Full Computational Fluid Dynamics Research Articles

An efficient strategy based on proper orthogonal decomposition for rapid determination of the location and rate of water vapor humidification in aircraft cabins

Currently, the air in commercial aircraft cabins during flights routinely exhibits extremely low relative humidity. But none of the previously published studies have addressed optimization of the humidification parameters in order to, create the desired humidity distributions in aircraft cabins at minimal cost. This study proposed an efficient strategy to accomplish the task. Under this strategy, a database of various potential humidification locations for aircraft cabins is first constructed. One scenario in the database is then selected by computational fluid dynamics (CFD)-based enumeration. Next, with the humidification location as prior information, the humidification rate and the air-supply temperature from the main ventilation system can be inversely designed by proper orthogonal decomposition. The above strategy was applied to a three-dimensional aircraft cabin model for the validated CFD cases. The results showed that an appropriate humidification for the cabin might both improve the thermal comfort of passengers and increase the humidity levels in the cabin; a decrease of the air-supply temperature from the main ventilation system would be beneficial in reducing the water vapor humidification rate required for passengers and allowing a higher humidification rate to be tolerated for the cabin walls. The parameters that improved the cabin air environment without significant moisture accumulation on walls were also provided by the proposed strategy. Compared with the full CFD simulation method, the proposed strategy is capable of providing accurate ranges of parameters while reducing the computing time by more than 95%, which demonstrates its competitive prospects.

Back to Bernoulli: a simple formula for trans-stenotic pressure gradients and retrospective estimation of flow rates in cerebral venous disease

BackgroundVenous sinus stenosis can be associated with cerebrovascular disorders. Understanding the role of blood flow disturbances in these disorders is often hampered by the lack of patient-specific flow rates. Our...

A shape-predictive model for the spreading of photo-curable polymers in material extrusion additive manufacturing

Recently, there is a growing interest in material extrusion to print thermoset resins to manufacture large format and high-performance parts. However, the fidelity and mechanical integrity of the printed parts are limited by challenges such as uncontrolled spreading of individual beads (filament/droplet) after deposition and during cure. There is a considerable lack of experimental and theoretical studies on the spreading of reactive beads on solid substrates. In this work, we studied the simultaneous spreading and photo-curing of the photopolymerizable thermoset beads via experiment and numerical simulations. We used a novel experimental setup to track the spreading of droplets and filaments during photopolymerization and validate a moving mesh computational fluid dynamics (CFD) model. The CFD model was used to develop an approach (predictive model) to accurately predict the final spreading coefficient of cured resin beads without the need for full numerical simulations. The predictive model combines the generalized theory of a Newtonian spreading filament with a characteristic viscosity μave and time to gelation, τgel. Interestingly, μave is shown to be a material parameter that does not depend on processing conditions, but only on the material’s chemorheology. The predictive model is tested against a wide range of chemorheology and cure kinetic parameters and found to be in excellent agreement with the full numerical CFD simulations. This work will be very useful in estimating the final shape of beads during the material extrusion printing process, as well as a model to successfully parameterize extrusion-based 3D printers to control the shape of printed beads a-priori.

Analysis of the air stream flow parameters generated by the positive pressure ventilator—full scale experiment and CFD simulation

Positioning the positive pressure ventilator in front of the door opening affects the effectiveness of the rescue operation carried out during a fire. An important factor determining the effectiveness of the positive pressure ventilator is also the layout of the rooms within the gas exchange path and the obstacles present there. The purpose of this article is to assess the feasibility of using analyses such as large eddy simulation (LES) to verify the efficiency of mobile fans under simulation conditions, without the need for time-consuming experimentation (also for complex room volumes of buildings). The article presents a comparative analysis to assess the degree of convergence of flow parameters obtained during an experiment (in a multi-story building) and computational fluid dynamics (CFD) simulations. For volumetric flow rate, convergence was achieved at levels ranging from 0.4% (for 5 m) to 11.5% (1 m), and for pressure values, the differences achieved ranged from 0.6% (5 m) to 30.1% (4 m). This paper demonstrates that the LES model can be used to perform CFD simulations in the area of assessing the performance of a positive pressure ventilator. The article also describes a test methodology for determining the flow parameters of an air stream, which can be used to perform numerical simulations.

