Unsteady flow action on bluff bodies

UNSTEADY FLOW ACTION ON BLUFF BODIES

Dr. S.A. Spassov

Reportnr. 796

Juni 1988

Deift University of Technology Ship Hydromechan ice Laboratory Mekelweg 2

2628 CD DeIft The Netherlands

UNSTEAJY FLOW ACTION ON BLUFF BODIES

DR. S.A.SPASSOV'

1. INTRODUCTION

2,. SHORT REMARKS FOR FLOW ACTION ON CYLINDERS

UNSTEADY BOUNDARY LAYER 3.1.UNSTEADY STARTING MOTION

3.1.1 IMPULSIVE MOTION

3.1.2 FLOW SEPARATION IN STARTING MOTION 3.2 OSCILLATING MOTION

3.2.1 FLOW SEPARATION IN OSCILLATORY MOTION SHARP-EtCE SEPARATION

4.1 STARTING MOTION OF A FLAT PLATE 4.2 OSCILLATORY MOTION OF A FLAT PLATE 43 OSCILLATORY'MOTION OF SHARP EDGE BODY

CONCLUSIONS REFERENCES ACKNOWLEDGEMENT

BSHC ,Varna , Bulgaria ,now a research fellow in Shiphydromechanics

1. INTRODUCTION

The problem for the solution on unsteady flow action on bluff bodies as it is well known is still 'open.The main reason for that is a difficulties to

solve three dimensional Navier - Stokes

equationsIn

recent years a

succesful attempts for solving two-dimentional problems in

Japan,USA,Netherlands [

9],

[ 10 ] are made.Many experiiiental results are

recieved for model scale [ 4 ] also.in this case,the main problem is how to translate a model. scale results to full scale problems .From experimental

point of view the way to solve this problems is to graduate the dimensions from model to full scale probably in open water.area.Some of the very nice

experimental results for model scale. are still nice only for this particular

case.

in numerical point of view basicly it s' posible(autors opinion) to try to

solve a full scale Navier-Stokes equations using some of the developed

direct methods for two-dimensional case or using mathematical modelling to s:implify _{the problem and corresponding to real physical processes to try to}

solve some particular problems.

Considering some offshore engineering problems. it can be said that one of the most often 'and relatively succesfull used equations is the Morison equation.

l.Morison equation gives a reasonable prediction for in-line force at high

XC numbers,,but not successful at layer values.'The reason is that Morison

equation give. unreal represantation of the behaviour of the wake of the

body,which is essential for time dependent flows in which the fluctuations

in the streanr are small compared with some superimposed uniform mean flow.

2. An othe'r problem for flow inachanism is' the Importance of vortices and their velocitiy fields for uniform and planar oscillatory incident flows. It is found that there are differences' between the vortex phenomena observed

for uniform incident flow and those likely occur for oscillatory flow and a

flow with and without vortex shedding.In this case the force representation

F=

A

+ 'B + C

potential separated separated flow

attached steady additional effect

flow flow of finite vortices

Moreover except a uniform steady and oscillatory flow , there is a wave flow

where the vortex shedding additional effects are difficult to predict.

If the problem is practically oriented, some other effects have to be

considered':

3.The flow phenomenon around a circular cylinder is different and nead different approaches:

-for vertical cylinder

-for perpendicular horizontal cylinder

-for in-line cylinder -for inclined cylinder

4. Interference effects:

-'influence between f low and cylinder - the relative velocities

between cylindrical elements an4 particle motions ar,e affected by the response;

-at joints-the problem not been studied in any systematic way

-in' arrays of cylinders - if it has a steady flow; a planar

oscillatory flow; a wave action.There is not enough information even for

subcritical and pos'tcriticai Re' numbers regime.

Many attempts are made 'to simplify a mathematical model coresponding to flow

action physical process on the bluff bodies and to find an applied results' for practical purpose.In this report the autor try to cover some problems where Morison equation is not valid.

-3-2. SHORT REMARKS FOR FLOW ACTION ON CYLINDERS

As it was mentioned in the introduction the Moriàon equation give reasonable predictions for in-line force at high XC ,but not successful at lower values

(KC < 15 ) . This is partially due to the fact that for the most tddeiy used and studied section,the circular cylinder, the basic attached flow inertia

force dominates in the region in which Morison gives the worst prediction for this effects of separated flow.Because of this basic limitations it is difficult to find any rational method of improving the Morison equation.

At present two alternatives suggest themselves.The. first is a complete

numerical solution of the Navier-Stokes equations.

In. practice these methods are completely limited , because of the ratio of

scale lenghts involved and the resulting computer. storage required, to

moderate Reynolds number (Re < 1000) at laminar flows. The presence at high

Re of thin thin random vortex sheets,even of a two dimensional case is

difficulte to calculate.

The other alternative is designed exclusively to model high Re number flows

and is therefore more appropriate for use in large scale situations.

Inviscid methods of the latter type follow the classical approach of the

fluid mechanIcs to high Re number flows in which the shear layer are assumed

to be sufficently thin to be represented by vortex sheets of zero thickness,

which are shed, as neceswsary into the flow from certain designated

separation lines on the bodyThemethodshavebeen developedquite

_recently for unidirectional flows past bluff bodies. in which the separating vortex

sheets of the wake.are' represented by an array of discrete vortices.The

numerical results are compared with experimental results for planar

oscillatory flows of the type generated in U-tube water tunnels or by oscillating cylinders in still water.

The main advantage of the discrete vortex models over Morison equation for

wave induced flows should ultimately be the more rational representation of

the behaviour of the wake of the body whIch is essential for improved force and pressure prediction in flows where drag is important.But this methods

should yet be regarded as a desgn tools since many ma'jor problems still require resolution.

