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■ 1 
 
 CQMFIRENCI 
 
 \i'M\VE:^5iTy OF ^ 
 
 iiaaitsiaiiiuniifliSRi^ 
 
 ,nced Comp, 
 
 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 
 
 URBANA, ILLINOIS 61801 
 
 CAC DocTiment No. 62 
 
 EVALUATION OF NUMERICAL VISCOSITY 
 EFFECTS IN TRANSONIC FLOW CALCULATIONS 
 
 By 
 
 S. Raj an 
 
 January 1973 
 
CAC Document No. 62 
 
 EVALUATION OF NUMERICAL VISCOSITY EFFECTS IN 
 TRANSONIC FLOW CALCULATIONS 
 
 By 
 
 S . Raj an 
 
 Center for Advanced Computation 
 University of Illinois at Urb ana-Champaign 
 Urbana, Illinois 618OI 
 
 January 1973 
 
 This vork was supported in part "by the Advanced Research Projects 
 Agency of the Department of Defense and vas monitored by the U. S, 
 Ariny Research Office-Durham under Contract No. DAHC0i+-T2-C-0001 . 
 
«..'JiVi 
 
 v^NEOlltG UBStA8£ 
 
 ABSTRACT 
 
 The inviscid transonic compressible flow around a circiilar 
 cylinder is calculated using a time dependent formulation that yields 
 the steady state solution as its asymptotic long time limit. The 
 conservation equations in polar co-ordinates are solved in finite 
 difference form using the explicit, two-step variation of the Lax- 
 Wendroff scheme. The total numerical viscosity of the difference 
 approximation is varied "by altering the value of an explicit artifi- 
 cial viscosity parameter incorporated in the finite difference scheme. 
 The far field boundary condition is satisfied exactly by using an 
 inverse transformation in this region. Two different finite differ- 
 ence formiilations of the cylinder wall boundary condition are studied 
 in the presence of different amounts of artificial viscosity. The 
 influence of numerical viscosity as it interacts with aerodynamic and 
 numerical criteria is discussed. In transonic flow regimes, where 
 the shock structure is weak, artificial viscosity effects are shown 
 to influence both the strength and location of the shock. At higher 
 Mach numbers, results sliow that a higher threshold value of the arti- 
 ficial viscosity is required. The importance of choosing an optimum 
 value for the viscosity parameter consistent with accuracy and stabi- 
 lity requirements is discussed. 
 
Nomenclature 
 
 p pressure 
 
 u, V radial and tangential coinponents of velocity 
 
 u free stream velocity 
 
 M free stream Mach number 
 
 00 
 
 r, radial and tangential co-ordinates 
 
 t time 
 
 ^ artificial viscosity parameter 
 
 s entropy 
 
 r reference radius 
 
 AR, AG finite difference increments in radial and 
 tangential directions 
 
 At incremented time step 
 
 Subscripts 
 
 °° free stream condition 
 
 values on the cylinder tody surface 
 
TABLE OF CONTENTS 
 
 Page 
 
 I. INTRODUCTION 1 
 
 II. THE PROBLEM 5 
 
 III. THE GOVERNING EQUATIONS 7 
 
 IV. THE FINITE DIFFERENCE APPROXIMATION 9 
 
 V. THE INITIAL AND BOUNDARY CONDITIONS 12 
 
 VI. METHOD OF CALCULATION lU 
 
 VII. DISCUSSION OF RESULTS l6 
 
 VIII. CONCLUSIONS 20 
 
 REFERENCES 21 
 
 FIGURES 22 
 
1. INTRODUCTION 
 
 Numerical solutions of flow problems involving discontinui- 
 ties invariably employ some form of artificial viscosity to provide 
 smooth and stable solutions. This artificial viscosity which is a 
 purely numerical phenomenon, has an effect on the computed solution 
 similar to a physical molecular viscosity, namely it smooths out the 
 discontinuity over a small number of mesh intervals, which is still 
 a small fraction of the total number of meshes in the computational 
 space. An alternative to employing numerical, artificial viscosity 
 in computing flows involving discontinuities such as a shock wave, 
 contact surface, a slip stream or a vortex sheet, etc., is to locate 
 the discontinuity exactly in a space-time domain and apply the actual 
 jump conditions that apply at the discontinuity. However in all but 
 the simplest of cases, this latter procedure is quite difficult and 
 more often than not unfeasible, in a computation of any length. Thus 
 one is obliged to resort to the introduction of an artificial viscosity 
 into either the governing differential equations or the approximating 
 finite difference equations. This expedient now circumvents the 
 necessity of applying the exact jump conditions at the transition 
 boundary, and enables the computation to proceed in the same manner 
 in the region of discontinuity as in other smooth parts of the flow 
 field. 
 
