THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING THE DEVELOPMENT OF DISTURBANCES IN SUPERCRITICAL FLOWS James M. Wiggert A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University Df Michigan Department of Civil Engineering 1962 April, 1962 IP-558

Doctoral Committee: Professor Ernest Fo Brater, Co-Chairman Professor Victor L. Streeter, Co-Chairman Assistant Professor Walter R. Debler Professor Maxwell Oo Reade Professor Chia-Shun Yih

PREFACE To the co-chairmen of the committee goes the writer's gratitude for their ever ready counsel, guidance and support, To the committee members, for their valued criticism and patience, the writer expresses his thanks. The University of Michigan and the Ford Foundation both contributed financial support. In addition, the Computing Center of the University was made available. All this support is gratefully acknowledged, In recognizing the extent of the contribution of his family, who shared the burden without complaint, the writer finds that words are inadequate, ii

TABLE OF CONTENTS Page PREFACE o o o o.. o.......................................... o. ii LIST OF FIGTJRES.................*.....ooo..................... iv I. INTRODUCTION.... o................oo........................ 1 II. HISTORY OF INSTABILITY INVESTIGATIONS.................. 2 III. PHYSICAL EXPERIMENTS.o.. o o..o o o o......................o 8 IV. BASIS FOR COMPUTER METHOD............ o..................oo 13 The Equations of Flow...............,.......,.......... 13 The Method of Characteristics........................oo 17 The Formulation for the Computer Solution............... 23 The Convergence of the Solution........o........o...... 29 V. RESULTS OF COMPUTER MPUTATION..... o.................. 32 VI. CONCLUSIONS.......... o o o o o................. o............. 39 BIBLIOGRAPHY o................................................... 59 APPENDIX.......................................... o............. 61 iii

LIST OF FIGURES Figure Page 1 Plot of Depth Reynolds Number Versus Length Reynolds Number for the Occurrence of Roll Waves..... 11 2 Definition Sketch for Open Channel Flow............... 13 3 The Characteristic Curves and the Domain of Dependence o...... o o................ o o o........... o o o o o 21 4 The Range of Influence...o..............o............ 22 5 The Range of Influence of a Set of Initial Values..... 23 6 The Intersection of Characteristic Curves and Initial Data.........o o o.......................... 25 7-14 Plots of the Computer Solutions of Non-Dimensional Depth and Velocity Versus Channel Length for Various Flow Parameters..........................o 41-58 iv

Io INTRODUCTION Roll waves are phenomena associated with steep channels, high velocities and small depths. Early observations were made by Cornish(20) of roll waves in drainage channels in the Alpso Photographs taken by Cornish remain one of the best illustrative examples of a periodic train of waves. Roll waves are the result of the introduction of disturbances into an unstable uniform flow. These disturbances grow until they eventually break on the downstream side, forming small amplitude boreso Then the bores coalesce, the larger overtaking and incorporating the smaller, until a series of relatively large, periodic bores is the result. Bores of this type have come to be known as roll waveso A roll wave configuration is characterized by a wave speed greater than the uniform flow velocity in the downstream direction. The velocity in the cross section of the wave is considerably higher than the velocity in the section of shallower water. Nearly all studies of the instability of flow in open channels start with consideration of a uniform flow, into which a disturbance is introduced. The disturbance is said to be a result of a slight protuberance in the channel itself, or some other factor affecting the uniform flow. This study pertains to the development with time of a disturbance introduced into an idealized uniform flow. The effects of changing the parameters describing the uniform flow are examined, as are the effects of the varying of parameters describing the disturbances. -1 -

IIo HISTORY OF INSTABILITY INVESTIGATIONS The stability of uniform flow can be examined by the introduction of a perturbation to the uniform flow. Consideration of existing formulas for hydraulic losses leads to the conclusion that the losses can be represented generally by the form un hf Rn where u is the velocity and R the hydraulic radiuso If consideration is limited to a wide open channel, the hydraulic radius is approximated by the depth, and the dynamic equation becomes (Figure -) Px um1 - - + g sing - K -uux + ut P hn Introducing a hydrostatic assumption, Px = 7 cosG hx, the dynamic equation can be written as um -g cosG hx + g sin - uux + ut o The continuity equation is hux + uhx + ht =0 If disturbances up = u (x,t), are introduced to the uniform u =u' + U, h where U and Y are uniform hI = h?(x,t) flow, then = h + Y flow velocity and depth respectivelyo -2 -

Substitution of the perturbed flow variables into the dynamic and con tinuity equations gives -g cosG h + g sinG - K ( u +U)u (u + )u i +, (h +Y)n (h' + Y)uI + (uP + U)hK + h = 0 x x t Linearizing by dropping products of perturbations and their derivatives, and then subtracting the equation of uniform flow, the equations become -g cos YnhT + g sinG nYnlhv - Kmum- Ynuu + ynu, Yu' + Uh' + h; = 0o x x t Assuming a potential exists for the perturbation velocities of the form = f(h)exp(at + ipx); then u = - -iPf(h)exp(at +I'x), ux 2f(h)exp(at + ipx), ut -iyPf(h)exp(at + ipx)o t If the depth perturbation is of similar form h' = f2(h)exp(rt + ipx), then h' = iff2(h)exp(at + ipx), h - y7f2(h)exp(at + ipx)o If the vertical perturbation velocity vU is introduced, v' = A = -f (h)exp(at + ipx)o oh

But v? = D-h h + Uht = (a +iPU)f2(h)exp(at + ipx)o Dt t X Equating expressions of vI, f'(h) f2(h) = - + i ). (a + ipu) Then, substitution of the values of f2(h) into the expression for the depth perturbation and its derivatives, and substitution of the values of the depth and velocity perturbations and their derivatives into the continuity equation gives f7(h) = 2Yf(h)o The same substitutions, along with the above value of f'(h), when made into the dynamic equations, result in g cos i2y _ g sing npYn + mium-1 =ynuf2 _ ynia a+ iPU a + iPU Simplifying, and making the uniform flow substitution yng sing = Um, the dynamic equation is written,?2 + (2i5U + mg sing)a - p22 + g cosG yp2 + i(m+n)pg sing = 0 U Solving for a, = iU - mg sin + 1 m2g2 sin29 4g cosG y02 4ing sinGl1/2 2U -2 TU2 Requiring the real part of a to be equal to zero for the neutral case gives U2 m2 s - gY n2

-5 - Introducing the Froude number F - lVgY the stability criterion is written F _ mrn-0i (1) > n The flow is stable for the top two signs, unstable for the bottom sign. The method of analysis employed above is essentially that of Jeffreys(l) and Dressler,(4) and was performed by the writer before knowledge of Dressler's paper. While Jeffreys considered the case of Chezy resistance, the above treatment and that of Dressler permit use of general exponents in the resistance function. Dressler left the instability criterion in the form m2ng2-ny2m-n sin29 n2nK2 cosn Introduction of the uniform flow relationship to Dressler's criterion results in Equation (1). Substitution of a Chezy resistance form (m=2, n=-l) into Equation (1) gives the familar criterion for instability F > 2, since -Jcos --- 1. Thomas(2) applied the concept of translating axis in order to obtain a steady flow configuration. This resulted in an equation of gradually varied flow. By categorizing profiles obtainable from his:tion, Thomas found that only two of the possible profiles met necessary conditions at critical depth. From this he found the criterion of instability to be i = 1 sin-1 8g 2

where C2 is the Chezy resistance coefficient (U2. C2RS) For wide channels for which the angle of inclination 9, of the bottom is small, sin 29 G 29, S g 9, R S Y, and the criterion reduces again to F = 20 Vedernikov(5) combined the dynamical and continuity equations, and considered the propagation velocity of the wavefront and the trajectory of a translation wave. From this he determined the variation of the propagation velocity of a given depth of the free surface. Considering the case when the variation of velocity of a given depth would represent an instability, he arrived at the criterion (1-R I)nUo dA,>1 (Wo Uo)m where R - hydraulic radiuso P: wetted perimeter. A L cross-sectional area. exponents in the law of resistance, Vm KSRn, mf where S is the slope. Wo p propagation velocity of the wave part in uniform flow. Uo uniform flow velocity. For a wide, open-channel, dP _ 0, w u ~ dP A 09 W - U + V/ dA

lnd for Chezy resistance m = 2, n = 1 So Vedernikov's criterion becomes F > 2. Schnfeld,(6) using the concept of the characteristic directions of the wave components, found as the criterion of neutral stability, 2 gho 0 + 2h dC 1+ - C dh where C is the Chezy coefficient and Uo, ho are uniform flow values of velocity and depth. If C is a constant, then Sch'nfeld's criterion takes on the form of the ':qa5iio-on () Lighthill and Whitham(21) introduced the concept of the kinematic wave, one which arises from a functional relationship between discharge and concentration (quantity per unit distance)o Comparison of the kinematic wave with the dynamic wave, where Newton's second law of motion governs, also led to a Froude number of two as critical in the case of Chezy resistance. Lighthill and Whitham obtained a critical Froude number of 3/2 for Manning resistance as did Dresslero(4) Other investigators(9-18) have worked on the instability problem, both analytically and experimentally. The formulation of the laminar flow problem has been made by the introduction of perturbations and considerations of the velocity profile (15,16) The general problem of a flow with a uniform cross section also has been considered by Craya,(l7) Escoffier,(l4) and Kuelegan and Patterson (12) The results are in agreement with those earlier citedo Craya also obtained the shape correction factor present in Vedernikov's work,

IIIo PHYSICAL EXPERIMENTS Minimum physical experimentation was performed, using the facilities of the Fluids Lab of the University of Michigan, The flume consisted of a masonite false bottom, placed in an existing flume of 50 ft, length, 1 fto width and 2 fto depth. The existing flume was of plexiglass framed by aluminum members. The flume was supported on three screw jacks,, linked to an electric motor which thus provided a means of changing slope. The slope attainable on the flume was initially considered inadequate (approximately 4%). The considerable roughness of the plexiglass flume, due to construction methods, was quite unacceptable. Therefore, the masonite false bottom was constructed, The masonite had a slope relative to the plexiglass flume of 6%. making the greatest slope attainable approximately 10%. The length of the false bottom was 25 ft. and the width 11-1/2 inches. Sides of masonite were also constructed to alleviate the roughness problem. The masonite surfaces had all screw holes filled, and the entire channel sanded and varnished to provide a smooth surface. Upstream of the 25 fto sloping section was a 7 ft. section of masonite flume parallel to the bottom of the plexiglass channel, and thus at a flatter slope than the 25 fto masonite flume. The transition from the 7 ft, section to the 25 ft. section was abrupt Water entered the upstream section from the head box of the 50 ft. plexiglass flume. Metering was performed by a venturi meter in a 2 inch supply line. The meter was calibrated in place by means of a weighing tanko Water was supplied from the constant head tank of the laboratory buildingo

-9 - In order to verify experimentally the criterion of instability, the works of the investigators cited, indicate that Froude numbers of two should be obtained in a uniform flowo Measurements in the channel described showed that no Froude numbers as low as two were obtainable for any measurable dischargeso Due to the length of the flume, and. aggravated by the presence of the abrupt change in slope, uniform flow could. not be obtainedo instability was present however, and roll'.waves were very much in evidenceo Further difficulties became apparent. The width of the flume (11-1/2 inches) was so narrow that the depths required were very small (less than.03 fto), if roll waves were to occur in the channel. The tilting mechanism. which2 while providing varying slope of the channel, caused a twisting of the flume so that the bottom of the channel was seldom level in cross section. This caused a speeding up of the waves in one area of a cross section over another area and resulted in distortion of the wave frontso However, several beneficial items resulted. from the physical studyo Suggestion that one cause of roll waves might be the manifestation of eddies sloughing from a change in slope inspired determination of the point of appearance of roll waves down the channel from the change in slope. By visual observation, the point along the channel was determined at which the roll waves were first noticeableo The position was quite simple to ascertain for the slower velocities where it was quite close to the change in slopeo For higher velocities, that is greater discharges, the occurrence of discernible roll wave beginnings was quite

