Pressure And Entropy Variations Across The Weak Shock Wave Due To Viscosity Effects


 Edward Waters
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1 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 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. KeyWrds: Pressure, Entrpy, Shck wave, Fluid, Apprimate slutin. 1 Intrductin The structure f nedimensinal 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 [19] 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
2 (1). The effect f the viscsitytemperature 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 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) ( 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 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 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 = (A1)/, 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. 0 = 1.3 Eact Prpsed Eact Prpsed slutin slutin slutin slutin Eact slutin Prpsed slutin
5 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 Prpsed slutin
6 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 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 nndimensinal 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 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 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 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, [] 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, [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 [9] Gilbarg, D.& Palucci, D.T.: the structure f shck waves in the cntinuum thery f fluid, J. Rat. ech. Anal., 1953, [10] Hamad, H.: Effect f viscsity n the structure f shck waves. Prceedings f the Ryal Sciety A 45,1996, [11] Pai, S.I.:Intrductin t the thery f cmpressible flw,van Nstr and Reinhld C., New Yrk, Lndn, Trnt, elburne, P [1] Thmpsn, Philip, A., Strck, Thmas W., and Lim, David S. : Phys. Fluids 6, 48, 1983.
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