A Cartesian based body-fitted adaptive grid method for compressible viscous flows

Transcription

1 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition 5-8 January 2009, Orlando, Florida AIAA A Cartesian based body-fitted adaptive grid method for compressible viscous flows Munikrishna N. and Meng-Sing Liou NASA Glenn Research Center, Cleveland 44135, Ohio, USA munikrishnanagaram@oai.org This work presents an adaptive Cartesian-based hybrid grid method for computing viscous compressible flows past complex bodies. While the near body region is filled with body-fitted grids to resolve the boundary layers, the remaining computational domain is tessellated with recursively generated Cartesian mesh. The problem of handling the interface region between near body mesh and Cartesian mesh is alleviated by allowing the grid blocks to overlap. An unstructured data based cell center finite volume method is employed for updating the solution even in the overlap region. The conservation property of the present overlap mesh scheme is numerically demonstrated even for the flows involving discontinuities. The robustness, solution accuracy and the potential of the present approach in dealing with viscous flow over complex multi-component configurations is demonstrated by solving a number of test flow cases. Nomenclature W F Ω F ˆn ˆt A σ M Re α C L C DP C Dv C D θ s b Vector of conserved variables Flux vector Cell volume Flux normal to the finite volume interface Unit normal to the finite volume interface Unit tangent to the finite volume interface Nozzle cross section area Estimate of mass loss Freestream Mach number Freestream Reynolds number Angle of incidence Lift coefficient Coefficient of pressure drag Coefficient of friction drag Coefficient of drag Flow separation angle Wake bubble length I. Introduction Inspite of the advancements made in the unstructured and Cartesian mesh-based algorithms, grid generation still remains a central problem in simulating viscous flows past complex geometries. The success of the numerical simulations hinges on the quality of the grid that models the geometry and flow, especially in viscous regions. An important issue arising with viscous flow calculations around complex configurations Post-Doctoral Fellow, NASA Glenn Research Center Senior Technologist, Aeropropulsion Division, NASA Glenn Research Center 1 of 16 Copyright 2009 by the American Institute of Aeronautics and American Astronautics, Institute Inc. All of rights Aeronautics reserved. and Astronautics

2 is how fast a suitable grid with reasonable number of computational elements can be generated? While it is a tedious task to discretize the complex flow domains with multi-zone structured meshes, the use of all tetrahedral mesh 1 leads to poor solution accuracy for simulating viscous flows. The level of automation associated with Cartesian mesh-based approaches have made them attractive for such class of flows. Despite the success shown by these methods in terms of inviscid flow computations, 2 their usage in viscous flows is yet to be realized in practice because the small cut-cells present in the viscous region close to the body 3 remains a major barrier for the Cartesian grid methods. Though the Cartesian like grid 4 generation strategy is an important development in the context of all Cartesian grid viscous calculations, it is useful to simulate laminar viscous flows past streamlined bodies. 4 Hybrid mesh-based algorithms have become a more practical option because most suitable methods can be used for tessellating both viscous and inviscid regions and merge them together into one single grid block. Within unstructured hybrid mesh approach, the viscous near-body region is filled with high aspect ratio body-fitted structured mesh involving hexahedral/prismatic elements and potential flow region is filled with tetrahedral elements. 5 8 However, in case of multiply connected domains, special efforts are required to fill the arbitrarily shaped regions with a quality tetrahedral mesh and to merge neatly with the viscous near-body mesh. 7, 8 In this context, hybrid Cartesian approach is attractive because the arbitrarily shaped regions can be filled with Cartesian mesh more efficiently rendering higher levels of automation to the grid generation process. Also, the mesh resolution near the outer boundary of the viscous near-body grid can be matched easily with the adaptive Cartesian grid. The hybrid Cartesian grid generation methods for two dimensional 9, 10 problems have existed, their extension to three dimensions is still problematic in terms of handling the non-conformal block boundaries present in the interface region between structured and Cartesian mesh blocks. In addition, referring to figure 1, tracking the grid data associated with arbitrarily shaped polyhedral volumes and the treatment of the small cut-cells in the interface region can be quite intensive for complex 3D domains. This problem may be circumvented by employing a hybrid like strategy where the viscous near-body mesh can be generated by using a projection and marching method from an undulated Cartesian grid front. 11, 12 However, in order to get good quality solution particularly close to solid boundaries, the use of conventional body-fitted grid generation 13 tools is a more suitable option for obtaining required viscous near-body mesh. Figure 1. Interface region between Cartesian and Structured grid blocks(taken from Delanaye 10 ) Motivated by the above mentioned limitations of the state of the art pertaining to Cartesian-based viscous calculations, we propose a Cartesian-based body-fitted grid strategy allowing the structured body-fitted and Cartesian mesh blocks to overlap in the interface region. We employ an unstructured data based cell center finite volume solution methodology everywhere in the domain including the overlap region, treating the entire domain as a single grid block. This involves the determination of interfacial fluxes at the faces present on the outer boundaries of Cartesian and body-fitted grid blocks. This can be accomplished by employing any of the conventional solution reconstruction procedures 14 associated with unstructured data based finite 2 of 16

