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Numerical simulation of a circulated gas flow based on the complete and simplified Navier-Stokes equations
91
3. BAUMGARTEL Ii.,Analytic perturbation theory for matrices and operators, Birkhauser, Basel, 1985. Translated by S.R.
U.S.S.R. Comput.Maths.Math.Phys .,Vo1.27,No.6,pp.91-94,19&7 Printed in Great Britain
OC41-5553/87 510.00+0.00 01989 Pergamon Press plc
NUMERICAL SIMULATION OF A CIRCULATED GAS FLOW BASED ON THE COMPLETE AND SIMPLIFIED NAVIER-STOKES EQUATIONS*
YU.P. GOLOVACEEV and N.V. LEONT'YEVA Using the example of stationary axisymmetric flow near any surface of a sphere situated in a supersonic wake region, the applicability of simplified mathematical models for the description of gas flows with a developed recirculation zone is studied. A comparison is made of the results of numerical solutions of the complete and simplified NavierStokes equations with two different boundary conditions at the head end shock wave. At the present time, the simplified ("parabolized")Navier-Stokes equations are used to simulate viscous gas flows. These include all terms of Euler's equations and the boundarylayer equations and do not contain second dervivatives of unknown functions with respect to the coordinate which coincides with the basic direction of flow. Mathematical models were formulated in /l-3/ for problems of supersonic flow over solids, as a result of evaluating a number of terms of the Navier-Stokes equations describing the molecular transfer of momentum and energy. These evaluations were based on concepts of classical boundary layer theory and hold for continuous flow at large Reynolds numbers. Practical experience in using the models under examination has shown that the region where they are applicable is wider than the formal limits expected from the evaluations specified above. Simplified equations are used, in particular, for calculating flows with flow separation from the surface of the solid and with the formation of closed regions of backward-circulationflow (see, for example, /4-6/I. The study of errors connected with the description of their simplified equations in the simulation of such flows is of interest. With this object in view, in this paper the results of a numerical solution of the complete and simplified Navier-Stokes equations are compared for one example of gas flow with a developed recirculation sane. 1. Statement of the problem and numerical method. Stationary axisymmetric flow over any surface of a sphere by a non-uniform supersonic stream, whose parameters correspond to flow in a remote trace is examined. The distributions of the gas-dynamic functions in the oncoming stream are described by formulae taken from /7/: V(y)=V(~)[l-oeap (-bu’)l,
h~~)-~(o)(~+c~i-~‘(y)/~‘(O)11, p(y)-const.
(0
Here, y is the distance from the axis of symmetry referred to the radius of the sphere, V is the modulus of the velocity vector, h is the specific enthalpy, p is the pressure, and a, b, and c are numerical parameters describing the shape of velocity and enthalpy profiles in the oncoming stream. using (l), flow in the wake is simulated by a parallel axisymmetric stream with constant pressure, axial minimum velocity and maximum enthalpy. It is assumed that the density of the gas is sufficient to form a head end shock wave in front of the sphere. Flow between the head end shock wave and the surface of the solid is described by the simplified Navier-Stokes equations /3/:
au a0 bP v an u--f---+P r -&-fd8+2u+vctg6 or r86 T( ea ” 80 ll= OP ;i;-+---+--0, Pa ( rai3 r ) ar av Y av pi-+---+( ar rae
ah
D
uv
tap a +---_-p-~_o, r ) F a8 ar (
1
au
-0,
(2)
ar j
Bh
s,+--r de ) Eqs.(Z) are written usingthegenerally accepted notation in a spherical system of coordinates which has its pole at the centre of the sphere in the flow. +Sh.vychisl.Mat.mat.Fiz.,27,11,1739-1744,1987
92
The flow region under examination is bounded by the surfaces of a solid, the head end shock wave, the axis of symmetry and the conical surface 6-S. which is placed so that the normal component of the gas velocity on the surface is supersonic in all.but the boundary region. Taking the flow axis of symmetry into account, calculations are performed for only one meridional plane which includes the axis of symmetry. The boundary conditions of the problem are specified in the same way as for a uniform flow /3/. The surface of the solid is considered to be heat-insulated. The adhesion and impermeability conditions are used here as velocity components. The gas parameters behind the receding shock wave are defined from the generalized /2/ or classical Rankine-Hugoniotrelations. Symmetry conditions are used on the line 6-O and approximated "soft" conditions of type a?fiaw=-0 on the radius 6-6 where f is any unknown function. The boundary conditions at 6-O and 6-e are specified for any function in the context of using secondary difference equations with respect to the angular coordinate. The use of simplified equations ensures maximum enconomy of computer resources when they are integrated by marching procedures or global iterations, which is possible for flows where the effects of turbulence propagation against the current are insignificant. In the class being considered, these effects do have a strong influence on the structure of the flow, which is subsonic over most of the region of calculation. Bearing this in mind, the solution of Eqs.(2) is found using appropriate non-stationary equations established as a result of integration over time. Preliminary values of the unknown functions in the majority of cases, are borrowed from available solutions for a related set of initial data.
