Machine Learning for RANS Turbulence Modelling of Variable Property Flows

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arXiv:2210.15384v1 [physics.flu-dyn] 27 Oct 2022

Rafael Dieza,∗, Stephan Smita, Jurriaan Peetersa, Rene Pecnika

aProcess and Energy Department, Delft University of Technology,

Leeghwaterstraat 39, 2628 CB Delft, The Netherlands

Abstract

This paper presents a machine learning methodology to improve the predictions of traditional RANS turbulence

models in channel ﬂows subject to strong variations in their thermophysical properties. The developed formulation

contains several improvements over the existing Field Inversion Machine Learning (FIML) frameworks described

in the literature, as well as the derivation of a new modelling technique. We ﬁrst showcase the use of eﬃcient

optimization routines to automatize the process of ﬁeld inversion in the context of CFD, combined with the use of

symbolic algebra solvers to generate sparse-eﬃcient algebraic formulas to comply with the discrete adjoint method.

The proposed neural network architecture is characterized by the use of an initial layer of logarithmic neurons followed

by hyperbolic tangent neurons, which proves numerically stable. The machine learning predictions are then corrected

using a novel weighted relaxation factor methodology, that recovers valuable information from otherwise spurious

predictions. The study uses the K-fold cross-validation technique, which is beneﬁcial for small datasets. The results

show that the machine learning model acts as an excellent non-linear interpolator for DNS cases well-represented in

the training set, and that moderate improvement margins are obtained for sparser DNS cases. It is concluded that the

developed machine learning methodology corresponds to a valid alternative to improve RANS turbulence models in

ﬂows with strong variations in their thermophysical properties without introducing prior modeling assumptions into

the system.

Keywords: Turbulence modelling, Machine learning, Variable Properties

1. Introduction

1.1. Turbulence modelling

The governing equations of ﬂuid ﬂow have long been established, yet modelling turbulence remains one of the

biggest challenges in engineering and physics. While it is possible to resolve the smallest scales of turbulent ﬂows

using direct numerical simulations (DNS), DNS is still unfeasible for real-world engineering applications. Due to

this reason, engineers must rely on RANS turbulence models to describe turbulent ﬂows. However, most of the de-

velopment of turbulence models has focused on isothermal incompressible ﬂuids. Therefore, these models can be

inaccurate when applied to ﬂows with strong variations in their thermophysical properties [1, 2], such as supercritical

ﬂuids or hypersonic ﬂows. Understanding the behaviour of ﬂows subject to strong property gradients is critical for

several engineering applications, such as heat exchangers, supersonic aircraft, turbomachinery, and various applica-

tions in the chemical industry [3, 4, 5, 6, 7]. Even incompressible ﬂuids, such as water, can present large changes in

viscosity when subjected to temperature variations.

For incompressible constant-property ﬂows, the governing parameter in the description of turbulent boundary

layers is the Reynolds number. For compressible ﬂows, the Mach number and the associated changes in properties

become additional parameters that characterize turbulent wall-bounded ﬂows. From past studies, it is known that dif-

ferences between a supersonic and a constant-property ﬂow can be explained by simply accounting for the mean ﬂuid

property variations, as long as the Mach number remains small [8]. This result is known as Morkovin’s hypothesis

∗Corresponding author

Email address: R.G.DiezSanhueza-1@tudelft.nl (Rafael Diez)

Preprint submitted to Computers &Fluids October 28, 2022

[9]. DNS of compressible channel ﬂows [10] also suggest that in the near-wall region most of the density and temper-

ature ﬂuctuations are the result of solenoidal ’passive mixing’ by turbulence. Previous work by Patel et al. [11] has

provided a mathematical basis for the use of the semi-local scaling as proposed by Huang et al. [12]. It was concluded

that under the limit of small property ﬂuctuations in highly turbulent ﬂows, a change in turbulence is governed by

wall-normal gradients of the semi-local Reynolds number, deﬁned as:

Re⋆

τ≡pρ/ρw

µ/µw

Reτ,(1)

where ρis the density, µdynamic viscosity, the bar denotes Reynolds averaging, the subscript windicates the value at

the wall, and Reτis the friction Reynolds number based on wall quantities and the half channel height, h. Thus, Re⋆

provides a scaling parameter which accounts for the inﬂuence of variable properties on turbulent ﬂows.