3 volume methodology. The required connectivity information in the interface region is established during the pre-processing stage. It should be remarked that the present approach enjoys the advantages of both the Cartesian and overset methodologies in terms of geometric flexibility. Also, the solution is obtained using an update procedure even in the overlap region and do not require special treatment as in case of Cartesian cut-cell method or intensive interpolation involved in classical overset methodology. Apart from this, the use of unstructured data based solution methodology in the entire domain naturally permits local mesh refinement/derefinement. The two important issues associated with any overlap mesh strategy are accuracy and conservation property of the numerical solution. Systematic numerical experiments in one and two dimensions are conducted in order to study the conservation and the solution accuracy of the present Cartesian-based overlap mesh strategy. Also, the robustness and geometric flexibility of the present approach are demonstrated by solving representative test flow cases. The paper is organised as follows. The grid generation and pre-processing is summarized in section 2. The solution methodology is presented in section 3. While the analysis pertaining to the conservation property of the present approach is presented in section 4, the numerical results are presented in section 5. Concluding remarks along with the future directions are presented in section 5. II. Hybrid Cartesian Grid Generation The hybrid Cartesian mesh generation technique consists of the following steps: a. Overlapped structured grid blocks b. Deleting certain region of overlap c. Recursively generated Cartesian grid d. Hybrid Cartesian overlap grid Figure 2. Stages in generating hybrid Cartesian overlap grid 1. High aspect ratio body-fitted structured grid blocks are generated around each component of the geometry using any of the conventional structured mesh generators, 13 allowing them to overlap, as 3 of 16

4 shown in figure 2a. 2. The regions of overlap between different structured grid blocks are identified and certain portion of overlap is deleted using a gap cutting algorithm based on some geometric criteria, 9, 10 as shown in figure 2b. 3. A Cartesian mesh is generated around the outer layer of the structured grid block(hereafter termed as structured grid front) by recursively dividing a master cell enclosing the entire computational domain as shown in figure 2c. In the figure, green line indicates the structured grid front. 4. Final grid is obtained by deleting the Cartesian cells that are completely submerged inside the structured grid front as shown in figure 2d. The grid data corresponding to whole domain is collected in an unstructured format suitable for the cell center finite volume flow solver. As stated earlier, the body-fitted and Cartesian grid blocks communicate through the interfacial flux computation at the faces present on the outer boundaries. For this purpose, the required connectivity information is established as follows: A face falling on the Cartesian grid front and structured grid front is shared by only one cell. For the face present on the Cartesian grid front, a donor cell is identified from the structured grid block in the following way, for defining the other state. Referring to figure 3.a, a set of cells where the unit vector along the line joining face mid-point and cell centroid, Jj makes an angle less than pre-defined value(30 o 50 o ) with the unit normal ˆn to the face J are identified. From this set, a cell centroid j which is closest to the mid-point Cartesian face is chosen. Similarly, for the face falling on the structured grid front, a cell centroid is identified from the Cartesian grid block. Also, the solution reconstruction procedure employed in the present work requires to obtain the solution values at the mesh vertices using the values available at the cell centroids. This necessitates the identification of support cells for the vertices as shown in figure 3.b. For the vertices falling on the outer boundary of a given block, the donor cells associated with the faces passing through the vertex under consideration are also included in the support set. a. Donor Cell identification for Cartesian grid front face b. Support cells for mesh vertex Figure 3. Neighbourhood identification III. Solution Methodology The compressible Navier-Stokes equations in conservation form can be written as, W t +. F = 0, (1) 4 of 16