An analytical linear two-dimensional actuator disc model and comparisons with computational fluid dynamics (CFD) simulations

Abstract. The continuous up-scaling of wind turbines enabled by more lightweight and flexible blades in combination with coning has challenged the assumptions of a plane disc in the commonly used blade element momentum (BEM)-type aerodynamic codes for the design and analysis of wind turbines. The objective with the present work is thus to take a step back relative to the integral 1-dimensional (1-D) momentum theory solution in the BEM model in order to study the actuator disc (AD) flow in more detail. We present an analytical, linear solution for a two-dimensional (2-D) AD flow with one equation for the axial velocity and one for the lateral velocity, respectively. Although it is a 2-D model, we show in the paper that there is a good correlation with axis-symmetric and three-dimensional (3-D) computational fluid dynamics (CFD) simulations on a circular disc. The 2-D model has thus the potential to form the basis for a simple and consistent rotor induction model. For a constant loading, the axial velocity distribution at the disc is uniform as in the case of the classical momentum theory for an AD. However, an important observation of the simulated flow field is that immediately downstream of the disc the axial velocity profiles change rapidly to a shape with increased induction towards the edges of the disc and less induction on the central part. This is typically what is seen at the disc in full non-linear CFD AD simulations, which is what we compare with in the paper. By a simple coordinate rotation the analytical solution is extended to a yawed disc with constant loading. Again, a comparison with CFD, now with a 3-D simulation on a circular disc in yaw, confirms a good performance of the analytical 2-D model for this more complicated flow. Finally, a further extension of the model to simulate a coned disc is obtained using a simple superposition of the solution of two yawed discs with opposite yaw angles and positioned so the two discs just touch each other. Now the validation of the model is performed with results from axis-symmetric CFD simulations of an AD with a coning of both 20 and −20∘. In particular, for the disc coned in the downwind direction there is a very good correlation between the simulated normal velocity to the disc, whereas some deviations are seen for the upwind coning. The promising correlation of the results for the 2-D model in comparison with 3-D simulations of a circular disc with CFD for complicated inflow like what occurs at yaw and coning indicates that the 2-D model could form the basis for a new, consistent rotor induction model. The model should be applied along diagonal lines on a rotor and coupled to an angular momentum model. This application is sketched in the outlook and is a subject for future research.

Cylinder Liner Velocity Calculation under Dynamic Condition in the Pursuit of Liner Cavitation Investigation of an Internal Combustion Engine

<div>An analytical method for nonlinear three-dimensional (3D) multi-body flexible dynamic time-domain analysis for a single-cylinder internal combustion (IC) engine consisting of piston, connecting rod, crank pin, and liner is developed. This piston is modeled as a 3D piston that collides with the liner as in a real engine. The goal is to investigate the piston slap force and subsequent liner vibration. Liner vibrational velocity is directly responsible for pressure fluctuations in the coolant region resulting in bubble formation and subsequent collapse. If the bubble collapse is closer to the liner surface, cavitation erosion in the liner might occur. The mechanism of liner cavitation is briefly explained, which would take a full computational fluid dynamics (CFD) model to develop, which is out of scope for the present work. However, as a first step, the present method focused on a comprehensive and accurate estimation of the highest inward and outward liner velocities, which are directly related to the bubble formation and collapse, respectively. Sensitivity of liner velocity to different engine-operating conditions (warm and hot, with highest skirt temperatures of 178 and 130°C), piston pin bore offsets (thrust side, anti-thrust side directions in the amounts of 0.6 mm, and the nominal no offset case), and liner thicknesses are determined. Piston thermal growth is considered as part of the analysis resulting in interference condition between piston skirt and liner under the hot operating condition and low minimum clearance under the warm condition. Correlation of liner velocity contour plots with real engine liner cavitation erosion is presented. Analytical model showed a maximum liner inward velocity of 55 mm/s with no piston pin offset under nominal engine-operating configuration. A correlation has been found between location of this highest liner velocity and location of the actual cavitation erosion in the field.</div>

Validation of a full scale CFD simulation of a self-propelled ship with measured hull roughness and effect of welding seams on hull resistance