 Finite difference schemes introduce a truncation viscosity 
 which is a natural consequence of approximating a plecewise contin- 
 uous and single valued function with the aid of known values at 
 discrete points. This truncation viscosity, while introducing errors 
 
into the computation is nonetheless necessary to provide computa- 
 tional stability. In addition it oftentimes becomes necessary to 
 add an explicit artificial viscosity term to provide stability. 
 In the calculation of shock discontinuities by Von Neumann and 
 Richtmyer [l] this explicit artificial viscosity is represented as 
 an additional quadratic term in the momentum and energy conserva- 
 tion equations. An artificial viscosity term which is linear in 
 the velocity gradient, instead of quadratic as in [l] has been 
 employed by Landshoff [2], 
 
 Another mode of introducing an explicit artificial visco- 
 sity into the computation is by an added term in the finite differ- 
 ence approximation of the governing partial difference equations as 
 in the computations of Kentzer [3] and Magnus and Yoshihara [h]. 
 This explicit artificial viscosity in the difference approximation 
 is oftentimes required to provide computational stability although 
 a linear analysis of the governing equations shows that the differ- 
 ence scheme without the artificial viscosity should be stable [5]. 
 The effect of this artificial viscosity is to attenuate the short 
 wave length components of the solution as the computation proceeds. 
 To do this the pseudo-viscosity in the difference approximation 
 should be dissipative. 
 
 It has been shown [5] that for fluid flows without any 
 discontinuities the addition of artificial viscosity terms does not 
 alter the stability of the difference equations. In practice, one 
 assumes the same to be true for piecewise continuous flows. Should 
 the computation prove unstable even in the presence of an artificial 
 viscosity term it is fairly common to increase the amount of added 
 
artificial viscosity until stability is assirred. While it is 
 generally known that this increased added artificial viscosity 
 renders the computation more inaccurate, no attempt has heen made 
 to study systematically its interaction with the aerodynamic flow 
 variables. In a series of numerical experiments Taylor, Ndefo and 
 Masson [6] have studied the effect of the truncation viscosity on 
 a rarefaction fan, a contact discontinuity and a shock wave. Their 
 purpose was to compare the five difference schemes which they tested 
 in regard to their suitability for calculating flows containing 
 these types of discontinuities. Cameron [7] has shown that introduc- 
 ing an artificial viscosity term into the differential equation after 
 the manner of Von Neumann and Richtmyer [l] or Landshoff [2] also 
 introduces errors into the computation if the shock discontinuity 
 encounters a change in material or mesh size. He has shown that the 
 errors can be substantially reduced by a suitable choice of mesh for 
 the second material or by modifying the definition of the artificial 
 viscosity term. 
 
 Very little effort has been expended in actually evaluating 
 the effects of artificial viscosity as it interacts with the aero- 
 dynamic flow variables, since the numerical viscosity has been looked 
 upon as a necessary evil in the process of obtaining a stable solution. 
 Especially in transonic flow regimes where the shock structure is 
 weak and easily modified, the pseudo- viscosity effects take on added 
 significance. It now becomes necessary to distinguish fluid dynamic 
 effects from numerical effects. The present investigation was there- 
 fore undertaken to understand and evaluate the effects of artificial 
 viscosity in a transonic flow situation. An evaluation of its inter- 
 
action with other aerodynamic and numerical criteria in an actual 
 physical situation serves as a guideline for future calculations 
 involving artificial viscosity. Accordingly a piarely mathematical 
 approach has been abandoned in favor of a numerical experiment 
 approach. 
 
II. THE PROBLEM 
 
 The steady state transonic flow around a circiilar cylinder 
 is computed using a time dependent finite difference formiilation. 
 An artificial viscosity term is introduced into the finite difference 
 approximation of the governing differential equations, and its inter- 
 action with other aerodynamic and numerical criteria is studied. 
 The total artificial viscosity is comprised of the implicit trunca- 
 tion viscosity plus the explicit numerical viscosity added to the 
 finite difference formulation. The truncation viscosity is a basic 
 property of the particiilar finite difference scheme used and cannot 
 "be changed without altering the difference scheme itself. However, 
 the explicit artificial viscosity can be changed and this is the 
 parameter that is varied in the present investigation. The well known 
 two-step, second order accurate Richtmyer variation of the Lax-Wendrof f 
 explicit finite difference scheme is used. 
 
