The effect of wall proposity on the stability of parallel flows over compliant boundaries

NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER

DO Cli N EN TAT I

liotheek van

February 1970

- C. 20007

THE EFFECT OF WALL POROSITY ON THE

STABILITY OF PARALLEL FLOWS OVER

COMPLIANT BOUNDARIES

This document has been approved for public

release and sale; its ditxibUtion is unlimited.

DEPARTMENT OF AERODYNAMICS RESEARCH AND DEV.. _REPORT

TMthT

70

Report 3330

The Naval Ship Research and Development Center is a U.S. Navy center for laboratory

effort directed at achieving improved sea and air vehicles.

It was formed in March 1967 by

merging the David Taylor Model Basin at Carderock, Maryland and the Marine Engineering

Laboratory at Annapolis, Maryland. The Mine Defense Laboratory, Panama City, Florida

became part of the Center in November 1961.

Naval Ship Research and Development Center

Washington, D.C. 20007

DEPARTMENT OF THE NAVY

NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER WASHINGTON, D. C. 20007

THE EFFECT OF WALL POROSITY ON THE

STABILITY OF PARALLEL FLOWS OVER

COMPLIANT BOUNDARIES

by

HarveyW. Burden

Prepared under the direction of Dr. Hsuan Yeh

Professor of Mechanical Engineering Director of the Towné School of Civil and

Mechanical Engineering The University of Pennsylvania

This document has been approved for public

release and sale; its distribution is unlimited.

February 1970 Report 3330

/c72,

oi/7f

Foreword

This repolt was prepared as a dissertation,

under the direction of, Dr..Hsuan Yeh, Professor

of Mechanical Engineering and Director of the Towne School of Civil and Mechanical Engineering,

The University of Pen'isylvania. It was approved

by D. I. M. Cohen and Dr. A. M. Whitman as

readers. The author is indebted to them for their encouragement and suggestions and to Mr. Cianni Gaetano of the NASA Langley Research Center and Mr. D. w. Pullen of the NSRDC Aerodynamics Laboratory for their invaluable assistance in computer programming and processing.

TABLE OF CONTENTS

Page

Introduction 1

1.1 Background 1

1.2 Purpose of this Investigation 3

2 Summary of Results 5

3 The Laminar Shear Flow Over a Plane Boundary 7

3.1 Basic Equations of Motion 7

3.2 Non-dimensionalization of Wall Parameters 10

3.3 Boundary Conditions 14

3.4 Spatial vs. Temporal Amplification of Disturbances 22

4 The Compliant, Porous Boundary 24

4.1 General Considerations 24

4.2 Spring Supported, Damped Porous Membrane 25

4.3 Visco-elastic Surface 27

5 Matching of Boundary Conditions 34

5.1 Boundary Conditions Far from the Wall 34

5.2 Boundary Conditions at the Wall 36

5.3 Shear Stress at the Wall 37

6 Computational Methods 39

6.1 General Solution Methodology 39

6.2 Numerical Integration 40

6.3 Purification of the Eigenfunction 40

6.4 Determination of the Eigenfunction 42

6,5 Velocity Perturbation, Reynolds Stress, and Vorticity 43

Distributions

V

Page

7 The Rigid Porous Surface 45

7.1 The Rigid Non-Porous Wa-1-1 45

7.2 The Effect of Porosity Upon Rigid Wall Stability 49

8 Porous Membranes 55

8.1 The Non-Porous Membrane Surface 55

8.2 Effect of Porosity Upon Membrane Stability . 58

9 Visco-Elastjc Porous Surfaces . 63

9.1 Visco-Elastic Surfaces 63

9.2 Highly Compliant Visco-Elastic Surface 63

9.3 Effect of Varying Wall Parameters of Highly Compliant

Visco-Elastic Wall 65

9.4 Effect of Porosity of. Highly Compliant Surface 71

9,5 Firm Visco-Elastic Surface 74

9.6 Effect of Varying Wall Parameters of Firm Visco-Elastic Wall 74

9.7 Effect of Porosity of Firm Visco-Elastic Surface 79

9.8 Nearly Incompressible Firm Visco-Elastic Surface 79

9.9 Rubber Surface 85

9.10 _{Effect of Porosity of Rubber Surface 89}

9.11 Effect of Varying Elastic Constants 89

9.12 _{Velocity Perturbation and Reynolds Stress Profiles 94}

10 Conclusions . 96

10.1 General Aspects of Boundary Layer Along a Compliant Plane

Surface 96

10.2 The Effect of Wall Porosity 97

103 106 111 113

Page

10.4 Effect of Additional ProbI Variables 100

]0.5 Direction of Future Investigations 101

Appendix A Three-Diensional Dis türbance in a To-D1menSiona1 Mean Flow

/ppendix B Asymptotic Solution with Porosity

Appendix C Numerical Integration Formi1as

Appendix D Digital Computer Progr USed

AppendIx A*isymmetric Laminar Shear Flow with Compliant,

Porous Boundaries

References

140 151

LIST OF FIGURES

Figure Title Page

3-1 Biasius Boundary Layer Sketch 8

3-2 Sketch of Compliant Visco-Elastic Wail 16

4-i Sketch of Membrane Wail 25

5-i Illustration of Integration Technique 35.

7-1 Rigid WaLl E.igenváiues 46

7-2 Rigid Wail EigenváIues and Experitental Results

(spatIal amplification atês) 47

7-3 Rigid Wall, Effect of Porosity 50

7-4 Rigid Wall, Effect of Porosity (spatial

amplificatton rates) 51

7-5 Rigid Wall, Perturbation Amplitude Functions 53

8-1 Membrane Wall Effect of Computational

Variations, c0 0.9 56

8-2 Membrane Wall Effect of Computational Variations,

c01.2

57

8-3 Membrane Wall Effect of Wall Mass, C0 = 0.9 59

8-4 Membrane Wall Effect of Porosity, rn0 5 60

8-5 Membrane Wall Effect of Porosity, m0 = 50 61

9-i Soft Sliding Visco-Elastic Wall Eigenvalues 64

9-2 Soft Sliding Visco-1astIc Wall,Effect of

Damping 66

9-3 Soft Fixed Visco-Elastic Wall,Effect of

DampIng 67

9-4 Soft Visco-Elastic Wall,Effect of Bottom

Fixity 68

9-5 Soft Sliding Vsçq-E1astLc Wall,Effect of

Depth 69

9-6 Soft Fixed Visco-Elastic Wail, Effect of

Depth 70

vi"

Figure Title Page

9-7 Soft Sliding Visco-Elastic Wail, Effect of

Porosity 7.2

9-8 Soft Fixed Visco-astic Wall., Effect of

Porosity - 73

9-9 Firm Sliding Vlâcó-ElàAtic Wall Eigenvaluês 75

9-10 Firm Fixed Visco-Elastiä Wall Eigtivalues 7.6

9-il Firm Sliding Visco-Elastic Wail, Effect of

Density and Damping : 77

9-12 Firm Fixed Visco-Elastjc Wall, Effect of Density and Damping .

. 78

9-13 Firm Visco-Elastic

Wa-1,

Effect qf Bottom

Fixity . 80

.9-14 Firm Sliding Visco-Elastic Wall, Effet of

Depth ., 81

9-15 Firm Fixed Viscà-ElastiO

Wail,

Effect of. Depth 82

9-16- Firm Sliding Visco-Elastic Wall, Effect of

Porosity .

