This paper investigates the application of data-driven approach to the optimization of cavitating flow simulations. An evaluation of the performance of commonly used RANS models (k-e, k-w and k-w SST) is presented by comparison with high fidelity data (DNS solution and X-ray experimental measurements). An ensemble based variational method is introduced and used to reconstruct the inlet velocity and calibrate the empirical parameters in the turbulence model and the cavitation model. Machine learning approach is discussed to construct a discrepancy function of the Reynolds stresses to address the RANS model-form uncertainty

COUTIER-DELGOSHA, Olivier

GOMEZ, Thomas

XIAO, HENG

ZHANG, Xinlei

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Science Arts & Métiers (SAM)
is an open access repository that collects the work of Arts et Métiers ParisTech
researchers and makes it freely available over the web where possible.
This is an author-deposited version published in: https://sam.ensam.eu
Handle ID: .http://hdl.handle.net/10985/15458
To cite this version :
Xinlei ZHANG, Thomas GOMEZ, HENG XIAO, Olivier COUTIER-DELGOSHA - Optimization of
cavitating flows simulation with data driven approach: from data assimilation to machine learning -
In: The 10th International Symposium on Cavitation, Etats-Unis, 2018-05 - The 10th International
Symposium on Cavitation (CAV 2018) - 2018
Any correspondence concerning this service should be sent to the repository
Administrator : archiveouverte@ensam.eu
*Corresponding Author, Xinlei Zhang: xinlei.zhang@ensam.eu 
  
Optimization of cavitating flows simulation with data driven approach: from data 
assimilation to machine learning 
1
Xinlei Zhang*; 
2
Thomas Gomez, 
3
Heng Xiao, 
1,3
Olivier Coutier-Delgosha 
1 Arts et Métiers ParisTech, Lille, France ; 2 Université Lille Nord de France , Villeneuve d’Ascq, France; 3Virginia Tech, 
Aerospace & Ocean engineering, Blacksburg, VA, USA 
Abstract  
This paper investigates the application of data-driven approach to the optimization of 
cavitating flow simulations. An evaluation of the performance of commonly used 
RANS models (k-e, k-w and k-w SST) is presented by comparison with high fidelity 
data (DNS solution and X-ray experimental measurements). An ensemble based 
variational method is introduced and used to reconstruct the inlet velocity and calibrate 
the empirical parameters in the turbulence model and the cavitation model. Machine 
learning approach is discussed to construct a discrepancy function of the Reynolds 
stresses to address the RANS model-form uncertainty. 
Keywords: RANS models; cavitating flows; data assimilation; machine learning 
1. Introduction 
Turbulent cavitating flows occur in many engineering practical applications such as pumps, propellers and 
nuclear reactors. The collapse of the cavitation bubbles in these devices can produce major detrimental effects, 
such as flow rate fluctuations, noise, vibrations, and erosion. It is thus essential to accurately predict the 
behavior of unsteady cavitation, to reduce their consequences for the machinery. 
After decades of efforts, the most commonly used approach to simulate these turbulent cavitating flows is still 
the Reynolds Averaged Navier Stokes (RANS) method combined with homogeneous cavitation models, thanks 
to its computational tractability. However, it is a consensus that RANS models perform poorly in cases of 
complex flows with separations, mean pressure gradients and/or curvatures. This limitation also leads to the 
poor prediction of the interactions between cavitation and turbulence in the specific case of cavitating flows. To 
address this issue, diverse data-driven approaches [1][2][3] have been developed in the turbulence community to 
improve the predictive accuracy of the RANS method from the perspective of quantifying and reducing the 
uncertainty due to boundary condition inconformity or model inadequacy. Especially, data assimilation methods 
and machine learning are both promising approaches and have demonstrated successfully their merits in the 
improvement of RANS model predictions. Accordingly, the goal of the present work is to investigate the 
application of the data-driven approaches (data assimilation and physics-informed machine learning approach) 
to the simulation of turbulent cavitating flows, thus contributing to an enhanced understanding of the complex 
flow physics. 
2. Limitations of the current RANS models 
2.1 CFD solver 
A 2D unsteady compressible solver is applied to conduct the numerical simulation. The code is based on the 
coupling of a homogenous one-fluid cavitation model with a two-equation RANS turbulence model. The 
governing Navier-Stokes equations and transport equation of void fraction can be expressed as: 
  
