KUNDU Fluid Mechanics 2 Episode 9 doc

... Mapping 14 8 15 0 1 52 15 4 15 6 15 7 15 7 15 9 16 0 16 3 16 6 17 0 17 1 17 3 17 5 17 6 18 1 18 4 18 5 18 7 18 8 .I 89 19 0 1 92 1 92 chi!pter 7 Gravity Waves 1 . Introduction 19 4 2 . ... 2 . Analogy between Heat and Vorticity Diffusion 3 . Pressure Change Due to Dynamic Effects 20 9 21 3 21 6 21 8 22 1 22 5 22 7 23 0...

Ngày tải lên: 13/08/2014, 16:21

... whcrc the two are related as (2. 26) (2. 27) As a check, Eq. (2. 27) givcs R11 = 0 and R 12 = -~ 123 ~3 = -w, which is in agree- ment with Eq. (2. 26). (In Chapter 3 we shall call ... (Figure 2. 7). Using Eq. (2. 12) , the components of he stress tensor in the rotated frame are 431 I& -43 .;I = CllC21t 12 + C2ICllt21 = TZU + TlU - TU, ti...

Ngày tải lên: 13/08/2014, 16:21

... il, i2, and i3 changc with time. To this observer the time derivative of P is d ( $)F = Z(plil+ P2i2 + - . dPi . dP2 . dP3 dir di2 di3 - 11 - + 12- + 1 .3 - + ... that a@ aY u -, (3. 34) a$ ax ' v E then Eq. (3. 33) is automatically satisfied. Therefore, a streamfunction JI can be defined whenever Eq. (3. 33) is valid. (A...

Ngày tải lên: 13/08/2014, 16:21

... 'Itucsdcll, C. A. (19 52) . Stokes' principle of viscosity. JoumZ oJRationol Mechanics ud Analysi.s 1: physical Journul131: 4 42 4 47 . Univemity Press. 22 8 -23 1. Supplernim?al Reading ... (5 .22 ) can be written as -V&nqi&ijkWk,jq = -V(&jSqk - &~haqj)wk.jq = -vWk,nk + vOn,jj = v%,jj. (5 .25 ) If we use Eqs. (5 .23 H5 .25 ),...

Ngày tải lên: 13/08/2014, 16:21

... ha2, 25 r 2n 2n m m 25 r 2? r =- 1n(x~-y*-u~+i2xy) 11na~. (6.48) Wc know that the logarithm of any complex quantity C = I< I exp (iQ) can be written as In 5 = In 15 ... 22 1 Logcr of ( hsiant Depth 22 3 Laycr of Vnriahle Depth H (.r) 22 4 Yotilir!fnr Stwpeniirg it1 CI .Voriili'pmii .v Mi~iiinti 22 5 H!rlruulir Jiunp 2...

Ngày tải lên: 13/08/2014, 16:21

... 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 ... the flow. To see this, consider the mechanical energy of a fluid particle at the surrace. E = u2 /2 + gH = Q2/2H2 + gH. Eliminatjng Q by Eq. (7.88) we obtain,...

Ngày tải lên: 13/08/2014, 16:21

... 26 8 Intcrd Fmiidt: Number 26 8 RiclmnLqm Aimher. 26 9 Mwh Niuihs 27 0 I’ra~idtl R’i~nhrx. 27 0 ficrc&es 27 0 Litemlure Cited 27 0 Siipplementul Reading 27 0 ... usccl p” = 6’ N4p ,2/ 2w2g2, found from Eq. (7. 166) after taking its real part. Use of thc dispersion rclation w2 = k2N2/(k2 + m’) shows that Ek = Ep. (7. 169) w...

Ngày tải lên: 13/08/2014, 16:21

... derivatives from Eq. (10 .22 ): rlll (10 .26 ) (10 .27 ) (10 . 28 ) (10 .29 ) (1 0.30) dS 3 Uqf" dS - - - u- -[f - fq] = a2+ a+ - = Uf', ay i 12+ Uf" a$ 6 ' ... = p(a2+/ay2)oY where the subscript zero stands for y = 0. Ushg a2$/ay2 = Uf"/S given in Fq. (20 .29 ). we obtain to = pVf "(0)/6, and finally 0.3...

