SAROD 2009 142 Symposium on Applied Aerodynamics and Design of Aerospace Vehicles (SAROD 2009) December 10-12, 2009, Bengaluru, India Store Separation Simulation using Oct-tree Grid Based Solver Saurabh Pandey and Bharat R. Agrawal Zeus Numerix Private Limited E-mail: saurabh.pandey@zeusnumerix.com, bharat.agrawal@zeusnumerix.com ABSTRACT Simulating moving solid body in a static Cartesian Oct-tree grid is the topic of discussion. Previously many implementations of oct-tree based adaptive grid to static steady state problems have been demonstrated. This work enhances the scope of oct-tree grid implementation to unsteady problems. Initially a compressible Euler inviscid solver is validated over some standard test cases. Grid is then modified to take into account moving geometry with impermeable boundaries and maintaining the conservation because of the movement. Dynamics of solid body is simulated wherein the trajectory of the body is predicted using aerodynamic forces estimated from the CFD simulation. Solver s results are discussed at the end for the well known store separation problem. Keywords: Cartesian Grid, Compressible Solver, Oct Tree, Store Separation, Unsteady NOMENCLATURE CV = Control Volume CS = Boundary of the control volume U = Vector of conserved variables F = Vector of flux of conserved variables ρ = Density within a control volume u = x component of velocity of fluid in control volume v = y component of velocity of fluid in control volume w = z component of velocity of fluid in control volume e o = Total energy of the fluid in control volume v n = Velocity normal to the boundary of a control volume
Store Separation Simulation using Oct-tree Grid Based Solver 445 p = Pressure n x,n y,n z = x, y and z components of outward normal unit vector to the boundary h o = Total enthalpy e = Static internal energy h = Static enthalpy a = Speed of sound γ = Ratio of specific heat capacities C p = Specific heat capacity of a gas at constant pressure C v = Specific heat capacity of a gas at constant volume 1. INTRODUCTION CFD has started playing a crucial role in arriving at designs with required expectations. Its extensive use has still not seen the horizon because of the time constraints of advanced CFD analysis. Over and above if the analysis has to be done on large number of possible design configurations the problem of time constraints of simulation makes it impossible in the existing scenario. If the simulation time could be reduced to make such numerous analyses possible then non-optimum configurations could be easily discarded in an early stage. This would save a substantial amount of human and machine time as other analysis such as structural, etc. would be done on lesser number of possible configurations. The most time consuming step in a CFD analysis of a complex geometry is the grid generation process as the solver generally are already utilizing power of parallel computing. Also, this time required increases with the complexity of geometry and after a certain level of complexity grid generation becomes almost impossible. This creates a need for an automatic grid generation tool which can be readily utilized in creating grids over various configurations under consideration in the design process. Furthermore, this grid generation should be such that the time taken in creating grid is independent from geometric complexity. Current methodologies of grid generation are largely based on boundary fitted methods. These methods are highly dependent on geometry surface and hence limited by the complexity of geometry. In order to remove this dependence, the other method which is boundary immersed method is prescribed. These methods start with a domain which is filling the space where the analysis needs to be done and progressively creates discretized cells towards the geometry surface. This creates the flexible framework where a small change in geometry configuration does not require restart of whole grid generation process [4]. Oct-tree based grid generation is a popular boundary immersed method [1-4]. The hallmark of oct-tree grids is that they are recursive in nature. Hence cells are created by division of bigger cell only in a region where smaller cells are required to capture the gradients. Other significant features of oct-tree grids includes simplifying the advanced grid generation algorithms like adaptive mesh refinement (AMR), domain partitioning, changing cells near the geometry on the fly, etc [6].