-83

9-17 Firm Fixed Visco-Elastic Wall, Efféc.t Of

Porosity 84

9-18 Firm Fixed Neárl 1ncompresib1è Visco-Elãstic

Wa-li, Effect of Po..osiy 86.

9 l9 Rubber Sliding Wall, Effectof Density,

Damping, and Depth -. 87

9-20 Rubber Fixed Wall, Effect of Density, Daiping,

and Depth - 88

9-21 Rubber Sliding Wall, Effect of Porosity 90

9-22 Rubber Fixed Wall, Effect of Porosity 91

9-23 Sliding. Visco-Elastic Wall, Effect. of Elastic

Constants 92

9-24 Fixed Visc9-Elastic Wall, Effect of Elastic

Constants - - 9.3

9-25 Fixed Vlsco-Elastic Walls, Perturbation

Figure. Title Page

E-1 Cylindric Poiseuille Flow Coordinate System 145

Sketch

E-2 Cylindric Membrane Surface 147

Table

D-1 Source Program in Fortran IV ₁₁₆

LIST OF SYOLS

= coefficients of 'isco-elast-ic wall displacEment functions

= porosity (admittance) factor of wall

C _{= wave speed Of disturbance}

C = natural wave speeds of

soli4 wall

C _{volume flow coficient}

d)d

viscous damping coefficient

£ = (Young's) modulus of elasticity

= shEar modulus

step size in numerical Jntegration schemE

H

v1sco-elastc wall depth

k

= proportiOnality coefficient of boundary layer growth with downstream distance

P3i = minbrane wall masS per unit length

I

(hydrostatic) flui4 pressure

P

= mean flow (hydrostatic) fluid pressurE

P

= combination coefficient of eigenfunction

= normalized wave speed differences

R = Reynolds number

= membrane wall spring factor per unit length

time

= membrane wall tension

= fluId velocity component in tangential direction

(rectangular) or radial direction (cylindrical)

U mean fluid velocity componr1t in tangential or radial

direction

= flud velocity component in normal direction (rectangular)

xi

V = mean fluid velocity component in normal or azimuthal direction

v'J _{= intermediate eigenfunction components in numerical}

integrat:ion stheme

= fluid velocity component normal to plane of u - V (rectangular) or along axis (cylindrical)

W

= mean fluid velocity component normal to plane of U - V or

along axis

= coordinate parallel to wall (rectangular)

y = coordinate normal to wall (rectangular)

y..., = wall admittance

= coordinate normal to x - y plane (rectangular) or along

axis (cylindrical)

2' = mean flow vorticity

= wave number in mean flow direction = wave frequency

= wave member in cross flow direction 6 _{= boundary layer depth}

= fluid divergence

C = small parameter defined in asymptotic solution

strain tensor

= vorticity or cylindrical wall displacement ±n axial ( )

direction

= normal (y) direction wall displacement

= azimuthal coordinate in cylindrical frame

= first Lám (isotropic elastic) constant

= fluid dynamic viscosity coefficient

= fluid kinematic viscosity coefficient

= tangential (x) direction wall displacement

density, fluid (unsubscripte4) or solid (subsrpted), or wall displacement In radial direction (cylindrical)

= stress tensor

= shear stress

'P = fluid.perturbatiOn stream function ampltuefuiction

= fluid perturbation strea. f eton amplitude function evaluated atboundàries

31' fluid (perturbation) st-re function

= fluid mean flow stream function

= wall characteristic (natural) frequency

Superscripts

* = complex conjugate or, in *, displacement thickness

= dithensiOnal. quantity

I

= perturbation quantIty, , as indicated

A

= amplitude funct{oxi of basic quantity

= condition without viscous damping

Subscripts

= critical

= fluid

= imaginary part .

=base condition .or free stream condition

= due to porosity

r

= real part .

-S _{= solid surface}

" wall

I 1ongitudna1 o _{dUataUonal wave ( c d and}

= transverse or shear wave ( c d and p-. )

1

IRODUCTION

1.1 Background

For years seamen and ships' passengers have reported seeing porpoises

or whales playing: in the bow waves of their ships and then racing away

from the ships at speeds of 20-30 knots. Drag calculations of bodies of

shapes similar to these animals with the normally expected turbulent

boundary layer indicate that the power required is considerably n excess

of that which can be expected of the animal (Kramer (33)). If such speeds

are acthal]y obtained by the porpoise, it was Soon apparent that some form

of hydrodynamic drag reduction is taking place rather than any unrealistic

physiological phenothenOn. Kramer (29, 30) suggested that the laminar

boundary layer about the porpoise is stabilized and therefore maintained to

higher Reynolds number by the compliant motion of the animal's skin. He

subsequently conducted experiments (31) which indicated that a compliant

coating on an underwater body n4eed could have significant drag reductiqn

characteristics. In addit:ion, Krer (33) analyzed the porpoise's ski-n from

the struótural viewpoint and conducted experiments with artificial skins

approximating the hide.

Spurred by these initial investigations and results, many

investiga-tors attempted a Satisfactory analytical explanation. Benjamin's (5)

analysis showed that a flexible boundary with damping did affect the

sta-bility of the fluid flow, but in a more cplicated fashion than that

sug-gested by Kramer. amr had hypothesized that the compliant damped

boUnd-ary simply absorbed energy from the flow, thus preventing it from becoming

turbulent. Benjamin's analysis showed that the compliance of the boundary

does have a stabilizing effect: On the Tollmien-Schlichting waves which are

promotes a type of instability not found with a rigid wall. nj.aifln

(5, 6, 7) and Landahl (35) named these two types of waves Class A and

Class B; Class A waves corresponding, to .TolJen-$chlicht1ng waves...

modified by the flexible boundary .aid Class. B waves analogous to panel.

flutter or water waves generated by.the fluid flow. .In..additipn they

identified a third type of instability called Cass. or

Kelvin-Hel.mholtz, which arises with a stiffness. of the,wall s,o'low that 'it

freely follows the pressure pertubations, These Class ç waves.are not

found in practice for any reasonable. flexible surface. . .,

Many other investigators,, inspired by'Krame's;init:ia1 results,

also have contributed recentlyto the body of knowledge of this problem:

for example, Betchov. (1), Rains and Price (2., 22, .23), Nonwejler (57).,.:

To date, experiments to verify the above ideas with iiing sea

animals have produced results varying from promising, through

inconc]u-sive to discouraging (Kramer (30) Lang (37), Lang and :Daybell (38),.

Lang and Norris (39), Lang and Pryor (4Q.), Gero (20), Rosen (60),.ang.

(private communication)). In additjpn, specific laboratory eperiments

with appropriate models have thus far had mixed results

(Mollp-Christensen & Landahl (56), Taneda & Honji (66)). - . . .

in spite of the lack of any conclusive, natural. ei4dence todate,

if.an "unnatural" method of boundary. layer stabilization can be devised,

it can b,e of very great engineering importance. The advantages of,

reducing skin friction drag on ships' hulls or on aircraft areobvious.

In addition, since the . turbulent boundary layer is considerably noisier

than is the laminar boundary layer, the advantages to submarines and

for surface ship sonar domes would be of considerable military

significance. . . . .. . . ..',

All the early analytical calculations referred tO above were

carried out for only the simplest of cases, such as the Biasius or plane

Poiseullie profiles.. More recently, Kaplan (26) has developed a numerical

method for integrating the Orr-Sommerfeid equation and utiliziiig boundary

conditions desèribing the coupling. between the fluid flow aid a compl±ant

boundary as well as the simple rigid wall boundary conditions. He was

able thereby to present numerical/graphical indications of the effects

of varying the physical parameters of the wall in a number of

combina-tions. The results Obai.ned with this nierical technique compared quite well with previous purely analytical as.well as with experimental results.

