Linear and nonlinear instability of shear driven liquid films

2805241776

Linear and nonlinear instability

of shear driven liquid films

Paul Caporn

P h. D. Thesis

1998

D ep artm en t of hlathem atics

U niversity College London

ProQuest Number: 10010135

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A c k n o w le d g e m e n ts

F irs t and fo re m o st I w o u ld like to express the deepest g ra titu d e to th e best S upervi­ sors I could have wished fo r, D rs Sergei T im o s h in and R o b e rt Bowles. W ith o u t y o u r tim e , patience and good h u m o u r none o f th is w ould have been possible. I w ould also lik e to th a n k P ro f. F ra n k S m ith fo r g ivin g me the o p p o rtu n ity to do a P h. D , P ro f. Susan B ro w n and D r T o m A lle n fo r th e ir interest in m y w o rk.

To a ll th e m a th e m a tic a l su p p o rt team , th a n k you, especially to P a u l, Simon and R ich a rd fo r keeping th e com puters at bay, and the basics backup boys in th e p o stg ra d ro o m (yo u k n o w w ho you are). To aU the m any friends w ho have been such good com pany over th e la st few years: Look w h a t Pve been up to !

F in a lly , M u m and D a d , T H A N K Y O U . I f i t h a d n ’t been fo r y o u r co n sta n t s u p p o rt th ro u g h e v e ry th in g I w o u ld n ’t be w h a t I am today. I owe you e v e ry th in g .

A b s t r a c t

T he g overning equations fo r a h igh Reynolds num ber flo w in a b o u n d a ry la ye r over a film coated w a ll are derived fro m the fu ll tw o dim ensional N avier Stokes equations o f m o tio n fo r a tw o flu id flo w . N u m e rica l studies o f th e p ro p e rtie s o f th e base flo w and its s ta b ility are described fo r the case o f th e flo w over an isolated surface roughness on an o therw ise fla t surface. In ve stig a tio n s o f b o th short and lo n g obstacles are u n d e rta ke n in term s o f th e flo w in a viscous-inviscid in te ra c tio n region.

A n in v e s tig a tio n o f s tro n g ly no n -lin e a r v o rte x wave in te ra c tio n in a la m in a r b o u n d a ry layer w ith tw o pairs o f obUque waves is carried o u t. F or a p a rtic u la r choice o f flo w param eters a resonance is fo u n d U nking the tw o pairs o f waves, and the governing am pU tude equatio n fo r the leading o rder d istu rb a n ce is derived and in ve stig a te d .

W ave-am pU tude equations are derived fo r the non-Unear m o d u la tio n o f ToUmien- SchUchting (T S ) ty p e disturbances at high R eynolds num bers. A n in v e s tig a tio n o f th e in s ta b iU ty o f R eynolds-stress generated m ean flo w to short w aveleng th secondary disturbances is carried o u t. A regim e w ith Unear T S /c a p iU a ry wave resonance is exam ined and the governing am pUtude equatio n fo r non-Unear wave in te ra c tio n is derived. T w o in te rm e d ia ry regimes are also studied.

T h e Unear in s ta b iU ty o f high Reynolds n um ber b o u n d a ry layer flo w over a film - coated waU is stu d ie d b o th num ericaUy and analyticaU y fo r th e p racticaU y im p o rta n t U m it o f h ig h film viscosity. We exam ine th e various instabiU ties present and re la te th e m to th e in s ta b iU ty classifications o f B e n ja m in (1963) and L a n d a h l (1962).

C ontents

1 I n t r o d u c t i o n 7

1.1 T h e dim ensiona l governing e q u a tio n s ... 15

2 F lo w o v e r a s u rfa c e m o u n t e d o b s ta c le 18 2.1 Blasius b o u n d a ry layer on a film -co a te d w a l l ... 19

2.2 T rip le deck on a waU m o u n te d o b s ta c le ... 21

2.3 B o u n d a ry layer on elongated o b s ta c le s ... 23

2.3.1 T he num e rica l m e t h o d ... 25

2.3.2 Results ... 29

2.4 In v is c id in s ta b ilitie s in film -co a te d f l o w s ... 30

2.4.1 T he long-w ave in s ta b ility ... 31

2.4.2 R ayleigh-w ave in s ta b ility ... 32

2.4.3 T he nu m e rica l R ayleigh in s ta b ility c a lc u la tio n ... 36

2.4.4 N u m e rica l results fo r i n s t a b ilit y ... 38

2.4.5 A p a rtic u la r tw o -flu id f l o w ... 39

2.5 T h e condensed flo w p r o b le m ... 40

2.5.1 T he nu m e rica l m e t h o d ... 41

2.5.2 Results fo r condensed f l o w ... 43

2.6 Tables and figures ... 45

3 V o r t e x - w a v e in t e r a c t io n in a t w o - f lu id f lo w 63 3.1 I n t r o d u c t io n ... 63

C O N T E N T S

3.1.2 A sufficient c o n d itio n fo r re s o n a n c e ... 67

3.2 V W I w ith tw o pairs o f waves... 69

3.2.1 T h e core flo w ... 69

3.2.2 T h e b u ffe r la y e r... 71

3.2.3 T h e viscous in te rfa c ia l l a y e r s ... 76

3.2.4 T h e viscous Stokes layer on th e w a l l ... 78

3.3 T h e a m p litu d e e q u a tio n s ... 79

A A p p e n d ix A : T h e C ritic a l l a y e r ... 86

3.2 Tables and figures ... 91

4 N o n lin e a r s h o r t - w a v e T S i n s t a b i l i t y 98 4.1 T h e trip le -d e c k equations and s c a lin g s ... 100

4.1.1 T h e in s ta b ih ty scalings ... 102

4.2 S h o rt scale disturbances. T h e in v iscid r e g io n s ... 104

4.2.1 T h e viscous Stokes layer on th e w a l l ... 109

4.2.2 T he in te rfa c ia l viscous l a y e r ... 110

4.2.3 T h e am pUtude e q u a t io n ... I l l 4.3 S o lu tio n p ro p e rtie s and secondary in s ta b iU tie s ... 114

4.3.1 In te rm e d ia te secondary d is tu r b a n c e s ... 115

4.3.2 In s ta b iU ty o f weak p rim a ry w a v e s ... 116

4.3.3 W eak surface t e n s io n ... 118

4.4 N onUnear TS-capiU ary wave re s o n a n c e ... 120

4.4.1 D e riv a tio n o f the am pU tude e q u a tio n ... 120

4.4.2 A n a lysis o f the am pU tude e q u a t io n ... 124

4.5 T h e near resonant r e g im e s ... 128

4.5.1 T h e firs t near-resonant regim e ... 128

4.5.2 T h e viscous diffu sio n la y e r ... 131

4.5.3 T h e in te rfa c ia l viscous l a y e r ... 132

4.5.4 T h e am pU tude e q u a t io n ... 133

C O N T E N T S

4.5.6 T he flo w e x p a n s io n s ... 137

4.5.7 T he in te rfa c ia l viscous l a y e r s ... 140

4.5.8 T h e a m p litu d e e q u a t io n ... 141

4.6 C o n clu d in g rem arks ... 146

4.4 Figures ... 148

5 I n s t a b i l i t y o f f lo w o n a v e r y v is c o u s f i l m 161 5.1 I n t r o d u c t io n ... 161

5.2 T h e p ro b le m f o r m u l a t i o n ... 163

5.3 T h e th ic k in v is c id film l i m i t ... 167

5.4 T h e n u m e rica l s o lu t io n ... 170

5.4.1 Long-w ave lim it ... 171

5.5 T h e th in - film lim it case ... 174

5.5.1 TS in s ta b ility on a viscous th in f i l m ... 174

5.5.2 T h in film s o f sm aller v is c o s ity ... 177

5.6 D is c u s s io n ... 181

5.7 Figures ... 184

C h a p te r 1

In trod u ction

The effect o f th in liq u id film coatings in a high R eynolds n u m b e r b o u n d a ry layer flow on b o th sep a ra tio n and tra n s itio n to turb u le n ce is a pro b le m o f great p ra c tic a l im p o rta n c e in m a n y real life situ a tio n s . These include th e flo w over ra in w e tte d planes and cars, th e de-icing o f plane wings and th e use o f lu b ric a n ts in m any engineering a p p lica tio n s. We begin th is Thesis w ith a b rie f review o f some o f th e developm ents in th e relevant theories fo r homogeneous flows before exa m in in g the specific role o f a film and th e a lte ra tio n s its presence entails.