446 Saurabh Pandey and Bharat R. Agrawal Having created the grid, a new solver is required to extract the maximum advantages from this grid. Finally, to completely solve the problem of store separation, a coupled system is needed which must comprise a highly dynamic grid generation module, a compressible solver for CFD and a 6DOF model for dynamics [8]. 2. GRID GENERATOR A grid generator module previously developed and tested with Euler incompressible solver is used in the current work [11]. Grid generation using oct-tree method over a complex geometry is constrained with three requirements from the solver. Firstly, the ratio of volume between two neighbouring cells should be at most eight or at least one-eight. This constraint is implemented to ease the gradient calculations. The second important constraint in the oct-tree grid is that the uncut volume of all neighbouring cells around a cut-cell should be same as uncut volume of the cut-cell itself. This is an essential requirement for quick merging of small cut-cells. Finally, volume and shape of cut-cell needs to be extracted with maximum possible accuracy so that the volume conservation is held near the geometry and the computations done at the boundary are accurate [5]. 2.1 Merging of cut-cells Oct-tree grid is infected with very well known Small Cell problem wherein as the geometry is inserted inside the grid it can intersect some cells in such a way that a very small cell is left in the computational domain. Such cells being very small in volume adversely affect the time stepping of the solver hence greatly increasing the simulation time [7]. Merging of such small cells is one of the well known remedies. It is done by choosing cells around the selected small cell and merging with it. The basis of selection is the area of the face between the cell selected for merging and the small cell. Neighbouring cell sharing the largest area with small cell is selected and merged. This process is continued as long as the merged cell s volume is above a threshold limit. 2.2 Maintenance of volume conservation Another important concern for simulating a moving solid boundary problem in a static Cartesian mesh is the conservation of volume. In order to get a judgment of change in volume of cell on its division volume of cell pre as well as post division is calculated and an error is evaluated. It is in general found that these operations don t produce significant change in the volume. Therefore cell division operations indeed conserve the volume. 3. COMPRESSIBLE SOLVER Finite Volume Method (FVM) based compressible solver is developed on these grids. The governing equation used is Euler inviscid which for a cell whose faces are not necessarily aligned along the x, y or z axis can be written as follows
Store Separation Simulation using Oct-tree Grid Based Solver 447 where,, and The velocity at the cell boundary is defined as the dot product of local velocity vector and unit outward normal to the boundary The total energy and enthalpy per unit mass in the above equations is composed of static and dynamic parts Static energy, enthalpy and pressure can be expressed as a function of local speed of sound, temperature of the fluid and the ratio of specific heats. where,,,, Flux schemes due to van Leer [9] and HLLC [10] are written and validated. The solver is made capable to account for AMR by maintaining conservation while such adaptive refinements are performed on the grid. Grid refinement is based on the difference in one of the primitive variables in neighbouring cells. If this difference in primitive is above a preset threshold value refinement is performed while maintain conservation. De-refinement on the other side is done by calculating the variance of a primitive variable in all child cells. If the variance is not large, again depending on a preset limit, then all the child cells are merged together to results in its parent cell. Needless to say that all the conserved distributed among children are summed up and given to the parent on derefinement. 4. DYNAMICS IMPLEMENTATION Final implementation in the setup to simulate store separation specifically or any unsteady flow phenomena involving motion of geometry with impermeable boundary is to incorporate dynamics. It involves calculation of aerodynamic forces on the solid body in consideration using CFD solver and thereafter moving the geometry with respect to the grid under the action of the aerodynamic and/or other body forces.