Fig.1
l;i
l----,,oj$?.__;;~ ._
a
;I;
0.002
0.002
o.oor
Fig.2 The numerical method used is described in detail in /3/. It was based on the application of a two-layered time implicit finite-differencescheme with second-order approximation in the spatial coordinates. The solution of the non-linear system of difference equations for values of functions in the new time layer was found by iterations, in each of which we solved independent linear subsystems at radii 6-coast using vector pivotal condensation.Thesolutions of the complete Navier-Stokes equations were obtained by this method in /a/. The establishment of a stationary solution was monitored from the relative values of the time derivatives of temperature and pressure. Time integration continued as long as the 3X10_? maximum value of these derivatives over the entire flow field remained greater than Most calculations were completed on a uniform difference grid MXN-50x25 where M is the number of calculation nodes on the radius B-coast and N is the number of radii. Controlcalculations were performed on grids with doubled and halved node numbers for each coordinate. Some results of these calculations are shown by the triangles in Figs.2 and 5. The accuracy of the results was also checked by the realization of the integral law of conservation of mass. In the solutions established, the corresponding integral relation is satisfied with an accuracy to a tenth of a per cent. 2. Results of calculations. Calculations were made for flow over a sphere by an ideal gas with a constant ratio of the specific heats I-l.4 a coefficient of viscosity p-F% and Prandtl number Pr-0.71. The values of numerical parameters in (1) were taken as the following: a-0.1, b-7.2, and c-3.0.The results presented below were obtained for Mach number 6 on the axis of symmetry, in front of the head-end shock wave and Reynolds number 177cRec5000.The Reynolds number was calculated from the parameters of the oncoming flow on the axis of symmetry and along the radius of the
93
sphere. This value was varied by changing the density of the oncoming flow. Nhere R&SOoo, the stationary solution of the complete Navier-Stokes equations with the accuracy given above was not reached.
a
I.1
1.2
f. J
Fig.3
1.4P !.5 Fig.4
The change of flow parameters acquires a fluctuating form which depends on the time and the spatial coordinate steps of the difference grid. A solution fails to be reached forsimplified equations at rather lower Reynolds numbers. The position of the head end shock wave and flow lines is shown in Fig.1. Obviously, in the shock layer, fairly complex flow with a closed recycled region occurs. This flow pattern is derived from the solution of both the complete and simplified Navier-Stokes equations. Figs.Z-5 show a comparison of certain flow features of any spherical surface obtained from the solution of these equations. The linear dimensions were related to the radius of the sphere R, the velocity values of VO, the pressure to PPVO',and the velocity vortex to
VolR. The coefficient of friction Cfwas defined as the ratio of the friction. stress on the surface of the solid to the value povo' (thevalues of oncoming flow parameters on the axis of symmetry are denoted by the subscript 0). The complete Navier-Stokes integrated with generalized Rankine-Hugoniot conditions at the head end shock solid lines correspond to these results). The dashed lines are the solutions
equations were wave (the of Eqs.(2)
with classical Rankine-Hugoniotconditions. Fig.2 shows the parameters of the basic reverse circulated flow as a function of the Reynolds number. Thequantity 7'isthecoordinateoftheouterboundary ofthis flow on the axisof symmetry and Q is the value of the velocity vortex at its centre. Fig.2a shows that, even with minimum Reynolds number, the positions of the outer boundary of the basic reverse circulated flow, found from the solution of the complete and simplified equations, agree well. The values of the shock wave as it recedes from the surface of the sphere also agree well. At the same time, the calculations show considerable disparity in the positions of lines which divide the recirculation rones, connected with the smallness of
the gas velocity in the neighbourhood of this line. When the simplified equations are in use, secondary recirculated flow occurs at smaller Reynolds numbers. The coordinates of the centre of the basic circulated flow and the vortex velocities at this point were obtained by means of graphic processing and the differentiation of numerical solutions, which led to a loss of accuracy in these values. Fig.lb clearly shows that the use of the simplified Navier-Stokes equations leads to a excessive vortex velocities and, unlike the geometrical characteristicsof circulated flow, the difference in the vortex values increases as the Reynolds number increases. Fig.2 also shows results which illustrate the convergence of the numerical solutions when the number of cells in the difference grid decreases. For He-1700, the results of the solution of the complete are shown by the symbols 2.3 -
Navier-Stokes equations on grids with .WQ =-33x12 and lOOxj0 nodes A and v, respectively (also in Fig.5). Fig.3 shows the profiles of the longitudinal component Re=1?70, at radii 8-coast:I for of the gas velocity for t?-7.5". 2 for 8-30'. and 3 for It can be seen that 6-60". velocities outside the recirculation region agree much better.
