Changju Thomas Wey and Nan-Suey Liu

Glenn Research Center, Cleveland, Ohio

Updates to Simulation of a Single-Element

Lean-Direct Injection Combustor Using a

Polyhedral Mesh Derived From Hanging-Node

Elements

NASA/TM—2014-218108

March 2014
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Changju Thomas Wey and Nan-Suey Liu

Glenn Research Center, Cleveland, Ohio

Updates to Simulation of a Single-Element

Lean-Direct Injection Combustor Using a

Polyhedral Mesh Derived From Hanging-Node

Elements

NASA/TM—2014-218108

March 2014

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

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NASA/TM—2014-218108 1 

Updates to Simulation of a Single-Element Lean-Direct 

Injection Combustor Using a Polyhedral Mesh Derived 

From Hanging-Node Elements 

Changju Thomas Wey and Nan-Suey Liu 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

This paper summarizes the procedures of inserting a thin-layer mesh to existing inviscid polyhedral 

mesh either with or without hanging-node elements as well as presents sample results from its 

applications to the numerical solution of a single-element LDI combustor using a releasable edition of the 

National Combustion Code (NCC). 

Introduction 

The lean direct injection (LDI) concept has the potential for low emissions under operational 

(compression ratios up to 60:1 and peak temperature 3000 F) conditions. For a LDI combustor, most of 

the air directly enters the combustor through the swirler. In this concept, the liquid fuel is injected from 

fuel injectors directly into the incoming swirling airflow, and the swirling air stream is used to atomize 

the injected liquid as well as to promote fuel-air mixing. The flame structure can be very complex and 

locally range from non-premixed to premixed burning. 

Recently, a single-element LDI combustor experiment has been used as a test bed for assessing, 

further developing and validating the capability of two-phase turbulent combustion modeling and 

simulation. This combustor consists of an air passage with a 60 six-bladed air swirler, a convergingdiversion section and a chamber with a square cross-section. The fuel is injected through the center of the 

swirler and exit at the throat of the venture. 

A series of numerical calculations have been performed by using (1) the time filtered Navier-Stokes 

(TFNS) methodology and (2) the large eddy simulation (LES) methodology. The sub-grid models 

employed for turbulent mixing and combustion include the well-mixed model, the linear eddy mixing 

(LEM) model; the Eulerian filtered mass density function (EUFDF/EUPDF) model, and the flameletbased model. Results from these methodologies invoking various sub-grid models are summarized in 

Reference 1, and a more detailed description of the TFNS approach can be found in Reference 2. It should 

be pointed out that TFNS is not LES, nor hybrid RANS/LES, nor, in general, unsteady Reynoldsaveraged Navier-Stokes (URANS). Like the LES, TFNS is capable of capturing the dynamically 

important, unsteady turbulent flow structures, even when RANS-grade meshes are used. Unlike the LES, 

the grid resolution and the turbulence model fidelity are not formally linked, therefore, in principle; a grid 

independent solution can be unambiguously attained in the TFNS approach. 

Species transport equation in the multiple rotating frames of reference (MRF) is written as: 

''' ( ) () m

m m g eff m mls VA A V V

d dV u U dA D dA w dV S dV

dt

  



                (1)

wm

  , the production rate of species m per unit volume, is the main source of the turbulence-chemistry 

interaction. The sub-grid, turbulence-chemistry model, either the linear eddy mixing (LEM) or the 

Eulerian filtered mass density function (EUFDF/EUPDF) model, needs finer volume lumps or Monte 

Carlo particles been generated for each CFD cells for entire domain as illustrated in Figures 1 and 2. 
NASA/TM—2014-218108 2 

Figure 1.—Schematic illustrating the splicing algorithm used for scalar convection. (Assuming number of 

LEM lumps is 12. Each lump contains volume and species density). 

Figure 2.—Schematic illustrating the process of the scalar convection and diffusion for the Eulerian scalar 

Monte Carlo FDF/PDF model. 