448 Saurabh Pandey and Bharat R. Agrawal Dynamics cycle is discussed here on store separation problem for simplicity and coherence but it is in general applicable for any relatively moving solid body simulation problem. The whole process is started by first simulating the aircraft with stores in place to its steady state. With the steady state results in hand the process of separation of store is started by detaching store geometry from the grid. This is achieved by removing cut information of store geometry from the grid. Steady state result obtained initially is used to calculate the aerodynamic forces on the store which further is used to move the store in 3D space. Grid is regenerated at this new moved location of the store and solver is now executed over this new configuration for few iterations. This in turn gives new forces on the store and it is again transported to a new location based on the dynamics. This cycle is repeated for a specified amount of time. Thus solving store separation problem. The main restrictive criterion here is the time for which the store is displaced is such that no part of the store geometry crosses more than one face in the computational domain. This is implemented to make the process of maintenance of conservation simple. It can also be viewed as a CFL type criterion for simulating dynamics. 4.1 Maintenance of mass, momentum and energy conservation Solid body movement relative to the grid brings new cells in the computational domain at the same time erases few cells from the fluid domain. Also adaptive refinement of cells as well as a cell s de-refinement creates and deletes cells from the domain. Mass, momentum and energy conservation has to be maintained in cells like these to correctly capture the physics of the problem. Cells which go inside the solid body and hence out of the fluid domain have to distribute the conserved quantities in the neighbouring cells which will remain in the domain even after the motion of the solid body. Hence conserved quantities from such cells are distributed among neighbouring cells in the ratio of interface area. Similarly for newly created cells, in order to maintain global conservation, are required to have ideally no conserved quantities. But zero initialization of conserved quantities will cause singularity in the solver as a result of which such newly born cells are initialized with infinitesimally small amount values. Refinement and de-refinement operations on cells are made conservative by interpolating the conserved data from old cell(s) to the new ones and then converting them to their primitive form. 5. DISCUSSION OF TEST RESULTS In this section 4 test cases are discussed. Starting with the validation of supersonic flow over a wedge and a forward step the discussion would move to the results obtained over a realistic geometry of an aircraft and the discussion ends with the dynamic simulation of store separating from the same aircraft.
5.1 Supersonic flow over a wedge of 10 angle Store Separation Simulation using Oct-tree Grid Based Solver 449 One of the basic validation cases for any compressible solver is how close the angle of a capture shock wave is. For this purpose, a flow of Mach number 2 is considered over a wedge of 10 angle with the flow direction. This problem is solved using flux splitting scheme due to van Leer [9]. The results are shown in Figures 1 and 2. Figure 1 shows the grid for solution. This case was initialized with a uniform Cartesian base grid and the clustering along the shocks appears completely due to the adaptive framework of the solver. Figure 2 shows the contour plot of the Mach number in the flow field. Comparison of obtained data with theoretical is as follows: Theoretical solution: shock angle = 39.31, pressure ratio across the shock wave = 1.7067. Computed solution: shock angle = ~40, pressure ratio across the shock wave = 1.7063. Figure1: Solution grid for wedge test case Figure2: Contour plot of Mach number for wedge test case 5.2 Supersonic flow over forward facing step In the next validation case, a flow of Mach number 3 inside a 2D duct with a forward facing step was considered. This test case was solved using HLLC scheme. Figure 3 shows the pressure contour plot from literature [9] and Figure 4 shows the pressure contour plot obtained using the present solver. In the obtained solution the maximum pressure is reached inside the Mach disc at the top boundary and the ratio of this pressure with upstream pressure is obtained as 12.23 which match closely with the literature results.
450 Saurabh Pandey and Bharat R. Agrawal Figure3: Contour plot of pressure from literature for forward step problem Figure4: Contour plot of pressure for forward step problem 5.3 Supersonic flow over an aircraft To demonstrate the speed of the solver, supersonic flow at Mach number 2.0 is considered over aircraft geometry at angle of attack of 5. Figure 5 and 6 shows the pressure profile over the upper and lower surfaces of the wing respectively. It can be noted that red and blue represents highest and lowest values respectively. Thus, the differences in colors on these surfaces suggest a positive lift over this geometry. This is a converged solution and was obtained in less than 15 minutes on a standard modern day desktop machine. Figure5: Pressure over the upper surface of the aircraft
Store Separation Simulation using Oct-tree Grid Based Solver 451 Figure6: Pressure over the lower surface of the aircraft 5.4 Store separation Aircraft and generic store geometry was simulated at supersonic flow of Mach number 2.0. Figure 7 shows the pressure variation over both the geometries after the steady state has been reached by the solver. In this figure a blue patch is clearly seen on the lower surface of the aircraft with. It can be noted that most of the surface of aircraft is blue which is due to the fact that this region lies outside the actual computational domain and does not have any pressure values associated with it. Figure7: Pressure contours after reaching steady state Figure 8, 9 and 10 shows the pressure variations and the location of the store at 10 iterations of the dynamics solver. It can be seen that from Figure 7 to figure 8, the blue patch which is a low pressure area grows in size and then in subsequent figures it moves towards the trailing edge and fades.