With the semi-local scaling framework and the fact that variable property turbulent ﬂows can be successfully

characterized by Re⋆

τ, two main developments followed. First, in Patel et al. [13], a velocity transformation was

proposed which allows to collapse mean velocity proﬁles of turbulent channel ﬂows for a range of diﬀerent density

and viscosity distributions. Although following a diﬀerent approach, this transformation is equivalent to the one

proposed by Trettel and Larsson [14]. Second, this insight has later been used in Pecnik and Patel [3] to extend the

semi-local scaling framework to derive an alternative form of the turbulent kinetic energy (TKE) equation. It was

shown that the individual budget terms of this semi-locally scaled (TKE) equation can be characterized by the semi-

local Reynolds number and that eﬀects, such as solenoidal dissipation, pressure work, pressure diﬀusion and pressure

dilatation, are indeed small for the ﬂows investigated. Based on the semi-locally scaled TKE equation, Rodriguez

et al. [1] derived a novel methodology to improve a range of eddy viscosity models. The major diﬀerence of the new

methodology, compared to conventional turbulence models, is the formulation of the diﬀusion term in the turbulence

scalar equations.

While these corrections improve the results of RANS turbulence models signiﬁcantly, they can still be subject to

further improvements. Due to these reasons, the present investigation will focus on building ML models to improve

the performance of existing RANS turbulence models.

1.2. Machine Learning

In recent years, machine learning has been successfully applied in ﬂuid mechanics and heat transfer due to its

inherent ability to learn from complex data, see for instance Chang et al. [15]. While diﬀerent ML methods are avail-

able, deep neural networks have emerged as one of the most promising alternatives to improve turbulence modelling

[16]. These systems are able to approximate complex non-linear functions by using nested layers of non-linear trans-

formations, which can be adapted to the context of every application to optimize the usage of computational resources

and to mitigate over-ﬁtting. Diﬀerent types of neural networks currently hold the state-of-the-art accuracy record in

challenging domains, such as computer-vision or natural-language processing [17]. During the last decade, one of the

main reasons behind the success of deep learning has been the ability of neural networks to both approximate general

non-linear functions while providing multiple alternatives to optimize their design.

Signiﬁcant works in the context of deep learning applied to CFD can be found in the studies of Ling et al. [18],

who developed deep neural networks to model turbulence with embedded Galilean invariance, or in the work of Parish

and Duraisamy [19], where ﬁeld inversion machine learning (FIML) is proposed in the context of CFD. Despite the

abundance of recent works, signiﬁcant research is still required regarding the application of ML in the context of

CFD, and rich datasets to study turbulence in complex conditions must still be outlined. The future availability of

datasets to study turbulence in complex geometries is particularly promising, as this could yield new models with

strong applications to industrial and environmental problems.

The methodology for the present study is based on the FIML framework proposed by Parish and Duraisamy [19].

This methodology focuses on building corrections for existing RANS turbulence models instead of attempting to

rebuild existing knowledge entirely. In the FIML framework, the process of building machine learning models is

split into two stages. In the ﬁrst stage, a data gathering process known as ﬁeld inversion is performed, where the

objective is to identify an ideal set of corrections for the RANS turbulence model under study. Then, in the second

stage, a machine learning system is trained in order to replicate the corrections identiﬁed. The main advantage of

this procedure is that the training process of a neural network is eﬀectively decoupled from the CFD solver, thereby

improving the eﬃciency of the procedure by several orders of magnitude.

For the present work, several modiﬁcations are proposed with respect to the study made by Parish and Duraisamy

[19] and the subsequent publications of Singh et al. [20, 21, 22]. The modiﬁcations considered cover diﬀerent stages

of the problem; such as the optimization methods employed in ﬁeld inversion, the generation of automatic formulas

to compute the gradients of the CFD system, the possibility to automate the process of generating feature groups for

the ML system, and novel methods to improve the stability of the FIML methodology while making predictions.

2. Fully developed turbulent channel ﬂows

In this work we consider fully developed turbulent channel ﬂows for which a large number of available DNS

studies exist, and for which the time and space averaged conservation equations can be substantially simpliﬁed.

2.1. DNS database

The DNS database of turbulent planar channel ﬂows consists of three diﬀerent sets of simulations. The ﬁrst set

represents variable property low-Mach number channel ﬂows with isothermal walls, heated by a uniform volumetric

source to induce an increase of temperature within the channel [3, 13, 23]. Using diﬀerent constitutive relations for

viscosity µ, density ρand thermal conductivity λas a function of temperature, diﬀerent DNS cases are used to study

the eﬀect of varying local Reynolds and Prandtl number on near wall turbulence. The cases with their respective

relations for the transport properties and their corresponding wall-friction velocity based Reynolds number and local

Prandtl number are summarized in table 1 (low-Mach number cases). Most of the cases have a friction based Reynolds

number at the wall of Reτ=395. Depending on the distribution of density, viscosity, and conductivity, the semi-local

Reynolds number Re⋆

τand the local Prandtl number are either constant, increasing or decreasing from the walls to the

channel center. More details on the cases can be found in Refs. [13, 23, 3]. The second set of DNS consists of high-

Mach number compressible channel ﬂow simulations with air modeled as a calorically perfect gas [14] (high-Mach

number cases). The Mach number ranges from 0.7 to 4 and the corresponding constitutive laws for the transport

properties, Reτand Prandtl number Pr are summarized in table 1 as well. The third set of simulations contains

incompressible channel ﬂows [24] (incompressible cases). These cases been added as an additional set to train the

FIML framework to account for a large range in Reynolds numbers for incompressible ﬂows.