Figs.4and5 show the change of the coefficient friction and pressure on the surface of the sphere. Curve 1 corresponds to He-177 and curve 2 corresponds to Rc-iii0 A large difference in these values occurs in regions of recirculated flow. They decrease as the Reynolds number increases. In the description of flow by simplified equations the increase in pressure on the surface of the sphere begins at large
Fig.5
values of the angular coordinate. Maximum pressure values only vary by a few per cent. A sharp change in the functions near the head end of the shock wave may occur in the flow under examination (see Fig.3). Therefore, it should be expected that an account of
94
moleculartransferprocesses in boundary conditions at the head end of the shock wave exert a greater influence on the results of the calculation of the shock layer than during flow over a solid by a uniform stream. For an explanation of this question the solutions of the complete Navier-Stokes equations were obtained with classical Rankine-Hugoniotconditions together with the solutions of Eqs.(2) with generalized conditions. For small Reynolds numbers, both these solutions vary significantly from the results obtained using the two mathematical models examined above and are in opposition. The best agreement withtheresults of applying the fullest of the models examined (the complete Navier-Stokes equations with generalized Rankine-Hugoniotconditions), give solutions of the simplified equations with classical conditions. As an example, the dot-dash lines 1 and 2 in Fig.5 show the pressure distribution obtained from the solution of the complete Navler-Stokes equations with classical Rankine-Hugoniotconditions and Eqs.(21 with generalized conditions, respectively. As the Reynolds number increases, so the difference in pressure distributions obtained using all the mathematical models described decreases. Thus, the results of calculations indicate that for fairly large Reynolds numbers the simplified mathematical model, including Eqs.(2) and the classical Randkine-Hugoniotconditions can be used to simulate complex flows with developed recirculation zones. This model quite accurately describes the change in the geometrical characteristicsof such flows, the distribution of pressure over the surface of the solid and the change in friction stress beyond the joining of circulated flow. At the same time, the simplified model is not suitable for a quantative analysis of the components of circulation flows. REFERENCES 1. DAVIS R.T. and FLUGGE-LOTZ I., Second-order boundary layer effects in hypersonic flow past axisymmetric blunt bodies. J. Fluid Mech., 20, 4, 593-623, 1964. 2. TOLSTYKH A.I., On the numerical calculation of supersonic flow past blunt bodies by a stream of viscous gas. Zh. vychisl. Mat. mat. Fiz., 6, 1, 113-120, 1966. 3. GOLOVACHEV Y0.P. andPOPOVF.D., The calculation of supersonic flow past blunt bodies by a viscous gas at large Reynolds numbers. Zh. vychisl. Mat. mat. Fiz., 12, 5, 1292-1303, 1972. 4. TANNEHILL J.C., VENKATAPATHY E. and P.AKICHJ.V., Numerical soltuion of viscous flow over blunt delta wings, AIAA Paper 81-0049, 1981. 5. RUBIN S.G., A review of marching procedures for parabolized Navier-Stokes equations. Numer. and Phys. Aspects of Aerodynamic Flows. N.Y. etc., Springer, 171-185, 1981. 6. MELESHKO S.V. and CHEWY1 S.G., An investigationof viscous compressible flows based on parabolized Navier-Stokes equations: Preprint 32. Novosibirsk, ITPM Siberian Dept., Akad Nauk SSSR, 47, 1985. 7. LIN T.C., REEVES B.L. and SIEGELMAN D., Blunt-body problem in non-uniform flow-fields. AIAA Journal, 15, 8, 1130-1137, 1977. 8. GOLOVACHEV YU.P. and LEONT'YEVAN.V., Circulated flow near any surface of a sphere by a supersonic trace-type stream. Izv. Akad. Nauk SSSR, Mekh. Zhidkosti i gaza, 3, 143-148, 1985. Translated by C.M.
