction which obeys the right-hand rule

Note that the vector product of two vectors is a vector

cos|||| BABA

GeBABA )sin|||(|


21/42

Orthogonal Coordinate System

A point P in space is located by specifying the threecoordinates (x,y,z)

The point can also be located by the position vector r

Where i, j, and k are unit vectors

If A is a vector, then

kjir zyx

kjiA zyx AAA


22/42

Scalar and Velocity Fields

Pressure, density, and temperature are scalarquantities

A scalar quantity given as a function ofcoordinate space and time t is called a scalarfield

),,,(

),,,(

),,,(

1

1

1

tzyxTT

tzyx

tzyxpp


23/42


Velocity is a vector quantity

Where

is the vector field for V in the Cartesian space

kjiV zyx VVV

),,,(

),,,(

),,,(

tzyxVV

tzyxVV

tzyxVV

zz

yy

xx


24/42


The scalar and vector products can be written interms of the components of each vector

Having

AndThen

And

kjA zyx AAA i

kjiB zyx BBB

zzyyxx BABABABA

)()()( xyyxzxxzyzzy

zyx

zyx BABABABABABA

BBB

AAA kji

kji

BA


25/42

Gradient of a Scalar Field

The gradient of p, , at a given point in spaceis defined as a vector such that

Its magnitude is the maximum rate of change of p

per unit length of the coordinate space at the givenpoint

Its direction is that of maximum rate of change of p

at the given point

p


26/42


The magnitude of is the rate of change of p perunit length in the direction from this point along whichp changes the most.

A line along which sigma p is tangent at every point isdefined as a gradient line.

At any point, the gradient line is perpendicular to theisoline.

p


27/42


Consider a gradient of p at a point (x,y). The rate ofchange of p per unit length in some arbitrary s directionis

Here n is a unit vector in the s direction

The expression for the gradient of p is

np

ds

dp

),,( zyxpp

kjiz

p

y

p

x

pp


28/42

Divergence of a Vector Field

Consider a fluid element of fixed mass movingalong a streamline with velocity V

As the fluid element moves through space, itsvolume will, in general, change


29/42

Divergence of a Vector Field

The time rate of change of the volume of amoving fluid element of fixed mass, per unitvolume of that element, is equal to thedivergence ofV

kjiVV zyx VVVzyx ),,(

zV

yV

xV zyxV


30/42

Curl of a Vector Field

Consider the same fluid element of fixed massmoving along a streamline with velocity V

It is possible that this fluid element is rotating

with an angular velocity as it translates alongthe streamline


31/42

Curl of a Vector Field

The angular velocity, , is equal to one-half ofthe curl of V

kjiVV zyx VVVzyx ),,(

y

V

x

V

x

V

z

V

z

V

y

V

VVV

zyx

xyzxyz

zyx

kji

kji

V


32/42

Line Integrals

Consider a vector field Consider a curve C connecting two points a and b

Let ds be the elemental length of the curve and n be aunit vector tangent to the curve

Then the line integral from point a to b is

),,( zyxAA

dsnds

a

b dsA


33/42

Line Integrals

If the curve C is closed

Where the counterclockwise direction around Cis positive

CdsA


34/42

Surface Integrals

Consider an open surface S bounded by a closed curve C At point P, let dS be an elemental area of the surface and n

be a unit vector normal to the surface

The orientation of n is in direction according to the right-

hand rule for a movement along C The vector elemental area is dSndS


35/42

Surface Integrals

The surface integral can be defined in three ways

1) Surface integral of a scalar p over the opensurface S (the result is a vector)

2) Surface integral of a vector Aover the opensurface S (the result is a scalar)

3) Surface integral of a vector Aover the opensurface S (the result is a vector)

dSS

p

dSAS

dSA

S


36/42

Surface Integrals

If the surface S is closed (spheres, cubes, etc.), nwill point out of the surface, and

dSS

p

S

dSA

S

dSA


37/42


38/42

Relations

Stokes theorem

Divergence theorem

Gradient theorem

dSAdsAS

C)(

VS

dV)( AdSA

VS

dVpp dS


39/42


40/42


41/42

Models of Fluid

Infinitesimal Fluid Element Infinitesimal fluid element fixed in space with the

fluid moving through it

Infinitesimal fluid element moving along astreamline with the velocity V equal to the local flow

velocity at each point


42/42

Documents

Viscous Flows Introduction