  
            = S                                     (1) 
   
  
   
   
 
         
   
   
   
    
  
  
   
    
   
   
  
        
 
   
   
 
  
 
   
   
 
  
where                is the mixture density,   is void fraction,    and    denote the convective and 
viscous flux densities, and S is the source term. 
2.2. Comparison of RANS model prediction with high fidelity data 
To eliminate the effects of the cavitation model, the comparison is performed here with non-cavitating flows. 
The performance of three RANS models (k-e, k-w & k-w SST) is investigated in two different canonical flows, 
namely i) the non-cavitating turbulent flow in a 2D Venturi type section, ii) the turbulent flow in the 
configuration of the Wallturb Bump [5]  
A. non-cavitating flow in the Venturi type section 
In the venturi-type section, the prediction on tangential velocity u with current RANS models is compared with 
experimental measurements based on X-ray imaging. The comparison is focused on the area immediately 
downstream the venturi throat along the bottom wall. The experiment set-up is detailed in [4]. 
 
Figure 1: Top: experimental observation of velocity u; Bottom: comparison of velocity u at different positions where ‘k-w wf’ indicate k-w 
model associated with wall function and ‘k-w lr’ indicate k-w model associated with low Reynolds method. 
 
From Figure 1, it can be seen that in the region far from the wall, the results of all these RANS simulations can 
fit well with experimental data; however, a quite large discrepancy can be figured out in the region near the wall, 
which indicates that the adverse pressure gradient(APG) is poorly predicted. Generally, these RANS models 
cannot predict the velocity u near the wall with confidence. 
B. turbulence flow in configuration of Wallturb bump 
The Wallturb bump geometry used in this work is presented in Figure 2. The reference DNS data [5] is used to 
estimate the predictive accuracy of Quantities of interest (QoIs) at wall with the RANS simulations (k-w with 
low Reynolds correction and k-w SST). 
 
Figure 2 Profile of the Wallturb Bump 
 
Figure 3. Comparison results of friction coefficient cf and pressure coefficient cp at the bottom wall 
The friction and pressure coefficient evolutions at the bottom wall are presented in Figure 3. In the favorable 
pressure gradient region, it has a good agreement with DNS result with both k-w SST and k-w model; However, 
in the APG region, the discrepancies tend to increase for both the friction coefficient and the pressure coefficient, 
although the k-w SST model performs quite better than the k-w. It indicates that current RANS models perform 
poorly regarding the prediction in the APG region. 
3. Data assimilation 
In light of these limitations of the current RANS models, a data assimilation technique is conducted to improve 
the predictions. It has been previously demonstrated [6] that quantifying the parameter uncertainty related to the 
cavitation & turbulence models and the inlet conditions is feasible with the data assimilation technique. In this 
work, we apply an Ensemble-based variational data assimilation scheme (4DEnVar) [1] to unsteady cavitating 
flows. 4DEnVar is a non-intrusive method combining the variational method and an Ensemble Kalman Filter. 
Specifically, the formulation of the variation method uses a Monte Carlo ensemble instead of adjoint operators 
to estimate the prior statistics. The control vector can be expressed as: 
              
     
where   , in each cell, refers to the initial state vector formed with the components of   in equation (1), and 
   refers to the inlet boundary condition and/or the model parameters at time n. After making a first guess 
about the optimal vector to initiate the background (prior knowledge)     , stochastic modeling can be 
performed to sample     ensemble of realizations around the background. Then the control vector   
(posterior knowledge) is expressed as:  
                                                                   (2) 
As for the observation (reference data), the stochastic model extension can be expressed as  
                                              (3) 
Where h is the observation operator that maps the state space to the observation space, and    is the possible 
measurement error, which is assumed to be a zero-mean, uncorrelated Gaussian random field. 
Based on the Bayesian inference, maximizing a posterior (MAP) estimation amounts to minimizing a cost 
function (4). This cost function is composed of two parts: (i) the difference between the control vector and the 
background and (ii) the difference between the CFD results based on the prior knowledge and the observation, 
weighed by the inverse of background covariance B and observation covariance C, respectively. 
  