 In computing the inviscid compressible flow around a circular 
 cylinder it is convenient to employ polar co-ordinates. The initial- 
 boundary value problem yields the steady state flow field as the long 
 time limit of the temporal formulation. Accordingly the solution 
 marches forward in time until the change in the flow variables with 
 time is less than a small specified amount. This is the condition 
 used to approximate the steady state. 
 
 The computational mesh layout is shown in Figure 1. The 
 discretised values of the flow variables p, u, v and p are computed 
 at each mesh point. The mesh size is varied to secure greater 
 accuracy in regions of steep gradients in the flow variables, such 
 
as for example near the forward, stagnation point. A coarser mesh 
 is employed in the far field, region away from the cylinder surface, 
 as the change in the flow variables in this region is small. The 
 free stream Mach n-umher of the flow is varied to examine the influ- 
 ence of the artificial viscosity on the flow in the transonic 
 regime. The effect of different finite difference formulations of 
 the wall boundary condition in the presence of artificial viscosity 
 is studied. The results are compared with those obtained by Holt 
 and Masson [8] by the method of integral relations. 
 
III. THE GOVERMING EQUATIONS 
 
 The partial differential ecLuations governing the invlscid 
 compressible flow around a circular cylinder are the conservation 
 laws for mass, momentum and energy expressed in polar co-ordinates, 
 and can he expressed as the following. 
 
 Conservation of mass: 
 
 9p.+ u9£+Z9^+o— +^— +^=0 (1) 
 
 9t dr r 30 ^ 3r r 30 r 
 
 Conservation of momentum: 
 
 r direction: 
 
 2 
 3t ^ Sr r SO r p 3r 
 
 dii'ection: 
 
 8v 3.'/ ^ V 3v ^ uy. ^ 1... 3£. ^ ^ (3) 
 
 9t 3r r 30 r pr 30 
 
 Conservation of energy: 
 3t 3r r 30 
 
 The equation of state: 
 
 p = PRT. (5) 
 
 The physical variables are normalized, using 
 
 •n r T> 
 
 P ' — ; — P 
 
 ^~ VV) /p » 
 
 - 00 00 
 
 as reference values for velocity, time and temperature respectively, 
 
 Also the normalized entropy is given by 
 
 (6) 
 
 = - fc) - ^ - (a 
 
The conservation equations (l ) - (4) can also be written In 
 matrix vector form as 
 
 where. 
 
 F, + AF + BF^ + c = 
 t r 
 
 F = 
 
 , A = 
 
 u 
 
 Y 
 
 
 
 
 
 
 
 p/p u 
 
 
 
 u 
 
 
 
 
 B = 
 
 . 1 
 r 
 
 
 
 
 
 u 
 
 
 
 
 
 e«i 
 
 
 "^ 
 
 
 
 
 
 yu 
 
 
 
 
 
 2 
 
 
 
 
 
 
 V Y 
 
 V 
 
 p/p V 
 
 V 
 
 C = 
 
 -V 
 
 uv 
 
 Lo J 
 
 (7) 
 
 Here P = 1 
 
 "(y 
 
 (8) 
 
IV. THE FINITE DIFFEEENCE APPROXIMATION 
 
 Finite difference equations corresponding to the governing 
 differential equations are employed at the mesh points in the compu- 
 tational space. The solution progresses from known specified values 
 at an initial time level to computed values at the next time level, 
 separated by a time increment At. A linear stability analysis of 
 the particular variation of the two step Lax-Wendroff scheme used in 
 the present analysis indicates that the scheme is stable if the 
 Courant-Friedrichs-Lewy criterion is satisfied. The value of the 
 time step At used is therefore given by 
 
 Ar 
 
 At = --—-' 
 
 i.s fjlTl + al 
 
 (9) 
 
 where a safety factor of 1.5 is used. 
 
 Values of the physical variables p, u, v and p are computed 
 in the first step at time level (n+— ) At. For the differential equa- 
 tion 
 
 F^ + AF + BF^ + C = 
 t r 
 
 the intermediate values are given by 
 
 nr- 
 
 .e.,in 
 
 f'"^ ■*■ -b^ + f" + f" 
 
 £-H,m Ji--l,m £,m+l H 
 
 ,m-lj 
 
 2Ar ^'.f.,!:! 
 
