Mechanisms of Pressure Loss in Chamfered Orifice Plates: Coupled Effects of Plate Thickness and Reynolds Number

In this study, a single-hole throttle orifice plate with a diameter of D = 50 mm was chosen as the research object. The fluid medium used was liquid water at standard temperature and pressure (300 K, and 0.1 MPa), with a density ρ = 1.0 × 10³ kg/m³ and dynamic viscosity μ = 1.01 × 10⁻³ Pa·s. The inlet boundary was set as a uniform velocity inlet, with velocities of 0.25 m/s, 0.5 m/s, 1.0 m/s, 1.5 m/s, and 2 m/s, corresponding to Reynolds numbers of 12376, 24752, 49505, 74257, and 99010, respectively. The outlet boundary was specified as a pressure outlet with a gauge pressure of 0 Pa. The wall boundary was treated as a no-slip condition, with a turbulence intensity of 5 % and a turbulence viscosity ratio of 10 %.

As a mature commercial solver software, Fluent finds widespread application in the field of fluid dynamics and delivers outstanding performance in numerical simulations of orifice plates. Consequently, Fluent was selected for the numerical analysis of the flow field in this study. Assuming incompressible flow, the SIMPLE algorithm was employed for the solution procedure. For turbulence modeling, the Standard k-ɛ model was chosen. This model introduces an additional equation for the turbulent dissipation rate ɛ, based on the equation for turbulent kinetic energy k, thereby forming a two-equation model. The Standard k-ɛ model has been extensively used in CFD studies of throttle orifice plates and has produced reliable results. Therefore, it was deemed appropriate for the numerical analysis in this study. The governing equations are as follows:

• Continuity equation: (1) ∂ρ∂t+∇⋅(ρU)=0 {{\partial \rho} \over {\partial t}} + \nabla \cdot (\rho U) = 0

• Momentum equation: (2) ∂(ρU)∂t+∇⋅(ρU×U)=∇⋅(−pδ+μ(∇U+(∇U)T))+SM {{\partial (\rho U)} \over {\partial t}} + \nabla \cdot (\rho U \times U) = \nabla \cdot (- p\delta + \mu (\nabla U + {(\nabla U)^T})) + {S_M} where ρ denotes the fluid density, T represents time, U is the fluid velocity vector, δ is the Kronecker delta, μ is the fluid dynamic viscosity, p is the static pressure, and S_M refers to the momentum source term.

k-ɛ equation: (3) ∂(ρk)∂t+∇⋅(ρUk)=∇⋅μ+μtσk∇k+Pk−ρε {{\partial (\rho k)} \over {\partial t}} + \nabla \cdot (\rho Uk) = \nabla \cdot \left[ {\left({\mu + {{{\mu_t}} \over {{\sigma_k}}}} \right)\nabla k} \right] + {P_k} - \rho \varepsilon (4) ∂(ρk)∂t+∇⋅(ρUε)=∇⋅μ+μtσε∇ε+εk(Cε1Pk−Cε2ρε) {{\partial (\rho k)} \over {\partial t}} + \nabla \cdot (\rho U\varepsilon) = \nabla \cdot \left[ {\left({\mu + {{{\mu_t}} \over {{\sigma_\varepsilon}}}} \right)\nabla \varepsilon} \right] + {\varepsilon \over k}({C_{\varepsilon 1}}{P_k} - {C_{\varepsilon 2}}\rho \varepsilon) (5) Qk=μt∇V⋅(∇V+∇VT)−23∇⋅V⋅(3μt∇⋅V+ρk)+Qkb {Q_k} = {\mu_t}\nabla V \cdot (\nabla V + \nabla {V^T}) - {2 \over 3}\nabla \cdot V \cdot (3{\mu_t}\nabla \cdot V + \rho k) + {Q_{kb}}

In this equation, k represents the turbulent kinetic energy, ɛ is the turbulent dissipation rate, and C_ɛ1, C_ɛ2, σ_k, and σ_ɛ are the constants of the standard k-ɛ turbulence model, with values of C_ɛ1 = 1.44, C_ɛ2 = 1.92, σ_k = 1.00, and σ_ɛ = 1.3, respectively. The turbulent viscosity, μ_t, is expressed as: (6) μt=Cμk2ε {\mu_t} = {C_\mu}{{{k^2}} \over \varepsilon} where C_μ is the turbulent viscosity constant, with a value of C_μ = 0.09. P_k represents the turbulence production due to viscosity and buoyancy, and is expressed as follows: (7) Pk=μt∇U⋅(∇U+(∇U)T)−23∇⋅U⋅(3μt∇⋅U+ρk)+Pkb {P_k} = {\mu_t}\nabla U \cdot (\nabla U + {(\nabla U)^T}) - {2 \over 3}\nabla \cdot U \cdot (3{\mu_t}\nabla \cdot U + \rho k) + {P_{kb}}

The mesh density plays a critical role in the accuracy of CFD simulations [18]-[19]. In general, the numerical accuracy improves with increasing mesh resolution provided that adequate mesh quality is maintained; however, beyond a certain refinement level, the gain in accuracy becomes marginal while the computational cost increases significantly. Therefore, an appropriate meshing strategy balancing accuracy and efficiency is required.