1.2 Puryose of this Investigation

Kramer (33) has pointed out that the analyses conducted to date

indicate only a moderate stabilization of the boundary layer by a

compliant wail. He also emphasized that the wall characteristics

indicated by such studies as those of Benjamin (5,6,7) and Landahl (35)

ate. at variance with the observed propert:ies of the porpoise skin. . From

this he stressed that other not heretofore included wall characteristics

should be investigated. Among these additional properties is wail

porosity, briefly mentioned by Benjamin (7). With the exception of this

one comment, no analytical or experimental work of any kind appears to

have been done on the effect of wall porosity on the flow stability.

It is well known that a very small arno nt Of wall suction has a

highly stabilizing influence on the boundary layer. Schlichting (61)

outlines the appropriate analysis for the Blasius profile. He shows

that the critical Reynolds number (based on dispiaceinnt thickness)

increases from R=575 for the flat plate with no suction to Rc7O,000

at the optimum amount of suction. This optimum amount of suction.

corresponds to a volume coefficient of suction of only

Cq=_V0/U=

i.ixio4.

It is with this result in mind that the present

investiga-tion has been pursued. Previous analyses of the small velocity and

pressure perturbations jn tie laminar boundary layer flow over either

a rigid or a compliant boundary show that such perturbations vary

rythmically along the wall. With a wail of.some small porosity, the

pressure perturbations producing a slight over-pressure will force fluid

into the wall - a "passive suction" effect. This inflow must be

balanced by an equal outflow when the cyclic slight underpressure reaches

that wall posit-ion.

tt is hypothesized here that the stabilizing effect of the fluid

flow into the wall is greater than the destabilizing effect of the fluid

flow from the wall back into the stream. The purpOse of this

investiga-tion is to determine the integrated effect of such fluid transpirainvestiga-tion

due to wall porosity of various amounts on the rigid wall and on compliant

surfaces of several types hicb have been previously investigated without

-2 StTh4MARYOF RESULTS

An analytical investigation is made of the effect of wall porosity

upon the stability of- quasi-parallel laminar flOws along plane cOmpliaiit

boundare5 Of everal types. A nteticà]. technique is employed for

solution of the coupled Orr-Sommerfeld/stress--strain/strain rate

equa-tions for incompressible, iscou fluid flowing over a porous, compliant,

visco-elastic boundary. The analysis shOwS that Small amount of porosity

have the destabilizing effect (overridtng those stabilizing effects

pre-viously identified for wal-1 pliability and -internal damping) Of ptomot-ing

transition and expanding the range of unstable disturbances. Both the

temporal and spatial amplif-icadon rates are increased for all types of

boundaries.

In general, the boundary material should be such that the wall density

is of, the order of the fluid density, the wall depth is much larger than

the boundary layer thickness, the equivalent shear modulus is of the

order of the fluid dynic pressure, there should be as little internal

wall dissipation as possible above a certain very small lower limit, and

the wall should not be porous. An inju4icious selection of wall

para-meters-namely,, too muëh flexibility or too much dissipation-will allow

additional type of flow instabilities not present in the simpler rigid

wall (Tol]1en-Schlichting) case. With the proper choice of parameters,

it is shown that some stabilization can be achieved.

The anajyt:ical and nter1cal methods are described in detail. The

results are compared with those of comparable situations without 'porosity,

which, in turn, are compared with results of previous investigations,

both analytical and empirical. The correspondence between previous

good. This lends confidence to the analytical results concernIng

porosity effects which have not yet. been .Investigat4 aRa1ytica1]y or

experimentally. Spatial plific.atiQn rates, velocity perturbation

profiles, and Reynolds stress profiles are shown for a elected number

of conditions.

Basic análi

.s presented for the case of parallel flow through

a un form circular cylindrical tube with simi1a wallcharacteitics

Numerical results are not presented; however, the direction of such

futurç investigation is jndjcated.

3 THE LAMINAR SHEAR FLOW OVER A PLANE BOUNDARY

3.1 Basic Equatipns of Notion

The laminar shear flow of an incompressible fluid moving near a

plane boundary must satisfy the basic equations of motion, i.e., the

momentum equations and the equation of continuity. For the case of small

perturbations of this flow, it is sufficient to consider only two

dimen-sional disturbances in general accord with Squire's theorem as described

in Appendix A.

The two-dimensional continuity equation., in teri of dimensional

quantitiesis, in rectangular coordinates(the tilde denotes dimensional

quantities):

where

be made. The quantities

and

LJ()

is the mean flow which is assumed to depend upon the vertical distance from the wall ( ) only and there is no vertical

velocity, V . This basic assumpt-ion of the parallel nature of the mean

flow will be justified later when an estimate of boundary layer depth can

the tangential and normal components of the velocity perturbation

respectively, and X

y

and t _{are the normal space and time}

coordinates (see Figure --- 3-1).

With the continuity equation can be associated a stream function

defined by

y) =fü

+

Z2 such that: 3-1 3=2 and (x,'',t) are --- 3-3 3-4

where

n

is the vorticity

8

Blasius Bouxidãry Layer

igure 3-i

If ths stréám function is introduced into the two-dimensional

version Of the rectangular mornentt (Navler-Stókes) equations and the

pressure

(,

,

1)

is eliminated between the resulting equations,

the vorticity equation results,

and .) is the fluid kinematic visèosity.

At this point, it is appropriate to non-dimensionalize the

vorticity equation with respect to TJ0 for velocities; S , the boun4ary

layer thickness, which Is the characteristic length of

the

problem, for

lengths; and

6/tJ0

for time. The Reynolds number,R, then appear as

U '/t) and the vorticity equation becomes,after linearIzing an ropping

the primes:

frc7z\7zT

3-7

Applying the non-dimensionalized express1on for

(xj, 'y, t)

produces

v2'

11RiI,

tYW

I ç..,ri

1)

+

- R

v

3-8

3-5 3-6

As

discussed

by Un (46), this vortiity equation Is ow linear with

coefficients which are functions.on1yof:.y.Wé then.may-expect a

solution of the form

(,j;t):

((XCt)

-

3-9

Entering, this into the vorticity equation

.3-8

produces (where prime

denotes

a

This is known

asthe Orr-Sonunerfeld equation for linearized infinitesimal

disturbances of a two-dimensional quasi-parallel Incompressible mean flow.

For the purposes of this analysis the Reynolds number, R, has been assumed

to be constant.

(

is known as the wave number of the disturbance and

C

is the wave speed; either or both may be complex constants (see Section

3.5).

LIn (46) and Cheng (8) present thorough discussions' of the

justifica-tions for retention of those terms remaining

in 3-10 after the process

of linearization.

In general, 'the rather standard boundary layer type

of approximation is made that the perturbation stream function varies

much more rapidly In the y direction than any variations caused by

boundary layer growth.

Since R

= U

s/i.),

and we take

as the

characteristic boundary layer depth, this is, equivalent to stating that

the Reynolds number is constant.

Purely analytical solutions of 'the Orr-Sommerfeld equation have

been obtained for certain restricted cases of flows and simple boundaries

as discussed by

Tolimien

(69),'Heisenberg (24),Lin .(46),.Schlichting

(61) and others. The basic concepts of setting up the solution for the

elgenvalues of the problem are quite straightforward, but the

details

9

of actual analytical solutions are'extremelyvexing'(Schlichting (61)).