This study presents the verification and validation of a full scale self-propulsion computational fluid dynamics (CFD) model including measured hull roughness and welding seams. The reference for validation is comprised of speed trial data from five sister ships. To ensure numerical accuracy, resistance and propeller open-water CFD simulations are verified and validated against model tests from two different towing tanks and discrepancies are found to be within the experimental uncertainty of model scale testing. The self-propulsion CFD results emphasise the necessity of measuring hull roughness for accurate performance calculations with discrepancies less than 3.4% on the delivered power relative to the fitted speed trial results. It is found that the empirical Townsin/Mosaad roughness formulation with measured average hull roughness, as recommended by ITTC, provides similar results, as modifying the wall functions using the Colebrook/Grigson roughness function. Homogeneous and heterogeneous roughness modelling methods are found to provide similar results for the studied ship. Modelling of the transverse welding seams on the hull surface is found to increase the total resistance by approximately 0.1% per seam, resulting in a total estimated increase in resistance of 1.5–1.8%. Finally, adding a fairing is found to significantly reduce the resistance penalty from welding seams.

Steady and transient sensitivity investigations on a passive battery thermal management system coupling with phase change materials and heat pipes: Full numerical modeling and orthogonal tests

For safety reasons, the temperature distribution of the battery pack should be kept within a reasonable range. Therefore, in the present research, a passive battery thermal management system (BTMS) coupling with phase change material (PCM) and heat pipes (HP) was proposed. Full CFD (Computational Fluid Dynamics) models validated well by the experimental results were adopted for this research. Firstly, the sensitivity of parameters such as PCM type, PCM thickness and ambient temperature on the thermal performance of BTMS was obtained by orthogonal tests. Numerical results show that the greatest influence on the maximum temperature is the ambient temperature. In the variance analysis showed that the PCM thickness has a significant effect on decay the battery temperature difference. Furthermore, operational range and variance analysis confirmed the optimal BTMS, i.e., with PCM3 thickness of 3 mm and ambient temperature of 30 °C. Subsequently, transient modeling and analysis of the battery discharge process were carried out. Unsteady results showed that the maximum temperature of the battery was reduced from 68.90 °C to 43.19 °C, and simultaneously, the maximum temperature difference dropped from 16.58 °C to 3.66 °C. In the process of battery discharge, PCM was properly adjusted regarding of its thickness; whereas, its melting temperature should be kept approaching to the ambient temperature, which was beneficial to the operational safety of the battery. In addition, temperature distribution of the battery has a linear relationship to the ambient temperature; when every 5 °C increase was observed in ambient temperature, the maximum temperature of the battery was boosted by 2.3 °C and the temperature difference was reduced by 1.2 °C, separately. Present research, straightforwardly maintaining the temperature distribution of the power battery in a safe and efficient range, could be significant for improving energy utilization and reducing extra energy consumption of the power battery, simultaneously.

3D CFD model for the analysis of the flow field through a horizontal axis wind turbine (HAWT)

With the world’s growing demand for energy, renewable energy production has become important in providing alternative sources of energy and in reducing the greenhouse effect. This study investigates the aerodynamics and performance of the WG/EV100 micro–Horizontal Axis Wind Turbine (HAWT) using Computational Fluid Dynamics (CFD). The complexity of VAWT aerodynamics, which is inherently unsteady and three-dimensional, makes high-fidelity flow models extremely demanding in terms of computational cost, limiting the analysis to mainly 2D Computational Fluid-Dynamics (CFD) approaches. This article explains how to perform a full 3D unsteady CFD simulation of HAWT. All main parts of the WG/EV100 HAWT were designed in SOLIDWORKS. Only the blade design was reverse engineered due to the unavailability of the CAD model and the complexity of its geometric characteristics. The impeller blade is scanned using a Coordi-nate Measuring Machine (CMM), and the obtained 3D scan data are exported from the PC-DMIS software to GEOMAGIC design X to obtain a CAD model of the blade.

CFD-assisted linearized frequency-domain analysis of motion and structural loads for floating structures with damping plates

Damping plates are widely used on floating offshore structures to reduce the dynamic responses in waves. Linearization of the nonlinear drag loads are needed to solve the floater motions by frequency-domain methods, which, to date, still dominate the analysis of wave-frequency responses of floating structures in engineering practice. While existing studies focus on damping effects on heave motions, there is a lack of understanding on how the drag loads on damping plates should be modeled to predict the angular motions and structural loads of the floaters. Two different configurations of Morison elements are investigated in this paper. The first considers the entire damping plate as a whole, while the second locates Morison elements at the estimated ‘points of acting’ of drag loads for different parts of the damping plate. Computational Fluid Dynamics (CFD) analysis is utilized to obtain the required drag coefficients and the ‘points of acting’ as function of Keulegan–Carpenter number. In comparison to full CFD analysis of the same floater in regular waves, a hybrid approach is shown to give satisfactory wave-frequency results, not only for motion responses, but also bending moments and shear forces on the damping plate. However, non-negligible second-harmonic structural loads that cannot be properly accounted for by the existing Morison-drag formulation are also identified in CFD analysis, calling for special attention to structural assessment under extreme wave conditions. This study also demonstrates that locating the Morison elements at ‘points of acting’ improves the prediction of motion responses and structural loads at resonance.