In this study, ANSYS Fluent Meshing was used to generate the computational mesh using a poly-hexcore approach. The meshing procedure is illustrated in Fig. 4. First, a global mesh was generated for the entire fluid domain. Then, local refinement was applied to regions with strong gradients and complex flow features using the body of influence (BOI) method, with particular emphasis on the vicinity of the orifice plate and the near-wake region where jet contraction, separation, and recirculation occur. In addition, since steep velocity and pressure gradients develop near solid boundaries, prism layers were generated along the pipe wall and the orifice bore to improve near-wall resolution and capture boundary layer behavior. The simulations employed the standard k-ɛ turbulence model with standard wall functions, the inflation layer parameters were designed to maintain wall y⁺ values within the recommended range across the investigated Reynolds numbers. A mesh independence study was conducted to determine the final mesh resolution and is reported in the next section.

Fig. 4.

Poly-hexcore mesh and local refinement strategy for the orifice plate simulations.

The credibility of CFD predictions depends on both numerical resolution and the capability of the adopted model to reproduce benchmark flow metering characteristics. Accordingly, numerical credibility is established through a mesh sensitivity assessment and an independent benchmark of the discharge coefficient C_d against ISO 5167 reference values. A mesh sensitivity study was carried out by progressively refining the grid and tracking the differential pressure ΔP across the orifice plate. The comparison of ΔP obtained with different cell numbers is shown in Fig. 5. When the total number of cells exceeds 900000, the variation of ΔP becomes negligible, indicating that the solution is insensitive to further mesh refinement. Considering both accuracy and computational cost, the mesh with approximately 900000 cells was selected for all subsequent simulations. In addition, the predictive fidelity for metering performance was examined by comparing the simulated discharge coefficient C_d with the ISO 5167 reference values over the Reynolds number range considered in this work. Fig. 6 presents the comparison between the ISO based C_d and the CFD-predicted C_d. The CFD results exhibit the same decreasing trend of C_d with increasing Re, and the deviation from the ISO reference is consistently small: the mean absolute deviation is 3.41 × 10⁻³, and the maximum relative deviation is 0.66 % within the investigated Reynolds number range. This agreement, together with the grid independence results, supports the numerical reliability of the present CFD methodology for evaluating the hydraulic characteristics of chamfered orifice plates under the investigated conditions.

Fig. 5.

Grid independence assessment based on the differential pressure ΔP across the orifice plate.

Fig. 6.

Comparison of the discharge coefficient C_d between the ISO 5167 reference values and CFD predictions.

To investigate the effect of thickness on the flow field characteristics of the throttle orifice plate, this study examined the flow behavior of throttle orifice plates with thicknesses ranging from h = 3mm to 36 mm at an inlet flow velocity of 1 m/s. The structural parameters of the throttle orifice plate are summarized in Table 1, where the relative thickness of the orifice plate is defined as t = h/d.

Fig. 7 illustrates the axial velocity distribution across the throttle orifice plate. From the graph, it is evident that as the thickness of the orifice plate increases, the maximum jet velocity of the fluid passing through the orifice plate gradually decreases. This effect is particularly pronounced for thinner plates, while for plates with a relative thickness t > 0.5, the influence of plate thickness on the maximum jet velocity becomes negligible. This behavior can be attributed to the differences in jet dynamics as the fluid passes through thin versus thick plates. As shown in Fig. 8, for thin plates (with h = 3 mm and h = 7 mm), the maximum jet intensity occurs immediately behind the orifice plate. In contrast, for thicker plates (with h = 30 mm and h = 36 mm), the maximum jet intensity is observed inside the orifice plate. For thin plates, the jet expands only once as the fluid passes through, whereas for thick plates, the jet initially expands inside the orifice and then undergoes a second expansion after exiting the orifice. As a result, the fluid's motion becomes smoother after the two expansions, leading to a lower maximum jet velocity for thick plates compared to thin ones.

Fig. 7.

Axial velocities of throttled orifice plates of different thicknesses.

Fig. 8.

Velocity clouds for different thicknesses of throttled orifice plates.