 
 
        
   
 
  
 
 
 
             
                           (4) 
Where       
       
         
       
   
      
   
        
           
   
   
with       
        ;       
        ;  
 
     
      ;       
  
  
            
                 
   
   
            
       
      (5) 
In order to minimize the cost function (4), one iteration of Newton CG method is performed with (5). The 
obtained   is used to update the optimal vector   according to (2). This iterative process is continued until the 
maximum iteration number is reached. 
4. Machine learning approach 
The data assimilation approach can infer improved boundary conditions and empirical parameters for the 
turbulence and cavitation models. However, it can only provide the uncertainty within the framework of the 
RANS approach, while the main source of uncertainty in RANS simulations is due to the constitutive 
assumptions of the RANS modeling. Therefore, over the past few years, machine learning approaches have been 
developed to address these model-form uncertainties.  
The physics informed machine learning approach (PIML) [2] has been applied to construct the discrepancy 
function in terms of Reynolds stress. In cavitating flows, by taking the X-ray experimental data as training data 
set, it is expected to extend the application of the PIML framework from steady flow at low Reynolds numbers 
to unsteady flow at high Reynolds numbers, and further to cavitating flows. It is of primary importance to define 
the input features in cavitating condition. Besides the ten features suggested in [2], some other features like the 
homogeneous density, the void fraction and void fraction gradient should also be embedded. Moreover, the 
challenge of propagation from Reynolds stress to the QoIs (mean velocity and pressure) needs to be tackled.  
References 
[1] Mons, V., Chassaing, J.-C., Gomez, T., & Sagaut, P. (2016). Reconstruction of unsteady viscous flows 
using data assimilation schemes. Journal of Computational Physics, 316, pp.255–280.  
[2] J Wang, J Wu and H Xiao. (2017). Physics-Informed Machine Learning Approach for Reconstructing 
Reynolds Stress Modeling Discrepancies based on DNS Data, Phys. Rev. Fluids 2, 034603. 
[3] Ling, J., Kurzawski, A., & Templeton, J. (2016). Reynolds averaged turbulence modelling using deep 
neural networks with embedded invariance. Journal of Fluid Mechanics, 807, pp.155–166. 
[4] Khlifa, I., Vabre, A., Hočevar, M., Fezzaa K., Fuzier S., Coutier-Delgosha O. (2017). Fast X-ray imaging 
of cavitating flows, Exp in Fluids 58: 157. 
[5] Marquillie, M., Laval, J. P., and Dolganov, R.(2008). Direct Numerical Simulation of Separated Channel 
Flows with a Smooth Profile, Journal of Turbulence, Vol. 9, No. 1, pp.1–23. 
[6] P. M. Congedo, E. Goncalves, M. G. Rodio.(2015). About the uncertainty quantification of turbulence and 
cavitation models in cavitating flows simulations. European Journal of Mechanics - B/Fluids, 53, pp.190-204. 


English

Optimization of cavitating flows simulation with data driven approach: from data assimilation to machine learning

