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Procedia Engineering 00 (2014) 000–000
www.elsevier.com/locate/procedia
“APISAT2014”, 2014 Asia-Pacific International Symposium on Aerospace Technology,
APISAT2014
Unstructured Adaptive Grid Refinement for Flow Feature Capture
Liu Zhou*, Yang Yunjun, Gong Anlong, Zhou Weijiang
China Academy of Aerospace Aerodynamics, Beijing, 100074, China
Abstract
An unstructured grid adaptation technology has been developed for capturing flow features effectively. The face-based
hierarchical data structure is used at grid refinement procedure. The difference of suitable flow properties between cell and its
neighbors for flow features detection. The results of different simulation cases show the strengths of unstructured grid adaptation
technology. The flow features such as shock wave and vortex are resolved better with grids refinement. The grid adaptation also
improves the computational accuracy of aerodynamic characteristics and loads.
© 2014 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of Chinese Society of Aeronautics and Astronautics (CSAA).
Keywords: Grid adaptation; Hierarchical data structure; Flow features detection; Computational accuracy; aerodynamic characteristics
1. Introduction
The main uncertainty source is discretization error on Computational Fluid Dynamics (CFD). Some methods can
be used to eliminate or decrease the discretization error. The commonly used method is the uniform refinement of
computational grid. For 3D simulation, the uniform refinement approach causes rapid increase of the grid numbers,
and achieving solution for this huge grid refined is often difficult on current computer hardware. So decrease of
discretization error using uniform refinement of grid is impossible or expensive at least, for real 3D configuration
CFD simulation.
An alternative approach which is known as grid adaptation[1-6] is the better strategy to decrease the discretization
error. An initial solution is obtained on a coarse grid, and then the elements of grid which major discretization error
* Corresponding author. Tel.: +086-010-68743745.
E-mail address: zhou_liu@foxmail.com
1877-7058 © 2014 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of Chinese Society of Aeronautics and Astronautics (CSAA).
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Liu Zhou / Procedia Engineering 00 (2014) 000–000
occurs are refined. A new solution is obtained on this local refinement grid. The procedures above can be run
repeatedly to achieve the error level required.
The error indicator is the key of successful grid adaptation. A kind of error indicator which is used widely is the
flow features[1,2,6]. The flow features often include the derivatives of flow variables, or difference of the
parameters. The Adaptation based flow feature implies that the major errors of solution occur on regions of higher
solution gradients which are detected as the flow features. In general, this implication is approximate right for most
of the flow. The feature based grid adaptation is suitable for detection of shock waves, expansion fans, vortices and
stagnation zones. In this paper, we choose flow features as error indicator to drive grid adaptation.
Another important aspect in adaptive grid is the grid refinement method. For h-adaptation of unstructured grid,
the major partition strategy of grid's elements that include tetrahedron, hexahedron, prism, pyramid is based on
specific divided templates[4,6]. This method is used widely, but its disadvantage is obvious too. The method based
templates is not flexible to complex configuration and the additional, unnecessary elements have to be introduce to
match the specific templates. In this paper, a more flexible partition strategy which uses face-based hierarchical data
structure[2] is used to simplify the complicate templates manipulating.
2. Unstructured Grid Adaptation Method
2.1. Grid Refinement strategy
The face-based hierarchical data structure is used at grid refinement procedure. Figure. 1 shows the refinement of
prism, tetrahedron, hexahedron and general cell[2]. This refinement approach is similar to Cartesian grid generation,
the new cells quality is maintained, and the neighbor stencil numbers of new cells don't reduce. The disadvantage of
this approach is that the increase of cell number after adaptation is remarkable. Another disadvantage is that the
additional hanging nodes are introduced into grid refined, but for our cell-center unstructured grid CFD solver using
here, this is not a critical problem and only a little codes need be modified.
Fig. 1. Subdivision of different cell types.
2.2. Error Indicator and Flow Feature Detection
The grid adaptation requires a function to detect and locate flow features. An error indicator is used to identify
the regions of higher solution gradients. The error indicator of cell i is evaluated by difference of flow properties
with volume weight in present study.
ei  volir  max q ij
(1)
Liu Zhou / Procedia Engineering 00 (2014) 000–000
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Here q is the suitable flow properties such as density, pressure, or velocity, and j represents the cells adjacent to
cell i. The symbol vol is the cell volume and r is the weight factor. The value of r is the reciprocal of the dimension
of problem in this study.