452 Saurabh Pandey and Bharat R. Agrawal Figure8: Pressure contours after running 10 iterations of dynamics solver Figure9: Pressure contours after running 20 iterations of dynamics solver Figure10: Pressure contours after running 30 iterations of dynamics solver 6. CONCLUSION AND FUTURE SCOPE An oct-tree based adaptive compressible solver is validated against standard test cases of flow over a wedge and forward facing step problems. A technique for simulating moving solid body in a static Cartesian grid is demonstrated. The whole unsteady simulation is
Store Separation Simulation using Oct-tree Grid Based Solver 453 broken down into four main steps starting with the CFD simulation of the body and calculating the aerodynamic forces. Next is detaching the moving solid body from the grid. Third step involves movement of the body by a small amount based on the forces calculated in first step. Finally the body is re-immersed in the grid after the displacement. Most obvious step ahead from the given point is implementing Arbitrary Lagrangian Eulerian (ALE) formulation to account for the moving boundary of a solid body with respect to the computational domain. Further the flux calculation scheme has to be modified accordingly to account for moving boundaries. ACKNOWLEDGMENT We would like to thank Prof. G. R. Shevare of IIT Bombay for his valuable guidance and Zeus Numerix for providing support for the development. REFERENCES [1] Aftosmis, M.J., Melton, J.E., and Berger, M.J., Adaptive Cartesian Mesh Generation. Chapter 22 in Handbook of Grid Generation, Thompson, J, Weatherhill, N., and Soni, B. eds. CRC Press 1998. [2] Berger, M.J., and Aftosmis, M.J., Aspects (and aspect ratios) of Cartesian mesh methods. Proceedings of the 16 th International Conference on Numerical Methods in Fluid Mechanics, Arcachon, France, July 1998. [3] Aftosmis, M.J., Solution adaptive Cartesian grid methods for aerodynamic flows with complex geometries, von Karman Institute for Fluid Dynamics, Lecture Series 199702, Rhode- Saint-Genèse, Belgium, Mar. 3-7, 1997. [4] Aftosmis, M.J., Berger, M.J., Melton, J.E., Robust and efficient Cartesian mesh generation for component based geometry, AIAA Paper 97-0196, Jan. 1997. [5] Berger, M.J., and Melton, J.E., An Accuracy Test of a Cartesian Grid Method for Steady flow in Complex Geometries, Proc. 8thrIntl. Conf. Hyp. Problems, Uppsala, Stonybrook, NY, Jun., 1995. also RIACS Report 95-02. [6] De Zeeuw, D., and Powell, K., An Adaptively Refined Cartesian Mesh Solver for the Euler Equations, AIAA Paper 91-1542, 1991. [7] Popinet, S., Gerris: A tree based adaptive solver for the incompressible euler equations in complex geometries. J. Comp. Phys. 190, 572 600, 2003. [8] Murman, S.M., Aftosmis, M.J., and Berger, M.J., Implicit Approaches for Moving Boundaries in a 3-D Cartesian Method, AIAA-2003-1119, 2003. [9] van Leer, B., Flux-vector splitting for the Euler equations, Lecture Notes in Physics.170. Springer, Berlin. pp. 507-512, 1982. [10] Toro, E.F., Spruce, M., and Speares, W., Restoration of the contact surface in the HLL- Riemann solver, Shock Waves, Volume 4, pp. 25-34, 1994. [11] Agrawal, B.R., and Pandey, S., Revisiting Projection Methods over Automatic Oct-tree Meshes, IISc Centenary International Conference and Exhibition on Aerospace Engineering, pp 267-274, 2009.