An outline of the'pröcedure whi wou'1dbe'used forananalytical solutIon

of the present problem is presented in Appendix B. The technique used

follows that of Tollmiem and Schlichting (Sthlichting (61)).

In the present analysis, advantage is taken of: the linearity of

the Orr-Soterfe1d equation to build upa completesolution of its Fourier

components, which components are obtained by solving the appropriate

boundary value problem... The :approprlate boundary, conditions will be

developed in Section 3.3. Since the physics of this problem clearly

indicates that large Reynolds rn.bers are involved (R> 1000), the

approxi-mate solution obtained herein will make full use of this fact (See

Section 3.2).

3.2 Non-dimensioña]jzation of Wall Parameters

The primary characteristic number of the fluid flow considered

here is, as expected, the Reynolds ntunber

R = Li0

8/i..

---

3-li

One of the basic assumptions of this analysis is that R is constant; but,

in reality, it can be seen that it varies in x through the factor

6 (x).

For this incompressible quasi-parallel flow over a semi-infinite

flat-plate, U0 and ) are constant.

As mentioned above, & is: a function of x and obeys the following

law for zero pressure gradient

K(t):x/Uo)*

_{(x R}

3-12"

where K depends upon 'the definition of .S and R

is the Reyolds nter

with respect to the horizOntal distance'doWnstream,, x. If

6

is defined to be that verticaidigtance at which the local. horizontal fluid velocity'

is 0.999 Ui,, then 6.02.' From.the.above defining equations (3-li

nd 3-12), it can be seen how R is related to;:

1) - b - I' D ±

- -I,) r\i-\1

11

--- 3-13

It is seen clearly that, in the present case, R is not constant, but

.does.vary with x. The range of Reynolds number, R, of interest here is

of the order l03R< io; and it is seen that the variation pith x is slow

as Ro(R½. In addition, equation 3-10 will still be valid if

-s.>

____z

ot'

RtU/ -

_-

_2R

Specifically, if R is large, as it is, 3-10 Is sufficiently accurate.

These arguments, modified for cylindric coOrdinates, are equally

valid for cylindric Poiseuille flow, h-ich is briefly discussed in

Appendix E.

AS pointed out by Kaplan (26), a proper analysis of the problem

under consideration requires that appropriate dimensionless variableé be

defined. Since. g varies with x and is the only length. reference avail-able here, it is important that variavail-ables normalized with in any

combi-nation not reflect this slow growth of the boundry layer. As will be

shown below, the boundary (wall) parameters must vary in certain ways

with R .(

S (x)).

For a wall with a depth H (the tilde denotes dimensional quantities),

thiè requires

-V I

H in

- -

ucI -

-LI

where (

U0/..3)

is the above mentioned properly non-dimensionalized para-meter which will. be independent of boundary layer growth. H may be

eaily found by defining (HU01J) to be a quantity HR with HH0 at

3-14

R=R0; thus -.

(-?--'

U

-(-'C R The term is c::?Q (-J. Ui2. -which is identified as

might be tentioned here

dimensionless parameter

thickness).

:: i.)

S()

LI 1) c-ia

R/),

seen tO be the dimensionless frequency and is

related to the temporal parameter by

1?

the frequency of the disturbance in the fluid. It that Lin (46) an4 Schlichting (61), use the

12

(.S*

is the displacement

H = H0 R0/R 3-16

Dissipation terms appearing in the characterization, of the compliant

boundary will be shown to enter in the combination

cL)

where the dimension of ' would normally allow '1 cL

However, the term! introduces &(x) through the time

non-dimensionaliza-tion parameter c)/L1,. We thus obtain:

-u J

C)

l--

---d ) t Cx) t t

since the dimension is the same as that of Further

2.

'-

__

cL

UL) -

R(ic)

where, as before, we may say is the proper non-dimensional

dissipation pareter and

d = d0R0/R

In Chapter 4, it will be shown that still another wall characteristic

is the resonance frequency, c.'-'. For this, define

-3-17

3-18

3-19

3-20

The e1ástc and shear moduij of the wall material are

nOn-dimensionálized by twice the mean st:ream dynamic pressure

G:

/1u

3-22

A characteristic wave speed of the boundary, , will be

non-dimensionalized only by

U0,

c=/Uo

--- 3-23

and no variation with R is required.

A two-dimensional membrane Surface mass parfflete-r may or may not be

scaled, dependiüg upon considerations to be described later, whereas a

visco-elastic wall density Is non-dimenstonalized only by

f .

We obtain

for these:

uc. ,

p(x) -

pUSti)

where is membrane mass per unit length,

_frs

Is the visco-elastic wall

density,, is the fluid dynamic viscosity, andy -is the fluid densIty

(constant).

One of the primary normalizing parameters has been U0. AS stated

above, this has been assumed to be constant in this analysis. In

addlt'ion, the classical Blasius boundary -layer flow Is assumed to exist

as the. mean flow and is assumed not modified In the mean. In Appendix E,

a similar assumption is stated regarding the meai axiSymmetric Poiseulile

flow in the case discussed there. In both cases, of course, any

st-te-wise pressure gradient will alter the mean velocity profiles, as

dis--cussed by Schlichting (61) and several other authors. A zero pressUre

gradient in the streamwise direction has been specified in this analysis

Landahi (35) has discussed techniques which may be used with such

pressure gradients.

3.3

Boundary Conditions

Equation 3-10 is a fourth order differential equation and, as such,

requires four boundary conditions for a solution. The first

two physical

conditions are that the normal and tangential velocity perturbations

vanish as y

.

The second

two physical conditions are the matching

of the normal and tangential velocities of the fluid with those of the

wall's outer surface with the modification that the vertical fluid

velocity perturbation may differ slightly (of the order of 1-10 per cent)

from that of the surface to account for a small amount of porosity of the

wall's outer surface. These four boundary conditions are sufficient to

assure mathematically unique solutions of the equations of motion. The

complete solution is then made up of a linear combination of the four

partial solutions to within an arbitrary constant which is unobtainable,

but unneeded in

any

case.

The first two boundary conditions are expressed in terms of the

stream function:

(p.o

q:'_-3

0

as y

3-25

The Orr-Sommerfeld equation 3-10, sImplifies somewhat as y

,

and even at y

. & . At that point UU0 and U" = 0, producing

422cP'

-

i.cR(Ue)( 47"

01Z4)

0

3-26

which may be rewritten

14

where

o(,z

f

( u

-

3-28

The solution of equation 3-27 is

v)

,et

yi.I

4) -

C, & + C

f C3 e

₄

C,

---where

y0

is an appropriately chosen value which insures satisfactory

matching between this solution and that generated when the boundary layer

is entered. The determination of the value of y0 is described in Chapter

5. In order to insure 4) does not grow without limit as y

w ,

C2=C4=0 and we have q7=

_C1e(YYo)

c3e(YYo)

3-30

The. method

of utilization of these first two

boundary cotidltlOns is also discussed in Chapter 5.

The second two boundary conditions involve the response of a

com-pliant wall to the. pressure perturbation in the fluid flow.

Any response

of the wall to shear stress IS assied to be of higher order and is

neglected.