3. BAUMGARTEL Ii.,Analytic perturbation theory for matrices and operators, Birkhauser, Basel, 1985. Translated by S.R.
U.S.S.R. Comput.Maths.Math.Phys .,Vo1.27,No.6,pp.91-94,19&7 Printed in Great Britain
OC41-5553/87 510.00+0.00 01989 Pergamon Press plc
NUMERICAL SIMULATION OF A CIRCULATED GAS FLOW BASED ON THE COMPLETE AND SIMPLIFIED NAVIER-STOKES EQUATIONS*
YU.P. GOLOVACEEV and N.V. LEONT'YEVA Using the example of stationary axisymmetric flow near any surface of a sphere situated in a supersonic wake region, the applicability of simplified mathematical models for the description of gas flows with a developed recirculation zone is studied. A comparison is made of the results of numerical solutions of the complete and simplified NavierStokes equations with two different boundary conditions at the head end shock wave. At the present time, the simplified ("parabolized")Navier-Stokes equations are used to simulate viscous gas flows. These include all terms of Euler's equations and the boundarylayer equations and do not contain second dervivatives of unknown functions with respect to the coordinate which coincides with the basic direction of flow. Mathematical models were formulated in /l-3/ for problems of supersonic flow over solids, as a result of evaluating a number of terms of the Navier-Stokes equations describing the molecular transfer of momentum and energy. These evaluations were based on concepts of classical boundary layer theory and hold for continuous flow at large Reynolds numbers. Practical experience in using the models under examination has shown that the region where they are applicable is wider than the formal limits expected from the evaluations specified above. Simplified equations are used, in particular, for calculating flows with flow separation from the surface of the solid and with the formation of closed regions of backward-circulationflow (see, for example, /4-6/I. The study of errors connected with the description of their simplified equations in the simulation of such flows is of interest. With this object in view, in this paper the results of a numerical solution of the complete and simplified Navier-Stokes equations are compared for one example of gas flow with a developed recirculation sane. 1. Statement of the problem and numerical method. Stationary axisymmetric flow over any surface of a sphere by a non-uniform supersonic stream, whose parameters correspond to flow in a remote trace is examined. The distributions of the gas-dynamic functions in the oncoming stream are described by formulae taken from /7/: V(y)=V(~)[l-oeap (-bu’)l,
h~~)-~(o)(~+c~i-~‘(y)/~‘(O)11, p(y)-const.
(0
Here, y is the distance from the axis of symmetry referred to the radius of the sphere, V is the modulus of the velocity vector, h is the specific enthalpy, p is the pressure, and a, b, and c are numerical parameters describing the shape of velocity and enthalpy profiles in the oncoming stream. using (l), flow in the wake is simulated by a parallel axisymmetric stream with constant pressure, axial minimum velocity and maximum enthalpy. It is assumed that the density of the gas is sufficient to form a head end shock wave in front of the sphere. Flow between the head end shock wave and the surface of the solid is described by the simplified Navier-Stokes equations /3/:
au a0 bP v an u--f---+P r -&-fd8+2u+vctg6 or r86 T( ea ” 80 ll= OP ;i;-+---+--0, Pa ( rai3 r ) ar av Y av pi-+---+( ar rae
ah
D
uv
tap a +---_-p-~_o, r ) F a8 ar (
1
au
-0,
(2)
ar j
Bh
s,+--r de ) Eqs.(Z) are written usingthegenerally accepted notation in a spherical system of coordinates which has its pole at the centre of the sphere in the flow. +Sh.vychisl.Mat.mat.Fiz.,27,11,1739-1744,1987
92
The flow region under examination is bounded by the surfaces of a solid, the head end shock wave, the axis of symmetry and the conical surface 6-S. which is placed so that the normal component of the gas velocity on the surface is supersonic in all.but the boundary region. Taking the flow axis of symmetry into account, calculations are performed for only one meridional plane which includes the axis of symmetry. The boundary conditions of the problem are specified in the same way as for a uniform flow /3/. The surface of the solid is considered to be heat-insulated. The adhesion and impermeability conditions are used here as velocity components. The gas parameters behind the receding shock wave are defined from the generalized /2/ or classical Rankine-Hugoniotrelations. Symmetry conditions are used on the line 6-O and approximated "soft" conditions of type a?fiaw=-0 on the radius 6-6 where f is any unknown function. The boundary conditions at 6-O and 6-e are specified for any function in the context of using secondary difference equations with respect to the angular coordinate. The use of simplified equations ensures maximum enconomy of computer resources when they are integrated by marching procedures or global iterations, which is possible for flows where the effects of turbulence propagation against the current are insignificant. In the class being considered, these effects do have a strong influence on the structure of the flow, which is subsonic over most of the region of calculation. Bearing this in mind, the solution of Eqs.(2) is found using appropriate non-stationary equations established as a result of integration over time. Preliminary values of the unknown functions in the majority of cases, are borrowed from available solutions for a related set of initial data.