C h a p te r 1: In tro d u c tio n

fo r a lm o st a ll a p p lic a tio n s , by G o ld ste in (1948), at the p o in t o f flo w reversal, and hence se p a ra tio n , v ia w h a t became kn o w n as a G oldstein s in g u la rity in th e slope o f b o th th e skin fr ic tio n on the b o d y surface and the b o u n d a ry layer displacem ent. A new th e o ry was re q u ire d to cope w ith separation. T h is was p ro vid e d by a tr ip le ­ deck scheme, a v is c o u s /in v is c id in te ra c tio n th e o ry w hich allowed fo r an unspecified pressure g ra d ie n t, as developed by S tew artson & W illia m s (1969), N e ila n d (1969), M essiter (1 9 7 0 ), S tew artson (1970). T h e th e o ry spht the b o u n d a ry layer in to tw o so-called decks w ith a n o n lin e a r viscous low er deck on th e b o d y surface d riv e n b y an e x te rn a l in d u ce d pressure and a ro ta tio n a l in viscid m a in deck, w hich rem ains la rg e ly passive, s h ifte d v ia th e displacem ent caused by the low er deck. A th ir d ’p o te n tia l- fio w ’ u p p e r deck com pleted the d e scrip tio n , w ith the lo ca l displacem ent fro m the low er deck a ffe c tin g th e induced lo c a l pressure, o f th e order o f the slope o f the stre a m ­ lines in th e b o u n d a ry layer, and hence affecting the low er deck. T h is ’in te ra c tiv e ’ approach avoids th e fa ilu re o f th e classical th e o ry due to the unspecified pressure g ra d ie n t, and hence u n kn o w n displacem ent. The th e o ry also does n o t depend on a p a rtic u la r set o f w a ll b o u n d a ry c o n d itio n s, w hich makes it applicable to a w ide range o f d iffe re n t pro b le m s e.g. flows over b lu ff bodies, plates w ith a lo ca l w a ll roughness o r flow s w ith w alls c o n ta in in g flu id in je ctio n s. For weak d is to rtio n s , flo w separa­ tio n leads to a fu lly viscous eddy and re -a tta c h m e n t fu rth e r dow nstream w ith in the trip le -d e c k region. L a rg e r d is to rtio n s lead to global (breakaw ay) sep a ra tio n w ith the viscous shear la y e r centering around an algebraic curve o f increasing distance fro m the b o d y d o w n stre a m o f th e separation p o in t, see Sychev (1972), M essiter (1975), S m ith (1977). T h e re are however some p a rtic u la r cases in w hich classical b o u n d a ry layer th e o ry is stiU applicable, as in th e m a rg in a l separation regim e exam ined in R uban (1 9 8 1 ), com plem ented by a lo c a l in te ra c tiv e s tru c tu re in R u b a n (1982) and S tew artson, S m ith & K aups (1982) o r in th e condensed flo w o f S m ith & D aniels (1981). A d e ta ile d re vie w o f these issues can be found in M essiter (1979) and S m ith (1982).

C h a p te r 1: In tro d u c tio n

T h e second im p o rta n t to p ic o f research relevant to th is Thesis is on th e in s ta ­ b ilit y o f flu id flows to in fln ite s im a l disturbances and the tra n s itio n o f la m in a r flo w to tu rb u le n c e . In s ta b ility th e o ry developed fro m a need to u n d e rsta n d w h y m ost h ig h speed flow s are o f a tu rb u le n t ra th e r then la m in a r n a tu re . W it h aU th e early w orks based on an in v is c id tre a tm e n t, i t has its beginnings in th e a n a ly tic a l studies o f H e lm h o ltz , K e lv in and L o rd R ayleigh and in the e x p e rim e n ta l w o rk o f R eynolds. T h is in s ta b ility th e o ry fo r in v iscid waves was extended to include th e effects o f v is ­ co sity in th e w orks o f O rr, Som m erfeld, T a ylo r, P ra n d tl, ToUmien and S chlichting. T h e reader is re fe rre d to D ra z in & Reid (1981) fo r a review o f th e e a rly th e o ry. ToUm ien and S ch lich tin g showed th a t viscous effects could p ro v id e th e m echanism fo r in s ta b iU ty , an essentiaUy counter in tu itiv e effect. AU these theories were based on Unear a p p ro x im a tio n s , and i t was n o t u n til L anda u (1944), in a q u ite general p o s tu la tio n , th a t a nonUnear theory, now term ed w eakly nonUnear, was proposed. L a n d a u ’s ideas were confirm ed in an e xa m in a tio n o f plane paraUel flow s by S tu ­ a rt (1960) and W a tso n (1960), and these firs t three a u th o rs lend th e ir nam e to th e ty p ic a l L a n d a u -S tu a rt-W a ts o n am pUtude e q uatio n governing w e a kly nonUnear in s ta b iU ty waves. I t oh (1974), w ho derived the same fo rm o f e q u a tio n fo r th e B la ­ sius la ye r, and m a n y subsequent w orks were aU appUed to flows at fin ite R eynolds num bers w here th e effects o f th e flo w non-paraUeUsm are non-negUgible. S m ith (1979a,b ) began m ore rigorous investigatio ns o f th e Unear and nonUnear in s ta b iU ty o f b o u n d a ry -la y e r flow s, a t large Reynolds num bers, by pla cin g th e base flo w and disturbance s w ith in th e trip le -d e c k scaUngs (an account o f earUer a s y m p to tic ap­ proaches to th e viscous-flow in s ta b iU ty is given in L in (1 9 5 5 )). M a n y o th e r h igh R eynolds n u m b e r studies foUowed, exa m in in g a w ide v a rie ty o f in s ta b iU ty m echa­ nism s and d is tu rb a n c e scaUngs (see review articles by S m ith (1993), HaU (1990), C ow ley & W u (1 9 9 3 )).

C h a p te r 1: In tro d u c tio n

10

th e wave d istu rb a n ce and the p ro p o rtio n o f the 3D v o rte x in the m ean flo w . W hen th e v o rte x p a rt o f th e flo w was sim p ly a sm all co rre ctio n to th e m ean p ro file th e in te r­ actions w ith th e disturbances were te rm e d w eakly n o n lin e a r, w h ils t those w here th e v o rte x com prised th e e n tire mean flo w were term ed s tro n g ly n o n lin e a r. In te ra c tio n s w ith sm a ll disturbance s o f the m in im u m m a g n itu d e to in s tig a te n o n -lin e a r in te ra c ­ tio n s , were in v e stig a te d fo r b o th in viscid R ayleigh and viscous T o U m ien-S chlichting (T S ) waves b y HaU & S m ith (1988, 1989, 1990), S m ith & W a lto n (1989), B lackaby (1991), S m ith & B lennerhassett (1992) fo r ’w e a k ly ’ non-Unear in te ra c tio n s , and by S m ith & W a lto n (1989) W a lto n & S m ith (1992), HaU & S m ith (1991), Seddougui & Bassom (1991) fo r ’s tro n g ly ’ non-Unear in te ra c tio n s . T h e w o rk o f HaU & S m ith (1991), a s tu d y o f b o th compressible and incom pressible flow s, reUed on th e exis­ tence o f a s a tu ra te d n e u tra l wave at some upstream p o s itio n , a t w h ich th e in te ra c tio n was in itia te d w ith th e v o rte x then developing dow nstream in o rd e r to keep th e wave n e u tra l. B ro w n et al. (1993) studied sh o rte r scale events fo r th e in itia tio n o f th is be­ h a v io u r in th e incom pressible case fo r R ayleigh waves, a lth o u g h stiU w ith an a b ru p t s ta rt to th e in te ra c tio n . S m ith , B ro w n & B ro w n (1993) exam ined even sh o rte r scale events, w ith th e v o rte x /w a v e in te ra c tio n o ccu rrin g chiefly th ro u g h th e ju m p in transverse shear stress across a c ritic a l layer. T h e y derived a w ave-am pU tude e q u a tio n g overning the wave disturbance and fo u n d various solu tio n s fo r th e d ow n­ stream b e h a v io u r in c lu d in g wave decay, a fin ite -d is ta n c e w ave-am pU tude b lo w -u p and p e rio d ic so lu tio n s, w hich th e y conjectured were m ore Ukely to occur th a n the d o w n stre a m m a tc h to a constant wave am pU tude required by B ro w n et al. (1993), HaU & S m ith (1991).

