Unsteady Problems & Separation Studies @ Zeus Numerix

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Unsteady problems &
Separation Studies in CFD
Compute heavy simulations, will require HPC

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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
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What is Unsteady Flow?
Flow field at given point is changing with time
Unsteady problems & Separation Studies in CFD15-Jul-2019

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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.
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V = 0V = 10
V = 10V = 0
flow changes with time
flow does not change

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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
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V = 100
V = 100 – r 
flow does not change

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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
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V = 100 m/s

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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

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Equations in ALE
▪ Inviscid compressible flow (Euler Equations)
Vs
Vs is velocity of cell face
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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
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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
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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
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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
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Basic Requirement – Create a New Mesh III
▪ Oct-tree approach
 Divide the cells to capture geometry and flow features
 Staircase approximation reduces the accuracy
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Basics for Oct-tree Meshes
▪ 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
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Grid Refinement to Capture Geometry
▪ NumerixExpertTM: An Oct-tree Cartesian Based Solver
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Solution Adaptive Mesh Refinement
NumerixExpertTM: An Oct-tree Cartesian Based Solver
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Basics of Oct-tree Meshes
▪ 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
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2D Oct-tree (Quad-tree) without cut
cells with Adaptive Mesh Refinement
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Oct-tree Ideal for Multi-bodies Relative Motion
▪ 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)
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Kinematics Affecting the Properties of Cells
▪ 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
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Kinematics Affecting the Properties of Cells
▪ 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
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Re-triangularisation of geometry
▪ Surface triangles get split due to oct-tree cells further
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Possible Direction for RANS Simulations
▪ To accommodate boundary layer
prismatic layers are required
▪ Instead of cutting cells along xy, yz
and zx planes, divide the cells along
diagonal
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Basic Validation: Is the Formulation true ALE?
▪ Both the sphere and free stream with the same velocity V=100 (i + j + k)
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Solution after large number of iterations shows only 1% error in veloicity and pressure
Solver is able to handle Arbitrary Lagrangean Eulerian simulations
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Dynamic Geometry
Solution of one or more objects either stationary of moving with time does not pose
any problems
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Doability on a Complex Geometry with two
Objects in Relative Motion

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Validation Case
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Rigid body dynamics with 6
Degrees of Freedom
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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

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FormulationWritten explicitly
After integration
produces position for the c.g. of the body

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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

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Formulation
Integration of these equations produces Euler angles for the body

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References
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
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www.zeusnumerix.com
+91 72760 31511
Abhishek Jain
abhishek@zeusnumerix.com
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Unsteady Problems & Separation Studies @ Zeus Numerix

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