-10 - far down the channelo Velocities were easily obtained, which were so high that the waves did not appear in the channel lengtho The depths at the point of occurrence of the roll waves were measured, by a dial gage drive to which was attached a pointed, brass probe. An electrical circuit was made from the probe to an oscilloscope, then to a potential source, and then back to a ground trailing in the water slightly downstream of the probeo When the probe was lowered to touch the moving water, the circuit was completed and a signal registered on the oscilloscope. Then the movement of the probe could be read on the dial gageo In practice, the probe was bottomed on the channel floor, raised above the water surface, and then brought back down to make contact. Due to the turbulence of the stream the surface presented a very rough, fluctuating aspect quite apart from any roll waves. For this reason the signal on the oscilloscope was a series of pulses, all of the same height but varying in duration when the probe was at some mean deptho A visual time average of the pulses gave a highly reproducible value within several percent of the depth (from comparing that value to that of the cases of no contact and of continual contact)o From the depth measurements so obtained at the position at the onset of the roll waves and from the discharges, the velocities, Froude numbers, and Reynolds numbers with both length from the change in slope and depth as the distance parameters were calculated. Figure 1 is a plot of the "depth" Reynolds number on the abscissa versus the "length" 'Reynolds number on the ordinateo The result is, in effect; a plot of the depth of the flow at the appearance of the waves versus the

100 - - 60G Z o 0 SLOPE =.0910 40 --- -- -- -- -- -- -- -- A -A - -- -- — I.l SLOPE =.0850 -0- SLOPE =.0772 Figure___ __ ___ __ 1, Plot ofDphRenlsN SLOPE -.0o564 -- - for ^ the A Occurrence SLOPE =.0617 20. 0 ---~ --- —- ---- 0.02.04.06.08.10.12.14.16.18 N x105 R(h) Figure 1. Plot of Depth Reynolds Number Versus Length Reynolds Number for the Occurrence of Roll Waves. H H I

-12 -downstream distance from the change of slope of that location The points show considerable linearity and indicate that the roll waves generated at a common locationo The divergence of the points from one another, especially in the larger values, probably reflects observational errors. Plotting of the Proude numbers against any of the other parameters proved. fruitless, This possibly was the result of small variations in the Froude number, and the considerable observational errorso

IVo BASIS FOR COMPUTER METHOD In order to examine in detail the development of a disturbance introduced, into a uniform flow, a finite difference scheme with a specified interval based upon the method of characteristics was used. The method of characteristics has been developed in detail by Courant and. Friedrichs.o7) This discussion will deal with the particular equations pertinent to this study. The Equations of Flow The dynamic equation for flow in an open channel is, assuming a uniform velocity distribution, -px + 7 sing - hf p- o(uux + ut) where the positive sense of x, y and u are as indicated in Figure 2 z Figure 2. Definition Sketch for Open Channel Flow. -13 -

-14 - and hf is a resistance term. The subscripts x and t represent partial differentiation with respect to x and t. If Chezy resistance is introduced and consideration limited to a wide, open-channel, so that the hydraulic radius is approximately equal to the depth, the dynamic equation becomes u2 -px + y sing - Kp - p(uu + ut) h From Figure 2, a hydrostatic pressure distribution assumption gives, p yh cosQ, 7(z+Y) cos, where Y is uniform flow depth. Then Px -- YZx cos9 yhx cosO Multiplying the dynamic equation by p, transposing and making the indicated substitution for Px, u2 ghX cosg - g sinG + h -+ uux + ut -0 o (2) The equation of continuity is [(uh)x]dx + (ht)dx - 0, or uhx + hu h + ht =. (3) Introducing the speed of propagation, or the wave speed, c, from a shallow water assumption, c2 = gh,

-15 - and so, ghx = 2ccx, (4) ght = 2cct. Substituting Equations (4) into (2) and (3), c2 2ucx + cux + 2ct = 0. (3a) If the dimensionless variables x t x U t x,,, T t Y U Y are employed (where Y and U are uniform flow depth and velocity), oie modified dynamic and continuity equations become, -c s U2 U2 U2 2gcc- cosQ - g sinG + gK2 + uux - + u - = 0, xc gY x Y Y 2ucx + cux + 2c- = 0 O Writing the Froude number F = and dropping the bars on the dimensionless variables for convenience, the equations become 2 cosO ccx - sinG + IF2 U + F2uux + F2ut = 0 c2 (5) 2ucx + cux + 2ct = 0 Equations (5) are the nonlinear shallow water equations of flow in a wide, open channel. The shallow water assumption of Equations (4)

-16 - requiresa large radius of curvature of any variation in height of the depth h, The assumption of hydrostatic pressure distribution is equivalent to the assumption that the vertical acceleration of the flow is negligible, This has been shown by Stoker,(8) to be correct in the first order as a consequence of the assumption of shallowness. It is assumed that the resistance to unsteady flow can be represented by values obtained from uniform flow. For uniform flow u2 sinG = F2 _ in dimensionless quantities. In dimensionless terms, the case of uniform flow is represented by the variable u and c as - u u = = 1 U c = - -1 /gY (where the bars are again used to denote dimensionless variables). Substituting the values of u and c into the result of the assumption of applicability of uniform flow resistance, sing (6) F2 Substitution of (6) into the dynamic equation then gives, along with the equation of continuity, u2 2 cosG ccx - sin(l - ) + F2uux + F2ut = 0, (7) 2ucx + cux + 2ct = 0

-17 - Equations (7) then are the equations for study. They are called quasi-linear partial differential equations of the first order for functions of two independent variables by Courant and Friedrichs.(7) The Method of Characteristics Equations (7) are a system of the hyperbolic type, and as such are amenable to solution by the method of characteristics. To this end we make a linear combination of the Equations (7), viz., 2 cosQ ccx - sin(l - u2) + F2uux + F2ut c2 + 2Xucx + Xcux + 2Xct = 0 Rewriting the combination, (c cos + u) 2Xcx + 2Xct + (xc + u)F2ux X F2 (8) c2 + F2ut - sing(l - 2) = 0o Recognizing du _. u dx + u dt ax dt at and d _ 6c dx + ac dt ax dt at Equation (8) can be written P2 du + 2X _ sinG(l - ) = (9) dt dt c2 This equation involves the derivatives of u and c in only one direction

(in an x-t plane) when the direction is given by dx. c cosO - -- + u dt X and. dx -t + u. dt F2 Equating values of dx dt.X,= + F/- os. So the characteristic directions are given by dx. u + c dt F dx _ - -r Iu - --- c 9 dt F and the characteristic equations become from Equation (9) (10) du I.. 2 / cosG dc dt F dt du 2 WsQ dc dt F dt sing F2 sin (1 - u) C2 c2~ (11) (1 - u) 0. o c2r Notice that the derivatives of to of u and c appear as total derivatives Mention should be made here that the choice of dependent variables, or the choice of parameters for non-dimensionalizing is not the only one that can be made. For example, if the dependent variables u and h had been chosen, the equations corresponding to (10) and (11)

-19 - would be dx dt — Vh cos(3 F dt F du 1 cos dh dt F 1 h dt sinG (1 -F2 2) =0, h du 1 cos dh dt F h dt sinG (1 -F2 u2 h) = 0 h in dimensionless terms. Uniform flow conditions for this case would be u -, y =1 Similarily, if the relation - C U had been the choice to make Equations (2a) and (3a) dimensionless, the result would have been dx: u + c - - OS dt dx dt *u - c -Ic C du dt du dt + 2VC d1 - sinG dt F2 - 2fos de sinG dt F2 [1,u2 c2 [1 - U2] - c2

-20 - with uniform flow conditions of u - 1, c =-. U =l, c. F The particular set of equations used would seem to be merely a matter of choice, except perhaps for the last case, where, for uniform flow, the ratio U=Fo This case could be considered to represent the dimensional flow more correctly on a descriptive basis, since u = U, h = Y, c =- y in dimensional terms, and u U c WIY Equations (10) and (11) are a system whereby, in the x-t plane, the direction of the derivatives of u and c in the first of Equations (11) is given by dx of the first of Equations (10); and similarly dt for the second equations of (11) and (10). Or alternatively, the first of Equations (11) is a constant along a curve represented by the first of Equations (10); and similarly for the second equations. The first equations of (11) and (10) correspond to the "forward" direction of calculation, while the second equations correspond to the "backward" direction. The curves described by the Equations (10) are called the

-21 - characteristic curves. If values of u and c are prescribed on a non-characteristic curve r, see Figure 3, values of u and c P C+ t B Figure 3. The Characteristic Curves and the Domain of Dependence. within the area bounded by the characteristic curves C+ and C_ depend upon those initial values of u and c on r between a and B. The values of u and c on the curve r outside the interval AB do not affect the computation of u and c within the shaded portion as the disturbances, or variances in u and c are propagated along the characteristic curves of which C+ and C_ are representative. The characteristic curves C+ and C_ through point P delineate the domain of dependence AB of the point P.(7) Similarly, the range of influence of Q on r is bounded by the characteristic curves C+ and C_ through Q, (Figure 4). Unless a shock exists (a bore, or jump, in an open channel flow), characteristic curves of the same "forward" or "backward" direction do not cross on another, so the validity of the domain of dependence is assured.

-22 -Range of - Influence Ct F x Figure 4, The Range of Influence. If the values of u and c are known for initial conditions, that is, if u(x,O), c(x0O), L1 < x < L2, are given, the solution can be computed in the area of the x-t plane bounded by C,, C- and the line t = O, L1 < x < L2, for that area is the range of influence of L1L2. See Figure 5. If specified finite intervals of x and t are taken in that area, and the intervals are small enough that the characteristic curves emanating from points in L1L2 can be represented by straight lines, a method of solution of Equations (7) becomes clear. Values of the characteristic directions, Equations (10), are computed from known values of u and c at A1,A2.,... These values of the directions are ns then used at the corresponding points to be computed, Bl,B29,,., to provide a means for interpolation between points

-23 - t C+ B B2 B3 B4 /~ * * * /C L1 Al A2 A3 A4 A5 L2 x Figure 5. The Range of Influence of a Set of Initial Values. on the line of A1,A2,.... Then Equations (11), in a finite difference form, are solved simultaneously for c and u at the point in the next row (of B1,B29..o). This use of the theory of characteristics has been called a method of specified time intervals by Lister,(19) The Formulation for the Computer Solution In the study at hand, the characteristic directions, from Equations (10), are C -- u A. 4icos+ F =u - u F For values of u and c close to unity, Froude numbers greater than one, and for moderate slopes, 1 F+1 +- -- >1 F F 1 F-l i F l < F F

So if the increments of non-dimensional t and x, At and Ax respectively, are equal, the curves C+ and C-, of which ~+ and 5_ approximate the slopes, will intersect the line t = t1 of known values as is shown in Figure 6. Then, to calculate the values of u and c at P, the initial values, UB, cB, are used to compute + and._, for only point B is in the domain of dependence of P. D+ and _ are used to interpolate for values of UR, cR and us, cS, and the values UR, CR, us, cS are employed as the initial values for solution of up, cp. To this end, Equations (10) and (11) are written as difference equations using the notation indicated by Figure 6. (xp - XR) (uB + cB)(tp - tR) -0, F (xp - xS) - (up - UR) + (up - u) - Writing ~+ and f_ two of Equations (12) (UB - CB)(tp - ts) - 0, F 2 —~-o~-~ 2 2 -Vo (cp - CR) (1 - = 0 F.2 CB2 2 9L (cp - CS) - in (1 - B2) 0. F F2 CB2 (12) for what they are give, in difference terms, the first + - xp - xR U + c9, C+ B XC tp - tR F xp - XS /e Cs - UB - - CB. tp - tS F (13)

