ADDITIONAL RESEARCH ON INSTABILITIES IN ATMOSPHERIC FLOW SYSTEMS ASSOCIATED WITH CLEAR AIR TURBULENCE

Transcription

1 ?wsw *>*> NASA CONTRACTOR REPORT M NASA CR-1985 ON ADDITIONAL RESEARCH ON INSTABILITIES IN ATMOSPHERIC FLOW SYSTEMS ASSOCIATED WITH CLEAR AIR TURBULENCE by Richard C. Steffler Prepared by UNITED AIRCRAFT CORPORATIOI "i East Hartfrd, Cnn. fr NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. APRIL 1972

2 1. Reprt N. NASA CR Title and Subtitle 2. Gvernment Accessin N. 3. Recipient's Catalg N. ADDITIONAL RESEARCH ON INSTABILITIES IN ATMOSPHERIC FLOW SYSTEMS ASSOCIATED WITH CLEAR AIR TURBULENCE 5. Reprt Date April Perfrming Organizatin Cde 7. Authr(s) Richard C. Steffler 9. Perfrming Organizatin Name and Address UNITED AIRCRAFT CORPORATION East Hartfrd, Cnnecticut 12. Spnsring Agency Name and Address Natinal Aernautics and Space Administratin Washingtn, D. C Perfrming Organizatin Reprt N. 10. Wrk Unit N. 11. Cntract r Grant N. NASW Type f Reprt and Perid Cvered Cntractr Reprt 14. Spnsring Agency Cde RAA 15. Supplementary Ntes 16. Abstract Additinal analytical and experimental fluid mechanics studies were cnducted t investigate instabilities in atmspheric flw systems assciated with clear air turbulence. The experimental prtin f the prgram was cnducted using the UARL Open Water Channel which allws investigatin f flws having wide ranges f shear and density stratificatin. The prgram was primarily directed tward studies f the stability f straight, stratified shear flws with particular emphasis n the effects f velcity prfile n stability; n studies f three-dimensinal effects n the breakdwn regin in shear layers; n the interactin f shear flws with lng-wavelength internal waves; and n the stability f shear flws cnsisting f adjacent stable and unstable layers. The results f these studies were used t evaluate methds used in analyses f CAT encunters in the atmsphere invlving wave-induced shear layer instabilities f the Kelvin- He lmhltz type. A cmputer prgram was develped fr predicting shear-layer instability and CAT induced by muntain waves. This technique predicts specific altitudes and lcatins where CAT wuld be expected. 17. Key Wrds (Suggested by Authr(s)) Clear Air Turbulence; Atmspheric Fluid Mechanics 18. Distributin Statement Unclassified - Unlimited 19. Security Classif. (f this reprtl 20. Security Classif. (f this pagel 21. N. f Pages 22. Price* Unclassified Unclassified 15 $3.00 Fr sale by the Natinal Technical Infrmatin Service, Springfield, Virginia i * 0 $<&sr-* j

3 FOREWORD Analytical and experimental fluid mechanics investigatins were perfrmed t investigate instabilities in atmspheric flw systems assciated with clear air turbulence. This wrk was a cntinuatin f investigatins reprted in NASA Cntractr Reprt CR-I60U, "Research n Instabilities In Atmspheric Flw Systems Assciated With Clear Air Turbulence," by J. W. Clark, R. C. Steffler, and P. G. Vgt (June 1970). The prgram was cnducted by United Aircraft Research Labratries under Cntract NASW-1582 with Natinal Aernautics and Space Administratin Headquarters, Washingtn, D. C, 205U6. The prgram was under the technical directin f the Chief, Aerdynamics and Fluid Dynamics Branch, Cde RAA. iii

4 TABLE OF CONTENTS Page RESULTS AND CONCLUSIONS 1 INTRODUCTION 3 STABILITY OF TWO-DIMENSIONAL, STRAIGHT, STRATIFIED SHEAR FLOWS HAVING "S-SHAFED" VELOCITY PROFILES 4 Review f Tw-Dimensinal Flws Having Hyperblic Tangent Prfiles... 4 Stability Criteria f Hazel fr "S-Shaped" Velcity Prfiles 7 Summary f Experiments with "S-Shaped" Velcity Prfiles 9 Cncluding Remarks 10 STABILITY OF "THREE-DIMENSIONAL", STRAIGHT, STRATIFIED SHEAR FLOWS 13 Summary f Experiments with "Three-Dimensinal" Flws 13 Cncluding Remarks 14 INTERACTION OF LONG-WAVELENGTH WAVES WITH TWO-DIMENSIONAL, STRAIGHT, STRATIFIED SHEAR FLOWS 15 Thery fr Wave-Induced Shear-Layer Instabilities 15 Summary f Wave-Induced Shear Layer Instability Experiments 17 Cncluding Remarks 20 APPLICATION TO ATMOSPHERIC SHEAR FLOWS 22 Review f the Fundamental Flw Phenmenn 22 Imprved CAT Predictin Mdel 25 Cncluding Remarks 30 REFERENCES 31 LIST OF SYMBOLS 34 APPENDIX I: METHOD FOR FITTING HAZEL'S THEORETICAL VELOCITY PROFILE TO DATA FOR "S-SHAPED" VELOCITY PROFILES 37 II: SUMMARY OF OTHER FLUID MECHANICS INVESTIGATIONS 39 FIGURES 45 v

5 RESULTS AND CONCLUSIONS 1. The water channel experiments t investigate the stability f straight, tw-dimensinal, stratified shear flws having "S-shaped" velcity prfiles cnfirmed that the shape f the velcity prfile affects the dimensinless wavenumber, ad*, f the instabilities which can ccur. The values f ad bserved in experiments with "S-shaped" prfiles were significantly larger than thse bserved previusly in experiments with hyperblic tangent prfiles. This result is in agreement with the theretical stability criteria f Hazel fr "S-shaped" prfiles and Drazin fr hyperblic tangent prfiles. 2. The experimental results and the theretical stability criteria fr twdimensinal flws were als in gd agreement regarding the critical value f Richardsn number. Bth "S-shaped" and hyperblic tangent prfiles are stable fr Richardsn numbers greater than Althugh the values f dimensinless wavenumber, ad, at which instabilities ccur at a Richardsn number f 0.25 a^ce larger fr "S-shaped" prfiles than fr hyperblic tangent prfiles, the actual wavelengths, A, are abut the same (assuming that the shear layer thicknesses, the velcity differences acrss the shear layer, and the mean shears at the center f the shear layers are apprximately equal). The thery indicates that the higher values f ad are due mainly t higher values f the parameter d necessary t describe "S-shaped" prfiles. The experimental results tend t cnfirm this. k. The preceding cnclusin als applies t the wavelengths f instabilities that can be expected when stable shear layers in the atmsphere are destabilized by lng-wavelength waves (i.e., wave-induced instabilities), such as muntain lee waves. If the shear-layer thickness, the velcity difference acrss the shear layer, and the mean shear at the center f the layer are fixed, then the wavelength f the instability ccurring when the Richardsn number decreases t 0.25 will be abut the same, whether the prfile is an "S" r a hyperblic tangent. This wavelength is clsely apprximated by the wavelength given by Drazin's thery, Ag = (2TT\2) *2d, where d is taken as half the shear-layer thickness. *The characteristic breakdwn flw pattern f Kelvin-Helmhltz-type instabilities cnsists f waves which develp int vrtices and turbulence. The wavenumber a is 2n\, where A is the wavelength f the instability; d is ne f several parameters which describe the velcity prfile (d is apprximately half f the shear-layer thickness fr hyperblic tangent prfiles, but this is nt a gd apprximatin fr "S-shaped" prfiles).

6 5. Water channel experiments and theretical studies were cnducted t investigate the stability f straight, stratified shear flws in which the width f the shear layer was frm abut h t ko times its thickness. Thus, these shear layers were "three-dimensinal" as ppsed t the usual tw-dimensinal shear layer which is assumed t extend unifrmly t infinity in the directin transverse t the flw. The results indicate that "three-dimensinal" flws are mre stable than twdimensinal flws. The measurements als indicate that, althugh the velcity prfiles were significantly different frm hyperblic tangent prfiles analyzed by Drazin fr tw-dimensinal flws, the wavelengths f the initial Kelvin-Helmhltztype instabilities that ccur are adequately predicted by his criterin. 6. Shear layer instabilities induced by lng^wavelength internal waves were als investigated in water channel experiments. Kelvin-Helmhltz-type waves which grew in amplitude and transitined t vrtices and turbulence were bserved in the thin shear layers. These disturbances mved at the mean flw velcity and were superimpsed n the statinary lng-wavelength internal waves. Occurrence f the instabilities was predicted quite well using the methd used previusly in this prgram t predict CAT resulting frm shear-layer instabilities induced by muntain lee waves. 7. Water channel experiments were cnducted t investigate the stability f cmbined shear layers cnsisting f adjacent stable and unstable layers. The results indicated that instability is initiated in the individual unstable layers at wavelengths that wuld be predicted by thery based n the thickness f these individual unstable layers. Hwever, in the final stages f vrtex grwth and turbulent breakdwn, the wavelength increases t abut that which wuld be predicted based n the thickness f the cmbined layer. Thus, in cmbined shear layers, small-amplitude, shrt-wavelength disturbances assciated with initial instabilities in individual layers can result in large-amplitude, lng-wavelength disturbances. 8. A methd was develped, and included in a cmputer prgram, fr predicting the ccurrence f CAT in muntain waves. The methd cmpares the amplitude f a lee wave required t destabilize an initially stable layer (i.e., t reduce the Richardsn number t 0.25) with a predicted lee wave amplitude. The layers are identified using rawinsnde data. Lee wave activity is predicted using the United Air Lines nmgram which is based n the sea level pressure difference between tw grund statins and maximum wind velcity assciated with the wave zne f interest. Lee wave amplitude is estimated frm a crrelatin f predicted lee wave activity (using the UAL nmgram) with wave amplitudes deduced frm recnstructed muntainwave flw fields. Turbulence is predicted at altitudes where the lee wave amplitudes required fr instability are less than the predicted lee wave amplitudes.

7 INTRODUCTION The general bjectives f the present prgram were: (L) t gain increased understanding f the nature and causes f turbulent atmspheric phenmena, particularly clear air turbulence; (2) t develp imprved criteria fr predicting neutrally stable states in atmspheric flw systems; and (3) t cmpare the results f this research with available meterlgical data and attempt crrelatins. The results f all fluid mechanics analyses and experiments cnducted under this prgram are reprted in Refs. 1 thrugh 5. The initial wrk (Refs. 1 and 2) prvided evidence that lng-wavelength waves, such as muntain waves, culd destabilize initially stable shear layers which ccur in the atmsphere. Since lng-wavelength waves may ccur quite ften in the atmsphere, the breakdwn f these layers culd accunt fr an appreciable fractin f CAT which is encuntered. The present reprt summarizes further research reprted in detail in Refs. 3 thrugh 5«The specific bjectives f the latter research were: (l) t cnduct further experiments n the effect f the shape f the velcity prfile n the stability f straight, twdimensinal, stratified shear flws; (2) t investigate the stability f "threedimensinal" flws by cnducting experiments in the UARL Open Water Channel n the stability f straight, stratified shear flws in which the width f the shear layer was less than the width f the channel; (3) t further investigate experimentally the interactin f lng-wavelength waves with stable shear layers; and (h) t develp a cmputer prgram which predicts shear-layer instability and CAT induced by muntain lee waves. Appendix I cntains the derivatin f equatins used t cmpare theretical with experimental "S-shaped" velcity prfiles. Appendix II cntains a discussin f an investigatin f the stability f shear flws cnsisting f adjacent stable and unstable layers and a descriptin f the methd used t btain a stability criterin fr "three-dimensinal", straight, stratified shear flws.

8 STABILITY OF TWO-DIMENSIONAL, STRAIGHT, STRATIFIED SHEAR FLOWS HAVING "S-SHAPED" VELOCITY PROFILES The primary purpse f this part f the fluid mechanics prgram was t investigate the effects f the shape f the velcity prfile n the stability f straight, tw-dimensinal, stratified shear flws. First, experiments were cnducted t btain data n the stability f flws having "S-shaped" velcity prfiles; these data were als cmpared with theretical stability criteria. The results were then cmpared with results frm an earlier investigatin f the stability flws having hyperblic tangent prfiles. The prgram was directed tward three areas: (l) identifying the cnditins under which such flws becme unstable, (2) determining the characteristics f the flw during the initial phases f breakdwn, and (3) evaluating existing theretical stability criteria fr subsequent use in studying atmspheric shear flws. Review f Tw-Dimensinal Flws Having Hyperblic Tangent Prfiles The stability f shear flws having hyperblic tangent velcity prfiles was investigated in detail in earlier wrk under this prgram (Refs. 1 thrugh h). Flws f this type were studied using the UARL Open Water Channel, shwn in Fig. 1. This facility prvides a 2-ft-wide by 10-ft-lng by 12-in.-deep, nn-recirculating, pen channel flw. Filter beds made frm prus fam material are used t intrduce desired vertical and transverse velcity prfiles in the flw. Ht^water nzzles in the plenum are used t intrduce vertical temperature gradients and, hence, density stratificatin. Dye tracing and hydrgen bubble wire techniques are used fr flw visualizatin and fr measurement f the inlet velcity prfile; standard submersible mercury thermmeters were used t measure temperatures. This facility and its assciated instrumentatin are described in detail in Refs. 1 and 2. Velcity prfiles apprximating hyperblic tangent prfiles were btained by shaping the fam material in the filter bed as shwn in «n Sketch A. The stability f flws having such prfiles was studied theretically by Drazin (Ref. 6) and thers. The hyperblic tangent velcity prfile which mst clsely apprximates the prfiles in the channel is given by FILTER BED VELOCITY ] PROFILE, V (dvdz), SKETCH A. FILTER BED AND VELOCITY PROFILE FOR HYPERBOLIC TANGENT VELOCITY PROFILE

