Unsteady Problems & Separation Studies @ Zeus Numerix
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The document discusses unsteady flow problems and separation studies in computational fluid dynamics (CFD), including the definitions, analysis methods, and specific challenges associated with oct-tree based solvers. It emphasizes the use of Arbitrary Lagrangian-Eulerian (ALE) formulations for solving complex motion scenarios in turbo-machinery and flying objects. Additionally, it covers mesh generation techniques and the importance of adaptive mesh refinement for accurate simulations.
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©ZeusNumerixPvtLtd:ConfidentialDocument
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Formulation
M dV/dt + V dM/dt = Fexternal (In inertial frame of
reference)
M dV/dt + M [Ω]BE [u v w] =
= Fa,p + [T]BLMg - 2[Ω]EI [u v w] - [Ω]EI[Ω]EISBI (in body frame of
reference)
where
Unsteady problems & Separation Studies in CFD15-Jul-2019](https://image.slidesharecdn.com/unsteadyflowscfdsimulation-201203131839/85/Unsteady-Problems-Separation-Studies-Zeus-Numerix-28-320.jpg)
![30
©ZeusNumerixPvtLtd:ConfidentialDocument
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Formulation
ω dI/dt + I dω/dt = Ma,p (Inertial frame)
I dω/dt + I [Ω] [p q r] = MBa,p (body frame)
[dp/dt dq/dt dr/dt] = [I]-1 ( - [I] [Ω] [p q r] + Ma,p )
where
Unsteady problems & Separation Studies in CFD15-Jul-2019](https://image.slidesharecdn.com/unsteadyflowscfdsimulation-201203131839/85/Unsteady-Problems-Separation-Studies-Zeus-Numerix-30-320.jpg)
Unsteady Problems & Separation Studies @ Zeus Numerix
- 1. 1Built-to-Spec | Product Approval | Engineering Services | Software Development Unsteady problems & Separation Studies in CFD Compute heavy simulations, will require HPC
- 2. 2 ©ZeusNumerixPvtLtd:ConfidentialDocument Contents Unsteady problems & Separation Studies in CFD ▪ Unsteady vs. unsteady flows ▪ Methods of analysis of unsteady flows ▪ Oct-tree based solvers for solution unsteady problems ▪ Some insight in to difficulty with Oct-tree based solvers ▪ Some examples ▪ 6 DoF equations 2 15-Jul-2019
- 3. 3 ©ZeusNumerixPvtLtd:ConfidentialDocument 3 What is Unsteady Flow? Flow field at given point is changing with time Unsteady problems & Separation Studies in CFD15-Jul-2019
- 4. 4 ©ZeusNumerixPvtLtd:ConfidentialDocument Steady vs. Unsteady Problems ▪ If temporal derivatives (i.e. /t) at a fixed point is nonzero the flow is unsteady ▪ In some cases unsteady flow can be converted to steady flow by changing the frame of reference. 4 V = 0V = 10 V = 10V = 0 flow changes with time flow does not change Unsteady problems & Separation Studies in CFD15-Jul-2019
- 5. 5 ©ZeusNumerixPvtLtd:ConfidentialDocument Steady vs. Unsteady Problems ▪ In rotating flows the field can be made stationary in the rotating frame ▪ Rotating frame of reference is not inertial frame. But valid equations can be written 5 V = 100 V = 100 – r flow does not change Unsteady problems & Separation Studies in CFD15-Jul-2019
- 6. 6 ©ZeusNumerixPvtLtd:ConfidentialDocument Unsteady Problem ▪ If the motion of object is known, appropriate eqns of motions and boundary conditions can be arrived at; e.g. turbo-machinery flows ▪ If motion of object mostly depends on flow the fluid domain or the boundary conditions or both change with time ▪ Arbitrary Lagrangean-Eulerian (ALE) description of flow 6 V = 100 m/s Unsteady problems & Separation Studies in CFD15-Jul-2019
- 7. 7 ©ZeusNumerixPvtLtd:ConfidentialDocument 7 Arbitrary Lagrangean-Eulerian (ALE) ▪ Elements are stationary in space and flow flows through the surface of the elements ▪ Find fluid dynamic properties in cells ▪ Elements are made of fluid. The elements change their position, shape, size, currently un- knows are required to be found out ▪ The elements are neither stationary nor moving with fluid, but their motion is known ▪ Find out the fluid dynamic properties of the moving elements Unsteady problems & Separation Studies in CFD15-Jul-2019
- 8. 8 ©ZeusNumerixPvtLtd:ConfidentialDocument Equations in ALE ▪ Inviscid compressible flow (Euler Equations) Vs Vs is velocity of cell face Unsteady problems & Separation Studies in CFD 8 15-Jul-2019