 F^- - F^ 
 
 2A0 i,m 
 
 f" - f"" 
 
 ] 1 
 
 t,m+" ^j^'p" — 
 
 2 Jl,m 
 
 (10) 
 
 Final values at the next time level (n+l)At are computed from the 
 difference equation 
 
10 
 
 
 'i-*K,.^? 
 
 f" + f'^ + f^ 
 
 l,m+l 2.,in-l £+l,m 
 
 + F. 
 
 i0 i.m 
 
 Ar il,in 1^ j,_^l^ 
 ■' 2 
 
 
 F - F 
 
 11 
 
 4 
 
 - At C„ 
 
 i2,,in 
 
 (11) 
 
 A nine point lattice is involved in the computation of the variables 
 at one point at the next higher time level. The parameter <t) is 
 introduced into the difference scheme at the second step. It pro- 
 vides additional numerical viscosity as it assumes values between 
 and 1, and provides a means of varying the total niimerical visco- 
 sity of the computation. The generation of an artificial viscosity 
 by the parameter ^ is illustrated using the simplified equation 
 
 F^ + AF =0 
 t r 
 
 (12) 
 
 in one dimension, without any loss of generality. Let F denote th 
 value of F at a mesh point (£Ar, nAt ) in the computational space. 
 Assuming A is constant, we have 
 
 „"*2 1 
 \*1 ' 2 
 
 
 - A^^ 
 ^ 2Ar 
 
 F^ - F^ 
 Jl+2 il 
 
 2 
 
 '^. ^ ^.-2^ 
 
 ^ 2Ar 
 
 f'^ - f" 
 
 J a -1-2 
 
 G 
 
 (13) 
 
 (IM 
 
 
 L A+i S'-i J 
 
 l+l ■*■ ^£-1 
 
 AAt 
 2Ar 
 
 (15) 
 
11 
 
 Let D denote equation (15) equated to zero. Assuming that the 
 F 
 
 function F is continuous at (5.Ar, nAt ) and has continuous deriva- 
 tives, we obtain values of F in the vicinity of (£Ar, nAt) by 
 Taylor series expansions around that point. Then 
 
 (16) 
 
 f-f-. = *liJft-* 
 
 8r 
 
 -2 2 -\ 
 
 , 3 F 1 9 F 
 A 
 
 2 2 2 
 8r at 
 
 We thus have two additional terms on the right hand side of (l6) 
 which represent a diffusion effect. The first diffusion term con- 
 tains a second order space derivative with a viscosity coefficient 
 proportional to (f). The second component is also comprised of second 
 order derivatives, but with a coefficient of viscosity proportional 
 to At. Thus it is seen that in any computation where all other 
 factors remain the same, the variation of cj) enables us to enhance 
 or decrease the numerical viscosity effects on the flow field. It 
 is also seen that larger values of tj) imply larger truncation errors. 
 Since transonic flows contain only a weak shock structure 
 the influence of a large artificial viscosity parameter can result 
 in serious errors in the computed solution. Also transonic flows 
 can be considered as the limit situation of smooth flows. Since the 
 introduction of an artificial viscosity into finite difference cal- 
 culation of smooth flows has been shown [5] to leave the stability 
 of the scheme unaltered, the same is assumed to hold true in transonic 
 flow calculations. 
 
12 
 
 V. THE INITIAL AND BOUNDARY CONDITIONS 
 
 The steady state solution computed as the long time limit 
 of the time dependent formulation is independent of the initial 
 conditions. Accordingly, the initial values of p, u, v and p that 
 are specified at each mesh point in the computational field, are 
 the free stream values p , u , v and p . On the cylinder "body 
 surface the radius r_ is taken as unity. At time t =0, at the 
 start of the computation, the radial component u,, of the cylinder 
 surface velocity is impulsively reduced to zero. 
 
 The flow is assumed to he symmetrical about the 0=0 
 and = TT boundaries (Figure l). Therefore, for mesh points situa- 
 ted at ± A0 from these boundaries we have 
 
 p(A0) = p(-A0) p(7T-A0) = p(7r+A0) 
 
 u(A0) = u(-Ae) , u(tt-A0) = u(rf+AO) 
 ^ ' and 
 
 v(A0) = _v(-A0) v:--A0) = -v(,7+A0) 
 
 p(A0) - p(A0) p(7T-A0) = p(7T+Ae) 
 
 for all r. Also v(0) = v(7t) = 0. 
 
 On the cylinder body surface, the radial velocity is zero. 
 Two finite difference formulations on the wall boundary condition 
 were experimented with to study the effect of this numerical criter- 
 ion on the computed flow field in the presence of various amounts 
 of artificial viscosity. In one case a centered scheme was employed 
 to compute the dependent variables at time t + At , and in the other 
 a one-sided finite difference formulation was used. For the two step 
 Lax-Wendroff scheme, a fictitious row of reflection points below the 
 body surface are required if a centered scheme is used. Therefore, 
 in this case we have 
 
13 
 
 p(rQ+Ar) = p(rQ-Ar) 
 
 u(r +Ar) =■■ -u(rQ-Ar) 
 
 vd-j^^-Ar) = v(rQ-Ar) 
 
 p(r +Ar) = p(rQ-Ar) 
 
 for all 0. 
 
