KUNDU Fluid Mechanics 2 Episode 6 ppsx

of a wave is Doppler.shifted by an amount U 9 K due to the mean flow. Equation (7.20) is easy to uiderstand by considering a situation in which the intrinsic frequency w is zero and the flow pattern has a periodicity in the x direction of wavelength 2n/k. If this sinusoidal pattern is translated in the x direction at speed U, then the observed frequency at a fixed point is OJO = Uk. The effects of mean flow on frequency will not bc considered further in this chaptcr. Consequently, thc involved frequencies should be interpreted as thc intrinsic frcquency. In this section we shall discuss gravity waves at the free surface of a sca of liquid of uniform depth H, which rmiy be large or small compared to the wavelength h. We shall assume that thc amplitude a of oscillation of the free surrace is small, in the sense that both a/h and a/H are much smallcr than one. The condition a/h << 1 implies that the slope of the sea surface is small, and the condition u/H << 1 implies that the instantaneous depth does not differ significantly from the undisturbed depth. Thesc conditions allow us to linearize the problem. The frequency of the waves is assumed large compared to the Coriolis frequency, so that the waves are unaffected by he earth's rotation. Hem, we shall neglect surface tension; in water its effect is limited to wavelengths (7 cm, as discussed in Section 7. The fluid is assumed to have small viscosity. so that viscous effects are confined to boundary layers and do not affect the wave propagation significantly. The motion is assumed to be generated from rest, say, by wind action or by dropping a stone. According to Kelvin's circulation theorem, rhe resulting motion is irivtariontil, ignoring viscous effects, Coriolis forces, and stratification (density variation). Formulation of the Problem Consider a case where the wavcs propagate in the s direction only, and that the motion is two dimensional in the xz-planc (Figure 7.4). Let the vertical coordinate z be measured upward froin the undisturbed free surface. The free surface displacement is q(x. r). Because the motion is ii-rotational, a velocity potential 4 can be defined 't H 1 2 =-H Figure 7.4 Wave nommnclaturc. such that Substitution into the continuity equation gives the Laplace equation a%p a%p -+-=o. ax2 az2 (7.21) (7.22) (7.23) Boundary conditions are to be satisfied at the [ne surface and at thc bottom. The condition at the bottom is zero nod velocity, that is at z = -H. (7.24) At the free surface, a kinematic boudui? condition is that the fluid particle never leaves the surface, that is D4J -= w,, at z=q, Dr where D/Dr = a/ar + u(a/a.r), and tu,, is the vertical component of fluid velocity at the free surface. The forementioned condition can be written as (7.25) For small-amplitude waves both it and aq/a-r are small, so that the quadratic term u(aq/ax) is one order smaller than other terms in Eq. (7.25), which then simplifies to (7.26) We can simpllfy this condition still further by arguing that the righl-hand side CN~ be evaluated at z = 0 rather than at lhc free surface. To justify this, expand 8qb/az in a Taylor scries around z = 0: Therefore, to the first order of acciuacy desired hen, a$/az in Eq. (7.26) can be evaluated at z = 0. We then have aq a4 at az at z = 0. - =- (7.27) The error involved in approximating Eq. (7.26) by (7.27) is cxplained again later in this section. In addition to the kinematic condition at the surface, there is a dyncimic condition that the pressure just below the free surfacc is always equal to the ambient pi~mre, with surface tension neglected. Taking the ambient pressurc to be zero, the condition is p=O at z=v. (7.28) Equation (7.28) follows from thc boundary condition on t n, which is continuous across an interface as established in Chapter 4, Section 19. As before, we shall simplify this condition for sinall-amplitude waves. Since the motion is