Pressure And Entrpy Variatins Acrss The Weak Shck Wave Due T Viscsity Effects OSTAFA A. A. AHOUD Department f athematics Faculty f Science Benha University 13518 Benha EGYPT Abstract:-The nnlinear differential equatins describing the structure f shck waves are reduced t a system f tw cupled nnlinear differential equatins. An apprimate analytical slutin fr such system is btained. This slutin enables us t btain the flw variables and the entrpy as eplicit functins f the dimensinless crdinate. The effects f viscsity and ach number n the velcity, the pressure and the entrpy acrss the shck wave are then investigated. Key-Wrds:- Pressure, Entrpy, Shck wave, Fluid, Apprimate slutin. 1 Intrductin The structure f ne-dimensinal shck relatin fr this slutin, but actually n waves is ne f the imprtant practical rigrus prcedure eists t determine such prblems in gas dynamics. This prblem inverse. interested many authrs [1-9] fr sme years, and the search fr slutins f Navier Stkes equatins has always been their cncern. Early, Taylr [] btained an eplicit slutin f the Navier Stkes equatins, by assuming a cnstant cefficient f viscsity while neglecting the heat cnductivity. Hamad [10] fr a variable cefficient f viscsity that is temperature dependent. The slutins btained The aim in this paper is t suggest an analytical slutin t the prblem in the frm u = u (). The ther flw variables are therefre described as eplicit functins f. Fr the purpse f cmparisn with the results that were btained by Hamad [10] and Taylr [], the cnstants f integratin in equatin (4.7) and (4.8) (Hamad, [10]) are recalculated at the inflectin pint where we set the rigin fr the variable. by Taylr and Hamad are given in the frm = Fr the distributins f velcity, (u) where is the dimensinless distance crdinate and u is the dimensinless velcity. Hwever, it is mre useful t have the inverse pressure and entrpy inside the transitin regin we prpse in the present paper the analytic epressin given in equatins (17), (19) and
(1). The effect f the viscsity-temperature relatin n the flw acrss the shck wave can therefre be studied and discussed easily. Basic Equatins The fundamental equatins describing the steady, ne dimensinal flw, parallel t - ais, f viscus, heat cnducting cmpressible fluid may be written in the dimensinal frm (cf. Pai, [11]). d ( ρ u ) = 0, (1) d du dp 4 d du ρ u + ( µ ) = 0, () d d 3 d d d u 4 = 0. d kh u ρ u h + µ (3) d d 3 µ cp Fr a perfect gas, the equatin f state is given by p = ρ RT, (4) where ρ is the density, u the velcity, h the enthalpy, p the pressure, T the abslute temperature, µ the cefficient f viscsity, k the heat cnductivity, R the universal gas cnstant, c p and c v the specific heats at cnstant pressure and cnstant vlume, respectively. It is nw cnvenient t intrduce the unprimed nndimensinal variables as ρ u ρ =, u =, ρ µ µ =, µ u h h =, h p p =, p = where L = 1.55 L µ γ ρ u is the mean free path ahead f the shck frnt and while u = is the ach number as -, c γp c = is the sund velcity, ρ cp γ = is c the rati f specific heats and P r is the Prandtl number. It shuld be mentined that the subscript crrespnds t - subscript 1 crrespnds t +. v and If we integrate equatins (1)-(4) we find in dimensinless frm that ρ u = 1, (5) 4 du γ u ( ) γ µ u = 1.55 γ, (6) 3 d 1 + γ u + h µ dh = 1.55 d P r 4 du ( γ 1) µ u. 3 d ( γ 1) ( γ 1) γ h+ u 1( + ). (7) We shall assume the viscsity temperature relatin in the frm µ = h. (8) Far in frnt f the wave and far behind the wave all gradients f the variables f state becme zer. T find the velcity and the enthalpy as du d dh = d = 0 +, we cnsider in equatins (6) and (7). Cnsequently ( γ 1) 1 + u 1 = A =, (9) u ( γ + 1)
