Reynolds-averaged Navier-Stokes Simulations Research Articles

Numerical investigation of scramjet inlet models for side spillage reduction

This study introduces and evaluates five models of Mach 7 scramjet inlets designed to reduce side spillage. The baseline model is characterized as a 2D-wedge type inlet without sidewalls. To mitigate interference from expansion waves, an alternative model featuring a converging angle identical to the freestream Mach angle is proposed. Another inlet model presents each ramp's converging angle matched to the local Mach angle of its respective ramp. Subsequently, an inlet model with sidewalls attached to the baseline model is introduced, and an additional inlet model is presented with reduced-height sidewalls. Three-dimensional Reynolds-Averaged Navier-Stokes simulations are conducted for each inlet model, and the performance of the inlets is compared. The mass flow averaged properties at the isolator exit for each inlet model are computed. To evaluate the performance of engines equipped with each inlet model, thrust and specific impulse are determined using thermodynamic cycle analysis. Ultimately, net thrust is calculated by subtracting drag from thrust, and the results are compared. The inlet with converging ramps, which has the largest captured area, indicates the lowest net thrust, amounting to 38% of that of the baseline model. In contrast, the inlet with reduced height sidewalls represents the highest net thrust, which is 3.5 times that of the inlet with converging ramps. These results demonstrate that for scramjet inlet design, using sidewalls to reduce drag is a more effective method for reducing side spillage compared to employing converging ramps with a large captured area.

Numerical simulation of deep-water wave breaking using RANS: Comparison with experiments

Wave breaking is a multifaceted physical phenomenon that is not fully understood and remains challenging to model. An effective method for investigating wave breaking involves utilising the two-phase Reynolds-averaged Navier–Stokes (RANS) equations to directly simulate breaking waves. In this study, we apply a RANS model with an adaptively refined mesh to simulate breaking waves in deep water using the stabilised RANS model proposed by Larsen and Fuhrman. This approach enables a more efficient simulation of the physics of breaking waves compared to Direct Numerical Simulations, as it places less stringent demands on grid resolution. Our findings demonstrate that the RANS model compares well with deep water wave breaking experiments in terms of surface elevation. We also give estimates of the breaking strength parameter of our RANS simulations and compared them with the literature.

Large-eddy simulation of vortex interaction in pitching-fixed tandem airfoils

In this study, the interaction of vortices generated from an oscillating airfoil with a hindfoil placed downstream of the oscillating forefoil at low-Reynolds-number flow was investigated numerically. The forefoil entered a deep dynamic stall induced by large-amplitude pitching oscillation. The dynamic stall process is characterized by unsteady separation and the formation of a strong clockwise vortex. A wall-resolved large-eddy simulation approach was applied to compute the flowfield. The numerical measurements were performed for an incompressible flow at a Reynolds number of Re = 30 000 based on chord length with a pitching reduced frequency of K= 0.5, and amplitude of A = 14.1° over Selig–Donovan 7003 airfoils. A single-airfoil case was validated against numerical and experimental measurements. In the present study, we investigated the flowfield and aerodynamic coefficients resulting from the deep dynamic stall of the pitching forefoil and the vortex interaction in tandem-airfoil configuration related to micro-air vehicle applications by employing large-eddy simulation approach. Large-eddy simulation was also compared to two-dimensional unsteady Reynolds-averaged Navier–Stokes simulation to determine the accuracy and validity of the low-fidelity approach in prediction of deep dynamic stall and vortex interaction at low-Reynolds-number flow.

Numerical simulation of involute-plate research reactor flow behavior using RANS, LES and DNS