-25 - t tl I -Ax' A r * /T~~~~~ At IL A C x Figure 6. The Intersection of Characteristic Curves and Initial Data. The values of uR, cR, us, CS are now calculated by a linear interpolation using.+ and. _o 1 _At, 1;+ 2Ax uC - uR uC - uA 1 + 1 1 At 2Ax 1 CC - CR cC - CA., At- _ 1 Ax uC - uS UC - UB At 1 Ax C - CS CC - CB

-26 - Solution of these equations for the desired values gives, UR = U - t+ (UC - UA) 2tLx CR = CC - + 2 — (CC - CA) US =C - c ( - u B) At CS - - C - _C B) Solving the last two of Equations (12) simultaneously for up and Cp, (after first recognizing (tp - t) = (tp - ts) = At). up 1 ~ + + sg uB2 up = (uR + us) + (CR CS) + (1 - ) Ait (15) Cp F (uR - US) + 1 (cR + CS) The procedure of computation then is to use known values of u and c to solve for 5+ and _ by Equations (13). Using known values of u and c and the values of + and _, Equations (14) are used to find uR, CRp us; cS. Then Equations (15) are applied to calculate Up, Cp. In such a fashion, values of u and c can be computed ranging over the entire area shaded in Figure 5. The solution thus found converges to the true solution as the mesh size of At, Ax reduces to zero, or as At - 0, Ax > O. In general; if the functions u(Llt), c(Ll,t), u(L2,t), c(L2,t), u(x,0), c(x,0) for L1 < x < L2 are known, then the computation can proceed through the entire domain t > 0, L1 < x < L2

-27 - (see Figure ) In the present problem, the "backward" characteristic curve, C_, in reality has a positive slope. Any values of c and u to the left of C+ in the x-t plane (Figure 5) would have to be continuous with values of c and u inside the shaded region. If the values were not continuous, the discontinuities themselves would be disturbances which would be propagated into the flow, and would tend to distort the wave shape changes. For this reason, and because of the difficulty in attaining continuous variables across C+, it is better to compute from the C+.urve. In other words, to compute only in the range of influence of initial values of the form u = u(x,O), L1 < x < L2 c - c(x,0), The computations were performed on an IBM 709 digital computer at the Computing Center of the University of Michigan. The Computing Center has developed a machine language for use on its 709 equipment called the Michigan Algorithm Decoder, or MAD. The program designed to perform the computations was written in the MAD language, and is reproduced in the Appendix. The variable notation in the program follows that used in this thesis as far as is possible. The various major portions of the program are labelled for guidance, Variation of several parameters was permitted; they were: Froude number (F), Slope (SL), Maximum amplitude of disturbance (A), Wave length (WL), Exponential decay factor (K1).

-28 - The first two parameters, the Froude number and slope, defined a flow, The second two, the maximum amplitude and wave length, permitted variations so that a large radius of curvature of the surface relative to the magnitude of the disturbance was assured. The last, the exponential decay factor, permitted changing the amplitude downstream so that the effect of different wave heights on one another could be observed. It was thought that a change of wave length might result from differences in wave height. The disturbances introduced were generated by the part of the program called the external function, The disturbances were sinusoidal variations of the depth, with the exponential term to permit a change of depth along the stream, thus in dimensionless terms, h - 1 + A exp(-K1 - x) e sin(WL ) o WL Initial values of c and u were computed from the depth for various values of x by c,- T (wave speed in shallow water), u 1 (continuity) h again in dimensionless variables, The results of computation were printed for every time increment. Lists were printed of all the velocities, then all the wave speeds, then depths, and finally the corresponding x-distances to the locations of the resultso Prior to the printing of the lists a general

-29 - heading for each set of variables was printed, along with the flow parameters. An example of the output is shown in the Appendix. The results obtained from the computer were plotted, the features chosen as most representative being the depth and velocity. The Convergence of the Solution The method as presented gave accuracy only in the first order of the dimensions of the mesh. Higher orders of accuracy could have been obtained by extrapolation methods, as developed by Lister.(19) The extrapolation methods for second order accuracy would have required twice the variable storage, Three cycles of computation in the computation block would have been necessary plus computation of an extrapolation formula, This would have meant somewhat more than five times the computer time required for first order accuracy. Note that dimensional time t o- t by definition of dimensionless time t. For this study U2 > Y 128.8 in nearly all cases9 because F > 2. Therefore t< t- 12808 Then, in a real flow case, where U might be approximately 2,5 feet/second, the dimensional time corresponding to a non-dimensional time of unity would be about.02 seconds. For that reason, saving of computer

-30 - time was highly desirable, since twenty cycles through the computation and output blocks took the IBM 709 about twelve minutes. For the time increment chosen, the twenty computation cycles would have resulted in a total non-dimensional time of 2, Greater accuracy could also have been attained by reduction in mesh size, but for the same the same reasons, the degree of this reduction was limited. The problem of staying in the range of influence of the initial data was met by dropping the upstream two points after each computation cycle. This was accomplished in the program by means of incrementing the index integer labelled J. The effect of the point which is most upstream in the row of known values, reaches only the third point in the row being computed, hence computation of the third point and all subsequent points can be made without fear of introducing an unwanted disturbance. No points other than the first two in the row of known values are involved in the computation of the third point of the row being computed. This condition is assured by the interpolation formulas for uR and cR involving ~+o When computation in the next row is begun, that which was the third point in the preceding row is now the.first point in the newly computed row of known values, and the argument is repeated, As the changes in value of u and c develop, it is possible that the intersection of C+ and C_ with the row of known values will fall outside the predicted intervals, It would be quite possible to expand the computer program to encompass this condition, but several obstacles arise. First, the program would become quite a bit more complex,

-31 - with all the attendant difficulties of clearing errors, checking for accuracy of logic, and so on. Second, the computation time would have been increased, since for each point a check of the intersection would have to be made and corrections performed before computation could be made at the point. An increase in variable storage would also be needed, but this probably could be kept at a minimum and would not be a factor, Because of the demand of these obstacles upon computing time, no attempt was made to improve the interpolation approximation. It is considered that such an omission is not a major source of error. This is because the interpolation is an approximation; the point of interception will not be far from the interval predicted; and the interpolation formulas hold even when the point of interception is outside the interval,

V. RESULTS OF COMPUTER COMPUTATION The results of the computer calculation were plotted to facilitate observation. Partial results of the plotting appear in Figures 7 to 14. The features chosen for plotting were non-dimensional depth and flow velocity versus length down the channel. Depth and velocity are measured along the ordinate, while channel length is measured along the abscissa. The vertical scale is magnified twenty times with respect to the horizontal scale. The curve of depth is a solid line, while that of velocity is dashed. Each set of curves represents the evolution with time of depth and velocity for a certain set of the parameters governing the flow and wave size. These parameters lab-:J each set of curves. The symbols representing the parameters are: F, Froude number; SL, Slope; K1, Exponential decay factor; WL, Wave length; A, Maximum wave amplitude. The non-dimensional time of occurrence accompanies each pair of depth and velocity curves, Since all the curves represent the results of data which included a decay factor other than zero, the curves of the initial disturbance show a reduction in magnitude along the channel length. Because of the requirement of dropping the first two points of computation -32 -

-33 - in each row, the plotting was performed at various downstream channel locations, depending upon the total time the problem was generated. in the computer. Accordingly, the initial data of various sets of parameters show different amplitudes. Some general comments can be made about the curves: 1. All the surface profiles exhibit a tendency for the upstream side of the wave to steepen more than the downstream side. That is, the distance along the channel, going downstream, from trough to crest is less than the distance from crest to trough. Sch3nfeld(6) indicates that the upstream side of a disturbance will steepen and then begin to flatten. No upstream flattening is apparent in these profiles. 2. The wave lengths of the disturbances do not show any change. This is true despite variation of any of the parameters describing the flow or the disturbance. 3o A pulsation, or beat, occurs in the velocity and depth. This pulsation is best seen in those sets of curves which are for the larger times, but the beginning can be noticed in most of the others. 4o The variations of depth and velocity tend. to stay about 90 degrees out of phase. When the depth variations are greatest, the velocity curve has experienced its greatest phase shift relative to the depth curve. When the depth variations have lessened the phase of the velocity curve relative to the depth curve is again near 90 degrees.

The pulsation of the amplitudes of both depth and velocity variations appears to be the result of the reciprocal nature of the depth and velocity in the initial disturbance. The fluid particles in the shallower depths are carried along faster by the higher velocities, while particles in the greater depths are moving more slowly relative to a reference velocity. This results in a piling up of fluid, and a lessening of velocity variation. When the velocity variation is no longer sufficient to support further growth of depth variations, dispersion of the wave begins, and causes a reduction in wave height along with an increase in velocity variation. This reduction continues until a period of renewed growth begins. The eventual changes in and role of the pulsations must be a subject for further study employing much longer effective computing times. The only observation that can be made of the pulsations is that the period of pulsations seems to be longer for cases of greater depth variations, in general, higher Froude numbers. A test of the validity of the method must be concerned with stability and Froude number. For the equations at hand, Dressier's instability criterion(4) can be written as F > 2 rosQ assuming uniform flow can be used to evaluate the resistance coefficient. For the slope most commonly used in the graphs,

-35 - Therefore, a flow with F - 1.5 should be stable, one with F = 3 should be unstable. Figure 11 represents flow with F 1,5, Figure 7 represents flow with F -- 3, The flow with F:- 1.5, A =.05 (Figure 11) shows a definite tendency for the amplitude of the waves to decrease after an initial period of growth. This period of growth ends at approximately time 1.5, and then a decrease begins which continues until time 3.0. At time 3o0, a new cycle starts which has not yet reached a low point at time 5 5. This observation is based upon the shape of the profiles at time 5,5 which are quite similar, except for amplitude, to those of time 2.5o On the other hand the profiles of time 3.0, where the decrease of the heights of the waves momentarily are halted, show a pattern of rise and fall similar to the initial disturbance of time 1.0. It could be expected that the pulsation will be repeated, and such is the case, with an increase in variation to time 4.0, and then a decrease to about time 6.0o The wave heights at time 4.0, which represents the peak of the second period of growth, show a maximum amplitude nearly exactly that of the initial disturbance. This is considered to be a coincidenceo Flow with F - 3 should show a growth in wave heights leading eventually to a discontinuous water surface, or breaking of the waves. Prior to that point, it is expected that the assumption of no vertical acceleration would become invalid, and that the intersection of the characteristic curves with the line of computed data would fall so far outside the interval assumed that the interpolation made there

-36 - would have little merit. Profiles of velocity and depth for F = 3, A -.05 (Figure 7) show a rapid growth of depth variation, and a corresponding decrease in variation of velocity up to time 2.5. Then there is a decline in the amplitude of the waves until time 60.0, when a period of considerable growth begins again. The question is whether the flow is unstable for F - 3 Figure 7 does not offer conclusive proof of instability, but there are differences between the curves for F - 3 and F -- 1o5 which indicate instability. 1o The first increase in depth variation for F = 3 is much greater relative to the initial disturbance than the increase in depth variation for F = 1.5. 2. A considerable increase in depth variation continues for a longer time for the higher Froude numbero 3. The time of maximum decline of amplitude, time 6.0, for F - 3 still exhibits amplitudes of the same magnitude as the original disturbance. These differences can lead to a conclusion that the flow is unstable for a Froude number of 39 but a definite statement would require a much longer period of computation. This must wait for the advent of a faster compute:ro Opposing the instability conclusion is the fact that the second increase in depth variation in the case of F = 3, which occurs at time 8.5 shows less variation in depth and velocity than time 2.5, This could be the result of another pulse or beat with a still longer

-37 - period than the first, which has a non-dimensional period of about 3,0. A longer computation would also resolve this question. Other curves support the instability conclusion. Figure 10 shows the development for a short time of flows with F = 3 and F - 4 for comparison. The initial disturbances are identical, with a maximum amplitude of 0.05. The flow with F = 4 shows considerably more variation than the flow with F - 3, which would indicate a higher degree of instability. Figure 9 represents the state of the flow for F = 2.5 and A - o.1l It is to be expected that the larger maximum amplitudes (values of A) would cause an earlier indication of instability. Figure 9 shows an extremely large increase in variation of depth up to time 2.0, then a decrease to an extremely distorted, pattern at time 5.5, where the depth variations are still of almost the same magnitude as those of the initial disturbance. This condition gives evidence of continued instability. Figure 12 is for the case of F 1. A.1, SL.1. This case was computed because the intersections of the characteristic curves with the line of known points would not be very far outside the assumed intervals. A decline in the depth variation is evident. Also of interest is the fact that the wave does not progress down the channel as is the case with the flows of F > 1. What would. happen to the profiles after time 4.0 is conjecture, but it appears that the wave is being dispersed by a drawing out of the fluid on the downstream side of the wave. The effect of sharply decreasing the amplitude of successive waves by increasing the decay factor is shown in Figure 13. Any

-38 - expectation of the larger waves overtaking smaller waves is fruitless, as the interval from crest to crest remains the same. Development of roll waves by incorporation of smaller, slower waves by larger, faster waves must not occur until the assumptions governing Equations (7) are invalid. Figure 8 shows the development of the disturbance for consecutive time intervals used in computer computations. This figure serves to illustrate the changes occurring between time 0. and time 0.5 which the other figures omit. The effect of changing the slope is illustrated by Figure 14. The depth variation resulting in the case of shallower slope are greater than the depth variations when the slope is steeper, Comparison of Figure 14 and Figure 7 illustrates the condition. Thus, a particular Froude number in the unstable range for a given slope represents a higher degree of instability than the same Froude number when the flow is on a steeper slope. As might be expected, the effect of a change is not as great as the effect of a change of Froude number.

