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Recent developments in transonic Euler flow over a circular cylinder
232
Mathematics
RECENT DEVELOPMENTS CYLINDER *
Manuel NASA
IN TRANSONIC
and Computers
in Simulation
XXV (1983) 232-236 North-Holland
EULER FLOW OVER A CIRCULAR
D. SALAS
Langley Research Center, Hampton,
VA 23665, U.S.A.
Numerical solutions to the Euler equations for transonic flow over a circular cylinder indicate that the inviscid flow separates ahead of the rear stagnation point. Our understanding of this phenomenon and various solutions presented at a workshop on this subject are discussed. INTROOUCTION At high speeds, as air rushes past a circular cylinder. a Docket of suoersonic flow terminated by a recompression shock'forms near the top of the cylinder. Recent numerical calculations made by the author, based on the inviscid Euler equations, also showed a bubble of recirculating flow at the rear of the cylinder. Since separation is usually associated with the vorticity generated at solid boundaries by viscosity, its occurrence in these inviscid calculations was at first thought to be an artifact of the numerical simulation. However, after many careful calculations to determine the effects of the initial conditions, the boundary conditions, and the artificial viscosity inherent in the numerical scheme, the phenomenon appeared to be real (i.e., consistent with the Euler equations). The strongest theoretical support for the validity of the recirculation bubble as a solution of the Euler equations comes from the work of Fraenke1.I For incompressible flow past a circular cylinder, assuming constant vorticity, Fraenkel has obtained the exact solution to the steady-state equations. The solution shows the occurrence of symmetric recirculation bubbles at the front and rear of the cvlinder. The analytic solution further shows how the level of vorticity controls the location of the separation point on the stagnation streamline. Similar theoretical results have also been obtained by Kiichemanr? for inviscid shear flow near the trailing edge of an airfoil. For the transonic flow investigated here, unlike the problems investigated by Fraenkel and Kiichemann, the vorticity is not carried by the free stream, but is generated by the recompression shock wave. However, after being generated at the shock it produces the same results. It prematurely retards the surface velocity on the stagnation streamline, leading to separation ahead of the rear stagnation point. Thus far, we have discussed the occurrence of recirculation bubbles in rotational incompressible and transonic flows. For supersonic flow past a circular cylinder, a recirculation bubble can also be observed by shearing the free-stream Mach number distribution. An * This paper is an extended version of a paper presented at the 10th IMACS Congress, Montreal,
originally 1982.
example of this type of flow pattern is shown in figure 1.
In order to gain further insight and to have the numerical results independently verified, a workshop was held at NASA Lanslev Research Center on'september 1, 1981. The main purpose of this paper is to review the results presented at the workshop. Mm=
18
-SHOCK WAVE STREAM LINES --- SONIC LINE VELOCITY DIRECTION Mm=10
Figure l.- Supersonic shear flow impinging on a circular cylinder as calculated by the author. PARTICIPANTS AT THE WORKSHOP Of a total of eight presentations given at the workshoo. two were in a oreliminarv state of development and will not'be discussed here. The remaining six talks described the work of Mohamed Hafez of George Washington University; Ron-Ho Ni of Pratt and Whitney; Joseph Steger of Stanford University; Eli Turkel of Tel Aviv University, Israel; Bram van Leer of Leiden State University, The Netherlands; and the author.S-B (References cited give details of the numerical schemes used, but they do not necessarily address the problem discussed here.)