The main feature of the bluff body flows is the subsequent roll up of these sheets which as a result of interaction with other vortices in the flow

(previously formed or from the opposite side of the wake) are then shed as

large scale lumps of vorticity and. involves considerable turbulent diffusion

and mixing with cancelation of vorticity.,So far no completely rational

method of representing this numerically has been achieved although some

amalgamation of vortices is often done for numerical convinience and stability. Another serious problem , particularly for circular cylinders concerns the prediction and representation of separation.The point at which

vorticity is shed from a.continues surface into the wake is usually taken to

be. the point at. which the boundary .layer:.separates and this- is controled by Re number and pressure gradient.In oscillatory flow.this point is.unlik&iy

to remain even approximately stationary.The prediction of. the. separation point in flows containing any element of randomness must be regarded as not

yet possible and an important field for research.if .the body has sharp edges- the sepa.ra.tion points are fixed by .the geometry.

The force induced. on the

body by

an isoene.rgetic two-dimensional

accelerating flow if the wake can be represented by a distribution of

inviscid vortices is expressed by Blasius theorem.The results show different

degrees of damping due to vortex shedding when the bodies make small

oscillations relative to fluid.The strenght by the vortex force on .the flat plate for example, which is only of order KC*(2/3) with respect to the inertia force, does .also rise even in the diffraction regime in this case The forgoing analysis suggests that all.. cylindrical bodies

may have

qualitatively similar flow patterns at low XC numbers.

The basic flow pattern in this range appears to consist of one vortex shed

from each. edge (or separation point ) on the. body during the first half

cycle.. Due to the ifluence of this vortex ,the sign of the vorticity being

shed changes before the free stream reverses and a vortex sheet of opposite

sign starts to roll up in the opposite drec.tion.The combined effect of this new vortex and the eventual free stream reversal sweeps the previuos vortex

back round the edge where it forms a pair with the previuos vortex.The

vortex pair is usually observed to convect rapidly away leaving some residual vortic.ity , although sometimes the new vortex appears to wind the

-5-old vortex into itself with cancelation of vor.ticity.The latter process

however may

be a feature

of the

comparatively low Re number of these

observations.This pairing process Is asymmetric and usually resuts in vortex

pairs. convecting away from the body along opposing diagonal. lines at about

45 degree to flow direction.At higher KC numbers ( > 15 ) the pair of the

vortices shed from the two edges or separation points start to interact with each other , rather than just with previous vortices from the some edge.This

developes ,lead'ing to the shedding of the thrid vortex in the half cycle and eventually at higher:KC to. a partial Karman vortex sheet.The convection of this

wake back

past,, usually .one.sizeonlyof ,the.cylindercan lead to

quite large' transverse (lift .). .forces.Flow'visualization therefore different

characteristics on either side of KC. 12-15 .It is noticeable that the

forcecoeffitlents show quite a lot of variation in this.:region.

The numerical methods which are used. are single pint vortex method and

multipoint vortex method'. In the first one each vortices is

replaced by

a

single point vortex.The method tends to overestimate vortex strength since

the requirement of smoothly separating flow , which can only be approximated

by a full stagnation point, is controled by the whole of the vorticity

centered on one point,, rather than distributed in a continues sheet right up

to separation point.However it yields qualitatively good results ad can be made to predict the observed low ICC number flow patternsBut the method does

not seem to be applicable to flows with ICC > l5since it has not yet been shown to be capable of reproducing naturally the .Karman vortex sheet.

In this case multipoint vortex.method is. more convenient nevertheless that that shedding an array of point vortices in. a way which is representative' of

vortex sheet leaving a separation point problem again.Mostmethods. of vortex release now satisfy the full stagnation (Kutta-Joukowsky) condition and then

close the equations specifying the vortex strenght either be. releasing the vortex at an empirically determined point or by relating the strenght. to the

convection velocity of previous vortices.The circular cylinder however still remains more difficult (in comparison with plates and spheres) ,even

assuming that the position of separation is known. and further analytical

work is required.

-6-The second major numerical problem concerns the development of randomisation in numerical vortex arrays.The increasing chaos of vortices swirling back

and forth past the body does not look very representative of the observed

flow paterns.At realistic flow patterns due to the disruptive effect on a

subsequent vortex sheet when a randomised cloud of previously shed vortices

is swept back past it.

More stable alternatives to the point vortex array , such as continuous line

vortex elements or different methods of numerically integrating the Blot-:Savart _integral. ,,have been' suggested. and used in certain unidirectional

similarity flows and theseshouldbe.investigated.for.use in wave flow also.

Summarizing::.forgoing. remarks it method.of.discrete

vortices is a very promising approach.to the . solution of oscillatory and

wave flows about two dimensional. structures. ,but a' number of important

problems (particularly concerning separation ) still require solution before

ft

can be

used as design method.The general idea of modelling discrete vortex structures also leads to useful analytical results particularly in KC

3. UNSTEADY BOUNDARY LAYER

As it is well known in the theory of unsteady boundary layer two main flow

classes exists.:

a)unsteady starting motion: b).oscillatory motion

.3.1 UNSTEADY STARTING NOTION

The forming boundary layer is very thin inthe.beginti.ing of the motion . and

the velocity of the flow is changedfosterthan'the velocityof thebody,, and rich the flow velocity in a distance.Then the separation appears: and

resistance increased significantly.This is the reason why the location and time of separation is so important.

Blasius used in 1908 the unsteady boundary layer ,theory for investigation

of the impulsive and the harmonic motion of the circular cylinder in the

unboundary incompressible viscous

fluid with

the sequence extension

method.Let the flow is perpendicular to the cylinder , which radius is R.Let

the coordinate system Oxy is fixed with the body and 0 is the front critical

point of the body (fig.l).

fig.l Coordinate system

-8-Let the velocity components

u

and V has same direction as Ox and Oy.Then the

equation of two dimensional unsteady boundary layer are:

+vU

8t Sx .Sy 6t Sx

6y2

+ 0

6x 6y

where

u

= U(x,t) is the velocity in the potential flow outside of the

boundary layer..