VI. CONCLUSIONS The conclusions of this study are: 1. The method of characteristics can be used to predict the development of disturbances in an unstable supercritical flow within the assumptions made. 2. While the Equations (7) still govern the flow, and the assumptions made are still valid, no change in wavelength of a decaying sinusoidal disturbance is indicated, 3. A pulsation in the variation of depth and velocity with respect to time is noted for all flows investigated. 4. The role of the Froude number as the most significant parameter in instability growth is verified. Further study is suggested, principally in the shape of disturbances, initial velocity and depth relationships, and in greater nondimensional times. -39 -

FIGURES 7 THRU 14 PLOTS OF THE COMPUTER SOLUTIONS OF NON-DIMENSIONAL DEPTH AND VELOCITY VERSUS CHANNEL LENGTH FOR VARIOUS FLOW PARAMETERS

-41 - >w 3L 0 -J 0 z I0. a -J Q 0 LI) W Z I 0 z 1.05 1.00 0.95 1.05 1.00 Q95 1.05 1.00 0.95 1.05 1.00 0.95 1.10 1.05 1.00 Q95 Q90 1.10 1.05 1.00 0.95 0.90 - -7.- ~ Js,/ /:=Z.J I.=Z,./ _.\' I '_ _ I T1-. TIME 0 TIME 0.5 TIME 1.0 TIME 1.5 TIME 2.0 TIME 2.5 18 19 20 21 22 23 24 25 26 NON- DIMENSIONAL DISTANCE ALONG THE CHANNEL F=3., SL=O.I., A=0.05., WL=4., K =0.02308. DEPTH VELOCITY ----- Figure 7a.

* q aznSij A113073A --- Hld3Q 809ZO'O: 1 I ' *:=IM ' SOO:V ' 0I-O=S ' ~:= 13NNVHO 3H1 9NOIV 3:NVISIO 1VNOISN3WIa -NON 9Z GZ f_ Z ZZ IZ OZ 61 81 r T I1 I I 1 I I I I I I I l I I gS 31a11 as 3W1I St 31W11 0' 3W1ll gS 3W1 0' 3WIll / ~./ ___ -,__ _ \- ~ I.-. - / \ ^ _/-v _ \^ _ ^,., r j ~ f ^ E XEES 7 f ---'.~_-'1 1.1 1 — ~~~~___ ~~\ / I ITI \/ S60 00'1 SO' 00-1 SO'I g60 00'1 o 060 f I SgOl - 060 m o I060 S0 -I 001 061 0 060 MI' I -_a+

03L njirnTff.. AO113013A - Hld30 '18000 ZO'O ' 'tI:M 'G S0'OV ' I'O: 1S ' ~:S13NNVHO 3H1 ON0IV 33NVISIO '1VNOISN3WlIO- NON 9Z SZ ~Z ~Z ZZ IZ OZ 61 81 '06 3W11 08' 3W11 OS 3W11 OL 3YII1 0*9 3111 -, - j - -. "" -. I -- / I. -- 960 00'1 go'l 960 001 9O'l 960 00'1 960 001 gO'I 960 001 SO I 960 00'1 gool 96'0 901 z 0 z I z 0 r o -n m -4 z z o I' 3E m ^ ^ -— ^...- --, —: ---7 — ^ — ^ ---- I * —_ I I ^^~Z^^^^_K I - ~, I T T T~ -~t

-44 - 1.05 1.00 0.95 1.05 1.00 0.95. 1.05 "- 1.00 o 0.95 L. o 1.05 z a 1.00 z 1.05 wi a 1.00 I z o 0.95 1.05 1.00 0.95 -^7. 1 <-N X LT"'^/-N -.i.t~,'kl / " / '. TIME 0 TIME 0.1 TIME 0.2 TIME 0.3 TIME 0.4 TIME 0.5 18 19 20 21 22 23 24 25 26 NON- DIMENSIONAL DISTANCE ALONG THE CHANNEL F:3., SL 0.1., A0.05., WLt4., KI O.02308. DEPTH VELOCITY... Figure 8.

-45 - 1.10 1.05 1.00 0.95,/ \ /.7 _'_^ ^7/0 ___ / ~~~~~/ \~~~% i'-' TIME 0 IAa z 4l Jr 4 z 0 z w z 0 Z 1.10 1.05 1,00 Q95 90 1.10 1.05 1.00 095 0.90 1.15 1.10 1.05 1.00 0.95 0.90 0.85 I I /' \ L.I,~ /.. \ / /__ < =I\/ I" \I / 37 —p-_SZ-%. / TIME 0.5 TIME 1.0 TIME 1.5 14 15 16 17 18 19 20 21 22 NON-DIMENSIONAL DISTANCE ALONG THE CHANNEL F:3., SL O.I., A:O. I., WLT=4., KI z.02308. DEPTH ----- VELOCITY --- Figure 9a.

-46 - 1.20 1.15 1.10 105 1.00 0.95 w 0.90 Q85 a Z 4 x 1.20 I-. 1.15 -J 4 z 2 1.10 ~ z 5 1.05 I z z 1.00 0.95 0.90 0.85 I=I = I________ l-klE5 TIME 2.0 TIME 2.5 14 15 16 17 18 19 20 21 22 NON- DIMENSIONAL DISTANCE ALONG THE CHANNEL FT3., SL 0.1., ACO. I, TWl4. K l0.0308. DEPTH ----- VELOCITY ------ Figure 9b.

-47 - 1.20 1.15 1.10 1.05 - 1.00 -- F- TIME 3.0 2 0.90 z j 1.15 o 1.05 1.00 - - - ME 3.5 0.95 0.90 0.85 14 15 16 17 18 19 20 21 22 NON-DIMENSIONAL DISTANCE ALONG THE CHANNEL F'3., SL:0.1., A O.I., WLt:4., KI:0.02308. DEPTH ---- VELOCITY -- Figure 9c.

-48 - 1.15 TIME 4.0 w J IL II. a 4 Z l-J 4 2 I o a I z 0 z 1.10 105 -% I / c \ I X 17 5 I=7 -.-~~~/-_^ _-_\i \ I / A \ \ I too.95 \.90.85 1.10 1.05 o.- -%E ^ =_V. \ / _ 1.0........................95 \\, _ /\ I I 00,.95 QAcrJ _ ^ -ni.,1s,1,z\ TIME 4.5 TIME 5.0.85. _ _ - %FO 18 19 20 21 22 23 24 25 26 NON-DIMENSIONAL DISTANCE ALONG THE CHANNEL F=3., SL:I.O., A:O.I., WL4., Kl=0.02308. DEPTH VELOCITY -- Figure 9d.

-49 - 1.05 1.00 0.95 090 z 4 Il I 1.05 1.00 Q95 1.10 1.05 1.00 0.95 1.10 I/ \ \ /___ ~x ZZ:^^_S// /' "-/ ' ---/ Pi8 ~~/ - \~~~~~~~~~,,__I '\,\ I I I I 1 1 1 1 1 1 1/__/ / 1 1 A \, ' ----: - VPo WF He In =~~~~~~~~~~~~~~~~~~~~~~~~to TIME 5.5 TIME 6.0 TIME 6.5 1.05 LOO Q95 0.90 14 15 16 17 18 19 20 21 22 NON-DIMENSIONAL DISTANCE ALONG THE CHANNEL F=3., SL:O.1., A:O.I, WL4., Kl O0.02308. DEPTH - VELOCITY. Figure 9e.

-50 - 1.0 0.95 --- -- --- -'- - /* \/ I.0_9 [0 1 - -; / I.,,,.95 1.05 t w 0 a o z 4 I I o4 z TIME 0 TIME 0.5 TIME 1.0 I- - 1.10 TIME 1.5 6 7 8 9 10 II 12 13 14 NON- DIMENSIONAL DISTANCE ALONG THE CHANNEL F=3., SL=O.I., A:O.05., WL4., KI O.02308. DEPTH VELOCITY - Figure 10a.

I8 'al J z 4 z J 4 z z w a I 1.05 1.00 0.95 1.05 1.00 0.95 1.10 1.05 1.00 0.95 Q90 1.10 1.05 1.00 0.95 0.90 TIME 0 TIME 0.5 TIME 1.0 TIME 1.5 // / /. \ / / ==- =% 6 7 8 9 10 1 12 13 14 4 NON- DIMENSIONAL DISTANCE ALONG THE CHANNEL F 4., SL O.1., AO.05., WL4., KI O.02308. DEPTH " VELOCITY Figure lOb.

-52 - 1.05 1.00 0.95 1.05 1.00 095 1.05 J1.00 ~ 0.95 oi 1.05 IU. o z 4 1.00 x & 0.95 - 1.05 4 z 1 c 1.00 0 Z I a 0.95 Z o 1.05 z 1.00 095 1.05 1.00 0.95. 0, k, /\ /, IZ I- I J_1 _ I t 4,-r / -... 1 '\ >/ / \' -... 415 16 17 18 19 20 21 2; Nt- -fs X l \t ---St / l op I I I I TIME 0 TIME 0.5 TIME 1.0 TIME 1.5 TIME 2.0 TIME 25 TIME 3.0 I 2 NON-DIMENSIONAL DISTANCE ALONG F 1.5, SL O.I., A 0.05., WLt4. DEPTH VELOCITY THE CHANNEL KI =O.02308. _ _m _mm_ Figure lla.

-53 - 1.05 1.00 Q95 1.05 100 I Iz 4 I4 z v) z w a Q95 1.05 0.95 0.95 1.05 1.00 0.95 1.05 1.00 IN - %:^^^^^ -,_ —k_/^- - V_7 —_, 1,,, ' ool a_- - _ L ^ — 1 N —1 1EE, II, TIME 3.5 TIME 4.0 TIME 4.5 TIME 5.0 TIME 55 TIME 5.8 Q95 1.05 1.00 0.95. 14 15 16 17 18 19 20 21 22 NON-DIMENSIONAL DISTANCE ALONG THE CHANNEL F:I.5, SL:O.I., AO0.05. WL:4., KlI0.02308. DEPTH VELOCITY Figure llb.

