1. Turbulence Modelling for Aerospace
January 2007
Alistair Revell 1

2. Outline 2 Review of EU Project work An Outline of recent research
Derivation of a URANS model for stress-strain misalignment
Application to industrial cases
Future Work
Further model development
Application to aerospace 2

3. CFD in Aerospace: A comparison of methods
There is a range of numerical methods applicable to prediction of turbulent flow.
trade-off between calculation cost and accuracy/detail
the suitability of a method is dependent upon the application and the required results
Spalart (2000) compared these methods based upon the complete calculation of flow around a typical commercial jet aircraft at cruise conditions
DES was put forward as a compromise between URANS and LES
Increasing cost
URANS : Unsteady Reynolds Averaged Navier-Stokes DES : Detached Eddy Simulation
LES : Large Eddy Simulation DNS : Direct Numerical Simulation 3

4. DESider 4 Detached Eddy Simulation for Industrial Aerodynamics
Previous EU project, FLOMANIA ( ), attempted to identify if any one turbulence modelling scheme was capable of tackling the range of challenges in an industrial case. (separation, reattachment, vortex shedding)
- the answer was NO
DESider ( ), has looked at improved URANS and hybrid LES-RANS approaches.
Aims to overcome deficiencies in RANS whilst retaining realistic computational requirements.
Objectives include:
Improved URANS modelling
Improved and standardised DES
Novel approaches (eg. SAS, Hybrid RANS-LES)
Embedded LES-RANS (inlet methods)
Involved in all activities 4

5. Testcase examples 5 Wing in deep stall Full aircraft EFA Landing Gear
DES
Expt
URANS
Wing in deep stall
Landing Gear
Ducted Cylinder 5

6. Derivation of new 3-equation model: SST-Cas
Objective: develop a turbulence model for industry that is able to accurately predict transient flow effects without compromising cost and stability
Reynolds Averaged Navier Stokes equations are unclosed; require turbulence models
Turbulent models predict evolution of one or more quantities in order to approximate the Reynolds Stresses ( ) or the turbulent viscosity ( )
Increasing complexity
1 eqn.
Eg. Spalart Almaras
2 eqns.
Eg. k-e,
k-w SST
7 eqns.
RSM
uiuj - e
3 eqns.
SST-Cas
The parameter, Cas, provides a measure of the misalignment of the tensors of stress anisotropy, and strain, (Revell, Craft, Laurence 2006)
New SST-Cas model reproduces some effects of the Reynolds Stress Model (RSM) 6

7. Advantages of SST-Cas model
The widespread success of two equation turbulence models is due in part to their compromise between complexity and numerical stability
But, in standard 2 equation Eddy viscosity models, the stresses are directly linked to the strains.
When strain Sij= 0 , stress anisotropy aij = 0
No transport or history effects
This is not the case with a 7 equation RSM, where transport equations are solved for the six individual stress components.
In some cases, the additional equations have caused convergence problems
The SST-Cas model inherits stability advantages of eddy viscosity while accounting for some of the stress transport effects 7

8. Implementation in Code_Saturne
The misalignment parameter is defined as:
Which will be zero when aij and Sij are mutually perpendicular (for 2D)
Where the implemented form of the transport equation is:
Which fits easily into the SST implementation with a modification of viscosity as: 8

9. Implementation of SST_Cas model
Model implemented into Code_Saturne
- 3D unstructured finite volume code
Validated for simple 1D and 2D unsteady flows:
- homogenous cyclic strain, oscillating channel
extra equation found to add a 10-15% cost compared to a 2-equation model.
RSM is around % more expensive than a 2-equation model
Flowfield around NACA0012 at 20o, Re= 105
Contour plots of long-time averaged Pressure with mean flow streamlines
(2-eqns)
(3-eqns)
(7-eqns)
Reproduces similar results to RSM: 9