3.1.. 1 IMPULSIVE MOTION

Impulsive started flow Is one of those unsteady fio situations for which analytical and numerical solutions exist for small time and relatively low Reynolds numbers. [5J.Let the jumping velocity is Uo(x).For the potential flow velocity It has:

U(xt)

0 at t=0

U(x,t) = Uo(x) at t>0

The initial and boundary conditions are:

u=Uo(x) v.O at y=O , t=O

u=0 v=0 at 0 t>0

at

t0

-9.-The solution of this problem is

found by

an iterative approximation method.At the beginning stage V 0, U Uo(x) the boundary layer is very

thin and tis significantly greater, compared with convective terms

and is balanced by local acceleration.Then the first approximation solution satisfied:

= p

.6t _&y2.

and

u1=O

at y=0

at y=O

After the changing of the variabie

and u1 = Uo(x)(,i)

2

where =

From the continuum equation it can be found

du° - 2

vi

2fit

j [ - ( 1 - e )] The equation, for second approximation is:

62u3

= dU0 -

-6t

°dx

' Sx '&,y

and the solution is. given by:

dU°

u2 = tu0

F2( q )

F2 ( q ) = (2,,21),2 + + (1- + 2

e22

-dU°

<0

dy

1.2(2,12+i) + O.804[ (2,2+1) +

Analogous is found v1 and v2

The abscissa of the separation point and the time of the beginning of the

separation t8 can be received from the condition.

(

_),:°

- 1+(1+T)()5t5

o

For t>O is necessary to have

-,

=

dx max

For circular cylinder in ideal fluid

I U0(x)=2U0sin () and = 2 llOcc

X.

dx a

(;)

Then

t = 0.351

S U0

and the distance is

a=u0 t

=O.351a

S

Taking into account a third order approximation Rosenhead found _[ 5 ]:

3.1.2 FLOW SEPARATION IN STARTING NOTION

For, any arbitrary shape of the body without sharp-edge corners,the

separation point

can be determined by boundary-layer theory.Suppose the:

velocity at the outer edge of the boundary layer , U(x,t),.is given in the

form

U(x,t) =0

t0

3

U0 (x)

(t +

t )

t0

where x is measured from the front stagnation point along the surface of the body , and

t

represents time.The coefficient represents an additional

acceleration (, >0) or deceleration(fl<O) .Without the term _{fi the flow with}

correspond to the case of a starting motion with constant acceleration

which was treated

by

Blasius [ 6 ] .The: objective of this section is to

extend Blasius treatment to the case with an additional deceleration

term

3

t

.The solution can 'be obtained by an extension in time series.Suppose

the

stream function

_(x.1y.,t)

is expended in the form:

A(x,y,t)

= 2Ji [ U +

= 2Ji [ _o +

t(U0

e0 +.

Ui2

+ ]

where y is measured normal to the wall., ' is the kinematic viscousity

and the quantities are functions of a normalized coordinate v=

yj#t. Then the velocity components u and v with respect to x and _y are

expressed in the form:

)

-12-(1)

Lu 2

6

-2.fi[10 +t1[() +ui]+...i

The present governing equation is the boundary la)er equation for unsteady

motion ,, which can be written in the form:

Su

U-

6u P 52.0

-i'

=0

St Sx Sx St 6y

with boundary conditions

at r = 0

u=U

at ,-'

Substituting u

and v into

eq ( 1 )and noticing that:

ILL

= const 6t = const - 2t 6i

we obtain the following time series:

(2) U0 ( -

-

'' - 1 - _{e''' )}

2

0

₍

1 ,',

,2

1

1+o

o

)

(3e.0

-The equation of the first Order solution' is follows

-13.-'I,

- 3)]+.

with the boundary conditions::

e0 (0)

(0) =0

and

)=1

with

2

r

e

'1

-

(1. + 2

2

)

The

second

derivative

of

e0at the

waiil.,which

corresponds

to the skin

friction

stress' ,

has the value e0 (0)=4/jir.The second order

equation

can

also be derived by rearranging eq (2:) and defining

= 2

/

u0

, , '2

(3)

+ 2

- 12

= -4 + 4 (

in. which theboundary conditions are homogeneous Eq.' ( 3 ) can be solved by

dividing the quantity

into two terms:

-

+ Y 12

The solution for the first term

,

is the same as

the

Blasius

second-order solution, which gives the value

at the wall.:

fl

256

(0)

_j

_{15 - 225ir}

The equation for

12can be expressed in the form

+ 2

t - 12 _{i2 =}

-1 +

12

(1°) =

e12

(0) =

i2.( )

= 0

This is newly obtained here and its solution can be written:

and

L.

_2

3

=je

7

1.

+ r

+

_'

₎

-14-1 - erfc('7) ( '72 +

Summing up all of these solutions it is obtained the velocity profile in the

boundary layer:

2 _dU

u

U0 t

_[ +

(

+

(

O + 8

12) ) +

The. skin friction stress, , is defined by.the equation:

I,iiI

p

6y y=O'

The separation point can be found by putting r, =0 in eq.( 4 ) so that the

condition of separation is expressed as follows;

1 + t2

₍

0.427 + 1.6

. ) - 0

Blasius original solution contained only the first two terms in ( 5 )

-the term is new here.Eq. ( 5 ) can be interpreted as determining the

separation time at the

point where

the velocity gradient and the

deceleration rate

_fi

are prescribed.Another interpretation is that it gives

the location of the separation point where eq.( .5 ) is satisfied.It can be said from eq. ( 5 ) that the separation occurs when the' pressure gradient

becomes. adverse,and that the. deceleration <0. makes .the separation. occur

-earlier in time or. further upstream in space,since a .negati\e value of

corresponds to an adverse pr.es sure gradient ( 0 _{< 0}

This . results can be applied to a couple of cases.. First, consider a

circular cylinder of

radius R

oscillating with amplitude x and

frequency w .The outer - edge velocity U(x,t) o.f the boundary

layer is given 'by potential-flow theory:.