Fig. 9 shows the dependence of the pressure loss coefficient ξ on the Reynolds number for orifice plates with different thicknesses. A pronounced thickness effect is observed: increasing plate thickness leads to a lower ξ over the investigated Reynolds number range. Moreover, opposite Reynolds number trends appear for thin versus thick plates — ξ increases with Re for thinner plates, whereas it gradually decreases with Re as the plate becomes thicker. This reversal is attributable to the different development of the near-wall boundary layer inside the orifice bore, which modifies the effective contraction process and, consequently, the irreversible loss. From a design perspective, increasing the plate thickness can be an effective way to reduce pressure loss, particularly at higher Reynolds numbers.

Fig. 9.

Effect of orifice plate thickness and Reynolds number on the orifice plate pressure loss coefficient.

Fig. 10 illustrates the streamline distribution near the throttle orifice for both thin and thick orifice plates. From the graph, it can be observed that for the thin orifice plate with h = 3 mm, the short length of the throttle orifice results in the entire wall area being within a low-pressure zone. Consequently, vortices formed behind the orifice plate penetrate the interior of the orifice. In contrast, for the thick orifice plate with h = 30 mm, the low-pressure zone and vortices generated by the jet are confined to the region near the orifice entrance. After passing through the low-pressure zone, the fluid begins to expand inside the throttle orifice, forming a new boundary-layer flow.

Fig. 10.

Trace distribution of thin and thick orifice plates.

As a result, for thin orifice plates, an increase in Reynolds number exacerbates vortex penetration into the region near the orifice plate wall, leading to higher pressure loss. On the other hand, for thick orifice plates, the increasing Reynolds number enhances the fluid's momentum, effectively suppressing the vortices near the entrance. This allows the boundary-layer flow inside the orifice to develop more quickly, thereby reducing pressure loss.

In throttling orifice flows, the pressure loss reflects an irreversible conversion of mechanical energy into internal energy, driven by viscous dissipation. In turbulent orifice jets, dissipation is promoted by separated shear layers, jet recirculation interaction, and near-wall high-shear regions. From a loss-budget standpoint, the total loss can be interpreted as the combination of a form-loss component (contraction, separation, recirculation, and mixing) and a friction-loss component (wall shear associated with boundary-layer development within the bore). For smaller plate thickness h, the bore length is short and the loss is primarily controlled by separation induced form loss and downstream mixing; thus, when the Reynolds number increases, the strengthened shear-layer dynamics and vortex activity can expand the effective mixing region (as evidenced in Fig. 10), leading to higher dissipation and increasing ξ (Fig. 9). For larger h, the bore behaves more like a short channel where the near-wall layer can develop and the shear layer can reorganize; this tends to mitigate the penetration of downstream recirculation into the bore and reduces the dominance of downstream mixing losses, yielding a decreasing or weakly varying ξ with Reynolds number within the investigated range.

To investigate the effect of chamfered structures on the flow field characteristics of orifice plates, this study designed a standard orifice plate with a thickness of h = 3 mm, along with chamfered orifice plates as detailed in Table 2. The flow field characteristics of these plates were analyzed under an inlet flow velocity of 1 m/s.

Fig. 11 presents the velocity profiles along the centerline for both the standard orifice plate and the various chamfered orifice plates. The data indicate that the maximum jet velocity of the chamfered orifice plates is notably lower than that of the standard orifice plate. Specifically, when the chamfer angles are 15°, 30°, and 45°, the reduction in the maximum jet velocity is particularly pronounced, with a maximum decrease of 17.67 %. This suggests that incorporating an appropriate chamfer structure can effectively enhance the flow field characteristics of the orifice plate.

Fig. 11.

Velocity change curves for standard and chamfered orifice plates.

The pressure loss coefficients for orifice plates with different chamfer angles, measured at various Reynolds numbers, are shown in Fig. 12. The graph reveals that for chamfer angles of 15° and 60°, the pressure loss coefficient increases with the Reynolds number. In contrast, for chamfer angles of 30° and 45°, the pressure loss coefficient decreases with increasing Reynolds number. This behavior can primarily be attributed to the penetration of vortices formed behind the orifice plate into the interior of the orifice.

Fig. 12.

Pressure loss coefficient versus chamfer angle at different Reynolds numbers.

From a practical design perspective, the results indicate that the chamfer angle should be selected to avoid the high loss regime observed at large angles. Within the investigated range, a moderate chamfer angle ranging from 30° to 45° consistently provides lower ξ than α = 60° across the Reynolds numbers considered. This benefit is attributed to the wall-attached jet and the suppression of vortex penetration into the bore region, as evidenced by the streamline patterns in Fig. 13. Therefore, α ≈ 30°–45° can be recommended as a robust choice for reducing pressure loss for the thin-plate case studied here (h = 3 mm), while the effect of bore thickness e is further quantified below.

