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

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

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:

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

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

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

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

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

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

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

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

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

Validation of the complete structure-based model for scalar transport

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

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

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

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

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

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

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

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

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

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

(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

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

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

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

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

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

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

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