The cavitating flow involves liquid and vapor phases. In this work, the homogeneous mixture model was adopted, assuming the multiphase fluid components share the same velocity and pressure. The mass and momentum conservation equations for the mixture flow are shown as follows: ∂ ρ m ∂ t + ∂ ρ m u j ∂ x j = 0 (1) ∂ ρ m u i ∂ t + ∂ ρ m u i u j ∂ x j = − ∂ p ∂ x i + ∂ ∂ x j μ m ∂ u i ∂ x j + ∂ u j ∂ x i − 2 3 ∂ u k ∂ x k δ i j (2) ρ m = ρ ν α ν + ρ l 1 − α ν (3) μ m = μ ν α ν + μ l 1 − α ν (4) where u i represents the velocity in the i direction; p is the pressure; ρ m and μ m represent the mixture density and dynamic viscosity respectively; the subscripts l and ν denote the water and vapor phases respectively; α ν is the vapor volume fraction.

Solving the above unsteady Navier-Stokes equations directly raises an excessive demand on computing resources, which makes it impossible to be implemented in practical applications. Although Reynolds-Averaged Navier-Stokes (RANS) methods need reasonable computational cost, they fail in the situations dominated by coherent large-scale anisotropic vortical structures. Aiming at better simulating at an affordable cost, the LES approach was used, carrying out filtering in a small-space area, and dividing the turbulent flow into large-scale motion and small-scale motion. The large-scale motion is solved directly while the small-scale motion is modelled using the Smagorinsky-Lilly subgrid-scale model.

The mass transfer between the vapor and liquid phases in cavitating flows is depicted using the following governing equation: ∂ ∂ t ρ ν α ν + ∂ ∂ x j ρ ν α ν u j = m e − m c (5)

The source terms m e and m c represent the evaporation and condensation rates during the phase change. The source terms have different expressions. In the present study, the Schnerr–Sauer cavitation model is used, expressed as, m e = 3 ρ v ρ l ρ m α v 1 − α v R B 2 | p v − p 3 ρ l , p ≤ p v (6) m c = 3 ρ v ρ l ρ m α v 1 − α v R B 2 | P − P v 3 ρ l , p > p v (7) where p v is the saturated vapor pressure; R B is the nucleus radius, expressed as R B = α v / ( 1 − α v ) 3 / 4 π n 1 / 3 , where n is the given empirical constant and the default value is n = 1 × 10¹³ (Schnerr et al., 2001).

The computational model is shown in Figure 1. The cylinder diameter D is 9.5 mm, and the two-dimensional computational domain is 60D × 30D with an upstream dimension 10D and a downstream dimension 50D. The computational domain is discretized using structured grids, and the grids in the cylinder surface and wake region are locally refined. The total number of elements is about 336,000 and the average wall y + value is less than 4.6.

Boundary 1 of the computational domain is set as a velocity inlet with a value V ∞ = 1 m/s, and the corresponding Reynolds number is R e = 9,500. Boundary 3 is set as a pressure outlet, and the value p ∞ is set according to the cavitation number, defined as σ = p ∞ − p ν / 0.5 ρ l V ∞ 2 . A no-slip boundary condition is imposed on the cylinder surface and the free slip condition on the up and down walls (Boundary 4 and Boundary 2). The solution method adopts pressure base solver and a coupled pressure–velocity coupling algorithm. The pressure dispersion mode is second-order discrete mode, and the momentum equation is discrete by finite central difference scheme. The transient scheme is a second-order backward Euler algorithm. The time step is set to 5 × 10⁻⁵ s such that the courant number does not exceed 1.0 in the computations. The convergence standard of all residuals is 10^–6. The validation of the present numerical simulation can be found in our previous work (Gu et al., 2021). It should be mentioned that in the following analyses, all parameters shown in the figures are nondimensionalized via the cylinder diameter and the inlet velocity.

Entropy production arises from irreversible processes, during which the conversion of mechanical energy into internal energy occurs, resulting in energy dissipation. Entropy, as a state parameter, is governed by an equilibrium equation in a single-phase, incompressible flow, which can be expressed as: ρ ∂ s ∂ t + u ∂ s ∂ t + v ∂ s ∂ t + w ∂ s ∂ t = div q T + Φ T + Φ Θ T 2 (8)

Kock and Herwig, (2004) proposed that the Reynolds-averaged process can be dissected into two components, namely the averaged and fluctuating components. By extending the traditional Reynolds-averaged process, they formulated the entropy balance equation as follows, ρ ∂ s ¯ ∂ t + u ¯ ∂ s ¯ ∂ t + v ¯ ∂ s ¯ ∂ t + w ¯ ∂ s ¯ ∂ t = div q → T ¯ − ρ ∂ u ′ s ′ ¯ ∂ x + ∂ u ′ s ′ ¯ ∂ y + ∂ u ′ s ′ ¯ ∂ z + Φ T + Φ Θ T 2 (9)

In the equation, Φ T and Φ Θ T 2 represent entropy production. The first term signifies entropy production due to dissipation in the flow process, while the second term accounts for entropy production related to heat transfer during the heat transfer process.