-54 - I OJ 0 z 4 Z -J 4 z 0 v) z w I z 0 z 1.10 1.05 1.00 Q95 Q90 1.10 1.05 1.00 0.95 Q90 1.05 1.00 095 1.05 1.00 0.95 1.05 1.00 0.95 \0, 0, \ ' 0 0, m / eI I A —^-/-^ A- -\\ 7_\ _ / _ f <7 '/T\ 7 '/? I// \ /,- \ I i \ +- / | r\ f - 5, L-", \ _'S5' 1'.^^ x /-\7z TIME 0 TIME 0.5 TIME 1.0 TIME 1.5 TIME 2.0 10 II 12 13 14 15 16 17 18 NON- DIMENSIONAL DISTANCE ALONG THE CHANNEL F:l.O, SL:0.I., A=0.I., WL:4., KlI0.02308. DEPTH -- VELOCITY Figure 12a.

-55 - 1.05 1.00.95 w CI -- o 0 Z w a -I 4 z 0 z w m.a o I z 0 z TIME 2.5 TIME 3.0 TIME 3.5 TIME 4.0 TIME 4.5 1.05 I,.95 l 1.05 - 1.00 -.95 10 11 12 13 1I NON- DIMENSIONAL DISTAN( F:l.O, SL:.I., A:O.I DEPTH 4 15 16 17 18 CE ALONG THE CHANNEL, WL:4., KI l0.02308. VELOCITY Figure 12b.

-56 - 1.05 1.00 095 1.05 1.00 J 0 I z 4 I -J o z z w z 0 z 095 1.10 1.05 1.00 0.95 IJO 105 1.00 095 -./ / `\ I TIME 0 TIME 0.5 TIME 1.0 TIME 1.5 TIME 2.5 NON-DIMENSIONAL- DISTANCE ALONG THE CHANNEL F=3., SL=I.O., A:O.I,WLt2.0, K 0O.2308 DEPTH - VELOCITY -- Figure 13.

-57 - IL z 4r eL I -J 4 z (n z L, 5 o I z o z 1.05 1.00 Q95 1.05 1.00 095 1.05 1.00 0.95 1.05 1.00 Q95 1.10 1.05 1.00 0.95 0.90 1.10 1.05 1.00 0.95 0.90. 4, ~ 7-', ^^- t- a —t 0,. 1 / 1' I\ I_11 /1 1 1 \1/71 1 1 '1'1 TIME O TIME 0.5 TIME 1. TIME 1.5 TIME 2.0 TIME 2.5 I I::. _ I - I 1 _ 1 I 18 19 20 21 22 23 24 25 26 NON- DIMENSIONAL DISTANCE ALONG THE CHANNEL F:3., SL:0.01, A 0.05., WL:4., KI 0.02308. DEPTH - VELOCITY Figure 14a.

-58 - 1.10 L05 1.00 Q95 0.90 1.10 1.05 1.00 Q95 w > 090 0. 1.05 4 x 1.00 ICL 0.95 -j 4 ~ 0.90 z w 5 1.05 z 1.00 0.95 1.05 1.00 0.95 r-== X I I / 1.01" = = low 1 1 1 I -=Z —I_ _--- I I I —< I I Y I I I-~c TIME 3.0 TIME 35 TIME 4.0 TIME 4.5 TIME 5.0 18 19 20 21 22 23 24 25 26 NON- DIMENSIONAL DISTANCE ALONG THE CHANNEL F:3., SL =0.01, A:O.05., WL4., KI 00.02308. DEPTH VELOCITY. Figure 14b.

THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING THE DEVELOPMENT OF DISTURBANCES IN SUPERCRITICAL FLOWS James M. Wiggert A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University &f Michigan Department of Civil Engineering 1962 April, 1962 IP-558

Doctoral Committee: Professor Professor Assistant Professor Professor Ernest F. Brater, Co-Chairman Victor L, Streeter, Co-Chairman Professor Walter Ro Debler Maxwell O0 Reade Chia-Shun Yih

-60 - 12. Keulegan and Patterson. "A Criterion for Instability of Flow in Steep Channels." Trans. A.G.U., 21st Meeting, Part II, (1940) P. 594. 13. Mayer, P. "Roll Waves and Slug Flows in Inclined Open Channels." Trans. AoS.C.E., Vol. 126, Part I, (1961) p. 505. 14. Escoffier, F. F. "A Graphical Method for Investigating the Stability of Flow in Open Channels or in Closed Conduits Flowing Partly Full." Trans. A.G.U., Vol. 31, (1950) p. 583. 15. Yih, C-S. Stability of Parallel Laminar Flow With a Free Surface. Proc. 2nd U.S. Nat'l Congress of Appl. Mech. (Publ. by A.S.M.oE) (1954) po 623. 16. Benjamin, T. B. "Wave Formation in Laminar Flow Down an Inclined Plane." J. Fluid Mech., Vol 2, (1957) p. 554. Corrections, Vol. 3, (1958) p. 657 17. Craya, A. The Criterion for the Possibility of Roll Wave Formation. U.S. Nat'l. Bureau of Standards, Circ. No. 521, 1952. 18. Binnie, A. Mo "Tnstability in a Slightly Inclined Water Channel." J. Fluid Mecho. Vol. 5, (1949) p. 561. 19. Lister, M. "The Numerical Solutions of Partial Differential Equations by the Method of Characteristics." Mathematical Methods for Digital Computers. Ralston, A. and Wilf, H. S., editors. John Wiley and Sons,9 New York, 1960. 20. Ccrnish, Vaughn. Ocean Waves. Cambridge Univ. Press, 1934. 21. Lighthill, M. J. and Whitham, G. B. "On Kinematic Waves. I. Flood Movement in Long Rivers." Proc. Royo Soc. of London, Vol. 229A, (May, 1955) p. 281.

APPENDIX -61 -

-62 - S COMPILE MAD, EXECUTE. DUMP* PRINT OBJECT. PUNCH OBJECT PRODOO........................... R NON-DIMENSIONAL CHARACTERISTIC SOLUTION. - R (FIRST ORDER ACCURACY) - R A PROGRAM TO EXAMINE THE EVOLUTION WITH TIME OF AN ARBITRARY R WAVE SHAPE... START READ DATA - - R-OUTPUT FORMAT FOR GENERAL HEADING" PRINT -COMMeNT -SCHARACTE-ISTIC M THOD- F 'SOLUTION OF THE NON1DIMENSIONAL FLOW EQUATIONSS - ----------- PRINT COMMENT SO F*F*DU/DT+F*F*U*DU/DX+C*DC/DX*COS(T 1)-SIN(T)+KF*F*U*U/( C*C) OS - - -—.. — -. —' -—. --- ---------—. --- -PRrNT- COMM-ENT'* 'S0. ' ' S ---- -—.C*DUiOX2 CDX+2D-/D2DC/DT0.O- ' ' ' ' ' -. --- —-- - - '- --— ''' -— ' —' ---- -' P - RINT COMMENT SOMULTIPLE DISTURBANCES ARE INTRODUCED FOR THE --- — 1PURPOSE OF EXAMINING THE EVOLUTION OF WAVES WITH TIME-. PRINT COMMENTS THE DISTURBANCES EXTEND OVER THE ENTIRE' INITIA iL- DATA-. THE UPSTREAM -TWO —POIhNTS- IN EACH COMPUTED ROW ARES- - - ------------------------------------------------------------- PRINT COMMENT S DROPPED IN THE SUCCEEDING ROWS - R --- —- ' BOOKKEEPING DECLARATIONS* DIMENSION U1(1000). Cl(1000), BIN(1000) INTEGER It Jo K L L -1000 --- ----- --- - R INITIAL DATA OUTPUT. __. __.. _...,.,. _.__,,.,,.................................

-63 - ----------- -YjffOSffNfSADIIALZG- --------- - 3 " ---- Z PRINT FOPAT-lFo- -L — XL- E* - TL -L --- —--------------------------------------------------------------- _ _ ----— ' —" --- —----------— ~ --- --------- ------- --------------------------------- R BFINITION STATEMENTS AND IiIiT ING SQC-SQRT ( 1e-SLPe2) TSUNMO. J-aO TH=TDEL/XDEL - - - - - - - ------ - - ----- --------------------- -- - EXECUTE Aile6Ui* - X-EL- A. L- -f. R OUTPUT BLOCK FOR PRINTING RESULTS. -TOTAT.........I,.,#? - -FOII~A'F'-T'J^a --- — — T6 --------- Tt-Ar — ---------------- ------------------------------------------------------------------------------------------- ---------- PRINT COMNENT SOVELOCITIESS pltT~- y -- r-4AT-r.; rj'o - -_r — -----— _-_ -- -- ------- -----------------— T- -------------------------- --------- -- - -- ---- PRINT FORMAT T2t C1(J).**Clf000) --------------— IT — MT SOD Cl(J)THCl) ---------------------------------------------------------- PRINT COMMENT SODEPTHSS - - - -----------— F T ---- - ---- - ---------------------------------------------------- --- --- PRINT FORHAT T2e BIN(Jl —.-Bi(1OOO--------- t --- —----------------------------------------------------------------- -- ---------------- PRINT COMM1ENT SOX-DISTANCES$ THROUGH BETA. FOR ItJt 1, I.G.1000-J BETA Bt IN(IXDEL* (I+JI -— ~- t - Ii i L- i i ------------------------------------------------------------------------------------------------------------------------- PRINT FORMAT TJ. BIN(J).OIN(O1000) ----------------------------- ----------- ------------------------ --------------------- --....R PROGRESSIVE --- SUBSTITIO — STATEM-EN-....TS.*... TSULMTSUM+TDEL.................................................................................................. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - — ___ _ _ _ _ _

-64 - 2-13-62 3 UPlUlIJ) CPlC1J ) UPiU1CJ+1) CPSC1 (J+1) R COMPUTATION1 BLOCK. R" — CO"P-~~' tAT' ION --- BLOCK ---' -~ --- —-- — ~ - - ------------------- -------------------- -- -- THROUGH ETA. FOR Ks2+J~ 1t.K6G1000 CHPIU1(-K-1)+C1IK-1 )*SQC/F CHMNU1l(K-1)-C1 (K —)*SQC/F UR-U1(K)-TH*CHP*(U1(K)-U11(K-2) /2 US-U1(K -TH*CHM 1(U1(K)-Ui(K1-1.. CRC1I(K)-TH*CHP*(C1(K)-C1(K-2i /2. CSC1(K)-TH*CHI*(C1(K)-C1(K-1) ) U1(K-2)iUP1 C1(K-2)nCPi UPI-UP CP1iCP.....P-Rs) 2+StCR-C) ---/F-EL*$L~-l' --- —-('l ( K — /- (1I-J --- —-- - -P — 1)/F.P.2 ETA CPi(UR-US)*F/ 4+.SQC)+(CR+CS )/2 R FINAL SUBSTITUTION STATEMENTS* U1(999)-UP1 C1(9)"CP1 Ui 1000! UP C1( 1000)CP

-65 - R CO ----P ---UTA-T ---- -ION — IN-EX INCREM-ENT --- —---- ------------------------------------ R COMPUTATION INIDEX INCREMENT J*J+2 R COMPLETION STOP AND LOCATION ASSIGNENT — ---------------------------------------- "~~~ —'-~~ — & -. - T ----------------------------------------------------------------------------------------------- WNREVtER TFUM*4TC# TRANSFER TO IOTA OMECA PRINT COMMENt SbPROGRAM OMPCET[ — ----- -R T --- ------------------------------------------------------------------------------------------------ TRANSFER TO START VECTOR VALUES TlSlISH4WAVE FORM AT TIME F7#3, 21H * STARTING --------------— T --- —---------------------------------------------------------------------------------------------------------- VECTOR VALUES T3SS(20F6*Tl)S ------- 7 ---- -— t ----N —S-N --- —---— sIFTU --- ------------------------------------------------------------------------ -------------- t S -' ZFD4WszH --- —---------- 2/24H TIME INCREMENT -F10O3,/24H AMPLITUDE ----------------- -H EAI T74 LII - ----------------------------------------------------------- 5CTOR *F1lg3*S L FUNCT --- --- ------ ----------------------------------------------------------------------- """'""""'n'DUMwit - MyAYr SKgWKET LrIT' WDuHCCTON LABEL4 ------------------------------ EXTERNAL FUNCTION (UUWi CC1~ XD~ AA# LL9 WWL9 KK1)