M.D. Salas / Recent developments in transonic Euler flow The talks concentrated on three cases corresponding to free-stream Mach numbers (M,) of 0.4, 0.5, and 0.6. The technique used by Hafez was unique in that it tried to solve the steady-state equations by intr0ducing.a stream function. The secondorder partial differential equation that results is very similar to the full-potential equation but with the vorticity acting as a source term. The equation is thus solved by standard relaxation methods with the additional complication that the source term must be evaluated at any point downstream of the shock wave by tracing the streamline at that point back to the shock wave. Hafez imposed far-field boundary conditions at about 15 radii from the cylinder. His far-field boundary condition required uniform free stream in front of the cylinder, while the outflow was required to be parallel and have the free-stream pressure level. The remaining participants all solved the timedependent Euler equations. Ni used a secondorder finite-difference scheme of the LaxWendroff type coupled with a multiple grid technique designed to accelerate convergence to steady state. In addition, Ni replaced the time-dependent energy equation by the assumption that the total enthalpy is constant during the transient. His far-field boundary condition used the idea of non-reflection of waves and was imposed at only 8 radji from the cylinder. Steger solved the four conservation equations using an Implicit approximate factorization method. Two spatial operators were tested-one consisting of second-order central differences and the other being a second-order upwing differencfng for the split flux vectors. Steger, in order to simplify the matrix inversion problem, overspecified the uniform free stream at approximately 36 radii away from the cylinder. Turkel solved the four conservation equations using a finite-volume Runge-Kutta integration method recently introduced by Jameson. The far-field boundary conditions were imposed at about 100 radii from the cylinder and the idea of nonreflecting waves was also used. To accelerate convergence to a steady state, the equations were modified to include a forcing term which should go to zero when the steady state is reached. In addition, the solution at each computational cell was advanced by its own local time step. Van Leer used his upwind flux splitting scheme with no attempt to accelerate the convergence to a steady state; thus, a true transient was followed. His far-field boundary condition consisted of imposing the Prandtl-Glauert solution at approximately 23 radii from the cylfnder. Finally, the au hor used the h-scheme introduced by Moretti 8 with the equations in nonconservation form. Like Ni, the timedependent energy equation was replaced by the assumption of constant enthalpy; and, like Turkel, local time steps were used to accelerate convergence. The far-field boundary conditions were imposed at infinity by means of an
233
inverse radial transformation. Uniform freestream values were imposed everywhere at infinity except on the rear symmetry line; there, the entropy is convected from inside the region, and the pressure is required to decay to the free-stream level. RESULTS PRESENTED AT THE WORKSHOP The M, = 0.4 case was chosen as a test case because it corresponds to the incipient formation of the supersonic bubble and the flow is still potential-like with fore and aft symmetry. All results presented for this case showed good qualitative and quantitative agreement. A slight fore/aft asymmetry could be observed in the results of Ni, Turkel, and van Leer. Figure 2 shows the streamline pattern for this case as computed by the author and figure 3 shows the Mach number contours from van Leer's calculation. None of the numerical calculations showed the occurrence of a recirculation bubble at this Mach number, which is consistent with the potential-like behavior of the flow.
_
_
~___~___
-~
Figure 2.- Streamline pattern for M, = 0.4 calculated by the author.
Figure 3.- Mach number contours for M, = 0.4 calculated by van Leer. Notice the slight fore/aft asymmetry.
234
M.D. Salas / Recent developments At M, = 0.5, all calculations found a recirculation bubble of approximately a diameter in width except van Leer's which was much smaller. (See figs. 4 and 5.) They were in qualitative agreement; some quantitative differences were, however, observed at this Mach number. The various calculations disagreed, particularly
Figure 4.- Comparison of recirculation bubble shape at Mm = 0.5.
Figure 5.- Streamline pattern for M, = 0.5 calculated by the author. Sonic line shown as dashed line.
in transonic Euler flow
+
CONSERVATIVE
POTENTIAL
Figure 6.- Pressure coefficient at M, = 0.5 computed by Ni and the author. Results from a conservative potential calculation are included for comparison.
Figure 7.- Streamline pattern for M, = 0.6 by the author. Sonic line shown as dashed line.
in the details of the recirculation bubble. For example, in figure 6 the computation by Ni shows the pressure within the recirculation bubble to be essentially constant; while the author finds a substantial variation in the pressure in this region. The differences in the results should not be very surprising considering the differences in the numerical methods, boundary conditions, and number of mesh points used by the participants. At the higher free-stream Mach number, M, = 0.6, the calculation of Turkel, Steger, van Leer, Hafez, and the author showed a very long recirculation bubble, possibly extending to infinity. (See figs. 7-10.) At this Mach number the calculation of Ni failed to converge.
______1..___.._..__.....__.....................
11
Figure 8.- Streamline pattern for M, = 0.6 calculated by Hafez. Sonic line shown as dashed line.
M.D. Salas / Recent developments
in transonic Euler
flow
SOME REMARKS ON THE UNIQUENESS QUESTION Granted the existence of regions of recirculating flow as solutions of the Euler equations, some remarks should be made concerning the uniqueness of these solutions. Here we must proceed with extreme caution.
Figure 9.- Mach number contours for M, = 0.6 calculated by the author.
Figure lo.- Mach number contours for calculated by Hafez.
