Inferring temperature fields from concentration fields in channel flows using conditional generative adversarial networks

Mixed convection flows occur in a wide range of industrial processes and natural phenomena.^1–3 Free and forced convection effects are comparable in mixed convection, leading to buoyancy-driven secondary flows as well as unstable and inhomogeneous flow fields. The flow characteristics of mixed convection are known to enhance heat transfer up to 4–5 folds as compared to pure forced convection.^4–8 Thus, a better understanding of heat and mass transfer in mixed convection flows is necessary to optimize the design of cooling systems for electronic devices, compact heat exchangers, cooling cores of nuclear reactors, and enhanced geothermal systems. For this purpose, extensive investigations have been performed to study heat transfer under different channel geometries and operating conditions.^9–12 

In mixed convection, both free convection and forced convection contribute to the heat transfer process, where the interplay between bulk fluid flow caused by forced convection and buoyancy-induced flow caused by temperature gradients can lead to complex flow patterns. Applications involving heat transfer in channel flows often involve not only heat transport but also mass (solute) transport. Thus, buoyancy-induced flow, caused by temperature differences, leads to a complex coupling among fluid flow, mass and heat transport, and mixing in single 3D channels. Obtaining accurate 3D temperature fields is essential for understanding this complex coupled system and determining how buoyancy-induced flow affects heat transfer in channel flows.

A number of previous studies have attempted to measure and visualize the temperature fields in 3D channels. For example, Patil and Vijay Babu¹³ experimentally studied mixed convection through a plain square duct in a laminar flow. However, the thermocouples installed at axial locations along the wall of the duct could only measure the inner wall temperature and did not capture the temperature fields at cross sections. Sandhu and Siddiqui¹⁴ successfully measured temperature fields at cross sections of a flat-plate solar collector tube, albeit with limited resolution. Even in controlled laboratory settings, measuring and visualizing the temperature fields in 3D channels pose significant challenges, especially when mixed convection flows are involved.

The transformation between temperature and concentration fields is feasible when the dimensionless governing equations for heat and mass transfer processes are analogous. Thus, one may envision inferring temperature fields from concentration fields because measuring the 3D solute concentration fields is typically easier than measuring the 3D temperature fields. In heat and mass transfer research, analogy functions have been widely used. When heat and mass transfer processes follow dimensionless governing equations of the same form, the temperature and concentration fields are correlated and can be interconverted, a phenomenon referred to as the heat and mass transfer analogy.¹⁵ For example, the temperature profile of the boundary layer can be derived from the concentration profile using the dimensionless governing equation or, in some cases, empirical analogy functions.¹⁶ The heat and mass transfer analogy has found extensive applications in determining heat transfer characteristics, such as heat transfer coefficients, in a variety of convection systems.^17–21 However, it is difficult to mathematically derive governing equations or analogy functions when multiple nonlinear effects should be taken into consideration in a heat transfer problem, such as mixed convection flows.

Recently, a variety of fields, including medicine, health care, robotics, and basic research, have benefited from machine learning (ML).^22–26 Due to the adoption of nonlinear activation functions, ML algorithms have the advantage of expressing highly nonlinear functions without being restricted by mathematical functional forms. In the area of fluid mechanics and heat transfer, several ML algorithms have been effectively implemented to predict and analyze mass and heat transfer characteristics in variable systems.^27–29 For example, convolutional neural networks (CNNs) were trained to predict the 2D distribution of local heat flux in a fully developed turbulent channel flow when the model inputs were the 2D images of stress and pressure fields.³⁰ CNNs leverage convolutional operations to extract features and expressions from high-dimensional data, making them well-suited for regression tasks using images.^31,32 To handle sequential data or time series data, recurrent neural networks (RNNs) with a recurrent mechanism have been widely used.^33,34 Recently, physics-informed neural networks (PINNs) have been introduced to infer the flow and pressure fields from limited snapshots of solute concentration fields.³⁵ PINNs fit their solutions to the governing equations (e.g., conservation of mass, momentum, and energy), thereby reducing the dependence of neural networks on the size of the dataset.^36–40 Generative adversarial networks (GANs) is another promising approach. With their unique network architectures, GANs can generate realistic and diverse data samples similar to those in the training, even without access to the original samples.^41–44 For example, a previous study successfully employed GANs to generate the temperature fields in a 2D domain induced by heat conduction.⁴⁵ Most importantly, GANs can effectively reduce the output space using conditional input, while CNNs can constrain their output only by adjusting their network parameters based on sophisticatedly designed loss functions. Therefore, GANs are often easier to train than CNNs and other deep neural networks.

