12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022

PREDICTION OF FRICTION DRAG IN PULSATING TURBULENT PIPE FLOW BY DEEPLEARNING FOR IMPROVEMENT OF GENERALIZATION CAPABILITY

Kazunori Matsubara, Akihiko Mitsuishi, Kaoru Iwamoto*, and Akira Murata
Department of Mechanical Systems Engineering
Tokyo University of Agriculture and Technology
2-24-16 Nakacho, Koganei, Tokyo 184-8588, Japan* Corresponding author: iwamotok@cc.tuat.ac.jp

ABSTRACTA new deep learning model was proposed to predict thespatiotemporal development of pulsating turbulent pipe flow.
The model is referred to as the time delay NN-RNN model
which is the combination of the previous long short-term
memory (LSTM) encoder-decoder model and time delay
neural network (TDNN). The flow field obtained by direct
numerical simulation (DNS) was used to train the model.
Being trained on data with different pulsation parametersfrom the test data, the time delay NN-RNN model predicts
the skin friction coefficient more accurately than the previousmodel. Therefore, the generalizability of the model isimproved.

with satisfying precision the global dynamics of the flow. On
the other hand, Yamaguchi et al. (2019) constructed a LSTM
encoder-decoder model which could predict the time
evolution of the flow field in pulsating turbulent pipe flow.
The model successfully predicted a pulsatile flow when its
pulsation parameters are the same as those of training data,
but it failed when they are different.
The objective of this study is to improve thegeneralizability of the model. For that purpose, TDNN isintroduced to the model of Yamaguchi et al. (2019). Then,the predictive performance of the model is evaluated whenthe training and predicting waves are different.
CALCULATION METHODBACKGROUND AND OBJECTIVES
Energy loss due to friction drag is a problem in thetransportation of oil and natural gas through pipelines. Oneof the methods to reduce the friction drag is pulsating control.
Considering that the turbulent vortices decrease/increaseduring acceleration/deceleration, the pulsation can yield dragreduction effect by carefully choosing the relevant
parameters.
There have been many studies on drag reduction forpulsating turbulent flow. Manna and Vacca (2008) confirmeddrag reduction rate up to 27% by large eddy simulation(LES) of pulsating turbulent pipe flow in the case where theamplitude of a sinusoidal wave of the streamwise pressure
gradient to make the pulsation is very large. Lodahl et al.
(1998) conducted experiments on pulsating turbulent pipe
flow to investigate the variation of turbulence. The pulsating
flow is regarded as a combination of oscillatory flow andcurrent. It is found that the pulsating flow relaminarizes if thepulsating flow is oscillatory flow dominated and theoscillatory component of the flow is in the laminar flow
regime.
In recent years, machine learning has been increasingly
applied to the various problem of fluid mechanics (Bruntonet al., 2020). Fan et al. (2021) proposed the CNN+LSTM
module as an efficient alternative to the synthetic eddy
method (SEM) which is time-consuming. The module could
accurately predict turbulence statistics even after a long timeand achieve over 15 times speedup compared with SEM.
Giannopoulos and Aider (2020) predicted the full-field
dynamics of backward-facing step (BFS) flow by focusedtime-delay neural network (FTDNN). Different upstream
local visual sensors, based on the velocity fields measured bytime-resolved particle image velocimetry (PIV), were tested
as inputs for the NN system. The NN system could predict

DNS of Pulsating Turbulent Pipe Flow
In this study, the flow field of pulsating turbulent pipe
flow was calculated by DNS. Governing equations are thecontinuity and Navier-Stokes equations. A highly energyconservative second-order-accurate finite difference method
for the cylindrical coordinate system (Fukagata and Kasagi,2002) is employed for the spatial discretization scheme. The
memory-saving third-order accurate Runge-Kutta method(Spalart et al., 1991) for the advection term and the CrankNicolson method for the diffusion term are employed for thetime integration. The friction Reynolds number is set to Reτ =uτR/ν = 180, where uτ is the friction velocity based on mean
wall shear stress, R the pipe radius, and ν the kinematic
viscosity. The computational domain is shown in Fig. 1. Here,r, θ, and z denote the radial, circumferential, and axial
directions, respectively. The number of computational cellsis (Nr, Nθ, Nz) = (96, 128, 256), and grid resolution is (∆r+,R+∆θ, ∆z+) = (0.35−2.24, 8.83, 8.83). The radial grid width issmall near the wall and large at the center of the pipe. Here,()+ denotes the value normalized by uτ and ν. The periodic
boundary condition is used at θ and z directions, and the noslip boundary condition is used at the pipe wall.
In the pulsating flow, the time change of spatiallyaveraged streamwise pressure gradient is controlled toaccelerate and decelerate the flow. In this study, the flow
field is calculated by giving a sinusoidal pulsating wave as
follows:(1)

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022

Figure 1. Computational domain.