M, = 0.6
The drag coefficient was available from four of the calculations reported here. In Table 1 they are compared to the drag coefficient from a conservative potential calculation. Although there is a considerable spread in the values for the Euler calculations, they all indicate a lower drag than predicted by the potential calculation. This comes about because in the Euler calculation the shock wave occurs ahead of the potential shock and is therefore weaker, and also because the recirculation bubble occurring in the Euler calculations, unlike the separation bubble in a typical viscous flow, is a region of high near-stagnation pressure which produces more thrust on the aft portion. Table l.- Drag coefficient computed by various methods. Euler
MC0
Cons. Pot.
Ni
Salas
Steger
0.40
-0.0000
0.0124
-0.0013
0.0062
0.0033
0.50
0.3799
0,1733
-0.0041
0.1794
0.0617
0.60
1.1293
0.0618
0.6066
0.1628
Not available
Turkel
First, it is well known that the steady-state Euler equations admit multiple solutions; see, for example, references 9-11. Hafez. for example, believes that the existence of multiple solutions is indicated in his steadystate calculations by the fact that he cannot evaluate the vorticity level for the streamlines inside the recirculation bubble since they form closed paths which do not go back to the shock wave where the vorticity is generated. In his calculations, he is therefore forced to model the vorticity behavior inside the bubble by invoking analytic continuation. However, if we consider the steady Euler solution as a limit of the unsteady equations, then the numerical experiments seem to indicate that the solution has very little dependence on initial conditions. For example, if we start with initial conditions corresponding to the converged solution for M, = 0.5 which has a recirculation bubble and we decrease the freestream Mach number to 0.4, we obtain the same bubble-free solution that we get if we start the calculation from initial conditions corresponding to incompressible flow. Many other experiments like this indicate the same behavior; however, there is little theoretical work on which to decide this issue conclusively. The question of uniqueness is further complicated when we ask: How is the steady Euler solution related to the limit solution of the Navier-Stokes equations when the Reynolds number goes to infinity?" Here again, it is ifportant to decide what the limiting order . Do we take the limit as time goes to infinity first, followed by Reynolds number going to infinity, or vice versa? The first limiting sequence has received a great deal of attention and is still a controversial subject. Saffman, for eitample, considers it as one "of the challen fng unsolved problems of fluid mechanics." 92 In performing the first limiting sequence, it is usually assumed that a steady state exists. This, however, might not be the case as is well known from experimental observations. Concerning this last point, an interesting discovery was made by Steger while Performing his calculations. Rather than limiting his computations to the upper half plane and imposing flow symmetry along the center line, as the other investigators did, Steger computed the full 3600 circle. For these calculations, Steger noticed that the flow at the rear of the cylinder oscillated, preventing convergence to a steady state. (See figs. 11 and 12.) This phenomenon has since been reproduced by Turkel and the author. In the calculations of Steger, it is triggered by an inherent asynsnetry in the approximate
235
236
M.D. Salas / Recent developments factorization technique used; while in the calculations by Turkel and the author, it is necessary to introduce an asymmetry into the initial conditions in order to observe it. The oscillations are only observed for supercritical cases and seem to be sustained by reflection of waves from the upper and lower shock waves.
Figure ll.- Instantaneous streamline pattern for unsteady case at M, = 0.5 calculated by Steger.
Figure 12.- Instantaneous streamline pattern for unsteady case at M, = 0.5 calculated by Steger. Figure corresponds to a slightly larger time than figure 11. CONCLUDING REMARKS All the evidence now available indicates that inviscid separation is a feature of the Euler equations induced by vorticity and/or stagnation pressure loss in the flow. The steadystate solutions seem to be independent of the initial conditions and the numerical experiments indicate that the time-dependent Euler equations converge to a unique solution. The relevance of this solution to the Navier-Stokes equations in the limit of vanishing viscosity is an open question. Furthermore, the oscillatory behavior observed by Steger for the full circle needs further investigation.
in transonic Euler j&w
REFERENCES 1.
Fraenkel, L. E.: On Corner Eddies in Plane Inviscid Shear Flow. J. Fluid Mech., vol. 11, 1961, pp. 400-406.
2.
Kiichemann, D.: Inviscid Shear Flow Near the Trailing Edge of an Airfoil. Z. Flugwiss, vol. 15, 1967, pp. 292-294.
3.