This paper investigates the flow behavior of involute-plate research reactors by performing Reynolds-Averaged Navier Stokes simulation (RANS), Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS) of the channel flow between fuel plates. By modeling turbulence with different numerical approaches, this study provides data with three levels of fidelity. For the RANS simulation, three widely used turbulence models, i.e., k-ε, k-ω, Reynolds Stress Turbulence model (RST) are applied by using the commercial CFD code STAR-CCM + . For LES and DNS, the open-source CFD code, Nek5000, is used given its outstanding scalability on High Performance Computer (HPC) and high-order technique. The results from RANS simulations are compared with that from LES and DNS for benchmarking. Both macroscale parameters and turbulence statistics, such as velocity magnitude, lateral velocity and turbulence kinetic energy, are presented and analyzed.The results from RANS simulation achieve good agreement with LES and DNS on velocity and turbulence kinetic energy prediction. The RST turbulence model predicts the most similar flow pattern of lateral velocity as compared to LES and DNS. The Lambda-2 (λ2) criterion with a reasonable threshold is used to demonstrate the instantaneous vortices distribution in the involute channel from both LES and DNS calculation. The DNS simulation captures more detailed turbulence especially near the corner, which explains the discrepancy between LES and DNS results near the corner. The normalized RMS error are defined and calculated to assess the performance of those turbulence models. The RST model captures the anisotropic feature of turbulence, which enable it to outperform other turbulence models for predicting the flow behavior in an involute channel. Although some discrepancies are found between LES and DNS results in the corner, the overall deviations between LES and DNS are found to be small. Given that the computational cost of DNS calculation is an order of magnitude higher, using LES data for benchmarking RANS model is a cost-effective approach.

Development and application of turbulent heat flux model for lead-bismuth eutectic based on deep learning

Existing turbulent heat flux (THF) closure models in Reynolds-averaged Navier–Stokes simulation of scalar transport are insufficient on accuracy for fluids with a low Prandtl number (Pr), such as lead–bismuth eutectic (LBE), especially under complex operating conditions like in heat exchangers of nuclear reactors. This paper aims to propose a THF model suitable for such conditions. Firstly, we established a high-fidelity database of forced convection flow past two tandem cylinders under a Reynolds number (Re) of 1 × 105 and various low Pr conditions by hybrid large eddy simulation and direct numerical simulation algorithm. Based on the database, a deep learning architecture was then built and trained to predict dimensionless THF. Finally, a posteriori test proved that the proposed THF model helps to improve the prediction performance for temperature fields and also performs well when extrapolated to cases of LBE past the tube bundles, especially under the case with large Peclet numbers.

Usage of high-fidelity large eddy simulation to improve the turbulence modeling of Reynolds averaged navier stokes simulation in film cooling applications via a neural network

In this study, high-Fidelity Large eddy simulation (LES) data was used to create a surrogate turbulence model to enhance the estimation of turbulent heat flux in a Reynolds averaged Navier Stokes (RANS) solver for film cooling applications. The LES simulation (A flat plate film cooling application with a Reynolds number of 17,382, a blowing ratio of 1 and a density ratio of 2) was validated by comparison with experimental data. A correlation analysis was performed to identify the most influential parameters, and temperature gradient, velocity gradient, turbulent dissipation rate, shear strain rate, and turbulent viscosity ratio were selected as the most effective parameters. A second model was developed to create a Galilean-invariant model that remains unaffected by changes in the coordinate system. Neural network was applied to the LES training data to propose a modified surrogate dynamic turbulent Prandtl number to be applied to the k − ε realizable RANS simulation. The first surrogate model was implemented to two different geometries, one geometry was a flat plate film cooling similar to the training data but with different flow conditions, and another one was a more complex geometry with pressure gradient. The second model was only implemented on the second geometry and produced similar results. The surrogate models predicted a more accurate thermal field near the cooling wall (up-to 58 % error reduction) with a computational time complexity similar to the base RANS solver.

Experimental and numerical investigation of the effects of porous bleed systems on a model scramjet inlet under high backpressure conditions

In this study, we conducted experimental and numerical investigations to assess the effects of backpressure and configurations of porous bleeds on a scramjet inlet's performance at Mach 5. Experiments in a high-enthalpy wind tunnel, using a backpressure controller and a porous bleed with 1 mm holes and 10 mm lateral spacing, were paired with Reynolds-averaged Navier-Stokes simulations to explore the mitigation of flow separation and the prevention of inlet unstart under various backpressure conditions. The simulations provided insights into how backpressure levels and porous bleed geometries impact internal flow structures, as well as the critical backpressure ratio at which inlet unstart occurs. Additional simulations varied the bleed system's hole diameter and spacing, identifying configurations that optimize aerodynamic performance by enhancing total pressure recovery and Mach number at the isolator exit, while reducing bleed mass flow rates. The study also quantified the impact of these configurations on engine thrust, determining that certain bleed system designs significantly improve thrust by efficiently suppressing the interaction between the core flow and corner recirculation zones. This study examined the internal three-dimensional flow structure characteristics under high back pressure and provided insights into porous bleed configuration capable of efficiently suppressing inlet unstart.