A hybrid method for modelling wake flow of a wind turbine

A fast and accurate prediction of the wake of an upwind turbine is very important to quantify the performance of downwind turbines in offshore wind farms, which become larger and larger. The wake flow and dynamics may be quite accurately simulated by high-fidelity computational fluid dynamics (CFD) software but its computational costs are too high, in particular to simulate a long wake flow often required in engineering practice. Therefore, the wake is often modelled by simplified dynamic wake models in design practices. They are computationally efficient but could not catch all physics, depend on pre-specified empirical parameters, and are not suitable for flow near the turbine. This paper proposes a new hybrid method, in which the near wake flow is simulated by a CFD model based on Navier–Stokes equations with the turbine represented by actuator lines while the far wake flow is modelled by an improved simplified CFD-based dynamic wake model. The two models are two-way coupled at a section downwind the turbine. The newly formulated method is validated by the results of full CFD simulations in the whole domain. Its performances are investigated under different conditions. It will be demonstrated that the new method takes considerably less computational time than the full CFD tool to produce similar results.

Rapid virtual fractional flow reserve using 3D computational fluid dynamics.

Over the last ten years, virtual Fractional Flow Reserve (vFFR) has improved the utility of Fractional Flow Reserve (FFR), a globally recommended assessment to guide coronary interventions. Although the speed of vFFR computation has accelerated, techniques utilising full 3D computational fluid dynamics (CFD) solutions rather than simplified analytical solutions still require significant time to compute. This study investigated the speed, accuracy and cost of a novel 3D-CFD software method based upon a graphic processing unit (GPU) computation, compared with the existing fastest central processing unit (CPU)-based 3D-CFD technique, on 40 angiographic cases. The novel GPU simulation was significantly faster than the CPU method (median 31.7 s (Interquartile Range (IQR) 24.0-44.4s) vs. 607.5 s (490-964 s), P < 0.0001). The novel GPU technique was 99.6% (IQR 99.3-99.9) accurate relative to the CPU method. The initial cost of the GPU hardware was greater than the CPU (£4080 vs. £2876), but the median energy consumption per case was significantly less using the GPU method (8.44 (6.80-13.39) Wh vs. 2.60 (2.16-3.12) Wh, P < 0.0001). This study demonstrates that vFFR can be computed using 3D-CFD with up to 28-fold acceleration than previous techniques with no clinically significant sacrifice in accuracy.

An efficient method for helicopter fuselage shape optimization

PurposeThis paper aims to develop a new method of fuselage drag optimization that can obtain results faster than the conventional methods based on full computational fluid dynamics (CFD) calculations and can be used to improve the efficiency of preliminary design.Design/methodology/approachAn efficient method for helicopter fuselage shape optimization based on surrogate-based optimization is presented. Two numerical simulation methods are applied in different stages of optimization according to their relative advantages. The fast panel method is used to calculate the sample data to save calculation time for a large number of sample points. The initial solution is obtained by combining the Kriging surrogate model and the multi-island genetic algorithm. Then, the accuracy of the solution is determined by using the infill criteria based on CFD corrections. A parametric model of the fuselage is established by several characteristic sections and guiding curves.FindingsIt is demonstrated that this method can greatly reduce the calculation time while ensuring a high accuracy in the XH-59A helicopter example. The drag coefficient of the optimized fuselage is reduced by 13.3%. Because of the use of different calculation methods for samples, this novel method reduces the total calculation time by almost fourfold compared with full CFD calculations.Originality/valueTo the best of the authors’ knowledge, this is the first study to provide a novel method of fuselage drag optimization by combining different numerical simulation methods. Some suggestions on fuselage shape optimization are given for the XH-59A example.