C h a p te r 1: In tro d u c tio n

11

a n a ly tic solutions and showed th a t a lin e a r p ro file in th e w a te r and B lasius p ro file in th e a ir are reasonable base-flow profiles fo r in s ta b ility calculation s. C ow a rd & H a ll (1996), in a s tu d y o f th e s ta b ility o f th in coatings o f w a te r on a porous w a ll in a ir flo w , co n stru cte d s im ila rity solutions fo r th e base profiles, m ade possible th ro u g h the w a ll c o n d itio n . We could n o t fin d o th e r works w h ich exam ined th e fo rm o f base profiles fo r b o u n d a ry layers w ith th in liq u id film coatings, and in C h a p te r 1 we aim to shed some lig h t, c o m p u ta tio n a lly , on th e b e h a vio u r and shape o f possible base flows.

C h a p te r 1: In tro d u c tio n

12

problem s was given by B e n ja m in (1960, 1963) and L a n d a h l (1962) w h o a tte m p te d a physical e x p la n a tio n o f in s ta b ilitie s based on energy levels. T h e y categorized three d iffe re n t in s ta b ilitie s appearing in flow s w ith fle xib le boundaries re fe rrin g to these as Class A , Class B and K e lv in -H e lm h o ltz (K - H ) waves. T h e fir s t o f these, the Class A in s ta b ility , w h ich includes T oU m ien-S chlichting waves m o d ifie d b y the fle x ­ ib le b o u n d a ry, is destabiUzed b y d issip a tive forces. T h e y d e m o n stra te d , v ia the energy consideratio ns, th a t an essentiaUy cou n te r in tu itiv e destabiU za tion, w ith the wave g ro w th accom panied by a tra n s fe r o f energy fro m th e wave to th e m a in flo w , is present in th e system . T he Class B in s ta b iU ty on th e o th e r hand is stabiUzed by the d issip a tive forces and grows v ia an energy tra n s fe r fro m th e m ean flo w to th e wave, a m o re in tu itiv e m echanism . T h e fin a l class, the K -H in s ta b iU ty is d rive n by v e lo c ity d is c o n tin u itie s. C lassifica tio n in these papers is based u p o n n e a r-n e u tra l calcu la tio n s and we aim in th is Thesis to v e rify the general classifications by d ire ct c o m p u ta tio n o f th e instabiU ties.

M o re recent in s ta b iU ty studies in v o lv in g tw o phase flows in clu d e those m ade by H ooper & B o y d (1986), M o rla n d & S affm an & Yuen (1991), S h rira (1 9 9 3 ), M o rla n d & S affm an (1993), C ow ard & HaU (1996) and T im o s h in (1997). S h rira (1993) ex­ am ined insta b iU tie s o f disturbances fo u n d in deep w a te r w ith a c u rre n t and a free surface, w h ils t M o rla n d & Saffm an (1993) carried o u t Unear s ta b iU ty analysis o f an in viscid p a ra lle l a ir flo w over w a te r and m ade n u m e rica l com parisons, fin d in g fa ir agreem ent w ith th e a n a ly tic s o lu tio n o f M iles (1957). C ow ard & H a ll (1996) studied the th re e -d im e n sio n a l flo w over a porous fla t p la te , w ith su ctio n o r b lo w in g chosen to m a in ta in a co nstant low er flu id d e p th . T h e ir sta b iU ty analysis showed, as in H ooper and B o y d (1986), th a t d is c o n tin u itie s in the visco sity a n d /o r d e n sity o f th e tw o im m is c ib le flu id s g re a tly enhanced in sta b iU ty. T im o s h in (1997) exam ined Unear instabiU ties w ith in the trip le -d e c k fo rm u la tio n o f a tw o -flu id flo w in c lu d in g th e case o f a ve ry viscous film and derived g ro w th rates fo r TS and in te rfa c ia l waves.

C h a p te r 1: In tro d u c tio n

13

1977). T h is effect is due to the ju m p in shears across th e viscous layers s u rro u n d ­ in g th e in te rfa c e as opposed to a s h ift in velocities due to viscous layers on solid boundaries. D ore (1976, 1977) in c o rp o ra te d o u te r viscous layers a b o u t th e in te rfa ­ cial layers, based on th e double b o u n d a ry layer theories developed by R ile y (1965) and S tu a rt (1966), th ro u g h w h ich th e induced m ean flo w is diffused. T h e effect o f th is s tro n g e r m ean flo w on the wave in s ta b ility is fe lt th ro u g h in te ra c tio n s at a low er o rd e r th a n those w ith w a ll induced m ean flow . A second flu id also allows fo r reso­ n a n t in te ra c tio n s between the various in s ta b ifltie s th a t m ay be present such as those classified b y B e n ja m in -L a n d a h l, o u tlin e d above. One such case is studied in A k yla s (1982), A k y la s & B enney (1982) w ho id e n tify a resonance between ’a ir ’ (Class A ) and ’w a te r’ (Class B ) modes in the case o f w in d on deep w a te r.

W e see th e n th a t th e s tu d y o f b o u n d a ry layers w ith th in film s is a com plex and fa s c in a tin g fie ld , w ith very lit t le k n o w n a b o u t th e effect o f film s on separation, along w ith th e a p p a re n tly s tro n g effect on s ta b ility provoked by th e presence o f an interface.

C h a p te r 1: In tro d u c tio n

14

base flo w developm ent. T h is provides an im p o rta n t in sig h t in to th e role o f a film in th e onset o f se p a ra tio n , an area w hich up to now has received scant a tte n tio n to th is a u th o r ’s best know ledge. A s ta b ility analysis is th e n carried o u t on the calculated p rofiles.

In C h a p te r 3 we in ve stig a te w e a kly-n o n lin e a r v o rte x /in v is c id wave in te ra c tio n s , in th e e a rly stages o f tra n s itio n fo r a tw o flu id flo w , based on th e single flu id s tu d y o f S m ith , B ro w n & B ro w n (1993). B y u tiliz in g th e assum ptions o f a p a ra lle l flo w and a base p ro file co nsisting o f tw o constant shears in o u r region o f study, we o b ta in wave- a m p litu d e equatio ns governing th e e v o lu tio n o f tw o pairs o f oblique waves tra v e llin g w ith id e n tic a l phase speed. We show th a t the n o n lin e a r developm ent o f th is flo w can lead to fin ite -d is ta n c e b lo w -u p o f th e wave disturbances. In the case o f non-resonant waves we show th a t th e am p h tu d e equations s im p lify to those derived in S m ith , B ro w n & B ro w n (1993).

T h e s ta b ility o f n o n lin e a r ToU m ien-S chlichting (T S ) waves is studied w ith in a trip le -d e c k fra m e w o rk in C h a p te r 4, an extension o f the lin e a r analysis o f T im o s h in (1997). In a w e a kly n o n -lin e a r analysis, th e te m p o ra l e vo lu tio n o f tw o dim ensional disturbance s is m o defied v ia an a m p litu d e e v o lu tio n equatio n coupled w ith equations gove rn in g th e R eynolds stress induced m ean flo w . A tw o fo ld in v e s tig a tio n o f b o th th e s ta b ility o f th e m uch altered mean flo w to R ayleigh scale disturbances and the developm ent o f th e wave d isturbance is carried o u t. T h e a m p litu d e e q u a tio n is fo u n d to co n ta in a s in g u la rity centered around a specific c o m b in a tio n o f the surface tension, g ra v ity , d e n s ity ra tio s and film thickness. A close analysis is p e rfo rm e d w ith in th is p a ra m e te r space and a resonant s tru c tu re fo u n d w ith m agnified disturbance a m p litu d e s. T h is n o n -lin e a r resonance is d ire c tly re la te d to the lin e a r resonance between g ro w in g TS and decaying c a p illa ry waves o u tlin e d in T im o s h in (1997). A fu ll in v e s tig a tio n o f th e p ro p e rtie s o f the governing a m p litu d e equ a tio n is carried o u t, w ith th e n o n lin e a rity appea rin g in an unusual diff'e rentiated fo rm . T h e properties o f th e a m p litu d e equations are q u ite disparate and tw o fu rth e r in te rm e d ia te regimes are stu d ie d , g iv in g a f u ll account o f the possible distu rb a n ce developm ent schemes.

ayleigh-C h a p te r 1: T h e dim ensiona l gove rn in g equations

15

scale s ta b ility p ro p e rtie s o f flo w over a very viscous fllm in C h a p te r 5. T h e d iffe r­ ent classes o f in s ta b ility suggested by B e njam in (1960,1963) and L a n d a h l (1962) are show n to be present as d iffe re n t lim its o f ou r general fo rm u la tio n , and we d e m o n s tra te th a t various cases studied previously, (e.g. K e lv in -H e lm h o ltz , ToUmien- S ch lich tin g , M ile s, c a p illa ry /T S wave resonance) are aU co n tin u o u sly lin k e d in the p a ra m e te r space stu d ie d here.