Hafez, M.; and Lovell, D.: Numerical Solution of Transonic Stream Function Equation. AIAA Paper 81-1017, A Collection of Technical Papers, AIAA 5th Computational Fluid Dynamics Conference, 1981, pp. 364-372.
4.
Ni, R. H.: A Multiple Grid Scheme for Solving the Euler Equations. AIAA Paper 81-1025, A Collection of Technical Papers, AIAA 5th Computational Fluid Dynamics Conference, 1981, pp. 257-264.
5.
Buning, P. G.; and Steger, J. L.: Solution of the Two-Dimensional Euler Equations With Generalized Coordinate Transformation Usino Flux Vector Solittina. AIAA Paper 82-0971 presented at"the Fluids, Plasma and Heat Transfer Conference, June 7-11, 1982, St. Louis, Missouri.
6.
Jameson, A.; Schmidt, W.; and Turkel. E.: Numerical Solutions of Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes. AIAA Paper El-1259 presented at AIAA 14th Fluid and Plasma Dynamics Conference, June 23-25, 1981.
7.
van Albada. G. 0.; Roberts, W. W., Jr.; and van Leer, 6.: A Comparative Study of Computational Methods in Cosmic Gas Dynamics. Astronomy and Astrophysics, vol. 108, 1982, pp. 76-84.
a.
Moretti, G.: The A-Scheme. Comp. and Flds., vol. 7, 1979, pp. 191-205.
9.
Lagerstrum. P. A.: Solutions of the Navier-Stokes Equation at Large Reynolds Number. Proc. Int'l. Symp. on Modern Developments in Fld. Dyns., J. Rom, ed., Society for Industrial and Applied Mathematics, 1977, pp. 364-376.
10.
Salas, M. 0.: On the Instability of Shock Waves Attached to Wedges and Cones. AMA Paper 82-0288. AMA 20th Aerospace Sciences Meeting, Jan. 11-14. 1982, Orlando, Florida.
11.
Chfldress, Stephen: Solutions of the Euler's Equations Containlng Finite Eddies. The Physics of Flds.. vol. 9. no. 5, May 1966, pp. 860-872.
12.
Saffman. P. G.: Dynamics of Vorticity, J. Fld. Mech.. vol. 106, 1981, pp. 49-58.
Mathematics
RECENT DEVELOPMENTS CYLINDER *
Manuel NASA
IN TRANSONIC
and Computers
in Simulation
XXV (1983) 232-236 North-Holland
EULER FLOW OVER A CIRCULAR
D. SALAS
Langley Research Center, Hampton,
VA 23665, U.S.A.
Numerical solutions to the Euler equations for transonic flow over a circular cylinder indicate that the inviscid flow separates ahead of the rear stagnation point. Our understanding of this phenomenon and various solutions presented at a workshop on this subject are discussed. INTROOUCTION At high speeds, as air rushes past a circular cylinder. a Docket of suoersonic flow terminated by a recompression shock'forms near the top of the cylinder. Recent numerical calculations made by the author, based on the inviscid Euler equations, also showed a bubble of recirculating flow at the rear of the cylinder. Since separation is usually associated with the vorticity generated at solid boundaries by viscosity, its occurrence in these inviscid calculations was at first thought to be an artifact of the numerical simulation. However, after many careful calculations to determine the effects of the initial conditions, the boundary conditions, and the artificial viscosity inherent in the numerical scheme, the phenomenon appeared to be real (i.e., consistent with the Euler equations). The strongest theoretical support for the validity of the recirculation bubble as a solution of the Euler equations comes from the work of Fraenke1.I For incompressible flow past a circular cylinder, assuming constant vorticity, Fraenkel has obtained the exact solution to the steady-state equations. The solution shows the occurrence of symmetric recirculation bubbles at the front and rear of the cvlinder. The analytic solution further shows how the level of vorticity controls the location of the separation point on the stagnation streamline. Similar theoretical results have also been obtained by Kiichemanr? for inviscid shear flow near the trailing edge of an airfoil. For the transonic flow investigated here, unlike the problems investigated by Fraenkel and Kiichemann, the vorticity is not carried by the free stream, but is generated by the recompression shock wave. However, after being generated at the shock it produces the same results. It prematurely retards the surface velocity on the stagnation streamline, leading to separation ahead of the rear stagnation point. Thus far, we have discussed the occurrence of recirculation bubbles in rotational incompressible and transonic flows. For supersonic flow past a circular cylinder, a recirculation bubble can also be observed by shearing the free-stream Mach number distribution. An * This paper is an extended version of a paper presented at the 10th IMACS Congress, Montreal,
originally 1982.
example of this type of flow pattern is shown in figure 1.
