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
CS
U
F
ρ
u
v
w
eo
vn
=
=
=
=
=
=
=
=
=
=
Control Volume
Boundary of the control volume
Vector of conserved variables
Vector of flux of conserved variables
Density within a control volume
x component of velocity of fluid in control volume
y component of velocity of fluid in control volume
z component of velocity of fluid in control volume
Total energy of the fluid in control volume
Velocity normal to the boundary of a control volume
Store Separation Simulation using Oct-tree Grid Based Solver
p
nx,ny,nz
ho
e
h
a
γ
Cp
Cv
=
=
=
=
=
=
=
=
=
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Pressure
x, y and z components of outward normal unit vector to the boundary
Total enthalpy
Static internal energy
Static enthalpy
Speed of sound
Ratio of specific heat capacities
Specific heat capacity of a gas at constant pressure
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
where,
,
447
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.
Store Separation Simulation using Oct-tree Grid Based Solver
449
5.1 Supersonic flow over a wedge of 10° angle
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.
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