9 V = v 0 + f- tanh (^ ) (!) where V is the lcal velcity at height z abve the channel flr; V Q = (V^ + VgOA? is the velcity at the center f the shear layer at height z Q ; AV = (Vn - v>>); and d = (AV)2((9V(?z) 0. The parameter d is a scale length and is apprximately half the thickness f the shear layer. Drazin als used an expnential variatin f density with height: Ri-d-((3vdz) z-z 0 PP 0 -e *m s - RI T z) m <*> where Ri is the Richardsn number and g is the gravitatinal cnstant. Since the change in density acrss the shear layer is small, a gd apprximatin t Eq. (2) is pp = l _ Ri-diav^l (^ (3) Drazin derives a criterin fr stability in Ref. 6 by intrducing a perturbatin stream functin <' = <f>(z) e< a( *- ct) (I*) int the equatins gverning the mtin f the fluid. Here, a is the wavenumber, a= 2-nx, and c is the cmplex wave velcity, c = c r + i«cj_. The equatins f mtin then yield a single stability equatin. Making use f the fact that the perturbatins neither amplify nr decay when c^ = 0, Drazin slves fr the fllwing equatin fr neutral stability n the ad - Ri plane: ' d -VlVp (5)

10 Sketch B shws this bundary which separates stable and unstable regins. The criterin indicates that the flw wuld be stable fr disturbances f all dimensinless wavenumbers,ad, fr Ri > Fr Ri < 0.25, the flw wuld be unstable fr dimensinless wavenumbers which lie inside the bundary and a Kelvin-Helmhltz (K-H) type f instability wuld ccur. NEUTRAL BOUNDARY Ri = 0.25 ad = A Ri SKETCH B. DRAZIN'S NEUTRAL STABILITY BOUNDARY FOR HYPERBOLIC TANGENT PROFILES The results f UARL Open Water Channel investigatins f the stability f flws having hyperblic tangent velcity prfiles are summarized and cmpared with Drazin's theretical criterin in Fig. 2. Fr each flw cnditin, the Richardsn number was calculated using the slpes (dvdz) Q and dldz frm the measured prfiles. The scale length, d, was calculated using the slpe (dv<9z) 0 and the velcity difference AV frm the velcity prfile; Av was based n the maximum and minimum velcities in the vicinity f the edges f the shear layer. The wavenumber, a = 2TTA, f instabilities bserved in the shear layer was calculated using wavelengths determined frm phtgraphs f dye traces. Thus, each flw cnditin at which waves were bserved is identified by a pint n the plt f ad vs Ri. The symbls in Fag. 2 dente different flw characteristics that were bserved. The pen circle symbls dente cnditins at which nly Kelvin-Helmhltz-type waves were bserved in the shear layer; that is, the waves extended the entire length f the channel withut breaking dwn. The wavelengths f these waves ranged frm abut 3 t 6 in. The pen circle symbls with flags indicate the nature f the disturbances bserved fr example, small-amplitude waves which persisted, waves which seemed t grw in amplitude t a certain pint and then nt grw further as they prgressed dwnstream, and waves which appeared in the flw nly intermittently. The half-slid symbls dente flw cnditins in which the waves transitined t vrtices but did nt transitin t turbulence befre reaching the dwnstream end f the channel, The full-slid symbls dente flw cnditins at which the full sequence f events assciated with cmplete shear-layer breakdwn ccurred waves, vrtices, and turbulence. The crsses indicate cnditins at which n waves f the type assciated with instability ccurred.

11 Examinatin f Fig. 2 indicates that mst f the bservatins are in gd agreement with Drazin's bundary. All cases in which full transitin was bserved fall in the unstable regin. Six cases in which waves were bserved fall in the stable regin. The intermittent small-amplitude waves indicated at Ri = 0.^3 and steady small-amplitude waves at Ri = O.38 were unexpected; in subsequent tests at apprximately the same cnditins, n waves were bserved. Fur cases were bserved which fall abve the bundary but at Ri<0.25. These fur cases were, at the time f the tests, suspected t be attributable t differences between the experimental velcity prfile and Drazin's hyperblic tangent prfile. This hypthesis was based n a theretical study by Hazel (Ref. 7) which shwed that instabilities assciated with flws having "S-shaped" velcity prfiles culd have dimensinless wavenumbers greater than 1.0. Stability Criteria f Hazel fr "S-Shaped" Velcity Prfiles "S-shaped" velcity prfiles are develped in the Water Channel by shaping the prus fam filter bed as shwn in Sketch C. VELOCITY PROFILE, V SKETCH C. FILTER BED AND VELOCITY PROFILE FOR "S-SHAPED" VELOCITY PROFILE The velcity prfile used by Hazel in this theretical study is AV V = V 0 +~2~ 5ech> (i^). fnh (^,) (6) where b is an expnent that affects the shape f the "S" and, as befre, d = (AV)2(<9Vaz) 0. The relatinship f the velcity difference AV t (Y 1 - V 2 ) and b is (see derivatin in Appendix i)

12 b»i 2 Av = (b + 0 (V, - V? ) (7) The theretical density prfile used by Hazel is P 2 Ri- d JdVdA tanh (^-) (8) Hazel derives criteria fr stability in Ref. 7 fr several velcity and density prfiles. This is dne by slving (using numerical techniques) a differential equatin which is satisfied by ne Furier cmpnent f the velcity perturbatin fr a plane, tw-dimensinal, Bussinesq shear flw. The full perturbatin velcity is given by w (x.z.t) = J-m w(z) la(x-ct) da (9) Like Drazin's criterin, Hazel's criteria (see Sketch D) indicate that the flw wuld be stable fr disturbances f all dimensinless wavenumbers fr Ri > Fr Ri < 0.25, the flw wuld be unstable fr dimensinless wavenumbers which lie inside the bundaries. The lcatins f the neutral stability curves n the ad - Ri plane are dependent upn the value f the expnent b (see sketch). ad b INCREASING STABLE NEUTRAL BOUNDARY FOR FIXED VALUE OF b SKETCH D. HAZEL'S NEUTRAL STABILITY BOUNDARIES FOR "S-SHAPED" PROFILES

13 Summary f Experiments with "S-Shaped" Velcity Prfiles Characteristics f Breakdwn f Flw Figure 3 illustrates the stages bserved as the flw in the shear layer breaks dwn. The breakdwn characteristics were very similar t thse fr hyperblic - tangent-type flws. There are fur very distinct and repeatable stages which ccur. The phtgraphs in Fig. 3 were taken thrugh the lucite channel side wall with the flw frm left t right. The scale in the phtgraphs was in the flw clse t the dye traces. In Fig. 3(a)j the flw appears undisturbed. In Fig. 3(b), 28 in. further dwnstream, the center dye trace indicates the presence f a K-H wave amplifying as it prgresses dwnstream. The wave has a wavelength f abut A = 5 in. and an amplitude (half the distance frm trugh t crest) f abut a = 0.25 in. at this pint. By placing dye traces at several transverse lcatins acrss the channel, it was verified that the flw was apprximately tw-dimensinal, i.e., the wave extended acrss the channel. In Fig. 3(c), anther 2k in. dwnstream, the waves have rlled up int vrtices. The circulatin f the vrtices has the same sense as the vrticity intrduced by the shear the shear is negative in this flw cnditin, and all f the vrtices rtated cunterclckwise. These vrtices grew slightly in size as they drifted dwnstream. Their dwnstream drift velcity was apprximately V 0, the velcity upstream at the center f the shear layer. The flw was als tw-dimensinal at this stage. In Fig. 3(d), anther 10 in. dwnstream 70 in. dwnstream f the filter bed the vrtices have "burst" and the flw appears turbulent. The fluid mtins were three-dimensinal at this stage. Velcity, Temperature, and Density Prfiles Velcity, temperature, and density prfiles fr three flw cnditins are shwn in Fig. h. These data are fr three different Richardsn numbers and three different values f the expnent b_used in Hazel's theretical velcity prfile. The crrespnding velcity and density prfiles in Hazel's thery are shwn by the dashed lines. Once a velcity prfile had been measured, the value f the expnent b was derived s as t prvide a reasnably gd match between theretical and experimental velcity prfiles. The theretical prfile was chsen by matching t the data the fllwing (see Appendix I and Sketch c): (l) the velcity difference, V-^ - V2, (2) the vertical distance between V^ and V2, z^ - zg, and (3) the mean velcity gradient, (av3z) 0, at z Q. The crrespnding theretical density prfile was chsen by matching the density gradient and mean density f the experimental prfile at the center f the shear regin.

14 Cmparisn f Experimental Results with Hazel's Theretical Stability Criteria Figure 5 is a summary f the results and a cmparisn with Hazel's theretical criteria. As in Fig. 2, the symbls dente different flw characteristics that were bserved. The values f the expnent b which prvide the best match between the theretical and experimental velcity prfiles are given next t the symbls. Mst f the experimental results are in agreement with Hazel's stability bundaries. Fr all cases in which instability was bserved, the Richardsn number was less than 0.25, and nly ne stable case was bserved fr which the Richardsn number was less than 0.25 («d = 0, Ri = 0.2). Fr all but ne f the cases fr which instability was bserved, the dimensinless wavenumbers, ad, were greater than 1.2. This is in cntrast t the results btained fr hyperblic tangent velcity prfiles (see Fig. 2) where the ad's assciated with instabilities were less than 1.2 fr all but ne case. Hazel's thery shws that as the value f b increases, the range f wavenumbers, ad, f instabilities which can ccur als increases. The data appear t cnfirm this trend, althugh insufficient data were btained t make detailed cmparisns with Hazel's criteria fr lw values f b. Only tw cases are nt in agreement with Hazel's neutral stability bundaries. They are (l) Ri =0, ad = 2.U, b = 0.8, and (2) Ri =0, ad = 2.07, b = 0.3. Hwever, these values f b are questinable since the agreement between the theretical and experimental velcity and density prfiles was cmparatively pr fr these cases. Cncluding Remarks The mst imprtant result f these experiments is that the data prvide sufficient evidence t cnfirm the hypthesis that instabilities which ccur in flws having "S" prfiles generally have values f ad cnsiderably larger than thse in flws having hyperblic tangent prfiles. As explained belw, hwever, the actual wavelengths f instabilities at Ri = 0.25 are abut the same due t cmpensating changes in d. The theretical criteria f Hazel differ frm Drazin's criterin nly in that ne might expect t bserve instabilities having larger dimensinless wavenumbers in unstable shear layers with "S" prfiles. Therefre, ne might be inclined t expect shrter wavelengths fr instabilities assciated with "s" prfiles than with hyperblic tangent prfiles. Hwever, it can be shwn that fr an experimental velcity prfile having a maximum velcity V^ at Zi, a minimum velcity V 2 at zg, and a velcity gradient (av3z) 0 at the center f the shear layer, the wavelength, A, at Ri = 0.25 fr the best fitting "s" prfile is generally abut the same as that fr the tanh prfile. This is because the value f d fr the best fitting "s" prfile 10

15 is larger than that fr the tanh prfile. Thus, the increase in the critical value f ad is primarily due t an increase in d and, therefre, the wavelengths f instabilities in flws having "S" prfiles are apprximately the same as thse fr hyperblic tangent prfiles, This is illustrated in Sketch E which shws, fr instabilities assciated with ad at Ri = 0.25, the 1.6 theretical variatin f USING HAZEL'S THEORETICAL RESULTS -EXTRAPOLATED the rati f A fr "Sshaped" prfiles t A fr hyperblic tangent prfiles with the parameter b (b is related t z^ - zg, V, - V. '2' and (9V9z) 0 in Eq. (25) in Appendix I). Fr the range f b shwn, and fr b nt near zer, this rati is nt greatly different frm 1.0. Data frm the water channel btained near Richardsn number f 0.25 tend t cnfirm this: 0.8 SKETCH E. THEORETICAL RATIO OF WAVELENGTHS FOR Ri =0.25 FOR "S-SHAPED" AND HYPERBOLIC TANGENT PROFILES 2 b Prfile b Ri ad d,in. A, in. "S-Shaped" > "S-Shaped" > Tanh - 0.2U O.7I+ O.58 5 Tanh U 0.52 k Tanh O.hh k 11

16 (is a result, Drazin's value f A= (2irv^)*2d (Fig. 2), where d is apprximatelyequal t half the shear-layer thickness, wuld prvide reasnably gd estimates f the wavelengths that might ccur in the atmsphere regardless f whether the prfiles are "s" r tanh in shape. The experiments and theries als prvide further evidence that 0.25 shuld be used as the critical Richardsn number, and that several distinct waves might be bserved in the isentrpes when instabilities ccur in atmspheric shear layers. 12

17 STABILITY OF "THREE-DIMENSIONAL", STRAIGHT, STRATIFIED SHEAR FLOWS A theretical criterin fr predicting the stability f "three-dimensinal", straight, stratified, shear flws is apparently nt available. Because the magnitude f the task f develping rigrus theretical criteria was beynd the scpe f this prgram, a simple analysis based n energy cnsideratins (similar t a methd used by Chandrasekhar; Ref. 8) was made; see Appendix II. Criteria btained using such simple techniques are admittedly questinable and may nt be t significant. Hwever, the criterin which was btained was in qualitative agreement with experimental results. The analysis in Appendix II predicts that the critical Richardsn number fr "three-dimensinal" stratified shear flw is Q T 1PS. A phtgraph f a shaped filter bed f the type used t develp velcity prfiles in experiments in the Water Channel is shwn in Fig. 6. The filter prvided n velcity gradient (i.e., the same velcity at all heights abve the channel flr) acrss the 2-ft width f the channel except fr a sectin in the center f the channel span where a hyperblic tangent vertical velcity gradient was prvided. The spanwise distance allwed fr transitin frm hyperblic tangent velcity gradient t zer velcity gradient was large enugh s that hrizntal velcity gradients were very small cmpared with the vertical gradients. Summary f Experiments with "Three-Dimensinal" Flws Characteristics f Breakdwn f Flw The fur distinct and repeatable stages which ccurred during breakdwn f the "three-dimensinal" flws were similar t thse bserved in tests f tw-dimensinal flws having hyperblic tangent and "S-shaped" velcity prfiles. Dye traces illustrating the phenmenn are shwn in the sketch and in phtgraphs in Fig. 7- At the lcatin in Figs. 7(a) thrugh 7(c), dye frm prbes at several transverse lcatins indicated that the disturbance was cnfined t the flw cntaining the shear layer in the central prtin f the channel. At the lcatin in Fig. 7(d), the vrtices have "burst" and the flw appears turbulent. Dwnstream f this lcatin the turbulence began t spread transversely. Velcity, Temperature, and Density Prfiles Velcity, temperature, and density prfiles are shwn in Fig. 8 fr three different values f Richardsn number. It was fund that neither a hyperblic tangent nr an "S-shaped" prfile was particularly representative f the experimental velcity prfiles. The temperature and density gradients shwn in Fig. 8(b) and (c) indicate that the temperature and density varied apprximately linearly thrugh the thermcline which separated the tw regins f the flw having apprximately unifrm temperatures. 13

