Numerical Investigation of the Coupled Effects of External Wind Directions and Speeds on Surface Airflow and Convective Heat Transfer in Open Dairy Barns

Abstract

Natural ventilation is a common cooling strategy in open dairy barns, but its efficiency largely depends on external wind directions and speeds. Misalignment between external airflow and fan jets often led to non-uniform air distribution, reduced local cooling efficiency, and an elevated risk of heat stress in cows. However, few studies have systematically examined the combined effects of wind directions and speeds on airflow and heat dissipation. Most research isolates natural or mechanical ventilation effects, neglecting their interaction. Accurate computational fluid dynamics (CFD) modeling of the coupling between outdoor and indoor airflow is crucial for designing and evaluating mixed ventilation systems in dairy barns. To address this gap, this study systematically analyzed the effects of external wind directions (0°, 45°, 90°, 135°, 180°) and speeds (1, 3, 5, 7, 10 m s⁻¹) on fan jet distribution and convective heat transfer around dairy cows using the open-source CFD platform OpenFOAM. By evaluating body surface airflow and regional convective heat transfer coefficients (CHTCs), this study quantitatively linked barn-scale airflow to animal heat dissipation. Results showed that both wind directions and speeds markedly influenced airflow and heat exchange. Under 0° wind direction, dorsal airflow reached 6.2 m s⁻¹ and CHTCs increased nearly linearly with wind speeds, indicating strong synergy between the fan jet and external wind. Crosswinds (90° wind direction) enhanced abdominal airflow (approximately 5.2 m s⁻¹), whereas oblique and opposing winds (135–180°) caused stagnation and reduced convection. The dorsal-to-abdominal CHTCs ratio ($R_{d/a}$) increased to about 1.6 under axial winds but decreased to 1.1 under cross-flow, reflecting reduced thermal asymmetry. Overall, combining axial and lateral airflow paths improves ventilation uniformity in naturally or mechanically ventilated dairy barns. The findings provide theoretical and technical support for optimizing ventilation design, contributing to energy efficiency, animal welfare, productivity, and the sustainable development of dairy farming under changing climatic conditions.

1. Introduction

Heat stress was recognized as a major challenge in modern dairy production, leading to reduced feed intake, milk yield, reproductive performance, and overall animal welfare [1,2]. Ventilation was the most common measure to reduce heat stress levels in dairy cows, with natural and mechanical ventilation being the two most prevalent methods. In temperate and subtropical regions, naturally ventilated barns were widely used, depending primarily on external wind for air exchange [3]. However, internal airflow patterns were strongly influenced by external meteorological conditions [4,5]. To overcome this limitation, turbulent fans were installed above feeding alleys or resting areas to provide supplementary airflow. These fans increased airflow velocities and ventilation rates within the animal occupied zones, enhanced airflow over the cow’s body surface and promoted convective heat dissipation, thereby mitigating heat stress [6,7].

The cooling effectiveness of dairy barns was influenced not only by installation parameters such as the angle, height, and spacing of the fans [7,8,9], but also by the coupled effects of external wind directions and speeds. Previous studies by Saha [10] and Fiedler [11] demonstrated that these external factors significantly affected airflow patterns in naturally ventilated dairy barns. Although the fans increased local airflow velocities and were less dependence on external airflow, mismatches between internal fan-driven flows and external wind flows could lead to inadequate local cooling or even cause back-flow and counter-flow. Therefore, combining the advantages of natural and mechanical ventilation represented the most effective approach for optimizing dairy barn ventilation [12,13]. Mondaca et al. (2019) [14] reported that an airflow velocity of 0.85 m s⁻¹ increased the threshold temperature at which dairy cows reached 53% of their maximum respiratory rate. Increasing the velocity to 1.5 m s⁻¹ further elevated this threshold and nearly eliminated the adverse effects of high humidity. A significant difference in air temperature was observed between still air and 1 m s⁻¹ airflow, whereas no significant difference occurred between 1 m s⁻¹ and 2 m s⁻¹. Therefore, a minimum cooling airspeed of 1 m s⁻¹ is proposed for dairy cows. However, systematic studies investigating the effects of external natural ventilation conditions, during the operation of the fans, on the surface airflow distribution and convective heat transfer characteristics of dairy cows were still lacking.

In recent years, computational fluid dynamics (CFD) has been widely applied to the study of livestock and poultry housing environments, allowing the virtual evaluation and optimization of ventilation strategies before practical implementation [15,16,17]. Currently, commercial CFD software such as ANSYS Fluent (Release 2022 R1), STAR-CCM+, and COMSOL Multi-physics (v6.4) were widely used due to robust solvers, intuitive interfaces, and comprehensive technical support [18,19]. However, the high licensing costs and limited customization options imposed significant constraints on large-scale agricultural research. In natural ventilation studies, the computational domains included both the dairy barn and the surrounding environment. The complex architectural structures of dairy barns, often consisting of irregularly shaped components at different scales, made mesh generation particularly challenging. This substantially increased computational resource consumption and posed major challenges for modern agricultural simulation research [15,20]. In contrast, the open-source platform OpenFOAM provided greater flexibility and lower costs, allowing researchers to develop customized solvers for specific research objectives. These advantages made it well suited for investigating complex airflow patterns in livestock and poultry housing and for optimizing the design of agricultural buildings [21,22]. However, its application is still associated with certain limitations, including a steep learning curve and potential numerical stability issues. Previous CFD studies primarily focused on optimizing indoor airflow in dairy barns under single ventilation modes, either natural or mechanical ventilation [23,24,25]. Nevertheless, systematic studies on the coupled effects of natural environmental conditions and fan configuration parameters on whole barn air exchange were still limited. In this study, OpenFOAM was used to simulate the influence of external wind directions and speeds on fan jets under natural ventilation. This approach reduced research costs and enabled the evaluation of their effects on dairy cow thermal comfort, providing a basis for further analysis of multi-scale airflow patterns.

Therefore, this study used numerical simulations with the open-source OpenFOAM platform to achieve the following objectives: (1) to investigate the coupled effects of external wind directions and speeds on airflow distribution in a naturally ventilated dairy barn; (2) to analyze airflow speed variations and convective heat transfer over different body regions of dairy cows. This study aims to support the development of ventilation strategies that are adaptable to changing climatic conditions. Rather than reproducing specific heat-stress events, the focus is placed on elucidating how external wind conditions influence indoor airflow and convective heat transfer, thereby providing a basis for optimizing barn design and mitigating heat stress through improved natural ventilation.

2. Materials and Methods

2.1. Geometric Model and Computational Domain

To reduce computational cost while preserving the representative airflow characteristics of the full-scale structure, the barn length (L) was reduced from 144 m to 36 m and the width (W) remained at 30 m, with the same cross-sectional geometry and ventilation openings retained. The eave height was 3.9 m, and the ridge height (H) was 9.2 m. The lower 1.6 m of the northern wall was constructed of solid brick, while the upper section remained open to promote natural air exchange. The southern wall was fully open to further enhance natural ventilation. The model was constructed without a symmetry plane to capture the full three-dimensional airflow behavior and potential asymmetries induced by external wind directions and fan operations. The detailed structural configuration of the dairy barn is shown in Figure 1.