U = 2 X w sin (x/R) sin wt (0)

(4)

-15-if we assume that this is a starting motion ,- then it is obtained

-16-U0 = 2 x w2 sin (ic/R) _-

_{p0 w2}

The deceleration constant can be

obtained by

applying the least squares method over a swing of. the motion , which gives

p0 =

Substituting eq.( 5 ) into ( 6 ), we. can obtain a separation point at an arbitrary. time.instant.For example,. at- the.time , -t =

ii /

2w when the

velocity-of the:.cyiinder is maximum,we:Obtain -the. separation--point'angle O measured -from the top of the cylinder:

(7 sin 0.822 /KC

where

O - x / ft

- ir /2.

Eq ( 7 ₎ shows that the separation depends on the amplitude of the

motion but not on the Reynolds numbed or frequency -according. to

present analysis.:However , the value of the coefficient in eq-.(7) is

somewhat small compared with the experimental value of ikeda [2] - as- shown in

fig.2.The experimental- data seem to- be expressed as-: (7a)- sin 0 = 1.49 / KC

Eq;. ( 5 ) can also be applied to the case of - a flat plate oscillating in its plane, if.. assume that dU0/dx=0: everywhere . on-

the--plate.From-( 5 ).and approximation for we-find that t=0 when wt.= 2.345.

in this case the conditionS t = 0 - does not - indicate, separation but merely the: turning point of. the osciilation.The value in- for wt - is in: quite good

2

4

it corresponds to phase

[ ' the velocity of the body.

là'a of the flow

pattern with

respect to

-'17-o _{: measured by Ikeda et a].}

1

_sinO9O,822/K

2 : sinG5 =l.98'I/Kc

3 : sinG5 = 1. 49/K

Kc'71

'fig. 2 Separation angle at t/2w for osci]il'atying circular cylinder.

ThIs corresponds to the phase lead r/4 , of the skin 'friction with

respect to the velocity of the body,which means that the body will

experience not only a damping force but also an added mass force due

to the fluid-viscosity effect,alihough 'the

phase is

independent of viscosity.

Through the present analysis, it can 'be concluded that the flow

deceleration makes the separation point move upstream

and makes

the

separation occur at earlier tiine.For the case of periodic oscillation

3..2 OSCILLATING NOTIONS

There are several fundamental differences between the unidirectional flow

and harmonically

oscillating

flow and between

the harmonically oscillating flow and the wavy flow over cylinder.When a cylinder is

subjected to a harmonic flow normal to. its axis , the flow does not

only accelerate well during each downstream to

sign.The separ layer over the

fully turbulent

subcritical to

have been or are

from and decelerate. tO zero .but changes direction as

cycle.Thi's. produces a reversal of the . wake from the

the upstream side . whenever the. velocity changes ation points undergo large ' excursions,. The boundary

cylinder may change. from fully laminar to partially or states and the Reynolds number. may change from

postcritical over a given cycie,..The vortices which

being formed or shed during the first half of the

flow period are also reversed around the cylinder during the wake

re'ersal giving rise to a transverse force with or without addItional

ortex shedding.This is particularly .pronounced for amplitudes of

flow oscillation for,which number of newly formed. vortices, which. have

survived dissipation and convected around the cylinder during the

wake reversa'l.Particülar.ly significant are changes in the lift,, drag

and inertia forces when the reversely-convected vortices are not symmetric.

The wavy flow are relatively more:.complex.Aside from the.. effects of

the free surface.,the orbital motion of the fluid particles give rise

to three dimensional flow . .effects.The. rotation of the wake about a

horizontal cylinder and the exponential decay of the representative wave velocity long a vertical cylinder further complicate the

matters In fact , it, is because. of these reasons that a number of

investigators preferred to separate the additional effects brought

about by waviness of the flow from those resulting from the periodic

As in the case of the other time-dependent flows, the most serious

difficulty with the harmonic flows lies in the description of the

time dependent force itself:.,

-19-3.2.1 FLOW SEPARATION IN OSCILLATORY MOTiON

We discussed already the Starting

motion by

expanding the equation in

time series,. Therefore the results should be

valid only

in a 'small

time period, and its application to oscillatory flow is also restricted

within a small period of the oscIllation i.e. ,high frequency or small

displacement of the body. In this case we shall follow SchIichting approach

[6] to the periodic-oscillation problem of' the boundary-layer 'equation. The

expansion is based on the assumptions

Uf/<U/wD<x/D<1

where x is' the 'amplitude of oscillation and' D is the reference dimension of

the body.Thus this is basically a small amplitude expan'sion.Suppose the velocity U(x,, t)outslde the boundary layer ₍ the 'edge velocity ' ) is

expressed in the form

U (x,t) = U (x)

et

Expanding the velocity u in the boundary layer in the 'series

(x,y,t) + 1a

(x,y,t)

+

substituting this into the boundary-layer .equation ( 1 ) and applyi'ng the

assumption ( 8 ) we can obtain the equations for u0 and u1

&2u _fiM '6t ' &y2 6t - v = U - u - v St 6y2 ' Sx ° 6x ' ° 6y

wIth the usual boundary conditions.

The solutions of eq. ( 9 ) and (, 10 ) were given by Schlichti.ng in the form

iwt

u0=Ue

(1-e

'

-20-2,,

uu

3 e ' -, I j _-,

(11) u10= - + + e ₍ cos , + 2sin,7) + (sin , - cos , ) ]

(12) IA1

where vj

_{= yJw/2L1}

and udu / dx

The most important feature of these solutions is that there appear

a steady flow term in eq.( i1);and a second harmonic term in eq.(12)

both due to higher-order effects of the oscillation.