In this study, we investigate the possibility of reconstructing the temperature fields directly from the concentration fields using a ML model. Considering the attractive characteristics of GANs, we employ the cGAN strategy. We trained the cGAN model using a series of concentration fields and corresponding temperature fields (training dataset). The cGAN model is trained to recognize the analogy function between two correlated scalar fields, concentration fields, and temperature fields. After training, the cGAN model is shown to successfully reconstruct the temperature fields from the input concentration fields (testing dataset).

Here, we consider heat and mass transfer in laminar channel flows. We consider a 3D channel with a width (W) of 60 mm, height (H) of 15 mm, and length (L) of 130 mm, as shown in Fig. 1. At an inlet, water at 35 °C (⁠ $T 0$⁠) enters with a uniform velocity distribution of $3 mm/ s$⁠. The bottom wall temperature $( T b)$ is maintained constantly at 43 °C, while the other walls are thermally insulated. Although the geometry and flow conditions are simplified and idealized, they serve as a good proxy for identifying critical flow and fluid-related factors affecting the mixed convection flow in a horizontal channel or fracture. A passive solute tracer is injected along with water, only at the bottom half of the inlet, to provide solute concentration in the channel. The Reynolds number $( R e = u l ν )$ of the reference case in this study is 72, which is obtained using the characteristic length of the channel $( l = 2 W H W + H )$⁠, the injection velocity of water $(u)$⁠, and the kinematic viscosity of water $(ν)$⁠. The Rayleigh number $( R a = ρ β Δ T l 3 g η α )$ of the reference case in this study is around $2.82× 10 6$⁠, where $ρ$ is the density of water, $β$ is the thermal expansion coefficient of water, $ΔT$ is the temperature difference between the bottom wall and the inlet, $η$ is the dynamic viscosity of water, and $α$ is the thermal diffusivity of water.

To reduce computation costs, only the half-section of the channel is modeled by using the yz-plane at x = 0 as a symmetric plane. Also, we used a structured mesh considering the domain geometry and computational efficiency. The time step size of the simulation model is 0.05 s, and the number of time steps is 7000. Using supercomputing resources at Minnesota Supercomputing Institute (MSI), it took about 4 days to simulate one case using 128 cores and 400 GB RAM. The simulation outputs are the solute concentration fields and temperature fields in the simulation domain obtained at every 20 time steps (1 s). Although the simulation is 3D, training and testing datasets for the ML model are created using 2D cross-sectional images of the concentration fields and the corresponding temperature fields across xy-planes to reduce the model training time. These 2D cross-sectional images are obtained every 10 mm along the z direction. Therefore, the total simulation results are 4550 pairs of cross-sectional concentration and temperature images with a resolution of 180 × 360. Note that not all of the simulation results need to be employed to create training and testing datasets for the ML model. There are three cases that do not require many data samples. First, at the early stage of simulation, when the dye has not passed through the channel (first 10 s), most parts of the channel have no concentration variation. Second, at the late stage of simulation, the fluid flow reaches a steady state (after 190 s), leading to minimal changes in concentration and temperature fields. Last, changes in the concentration and temperature fields at the cross sections near the inlet (z < 20 mm) and near the outlet (z > 120 mm) are not significant. For these cases, a small number of data samples are needed for training and testing. With this consideration, the final datasets consist of 1980 pairs of cross-sectional concentration images and temperature images with a resolution of 180 × 360 obtained across the xy-planes. This dataset is denoted as the entire dataset in the following sections. The dataset obtained at cross sections where z = 90 and 110 mm are used for testing, while the other images are used for training.