10. NACA0012 @ 20o 10 1 2 3 1 2 3 Unsteady flow in wake of aerofoil:
misalignment of stress and strain is shown by eigenvectors, and elements are coloured by values of Cas 1 2 3
Phase averaged velocity profiles <U>
Good agreement of SST_Cas with RSM (SSG) 10

11. Cylinder in Square Duct
Experimental setup showing location of PIV data planes
work done at IMFT
Re = 1.4x106
3D Calculations with ~2x106 cells, where long-time averaged solutions obtained after:
e.g ~ 2 weeks calculation on 16 processors 11

12. Cylinder in Square Duct
Streamlines
<U>
Iso-Q contours
SST
SST-Cas
SST-DES
In Streamlines and <U>, top half is experimental data: bottom half is numerical results
‘Q’ criterion: a parameter used to visualise structures in the flow = (Sij2-Wij2) 12

13. Wingtip Vortex 13 CTR Summer Program 2006
Since standard RANS models over-predict the levels of turbulence in a vortex, they also tend to over-predict the decay rate of the axial velocity of a vortex.
Results from Linear and Non-linear EVM are found to exhibit a far too rapid decay of the vortex core.
A Reynolds stress transport model (RSM) reproduces the principal features found in the experimental measurements. (Craft et al 2006) 13

14. Isolated Vortex 14 Uq Ux CTR Summer Program 2006
The case of an isolated decaying vortex was studied, using DNS data for initial conditions, to study the response of RANS models in closer detail.
On the graph to the right, the black lines are DNS data (reference) at fixed time intervals after the start of the calculation. (Numerical results are in red)
The RSM and new SST-Cas are able to capture correctly this behaviour, while the SST model over-predicts the decay-rate as expected. Uq Ux 14

15. Wingtip Vortex 15 CTR Summer Program 2006
Many cells are required to correctly resolve both the very thin boundary layer over the wing, and the core of the vortex behind the wingtip (Re=4.35x106).
Both regions are crucial in order to correctly model the flow.
This presents an opportunity for the use of unstructured meshes, which have the potential to reduce grid sizes by over 90%.
Initial results are encouraging
Hanging Nodes: Total Mesh ~ cells
Fully Structured: Total Mesh ~ cells 15

16. CTR Summer Program 2006
Wingtip Vortex
Contours of mean velocity downstream of the wingtip (left: axial, right: crossflow)
Initial results are encouraging, showing a slower rate of decay of the vortex from the SST-Cas compared to the SST
Further work is required to test grid convergence: an important issue with highly unstructured grids. 16

17. Conclusions 1
Equation derived to allow for stress-strain misalignment in URANS
Only solving 3 equations
Only small additional expense compared to EVM (additional ~15%)
Remains ~40% cheaper than a full RSM
Retains stability advantages of eddy viscosity over individual stress components.
Fully implemented into Code_Saturne
Good practise is to start the SST-Cas calculation from a converged SST flow field.
Further work
The possibility to use the SST-Cas model in a Detached Eddy Simulation (DES) framework should be explored
Development required for near-wall effects
Investigate inclusion of Reynolds stresses in momentum eqn.
Secondary flow effects 17

18. Conclusions 2 18 Code_Saturne is now an open source code
University of Manchester will aim to become the hub for development
We will benefit from the implementation of further capabilities from 3rd parties
New version has ‘Chimera’ moving-grid feature
allowing for calculation of Fluid-Structure Interactions
Aerospace applications
Ongoing work on wingtip vortex (paper for TSFP5)
Transient calculations can be used to study aeroacoustics
Investigate noise reduction e.g. jagged flap edges, casing around landing gear, cavities
Develop cross-discipline topics in aerospace areas
Fluid-Structure interactions: eq. aeroelasticity
Combustion in complex cases: eg. Entire jet engine flow
Drag reduction calculations can examine effect of bumps, grooves, dimples…
Unstructured code used to tackle complex geometries. 18