SAM : Science Arts et Métiers

Science Arts & Métiers (SAM)is an open access repository that collects the work of Arts et Métiers Institute ofTechnology researchers and makes it freely available over the web where possible.This is an author-deposited version published in: https://sam.ensam.euHandle ID: .http://hdl.handle.net/10985/15458To cite this version :Xinlei ZHANG, Thomas GOMEZ, HENG XIAO, Olivier COUTIER-DELGOSHA - Optimization ofcavitating flows simulation with data driven approach: from data assimilation to machine learning -In: The 10th International Symposium on Cavitation, Etats-Unis, 2018-05 - The 10th InternationalSymposium on Cavitation (CAV 2018) - 2018Any correspondence concerning this service should be sent to the repositoryAdministrator : archiveouverte@ensam.eu*Corresponding Author, Xinlei Zhang: xinlei.zhang@ensam.eu   Optimization of cavitating flows simulation with data driven approach: from data assimilation to machine learning 1Xinlei Zhang*; 2Thomas Gomez, 3Heng Xiao, 1,3Olivier Coutier-Delgosha 1 Arts et Métiers ParisTech, Lille, France ; 2 Université Lille Nord de France , Villeneuve d’Ascq, France; 3Virginia Tech, Aerospace & Ocean engineering, Blacksburg, VA, USA Abstract  This paper investigates the application of data-driven approach to the optimization of cavitating flow simulations. An evaluation of the performance of commonly used RANS models (k-e, k-w and k-w SST) is presented by comparison with high fidelity data (DNS solution and X-ray experimental measurements). An ensemble based variational method is introduced and used to reconstruct the inlet velocity and calibrate the empirical parameters in the turbulence model and the cavitation model. Machine learning approach is discussed to construct a discrepancy function of the Reynolds stresses to address the RANS model-form uncertainty. Keywords: RANS models; cavitating flows; data assimilation; machine learning 1. Introduction Turbulent cavitating flows occur in many engineering practical applications such as pumps, propellers and nuclear reactors. The collapse of the cavitation bubbles in these devices can produce major detrimental effects, such as flow rate fluctuations, noise, vibrations, and erosion. It is thus essential to accurately predict the behavior of unsteady cavitation, to reduce their consequences for the machinery. After decades of efforts, the most commonly used approach to simulate these turbulent cavitating flows is still the Reynolds Averaged Navier Stokes (RANS) method combined with homogeneous cavitation models, thanks to its computational tractability. However, it is a consensus that RANS models perform poorly in cases of complex flows with separations, mean pressure gradients and/or curvatures. This limitation also leads to the poor prediction of the interactions between cavitation and turbulence in the specific case of cavitating flows. To address this issue, diverse data-driven approaches [1][2][3] have been developed in the turbulence community to improve the predictive accuracy of the RANS method from the perspective of quantifying and reducing the uncertainty due to boundary condition inconformity or model inadequacy. Especially, data assimilation methods and machine learning are both promising approaches and have demonstrated successfully their merits in the improvement of RANS model predictions. Accordingly, the goal of the present work is to investigate the application of the data-driven approaches (data assimilation and physics-informed machine learning approach) to the simulation of turbulent cavitating flows, thus contributing to an enhanced understanding of the complex flow physics. 2. Limitations of the current RANS models 2.1 CFD solver A 2D unsteady compressible solver is applied to conduct the numerical simulation. The code is based on the coupling of a homogenous one-fluid cavitation model with a two-equation RANS turbulence model. The governing Navier-Stokes equations and transport equation of void fraction can be expressed as:                 = S                                     (1)                                                                                  where                is the mixture density,   is void fraction,    and    denote the convective and viscous flux densities, and S is the source term. 2.2. Comparison of RANS model prediction with high fidelity data To eliminate the effects of the cavitation model, the comparison is performed here with non-cavitating flows. The performance of three RANS models (k-e, k-w & k-w SST) is investigated in two different canonical flows, namely i) the non-cavitating turbulent flow in a 2D Venturi type section, ii) the turbulent flow in the configuration of the Wallturb Bump [5]  A. non-cavitating flow in the Venturi type section In the venturi-type section, the prediction on tangential velocity u with current RANS models is compared with experimental measurements based on X-ray imaging. The comparison is focused on the area immediately downstream the venturi throat along the bottom wall. The experiment set-up is detailed in [4].  Figure 1: Top: experimental observation of velocity u; Bottom: comparison of velocity u at different positions where ‘k-w wf’ indicate k-w model associated with wall function and ‘k-w lr’ indicate k-w model associated with low Reynolds method.  From Figure 1, it can be seen that in the region far from the wall, the results of all these RANS simulations can fit well with experimental data; however, a quite large discrepancy can be figured out in the region near the wall, which indicates that the adverse pressure gradient(APG) is poorly predicted. Generally, these RANS models cannot predict the velocity u near the wall with confidence. B. turbulence flow in configuration of Wallturb bump The Wallturb bump geometry used in this work is presented in Figure 2. The reference DNS data [5] is used to estimate the predictive accuracy of Quantities of interest (QoIs) at wall with the RANS simulations (k-w with low Reynolds correction and k-w SST).  