LSTM Encoder
Input
ur** t* − 9∆t* ur** t*uu

LSTM Decoder
Outputur** t* + ∆t* ur** t* + 10 ∆t*uθ*uz *pθθuz**puz**p・・・

Encoder
TDNN

Decoder

Encoder

・・・・・・

TDNN

LSTM

・・・

uθ*uz *p・・・

Decoder

・・・

LSTM

LSTM

・・・

LSTM

TDNN

TDNN・・・

Figure 2. Time delay NN-RNN model.
TimeActivation function is
hyperbolic tangent

t*・・・・・・・・・

t* －∆ t*Output(t*)

t* － k ∆ t*
k : time step
Input

HiddenOutputFigure 3. Structure of TDNN.

where A and T are amplitude and period, respectively. The
angle brackets < > denotes spatial averaged value, and ()*denotes the value normalized by uτ and R.

dimensional flow field. Then, the output of LSTM is
deconvoluted by the decoder to the original size.
In this study, TDNN was added to the model of
Yamaguchi et al. (2019) to improve the generalizability ofthe model. TDNN is composed of three layers: input, hidden,
and output layers as shown in Fig. 3. In addition to thepressure gradient at the current time, the pressure gradient up
to k time steps before is also input into TDNN. Therefore,
extracted features can be related to the time variation of thepast pressure gradient. Several time steps k was tested andfound k = 2 was the highest prediction accuracy. Thus, in thispaper, we report the results of k = 2. The Adam algorithm isapplied as the optimizer. The learning rate, the number ofepochs, and the minibatch size are set to 0.001, 100, and 32,respectively. Here, the training data consist of 14,000 steps

Deep Learning for DNS Data
The time delay NN-RNN model shown in Fig. 2 wasconstructed to predict the time evolution of flow fields:
velocity and pressure (ur*, uθ*, uz*, p*) in r − z cross-section
obtained by DNS. Here, the time interval of flow field ∆t* is
set to 5×10−3. The time delay NN-RNN model consists of theencoder, TDNN, LSTM, and decoder. Inputs to LSTM areflow fields convoluted by the encoder to 1/32 and features ofspatially averaged streamwise pressure gradient extracted byTDNN. LSTM predicts the time evolution of the low

212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022

Figure 4. Time variation of spatially averaged pressure gradient, bulk velocity u b∗, and skin friction coefficient Cf.

Deceleration

Acceleration

Red: u’+ > 3
Blue: u’+ < −3

White: Q+ = 0.03

Figure 5. Visualization results of pulsating turbulent flow with high-speed streak (red), low-speed streak (blue), isosurface of thesecond invariant of velocity gradient tensor (white).

spanning 10 < t*< 80. The test data consist of 14,000 steps
spanning 80 < t*< 150. Table 1 shows the amplitude andperiod of the sinusoidal pulsating wave for training data andtest data. For each pulsating flow, one r − z cross-section isused for training and prediction.

Table 1. Pulsation parameters for training and test data.
(A*, T*)
Training data
Test data(4, 5), (5, 4), (6, 5), (5, 6)(5, 5)RESULTS
Figure 5 shows visualization results of the high-speed
and low-speed streaks, and vortex defined by the second
invariant of velocity gradient tensor during acceleration anddeceleration. It can be seen from the figure that the numberof vortices decreases during acceleration and increases
during deceleration. Comparing Figs. 4 and 5, it is found thatthe friction drag reduces by decreasing vortices during
acceleration, and increases by increasing vortices duringdeceleration.
Figure 6 shows the radial profiles of the phase-averaged
streamwise velocity, RMS value of phase-averaged
streamwise velocity fluctuations, and Reynolds shear stressrespectively. The red and blue line indicate acceleration anddeceleration. Here, each line is mean value of the phaseaverage in each interval shown in the legend of the figure.
The phase for 0 < t*/T* < 1/6, 1/6 < t*/T* < 1/3, and 1/3 <t*/T* < 1/2 represent the beginning, middle, and end ofacceleration, respectively. The phase for 1/2 < t*/T* < 2/3, 2/3< t*/T* < 5/6, and 5/6 < t*/T* < 1 represent the beginning,middle, and end of deceleration, respectively.