This investigation is limited to compliant boundaries viherein

the stress tensor is related linearly to the strain and strain rate

tensors, thus excluding any hysteresis effects. In view of the

lineariza-tion of the equalineariza-tions of fluid molineariza-tion, this linearizalineariza-tion is consistent

and not considered unduly restrictive. The velocity perturbations are

assumed very small as are the wall's deflection components from its mean

(plane) position. For this reason., both the velocity perturbation

express-ions and those for the wall's deflect-Ions will be Taylor series expansexpress-ions

about the mean values and

such series will be truncated

after the. linear

terms.

The horizontal

and

vertical velocity perturbations

and the pressure

Figure 3-2 is a detail view of a

point of the deflected wall. It is abput

the point 0 that the Taylor series

expan-sion of both wall deflection ad fluid

motion are taken. The horizontal and

vertical displacements of. the wall are

as in Section 3.2. Y 16

XYY ()(XXX

3-33

PS described y - Figure 3-2 - -,

(x,7,t)

7(y) e

-It is- thus -seen that the linearized series expansion for

,(x,y,d;

-.

334

in

line-

with usual boundary layer apprd*imation and the very small

deflections considered.

The velocity components of the wall are:

- cY,p0

( . -

_3-35

LL(X,X,t)

A

17(X,,1t): tx(>J e

p(xy,tJ

- C 1% 3-31

The velocities of: the surface elements (subscript s) are assted to

be uniquely related to the pressure at the surface by

cA_s Y,.

(ix1c.)

p

/2L((,t)

_p,

3-32

where

_y

and Y22 are dimensionless wall aittances, horizontal and

vertical, respectively. They are assted to descr±be the reaction of the

wall to any pressure perturbations present and their development is

cartie4 out in Chapter 4. They are related, in turn, to dimensional

The mean fluid velocity components at the wall (denoted by U and V

with subscripts f) are

V(x,y,t)

3-36

The perturbation fluid velocity components are e*pressed in terms of

the stream function (3-9) as

u

(

x, y, L) =

q91(y) e''

' e

.,c(x-ct)

v' (

x, y,

) = -

o e

--

e

3_37

where similar linearizations have been applied.

The small velocity components at the wall that reflect the porosity

of the wall are (linearized as above):

- ::

0

lYfp

-

-

B2

_i-

_3-38

where it is recognized that the pressure perturbation in the fluid and in

the wall are essentially the same as discussed below.

The tangential velocity term due to porosity is extremely small since

the wall deflection is of very small magnitude and B12 is directly related

to sin (arctan± ) 0 whereas B22 is related to cos (arctan4 ) - 1.

The small change in horizontal velocity as

a

result of porosity is then neglected Since the porosity effect at its greatest is assumed to be of

the order of only ten per cent of the wall pressure perturbation (i.e.,

B22 0.10). The horizontal velocity component due to porosity is thus

approximately zero.

The effect of the flow through the pores on the wall pressure Itself

The normal stress in. the fluId. is

-

+(

3-39

where

- 3

A - +---

-is the divergence of the perturbation velocity components and van-ishes

for this incompressible fluid The pressure perturbation and velocity

perturbation componefits -.

18

are assumed of order (10-2) _{in this analysis The streamwise derivative} -

-is at highest of order (l0) since S is of order (IO) and under the

basic assumptions of this analysis the stteãw1sevariãt1on is of. Order

(1/R). -

.-The noriña-1 derivative

-is assufed- to-be at the highest of-ordët (101). The notidiensionaUzed

is seen to be of order (10-5), and

2,-

is of order (10-4) for a water flow of 30 ft/sec.

The order of magnitude of the terms on the right hand side of

equation --- 3-39 are thus

--p

(10) (l0) (i0)

and it is seen.that the viscous stress can be neglected in this analysis

when compared with the average stress or hydrostatic pressure. The fluid

stress at the wall is then closely approximated by

,yl

= - -pw

-

3-40

which justifies the form of the pressure terrn in equations 3-38.

The inertial effect of the motion of the fluid through the pores upon

the wall pressure must also be considered. Since the wall pressure

pertur-bation is of order (10-2) and B22 is of order (10-1), it is seen that we

are dealing with very low Reynolds number flow through the wall pores.

The momentum equation for this very small normal velocity perturbation

component due to the porosity effect is closely approximated by

.L:-

_LàP

-

J,

)y

P

where 4p is the pressure drOp acrOss the membrane or outer wall layer

of depth y which actually causes the fluid to mOve through the pores.

This p is thus the correction term to the wall pressure perturbation

.-to account for the inertia of the fluid moving through the pores. With

this in mind, the normal fluid velocity perturbation component due to

For the low pore velocity coas idered, the fluid acceleration is assted

to be no larger than order (10). in a membrane A y

0

. The

vlsco-elastic wall Is assi.nned

to

be Of such material that the f1ud receiving

plenum below the pores is a characteristic of the wall materIal itself

and the "outer skin" over which the occurs IS also vanishingly small.

Ay effect upon wall pressure due to the inertia of the fluid moving

through the pores is thus seen to be of higher Order and negligible in

thi case!

The fluid velocity perturbation componentS due to wall porosity are

thus approximated by equation 3-38 sufficiently closely for this analysis a

By combining equations .3-36 and 3-37, the total fluId velocity

components at the wall, Iinear{zed, are:

1 4

LLç

(u0

+

cP0)

e

f_oce

---3-41

Applying the third boundary condition of. continuity between the

vertical velocity components of the fluid and the wall and accounting for

the small effect of porosity (3-38), we obtain.

=

i.r

c1=,

_cIi,-

-From the fourth. boundary condition Ø. continuity between the

horl-zontal velocity compOnents of the fluid.,and .the wall we get

Uç

'/, _-s L/

EL

çt2 -' - 3-43 or, rearranging,

UaD \-i

+ Z7

°e )-p0

20 ---j 3-42

From the linearized x-mOmentum equation, evaluated at the wall and

again linearized as above, the following expression for wall pressure

perturbation is obtained

When 3-44 is entered into 3-43:

C Cf L)

I''

- o

C/"

,' \

J ___ 3-44

u 8

(I-

'/,

-l- u0 c )-

(q "-

cp).

This equation will be used in the computation as the compatabi.lity between

the fluid and wail perturbations to evaluate the constant of coination,.

P, in equation 3-30.

Actually only pat of the third boundary condition wa Utilized in

3-42 in that no mention was made of the compliance of the wall with the

pressure, through Y22. This last factor will be used

ih

the following manner. In the computational technique, which is that developed by

Kaplan (26), the fourth order differential equation 3-10, which is truly

part of a boundary value problem, is solved as a combination boundary

value/iterated initial value probl, the iterations being continued

until the 1st boundary condition is satisfactorily met. This is

necessitated by the digital nterical method employed. Further details

of this aspect are presented in Chapter 6. The remaining factor referred

to here is in matching further the vertical velocities of the wall and

the fluid through their ad.ttances.

With the stream function defined to ithin an arbitrary complex

cons tànt., the normal admittance of the boundary can be calculated in

terms of the flUid (linearized) as

- - c. - .o'

- - -

-z -.

-

Rq'

22

This normal admittance of the boundary is then compared with that

calculated from the wall characteristics (Chapter.4). When they.match

to within an acceptable tolerance, a solution is said to exist aid the

disturbances in the wall and in. the fluid are assured of being coupled

as they are in actual conditions.

The method just described is, of course, at variance with the

asymptotic approximation approach utilized by Heisenberg (24), Tolimien

(69),-and described by Schli.chting (6l). A brief description of this.

method of analysis ji this case with porosity Is presented in Appendix

B.

TJ. to this point, ithas been asted that the tangential motion of

the wall was so small that it cou-ld be ignored in this linear approach.

