DOI 10.1007/s10891-015-1263-x
Journal of Engineering Physics and Thermophysics, Vol. 88, No. 4, July, 2015
SIMULATION OF SUPERSONIC FLOW IN A CHANNEL WITH A STEP ON NONSTRUCTURED MESHES WITH THE USE OF THE WENO SCHEME P. V. Bulata and K. N. Volkova,b
UDC 532.529
A numerical simulation of flow in a channel with a right-angled step has been performed on nonstructured meshes having different resolving powers with the use of high-accuracy WENO schemes. The calculated shock-wave structure of this flow was compared with analogous structures described in the literature. Accuracy criteria of numerical calculations, associated with the position of shock-wave structures, are discussed. Recommendations on practical use of high-accuracy difference schemes for calculations on nonstructured meshes are given. Keywords: channel, step, supersonic flow, numerical simulation, WENO scheme. Introduction. The development of new-generation gasdynamic systems calls for comprehensive information on the local gasdynamic processes arising in the case of supersonic flows in the channels of such a system and on the range of its stable work. To exactly calculate supersonic flows and reproduce real physical processes proceeding in them, it is necessary to use highly accurate methods, methods minimizing the numerical viscosity to the level smaller than the physical viscosity, or methods that do not give numerical dissipation [1]. In the case where a low-accuracy scheme is used for calculating a supersonic flow, its pattern is smoothed due to the numerical effects and does not represent the real structure of the flow [2]. Unlike subsonic flows, a characteristic of supersonic flows is the existence of abrupt changes and discontinuities in their parameters. To avoid the nonphysical oscillations of the solution of the problem on a supersonic flow, it is necessary to calculate the convective terms on adaptive templates providing automatic analysis of the smoothness of the numerical solution and to use high-accuracy schemes of the ENO and WENO types. Calculations of a nonstationary supersonic flow in a channel with a forward-facing step at a Mach number M = 3 are used widely for comparison of the accuracies of difference schemes and verification of the efficiency of use of numerical methods for calculations on different meshes. In the problem of supersonic flow in a channel with a step, of interest is the accuracy of calculating the complex gasdynamic structures formed in this flow as a result of its interaction with the step, the nonstationary interaction of the rarefaction and compression waves, and the interaction of the Mach disks arising in the case of nonregular interaction of the shock waves with each other and with the step by a selected numerical method. The problem on supersonic flow in a channel with a step was formulated in [3], was used in [4], and became widely known due to work [5]. In [6, 7], the calculations of a supersonic flow in a channel with a step on meshes having different topologies with the use of different difference schemes were used for estimating the accuracy of the commercial computational Ansys Fluent and CFX packages . In [8–10], the problem on the indicated flow was solved by the Galerkin method with the use of discontinuous basis functions and different methods for calculating flows, and, in [11–19], this problem was solved using the ENO and WENO schemes having different orders of accuracy on structured and nonstructured meshes. The spectral volume method was used in works [20, 21]; in them, the problem on the collapse of an unconditional discontinuity was solved by the Roe method with the use of different TVD limiters. It is shown in [16] that calculations performed with the use of difference schemes of the second, third, and fourth orders of accuracy on nonstructured meshes having different numbers of triangular and rectangular cells give qualitatively different patterns of flow. The calculations of supersonic flow in a channel with a step on meshes with rectangular cells give a flow region with closed streamlines formed between the reflected shock wave and the step. The use of the second-order a
St. Petersburg National Research University of Information Technologies, Mechanics, and Optics, 49 Kronverkskii Ave., St. Petersburg, 197101, Russia; bSchool of Mechanics and Automotive Engineering, Faculty of Science, Engineering and Computing, Kingston University, Friars Ave., Roehampton Vale, London, SW15 3DW, UK; email:
[email protected]. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 88, No. 4, pp. 848–855, July–August, 2015. Original article submitted September 9, 2014. 0062-0125/15/8804-0877 ©2015 Springer Science+Business Media New York
877
Fig. 1. Computational region. scheme leads to the appearance of a flow structure downstream of the Mach disk, which is characteristic of the interaction of a shock wave with a boundary layer. These effects are less pronounced in the calculations on meshes with triangular cells. The WENO scheme of the third order of accuracy gives a better resolution of the elements of a flow as compared to the WENO schemes of the second and fourth orders of accuracy [16]. It should be noted that different smoothness indicators are used in the calculations with the WENO schemes of the third and fourth orders of accuracy. The smoothness indicator proposed in [22] is used in calculations based on the WENO scheme of the third order of accuracy, and the smoothness indicator proposed in [23] is used in the calculations based on the WENO scheme of the fourth order of accuracy. The results of calculations performed with the use of linear schemes and the WENO schemes of the third and fourth orders of accuracy on triangular meshes were compared in [13, 24]. In these works, the real order of accuracy of calculations performed with these schemes on meshes having different resolving powers is discussed. The problem being considered can be also solved with the use of a family of Gamma schemes on a nonstructured adaptive mesh [25]. In the present work, the possibility of calculating a flow with high accuracy on the basis of the ideas of the WENO method is demonstrated by a numerical simulation of a supersonic flow in a channel with a right-angled step. The results of calculations of this flow on nonstructured meshes having different resolving powers are presented and are compared with available literature data. Geometry of the Computational Region. The computational region represents a channel with an abrupt narrowing. The length of the channel is equal to 3, and its height at the input cross section is equal to 1 (the sizes of the channel are given in relative units). A step of height 0.2 is positioned at a distance of 0.6 from the left boundary of the computational region and extends to its right boundary (Fig. 1). Boundary Conditions. At the input boundary of the channel, the Mach number (M = 3) is prescribed, and, at its output boundary, the condition of free flow is set. The flow at the output cross section of the channel is supersonic; therefore, the boundary conditions at the output of the channel do not influence the structure of the flow inside the computational region. The conditions of mirror reflection are set at the lower and upper boundaries of the channel. The calculations were performed for an ideal perfect gas with a specific heat ratio γ = 1.4. For simplification of the formulation of the problem and estimation of the accuracy of its numerical solution, it is assumed that ρ = 1.4 and p = 1 at the left boundary of the channel. In this case, the local velocity of sound c is equal to unity and the Mach number becomes numerically equal to the velocity of a nondisturbed flow (u = M, v = 0, e = 6.286). It is assumed that, at the initial instant of time, the gas in the channel is at rest (ρ = 0.5, u = v = 0, e = 0.125). Computational Procedure. The Euler equations were digitized by the finite-volume method on a nonstructured mesh. Details of the general computational procedure are described in [26]. For digitization of nonviscous flows, the WEMO scheme of the third order of accuracy with splitting of flows by Roe was used. The digitization of the flows with respect to time was performed by the third-order Runge–Kutta scheme. It is assumed that the Courant number is equal to 0.55. A stationary pattern of a flow was obtained for 3·104 time steps. The calculations were performed with a smoothness indicator proposed in [22]. Computational Meshes. The calculations were carried out on nonstructured meshes with different numbers of cells representing nearly equilateral triangles with a side h bunching near the angle point of the step where the cells have a decreased characteristic size (Fig. 2).