In order to gain further insight and to have the numerical results independently verified, a workshop was held at NASA Lanslev Research Center on'september 1, 1981. The main purpose of this paper is to review the results presented at the workshop. Mm=
18
-SHOCK WAVE STREAM LINES --- SONIC LINE VELOCITY DIRECTION Mm=10
Figure l.- Supersonic shear flow impinging on a circular cylinder as calculated by the author. PARTICIPANTS AT THE WORKSHOP Of a total of eight presentations given at the workshoo. two were in a oreliminarv state of development and will not'be discussed here. The remaining six talks described the work of Mohamed Hafez of George Washington University; Ron-Ho Ni of Pratt and Whitney; Joseph Steger of Stanford University; Eli Turkel of Tel Aviv University, Israel; Bram van Leer of Leiden State University, The Netherlands; and the author.S-B (References cited give details of the numerical schemes used, but they do not necessarily address the problem discussed here.)
M.D. Salas / Recent developments in transonic Euler flow The talks concentrated on three cases corresponding to free-stream Mach numbers (M,) of 0.4, 0.5, and 0.6. The technique used by Hafez was unique in that it tried to solve the steady-state equations by intr0ducing.a stream function. The secondorder partial differential equation that results is very similar to the full-potential equation but with the vorticity acting as a source term. The equation is thus solved by standard relaxation methods with the additional complication that the source term must be evaluated at any point downstream of the shock wave by tracing the streamline at that point back to the shock wave. Hafez imposed far-field boundary conditions at about 15 radii from the cylinder. His far-field boundary condition required uniform free stream in front of the cylinder, while the outflow was required to be parallel and have the free-stream pressure level. The remaining participants all solved the timedependent Euler equations. Ni used a secondorder finite-difference scheme of the LaxWendroff type coupled with a multiple grid technique designed to accelerate convergence to steady state. In addition, Ni replaced the time-dependent energy equation by the assumption that the total enthalpy is constant during the transient. His far-field boundary condition used the idea of non-reflection of waves and was imposed at only 8 radji from the cylinder. Steger solved the four conservation equations using an Implicit approximate factorization method. Two spatial operators were tested-one consisting of second-order central differences and the other being a second-order upwing differencfng for the split flux vectors. Steger, in order to simplify the matrix inversion problem, overspecified the uniform free stream at approximately 36 radii away from the cylinder. Turkel solved the four conservation equations using a finite-volume Runge-Kutta integration method recently introduced by Jameson. The far-field boundary conditions were imposed at about 100 radii from the cylinder and the idea of nonreflecting waves was also used. To accelerate convergence to a steady state, the equations were modified to include a forcing term which should go to zero when the steady state is reached. In addition, the solution at each computational cell was advanced by its own local time step. Van Leer used his upwind flux splitting scheme with no attempt to accelerate the convergence to a steady state; thus, a true transient was followed. His far-field boundary condition consisted of imposing the Prandtl-Glauert solution at approximately 23 radii from the cylfnder. Finally, the au hor used the h-scheme introduced by Moretti 8 with the equations in nonconservation form. Like Ni, the timedependent energy equation was replaced by the assumption of constant enthalpy; and, like Turkel, local time steps were used to accelerate convergence. The far-field boundary conditions were imposed at infinity by means of an
233
inverse radial transformation. Uniform freestream values were imposed everywhere at infinity except on the rear symmetry line; there, the entropy is convected from inside the region, and the pressure is required to decay to the free-stream level. RESULTS PRESENTED AT THE WORKSHOP The M, = 0.4 case was chosen as a test case because it corresponds to the incipient formation of the supersonic bubble and the flow is still potential-like with fore and aft symmetry. All results presented for this case showed good qualitative and quantitative agreement. A slight fore/aft asymmetry could be observed in the results of Ni, Turkel, and van Leer. Figure 2 shows the streamline pattern for this case as computed by the author and figure 3 shows the Mach number contours from van Leer's calculation. None of the numerical calculations showed the occurrence of a recirculation bubble at this Mach number, which is consistent with the potential-like behavior of the flow.
