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Data-Driven Turbulence Modeling: Physics-Informed Machine Learning Framework Jinlong Wu, Jianxun Wang, Heng Xiao

Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University

J-X. Wang, J-L. Wu, H. Xiao, “A Physics Informed Machine Learning Approach for Reconstructing Reynolds Stress Modeling Discrepancies Based on DNS Data”, Accepted, Physics Review Fluids, 2017. J-X. Wang, J-L. Wu, J. Ling, G. Iaccarino, H. Xiao, “A Comprehensive Physics-Informed Machine Learning Framework for Predictive Turbulence Modeling”, Submitted, available at: arXiv:1701.07102, 2017.

Motivation

Objective and Approach

Application II. High Mach Number BL

Bibliography

For more information, please contact Heng Xiao (hengxiao@vt.edu)or Jinlong Wu (jinlong@vt.edu). More information about the authors are also available online at https://sites.google.com/a/vt.edu/hengxiao/home.

Further Information

Physics-Informed Machine Learning Framework

Application I. Flow in A Square Duct

Application III. Flow over Periodic Hills

Conclusions

The objective of this work is to demonstrate that the RANS simulated Reynolds stress of a new flow can be improved via our PIML framework with existing LES/DNS database. Three essential parts of our PIML framework is: (1) representation of Reynolds stresses discrepancies as responses; (2) identification of mean flow features as inputs and (3) construction of regression functions from training data.

Reynolds Averaged Navier-Stokes Equations:

⌧ = �u0iu

0j

Hub of RANS models

Our Motivation: q  RANS modeled Reynolds stresses are known to be unreliable for many flows. q  LES/DNS simulations are still infeasible for many industrial flows. q  Is it possible to employ existing LES/DNS database to enhance the RANS

simulations?

⌧ = 2k

✓1

3I+A

◆= 2k

✓1

3I+V⇤VT

◆

(1)  Representation of Reynolds Stresses as Responses

�⌧ ) (� log2 k,z }| {�⇠,�⌘,

z }| {�'1,�'2,�'3)

(2) Identification of Mean Flow Features

) qi (i = 1, 2, ..., 47)Integrity Basis of

(3) Construction of Regression Functions

Random Forest Neural Network

−1 0 1−1

0

1

Reb=2200

z

y

−1 0 1−1

0

1

Reb=3500

z

y

Training Flow

(Re=2200)

Test Flow

(Re=3500)

⌧zz

⌧yy

0 0.25 0.5 0.75 10.0

0.2

0.4

0.6

0.8

1.0

RANS

DNS

PIML

Procedures of PIML Framework: q  Perform baseline RANS simulations on both the training flows and the test flow. q  Compute the mean flow feature vector q(x) based on the RANS simulations. q  Compute the discrepancies field between RANS modeled Reynolds stresses

and the high-fidelity simulation data for the training flows. q  Construct regression functions q  Compute the Reynolds stresses discrepancies for the test set by evaluating the regression functions.

�⌧↵ = ⌧RANS↵ � ⌧ truth↵

f↵ : q 7! �⌧↵

2-Comp 1-Comp

3-Comp

C1 C2

C3

Training Flow: q  Test Flow: q  M = 8, Tw = 0.53

M = 2.5, Tw = 1

general flow direction

recirculation zone

Training Flow Configuration: q  Dashed Line

Test Flow Configuration: q  Solid Line

In this work, we proposed a physics-informed machine learning (PIML) approach to predict RANS modeled Reynolds stresses discrepancies by utilizing DNS database. The potential impacts include: q  Utilizing current high-fidelity simulations database to improve the accuracy

of RANS simulations q  Assisting turbulence modelers to derive better RANS models q  Inspiring other data-driven modeling approaches in computational mechanics

⌧xy

k

Ux

Re = 5600

RANS

DNS

PIML

RANS DNS PIML

⌧xy k

RANS DNS PIML

{S,⌦,rp,rk}

Tinoco et al., 55th AIAA Aerospace Sciences Meeting, AIAA2017-1208

@Ui

@t

+@(UiUi)

@xj+

1

⇢

@p

@xi� ⌫

@

2Ui

@xj@xj= r · ⌧

@Ui

@t

+@(UiUi)

@xj+

1

⇢

@p

@xi� ⌫

@

2Ui

@xj@xj= r · (⌧RANS +�⌧)