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1 k D-i~ 64 IPLIFIED NR VIER-STOKES EQUATIONS AND COMBINED i/i ADR4 63 SOLUTION OF NON-VISCOUS A.(U) FOREIGN TECHNOLOGY DIV I NWRIGHT-PATTERSON AFB OH Z GAO 10 PlAY 84 p NCLASSIFIED FTD-ID(RS)T-03±8 84 F/G 28/4 NL In

2 ,4 S.L. * I25 all- IIII.51fll1.4 4 _-"_ MICROCOPY RESOLUTION TEST CHART NATIONAL BUREAU OF STANDARDS 1963 A S., ~.%

3 SI FTD-ID(RS )T FOREIGN TECHNOLOGY DIVISION.4. I. -- SIMPLIFIED NAVIER-STOKES EQUATION kand COMBINED SOLUTION OF NON-VISCOUS AND VISCOUS BOUN1DRY LAYER EQUATIONS by Gao Zhi 0 C.L14 ' ote QMAY3 o 984fl distribution unlimited. *1..:," f

4 FTD-ID(RST-o EDITED TRANSLATION FTD-ID(RS )T May 1984 MICROFICHE NR: FTD-84-C SIMPLIFIED NAVIER-STOKES EQUATIONS AND COMBINED SOLUTION OF NON-VISCOUS AND VISCOUS BOUNDRlY LAYER EQUATIONS By: /Gao Zhi English pages: 11 Source: Lixue Xuebao, Nr. 3, November 1982, pp )1._ Accession For Country of origin: China NT1V3 Translated by: LEO KANNER ASSOCIATES DICT3 CrA&I Requester: F D0264 F FTD/TQTD ut' Approved for public release; distribution unlimited. k.'±v 1 11 Codes Dist IA '-pocial THIS TRANSLATION IS A RENDITION OF THE ORIGI. NAL FOREIGN TEXT WITHOUT ANY ANALYTICAL OR EDITORIAL COMMENT. STATEMENTS OR THEvjRIES ADVOCATED OR IMPLIED ARE THOSE OF THE SOURCE PREPARED BY: AND DO NOT NECESSARILY REFLECT THE POSITION TRANSLATION DIVISION OR OPINION OF THE FOREIGN TECHNOLOGY DI. FOREIGN TECHNOLOGY DIVISION VISION. WP.AFB, OHIO0. p1 FTD-ID(RS )T Date 10 May 19 84

5 -A- U. S. BOARD ON GEOGRAPHIC NAMES TRANSLITERATION SYSTEM Block Italic Transliteration Block Italic Transliteratioc. A a A a A, a P p p p R, r 6 6 S 6 B, b C c C c S, s Bs B a V, v Tr Tm T, t Fr ra G, g Y y Y y U, u S9 D, d 0 F, f E e E a Ye, ye; E, e* X X x Kh, kh hi W X a Zh, zh LA 4 i f Ts, ts s Z, z 4 V Ch, ch H H H u I, i W w. IV Sh, sh P A U Y, y L U4 L w Shch, shch H K Kx K, k 'b -6., 1 J7 L, 1 b W W Y, y M M M, m b b ' H H H ) N, n 3 a 0 i E, e o 0 0 O, 0 k 3 0 o Yu, yu F1 n /7 x P, p F1R a Ya, ya *ye initially, after vowels, and after b, b; e elsewhere. When written as 9 in Russian, transliterate as y9 or 9. RUSSIAN AND ENGLISH TRIGONOMETRIC FUNCTIONS Russian English Russian English Russian English sin sin sh sinh arc sh sinh-{ cos cos ch I cosh arc ch coshtg tan th tanh arc th tanhctg cot cth coth arc cth cothsec sec sch sech arc sch sechcosec csc csch csch arc csch csch Russian rot ig English curl log GRAPHICS DISCLAIMER All figures, graphics, tables, equations, etc. merged into this translation were extracted from the best quality copy available. j