5 where, W is the state vector and F is the flux vector. Expressing the above equation in integral form over an arbitrary finite volume Ω i, the following space discretized equation can be obtained: dw dt = 1 F J S J. (2) Ω i In the above equation, J refers to the interface between the cell i and its neighbouring cell j, F J is the flux normal to the J th interface and S J is the area of the J th interface. The solution is updated using a matrix-free implicit time integration procedure. 15 Inviscid flux discretization In the finite volume method, for the purpose of solution reconstruction, we need the gradient information at the cell centroids. In the present work, Green-Gauss theorem based linear reconstruction procedure 14 is employed. This choice has been made due to the higher levels of robustness the reconstruction procedure offers in case of meshes involving high aspect ratio and stretched cells typically encountered in resolving turbulent boundary layers. 14 It involves the solution values at the mesh vertices besides the solution values available at the cell centroids. The solution values at the mesh vertices are obtained by employing a linearity-preserving weighted average procedure, pseudo-laplacian. 16 Assuming a linear variation of the solution in the cell, the left and right states are defined at the face mid-point. A suitable upwind scheme can be used for computing inviscid fluxes. Viscous flux discretization The interfacial viscous fluxes require the solution gradients at the face mid-point besides the solution val- J Figure 4. Typical Finite Volume Interface: Viscous Discretization ues. Here we employ a gradient interpolation strategy which is also linked to the solution reconstruction at the cell centroids. The gradients are computed as a vector sum of the normal and tangential components. Consider a finite volume interface J shared by two cells represented by their centroids i and j as shown in figure 4. Let r be the vector connecting the left and right cell centroids of the face J. Also, ˆn and ˆt are the unit vectors in the normal and tangential directions to the face respectively. The gradient(say for any scalar φ) in the normal direction is computed using Strang s strategy 17 as follows: φ (n) = φ j φ i. (3) ˆn. r An interesting feature of this procedure is that the method is found to be positive and satisfies discrete maximum principle when applied to a Laplace equation. 14 The use of a positive procedure renders the code required robustness when simulating turbulent flows. The tangential component is computed as an average of the already available cell wise gradients(from linear reconstruction) 18 i.e., φ (t) = 1 2 [ φi. t + φ j. t ]. (4) 5 of 16

6 The face gradients are given by the relation, φ J = φ (n)ˆn + φ (t)ˆt. (5) The solution values at the face mid-point are simply obtained by averaging the values available at the vertices constituting that face. The faces falling on the outer boundary of the body-fitted and Cartesian grid blocks in the overlap region are also treated similar to the faces present in the interior. The interfacial fluxes are computed using the procedures described above. IV. Conservation Study One of the important issues associated with overlap mesh-based numerical scheme is conservation property. While finite volume scheme on non-overlapping mesh is conservative because of the telescopic collapse of the flux integral on the volume interface, it is not the case with overlap mesh. Conservation property of the hybrid Cartesian overlap mesh is studied here by systematically employing a series of numerical experiments. Two internal flow test cases are considered for the analysis. Conservation implies, mass inflow into the domain is same as the mass outflow at steady state. Any difference in the mass flow indicates the numerical source/sink associated with the numerical scheme. This difference(σ) is employed as an indicator for the loss of conservation of the numerical scheme and monitored with grid refinement. It should be remarked that this difference would be comparable to the residue defined on the mass(at the time of declaring convergence) in case of finite volume scheme on non-overlapping mesh. Starting from a coarse grid, fine grids are obtained by successive isotropic refinement of all the cells present in the computational domain. Numerical results are obtained using the van Leer 19 scheme. Venkatakrishnan s limiter 20 is employed for preserving solution monotonicity. IV.A. Case A: Quasi 1D flow through a nozzle This is an ideal test case for studying the conservation of the numerical scheme. Quasi 1D Euler equations are solved for computing flow through a nozzle with area variation: A = x x 0.5, (6) with stagnation pressure = Pa and stagnation temperature = 300 K. The exit pressure is specified such that a normal shock forms in the divergent portion of the nozzle at x = The domain is filled with two overlapping grids such that the normal shock forms in the overlap region. The base grid(level 0) has 20 cells. Fine meshes are obtained by successive refinement of this base grid. At the inflow boundary, subsonic inlet boundary condition is imposed. At the outflow boundary, exit pressure is specified, density and velocity are obtained from within. The quantity, log(σ) is plotted against log(dx), where dx is a mean grid spacing, in figure 5a. The slope of the best line fit is: σ O( x 2.6 ) as x 0. (7) This result amply demonstrates the conservation property of the overlap mesh-based numerical scheme because the value of σ decays to zero with grid refinement. The L 2 norm of the error in Mach number distribution in the nozzle also decreases with grid refinement as shown in figure 5b. The convergence rate associated with overlap mesh is comparable with the convergence rate corresponding to the non-overlapped mesh with similar resolution. Figure 5c presents the Mach number variation. The solution on the base(level 0) grid has error of O( x) = O(0.1), giving an inaccurate representation of shock location and strength. It can be seen that the numerical solution obtained using overlapped mesh converges towards the exact solution with grid refinement, re-assuring the conservation property of the overlap mesh scheme. IV.B. Supersonic Flow Case B: Flow through a channel This test case involves a supersonic flow(m = 1.4) in a channel with a circular arc bump on the lower wall and this case is used in the work of Ni. 21 The height of the channel is equal to the length of the bump 6 of 16