Basic Characteristics of DNS Data
The flow fields with the sinusoidal pulsating wave of its
amplitude A* = 5 and period T* = 5 were calculated by DNS.
Figure 4 shows the time variation of spatially averagedpressure gradient, bulk velocity ub ∗ , and skin friction
coefficient Cf calculated by DNS. Here, the skin friction
coefficient Cf is defined by the bulk velocity ub* and the wall
shear stress τw*, as shown in Eq. 2.

(2)

The phase of the bulk velocity delays by π/2 than thespatially averaged pressure gradient. The skin friction
coefficient decreases in acceleration, and vice versa. Thus,
the pulsating turbulent pipe flow is the unsteady turbulent
flow, in which bulk velocity and skin friction coefficient
temporally change.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022

(a) Phase-averaged value(b) RMS value

Acceleration

Deceleration(c) Reynolds shear stress
Figure 6. Radial profiles of phase-averaged streamwise velocity, RMS value of phase-averaged streamwise velocity fluctuations,and Reynolds shear stress.

The phase-averaged streamwise velocity has in all phasesa minimum value near the wall and a maximum value at thepipe center and increases from near the wall to the pipe
center. In the entire region from the wall to the pipe center,the streamwise velocity increases the beginning, middle, andend of acceleration, in that order. On the other hand, thatdecreases the beginning, middle, and end of deceleration, inthat order. In other words, the streamwise velocity increases
and decreases with the acceleration and deceleration of thepulsating flow.
In the RMS value, it has a minimum value near the wall,increases from the viscous sub-layer to the buffer layer, anddecreases from the buffer layer to the pipe center in allphases. The maximum RMS value increases from thebeginning of acceleration to the end, while it decreases fromthe beginning of deceleration to the end. Therefore, inpulsating flow, turbulence increase with acceleration anddecreases with deceleration.
In the Reynold shear stress, the values are zero at thewall and the pipe center, and the values increase rapidly andthen decrease slowly from the wall to the pipe center for allphases. The Reynolds shear stress during acceleration islower than that during deceleration. Comparing Figs. 5 and 6,
there is a high correlation between the increase or decrease in

the vortex and the increase or decrease in the Reynolds shear
stress due to the acceleration and the deceleration. Therefore,in pulsating flow, the vortices increase during thedeceleration term, and the turbulent momentum transfer
becomes more active.

Generalizability of Time Delay NN-RNN Model
Figure 7 shows the prediction results of phase-averaged
streamwise velocity during the acceleration and deceleration
terms along with the prediction results by the previous model
without TDNN (Yamaguchi et al., 2019). The prediction
results of both models reproduce the monotonic increase of
the streamwise velocity from zero at the wall to themaximum at the center of the pipe. In the outer layer, thepresent model with TDNN has higher prediction accuracy
than that without TDNN during the acceleration term.
Figure 8 shows the prediction results of the RMS valueof phase-averaged streamwise velocity fluctuations during
the acceleration term and deceleration term. Focusing on theviscous sub-layer and the buffer layer, the accuracy of boththe models is high during the acceleration term, however, theaccuracy of the present model with TDNN is higher than thatwithout TDNN during the deceleration term. On the otherhand, focusing on the outer layer, the accuracy of bothmodels is low in all phases.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022(b) Deceleration(a) Acceleration

Figure 7. Radial profiles of phase-averaged streamwise velocity.

(b) Deceleration(a) Acceleration

Figure 8. Radial profiles of RMS value of phase-averaged streamwise velocity fluctuations.