Both of the turbulence-chemistry sub-grid models mentioned above does not solve components of the 

momentum in the sub-grid elements or particles. In addition to the physics-based sub-grid models for the 

entire domain, an adaptive mesh-based sub-grid model was developed and implemented recently. 

In previous work (Ref. 3), a mesh-based enhancement for the scalar mixing was proposed. It was 

speculated there that greater number of the flux-exchange between solution elements, i.e., using 

polyhedrons as control volumes, may have the potentials to enhance the mixing of the scalar for the 

modeling of the turbulence-chemistry interaction. In Reference 3, a honey-comb like polyhedral mesh 

was generated from a regular unstructured mesh, which could be the combinations of tetrahedrons, 

hexahedrons, prisms and pyramids, through the processes of refinement, reconnection and agglomeration. 

In Reference 4, a polyhedral mesh is generated or more precisely post-processed from a set of 

hanging-node or conforming unstructured mesh. A LDI combustor mesh refined from a set of allhexahedral mesh is shown in Figure 3 with the temperature as the indicator for the criterion of refinement. 

In Figure 4, the region of the refinement is identified by prescribing the desired location. Those subgrid refinements are done in all directions for the selected cells. For the cases of high Reynolds number 

flows, the sub-grid refinements normal to wall are desirable and necessary. In the present work, a thinlayer, normal to wall refinement scheme is developed. It is well known that inviscid tetrahedral mesh or 

hanging-node mesh, such as Cartesian mesh, can be generated very quickly with minimal user interfaces. 

They are ideal for the rapid prototyping of the combustor geometry. However in most of cases, those 

meshes are not suitable for the high Reynolds number turbulent flows due to lack of well-behavior layer 

mesh near the walls. Even in the conceptual design stages of the combustors, the viscous effects of the 

solid walls should be treated as one of the design variables. 

Arrows—Convection direction 

Initial number of lumps—12 

Light Green—inward mass (1.5) 

Dark Green—inward mass (4.5) 

Red—outward mass (1.5) 

Magenta—outward mass (2.5) 

Current number of lumps—14 

Final number of lumps will became 12 after 

re-gridding via equalizing the volume. 

Arrows—Convection direction 

Light Green—inward mass (15% = 3) 

Dark Green—inward mass (25% = 5) 

Red—net inward mass (5% = 1) 

Magenta—net inward mass (10% = 2) 

Number of Monte Carlo particles–20 

Contents of randomly selected particles are 

replaced by those of randomly selected 

particles from adjacent cells. 
NASA/TM—2014-218108 3 

Figure 3.—A plane cut of polyhedrons around the mid-plane of the 

combustor with the temperature as the criterion scalar for refinement. 

Figure 4.—A plane cut of polyhedrons around the mid-plane of the 

combustor with specified region refined. 
NASA/TM—2014-218108 4 

Generation of a Thin Layer Grid Normal to the 

Wall Boundary of an Inviscid Mesh 

A set of inviscid cells that represents a LDI combustor is shown in Figure 5. The process starts by 

identifying each node’s level of connectivity from the wall for the entire mesh. The nodal level of 

connectivity from the wall will be used later to determine the sequence of the displacement of the grid 

nodes. There is an analogy between the nodal level of connectivity of an unstructured grid and 

K = constant surfaces of an I*J*K structured grid. For example a plane cut through the center of an 

inviscid mesh representing a LDI combustor geometry is shown in Figure 3. In Figure 6, iso-surfaces of 

level of connectivity of the nodes in the center of the combustor are shown. 

The next step will be marching-grid generation from the wall boundary of the mesh. The original 

method of the marching grid generation was described in Reference 5. The process will be described here 

very briefly. Determination of marching distances per layer is based on the rate of area change due to 

surface virtual offset. It yields relatively large marching steps in the convex regions where dihedral angle 

is less than 180, and small steps in the concave regions where dihedral angle is larger than 180. 

Figure 5.—A plane cut of an inviscid all-tetrahedral grid around the mid-plane 

of the combustor. 