_
_
~___~___
-~
Figure 2.- Streamline pattern for M, = 0.4 calculated by the author.
Figure 3.- Mach number contours for M, = 0.4 calculated by van Leer. Notice the slight fore/aft asymmetry.
234
M.D. Salas / Recent developments At M, = 0.5, all calculations found a recirculation bubble of approximately a diameter in width except van Leer's which was much smaller. (See figs. 4 and 5.) They were in qualitative agreement; some quantitative differences were, however, observed at this Mach number. The various calculations disagreed, particularly
Figure 4.- Comparison of recirculation bubble shape at Mm = 0.5.
Figure 5.- Streamline pattern for M, = 0.5 calculated by the author. Sonic line shown as dashed line.
in transonic Euler flow
+
CONSERVATIVE
POTENTIAL
Figure 6.- Pressure coefficient at M, = 0.5 computed by Ni and the author. Results from a conservative potential calculation are included for comparison.
Figure 7.- Streamline pattern for M, = 0.6 by the author. Sonic line shown as dashed line.
in the details of the recirculation bubble. For example, in figure 6 the computation by Ni shows the pressure within the recirculation bubble to be essentially constant; while the author finds a substantial variation in the pressure in this region. The differences in the results should not be very surprising considering the differences in the numerical methods, boundary conditions, and number of mesh points used by the participants. At the higher free-stream Mach number, M, = 0.6, the calculation of Turkel, Steger, van Leer, Hafez, and the author showed a very long recirculation bubble, possibly extending to infinity. (See figs. 7-10.) At this Mach number the calculation of Ni failed to converge.
______1..___.._..__.....__.....................
11
Figure 8.- Streamline pattern for M, = 0.6 calculated by Hafez. Sonic line shown as dashed line.
M.D. Salas / Recent developments
in transonic Euler
flow
SOME REMARKS ON THE UNIQUENESS QUESTION Granted the existence of regions of recirculating flow as solutions of the Euler equations, some remarks should be made concerning the uniqueness of these solutions. Here we must proceed with extreme caution.
Figure 9.- Mach number contours for M, = 0.6 calculated by the author.
Figure lo.- Mach number contours for calculated by Hafez.
M, = 0.6
The drag coefficient was available from four of the calculations reported here. In Table 1 they are compared to the drag coefficient from a conservative potential calculation. Although there is a considerable spread in the values for the Euler calculations, they all indicate a lower drag than predicted by the potential calculation. This comes about because in the Euler calculation the shock wave occurs ahead of the potential shock and is therefore weaker, and also because the recirculation bubble occurring in the Euler calculations, unlike the separation bubble in a typical viscous flow, is a region of high near-stagnation pressure which produces more thrust on the aft portion. Table l.- Drag coefficient computed by various methods. Euler
MC0
Cons. Pot.
Ni
Salas
Steger
0.40
-0.0000
0.0124
-0.0013
0.0062
0.0033
0.50
0.3799
0,1733
-0.0041
0.1794
0.0617
0.60
1.1293
0.0618
0.6066
0.1628
Not available
Turkel
First, it is well known that the steady-state Euler equations admit multiple solutions; see, for example, references 9-11. Hafez. for example, believes that the existence of multiple solutions is indicated in his steadystate calculations by the fact that he cannot evaluate the vorticity level for the streamlines inside the recirculation bubble since they form closed paths which do not go back to the shock wave where the vorticity is generated. In his calculations, he is therefore forced to model the vorticity behavior inside the bubble by invoking analytic continuation. However, if we consider the steady Euler solution as a limit of the unsteady equations, then the numerical experiments seem to indicate that the solution has very little dependence on initial conditions. For example, if we start with initial conditions corresponding to the converged solution for M, = 0.5 which has a recirculation bubble and we decrease the freestream Mach number to 0.4, we obtain the same bubble-free solution that we get if we start the calculation from initial conditions corresponding to incompressible flow. Many other experiments like this indicate the same behavior; however, there is little theoretical work on which to decide this issue conclusively. The question of uniqueness is further complicated when we ask: How is the steady Euler solution related to the limit solution of the Navier-Stokes equations when the Reynolds number goes to infinity?" Here again, it is ifportant to decide what the limiting order . Do we take the limit as time goes to infinity first, followed by Reynolds number going to infinity, or vice versa? The first limiting sequence has received a great deal of attention and is still a controversial subject. Saffman, for eitample, considers it as one "of the challen fng unsolved problems of fluid mechanics." 92 In performing the first limiting sequence, it is usually assumed that a steady state exists. This, however, might not be the case as is well known from experimental observations. Concerning this last point, an interesting discovery was made by Steger while Performing his calculations. Rather than limiting his computations to the upper half plane and imposing flow symmetry along the center line, as the other investigators did, Steger computed the full 3600 circle. For these calculations, Steger noticed that the flow at the rear of the cylinder oscillated, preventing convergence to a steady state. (See figs. 11 and 12.) This phenomenon has since been reproduced by Turkel and the author. In the calculations of Steger, it is triggered by an inherent asynsnetry in the approximate
235
236
M.D. Salas / Recent developments factorization technique used; while in the calculations by Turkel and the author, it is necessary to introduce an asymmetry into the initial conditions in order to observe it. The oscillations are only observed for supercritical cases and seem to be sustained by reflection of waves from the upper and lower shock waves.