Numerical analysis of static and dynamic wind turbine airfoil characteristics in transonic flow

This study performed an aerodynamic characterization of the FFA-W3-211 wind turbine tip airfoil in transonic flow using Unsteady Reynolds-Averaged Navier-Stokes (URANS) simulations, for both steady and dynamic operational conditions. First, the boundary between subsonic and supersonic flow in static conditions was identified, depending on the angle of attack, the approach flow Mach number, and the Reynolds number. The analysis points out that higher Reynolds numbers promote the occurrence of local supersonic flow. Thereafter, to investigate the dynamic behavior in the transonic flow regime, a sinusoidal pitching motion with representative values was imposed. A hysteresis, similar to but distinct from dynamic stall, was observed for entering and leaving the supersonic and subsonic regions. Elevated reduced frequencies widened the hysteresis loop, resulting in increased normal forces on the airfoil. The study indicated that an increase in reduced frequency leads to an earlier onset of transonic flow. In conclusion, the risk of transonic flow occurring during normal operation of the next generation wind turbines predicted in earlier studies could be corroborated. Moreover, dynamic effects and Reynolds number dependencies can be significant.

Study of surface roughness on flow past a wind turbine tower section

The flow around wind turbine towers usually reaches very high Reynolds numbers greater than a million. Understanding the flow around the towers under these conditions is crucial, as it may lead to vibrations due to the vortices formed. Investigating aerodynamic characteristics at such high Reynolds numbers, both numerically and experimentally, is challenging. The current study validates such an experimental study, where a rough surface is employed to increase the effective Reynolds numbers and accelerate the laminar-turbulent transition in the boundary layer. Unsteady Reynolds-Averaged Navier-Stokes (RANS) simulations are carried out using OpenFOAM for a Reynolds number range of 1.36·105 to 6.8·105. The constant (a 1) used to calculate the eddy viscosity is varied to simulate the flow separation during adverse pressure gradients. A force partitioning method is implemented in OpenFOAM and various force contributions are analysed for this Reynolds number range. It is seen that the RANS simulations overpredict the aerodynamic characteristics and the extent of flow separation unless the value of a 1 is varied as a function of the Reynolds number. Furthermore, it is observed that the only force contributor is the vorticity-induced force, as the simulations are performed for a fixed cylinder.

The impact of blockage and wakes on seven power performance tests conducted at two wind farms

Blockage and wakes can potentially influence wind turbine power performance measurements (PPM), distorting the measured power curve relative to the true performance of the test turbine. In this study, we take a previously proposed method to correct for the impact of blockage and wakes and test it on seven PPM conducted in simple terrain at two wind farms. Each PPM was completed according to the prevailing standard from the International Electrotechnical Commission. The correction factors derive from Reynolds-Averaged Navier-Stokes simulations of the two wind farms, and we apply these factors to the measured wind speed data on a record-by-record basis. The wind speed corrections clearly reduce variability in the measured power coefficients for the three PPM affected by long-distance wakes. The corrections do not reduce performance variability within the other four PPM. They also do not reduce differences in measured performance between the turbines at each wind farm, though numerical site calibrations do. Given these results, and considering uncertainty in the measurements, including confounding influences like terrain and turbine differences, validation against many more PPM is likely needed to adequately assess the reliability and utility of the correction method.

Net escape velocity, transfer probability, and travel time distributions within a cross-ventilated room model sheltered by urban-like block array

Natural ventilation offers significant advantages, especially in terms of energy conservation. However, most published articles have focused on the ventilation flow rate to determine the average contaminant concentration, while few have examined local ventilation distributions. Therefore, isothermal steady-state Reynolds-averaged Navier–Stokes simulations were conducted to analyze the airflow and scalar concentration fields of a cross-ventilation model sheltered by buildings. Based on these fields, we generated the spatial distributions of ventilation indices, namely, the net escape velocity (NEV), point-to-point indices transfer probability (TP), and travel time (TT), to discuss the dispersion of scalars emitted from local points in a room with natural ventilation. The NEV, which indicates the direction of the scalar discharge from each cell, exhibited a distribution distinct from the advection velocity because of concentration gradient effect. Higher TPs were observed when evaluated from a contaminant source to a target point following the NEV streamlines. Meanwhile, lower TTs were observed when evaluated in a point near the source and also from a contaminant source to a target point following the NEV streamlines. Although TP and TT are independent ventilation indices, a negative correlation between them was observed. Instead of the ventilation flow rate, the detailed structure of scalar dispersion can now be described by simultaneously analyzing the ventilation efficiency indices, which provide information on the general contaminant transport direction, and the probability and duration of transfer from a source to a target point. This potentially helps in the design of ventilation system especially in room lay-out to avoid cross-contamination.