Figure 9 shows the prediction and ground truth of phaseaveraged bulk velocity u*b. The relative error and correlation
coefficients are 3.8% and 0.98 without TDNN, −3.4%, and0.99 with TDNN, respectively. The correlation coefficientsof both with TDNN and without TDNN are more than 0.95.
The magnitude of the relative errors of both are less than 5 %.
Hence, both models can predict the periodic change and theaverage value for one period with high accuracy. In the bulkvelocity, the phase difference between ground truth andprediction with TDNN is smaller than without TDNN. It isbecause the time change of pulsating waves can be trained byTDNN.
Figure 10 shows the prediction and ground truth ofphase-averaged wall shear stress τ*w. The relative error andcorrelation coefficients are −13.0% and 0.41 without TDNN,−6.1%, and 0.75 with TDNN, respectively. The correlation
coefficient with TDNN is larger than that without TDNN,
and the magnitude of the relative error with TDNN is smaller
than without TDNN, which means that the prediction
accuracy of the wall shear stress is improved when theTDNN is introduced. In addition, the prediction results ofwall shear stress correspond with the ground truth at t*/T*=0.0 ~ 0.3 without TDNN, at t*/T* =0.0 ~ 0.5 with TDNN. In
other words, it can be said that the introduction of TDNN
improves prediction accuracy. However, the prediction
accuracy is low at t*/T* =0.5 ~ 1.0 during deceleration. Thisis suggested by the lower accuracy of the RMS value near

the wall during the deceleration term compared to theacceleration term.
The phase-averaged bulk velocity and wall shear stressfor each sinusoidal pulsating wave included in the training
data are calculated, and then the averaged values for each ofthem are calculated. As a result, it is found that these values
are consistent with the true values. Therefore, in the current
parameter range, the response of the pulsating turbulent pipe
flow is high linearity. Nevertheless, the generalizability ofthe training model is not sufficient because it does not predict
statistics with very high accuracy.
Figure 11 shows the prediction and ground truth ofphase-averaged skin friction coefficient Cf. The relative error
and correlation coefficients are −18.6% and 0.97 without
TDNN, −1.0%, and 0.95 with TDNN, respectively. The
correlation coefficients of both without TDNN and withTDNN are more than 0.95, which indicates that both models
can predict the periodic change of the skin friction
coefficient. On the other hand, because the absolute value ofthe relative error of the model with TDNN is smaller thanthat without TDNN, the model becomes to accurately predict
the average of the skin friction coefficient to introduce
TDNN. In other words, compared to the previous study, thisstudy can accurately predict the skin friction coefficient,which determines ultimately the drag reduction effect.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022

Figure 10. Prediction and ground truth ofphase-averaged wall shear stress τ*w.

Figure 9. Prediction and ground truth ofphase-averaged bulk velocity u*b.

Figure11. Prediction and ground truth ofphase-averaged skin friction coefficient Cf.

CONCLUSIONS AND OUTLOOKThe deep learning model predicted the spatiotemporaldevelopment of pulsating turbulent pipe flow. To improve
the generalizability of the model, the time delay NN-RNN
model was constructed by combining TDNN with theprevious LSTM encoder-decoder model. The model has
predicted flow fields for pulsating flow which are training
data (A*, T*) = (4, 5), (5, 4), (6, 5), and (5, 6), and test data
(A*, T*) = (5, 5). The new model can predict streamwise
velocity and RMS value of that fluctuations more accurately
than the previous model. Therefore, the generalizability ofthe model is improved. However, despite the high linearity ofthe pulsating flow response, the prediction of statistics for thebulk velocity and wall share stress is low accuracy. Hence,
the generalizability of the training model for this study is not
sufficient.

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with Link-Wise Artificial Compressibility Method",
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Velocimetry Data-Sets", International Journal of Heat andFluid Flow, Vol. 82, 108533.
Yamaguchi, R., Mitsuishi, A., Shimura, T., Iwamoto, K.,
and Murata, A., 2019, "Decomposition of Friction Drag andHeat Transfer in Pulsating Turbulent Pipe Flow",
Proceedings, 2nd Pacific Rim Thermal Engineering
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REFERENCES
Manna, M., and Vacca, A., 2008, "Spectral Dynamic ofPulsating Turbulent Pipe Flow", Computers and Fluids, Vol.
37, Issue 7, pp.825-835.
Lodahl, C. R., Sumer, B. M., and Fredsøe, J., 1998,"Turbulent Combined Oscillatory Flow and Current in aPipe", Journal of Fluid Mechanics, Vol. 373, pp. 313-348.

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