1. A Structure-Based Model for the Transport of Scalars in Homogeneous Turbulent Flows
C. F. Panagiotou and Y. Hasegawa
(Institute of Industrial Science, The University of Tokyo)
S. C. Kassinos
(University of Cyprus)
UMich/NASA Symposium 2017
Michigan, USA

2. Outline Structure-Based Models (SBMs)
Development of scale equations for scalar transport
Validation cases
Conclusions

3. Idea What exactly is the information that gives
us and is it indeed all we need?
Reynolds (1991), Kassinos & Reynolds (1994):
-componentality information- not sufficient to fully
characterize turbulence anisotropy
Idea
Information about the turbulence structure is crucial for the
description of anisotropy:

4. Basics Stream-function: Set of structure tensors:
Connection between the structure tensors:

5. Applications of SBMs: Diagnostic purposes:
e.g. canonical flows, stably stratified flows, etc
Modelling purposes:
e.g. pressure-strain rate tensor, turbulent scales, etc

6. Development of scale equations for
scalar transport
Predominant Theory: Kolmogorov (1941), Obunkov (1949)
-Small scales remain mostly isotropic
-No dependence on large-scale structure
Tong & Warhaft (1994), Sreenivasan (1977):
Deviation from small-scale isotropy
- especially for higher order statistics
Small-scale anisotropy linked to effect of large-scale structure

7. Motivation Objectives
Significant effect of large-scale structures on the
small-scale scalar statistics
Objectives
Construct an SBM for scalar transport
Develop a set of transport equations for the turbulence
Scales of the SBM, which are sensitized to the structure of
the large-scales

8. Homogeneous turbulence
Statistics of the turbulent field are independent of position
Mean field parameters:

9. Two-equation models for scalar scales
Traditional formulation:
SBM formulation:
large-scale field

10. Current formulation
Triple Decomposition Scheme: Reynolds & Hussain (1972)
Reynolds et al. 2002
Bracket averaging: Extract large-scale coherent motions

11. Current formulation Model in terms of one-point structure tensors
Proposed turbulence scales for the passive scalar transport:
Half fluctuating scalar-variance
Large-scale scalar-gradient variance
Model in terms of one-point structure tensors

12. Panagiotou and Kassinos, 2016 (PK16)
Large-scale model (LSM) for turbulence scales
Reynolds et al., 2002 (RLK02)
PK16
Panagiotou and Kassinos, 2016 (PK16)

13. Validation of the complete structure-based
model for scalar transport

14. Homogeneous shear in fixed and rotating frames
Mean flow Configuration:
Set up parameters:
Shear parameter ratio:
Frame rotation parameter ratio:
Richardson number:

15. Homogeneous shear in the presence of streamwise mean scalar gradient
DNS computations (symbols): Brethouwer (2005)
Set up parameters:
Performance indices:
Sensitised to the direction of the mean scalar gradient

16. Homogeneous shear in the presence of streamwise mean scalar gradient
-Good agreement for
- Fair predictions for , while it underpredicts

17. Homogeneous shear in the presence of transverse mean scalar gradient
DNS (symbols) and LES (thin lines): Kaltenbach (1994)
Set up parameters:
Captures critical Richardson number

18. Turbulence tends towards pancake-like structure
Homogeneous shear in the presence of
transverse mean scalar gradient 2C 1D
Turbulence tends towards pancake-like structure

19. Conclusions Future work
An structure-based model is proposed for the scalar transport
A two-equation model is proposed in order to provide the turbulence scales
Validate the complete SBM in a large number of cases, for both stationary and rotating frames, exhibiting encouraging results
Verify the ability of the structure tensors to be used as
diagnostic tools for stably stratified flows
Future work
Compare the complete SBM to other models
Extend the proposed SBM model to inhomogeneous flows

21. Structure-Based Models
Personal concerns: Part 1
Reynolds (1895): Split between mean and fluctuating parts
Closure problem
Model Reynolds stress and scalar-flux to bring closure

22. Personal concerns: Part 2
Lumley Invariant Map
Traditional models (such as EVM) use only the Reynolds stresses to:
describe the anisotropy of the turbulence
check if they are realisable

23. Common approach Objective Tasks
Construct an engineering model for homogeneous turbulence
Extend the model to account for inhomogeneous effects
Objective
Construct a turbulence model for the transport of scalars
in homogeneous turbulent flows
Tasks
Develop evolution equations for turbulence scales
Evaluate a set of statistical quantities, including
Validate the model for a large number of test cases

24. Structure-Based Turbulence Modeling
Main features of SBM:
- complete tensorial representation of turbulence structure
- family of one-point models of varying complexity already developed
DSBM: Differential Structure-Based Model
ASBM: Algebraic Structure-Based Model
IPRM: Interacting Particle Representation Model
Key Idea:
Follow the evolution of an ensemble of particles
Determine the statistics of the ensemble
Use them to evaluate the one-point statistics

25. (Advection-Diffusion eq.) Turbulence Engineering Models
Why turbulence models?
Direct Solution of time-dependent, three dimensional Navier-
Stokes equations through Direct Numerical Simulations (DNS)
Navier-Stokes equations for Newtonian, incompressible flow:
(Advection-Diffusion eq.)
Provides accurate predictions, however is not a practical option
for aerodynamic flows due to high computational cost
Turbulence Engineering Models

26. Large-scale statistics
Modeling the fluctuating passive scalar variance equation
Richardson notion (1920):
Small-scale eddies are in dynamical equilibrium with the large-scale eddies
Alternative expression:
Modeled Expression:
Large-scale statistics

27. Validation of the complete structure-based
model for stably stratified flows

28. Homogeneous shear in stably stratified flow
Configuration of the mean velocity flow:
Stress correlation coefficient:

29. Homogeneous vertical shear
DNS (symbols), LES (thin lines): Kaltenbach et al. (1994)
Initial conditions:
- Excellent agreement is achieved for Richardson number
- Fair agreement achieved for
- Accurately captures critical to be around 0.13

30. Homogeneous vertical shear
-Fair predictions for
-Qualitative agreement for until

31. Homogeneous non-vertical shear
DNS (symbols) : Jacobitz and Sarkar (1998)
Initial conditions:
- Captures evolution profile for (except peak magnitudes).
- Qualitative agreement for for

32. Homogeneous non-vertical shear
-Fair predictions for angles
-Qualitative agreement for
Shear parameter insensitive to the inclination angle (also seen in DNS )