irrotational, Bernoulli's cquation (see Eq. (4.81)) 22 + i(u2 + 102) + + gz = F(t), at - P (7.29) is applicable. Here, the function F(t) can be absorbed in a#/at by redefining 4. Neglecting the nonlinear term (u' + w') for small-amplitude waves, the linearized form of the unsteady Bernoulli equation is a4 P - + - + gz = 0. at P Substihition into thc surface boundary condition (7.28) gives a4 at -+gq=O at z=r]. (7.30) (7.31 ) As behe, for small-amplitude waves, the term &$/at can be evaluated at z = 0 rather than at z = r] to give gr] at z=0. a4 at _- Solution of the Problem Recapitulating, we have tq solve a24 a'# -+ 0. i)~' subject to the conditions gr] at z=0. a4 at _- (7.32) (7.22) (7.24) (7.27) (7.32) IJI order to apply the boundary conditions, we need to assume a form for q(x. 1). The simplest case is that of a sinusoidal component with wavenumber k and frequency w, lor which q = COS(kx - wt). (7.33) One motivation for studying sinusoidal waves is that small-amplitude waves on a water surface become roughly sinusoidal some time after their generation (unless the water depth is very shallow). This is due to the phenomenon of wave dispersion discussed in Section 10. A second, and stronger, motivation is that an arbitrary disturbance can be decomposed into various sinusoidal components by Fourier analysis, and the mpoiise of the system to an arbitrary small disturbance is the sum of the responses to the various sinusoidal components. For a cosine dependence of q on (kx - ot), conditions (7.27) and (7.32) show that q5 must be a sine function of (kx - at). Consequently, we assume a separable solution of the Laplace equation in the form q5 = f(z) sin(kx - ut), (7.34) where f (z) and w(k) are to be determined. Substitution of Eq. (7.34) into the Laplace equation (7.22) gives k2f d2 .f =0, dz2 whose general solution is f(e) = Aek' + Be-kz. The vclocity potential is then q5 = (Ae" + Be-") sin(kx - wt). (7.33 The constants A aud B are now determined from the boundary conditions (7.24) and (7.27). Condition (7.24) gives B = Ae-2k". (7.36) Before applying condition (7.27) in the linearized form, let us explore what would happen if we applied it at z = q. From (7.35) we get Here we can set e kq 21 e 2: 1 if kq << 1, valid for small slope of the free surface. This is efkctively what we are doing by applying the surface boundary conditions Eqs. (7.27) and (7.32) at z = 0 (instead of at z = q), which we justified previously by a Taylor serics expansion. Substitution of Eqs. (7.33) and (7.35) into the surface velocity condition (7.27) gives (7.37) k(A - E) = (IO. The constants A and B can now be determincd from Eqs. (7.36) and (7.37) as The vclocity potential (7.35) then becomes from which the velocity components are found as sinhk(z + H) sinh kH 111 = UW sin(ks - or). (7.38) (7.39) We have solved the Laplace equation using kinematic boundary conditions alone. This is typical of irrotational flows. In the last chapter we saw that the equation of motion, or its integral, thc Bernoulli equation, is brought into play only to find the prcssurz distribution, after he problem has bcen solved from kincinatic considerations alonc. In the present case, we shall find that application of the dynamic free surface condition (7.32) gives a relation between k and w. Substitution of Eqs. (7.33) and (7.38) into (7.32) gives thc dcsired relation w = J (7.40) Thc phase speed c = w/k is related to the wave sizc by I ~ This shows that the speed of propagation of a wave component depends on its wavenumbcr. Waves for which c is a function of k arc called dispersive because waves of different lengths, propagating at dZFerent spmds, “dispersc” or separate. (Dispersion is a word borrowed from optics, whcrc it sigilifies separation of different colors due to the speed of light in a medium dcpending on thc wavelength.) A relation such as Eq. (7.40), giving w as a function of k, is called a dispcwion relation because it expresses the nature of the dispersive process. Wave dispersion is a €undamental pmccss in many physical phenomena; its implications in gravity waves are discussed in Scctions 9 and 10. 