-66 - 2-13-62 5 ENTRY TO AMP2 RNOTE THAT THIS EXTERNAL FUNCTION DEMANDS THAT WWL BE -- R DIVISIBLE BY XD -AN —INTEGRALN-UMBER OF liMS AN TATE ------ II- -.ME' RTOTAL NUMBER OF POINTS- BE- WITHIN- THE STORAGE OF THE VECOR --- - -- R -STORE.~ R FUNCTION BOOKKEEPING. DECLARATIONS..._.. --., ------— ~'~` —~~~`~ `-.- — ^ ^ '-^'^ ^ — ~" ----^ — ^ ^ ^ ^ ^ ^^ ------— ~~ ---- ----- -~^- — ~~ —.' ~' - ~. --- —.~`~' —~~ " — - ~.- ----- - -- ` ~' ~~` ~ ~- -- -. --- —------------—, --- —-- -_ --- DIMENSION STORE(2000 -INTEGER Is LL '~~~ — - -— ` --- —-"~~~~- PRINT COMMENT'SOTHE~~ ---- WAVE —~~~'~~-~' —~ -po^ ^-^~- ^^ ^^ ^ ^ -^ ^ ^ -^~~'~~ ~ ~ ~ ~~ ~ ----------------------- ---------- --------------------------------. R - FUN-TON- F U NIDENTIFICATION — OU-TPUT - - FORM —AT. -- PRINT C ''OMMENT — XPNETEALDANRA L --- RUSOIAL WAVE3OF V AE E A ~IN AATEDA - -WAVE -LENGTH —ANDDECAFACTORS --- —------------ --------------------------------------- - R INITIALIZ6 T OF VARIABLE.--- -X. --------—. --- —.. --- —-- R VARIABLE COMPUTATION BLOCK* -i-i -------— FO, ---i., --- -------— F --- —-------------— V --- —---- STORE(I)ul*+AA*EXP.(-KK1*X)*SINe(6*2831853*X/WWL) UU(........ /STORE ).. CC1(i-).S RT.(STORE(I)) Fl XX+XD FUNCTION RETURN END OF FUNCTION ---- CC-C. -.-.. 0-42 -

-67 - CHARACTERISTIC METHOD..OF.SOLUTION OF THE NON-DIMENSIONAL FLOW EQUATIONS F*F*nU/nT+F*F*I1*I./nx+r*nr/ny*rn CT'T)-c TNfTa+*F*_FiU*U/.Cfr,*r=n. ___ ______ __._...................... C*DU/DX+2*U*DC. DX+2DC T=.......................MULTIPLE DISTURBANCES$.ARE INTRODUC.ED.FOR THE PURPOSE OF EXAMINING THE EVOLUTION OF WAVES WITH TIME. THE DISTURBANCES EXTEND OVER THE ENTIRE INITIAL DATA. THE UPSTREAM TWO POINTS IN EACH COMPUTED ROW ARE DROPPED IN THE SUCCEEDING ROW. ________ FROUDE NUMBER = 3,0..... SLOPE =.100 DISTANCE INCREMENT =.10O... TIME INCREMENT =.100 AMPLITUDE.100 GENERATION TIME = 7.000 LIMIT ON STORAGE = 1000 INITIAL WAVE LENGTH= 4.000 DECAY FACTOR =.023 THE WAVE FORM IS AN EXPONENTIALLY DECAYING SINUSOIDAL WAVE, OF INDICATED WAVE LENGTH, AND DECAY FACTOR. WAVE FORM AT TIME.000; STARTING INDEX IS 0 VELOCITIES — 1.000.985.970.957.945.935.926.919.915.912.911.912.915.920.927.936.946.958.971.985 1.000 1.015 1.030 1.045 1.059 1.072 1.082 1.091 1.098 1.102 1.103 1.101 1.097 1.090 1.081 1.070 1.057 1.043 1.029 1.015 1.000.986.973.961.950.940.932.926.922.919.918.919.922.927.933.941.951.962.974.987 1.000 1.014 1.028 1.041 1.05.3 1.065 1.075 1.083 1.088 1.092 1.093 1.092 1.088 1.081 1.073 1.063 1.052 1.040 1.026 1.013 1.000.987.975.964.954.945.938.932.928.926.925,.926.929.933.939.946.955.965.976.988.000 -1.013 1.025 1.037 1.048 1.059 1.068 1.075 1.080 1.083 1.084 1.083 1.079 1.074 1.066 1.057 1.047 1.036 1.024 1.012 1.000.988.977.967.958.950.943.938.934.932.931.932.934.938.944.951.959.968.978.989 1.000 1.011 1.023 1.034 1.044 1.053 1.061 1.068 1.072 1.075 1.076 1.075 1.072 1.067 1.060 1.052 1.043 1.033 1.022 1.011 1.000.989.979.970.961.954.948.943 939.937,937.938,940.944.949.955.962.971.980.990 1.000 1.010 1.021 1.031 1.040 1.048 1.056 1.061 1.066 1.068 1.069 1.068 1.065 1.061 1.055 1.047 1.039 1.030 1.020 1.010 1.000.990.981.972.965.958.952.948.944.943.942.943.945.948.953.959.966.973.982.991 -—;- 1.009 1.019 1.028 1.036 1.044 1.0501.056 1.060 1.062 1.062 1.062 1.059 1.055 1.049 1.043 1.035 1.027 1.018 1.009 1.000.991.983.975.968.961.956.952.949.947.947__ 48.950.953.957.962.968.976.983.991 1.000 1.009 1.017 1.025 1.033 1.040 1.046 1.051 1.054 1.'056 1.057 1.056 1.053 1.050 1.045 1.039 1.032 1.025 1.017 1.008 1.000.992.984.977.970.965.960.956.953.952.951.952.954.957,.961..965.971.978.985.992 1.000 1.008 1.016 1.023 1.030 1.036 1.042 1.046 1.049 1.051 1.051 1.051 1.049 1.045 1.041 1.035 1.029 1.022 1.015 1.008 1.000.993.986.979.97.3.968.963.960.957.956.955.956.958.960.964.968.974.980..986.993 1.000 1.007 1.014 1.021 1.027 1.033 1.038 1.042 1.044 1.046 1.047 1.046 1.044 1.041 1.037 1.032 1.027 1.020 1.014 1.007 1.000.993.987.981.975.970.966.963.961.960.959.96_,.961.964.967.971.976.981.987.994 1.000 1.007 1.013 1.019 1.025 1.030 1.034 1.038 1.040 1.042 1.042 1.042 1.040 1.037 1.034 1.029 1.024 1.018 1.012 1.006 1.000.994.988.982.977.973.969.966.964.963.963.963.965.967.970.974.978.983.988.994. 1.000 1.006 1.012 1.017 1.023 1.027 1.031 1.034 1.037 1.038 1.038 1.038 1.036 1.034 1.031 1.027 1.022 1.017 1.011 1.006 1.000.994.989.984.979.975.972.969.967.966.966.966.968.970.972.976.980.984.989.995 1.000 1.005 1.011 1.016 1.021 1.025 1.028 1.031 1.033 1.035 1.035 1.034 1.033 1.031 1.028 1.024 1.020 1.015 1.010 1.005 1.000.995.990.985.981.977.974.972.970.969.969 99.970.972.975.978.982.986.990.995 1.000 1.005 1.010 1.014 1.019 1.023 1.026 1.028 1.030 1.031 1.032 1.031 1.030 1.028 1.025 1.022 1.018 1.014 1.UUY 1.UU5 1.000.995.991.987.983.979.977.974.973.972.971.972.973.975.977.980.983.987.991.996 1.000 1.005 1.009 1.013 1.017 1.021 1.023 1.026 1.028 1.029 1.029 1.028 1.027 1.025 1.023 1.020 1.017 1.013 1.009 1.004