18 Cmparisn f Results with Theretical Stability Criterin fr Tw-Dimensinal Flws Figures 9 thrugh 11 present a cmparisn f the test results with Drazin's criterin fr tw-dimensinal flws. The data were btained fr three different ranges f shear layer width, w, t thickness, (2d) m, ratis. See Fig. 6 fr w. During these tests, w s in Fig. 6 was varied between 0 and 12 in., and w-t between 2 and 5 in. The shear layer thickness (2d) m was the measured distance between the maximum and minimum velcities f the shear layer and fr mst cases was apprximately equal t twice the scale length, d. Data was presented in Figs. 9» 10» and 11 fr w(2d) m between U.8 and 8.6, 9«7, and lu.3> and lu.5 and k, respectively. Except fr five cases (all n Fig. 9)> the dimensinless wavenumbers fr cases where instability was bserved are in agreement with Drazin's stability criterin fr tw-dimensinal flws having hyperblic tangent velcity prfiles (i.e., the data pints are inside the stability bundary). Fr the five exceptins the dimensinless wavenumbers were greater than 1.0. This is nt unexpected since the velcity prfiles were smewhat "S-shaped". By inspecting Figs. 9 thrugh 11 it can be seen that as the width f the layer cntaining the vertical velcity gradient decreases, the Richardsn numbers at which stable flws are bserved generally decrease, and the Richardsn numbers abve which instabilities are nt bserved als generally decrease. It appears, then, that the critical Richardsn number decreases ~as the width f the vertical velcity gradient decreases. This trend is shwn by the shaded bundary in Fig. 12. The trend is cmpatible with the result btained by extending Chandrasekhar's technique fr predicting the stability f tw-dimensinal stratified, shear flws t "threedimensinal" flws (see Appendix II); i.e., the simplified extensin f the thery predicts that the critical Richardsn number, Ri, fr "three-dimensinal" flws is Cncluding Remarks- The mst imprtant result is that data and thery prvide evidence that "three - imensinal" stratified shear flws are njjje^^jajjle than tw-dimensinal flws. The ritical Richardsn number is less than 0*25; it decreases with decreasing shear ayer width t thickness ratis. The results als indicate that, althugh the experimental velcity and density prfiles were quite different frm theretical prfiles used by Drazin in his analysis, Drazin's criterin fr predicting the wavelength f instabilities in tw-dimensinal flws culd be used t predict the wavelength f instabilities in "three-dimensinal" flws. Finally, as many as fur r five wavelengths were ften bserved upstream f the first discernable vrtex, as in tw-dimensinal flws; thus, it is reasnable t expect that several waves will als be bserved in isentrpes when instabilities ccur in "three-dimensinal" atmspheric shear layers. Ik

19 INTERACTION OF LONG-WAVELENGTH WAVES WITH TWO-DIMENSIONAL, STRAIGHT, STRATIFIED SHEAR FLOWS The primary purpse f this part f the fluid mechanics prgram was t investigate, using the Water Channel, the destabilizatin f initially stable shear flws by lng-wavelength waves. Evidence that this phenmenn ccurs in the atmsphere and may be the cause f an appreciable fractin f clear air turbulence has been dcumented previusly under this prgram (Refs. 1 thrugh k). In Ref. 1, an analytical technique was develped t predict when lng-wavelength waves, such as muntain lee waves, wuld destabilize initially stable shear layers in the atmsphere. In using the technique, Ri = 0.25 was used as the critical Richardsn number fr neutral stability. The increment in shear induced by the presence f lng waves was estimated using an extensin f a thery develped by Phillips (Ref. 9) and the lng^wave wavelenths were estimated using Haurwitz 1 thery (Ref. 10). The bjectives f the Water Channel experiments and analyses reprted in this sectin were: (l) t investigate the cnditins fr wave-induced instability and the nature f the initial disturbances, (2) t btain experimental verificatin f Phillips' and Haurwitz' theries, and (3) t verify the validity f the cmbined theretical apprach develped t predict the ccurrence f wave-induced shear-layer instabilities. Thery fr Wave-Induced Shear-Layer Instabilities In Ref. 9> Phillips derives an equatin fr the increment in shear induced by the presence f a wave traveling alng the thermcline in a fluid. The expressin he btained fr the wave-induced shear is (10) where gdp\ Nj^ = Brunt-Vaisala Frequency = v~~p\t~ 1 n = Wave Frequency = 2 TT V Q ALW a = Wave amplitude V Q = Wave velcity Fr statinary waves, the wave velcity, V Q, is equal t the mean velcity thrugh the wave, 15

20 In Ref. 10, Haurwitz derives an expressin fr the wavelength f waves traveling n a discntinuity f density and velcity at a velcity equal t the mean flw velcity and in a directin ppsite t the flw directin (thus, the wave is statinary relative t the bserver). The velcity and density gradients were cnsidered t be zer n either side f the discntinuity. The expressin btained by Haurwitz is 2. 2m * ((*$ L +,\ J (ii) A LW g (p 2 -p)\\sj 2! p 2 7 K J where V-, and p-^ are the velcity and density, respectively, f the light fluid which flws ver the heavy fluid having a velcity and density Vg and pg, respectively. Using estimated values f wave-induced shear, it is pssible t predict the effects f lng-wavelength waves n lcal Richardsn number in shear flws in the Water Channel (this was dne fr atmspheric flws in Ref. 1). A schematic diagram f the flw cnditin is shwn in Fig. 13(a). At the left are shwn upstream velcity and temperature prfiles with a stable shear layer having a thickness 2d. Within this layer, the mean velcity is V 0 and the mean temperature is T 0 ; the mean shear is (9V9z) 0 (all mean values are at the center f the shear layer). At the right in Fig. 13(a) is shwn a prtin f a lng-wavelength wave having an amplitude (which might be 0.1 t k.o in.) and a wavelength A LW (which might be 6 t 30 in.). It is assumed in this analysis that the thickness f the shear layer, the mean temperature, and temperature gradient remain cnstant as the flw within the shear layer experiences the undulating mtin. The minimum lcal Richardsn number in the flw is calculated with the waveinduced shear, A (dvdz), added t the initial mean shear. An expressin fr the minimum Richardsn number (which ccurs lcally at the crest in the example given, but wuld ccur at a trugh if the initial shear were negative) is 2 M,N " (\(dvdz) 0 \ + \A(dWdz)\f < 12 ) Using Eq. (10) fr the wave-induced shear, this becmes R ' M,N " ( (dvdz) 0 + (N^-n^HaV))* K ' 16

21 Frm Eq. (13) it can be seen that small values f Ri^iN are assciated with large initial shears, (<?V"dz) 0 ; with large wave amplitudes, a; with small flw velcities, V 0 ; and with lng wavelengths,alw( n = 2TTV 0 \L W ). The effect f the temperature gradient, dtdz, n RiMijj can be seen in Fig. 13(b). With increasing initial temperature gradient, Ri>HN first increases t a maximum value, and then ' decreases. Since the flw is unstable fr Ri < 0.25, weakly stabilized layers (dldz near zer) as well as very strngly stabilized shear layers (<9Tdz large) culd be destabilized in the presence f a lng-wavelength wave. The dashed curves in Fig. 13(b) were btained by assuming that X TU =00, s that 2 ^ _2 LW %» n. This reduces Eq. (10) t A(dVdz)= "»(-%) (1*0 The slid curves in Fig. 13(b) were btained by using the equatin fr finite wavelength waves, Eq. (10). A cmparisn f the dashed and slid curves shws (l) that fr the flw and wave cnditins f Fig. 13(b), use f the simple expressin fr wave shear, Eq. (1*0, i n the calculatin f RijvnN des nt cause large errrs in RiMiN> and (2) that these small errrs in Ri^iN which d ccur decrease with increasing value f initial shear and are least fr very small and very large initial temperature gradients. Summary f Wave-Induced Shear Layer Instability Experiments Example f Effects f Lng-Wavelength Wave n Lcal Richardsn Number A lng-wavelength wave typical f thse studied in the Water Channel is shwn in Fig. lb (X L = Ik in. and a = l.k in.). The statinary gravity wave is represented by the slid wavy line in the sketch at the tp f Fig. lb. A wave-induced instability is represented by the dashed line superimpsed n the gravity wave. The wave-induced instability first became apparent as a small amplitude wave just upstream frm the trugh at x = 7 in. This wave then grws in amplitude, rlling up int vrtices near the first crest at x = Ik in. The vrtices subsequently transitin t turbulence dwnstream frm the crest at x = lu in. Further dwnstream (x greater than apprximately 30 in.) the turbulence appears t decay and the flw restratifies. Phtgraphs f examples f instabilities which were bserved are shwn in Fig. 15 (discussed subsequently). 17

22 i The mean velcities, V, measured at the center f the shear layer thrugh the trugh and crest were apprximately 0.08 and 0.06 ftsec, respectively the average being 0.07 ftsec (velcity, temperature, and density prfiles fr this test are shwn in Fig. 16). Based n an average f the lcal flw cnditins (measured at the center f the shear layer) at the trugh at x = 7 in. and at the crest at x = lu in., the initial shear, (8V3z) 0, was sec, the initial Richardsn number, Ri 0, was 0fO, and the wave-induced shear, A(9V3z), was sec"-'-. The data f Figs. 16(a) and 16(b) shw that the magnitude f the shear was greater at the trugh than at the crest. The minimum Richardsn number, RiMXN, was 0.37 and ccurred at the trugh. This was 53 percent f the initial Richardsn number but was greater than the minimum Richardsn number fr instability, Ri = One pssible explanatin fr this discrepancy is that the velcity prfile was distrted at the time f its measurement by the wave-induced instabilities which were first apparent just upstream frm the trugh at x = 7 in. Attempts were made t measure the velcity prfile at times when the prfile was least disturbed, i.e., between crests and trughs f the wave-induced instabilities. In mst ther tests in which wave-induced instabilities were bserved the minimum Richardsn numbers were less than O.25. The Richardsn number which was bserved at the crest at x = 1^ in., Ri = 1.8, was 260 percent f the initial Richardsn number. Example f Instability Induced by Lng-Wavelength Wave Phtgraphs f an instability induced by a lng-wavelength gravity wave are shwn in Fig. 15. The upper dye traces shw the instability superimpsed n the statinary gravity wave. The estimated shape f the undisturbed gravity wave is sketched in white n the phtgraph. The lwer trace appears t be undisturbed by the wave instability. The phtgraphs presented in Fig. 15(b) shw that the waves have transitined t vrtices, and the trace in the phtgraph n the right side f Fig. 15(b) gives evidence that sme turbulence exists dwnstream f the vrtices. Cmparisn Between Measured and Predicted Wave Shears in Lng-Wavelength Waves Measured values f wave-induced shear caused by lng-wavelength waves are cmpared with values calculated using Phillips' thery (Ref. 9) injj^ju^iz T^e measured wave shear, A(9V"9Z), was determined by taking ne-half the difference between the shears at the crest and trugh f the lng-wavelength wave. The predicted wave shears were calculated using Eq. (10) with measured values f wave amplitude, wavelength, and velcity. The Brunt-Vais'ala frequency, NM, was determined frm measured temperature prfiles. The results shwn in Fig. 17 indicate that in mst cases the agreement between measured and predicted values f wave shear was gd, The results f these tests prvide sme experimental evidence that Eq. (10), which is used in this prgram t predict wave-induced shear in the atmsphere is adequate. 18