7 a. Estimate of mass loss(σ) b. l2 error in Mach number c. Mach number variation Figure 5. Quasi 1D Nozzle Flow and the thickness-to-chord ratio of the bump is 4%. Mirror boundary condition is imposed on the lower and upper walls. Supersonic inlet and supersonic outlet boundary conditions are imposed. Figure 6a shows the base grid(level 0) which has 638 number of cells. A structured mesh with three layers is generated around the bump and the rest of the domain is filled with recursively generated Cartesian mesh, allowing them to overlap. The meshes are overlapped such that the region of overlap intersects with the important features of the flow. A zoomed view of the grid in the overlap region is shown in figure 6b. It can be seen that the mesh is quite complex in the region of overlap. This grid is subjected to four levels of refinement and the last level grid has 163,328 cells. For solution comparison, a non-overlap mesh with similar resolution as overlap one is generated as shown in figure 6c. This grid has 717 cells and it is also subjected to four levels of refinement. Mach number contours obtained using different levels of overlap grid are compared with the solutions obtained on different levels of non-overlap grid in figures 6d to 6i. Solutions look identical on the overlap and non-overlap meshes. It can be clearly seen that the flow features on the coarse grid becomes sharper on the finer ones. The Mach number variation along the lower wall for different levels of overlap mesh is compared with the solution obtained using non-overlap meshes in figure 7a. In case of overlap mesh also, the shocks are predicted with right strength and at correct locations. In figure 7b, the value, log(σ)( estimate of mass loss) is plotted against a mean grid spacing, log(1/nc), where nc is the number of cells in the computational domain at a given level. The estimate of mass loss is comparable with an estimate of the discretization error(associated with the interpolation in the overlap region) h 2, where h is the length of the largest edge falling in the overlap region. This indicates that the error associated with conservation is proportional to the discretization error. This result also, clearly demonstrates the conservation property of the present hybrid Cartesian overlap mesh scheme because the 7 of 16