5. Sornc? l~bt~~r~rurx of Sutfacc CmLiiQ- H%t~.?t?s Scvcral featurcs 01 surface gravity wavcs are discussccl in tlus scction. In particular, we shall examine thc nature of pressure change, particlc motion, and the energy flow duc to a sinusoidal propagating wave. Thc water depth H is arbitrary; simplitications that result from assuming the depth to be shallow or deep arc discussed in the next scction. Pressure Change Due to Wave Motion It is sometimes possible to measure wave parameters by placing pressure sensors at the bottom or at some other suitable depth. One would theEfore like to how how deep the pressure fluctuations pcnetrate into the water. Pressure is given by the linearized Bernoulli equation a@ P - + - + gz = 0. at P If we define PI = p + pgz, (7.42) as theperturbation pressure, that is, the pressure change fromthe undisturbed pressure of -pgz, then Bernoulli’s equation gives a4 pl= -p at On substituting Eq. (7.38), we obtain Paw2 cash k(z + H) cos(kx - p’ = - k sinhkH which, on using the dispersion relation (7.40), becomes p’ = pga cosh k(z + H) cosh k H COS(kX - wt). (7.43) (7.44a) (7.44b) The perturbation pressure therefore decays into the water column, and whether it could be detected by a sensor depends on the magnitude of the water depth in relation to the wavelength. This is discussed further in Section 6. Particle Path and Streamline To examine particle orbits, we obviously need to use Lagrangian coordinates. (See, Chapter 3, Section2foradiscussionof theLagrangiandescriptionJLet (xo+~, ZO+ f) be the coordinates of a fluid particle whose rest position is (XO, ZO), as shown in Fig- ure 7.5. We can use (XO, ZO) as a “tag” for particle identification, and write &o, ZO, t) and ((.TO, zo, r) in the Lagrangian form. Then the velocity components are given by a6 at ’ =- ar at ’ w=- (7.45) where the partial derivative symbol is used because the particle identity (XO. ZO) is kept fixed in the time derivatives. For small-amplitude waves, the particle excursion (6, () is small, and the velocity of a particle along its path is nearly equal to the fluid velocity at the mean position (XO. ZO) at that instant, givcn by Eq. (7.39). Therefore, 'f Figure 7.5 Orbit oFa Ruid particlc whose mean position is (q). zn). Eq. (7.45) gives Inlegrating in time, we obtain cash k(~o + H) 6 = -(I sin(kx0 -or). sinh kH sinh k(z0 + H) shh k H <=a cos(kxo - wt). (7.46) Elimination of (kxn - ut) givcs ~~~hk(io+H)]' /[usinhk(z(l+ H)I2 = 1. (7.47) sinhkH + 5'' sinhkH which rcpresents cllipses. Both the semimajor axis n coshIk(z0 + H)]/sinh kH and the semiminor axis a sinh[k(zo + fl)]/siiih RH decrcase with dcplh, the minor axis vanishing at LU = -H (Figurc 7.6b). Thc distance between foci remains constant with depth. Equation (7.46) shows that thc phase of the motion (that is, thc argument of thc sinusoidal term) is independent of zo. Fluid particles in any vertical column arc therefore in phase. That is, if onc of !hem is at the top of its orbit, then all particles at the same .VI) are at the top of their orbits. To find thc streamlinc pallern. wc need to dctermiue thc streamfunction @? related to the velocity components hy il @ coshk(z + H) az sinhkH - = 11 = CIW COS(k.r - UJf). sinh k(z + H) sin(kx - of). w BX sinh kH - = -711 = -1IW (7.48) (7.49) Figure 7.6 Particle orbits of wavc motion in deep, intermediate and shallow seas. where Eq. (7.39) has been introduced. Integrating Eq. (7.48) with respect to z, we obtain ao sinh k(r + H) cos(kx - ot) + F(x, t), '=T sinhiiH where F(x,t) is an arbitrary function of integration. Similarly, integration of Eq. (7.49) with respect to J gives '=- a" sinh k(r -k k sinhkH cos(kx - ot) + G(z, t), where G(z, t) is another arbitrary function. Equating the two expressions for @ wc see