NPS ARCHIVE 1968 KLINEDINST, P. GENERATION OF WATER WAVES BY TURBULENT WIND FLOW by Paul Richard Kl i nedi nst, Jr, LIBRARY NAVAL POSTGRADUATE SCHOQfc CJLIF. 93940 DUDLEY KNOX LIBRARY NAVAL POSTGRADUATE SCHOOL MONTEREY, CA 93943-5101 UNITED STATES NAVAL POSTGRADUATE SCHOOL THESIS GENERATION OF WATER WAVES BY TURBULENT WIND FLOW by Paul Richard Klinedinst, Jr. June 1968 Approved for public release; distribution unlimited LIBRARY NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIF. 93940 GENERATION OF WATER WAVES BY TURBULENT WIND FLOW by Paul Richard Klinedinst, Jr. Lieutenant Commander, United States Navy B.S., U. S. Naval Academy, 1958 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN OCEANOGRAPHY from the NAVAL POSTGRADUATE SCHOOL June 1968 icuNewNsr, p. ABSTRACT The paper is concerned with a theoretical study of shearing flow bounded by a wavy surface with consideration of turbulent flow above the air-sea interface. Account is to be taken of turbulence in the flow through the use of Reynolds stresses associated with turbulent flow. An adaptation of the mixing length theory as applied to pipe flow is made for channel flow and the resulting mixing length versus energy relation- ship is incorporated in the Reynolds stress term. Finally, the rate of wave growth is calculated from the normal surface pressure in phase with the wave slope. A curvilinear coordinate system which follows with the wave train is used in order to simplify the formulation of the problem. All para- meters are non-dimensionalized and the analysis is made considering a velocity profile adapted from pipe to channel flow. LIBRARY E SCH00L NAVAL P°sT^p 93940 DUDLEY KNOX LIBRARY MONTEREY, CALir . NAVAL POSTGRADUATE SCHOOL MONTEREY, CA 93943-5101 TABLE OF CONTENTS Section Page 1. Introduction 13 2. Formulation of the Model 17 2.1 The Zero-Order Flow 17 2.2 The First-Order Flow 22 2.3 Normal Pressure, Shear Stress and Wave Growth 28 3. Solution of the First-Order Problem 31 4. The Computer Solution 34 5. Results and Conclusions 35 6. Bibliography 4 6 7. Appendix I 4 7 . LIST OF TABLES Table Page 1. Wall pressure, shear stress and wave growth rate as a function of perturbation surface roughness. 39 2. Stream function values near the critical level. 45 LIST OF ILLUSTRATIONS Figure Page 1. Channel and velocity profile. 19 2. Typical velocity profile and wave form. 22 3. Inverse hyperbolic tangent velocity profile. 35 4. Wall pressure versus d, various cases. 37 5. Wall pressure versus d, converging case. 38 6. Wall pressure versus d, non-converging case. 41 7. Stream function. 43 SYMBOLS AND ABBREVIATIONS a non-dimensional wave amplitude A constant parameter c non-dimensional wave celerity C constant d position at which upper boundary conditions are applied E' turbulent energy E wave energy E rate of energy input to wave F function determined by solution Froude number r g acceleration of gravity (980 cm sec ) H channel height J Jacobian K kinematic eddy viscosity mixing length mixing length (perturbation term) mixing length (zero-order term) P pressure q velocity of main flow above boundary layer U velocity of main flow in the channel u non-dimensional velocity parallel to \ axis u* friction velocity SI 2, L v non-dimensional velocity parallel to V\ axis °<^ constant X real component of pressure o imaginary component of pressure € real component of shear stress £ imaginary component of shear stress o constant parameter X mixing length constant (0.4) A non-dimensional wave length ^ stream function AU. molecular viscosity B^ air density (0.0013 g cm"^) JJj water density (1.027 g cm-3) L shear stress 1* 7) curvilinear coordinates 10 ACKNOWLEDGMENT Professor Theodore Green III of the Naval Postgraduate School is responsible for the turbulent equations of motion and the differential equations which provide the basis for the theoretical development of the model. He also suggested the application of Zagustin's work, to channel flow. The author wishes to express his appreciation to Professor Green for the assistance he has given