-68 - 1.000. 996.992..9898 94.981.979.976.975.974.974.974.975.977.979.982.985.988.992.996 1.000 1.004 1.008 1.012 1.016 1.019 1.021 1.024 1.025 1.026 1.026 1.026 1.025 1.023 1.021 1.018 1.015 1.012 1.008 1.004 1.000.996.992.989.986.983.80.979.977.97A Q.7ji..977 6 2.979.9J81.983_S6..._98..293.996 1.000 1.004 1.007 1.011 1.014 1.017 1.019 1.021 1.023 1.024 1.024 1.024 1.023 1.021 1.019 1.017 1.014 1.011 1.007 1.004._1.4_._ 9__93_96_.,_._99073..__^>.._9-7 5848.9....-982... 980.979..978.978.978.979.981.982.985.987.990.993.997 1.000 1.003 1.007 1.010 1.013 1.015 1.018 1.019 1.021 1.022 1.022 1.021 1.021 1.019 1.017 1.015 1.013 1.010 1.007 1.003 -.O,.97,.4,.5.1.8.-,28.6...9.4._.....8.2 82 981.980.980.980.981.982..984.986.988.991.994.997 1.000 1.003 1.006 1.009 1.012 1.014 1.016 1.018 1.019 1.020 1.020 1.020 1.019 1.017 1.016 1.014 1.011 1.009 1.006 1.003 1.000.997.994.992.989.987.985.984.983.982.982.982.983.984.985.987.989.992.994.997 1.000 1.003 1.006 1.008 1.011 1.013 1.015 1.016 1.017 1.018 1.018 1.018 1.017 1.016 1.014 1.013 1.010 1.008 1.005 1.003 _1_.-__997..-995...95>2..9.9Q9____8. x96.985.984.9.84.983.984.984.985.987.988.990.992.995.997 1.000 1.003 1.005 1.008 1.010 1.012 1.013 1.0151.016 1.016 1.016 1.016 1.016 1.014 1.013 1.011 1.009 1.0071.005 1.002.1.000 98.. _..., 9.93_....91..9898.988.986.985.985.985.985.986.987.988.989.991,993.995.998 1.000 1.002 1.005 1.007 1.009 1.011 1.012 1.013 1.014 1.015 1.015 1.015 1.014 1.013 1.012 1'.010 1.009 1.007 1.004 1.002 1.000.998.996.994.992.990.999.988.987.986.986.986.987.988.989.990.992.994.996.998 1.000 1.002 1.004 1.006 1.008 1.010 1.011 1.012 1.013 1.013 1.014 1.013 1.013 1.012 1.011 1.009 1.008 1.006 1.004 1.002 1.000..998 3.996.994.992....9.9.1...990 989.988....,987.987.,988 988._._,989.,990.991.993.994.996.998 1.000 1.002 1.004 1.006 1.007 1.009 1.010 1.011 1.012 1.012 1.012 1.012 1.012 1.011 1.010 1.0091.007 1.006 1.004 1.002 1.000.998.996..995.993.992 _991.990.989.989.88.989.989.990.991.992.993.995.996.998 1.000 1.002 1.004 1.005 1.007 1.008 1.009 1.010 1.011 1.011 1.011 1.011 1.011 1.010 1.009 1.008 1.007 1.005 1.003 1.002 1.000.998.997.995.994.992.991.991.990.990.989.990.990.991.992.993.994.995.997.998 1.000 1.002 1.003 1.005 1.006 1.007 1.008 1.009 1.010 1.010 1.010 1.010 1.010 1.009 1.008 1.007 1.006 1.005 1.003 1.002 1i.000. WRVE VELOCITIES 1.000 1.008 1.015 1.022 1.0291.034 1.039 1.043 1.046 1.047 1.048 1.047 1.045 1.042 1.038 1.034 1.028 1.022 1.015 1.007 1.000.993.985.978.972.966.961.957.954.953.952.953.955.958.962.967.973.979.986.993 1.000 1.007 1.014 1.020 1.026 1.031 1.036 1.039 1.042 1.043 1.044 1.043 1.041 1.039 1.035 1.031 1.026 1.020 1.013 1.007 1.000.993.987.980.974.969.965.961.958.957.957.957.959.962.965.970.975.981.987.993 '1.000 1.006 1.013 1.019 1.024 1.029 1.033 1.036 1.038 1.039 1.040 1.039 1.038 1.035 1.032 1.028 1.023 1.018 1.012 1.006 1.000.994.988.982.977.972.968.965.962.961.960.961.963.965.968.973.977.983.988.994 'T:08 1.d-006 1.012.017 1.0221.026. 0301. 1.035 1.036 1.036 1.036 1.034 1.032 1.029 1.026 1.021 1.016 1.011 1.006 1.000.994.989.984.979.974.971.968.966.964.964.965.966.968.971.975.979.984.989.995 1.000 1.005 1.011 1.015 1.020 1.024 1.027 1.030 1.032 1.033 1.033 1.033 1.031 1.029 1.027 1.023 1.019 1.015 1.010 1.005 1.000.995.990.985.981.977.973.971.969.968.967.968.969.971.974.977.981.985.990.995 1.000 1.005 1.010 1.014 1.018 1.022 1.025 1.027 1.029 1.030 1.030 1.030 1.029 1.027 1.024 1.021 1.018 1.014 1.009 1.005 1.000.995.991.986.982.979.976.973.971.970.970.971.972.974.976.979.983.987.991.995 1.000 1.004 1.009 1.013 1.0"17 1.020 1.023 1.025 1.026 1.027 1.028 1.027 1.026 1.025 1.022 1.019 1.016 1.012 1.008 1.004 1.000.996.992.988.984.981.978.976.974.973.973.973.974.976.978.981.984.988.992.996 1.000 1.004 1.008 1.012 1.015 1.018 1.021 1.023 1.024 1.025 1.025 1.025 1.024 1.022 1.020 1.018 1.015 1.011 1.008 1.004 1.000.996.992.989.985.982.980..978.976.975.975.976.977.978.980.983.986.989.993.996 1.000 1.004 1.007 1.011 1.014 1.017 1.019 1.021 1.022 1.023 1.023 1.023 1.022 1.020 1.019 1.016 1.013 1.010 1.007 1.004 1.000.996.993.990.987.984.982.980..978.978.977.978.979.980.982.984.987.990_.993.997 1.'000 1.003 1.007 1.010 1.01.3 1.015'1.017 1.0191.02011.021.1.02.021 1.020 1.019 1.017 1.015 1.0121.0091.0061.003 1.000.997.994.991.988.985.9L.982.980.980.979,80.981.982.984.986.988.991.994.997 1.000 1.003 1.006 1.009 1.012 1.014 1.016 1.017 1.018 1.019 1.019 1.019 1.018 1.017 1.015 1.013 1.011 1.009 1.006 1.003 1.000.997.994.991.989.987.985.983.982.981.981.982.982.983.985.987.989.992.994.997 1.000 1.003 1.006 1.008 1.010 1.013 1.014 1.016 1.017 1.017 1.018 1.017 1.017 1.016 1.014 1.012 1.010 1.008 1.005 1.003 1.000.997.995.992.990.988.986.985.984.983.983.983.984.985.986.988..990.992_.995..97 1.000 1.003 1.005 1.007 1.010 1.011 1.0131.014 1.015 1.016 1.016 1.016 1.015 1.014 1.013 1.011 1.009 1.007 1.005 1.002 1.000.998.995.993.991.989.937.928 6.985..99885...998 4-8 - -.986.988 989.991.993.995.998 1.000 1.002 1.005 1.007 1.009 1.010 1.012 1.013 1.014 1.014 1.015 1.014 1.014 1.013 1.012 1.010 1.008 1.007 1.004 1.002 1.000.998.996.993.992.990.988.987.986.986.986.986.987.987.989.990.992.994.996.998 1.000 1.002 1.004 1.006 1.008 1.010 1.011 1.012 1.013 1.013 1.013 1.013 1.013 1.012 1.011 1.009 1.008 1.006 1.004 1.002 1.000.998.996.994.992.991.989.988.988.987.987.987.988.989.990.991.993.994.996.998 1.000 1.002 1.004 1.006 1.007 1.009 1.010 1.011 1.012 1.012 1.012 1.012 1.012 1.0111.010 1.009 1.007 1.005 1.004 1.002 1.000.998.996.995.993.992.990.989.989.988.988.988.989.990.991.992.993.995.996.998 1.UUU 1.UU~ 1.UU4 1.UU~ 1.UU(1.UU8 1.LILI') 1.U1U 1.U11 1U11 1.U11 lull 1.Ull l.L'IU.ULr liUUn q.U nriI.UUnn.UV. 1.000 1.002u 1.004 1.005 1.007 1.008 1.009l 1.010 1.0u 1 1.011 1.011 1.011 1.Ul. 1.01u 1.u 00 1. UA 1.Uu Il.uuZ. uu.5. uuze 1.000.998.997.995.994.992.991.990.990.989.989.989.990.991.991.993.994.995.997.998 1.000 1.002 1.003 1.005 1.006 1.007 1.008 1.009 1.010 1.010 1.010 1.010 1.010 1.009 1.008 1.007 1.006 1.005 1.003 1.002

-69 - 1.00 ___9 _. 9__.0.?__ 5__ 99 __^ _ 9. __ i- -9-_ S^^.~.~992 *9 _z9_93.994. 996. 997. 999 1.000 1.001 1.003 1.004 1.006 1.007 1.008 1.008 1.009 1.009 1.009 1.009 1.009 1.008 1.007 1.006 1.005 1.004 1.003 1.001 1.000.999.997.996.995.994.993.992.992.991.991.991.992.992.993.994.995.996.997.999 1.000 1.001 1.003 1.004 1.005 1.006 1.007 1.008 1.008 1.008 1.008 1.008 1.008 1.007 1.007 1.006 1.005 1.004 1.003 1.001 1.-000 *.999?997_ —.9 _9-5- _9 —9_,93__2-_9 998. 999 1.000 1.001 1.002 1.004 1.005 1.005 1.006 1.007 1.007 1.008 1.008 1.008 1.007 1.007 1.006 1.005 1.004 1.003 1.002 1.001 1.000.99_9__.9._.9_97_.996. 995 -— 4.99 993_ 93_993_.993_._9 _...993.994,_995.996.997.998 _ 1.000 1.001 1.002 1.003 1.004 1.005 1.006 1.006 1.007 1.007 1.007 1.007 1.007 1.006 1.006 1.005 1.004 1.003 1.002 1.001 1.000.999.998.997.996.995.995.994 994.993.993.993.994.994.995.995.996.997.998.999 1.000 1.001 1.002 1.003 1.004 1.005 1.005 1.006 1.006 1.006 1.006 1.006 1.006 1.006 1.005 1.004 1.004 1.003 1.002 1.001 1.000.999.998.997.996.996.995.994.994.994.994.994.994._995.9__95__.996.996.997.998.999 1.000 1.001 1.002 1.003 1.003 1.004 1.005 1.005 1.006 1.006 1.006 1.061006 1.006 5 1.005 1.004 1.003 1.003 1.002 1.001 1.000.999.998.997.997.996 _.999 __95___.995.995.994 _.994.994.995.9__95___.996.__996._997.____998_.998.999 1.000 1.001 1.002 1.002 1.003 1.004 1.004 1.005 1.005 1.005 1.005 1.005 1.005 1.005 1.004 1.004 1.003 1.002 1.002 1.001 1.000.999.998.998.997.996.996.995.995.995.995.995.995.995.996.996.997.998.998.999 1.000 DEPTHS --- —------------------- -------------------------------- 1.000 1.016 1.031 1.045 1.058 1.070 1.080 1.088 1.093 1.097 1.098 1.096 1.093 1.086 1.078 1.068 1.057 1.044 1.030 1.015 1.OQO.985.971.957.944.933.924.916.911.908.907.908.912.917.925.935.946.958.972.986 1.000 1.014 1.028 1.041 1.053 1.064 1.073 1.080 1.085 1.088 1.089 1.088 1.084 1.079 1.071 1.062 1.052 1.040 1.027 1.014 1.000.986.973.961.949.939.931.924.919.916.91.5.916.919.925.932.941.951.962.974.987 1.000 1.013 1.026 1.037 1.048 1.058 1.066 1.073 1.078 1.080 1.081 1.080 1.077 1.072 1.065 1.057 1.047 1.036 1.025 1.012 1.000.988.976.964.954.945.937.930.926.923.922.924.927.931.938.946.955.965.976.988 1.000 1.012 1.023 1.034 1.044 1.053 1.060 1.066 1.071 1.073 1.074 1.073 1.070 1.066 1.059 1.052 1.043 1.033 1.022 1.011 T.-btid'-89.978.96?.958.949.942.937.932.930.929.930.933.937.943.951.959.968.979.989 1.000 1.011 1.021 1.031 1.040 1.048 1.055 1.061 1.065 1.067 1.068 1.067 1.064 1.060 1.054 1.047 1.039 1.030 1.020 1.010 1.ooo.990.980.970.962.954.947.942.938.936.936.936.939.943.948.955.963.971.980.990 1.000 1.010 1.019 1.028 1.037 1.044 1.050 1.055 1.059 1.061 1.062 1.061 1.058 1.054 1.049 1.043 1.036 1.028 1.019 1.009 T:Wb —:ftr ---.973.965.958.952.947.944.942.941.942.944.948.953.959.966.974.982 1.000 1.009 1.018 1.026 1.033 1.040 1.046 1.050 1.054 1.056 1.056 1.055 1.053 1.050 1.045 1.039 1.033 1.025 1.017 1.009 1:Z.9W-~ r ---.9-ff3 ---.-975 ---.-968-.9~62.956.952.949.947.946.947.949.953.957.963.969 1.000 1.008 1.016 1.024 1.031 1.037 1.042 1.046 1.049 1.051 1.051 1.050 1.048 1.045 1.041 1.036 1.030 1.023 1.016 1.008 1.000.992.985.977.971.965.960.956.953.952.951.952 954.957.961.966.972.978.985.993 1.000'1.007 1.015 1.022 1.028 1.033 1.038 1.042 1.045 1.046 1.047 1.046 1.044 1.041 1.037 1.033 1.027 1.021 1.014 1.007 1.000.993.986.979.973.968.964.960.957.956.955.956.958.961.964.969.974.980.986.993 1.000 1.007 1.013 1.020 1.025 1.030 1.035 1.038 1.041 1.042 1.043 1.042 1.040 1.038 1.034 1.030 1.025 1.019 1.013 1.007 i~~aoO;994.987.981.976.971.967.964.961.960.959.960.962.964.967.972.976.982.988.994 1.000 1.006 1.012 1.018 1.023 1.028 1.032 1.035 1.037 1.038 1.039 1.038 1.037 1.034 1.031 1.027 1.023 1.017 1.012 1.006 1.000.994.988.983.978.973.970.967.965.963.963.963.965.967.970.974.979.983.989.994 1.00O 1.006 1.011 1.016 1.021 1.025 1.029 1.032 1.034-1.035-1.035 1.035 1.034 1.031 1.028 1.025 1.021 1.016 1.011 1.005 1.000.995.989.984.980.976.972.970.968.967.966.967.968.970.973.976.980.985.990.995 1.000 1.005 1.010 1.015 1.019 1.023 1.026 1.029 1.031 1.032 1.032 1.032 1.031 1.029 1.026 1.023 1.019 1.014 1.010 1.005 1.000.995.990.986.982.978.975.972.971.969.969.970.971.973.975.978.982.986.991.995 1.000 1.005 1.009 1.014 1.018 1.021 1.024 1.026 1.028 1.029 1.029 1.029 1.028 1.026 1.024 1.021 1.017 1.013 1.009 1.005 1.000.996.991.987.983.980.977.975.973.972.972.972.973.975.977.980.984.987.991.996 1.000 1.004 1.008 1.012 1.016 1.019 1.022 1.024 1.026 1.027 1.027 1.026 1.025 1.024 1.022 1.019 1.016 1.012 1.008 1.004 1.000.996.992.988.985.982.979.977.976.975.974.975.976.977.979.982.985.989.992.996 1.000 1.004 1.008 1.011 1.015 1.017 1.020 1.022 1.023 1.024 1.024 1.024 1.023 1.022 1.020 1.017 1.014 1.011 1.007 1.004 1:~0<66~~~.996~~~.~993 ~~.'989.986 -.983-.9-81.979.978.977.977.'977.978 -.979.981 "7.984.986.990.993.996 1.000 1.004 1.007 1.010 1.013 1.016 1.018 1.020 1.021 1.022 1.022 1.022 1.021 1.020 1.018 1.016 1.013 1.010 1.007 1.003 1.000.997.993.990.987.985.983.981.980.979.979.979.980.981.983.985.988.990.994.997 1.000 1.003 1.006 1.009 1.012 1.015 1 07_1 01_1_ 019 17 020- 1_ 020 1,020 1.019 1.018 1.016 1.014_1.012 1.009 1.006 1.003 1.000.997.994.991.988.986.984.983.981.981.981.981.982.983.984.986.989.991.994.997 1.000 1.003 1.006 1.009 1.011 1.013 1.015 1.017 1..16___ 1J1_9_1.1 10Q113 1.008 1.006 1.003 1.000.997.994.992.989.987.986.984.983.982.982.983.983.984.986.988.990.992.995.997 1.000 1.003 1.005 1.008 1.010 1.012 1.014 1.015 1.016 1.017 1.017 1.017 1.016 1.015 1.014 1.012 1.010 1.008 1.005 1.003 i RNAU 001( CNOW OO ekon.9191.-9!.29b.292.29.29.29 n.29.29.29!.22.22 -.226.22.22 I. UUU0.997.995.993.99.988.987.986.985.984.84.984.983.9986.987.989.991.993. 993.998 1.000 1.002 1.005 1.007 1.009 1.011 1.013 1.014 1.015 1.015 1.015 1.015 1.015_1.014 1.012 1.011 1.009 1.007 1.005 1.002 1.000.998.995.993.991.989.988.987.986.985.985.985.986.987.988.990.991.993.996.998