For all of the variable property DNS cases, it is possible to show that Morkovin’s hypothesis applies [11]. This

hypothesis establishes that only the averaged values in molecular properties can be used to characterize the changes

in turbulence, and that any higher-order correlations of turbulent ﬂuctuations observed in these properties have a

negligible impact in the mean balances [11, 10].

2.2. RANS equations

To model the turbulent channel ﬂows described above, we use the Reynolds/Favre averaged Navier-Stokes equa-

tions. For a fully developed turbulent channel ﬂow, the only in-homogeneous direction of the averaged ﬂow corre-

sponds to the wall-normal coordinate, leading to a set of one-dimensional partial diﬀerential equations for the mean

momentum, mean energy and any additional transport equations for the turbulence quantities used to close the RANS

equations. The Reynolds/Favre averaged streamwise momentum and energy equations for a fully developed turbulent

channel ﬂow read

∂

∂y" µ

Reτ,w

+µt!∂u

∂y#=−1,(2)

∂

∂y" λ

Reτ,wPrw

+cpµt

Prt!∂T

∂y#=−Ecτ,w µ

Reτ,w

+µt! ∂u

∂y!2

−φ

Reτ,wPrw

,(3)

with the variables uand Treferring to the Favre-averaged streamwise velocity and the cross-sectional temperature

proﬁles, respectively. The variables cp,Prtand Sφrefer to the isobaric heat capacity, the turbulent Prandtl number, and

an arbitrary volumetric heat source term. The coordinates xand yfurther refer to the streamwise and the wall-normal

Number Case ID ρ µ λ Reτ,wPrwEcτ,wφ

Low-Mach number cases [13, 23, 3]

1CP150 1 1 1 150 1 0 0

2 CP395 1 1 1 395 1 0 17.55

3CP550 1 1 1 550 1 0 0

4CRe⋆

τT−1T−0.51 395 1 0 17.55

5S Re⋆

τGL 1T1.21 395 1 0 18.55

6GL T −1T0.71 395 1 0 17.55

7LL1 1 T−11 150 1 0 29

8S Re⋆

τLL T0.6T−0.75 1 150 1 0 31.5

9S Re⋆

τCν1T−0.51 395 1 0 17.55

10 CνT−1T−11 395 1 0 16

11 LL2 1 T−11 395 1 0 17.55

12 CRe⋆

τCPr⋆T−1T−0.5T−0.5395 1 0 17.55

13 GLCPr⋆T−1T0.7T0.7395 1 0 17.55

14 VλS Pr⋆

LL 1 1 T1395 1 0 17.55

15 CP395Pr41 1 1 395 4 0 34

16 JF M.CRe⋆

τT−1T−0.51 395 1 0 95

17 JF M.GL T −1T0.71 950 1 0 75

18 JF M.LL 1T−11 150 1 0 62

High-Mach number cases [14]

19 M0.7R400 437 5.736·10−4

20 M0.7R600 652 5.190·10−4

21 M1.7R200 322 2.804·10−3

22 M1.7R400 663 2.394·10−3

23 M1.7R600 ∝T−1T0.75 T0.75 972 0.7 2.135·10−30

24 M3.0R200 650 4.751·10−3

25 M3.0R400 1232 4.185·10−3

26 M3.0R600 1876 3.752·10−3

27 M4.0R200 1017 5.574·10−3

Incompressible cases [24]

28 IC.Re180 - - - 180 - - -

29 IC.Re550 - - - 550 - - -

30 IC.Re950 - - - 950 - - -

31 IC.Re2000 - - - 2000 - - -

32 IC.Re4200 - - - 4200 - - -

Table 1: DNS database of turbulent channel ﬂows with variable properties (low-Mach)[13, 3], with ideal gases at high-Mach numbers [14], and

with constant properties (incompressible) [24].

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arXiv:2210.15384v1[physics.flu-dyn]27Oct2022MachineLearningforRANSTurbulenceModellingofVariablePropertyFlowsRafaelDieza,∗,StephanSmita,JurriaanPeetersa,RenePecnikaaProcessandEnergyDepartment,DelftUniversityofTechnology,Leeghwaterstraat39,2628CBDelft,TheNetherlandsAbstractThispaperpresentsamachinelea...

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Machine Learning for RANS Turbulence Modelling of Variable Property Flows

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