in formulating the model, in checking the results and in improving the presentation of the work. 11 1. INTRODUCTION A semi-empirical study of turbulent air flow over a wavy surface is made in which normal and tangential stresses on the surface are examined with the object of estimating the growth of water waves due to a turbulent wind . Ursell (195 6) gives a summary and a critical review of the theories of wind-wave formation extant in 1956. The review gave the necessary impetus for further investigations, both theoretical and experimental, in this area. Since 1956 there has been much work done on the generation of water waves by surface stresses induced by the wind. Miles (195 7) and Benjamin (195 9) considered laminar air flow over a wavy water surface and deduced formulae for exponential wave growth due to an instability mechanism. Later Phillips (1966) included air-stream turbulence in the instability model from a semi-empirical point of view. In his initial work, Miles (1957) assumes an inviscid, incompress- ible, two-dimensional, parallel shear flow coupled with a prescribed two-dimensional deep-water gravity wave. Turbulent fluctuations in the perturbation equations of motion are neglected, even though they are decisive in maintaining the mean shear flow. Furthermore, the wave- induced velocities and pressure are small, which justifies linearization of the equations. The water is also assumed to be inviscid, incompress- ible, and irrotational, and the slope of the displaced surface is assumed to be small. Only that component of the aerodynamic force in phase 13 with the wave slope is important in the wave-generation mechanism. The viscous drag forces of the air are assumed to be negligible compared with the normal pressure in their effect on the surface wave. This model is used to determine a first-order approximation to the disturbed motion of the air and the consequent energy transfer to the wave. The velocity distribution is taken to be that for turbulent flow over an aerodynamically smooth water surface. The result rests on the solution of the inviscid Orr- Sommerfeld equation which has a singularity at the height where \J = c (the critical level) . An exponential increase in amplitude of water waves with phase velocity c occurs if the mean air flow velocity U (-y) in the direction of wave travel varies with height above the surface such that U at the critical level is negative. This is almost always the case. Phillips (1966) attempts to include wave-induced turbulence in the instability model by including in the momentum flux equation an appropriate integral over the entire layer of air of the contribution due to perturbation eddy viscosities. This integral is multiplied by an unknown constant A which is estimated from experimental results from flow over fixed wavy surfaces. The uncertainty in A is estimated to be + 50 per cent. Phillips concludes that even though there exists a high degree of uncertainty in A the results do provide a framework in which experimental measurements may be interpreted. According to Miles (1967) , experimental results indicate that a theoretical model based on laminar flow may be adequate in the 14 laboratory but not on an oceanographic scale. The inviscid, laminar model underestimates the energy transfer from wind to waves over at least a significant portion of the spectrum for an open sea. This suggests that the significance of wave-induced perturbations in the turbulent Reynolds stresses for momentum and energy transfer from wind to wave must increase with an appropriate scale factor. Miles concludes that further theoretical work in this area appears to demand some 'ad hoc' hypothesis to describe these perturbation Reynolds stresses. This can be done by considering a slightly generalized form of the classical mixing-length hypothesis. A means must be found to predict the value for the change in mixing length upon which the perturbation Reynolds stresses are based. Zagustin (1963) has given a semi-empirical solution for turbulent flow in a pipe