-70 - 1.000 1O^_._002 l.0_Q4 _1.0Q0 _4..08_.A.i __4-Q14__._.Q14 1.00 1.021.006 1-1. 004 1.002 1.000.998.996.994.992.990.989.988.987.987.987.987.987.988.989.991.992.994.996.998 1.000 1.002 1.004 1.006 1.008 1Q009 1-010 1.012 1.012 1.013 1.013 1.013 1.012 1.011 1.010 1.009 1.007 1.006 1.004 1.002 1.000.998.996.994.993.991.990.989.988.988.988.988.988.989.990.991.993.995.996.998 -J,1~~-.JOlQOlJ^QQ^.J^~QO^.__tLPP^OiLD^-l^OILTL-l^ ~ia-l^JLLQ-J^JlLO-J^Qll-JL^QlZ.-L^OJ1j002_ 1.000.998.996.995.993.992.991.990.989.989.989.989.989.990.991.992.994.995.997.998.J-^Op1,~~~~~~~Ol,___L^OO^3_l1~~~~~jD ~,Q~~~~~ll~~~~~,QSl^Jltt51~~~ 00 0031310_0Q2 1.000.998.997.995.994.993.992.991.990.990.990.990.990.991.992.993.994.995.997.998 1.000 __000~l~_r_c_N__________________________________________________________ X-DISTANCES.0.1.2.3.4.5.6.7.-8.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2_.0 ___2_.___1 2.2 ___2.3 2.4 2.5 2.6 2.7.8 29 3.0 3.1 32 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4.4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9.0 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10.0 10.1 19410 1 QQ — l L_!_l A_1.9_9 -101- 1 L2iI AL411.5 11.6 11.7 11.8 119. —1-.__1 ____P..___0 _1_Q.4___0 — L0^6 —107 —1_.A — __ _ _.J_1_._^~... __ 12.0 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 13.0 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 14.0 14.1 14.2 14.3 14.4 14.5 ___14.6 14.7 14.8 14. 15.0__ 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 16.0 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 17.0 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.9 18.0 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8 18.9 19.0 19.1 19.2 19.3 19.4 19.5 19.6 19.7 19.8 19.9 20.0 20. 20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 21.0 21.1 21.2 21.3 21.4 21.5 21.6 21.7 21.8 21.9 22._0 __22.1 __22.2 22.3 22.4 22.5 22.6 22.7 22.8 22.9 23.0 23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8 23.9 24.0 24.1 24.2 24.3 24.4 24.5 24.6 24.7 24.8 24.9 25.0 25. 25.2 25.3 25.4 25.5 25.6 25.7 25.8 25.9 26.0 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9 27.0 27.1 27.2 27.3 27.4 27.5 27.6 27.7 27.8 27.9 28.0 28.1 28.2 28.3 28.4 28.5 28.6 28.7 28.8 28.9 29.0 29.1 29.2 29.3 29.4 29.5 29.6 29.7 29.8 29.9 30.0 30.1 30.2 30.3 30.4 30.5 30.6 30.7 30.8 30.9 31.0 31.1 31.2 31.3 31.4 31.5 31.6 31.7 31.8 31.9 32.0 32.1 32.2 32.3 32.4 32.5 32.6 32.7 32.8 32.9 33.0 33.1 33.2 33.3 33.4 33.5 33.6 33.7 33.8 33.9 34.0 34.1 34.2 34.3 34.4 34.5 34.6 34.7 34.8 34.9 35.0 35.1 35.2 35.3 35.4 35.5 35.6 35.7 35.8 35.9 36.0 36.1 36.2 36.3 36.4 36.5 36.6 36.7 36.8 36.9 37.0 37.1 37.2 37.3 37.4 37.5 37.6 37.7 37.8 37.9 38.0 38.1 38.2 38.3 38.4 38.5 38.6 38.7 38.8 38.9 39.0 39.1 39.2 39.3 39.4 39.5 39.6 39.7 39.8 39.9 40.0 40.1 40.2 40.3 40.4 40.5 40.6 40.7 40.8 40.9 41.0 41.1 41.2 41.3 41.4 41.5 41.6 41.7 41.8 41.9 42.0 42.1 42.2 42.3 42.4 42.5 42.6 42.7 42.8 42.9 43.0 43.1 43.2 43.3 43.4 43.5 43.6 43.7 43.8 43.9 44.0 44.1 44.2 44.3 44.4 44.5 44.6 44.7 44.8 44.9 45.0 45.1 45.2 45.3 45.4 45.5 45.6 45.7 45.8 45.9 46.0 46.1 46.2 46.3 46.4 46.5 46.6 46 —7 - -_ 4J^ 47.1__47._ 47.347 4. 4647.7 47.8 47.9 48.0 48.1 48.2 48.3 48.4 48.5 48.6 48.7 48.8 48.9 49.0 49.1 49.2 49.3 49.4 49.5 49.6 49.7 49.8 49.9 50.0 50.1 50.2 50.3 50.4 50.5 50.6 50.7 5 09__5J___51.15.2 51.3 51.4 51.5 51.6 51.7 5.8 51.9 52.0 52.1 52.2 52.3 52.4 52.5 52.6 52.7 52.8 52.9 53.0 53.1 53.2 53.3 53.4 53.5 53.6 53.7 53.8 53.9 54.0 54.1 54.2 54.3 54.4 54.5 54.6 54.7 54.8 54.9 55.0 55.1 55.2 55.3 55.4 55.5 55.6 55.7 55.8 55.9 56.0 56.1 56.2 56.3 56.4 56.5 56.6 56.7 56.8 56.9 57.0 57.1 57.2 57.3 57.4 57.5 57.6 57.7 57.8 57.9 — 58.0 58.1 58.2 58.3_5.4. 4_6 j _ —__9 —j ---.J_- -.4_ 5 59.6 59.7 59.8 59.9 60.0 60.1 60.2 60.3 60.4 60.5 60.6 60.7 60.8 60.9 61.0 61.1 61.2 61.3 61.4 61.5 61.6 61.7 61.8 61.9 _62.0_ 62.1_ 62.2 2_.62_._3.._.___ __.6.^6___ ^J__.62 _ —6 2^ __6 3 ---63. —~-2-5 63.6 63.7 63.8 63.9 64.0. 64.1 64.2 64.3 64.4 64.5 64.6 64.7 64.8 64.9 65.0 65.1 65.2 65.3 65.4 65.5 65.6 65.7 65.8 65.9 66.0 66.1 66.2 66.3 66.4 66.5 66.6 66.7 66.8 66.9 67.0 67.1 67.2 67.3 67.4 67.5 67.6 67.7 67.8 67.9 68.0 68.1 68.2 68.3 68.4 68.5 68.6 68.7 68.8 68.9 69.0 69.1 69.2 69.3 69.4 69.5 69.6 69.7 69.8 69.9 70.0 70.1 70.2 70.3 70.4 70.5 10~6 70.7 _70.8 70.9_71.Q 71.41 127. 71.4 71.5 71.6 __.7 71.8 71.9 72.0 72.1 72.2 72.3 72.4 72.5 72.6 72.7 72.8 72.9 73.0 73.1 73.2 73.3 73.4.73.5 73.6 73.7 73.8 73.9 74.0 74.1 74.2 74.3 74.4 74.5 74.6 74.7 74.8 74.9 75.0 75.1 75.2 75.3 75.4 75.5 75.6 75.7 75.8 75.9 76.0 76. 76.2 76.3 76.4 76.5 76.6 76.7 76.8 76.9 77.0 77.1 77.2 77.3 77.4 77.5 77.6 77.7 77.8 77.9 78.0 78.1 78.2 78.3 78.4 78.5 78.6 78.7 78.8 78.9 79.n 79.1 79.2 79.3 79.4 79.5 79.6 79.7 79.8 79.9 80.0 80.1 80.2 80.3 80.4 80.5 80.6 80.7 80.8 80.9 81.0 81.1 81.2 81.3 81.4 81.5 81.6 81.7 81.8 81.9 82.0 82.! 82.2 82.3___82__A 83. 7__7838.1_ —32L ---83.J- A5_3__._6 ~. 83.9 84.0 84.1 84.2 84.3 84.4 84.5 84.6 84.7 84.8 84.9 85.0 85.1 85.2 85.3 85.4 85.5 85.6 85.7 85.8 85.9 86.0 86.1 86.2 86.3 86.4 86.5 __ 6 -6.__86,.i__86.-89__6LQ__.7_.l1_1.2_-7~3___6..._7_3.._7A87. 88 9 88.0 88. 88.2 88.3 88.4 88.5 88.6 88.7 88.8 88.9 89.0 89.1 89.2 89.3 89.4 89.5 89.6 89.7 89.8 89.9 90.0 90.1 90.2 90.3 90.4 90.5 90.6 90.7 90.8 90.9 91.0 91.1 91.2 91.3 91.4 91.5 91.6 91.7 91.8 91.9 92.0 92.1 92.2 92.3 92.4 92.5 92.6 92.7 92.8 92.9 93.0 93.1 93.2 93.3 93.4 93.5 93.6 93.7 93.8 93.9 94.0 94.1 94.2 94.3 94.4 94.5 94.6 94.7 94_.894.95.0 95.1. 95.2 95.3 95.4__95.5 95.6 95.7 95.8 95.9 96.0 96". 1 "96.2 ~96.3 96.4 96.5 96.6 96.7 96.8 96.9 97.0 97.1 97.2 97.3 97.4 97.5 97.6 97.7 97.8 97.9 - 98.0_98!!-_ 96.2_96 _98_9896__ 9 78-58.-.9...99. 1..992.._99. _99~4 9.9.5...99.6 99.7 99.8 99.9 100.0