We begin o u r in v e s tig a tio n by o u tlin in g the dim ensional governing equations and b o u n d a ry c o n d itio n s fo r tw o -flu id flows w hich we w ill use th ro u g h o u t th is s tu d y w ith th e specific n o n -d im e n sio n a liza tio n s given at th e s ta rt o f each chapter.

1.1

T h e dim ensional governing equations

T h e tw o flu id flow s stu d ie d in th is thesis are governed by the incom pressible N avier- Stokes equatio ns. W e deflne z *, y* to be the dim ensional coordinates p a ra lle l and n o rm a l to th e flo w d ire c tio n and z* to be the spanwise coo rd in a te p e rp e n d ic u la r to æ* in th e plane ?/* = 0. T h e n u f , v f , w f , p f represent th e stream w ise, n o rm a l and cross-flow velocities and the pressure respectively, w ith the superscripts + / — d e n o tin g th e regions above o r below the interface separating the tw o flu id s at y* = /♦ (x + , z*, f* ) . A ll th e flows stu d ie d take place w ith in a b o u n d a ry layer w h ich develops over a surface defined b y ?/* = h * ( z * ) , placed in the flo w . T he density and viscosity o f the flu id in th e fllm are denoted by p " , p~ and in th e m a in b o u n d a ry layer flu id b y p ^ , p j". W it h p* representing the dim ensional g ra v ita tio n a l acceleration, the gove rn in g equations are

± / a2„.,± a2„„±

D w ^ —1 d p f p f f w:

C h a p te r 1: T h e dim ensional g overning equations

16

T he b o u n d a ry co n d itio n s fo r these equations are fir s tly those o f no slip on the surface, c o n tin u ity in the stream wise and spanwise velocities at th e in te rfa ce and th e k in e m a tic c o n d itio n a t the interface

u~ = v~ = 0 , a t (1.1.2)

j } 3-t y* — Z*, t * ) ; (1.1.3)

where th e m a te ria l d e riva tive is defined by D f D t ^ = d/dt^, - \ - u f d / d x ^ + +

w ^ d j d z ^ . Secondly, at th e interface, defined by ?/* = / * , between th e tw o flu id s the e q u a tio n

l a . n ] t - H7* = 0 at 2/* = (1-1-4)

m ust be satisfied, where a is the stress tensor, 7* is th e surface tension, th e square brackets [] denote a ju m p across the interface,

1

1

R i ( i + ( 0 / . / a x . ) 2 ) 3 / 2 ’ Æ2 { i + { d f . / d z , y f / ^ ’

are the ra d ii o f c u rv a tu re , and n is the u n it n o rm a l to th e in te rfa ce given by (1.1.5)

We w rite th e u n it ta n g e n t vectors to n as

C h a p te r 1: T he dim ensional g overning equations

17

and, ta k in g th e dot p ro d u c t o f (1.1.4) w ith t^ * , t^ , n , respectively we o b ta in three in te rfa c ia l ju m p condition s

- B ,

( dv^

d x ^ )

d h d u

_{_}

_n

dz^ dx^

(

ÊA

dz^ \ 9 y .

- B ,

_{f i}

-d f * dx^ dz^

1

]

/ J

= 0 (1.1.9)

d f * dz^

% % \

dz^ dx^, dx ^

( du^ f df^ \ ^

dv^

dw^ f

^

dx^ \ d x ^ - 2 ^ B . +

U C / Z ^

^

^

=

0

(

1

.

1

.

10

)

J \

+7#

+

=

0 (1.1.11)

(1 + ( a A / ^ z , ) 2 ) 3 / 2 (1 + ( a A / a z , )2) " / \

where square brackets in d ica te a ju m p across the interface, p aram eters in th e b o u n d ­ a ry layer and film are denoted by + / — respectively (a d iffe re n t n o ta tio n is used in chapter 2 fo r these layers) and

I ( du^ d v ^ \ I f d v ^ dw^

18

C h a p te r 2

Flow over a surface m ou n ted

ob stacle

In th is ch a p te r we n u m e ric a lly tackle the flo w w ith in a b o u n d a ry la ye r on a w a ll, fo r homogeneous flows and fo r those w ith a th in film co a tin g on th e w a ll. T h e aim s here are to prepare the g ro u n d fo r e xa m in a tio n o f fllm coated flow s in th e subsequent chapters. F irs t and forem ost we provide a re a lis tic m odel fo r th e base flo w , w h ich we w ill use fo r a ll th e subsequent w o rk. Secondly we in ve stig a te th e effects o f a c e rta in prescribed w a ll roughness on the base flo w , and the fo rm o f th e s in g u la ritie s w h ich we expect to fin d in th e b o u n d a ry layer so lu tio n w hen the pressure on th e b o u n d a ry layer and in th e film is given. We show th a t fo r a ll cases considered th e s in g u la ritie s are always due to zero w a ll shear, as fo u n d in th e w orks o f G o ld s te in (1 9 4 8 ), S tew ­ artso n , S m ith & K aups (1982), R uban (1981,1982) ra th e r th a n to flo w reversal in the m id d le o f the flo w region as in Sychev (1980), E U io tt, S m ith & C ow ley (1983), T im o s h in (1996). O u r fin a l aim in th is chapter is to in ve stig a te th e d e s ta b iliz in g effect o f th e waU roughness, o r indeed o f any o th e r m echanism w h ic h affects co nstant shear profiles in th in film s . We fin d in viscid in s ta b ih ty w h ich is s tro n g ly influence d by th e p ro p e rtie s o f the interface.

C h a p te r 2: B lasius b o u n d a ry la y e r on a h lm -co a te d w a ll

19

arise w hen th e w a ll roughness is long com pared to th e tr ip le deck. In §2.5 we also exam ine th e opposite lim itin g case o f sh o rte r obstacles, leading to a condensed flo w fo rm u la tio n . T h e second regim e is irre le va n t to the base flo w in v e s tig a tio n , w hich is one o f th e p rim a ry interests in th is chapter, b u t is a lo g ic a l a n a ly tic c o m p lim e n t to th e s o lu tio n we derive fo r the longer obstacle, re q u irin g o n ly a s tra ig h tfo rw a rd change in th e p ro b le m fo rm u la tio n . A n analysis o f th e sin g le -flu id trip le -d e c k p ro b ­ lem fo r obstacles in b o th these lim its and o f those on th e trip le -d e c k scale is given by, fo r exam ple. S m ith , B rig h to n , Jackson, H u n t (1981); see also a review a rtic le by S m ith (1982).

2.1

B lasius boundary layer on a film -coated wall

In the fo llo w in g chapters th e in ve stig a tio n s u tiliz e an in it ia l u n p e rtu rb e d up stre a m flo w consisting o f piecewise lin e a r profiles in th e n e a r-w a ll p a rt o f th e b o u n d a ry layer. In th is section we o u tlin e the general assum ptions and scalings we wiU use fo r ta c k lin g film coated flow s, in c lu d in g the base flo w profiles. Once th e fo rm o f base flow has been ve rifie d , th e fo llo w in g subsection o utlines th e trip le -d e c k scalings used to exam ine flo w over a w a ll m ounted obstacle.