3 ( γ 1) ( γ 1) 1 + γ h 1 = B = (10) h γ + 1 Fr P r = equatins (6) (8), with k = 0, are reduced t γ( γ 1) ( γ 1) h = u ( γ 1)( 1+γ ) u+γ 1, + (11) 8 3 1.55 ( γ + 1) du ( u 1)( u A) =. γ d uh 3 Shck Wave Thickness (1) Discussin f shck wave structure has been based n a single parameter characteristic f the prfile. This parameter δ is the maimum slpe thickness f the prfile f the velcity u and it is defined by Prandtl (cf. rduchw and Libby, [5]) as du δ = (1 A)/ (13) d ma It is cnvenient t take the rigin at the inflectin pint which is essential fr the eistence f a shck wave, where we have d u = 0, at = 0. (14) d The relatin between the velcity u c at the inflectin pint ( = 0) and the velcity A at plus infinity (cf. eyrhff, [6]) is u c = A, (15) fr µ = cnstant. If µ = h, the velcity u c at the rigin is the slutin f the quartic algebraic equatin u 4 + a 1 u 3 + a u + a 3 u + a 4 = 0, (16) where a 1 =-(A+1), a = + (1 α) (3 + α) A ( γ 1) a, 4 4A a 3 =, ( 1 + γ ) = α γ ( γ 1) A 1 + =. ( γ 1) α, Equatin (16) can be slved eactly psitin t evaluate the shck wave thickness δ as given by equatin (13). The calculatins fr the case µ = cnstant and the case µ cnstant with different values f are then given in Table I. Table I: The shck wave thickness δ µ = cnstant µ cnstant 1.1 1. 1.3 1.4 1.5 16.10 7.9 5.1 3.87 3.07 16.95 8.64 5.90 4.54 3.73 4 Structure f Weak Shck Wave If instead f cnsidering the shck wave as a surface f discntinuity we shall lk at it in the present wrk as a transitin regin in which the variatin f the flw variables is cntinuus. The width f the transitin regin is
4 cnsidered t etend frm minus infinity t plus infinity, but it culd be shwn that the main change takes place in a regin with finite length. Althugh n eact analytical slutin f equatin (9) and (10) in the frm u = u () is knwn, we prpse an apprimate analytical slutin similar t that cnsidered by Thmpsn et al. ([1]). The frmula t be suggested fr the velcity is the fllwing u = β1 + β tanh ( β 4 ) + β 3 sec h ( β 4), (17) where β 1, β, β 3 and β 4 are fur unknwn cnstants t be determined frm the bundary cnditins n u at ± and at = 0. These cnditins then give β 1 = (A + 1)/, β = (A-1)/, β 3 = (u c - β 1 ), du c d β 4 =, (18) β where u c is the eact slutin f Equatin (16) and du c /d is given frm Equatins (11) and (1) by putting u= u c. Table II Cmparisn f the prpsed slutin and that btained eactly by Taylr. 0 = 1.1 0 = 1. 0 = 1.3 Eact Prpsed Eact Prpsed slutin slutin slutin slutin Eact slutin Prpsed slutin 5.11 4. 0.79 0.79 4.5 9.3.73 0.81 0.81.60 0.7 0.7 6.38 1.77 0.83 0.83 1.60 0.75 0.75 4.45 0.85 0.85 0.78 0.78.93 0.8 0.39 0.81 0.81 1.59 0.9 0.9 0.83 0.83-0.4-0.1 - -1.14-0.63 -. -1.93-1. -3.6 -.89-1.9-5.4-4. -.91-7.34-6.85-5.18-10.65
5 T cmpare ur results with that btained by Talyr [] and Hamad [10], the cnstants f integratins are nt taken t be zer, but we shuld determine them such that the rigin f the crdinate is t be lcated where the velcity prfile has an inflectin pint. A cmparisn between the velcity distributin due t ur suggested slutin in Equatin (17) and the eact slutin btained by Taylr [] are tabulated in Table II. Fr µ cnstant, a cmparisn between the velcity distributin due t ur suggested analytical slutin (17) and that btained by Hamad [10]are tabulated in Table III. Table III Cmparisn f the prpsed slutin and that btained by Hamad = 1.1 = 1. = 1.3 Eact slutin Prpsed slutin Eact slutin Prps ed sluti n Eact sluti n 7.5 10.19 6.99 4.9 3.9 1.87-0.76 -.09-3.54-5.1-7.35-10.7 0.9 0.9 5.44 3.33.4 1.80 1.9 0.85 0.45-0.33-0.73-1.15-1.63 -.19 -.93-4.11-5.3 0.7 0.74 0.76 0.78 0.80 0.8 0.86 0.9 0.7 0.74 0.76 0.78 0.80 0.8 0.86 0.9 3.61.06 1.63 1.31 1.04 0.8 0.61 0.4 0.3-0.31-0.50-0.69 - -1.14-1.57-1.96 -.51-3.63 0.59 0.6 0.64 0.66 0.68 0.7 0.74 0.76 0.79 0.8 0.86 Prpsed slutin 0.59 0.6 0.64 0.66 0.68 0.7 0.74 0.76 0.79 0.8 0.86