which gives good agreement with experimental data and which predicts the value of the mixing length. This model is generalized below to include flow over a wavy surface in a manner analogous to that of Miles and Benjamin in that it assumes the flow to follow the wavy surface. Viscous stress terms are replaced by the Reynolds stresses associated with turbulent flow. Normal Reynolds stresses are neglected. Zagustin' s method of closing the system of mean turbulent equations of motion is not unique (see, for example, Squire, 195 9) , but there seems to be no good reason to use any other model in its place. The investigation below details the application of Zagustin' s work to this model and outlines the solution of the differential equations 15 using numerical methods and appropriate boundary conditions. The unknown parameters in the model are fitted to experimental data describing the normal pressure in phase with the wave slope. 16 2. FORMULATION OF THE MODEL The zero-order flow is taken as flow in a two-dimensional channel. All lengths are made non-dimensional using the channel halfwidth, H. Velocities are made non-dimensional using the midstream velocity, Uc All stresses are made non-dimensional by dividing by o \J0 , where o is the density of air. 2.1 THE ZERO-ORDER FLOW The velocity is expressed in terms of a stationary and a fluctuating (turbulent) part: (2.1) ttJL = "0" ■+■ WL Following Zagustin, the Reynolds shear stresses L are written as 2. (2.2) "f = L (qrcLcLq) where Q is the mean speed. This can be considered a defining equation for L (although here it appears as the usual mixing length) in that L will not be prescribed, but will be part of the solution. Then the two-dimensional mean equation of motion is (2.3) ( "0" • grad) \J = -grad P + |K x grad T A where IK is a unit vector normal to the plane of the flow, P is the pressure, and K = |_ grad q is the eddy viscosity. The turbulent 17 1 I kinetic energy, E' = HL* UA. , is assumed to have the form (2.4) E' = C |_2(grad q)2 where C is an unknown constant. It is assumed that the rate of dissipation of E' is proportional to E'. A flux of turbulent kinetic energy into any volume element is assumed to occur when the quantity U varies over the element. Thus, the rate of inflow of E1 is (2.5) <=< (grad L , n)dS where n is the normal to any surface and °^_ is an unknown constant depending on the turbulent intensity. The turbulent kinetic energy is constant at any point, and the above assumptions then imply that (2.6) J ) (grad |_ , n)dS + B 1 E' clV=0, where B is an unknown constant. Applying the divergence theorem, considering V to be an arbitrary volume, and using equation (2.4), equation (2.6) becomes (2.7) V2 L + B L2(grad q)2 = 0. Equations (2.3) and (2.7), together with the continuity equation, form a closed set for the mean velocities and the quantity |_ . The constant B must be determined experimentally. For flow between two parallel walls as shown in Figure 1, the continuity equation is trivial, and equations (2.3) and (2.7) become (2.8) -grad P + i. |"L? dy L {- V dy -) - 0 and (2.9) dy + B LV v )2 =0 dy Figure 1 y= 1 i y= -1 Defining B = A for the zero-order flow case and setting 2 equation (2.9) becomes = | grad P | , (2.10) dy' J 4 A L (d T7 dy 0, and equation (2.9), after integrating, becomes (2.11) hy 2,H V L> dy Equations (2.10) and (2.11) give (2.12) dy2 -fhy. 19 which has the boundary conditions: d L dy L_ o i£ = 0 and L L y = 0 y = 1 where i o is a quantity analogous to the roughness length in mixing- length theory. Integrating equation (2.12) twice and applying these boundary conditions results in (2.13) L -"ftV-a + lo. u Next, solving equation (2.11) for d \J and substituting for dy L gives (2.14) TJ dy 7h7 L + ^d -y3) O 1^ By defining = l + 12 L Ah and z = -*- VT equation (2.14) simplifies to (2.15) 12 -1 A/hF iTJ JT dz 1 - z3 Integrating equation (2.15), applying the no-slip condition at z and returning to the y coordinate gives = - N/~b7 20 (2.16) U(y) = A/hF tanh"1 (I) - tanh*"1/ v3/2 , The mean velocity at