23 Cmparisn Between Measured and Predicted Wavelengths f Lng-Wavelength Waves Measured wavelengths f lng-wavelength waves are cmpared with values predicted using Haurwitz 1 thery (Ref. 10) in Fig. 18. The measured wavelengths, (^i\j) m, were determined by measuring the distance between crests and trughs f the lng-wavelength waves frm phtgraphs f the dye traces. The predicted wavelengths, (ALw)p> were calculated using Eq. (ll) with densities and velcities frm measured temperature and velcity prfiles, respectively. The data in Fig. l8 indicate nly fair agreement between measured and predicted wavelengths. The differences between measured and predicted values increase with increasing wavelength. Part f the reasn fr this can be seen by examining the equatin. One way t increase wavelength, Xj^, is t decrease the density difference, pg - P]_. Hwever, this tends t make the predicted wavelengths mre sensitive t errrs in p-, r P2- This may explain sme f the differences in Fig. 18 since in this series f experiments it was easier t btain changes in A^ by changing density gradient than velcity gradient. Hwever, this still des nt accunt fr the general trend in which the measured wavelengths were generally less than predicted values. Subsequently, the methd f Haurwitz (Ref. 10) was examined t determine the pssible effect f having a finite depth in the Water M*- Channel n the measured wavelengths. The influence n wavelength f having a free surface abve the shear layer and a bundary belw the shear layer (applied by Phillips in Ref. 9 t flws in the cean) was calculated. This methd predicted a trend that did nt accunt fr the discrepancy in the data. That is, it predicted that wavelength shuld increase with decreasing depth while the wavelengths measured in the water channel were generally less than thse predicted fr infinite-depth flws. Althugh these differences between thery and experiment at lng wavelengths were nt reslved, they have n bearing n calculatins f atmspheric shear layer instabilities. This will be apparent later in the reprt. Predictin f Wave-Induced Shear Instabilities in Water Channel Figure 19 is a plt f initial shear, (9V3z) 0, versus the Brunt-Vaisal'a frequency, NM- The bundaries define regins where wave-induced instabilities can ccur. They were calculated using Eq. (l^) with the cnditin N M» n. The bundaries are lci f RiMIN = Q» 5_ f r cnstant values f av 0. T the left f the bundary fr av = 0, the presence f a lng-wavelength wave is nt required fr the flw t be unstable, and any instabilities bserved in this regin wuld nt be wave-induced. In the regin t the right f the bundary fr av 0 = 0, the bundaries define the lwer limit f av 0 required fr wave-induced instability t ccur. Fr example, a lng-wavelength wave having av 0 > O.75 wuld cause wave-induced instability t ccur anywhere in the regin between the bundary fr av 0 = 0 and av 0 = 0.75; a lng-wavelength wave having av < 0.5 wuld nt cause wave-induced instability anywhere in the regin t the right f the bundary fr av 0 =

24 The pen and slid symbls in Fig. 19 dente cases fr which lng^wavelength waves were and were nt bserved, respectively. Fr each flw cnditin fr which a lng-wavelength wave was present (the pen symbls) (l) the Brunt-Vaisala frequency was determined frm an average f the crest and trugh temperature gradients (in these tests, measurements were made at the mst upstream crest and trugh fr which measurements culd be made), and (2) the abslute value f initial shear (9V9.z) 0 was determined frm an average f the shears measured at the crest and trugh. Fr cases in which lng^wavelength waves were nt bserved (the slid symbls), the Brunt-Vaisala frequency and the abslute value f initial shear were btained frm the temperature gradient and the velcity gradient, respectively, at the center f the shear layer (measurements were made at apprximately the same distance dwnstream frm the filter as the measurements that were made when waves were present). The numbers near the symbls dente values f av Q. The letters near the symbls dente the type f instability that was bserved W - wave, V - Vrtex, T - Turbulence. The slid symbls shw that lng-wavelength waves were nt bserved in the channel fr values f Brunt-VaisaTa frequency, Njjj less than abut 0.3 sec. Therefre, it was nt pssible t investigate wave-induced instability fr values f % less than abut 0.3 sec" Examinatin f Fig. 19 indicates that mst f the bservatins are in agreement with the stability bundaries. Instability f sme type was bserved fr all cases which are t be left f the bundary fr av Q = 0. This is expected since the flw is predicted t be unstable in this regin even withut the additinal shear frm a lng-wavelength wave. Suspected wave-induced shear instabilities were bserved in six tests (see data marked with asterisks) which are t the right f the lcus av 0 = 0. In these cases, lng-wavelength waves were present. Based n measured values f av 0 fr the six cases fr which wave-induced instability was suspected and the values f av 0 fr the cases where wave-induced instability was nt bserved, the data agree with the stability bundaries. Cncluding Remarks These labratry experiments tend t cnfirm the analytical methds that have been used t estimate wave-induced shear and t predict the nset f shear-layer instabilities in analyses f CAT encunters in the atmsphere. The mst imprtant results are shwn in Figs. 17 and 19. Figure 17 shws that the wave shear A(3v9z) the change in shear that ccurs when a thin shear layer flws thrugh a lng-wavelength wave can be reasnably well predicted using Eq. (10). The inputs needed fr this calculatin are the lng-wave amplitude (a), the mean flw velcity (V 0 ), the Brunt-Vaisala frequency (1%), and the lng-wave frequency (n). 20

25 Figure 19 shws that the ccurrence, r lack f ccurrence, f wave-induced shear-layer instabilities in the water channel experiments culd be predicted fairly cnsistently using Eg.. (13). This equatin incrprates Eq. (10) fr estimating the wave shear. The criterin fr instability that was used is that the shear layer will becme unstable when the predicted minimum Richardsn number is less than

26 APPLICATION TO ATMOSPHERIC SHEAR FLOWS In earlier wrk (Refs. 1 thrugh h) in this prgram, clear air turbulence was assciated with the destabilizatin f initially stable shear layers by lng gravity waves. Several cases were presented in which the lng gravity waves were induced by muntains. Als, a few cases were presented in Ref. 3 in which the lng wave-like undulatins were induced by thunderstrms. These findings appear t be in agreement with evidence assembled by ther investigatrs ntably Wds (Ref. 11, a study f wave-induced instabilities in the cean), Ludlam (Ref. 12, a study f billw clud frmatin), Mitchell and Prphet (Ref. 13, an analysis f USAF Prject HICAT flight data), Spillane (Ref. Ik, an analysis f high-altitude CAT ver the Australian desert regin), Hardy (Ref. 15, radar measurements that indicated wave-like mtins in regins f CAT), Hicks (Ref. 16, radar bservatins f gravitatinal waves assciated with CAT near the trppause), Bucher (Ref. 17, radar bservatins f waves assciated with CAT at a subsidence inversin), Gssard, Richter and Atlas (Ref. 18, radar bservatins f internal gravity waves and billws at an ceanic inversin), Brwning find Watkins (Ref. 19, radar bservatins f billws assciated with CAT near a frntal zne beneath the jet cre), Brwning and Watkins (Ref. 20, radar bservatins f billws assciated with CAT in muntain lee waves), Rach (Ref. 21, aircraft reprts f gravity waves assciated with CAT ver the Atlantic Ocean), Perm and Thmpsn (Ref. 22, turbulence measurements assciated with stable layers which are extensive in area, persist in time and have large vertical wind shears), and Axfrd (Ref. 23, aircraft bservatins f gravity waves assciated with CAT in the lwer stratsphere). The purpse f the present effrt was t develp and evaluate a CAT predictin prcedure based n the mechanism f wave-induced instability f initially stable shear layers. The analysis used in the predictin prcedure was develped previusly during this prgram and is reviewed in this sectin. It shuld be mentined that sme f the wave and turbulence frecasting infrmatin used in the develpment and evaluatin f the predictin prcedure was prvided by United Air Lines and the Air Frce Glbal Weather Central. Als, muntain wave data used t evaluate the UAL wave predictin prcedure was btained frm I968 and 1970 Lee Wave Observatinal prgrams at the Natinal Center fr Atmspheric Research. Review f the Fundamental Flw Phenmenn A schematic diagram f the flw cnditin cnsidered here ia shwn in Fig. 20(a). This is analgus t the situatin shwn in Fig. 13(a) which was used t describe wave-induced shear layer instabilities in the Water Channel. At the left in Fig. 20(a) are shwn upstream wind and temperature prfiles with a stable shear layer having a thickness 2d. Within this layer, the mean wind is V 0 and the mean temperature is TQ; the shear is (9V8z) 0 and the envirnmental lapse rate is 3T3z. 22

27 At the right in Fig. 20(a) is shwn a prtin f a lng^wavelength wave having an amplitude, a (which might be 2000 r 3000 ft), and a wavelength A LW (which might be 10 r 20 nmi). It is assumed in this analysis that the thickness f the shear layer, the mean temperature and the lapse rate all remain cnstant as the flw within the shear layer experiences the undulating mtin (2d, T 0, and 3T3z are cnstant). The increase in shear that ccurs at the crest can be calculated frm Eq. (10) given previusly. Nw, fr flws in the atmsphere, N M = Brunt-Vaisala frequency = yj(gt 0 ) ' (3T9z)-(9Taz) a(i n = wave frequency = 2TTV 0 AL^ (3Taz) a a = adiabatic lapse rate, x 10"3 deg cft* ' This increase in shear is added t the initial shear, and Eq. (13) can be used t calculate the minimum Richardsn number (which ccurs lcally at the crest in Fig. 20(a), but wuld ccur at the trugh if the initial shear were negative). Since %j > n under mst cnditins f interest (this is nly untrue fr weakly stable lapse rates, i.e., when 8T9z =(3T9z) ad ), Eq. (13) can be further simplified t Rl MIN K M (15) il.wj,! I. K, 2» w ( fvdz) 0 + N 2 u2 M (av 0 )) z Frm Eq. (15) it is evident that lw values f RijvuN are assciated with large initial shears, (avdz) 0 ; with large lng-wave amplitudes, a; and with lw winds, V 0. The latter tw parameters are nt independent, hwever, since large amplitude waves d nt usually ccur under lw-wind cnditins. The effect f the envirnmental lapse rate, dtdz, n Ri^iN is nt evident frm Eq. (15) but can be seen in Fig. 20(b). This figure is based n typical cnditins under which lng-wavelength waves, such as muntain lee waves, are bserved in the lwer stratsphere. Curves are shwn fr three shears small shears (l and 2 kts1000 ft) and a mderately large shear (12 kts1000 ft). The curves shw that the greater the initial stability frm a cnvective standpint (i.e., the greater dtdz), the lwer Rij^iN will be. They als shw that shear layers which have small *Nte use f minus sign t dente temperature decreasing with increasing altitude. ' 23

28 ' values f initial shear are stable when the envirnmental lapse rate is near zer. Thus, it is primarily the mst stable layers appearing in the temperature prfile that are f interest. The curves als shw that as the envirnmental lapse rate appraches the adiabatic lapse rate (-3 cleg cft), all shear layers becme unstable. Previus theretical and experimental studies which have been discussed in this reprt have shwn that the critical Richardsn number belw which tw-dimensinal flws are unstable is Fr RijvUN = 0.25 and fixed values f wave amplitude, a wind velcity, V Q, and wavelength, Aj^, Eq. (13) can be used t determine the neutral stability bundaries n a plt f abslute value f initial wind shear, (dvdz) 0, versus vertical temperature gradient, STdz. The stability bundaries are shwn fr wave amplitudes f 500 and 3000 ft in Figs. 21 and 22, respectively. The bundaries are shwn fr wind velcities, V OJ f 10, 25, 50, 75 } and 100 kts (A^ = 15 nmi). Atmspheric flws having cmbinatins f vertical temperature gradient and initial wind shear which plt under the bundary fr a given wind velcity wuld remain stable as they passed thrugh the undulatins f the lng-wavelength waves. Flws having cmbinatins f dtdz and (dv8z) 0 which plt abve the bundary wuld becme unstable. Flws having 9T9z < deg cft wuld be cnvectively unstable. It is seen in Figs. 21 and 22 that the stability f the flw decreases with decreasing wind velcity. Figures 21 and 22 culd be used with rawinsnde data t predict whether r nt initially stable shear layers will be destabilized by lng-wavelength waves. Fr example, V 0, (9V9z) 0, and 9T9z wuld be determined frm rawinsnde data and used with Figs. 21 and 22 fr weak and strng wave activity, respectively, t make predictins abut the stability f the flw. Where flws are predicted t becme unstable, CAT culd be expected t ccur. A mre cmprehensive apprach t CAT predictin is presented in the next sectin, hwever. The variatins in maximum allwable initial shear with wind velcity fr wave amplitudes f 500, 1000, 2000, and 3000 ft are shwn in Fig. 23. Fr values f initial shear greater than this maximum, the flw fr the given wave amplitude wuld be unstable fr all values f vertical temperature gradient. The curves shwn in Fig. 23 were btained using \idvdz) 0 \ m% = [i + (27raA LW f] [va] (l6) Equatin (l6) was btained by slving Eq. (13) fr (9V"9z) 0, differentiating with respect t the Brunt-VaisaTa frequency, and slving fr the maximum value f (9V9z). Fr lng wavelengths, i.e., Aj^ >2-rra, Eq. (l6) can be simplified t 2k

29 (dvdz) 0 MAX 0' u The variatin f (av3z) 0 j^ with V 0 fr a = 3000 ft was determined using Eq. (17) and is cmpared in Fig. 23 (dashed curve) with that btained using Eq. (l6). Fr X LW >15 nmi and a = 3000 ft, the difference in values f (3V3z) 0 MAX is quite small. This difference wuld be even less fr smaller wave amplitudes. Imprved CAT Predictin Methds. ) -,! A. In general, mst CAT predictin techniques are based n synptic features which are present and n a cmparisn f the values f ne r mre atmspheric parameters (such as hrizntal wind shear, vertical wind shear, hrizntal temperature gradient, streamline curvature, etc.) with empirically determined critical values. In mst cases, there is n physical mdel fr the breakdwn mechanism. An example f such a * predictin technique is given in Ref. 2k. The technique is based n hrizntal and - «L vertical temperature gradients (determined frm rawinsndes alng the flight rute) and is quite simple t use. In Ref. 3 it was recmmended that cnsideratin be given t develpment f a predictin prcedure which wuld predict the altitudes where CAT wuld be encuntered in muntain waves and which is based n the mechanism f wave-induced instability f initially stable shear layers. During the present prgram, such a prcedure was develped and evaluated. Evaluatin was limited, hwever, because f the limited amunt f available data. This prcedure was cmprised f three distinct peratins cntained in a single cmputer prgram which is discussed in detail in Ref. 5: (l^the rawinsnde data frm a given statin is prcessed t predict the lng^wave amplitude necessary t destabilize each shear layer appearing in the prfiles; (2)meterlgical data frm muntain wave znes are used with wave predictin methds t predict lee wave activity and amplitudes; and (3} the results frm (l) and (2) are cmpared t see if wave-induced instabilities wuld ccur and t identify the assciated altitudes and lcatins. A blck diagram f the cmputer prgram lgic is shwn in Fig. 2.k. Calculatin f Required Lng-Wave Amplitudes This prtin f the prcedure is relatively straight-frward. ^Hlfttiildtt? ilj} n " n be rewritten t yield the minimum lng-wave amplitude necessary t destabilize (RiMIN = 0.25) a shear layer: 25