To characterize airflow distribution within the barn and to evaluate convective heat transfer between different body regions of dairy cows and the surrounding air under a standing feeding posture, a simplified geometric model of the cows was developed. Each cow was represented by two cylindrical elements: a torso cylinder (0.66 × 0.66 × 1.60 m) and a head cylinder (0.32 × 0.32 × 0.60 m). The limbs were omitted, as their surface area accounts for approximately 16% of the total skin area and has a negligible influence on the overall convective heat transfer of dairy cows. A total of 40 cows were arranged in the computational domain according to typical barn layouts, and the total body surface area was calculated using Equation (1) [26]. Fifty-inch fans were installed above the central axis of the lying area and the headlock area in a 4 × 4 layout, with 6.0 m spacing between adjacent units, resulting in a total of sixteen fans. The fan centers were positioned 3.0 m above the ground, with an installation angle (defined as the angle between the airflow direction and the horizontal plane) of 30°. The cows were spaced at approximately 1.2 m between individuals. The layout of the cows and fans within the barn is shown in Figure 2.

$$S_{skin}=0.14W_{cow}^{0.57}\times \left(1-16\%\right)$$

(1)

where $S_{Skin}$ is the surface area of the simplified cow model, m²; $W_{cow}$ is the weight of the cow (about 625 kg), kg.

The computational domain consisted of the interior space of the dairy barn and the surrounding external environment, forming an overall cylindrical domain. The domain had a radius of approximately 124 m and a total height of 54.2 m, corresponding to overall dimensions of 248 × 248 × 54.2 m. The barn was located at the geometric center of the domain to minimize the influence of boundary conditions on airflow development and to reduce potential boundary effects. Geometric modeling was completed using Blender 4.1. The coordinate system was defined as follows: the X-axis represented the barn width, the Y-axis represented the barn length, and the positive Z-axis pointed vertically upward (Figure 3). The lowest point inside the barn was designated as the origin [27]. The radius (R) and total height ($H_{total}$) of the external domain satisfied the recommended domain size, specifically [28]:

$$H_{total}=H+45$$

(2)

$$R=\sqrt{2W^2+2L^2}+5\sqrt{2}+50$$

(3)

where $H_{total}$ is the height of the entire computation domain, m; $H$ is the ridge height of the dairy barn, m; $R$ is the radius of the entire computational domain, m; $W$ is the width of the dairy barn, m; $L$ is the length of the dairy barn, m.

2.2. Meshing Strategy and Independence Test

The computational mesh was generated using the BlockMesh and SnappyHexMesh utilities in OpenFOAM. First, BlockMesh was used to create a structured hexahedral background mesh around the STL (stereolithography) geometry of the dairy barn, defining the external computational domain and establishing the base mesh size. Subsequently, SnappyHexMesh was applied to further refine critical geometric regions on the pre-partitioned background mesh. Local refinement regions included the barn walls, cow surfaces, and fan surfaces. The refinement was controlled by setting minimum and maximum refinement levels [29], as summarized in

Table 1. Local mesh refinement strategy and refinement regions.

(a)
Mesh level	0	1	2	3	4	5	6
Mesh size (unit: m)	1.6	0.8	0.4	0.2	0.1	0.05	0.025
(b)
Refinement region	Mesh level of refinement region (Min level; Max level)
Inlet	(0; 0)
Outlet	(0; 0)
Sky	(0; 0)
Ground	(0; 0)
Building walls	(3; 5)
Cow	(5; 6)
Fans	(5; 6)

.

To maintain high mesh resolution near the dairy barn while reducing the overall cell count, three nested refinement regions were established at the center of the computational domain, with dimensions of (L + 10 m) × (W + 10 m) × (H + 3 m), (L + 18 m) × (W + 18 m) × (H + 7 m), and (L + 34 m) × (W + 34 m) × (H + 23 m), where L, W, and H denoted the length, width, and height of the dairy barn, respectively. The corresponding refinement levels were set to 3, 2, and 1, with cell sizes of 0.20 m, 0.40 m, and 0.80 m, respectively. To improve prediction accuracy in the near wall regions, two boundary layers were generated along each wall of the dairy barn, while three boundary layer meshes were applied to the cow’s body surface. This approach more accurately captured the airflow characteristics and convective heat transfer near the cow’s surface. Finally, mesh quality was verified using checkMesh, with the maximum skewness constrained below 4 and the maximum non-orthogonality limited to less than 65°, ensuring numerical stability.

To verify that mesh size did not influence the simulation accuracy, a mesh independence test was conducted. The analysis focused on evaluating the sensitivity of the results to mesh resolution rather than to turbulence model selection. Under a reference wind speed of 3 m s⁻¹ and a wind direction of 0°, simulations were conducted using three mesh configurations: fine (6,606,291 cells), medium (4,092,959 cells), and coarse (2,290,087 cells). The results are summarized in

Table 2. Dairy barn structure grid independence test.

Terms	Case1	Case2	Case3
Number of Meshes	6,606,291	4,092,959	2,290,087
Average speed at Z = 1.42 m (m s−1)	5.28	5.24	5.13
Relative Error	0.0%	0.76%	2.84%
CHTCs (W m−2 K−1)	7.78	7.72	7.48
Relative Error	0.0%	0.77%	3.86%

. Compared with the fine mesh (Case 1), the relative errors in dorsal airflow velocity (z = 1.42 m) and the CHTCs on the cow body surface within the animal occupied zone for the medium and coarse meshes (Cases 2 and 3) remained below 3% and 4%, respectively, indicating that numerical convergence had been achieved. Considering both accuracy and efficiency, the medium mesh configuration, containing approximately 4.1 million cells, was adopted for subsequent simulations.

2.3. Numerical Simulation

Numerical simulations were conducted on a workstation equipped with an AMD Ryzen Threadripper 3970X (32 cores, 3.69 GHz) and 128 GB RAM, running OpenFOAM-v2306 under a 64-bit Linux operating system. Parallel computations were carried out using 32 processor cores. A steady-state Reynolds Averaged Navier–Stokes (RANS) approach was applied to solve airflow within the atmospheric boundary layer, a method widely applied in numerical studies of airflow and convective heat transfer in building environments [20,30]. The buoyantSimpleFoam solver based on the Boussinesq assumption was selected. Under the assumption of small temperature variations, air density was assumed to vary only with temperature, and air was treated as a weakly compressible fluid. Consequently, density variations affected only buoyancy, with negligible effects on other flow characteristics. Chemical reactions and phase change phenomena were neglected. Based on the $\mathrm{B}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{q}$ formulation, the continuity, momentum, and energy equations for a weakly compressible flow were solved simultaneously, as shown below [31]:

$$\nabla ·\left(\rho U\right)=0$$

(4)

$$\nabla ·\left(\rho UU\right)=-\nabla p+\rho \mathrm{g}+\nabla ·\left(2{\mu }_{eff}D\left(U\right)\right)-\nabla \left({\displaystyle \frac{2}{3}}{\mu }_{eff}\left(\nabla ·U\right)\right)$$