Design, Multi-Point Optimization, and Analysis of Diffusive Stator Vanes to Enable Turbine Integration Into Rotating Detonation Engines

Abstract Pressure gain combustion is considered a possible path toward improved thermal cycle efficiency and reduced carbon emissions. However, ad hoc turbine designs are required to maximize the thermodynamic potential; the new turbomachinery should be suited for the oscillations in flow conditions generated by the detonation combustor, which are very different from conventional gas turbines. This paper investigates the design, optimization, and analysis of diffusive stator vanes operating under large inlet flow angles in the high-subsonic regime. First, the design methodology is outlined, focusing on the geometric requirements to ingest high Mach number flow and the parametric modeling of the endwall and the 3D vane pressure and suction sides. Then, the impact of the inlet flow angle on the flow field and vane design is studied through a multi-point, multi-objective optimization with three different inlet angles, performed using steady Reynolds-averaged Navier–Stokes simulations. Remarkable reductions in pressure loss and stator-induced rotor forcing are attained while maintaining an extensive operating envelope and high flow turning. Moreover, several design guidelines are provided based on the analysis of the optimized geometries. Finally, the effectiveness of the proposed methodology is verified with an unsteady assessment of the baseline and optimized vane designs.

The Impact of Upstream Static Deformation on Flow Past a Cylinder/Flare

Reynolds-averaged Navier–Stokes simulations are performed for supersonic turbulent flow over a cylinder/flare with upstream surface distortion representative of structural deformation induced via fluid–structural and fluid–thermal–structural behavior. Broad parametric analysis is carried out through the generation of Kriging-response surfaces from a database of general simulations. A posteriori simulations are then carried out at parametric combinations that correspond to extrema in the Kriging response surfaces to gain deeper insights into the interaction between the surface distortion and flow responses. Upstream distortions tend to decrease, rather than increase, the peak pressure and heat flux loads on the flare compared to an undeformed cylinder. Furthermore, decreases in these quantities reach up to O(10%) compared to up to O(1%) for increases. Integrated quantities over the flare are relatively insensitive to upstream distortion. The corner separation length is the most sensitive quantity to upstream distortion, with protrusions tending to increase the separation length and recessions reducing the separation length. Modifications in the separation length of up to 40% are observed. Reductions in peak loads tend to correspond to increases in the corner separation length. The movement of the surface distortion relative to the corner indicates a negligible impact beyond 1.5 distortion lengths from the corner, and the largest impact on the corner separation length occurs when distortion is directly adjacent. These results are an important step toward understanding and quantifying the impact of surface deformations on downstream components.

Flows past airfoils for the low-Reynolds number conditions of flying in Martian atmosphere

Fixed-wing aircraft with potential for long-duration flight and efficient manoeuvrability is expected to be the next frontier of Mars surface exploration. However, the feasibility of such an aircraft demands for wing airfoil suitable for low Reynolds flight conditions. In this framework, the paper deals with the computational study of two wing sections specifically designed for Mars exploration aircraft. Two-dimensional steady Reynolds-averaged Navier–Stokes simulations, using the computational fluid dynamics tool SU2 with the γ−Reθ transition model and the XFoil panel flow solver, are performed to address airfoils’ aerodynamics. Computational tools are validated by simulating flow past the Eppler 387 airfoil at Re∞=60×103, and comparing the results with experimental data collected in wind tunnel experiments. Aerodynamic performances of optimal wing sections are investigated at Re∞=3.4×104 by considering pressure coefficients and skin friction coefficients. The application of a literature-based correlation, specifically tailored to identify transition and reattachment locations, is extended to separation point detection. Computational Fluid Dynamics analysis conducted on the studied airfoils revealed that separation occurs within the laminar regime. Additionally, examination of turbulent shear stresses highlighted the role of airfoil curvature in counterbalancing the suction defect produced by the laminar separation bubble. This curvature-induced effect was found to play a crucial role in optimizing airfoil efficiency.