30 2N M - (a v a 2 )J, VQ (l8) N 2 M -n 2 ( The rawinsnde data are prcessed by the cmputer prgram t identify stable shear layers (using temperature and velcity prfiles) and, fr each layer, t calculate the mean flw parameters V 0, (8V8z) 0, Hju, and n. In calculating n = 2-nY 0 Xjj^, it is necessary t assume a value fr the wavelength, A^y. In general, X-^j is f the rder f _15_nmi and (fr atmspheric flws) n is small cmpared t Nj^ s that large errrs in n have little effect n the cmputed amplitude, a. Using the mean flw parameters and an assumed lng-wave wavelength, the required amplitude fr instability is calculated fr stable shear layers which have temperature gradients greater than sme preselected minimum value (fr a given wave amplitude, the induced wave shear, A(3v3z), increases with increasing sfrabjll^frv s that the mst stable layers are the mst likely t experience instabilities). Lng-Wave Amplitude Frecast At this pint, the characteristics f waves in the lcal zne f the frecast must be cnsidered. T frecast- f.at -inrhmpr] by fflninpfcai-n l qg wavpg.) gd techniques fr frecasting the ccurrence and amfiiiiucles f lee waves are required. The techniques which are used by thse active in frecasting muntain wave CAT, such as United Air Lines (Ref. 25), Nrthwest Airlines, and Glbal Weather Central at Offutt Air Frce Base, are empirical and rely n such parameters as wi.nd directin V and strength abve the muntains, sea level pressure difference acrss the muntains, \ lapse rate and synptic cnditins in the lcal zne f frecast. When significant wave activity is frecast, attempts are made t avid flying within apprximately ft f the trppaug 7 United Air Lines uses synptic data in the frecast zne t frecast the ccurrence f muntain waves and CAT. During the frecast perid, they check their frecast with a nmgram which uses sea level pressure difference acrss the muntains and maximum wind velcity between 10,000 and 20,000 ft (perpendicular t the muntains) t predict muntain wave activity fr 20 wave znes in the United States. This nmgram, which is shwn in Fig. 25, was first used t predict the Denver wave. The pressure difference used was the sea level pressure at Grand Junctin minus the sea level pressure at Denver. In using the nmgram fr ther wave znes, where the distance between the statins may be different than the distance between Grand Junctin and Denver, a pressure crrectin is required. The magnitude f the crrectin depends n the difference in distance, and the crrectin is plus if the distance is less and minus if the distance is greater. The wave znes and pressure crrectins are given in Ref. 25. Pressure tendency is ften used t extend the perid f validity f the nmgram :.', ' (

31 The nmgram (Fig. 25) indicates nn Ifi^^vfis wuld be expected when the maximum wind velcity perpendicular t the muntain was ip^ than Pfl fctg _ p 0 r maximum wind velcities abve 20 kts, the sjy^gnjjy^ f the wave depends n the sea level pressure difference. Nrthwest Airlines als uses the nmgram shwn in Fig 25^t predict the presence f muntain waves; hwever, they use the wind velcity perpendicular t the muntains at 300 mb altitude rather than maximum wind velcity between 10,000 and 20,000 ft. Glbal Weather Central has crrelated the crdinates n the nmgram with the severity f turbulence and uses the pressure -difference and maximum winds (between 10,000 and 20,000 ft, perpendicular t the muntains) t predict the ccurrence and severity f CAT. Sme data btained by the authr during the 1970 Lee Wave Observatin Prgram at the Natinal Center fr Atmspheric Research fr the Grand Junctin-Denver wave zne are pltted n Fig. 25 fr cmparisn with the nmgram. The data btained abut 0500 MST between February 11 and February 27. The numbers next t the data pints dente the date. Flights were made n February 13, 17 } 18, and 26 t make measurements in the wave field (flights were nly made n days when wave activity was frecast by ther techniques). Results f these flights indicated that weak waves were present n February 13 and 26, and weak t mderate r strng waves were present n February 17 and 18. Isentrpes recnstructed frm flight and radisnde data which shw the muntain wave patterns which ccurred n February 13, 17, and 18 are shwn in Figs. 19, 21, and 23 f Ref. 3» Th e data pints which crrespnd t days n which flights were made are flagged (a weak wave bservatin is indicated by a single flag and a mderate wave by a duble flag). The data fr which wave activity was cnfirmed by flights is in fairly gd agreement with the nmgram. Fr instance, the nmgram indicates (l) the presence f weak waves n February 12, 13, and 1^ weak waves were bserved n February 13, and (2) the presence f weak waves n February 16 and 18 and mderate t strng waves n February 17 mderate waves were bserved n February 17 and 18. Wave amplitudes up t 1000, 1350, and 1800 ft were bserved n February 13, 17, and 18, respectively. Als, n days fr which flights were nt made (because n wave activity was frecast by ther mre cmplicated techniques), mst f the data (February 15, 19, 21, 22, 23, and 2U) fall int the "n wave" regin f the nmgram. Althugh negative pressure differences were nt cnsidered in Ref. 25, it is felt that waves wuld nt ccur under such cnditins. It appears then that if such a nmgram culd be mdified t prvide wave amplitude infrmatin it culd be used with rawinsnde data t predict the altitude where CAT wuld ccur frm destabilizatin f initially stable layers by muntain waves. Pressure tendency and frecast winds might be used with the nmgram t frecast the ccurrence f muntain lee waves and the altitude where CAT wuld ccur frm destabilizatin f stable shear layers by the lee waves. 27

32 This nmgram was used, in mdified frm, t prvide vave amplitude infrmatin and was incrprated in the cmputer prgram fr predicting muntain-wave induced shear-layer instability resulting in CAT. A crrelatin f wave amplitude with nmgram-predicted wave activity was attempted by the present authr using bserved wave amplitudes fr the three muntain wave patterns fr February 13, 17, and 18 (presented in Ref. 3) with predicted wave activity fr thse days. Nmgram-predicted wave activity and wave amplitude are cmpared in Sketch F. In lee wave patterns presented previusly in Ref. 1, wave amplitudes up t 3000 and 3500 ft are apparent. Althugh nmgram-predicted wave activity is nt available fr these cases, the data were pltted in Sketch F in the regin f mderate r strng wave activity w Q 0- < > < CORRELATION USED IN COMPUTER 3000 PROGRAM FOR CAT PREDICTION k FEB FEB \ > FEB 15 O 1968 FEB FEB NO WAVE WEAK WAVE MODERATE OR STRONG WAVE PREDICTED WAVE ACTIVITY SKETCH F. CORRELATION OF WAVE AMPLITUDE WITH PREDICTED WAVE ACTIVITY Based n the data presented in Sketch F, wave amplitudes f 1QQQ. and 2500 ft_ were selected t crrelate weak and mderate r strng wave activity, respectively. Because f the scarcity f data, the accuracy f this crrelatin is admittedly questinable. Als, since amplitudes f 1000 and 2500 ft are less than the maximum amplitudes bserved fr the crrespnding wave activities, CAT may be underpredicted by the cmputer prgram (use f amplitudes equal t 1^00 and U000 ft fr weak and mderate r strng wave activities, respectively, wuld prbably be mre cnservative). Hwever, the selected crrelatin between wave amplitude and activity culd be changed based n experience and new data. 28

33 The cmputer prgram (Fig. 2k) then, predicts lee wave amplitudes f 1000 and 2500 ft when weak and mderate r strng wave activity is predicted, respectively. This prvides a means fr estimating lee wave amplitude frm sea level pressure difference and maximum wind velcity perpendicular t the muntains in the altitude range frm 10,000 t 20,000 ft. Screening and CAT Frecast At this pint, the cmputer prgram (Fig. 2U) cmpares the wave amplitudes required t destabilize each initially stable layer with the predicted amplitude f the muntain lee wave. Layers requiring amplitudes smaller than the lee wave amplitude wuld be expected t becme unstable and prduce CAT. Then, when weak waves (a = 1000 ft) were frecast, turbulence wuld be frecast within ft f the altitude f layers which were predicted t becme unstable and when strng waves (a = 2500 ft) were frecast, turbulence wuld be frecast within ft f the altitude f unstable layers. Cmputer Prgram fr Frecasting Muntain Lee Wave CAT Referring t Fig. 2U, the prgram first cmputes the vertical velcity and temperature prfiles and the velcity and temperature gradients fr each layer frm rawinsnde data btained upwind frm the muntains (blck (l)). The prgram then identifies stable shear layers frm the velcity and temperature prfile data and selects thse which are f interest fr further analysis (temperature gradient abve preselected minimum value). Next, the prgram calculates the lee wave amplitude required t destabilize (reduce RiMIN t 0.25) these stable shear layers, using frm the rawinsnde data the initial shear, (W8z) OJ velcity, V 0, temperature gradient, (3T8z) 0, and temperature, T 0 (blck (2)). In the next step (blck (3)), the prgram uses the United Air Lines nmgram (with the amplitude mdificatin made by the authr) t (a) predict lee wave activity and amplitudes using values f pressure difference and maximum wind velcity r (b) frecast future lee wave activity and amplitudes using pressure tendency and frecast wind velcity. The prgram then (blck (k)) cmpares the predicted amplitude f the lee waves with the amplitudes required t destabilize the stable shear layers. CAT is predicted t ccur at altitudes crrespnding t layers which are predicted t becme unstable. 29

34 Cncluding Remarks Previus analyses f CAT cases invlving muntain waves (Refs. 1 and 3) have shwn that there is a fair t gd agreement between the altitudes at which CAT was detected and the altitudes f initially stable layers which were predicted t be destabilized by the wave-induced shear. Measured lee wave amplitude was used in these analyses and in general the amplitude varied with altitude. The UAL nmgram, which is based n airline experience, des quite well in predicting current wave activity using current pressure difference and maximum wind velcity. Data btained in the 1970 Lee Wave Observatin prgram tend t cnfirm this. Als, based n lee wave bservatins, it des nt appear unreasnable t crrelate wave amplitude with wave activity. Hwever, using the nmgram there is n way f estimating the variatin in wave amplitude with altitude. The success in using the nmgram with the cmputer prgram t frecast future wave activity will depend n the success in using pressure tendency t frecast pressure difference and success in frecasting the maximum wind velcity. The frecast might be quite gd fr shrt frecast perids. The majr advantage in using the nmgram is that it is quite simple t use with the cmputer prgram. Hwever, if the nmgram results in errneus wave frecasts r lng range frecasts are desired, then mre subjective techniques culd be used fr frecasting lee wave activity as suggested in the lwer part f Fig. 2k. Such techniques are presently in use by UAL, USAF, and thers and use synptic data, such as the change in height f the trppause upwind frm the muntains, wind velcity in the wave zne, wind and istherm patterns, and lcatin f warm tngues f air at 850 mb t the lee f the muntains t frecast muntain waves. It des nt appear that a technique such as this culd be included in the cmputer prgram; if such a technique were used, sme way f estimating the wave amplitude is again required. 30

35 REFERENCES 1. Clark, J. W., R. C. Steffler, and P. G. Vgt: Research n Instabilities in Atmspheric Flw Systems Assciated with Clear Air Turbulence. NASA Cntractr Reprt CR-160U, prepared under Cntract NASW-1582, June 1970; als United Aircraft Research Labratries Reprt H9IO563-9, June 1969«2. Clark, J. W. : Labratry Investigatins f Atmspheric Shear Flws Using an Open Water Channel. Paper presented at AGARD-NATO Specialists' Meeting n "The Aerdynamics f Atmspheric Shear Flws", Munich, Germany, September 15-17, 1969< AGARD Cnference Prceedings N. U8, February Steffler, R. C: Further Research n Instabilities in Atmspheric Flw Systems Assciated with Clear Air Turbulence. United Aircraft Research Labratries Reprt J U, prepared under Cntract NASW-1582, June Steffler, R. C. and J. W. Clark: Research n Instabilities in Atmspheric Flw Systems Assciated with Clear Air Turbulence. Abstracts f NASA Cntractrs Meeting n Aernautical Fluid Mechanics, June 15-17, 1971, Washingtn, D. C. 5. Steffler, R. C: Additinal Research n Instabilities in Atmspheric Flw Systems Assciated with Clear Air Turbulence. United Aircraft Research Labratries Reprt K9IO563-I9, prepared under Cntract NASW-1582, August Drazin, P. G.: The Stability f a Shear Layer in an Unbunded Hetergeneus Inviscid Fluid. Jurnal f Fluid Mechanics, Vl. k, 1958, pp. 2lU-22^. 7. Hazel, P.: Instabilities f Stratified Shear Flw. Paper submitted fr publicatin in Jurnal f Fluid Mechanics, Chandrasekhar, S.: Hydrdynamic and Hydrmagnetic Stability. Oxfrd University Press, Lndn, 1961, p. U Phillips, 0. M.: The Dynamics f the Upper Ocean. Cambridge University Press, Haurwitz, B.: Dynamic Meterlgy. McGraw-Hill Bk Cmpany, Inc., New Yrk, Wds, J. D.: Wave-Induced Shear Instability in the Summer Thermcline. Jurnal f Fluid Mechanics, Vl. 32, 1968, pp