A generalized zoning framework for development of CFD‐based reduced‐order compartment models

AbstractWhile computational fluid dynamics (CFD)‐based compartment models have gained popularity as a cost‐effective alternative to full CFD modeling of complex mixing systems, model development involves significant trial‐and‐error efforts. This work presents a generalized zoning framework for the streamlined development of CFD‐based compartment models with detailed characterization of the reduced‐order representation of the flow physics. With a stirred tank as an illustrative case, reduced‐order model for species transport and heat transfer with turbulent flow is derived, followed by introducing the generalized zoning framework to demonstrate how a reduced‐order compartment model can be constructed based on a simplified CFD simulation. A test case of mixing two miscible thermal fluids is used to evaluate the CFD‐based compartment model. The results demonstrate that the proposed zoning framework exhibits an accurate representation of the CFD simulation of hydrodynamics and demonstrates capabilities of capturing species and heat transfer in turbulent flow systems with complex geometric configurations.

Geodesic Convolutional Neural Network Characterization of Macro-Porous Latent Thermal Energy Storage

Abstract High-temperature latent heat thermal energy storage with metallic alloy phase change materials (PCMs) utilize the high latent heat and high thermal conductivity to gain a competitive edge over existing sensible and latent storage technologies. Novel macroporous latent heat storage units can be used to enhance the limiting convective heat transfer between the heat transfer fluid and the PCM to attain higher power density while maintaining high energy density. 3D monolithic percolating macroporous latent heat storage unit cells with random and ordered substructure topology were created using synthetic tomography data. Full 3D thermal computational fluid dynamics (CFD) simulations with phase change modeling was performed on 1000+ such structures using effective heat capacity method and temperature- and phase-dependent thermophysical properties. Design parameters, including transient thermal and flow characteristics, phase change time and pressure drop, were extracted as output scalars from the simulated charging process. As such structures cannot be parametrized meaningfully, a mesh-based Geodesic Convolutional Neural Network (GCNN) designed to perform direct convolutions on the surface and volume meshes of the macroporous structures was trained to predict the output scalars along with pressure, temperature, velocity distributions in the volume, and surface distributions of heat flux and shear stress. An Artificial Neural Network (ANN) using macroscopic properties—porosity, surface area, and two-point surface-void correlation functions—of the structures as inputs was used as a standard regressor for comparison. The GCNN exhibited high prediction accuracy of the scalars, outperforming the ANN and linear/exponential fits, owing to the disentangling property of GCNNs where predictions were improved by the introduction of correlated surface and volume fields. The trained GCNN behaves as a coupled CFD-heat transfer emulator predicting the volumetric distribution of temperature, pressure, velocity fields, and heat flux and shear stress distributions at the PCM–HTF interface. This deep learning based methodology offers a unique, generalized, and computationally competitive way to quickly predict phase change behavior of high power density macroporous structures in a few seconds and has the potential to optimize the percolating macroporous unit cells to application specific requirements.

Towards Dynamic Velocity Prediction of NACRA 17

NACRA 17 is a small foiling catamaran used in Olympic sailing. Sailing in the NACRA 17 is truly unsteady, especially in the foiling conditions. Dynamic velocity prediction has gained a lot of attention in recent years due to the unsteady nature of foiling sailboats. Different approaches have been applied to predict the motions of foiling or skimming boats in unsteady conditions. Typically, the aerodynamic forces are computed by the quasi-steady theory assumption. Static coefficients are used, and the applied velocities are corrected with boat motions. The hydrodynamic forces of a fully foiling boat can be computed in a similar fashion, but this method fails to include the effect of changes in the free surface as well as added mass. This paper proposes an approach where dynamic coefficients of the NACRA 17 hydrofoils are determined based on unsteady Computational Fluid Dynamics (CFD) simulations with a free surface. In the simulations the foil is moved in prescribed motions in one degree of freedom (DOF) at a time and dynamic coefficients are extracted for different frequencies of motion. All coefficients are combined in a coupled system of equations, and the equations can be solved in 6 DOFs in the time domain. Test scenarios with time-varying conditions are simulated and compared with full dynamic CFD simulations. The initial investigation of the approach shows promising results, but more investigations are needed to validate the approach and to identify possible limitations.