878
Fig. 2. Nonstructured meshes with different resolving powers in a channel with a step (the part of the computational region near the step is shown). Meshes 1 (Fig. 2a) and 2 (Fig. 2b) have triangular cells with sizes (h = 1/40) approximately equal everywhere over the computational region, excepting the neighborhood of the singular point, near which the sizes of the cells decrease to h/160 for mesh 1 and to 1/240 for mesh 2. The sizes of the triangular cells of mesh 3 (Fig. 2c) comprise h/80 at a distance from the singular point and about h/230 in the vicinity of this point (at the angular point of the step). The sizes of the triangular cells of mesh 4 (Fig. 2d) are equal to h/160 everywhere over the computational region, excepting the neighborhood of the singular point, and they decrease to h/320 in the vicinity of the angular point of the step. Mesh 1 contains 13,574 control volumes and 6949 nodes, mesh 3 contains 55,314 control volumes and 27,979 nodes, and mesh 4 contains 110,628 control volumes and 55,928 nodes. The numbers of nodes and cells of mesh 2 differ insignificantly from those of mesh 1. Accuracy Criteria. The criteria of accuracy of the numerical solution of the problem on a supersonic flow in a channel with a step is the space position of the triple point (the point of intersection of the upper Mach disk with the bow wave), the length of the upper Mach disk, and the structure of interaction of the shock waves with the upper walls of the step and the channel. The solution obtained in [5] is takes as a standard. The field of the flow density representing a parameter that is most difficult to calculate because of the presence of a small contact discontinuity caused by the interaction of the arched
879
Fig. 3. Lines of the flow density calculated on the meshes having different resolving powers. bow wave with the upper wall of the channel is used for analysis (for more clear visualization, the decimal density logarithm is also used). Results of Calculations. In a supersonic flow over the step in the channel being considered there arises a shock wave that reflects successively from the step and the walls of the channel. After the indicated wave is reflected from the step, a bell-like shock wave is formed. This wave interacts with the upper wall of the channel, with the result that a Mach disk and a contact discontinuity appear, and a Kelvin–Helmholtz instability develops in this discontinuity. Calculation methods were estimated by the quality of breakage of the indicated structures. The results of calculations carried out in [6, 7] with the use of the Ansys Fluent package show that in the case of use of the MUSCL scheme of the third order of accuracy for digitization of convective flows, a substantially nonmonotone numerical solution of the problem on a supersonic flow in a channel with a step and a distorted topology of this flow are obtained. In this case, the length of the Mach disk is overestimated by 24%, the triple point is shifted by 8% to the left in the longitudinal direction, and the points of interaction of oblique shock waves with the upper walls of the step and the channel are shifted by 8 and 6% as compared to the data obtained in [5]. To avoid the problems associated with the existence of singular points, the angular point was changed for an arc of a circle with a radius of 0.01. 880
Fig. 4. Lines of the flow pressure calculated on the meshes having different resolving powers. It should be noted that, because of the numerical errors, the flow calculated has some nonphysical features. Among them is a weak rarefaction wave appearing as a result of the interaction of a fan of rarefaction waves, originating from the angle of the step, with its upper wall and a Kelvin–Helmholtz instability propagating from the triple point along the upper wall of the channel. The main reason for the appearance and development of this instability is the existence of small entropy oscillations generated by the numerical schemes at the triple point (numerical boundary layer). The above-described features of a supersonic flow in a channel with a step are independent of the types of both the computational meshes and the numerical algorithms used [6, 7]. The results of calculations on meshes 1–4 were processed in the form of density and pressure lines of the flow at the instant of time t = 4 corresponding to the stationary flow regime; they are presented in Fig. 3 (the density changes in the range from 0.332 to 6.15) and in Fig. 4 (the pressure changes in the range from 0.5 to 12). About 20 density and pressure lines are presented in these figures. In the stationary flow upstream of the step there arises a reflected shock wave with a curvilinear bell-like front. The solutions obtained on the meshes having different resolving powers have given correct values for the position of the front of the reflected shock wave, the size of the Mach disk, and its position. The size of the Mach disk comprised about 10% of the height of the channel. The front of the shock wave extended for 3–4 cells on the coarse 881