It is known that the skin friction

variation on

the wall,which leads

to the condition of separation.The skin friction can be expressed

in the form

- pjii'w u

[ei)t

+ ff/4)

₊

₍

+ (.12-1)

e 1(2wt - ff14))]

For convenience in comparing this to the results of the previous section

suppose the edge velocity is prescribed in the form

U = U

sin wt

Then the skin friction can be written as follows:

r _{= p.Jl}

ii

_[ sin ( ct + ff/4)

+

fr (

-(J2

- 1 )sin(2wt + ir/4)) ] The first term coincides with that of flat-plate oscillation ,while

the second terin,which is. always positive,represents the modification

of the pressure gradient or of, the amplitude gradients of the

external flow along the wall.

in case rO into eq.( 13) it can derive the separation condition:

I dU f2 sin(wt + ff/4) - w dx = _1/2-(.f - 1)sin (2wt + n./4)

U1 = U10 + U11

=

._L

e

1+i)f,

_{+ '7.)}

e1'7

2 2 2

-21-Eq... ( 14 ) shows that the separation occurs when and where the va]ue

of the left-hand side exceeds that the right-hand side in one swing The nature of the separation condition is almost the same as in the

previous section in that it. mainly depends on the amplitude of the

motion but not on the frequency or the Reynolds number.

Eq.( 14 ) also

could be

applied to the sway motion of a circular

cylinder of radius R .Then the velocity amplitude is expressed as

U 2xw sin(x/R)

Let us consider again the _{problem..of.obtaining.theseparation.point xs}

or the angle O =x / R -ir/2 at the instant wt =ir/2.Substituting.( 15 )

and wt - ,r/2 into ( 14 ) we obtain

sin = 1.98/(ICC)

Eq.. ( 16 ) has the same form as _{( 7a )} ,but the coefficient differs much..

The experimental value in ( 7a ) is almost average of ( 7 _{) and (} 16 ₎ as

shown in fig.2..'Fig.3 shows the comparison of the separation condition in an

arbitrary time within one swing.The values from ( 14 ) and ( 5 ) have a

similar tendency in .the middle par.t of the one swing period..However,at both ends of the period there. is a diffe.rence due to the tome history effect.

Moreover an interesting thing is that both curves go to zero at t = 3r/4.

This means that at this time the. skin friction becomes zero at the

point where dU _/ dx =0 .In c.ase of .a flat plate

, r changes its

sign at ct = 3ir/4 .,while .in the case of a body with round..bluff corner the

separation poInt . moves upstream into the accelerated flow region (dU/d0,)

on the .body surface.

Fig.3 also shows an approximation curve which is determined in such a

way that it agrees with the experiment at the time wt = n/2 ( 7a ).The form of the curve is expressed. as. follows:

I dU 1 16

(.17) --

= 4.213 [

22 92

I

-22-Eq.. ( 17 ) would be useful to obtain the separation point of an

arbitrary shaped body at any instant of one swing period.

1 dÜ

wdx

10

I

Eq. C 5 )

starting motion (expanded Blasius' method with U -Eq. C q oscillatory motion U = Usinwt approximation Eq. ( 7 11/4 11/2 311/4

fig..3 Condition of separation in one swing

Throughout the preceding analysis ,we.have considered that the separation is

expressed by

the condition r =0 .However this condition may alsO imply

attachment or a simple turning of without separation of the flow.An

additional condttion to distinguish separation from the others can be found

in .the following way

Suppose the stream function 6(x,y,t) close to the wall is expanded in the form

(18) b(x,y,t) ay2 + by3 +

-24-The coefficient a can be related to the skin friction

r

,and the other

U

coefficient , b , can also be found from the leading order term of

boundary-layer equation ( 1 ) expanded near the wall.Then (18) becomes:

2pv

Near to the point r = , can be expanded in the form

rw(x-x)+ w(t-t

At the time t t , the stream function & becomes

-(

w (x-x) +

(t-t) +

The quantity in brackets shows that a dividing streamline s.tarts at the

point x=x

(r=O

),and its inclination tan a is expressed as:

6r

w

₆

tan a

= + u.6x

A positive

value of tana means that the flow is upwards,i.e.,.there

is a separation.A negative value corresponds to a .reattachment and

zero value corresponds to simple turning'of

r

For the separation of one swing period ( .u>O ) of oscillatory

motion, we can safely assume that

;

6x 6t 6x

Eq.(l9 ) is almost satisfied by the value of (14).Substituting(13)into (19)

an additional condition for separation in oscillatory flow follow simple

form:

On the

contrary,u >

0 shows an attachment.Eqs.. (17 ) and ( 20 ) can be

used to obtain the location of the separation point at a given instant.

From the present calculations and analysis of Schlichting1s solution for an oscillatory boundar.y layer , it

can be

concluded that the. separation point is determined by the derivative of, the edge velocity

along the surface,i..e. ,the pressure gradient..Be.fore the end of ne

swing,the separation point moves upstream across the point, of zero pressure. gradient.These features are the. same as those:. obtained in

the preceding section.. When the potential flow theory is used to

obtain the edge velocity distribution . .along . the surface ..,we can

roughly predict the

location of

the. separation. point by. applying the

criterion formula in this sectlon.However , rmOre. precise'treatment of

the outer and inner flows would be necessary to obtain, an, . accurate

prediction o .the separation. and the attachment .,points.The present

analysis is a possible starting step for the problem of flow separation from

smooth surface..

The second order steady part solution (which is appeared from nonlinear

convective, terms in boundary layer equations) recieved by schlichting,, shows that in large dis:tances. of the body i.e. out of boundary layer , the

stationary flow exist,. The velocity of this steady flow in direction of

oscillations in outher flow is equal to.:

St.

3 3irn

U2 = - U0(x) =

-The conclusion is that in unsteady flow ., creating 'byharmonic oscillations

of . fluid particles , not: only a boundary layer is appeared

, but a second order steady flow also.This additional flow is a nonlinear effect of

circular cylinder harmonic oscillations in viscous fluld.in, the boundary layer , a steady vortex is appeared .When we have a several _circular

cylinders oscillating in viscous flow , existance of this steady vortex

makes the phase shift determination very difficult.