mid-channel is U(y) = 1 y- 0 which, when applied to equation (2.16), results in the expression (2.17) aThf tanh m ' x Equation (2.16) therefore becomes (2.18) tanh^y^ U (y) = i - VF ih_1(I L The expression (2.13) for L_ can be simplified by recalling that von Karman's constant, #V , gives the rate of growth of I at the wall. Therefore, (2.19) where K = L dy y- - 1 K« it is found that (2.20) (2.21) (2.22) 0.4 is the accepted value. Then, from equations (2.13) Ah = 1.6 b = 1 + 7.5 L .3 L- b - v- 7.5 21 Following Benjamin (1959), the equations of motion in terms of \ ind ^7 are (2.30a) J + ^y^2 +Pt?2) = -i\ + £ where the subscripts denote partial differentiation. The stream function is composed of a zero-order term plus a periodic perturbation. When there is no wave, the stream function is '1r (2.31) H JJ(y[) d 1 where "*| + y = 1 . Vy (V7) is the velocity profile from equation (2.18) expressed in curvilinear coordinates (2.32) TJ"i' tanh' 1 - When the water surface is disturbed by a stationary wave, the stream function becomes (2.33) ^ (fe,>})=U> +aJF(>|)+ U (7j) ~c je-^le11^ where F(yi) is to be determined. The mixing length is expressed in a form similar to that for \y : (2.34) J[ = [_(>f) + aHi(rl)eik^ 24 where |_ (7f) is given above and *X. , ( V» ) is to be determined. Similarly, (2.35) B = \ + a^T oik) where A is given above and j )e~k>0- elkJ? b) v * 2r = -ilea J* F(T,) ♦ [TjT(IJ) " e] e"^ } eik* . To formulate the differential equation in -JL, to be solved, substitution is made for JC , B and q in equation (2.7) expressed in curvilinear coordinates: (2.37) vi + B(^_)2 = 0.. Simplification by elimination of all terms of 0(u2) and above yields (2.38) J[ + JJ((AE k2) + AHF" + ^D + Ae~kyl (G - kD) = 0 where *2 I 2 d - U 2 L E = IP L . and 25 Similarly, when equations (2.30) are expanded, cross differentiated to eliminate pressure terms, and linearized in a, one obtains (2.39) ik_ 2 i ( \J ' c)(F" - k2F) - "[J* F \ sgn( ~[T ) = (^—T + k2)(HF" + eJ?,) + d7| e"kY( (G" - 2kG' + 2k2G) where g=u'U"L2 H = rj'L2 and where U is positive for the lower half of the channel. Note that a singularity occurs at mid-channel ( \J =0). Equations (2.38) and (2.39) are to be solved for F and j(try . The boundary conditions are as follows. At the interface where ~*\ ~ 0, \j (0) = 0 to satisfy the no-slip condition, and v = 0 because the wave is stationary. Therefore, from equation (2.36b): (2.40) F(0) = c. The velocities in the water must take on negative values in order that the shearing stress across the interface be continuous. The variable component of the tangential water velocity associated with an irrotational wave is of the form (2.41) u' = a x/T /3H n 17, 2 e ik 26 The total tangential water velocity is (2.42) u - -c - a/T /sK_ eik£ . \J O I o Defining the Froude number 2 gH" tangential air velocity is TT 2 (2.43) H~ = % ° , and invoking the no-slip condition, the n h (2.44) u = -c - a l-^- elk) at >7 =0, or (2.45) u = -c - a 0 ^) « ?? = o where p = /~|p — . From equation (2.36a) the second lower boundary condition then becomes (2.46) F'(0) = - U'(0) + ft . The perturbation in Jc at the water surface, JCt (0) , must also be fixed. This must be regarded (along with cT ) as an unknown parameter in the problem, to be adjusted empirically. The perturbed flow must die away at some large distance (relative to k ) from the wavy surface. Thus, as V? becomes large, (2.47) Ft>|) >- (c -U )e"k7l (2.48) F'(*W ) H - \J* e~k7i (2.49) Jc,(>|) H 0. 