6 SIMPLIFIED NAVIER-STOKES EQUATIONS AND COMBINED SOLUTION OF NON-VISCOUS AND VISCOUS BOUNDARY LAYER EQUATIONS Gao Zhi * (Institute of Mechanics, Academia Sinica, Beijing) Abstract This paper presents a part of the technical report[2) in which the author studied simplified Navier-Stokes equations and the combined solution of non-viscous and boundary layer equations. From the full Navier- Stokes equations and an analysis of the combined solution of non-viscous and boundary layer equations, simplified Navier-Stokes equations were worked out. A perturbation analysis which differs slightly from the match-perturbation-expansions of inner-outer layers developed by Van Dyke[l] shows that the solution of the simplified Navier-Stokes equations in uniformly valid wtacuayoore-1i/2 with accuracy of O(Re, 1/2) in the whole flow field, where Re,,= is the density of free stream, U. the x-component of velocity, L the characteristic length, A4,, the dynamic viscosity of free steam. The simplified N-S equations possess the characteristics of parabolic-hyperbolic equations. We can use the forward advancing calculation method for stationary situations and it must be much simpler than numerically solving the elliptical complete N-S equations; by solving the simplified N-S equations,." we can simultaneously calculate the non-viscous outer flow and viscous boundary layer flow. Theoretically, they must be superior to the conventional method of first calculating the non-viscous flow and later calculating the viscous boundary - layer flow. The simplified N-S equations can truly reflect the mechanical appearances of many typical flow fields. For example, they can accurately calculate the complex flow chart etc. of the mutual perturbation between the viscous boundary layer, upper entropic layer and non-viscous outer flow in the hypersonic flow field around a body. ^. v 1 w.1

7 S Developments in the study of solving the simplified N-S equations are still being made. In reality, the problems of its mathematical character, steadiness, accurate formulation of the Cauchy problem, accuracy which can be reached in calcul- ating the flow field etc. have still not been perfectly resolved and the views of different authors are not in agreement. This paper presents part of the research contents done - by the uathor in 1967 investigating the simplified N-S equations, brings forth simplified N-S equations starting from analysis of the combined solution oi the non-viscous and viscous boundary layers and uses the perturbation method to prove that we can obtain a uniform solution with accuracy of O(Reco -1/2) quantitative level from these equations. It should be pointed out in passing that analagous analysis for deducing the simplified N-S equations were later seen in references such as Reference [3]. 1. Simplified Navier-Stokes Equations If x and y are separately the orthogonal coordinates along the wall surface and vertical wall surface (Fig. 1), u and v are the corresponding speed components, and P p, T, g and A, separately indicate the density, pressure, temperature, viscosity coefficient and thermal conductivity. The overtaking flow 2 parameters U, and p U, are separately the characteristic quantities of the speed and pressure and fixed wall length L is the length's characteristic quantity; we assume the quantitative levels of PO and A- can use Q, andj4 for balance :. ~~~~~ ,.,,- '-,-.... o,...-,..,., ',-*

8 * 4 Fig. 1 Based on the two views of the non-viscous outer flow and viscous boundary layer flow, we carry out the following quantitative estimates of each term in the entire N-S equations:... k + w u -1 ap I an P ON As-j -S.- "'T, +--"''& W ""-,;H- V P-T- -37(-, L L L L L L L L ' 2- -E '- 8'L ""L _- aw. LEII. - Re'-E- e - 4o Rq.W '-'U xx ol, Ol ' ' 5 I Fi g. AR ' L I L L L L L L (1.2) Key: (1) Non-viscous flow; (2) Boundary layer flow; (3) Non-viscous flow; (4) Boundary layer flow. L L I.L LL, Here, r=j 209+ k, j=o (two dimensional flow, 1 (axially.4 symmetric flow), H=l+ky, k is the wall surface curvature, :" ": '" : " " " " " '' ' " " Revm L-L - "' ' "' "e =OR1 I 2 ;"" "' 7 " i ' the ' " boundary " " " layer " " " " "" " thickness, I t and k are separately the quantitative levels in the tangential and normal equations which are equal to or 3

9 smaller than the terms of the O e- quantita tive level. We carry out the following discussion when the large Reynold's number Re0 >> 1) When y, Ye omit the terms with quantitative levels of of O(ReAP U ) in the entire N-S equations and obtain the Euler equations; when y.<5, we omit the terms with quantitative levels smaller than and equal to O(Re 1/2, and obtain the Prandtl boundary layer equations. 5tions 2) When jointly solving the Euler and boundary layer equa- and there exists mathematical singularity on the boundary line of y= 6, the asymptotic value of the solution of the Euler equations when y( >e )-)Ci and the asymptotic value of the solution of the boundary layer equations when y(<&) -- cannot be completelx the same and the singular real quantitative level is O(Re. / 2 r4l. It should be noted that when,y(>c5')--)f, the real quantitative level of the terms inertia in the normal Euler equations is also O(Re 1/2 U. 00 L 3) Therefore, in order to eliminate the singularity which appears when jointly solving the non-viscous outer flow and viscous boundary layer flow, we should select and omit terms in the N-S equations which are equal to and smaller than o(r-1 U1-) and afterwards obtain the equations. These are then the simplified N-S equations; by combining the solution of the simplified N-S equations and Euler equations we can eliminate the mathematical singularity and the obtained accuracy is the O(el U30 quantitative level which is suitable for the uniform solution of the entire flow field. N As mentioned above, the simplified N-S equations point to the set of equations obtained after omitting the terms with quantitative levels equal to and smaller than O(Rel1 so L ) in the entire N-S equations estimated according to the boundary N