(5)

$$\nabla ·\left(\rho Uh\right)+\nabla ·\left({\displaystyle \frac{1}{2}}\rho U{\left|U\right|}^2\right)=\nabla ·\left({\alpha }_{eff}\nabla h\right)+\rho Ug$$

(6)

$$\rho ={\rho }_{ref}{\rho }_k$$

(7)

$${\rho }_k=1-\beta \left(T_{air}-T_{ref}\right)$$

(8)

$$D\left(U\right)={\displaystyle \frac{1}{2}}\left(\nabla U+{\left(\nabla U\right)}^T\right)$$

(9)

where $U$ is the velocity field, ${\mathrm{m\; s}}^{-1}$; ∇· is the divergence operator; $p$ is pressure, Pa; $\rho$ is the air density, ${\mathrm{kg\; m}}^{-3}$; $g$ is gravitational acceleration, ${\mathrm{m\; s}}^{-2}$; ${\mu }_{eff}$ is the sum of molecular viscosity and turbulent viscosity (${\mu }_{eff}=\mu +{\mu }_t$), where ${\mu }_t$ is the turbulent dynamic viscosity, representing the additional momentum diffusion induced by turbulent eddies, $\mathrm{Pa\; s}$; ${\alpha }_{eff}$ is the effective thermal diffusivity, the sum of laminar and turbulent thermal diffusivity (${\alpha }_{eff}=\alpha +{\alpha }_t$), where ${\alpha }_t$ denotes the turbulent thermal diffusivity derived from the turbulent viscosity, m² s⁻¹; $D\left(U\right)$ is the rate of the strain tensor; $h$ is the sensible enthalpy, J; ${\rho }_k$ is a dimensionless parameter; ${\mathsf{\rho }}_{\mathrm{r}\mathrm{e}\mathrm{f}}$ is the reference density of the air, ${\mathrm{kg\; m}}^{-3}$; $T_{air}$ is the air temperature, $\mathrm{K}$; $T_{ref}$ is the air reference temperature, $\mathrm{K}$; $\beta$ is the volumetric expansion coefficient, ${\mathrm{K}}^{-1}$.

Because the objective of this study was not to compare the performance of different turbulence models but to analyze airflow behavior and convective heat transfer under different external wind conditions. The standard k–ε turbulence model was adopted due to its robustness, numerical stability, and widespread application in building ventilation studies [20,32,33,34,35]. The average $yPlus$ value was maintained within the recommended range (30 < $yPlus$ < 300). The governing equations were discretized using the finite volume method, and pressure-velocity coupling was achieved using the SIMPLE algorithm within $\mathrm{b}\mathrm{u}\mathrm{o}\mathrm{y}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{S}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{F}\mathrm{o}\mathrm{a}\mathrm{m}$. Gradients were discretized using the cellLimited Gauss linear scheme with a limiting factor of 1 to suppress numerical oscillations and non-physical extrema. Diffusion terms were discretized using the Gauss linear limited corrected scheme with a limiter coefficient of 0.33, which controlled non-orthogonal corrections to maintain numerical accuracy and stability [29]. Discretization schemes for other variables and the convection terms are listed in

Table 3. Discrete Control Format.

Terms	FvSchemes
div ($\varphi$,U)	bounded Gauss linearUpwind default
div ($\varphi$,h)	bounded Gauss linearUpwind default
div ($\varphi$,Ekp)	bounded Gauss linear
div ($\varphi$,K)	bounded Gauss linearUpwind default
div ($\varphi$,k)	bounded Gauss upwind
div ($\varphi$,$\epsilon$)	bounded Gauss upwind
div ((($\rho \times {\mu }_{eff}$) $\times$ dev2(T(grad(U)))))	bounded Gauss linear

. The pressure equation was solved using the generalized geometric–algebraic multigrid (GAMG) method, while velocity, enthalpy, turbulence coefficient k and ε were solved using the PBiCGStab algorithm. Relaxation factors were set to 0.7 for pressure, 0.2 for velocity and enthalpy, and 0.5 for k and ε to ensure stable convergence [29].

In the discretization of convective terms, the general form div ($\varphi$, X) represents the divergence of the flux of variable X, where U is the velocity vector in the momentum equation; h is the specific enthalpy used in the energy equation; Ekp and K are turbulence-related transported variables defined by the selected solver and turbulence model; k is the turbulent kinetic energy; and ε is the turbulent dissipation rate. Here, $\varphi$ = U⋅S denotes the face flux and dev2(T(grad(U))) represents the deviatoric part of the symmetric velocity gradient tensor, which corresponds to the viscous stress contribution in the momentum diffusion term. Different interpolation schemes were applied to different transported variables to balance accuracy and numerical stability. LinearUpwind schemes were used for velocity and enthalpy to improve solution accuracy, while first order upwind schemes were applied to the turbulence variables k and ε to reduce spurious oscillations and improve numerical robustness. All schemes were bounded to ensure numerical stability.

The convergence of simulation results was evaluated using the residuals and the mass flow rate difference between the inlet and outlet. Since the momentum predictor equation was disabled in the solver, convergence was primarily assessed based on the residuals of pressure ($p_{rgh}$), energy (h), turbulent kinetic energy (k), and turbulence dissipation rate (ε). The solution was considered converged when the initial residual of $p_{rgh}$ was less than 10⁻⁴, the initial residuals of h, k, and ε were less than 10⁻⁵, and the difference between inlet and outlet mass flow rates were less than 10⁻⁴.

2.4. Simulation Cases and Boundary Conditions

To evaluate the coupled effects of external wind directions and speeds on airflow over the cow’s body surface and the CHTCs, twenty-five simulation cases were conducted. These included five wind directions (0° aligned with the fan jet, 45°, 90°, 135°, 180° opposite to the fan jet; see Figure 3), and five wind speeds (1 ${\mathrm{m\; s}}^{-1}$, 3 ${\mathrm{m\; s}}^{-1}$, 5 ${\mathrm{m\; s}}^{-1}$, 7 ${\mathrm{m\; s}}^{-1}$, 10 ${\mathrm{m\; s}}^{-1}$). In addition to the longitudinal airflow generated by fans, barn airflow was also affected by natural ventilation driven by thermal buoyancy and wind pressure differences. Atmospheric boundary layer (ABL) conditions were defined using a ground-normal streamwise velocity profile. The inlet velocity and turbulence dissipation rate were expressed by Equations (10) and (11). At the sky top boundary, friction velocity ($u^{\ast }$) and tangential stress ($\tau$) were expressed by Equations (12) and (13), respectively. The sky top was set as fixedShearStress to maintain tangential stress across the domain [29]. In buoyancy-driven flow, pressure is typically decomposed into the actual pressure p and corrected pressure $p_{rgh}$, as expressed by Equation (14).

$$u={\displaystyle \frac{u^{\ast }}{\kappa }}ln\left({\displaystyle \frac{z-d+z_0}{z_0}}\right)$$

(10)

$$\epsilon ={\displaystyle \frac{{\left(u^{\ast }\right)}^3}{\kappa \left(z-d+z_0\right)}}\sqrt{C_{1}ln\left({\displaystyle \frac{z-d+z_0}{z_0}}\right)+C_2}$$

(11)

$$u^{\ast }={\displaystyle \frac{u_{ref}\kappa }{ln\left(\frac{z_{ref}+z_0}{z_0}\right)}}$$

(12)

$$\tau =\rho \ast {\left(u^{\ast }\right)}^2$$

(13)

$$p=p_{rgh}+\rho gh$$

(14)

where $u^{\ast }$ is the friction velocity, ${\mathrm{m\; s}}^{-1}$; $\kappa$ is the Karman’s constant (0.41); $z$ is the vertical height from the reference ground, m; $z_0$ is roughness length (0.01 m); d is the zero plane displacement height, m; $C_1$ is the empirical coefficient controlling the vertical growth of turbulent dissipation with height; $C_2$ is the empirical coefficient setting the baseline turbulent dissipation near the wall; $u_{ref}$ is the reference wind speed, ${\mathrm{m\; s}}^{-1}$; $z_{ref}$ is the reference height (10 m); $\rho$ is the air density, ${\mathrm{kg\; m}}^{-3}$; p is the air pressure, Pa; $p_{rgh}$ is the hydrostatic-corrected pressure, Pa.