that F = G = hction of time only; this can be set to zcro if we regard $ as due to wave motion only, so that 3 = 0 when a = 0. Therefore aw sinhk(z + H) e=- cos(kx - or). k sinhkH Let us examine the streamline structure at a particular timc, say, t = 0, when $ o( sinhk(z + H)coskx. (7.50) It is clear that $ = 0 at z = -H, so that the bottom wall is a part of the $ = 0 streamline. However, $ is also zero at kx = f17/2, f3n/2, . . . €or any z. At these C T uu =O Figure 7.7 Instantaneous strcanlinc pattern in ;I sdacc gravity wivc pmpagating LO thc right. values of kx, Eq. (7.33) shows that q vanishes. The resulting stremiline pattern is shown in Figure 7.7. It is seen that the vebcio is in the direction qfpmpugation (and horizontal ) ut all depths below the crests, rmd opposite to the direction qfpropagurioii at all depths below truugh. Energy- Considerations Surface gravity waves posscss kinetic encrgy due to motion of the fluid and potcntial energy due to dcIbnnation of the free surface. Kinetic energy per unit horizontal area is found by integrating over the dcpth and avcraging over a wavelength: Here the z-integral is taken up to : = 0, because the integral up to z = q gives a highcr-order tcrm. Substitution of thc velocity components from Eq. (7.39) gives 0 pw’ [ 1 li u2 cos’(k.r - wt) dx cosh’ k(z + H) dz 1, Ek = 2sinh2kH h 1 I. +,. Jd u2 sin2(kx - ut) dx lH sinh2k(z + H) dz] . (7.51) In tcrms of frcc su~facc displacemcnt q. the x-integrals in Eq. (7.5 I) can be written as a2 cos2(kx - wt) d.r = a’ sin2(kx - wt) dx where 3 is the mean square displacement. The z-integrals in Eq. (7.51) are easy to evaluate by expressing the hyperbolic functions in terms of exponentials. Using thc dispersion relation (7.40), Eq. (7.51) finally becomes - Ek = ipgq’, (7.52) which is the kinetic energy of the wave motion per unit horizontal area. Consider next the potenrid energy of the wave system, defined as the work done to deform a horizontal fixe surface into the disturbed state. It is therefore equal to the djference of potential energies of the system in the disturbed and undisturbed states. As the potential energy of an element in the fluid (per unit length in y) is pgz dx dz (Figure 7.Q the potential energy of the wave system per unit horizontal area is (7.53) (An easier way to arrive at the expression for E, is to note that the potential energy increase due to wave motion equals the work done in raising column A in Figure 7.8 to the location of column By and integrating over halfthe wavelength. This is because an interchange of A and B over half a wavclength automatically forms a complete wavelength of the deformed surface. The mass of column A is pq dx and the center of gravity is raised by q when A is taken to B. This agrees with the last form in Eq. (7.53).) Equalion (7.53) can be written in terms af the mean square displacement as (7.54) Comparison of Eq. (7.52) and Eq. (7.54) shows that the average kinetic and potential energies are equal. This is called theprinciple ofequipartition ofenergy and is valid in conservative dynamical systems undergoing small oscillations that are unaffected by Figure 7.8 Cdculation of potential cnergy of a fluid column. [...]... cos(k2x - W t ) Applying the trigonometric identity for cos A + cos B, we obtain r] = 24 2 cos [$(k - kl)X Writing k = (kl we obtain I - z(* + k2) /2, w = r] (01 - w , ) f ]cos [$il + k2)x - i(O1 + w2)t] + w2) /2, dk = k? - kl, and d o = wz - w1, = 24 J cos ( dk x - ; o t ) cos(kx - o r ) ; -d (7.74) Here, cos(kx - w f )is a progressive wave with a phase speed of c = w / k However, its amplitude 2u is... A = A,, and surface tension dominates for h A < A (Figure 7.1.3).Setting d c / h = 0 in Eq.(7 .66 ),and assuming the deep-water , approximationtanh(2aHlA) 2 I valid for H > 0 .28 A,we obtain (7 .67 ) For an &-water interface at 20 "Cythe surface tension is u = 0.074 N/in, giving = 23 .2 cm/s at A,, = 1.73cin (7 .68 ) Only small waves (say, A < 7 cm for an air-wakr interface), called ripples, arc therefore affected... presented in Section4, exceptthat the pressure boundary condition (7. 