- 9. 9 ©ZeusNumerixPvtLtd:ConfidentialDocument Statement of the Problem - I ▪ The motion of a rigid body / bodies is known ▪ For every instant of time Create a new mesh Execute CFD solver written ALE formulation Inspect the flow field ▪ Go to the next instant and repeat 2nd step ▪ Applications mixing flows, wind mills, Turbo-machinery flow 9 Unsteady problems & Separation Studies in CFD15-Jul-2019
- 10. 10 ©ZeusNumerixPvtLtd:ConfidentialDocument Statement of the Problem - II ▪ The forces other than fluid dynamic forces acting on the body / bodies are known ▪ For every instant of time Create a new mesh Execute CFD solver written ALE formulation Find out the position of bodies as all the forces including fluid dynamic forces are known Inspect the flow field ▪ Go to the next instant and repeat 2nd step ▪ Applications Motion of flying objects Store separation 10 Unsteady problems & Separation Studies in CFD15-Jul-2019
- 11. 11 ©ZeusNumerixPvtLtd:ConfidentialDocument Basic Requirement – Create a New Mesh I ▪ Un-structured grid approach Grid shearing in regions of large relative motion Grid generation is sensitive to body definition Unsteady problems & Separation Studies in CFD 11 15-Jul-2019
- 12. 12 ©ZeusNumerixPvtLtd:ConfidentialDocument Basic Requirement – Create a New Mesh II ▪ Chimera approach Data interpolation is difficult in regions between close objects Body-fitted grids are sensitive to body definition Unsteady problems & Separation Studies in CFD 12 15-Jul-2019
- 13. 13 ©ZeusNumerixPvtLtd:ConfidentialDocument Basic Requirement – Create a New Mesh III ▪ Oct-tree approach Divide the cells to capture geometry and flow features Staircase approximation reduces the accuracy Unsteady problems & Separation Studies in CFD 13 15-Jul-2019
- 14. 14 ©ZeusNumerixPvtLtd:ConfidentialDocument Basics for Oct-tree Meshes Unsteady problems & Separation Studies in CFD ▪ Start with a block of L x M x N cells ▪ The cells constitute a structured Cartesian grid ▪ Cells must be identified as one of three types: flow; solid; or intersected ▪ Intersected cells must be cut to produce cut-cells ▪ Some cells may become split cells 14 15-Jul-2019
- 15. 15 ©ZeusNumerixPvtLtd:ConfidentialDocument Grid Refinement to Capture Geometry Unsteady problems & Separation Studies in CFD ▪ NumerixExpertTM: An Oct-tree Cartesian Based Solver 15 15-Jul-2019
- 16. 16 ©ZeusNumerixPvtLtd:ConfidentialDocument Unsteady problems & Separation Studies in CFD 16 Solution Adaptive Mesh Refinement NumerixExpertTM: An Oct-tree Cartesian Based Solver 15-Jul-2019
- 17. 17 ©ZeusNumerixPvtLtd:ConfidentialDocument Basics of Oct-tree Meshes Unsteady problems & Separation Studies in CFD ▪ Each cell is divided in to 8 cells if better space resolution is required ▪ Resolution may be required to capture slope discontinuity curvature, etc. ▪ Resolution may be required to capture shock, separation, etc. ▪ Resolution can be carried out selectively to economise cost of simulation whenever and wherever required ▪ Can be parallelised 17 2D Oct-tree (Quad-tree) without cut cells with Adaptive Mesh Refinement 15-Jul-2019
- 18. 18 ©ZeusNumerixPvtLtd:ConfidentialDocument Oct-tree Ideal for Multi-bodies Relative Motion Unsteady problems & Separation Studies in CFD ▪ Runge-Kutta formulation permits a varying control (cell) volume ▪ Cell geometry needed at three instances during an update ▪ Time step calculation and flux computation must now include cell face velocities (damping derivatives required for 6DOF simulations are automatically get built in forces) 18 15-Jul-2019
- 19. 19 ©ZeusNumerixPvtLtd:ConfidentialDocument Kinematics Affecting the Properties of Cells Unsteady problems & Separation Studies in CFD ▪ Cell volumes can appear or disappear during a solution update ▪ Permit a cell transforming from solid to cut cell or vice versa and fluid to cut and vice versa ▪ Do not permit transformation from solid to fluid directly or vice versa 19 15-Jul-2019
- 20. 20 ©ZeusNumerixPvtLtd:ConfidentialDocument Kinematics Affecting the Properties of Cells Unsteady problems & Separation Studies in CFD ▪ Multiple simply connected cells are grouped together ▪ Each group of cells is treated as an individual composite cell during a solution update ▪ At the end of the update each member cell receives its appropriate portion of the updated solution 20 15-Jul-2019