 Since it vould be necessary to specify a large number of 
 mesh points to be able to compute deep into the far field, an inverse 
 transformation is used in this region away from the cylinder surface. 
 The circular computational space external to a given radius is mapped 
 by an inverse transformation such that infinity in the physical space 
 becomes the origin in the transformed space. The transformed conser- 
 vation equations (l) - (h) are applied in finite difference form in 
 this far field calculation. The conditions at infinity are applied 
 on a circle of small radius in the transformed space. Sufficient 
 overlapping of points in the physical and transformed spaces ensures 
 continuity between the two regions. In this way no approximations 
 enter into the calculation of the transonic flow field in fulfilling 
 the far field boundary conditions. 
 
Ik 
 
 VI. METHOD OF CALCUMTION 
 
 Initial values of the dependent variables equal to the 
 free stream values are specified at all points in the computational 
 space. The points just above and below the 0=0 and = tt lines 
 are made to correspond to the reflection boundary conditions speci- 
 fied. The radial velocity of all points on the cylinder body surface 
 is set to equal zero. Starting vith the initial values at time t = 0, 
 the dependent variables p, u, v and p are calculated at time t + At 
 on the circumferential row of mesh points corresponding to the cylin- 
 der surface; i.e., r = r_. For the two step explicit finite differ- 
 ence scheme used, intermediate values at time level — are first 
 computed before final values at time level At can be obtained. A 
 centered finite difference formulation of the wall boundary condi- 
 tion (u = O), and also a one-sided formulation have been used. On 
 obtaining the dependent variables at time level At, it is usually 
 found that the computed value of the radial velocity u at the wall 
 is not zero. A correction is then applied in the form of an impul- 
 sive wave to force the radial velocity of the wall to zero. 
 
 The radial increment Ar is increased and the dependent 
 variables p, u, v and p are computed at time level At for the row 
 of circumferential mesh points situated at radius r + Ar. Computa- 
 tion thus proceeds outwards with increasing radius until it is time 
 to begin the far field calculations extending to a specified large 
 radius. Dependent variables at mesh points in the far field are 
 computed from the finite difference analogue of the trans fonned 
 governing equations. Since an inverse transformation has been used, 
 the radial increment Ar' in this region is negative. Calculation 
 
15 
 
 proceeds in essentially the same manner in the transformed space as 
 in the physical space, with values of p, u, v and p being computed 
 on a row of circumferential mesh points at time level At, until a 
 specified small radius in the transformed space is reached. Here 
 the free stream conditions p , u , v and p are held to remain 
 
 -^oo' oo' OO 00 
 
 constant with time, so that this last boundary row of circumferen- 
 tial points is not computed. 
 
 Using these computed values at time level At as initial 
 values, the computational field is traversed again beginning at the 
 cylinder surface r = r^^, to compute the dependent variables at all 
 the mesh points at time level 2At. This marching procedure in time 
 is continued until the steady state is reached. 
 
 For an initially specified free stream Mach number in the 
 transonic range and viscosity parameter cf), the shock forms naturally 
 on the rearward side of the cylinder. Computation of the transonic 
 flow field is made for different values of the free stream Mach 
 number, for different values of the viscosity parameter ^ and for 
 two different finite difference formulations of the wall boundary- 
 condition. 
 
l6 
 
 VII. DISCUSSION OF RESULTS 
 
 Figiires 2, 3 and h show the computed values of cylinder 
 surface velocity, surface pressure and density respectively for a 
 free stream Mach number of 0.^25. The artificial viscosity parameter 
 <J) is varied from 0.05 to 0.175. The computed values of the pressure 
 jumps at the recompression shock decrease with an increase in the 
 amount of artificial viscosity. Therefore not only does the artifi- 
 cial viscosity smear out the shock over a larger number of meshes, 
 it affects the shock strength, as can be readily observed from 
 Figures 2, 3 and h. This in turn strongly influences the pressure 
 coefficient. It is seen that for the supercritical Mach number of 
 O.i+25 and a value of (|) = 0.05, there is still some velocity fluctua- 
 tion persisting in the flow on the downstream side (Figure 2). 
 Increasing the artificial viscosity, not only reduces the strength 
 of the recompression shock but also influences the shock location. 
 A variation of cj) from 0.05 to 0.175 at a free stream Mach number of 
 O.U25, moves the recompression shock upstream about 10 degrees on the 
 cylinder arc. 
 