A Full-Scale Tidal Blade Fatigue Test using the FastBlade Facility

Fatigue testing of tidal turbine blades requires the application of cyclic loads without the ability to match the natural frequency of the blade due to their high stiffness and the associated thermal issues of testing composite materials at those frequencies (i.e., 18–20 Hz). To solve this, loading the blades with an auxiliary system is necessary; in most cases, a conventional hydraulic system tends to be highly energy-demanding and inefficient. A regenerative digital displacement hydraulic pump system was employed in the FastBlade fatigue testing facility, which saved up to 75 % compared to a standard hydraulic system. A series of equivalent target loads were defined using Reynolds-Averaged Navier Stokes (RANS) simulations (based on on-site collected water velocity data) and utilised in FastBlade to demonstrate an efficient way to perform fatigue testing. During the test, a series of measurements were performed on the blade response and the Fastblade test structure itself, providing novel insights into the mechanical behaviour of a blade, and enabling improved testing practice for FastBlade. Without catastrophic failure, the blade withstood the principal tidal loading for 20 years (equivalent). This test data will enable FastBlade to identify improvements to the testing procedures, i.e., control strategies, load introduction, instrumentation layout, instrument calibration, and test design.

Extraction of flow features around a bridge pier with an evolving scour hole using Lagrangian coherent structures

Local scouring around a bridge pier poses a severe threat to the safety of the bridge. A better understanding of flow features around the bridge piers is necessary for accurate prediction of the scour depth. The ridges of the finite-time Lyapunov exponent, called Lagrangian coherent structures (LCSs), were used to extract the flow features around a circular bridge pier with an evolving scour hole. The velocity field required for the LCSs computation was obtained using a three-dimensional Reynolds-averaged Navier–Stokes simulation. The simulation results were validated with the published experimental and numerical findings. The computed LCS stretching field extracted all the flow features around the bridge pier that were previously reported in the literature. In addition, the LCSs extracted the region of flow acceleration on both sides of the pier. The forward LCSs upstream of the pier extracted a particle trapping region, providing insight into the volume of fluid converting into the downflow. They extracted anchor-like structures inside the scour hole upstream of the pier. The analysis of velocity variations along the width and depth of the flow domain revealed that a change in the velocity profile is triggering the formation of LCS. The behavior of non-inertial particles released and integrated into the flow field revealed the significance of LCSs in particle transport. Using the LCS method, the study extracted the flow features that were difficult to extract with traditional flow visualization methods.

Aerodynamic Prediction of the High-Lift Common Research Model with Modified Turbulence Model

Aerodynamic simulations were carried out in the study presented in this paper focusing on the stall performance of the High-Lift Common Research Model obtained from the fourth AIAA High-Lift Prediction Workshop. Various turbulence models of Reynolds-averaged Navier–Stokes simulations are analyzed. A modified version of the transitional k−v2¯−ω model was developed to enhance stall prediction accuracy for high-lift configurations with a nacelle chine. The vortex generator, three-element airfoil, and high-lift model are numerically simulated. The results reveal that implementing a k−v2¯−ω model with the separation shear layer fixed notably enhances the stall prediction behavior for both the three-element airfoil and high-lift configuration without affecting the prediction of the vortex strength of a vortex generator. Moreover, incorporating rotation correction into the SPF k−v2¯−ω model improves the prediction of vortex strength and further enhances stall prediction for the high-lift configuration. The relative error in predicting the maximum lift coefficient is less than 5% of the experimental data. The study also investigated the impact of the nacelle chine on the stall behavior of the high-lift configuration. The results demonstrate that the chine vortex can mitigate the adverse effects of the nacelle/pylon vortex system and increase the maximum lift coefficient.