Conditional generative adversarial networks (cGANs) have been used to predict transient temperature field images from numerical array inputs describing the channel geometry, thermal and hydraulic boundary conditions, location, and time.⁴⁸ cGAN is a type of GAN that involves the conditional generation of images by a generator model.⁴¹ cGAN relies on a generator that learns to generate new images, and a discriminator that learns to distinguish synthetic images from real images. In cGANs, the generator network learns to create an image close to the ground truth image to fool the discriminator network. The discriminator network, on the other hand, learns to distinguish ground truth images from fake images that the generator network creates. Various types of generator and discriminator networks have been developed for cGANs. Here, we use cGANs to estimate transient temperature fields from graphical inputs describing concentration fields in a three-dimensional (3D) channel. The details of the generator and discriminator architectures we used can be found in the supplementary material.

We optimized hyperparameters such as training epochs and $λ$ to balance the training quality and computation duration. It took about 8 h to train the cGAN model using the GPU of a high-performance workstation (Dell Precision 7920 Tower). Training the cGAN model takes a much shorter time than CFD simulations. Therefore, having such a ML model can significantly lower computational costs than conventional numerical simulation techniques. Figure 2 shows the training losses of the optimized (reference) model as a function of epochs. From Fig. 2, we can see that $L D$⁠, $L L 1$⁠, and $L T o t a l$ gradually decrease with the increase in epoch. $L c G A N$ slightly increases over the epoch but remains relatively stable in general. The hyperparameter $λ$ balances the two objectives for generator training. If $λ$ increases, the generator focuses more on minimizing the mean absolute difference between X and Y, thereby improving the low-frequency accuracy. Considering the training duration, we conducted all the subsequent model training with 1000 epochs and λ = 10⁴.

The image contrast, determined by concentration and temperature gradients, can affect model accuracy. A previous work reported that the cGAN model tended to generate blurred features in the temperature fields when the Michelson image contrast was below 0.011.⁴⁸ Additionally, concentration and temperature fields significantly change with location. For example, concentration and temperature gradients are smaller at the cross sections near the outlet because of flow mixing. Consequently, how the field images are sampled from the entire dataset will determine the model’s accuracy and generalizability. To investigate the sensitivity of data sampling on reconstruction accuracy, we trained two additional models and compared their results with those of the model trained by the entire dataset, denoted as the reference model. In the first additional model (denoted as model 1), pairs of concentration and temperature images sampled at z = 20–50 mm are used for training, while the remaining data samples are used for testing. In the second additional model (denoted as model 2), pairs of concentration and temperature images sampled at z = 90–120 mm are used for training, and the remaining images are used for testing. This means that in model 1, the training dataset is sampled closer to the inlet, while in model 2, the training dataset is sampled closer to the outlet.

In Fig. 3, the results of the different models are qualitatively compared at an early stage, t = 50 s, for selected z locations. The first row shows the concentration fields at different cross sections from simulation, which are used as input data in the ML model. The second row shows the temperature fields at different cross sections from simulation, serving as ground truth for comparison with the reconstructed results from the ML model. As expected, the concentration and temperature gradients are larger near the inlet (z = 40 mm) compared to those near the outlet (z = 100 mm). The third to fifth rows show the reconstructed results from different models.

Results show that the reference model performed the best because it was trained using the figures from both near the inlet and near the outlet. The reference model can reconstruct the temperature fields across the channel, regardless of the gradient. Model 1 was trained with the images near the inlet, where the concentration and temperature gradients are larger. Therefore, at the cross section near the inlet (z = 40 mm), the reconstructed result from model 1 is better than that from model 2. However, model 1 fails to reconstruct the temperature fields at the cross section near the outlet (z = 100 mm) because it did not have enough experience inferring the temperature fields from the concentration fields, when the gradient is small. Overall, model 2 performed better than model 1.