Fig. 5. Lines of the Mach number (a), the density (b), and the temperature (c) of the flow. meshes (meshes 1 and 2) and for 2–3 cells on the fine meshes (meshes 3 and 4). Unlike meshes 1 and 2, the calculations on meshes 3 and 4 allowed us to reproduce the shock waves reflected from the upper surfaces of the step and the channel with a sufficient degree of accuracy (in the case where these shock waves were reproduced on the coarse meshes, they had a low intensity). The results of numerical calculations on more detail meshes demonstrated the tendencies obtained in work [18] on the nonstructural meshes with triangular cells. A characteristic feature of the solutions obtained is the absence of nonmonotonicity in the density and pressure distributions (the solution was slightly nonmonotone only in the case where it was obtained on a coarse mesh for the flow in the region between the step and the front of the reflected shock wave). In this case, the average difference between the numerical and standard solutions [5] did not exceed 3%. A maximum disagreement was obtained on mesh 1, and a minimum one was obtained on mesh 4. In this case, the features of the flow of the type of a boundary layer or a numerical noise at the upper surface of the step, detected in works [18, 27], were absent. The lines of the Mach number, the density, and the temperature of the stationary flow at the instant of time t = 4, obtained on mesh 4, are presented in Fig. 5. The results of numerical simulation are in fairly good agreement with the calculation data of [11] obtained with the use of the ENO scheme of the third order of accuracy, the calculation data of [21] obtained with the use of the ENO scheme of the second order of accuracy, and the results of calculations [16] performed with the use of the ENO and WENO schemes of the third and fourth orders of accuracy on the mesh adapted to the numerical solution. In the indicated works, the computational region of original shape [5] was considered, and special approaches to the description of the field of the flow near the angular point of the step [6, 7] were not used. The results of investigations of the formation of shock-wave structures and their evolution with time show different stages of this process: the propagation of a shock wave moving from the left to the right and its successive reflection from the walls of the channel and the surface of the step at the instant of time t = 0.1, the appearance of a shock wave reflected from the step (t = 0.5), the movement of the shock wave to the right boundary of the computational region and the appearance of a shock wave reflected from the upper wall of the channel (t = 1.0), the further formation of shock waves in the channel (t = 1.5), the appearance of a Mach disk and a contact discontinuity (t = 2.0), and the termination of formation of the reflected shock wave and the contact discontinuity (t < 4.0).
882
At the instant of time t = 1, the shock wave reflected from the upper wall of the channel reaches the surface of the step. The main and reflected shock waves begin to interact at the point x = 1.45, while the reflected shock wave is formed at the instant of time t = 0.65 and reaches the point x = 0.95 at the instant of time t = 1. The angle between the incident shock wave and the wall of the channel increases as long as t = 1.5, and then it exceeds the maximum possible value corresponding to the regular reflection (this angle is equal to 40o at γ = 1.4) and the Mach (nonregular) reflection of the shock wave from the channel wall takes place. The point of interaction of the shock wave reflected from the surface of the step with the wall of the channel has a coordinate x = 1.65 at the instant of time t = 2. The shock wave reflected from the surface of the step reaches the upper wall of the channel at the instant of time t = 2.5. Conclusions. A numerical simulation of a supersonic flow in a channel with a step has been performed with the use of high-accuracy WENO schemes on nonstructured meshes having different resolving powers. The shock wave structure of the flow obtained is consistent with the available numerical solutions. In this case, the front of the shock wave reflected from the step, calculated on a fine mesh, extended for 2–3 its cells. The results of numerical simulation performed using the highaccuracy WENO schemes show that these schemes give no solutions with nonphysical oscillations characteristic of the TVD schemes having a low order of accuracy. The method developed for numerical solution of the Euler equations on nonstructured meshes is fairly common and can be used for numerical simulation of a supersonic flow in a space having a complex geometry. This work was carried out with financial support from the Ministry of Education and Science of the Russian Federation (Agreement No. 14.575.21.0057).
NOTATION c, velocity of sound, m/s; e, specific total energy, J/kg; h, mesh step, m; M, Mach number; p, pressure, Pa; t, time, s; u, v, components of the flow velocity, m/s; x and y, coordinates, m; γ, specific-heat ratio; ρ, density, kg/m3.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
8. 9. 10. 11. 12.