A New Axial Compressor Test Facility for the Investigation of Aerodynamic Damping Featuring an Electromagnetic Excitation System

Abstract Aerodynamic damping simulations involving detailed computational fluid dynamics (CFD) models are nowadays an integral part of turbomachinery design processes, thanks to advancements made in simulation tools and in computer hardware over the last decades. However, the test cases available in the open literature suitable for the validation and continued development of simulation tools is rather limited. On this background, a new test facility is introduced that allows acquiring relevant and high-quality aerodynamic damping test data on axial compressor rotor test objects in a controlled manner. The test facility is built as a closed loop and thus enables operation at variable pressure levels ranging from near-vacuum to overpressure and with different operating media. Overall damping test data are acquired at controlled synchronous and nonsynchronous vibration conditions. While vibrations of the blades at controlled nonsynchronous vibration conditions can be introduced in the test facility by means of a proprietary electromagnetic excitation system that is acting below the hub line, vibrations at controlled synchronous conditions can be introduced from a set of magnets integrated into the casing. Great care has been spent to create a clean test case without any intruding parts or openings in the test section. In addition, an axial compressor blisk is presented, which acts as test object to show the capabilities of the test facility as well as to allow for detailed investigations of aerodynamic damping. While the first part of the article discusses the setup and the instrumentation of the test facility to excite and measure blade vibration in a nonintrusive way, flow-field measurements and the validation accompanying steady-state Reynolds-averaged Navier-Stokes (RANS) simulations are shown in the second part. The third part of the article focuses on aerodynamic damping measurements for seven different modes at three different pressure levels in total. It can be shown that aerodynamic damping is by far the largest contributor to the overall damping.

Turbulent Particle Flow in Spent Nuclear Fuel Storage Systems

Deposition models are being developed with the commercial computational fluid dynamics software STAR-CCM+ to evaluate particulate deposition on spent nuclear fuel (SNF) canisters. The primary particulate of concern is chloride salts, which are dispersed in the atmosphere and then deposited onto the canisters. During dry storage, the primary degradation process is likely to be chloride-induced stress corrosion cracking (CISCC) at the heat-affected zones of the canister welds. It is known that stainless steel canisters are susceptible to CISCC; however, the rate of chloride deposition onto the canisters is poorly known, based on sparse field data from a small number of sites. This paper describes work looking at various approaches to modeling turbulence, such as Reynolds-averaged Navier Stokes (RANS) and large eddy simulation (LES), and its impact on particle flow and deposition within a ventilated SNF storage system. The deposition rate is determined for a vertical canister system and a horizontal canister system. LES has the potential to simulate turbulent flows more accurately versus RANS, but is much more computationally expensive. A k-omega version of the RANS turbulence model was used for this study. The computational efficient RANS steady-state model predicted a similar canister deposition result as the LES simulation for a vertical canister storage system. For a horizontal storage system, the RANS steady-state model predicted more particles depositing on the canister than the LES simulation, providing a conservative estimation for particle deposition. A wall correction factor was added to the RANS model to dampen the turbulence fluctuation normal to a surface that left undamped, leads to the RANS model overpredicting deposition along a surface for smaller particles. This work was done to further development of deposition models that could be used to plan and inform SNF canister aging management programs with predictive models for the timing and occurrence of canister CISCC. This work is part of a larger effort tasked with understanding the likelihood of canister degradation due to CISCC. These models are still under development, and testing is needed for validation. However, these models are being presented now to demonstrate to canister vendors, utilities, regulators, and stakeholders the value of this type of modeling.

Sparse learning model with embedded RIP conditions for turbulence super-resolution reconstruction

In practical engineering scenarios, constraints arising from sensor placement, quantity, and the limitations of current testing technologies often lead to turbulence data characterized by low resolution and irregular structures. Turbulence super-resolution reconstruction is crucial for extracting finer details from irregularly structured, low-resolution measurement data, thereby facilitating comprehensive flow field analyses. This article introduces a sparse learning model (Embedding Restricted Isometry Property Autoencoder (RIP-AE)) for achieving super-resolution reconstruction of flow fields by obtaining compressed representations and sparse transform domain information. In this model, we embed the Restricted Isometry Property (RIP) condition to ensure the effectiveness of compressed representations and sparse transform domains, thereby enhancing the super-resolution reconstruction accuracy of the model. To assess the performance of the RIP-AE model, we utilized data sets of flow around a cylinder at different Reynolds numbers generated by unsteady Reynolds-Averaged Navier–Stokes simulations. We sequentially validate the effectiveness of the RIP condition at different Reynolds numbers (ranging from 1,000 to 500,000) and compare the reconstruction results of the RIP-AE model with those of the downsampled skip-connection/multi-scale (DSC/MS) and MS-AE models under different sampling ratios. The results indicate that the RIP-AE model excels in terms of L2 relative error and demonstrates the capability to achieve high-precision flow field reconstruction even under high sampling ratios.