27 2.3 NORMAL PRESSURE, SHEAR STRESS AND WAVE GROWTH The pressure on the wavy surface is the sum of that due to normal atmospheric turbulence and that generated by air moving over the undulating surface. The latter pressure predominates in the generation mechanism considered here and is expressed as (2.50) P(0) = a(% + icf )eikj? where ^X/ and o are to be determined. Differentiating equation (2.50) with respect to | , substituting the result into equation (2.30a), and simplifying yields (2.51) P(0) = [THf " ( U " c)F' - ^ ^— (HF" + E^,) + (G' " Gk) 1 J -Y[ = 0 which gives ^ and o In the same manner, the shear stress at the wall is expressed as (2.52) 7T (0) = a(6 + i £ )eik Jf where £ and £, are to be determined. To evaluate the shear stress, equation (2.2) is expanded in terms of y '• (2.53) T = ^V (^ + ln iyn (y^). Substituting for JL , ], if/ , or their derivatives where required obtains 28 (2-54) T (0) = ( 2 |_U"[|_U" + L ]) which gives £ and £. . The shear stress seems to be relatively unimportant to wave growth, however, and will not be considered herein. To compute the wave growth rate the normal pressure in phase with the wave slope is required. Expansion of equation (2.50) and comparison with equation (2.25) for the wave form show that a positive o results in wave growth. The average energy input due to normal pressure is r* (2.55) E = - A I P^1- d£ 'o or, in dimensional terms, ~ a2 T72X (2.56) E = |jjlO fJJo <*, 2 where CO = gk is the wave frequency. The surface energy of the wave is (2.5 7) E = 1/2 O ga2 where 0 is the water density , so that (2-58) I - J*, ga^ . Equating equations (2.56) and (2.58) and solving for %$- gives dT a TT 2 r ^•by> dt P 2gH 29 Integrating produces the familiar exponential wave growth equation: (2.60) ao exP 1 Y^ " O ^ ^S -\ *" J The rate of energy input due to surface shear is r (2.61) 1 z% u L d\ / or (2.62) E = a2 £ -¥-^6. This energy is expected to go, at least initially, into a surface shear flow. 30 3. SOLUTION OF THE FIRST-ORDER PROBLEM A solution for the function F and its derivatives must now be found so that values for normal pressure, shear stress and wave growth rate may be determined. There are several numerical integration methods available for this purpose, having the common property of starting at one boundary and ending at the other. However, the problem as formulated has conditions at both boundaries, and a method of super- position of solutions must be used. Equations (2.38) and (2.39) can be rewritten as six linear first- order ordinary differential equations. That is letting: (3.1a) F =Yl b) r =y2 c) F" = Y 3 d) F" = Y e) £=*, f) X - \ gives six ordinary differential equations: (3.2a) Yi = Y2 b) Y* = Yo 2 3 c) K =Y4 31 r d) y;=hj 2 (U - c) (Y3 - k2Yx) - TJ" Yx - 2H'Y ,2_ . „£ _„„.„ „.. .,.2. - (H" + k^H)Y0 - E>L, - 2E'YC - (E" + k*E)Yc O DO -k - e ^ (2Gk2 - 2G'k + G") f and e) Y' =Y_ 5 6 f) Y' = -Y,- (AE - k2) - AHYo - 0 D 6 o 6 -Ae*"k?* (G - kD). There are now six equations, each requiring an initial input at the lower channel boundary. Linearly independent solutions are found by starting repeatedly at the lower boundary with different initial values, Y(0) . The final solution is obtained by superposing the independent solutions to fit the required conditions at the upper boundary. The independent solutions, denoted by the superscript j, have initial values given by (3.3) Y^+l(o) = j j at T( = 0, where is the Kronecker delta. The solution to the homogeneous form of equations (3.2) can be written as a sum of independent solutions to the homogeneous equation: 6 <3-4) Yi = y h Y- r-i J 1 32 where the J's are complex constants. Since F(0) = F'(0) = Xx (0) - 0 for the homogeneous system, the constants J-^, J?, and J_ in equation (3.4) must equal zero. The complete solution is achieved by adding to the homogeneous solution a particular solution of equation (3.2). For this particular solution, the initial conditions are (3.5a) Y^c b) Yp - $ - -g c) YP = 0 3 d) e) yp = 0 4 Y^ = i,(0) f) Yp = 0. Then the complete solution, satisfying all the boundary conditions at y\ = o, IS (3.6) Y.(>J) = T3^C|) + T4Y4(>() + T6Y6(^) + ^C?)' where the J's are to be determined. The position of the upper boundary, where conditions (2.47), (2.48), and (2.49) hold, is not known, and must be determined by a convergence criterion. The solution is determined by applying the upper boundary condition at various increasing values of "^ = d , solving the resulting equations (3.6) evaluated at d for the J's, and evaluating the wall pressure each time using equation (2.51). When the wall pressure determined in this manner converges, the procedure is stopped. 