Fig.1
l;i
l----,,oj$?.__;;~ ._
a
;I;
0.002
0.002
o.oor
Fig.2 The numerical method used is described in detail in /3/. It was based on the application of a two-layered time implicit finite-differencescheme with second-order approximation in the spatial coordinates. The solution of the non-linear system of difference equations for values of functions in the new time layer was found by iterations, in each of which we solved independent linear subsystems at radii 6-coast using vector pivotal condensation.Thesolutions of the complete Navier-Stokes equations were obtained by this method in /a/. The establishment of a stationary solution was monitored from the relative values of the time derivatives of temperature and pressure. Time integration continued as long as the 3X10_? maximum value of these derivatives over the entire flow field remained greater than Most calculations were completed on a uniform difference grid MXN-50x25 where M is the number of calculation nodes on the radius B-coast and N is the number of radii. Controlcalculations were performed on grids with doubled and halved node numbers for each coordinate. Some results of these calculations are shown by the triangles in Figs.2 and 5. The accuracy of the results was also checked by the realization of the integral law of conservation of mass. In the solutions established, the corresponding integral relation is satisfied with an accuracy to a tenth of a per cent. 2. Results of calculations. Calculations were made for flow over a sphere by an ideal gas with a constant ratio of the specific heats I-l.4 a coefficient of viscosity p-F% and Prandtl number Pr-0.71. The values of numerical parameters in (1) were taken as the following: a-0.1, b-7.2, and c-3.0.The results presented below were obtained for Mach number 6 on the axis of symmetry, in front of the head-end shock wave and Reynolds number 177cRec5000.The Reynolds number was calculated from the parameters of the oncoming flow on the axis of symmetry and along the radius of the
93
sphere. This value was varied by changing the density of the oncoming flow. Nhere R&SOoo, the stationary solution of the complete Navier-Stokes equations with the accuracy given above was not reached.
a
I.1
1.2
f. J
Fig.3
1.4P !.5 Fig.4
The change of flow parameters acquires a fluctuating form which depends on the time and the spatial coordinate steps of the difference grid. A solution fails to be reached forsimplified equations at rather lower Reynolds numbers. The position of the head end shock wave and flow lines is shown in Fig.1. Obviously, in the shock layer, fairly complex flow with a closed recycled region occurs. This flow pattern is derived from the solution of both the complete and simplified Navier-Stokes equations. Figs.Z-5 show a comparison of certain flow features of any spherical surface obtained from the solution of these equations. The linear dimensions were related to the radius of the sphere R, the velocity values of VO, the pressure to PPVO',and the velocity vortex to
VolR. The coefficient of friction Cfwas defined as the ratio of the friction. stress on the surface of the solid to the value povo' (thevalues of oncoming flow parameters on the axis of symmetry are denoted by the subscript 0). The complete Navier-Stokes integrated with generalized Rankine-Hugoniot conditions at the head end shock solid lines correspond to these results). The dashed lines are the solutions
equations were wave (the of Eqs.(2)
with classical Rankine-Hugoniotconditions. Fig.2 shows the parameters of the basic reverse circulated flow as a function of the Reynolds number. Thequantity 7'isthecoordinateoftheouterboundary ofthis flow on the axisof symmetry and Q is the value of the velocity vortex at its centre. Fig.2a shows that, even with minimum Reynolds number, the positions of the outer boundary of the basic reverse circulated flow, found from the solution of the complete and simplified equations, agree well. The values of the shock wave as it recedes from the surface of the sphere also agree well. At the same time, the calculations show considerable disparity in the positions of lines which divide the recirculation rones, connected with the smallness of
the gas velocity in the neighbourhood of this line. When the simplified equations are in use, secondary recirculated flow occurs at smaller Reynolds numbers. The coordinates of the centre of the basic circulated flow and the vortex velocities at this point were obtained by means of graphic processing and the differentiation of numerical solutions, which led to a loss of accuracy in these values. Fig.lb clearly shows that the use of the simplified Navier-Stokes equations leads to a excessive vortex velocities and, unlike the geometrical characteristicsof circulated flow, the difference in the vortex values increases as the Reynolds number increases. Fig.2 also shows results which illustrate the convergence of the numerical solutions when the number of cells in the difference grid decreases. For He-1700, the results of the solution of the complete are shown by the symbols 2.3 -
Navier-Stokes equations on grids with .WQ =-33x12 and lOOxj0 nodes A and v, respectively (also in Fig.5). Fig.3 shows the profiles of the longitudinal component Re=1?70, at radii 8-coast:I for of the gas velocity for t?-7.5". 2 for 8-30'. and 3 for It can be seen that 6-60". velocities outside the recirculation region agree much better.