 From the results it is seen that the optimum value of the 
 added explicit artificial viscosity as expressed by the parameter cf) 
 lies somewhere between 0.05 and 0.075' At values of cj) below 0.05, 
 oscillations in the velocity, pressure and density profiles became 
 stronger with time, so that it becaune difficult to obtain a steady 
 state solution. It is also seen from Figures 2, 3 and h that at the 
 high value of (}) = 0.175, the recompression shock has been smeared 
 out to such an extent as to make it almost indistinguishable. 
 
17 
 
 Figure 5 shows the cylinder wall pressure for a super- 
 critical free stream Mach number of O.i+5, with values of ^ ranging 
 from 0.075 to 0.175. Even at a value of (}) = 0.075 the oscillations 
 on the downstream side of the shock still persist, while below 
 <|) = 0.075 it was difficult to obtain stable steady state solutions. 
 It is possible to obtain stable, smooth solutions at (f) = 0.1 as 
 seen from Figure J. Thus for a free stream Mach number of 0.^25, 
 the optim-um value of (J) to obtain smooth stable solutions lies 
 between 0.05 and 0.075, while at a free stream Mach number of 0.1+5, 
 the optim\jm value of cj) lies between 0.075 and 0.1. Therefore we 
 can conclude that there is a threshold optim-um value of 4' foi" a 
 given particular value of the free stream Mach number, and that 
 this threshold optimum value increases with higher free stream Mach 
 numbers. 
 
 Comparing Fig\ires 3 and 5, it is seen that at higher Mach 
 numbers, the artificial viscosity does not influence the shock loca- 
 tion as strongly as at lower Mach numbers. 
 
 Figure 6 shows the influence of the finite difference 
 formulation of the wall boundaiy condition on the computed cylinder 
 wall pressure. A value of (}) = 0.10 ensures a smooth solution at a 
 Mach nijmber of O.U5. It is seen from Figure 6 and also from Figure 
 7 that the shock location is transported a little downstream with 
 a one-sided scheme at the wall. The peak pressure is higher with a 
 one-sided scheme. 
 
 Figure 7 compares the cylinder wall velocity profiles 
 obtained from the present method with that obtained by Holt and 
 Masson [8] by the method of integral relations. It is seen that the 
 
18 
 
 one-sided finite difference formulation of the wall boundary condi- 
 tion gives a closer approximation to the result of [8], While the 
 velocity jump is essentially the same with a one-sided scheme and 
 a centered scheme at the wall, the peak velocity is higher with a 
 one-sided scheme. The curve obtained from the present calculation 
 with 4> = 0.075 agrees more closely with that of reference [8] on 
 the upstream side of the recompression shock that on the downstream 
 side. 
 
 Figure 8 shows the number of iterations required for con- 
 vergence for different values of <^ at free stream Mach numbers of 
 O.J+25 and O.i+5. One possible explanation for the fact that at a 
 higher Mach number the number of iterations decreases with an increase 
 in artificial viscosity is that the shock is now steeper. This 
 reduces the severity of the convergence criterion in a point by 
 point application. The convergence criterion employed was that the 
 computed solution at each point should not differ by a total of one . 
 percent for ten successive time iterations. 
 
 The present work differs from, the work of Emery [9] in that 
 it applies to the transonic regime and the method of computation is 
 entirely different. Emery's calculations of a Mach 3 shock with 
 different finite difference schemes containing different amounts of 
 artificial viscosity, does not study the influence of artificial 
 viscosity as such. As compared with the work of Kentzer [3], the 
 present method of computing the transonic flow around a circular 
 cylinder is shown to yield meaningful results with low enough values 
 of the artificial viscosity, even when the free stream Mach number 
 is fairly high in the supercritical range. The time dependent 
 
19 
 
 formulation of computing the transonic flow discussed herein can be 
 readily extended to include more complicated geometries such as 
 airfoil sections. 
 
20 
 
 VIII. CONCLUSIONS 
 
 The method of computing the steady state transonic flow 
 aroTjnd a circular cylinder using a time dependent formulation is 
 shown to yield reliable results even at low values of explicit 
 artificial viscosity. For a given free stream Mach number, there 
 is an optimum value of artificial viscosity which enables a stable 
 solution to be realized. This optimum value increases with an 
 increase of free stream Mach number. Increasing the added artifi- 
 cial viscosity beyond this optimum value merely increases the 
 inaccuracies consequent to the introduction of higher truncation 
 errors. Results obtained by the present method compare favorably 
 with those obtained by other investigators. The study shows that 
 in the transonic regime the artificial numerical viscosity influ- 
 ences both the location and strength of the recompression shock, 
 thus altering the pressure coefficient. This influence is seen to 
 be mitigated at higher Mach numbers in the transonic range. 
 