8 value of σ decreases at least as fast as grid spacing does, even when discontinuities are present. a. Level 0 overlap grid b. zoomed view c. Level 0 non-overlap grid near leading edge of circular arc d. Level 0 overlap grid: 638 cells e. Level 0 non-overlap grid: 717 cells f. Level 2 overlap grid: 10,208 cells g. Level 2 non-overlap grid: 11,472 cells h. Level 4 overlap grid: 163,328 cells i. Level 4 non-overlap grid: 183,552 cells Mach number contours [ min=1.0, =0.03,max=1.7] Figure 6. Supersonic Flow over Circular Arc Bump: M = 1.4 Subsonic Flow In order to verify the solution accuracy and conservation property of the hybrid Cartesian overlap mesh strategy for a smooth flow, a subsonic flow(m =0.6) over circular arc bump is simulated. The grids used in the supersonic flow test case are employed. The Mach number variation along the lower wall obtained using level 3 overlap mesh is compared with the solution obtained using non-overlapping mesh in figure 8a and no discernible difference is found. The mass loss decreases with grid refinement as shown in figure 8b. Also, the mass loss is found to be smaller compared to the values obtained for supersonic case. Mach number contours obtained using both overlap and non-overlap mesh are displayed in figures 8c and 8d respectively. V. Results and Discussions The efficacy of the present approach is demonstrated by solving standard inviscid and viscous flow problems. The test cases are chosen with the aim of evaluating the robustness, accuracy of the solution methodology and efficacy of the present hybrid Cartesian approach in dealing with complex geometries. The 8 of 16

9 a. Mach number variation along the lower wall b. Estimate of mass loss(σ) Figure 7. Supersonic Flow over Circular Arc Bump: M = 1.4 a. Mach number variation along the lower wall b. Estimate of mass loss(σ) c. Level 3 overlap grid: 40,832 cells d. Level 3 non-overlap grid: 45,888 cells Mach number contours [ min=0.46, =0.011,max=0.7] Figure 8. Subsonic Flow over Circular Arc Bump: M = of 16

10 details of the test cases and grids used are shown in table 1. Inviscid interfacial fluxes are computed using van Leer 19 scheme in case of inviscid flows. Roe 22 scheme is used for viscous flow test cases. Spalart-Almaras 23 one equation turbulence model is implemented for computing turbulent flows. On the wall boundary, no slip, adiabatic boundary conditions are imposed and pressure is extrapolated from the interior. A Riemann invariant based characteristic boundary condition is employed at the farfield. Case Geometry M Re α Number of number of Reference cells wall points data 1 NACA o AGARD 24 2 Circular Tritton 25 Cylinder Ratnesh 26 3 NASA x o Omar 27 5 element Airfoil 4 Circular x (Level 0) 40 Bashkin 28 Cylinder 35301(Level 2) 240 Table 1. Test Cases: Details of Freestream Conditions and Point Distribution V.A. Case 1: Inviscid Transonic flow past NACA 0012 Airfoil This is a standard AGARD 24 test case where a transonic flow past NACA 0012 airfoil is simulated. The freestream conditions are M = 0.85, α = 1 o. A hybrid Cartesian overlap mesh is generated and it consists of 8,982 cells. A zoomed view of the grid is shown in figure 9a. The wall pressure distribution is compared with the pressure distribution obtained using a non-overlapping structured mesh(200x60 cells) in figure 9b. The Mach number contours for both meshes are displayed in figure 9c. The results bring out the fact that the jumps across the top and bottom shocks, their locations are all captured accurately using the hybrid Cartesian mesh. The values of the computed aerodynamic coefficients are in good agreement with the AGARD 24 values as presented in table 2. Solution C L C D Hybrid Cartesian mesh Structured mesh, 200x60 cells AGARD Table 2. case 1, Lift and Drag coefficients: Transonic inviscid flow over NACA 0012 V.B. Case 2: Laminar flow past Circular Cylinder This test case involves simulating laminar flow past Circular cylinder at Re = 40 based on the diameter of the cylinder. The importance of the test case stems from the presence of wake bubble behind the cylinder. The zoomed view of the hybrid Cartesian grid used for simulation is shown in figure 10a. In order to evaluate the solution methodology used in the overlap region for predicting strong viscous features, the grid is generated such that the overlap region intersects with the wake bubble. The computed pressure and friction coefficient distribution is in reasonably good agreement with the experimental values 25 as shown in figures 10.b and 10.c respectively. The streamline plot shown in figure 10.d indicate the presence of two counter rotating vertices behind the cylinder. The predicted wake bubble length b(measured from the center of the cylinder), angle of separation θ s and drag coefficient are in good comparison with the values reported in literature 25, 26 and the comparison is given in table 3. These results indicate the ability of the present hybrid Cartesian approach in computing flows with strong viscous features. 10 of 16