32) is replaced by (7 .64 ).This only changes the dispersion relation w(k),which is found by substitution of (7.33) and (7.38) into (7 .64 ), to give w =, / k (g + $)tanh kH (7 .65 ) Thc phase velocity is therefore c- = /(E + $) tanh kH = ,/( + %) tanh 21 rH (7 .66 ) A plot of Eq (7 .66 ) is shown in Figure 7.13 It is apparent that the eflect of surface... overtake the region of depression (Figure 7 .22 ) We shall call the front face AB a ‘%ompression region” because the elevation here is rising with h e Figure 7 .22 shows that the net effect of nonlinearity is a steepening m + V I U Figure 7 .22 Wave profilcs at four d u e s of lime At b the profilc has formcd a hydraulic jump The p f i l e at 13 is impossible 22 7 I% I ~ ~ ~ I I JurripI I ~ ~ vII of the... For these waves Eq (7 .60 ) shows that the wave speed is independent of wavelength and increases with water depth To determine the approximate form of particle orbits for shallow-water waves, we substitute the followi& approximations into Eq (7. 46) : + H) 2: 1 sinh k ( z + H ) = k(z + H ) coshk(z sinh k H 2: k H The particle excursions given in Eq (7. 46) then become a sin(kx - wr) kH 6 = < = (1 + LI 3... approximate expression is for small slopes Therefore, - p = cr-.a29 pa ax2 Choosing the atmosphericpressure Pa to be zero, we obtain the coiidition (7 .63 ) Using the linearized Bernoulli equation 34 - P + - +gz = 0, at P condition (7 .63 ) becomes (7 .64 ) As before, far small-amplitude waves it is allowable to apply the surface boundary condition (7 .64 ) at z = 0, instead at z = q Solution of the wave problem... -pgz) ( l[( p'rr d r ) - pg(u) = F= y'u dz) pu d r ) = (I: i dz (7. 56) whcrc i ) denotcs an averagc over a wavc period; wc have used the fact that ( u ) = 0 Substiluting for p' from Q (7.44a) and u from Eq.(7.39), Eq (7. 56) becomes 1' F = (cos2(kx- cot)) pu20" cosh2k(z ksinh'kH -H + H )dz The time average of cos'(kx - rut) is 1 /2. The z-integral can be carried out by writing it in tenns of exponcntials... slate because or the frec fall (Figure 7 .23 4 A tidal bore propagating into a river mouth is an example of a propagating hydraulic jump Consider a control volume across a stationary hydraulic jump shown in Figure 7 .23 The depth riscs from Hl to H2 and the vclocity falls from u1 to 111 If Q is a (a) Example @) Stationary (c) Propagating F r 7 .23 Hydraulic jump we 22 9 I.? l~~diviulic Juirtp thc voluine rate... conservationrequires Q = ~1 HI = 1i2Hz Now use thc momentum principle (Section 4.8), which says that the sum of the forces on a control volumc equals the momentum outflow rate at section 2 minus the momentum inflow rate at section 1 The force at section 1 is the avenge pressure p g HI /2 limes the area HI similarly,the force at section 2 is pgH,’ /2 If the distance ; betwccn sections 1 and 2 is small, then the force... such a solution violates the second law of therinodynsunics,because it implies an increase of inechanical energy of the flow To see this, consider the mechanical energy of a fluid particle at the surrace E = u2 /2 gH = Q 2 / 2 H 2 gH.Eliminatjng Q by Eq (7.88)we obtain, after some algebra, + + This shows that Hz e H1 implies E? > El, which violates the second law o therf modynamics Thc mechanical ciici-gy, . 't H 1 2 =-H Figure 7.4 Wave nommnclaturc. such that Substitution into the continuity equation gives the Laplace equation a%p a%p -+-=o. ax2 az2 (7 .21 ) (7 .22 ) (7 .23 ) Boundary. Recapitulating, we have tq solve a24 a'# -+ 0. i)~' subject to the conditions gr] at z=0. a4 at _- (7. 32) (7 .22 ) (7 .24 ) (7 .27 ) (7. 32) IJI order to apply the boundary. dc/h = 0 in Eq. (7 .66 ), and assuming the deep-water approximation tanh(2aHlA) 2 I valid for H > 0 .28 A, we obtain (7 .67 ) For an &-water interface at 20 "Cy the surface