In Chapter 5, a method will .be described, to. accomt for the tangential

motion dUe to shear stress if such should ever prove necessary.

3.4 Spatial vs. TemporalAmplificat-ion of Disturbances

In the analysis to this point, the characteristics of. the wave

nber,

,, and the wave speed.,c,, have not been specified. In general,

either or both may bdèömplex. quantities, but for clarity of analysis one

or the other is usually restricted to, be real. The classical approach

taken by Heisenberg (24), Tollmien (69), Schlichting (61), and others has

almost 'invariably been to consider real and 0CC. _{as complex.} This

specifies that the perturbations are perodic in space and in tjme, but

are amplified or dape.d. In time only.. is seen clearly by inspecting

theterm ,. .

(oCx-d)

t:

(0Xdr)

23

Conversely, one. may consider as doip'lex ndl as real, producing

e

e

-(O,-X

flt)

3-48

This second approach is different from the first in one important aspect.

The space variable, x, is symmetric whereas time, t, is unidirectional.

For this reason, a particular sign of x can correspond to damped or

amplified disturbances depending upon whather the downstream (X>O) or

the upstre .(XO> direction is considered. Normally, only. dowistrearn

traveling waves are considered since those ttaveling upstream are

pro-ceeding into an area of higher inherent stability and are not so likely

to cause further disturbance.

Early investigators recognized that the perturbations actually

observed were a combination of temporally and spatially amplified

dis-turbances. The experiments of Sëhubauer and Skrarnstad (62) were

investi-gations. of spatially amplified disturbance which were induced by a

vibrating ribbon in the boundary layer and elucidated the nature of such

disturbances. Spatial .atftplification was also Investigated by Klebanoff,

TidstrOm., and Sargent (27) and Kaplan (.26). .

- Gaster (18) developed a transformation, for deducing the spatial

amplification rate, O , from the temporal rate,

_fl.

, but his technique

is limited to very small rates.. In contrast to this, the. method utilized

in the preent investigation allows, direct computation of the spatial

rate w-ith no such restriction upon . The stability Thci presented in

this investigation will be predominantly in terms of temporally .amplif led!

damped disturbances. Several selected stability loci will be shown

4 ThE COMPLIANT, POROUS BOTflDARY

4.1 General Considetatons

A compliant boundary

will

respond to the pressue f1utuations of the

fluid passing over it where such fluctuations are characterized by thei

wave nber

O( , and wave speed, c, as hotn n Chapter 3.

Refering to Figure 3-2, it is seen .that the deflection of the

wall

is described by

A

(x,yt)

(y) e

1

x-ct)

from which we obtain, upon differentiating with respect to time, the

surface velocity components

CL5

As pointed out in Chapter 3, only those surfaces are considered here

which exhibit linear stress-strain or stressrate of, strain relationships

This encompasses the gteat majority of practical materials except those

exhibiting marked hysteresis effects.

The compliance of, such boundaries defined as above cart effectively

and advantageously be described ,br surface tangential and normal admittance.

These are the admittances defined in equation 3-32 and are repeated here

for ready reference

ylz

"

11

-'

o(s,)

G

in the following sections two general types of surfaces are

24

4-1

4-2

4-3

considered.: a spring supported and damped membrane and a rather general visco-elastic (Voigt) solid of variable thickness. These surfaces are

naturally idealizations of what actually obtain in nature, but their use

serves to simplify the analysis considerably and they are felt to be quite

satisfactory to Illustrate the effect of wall porosity which is the

central theme of this invest-igatiofl. The physical parameters have been

chosen, however, with a view to physically reasonable surfaces.

4.2 Spring Supported, Damped Porous Membrane

Figure 4-1 illustrates an inextensible (to first order) membrane of

mass, m, per unit length, under constant tension, T

U &

supported by sptinginess of factor s = /p Uc. , per unit length,

and damped by factor., d, per unit length.

(ks

I

/ c);: _)'

1-dx2

-

/

)cz

Membrane Wall

Figure 4-1

body diagram produces the following equation of motion of the ebtane.

(If

- _JX

-

_'

-

d

-

-- 45

Utilizing equations 4-1 and 4-2, thIs becomes

'

r Z

JI

s

/

17 c

- c

-

.t °' ( J -- 4-6

where c02 = T/m and &)O2

= S/ni

--- 4-7

The membrane aittances are immediately obtained from equations 4-4

where the tangential aittance vanishes by assiption under the type

boundary considered here:

= C)

-

_cJ

_{--- 4-s}

The quantities c0 and & above are idettified as the membrane's

natural wave speed and cut-Off frequency, respectively. They and the mass

per unit length must be properly scaled

as

described in Section 3.2. The

membrane thus appears to become lighter and damped less and its cut-off

frequency appears to inërease as the Reytiold.s ni.ber increases and the

boundary layer becomes thicker. This variation or scaling of surface

para-meters is required to insure that the surface is truly non-iensionalized

with respect to the boundary layer depth. If such scaling is not employed,

the surface must be considered nOt

a

"normal" or physically uniform surface, but a "spec±all tailored" one whose dimensional properties

change in x. Kaplan (26) has described the results of such surface

tailoring.

From equation 4-8 it can be seen that, for real values of and c

(eigenvalues, or neutral stability locus), is almost purely imaginary

since crd is generally quite small. The imaginary part of ₂₂ can b seen to change Sign as the effective wave speed of the membrane.,

Cc02

+J02/c)½,

is passed andthe effect of the cut-off frequency upon

this change is clearly indicated.

The porosity effect is not explicitly evident in these wall

aittances as the effect of the porosity Upon the flow is properly

accounted for in the boundary conditions for the equations descrIbing

the fluid. The porosity pareter IS, of course, a wall characteistic

and is entered as such in the calculations.

4.3 Visco-elastic Surface

The second compliant boundary type utilized is a more general

visco-elastic wall obeying a stress-strain/straIn rate relationship suggested

by Nonweiler (57). The equations of motion for an elastic solid in the

absence of body forces, using (Cartesian) tensor notation, are

sJ

4-9

where

9

is the density of the solid. The strain and stress tensors are

of theforms

* (

-,,) -.

'i,_ )

4-10

.

[c d1

4-11

where 2 =G (E-2G)/(3G-E) is the first Lam constant, E is the (Young's)

modulus of elasticity, and G is the shear modulus.

The appropriate non-dimensionálizatiOns are

(51;

6. /y

strain/rate of strain relationship is known as a Voigt solid (Kolsky (28))

and is a fairly reasonable representation of many actual visco-elastic

materials. Its mechanical representation is similar to that of the

A solid obeying this

membrane of Section 3.2 in that the springiness and damping are in

parallel. The factors d1 and d2 are damping coefficients, related

generally to the infinite bpdy longitudinal and transverse (shear) waves,

respectively, as will be seen later. If they are absent, equation 4-11

reduces to the familiar isotropic Hooke's Law.

- When the stress tensor (4-11) is put into the equations of motion

(4-9), the gener&1 soipn for the displacement amplitudes in terms of

the displacements described as 1-i 4-2 can be sbowii to be

l

c.s6

roLy

4-

I

cy

r2.nt PafJ

L

os6 r,oy

+

s.iLi r1y

r1

r,o(f

4- /1 c.si

r,acy]

--4-12

where

_r,

I-

c/C,

R/

2

C:

_o(Cd1

- L(CII,

c

---4-13

is seen to be the solid's longi-tudinal or dilatational wave speed

while c2 is the transverse or shear wave speed, both having been modife4

by damping factors. It can be seen from 4-13 that, for real O , a damped

fluid perturbation (cO) will decrEase c1 and c2 and will disperse

the waves by

± 17/4

for>O, cr>O, and d>O, which s the normal case.