33 4. THE COMPUTER SOLUTION The solution for the function F and its derivatives has been obtained by solving the problem as outlined in section 3 on an IBM 360- 67 computer using complex double-precision arithmetic. Major sub- routines are incorporated in the program to calculate the velocity profile, perform the integration using the Runge-Kutta Gill fourth-order method, compute the shear stress, calculate the wave growth rate and evaluate the stream function. Appendix I contains the format of the program and a set of results obtained. The accuracy of the Runge-Kutta Gill integration method depends on the size of the step used in the iteration. The value 10 was selected in order to determine whether the surface pressure converged on a particular value. Because surface pressure did not converge for this step size, it was decreased until convergence was achieved. A step size of 10 was necessary to obtain convergence of the wall pressure, and the program running time was about 112 minutes. A final optimum step size of 10 ° for >] less than 0.3 and 2.5 X 10~5 for V greater than 0.3 was then adopted. This step size gives convergence on a result essentially the same as that for 10 and a running time of 65 minutes . 34 RESULTS AND CONCLUSIONS The zero-order velocity profile is given by equation (2.18) which has I (0) as a parameter. A plot of U (y) for several values of I — (0) appears in Figure 3 . From the figure it can be seen that as the parameter | (0) decreases, the shape of the profile changes from nearly linear to almost a step function. A representative logarithmic profile is also shown in the same figure so that the two profile shapes may be easily compared. 1.0 f 0.8 - 0.6 0.4 . 0.2 - 1.0 Figure 3 . 35 The remainder of the analysis is concerned with the determination of suitable values for the parameters [__ (0) , Jl, (0) , and o . For the channel under consideration: H = 10 meters \J0= 10 meters/second which results in a Froude number of 1.018. For the wave number, k = 15, the instability mechanism considered is important. These values for the Froude number and the wave number have been used throughout the analysis. The following parameter values were first used: |__(0) = 2"22 X 10~2 J2,(0) = 0 ** = 10. The wall pressure failed to converge for these values. Convergence was assumed to have resulted if wall pressure values were nearly constant over a significant range of d (say 0.1) or reached a minimum value at some point in the channel. (This latter criterion is not completely defensible, but the singularity in equation (3. 2d) at 77 = 1, itself a peculiarity of the Zagustin assumptions, is probably responsible for the erratic behavior of the wall pressure as 77— KL.) 0 was then varied from -1000 through zero to +1000 and given imaginary values as well. The wall pressure changed only in the fourth or fifth decimal 36 which was considered insignificant, although ^-i Thus, the solution seems insensitive to 0 New values for the parameters were then used: ,-5 changed markedly, L (0) = 5X1 2t (0) = 0 & = 10. The wall pressure converged for these values, but the value was too high in magnitude and of the wrong sign. A value nearer to 5 (Miles, 195 7) was considered appropriate. With L (o) = 5 X 10 & (o) = 1 y = 10 -5 the resulting wall pressure was positive but still too high in magnitude. Figure 4 summarizes the results of varying | (0) and ->L, (0). 1 d (position where upper boundary 1 . 0 conditions are applied) -«* l_(0) = 2.22 X 10 J?,(0) = 0 y =iq 1 win "' ' > I (0) = 5 X 10 J2i(0) = o = 10 •— < — -io' -10J -10^ -io Wall Pressure, Figure 4 . 37 It appeared that the wall pressure was more directly dependent on Jit (0) than on any other parameter. Subsequent investigation showed that for any >? the wall pressure was a linear function of the real or imaginary part of *X.X (0) providing that the other component was held constant. For Leo 2,(0) 5 X 10"5 0.65788 10 the resulting minimum wall pressure was 3.83 at d = 0.48. Figure 5 shows the resulting wall pressure in this case versus d. 1.0 0.8 0.6 0.4 0.2 10 5 X 10~5 0.65788 +0i = 10 -w* J h 102 103 104 Wall Pressure, O Figure 5 . 