Thermal-hydraulic 0D/3D coupling in OpenFOAM: Validation and application in nuclear installations

The nuclear safety assessment involving large transient simulations is forcing the community to develop methods for coupling thermal-hydraulics and neutronic codes and three-dimensional (3D) Computational Fluid Dynamics (CFD) codes. In this paper a set of dynamic boundary conditions are implemented in OpenFOAMⓇ in order to apply zero-dimensional (0D) approaches coupling with 3D thermal-hydraulic simulation in a single framework. This boundary conditions are applied to model pipelines, tanks, pumps, and heat exchangers. On a first stage, four tests are perform in order to assess the implementations. The results are compared with experimental data, full 3D CFD, and system code simulations, finding a general good agreement. The semi-implicit implementation nature of these boundary conditions has shown robustness and accuracy for large time steps. Finally, an application case, consisting of a simplified open pool with a cooling external circuit is solved to remark the capability of the tool to simulate thermal hydraulic systems commonly found in nuclear installations.

Modeling and Characteristic Analysis of a Cylinder Block/Valve Plate Interface Oil Film Model for 35 MPa Aviation Piston Pumps

Oil film characteristics are critical to the high-reliability operation of high-pressure aviation piston pumps (APPs). However, there is still a lack of research on oil film modeling and characteristic analysis of high-pressure APPs. This paper takes the oil film at the cylinder block/valve plate interface of a 35 MPa high-pressure APP as the research object. By introducing a full oil film computational fluid dynamics (CFD) model considering non-isothermal and cavitation effects under multi-field coupling conditions, a cylinder block/valve plate interface oil film model is established, which includes a viscous wedge geometric model, multi-body dynamics model, and full oil film CFD model. The mesh independence test and force balance error analysis ensure the accuracy of the model calculation. Based on the established model, the oil film’s lubricating, sealing, load-bearing, and overturning characteristics are comprehensively and systematically analyzed, and the evolution law of different oil film characteristics with pressure changes is revealed. Moreover, suggestions for improving the structure and operating conditions of a 35 MPa high-pressure APP are proposed to optimize the oil film characteristics of the cylinder block/valve plate interface.

CFD-Modified Potential Simulation on Seakeeping Performance of a Barge

This study proposes a computational fluid dynamics (CFD)-modified potential (CMP) model. This hybrid model uses linear potential theory with a corrected damping ratio obtained from CFD simulations to analyze the seakeeping performance of a small vessel. According to the analysis procedure of the proposed model, a motion analysis, including the prediction of the roll and pitch damping ratio of a small barge, was conducted; to verify reliability; the results were compared to those of an experiment performed in a physical tank. The relative errors in the experiment for peak amplitude in the roll motion response amplitude operators (RAOs) using the CMP model were relatively small, whereas those obtained from only the potential analysis were large errors in all three conventionally used roll-damping ratios. In addition, the computational time consumed by the CMP model was longer than that consumed by the potential theory but faster than the full CFD simulation for all wave conditions. Subsequently, based on the motion analysis results, the seakeeping performance was evaluated in a real sea environment, and the results on the single significant amplitude (SSA) were discussed through comparison with the results of the potential analysis and experiment.

A Reduced Radiator Model for Simplification of ONAN Transformer CFD Simulation

In this paper, a reduced computational fluid dynamics (CFD) model is proposed for determining the total heat dissipation and the oil temperature distribution of transformer radiators in oil natural air natural (ONAN) cooling mode. In the reduced radiator CFD model, the air flow simulation is replaced with an optimized air heat transfer coefficient ( <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>h</i><i>_air</i></b> ) equation. An experimentally verified full radiator CFD model, which includes the air flow simulation, has been introduced as a benchmark model for evaluating the validity of the reduced model. It was found that one <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${{\boldsymbol{h}}_{{\boldsymbol{air}}}}$</tex-math></inline-formula> can provide sufficient accuracy in simulating the air convection effect of transformer radiators. An <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${{\boldsymbol{h}}_{{\boldsymbol{air}}}}$</tex-math></inline-formula> equation was derived through a parametric study using the full radiator CFD model. The performance of the proposed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${{\boldsymbol{h}}_{{\boldsymbol{air}}}}$</tex-math></inline-formula> equation was compared with the existing empirical equations, and its validities for different insulating liquids and different ambient temperatures were proven by additional CFD modelling. This <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${{\boldsymbol{h}}_{{\boldsymbol{air}}}}$</tex-math></inline-formula> equation can be transformed into a dimensionless form to help the radiator design and facilitate the complete-cooling-loop (CCL) based transformer CFD modelling.