Given the negligible temperature change in the examined cylindrical flow phenomenon, this study excludes the consideration of entropy production during heat transfer. The direct solution method is utilized to compute the entropy production Φ T during turbulent flow in this paper, incorporating viscous dissipation S pro , D ¯ and turbulent dissipation S pro , D ′ resulting from velocity fluctuations. The formula for the local entropy production rate per unit volume is presented as follows: Φ T = S pro , D ¯ + S pro , D ′ (10) S pro , D ¯ = μ T 2 ∂ u ¯ ∂ x 2 + ∂ v ¯ ∂ y 2 + ∂ w ¯ ∂ z 2 + ∂ u ¯ ∂ y + ∂ v ¯ ∂ x 2 + ∂ u ¯ ∂ z + ∂ w ¯ ∂ x 2 + ∂ v ¯ ∂ z + ∂ w ¯ ∂ y 2 (11) S pro , D ′ = μ T 2 ∂ u ′ ∂ x 2 ¯ + ∂ v ′ ∂ y 2 ¯ + ∂ w ′ ∂ z 2 ¯ + ∂ u ′ ∂ y + ∂ v ′ ∂ x 2 ¯ + ∂ u ′ ∂ z + ∂ w ′ ∂ x 2 ¯ + ∂ v ′ ∂ z + ∂ w ′ ∂ y 2 ¯ (12) S pro , D ′ cannot be directly calculated due to the inclusion of velocity pulsation terms. Kock assumed that it is related to turbulent dissipation rate ε and temperature T , and its calculation model is as follows: S pro , D ′ = ρ ε T ¯

The total entropy generation rate of the wall region is obtained by integrating over the wall region. Calculation formula for wall entropy generation: Δ S pro , W = ∫ S τ W ⇀ ⋅ v P ⇀ T ¯ d S (14)

Therefore, the total entropy production rate ψ inside the flow field is defined as follows: ψ = T ¯ × S pro , D ¯ + T ¯ × S pro , D ′ (15)

The entropy production rate near the cylindrical wall surface is: ψ W = τ W ⇀ ⋅ v P ⇀ (16)

The distribution of the entropy production rate and the dimensionless dynamic pressure in the wake flow field of a cylinder within a swirling flow is depicted in Figure 2. The dimensionless dynamic pressure is defined as follows: h = 0.5 ρ l V 2 0.5 ρ l U ∞ 2 (17)

The magnitude of the dimensionless dynamic pressure is representative of the fluid’s dynamic pressure in a specific region. The variation in color along isobars of nondimensional dynamic pressure demonstrates the dimensionless pressure drop in the flow field, with darker hues indicating more significant pressure drops.

In Figure 2A, in the non-cavitating case, energy losses in the vortex separation region near the upper and lower surfaces of the cylinder and in the wake region are revealed. The distribution trend of dimensionless dynamic pressure along the isobars is similar to that of the dimensionless pressure drop. In the region near the stagnation point at the leading edge of the cylinder, the obstructive effect of the cylinder results in lower fluid velocities, leading to a corresponding decrease in dynamic pressure at this location. From the stagnation point to the vicinity of the vortex shedding point on the cylinder surface, the flow separation caused by the acceleration and subsequent deceleration of flow due to the increasing flow velocity in this region results in a variation of dynamic pressure characterized by an initial increase followed by a decrease. In the near-wake region, as multiple-scale vortex structures detach and evolve, the pressure at the center of the vortices is relatively lower than the ambient pressure due to centrifugal forces. The velocity increases gradually from the vortex center to the vortex periphery, and the velocity gradient is significant. Consequently, the dimensionless isobars in the vicinity of the vortex periphery are densely distributed, with relatively higher dynamic pressure values. Upon comparison, it is evident that the regions of energy loss in the wake flow field around the cylinder are typically located in the low-pressure areas and their vicinity where the pressure drop is significant. In these regions, the dimensionless isobars are densely distributed, indicating a high velocity gradient in the flow field. Thus, the energy loss in the wake flow field of the cylinder in the non-cavitating model typically occurs in spatial ranges characterized by significant velocity gradients between the vortex center and its periphery.