Figure 6.—Iso-surfaces of nodal level of connectivity from the wall in the center of the combustor 

to be used for the orderly deformation of the mesh. 
NASA/TM—2014-218108 5 

The surface nodes are divided into two groups, one is the grid points with larger rate of area change 

due to surface virtual offset, and the other is the grid points with smaller rate of area change due to 

surface virtual offset. The former group will be marching first. 

The initial marching vectors are the normal vectors. However, this may not provide a valid grid since 

the grid lines may intersect each other in the convex regions. 

To prevent intersection, the weighted Laplacian type smoothing operation is applied to the 

components of the marching vectors. The weighting function is also based on the rate of area change due 

to surface virtual offset. The grid points in latter group will be marching after the first group. The 

marching vectors are constraint by the marching direction of the first group. 

The resulting grid is a set of semi-structured, distinct layer grid. It will be the source of the 

deformation for the inviscid grid. The process starts by inserting the semi-structured grid from the user 

selected layer upward. 

Previous computed levels of nodal-connectivity for the inviscid grid is used to guide the deformation 

of the mesh in such order that the grid points of level one is displaced first, the grid points of the level two 

is displaced second, and so on. To reduce the tangling of the mesh, spring analogy is applied to each edge 

of the mesh. That is that each edge of the inviscid grid is modelled as a tension spring. The spring 

constant is defined as the inverse of the length of the edge. The displacements of the grid points are 

adjusted for individual level initially. After the grid points have been deformed on current level, the 

displacements of the grid points will be conveyed into edge displacements in which they become the 

nodal displacement on next level. Once all the layer grid points have been inserted, the entire network of 

springs of the inviscid grid will adjusted again to untangle and smooth the grid. In Figure 7, it is shown 

that a cell-cutting body-captured inviscid mesh is refined and to be able to handle high Reynolds number 

turbulent flow. A node-blanking body-captured inviscid mesh with the inserted thin-layer refined mesh is 

depicted in Figure 8. 

For the case of LDI combustor shown in Figures 5 and 6, a thin-layer refined grid is generated 

between solid wall and the inviscid all-tetrahedral mesh. The result is shown in Figure 9. An expanded 

view near the corner of a swirler blade is shown in Figure 10. The former grid consists of 1,554,419 

tetrahedrons, 283,200 nodes. The latter grid consists of 2,593,215 cells (tetrahedrons and prisms) and 

803,852 nodes which is a whopping 67 percent increase in cell count. It demonstrates that a typical 

turbulent flow simulation costs much more than an inviscid flow analysis. 

Figure 7.—Left: a cell-cutting inviscid mesh; right: a thin-layer refined grid merged with original grid. 
NASA/TM—2014-218108 6 

Figure 8.—A node-blanking body-captured viscous grid. 

Figure 9.—A thin-layer refined grid is generated between solid wall and 

the inviscid all-tetrahedral mesh. The final mesh mounts to a whopping 

67 percent increase of cells. 

Figure 10.—An expanded view near the corner of a swirler blade; There are 

12 extra layers are inserted into the original grid. 
NASA/TM—2014-218108 7 

Sample Applications of Hanging-Node Polyhedral Meshes 

A hybrid prism and tetrahedron mesh describing a single-element LDI combustor is selected for the 

purpose of demonstration. This combustor consists of an air passage with a 60 six-bladed air swirler, a 

converging-diversion section and a chamber with a square cross-section. The fuel is injected through the 

center of the swirler and exit at the throat of the venture. The original mesh contains consists of 2,593,215 

cells (tetrahedrons and prisms) and 803,852 nodes. 

Three meshes are adaptively generated by selecting, (1) temperature; (2) unburned hydrocarbon 

(UHC); (3) fuel-vapor just right after evaporated, of the field solutions as the criterion for refinement. The 

first grid consists of 2,783,673 polyhedrons, 992827 nodes and a center plane cut is shown in Figure 11. 

The second grid consists of 2,632,317 cells and 840,482 nodes. A center plane cut is shown in 

Figure 12. The third grid consists of 2,636,377 polyhedrons, 844,422 nodes and a center plane cut is 

shown in Figure 13. The number of solution elements is compatible between these three grids. 

