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# Ocean Wave Mechanics EOC 4422

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This 5 page Class Notes was uploaded by Gino Zemlak on Monday October 12, 2015. The Class Notes belongs to EOC 4422 at Florida Atlantic University taught by Ananthakrishnan in Fall. Since its upload, it has received 30 views. For similar materials see /class/221660/eoc-4422-florida-atlantic-university in Engineering Ocean at Florida Atlantic University.

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Date Created: 10/12/15

Supplementary Notes P Ananthakrishnan EOC4422 OCEAN WAVE MECHANICS FALL 2007 Bernoulli39s Equation and Euler39s Integral In the following we will assume density p and acceleration of gravity 9 to be constants Case 1 Bernoulli39s Equation for Steady lnviscid Flow In this case7 the equation of motion is given by the Euler s equation 57 a a A 0 E u Vu Vpipgk VPPQZgt Where 37 0 because of ow being steady Therefore 97 VW p 992 The component of the above equation along a streamline direction 5 see gure below streamlines Fig 8 1 Steady lnviscid Flow can be written as pusa PP92 n pan 773 2 as 7 88PP92gt n a u 0 i i Z as P 2 P P9 Since streamwise velocity us is the same as the magnitude of velocity U ow is along streamlines a U2 0 i 7 Z as P 2 P P9 lntegrating and re arranging terms U2 1 0 pgz ConstantC along a streamline Note that the quantity on the left is constant only along the streamline Different streamlines will have different constants as denoted by 01 02 C3 etc in the gure on the previous page Keep this in mind Case 2 Bernoulli39s Equation for Steady Inviscid Irrotational Flow Next let us consider the case in which the ow is not only steady but also irrotational In this the Euler s equation of motion reduces as follows W a a A 0 E u Vu Vpipgk VPPQZgt where git 0 because of ow being steady Therefore 97 VW Wp 992 Using vector identity 12 WW E vm 7 V X a X a the above equation of motion can be written as 1 p gm 7 v x 22gt x 7vltppgzgt 2 As the ow is also assumed to be irrotational ie V X 12 0 the Euler s equation now reduces to 1 a 2 pivlul Wp 992 Or P a 2 7 V 13 ilul pgz 70 Since the gradient V ie in ALL directions of the the quantities in the paranthesis is zero and also the ow is steady p pgz Constant everywhere Denoting as U the equation can be written in a more familiar form as U2 p pgz Constant The difference between the above Bernoulli s equation for steady inviscid and irrotational ow and the Bernoulli s equation given in Case 1 for steady inviscid ow is that in the present case Case 2 the quantity p U2 if 2 P 2 2 P9 is constant everywhere whereas in Case 1 the quantity is constant only along a streamline In Case 1 the constant will have different values on different streamlinesi as denoted by C1 C2 etc in the gure on the previous page In Case 2 C1 C2 C3 C I hope difference in the Bernoulli s equation for steady ow and steady and irrotational ow is now clear Case 3 Euler39s Integral aka Unsteady Bernoulli39s Equation for the case of Unsteady Potential Flow Recall from lectures refer to Lecture Notes 1 the basic equation governing an ideal ow ie incompressible inviscid irrotational are given by 12 Vq where 45 is the velocity potential Equation of continuity yields V a I 0 because of incompressibility a V245 O Laplace equation Next let us consider the substitution of 7 di in the Euler s equation For this we consider the following version of the Euler s equation 8a 1 0 iVl l2 7 V gtlt 12 gtlt 12gt 7V3Dgz 3 Note that the second and third terms by vector identity are equal to 22 V22 Becauase of irrota tionality V X 2 O the above Euler s equation therefore becomes 82 1 22 7 plt vlulgt7 VOW092 Substituting 2 7V and combining terms we get 5175 1 2 P V 77 7 V 7 0 at 2 p92gt As the gradient of the quantities in the paranthesis is equal to zero and as the ow is NOT steady above implies 5175 1 2 P 77 7 V 7 C t at2 pgz ltgt which is known as correctly Euler s integral and incorrectly as Bernoulli s equation Note here C is function of time Recall that in cases 1 and 2 the Bernoulli s constant C is not function of time In the Bernoull s equation of Case 1 C is function of streamlines in the Bernoulli s equation of Case 2 C is same everywhere in the uid Those cases 1 and 2 are for steady ows In the present case of unsteady irrotational ow C C This time function Ct can be absorbed in 45 in the following manner Let a 345 5 t T at where 45 is a new potential lntegrating with respect to time t 175 CT d7 Since 45 and 45 differ only by a time function In other words both 45 and new 45 give the same ow velocity The Euler s integral can now be written as 345 1 2 P 7 7V 7gzgt0 lt at 239 I 0 In this form of Euler s integral the right hand side is zero The question now is what are the equations governing 45 7 As observed above as V V 722 Therefore because of incompressibility v ao HVZW 0 In other words 45 is also governed by the same Laplace equations In sum the equations of ow motion with the introduction of a new potential 175 are given by 12 Vq vzqr 0 at which are basically same as that for 45 except with the use of 45 the time function Ct appearing 8 1 lt7 75 v 2 gzgto in the Euler s integral has disappeared Hitherto we will de ne the potential ow with 45 and for convenience drop the superscript Absorption of Ct is not so possible if the ow is NOT unsteady Let us go back to the Euler s integral with Ct given earlier as 5175 1 2 P if 7 7 C t a 2V l pgz ltgt If the ow is steady then O and Ct C where is C constant Then 1 WMIEWC 2 p Observing Vq 2 U2 above can be written as 1 EU2 g 92 C constant everywhere in the ow 0 which is same as Case 2 Bernoulli s equation Here there is no way of getting rid of the constant C as we did before in Case 3 for the Euler s integral Of course we can rede ne pressure to get rid of C for example as 4 Camp P P in which case the above equation will reduce to 1 4lt U2Jigz0 p but on the free surface the new pressure 13 will not be equal to paw Wave Mechanics is an inter esting subject or what

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