Through comparative analysis, it is observed that under the non-cavitation model, energy losses in the wake flow field around the cylinder primarily manifest in low-pressure regions with significant pressure drops and their immediate surroundings. In these regions, a dense distribution of dimensionless dynamic pressure on equipotential lines is noted, indicating a notable velocity gradient in this flow area. Consequently, under the non-cavitation model, energy loss in the wake flow field around the cylinder typically occurs in spatial regions where a substantial velocity gradient exists between the center and the edge of the vortex.

In Figure 2B, it can be observed that the introduction of the SS cavitation model results in energy loss in the vortex separation region near the upper and lower surfaces of the cylinder, as well as in the wake region. Notably, the areas where energy loss occurs in the wake flow field are relatively dispersed, and the distribution area is comparatively small. The trend in the distribution of dimensionless dynamic pressure on equipotential lines generally aligns with that observed under the non-cavitation model.

Under the SS cavitation model, locations of energy loss in the wake flow field around the cylinder typically coincide with low-pressure regions experiencing substantial pressure drops. In these areas, a dense distribution of dimensionless dynamic pressure on equipotential lines is noted, indicating a significant velocity gradient in this flow region. Consequently, under the influence of the SS cavitation model, energy loss in the wake flow field around the cylinder generally occurs in spatial regions exhibiting a substantial velocity gradient between the center and the edge of the vortex.

When Figures 2A, B are compared, both display energy loss from the vortex separation region on the cylinder surface to the near-wake region. In contrast to the SS cavitation model, under the non-cavitation model, the energy loss region in the near-wake area is more concentrated, and the area is relatively larger. The primary reason for this difference lies in the generation of cavitation bubbles in the wake flow field when employing the SS cavitation model. These bubbles compress the upstream fluid, causing deceleration of the upstream flow and significant deformation of the detached multi-scale vortex system. Consequently, under non-cavitation model, the maximum dimensionless dynamic pressure in the wake flow field is higher than that under combined SS cavitation model conditions. Simultaneously, the compression of the multi-scale vortex system by the cavitation bubbles induces deformation and dispersion of the vortex system, resulting in a more scattered distribution of the regions experiencing energy loss in the wake flow field. However, in both conditions, energy loss in the wake flow field around the cylinder occurs in regions with a significant velocity gradient.

In Figure 3, under non-cavitation model conditions, the areas characterized by higher entropy production values in the wake flow field are larger when compared to those observed under the SS cavitation model. This distinction arises because, under non-cavitation model conditions, the multi-scale vortex system in the flow field remains relatively intact, featuring interactions solely between vortex systems. Consequently, the areas with significant changes in pressure gradient in the wake flow field, and hence higher entropy production values, are relatively larger, and there are nearly no points with exceptionally high entropy production.

Conversely, in the wake flow field under the combined SS cavitation model, the presence of cavitation bubbles results in the compression of the upstream fluid, inducing deceleration in the upstream flow. This leads to a reduction in velocity and velocity gradient within the boundary layer near the cylinder surface. Consequently, in the regions near the cylinder surface, most of the extreme points of entropy production are smaller than those observed under the non-cavitation model. Moreover, the generation of cavitation bubbles in the wake flow field compresses the multi-scale vortex system, causing significant deformation. The initially more concentrated vortices weaken in strength and disperse in the near-wake region due to the compression effect of the cavitation bubbles. This results in the emergence of some regions with significant changes in pressure gradient between interacting vortices, leading to localized extreme points of high entropy production.

For the facilitation of a quantitative analysis of energy loss in the near-wake flow field around a cylinder, the region is systematically divided into four zones: W0, W1, W2, and W3. These zones correspond to the leading-edge region, trailing-edge region, near-wake region, and far-field region of the wake flow field around the cylinder, as illustrated in Figure 4.

By integrating entropy production rates in each region, the calculation of energy losses is conducted and summarized in Table 1. The results emphasize a significant disparity in total energy loss, with the leading-edge region exhibiting notably greater losses compared to the other three regions, both in the absence of cavitation and when the SS cavitation model is incorporated. Notably, the energy loss in the leading-edge region without cavitation slightly exceeds that with the SS cavitation model. In both operational conditions, energy loss locations in this region concentrate near the upper and lower surfaces of the cylinder’s leading edge. In contrast, entropy production in the wake flow region is notably smaller than those in the leading-edge region. Specifically, losses in the trailing-edge region, near-wake region, and far-field region are compared, as illustrated in Figure 5. Observations indicate a gradual decrease in entropy production from the near to the far direction of fluid flow in the wake region. Furthermore, when the SS cavitation model is incorporated, energy losses in each region of the wake flow field are lower compared to the condition without cavitation. This reduction is primarily attributed to the generation of cavitation bubbles in the wake flow field, compressing the upstream fluid and leading to a decrease in flow velocity in the near-wake region. Consequently, the intensity of multiscale vortices diminishes, and the average velocity gradient between vortices decreases, thereby reducing entropy production in the wake flow field.