36 REFERENCES (Cntinued) 12. Ludlam, F. H.: Characteristics f Billw Cluds and Their Relatin t Clear Air Turbulence. Quarterly Jurnal f the Ryal Meterlgical Sceity, Vl. 93, 1967, pp. U19-U Mitchell, F. A. and D. T. Prphet: Meterlgical Analysis f Clear Air Turbulence and Its Detectin. Being Scientific Research Labratries Reprt DI-82-07UO, Seattle, Washingtn, August Xk. Spillane, K. T.: Clear Air Turbulence and Supersnic Transprt. Nature, Vl. 2lA, N. 5085, April 15, 1967, pp. 237" Hardy, K. R.: Radar Eches frm the Clear Air. Paper prepared fr NATO Advanced Study Institute n the Structure f the Lwer Atmsphere and Electrmagnetic Wave Prpagatin, Aberystwyth, Wales, September 2-15, Hicks, J. J.: Radar Observatins f a Gravitatinal Wave in Clear Air Near the Trppause Assciated with CAT. Jurnal f Applied Meterlgy, Vl. 8, August I969, pp Bucher, R. J.: CAT at a Subsidence Inversin; A Case Study. Jurnal f Applied Meterlgy, Vl. 9, June 1970, pp. 53^ Gssard, E. E., J. H. Richter, and D. Atlas: Internal Waves in the Atmsphere frm High-Reslutin Radar Measurements. Jurnal f Gephysical Research, Vl. 75, N. 18, June 20, 1970, pp. 3523" Brwning, K. A. and CD. Watkins: Observatins f Clear Air Turbulence by High Pwer Radar. Nature, Vl. 227, July 18, 1970, pp Brwning, K. A., et al. : Simultaneus Measurements f CAT by Aircraft and Radar. Nature, Vl. 228, December 12, Rach, W. T.: Sme Aircraft Reprts f High-Level Turbulence. The Meterlgical Magazine, Vl. 98, N. Il60, March 1969, pp Perm, S. and G. J. Thmpsn: Stable Laminae Assciated with Clear Air Turbulence. Paper Presented at Ryal Aernautical Sciety Internatinal Cnference n "Atmspheric Turbulence", Lndn, May 18-21,

37 REFERENCES (Cncluded) 23. Axfrd, D. N.: An Observatin f Gravity Waves in Shear Flw in the Lwer Stratsphere. Quarterly Jurnal f the Ryal Meterlgical Sciety, Vl. 96, 1970, pp k. Ashburn, E. V., D. E. Wac, and F. A. Mitchell: Develpment f High Altitude Clear Air Turbulence Mdels. Air Frce Flight Dynamics Labratry Reprt AFFDL-TR-69 _ 79> prepared by Lckheed Califrnia Cmpany, Nvember 1969* 25. Harrisn, H. T. and D. F. Swa: Muntain Wave Expsure n Jet Rutes f Nrthwest Airlines and United Air Lines. UAL Meterlgy Circular N. 60, February 1, Atlas, D. and J. I. Metcalf: The Amplitude and Energy f Breaking Kelvin- Helmhltz Waves and Turbulence. The University f Chicag (Department f Gephysical Sciences) and Illinis Institute f Technlgy (Department f Electrical Engineering) Labratry fr Atmspheric Prbing, Technical Reprt N. 19, Octber 26, G J { ( $ \, C>±) 27. Thrpe, S. A.: Kelvin-Helmhltz Instability and Turbulence. Paper Presented at Ryal Aernautical Sciety Internatinal Cnference n "Atmspheric Turbulence", Lndn, May 18-21,

38 LIST OF SYMBOLS a a Amplitude f wave (half the height frm trugh t peak), ft r in. Critical lee wave amplitude required t destabilize shear layer in atmsphere, ft b Expnent in theretical velcity prfile (Eq. (6)), dimensinless c Cmplex wave velcity, c = c r + i'c^, ftsec Ci Imaginary part f cmplex wave velcity, ftsec c r Real part f cmplex wave velcity, ftsec d Shear-layer scale length parameter fr describing velcity prfiles, d = (AV2)(9V8z) 0, ft E Kinetic energy per unit vlume assciated with shear (Eq. (27)), ft-lbft3 g Gravitatinal cnstant, 32.2 ftsec i Unit imaginary number, V-T", dimensinless n Wave frequency, 2TTV 0 A LW, sec N M Brunt-Vais'ala frequency, N M = V(gT 0 )' (3T3z)-(aT3z) ad in the atmsphere and -W(-gp)(3p3z) in the Water Channel, sec~l Ri Ri 0 RijYjjU t T Richardsn number, Ri = (gt 0 )' [(9Taz)-(8T8z) ad ](3V8z) 2 in the atmsphere and (-gp) (3p3z)(9V3z)2 in the Water Channel, dimensinless Initial r upstream Richardsn number, dimensinless Minimum Richardsn number caused by influence f lng-wavelength wave, dimensinless Time, sec Temperature, deg C, K, r F T Q Temperature at center f shear layer, deg C, K, r F 3h

39 LIST OF SYMBOLS (Cntinued) V Velcity r maximum velcity perpendicular t muntains between 10,000 and 20,000 ft, ftsec r kts V Velcity at center f shear layer r velcity f lng-wavelength wave, ftsec r kts Vn,Vp Velcities in upper and lwer streams bunding shear layer, respectively, fr hyperblic tangent prfiles, ftsec w(x,z,t) Perturbatin velcity (Eq. (9)) 5 ftsec w Width f "Three-Dimensinal" shear layer (Fig. 6), w = w s + 2wt, in. w s Width f part f "Three-Dimensinal" shear layer cntaining n hrizntal transverse velcity gradient (Fig. 6), in. Width f "Three-Dimensinal" transitin regin where shear prfile changes t unifrm prfile (Fig. 6), in. w B Buyant wrk per unit vlume (Eq. (26)), ft-lbft-* W F Kinetic energy per unit vlume assciated with lateral mtin (Eq. (32)), ft-lbft3 W T Ttal wrk per unit vlume (Eq. (33)), ft-lbft^ x Dwnstream crdinates in Water Channel (Fig. 1), ft r in. y Transverse crdinate in Water Channel (Fig. 1), ft r in. z Vertical crdinate in Water Channel (Fig. 1), ft r in. z zj_, z 2 Vertical crdinate f center f shear layer in Water Channel, r mean altitude f stable layer in the atmsphere, ft r in. Vertical distance t V]_ and V2, respectively, in. a Wavenumber f small-amplitude waves assciated with instabilities in shear layers, a = 2TTA, ft 35

40 LIST OF SYMBOLS (Cncluded) (3 Parameter in density prfile equatin (Eq. (2)), 3 = Ri # d'(9vbz) g, dimensinless An AV Sea level pressure difference, (Fig. 25), mb Velcity difference parameter fr describing velcity prfiles (Eq. (1)), ftsec 6 Increment in quantity, dimensinless A(3V"8z) change in shear caused by influence f lng-wave length wave, sec (8T8z) ad Adiabatic lapse rate, (8T3z) ad = x 10"3 deg cft (9T3z) 0 Initial upstream temperature gradient, deg Fft r deg c1000 ft (3V8z) 0 Initial r upstream shear, sec - -'- r kts1000 ft (2d) m A X LW Measured thickness f shear layer befre turbulent breakdwn, in. Wavelength f small-amplitude waves assciated with instabilities in shear layers, ft r in. Wavelength f lee wave r lng-wavelength wave in Water Channel, in., ft, r nmi v Kinematic viscsity, ft sec p p 0 P]_>P2 Density f water r air, slugsft3 Denisty f water r air at center f shear layer, slugsft^ Densities in upper and lwer streams bunding shear layer, respectively, slugsft3 <j>( z ) Perturbatin amplitude functin in shear-layer stability analysis, ft sec <P' Perturbatin stream functin in shear-layer stability analysis, ft sec 36

41 APPENDIX I: METHOD FOR FITTING HAZEL'S THEORETICAL VELOCITY PROFILE TO DATA FOR "S-SHAPED" VELOCITY PROFILES This Appendix describes the methd which was used t fit the theretical velcity prfile f Hazel (Ref. 7) t the "S-shaped" velcity prfiles btained experimentally in the UARL Open Water Channel. The theretical prfile was given previusly in Eq. (6). It will als be helpful t refer back t Sketch C in the main text. Fitting Eq. (6) t a measured prfile invlves determining the cnstants b,av, and d. In the fllwing develpment this will be dne by matching the velcities Vi and V2 at z^ and z 2, the shears at Z]_ and Z2 (which are zer), and the mean shear, (9V8z) 0, at z 0 f the measured prfile t thse f the theretical prfile. The shear can be btained by differentiating Eq. (6) with respect t the vertical crdinate z: *"* (&) {[»-' W(^)]-b}sech b (^) (19, In this equatin, the fllwing substitutin was made : At zi the shear is zer, s Eq. (19) yields (dvdz) 0 = j (20) sech(( 2 -Z)d) = V-5TT (21) Substitutin f Eq. (21) int Eq. (6) prvides an expressin fr V]_: V, = V 0 + (^) b*(b + l)~^ (22) Replacement f V 0 with (V-^ + V 2 )2 in Eq. (22) yields an equatin fr AV as a functin f b, V^, and V 2 : 4 ^LL (23) AV = (v.-vj b 2 (b+i) 2 Nw, substitutin f AV frm Eq. (20), (z-^ - z 0 )d frm Eq. (21) and V 0 (V x + V 2 )2 int Eq. (22) results in 2\ b b+l [(z -z 0 )(V l -V 2 )](avaz) 0 = Y (arccsh (-^y-) ) b 1 (b + l) 2 {2k) 37

42 Using a similar develpment but with the cnditin that the shear is als zer at Z2 yields an expressin that is similar t Eq. (2k) except that z-y is replaced by zg and the right side f the equatin is negative. Cmbining this result with Eq. (2k) prduces f(z,-z 2 )(V,-V 2 )l(dvdz) 0 = (arccsh (25) The left side f Eq. (25) can be determined entirely frm prperties f the measured prfile. The right side is nly a functin f b. Therefre, the value f b fr a given prfile can be btained by pltting b versus the right side f Eq. (25) and entering the plt at the value given by the left side f Eq. (25). After btaining b fr the prfile, AV can be btained frm Eq. (23) and, finally, d can be btained frm Eq. (20). Thus, all f the parameters needed in Eq. (6) t describe the velcity V as a functin f z have been determined. 38

43 APPENDIX II: SUMMARY OF OTHER FLUID MECHANICS INVESTIGATIONS Experimental Investigatins f Stability f Straight, Stratified, Shear Flws Having Multiple Shear Layers The purpse f these investigatins was t investigate the stability f a shear layer cnsisting f a cmbinatin f several thinner stable and unstable layers. Experiments were cnducted t investigate cnditins under which such flws becme unstable and the characteristics f the flw during the initial phases f breakdwn. Descriptin f Cmbined Shear Layer Experiments The tw types f velcity prfiles which were investigated are shwn in Sketch G. The prfiles are designated Type LSL and Type SLS (S fr small shear and L fr large shear). Type LSL velcity prfile cnsists f a layer having small shear sandwiched a. O O TYPE TYPE SLS Hi < I > CO < UJ I VELOCITY, V COMBINED SHEAR LAYER VELOCITY, V \ COMBINED SHEAR LAYER 1» SKETCH G. VELOCITY PROFILES USED IN STABILITY TESTS OF COMBINED SHEAR LAYERS between tw layers having large shears. Type SLS cnsists f a layer having large shear sandwiched between tw layers having small shears. Tests were made t determine if turbulent breakdwn f layers having large shears wuld cause a cmbined shear layer, cnsisting f unstable large-shear layers and stable small-shear layers, t becme unstable and break dwn. Accrdingly, attempts were made t establish density gradients which wuld stabilize the small-shear layers (Ri > 0.25) but nt the large-shear layers (Ri < 0.25). 39

44 Summary f Experimental Results Typical velcity, temperature, and density prfiles fr the cmbined shear layer experiments are shwn in Jig. 26. Prfiles fr a type LSL cmbined layer having a layer f small shear (dvdz = sec -1 ) sandwiched between tw layers f large shear (lwer layer 3VBz = -2.7 sec"l upper layer 3V3z = sec -1 ) are shwn in Fig. 26(a). The Richardsn numbers, Ri, fr the upper, middle, and lwer shear layers were O^OUU, 0.272, and 0.003, respectively. Thus, the upper and lwer layers wuld be expected t becme unstable and the middle layer t be marginally stable. The stages f breakdwn which were bserved during this test are sketched in Fig. 27(a). First, there is a regin where the flw appears t be undisturbed. Then waves appear in the unstable layers. The waves in the upper layer (V" equal t 0.02 ftsec) have a wavelength, \, f k in. and appear upstream frm the waves in the lwer layer (V 0 equal t 0.09 ftsec) which have a wavelength f 2 in. The waves grw in amplitude and then transitin t vrtices. Further dwnstream the vrtices interact with each ther causing turbulent breakdwn f the cmbined layer. In the turbulent regin there was sme evidence that the cmbined layer was influencing the breakdwn. There appeared t be an verall swirling" Uiulluil superimpsed n the turbulence. This swirling mtin was peridic and had a wavelength, A, apprximately equal t 8 in. This wavelength is mre characteristic f the length expected based n the cmbined layer thickness than n the thickness based n any f the individual layers and suggests, therefre, that the cmbined layer is influencing the breakdwn (frm Refs. 1 and 3, the expected wavelength is (V2uV2d). Typical prfiles fr cmbined layer type SLS having a layer f large shear sandwiched between tw layers f small shear are shwn in Fig. 26(b). The Richardsn numbers, Ri, fr the upper, middle, and lwer shear layers were 2.02, 0.01U, and 0.21, respectively. Thus, the upper layer wuld be expected t be stable, the lwer layer t be slightly unstable, and the middle layer unstable. The stages f breakdwn which were bserved during this test are sketched in Fig. 27(b). First, there is a regin where the flw appears undisturbed. Then waves appear which have a wavelength,a, equal t U_in. These waves grw in amplitude and then transitin t vrtices. Further dwnstream sme f these vrtices cntinue t grw in size and sme are sujjjnr^yyjed. The wavelength assciated with the grwing vrtices was 8 in. and, based n the thickness f the layers, was mre characteristic f the cmbined layer than any f the individual layers. Further dwnstream, turbulent breakdwn ccurred as the grwing vrtices interacted with each ther and the suppressed vrtices. Observatins f the stages f breakdwn during these tests indicated that the breakdwn f the unstable layers was'nt)independent f the adjacent stable layers. The grwth f instabilities initiated"in the unstable layers appeared t be fed in part by the shear energy frm the adjacent stable shear layers. The final stages f k