Fig. 13.

Flow field traces of throttled orifice plates with different chamfer angles.

Fig. 13 compares the streamline patterns for different chamfer angles and reveals a distinct change in the jet recirculation topology inside the orifice. For α = 15° and 60°, the jet remains concentrated in the pipe core, leaving a low momentum region near the orifice wall that facilitates the inward penetration of the downstream recirculation structures toward the bore wall. In contrast, for moderate chamfers (α = 30° and 45°), the incoming flow is redirected, causing the jet to attach to the wall, which stabilizes the near-wall shear layer and effectively suppresses vortex penetration into the bore region. This flow-structure difference provides a mechanistic explanation for the trends in ξ: as Re increases, the strengthened recirculation promotes vortex penetration and higher losses at α = 15° and 60°, whereas the wall-attached jet at α = 30°–45° becomes more robust and maintains lower losses.

As a result, for orifices with chamfer angles of 15° and 60°, an increase in Reynolds number intensifies the vortices behind the orifice, leading to greater vortex penetration and, consequently, higher pressure loss. Conversely, for orifices with chamfer angles of 30° and 45°, as the Reynolds number increases, the enhanced fluid momentum along the wall suppresses vortex penetration, resulting in lower pressure loss.

The chamfered structure of the orifice plate is determined not only by the chamfer angle but also by the orifice bore thickness. However, previous studies have focused mainly on the effect of the chamfer angle on the flow field characteristics of the orifice plate, while the impact of orifice bore thickness has received comparatively less attention. To further explore the influence of the chamfered structure on the flow field characteristics, this study also conducted a series of simulations examining both the orifice bore thickness e and chamfer angle α of the chamfered orifice plate. The experiments analyzed different orifice bore thicknesses (e = 0, 1, 2, and 2.5) and chamfer angles (α = 15°, 30°, 45°, and 60°) for a 3 mm thick orifice plate, with the results for the pressure loss coefficient summarized in Table 3.

It is worth noting that the same dissipation-based mechanism also underpins the chamfer related trends discussed in this section. The chamfer angle α and the orifice bore thickness e primarily affect the inlet contraction and the initial shear-layer orientation, thereby modifying whether the jet attaches to the bore wall and whether downstream recirculation can penetrate toward the bore interior. As shown in Fig. 13, cases where the jet remains more wall attached tend to suppress vortex penetration and confine strong recirculation downstream, reducing the effective mixing and dissipation near the bore; conversely, configurations that promote a core jet with weaker near-wall momentum facilitate vortex encroachment toward the bore wall and intensify shear-driven mixing, leading to larger losses. This provides a consistent physical explanation for the observed dependence of ξ on α and e in Fig. 12–Fig. 14 and connects the chamfer effects to the thickness-related mechanisms.

Fig. 14.

Pressure loss coefficients for different chamfered structure orifice plates.

Fig. 14 illustrates the pressure loss coefficients for orifice plates with different chamfer angles and orifice bore thicknesses. From Fig. 14(a), it is evident that when e = 0, the pressure loss coefficient increases with the chamfer angle. The minimum pressure loss coefficient occurs at a chamfer angle of 15°, while the maximum occurs at 60°, with a maximum difference of 39.4 %.

Fig. 14(b) and Fig. 14(c) show that for e = 1 and e = 2, the pressure loss coefficient initially decreases and then increases as the chamfer angle increases. The minimum pressure loss coefficient is observed at α = 30°. A notable difference between the two cases is that at α = 15°, the orifice plate with e = 2 exhibits a significantly greater pressure loss coefficient than the one with e = 1.

From Fig. 14(d), it can be seen that for e = 2.5, the pressure loss coefficient is minimized at a chamfer angle of 45° and maximized at 15°, with the maximum pressure loss coefficient increasing by 35.3 %.

Moreover, Fig. 13 shows that as the orifice bore thickness e increases, the chamfer angle corresponding to the minimum pressure loss coefficient also increases. Therefore, in the design of the chamfered orifice plate, it is essential to consider both the orifice bore thickness and chamfer angle in a comprehensive manner. For the investigated configurations, the optimal chamfer angle increases with bore thickness: 15° at e = 0 mm, rising to 30° for e = 1–2 mm, and reaching 45° at e = 2.5 mm. This trend provides a clear design guideline: increasing the bore thickness requires a larger chamfer angle to maintain low-loss flow attachment and suppress vortex penetration.