Probably in jack-up platforms case , the frame construction diminished the.

effects of steady votices and phase shift results correspondes better to the

-26-reaiity.On the other hand , this is one of the reasons for avoiding a

resonance effects which corresponde to a inviscid case .

-27-4. SHARP-EDGE SEPARATION

The problem of the separation at, a sharp edge corner is

_Elsa

important for

estimating hydrodynamic forces on an oscillating bluff body like a ship. In this

chapter,an attemp.t is made to obtain the forces in a simpler way,. Instead of

expressing the separated flow in terms of vortex sheets shed from the

e4ges, an.

assumption is made that one or two concentrated vortices.can be used.In the following sections ,the motions of a flat plate normal to its plane are

discussed.

4.1 STARTING NOTION OF A FIAT PLATE

To obtain the normal force on a flat plate starting from the rest with a

constant velocity,the following simple procedures are adopted:

.a).put a pair of vortices at the midpoints of the path lines of the separating edges (fig.4)

b)Determine the vortex stEength by KuttaJoukowski condition

To begin .,suppose that the flat plate of breadth. 2 in the Z plane is mapped

into a segment of length 2 in the Z plane, as shown in fig.4.The edge z=i corresponds to ç=ø .The mapping function is:

z - ç2

-i

Then assume that the vortex of strength r lies on each of the path lines of

the separating edges at an arbitrary distance x0 from the edge

zo - xo + i

y

!ath line of edge

1

0

xo, _{z plane} x.

fig.'I Assumed pair of voiticies behind

_{afiatplatestartjng from}

rest

28

-ri

(plane

L 1

in the plane ç .The two vortices have opposite directions of rotation.

If the flow is described with reference to the plate,there is an incident stream of velocity V(t) as in fig.4.The fluid velocity can be expressed in

terms of the (complex) conjugate velocity, that is,

q(z)=u(x,y)-iv(x,y) Q (:) /

where

1

r

F

Q() = v +

( -- -)

At the edge point z=i or ç=ø .,the Kutta condition is applied to determine

the vortex strength

(20a) F = -2,rv ( ), = - 2irv Jx2+4 [x0 + Jx2 + 4

For small values of x0 ,this becomes approximately

r - -

2irvJ0

The vortex is assumed to iie on the midpoint of the path length of the edge:

x0= = vt

The force on the plate is easily derived from the impulsive momentum I on the flow field,

dl d ₂

1? - = (ITpb v - 2py0F]

where b -half breadth =1 ,y,=l .Then

-29-dv d

F=1rpd -2P;

From eq.( 21 ) and ( 22 ) , the first approximation

of r

for small x0 is:

F

-2irv/2

The force can be calculated in the form

1

c

O.5p2bv2

= W

with t'=vt.Figs.S and 6 show results calculated from eqs(23 ₎ and (.24)Eq,. (23) becomes quite close to Wedermeyrs exact. solution and (24 ₎ is

acceptable compared to Kudos calculation [ 3 ].

0: , Eq. C :

Eq. (WJ

x: Eq. -1 211Vb A A

fig.5 Vortex generation in starting motion of flat plate of

half-breadth b1

o.

q.

(2L1

-:

Kudo's calculation 0 5 ₀ -fig.6 Norma-lforee--onf-ia-tp-1-ateinstart-jrig-motion -30-0 A Wedemeyer KudO's calculation 2 ₄

The success of this simple approach can be understood in the following

way:The velocity q,,at a vortex position can be obtained by dropping the term corresponding to the vortex.Thus,

dz / and

xoJ x+ 4

1-2(x,J x + 4

-x2

] So we find that (25) as

Eq.( 25) means that the vortex velocity is half of the velocity in the absence of that vortex pair.Although this does not mean that the location of

the vortex is on the half-way point of the path of the edge.

If again assume that the vortex lies on the line y=l and obtain the

distance x0 from the edge by solving the equation

= U, (x0)

Re(q)

=v/4

for small

x0

the result is

(3/8 _J vdt

2/3

Applying this to the case of starting motion, and substituting into (21)

we obtain.

1. 13

(:26) T --2irv(3/8

Vt

-31-Eq. ( 26 ) is also shown in fig..5 and gives slightly lower values than ( 23 ).

in the range vt>l .. On the other hand _,( 20a ),the original

expression,glves higher values when x0is taken as s/2-the m.idpoint.The.refore

we can realize that (23) compensates. two errors, one arising from the

assumption

of vortex

location (x0 = s/2).and the. other coming from the

simplification of the express:ion

There seems to be evidence for present assumption ,( 22 ),on vortex

Iocation.However,we can cite here some support,for instance numerical

calculation [ I ], [ 3 ]. and experimental

results. that the trailing vortex of a lo -aspect-ratio wing sheds at half

the angle of incidence.

In. conclusion, it can be said that the present simple approach predicts the force on a flat plate. with good accuracy.However ,it can not be expected.

that this method will also give a valid detailed desc.ription of the flow field..

4.. .2 OSCILlATORY NOTION OF A FLAT PLATE

In

oscillating motion

:of a flat plate a pair of vortex-sheet cores is

created during each s4ng of the body..However,toward the end of a swing

before the plate motion cases,the si.gn of the vorticity density near the edge becomes. opposite to that of the. vortex core created during the

swing,. This means that the strength of the shed vor,ticity has a phase lead

with respect to the velocity of the pia.te.The physical interpretation of

this fact seems to be .that the velocity at the rear side of the. edge induced .by the vortex core just created becomes greater than the

velocity on

the

front side near the edge. During the next swing, the previous vor.tex moves

downstream past the. plate and a new vortex core is generated behind the

-32-plate.According to the. calculations. [ 1 ] the location of previous vortex

is nearly above the newly-created vortex..During the continup.us oscillatory sway motion,the vortices thus created seem to flow away from the plate.

in order to formulate this kind of flow mathematically the discrete vortex method can be applied.With the coordinates fixed on the body,the upstream velocity can be expressed

VV sin wt

wx

Assume that the. flow field is. represented by two pairs of concentrated,

vortices which lie, somewhere. . on the horizontal. . lines through the

edges. Suppose their distances from the edge. are. x0.and .x and their

strengths I0and F1 ,(fig7).When the .flow.goes .right (V>O ).it is assumed

that r0represents' the previous vortex..., wh1!ch.-is. constant..and 'T1±he.. new one

which s growing with time.