Error metrics are commonly used to evaluate the performance of ML models by comparing predicted values with actual values. Typical error metrics include the mean squared error (MSE), root mean squared error (RMSE), mean absolute error (MAE), and R-squared (R²). Each error metric serves a distinct purpose and provides insights into different aspects of the model performance. Here, we focus on RMSE because it is a standardized error metric that offers a measure of the average magnitude of errors in the same units as the target variable. This facilitates the interpretation and comparison of different models.

Figure 4 shows how RMSE changes with locations at three different times for different models. Black lines indicate the reference model, blue lines indicate model 1, and red lines indicate model 2. Results generally show that model 2 exhibits smaller RMSE than model 1, indicating the better performance of model 2, consistent with our qualitative comparison with Fig. 3. For model 1, the reconstruction zone (or unseen zone) is when z = 50–120 mm (not including z = 50 mm), while for model 2, the reconstruction zone is when z = 20–90 mm (not including z = 90 mm). In the reconstruction zone, the maximum RMSE ranges from 0.19 to 0.24 for model 1 at different times, whereas for model 2, the maximum RMSE ranges from 0.1 to 0.19 at different times. Based on the quantitative and qualitative comparisons, model 2 is considered to have superior performance to model 1. This result suggests that using the figures with smaller concentration and temperature gradients but with more complex spatial patterns as the training dataset can help the ML model in better inferring the temperature fields from the concentration fields. For channel flows, one may benefit from better measurement of the concentration fields near the outlet in order to estimate the temperature fields across the channel.

From Fig. 4, we observed that the RMSE tends to increase with time for all three models. At later times, because of mixing, concentration and temperature gradients decrease. Therefore, models trained with data at early times, where the concentration and temperature gradients are large without exhibiting much unique features in fields, may not perform well when reconstructing the temperature fields at later times. However, the capability of reconstructing the temperature fields at later times is important for most applications. To investigate the possibility of using early-time data to reconstruct late-time temperature fields, an additional model (denoted as model 3) was trained using simulation results from the first 40 s as the training dataset, while results at 50 s and 100 s were used as the testing dataset. The reconstructed results are shown in Fig. 5. Based on qualitative analysis, the accuracy of model 3 when reconstructing the temperature fields at later times is relatively poor, especially near the outlet. At later times and near the outlet, due to increased mixing, the concentration and temperature gradients are smaller with more unique field features than at the early times. Therefore, a model trained with data at early times has a hard time reconstructing the temperature fields at later times.

Figure 6 shows how the RMSE changes with time for the reference model and model 3. At the cross section near the inlet, the RMSE of the reference model gradually increases with time. At the cross sections in the middle of the channel and near the outlet, the RMSE of the reference model stays constant at early times but increases at later times. For model 3, the RMSE increases dramatically at around 50 s. After 50 s, the change in RMSE with time is not very significant. Although the qualitative analysis in Fig. 5 shows that the accuracy of model 3 when reconstructing the temperature fields at later times is relatively not good, the RMSE at late times from Fig. 6 is about 0.14, which is acceptable. We can infer that the cGAN has the capability of reconstructing the temperature fields at later times even when we only have data at early times, although the result may not be highly accurate. The cGAN will exhibit better extrapolation capability in the time domain when there is a greater similarity in field features between early times and later times.

The results so far suggest that the ML approach can be a powerful tool for overcoming the limitations of traditional heat and mass analogy approaches. In particular, we confirmed the possibility of using an ML model to infer the temperature fields from the concentration fields for the laminar mixed convection in a 3D horizontal channel. However, ensuring the generalization of ML models to untrained conditions, such as different flow regimes or geometries, remains a major challenge. Developing models that are more adaptable to varying convection scenarios is an important topic. In this section, we test the applicability of our model in different convection scenarios.