Figure ll.- Instantaneous streamline pattern for unsteady case at M, = 0.5 calculated by Steger.
Figure 12.- Instantaneous streamline pattern for unsteady case at M, = 0.5 calculated by Steger. Figure corresponds to a slightly larger time than figure 11. CONCLUDING REMARKS All the evidence now available indicates that inviscid separation is a feature of the Euler equations induced by vorticity and/or stagnation pressure loss in the flow. The steadystate solutions seem to be independent of the initial conditions and the numerical experiments indicate that the time-dependent Euler equations converge to a unique solution. The relevance of this solution to the Navier-Stokes equations in the limit of vanishing viscosity is an open question. Furthermore, the oscillatory behavior observed by Steger for the full circle needs further investigation.
in transonic Euler j&w
REFERENCES 1.
Fraenkel, L. E.: On Corner Eddies in Plane Inviscid Shear Flow. J. Fluid Mech., vol. 11, 1961, pp. 400-406.
2.
Kiichemann, D.: Inviscid Shear Flow Near the Trailing Edge of an Airfoil. Z. Flugwiss, vol. 15, 1967, pp. 292-294.
3.
Hafez, M.; and Lovell, D.: Numerical Solution of Transonic Stream Function Equation. AIAA Paper 81-1017, A Collection of Technical Papers, AIAA 5th Computational Fluid Dynamics Conference, 1981, pp. 364-372.
4.
Ni, R. H.: A Multiple Grid Scheme for Solving the Euler Equations. AIAA Paper 81-1025, A Collection of Technical Papers, AIAA 5th Computational Fluid Dynamics Conference, 1981, pp. 257-264.
5.
Buning, P. G.; and Steger, J. L.: Solution of the Two-Dimensional Euler Equations With Generalized Coordinate Transformation Usino Flux Vector Solittina. AIAA Paper 82-0971 presented at"the Fluids, Plasma and Heat Transfer Conference, June 7-11, 1982, St. Louis, Missouri.
6.
Jameson, A.; Schmidt, W.; and Turkel. E.: Numerical Solutions of Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes. AIAA Paper El-1259 presented at AIAA 14th Fluid and Plasma Dynamics Conference, June 23-25, 1981.
7.
van Albada. G. 0.; Roberts, W. W., Jr.; and van Leer, 6.: A Comparative Study of Computational Methods in Cosmic Gas Dynamics. Astronomy and Astrophysics, vol. 108, 1982, pp. 76-84.
a.
Moretti, G.: The A-Scheme. Comp. and Flds., vol. 7, 1979, pp. 191-205.
9.
Lagerstrum. P. A.: Solutions of the Navier-Stokes Equation at Large Reynolds Number. Proc. Int'l. Symp. on Modern Developments in Fld. Dyns., J. Rom, ed., Society for Industrial and Applied Mathematics, 1977, pp. 364-376.
10.
Salas, M. 0.: On the Instability of Shock Waves Attached to Wedges and Cones. AMA Paper 82-0288. AMA 20th Aerospace Sciences Meeting, Jan. 11-14. 1982, Orlando, Florida.
11.
Chfldress, Stephen: Solutions of the Euler's Equations Containlng Finite Eddies. The Physics of Flds.. vol. 9. no. 5, May 1966, pp. 860-872.
12.
Saffman. P. G.: Dynamics of Vorticity, J. Fld. Mech.. vol. 106, 1981, pp. 49-58.