10 layer viewpoint. We concretely write them (continuity and energy equations) as: (P-( i,) (p;h,,) 0 (1.3) 0: by + u.. + A.. + F9 S. *Ho: y ay - V 6 T + M - 'H. () "), (.6) PC, - (.!r 2,, ap" *1. % In the formulas TY a y HaybyH c p is the constant pressure specific heat..2simplified N-S equations (1.3)-(1.6) do not include the 2 term and when k -fa0, it has four repeated characters and xtwo non-zero real characters. Therefore, it has parabolic-hyperbolic equation characteristics [4] and for constant situations we can use the forward advancing calculation method. At the same time, we should note that when y(( )-4, -- goes from O(Rel/2)--0(I) and the quant tativeylevels of the viscous * terms all change into O(Re1 U_._); L accuracy reaches the SO(Re,,.-1/2U.00 "L-) quantitative level and when y >/f, the simplified N-S equations will smoothly transit to be Euler equations..,.5 This is to say that in order to obtain complete flow field solutions with the accuracy of the O(Re-/ 2 11-) quantitative

11 level, we need not simultaneously solve the simplified N-A and Euler equations. In principle, it is only necessary to give the initial boundary value, which is the overtaking flow (or shock wave) and wall surface conditions, and independently solve the simplified N-S equations. 2. Discussion of Accuracy When using quasi-linear simplified N-S equations to calculate the entire flow field including the non-viscous outer flow and viscous boundary layer, we can obtain a uniform solution _ -1/2 U2.a with accuracy of O(Re 1 U-) In order to prove this conclusion, it is necessary to use slightly different perturbation expansion from common inner and outer lay matched expansion [1]. The solution of entire N-S equations (1.1) and (1.2), when in the O. y; c5 viscous boundary layer, can expand into the following power series of E =Re,-1/2 u/u. - u 1 o + axi + ae 2 U +. V/SU. - Vie + Eri, + ai + " p/p- - pi + Ep;, + e 2 p12 + (2.1) P/P. - P.0 + Fp's + 8 Pia + " [p(x, Y) - p.(x)]/ep.u 2. - p,. + +rij + 2 P"a + " In this series, the X - I/L, Y - y/sl, a- Re; ", 00); Ns, U quantities and their partial derivatives corresponding to X and Y are all of the 0(l) quantitative level. We substitute expansion formula (2.1) into N-S equations (1.1) and (1.2), and by comparing the homogeneous powers of we obtain: i.j.v

12 ** b 0(SO) Pie i's " + Ii. Bi.) + ( /L ex by) dx by b 0(8~.) Pea + k o.- + '- b Y (a (2.2) + +",a. +.,,. ove] Ui o +,O,. o).51) A. a :, + P( ax x 0-- ( a'. + d '(X)++ Y dx -a,) aoyby + S,,io - k "o ia (,o,-.+ vo - 2kU.,,,, + ly' - p - - ap ( x byab " + " + + (2.3) 3 BY by eux By J y In the formulas, kl=kl,, rl=r/l, pw(x)=pw(x)/p, U 2 k1 -r'd. similarly carry out expansion of the continuity and energy we can equations. It is necessary to explain that in the present expansion method, when y 5' the 0(I) quantitative levels of the N-S equations are close to being first order equation (2.2) and the 0(E ) quantitative level equation is second order equation (2.3). They are all different from the common inner and outer layer matched expansion method [1]. In the common expansion method, when y 6', the first order and second order normal momentum equations are separately: V-. 0' 01 '..4. Therefore, in the common expansi-r method, when using the first order equations of the inner and outer layers for simultaneous.4 solution as well as when using second order equations for simultaneous solution, there always exists the mathematical -A -A" 7 V' ''... a'.:'. < ",'%. ',-"... --;-,,,..--,--,.-.-,..:-, -,-