Fung (17) brings out this same effect of damping on the stress waves.

This i11üsrates clearly the coupling between the fluid and the solid

boundary.

Several physically reasonable sets Of four boundary conditions may

be chosen from the following:

both X, 0 ₎

- Pv and

cJ'(x,o)

0

plus

= b

or

7(x-/4,

t )

0

and

0

(x,-fi, t)

0

4-14

where H is the thickness of the wall. The first of these boundary

conditions is the statement of the assption discussed In Section 2.3.

The second equation is a further simplifying assi.ptf on in that it states

that the wall shear stress vanishes. This is not actually the case and,

as previously mentioned, a technique will be described in Chapter 5 to

match the shear stress in the wall with that in the adjacent fluid. It

will be recalled that the basic assi.ption of this analysis regarding

this is not the vanishing of the wall shear stress but that any wall

deflections due to wall shear stress are very small and negligible in

comparison with wall deflections due to normal stress.

The first two boundary conditions of equations 4-14 naturally must

be used in any cOmbination established. Of the remaining four, either

normal stress or normal deflection, but not both, and either tangential

stress or tangential deflection, but not both, may be specified.

Io

types of surfaces were investigated in this analysis. In the first model

the following remaining two boundary conditions were chosen

(i)(y ( X, -16/,

)

0

77 (x,

-/

fl

U

This choice was made to allow some measure of direct comparison with

previous calculations of Kaplan (26).

The second model chosen was

U

0

This

combination was felt to give the closest approximation of a coating firmly attached to a substructure at its bottom surface as would most

likely be used in practice.

4=15

The wall ptrbaticn velocity amplitudes are derived directly from

equations 4-12 as.

L . ₍

+ $:)

C

(

14, + IL)

4-17

Utilizing the first of. equations 4-14 and equations 4-11 ard 4=13, the

wall aittances can be expressed in terms of the displacement amplitude

coefficients as (J LLO

-

---'2 _-Ps 1'--

I

r

-

-c/?Lc,(r -I )I;-3

_2-.

a

-e

-A LL0 A

where c1, c2, r1, and r2 are as defined in equations 4-13.

These expressions are valid for both models investigated. The

remain-ing three boundary coiditiPns for, each model are then invoked to define

the ratios of amplitude coefficients A2/A1, A3/A1, and A4/A1.

When these ratios are entered into equations 4-18, the a±ttances of

the appropriate wall model are fUlly defined in terms of the wall's

physical properties.

The results for the first model, equation 4-15, are:

co+/i

rcxR

.of1, r,ot

1-1

r1

4-19

If the wall thickness, H, is very large compared with the length

scale of the problem,

g

-, the hyperbolic cotangents above approach uflity and the coefficient ratios of 4-19 become

30

/+Aq\ I

r 43 21

---

4-18

L-.3)/S'5LC, fri

-I)n+

,L7 /,

1-I

1,1J4,

4-20

Equations 4-19 or 4-20, as appropriate., are entered in the

express-ions for the ia1l admittances (4-18) and these admttances are utilized

in the solution of the general eigenvalue problem. Y12 is put c3irectly

in the compatibility expression (3-45) while Y22 will be compared with

the normal admittance as given by equation 3-46 to temie solution

convergence.

The forms of the adinittances when manipulated as indIcated above are

= -D = i c r1 (1-r22 4-21 D where

Al

(rf1 )

cotL

r,,q - 2

r v _{coth v o( H}

0

C c,th ron! -

'-1 v r

r

_'NJ

---

4-22

If the wall depth becomes vanishingly small, or H- 0, the admittances

approach

'I - -;$ + - R

i-'Ii

?Ca

I(VRi-I)-tr,--.4

These forms are as expected since the wall is allowed to slide tangentially

at its lower surface, but nOt deflect verticalIr there. $iflcè this is

quite unrealis tic physically, no calculations are performed for this case.

A simplification of this model suggested by plan (26) is to

consider the wall ater-ial to be incompressible. In this case

and r1 - 1. This is exactly equivalent to stating that

'

in the above equations implies consideration of an incompressible wall.

In the cases considered here, r1 is very close to unity, the small

differ-ences being due to the 41ff erence between the elastic and three times the

shear moduli and. the viscous damping term. This last term will be

com-pletely overridden, as ' since d1 is of small magnitude (refer

to equation 4-13). One set of data was obtained with 3G as close to E

as the computer would allow without exponent oerflow to illustrate the

effect of an (a1ost) incompressible wall.

The coefficient ratios resulting from application of the second

model's boundary conditions at y=-H, equations 4-16, are:

I?1I/

cJirfic,sirr'I4]

r1 rR r,oLH C

r+

-shVo(H

Csj,rcI-1

siIiroH

If the wall thickness, i, becomes very much larger than the boundary

layer depth, &, then the ratio of hyperbolic sine to cosine and cosine

to sine each vanish in the limit and the hyperbolic tangents and

cotangents each approach unity as before. In this case the three

coefficient ratios of equations 4-23 approach exactly the same forms as

before, equations 4-20. This Is as expected as extension of the wall

to a sj--infin-ite solid essentially overrides any detail. dJfferences

between the boundaries at the bottom (infinite) edge.

The expressions for the wall admittanëe correspondIng to equations

4-21 and 4-22 are quite lengthy and will not be reproduced here. They

are, however, calculated In the course of the computer program and are

given in Appendix D in computational format (Section D.3, Subroutine

4-23

D62SFV).

If the wall depth becomes vanishingly small in this case, the wall

adittances both vanish

Yl2 O'

)

asH -p 0

22

oJ

This again is quite predictable since the limit, as the wall depth

vanishes, of this model with its lower boundary fixed is simply a rigid

wall. The results for the rigid wall are thus directly applicable to

5 MATCHING OF BOUNDARY CONDITIONS

5.1

Boundary COnditions Fat From the Wall

in

Section 3.3 it

was shown that the Orr-Soerfeld equation (3lO)

can,

at and beyOnd.

y

, be expresed as

In this case, the solution of equation 5-1 is given by equation

330.

513

The precis

boundary conditions on

q

and

are that they vanish

far from the wall (y

oc).

Since a boundary value at infinity can only

be

approximated1 in a numerical method

used on _a digital computer, the

following technque

was

followed (Kaplan, private counnunication).

The

fp

of the eignfunct-ion was specified as in equations 3-30 and 5-.3

such that the boundary conditions and further derivatives of Eva1uatd

as Y

Yo can be etesed

as:

4

(yb)

I /

ci',' (

-

c; (V4)

(JL

-

_Q3"'(y0)

_{---; 5-4}

These values were then

used in the integration scheme

employed here,

he

general modus oparandi of which ±s described in Chapter 6. The form of

given in 5-3 insures the required exponential

decay of

and , thus A Lt

and

'...r ,

as y

_{._} 34

r

(o

.2

42)

_piI

with

7r

+ cR

(ui

c)

5-2

Initially, a value of ten times the boundary layer depth or y0' 10,

was tried and a solution of the problem attpted. A series of values

of y0 decreasing from 10 was assigned and the final

eigenvalUes/eigen-functions compared. It was found that the difference between the

solu-tions obtained with y0= L5 'and Yo1° was less than 0.1 percent 4th

this error haingbeen decreasing as y0 decreased from 10. A final

value' of y0=l was then selected and used in the remainder Of the

cälcu-lat-ions. This procedure essentially "empirically" deteined the

boundary layer depth as that y where 0.998U, in contrast with the

tJ = 0.999U mentioned in Chapter 3 (Kaplan (26) and private

communica-tion). Background for this type technique is presented in Shen (64).