Assume that the. total circulation in the upper half plane is of the form

= 1' cos (wt+ )

with. amplitude. I' and. phase lead .At the time wt there is only a pair

Of vortices. , the strength of which is

r0=r

Thereof .r0.. is'. constant.However a new pair of strength.F1grows ;with time

until it reaches its maximum value,

-'2 at t -

-At ,cot=ir .- e' the 'total circulation r equals -r ,that is

r

r0+r1.=

-However,the flow 'field should aga.in be expressed by one pair of vortices of strength r1, at wt=-.e, so that the prevous vortex should be mixed into the new vortex F1 'at wt=ir-e .in other words F0 has traveled along ' an unknown

route and joined with the new vortex r1 at

wt=ir-e,.

In this model we do not concern ourselves with the path of the. vortex r0.Instead the variation of _Q

is assumed. To determine the vortex strengths the Kutta-Joukowski condition

is applied at the edges of. the plate.At wt=-e ,there is a vortex pair. The

velocity at the the edge z=1. or =0 can be expressed in the plane by

using the same mapping, function as in starting motion.

1 i

Q=V+[--=

-2,ri

ç-The Kutta-Joukowski condition ( Q=O ) requires that

(26a) 1' -'2iriV

£010

For 'small values of x0,this becomes

Fo 2,r,ri V sinE = 1'

At times such that wt>-e,there are two vortexpairs r0and r1.The

Kutta-Joukowski condition for the growing vortex r1can be written in the form Q = V(t) + V0(t) + -i_1(1/(_.ç)1/()) - 0

or

r1

-

_-2ir[ (V + V0) for small x.1

Where V1 represents the induced velocity of the previous vortex,the strength

of which 1's, is now cOnstantBut the location of r0 is . now unknown.

For the growing vortex r1 , we assume.thevel'ocity.x1..as..follows: .

= 0.5 [ V + V0

_I

for

-c <w t

< ir

-The quantity V-s-V0 is the velocity at ç=o in the ç plane, which means that

the incoming velocity at the edge has increased by the amount V0due to the

effect of the previous vortex.This assumption is similar to the one that the vortex lies on 'the midpoint of the travel path of the edge.From (27),(28)

and

= r - r0 = F

(cós(wt+)-1) we can obtain

x1

-35-3 12

8/3 ,rx (wt + c - sin ( wt +e ))

It should be noted that we have not assumed the form of V0 .Atthe instant

wt - ir -e,( 29 ) becomes

8/3 3/2

and,from symmetry of the vortex locations between the times ,and - we obtain another relation from (26a):

2irV sin e

Eqs. (30) and (31) can be arranged into the forms:

(2r sin)" (3/8)

X w

fT

-

(3ir/4) sin

X

Note that x1 or x0 _{is now the vortex location at the time when ts strength}

becomes maximum,that is,at wt = e or

t

-The phase lead of the circulation F with respect to the velocity V plays an important role in this analysis.. It is introduced to express _{an effect of} the previous swing.However, it is still not clear what is a reasonable

condition to determine the value ofe.

Here ,as it was mentioned in previous section,we assume the value to be

e = ir/4

The support of this assumption can be found in several examples of viscous

flow theory.For instance,the shear stress in oscillatory boundary layer has

the same phase.The phase shift is a difficult problem and the assumption now

is that we can take it equal to r /4 .The force can be obtained, by taking the time derivative of the impulsive momentum of the flow field:

dV

dr

F = pir

- 2 p

Thus

F = p[ irL,2 + 2pXw2JX sin]coswt + 2pXw2JX cosEsinwt

where

3/2

1/2

(2irsin) '

(3/8) '

fi

= 5.735 for

= ir/4

in

the

analysis,

th

force

has

only

simple. '-harmonic. components

at

the

basic

frequency

osciliation.In.order. to :compare.the 'result with other

results we have transform the damping term into the usual nonlinear form

F = F1

cos cot + Fãsin cot

where.

_Fd

0.5 ,p C

Sb V2

Equating the work done in one swing period it is obtained:

Cd

0.75

r COS

(1/ji)

C

- 1. + (.2/ir)sinej

Figs.8 and 9 show the comparison of ( 34 ) and ( 35 ) with the numerical

calculations by Kudo and the Keulegan Carpenter experiments.The agreement of

the

_{Cdvalues is quite good, while C agrees with Kudo's calculation which is}

about twice as large as the value from the Keulegan Carpenter experiments.It

is dtscussing problem,but it is good indication that this method make a good

prediction for drag force.

V=Vs int

fig.7

C

.36 -y

z plane

x_ X1 _x

fig.8 Added mass coefficient of oscillating flat plate

- Present formula

Estimated value by Iudo

0 0 Experimental data by Keulegan

etal

-

Present formula

S lEstimated value by Kudo .0 Experimental data by

Keulegan et al

fig.9 Drag coefficient of oscillating _{flat plate}

-3.7-0 ° ₈ 0OSJ 0 0. ₀ 2

3X4

5 4 3 1

The conclus1ons of this

section can be

summarized as foliows:A simple

approach to the problem of the flat plate oscillating normal to its plane is attempted.The flow field is represented by two pairs of vortices.The

variation of the vortex strength is assumed to be sinusoidal and the vortex velocity to be half the flow velocity at the separating edgeAn additional assumption on the phase lead of the vortex strength is also made.With the

Kutta-Joukowski condition satisfied at:the edge,the results re expressed in

applied form and agree with other works,especially for Cd , as it was

naturally to expected following the advantages and

disadvantages of the numerical method.