- 21. 21 ©ZeusNumerixPvtLtd:ConfidentialDocument Re-triangularisation of geometry Unsteady problems & Separation Studies in CFD ▪ Surface triangles get split due to oct-tree cells further 21 15-Jul-2019
- 22. 22 ©ZeusNumerixPvtLtd:ConfidentialDocument Possible Direction for RANS Simulations Unsteady problems & Separation Studies in CFD ▪ To accommodate boundary layer prismatic layers are required ▪ Instead of cutting cells along xy, yz and zx planes, divide the cells along diagonal 22 15-Jul-2019
- 23. 23 ©ZeusNumerixPvtLtd:ConfidentialDocument Basic Validation: Is the Formulation true ALE? Unsteady problems & Separation Studies in CFD ▪ Both the sphere and free stream with the same velocity V=100 (i + j + k) 23 Solution after large number of iterations shows only 1% error in veloicity and pressure Solver is able to handle Arbitrary Lagrangean Eulerian simulations 15-Jul-2019
- 24. 24 ©ZeusNumerixPvtLtd:ConfidentialDocument Dynamic Geometry Unsteady problems & Separation Studies in CFD Solution of one or more objects either stationary of moving with time does not pose any problems 24 15-Jul-2019
- 25. 25 ©ZeusNumerixPvtLtd:ConfidentialDocument 15-Jul-2019 Unsteady problems & Separation Studies in CFD 25 Doability on a Complex Geometry with two Objects in Relative Motion
- 26. 26 ©ZeusNumerixPvtLtd:ConfidentialDocument Unsteady problems & Separation Studies in CFD 26 Validation Case 15-Jul-2019
- 27. 27 ©ZeusNumerixPvtLtd:ConfidentialDocument Rigid body dynamics with 6 Degrees of Freedom 27 Unsteady problems & Separation Studies in CFD15-Jul-2019
- 28. 28 ©ZeusNumerixPvtLtd:ConfidentialDocument 28 Formulation M dV/dt + V dM/dt = Fexternal (In inertial frame of reference) M dV/dt + M [Ω]BE [u v w] = = Fa,p + [T]BLMg - 2[Ω]EI [u v w] - [Ω]EI[Ω]EISBI (in body frame of reference) where Unsteady problems & Separation Studies in CFD15-Jul-2019
- 29. 29 ©ZeusNumerixPvtLtd:ConfidentialDocument 29 FormulationWritten explicitly After integration produces position for the c.g. of the body Unsteady problems & Separation Studies in CFD15-Jul-2019
- 30. 30 ©ZeusNumerixPvtLtd:ConfidentialDocument 30 Formulation ω dI/dt + I dω/dt = Ma,p (Inertial frame) I dω/dt + I [Ω] [p q r] = MBa,p (body frame) [dp/dt dq/dt dr/dt] = [I]-1 ( - [I] [Ω] [p q r] + Ma,p ) where Unsteady problems & Separation Studies in CFD15-Jul-2019
- 31. 31 ©ZeusNumerixPvtLtd:ConfidentialDocument 31 Formulation Integration of these equations produces Euler angles for the body Unsteady problems & Separation Studies in CFD15-Jul-2019
- 32. 32 ©ZeusNumerixPvtLtd:ConfidentialDocument References Unsteady problems & Separation Studies in CFD 1) Scott M. Murman, Michael J. Aftosmis, Marsha J. Berger, “Implicit Approaches for Moving Boundaries in a 3-D Cartesian Method” , AIAA-2003-1119 2) Scott M. Murman, Michael J. Aftosmis, Marsha J. Berger, “Simulations of 6-DOF Motion with a Cartesian Method” , AIAA 2003-1246 3) Alex Cenko, “Experience in the use of computational aerodynamics to predict store release characteristics” Progress in Aerospace Sciences 37, 477–495, (2001) 4) Elias E. Panagiotopoulos, Spyridon D. Kyparissis, “CFD Transonic Store Separation Trajectory Predictions with Comparison to Wind Tunnel Investigations” 5) Scott M. Murman, Michael J. Aftosmis, Marsha J. Berger, “Simulations of Store Separation from an F/A-18 with a Cartesian Method”, journal of aircraft, August 2004 6) A. Cenko, “Store Separation Lessons Learned During the last 30 years”, ICAS, 2010 7) Lawrence E. Lijewski, Norman E. Suhst., “Time-Accurate Computational Fluid Dynamics Approach to Transonic Store Separation Trajectory Prediction”, Jounal of Aircraft, July-Aug. 1994 8) Alex Cenko, Frank Taverna., “The United States Navy‟s Integrated Approach to Store Separation Analysis”, 1998 9) Finney. Luke Patrick., “Investigation of Cavity Flow Effects on Store Separation Trajectories”,2010 10) Marsha J. Berger, “Cartesian grids for Moving Geometries”, 2006 11) Z.J. Wang, Ravishekar Kannan, “An Overset Adaptive Cartesian/Prism Grid Method for Moving Boundary Flow Problems” 12) H. ¨Ozg¨ur Demir, “Computational Fluid Dynamics Analysis Of Store Separation” 13) T Mahmood, M N Aizud, S Zahir., “Aerodynamic Effects of the Store Release on the Roll Attitude of a Wing Configuration in Transonic Flight”, 2011 32 15-Jul-2019
- 33. 33 www.zeusnumerix.com +91 72760 31511 Abhishek Jain abhishek@zeusnumerix.com Thank You !