11 a. Hybrid Cartesian Overlap Mesh b. Pressure distribution c. Mach number contours for overlap mesh and 200x60 Structured mesh[min=0.05, =0.07,max=1.44] Figure 9. Inviscid Flow over NACA 0012 Airfoil: M = 0.85, α = 1 o V.C. Case 3: Subsonic turbulent flow past multi-element airfoil This test case involves simulating subsonic turbulent flow past a NASA supercritical airfoil along with high lift systems. The freestream Mach number is and the Reynolds number is 2.83x10 6 with an angle of attack of 8.21 o. The high lift system contains leading edge slat and three deflected flaps, total of five elements. This test case is chosen to demonstrate the capability of the hybrid Cartesian approach for simulating flow past complex bodies. Body-fitted structured grid blocks are generated around each component allowing them to overlap. The spacing of the first grid line off the wall is 5x10 6 times the chord of the airfoil resulting in average y + value of 0.3. The structured grid block consists of 45 layers with a stretching factor of The zoomed view of the hybrid Cartesian mesh close to the body is shown in figure 11.a. Also, another zoomed view of the grid in the slotted region around first flap is shown in figure 11.b. It can be seen that the Cartesian mesh is well resolved around the outer boundary of the structured grid blocks. For the purpose of comparison, solutions are also obtained employing a multi-block structured grid with 2,86,523 number of cells. The Mach number field computed using hybrid Cartesian grid and structured grids are identical, Table 3. Solution wake bubble length, b angle of separation, θ s ( o ) C D Hrbrid Cartesian mesh References 25, Case 2: Laminar flow over circular cylinder: Wake bubble length, angle of separation and drag coefficients 11 of 16

12 a. Hybrid Cartesian Overlap Mesh b. Pressure distribution c. Skin friction Coefficient d. Streamlines Figure 10. Case 2, Laminar Flow over Circular Cylinder: M = 0.1, Re = 40 as can be seen from figures 11.c and 11.d respectively. The streamlines shown in figures 12.a and 12.b indicate the presence of separation bubble in the lower surface of the airfoil in the slotted region between main airfoil and first flap. Figure 12.c shows the skin friction coefficient distribution on the main airfoil close to the flow separation region. From the zero crossover of the friction coefficient values, it can be seen that the flow separation and reattachment points are in accurate comparison with values corresponding to structured mesh. It should be emphasized that even when the meshes are allowed to overlap in the region where viscous features are dominant, the present approach can compute the flow features quite accurately. The wall pressure distribution is in good comparison with the profile obtained using multi-block structured grid and experiments 27 in figure 12.d. Also, the aerodynamic coefficients are in good agreement with the values obtained using structured grid and experiments 27 as shown in table 4. Table 4. Solution C L C Dp C Dv C D Hybrid Cartesian mesh Multi-Block Structured mesh Omar Case 3, Lift and Drag coefficients: Subsonic turbulent flow over multi-element airfoil 12 of 16

13 a. Hybrid Cartesian Overlap Mesh: Zoomed view b. Closed view in the slotted region c. Mach field: Hybrid Cartesian grid d. Mach field: Multi-block structured grid Figure 11. Case 3, Subsonic turbulent flow over multi-element airfoil: M = 0.201, Re = 2.83x10 6, α = 8.21 o V.D. Case 4: Supersonic turbulent flow past cylinder In this case, a supersonic turbulent flow past circular cylinder at freestream Mach number 1.7 and Reynolds number 2x10 5 has been considered as a formidable test case for the proposed overlap mesh strategy. A rectangular computational domain is considered with dimensions [ 8D, 9D]x[0, 8.5D], D being the diameter of the cylinder. A base grid(level 0) with 4137 number of cells is employed as shown in figure 13.a. Again, to test the accuracy of the solution procedure, the body-fitted and Cartesian grid blocks are overlapped in the region of strong flow features. This grid is further subjected to two levels of solution based refinement and the level 2 grid is shown in figure 13.c. The cells are identified for refinement by employing curl and divergence of velocity field as the sensors. Mach number field for level 0 and level 2 grids are shown in figures 13.d and 13.e. For the considered freestream M, a bow shock is formed upstream of the cylinder; the subsonic flow at the front stagnation region accelerates to supersonic flow region and envelops a subsonic recirculation region behind the cylinder. Mach number field indicates that these flow features are captured sharply on the adapted grid. The computed position of the separation point and drag coefficient are compared with experiment values 28 in table 5. In the computations, a delay in the flow separation is observed. This aspect is verified by employing a structured grid(21,600 cells) and the delay in the flow separation is noticed. The surface pressure distributions for hybrid Cartesian and structured meshes are compared with experimental values 28 in figure 14.a. While the two solutions overlap, as expected, the agreement with experiments is not satisfactory in the recirculation region. The friction coefficient distribution for hybrid Cartesian mesh is identical with the results obtained using structured grid as plotted in figure 14.b. The discrepancy in the flow separation needs to be verified with more sophisticated turbulence model because this test case is 13 of 16