 The method described herein for transonic flow calcula- 
 tions can be readily applied to more complicated geometries such 
 as ellipsoids of revolution and airfoils. 
 
21 
 
 REFERENCES 
 
 [l] VonNeumann, J., and Richtmyer, R. D. , "A Method for the 
 
 N\xmerical Calculation of Hydrodynamic Shocks", J. Applied 
 Physics, Vol. 21, March 1950. 
 
 [2] Landshoff, R. , Report No. 1930, Los Alamos Scientific 
 Laboratoiy, 1955 • 
 
 [3] Kentzer, C. P., "Computations of Time Dependent Flows on 
 an Infinite Domain", A. I. A. A. Paper No. TO-U5, 1970. 
 
 [h] Magnus, R. and Yoshihara, H. , "inviscid Transonic Flow over 
 Airfoils", A. I. A. A. Journal, Vol. 8, No. 12, December 1970. 
 
 [5] Richtmyer, R. D. and Morton, K. W. , Difference Methods for 
 
 Initial -Value Problems, Interscience Publishers, N. Y., 1967. 
 
 [6] Taylor, T. D. , Ndefo, E. , and Masson, B. S. , "A Study of 
 Numerical Methods for Solving Viscous and Inviscid Flow 
 Problems'; J. of Computational Physics, 9, pp. 99-119, 1972. 
 
 ,7] Cameron, I. G. , "An Analysis of Errors Caused by Artificial 
 Viscosity Terms to Represent Steady-State Shock Waves", 
 Journal of Computational Physics, 1, 1966. 
 
 [8] Holt, H. and Masson, B. S., "The Calculation of High Sub- 
 sonic Flow past Bodies by the Method of Integral Relations", 
 Proc. Second International Conference on Fluid Dynamics, 
 Berkeley, Springer-Verlag, N. Y., 1971- 
 
 [9] Emery, A. F. , "An Evaluation of Several Differencing Methods 
 for Inviscid Fluid Flow Problems", Joiirnal of Computational 
 Physics, 2, I968. 
 
22 
 
 -^UcD 
 
 u 
 
 
 .>• /" \ 
 
 /> 
 
 
 -- .^ / "^ 
 
 
 
 Figirre 1. Computational Mesh System 
 
23 
 
 CYLINDER SURFACE VELOCITY Vo/a 
 
 FigTire 2. Influence of artificial viscosity on cylinder 
 vail velocity at M = 0.ii25. 
 
2k 
 
 CYLINDER WALL PRESSURE P/P* 
 
 \ii 
 
 C) 
 
 
 O 
 O 
 
 O 
 
 cn 
 o 
 
 p 
 
 ro 
 
 Figure 3. Influence of artificial viscosity on shock 
 
 strength and location for a free stream Mach 
 number of 0.i;25. 
 
25 
 
 DENSITY AT WALL P„/P, 
 
 (O. 
 
 O 
 
 O 
 
 o 
 
 O' 
 
 o- 
 
 O 
 
 o 
 
 o 
 
 CT)- 
 
 o 
 
 to- 
 o 
 
 p 
 
 ro 
 
 Fig-ure \. Influence of artificial viscosity parameter 
 on cylinder surface density. 
 
26 
 
 WALL PF^ESSURE P/P, 
 
 00 
 
 o 
 
 o 
 
 4i 
 O 
 
 
 (^ 
 
 
 o 
 
 CO 
 
 
 o 
 
 
 o 
 
 
 c;j 
 
 
 
 C5 
 
 o 
 
 O 
 
 •-;• 
 
 
 
 
 )"-' 
 
 
 
 
 
 M 
 
 
 O 
 
 
 o 
 
 o 
 
 Of 
 O 
 
 O 
 
 p 
 
 r.v 
 
 ^ 
 
 Figure 5. 
 
 Influence of artificial viscosity on 
 cylinder vail pressure at M =0.^5. 
 
27 
 
 PRESSURE AT WALL P/P 
 
 00 
 
 O 
 O 
 O 
 
 ^( 
 
 o 
 
 Pi 
 
 o 
 
 !.•> 
 C; 
 
 o- 
 
 c- 
 
 
 ( ■ 
 
 (:■■' 
 
 
 ( ■ 
 
 01 
 
 o 
 
 p 
 
 1 - 
 
 p 
 
 p 
 
 p 
 
 a> 
 
 1 
 
 •-• 
 b 
 
 i 
 
 T-r h— 
 
 m 
 
 
 ' 
 
 2 
 
 to 
 
 2=H 
 
 o 
 
 x-m 
 
 m 
 
 r 33 
 
 C7 
 
 r ni 
 
 
 u 
 
 </) 
 
 
 o 
 
 CO 
 
 I 
 
 o 
 
 m 
 
 
 S 
 
 f '1 
 
 m 
 
 tTl 
 
 FigTjre 6, 
 
 Influence of finite difference formiilation of 
 vail boundary condition on cylinder wall 
 pressure for ^ = 0.10. 
 