0 was then given a value: S* = 1000. 38 -^.(0) Out-of-phase In-phase Wall Pressure ,i Shear Stres 0.0 4- Oi -6495.3 -29.78 0.2 + Oi -4519.4 -14.89 0.4 + Oi -2543.6 0.001 No significant change occurred in the wall pressure. Convergence, as defined above, always occurred near d = 0.5. Table 1 summarizes the values of the wall pressure, shear stress, and wave growth rate versus Jl, (0) at d = 0.5 for parameter values: L(0) = 5 X 10"5 r = 10. Wave Growth ar Stress, 6. Rate , j^ iE_ X ^ -1.61 -1.12 -0.63 0.6 + Oi -567.7 14.87 -0.141 0.349 0.838 0.001 -1.61 -1.58 -1.55 -1.52 -1.49 0.013 Table 1. Wall pressure, shear stress, and wave growth rate as a function of the perturbation surface roughness 0.8 + Oi 1408.2 29.76 1.0 + Oi 3384.0 44.64 0.65788+0i 3.83 19.19 0.0 4- 0.21 -6495.3 -29.78 0.0 + 0.41 -6401.8 -29.78 0.0 + 0.61 -6308.3 -29.78 0.0 + 0.81 -6214.8 -29.78 0.0 +- 1.01 -6221.3 -29.78 0.65788+0. li 50.9 -29.78 39 It can be seen that imaginary values for ^-x (0) have no effect on the shear stress. The required value for J., (0) is 0.65 788 + Oi for a wall pressure of 3.83, but the shear stress seems high. (However, the shear stress value is a first-order contribution to the total shear stress and is reduced by the fractional value of the wave amplitude.) Exponential wave growth rates, due to the wall pressure alone, appear reasonable. To reduce the shear stress to a more acceptable value in relation to the normal pressure, parameter values L (0) = 5X10 5 X, (0) = 0.4 + 5.45i 8* = 10 were used, and give at 77 = 0.5 Normal pressure = 5.0 Shear stress = 0.001 Growth rate = 0.001. However, the wall pressure failed to converge, and had the form shown in Figure 6. Equation (2.33) for the stream function has been evaluated with the values of F calculated from the constants J determined at d = 0.48 where the wall pressure was found to converge. Parameter values were |_(0) = 5X10 5 Jl,(0) = 0.65788 V = 10. 40 LO X LO LO + o II o \ o o eg o to s- ^ tL< CO O CO O o 41 Figure 7 shows a plot of the streamlines with the 'cats eye1 present and centered about 30 degrees ahead of the wave crest. The 'cats eye' or closed loop is indicative of the wave induced perturbations in the flow and characterizes the region of turbulence (Phillips, 1966). Higher in the channel the streamlines become less wavy and approach straight lines. Table 2 shows values of the stream function near the critical * layer. In view of the foregoing limited results, the following tentative conclusions can be made: 1) The inverse hyperbolic tangent velocity profile, a result of the theory of the model, can be fitted to almost any known profile type. The profile which has a large value for — VJ near the critical level and for which the parameter {_ (0) has the value 5X10 produces surface pressures which can be made to converge on the desired value by varying the wall perturbation in ~M. . 2) Wall pressure is extremely sensitive to very small changes in -^i(O) and insensitive to moderate changes in o . 3)Reasonable wall pressure (and wave growth rates) can be obtained with an appropriate choice of JCX (0). This indicates that small, unequal variations in the surface roughness, such as would be caused by parasitic capillaries, are very important in the wave-generation process. 4) Shear is independent of imaginary values of X, (0). Therefore, to vary the shear stress, 0\jfc- ( ^ , (0)), 0 or I (0) must be varied individually or in combination as indicated by further analyis of the model, 42 c u, CD -M 4-> (0 a, s CD CD c O •rH Cn s -r-l (0 CD CO 43 Based on the above results, the following suggestions are made: 1) I (0) should be increased slightly to L(o) = io"4 in order to reduce the shear stress relative to the wall pressure. One computer run indicates that the shear stress is halved for this value. 2) Additional wave numbers should be investigated, say, in the range 5 C k C 20, and their effect on wall pressure and shear stress noted. 