14 c. Streamlines: Hybrid Cartesian grid d. Streamlines: Multi-block structured grid c. Friction coefficient distribution d. Pressure distribution Figure 12. Case 3, Subsonic turbulent flow over multi-element airfoil: M = 0.201, Re = 2.83x10 6, α = 8.21 o computed in the literature 28, 29 using k ω turbulence model. Table 5. Solution angle of separation, θ s ( o ) C D Hybrid Cartesian mesh, Level Structured mesh Reference Case 4: Supersonic turbulent flow over cylinder: angle of separation and drag coefficients VI. Conclusions In the present work, we have proposed a Cartesian-based body-fitted adaptive method with an aim at computing viscous flows past complex industrial configurations. The viscous near-body region is filled with a high aspect ratio body-fitted grid and the inviscid flow region is filled with an adaptive Cartesian mesh, allowing them to overlap in the interface region. We have employed an unstructured data based finite volume methodology for the solution update even in the overlap region. The present approach may be considered as a blending of both Cartesian and overset methodologies. While harnessing the efficiency and higher levels of automation rendered by adaptive Cartesian meshes, the procedure also enjoys the geometric flexibility associated with overset methodologies. On one hand, the use of overlap strategy completely alleviates the problems in terms of handling interface region between viscous near body mesh and Cartesian grid blocks. On the other hand, the use of unstructured data based finite volume update procedure in the overlap region does not involve intensive interpolation method and makes local mesh refinement straightforward to carry 14 of 16

15 a. Level 0 b. Level 0: Zoomed View c. Level 2 Hybrid Cartesian mesh d. Level 0 e. Level 2 Mach field: Hybrid Cartesian mesh Figure 13. Case 4, Supersonic turbulent flow over cylinder: M = 1.7, Re = 2.0x10 5 out. The conservation property of the overlap mesh scheme is studied by conducting numerical experiments, which have confirmed grid convergence with a rate proportional to the discretization error and improvement in the conservation with mesh refinement. The accuracy and robustness of the present approach is tested by solving a number of representative test cases. The turbulent flow around a five-element airfoil is simulated to demonstrate the potential of the hybrid Cartesian approach to complex multi-component geometries. Logically next step is to extend the procedure for computing flow past 3D real life configurations. Acknowledgements The first author acknowledges the NASA Postdoctoral Program administered by Oak Ridge Associated Universities, Tennessee for funding the work presented in this paper. We thank Mr. Ravindra, Simulation and Innovation Engineering Solutions, Bangalore, India for providing us the multi-block structured grid for five element airfoil configuration. References 1 Marcum D., Generation of Unstructured Grids for Viscous Flow Applications, AIAA Paper , Aftosmis M.J., Berger M.J. and Melton J.E., Robust and Efficient Cartesian mesh generation for component-based geometry, AIAA Journal, Vol. 36, No. 6, pp , June Coirier W.J., An Adaptively-refined, Cartesian cell based scheme for the Euler and Navier-Stokes equations, Ph.D. Dissertation, The University of Michigan, Mondal P., N. Munikrishna and N. Balakrishnan, Cartesian like grids using a novel grid stitching algorithm for viscous flow computations, AIAA Journal of Aircraft, Vol. 44, No. 5, p.1598, Kallinderis Yannis, Khawaja Aly and McMorris Harlan, Hybrid prismatic/tetrahedral grid generation for viscous flows around complex geometries, AIAA Journal, Vol. 34, No. 2, pp , February of 16

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