To test the applicability of our model to different convection scenarios, we simulated various scenarios by changing the temperature differences $T h o t− T r e f$ and characteristic velocities $(v)$⁠. The goal is to test the applicability of the reference model, trained at specific Ra and Re, to scenarios with different Ra and Re. Table I shows the Ri, Ra, and Re for different cases. The Ri for the case we used to train our model (reference case) is marked in red. Subsequently, we employed the trained reference model to reconstruct the temperature fields of various cases and calculated the averaged RMSE. Table II shows the averaged RMSE for different cases, with the average RMSE for the reference case marked in red. To better visualize how the RMSE changes across different convection scenarios, Fig. 7 is plotted.

Figure 7 shows how the RMSE changes with Ra and Re. In Fig. 7, colors represent the value of RMSE, where yellow indicates a large RMSE while blue means a small RMSE. The reference case is marked by a red star. Four representative cases with different Re and Ra are shown in Fig. 7 as red circles and the corresponding images are presented. From Fig. 7, it can be observed that the model trained under mixed convection conditions can still be applicable for reconstructing the cases under natural or forced convection conditions. The maximum RMSE in this figure is about 0.1, which is acceptable. With an increase in Ra, the RMSE increases significantly, while changes in Re have less effect on the RMSE. The impact of changes in Re on the RMSE is more pronounced in regions with higher Ra compared to regions with smaller Ra. This indicates that model accuracy is more sensitive to changes in Ra than changes in Re in the corresponding regimes of Ra and Re. Thus, when using the trained model on a case with different $Δ T$ or velocities, the difference in $Δ T$ between the trained case and the test case will have a greater impact on model performance than the difference in velocity. In general, the results show that the ML approach can be used to infer temperature fields in unseen flow scenarios, especially when Ra does not vary significantly.

To understand the complex convective transport processes, both mass concentration and temperature fields are critical information. However, experimentally measuring the temperature field is more challenging and requires sophisticated experimental setups. Our results in this section show that the trained model can be applied to different flow conditions. This implies that if we have the concentration and temperature datasets of one experiment, we could use the dataset to train the model and reconstruct the temperature fields of different experiments under different flow conditions. The concentration fields of different experiments need only be measured, which is relatively easier than measuring the temperature fields.

Through the use of machine learning, we showed the possibility of inferring 3D temperature fields from 3D concentration fields for laminar mixed convection in 3D horizontal channel flows. Due to the scarcity of available experimental data, we conducted 3D numerical simulations to generate sufficient data on concentration fields and temperature fields for training and testing a cGAN model. Our study demonstrated that 3D temperature fields can be accurately estimated from 3D concentration fields. We further investigated the effects of a training dataset on model performance and found that the reconstruction accuracy of the ML approach is sensitive to the sampling of the training dataset. ML models trained with data exhibiting smaller concentration and temperature gradients, but spatially more complex patterns, are shown to perform better. Furthermore, we tested the applicability of our model to different convection scenarios by employing the trained model to reconstruct the temperature fields of various cases under different convection scenarios. The concentration and temperature fields at different convection scenarios are obtained by simulating the cases with varying degrees of temperature difference and injection velocity. When applying the trained model to different cases under different convection scenarios, the maximum RMSE obtained was 0.1, which is acceptable. Additionally, model accuracy was found to be more sensitive to changes in Ra than changes in Re. This study shows the promise of using machine learning for inferring 3D temperature fields, which is important for a wide range of applications.

The supplementary material describes how the reference model parameters were optimized and selected. In particular, we address the generator and discriminator architectures of conditional generative adversarial networks (cGANs).

The authors acknowledge the support of the National Science Foundation (NSF) under Grant No. CBET-2053413.

The author has no conflicts to disclose.

Hongfan Cao: Data curation (lead); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (lead); Writing – original draft (lead). Beomjin Kwon: Conceptualization (equal); Formal analysis (supporting); Funding acquisition (supporting); Investigation (equal); Project administration (supporting); Software (supporting); Supervision (equal); Visualization (equal); Writing – review & editing (equal). Peter K. Kang: Conceptualization (lead); Formal analysis (supporting); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (lead); Visualization (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.