The boundary conditions defined in this study are presented in

Table 4. Boundary condition.

Terms	Inlet	Outlet	Sky	Ground	Cow	Walls
${\alpha }_t$	calculated	calculated	calculated	atmAlphatkWallFunction	compressible::alphatJayatillekeWallFunction	compressible::alphatJayatillekeWallFunction
k	atmBoundaryLayerInletK	inletOutlet	zeroGradient	kqRWallFunction	kqRWallFunction	kqRWallFunction
$\mathsf{\epsilon }$	atmBoundaryLayerInletEpsilon	inletOutlet	zeroGradient	atmEpsilonWallFunction	epsilonWallFunction	epsilonWallFunction
U	atmBoundaryLayerInletVelocity	inletOutlet	fixedShearStress	noSlip	noSlip	noSlip
T	fixedValue, 298.15 K	inletOutlet, 298.15 K	zeroGradient	zeroGradient	fixedValue, 311.85 K	zeroGradient
p	calculated	calculated	calculated	calculated	calculated	calculated
$p_{rgh}$	fixedFluxPressure	uniformFixedValue	fixedFluxPressure	fixedFluxPressure	fixedFluxPressure	fixedFluxPressure
${\mu }_t$	calculated	calculated	calculated	atmNutkWallFunction	nutkWallFunction	nutkWallFunction

. Atmospheric boundary layer profiles were prescribed at the inlet to represent realistic natural wind conditions, while wall function approaches were applied on solid surfaces to represent turbulence and heat transfer in the near wall region. Outlet boundaries were treated with inletOutlet or zeroGradient conditions to maintain numerical stability under possible flow reversal.

At the inlet, the velocity (U), turbulent kinetic energy (k), and dissipation rate (ε) were prescribed using the atmospheric boundary layer (ABL) profiles implemented by atmBoundaryLayerInletVelocity, atmBoundaryLayerInletK, and atmBoundaryLayerInletEpsilon, respectively. These boundary conditions impose vertically varying inlet profiles derived from atmospheric boundary layer theory, ensuring consistency between mean wind speed, turbulence intensity, and surface roughness effects under natural wind conditions. At the outlet, an inletOutlet condition was applied for U, k, and ε, allowing flow to leave the domain freely while permitting weak inflow if local recirculation occurred. The temperature (T) at both inlet and outlet was fixed at 298.15 K to represent uniform ambient conditions.

The ground surface was treated as a rough wall using atmNutkWallFunction and atmEpsilonWallFunction to account for surface roughness effects on turbulence generation within the atmospheric boundary layer, while standard wall functions (nutkWallFunction, kqRWallFunction, and epsilonWallFunction) were applied on the cow bodies and structural walls to model near-wall turbulence behavior [29]. A noSlip condition was imposed on all solid surfaces for velocity.

For pressure, p was set as a calculated field, while the modified pressure $p_{rgh}$ was prescribed using fixedFluxPressure on solid boundaries and the inlet to ensure mass conservation under buoyancy-driven flow [29]. The turbulent thermal diffusivity (${\alpha }_t$) near solid surfaces was modeled using compressible::alphatJayatillekeWallFunction, which was formulated for compressible flow solvers and represents heat transfer in the near wall region by linking temperature gradients with local velocity and turbulence quantities. In the core flow region, ${\alpha }_t$ and ${\mu }_t$ were computed directly by the turbulence model. The initial values of k and ε were calculated, ${\alpha }_t$ and ${\mu }_t$ were initialized to zero.

In this study, fans were defined as fanMomentumSource in the fvOptions file [29], where fans were modeled as velocity-driven sources. The pressure drops induced by the fans was represented using internal fan boundary conditions combined with pressure–velocity characteristic curves. The corresponding static pressure data at different fan speeds were shown in

Table 5. Fan performance parameters.

Static Pressure (pa)	0	12	25	37	50	62	75
Flow rate (m3 s−1)	12.00	11.39	10.89	10.33	9.72	8.97	8.08
Fan speeds (${\mathrm{m\; s}}^{-1}$)	9.47	8.99	8.60	8.16	7.67	7.08	6.38

.

2.5. Convective Heat Transsfer

Although dairy cows dissipate body heat to the surrounding environment through multiple pathways including convection, conduction, radiation, and evaporation, this study focused on exclusively convective heat transfer between the cows and the surrounding airflow while neglecting other heat transfer processes. This approach aims to further evaluate the effects of wind directions, speeds, and fan arrangement parameters on heat exchange rates between cows and their environment. The convective heat transfer coefficients (CHTCs) of the cow’s body surface were selected as a key indicator of thermal comfort, as it integrates the combined effects of temperature gradients, body surface area, fluid properties, and airflow dynamics. The calculation formula is:

$$h_c={\displaystyle \frac{q_{conv}}{S_{skin}\left(T_{skin}-T_{air}\right)}}$$

(15)

where $h_c$ is the CHTCs, ${\mathrm{W\; m}}^{-2} {\mathrm{K}}^{-1}$; $q_{conv}$ is the convective heat transfer rate, W; $S_{skin}$ is the total surface area of the cow’s body, ${\mathrm{m}}^2$; $T_{skin}$ is the surface temperature of the cow’s body, K; $T_{air}$ is the air temperature, K; in this study, the convective heat transfer rate on the cow’s body surface is the heat flux on the cow’s body surface.

2.6. Statistic Method Used in This Study

To enhance the interpretability and robustness of the simulation results, this study incorporated lightweight statistical analyses. This study primarily examined the rate of change (Δ%) in the mean airflow velocity over the dorsal and abdominal regions of dairy cows in response to variations in wind speeds, as well as their sensitivity to wind directions, expressed as the coefficient of variation (CV). The formulae for calculating Δ% and CV were as follows:

$$∆\%={\displaystyle \frac{\left(V_i-V_1\right)}{V_1}}\times 100\%$$

(16)

$$CV={\displaystyle \frac{\sqrt{\frac{1}{4}\sum _{i=1}^5{\left(V_i-\overline{V}\right)}^2}}{\frac{1}{5}\sum _{i=1}^5{V}_i}}\times 100\%$$

(17)

where $V_i$ is the airflow velocity over each body region under different wind speeds, m s⁻¹; $V_1$ is the airflow velocity over each region at a wind speed of 1 m s⁻¹, m s⁻¹; and $\overline{V}$ is the mean airflow velocity across all wind directions at each wind speed, m s⁻¹. The statistical analysis of the CHTCs across the different body regions of the cow followed the same methodological principles as those used in Equations (16) and (17).