4.3 OSCILLATORY NOTION OF SHARP EDGE ELONGATED BODY.

One of the possible application of the solution for a flat plate is to extend it to the sharp edge body solution, considered the body as a number of flat plates.

it is well known that the low frequency damping cannot be determined by the potential theory.Ikeda's [ 2 ] investigations showed that in flat plate oscillation in low KC numbers,there is a symmetric vortex shedding.On the

other hand,these numbers correspond to 5-6 X horizontal displacement

restrictions for Semisübmersibles for exampie.The pontoons of these

platforms are usually slender bodies with almost rectangular cross sections

and that means that the points of vortex shedding are known .( as in the flat

plate).This allows modelling us to model the hydrodynamic interaction between the pontoon and the fluid,to calculate the low frequency

hydrodynamic damping, with much better accuracy than the potential theory

,using a discrete vortex method and its insertation after equivalent

linearization in the equations of motions Fig.lO shows a comparison between the experimentally and the theoretically determined damping coefficients, by

integration over the body, of an slender prismatic body - it can be

semisubmersibie pontoon.It can be pointed out that for 4 < KC

<

12 , the

-38-coincidences are completely satisfactory ( a symmetric wake pattern).We must emphasize that for floating facilities this zone is the greatest practical

importance at long-period oscillations.

exp It50y X3IcmI 10 0.35 0.50

-39-0.75 UI,odlsl 1.25

fig.lO Effect of amplitude of motion on the low frequency waying damping coefficient

In this way we can obtain the damping coefficient:

potential. vor. pot. vor.

B22 B22

+ B22 X(B22,

+ B22 )x2

In fig.11 an example for damping coefficient geometry dependance is shown

[8]and in fig.12 the results for sway motion of SR 192 are presented [9]This

sway damping has an influence to roll. damping coefficient B44,though no.t so

I 20 V3 Ti ₄₀

Eir

B37 koI to

strongly1since there the wave generation induced by the vertical motion exerts the basic influence.

0.2

0.I,

0.O 0.OB 0.12 0.16 WHo 0.20

fig llinfluence of pontoon height and motion frequency on low

frequency sway damping coefficients ( * - experimental results, -- theoretical prediction )

X=90°

-j

caLc:

_exp.

- X2 003 ' 0

IA

_"X2a0,06M

C -

' '

_{X' =010}

M

0

A 1 D - _.

-0 5

-40-2')

_{3.0 ru}

(sJ

Fig.a Sway t4otion.Characteristics.

exp.BSHC;-

-,

calc.BSHCwithLow Frequency

Damp-ing;

_{others -}

_{ITTC'84 ResUlts).}

5. CONCLUSIONS

Two different approaches have bean made. to the problem of flow separation in unsteady motion of the body.The separation from a smooth curved surface is' discussed from the view of. the boundary layer the9ry.I.t' is found through the

analyses of a starting motion and of .an oscillatory motion that the

separation io.int depends mainly on .the amplitude ratio , providing that

thEtratio is, 'small.

The problem of the sharp edge separation is 'discussed in a way 'through the

use of' a small number of vortex pairs.'This..method is ap,pliedxO the.. staring

and osciiiatory motions of a flat" plate,, the motion being.normal.'to 'the

plane of the pIate.The predicted force'agrees.well.w.i.th'.thexper.iments and with others' numerical computation..

This approach wa's successfully used for determination of low frequency

damping in a restricted range of body excursion,,whièh corresponds.. to .the

restriction for horizontal shifting from the point of positioning, and from

the other 'hand to a symmetrical eddy-making around the pontoons.The thus modelled, damping forces of potential and vortex nature,,whose action is

determined in a 'high and low frequency ranges can be determined with better

accuracy. ' '

The author 'believes that this attempt to used some analyt.icall solution for

applied' problems will take part together wi'th many. empirical "useful formulae.

-42-REFERENCES

1.FINK P.T., SOil W.K.

"CALCULATION OF VORTEX SHEETS IN USTEADY FLOW AND APPLICATIONS IN SHIP HYDRODYNAMICS"

10 TH SYMPOSIUM ON NAVAL HYDRODYNANICS,1984

2.IKEDA Y.,TANAKA N.

"ON VISCOUS DRAG OF OSCILLATING BLUFF BODIES"

13 SMSSH,BSNC,1983JVARNA

3.KUDO K,.

"AN INVISCID MODEL OF DISCRETE-VORTEX SHEDDING FOR TWO DIMENSIONAL OSCILLATING: FLOW ROUND A FLAT PLATE"

JSNAJ ,1979 , 145

4.SARPKAYA T. ,ISAACSON M.

"MECHANICS OF WAVE FORCES ON OFFSHORE STRUCTURES"

VAN NOSTRAND REYNUOLD,1981

5. SHKADOV V. ,ZAPRIANOV Z.

"DYNAMICS OF VISCOUS FLUIDS"

NAUKA ISKUSTVO, 1986 ,;SOFIA

6SHLICHTING H.

"BOUNDARY LAYER THEORY" ,, 1968

7.SHOW T.

"WAVE FORCES ON VERTICAL CYLINDERS!' 1.979,LONDON

8. SPASSOV S.A

"MATHEMATICAL MODELLING FOR CALCULATION OF HYDRODYNAMICS COEFFICIENTS OF SEMISUBMERSIBLE PLATFORMS"

16 SMSSH ,1986,VARNA,BSNC

9.1TTC ' 87 ,JAPAN,KOBE,1987

-43,-The author wanted to thank to prof.Cerritsma for his support for

this problem investigation in recent years.Special thanks to

Mr.Beukelman for his social support.

For the interest to this work the author thank to his frends-drPatarinsky,T.Nedkov(. - research fellows in TU Delft.

ACKNOWLEDGMENTS