45 the breakdwn appeared t be influenced by the cmbined layer at its characteristic wavelength. This was mst bvius in the test f cmbined layer type SLS. Here, then, is a case where small amplitude instabilities having shrt wavelengths result in larger amplitude instabilities having lnger wavelengths. These results are in agreement with yjgyr^yjgl wrk dne by Atlas (Re ^26) and experimental results btained by Thrny fref Atlas shwed that nce the minimum Richardsn number in a layer decreases t a value less than 0.25, any resulting instability wuld grw in amplitude until the layer Richardsn number was Q.»5 (lsses because f dissipatin wuld result in grwth t a layer Ri less than 0.5). Thus, a thin, unstable (Ri<0.25) shear layer in a thick slightly stable (Ri> 0.25) shear layer may result in the destabilizatin f the thick as well as the thin layer, prvided the layer Richardsn number is 0.5. The ^^^^ Richardsn number is an average Richardsn number based n the initial velcity and density prfiles and a layer thickness equal t tfi^the maximum amplitude f the instability. TJirne has bserved that when the Kelvin-Helmhltz type f instability ccurs in stratified shear flws the vlume f turbulent fluid grws until the layer Richardsn number is abut 0.^. i J l U 1 Cncluding Remark-' f OM'r The results f these experiments indicate that, fr the type f velcity and temperature prfiles tested, the breakdwn f cmbined shear layers cnsisting f stable and unstable layers is initiated by the instability f the unstable layers and that the final stages f breakdwn are influenced by the c mbined layer at its < characteristic wavelength. The bservatins indicated that, in cmbined shear layers, the small amplitude shrt-wavelength disturbances assciated with the initial instability can result in large amplitude, lng-wavelength disturbances. Stability Criterin fr "Three-Dimensinal", Straight, Stratified Shear Plws The value f stability criteria derived by energy methds is pen t questin. Hwever, since such criteria have been derived by ther investigatrs, a simple criterin fr the three-dimensinal case was derived and is presented in this ^Appendix. Chandrasekhar (Ref. 8) determined the critical value f Richardsn number fr tw-dimensinal stratified shear flws using energy cnsideratins. Chandrasekhar ^reasned that fr a stably stratified shear flw t becme unstable there must be enugh kinetic energy available frm the shear t wrk against the buyant stabilizing frces. Chandrasekhar determined the critical Richardsn number by equating the wrk required t interchange fluid particles (l) and (2), shwn in Sketch H, t the kinetic energy available frm the difference in velcity between particles (l) and (2). Ul

46 p V p V p + 8p p + 8p 1 (i) (2) 4 V v + Sv v +3 v p + Sp v +5v SKETCH H. DENSITY AND VELOCITY DISTRIBUTION FOR TWO-DIMENSIONAL STABLY STRATIFIED SHEAR FLOW ANALYSED BY CHANDRASEKHAR It was shwn (Ref. 8) that the wrk per unit vlume, Wg, required t interchange particles (l) and (2) is given by W B = -q -Bp-Bz (26) and that the kinetic energy per unit vlume, E, available frm the shear layer is given by E = (8V) 4 (27) Fr instability t ccur, E must be greater than Wg: W B E = q-bp-sz f (8Vr4 The critical Richardsn number is btained by dividing the numeratr and denminatr in Eq. (28) by Sz and is given by < 1.0 (28) g(spsz! D(SV8Z) 2 = Ri < 0.25 (29) Thus, fr tw-dimensinal stably stratified shear flws ne wuld expect instability t ccur when the Richardsn number was less than This result is in agreement with theretical stability criteria f Drazin, Hazel, and thers. It is als in agreement with experimental results btained in this prgram and reprted previusly in Refs. 1 thrugh 5. 42

47 The methd f Chandrasekhar was extended t investigate the stability f "three-dimensinal" stably stratified shear flws fr which the thickness f the shear layer was small cmpared t the width f the shear layer; i.e., the velcity prfile was three-dimensinal. A side view f the flw sectined thrugh the shear layer n the centerline f the "three-dimensinal" flw is the same as that shwn in Sketch H. A frnt view f the flw after fluid particles (l) and (2) have been interchanged is shwn in Sketch I. REGION OF SHEAR REGION OF [-«UNDISTURBED*-) LAYER \~* UNDISTURBED FLOW ZONE FLOW SHEAR LAYER ZONE SKETCH I. FRONT VIEW OF "THREE-DIMENSIONAL" STABLY STRATIFIED SHEAR FLOW ANALYSED USING METHOD OF CHANDRASEKHAR In Sketch I, the flw directin is ut f the paper. Fr this analysis it is assumed that hrizntal velcity gradients are negligible. The difference in velcity and density between particles (l) and (2) is SV and p, respectively. The critical Richardsn number f the three-dimensinal flw shwn in Sketch I can be btained, in a manner similar t the methd used fr tw-dimensinal flws, by equating the kinetic energy per unit vlume available frm the shear t the energy per unit vlume required t interchange particles (l) and (2). The wrk per unit vlume, W B, required t vercme buyant frces and the kinetic energy per unit vlume, E, available frm the shear are the same fr the three-dimensinal flw as fr the tw-dimensinal flw and were given previusly in Eqs. (26) and (231), respectively. Hwever, fr the three-dimensinal flw, as particles(l) and (2) are interchanged, pressure frces ccur which tend t make the fluid spread laterally. This lateral mtin des nt help d^yjjgmik ze the flw and, since the kinetic energy per unit vlume f this mtin must als cme frm the kinetic energy per unit vlume assciated with the shear, the three-dimensinal flw appears mre stable than the tw-dimensinal \ 1.3

48 flw. The kinetic energy per unit vlume assciated with the lateral mtin was assumed t be equal t the ptential energy per unit vlume assciated with the lateral pressure frces which wuld act n the particles if the particles were restrained frm mving laterally while they are being interchanged. The lateral pressure frces, which are equal t ptential energy per unit vlume, result frm the differences between the pressure at the center f particles (l) and (2) (at pints 2 and 5 shwn in Sketch I) and the pressure at the sides (at pints 1, 3, k, and 6) f the particles. These pressure differences, which are equivalent t ptential energy per unit vlume, are given by (p 2 -p,) (2) = (p 2 -p 3 > (2, = g-fy> -sz2 (30) (p 5 -P4) ( ir ( p 5-p 6 ) ( i) : g %> sz2 (31) These pressures result in frces which wuld tend t split the particles causing flw t bth sides. Assuming that the kinetic energy per unit vlume assciated with lateral mtin is equal t the ptential energy per unit vlume assciated with (and equal t) the lateral pressure differences (pressure equivalent t ptential energy per unit vlume), the kinetic energy per unit vlume is given by W F = -g Bp 8z (32) The ttal energy per unit vlume required t interchange the particles is the buyant wrk WB (Eq. (26)) plus the kinetic energy per unit vlume assciated with the lateral pressure differences, Wp (Eq. (32)): W T = - 2g-8v8z (33) Fr instability t ccur the kinetic energy per unit vlume, E, available frm the shear (Eq. (27)) must be greater than the ttal energy per unit vlume, W : W T 2g 8-8z,. E y(8v) 2 4 The critical Richardsn number is btained by dividing the numeratr and denminatr in Eq. fak) by <5z: g(88z) p(8v8z) 2 Ri < (35) Thus, fr three-dimensinal, stably, stratified, shear flws ne wuld expect instability t ccur nly when the Richardsn number was less than On the basis f this result, then, "three-dimensinal" flws wuld be mre stable than twdimensinal flws which have a critical Richardsn number f This result is cnfirmed t sme degree by experimental results btained during this prgram and discussed previusly. kh

49 FIG. 1 LU Q- O CC C0 h5

50 FIG. 2 LU CD >- K <C _l h- m O <c _l t OQ ce ca LU LL. Q_ >- z* 31 O ce CC O I- _l LU CC _J LU <c LU i- ce 1X5 "z: C Q 3: 3: ^ CC LU LU _l * 2 LL. - Ee - 1 Q_ > O P" 'D3awnN 3AVM SS31NOISN3WIC] k6

51 FIG. 3 TYPICAL STAGES OF BREAKDOWN OF FLOW IN SHEAR LAYERS HAVING "S-SHAPED" VELOCITY PROFILES V 0 = 0.07 FTSEC (dvd*) 0 = SEC " 1 0T^) 0 = 0 FLOW <TT77 11 I I I I I I I I I I I I I I I I I I I I I 111 I I I I I I I I I I 141 I I I I I I I I I > FLOOR -i = 0 () x = 8 IN. - UNDISTURBED (b) x = 36 IN. - WAVES * (c)x =60 IN. - VORTICES (d)x = 70 IN. - TURBULENCE ^7

52 TYPICAL "S" VELOCITY, TEMPERATURE AND DENSITY PROFILES FOR SHEAR-FLOW ' EXPERIMENTS IN WATER CHANNEL _ CORRESPONDING PROFILE IN HAZEL'S THEORY SEE EQS. (6) AND (8) FOR THEORETICAL VELOCITY AND DENSITY PROFILES - O _ ar 0 dz - - DEG FFT Jp. - dz (b) Ri , b = 0.3 VELOCITY, V - FTSEC TEMPERATURE, T - DEG F i i DENSITY, p - SLUGSFT' X 10-2 < \\ (SLUGSFT 3 )FT 6\ VELOCITY, V - FTSEC TEMPERATURE, (c) Ri = 0.5, b = 0.4 T - DEG F \^X> DENSITY, p - SLUGSFT SEC -1^ ^^ 0-0 W VELOCITY, V - FTSEC TEMPERATURE, T -DEG F DENSITY, p - SLUGSFT 3

53 COMPARISON OF WATER CHANNEL RESULTS FOR "S-SHAPED" VELOCITY PROFILES WITH HAZEL'S CRITERIA FOR STABILITY FIG. 5 FLOW DEVELOPED USING TWO-DIMENSIONAL CONTOURED FILTER BEDS SEE EQS. ( 6 ) AND (8 ) FOR THEORETICAL VELOCITY AND DENSITY PROFILES RICHARDSON NUMBER, Ri

54 FIG. 6 LU g rr LU O a: a. a. < * 9 X * CM CD _l LU > LU! : 1 CD 50

55 FIG. 7 TYPICAL STAGES OF BREAKDOWN OF FLOW IN SHEAR LAYERS HAVING "THREE-DIMENSIONAL" VELOCITY PROFILES V = 0.07 FTSEC dtdi = 0. SURFACE x=6 IN. ^mm^m. sf 77 7 i i) ) i i i i i > i i i i i I i > > i i i i i i i > i i i t i i 1 t in i i r 11 FL : LOOR i = 0 (a) x = 3 IN. - UNDISTURBED (b) x = 10.5 IN. - WAVES (e) x - 21 IN. - VORTICES (d) x 62 IN. - TURBULENCE '1

56 TYPICAL VELOCITY, TEMPERATURE AND DENSITY PROFILES FOR "THREE-DIMENSIONAL" SHEAR-FLOW EXPERIMENTS FIG. 8 ALL MEASUREMENTS 2 IN. DOWNSTREAM FROM TAPERED FILTER BED AT CENTER OF CHANNEL (a) Ri =0, w(2d) m = dr VELOCITY, V - FTSEC TEMPERATURE, T - DEG F (b) Ri =0.07,, w(2d) m = J.938 DENSITY, p - SLUGSFT 0 O UJ z < X <J UJ > < X UJ X VELOCITY, V - FTSEC TEMPERATURE, (c) Ri , w(2d) m =9.8 T ' DEG F DENSITY, p - SLUGSFT T _ (SLUGSFT3)FT-' VELOCITY, V - FTSEC TEMPERATURE, 52 T-DEGF DENSITY, P - SLUGSFT 3

57 COMPARISON OF RESULTS FOR FLOWS HAVING "THREE-DIMENSIONAL" SHEAR LAYER WIDTH-TO-THICKNESS RATIOS BETWEEN 4.8 AND 8.6 WITH DRAZIN'S CRITERION FOR TWO-DIMENSIONAL FLOWS FIG. 9 DRAZIN'S CRITERION FOR HYPERBOLIC TANGENT VELOCITY PROFILES SYMBOLS (j SMALL-AMPLITUDE WAVES q INTERMITTENT WAVES WAVES TRANSITION TO VORTICES O' WAVES TRANSITION TO VORTICES (SMALL-AMPLITUDE) WAVES TRANSITION TO VORTICES AND TURBULENCE 4 WAVES TRANSITION TO VORTICES AND TURBULENCE (SMALL-AMPLITUDE) X NO INSTABILITY OBSERVED I.B 4 3 f UJ 5 UJ > < * UJ O \n z UJ 1.6 I > r( <.*«* V UNSTABLE ad = V 22 \ = (2 irv T)-2d 7 STABLE \ DRAZI >I'S CRITERIO N^ 1 'H±VV4-Ri "f - 2d n ^ X -X X X X RICHARDSON NUMBER, Ri

58 COMPARISON OF RESULTS FOR FLOWS HAVING "THREE-DIMENSIONAL" SHEAR LAYER WIDTH-TO-THICKNESS RATIOS BETWEEN 9.7 AND 14.3 WITH DRAZIN'S CRITERION FOR TWO-DIMENSIONAL FLOWS FIG. 10 DRAZIN'S CRITERION FOR HYPERBOLIC TANGENT VELOCITY PROFILES SYMBOLS O" SMALL-AMPLITUDE WAVES Q. INTERMITTENT WAVES WAVES TRANSITION TO VORTICES ef WAVES TRANSITION TO VORTICES (SMALL-AMPLITUDE), WAVES TRANSITION TO VORTICES (INTERMITTENT) WAVES TRANSITION TO VORTICES AND TURBULENCE X NO INSTABILITY OBSERVED a d = \ 22 A = (2 it \ r 2~) 2 d, 1.0 DC UJ CO => z > < UJ _l z z UJ UNSTABLE < 1 5 Q * STABLE 0.4 <* e DRAZIN'S CRITERION 9 ^- a A = V J4 * It-RI =x y 2 d 0.2 5^ 0.04 XX X X* X X RICHARDSON NUMBER, Ri L