2.7. Verification of CFD Model

The jet flow generated by dairy barn fans had been validated in previous studies. Therefore, airflow velocity and the Nusselt number were selected as the primary verification parameters in this study.

2.7.1. Verification of Wind Speeds and Directions in Natural Domain

The wind speed validation data were identical to those used by Hong et al. [27] for validating open-source CFD codes for simulating natural ventilation in dairy barns, as the case shared comparable geometric and boundary conditions with the present model. The remaining cases involved only minor structural variations and lacked sufficient measurement data for quantitative comparison. Although experimental data on ambient temperature and humidity were unavailable, this did not substantially affect the validation, as the focus was on airflow velocities distribution rather than heat or moisture transfer. Consequently, the absence of these parameters was not expected to compromise the accuracy of the simulated velocity field. These data were obtained from a naturally ventilated dairy barn with two continuous side vents and one ridge vent [36]. The barn measured 96.15 m in length, 34.2 m in width, with an eave height of 5.01 m and a ridge height of 10.87 m. The detailed model and wind speed measurements were taken 2 m above ground level (Figure 4). The inlet wind followed a logarithmic profile with 6.5 m s⁻¹ at 10 m height and surface roughness of 0.007 m. Other boundary conditions and domain settings were defined as previously described.

2.7.2. Verification of Convective Heat Transfer

A cylindrical model with a diameter of 0.3 m and a length of 1.5 m was employed, consistent in dimensions with the model used by Wang et al. (2018) to investigate the effects of airflow velocity and direction on convective heat transfer in both standing and lying cows [23]. The computational domain was defined with a length of 16.5 m (11 × L, where L is the cylinder length), a width of 10.5 m (7 × L), and a height of 3 m (10 × D, where D is the cylinder diameter). The cylinder was located 4.5 m (3 × L) downstream of the inlet, elevated 1.5 m (5 × D) above the ground, and placed 5.25 m (3.5 × L) laterally from the side boundary. The detailed geometric configuration is shown in Figure 5. In all simulations, the ratio of the projected cross-sectional area of the cylinder in the airflow direction to the inlet area was 1.43%. Franke and Hellsten [37] noted that when the obstruction ratio was less than 3%, the obstruction effect can be neglected in this study.

The dimensionless Nusselt number is widely used to characterize the ratio of convective to conductive heat transfer at a boundary surface. It is described as follows:

$$Nu={\displaystyle \frac{h_{c}D}{k_{air}}}$$

(18)

where $h_c$ is the CHTCs obtained from CFD simulation calculations, ${\mathrm{W\; m}}^{-2} {\mathrm{K}}^{-1}$; D is the characteristic dimension, which is the diameter of the cylinder in this study, m; $k_{air}$ is the thermal conductivity of the air, ${\mathrm{W\; m}}^{-1} {\mathrm{K}}^{-1}$.

Similarly, the Reynolds number is also used to describe airflow in a dimensionless form, and is determined by the following factors:

$$Re={\displaystyle \frac{VD}{\upsilon }}$$

(19)

where $V$ is the free stream speed, ${\mathrm{m\; s}}^{-1}$; $\upsilon$ is the kinematic viscosity, ${\mathrm{m}}^2 {\mathrm{s}}^{-1}$.

To validate the accuracy of the CFD simulation, five different inlet velocities (1 ${\mathrm{m\; s}}^{-1}$, 2 ${\mathrm{m\; s}}^{-1}$, 3 ${\mathrm{m\; s}}^{-1}$, 4 ${\mathrm{m\; s}}^{-1}$, and 5 ${\mathrm{m\; s}}^{-1}$) were set, with the inlet temperature set to 293.15 K and the cylindrical surface temperature at 311.85 K. The CFD model was validated by comparing the predicted results with two empirical formulas for cylindrical bodies in cross-flow. The formula used to calculate the average Nusselt number was as follows:

$${Nu}_{zuk}=0.26{Re}^{0.6}{Pr}^{0.37}$$

(20)

$${Nu}_{chu}=0.3+{\displaystyle \frac{0.62{Re}^{1/2}{Pr}^{1/3}}{{\left[1+{\left(0.4/Pr\right)}^{2/3}\right]}^{1/4}}}\left[1+{\left({\displaystyle \frac{Re}{\mathrm{282,000}}}\right)}^{1/2}\right]$$

(21)

For 20,000 < Re < 400,000

where ${Nu}_{zuk}$and ${Nu}_{chu}$ are the Nusselt numbers calculated from the relevant studies by Zukauskas [38] and Churchill and Bernstein [39], respectively, and Re is the Reynolds number; $Pr$ is the Prandtl number, a dimensionless parameter defined as the ratio of the momentum diffusion coefficient to the thermal diffusion coefficient ($Pr=\nu /\alpha$, where $\nu$ is the momentum diffusion rate, ${\mathrm{m}}^2 {\mathrm{s}}^{-1}$; $\alpha$ is the thermal diffusion coefficient, ${\mathrm{m}}^2 {\mathrm{s}}^{-1}$). All air properties were evaluated at the film temperature, defined as the arithmetic mean of the inlet air temperature and the cylinder surface temperature [40].

To confirm a good agreement, the relative error (Error) is calculated:

$$Error=\left|{\displaystyle \frac{{Nu}_{CFD}-{Nu}_{EF}}{{Nu}_{EF}}}\right|$$

(22)

where ${Nu}_{CFD}$ and ${Nu}_{EF}$ were based on the Nusselt number predicted by CFD simulation and the Nusselt number calculated by empirical formulas.

3. Results and Discussion

3.1. Airflow Speed and Convective Heat Transfer CFD Verification

3.1.1. Natural Domain Wind Speed CFD Validation

Figure 6 showed the wind directions and speeds at 41 points at a height of 2 m, obtained from wind tunnel experiments and CFD simulations. Overall, the simulated wind speeds (Figure 6a) agreed well with the experimental data, with relative errors below 10% at most points. Larger deviations, up to 45%, were observed at points 22, 23, 37, and 38. The predicted airflow directions (Figure 6b) also corresponded closely with the measurements, although discrepancies exceeding 30% occurred at points 1, 5, 6, 8, 10, 35, 36, 39, and 40. These results indicated that the CFD model accurately reproduced the main characteristics of the airflow field and direction distribution.

3.1.2. Convective Heat Transfer CFD Verification

To validate the accuracy of the CFD model in simulating convective heat transfer around a cylindrical surface, the simulated Nusselt numbers were compared with values predicted by two established empirical correlations. Figure 7 illustrated the relationship between the Nusselt number and the Reynolds number. The differences between the simulated and empirically predicted Nusselt numbers ranged from 0.73% to 9.86%. Greater discrepancies were observed at lower inflow velocities, where the simulations deviated more from the empirical predictions. Overall, the CFD results demonstrated good agreement with the empirical data. The relative errors in the simulated Nusselt numbers were below 10%, indicating that the calculated CHTCs were sufficiently accurate and that this approach was suitable for simulating convective heat transfer in cow geometries approximating cylindrical forms [41].