Figs.4and5 show the change of the coefficient friction and pressure on the surface of the sphere. Curve 1 corresponds to He-177 and curve 2 corresponds to Rc-iii0 A large difference in these values occurs in regions of recirculated flow. They decrease as the Reynolds number increases. In the description of flow by simplified equations the increase in pressure on the surface of the sphere begins at large
Fig.5
values of the angular coordinate. Maximum pressure values only vary by a few per cent. A sharp change in the functions near the head end of the shock wave may occur in the flow under examination (see Fig.3). Therefore, it should be expected that an account of
94
moleculartransferprocesses in boundary conditions at the head end of the shock wave exert a greater influence on the results of the calculation of the shock layer than during flow over a solid by a uniform stream. For an explanation of this question the solutions of the complete Navier-Stokes equations were obtained with classical Rankine-Hugoniotconditions together with the solutions of Eqs.(2) with generalized conditions. For small Reynolds numbers, both these solutions vary significantly from the results obtained using the two mathematical models examined above and are in opposition. The best agreement withtheresults of applying the fullest of the models examined (the complete Navier-Stokes equations with generalized Rankine-Hugoniotconditions), give solutions of the simplified equations with classical conditions. As an example, the dot-dash lines 1 and 2 in Fig.5 show the pressure distribution obtained from the solution of the complete Navler-Stokes equations with classical Rankine-Hugoniotconditions and Eqs.(21 with generalized conditions, respectively. As the Reynolds number increases, so the difference in pressure distributions obtained using all the mathematical models described decreases. Thus, the results of calculations indicate that for fairly large Reynolds numbers the simplified mathematical model, including Eqs.(2) and the classical Randkine-Hugoniotconditions can be used to simulate complex flows with developed recirculation zones. This model quite accurately describes the change in the geometrical characteristicsof such flows, the distribution of pressure over the surface of the solid and the change in friction stress beyond the joining of circulated flow. At the same time, the simplified model is not suitable for a quantative analysis of the components of circulation flows. REFERENCES 1. DAVIS R.T. and FLUGGE-LOTZ I., Second-order boundary layer effects in hypersonic flow past axisymmetric blunt bodies. J. Fluid Mech., 20, 4, 593-623, 1964. 2. TOLSTYKH A.I., On the numerical calculation of supersonic flow past blunt bodies by a stream of viscous gas. Zh. vychisl. Mat. mat. Fiz., 6, 1, 113-120, 1966. 3. GOLOVACHEV Y0.P. andPOPOVF.D., The calculation of supersonic flow past blunt bodies by a viscous gas at large Reynolds numbers. Zh. vychisl. Mat. mat. Fiz., 12, 5, 1292-1303, 1972. 4. TANNEHILL J.C., VENKATAPATHY E. and P.AKICHJ.V., Numerical soltuion of viscous flow over blunt delta wings, AIAA Paper 81-0049, 1981. 5. RUBIN S.G., A review of marching procedures for parabolized Navier-Stokes equations. Numer. and Phys. Aspects of Aerodynamic Flows. N.Y. etc., Springer, 171-185, 1981. 6. MELESHKO S.V. and CHEWY1 S.G., An investigationof viscous compressible flows based on parabolized Navier-Stokes equations: Preprint 32. Novosibirsk, ITPM Siberian Dept., Akad Nauk SSSR, 47, 1985. 7. LIN T.C., REEVES B.L. and SIEGELMAN D., Blunt-body problem in non-uniform flow-fields. AIAA Journal, 15, 8, 1130-1137, 1977. 8. GOLOVACHEV YU.P. and LEONT'YEVAN.V., Circulated flow near any surface of a sphere by a supersonic trace-type stream. Izv. Akad. Nauk SSSR, Mekh. Zhidkosti i gaza, 3, 143-148, 1985. Translated by C.M.