Figure 11.—Temperature-based refined grid. 

(2783673 cells).

Figure 12.—UHC-based refined grid. (2,632,317 cells). 

Figure 13.—Fuel droplet vapor-based refined grid. (2,636,377 cells).
NASA/TM—2014-218108 8 

Preliminary Results 

Nonreacting Case 

At the inlet of the swirler, the inflow velocity, the static temperature and the density of the gas phase 

are specified as 20.14 m/s, 294.28 K, 1.19 kg/m3

, respectively. At the outflow boundary, the static 

pressure is imposed at 101325 Pa. The generalized wall function is applied to solid wall boundaries. The 

thermal boundary condition for all the solid surfaces of the combustor is set to adiabatic. Two sets of 

mesh are selected for this case, one is shown in Figure 5 which is all-tetrahedron inviscid grid, and the 

other is shown in Figure 9 which is a hybrid prism and tetrahedron grid. The RANS cubic turbulence 

model is used for both meshes. In Figure 14, the contours of the axial velocity in the center plane of the 

geometry in the upper panel are computed from inviscid mesh, while that of the viscous mesh is shown in 

the lower panel. The velocity pattern is quite different near the exit of the swirler cup. In Figure 15, the 

contours of the static pressure in the center plane of the geometry in the upper panel are computed from 

inviscid mesh, while that of the viscous mesh is shown in the lower panel. The low pressure contour 

represents the preprocessing vortex core (PVC) for the viscous mesh is noticeable compared with that of 

inviscid mesh. It confirms that the effect of the viscous grid should be part of the design parameters. 

Figure 14.—Upper panel shows axial velocity contours with inviscid grid, lower panel with 

viscous grid. 

Figure 15.—Upper panel shows static pressure contours with inviscid grid, lower panel with 

viscous grid. 

 
NASA/TM—2014-218108 9 

Reacting Case 

For reacting cases, beside the boundary conditions for the gaseous phase described above, the liquid 

fuel, C12H23, is injected into the computational domain from a pressure swirl injector. The diameter of 

the orifice is 0.0006 m. The fuel atomizer is set at a pressure 110 kPa and delivers a flow rate of 0.025 kg/ 

min. The spray has a 90 spray angle which is the same as the converging-diverging venture attached to 

the swirler. An initial droplet size distribution is prescribed to provide the liquid fuel injection condition, 

0.4

32

16.98( ) 6 3.5

32 32

4.21 10 [ ]

d

dn d dd d

e

nd d



  (2) 

where n is the total number of the droplets and dn is the number of droplets in the size range between d

and d dd  . This correlation also requires the specification of Sauter mean diameter, 32 d , and the number 

of droplet classes. The equivalence ratio computed from the gaseous and liquid inlet boundary conditions 

is about 0.72. The adiabatic flame temperature is around 2100 K. The chemical kinetics for the gaseous 

phase is represented by five reactions and seven species, which consist of C12H23, O2, CO2, CO, H2O, 

N2 and NO. 

TFNS results, obtained with the well-mixed combustion model for the meshes depicted in Figures 10 

to 13, are shown for axial velocity, static pressure, temperature and NO mass fraction in the center plane 

(i.e., z = 0 plane), respectively. 

The temperature contours in Figure 16(d) indicates the flame near the exit of the swirler cup is not 

sustainable computed from Figure 10 mesh. 

Figure 16.—Results from baseline mesh (Fig. 10): (a) axial velocity (b) pressure 

(c) temperature (d) NO mass fraction. 

(a) 

(b) 

(c) 

(d) 
NASA/TM—2014-218108 10 

The temperature contours in Figure 17(d) indicates the flame near the exit of the swirler cup is 

sustainable computed from Figure 11 mesh though the region. 

Figure 17.—Results from mesh in Figure 11: (a) axial velocity (b) pressure (c) temperature (d) NO mass 

fraction. 