We assume th e flo w to be tw o -d im e n sio n a l and, fu rth e r, th a t a t th e leading edge o f a fla t p la te in a u n ifo rm stream a b o u n d a ry layer on a surface is generated in w hich we have steady, incom pressible p la n a r flo w . D o w n stre a m o f th e leading edge we have a film generated by a slo t, in the fo rm o f a je t; see fig 2 :1(a ). G ra v ity and surface ten sio n are in clu d e d in th e p ro b le m fo rm u la tio n fo r th e base flo w ca lculation s, a lth o u g h th e y are discarded in th e n u m e rica l in v e s tig a tio n la te r as in a re la te d w o rk by Nelson et al. (1995).

coef-C h a p te r 2: Blasius b o u n d a ry layer on a film -co a te d w a ll

20

fic ie n t 7 = 7* / / ) j ^ T h e non-dim ensionalized stream w ise and n o rm a l ve lo c ity com ponents are u, v, we take x, y as the b o u n d a ry layer coordinates p a ra lle l and n o r­ m a l to th e flo w , w ith z * = 2 * ( z -f 1), y* = W ith p th e n o n-dim en sional pressure and / the in te rfa ce p o s itio n between the tw o flu id s , ta ke n in itia lly to be a , th e h eight o f the slo t, th e governing equations become

f -»■ ï + f -»■

<-^1

w here we deflne = 1, i/+ = 1, p~ = p * ~/ = f/* — Th e a p p ro p ria te b o u n d a ry co n d itio n s are

y -> 0 0 = 1, (2.1.3)

j D /

= U , = V = — = p

3/ = / : ' D r a y (2.1.4)

p+ - p - = j f î i - { p - - l ) f / F r ,

y = 0 : u “ =: 0, 7; “ = 0, (2.1.5) X = 0 : u'^ = Ub, = J y { à - ÿ), ; f = à (2.1.6)

where Ub = Ub{v — à) is the Blasius p ro file , J is a constant m easuring th e stre n g th o f th e je t and th e shape o f the interface is described by y = f { x ) .

C h a p te r 2: T rip le deck on

a

21

w ith th e m ain b o u n d a ry layer gro w in g like and th e in te rfa c ia l shear in th e b o u n d a ry layer decreasing like w ith the film flo w d rive n by th e applied in te rfa c ia l shear. I f Ÿg is the mass flu x at th e slot, given by th e value o f th e stream fu n c tio n Ÿ a t th e in te rfa ce , so th a t u~ = d ' ^ f d y , th e n th e stream w ise velocities and in te rfa c e p o s itio n in th e region y ~ are given b y

w here \ b = d U B l d y { 0 ) , and the b o u n d a ry layer flo w approaches th e B lasius p ro file d o w n stre a m in the region y x ~ ^ / ^ ~ 1. B y a lte rin g Ÿg , e ith e r v ia th e size o f th e in je c tio n slot o r the speed o f th e je t, we can set th e fllm thickness d o w n stre a m . A t z 1 th e fllm forgets about th e specific source. T h is allows us to p e rfo rm th e trip le -d e c k analysis a t a s ta tio n A* dow nstream in the n e x t section, w h ich requires a fllm o f thickness 0 ( E e “ ^/®), in te rm s o f th e lo ca l Reynolds n um ber (see fig 2 : l( a ) ) , i f

V “ ly/8 3/8__3/8

we have a flu x o f 0(TZe L j A* ). Hence we m ay assume fo r aU o u r subsequent analysis th a t o u r p redeterm ine d in itia l lo ca l base p ro file consisting o f tw o constant shears can be o b ta in e d , o r is indeed ty p ic a l, in a tw o flu id system . O th e r mechanisms can be tre a te d in a s im ila r fashion, fo r exam ple in je c tio n th ro u g h a porous w a ll, c f C ow a rd & H a ll (1996).

2.2

Triple deck on a wall m ounted ob stacle

In th is subsection we o u tlin e th e scalings used to in ve stig a te th e flo w w ith in a b o u n d ­ a ry la ye r w h ich develops over a lo c a l surface roughness defined by %/* = h * ( z * ) , see fig 2 ;2 (a ). W e quote the rescalings used fo r a short-scale analysis o f flo w over a fla t p la te , w h ic h lead to th e trip le -d e c k equations fo r fllm coated flow s, as derived by T im o s h in (1997), Tsao et al. (1996) fro m th e fu ll N avier-S tokes equations. A fte r n o n -d im e n sio n a lizin g we p e rfo rm a P ra n d tl s h ift, in tro d u c in g th e surface shape in to th e p ro b le m .

C h a p te r 2: T rip le deck on a w a ll m o u n te d obstacle

22

to th e lo c a l p o in t o f in v e s tig a tio n (ta ke n to be the centre o f an obstacle on th e w a ll la te r in th is section), w ith corresponding R eynolds n u m b e r Re = ( > 1),

Froude n um ber F r — and surface tension coefficient 7 = 7* //) + T he

non-dim ensionalized te m p o ra l and s p a tia l co-ordinates, velocities, pressures and in ­ terface shape are denoted by f, x , ÿ, û, v, p and / , W it h à d e n o tin g th e u n p e r­ tu rb e d in te rfa ce p o s itio n up stre a m fro m th e area under in v e s tig a tio n , we rescale to the trip le -d e c k variables in the viscous sublayer (zone I in fig 2 : l( a ) )

û , v , p , x , ÿ , i , à , f =

€ ^ (A + )-^ /7 , (2.2.1)

w ith €o = y representing the lo ca l n o rm a l co o rd in a te in th e viscous sublayer, and A+ d e n o tin g th e shear o f th e upper pro file . W e w rite p"*" = 1, z/"*" = 1, p “ =

p^ T! P * ^ ■> and defining y = h ( x ) to be th e non-d im e n sio n a l w a ll shape apply a P ra n d tl s h ift to the trip le -d e c k equations,

y = Y / i( x ) , V =■ V u——. (2.2.2) T his leaves us w ith

and th e a p p ro p ria te b o u n d a ry co n d itio n s are

y ^ 00 u'^ = Y + a( - 1) + A ( x ) -1- h { x ) -j- o ( l ) , (2.2.5)

y = 0 u~ = 0, v~ = 0, (2.2.6)

u + = Y - a + U , , ^ (2.2.7)

u~ = \ ~ Y

where A“ = Ug = a \ ~ and th e shape o f th e in te rfa ce is described by

C h a p te r 2: B o u n d a ry la ye r on elongated obstacles

23

•p’*’ ~ P — l { f x x + hxx) — {p — 1 ) ( / + h ) l F r (2.2.9)

w here 7 = F r = are th e rescaled surface tension coeffi­

cient and Froude n u m b e r respectively. F in a lly the in te ra c tio n c o n d itio n fo r subsonic flows com pletes th e tr ip le deck fo rm u la tio n ,

=

(

2

.

2

.

10

)

^ 7-00 3 - 5

2.3

B oun d ary layer on elon gated ob stacles

F rom now one in th is C h a p te r we exam ine th e steady case, d j d t = 0. T he firs t step now is to rescale th e p ro b le m ta k in g th e le n g th o f th e hu m p L as o u r ty p ic a l le n g th scale. W e consider lo n g hum ps w ith T 1 on th e tr ip le deck scale. T he procedure is s im ila r to th a t used in S m ith , B rig h to n , Jackson, H u n t (1981). W e take

y ~ Z f 3 ,

T h e n fro m th e balance ~ p ^ and, fro m (2.2.10) we know A ~ zp"*", we have

A L s .

We exam ine no w hum ps w ith a h e ig h t scale /i ~ T s ^ as we w a n t a c o n trib u tio n

{ A - \ - h ) 0 { L ^ ) i.e. A = - h - \ - 0 { L ï ) . T h is height produces a nonlin e a r response in th e viscous la y e r w ith th e induced pressure p ro p o rtio n a l to th e slope o f the obstacle

h / L .

To keep th e in te rfa c ia l effects in th e analysis we m ust ensure th a t th e fllm rem ains w ith in th e viscous sublayer so we ta ke

a ~ / ~ T 3 => f <. h. (2.3.1)

F in a lly in o rd e r to keep surface te n sio n and g ra v ita tio n a l effects in th e p ro b le m fo rm u la tio n we ta ke

C h a p te r 2: B o u n d a ry la y e r on elongated obstacles

24

In accordance w ith th e estim ates above we in tro d u c e new variables,

Y = X = L x ,

f = h = L ^ / % a = L ^ / %

7 = X7, Ft = ^ , A = - L ' ^ h [ x ) + L~^ À { x ) . (2.3.3)

T h e flo w scheme is show n in flg 2 :2 (a ) and th e governing equations expressed in the new variables are

w ith th e b o u n d a ry c o n d itio n s

ÿ 00 : Ü + = ÿ - â y U , + À { x ) + o { l ) , (2.3.6)

ÿ = 0 : û~ = 0, v~ = 0, (2.3.7)

x - > - o o f (2.3.8)

u - = X~y,

and also a t ÿ = f { x )

S+ = « - , 5 + = î - = ü ± / î ( x ) , ^ = (2.3.9)

p+ - p - = j h î - h { p - - 1)1 Ft, (2.3.10) (2.3.11)

7T X - S

T h e re is no pressure/ displacem en t re la tio n now w ith the in te ra c tio n c o n d itio n re­ placed b y a given pressure re la te d to th e surface roughness.