When error indicator of cells is larger than the threshold value, these cells need be marked to be subdivided. The
threshold value is:
ei  m    
(2)
Here, m is the average of the error indicator over the grid and σ is the standard deviation. ξ is the adjustable factor.
m
1 n
 ei
n i 1
n

 ei  m2
(3)
i 1
n
3. Numerical Simulation Method
The cell center based unstructured grid finite volume flow solver was used for simulations. Roe scheme[7] for
inviscid fluxes and Gauss theory for variables' gradients. Venkatakrishnan limiter[8] was used to suppress the
numerical oscillation. The unsteady computations use dual time stepping method. The second order back difference
scheme was used for physical time marching while LU-SGS method[9] for pseudo time sub-iteration. Delayed
Detached Eddy Simulation (DDES) based SST turbulence model[10,11] is used for massive separation flow. WeissSmith low Mach preconditioning method[12] is used for low speed flow.
4. Results
Two Cases were chosen to demonstrate the strengths of unstructured adaptive grid refinement. The 1st case is the
transonic computation of NACA 0012 airfoil[1], and the 2nd case is the unsteady simulation of slender delta wing at
high angle of attack[13].
4.1. Transonic adaptive computation of NACA 0012 airfoil
The case chosen to study is a NACA 0012 airfoil at a freestream Mach number of 0.95 and an angle of attack of
0°. This case is used widely to investigate the effect of error indicator[1]. We hope the adaptation can capture the
weak normal shock wave location Xs (Fig. 2). Figure. 3 shows the original grid and grid after 4 adaptations, and
Figure. 4 compares the result of Xs of adaptation with global refinement.
From Figure. 3, the trail edge oblique shock wave and the weak normal shock wave are refined with solution
adaptation. The location of the normal shock downstream of the trailing edge has been determined through global
uniform grid refinement. The grid convergence has achieved in uniform refinement and the 2nd order Richardson
extrapolation is used to obtain the correct weak normal shock wave location. The grid adaptation obtains the weak
normal shock wave location exactly with a little cell number increases. The difference of Xs location between
uniform refinement and grid adaptation is smaller than 1%.
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Liu Zhou / Procedia Engineering 00 (2014) 000–000
Fig. 2. Transonic flow feature of NACA 0012 airfoil.
(a)
Fig. 3. Grids comparison of NACA 0012 airfoil: (a) Original grid; (b) Adaptive grid.
(b)
Liu Zhou / Procedia Engineering 00 (2014) 000–000
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Fig. 4. Convergence of value of Xs with different refinement strategy.
4.2. Unsteady simulation of delta wing at high angle of attack
This case is an unsteady computation for ONERA 70° delta wing[13] at α = 27°, M = 0.069, and Rec = 1.56×106.
Figure. 5 shows the delta wing model. DDES method is used for this massive separation flow. The physical time
step chosen is 1.0×10-4s to capture unsteady flow phenomenon.
Fig. 5. ONERA 70° delta wing model.
The grid adaptation is base on steady simulation results. Figure. 6 compares the adaptive grids with original grid.
The elements of grid near the vortices region are refined through solution adaptation. Figure. 7 shows the time
averaged pressure coefficients at different x location. The solution adaptations improve the pressure distribution on
upper surface remarkably, especially for suction peak under vortex. The improvement of pressure distributions on
the rear of the upper surface (x >= 0.6m) is more remarkable, the reason is that the adaptations satisfy the
requirements to resolve the complicate flow structures at the region of vortex breakdown.
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Liu Zhou / Procedia Engineering 00 (2014) 000–000
Two stage
adaptive grid
Original grid
(a)
(b)
Fig. 6. Grids comparison of delta wing, (a) Surface grid; (b) x=800mm slice grid.
(a)
(b)
(c)
(d)
Fig. 7. Time averaged pressure coefficients at different x location, (a) x=0.5m; (b) x=0.6m; (c) x=0.7m; (d) x=0.8m.
Figure. 8 shows the Instantaneous flow structure with Q criterion on different adaptive grids level. The isosurfaces are colored by pressure. From Figure. 8, the vortex structures of massive separation flow are resolved better
with grids refinement, especially after vortex breakdown.
Liu Zhou / Procedia Engineering 00 (2014) 000–000
Adaptation
Stage 0
Adaptation
Stage 1
7
Adaptation
Stage 2
Fig. 8. Iso-surface of Q=100000.
5. Results
An unstructured adaptive grid refinement method has been developed in present study. The face-based
hierarchical data structure is used at grid refinement procedure. The high gradient flow features is used as error
indicator. Some cases with different flow features were simulated. The results demonstrate that unstructured grid
adaptation can resolve flow features better than original grids, while the computational accuracy of aerodynamic
characteristics and loads are also improved with adaptive grid refinement.
References
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