3.2. Effects of External Wind Direction and Speed on Cow Surface Airflow

The validated CFD model was used to simulate five wind directions (0°, 45°, 90°, 135°, and 180°) and five wind speeds at 10 m height (1, 3, 5, 7, and 10 m s⁻¹). Bottcher et al. (1998) also noted that, for evaluating airflow velocities at the animal level, the area-averaged velocity (AAV) was a more practical indicator than the total airflow supplied by fans, as it represented the actual airflow perceived by the animals [42]. To ensure accurate airflow data, the measurement planes were set 0.02 m above the cow’s dorsal (z = 1.42 m) and 0.02 m below the cow’s abdominal (z = 0.72 m). The airflow velocities distributions over the cow’s dorsal and abdominal under different wind directions and speeds were shown in Figure 8. The results indicated that both wind directions and speeds had a significant effect on the airflow velocities over the surface of dairy cows, but the magnitude and pattern of influence differed among body regions. As reported by Fiedler et al. (2013) neither wind speeds (WS) nor wind directions (WD) alone significantly affected the internal airflow distribution in dairy barns; however, their interaction (WS × WD) had a pronounced influence on airflow patterns [11].

As shown in Figure 8a, the average airflow velocities over the dorsal plane generally decreased and then increased as the wind direction varied from 0° to 180°. The dorsal region was directly influenced by both the fan jet and the external airflow, leading to pronounced variations in local velocity. Under 0° wind direction (downwind), the interaction between the fan jet and the approaching flow produced the highest dorsal velocities, with the average value reaching 5.41 ${\mathrm{m\; s}}^{-1}$ and exhibiting an approximately linear increase with wind speeds. At a wind speed of 10 ${\mathrm{m\; s}}^{-1}$, the maximum dorsal airflow velocities approached 6.2 ${\mathrm{m\; s}}^{-1}$, which was higher than those observed under other wind directions. At 90° wind direction, dorsal airflow velocities also remained relatively high, particularly at high wind speeds (≥7 ${\mathrm{m\; s}}^{-1}$), with average values exceeding 4.35 ${\mathrm{m\; s}}^{-1}$, indicating enhanced air exchange along the dorsal surface. In contrast, at 45°, 135°, and 180°, opposing or deflected inflows weakened the fan jet momentum, reducing dorsal velocities to 1.22–4.17 ${\mathrm{m\; s}}^{-1}$ and exhibiting limited sensitivity to wind speeds. Rong et al. [3] simulated airflow in a mixed-ventilation dairy barn under a reference wind speed of 3.86 ${\mathrm{m\; s}}^{-1}$ at 10 m height and reported that the air exchange rate decreased progressively as the wind directions shifted from 0° to 90°, which was consistent with the trend observed in the present study at 3 ${\mathrm{m\; s}}^{-1}$.

As shown in Figure 8b, airflow velocities in the abdominal region were generally lower than those over the dorsal region. The maximum abdominal velocities occurred at a wind direction of 90°, where lateral inflow interacted more effectively with the fan jet. As a wind speed increased to 10 ${\mathrm{m\; s}}^{-1}$, the velocities in this region approached 5.22 ${\mathrm{m\; s}}^{-1}$. Under 0° wind direction, abdominal velocities remained moderate (approximately 3.02–3.67 ${\mathrm{m\; s}}^{-1}$) and increased gradually with wind speeds. Mondaca et al. [25] reported convective heat transfer between dairy cows and the surrounding airflow under different airflow conditions and analyzed their thermal responses to heat stress. They reported that increasing airflow velocities to 1 ${\mathrm{m\; s}}^{-1}$ provided significantly greater cooling benefits than further increasing it to 2 ${\mathrm{m\; s}}^{-1}$. Accordingly, the minimum cooling airspeed (MCAS) for resting dairy cows was identified as 1 ${\mathrm{m\; s}}^{-1}$. However, at wind directions of 135° and 180° (oblique and headwind), abdominal velocities were lowest, with mean values below 1 ${\mathrm{m\; s}}^{-1}$ in some cases, indicating reduced air renewal beneath the cow due to weakened jet penetration. Compared with the dorsal region, the abdominal region exhibited greater susceptibility to flow stagnation because of its location within the lower recirculation region and partial shielding by the body. Wu and Gebremedhin [43] simulated airflow around multiple dairy cows and found that the local flow field was influenced by both animal positioning and orientation relative to the incoming flow. Vortices developed in the spaces between individuals, and airflow velocities were lower and more stable behind and beneath the abdominal regions. These observations were generally consistent with the present results.

The rate of change (Δ%) in the average airflow velocities with external wind speeds and the corresponding sensitivity to wind directions, quantified by the coefficient of variation (CV) to wind directions were shown in

Table 6. Rate of change (Δ%) of the average airflow velocity over different regions of the cow body with increasing wind speeds, and sensitivity to wind directions quantified by the coefficient of variation (CV).

Regions	Wind Directions	Δ% (Relative to 1 m s−1)					CV
		1m s−1	3 m s−1	5 m s−1	7 m s−1	10 m s−1
dorsal	0°	0.0%	19.9%	26.8%	29.5%	42.6%	11.3%
	45°	0.0%	−41.7%	−53.0%	−41.0%	−21.3%	27.2%
	90°	0.0%	−41.8%	−20.7%	4.3%	46.4%	30.1%
	135°	0.0%	−30.6%	−47.1%	−52.6%	−5.2%	29.3%
	180°	0.0%	−39.7%	−68.1%	−66.1%	−6.3%	44.9%
abdominal	0°	0.0%	1.6%	−5.3%	−5.3%	15.0%	7.4%
	45°	0.0%	−40.8%	−29.0%	−0.4%	40.8%	30.1%
	90°	0.0%	−12.8%	18.4%	54.7%	123.1%	35.7%
	135°	0.0%	−37.0%	−53.7%	−5.1%	64.4%	43.3%
	180°	0.0%	−52.1%	−64.5%	−14.7%	18.0%	40.3%

. For both the dorsal and abdominal planes, velocities declined markedly when the approach flow shifted to the leeward side (135–180°), with reductions of approximately 40–70% relative to the 1 m s⁻¹ condition. These wind directions also exhibited the highest CV values (e.g., 44.9% for the dorsal and 40.3% for the abdominal at 180°), indicating strong directional dependence and increased flow variability. From an engineering standpoint, 135° and 180° were therefore identified as the least favorable orientations, as the cow was located within a pronounced wind-shadow zone where convective cooling was constrained.

In contrast, 0° and 90° wind directions were identified as more favorable wind directions for natural ventilation. At 0° wind direction, the cow directly faced the incoming flow, resulting in a near-linear increase in local airflow velocities with increasing wind speeds and the lowest CV values (11.3% for the dorsal and 7.4% for the abdominal). At 90° wind direction, lateral inflow enhanced flow separation and reattachment around the torso, generating localized acceleration, particularly along the abdominal plane, where velocity increased exceeded 100% at high wind speeds. The abdominal region (z = 0.72 m) exhibited the highest sensitivity to wind directions, with CV values exceeding 35% under wind directions of 90–180°, indicating a higher risk of insufficient or unstable ventilation. This observation implied that design strategies for naturally ventilated dairy barns, including adjustment of inlet elevation, use of lower flow deflectors, and optimization of opening geometry, should explicitly consider the susceptibility of the abdominal region to weak airflow to ensure stable and effective convective heat removal.