F o r c o m p u ta tio n a l purposes th e pressure-hum p shape re la tio n (2,3.11) is r e w r it­ te n ta k in g F o u rie r tra n s fo rm s ,

/

C h a p te r 2: B o u n d a ry la y e r on elongated obstacles

25

so th a t

^ ^ / e'kc I I 3 ^ { h{ x) ) ] dk. (2.3.13)

dx I'K y_oo

F or th e pressure g ra d ie n t in the film th e re la tio n (2.3.10) can be d iffe re n tia te d and th e d e riva tive s o f h c a lcu la te d e x p lic itly fo r a chosen roughness.

2.3.1

T h e num erical m eth o d

N u m e ric a l s o lu tio n s are o b ta in e d to th e problem s setup in §2.1 and §2.3, by m a rch in g in th e a p p ro p ria te c o -o rd in a te x o r ï , using ite ra tio n s at each stream w ise s ta tio n . We o u th n e th e n u m e ric a l m e th o d in term s o f the variables in §2.3, however the rescalings ta k e th e same fo rm fo r b o th problem s, replacing aU variables x , . . . w ith X , ... and s e ttin g h = 0, w ith the o n ly difference in n u m e rica l representations being th e fa r-fie ld b o u n d a ry con d itio n s.

To c o n s tru c t th e a c tu a l so lu tio n we rescale the n o rm a l co o rd in a te in the film and m ake a fu r th e r P r a n d tl sh ift in th e b o u n d a ry layer w ith respect to th e u n kn o w n in te rfa c e p o s itio n . In th e b o u n d a ry -la y e r equations (2.3.4), (2.3.5) we w rite

(7+ = V + = Ü+ — ü " ^ ^ ( x ) , X = X, (2.3.1.1a)

P + = p + ,

r + = ÿ - / ( x ) ,

F = f,

(2.3.1.1b)

w h ich yields th e e quatio ns, vaHd fo r > 0,

^ + . « P .3 .1 .3 )

w ith b o u n d a ry c o n d itio n s

y + - o o : U+ = Y + + U , - â + À { X ) + F ( X ) + o { l ) (2.3.1.4)

r + = 0 : 7 + = 0 (2.3.1.5)

X - 0 0 : U + = Y + + U „ F (2 .3 .1 .6 )

C h a p te r 2: B o u n d a ry la y e r on elongated obstacles

26

T h e film re g io n is m apped o n to th e fin ite s trip 0 <

Y~

Y~

U~

f { ^ )

V - = V ~

P-=r,

F = f ,

w h ich give th e equations

and b o u n d a ry c o n d itio n s

Y ~ = 0 : U - = V ~ = 0,

y - = 1 : y - = 0, (2.3.1.11 )

X - ^ - o o :

U

UsY~.

T h e in te rfa c ia l c o n d itio n s o f c o n tin u ity o f ta n g e n tia l v e lo c ity and n o rm a l ve lo city, th e ju m p in pressures and shears and the viscous-inviscid in te ra c tio n c o n d itio n , w h ich becomes a given pressure re la tio n , are th e n expressed as

u + { Y + = 0) = u - { Y - = 1), ^ ( y + = 0) = ^ ^ ( y - = i ) , (2.3.1.13)

y + ( y + = 0) =

V - { Y ~

d P + d P - _ d ? H { p - - l ) d B

d X d X ' d X ^ Ft d X ' (2.3.1.15 )

P + = - - r (2.3.1.16 )

7T / __ _{— 00} X — S

where h [ x ) = H { X ) .

C h a p te r 2: B o u n d a ry la y e r on elongated obstacles

27

0. T h e o n ly differences, fo r nu m e rica l purposes, are the b o u n d a ry c o n d itio n s , (2.1.3), (2.1.6), w h ich become

00 : £/■+ -^ 1, (2.3.1.17)

X = 0 : U-^ = Ub( Y + ) , U ~ = J Y - ( l - Y - ) , F = a (2.3.1.18)

w here J = d?J.

A th re e -p o in t b a ckw a rd difference is used fo r derivatives and a tw o -p o in t c e n tra l difference fo r th e n o rm a l d ire c tio n :

d i { X ) U { X ) - A i { X - A X ) -h i { X - 2 A X )

d X 2 A X

a e ( y ± ) ^ ( y ± -h A y ± ) - ^ ( y ± - A y ± )

(2.3.1.19)

(2.3.1.20)

5 y ± 2A y ±

w here ^ is a re p re se n ta tive fu n c tio n .

P ro v id e d th e so lu tio n is know n at X — 2 A X and X — A X the m o m e n tu m equa­ tio n s a t th e n e xt X - s ta tio n are w r itte n in the fo rm

= d f (2.3.1.2 1)

w ith th e coefficients given by

— P A T T - _ O T T - P

V - ^ A Y - jy- _ 3 U - ^ A Y - 2 u

-“1 -

-

2 AX

+

(2-3-l-22a)

V T ^ A y

-9 - (2.3 .1 .2 2 b )

. . ,2 / 1 d P - u f

d - = ( A y - ) ^ ^

,

y - + P A y +

^

2

^

C h a p te r 2: B o u n d a ry la y e r on elongated obstacles

28

where the superscripts p, 2 ,1 and su b scrip t j represent, respectively, th e p re d ic to r value fo r th e cu rre n t X - s ta tio n , the value at the X — A X and a t X — 2 A X and the p o sitio n . O m ittin g references to th e specific flu id layer, to ca lcu la te th e new values fo r U, V, F , th e fu n c tio n is w r itte n in the fo rm

U j ^ i = P j U j + qj (2.3.1.23) w hich upon s u b s titu tio n in to (2.3.1.21 ) gives

^

From th e b o u n d a ry co n d itio n s, (2.3.1.10 ), (2 .3 .1 .4 ), along w ith th e c o n d itio n o f c o n tin u ity in at the in te rfa ce we fin d

Pj max- - 1 ~ ^ j m a x - - l ~ — 0, U}' = U^nt’ (2.3.1.25) For the p ro b le m in b o u n d a ry layer flo w in §2.1 the fa r-fie ld co n d itio n s re q u ire

P^Tnax+-l ~ (2.3.1.26a)

and fo r th e trip le deck p ro b le m described in §2.3 we have

^ ^ a x + - l “ ^t ma x +- l ~ > (2.3.1.26 b)

where th e subscripts j m a x ~ ^ , j m a x ~ , i n t refer to values being ta ke n a t th e fin a l points in Y + , Y " and a t th e in te rfa ce , respectively. is th e n fo u n d usin g th e in ­ com pre ssib ility condition s (2 .3 .1 .9 ), (2.3.1.3) w hich have also been discretized using the tw o- and th re e -p o in t difference fo rm s (2.3.1.19 ), (2.3.1.20).

To begin th e so lu tio n procedure, guesses are m ade fo r th e in te rfa c ia l ve lo c ity and p o s itio n Uint, F at th e new X —s ta tio n , to g e th e r w ith a p re d ic te d v e lo city d is trib u tio n across th e flo w . T h e n the re la tio n s (2.3.1.21) -(2 .3 .1 .2 5 ) w ith the a p p ro p ria te fa r-h e ld c o n d itio n , (2.3.1.26a) or (2.3.1.26 b) are used to calculate the ’c o rre c to r’ velocities T h is procedure is carried o u t w ith th re e pairs o f in itia l guesses fo r th e in te rfa c ia l speeds and p o sitions. T w o fu n c tio n s representing in te rfa c ia l b o u n d a ry co n d itio n s, A and B defined as

C h a p te r 2: B o u n d a ry la y e r on elongated obstacles

29

are used to ca lculate c o rre c to r values fo r Uint, F w ith tw o -p o in t N e w to n ite ra tio n s aim ed a t s a tis fy in g th e co n d itio n s A = 0, J9 = 0 and th e w hole procedure is ite ra te d u n til a convergence c rite rio n ,

A U < €

A F C e i ,

(2.3.1.28)

is satisfied, w here ei is a chosen tole ra n ce , ty p ic a lly o f 0 (10“ ®).