 

(a) 

(b) 

(c) 

(d) 
NASA/TM—2014-218108 11 

The temperature contours in Figure 18(d) indicates the flame near the exit of the swirler cup is also 

not sustainable computed from Figure 12 mesh. The region with higher temperature flame is about the 

same as that of Figure 10 mesh. 

Figure 18.—Results from mesh in Figure 12: (a) axial velocity (b) pressure (c) temperature (d) NO mas 

fraction. 

 

(a) 

(b) 

(c) 

(d) 
NASA/TM—2014-218108 12 

The temperature contours in Figure 19(d) indicates the flame near the exit of the swirler cup is quite 

sustainable computed from Figure 13 mesh. The region with higher temperature flame is much larger than 

previous cases. It could be because that the mesh is refined much near the fuel injector. The number 

density of spray droplets in each cell near the fuel injector will be much smaller than other cases such that 

evaporation could be more effective. 

Figure 19.—Results from mesh in Figure 13: (a) axial velocity (b) pressure (c) temperature (d) NO mas 

fraction. 

(a) 

(b) 

(c) 

(d) 
NASA/TM—2014-218108 13 

The time-averaged centerline axial velocity and the centerline temperature are presented in Figures 20 

and 21, along with the measured data. The discrepancy between the computed values and the 

experimental data are noticeable. 

Figure 20.—Comparison of the mean axial velocity along the center line. 

Figure 21.—Comparison of the mean temperature along the center line. 
NASA/TM—2014-218108 14 

Concluding Remarks 

Capability of generating a thin-layer mesh for a given polyhedral mesh either with or without 

hanging-node elements is now implemented in a preliminary version of the OpenNCC, which is intended 

as the self-contained, releasable edition of the National Combustion Code (NCC). It is the second part of 

the adaptive mesh refinement option in OpenNCC. The thin-layer refinement consists of a radiated 

marching grid generation and a node-connectivity controlled mesh deforming technique. 

A stand-alone single-element LDI combustor is selected for the demonstration. A viscous mesh is 

generated from an existing all-tetrahedron inviscid mesh. After a baseline viscous grid is built, three more 

refined grids are generated from the baseline grid. Their criterions are the temperature, the UHC mass 

fraction and the vapor of fuel droplets of the solutions. 

It is observed that the temperature-based and fuel droplet vapor-based refined meshes resulted in 

more stable flame than other meshes. 

References 

1. Liu, N.-S., “Assessment and Improvement of Engineering Simulation for Multiphase Turbulent 

Combustion in a Lean Direct Injection Combustor,” ISABE–2011–1108, 20th International 

Symposium on Air Breathing Engines, September 12–16, 2011, Gothenburg, Sweden. 

2. Liu, N.-S., Shih, T.-H., and Wey, C.T., “Numerical Simulations of Two-Phase Reacting Flow in a 

Single-Element Lean Direct Injection (LDI) Combustor Using NCC,” NASA/TM—2011-217031, 

July 2011. 

3. Wey, T., and Liu, N.-S., “Simulation of a Single-Element Lean-Direct Injection Combustor Using 

Arbitrary Polyhedral Mesh,” AIAA–2012–0204, 50th Aerospace Sciences Meeting and Exhibit 

January 9–12, 2012, Nashville, Tennessee. 

4. Wey, T.C. and Liu, N.-S., “Simulation of a Single-Element Lean-Direct Injection Combustor Using a 

Polyhedral Mesh Derived from Hanging-Node Elements,” AIAA–2013–0739, 51th AIAA ASM, Jan. 

7–10, 2013, Grapevine, TX. 

5. Wey, T.C., “The Applications of An Unstructured Grid Based Overset Grid Scheme to Applied 

Aerodynamics,” 8th International Meshing Roundtable, 1999. 

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14. ABSTRACT

This paper summarizes the procedures of inserting a thin-layer mesh to existing inviscid polyhedral mesh either with or without hangingnode elements as well as presents sample results from its applications to the numerical solution of a single-element LDI combustor using a 

releasable edition of the National Combustion Code (NCC). 

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