3.3. Effects of External Wind Direction and Speed on Convective Heat Transfer

Figure 9 presented the average CHTCs for the entire cow body surface (Figure 9a) and for specific body regions (head, dorsal, abdominal) under five wind directions (0°, 45°, 90°, 135°, and 180°) and five wind speeds (1, 3, 5, 7, and 10 m s⁻¹). The head region corresponded to the small cylindrical section (0.32 × 0.32 × 0.60 m) of the cow model, whereas the dorsal region referred to the upper portion of the large cylindrical section (0.66 × 0.66 × 1.60 m) above the z = 1.07 m. The remaining lower sections of the large cylinder was defined as the abdominal region.

As shown in Figure 9, both wind directions and speeds significantly influenced the CHTCs on the cow surface, with clear spatial differences among the head, dorsal, and abdominal regions. Overall, the mean CHTCs (Figure 9a) exhibited a non-monotonic variation with wind directions and increased with wind speeds, reflecting the combined effects of the fan jet and external airflow. The highest values occurred under 0° wind direction, where the approaching flow aligned with the fan-induced jet. As wind speeds increased from 1 to 10 m s⁻¹, the overall CHTCs increased approximately linearly from about 6.50 to 8.24 W m⁻² K⁻¹. In contrast, under oblique or opposing wind directions (45°, 135°, and 180°), CHTCs decreased markedly to 4.63–6.54 W m⁻² K⁻¹, indicating reduced flow attachment and weaker convective heat transfer efficiency. These trends are consistent with the observations of Wang et al. (2018), who reported that CHTCs for cows in both standing and reclining postures were highest under crossflow conditions, whereas head-on flow produced comparatively lower values [23].

The head region (Figure 9b) consistently exhibited the highest CHTCs among all body parts, reaching up to 52.81 W m⁻² K⁻¹ at 0° wind direction and wind speed of 10 m·s⁻¹. This behavior was attributed to direct exposure to the incoming flow and elevated local velocity gradients associated with the head geometry. When the wind directions shifted to 135° and 180°, the average CHTCs decreased to below 36.28 and 37.13 W m⁻² K⁻¹, respectively, corresponding to reduced stagnation effects and lower effective incidence angles. Under 90° wind direction, CHTCs remained relatively high (above 37.18 W m⁻² K⁻¹), indicating that lateral airflow maintained effective convective exchange over the head.

In the dorsal region (Figure 9c), CHTCs distributions followed trends consistent with the airflow velocity field, where higher local airflow velocities enhanced convective heat transfer efficiency around the animal [41,44,45]. The 0° and 90° wind directions yielded relatively higher CHTCs (especially at wind speeds ≥ 7 m s⁻¹), with mean values of 16.3–24.29 W m⁻² K⁻¹ at 10 m s⁻¹. These conditions promoted direct or lateral flow impingement on the dorsal surface and increased boundary-layer disruption. At 45°, 135° and 180°, dorsal CHTCs declined by nearly 35.02%, reflecting shielding effects and enhanced flow separation downstream of the body.

The abdominal region (Figure 9d) exhibited the lowest CHTCs among all regions, indicating comparatively limited ventilation beneath the body. The highest abdominal CHTCs (approximately 15.41 W m⁻² K⁻¹) occurred under 0° wind direction at high wind speeds. Under wind directions of 45°, 135°, and 180°, mean CHTCs decreased by 26.91% relative to the 0° condition, consistent with the formation of low velocity zones near the abdomen. The 90° crosswind condition produced slightly higher values than oblique or headwind cases, suggesting that lateral inflow promoted partial circulation beneath the body.

Table 7. Rate of change (Δ%) in the CHTCs cross cow body regions with wind speeds, and sensitivity to wind directions quantified by the coefficient of variation.

Regions	Wind Directions	Δ% (Relative to 1 m s−1)					CV
		1m s−1	3 m s−1	5 m s−1	7 m s−1	10 m s−1
head	0°	0.0%	21.8%	30.7%	28.7%	30.7%	9.5%
	45°	0.0%	−19.7%	−22.1%	−15.0%	−3.0%	10.1%
	90°	0.0%	−16.8%	−19.2%	−15.6%	−1.7%	9.1%
	135°	0.0%	−15.3%	−22.8%	−24.7%	−11.7%	10.4%
	180°	0.0%	−18.3%	−29.3%	−19.9%	−7.6%	12.0%
dorsal	0°	0.0%	18.3%	26.9%	27.4%	30.1%	9.1%
	45°	0.0%	−27.1%	−30.9%	−24.3%	−12.7%	13.9%
	90°	0.0%	−19.4%	−25.3%	−18.8%	−4.1%	11.2%
	135°	0.0%	−11.4%	−15.9%	−21.6%	−15.3%	8.3%
	180°	0.0%	−11.6%	−23.2%	−23.7%	−0.5%	11.7%
abdominal	0°	0.0%	8.9%	7.4%	5.9%	14.1%	4.2%
	45°	0.0%	−21.5%	−25.6%	−11.6%	9.9%	14.7%
	90°	0.0%	−10.3%	−7.7%	1.7%	29.1%	13.6%
	135°	0.0%	−13.3%	−22.1%	−9.7%	12.4%	12.7%
	180°	0.0%	−22.2%	−24.8%	−13.7%	−0.9%	11.8%

reported the rate of change (Δ%) and coefficient of variation (CV) of CHTCs for the head, dorsal, and abdominal regions. The analysis indicated pronounced directional asymmetry in convective heat transfer. The head region consistently exhibited the highest CHTCs and the largest positive Δ% under 0° wind direction, reflecting strong exposure to the incoming flow. In contrast, oblique and lateral winds (45–135°) led to substantial reductions in heat transfer over both the head and dorsal regions due to increased flow separation and shielding. Among all orientations, 135° produced the lowest Δ% values across regions, indicating the least favorable convective cooling conditions. CV analysis further showed that the dorsal and abdominal regions were more sensitive to wind directions than the head, with CV values exceeding 13%. Under 45–90°, reflecting increased flow variability near the torso. From an engineering perspective, these results suggested that natural ventilation designs should avoid dominant oblique inflow orientations that suppress convective cooling over critical body surfaces. Promoting more axial or near-axial airflow directions is expected to provide more stable convective heat removal and improved thermal comfort under warm conditions.

3.4. Asymmetric Heat Dissipation Between Dorsal and Abdominal Regions Under Different Wind Conditions

Previous results indicated significant differences in CHTCs between the dorsal and abdominal regions of cows. To further quantify and discuss this thermal asymmetry, we defined the dorsal–abdominal CHTCs ratio ($R_{d/a}$) as:

$$R_{d/a}={\displaystyle \frac{h_{dorsal}}{h_{abdominal}}}$$

(23)

where $h_{dorsal}$ and $h_{abdominal}$ were CHTCs over the dorsal and abdominal regions of cows.