2.3.2

R esu lts

H a vin g estabhshed a credible m odel fo r th e tw o -flu id flo w , w ith in th e film and fo r a com parable d e p th w ith in th e b o u n d a ry la ye r flu id , we lo o k at th e various param eters w h ich influence th e onset o f the G o ld s te in s in g u la rity /flo w b re a kd o w n fo r flo w over an obstacle. F or th e purposes o f th e n u m e rica l calculation s th e obstacle was defined to be

H { X ) = hoe~^^, -o o < X < oo.

(2.3.2.1)

T h e firs t and m ost obvious p a ra m e te r is /iq, the h u m p size coefficient. The v e lo c ity p ro file in th e film can be w r itte n in the fo rm

U ~ = X ~ Y ~ + h o t / - , (2.3.2.2) and since th e flo w breaks dow n w here d U ~ I d Y ~ —> 0, we expect th e he ig h t Hq to be im p o r ta n t, especially close to th e w a ll where the c o rre c tio n Ü~ is h ke ly to have its greatest influence. L o o k in g at o u r figures 2:2 and 2:3 we see th a t th e slope o f the skin fr ic tio n approaches th e G oldstein s in g u la rity , th ro u g h th e m a rg in a l s in g u la rity where d U ~ / d Y ~ {0) —> 0, as th e obstacle h e ig h t, |/io|, is increased. G raphs o f th e co m p a ra tiv e h u m p effects are shown in figs 2:2(b )-(e ), fo r a system o f w a te r in the film and an equal m ix tu re o f sihcone o il V 2 and 1 -2 -3 -4 -te tra h yd ro n a p h ta le n e in the b o u n d a ry la ye r, w ith th e param eters ta ke n fro m P ouhquen, C hom az & H uerre (1994) as an exam ple o f a real d y n a m ic a l system , and in Figs 2 :3 (a ),(b ) fo r a homogeneous system . T h e o th e r fa c to rs w h ich can affect th e onset o f th e m a rg in a l s in g u la rity are,

Chapter

30

S ta rtin g w ith th e effect o f th e d e n sity ra tio s on flo w separation , i t was found th a t i f th e film flu id is less dense th a n th a t in th e b o u n d a ry layer th e n separation is re ta rd e d , w h ils t denser film s cause sep a ra tio n fo r sm aller obstacles. T h is is because the shear is reduced in a denser film flu id and vice versa and so a film co a tin g o f a less dense flu id w ill cause an o therw ise separating flo w to re m a in a ttached .

In tro d u c tio n o f surface tension in to th e system appears to enhance flo w reversal as does an increased g ra v ita tio n a l influence, F r ^ oo. As w ith density, less viscous flu id s in th e film do n o t cause separated flow s fo r th e same obstacle h eight as the equivale nt hom ogeneous system , w h ils t m ore viscous fluids in th e film have the o p posite effect, enhancing separation .

T h e fin a l p a ra m e te r, th e no n -d im e n sio n a l film thickness, does n o t appear to affect th e b e h a v io u r o f th e system in te rm s o f flo w reversal, at least n o t fo r the chosen values o f â, a lth o u g h i t m u st be rem em bered th a t th e a ssum ption has been m ade in th e scaling o f th e p ro b le m th a t th e film thickness rem ains w ith in th e viscous sublayer o f th e tr ip le deck fo rm u la tio n . N o in te rn a l separation o f th e ty p e fo u n d in Sychev (1 9 8 0 ), E llio t t , S m ith , C ow ley (1993), T im o s h in (1996) was encountered in th e cases s tu d ie d here, w ith aU failu re s o f th e n u m e rica l m e th o d , i.e. th e occurrence o f sin g u la ritie s , b eing caused by zero waU shear.

Leavin g se p a ra tio n aside, and c o n c e n tra tin g on flow s whose stream w ise v e lo city profiles r e tu rn to th e ir o rig in a l lin e a r fo rm fa r dow nstream , we tu r n to th e graphs o f the displacem ent fu n c tio n À { x ) and th e skin fr ic tio n fo r a given obstacle and tw o flu id system . Figs 2 :2 (b ),(e ) and 2 :3 (a ),(b ) show in te rva ls o f x w ith decreasing w a ll shear b u t increased displacem ent and hence th e lik e lih o o d o f in fle x io n p o in ts developing in th e v e lo c ity p ro file s fig 2 :4 (a ),(b ), w h ich w ill fa c ilita te R ayleigh in s ta b ility . T h is w in be exam ined in th e n e x t section.

2.4

Inviscid in stab ilities in film -coated fiows

C h a p te r 2: In v is c id in s ta b ilitie s in film -c o a te d fiow s

31

layer flo w over an obstacle on th e trip le -d e c k scale, w ith L representing the obstacle le n g th , th e n th e new d is tu rb a n c e lengthscale scale is ta ke n as

P ^ ^ R e ~ < < < 2 '/ ^ (2.4.1) i.e. sh o rt com pared to th e tr ip le deck scale b u t a t least as lo n g as the c h a ra cte ristic

R ayleigh scale F ir s t we exam ine th e case <C fro m w h ich

we o b ta in a long-w ave (in te rm s o f the film thickness) in te g ra l co n d itio n fo r the d istu rb a n ce phase speed c. We th e n exam ine disturbances w ith = R e ~ î L ^ ^ ^ ,

v a ria tio n s o f th e pressure te rm in y th e n affect th e flo w and in s ta b ih ty is governed b y th e f u ll in v is c id R a yle ig h e q u a tio n .

We begin w ith an a n a ly tic a l e x a m in a tio n o f b o th regim es, solving fo r a s lig h tly p e rtu rb e d tw o shear stream w ise lin e a r v e lo c ity p ro file , such as th a t generated by the flo w over a sh a llo w obstacle, to o b ta in th e d istu rb a n ce g ro w th ra te expHcitly. T h e results are th e n com pared w ith those o b ta in e d n u m e ric a lly using a discrete ite ra tiv e m e th o d fo r th e f u ll R a yle ig h p ro b le m , as o u th n e d in a subsequent section, on th e profiles c a lcu la te d in §2.3.2.

2.4.1

T h e long-w ave in sta b ility

We in tro d u c e sm a ll te m p o ra l and s p a tia l wave p e rtu rb u rb a tio n s , 0(<?), to th e veloc­ itie s and pressure fields and define the wave as

E = exp i k ( æ — -— t (2.4.1.1) W it h th e g o v e rn in g equatio ns given by (2 .3 .4 )-(2 .3 .1 1 ) we w rite th e velocities and pressure as

= U ^ { ÿ ) + 6 { ü f E + c . c . ) + ... (2.4.1.2)

= ... H—-— E -{- c.c.) + ... (2.4.1.3)

Li n V

6 { p f E C . C . ) + ... (2.4.1.4) w h ich lead to th e re la tio n

C h a p te r 2: In v is c id in s ta b ilitie s in him -co a te d hows

32

T h e b o u n d a ry co n d itio n s a t the w a ll, in fin ity and an in te rfa c ia l c o n d itio n re la tin g th e n o rm a l v e lo c ity and pressure in th e film to th a t in the b o u n d a ry la ye r com plete th e p ro b le m fo rm u la tio n :

V (2/ = 0) = 0 , - g ^ i y = 00) = 0 (2.4.1.6)

and at y = f

( [ , . _ , ) ( ( W - 1" * - ‘ K . ) I . # . , .

\ - p ' K B r t f . - - ( y - /

w here 7 = 7/ Z ^ . W e solve th is and fin d a general s o lu tio n in the fo rm

5+ = Q t { U + - c) £ - 0) (2 4.1.8)

^ 1

V - = Q - ( U - - c) ^ _ _ dg + Q ^ { U - - c) (2.4.1.9)

T h e b o u n d a ry c o n d itio n s (2 .4 .1 .6 ) force = 0, Q2 = 0, fro m th e in te rfa c ia l

c o n d itio n (2 .4 .1 .7 ) we o b ta in th e re la tio n

Q Z = P ~ Q ï - ^ T c ’ (2.4.1.10)

and th e n o rm a liz a tio n c o n d itio n V i ( f ) = 1 gives

Q l — ~ Jo { U ~ — (2.4.1.11)

C o n tin u ity o f v a t th e in te rfa ce th e n gives th e dispersion re la tio n

2.4.2

R ayleigh -w ave in sta b ility

C h a p te r 2: In v is c id in s ta b ilitie s in film -c o a te d flows

33

We s ta rt by ta k in g an a n a ly tic a p p ro x im a tio n to the stream w ise base v e lo c ity profiles U ( ÿ ) in (2.4 .1.2), w r itin g th e m as piecewise-hnear profiles w ith a sm a ll co r­ re c tio n 0 ( e ) , here p ro p o rtio n a l to base flo w d e parture fro m lin e a rity and hence th e hum p size ho, where d <C e <C 1. A s im ila r analysis applies to th e enhanced m ean flo w profiles exam ined in C h a p te r 4. W e also expand th e in te rfa c ia l p o s itio n in powers o f e, and w rite

(7+ = ÿ - â + U , + eG+( ÿ) , (2.4.2.1)

U ~ = A - ÿ + e G - ( ÿ ) , (2 ,4 .2 .2 )

f — c - j - c f i .