28 
 
 CYLINDER SURFACE VELOCITY v«/a 
 
 o 
 o 
 o 
 
 rn 
 
 p 
 
 M 
 
 Figure 7. Comparison of present resxilts with those of 
 
 Holt and Masson for different values of (j) and 
 different formulations of wall boundary condition. 
 
29 
 
 NO. OF ITERATIONS FOR CONVERGENCE 
 
 (/? 
 o 
 o 
 (o 
 
 H 
 
 -< 
 
 •J) 
 
 > 
 m 
 m 
 
 P o 
 
 o 
 ro 
 
 <r. 
 
 O 
 O 
 
 Oi 
 
 > 
 
 o 
 
 •y^ 
 
 b 
 
 d 
 
 Ul 
 
 in 
 
 
 o 
 
 
 3> 
 
 
 r 
 
 p 
 
 
 !-• 
 
 <• 
 
 C) 
 
 ^w 
 
 r.> 
 
 o 
 
 
 p 
 I-' 
 (.1 
 o 
 
 o 
 
 O 
 fO 
 O 
 O 
 
 8 
 
 o 
 o 
 
 Figure 8. Effect of viscosity parameter on convergence 
 rates at different free stream Mach numbers. 
 
UNCLASSIFIED 
 
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 DOCUMENT CONTROL DATA R&D 
 
 (Smcurtty elmaalllemtlon »/ tHI; bodf ol mbatmet mnd In^mlng mmotmtltm mumi b« mntatmd mhmn Otm ovmrmll ruport I* clm»»lll»d 
 
 I. ORiaiNATlM6 ACTI VI TV fCeiperal* Mitfior^ 
 
 Center for Advanced Computation 
 University of Illinois at Urbana-Champaign 
 
 Urbana^ Illinois 618OI 
 
 im. REPORT SECURITY C I. A tsi P I C A TIO)> 
 
 UNCLASSIFIED 
 
 2b. CROUP 
 
 S. REPORT TITLE 
 
 Evaluation of Numerical Viscosity Effects in Transonic Flow Calculations 
 
 4. OCtCRlPTivc wOTK« (Typm el f p ort mnd htelualwm dmiua) 
 
 Rpsea.rp.h Report 
 
 S. AuTHOR(Si (FItmt nmmtu. mlddl» Intliml, Immt nmma) 
 
 S. Raj an 
 
 «. REPORT DATE 
 
 January 1973 
 
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 35 
 
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 10. DISTRIBUTION STATEMENT 
 
 Copies may be requested from the address given in (l) above. 
 
 II. SUPPLEMENTARY NOTES 
 
 12. SPONSORINC MILITARY ACTIVITY 
 
 US Army Research Office - Durham 
 Duke Station, Durham, North Carolina 
 
 13. ABSTRACT 
 
 The inviscid transonic compressible flow around a circular cylinder 
 is calculated using a time dependent formulation that yields the steady state 
 solution as its asymptotic long time limit. The conservation equations in 
 polar co-ordinates are solved in finite difference form using the explicit, 
 two-step variation of the Lax-Wendroff scheme. The total niimerical viscosity 
 of the difference approximation is varied by altering the value of an explicit 
 artificial viscosity parameter incorporated in the finite difference scheme. 
 The far field boundary condition is satisfied exactly by using an inverse 
 transformation in this region. Two different finite difference formulations 
 of the cylinder wall boundary condition are studied in the presence of dif- 
 ferent amounts of artificial viscosity. The influence of numerical viscosity 
 as it interacts with aerodynamic and numerical criteria is discussed. In 
 transonic flow regimes, where the shock structure is weak, artificial 
 viscosity effects are shown to influence both the strength and location of the 
 shock. At higher Mach numbers, results show that a higher threshold value of 
 the artificial viscosity is required. The importance of choosing an optimum 
 value for the viscosity parameter consistent with accuracy and stability 
 requirements is discussed. 
 
 DD ,'»'r..1473 
 
 UNCLASSIFIED 
 
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UNCLASSIFIED 
 
 Security Claaaiflcation 
 
 K C V WOROI 
 
 Inviscid transonic compressible flow 
 circular cylinder 
 artificial viscosity 
 
 MOLE 
 
 UWCLA.SSIFIED 
 
 Security Claaaificatton 
 
V^NO^