13 singularity of the O(Re; 2 quantitative level on the boundary line of y=6?. In the y>6 non-viscous outer flow area, the solution of N-S equations (1.1) and (1.2) can be expanded to the following power series of./u. - "I + Eu I-- VgD + EVAg + 2,, + I p(x, -)/p I EP + 12P + P(x, Y,)l.U'. - PA + SP. + IP, + In the formulas, Ye=y/L; the ueo'uel*'' quantities and their partial derivatives corresponding to X and Y are all of the 0(l) quantitative level. We substitute (2.4) into (1.1) and (1.2), compare the same order powers of E and obtain:.'naj H" /, axh :'"' [ E"I) + -ex- + " --.- i ' (23) \Hax BY, HI 061 KNOX + '.a +,, -+ ± ( +"-,e P1 [H ax A..,,+ yo +2 O o,. H I. 0P qa' o N--S + ean s I \HOX ay, H I bx PaL H a H OX ay, t.a Cu ) -I. We can see that in the non-viscous flow area of y > ', the * 0(l) quantitative levels of the N-S equations are close, that is, the first order equations in (2.5) are Euler equations. Accuracy reaches the approximation of the 0(9 ) quantitative level, that is, the second order equations in (2.6) which are the modified,-,.'. *ALI

14 solution after considering the boundary layer's displacement thickness effects. From the combined solution of equations (2.2) and (2.5), we can obtain a uniform solution in the entire flow field with the accuracy reaching the 0(l) quantitative level; from the combined and matched solutions of equations (2.2), (2.3) and (2.5), (2.6), we can obtain the uniform solution in the entire flow field with the accuracy reaching mm _ 1/2. the O(Re1/ ) quantitative level. By comparing simplified N-S equations (1.4) and (1.5) with equations (2.2) and (2.3) as well as with (2.5) and (2.6), we can prove that the solution of the simplified N-S equations satisfies the following relationship and using u as the example we have -U. (2.7 ) As regards the other variables, formula (2.7) is established in the same way. Therefore, the solution of the simplified N-S equations is suitable for the uniform solution of the entire flow field with accuracy of the O(Re,/2 quantitative level. 3. Concluding Remarks The specific form of the simplified N-S equations will be slightly different because of the differences in the selection of the O(Re, 2 ) quantitative level viscous terms. It is commonly considered that the actual difference caused by the 1/2 different selection of O(Re,, ) quantitative level viscous terms is not large [3-71. However, in order to obtain a uniform solution in the entire flow field with accuracy of the O(Re./ 2 ) quantitative level, use of simplified N-S equations (1.3)-(1.6) proposed by the author [2] is possibly suitable. -9 1%,

15 The simplified N-S equations can correctly calculate the non-viscous flow, high entropic layer and complex perturbation between the boundary layer flows [3-7] in the shock wave layer. They are also widely used for the calculation of other flow fields: for example, nonequilibrium flows, duct flows, distant wake flows, shock wave-boundary layer perturbation as well as two dimensional pointed and dull viscous flows around bodies etc. The simplified N-S equations are also used for the calculation of complex flows such as separated flows, hear wake flows and those with compressive corners etc. However, it should be noted that the simplified N-S equations only occupy a dominant position in viscous effects, that is, when AO, they possess parabolic-hyperbolic equation characteristics. When in y >5 viscous effects, they can be overlooked, that is, they transit to become Euler equations and the characteristic root is mv -cvm. Here, C is the speed of sound and M is the Mach number. When M > 1, 7.3, 4 is the real character, it is hyperbolic and the mathematical type does not change; when M< 1, 7-3,4 is the repeated character, it is elliptic and the type of the simplified N-S equations changes. The Cauchy problem does not pertain to the simplified N-S equations of the elliptic type. It is essential to use iterative solutions or carry out other special processing. References [1] Van Dyke, H., Hypersonic Flow Resea rch, Academy Press. N.Y. (1962) [2] Gao Zhi, Combined Solution of Non-Viscous Outer Flows and Viscous Boundary Layers, Work Report of the Institute of of Mechanics, Academia Sinica (1967). 3 Muars, K. K, V,=. AH. ccp, M)U, 2(19O). 44-m. 10

16 References (continued) [4] Wang Ruquan, Calculation Method of High Speed Aerodynamics (Collected Works), The 1978 Spacecraft Calculation of Aerodynamics Conference Materials, Hangzhou, (1978). [51 Wang Ruquan, Liu Xuezong, Jiao Lujing and Gao Zhi, Journal of Mechanics, 3(1980), [6] Davis, R.T., AIAA J., 8, 5(1970) [7] ronums, io. n. Ap.,.a AfBMMO., 13. 4(1973) S. " 4-. b a' 11 "S., 4..Q

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