3) The model should be checked using data from Stanton and Motzfeld as quoted in Ursel? (1956), and with recently obtained laboratory- data (see, for example, Zagustin, et al. , 1966). In summary, the model appears to be fairly well suited to a theoretical study of air-stream turbulence and wave generation. Unfortunately, the long time required for a computer run prevented the calculation of further data using appropriate parametric values. 44 k ChO 0_A (h_2 CLJ 0.4 0.0 0.0 0.0 0.0 0.0 0.0 0.002 -1.522 -1.256 0.508 0.434 0.216 0.005 -0.683 -0.571 -0.227 0.218 0.594 0.008 -0.299 -0.253 -0.080 0.154 0.359 0.010 -0.138 -0.116 -0.007 0.145 0.285 0.015 0.110 0.101 0.129 0.183 0.243 0.020 0.244 0.229 0.238 0.269 0.309 0.025 0.325 0.315 0.336 0.381 0.432 0.035 0.424 0.440 0.525 0.646 0.758 0.045 0.506 0.556 0.722 0.940 1.128 H 0.5 0.6 0.7 0.8 0.9 1 0.0 0.0 0.0 0.0 0.0 0.0 0.002 1.533 1.266 0.518 0.426 -1.522 0.005 0.757 0.645 0.300 -0.145 -0.520 0.008 0.456 0.410 0.236 0.003 -0.202 0.010 0.357 0.336 0.228 0.075 -0.065 0.015 0.284 0.293 0.265 0.211 0.152 0.020 0.343 0.359 0.349 0.319 0.278 0.025 0.470 0.480 0.459 0.414 0.363 0.035 0.817 0.801 0.716 0.594 0.482 0.045 1.212 1.162 0.996 0.778 0.591 Table 2. Values of the stream function near the critical layer 45 BIBLIOGRAPHY 1. Benjamin, T„B. "Shearing flow over a wavy boundary," Tournal of Fluid Mechanics, Vol. 6, 1959. pp 161-205. 2. Lighthill, M.J. "Physical interpretation of the mathematical theory of wave generation by wind, " Tournal of Fluid Mechanics, Vol. 14, 1962. pp 385-398. 3. Miles, J.W. "On the generation of surface waves by shear flows," Journal of Fluid Mechanics, Vol. 3, 1957. pp 185-204. 4. Miles, J. W. "On the generation of surface waves by shear flows. Part 2," Tournal of Fluid Mechanics, Vol. 6, 1959. pp 568-582. 5. Miles, J.W. "On the generation of surface waves by turbulent shear flows , " Tournal of Fluid Mechanics , Vol. 7, 1960. pp 469-478. 6. Miles, J.W. "On the generation of surface waves by shear flows. Part 5," Tournal of Fluid Mechanics, Vol. 30, 1967. pp 163-175. 7. Phillips, O.M. "On the generation of waves by turbulent wind," Tournal of Fluid Mechanics, Vol. 2, 1957. pp 417-445. 8. Phillips, O.M. The Dynamics of the Upper Ocean. London and New York: Cambridge University Press , 1966. 9. Phillips, O.M. "The Theory of Wind Generated Waves, " Advances in Hydroscience Volume 4. New York: Academic Press, 1967. pp 119-149. 10. Ralston, A. and H. Wilf. Mathematical Models for Digital Computers. New York: John Wiley and Sons , 1960. pp 110-120. 11. Roll, H.U. Physics of the Marine Atmosphere. New York and London: Academic Press , 1965. 12. Ursell, F. Article in Surveys in Mechanics, edited by G.K. Batchelor and R.M. Davies. London: Cambridge University Press, 1956. pp 219-249. 13. Zagustin, K. "Analytical Solution for Turbulent Flow in Pipes , " Central University of Venezuela, 1963. 14. Zagustin, K. , E. Y. Hsu, R. L. Street and B. Perry. "Flow over a moving boundary in relation to wind-generated waves , " Stanford University, Dept. of Civil Eng. Tech. Rep. No. 60, Stanford, (1966). 46 7. 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Defense Documentation Center 20 Cameron Station Alexandria, Virginia 22 314 2 . Library 2 Naval Postgraduate School Monterey, California 93940 3. Professor T. Green, III 10 Department of Meteorology & Oceanography Naval Postgraduate School Monterey, California 93940 4. Oceanographer of the Navy 3 The Madison Building 732 N. Washington Street Alexandria, Virginia 22314 5. Department of Meteorology & Oceanography 3 Naval Postgraduate School Monterey, California 93940 6. National Oceanographic Data Center 1 Washington, D. C. 20390 7. LCDR P. R. Klinedinst, Jr., USN 3 114 New Hyde Park Road Garden City, New York 11530 8. Naval Oceanographic Office 1 Attn: Library Washington, D. C. 20390 9. Director, Maury Center for Ocean Sciences 1 Naval Research Laboratory Washington, D. C. 20390 59 UNCLASSIFIED Si-runtv Cldssification DOCUMENT CONTROL DATA -R&D cir/M- i lassification ol title, body .