59 COMPARISON OF RESULTS FOR FLOWS HAVING "THREE-DIMENSIONAL" SHEAR LAYER WIDTH-TO-THICKNESS RATIOS BETWEEN 14.5 AND 40 WITH DRAZIN'S CRITERION FOR TWO-DIMENSIONAL FLOWS FIG. 11 DRAZIN'S CRITERION FOR HYPERBOLIC TANGENT VELOCITY PROFILES SYMBOLS d SMALL-AMPLITUDE WAVES C\ INTERMITTENT WAVES WAVES TRANSITION TO VORTICES efwaves TRANSITION TO VORTICES (SMAL L- AMPLITUDE) swaves TRANSITION TO VORTICES (INTERMITTENT) WAVES TRANSITION TO VORTICES AND TURBULENCE X NO INSTABILITY OBSERVED A «d >, T2 (2irV 2) -2 6 ~J 1.0 CD 5 > < Z 11 z UJ Q > * e «, * f ** < * UNSTABLE rf STABLE M \ DRA ZIN'S CRITEF ION: IK (-V4-Ri t. 2d A (1 x x RICHARDSON NUMBER, Ri 55

60 CO O FIG.12 c I LU LU CO <: LU < <: CO CO CO LU CO >- v) UJ > < a UJ 3 u H z _J Z UJ a. UJ -1 X K 3 < H I UJ h < (- Q 5 v» z Z < v) vi vi vi UJ UJ Ul UJ U u U u H i- H i- v> ce cc r : Q Ixl O O UJ > < > > > > > * v> O a: UJ UJ > 1-1- UJ a < z Z z z m 3 3: O H K F 1- t- H >- 1- _J Z VI VI v1 VI Q. UJ z z z z -J s H < < < < < CQ t- O: a: UL a: < 1 2 H H H t- -I C vi v> wi VI vi -I UJ UJ UJ UJ UJ z < 1- > > > > " s 2 < < < < VI * * 3: i z > cf ^ C x LU O UJ u. "(P3)* 'M3AVT UV3HS JO SS3N)I3IH1 01 HiaiM =10 OliVU 56

61 FIG. 13 EFFECT OF A LONG-WAVELENGTH WAVE ON THE MINIMUM RICHARDSON NUMBER IN A SHEAR FLOW (a) SCHEMATIC DIAGRAM OF FLOW CONDITION UPSTREAM VELOCITY AND TEMPERATURE PROFILES (b) EFFECT OF TEMPERATURE GRADIENT AND INITIAL SHEAR ON Ri M1N IN WATER V 0 =0.10 FTSEC Ri MIN [l&<"«-" -^ T 0 =60 DEG F = 0.05 FT TEMPERATURE GRADIENT, dldz - DEG FFT 57

62 WATER CHANNEL RESULTS SHOWING EFFECT OF A LONG-WAVELENGTH WAVE ON THE LOCAL RICHARDSON NUMBER IN A SHEAR FLOW FIG. 14 V =0.07 FTSEC Ri =0.70 (<?V<?i)0= SEC~' SEE FIG. 16 FOR VELOCITY, TEMPERATURE AND DENSITY PROFILES X \ FLOOR - x - 0 () x = 7 IN.-TROUGH <?V'di= SEC"', r9tr)i = 120 DEG FFT Ri = 0.37 (b) x - 14 IN.-CREST dv'dz SEC" ', r9t(9i - 82 DEG FFT Ri «1.8 58

63 PHOTOGRAPHS SHOWING EXAMPLES OF INSTABILITIES INDUCED BY LONG-WAVELENGTH WAVES SEE FIG. 14 FOR FLOW CONDITIONS AND FIG. 16 FOR VELOCITY, TEMPERATURE, AND DENSITY PROFILES DIRECTION OF FLOW FIG.15 (a) FIRST TROUGH TO SECOND CREST (X=7 TO 14 IN.hWAVES (b) SECOND CREST TO SECOND TROUGH (X= 14 TO 21 IN.)- VORTICES 59

64 TYPICAL VELOCITY, TEMPERATURE AND DENSITY PROFILES FOR FLOW IN A LONG-WAVELENGTH WAVE (a) VELOCITY SEE FIGS. 14 AND 15 FOR PHOTOGRAPHS OF THE FLOW V 0 = 0.07 FTSEC, 0Vdz) = SEC TROUGH-X = 7 IN. 1 A LW = 14 IN., a = 1.4 IN. CREST-X = 14 IN. FIG (b) TEMPERATURE VELOCITY, V-FTSEC (c) DENSITY TEMPERATURE, T-DEG F DENSITY, p-slugs FT

65 FIG. 17 COMPARISON BETWEEN MEASURED AND PREDICTED VALUES OF WAVE SHEAR IN A LONG-WAVELENGTH WAVE SHEAR PREDICTED USING PHILLIPS' THEORY (REF.9) 2, A(<9V«9Z) =(OV 0 )(NM -" 2 MEASURED WAVE SHEAR GENERALLY BASED ON FIRST HALF WAVELENGTH OF LONG-WAVELENGTH WAVE DATA FROM UARL OPEN WATER CHANNEL > "3 UJ X «> UJ > < UJ 2 PREDICTED WAVE SHEAR, la (dv<9i) p - SEC ' 61

66 FIG. 18 COMPARISON BETWEEN MEASURED AND PREDICTED VALUES OF WAVELENGTH OF LONG-WAVELENGTH WAVES DATA FROM UARL OPEN WATER CHANNEL WAVELENGTH PREDICTED USING HAURWITZ'S THEORY (REF. 10) ^0 2, <iv * 1 (H- -? <P 2 ~ P\ > W2 P 2 3 z UJ > < UJ a: < UJ 2 24 (A L w)m(alw)p= 1.0 -v - s -X 20 ^ ^ 16 " ^" "^ 12 CDO O Q.O 8 y 4 n ^ i i i i i i i PREDICTED VAVELENGTH, (A LW )p - IN. 62

67 EFFECT OF WAVE AMPLITUDE RATIO ON STABILITY BOUNDARIES FOR SHEAR FLOWS IN LONG-WAVELENGTH WAVES DATA FROM UARL OPEN WATER CHANNEL FIG. 19 SYMBOL O n, av 0 LONG-WAVELENGTh WAVE OBSERVED? YES NO LET- TER W V T INSTABILITY OBSERVED WAVES VORTICES TURBULENCE < )"-<*) u > CD X U. O < > s c < 3.2 WV < \ WV WVT w 0.42 NEUTRAL STABILITY BOUNDARY F0RV - WVT wv \y \\y 0.68 \y FLOW UNSTABLE EVEN WITHOUT WAVES (Ri < 0.25) wv A WV» 0.40 \Y Ay w* Ay A 1.32 \ DENOTES WAVE-INDUCED SHEAR INSTABILITY SUSPECTED J UNSTABLE FORaV 0 > 0.5, STABLE FOR av 0 -= 0 v\\)a^<^v " A>^ UNSTABLE FOR av, > 0.75, STABLE FOR av Q < 0.5 WV Ay wv t \V WV 0.51 AV' WVT*,\y A WVTQ ix ^UNSTABLE FOR av 0 > l.o* 4^ WVT*1.31^\\y,\\)^, Z^^ STABLE FOR a V ^V^, 1.0 ' WVT WVT W n xvvlllr'** 0 y%^wvt*ou67 ip+^ulh, O0.98^V XwuSO-?0 X<^ ^0.78 OOr _i an \J. \\JT VA WQO.95 > ^<Ulj 11 V JNSTABLE FOR V Q^ 2.0 \ STABLE FOR av 0 < ' 98 OI %. FLOW STABLE FOR 'V V V 0 ^ 2.0 Xy av 0 =- 1.0 V BRUNT-VAISALA FREQUENCY, N M - SEC " ' X, 63

68 FIG. 20 EFFECTS OF LONG-WAVELENGTH WAVES ON STABILITY OF ATMOSPHERIC SHEAR LAYERS () SCHEMATIC DIAGRAM OF FLOW CONDITION MAXIMUM SHEAR, Ri MIN AT CREST 2d- UPSTREAM WIND AND TEMPERATURE PROFILES (b) EFFECT OF LAPSE RATE AND SHEAR ON Ri MIN FOR TYPICAL WAVE CONDITIONS 1.0 KM) J * -- fe) a = 3000 FT, V = 30 KTS, T Q = 213 K SEC - ' (1 KT1000 FT) 6k ENVIRONMENTAL LAPSE RATE, dt<fe - DEG CFT 4X 10"

69 FIG. 21 EFFECT OF WIND VELOCITY ON STABILITY BOUNDARIES FOR ATMOSPHERIC SHEAR FLOWS IN A LONG-WAVELENGTH, 500-FT-AMPLITUDE WAVE ALTITUDE, i = 34,089 FT TEMPERATURE, T =-56.5C WAVELENGTH, A LW - 15 NMI NOTE: FLOW STABLE FOR (d Vd*) AND dt<?z BELOW BOUNDARY FOR GIVEN V 0 AND UNSTABLE FORJ(dVd*) 0 AND dtdi ABOVE BOUNDARY VELOCITY, V = 100KT^ - 75-v \ ^^S=^-""~^ Li. O s 40 I _ 20 > at < 10 8 ADIABATIC LAPSE RiTE \, A i 1 i f 10- """lo-' 25- a z VERTICAL TEMPERATURE GRADIENT, 3TS z - (DEG C)FT

70 FIG. 22 EFFECT OF WIND VELOCITY ON STABILITY BOUNDARIES FOR ATMOSPHERIC SHEAR FLOWS IN A LONG-WAVELENGTH, 3000-FT-AMPLITUDE WAVE ALTITIDE, i = 36,089 FT TEMPERATURE, T C WAVELENGTH, A w = 15 NMI NOTE: FLOW STABLE FOR <<9Vd*) 0 AND dtfa BELOW BOUNDARY FOR GIVEN V 0 AND UNSTABLE FOR ((9Vdz) 0 AND dtdi ABOVE BOUNDARY I ADIABATIC LAPSE RATE > 1 1 u. r VELOCITY, V 0 > A-75 -^ t 100 KT > n < r\ ~ 50 ^\ Z * < \ lr VERTICAL TEMPERATURE GRADIENT, dtdz - (DEG C)FT

71 FIG. 23 EFFECTS OF WIND SPEED AND LONG-WAVE WAVELENGTH ON MAXIMUM ALLOWABLE INITIAL WIND SHEAR FOR STABILITY (dvdx) ' MAX «V5x) 0 MAX.[1 + (2^A L l[v 0 al,a LW =15NMI x 200 < s > CO < v1 OH < 5 20 Q E 10 ^ x < _l CO < O WIND VELOCITY, V 0 -KT

72 1 r O I) N III I i t UJ I t J I L FIG i * > " a -J < Ul S LU i i CD N r. I O CL LU Uj Q Q UJ!H «N - i! Ul _j UJ f * 5 < < Ul < j 5 * Ul < J u. a g _l u < u f 2 ^ O H DC,_, U. l- >> 1 r O Ul I Ul Z a 1- O < z K r < Ul Ul a r * 151 O UJ f i «. J z C3 -I CK CD < z; ui «x *> V* a. m ce "IB z * J s z z 31 Ul r 3 ui z a I S z a»- " z O 1- * 5) H < 5 r i-2 < z * E 68 a z z * < r

73 FIG. 25 COMPARISON OF DATA FROM 1970 LEE WAVE OBSERVATION PROGRAM WITH UNITED AIR LINES NOMOGRAM FOR PREDICTING WAVE OCCURENCE NOTES: I. NUMBER NEAR SYMBOL DENOTES FEBRUARY, 1970 DATE 2. NO FLAG DENOTES FLIGHTS NOT MADE ASiNO WAVE ACTIVITY WAS PREDICTED BY OTHER METHODS 3. SINGLE FLAG DENOTES WEAK WAVE OBSEf^vn m PLJpHT - 4. DOUBLE FLAG DENOTES MODERATE WAV VAVE OBSERVED IN FLIGHT - 20 i < ^ Z NO V fave WEAK W VE O > 16 MODERATI : OR STRONG WAVE 10 ^ 18 at z UJ LU * UJ U z UJ at UJ Q UJ at =3 ul t> UJ at a I > 27 O <C O Ol2 8 2 ' 19 1 c O i MAXIMUM WIND VELOCITY PERPENDICULAR TO MOUNTAINS, V-KTS 80 69

74 TYPICAL VELOCITY, TEMPERATURE AND DENSITY PROFILES FOR COMBINED SHEAR LAYER EXPERIMENTS FIG.^6 (a) COMBINED LAYER TYPE LARGE-SMALL-LARGE 0- i O O z < x u LU > < X VELOCITY, V-FTSEC 40 "50 60 TEMPERATURE, T-DEG F r) pdz = (SLUGSFT 3 )FT DENSITY, p-slugsft 3 - Vs (SLUGSFT 3 )FT V, (SLUGS FT 3 ) - dpd* -XL \. (SLUGS FT 3 ) FT- 1 1 \l VELOCITY, V-FTSEC TEMPERATURE, T-DEG F DENSITY, p-slugsft 3 FT 70

75 FIG. 27 t LU >- t Q LU Q :«: < LU r CQ Q: LL. < _i CO _i LU _J O < <: 5 vo h- c LU _l O Q_ < _J LU >- 0. (- >- h- Ct LU > < _1 Q LU z c 5 u < I LU I -i LU Q. a: LU Q LU CQ 5 liiiinniuu lsl_ IJ-'T'tlll 11 f T ' ' ' * ' * -I 1L -y NASA-Langley, CR-1985 A