K. N. Volkov, Yu. N. Deryugin, V. N. Emel′yanov, A. S. Kozelkov, A. G. Karpenko, and I. V. Teterina, Increasing the Rate of Gas-Dynamical Calculations on Nonstructured Meshes [in Russian], Fizmatlit, Moscow (2013). A. M. Lipanov and S. A. Karskanov, Use of high-order approximation schemes in the simulation of the processes of retardation of supersonic flows in rectangular channels, Vychisl. Mekh. Splosh. Sred, 6, No. 3, 292–299 (2013). A. F. Emery, An evaluation of several differencing methods for inviscid fluid flow problems, J. Comput. Phys., 2, No. 3, 306–331 (1968). B. Van Leer, Towards the ultimate conservative difference scheme. V. A second order sequel to Godunov's methods, J. Comput. Phys., 32, No. 1, 101–136 (1979). P. R. Woodward and P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54, No. 1, 115–173 (1984). S. A. Isaev and D. A. Lysenko, Testing of the FLUENT package in calculation of supersonic flow in a step channel, J. Eng. Phys. Thermophys., 77, No. 4, 857–860 (2004). S. A. Isaev and D. A. Lysenko, Testing of numerical methods, convective schemes, algorithms for approximation of flows, and grid structures by the example of a supersonic flow in a step-shaped channel with the use of the CFX and FLUENT packages, J. Eng. Phys. Thermophys., 82, No. 2, 321–326 (2009). B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Analysis, 35, No. 6, 2440–2463 (1998). M. P. Galanin, E. V. Grishchenko, E. B. Savenkov, and S. A. Tokareva, Use of the RKDG Method in Numerical Solution of Gas Dynamics Problems, Preprint No. 52 of the M. V. Keldysh Institute of Applied Mathematics, Moscow (2006). M. P. Galanin, E. B. Savenkov, and S. A. Tokareva, Solution of the problems of gas dynamics in the presence of shock waves by the RKDG method, Mat. Modelir., 20, No. 11, 55–66 (2008). R. Abgrall, On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, J. Comput. Phys., 114, No. 1, 45–58 (1994). T. Sonar, On the construction of essentially non-oscillatory finite volume approximations to hyperbolic conservation laws on general triangulations: polynomial recovery, accuracy and stencil selection, Comput. Methods Appl. Mech. Eng., 140, No. 2, 157–181 (1997). 883
13. C. Q. Hu and C.-W. Shu, Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150, No. 1, 97–127 (1999). 14. D. Balsara and C.-W. Shu, Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. Comput. Phys., 160, No. 2, 405–452 (2000). 15. W. R. Wolf and J. L. F. Azevedo, High-order unstructured grid ENO and WENO schemes applied to aerodynamics flows, AIAA Paper, No. 2005-5115 (2005). 16. W. R. Wolf and J. L. F. Azevedo, High-order ENO and WENO schemes for unstructured grids, Int. J. Numer. Methods Fluids, 55, No. 10, 917–943 (2007). 17. W. R. Wolf and J. L. F. Azevedo, Implementation of ENO and WENO schemes for finite volume unstructured grid solutions of compressible aerodynamics flows, Therm. Eng., 6, No. 1, 48–56 (2007). 18. I. Christov and B. Popov, New nonoscillatory central schemes on unstructured triangulations for hyperbolic systems of conservation laws, J. Comput. Phys., 227, No. 11, 5736–5757 (2008). 19. S. Clain, S. Diot, and R. Loubere, A high-order finite volume method for hyperbolic systems: multi-dimensional optimal order detection (MOOD), J. Comput. Phys., 230, No. 10, 4028–4050 (2011). 20. Z. J. Wang and Y. Liu, Spectral (finite) volume method for conservation laws on unstructured grids. III. One-dimensional systems and partition optimization, J. Sci. Comput., 20, No. 1, 137–157 (2004). 21. Z. J. Wang, L. Zhang, and Y. Liu, Spectral (finite) volume method for conservation laws on unstructured grids. IV. Extension to two-dimensional systems, J. Comput. Phys., 194, No. 2, 716–741 (2004). 22. G.-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, No. 1, 77–99 (1996). 23. A. Harten and S. R. Chakravarthy, Multi-dimensional ENO schemes for general geometries, ICASE Report, No. 91–76 (1991). 24. J. Shi, C. Hu, and C.-W. Shu, A technique of treating negative weights in WENO schemes, J. Comput. Phys., 175, No. 1, 108–127 (2002). 25. H. Jasak, H. G. Weller, and A. D. Gosman, High resolution NVD differencing scheme for arbitrarily unstructured meshes, Int. J. Numer. Methods Fluids, 31, No. 2, 431–449 (1999). 26. K. N. Volkov and V. N. Emel′yanov, Computational Technologies in Problems of Fluid Mechanics [in Russian], Fizmatlit, Moscow (2012). 27. C. Berthon, Robustness of MUSCL schemes for 2D unstructured meshes, J. Comput. Phys., 218, No. 2, 495–509 (2006).
884