Figure 5-1 Is a sketch from Kaplan's report whiCh diagratica11y

describes the concept us,e here.

42, e

q9:

e y=l y=0 U (y) Figure 5-1 (exact solutions) initial conditions numerical integrations of COMPATIBILITY

Integration of the Orr-Sommerfeld Equation

5.2 Boundary.. Conditions at the Wall

The remaining two boUidary conditions require that the hOrizontal

and tangential velocity cc4mponents of the flud match thos.e of the wail's

outermost surface.

-I

The .mean fluid flow is that of the Blasius bowiclary layer profile

and both velocity cOmponents. vanish at the mean position of the wall.

The mean position of the wall is, as implied, a stationary plane atid its

velocity components vanish. This ptoides the proper match of the meai

flow with the mean wall.

The horjzpntal fluid perturbation v4ocity component is -shown by

equation 3-43 to be expressed as

u[(P0? -)

cpJe'

(xt)

--

5-5

while that of the wall

is

expressed in 332 in terms of the gall's aittance and the pressure perturbation at the wall as

us

Y,1 _-p

yIz i:

-f-- 5-6

The wall's tangential admittance, Y12, is then computed as In Chpter

4 and the wall pressure perturbation is expressed in terms of the

luj.d

velocities as described in Chapter 3. When 5-5 and 5-6 are equated,, the

general compatibility cond±tion of equation 3-45 results.

it should be borne mind that a basic asstpt1on of thts analysis

is that the wall's respons to any shear stress applied to it is

negligibly small compared with its resppse to the normal stress component

(the pressure perturbation). This is seen in the form of the two admittance

parameters of equation 3-32. This asstption does not imply that the shear

stress or the tangential velocity components vanish. The assption made

in Chapter 4 of the vanishing Of the shear stress is an additional assption

A

--37

5-9

ãnl

will be discussed in Section 5.3 The first assumption of wall response only to normal stress components is felt to be reasonable in

this linear approach since any surface response to

perturbation-produced shear stress is certainly of much lower magnitude than that

produced by the pressure perturbations.

The. normal fluid perturbation velocity component is shown by

equation

3-42

to be expressed as

--

_(-

o:

_(f

(1

5-7

while that of the wall is expressed in 3-32 in

terms

of the wall'

admittance and the pressure perturbation at the wall as

=

)1?e

1s

-

'Zz 5-8

In this case, contrary to the direct method employed above, the

normal admittance of the fluid is expressed wholly in terms of the

characteristics of the fluid

_flow

as shown in equation 3-46. The

normal admittance

of the wall to

the pressure perturbation is then

computed

from

the wal].'s characteristics as described in Chapter

4.

These two admittances are then compared and when they are. equal the

fourth boundary condition of normal velocity component matching is

satisfied.

5.3

Shear Stress at the Wall

The shear stress in the wall, at its outer surface and in terms of

the Volgt solid considered

in Chapter 4,

is

((1-

d2)(-

f7)

where the linearization, non-dimensionalizaton and form of dJ,sp1aceient.

functions are as in Chapter 4.

-The

wall

shear stress in the fluid is .

dy

-h--

,

(

+ ---)

..

and, non-dimensionalized and linear-izedas above, this becomes

-p- ( q'011 -f-- c'

(f, )

5-10

Either of the above two equations may be used to calculate the wall

shear stress if it is desired since the stress is continuous at the wall

In view of '(y) appearing as the eigenfunction of thi problem, whereas

' and are "buried" in the . calculat-ions, it appear that equation

5-10 wculd be the more natural choice.

The above two equations may also be used to satisfy mOre exactly the

boundary condition of continuity of wall shear if siichis deemed necessary..

As stated previously (equation 4-i et seq.) this boundary condition was

simplified by assuming that the tangential displacement due to wall shear

is negligible. The technique alluded to previously would consist of fjst

setting the

wall

shear to zero as in equations 4-17, solving for the

-eigenfunction (y) and its derivatives, and utilizing equation -5-10 to

solve for the first estimate of the

wall

shear stress.. This value is then

entered in the first of equatiOns 4-17 as an itihomogeneous term and the

complete calculations repeated This iterative scheme is followed until

two successive values for (or Yj2) agree sufficiently., _{The present}

analysis does not include this refinement; however, the value of

the wall shear stress amplitude function, is calculated in the subroutine

which produces the veloity perturbation, Reynolds stress, and wall

pressure perturbation amplitude functions (see Appendix. D).

6 COMPUTATIONAL METHODS

6.1 General SOlution Methodology

In Chapter 3, it was mentioned that the method of solution of this

boundary value problem is handled here as an iterated initial value/final

value comparison problem. The following scheme is employed and is that

developed by Kaplan (26):

Basic equation of motion (Orr-Sornmerfeld) is expressed in

terms Of pCrturbatiofl stream function.

Boundary (wall) admittance parameters are expressed in

terms of physical characteristics of the wafl.

Boundary conditions on the fluid flow are expressed in

terms of the perturbation stre function at the point where the boundary

layer joins the main flow.

The equation of motion is integrated toward the wall from

the boundary (treated. as initial) conditions of c. and utilizing certain

fixed input parameters and an initial estimate of the eigenvalue..

The condition of continuity of the tangential perturbation velocity component between the fluid and the wall is expressed, suitably

linearized, in terms of the perturbation st-ream function

ard

the wall's tangential admittance from b.

When the wall is reached in d., the equation resulting from

e. is utilized to define the constant of integration which is the

combina-tion coefficient in the eigenfunccombina-tion.

The normal admittance of the wall is then expressed solely

in terms of the perturbation stream function.

h.. The normal admittance computed in g. is then compared with

that computed in b. strictly in terms of the wall's physical

small amount, then the fourth boundary conditIon of continuity of the

normal component of the perturbation velocities in the fluid and in the

wall. is met.

i. If. the convergence criterion in h. is not met, then a new

estimate of the eigenvalue is made in d. and the process is repeated up

to an arbitrary maximum number of tithes until the required convergee is

attained.

After a satisfactory solution for each desired point is obtained,

the elgenvalues and, if esited, the eigenfunctions for the entire flow

field are available. Families of appropriate curves are then plotted.

6.2 Numerical Integration

The numerical integratiOn technique employed in the analysis is a

modified Runge-Kutta metiod. The formulas used are presented in Appendix

C. Kaplan (private commUnication) found that a step size of one-fiftieth

of the (non-dlmensionaiized) boundary layer depth gave, satisfactory

con-vergence within a reasonably small number (eight or less) of integration

iterations.. The rea].In o wave numbers and Reynolds numbers for which this

sche is sufficiently accUrate is defined by 0 o R . l0. Above this

limit,

4,

the so-calledviscous solution (see SectiOn 6..3), may

over-flow the single precision capacity of the computer' for certain wave speeds;

mother words,

J3tu)>io322

(for CDC 6600). This was not felt to be a

restrictIon in this casein view of the range of wave numbers (0.1< oC< 10)

and Reynolds rnbers.

(io

R'

l0) involved.

6.3 the Eigenfunction

The to components of the perturbation stre fimcton which result