Figure 2 Profile of the Wallturb Bump  Figure 3. Comparison results of friction coefficient cf and pressure coefficient cp at the bottom wall The friction and pressure coefficient evolutions at the bottom wall are presented in Figure 3. In the favorable pressure gradient region, it has a good agreement with DNS result with both k-w SST and k-w model; However, in the APG region, the discrepancies tend to increase for both the friction coefficient and the pressure coefficient, although the k-w SST model performs quite better than the k-w. It indicates that current RANS models perform poorly regarding the prediction in the APG region. 3. Data assimilation In light of these limitations of the current RANS models, a data assimilation technique is conducted to improve the predictions. It has been previously demonstrated [6] that quantifying the parameter uncertainty related to the cavitation & turbulence models and the inlet conditions is feasible with the data assimilation technique. In this work, we apply an Ensemble-based variational data assimilation scheme (4DEnVar) [1] to unsteady cavitating flows. 4DEnVar is a non-intrusive method combining the variational method and an Ensemble Kalman Filter. Specifically, the formulation of the variation method uses a Monte Carlo ensemble instead of adjoint operators to estimate the prior statistics. The control vector can be expressed as:                    where   , in each cell, refers to the initial state vector formed with the components of   in equation (1), and    refers to the inlet boundary condition and/or the model parameters at time n. After making a first guess about the optimal vector to initiate the background (prior knowledge)     , stochastic modeling can be performed to sample     ensemble of realizations around the background. Then the control vector   (posterior knowledge) is expressed as:                                                                     (2) As for the observation (reference data), the stochastic model extension can be expressed as                                                (3) Where h is the observation operator that maps the state space to the observation space, and    is the possible measurement error, which is assumed to be a zero-mean, uncorrelated Gaussian random field. Based on the Bayesian inference, maximizing a posterior (MAP) estimation amounts to minimizing a cost function (4). This cost function is composed of two parts: (i) the difference between the control vector and the background and (ii) the difference between the CFD results based on the prior knowledge and the observation, weighed by the inverse of background covariance B and observation covariance C, respectively.                                                              (4) Where                                                                   with               ;               ;              ;                                                                       (5) In order to minimize the cost function (4), one iteration of Newton CG method is performed with (5). The obtained   is used to update the optimal vector   according to (2). This iterative process is continued until the maximum iteration number is reached. 4. Machine learning approach The data assimilation approach can infer improved boundary conditions and empirical parameters for the turbulence and cavitation models. However, it can only provide the uncertainty within the framework of the RANS approach, while the main source of uncertainty in RANS simulations is due to the constitutive assumptions of the RANS modeling. Therefore, over the past few years, machine learning approaches have been developed to address these model-form uncertainties.  The physics informed machine learning approach (PIML) [2] has been applied to construct the discrepancy function in terms of Reynolds stress. In cavitating flows, by taking the X-ray experimental data as training data set, it is expected to extend the application of the PIML framework from steady flow at low Reynolds numbers to unsteady flow at high Reynolds numbers, and further to cavitating flows. It is of primary importance to define the input features in cavitating condition. Besides the ten features suggested in [2], some other features like the homogeneous density, the void fraction and void fraction gradient should also be embedded. Moreover, the challenge of propagation from Reynolds stress to the QoIs (mean velocity and pressure) needs to be tackled.  References [1] Mons, V., Chassaing, J.-C., Gomez, T., & Sagaut, P. (2016). Reconstruction of unsteady viscous flows using data assimilation schemes. Journal of Computational Physics, 316, pp.255–280.  [2] J Wang, J Wu and H Xiao. (2017). Physics-Informed Machine Learning Approach for Reconstructing Reynolds Stress Modeling Discrepancies based on DNS Data, Phys. Rev. Fluids 2, 034603. [3] Ling, J., Kurzawski, A., & Templeton, J. (2016). Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. Journal of Fluid Mechanics, 807, pp.155–166. [4] Khlifa, I., Vabre, A., Hočevar, M., Fezzaa K., Fuzier S., Coutier-Delgosha O. (2017). Fast X-ray imaging of cavitating flows, Exp in Fluids 58: 157. [5] Marquillie, M., Laval, J. P., and Dolganov, R.(2008). Direct Numerical Simulation of Separated Channel Flows with a Smooth Profile, Journal of Turbulence, Vol. 9, No. 1, pp.1–23. [6] P. M. Congedo, E. Goncalves, M. G. Rodio.(2015). About the uncertainty quantification of turbulence and cavitation models in cavitating flows simulations. European Journal of Mechanics - B/Fluids, 53, pp.190-204. 

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Optimization of cavitating flows simulation with data driven approach: from data assimilation to machine learning

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