Figure 10 showed the variation in the ratio of dorsal to abdominal convective heat transfer coefficients ($R_{d/a}$) under different wind directions and speeds. In all cases, $R_{d/a}$ remained greater than 1, indicating that the dorsal surface consistently experienced higher convective heat dissipation than the abdominal surface. This confirmed that the dorsal region of cows acted as the primary region for sensible heat exchange.

Under 0° and 180° wind directions, $R_{d/a}$ increased with wind speed suggesting that external airflow enhanced dorsal convection more effectively than abdominal heat transfer. For example, under 0° wind directions, $R_{d/a}$ rose from approximately 1.38 at 1 m·s⁻¹ to around 1.58 at 10 m·s⁻¹. This trend was attributed to intensified flow impingement on the dorsal surface, which increased local turbulence and reduced the thermal boundary layer thickness. In contrast, under 90° wind direction, $R_{d/a}$ decreased from about 1.46 to 1.08 as the wind speed increased. The reduction indicated that crossflow promoted convective heat exchange along the lateral and abdominal regions, thereby weakening the asymmetry between the dorsal and abdominal heat transfer. At 45° and 135° wind directions, $R_{d/a}$ exhibited non-monotonic variations, likely resulting from complex interactions among impinging jets, recirculating vortices, and the geometric shielding effect of the body.

Mechanistically, the elevated $R_{d/a}$ at 0° and 180° were attributed to stronger airflow impingement and momentum transfer on the dorsal surface. In contrast, the reductions observed at 45° and 90° were attributed to the limited space beneath the abdomen, which increased airflow resistance. Although higher airflow velocities generally reduced boundary-layer thickness, the extent of heat transfer enhancement was determined by local flow realignment and separation patterns. From a thermal comfort perspective, a high $R_{d/a}$ (>1.50) indicated insufficient convective cooling on the abdominal surface and the potential for localized heat accumulation in the abdominal region. In terms of barn ventilation design, incorporating crossflow components (e.g., adjusting fans orientation or adding side-wall air inlets) could improve airflow uniformity and reduce dorsal–abdominal disparities. Conversely, when rapid cooling of the dorsal surface was required, wind directions aligned with the main jet axis were more advantageous.

3.5. Research Limitations and Perspectives

3.5.1. Applicability and Limitations of the Modeling Scope

We acknowledged that cows occupied different positions within naturally ventilated barns. However, this study focused on standing cows near the feed trough, where airflow distribution and convective heat transfer over the head, dorsal, and abdominal regions could be assessed with higher anatomical resolution. Airflow behavior in the lying area was expected to follow similar trends to that in the feeding alley, as the effective fan height relative to the cow body decreased when cows were in a recumbent posture. As the central objective was to examine the influence of natural wind on mechanically assisted ventilation, future work will incorporate cows in the lying area. It should also be noted that modeling a widely dispersed herd could obscure consistent airflow and heat transfer patterns, reducing the interpretability of comparative analyses.

3.5.2. Applicability of the Findings to Real World Farming Conditions

It should be noted that the airflow patterns and CHTCs reported in this study were obtained from steady-state simulations under fixed wind conditions. In naturally ventilated dairy barns, however, air exchange rates and indoor airflow vary dynamically with changes in outdoor wind and thermal buoyancy. Consequently, the reported velocities represent quasi-stationary or short-term ventilation states rather than constant operational conditions. Nonetheless, the identified airflow distribution and heat transfer characteristics offer valuable guidance for barn design and orientation, supporting improvements in natural ventilation efficiency and thermal comfort under variable field conditions.

3.5.3. Future Research Directions

Nordlund et al. (2019) reported that the overall average heat loss rate for dairy cows standing in free stalls was approximately −0.25 °C h⁻¹, whereas Hillman et al. (2005) observed higher rates ranging from −0.59 to −0.75 °C h⁻¹ for cows standing under feedline sprinklers and fans [46,47]. However, the cows in this study were evaluated only under fan conditions within free stalls, without accounting for the combined effects of sprinklers and airflow. Collier et al. [48] also demonstrated that the integration of sprinklers with mechanical ventilation provided more effective cooling than either system alone. Future work will further investigate the effects of sprinklers and fan jets on convective heat transfer in dairy cows under different natural conditions.

4. Conclusions

This study conducted a numerical investigation of the airflow speeds and CHTCs in different regions of the cow’s body surface under different external wind directions and speeds. Based on the results, this study draws the following main conclusions:

(1)

Under 0° wind direction, the dorsal airflow velocities reached up to 6.2 ${\mathrm{m\; s}}^{-1}$ at 10 ${\mathrm{m\; s}}^{-1}$ wind speed, indicating strong dorsal ventilation, as characterized by elevated airflow velocities over the cow’s dorsal region. In contrast, oblique and opposing winds (45–180°) reduced dorsal airflow velocities to 1.22–4.17 m s⁻¹, reflecting weakened ventilation performance.

(2)

The abdominal region exhibited maximum airflow velocities (approximately 5.22 ${\mathrm{m\; s}}^{-1}$) under 90° wind direction, but fell below 1 ${\mathrm{m\; s}}^{-1}$ at 135–180° wind direction, reflecting stagnation and limited air renewal beneath the body due to shielding by the body and reduced penetration of the incoming flow.

(3)

Under a wind direction of 0°, CHTCs increased linearly with wind speed (6.50–8.24 W m⁻² K⁻¹), indicating strong forced convection, while oblique or opposing winds (45–180°) reduced CHTCs by up to 35%, mainly because the incoming flow was less aligned with the fan jet, leading to weakened airflow impingement and reduced air renewal near the body surface.

(4)

The dorsal region exhibited 16.3–24.3 W m⁻² K⁻¹, whereas the abdominal region remained lowest (approximately 15.4 W m⁻² K⁻¹), reflecting weak ventilation and recirculating flow beneath the body, particularly under 135–180° winds.

(5)

Both airflow velocity and convective heat transfer analyses showed strong wind direction dependence, with 135–180° producing the sharpest Δ% reductions and highest CV values, indicating unstable, ventilation-limited zones, especially over the torso, the airflow velocities and CHTCs were minimal, indicating the most compromised convective cooling and a decrease in local heat dissipation.

(6)

Under axial winds (0° and 180°), the dorsal-to-abdominal CHTCs ratio ($R_{d/a}$) increased from 1.38 to 1.58 with airflow acceleration, whereas crosswinds (45–90°) reduced $R_{d/a}$ to nearly 1.1, indicating enhanced abdominal convection and improved thermal symmetry.

These findings demonstrates that the interaction between external wind directions and animal geometry play a critical role in determining airflow distribution and convective heat exchange in naturally ventilated dairy barns. Axial winds promote directional jet penetration and effective dorsal cooling. In contrast, oblique and crosswind conditions result in reduced airflow velocities and more heterogeneous airflow and temperature distributions around the animal body. Therefore, optimizing barn orientation and ventilation openings relative to prevailing winds is essential to maintain balanced air exchange and mitigate thermal non-uniformity in naturally ventilated dairy barns.