6 Frm these Tables it is clear that the velcity distributins inside the transitin regin are identical with that btained by Taylr [] and Hamad [10]. The advantage f having u frm Equatin (17) is that we can directly find the density ρ frm Equatin (5) and the enthalpy h frm equatin (11) where bth are functins f the crdinate. The pressure can als be calculated where it is given by p=ρh (19) The velcity distributins inside the transitin regin fr =1., γ = 5/3 with µ = cnstant and µ cnstant are shwn in Fig. 1. The velcity distributins inside the transitin regin fr =1.5, γ = 5/3 with µ = cnstant and µ cnstant are shwn in Fig.. Als the velcity distributins inside the transitin regin fr the case µ cnstant with =1. and =1.5 are shwn in Fig. 3. The pressure distributins are presented in the fllwing cases : (i) = 1., γ= 5/3 with µ = cnstant and µ cnstant as shwn in Fig. 4. (ii) = 1.4, γ= 5/3 with µ = cnstant and µ cnstant as shwn in Fig. 5. 5 The Entrpy Distributin It will be f interest t determine the entrpy S as an eplicit functin f. In general, fr a perfect gas, S = c p ln ( T / T `) - R ln ( p / p `). (0) In a nn-dimensinal frm, equatin (0) may be written as S S = = ln( hu c v γ 1 ) (1) Using equatins (11) and (17), Equatin (1) gives the entrpy as an eplicit functin f. Fig. (7) shws the distributin f the entrpy when = 1.5 fr µ =cnstant and µ cnstant. T shw the effect f ach numbers n the entrpy distributins inside the transitin regin we carried the calculatins fr the = 1. and = 1.5 cases and these are shwn in Fig. 6. 5 Results The apprimate analytical slutin which we have established in this paper can prvide us directly with all calculatins f the shck wave. In fact, we summarize the main results btained as fllws: 1- The shck wave thickness becmes greater when µ cnstant fr cnstant as shwn in Table I. - The shck wave thickness decreases with the increase f as shwn in Table I. 3- The prpsed analytical slutin (17) gives results identical with thse btained via the eact slutin frµ =cnstant r µ cnstant and fr different values f as shwn in Tables II and III. 4- The transitin regin fr µ cnstant is greater than that in case µ = cnstant with cnstant as shwn in Figures 1and fr
7 velcity, Figures 4 and 5 fr pressure and Figure 7 fr entrpy. 5- The greater initial ach number will prduce a smaller transitin regin as shwn in Fig. 3 fr velcity, Fig 6 fr entrpy and Figures 4 and 5 fr pressure. 6- The increase in viscsity causes the transitin regin f the entrpy t increase fr a given as shwn in Fig. 3. 7- The entrpy increases with the increase f initial ach number as shwn in Fig. 4. References [1] Rankine, W.J..: On the thermdynamic thery f waves f finite lngitudinal disturbance. Transactins f the Ryal Sciety 160,1870, 77-88. [] Taylr, G.I.: The cnditins necessary fr discntinuus mtin in gases. Prc. R. Sc. Lnd. A 84,1910, 371. [3] Becker, R.: Stsswelle und Detnatin, hysik 8, 19, 31-36. [4] Thmas, I.H. :Nte n Becker's thery f the shck frnt, Jurnal f Chemical Physics 1,1944, 449. [5] rduchw,. and Libby, P.A. : J. Aernaut. Sci.,1949, 16: 674. [6] eyerhff, L. : J. Aernaut. Sci., 17,1950, 775. [7] ises, R. Vn : On the thickness f a steady shck wave, J. Aernaut. Sci., 17,1955, 551. [8] Wang Chang, C.S. : On the thery f the thickness f weak shck waves. University f ichigan, Dept. f Eng. Research APL/JHU, C-503. 1948. [9] Gilbarg, D.& Palucci, D.T.: the structure f shck waves in the cntinuum thery f fluid, J. Rat. ech. Anal., 1953,617-64. [10] Hamad, H.: Effect f viscsity n the structure f shck waves. Prceedings f the Ryal Sciety A 45,1996, 163-17. [11] Pai, S.I.:Intrductin t the thery f cmpressible flw,van Nstr and Reinhld C., New Yrk, Lndn, Trnt, elburne, P. 31.1959. [1] Thmpsn, Philip, A., Strck, Thmas W., and Lim, David S. : Phys. Fluids 6, 48, 1983.
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