Ngày tải lên: 13/08/2014, 16:21

... convcrgcnt series for a Bessel function Jo(x), given by x2 x4 X6 J()(X) = 1 - - + - - - + ~ 2& apos; 22 42 22 426 2 with the first term of its asymptotic expansion (10.64) (1 ... scheme of numerical integration. 2. A flat platc 4 m wide and 1 m long (in the direction of flow) is immerscd in kcroscnc at 20 'C (u = 2. 29 x 10-6m2/s, p = 800 kg/m...

Ngày tải lên: 13/08/2014, 16:21

... nonzero ones are pupcd as element matrices and vectors, At (1 1 .22 1) (I 1 .22 2) ( 11 .22 3) (1 1 .22 5) ( 1 1 .22 6) Thc prohlern can be nondirnensionalized using the djameter of the ... 1 .20 4) for all the vclocity nodes A and pressure nodes R. Equations (1 1 .20 2)-( I 1 .20 4) can be organized into a matrix .form, (1 1 .20 5) whcrc and (1 1...

Ngày tải lên: 13/08/2014, 16:21

... sinhq*z, (0‘ - K’)2W = A(qi + K’)’ COSqoZ + B(q2 - K2 )2~ ~~hq~ + C(q*? - K2j2 cosh 4*z. The boundary conditions ( 12. 27) then require 4* cosh - 2 4 cash - 2 4 q sinh - ... substitution K Equations (1 .2. 16) and ( 12. 17) bccome ( 12. 16) ( 12. 17) (1 2. 18) ( 12. I 9) where gord4 RaG-, KV The basic slate satisfies d2T dZ2 0 = K ( 12....

Ngày tải lên: 13/08/2014, 16:21

... 2. 38) Eq. (1 2. 38). and ( 12. 40), the cigcnvalue problem for determining the marginal stale (a = 0) is ( D2 - k2)’i,. = (1 + ax)&, ( D2 - k2)2he = -Fd k%,: (1 2. 87) ... Eq. ( 12. 72) : the iirst and third of Eq. ( 12. 72) are added; the rest are simply transformed. The result is iki + i,, = 0. These equations are exactly the sanc as Eq. (...

Ngày tải lên: 13/08/2014, 16:21

... eddies irrotational fluid irrotational fluid turbulent fluid turbulent fluid 4 Figure 13. 14 htrainmcnt of a nonturbulent fluid and its assimilation into turhuleni fluid by viscous ... reached after n - 1 steps. Using rule (13. 84) successively, we get R; = R;-~ + L~ = R' ,, -2 + 2L2 - = R: + (n - l)Lz = nL2. The rms distance...

Ngày tải lên: 13/08/2014, 16:21

... dz2 2 +A- =O. (14. 101 j (14. 1 02) In Eq. (14. 101) the term d2A/dz2 is negligible because its ratio wilh the second term is 1 d2 A/dz2 Am2 H2m2 "" Equation (14. 101) ... = - gp. az (14. 12) (14. 13) (14. 14) Eliminating p between Eqs. (14. 12) and (14. 14), and also between Eqs. (14. 13) and (1 4 .14) , we obtain, respectivel...

Ngày tải lên: 13/08/2014, 16:21

... = y/sin0, (1 5.6) where Eq. (15. 4) has been used. We now eliminate 0 between Eqs. (15. 5) and (15. 6). First note from Eq. (15. 6) that cos2e = (2btan fi - y)/2btan p, ... obtain (1 5.4) x2 sinze - y2 cosze = 4b2 sin% cos2e. (15. 5) To understand the shape of the curve represented by Eq. (15. 5) we must express 0 in terms of x, y,...

Ngày tải lên: 13/08/2014, 16:21