(2.4.2.3)

Here d is th e u n p e rtu rb e d in te rfa ce p o s itio n , A“ = l / p ~ , U a = A“ a a n d we n o rm a lize the flo w such th a t at th e in te rfa ce û ^ ( ÿ — f ) = Ug -{■ e.

T he shortened lengthscale, com pared w ith th a t o f §2.4.1, leads to th e fuU in v is c id R ayleigh equ a tio n fo r th e n o rm a l velocities v ^ ,

(Cr± - c )(ü ± J - k ^ v * ) = U ÿ ÿ ^ v f (2.4.2.4)

w ith b o u n d a ry co n d itio n s

v ~ ( y = 0) = 0 , v ' ^ (ÿ = 00) = 0. (2.4.2.5)

E x p a n d in g v i and c in powers o f e

Vi = Vq "b •••> c = cq "1“ eci -j- ..., (2.4.2.6)

and, s u b s titu tin g in to (2.4.2.4), we fin d the solutions

= (2.4.2.T )

We take these solu tio n s, w h ich satisfy th e b o u n d a ry co n d itio n s (2 .4 .2 .5 ) and p u t th e m in to th e in te rfa c ia l c o n d itio n (2.4.1.7). L in e a riz in g and ta k in g te rm s 0 ( 1 ) gives a dispersion re la tio n fo r th e leading order phase speed c q,

C h a p te r 2: In v is c id in s ta b ilitie s in film -c o a te d flows

34

w h ich has th e so lu tio n

1 - p A ± \ / ( l - p - A - ) ^ + 4 7*3(1 + p - cothfcS ) /o . o 01

2 fc (l + p - c o t h i 5 ) ■ ^

We n ote th a t, fo r a ll p o sitive wavenum bers

k, cq is

C o = U. o r C o = ! 7 , + ^ ' (2.4.2.10)

T h e g ro w th ra te w ill be fo u n d fro m c i. T h e n e xt o rder te rm s, 0 ( e ) , in (2 .4 .2 .4 ) give

(£1± - c o ) ( v 4 - - 1 : V ± ) = Fo±, (2.4.2.11 )

and we lo o k fo r a so lu tio n o f th e fo rm V- ^{ ÿ ) = F ^ { ÿ ) V ^ { ÿ ) . T h e s o lu tio n now depends on th e p o s itio n o f a c ritic a l layer w h ich fo rm s at ÿ = ÿc w here U{ ÿc) = cq.

We w ill fir s t pursue the case o f a c ritic a l layer in the film . We have fo r ÿ < ÿc,

V f = Vo / ; ^ V ; (2.4.2.12 )

fo r ÿ > ÿc,

= V

r

A I r

r

dye V

[dye

^

^0

J

dye: V

•• G +,Vq^ '

0

ÿ - â + u ,

,

^

.0

w here th e 6i ’s are constants o f in te g ra tio n . T he so lu tio n fo r V { ' contains a te rm o f the fo rm

V f

=

...,

(2.4.2.15)

and as ÿ —> ÿc

C h a p te r 2: In v is c id in s ta b ilitie s in film -c o a te d flows

35

so B = G ^ ÿ { ÿ c ) V ^ { ÿ c ) l ^ ~ and ÿc = c q/ \ ~ . T h e ju m p in th e d e riv a tiv e o f the

n o rm a l v e lo c ity across th e c ritic a l layer is p ro p o rtio n a l to th e lo g a rith m ic te rm in (2 .4 .2 .1 5 ),

y { ÿ { v t ) - y i ÿ i ÿ c ) = (2.4.2.17) w here ÿ f denotes th e lim it value taken as y ÿc above o r below th e c ritic a l layer. T h e re m a in in g b o u n d a ry condition s are th e n o rm a liz a tio n (2 .4 .1 .6 ) a t th e in te rfa ce , th e c o n tin u ity in th e n o rm a l v e lo c ity across the c ritic a l layer, i.e.

F ± ( a ) = -/iV o $ (â ), V f (

5

and th e pressure ju m p across the in te rfa ce w r itte n as (2.4.1.7).

O u r a im is to fin d th e im a g in a ry p a rt o f Ci so we need co n ce n tra te o n ly on th e im a g in a ry p a rts o f th e re la tio n s (2.4.2.18). S olving fo r Vi,- using th e b o u n d a ry c o n d itio n s (2 .4 .2 .1 7 ), (2.4.2.18 ), s u b s titu tin g in to (2 .4 .1 .7 ), and ta k in g th e im a g in a ry te rm s 0 (e) we fin d

^ ^ _____________f>_(t^ » -C o )V G j,j,(ÿ c )s in h ^ fc ÿ c _____________ (2.4.2.19 )

X~ sinh k a[ { Ug — co)^A:(sinh A:â + p~ cosh A:â) — sinh k à )

In th e case o f th e c ritic a l layer o c c u rrin g in the b o u n d a ry la ye r we fin d

{Us - co)^7rO ^(ÿc)e^(°"^=)

Cli = 7— T T T :---— --- r-7T I---ITTTT^. ., , , 2

(sinh k{yc - a) + cosh k{yc - a ) ) { k { l + p~ c o thka){Ua - cq) - 7/2^)

(2.4.2.20)

T a k in g th e lo n g wave lim it A; —> 0, th e im a g in a ry p a rt o f th e co m p le x wave speed is given b y th e fo rm u la

C i i i f < a, (2.4.2.21)

C l i ~ ' ^1 i f > a. (2.4.2.22)

p a ^

W e see th a t p o s itiv e c u rv a tu re a t th e c ritic a l h e ig h t ÿ = ÿc provokes in s ta b ility i f

C h a p te r 2: In v is c id in s ta b ilitie s in h lm -co a te d Sows

36

We also exam ine the sh o rt wave h m it A; oo o f (2.4.2.19 ), (2 .4 .2 .2 0 ) to see i f in s ta b ility persists. For b o th th e case o f th e c ritic a l la ye r in th e film and th e c ritic a l layer in th e b o u n d a ry la ye r flo w we see th a t surface tension becomes th e d o m in a n t effect fo r s h o rt waves and the d istu rb a n ce is s tro n g ly stabihzed. I f th e re is no surface tension c ^ —^ O'*' as A: —> oo.

2.4.3

T h e num erical R ayleigh in sta b ility calculation

Here we solve th e p ro b le m n u m e ric a lly fo r th e b o u n d a ry la ye r and film flows w ith b o u n d a ry c o n d itio n s (2 .4 .1 .6 ), (2 .4 .1 .7 ) w ith th e v e lo c ity p ro file , ü ^ , and in te rfa c ia l p o s itio n / ca lcu la te d using th e n u m e ric a l m e th o d o f §2.3.1. O u r n u m e ric a l m e th o d uses th e in v is c id R ayleigh e q u a tio n (2.4.2.4) re w ritte n w ith th e second o rd e r deriva­ tiv e in th e n o rm a l v e lo c ity in a c e n tra l difference fo rm ,

+ a j V p + v g t i = 0, (2.4.3.1)

where

a j = - { 2 + j , (2,4.3.2)

w ith th e base v e lo c ity profiles w r itte n in term s o f th e o rig in a l v e rtic a l co -o rd in a te

ÿ, subscripts j correspond ing to th e discrete ÿ p o s itio n and prim es d e n o tin g d iffe r­ e n tia tio n w ith respect to ÿ. We w rite

= P i (2.4.3.3)

w hich, u p o n s u b s titu tio n in to (2 .4 .3 .1 ), gives us the fo rm u la e

P ? + i - p f + a f ’

±

w ith th e b o u n d a ry co n d itio n s (2 .4 .1 .6 ), (2.4.1.7) re q u irin g

Q2 — P t ~ ^m a x + ~ (2.4.3.5)