Figure 6 displays the proportions of direct entropy production (left) and fluctuating entropy production (right) in the corresponding regions of the wake flow field. A comparison reveals that in the absence of cavitation, entropy production in the leading-edge region is primarily attributed to direct entropy production. In contrast, in the trailing-edge, near-wake, and far-field regions, the proportion of fluctuating entropy production gradually increases, becoming the primary source of entropy production in those areas. However, with the incorporation of the SS cavitation model, entropy production in the leading-edge, trailing-edge, and near-wake regions of the wake flow field are predominantly associated with direct entropy production, while in the far-field region, fluctuating entropy production assumes prominence. Remarkably, in both scenarios, entropy production in the leading-edge region is primarily due to direct entropy production. This is a consequence of significant pressure gradient variation in this region. Simultaneously, the presence of cavitation bubbles in the wake flow field compresses the upstream fluid, causing a decrease in the ratio of fluctuating entropy production to total entropy production in the trailing-edge and near-wake regions. In the far-field region, entropy production is primarily associated with fluctuating entropy production. Due to the influence of cavitation bubbles, the proportion of fluctuating entropy production in this region is slightly lower when the SS cavitation model is applied compared to the condition without cavitation. Combining this information with Figure 5, it is evident that under SS cavitation model conditions, the total entropy production in the trailing-edge, near-wake, and far-field regions of the wake flow field decreases, accompanied by a corresponding reduction in the proportions of fluctuating entropy production in each region.

In summary, under both operational conditions, entropy production in the flow field around the cylinder are predominantly governed by direct entropy production in the leading-edge region (W0) and fluctuating entropy production in the far-field region (W3). The impact of cavitation bubbles on entropy production is primarily manifested in the trailing-edge region (W1) and near-wake region (W2).

Figure 7 depicts wall entropy production on the surface of the cylinder for both the non-cavitation model and the combined SS cavitation model, with the 180° direction corresponding to the leading-edge stagnation point on the cylinder surface, and the 0° direction corresponding to the trailing-edge stagnation point. The distribution, magnitude, and trends of wall entropy production on the cylinder surface are generally consistent for both the non-cavitation model and the combined SS cavitation model, as revealed in the graph. In summary, wall entropy production on the cylinder surface primarily concentrates on the upper and lower surfaces near the leading edge. The leading-edge surface exhibits a slightly higher maximum wall entropy production in the non-cavitation model compared to the combined SS cavitation model. However, during minor wall entropy production fluctuations on the trailing-edge surface, the maximum wall entropy production in the combined SS cavitation model surpasses that in the non-cavitation model. According to wall entropy production theory, the magnitude of wall entropy production is primarily related to the velocity of the fluid in the vicinity of the wall and the wall shear stress induced by velocity gradients. Thus, the distribution trend of wall entropy production along the cylinder surface aligns generally with the velocity distribution trend in the nearby region. This suggests that wall entropy production is mainly concentrated in regions with higher velocity gradients, and as velocity gradients increase, the values of wall entropy production also increase.

Given the frequent incidence of energy losses in regions characterized by substantial velocity gradients, there arises a necessity for a meticulous investigation into entropy production rates within the velocity boundary layer on the surface of the cylinder. Circles C1, C2, and C3, centered around the cylinder’s axis, were deliberately selected within the velocity boundary layer at distances ranging from 0.5D to 0.525D. The visual representation of the distribution of entropy production rates for these circles is presented in Figures 8, 9.

Referring to Figure 8, it becomes evident that the distribution trends of entropy production rates on circles C1, C2, and C3 remain consistently similar for both the non-cavitation model and the combined SS cavitation model. At the leading-edge stagnation point (180°), the entropy production rate is nearly zero for all three circles. Proceeding clockwise from the leading-edge stagnation point to the region near the upper surface flow separation point (ranging from 180° to 90°), in both scenarios, the entropy production rate rapidly increases to a maximum value and then experiences a rapid decrease. On the trailing-edge surface (ranging from 90° to 270°), there is a pronounced fluctuation of entropy production rate only in the vicinity of the trailing-edge stagnation point (0°). Near the lower surface flow separation point and back to the leading-edge stagnation point, the entropy production rate again rapidly increases to a certain maximum value before decreasing rapidly.

Upon comparing C1, C2, and C3, it is apparent that the locations of the maximum entropy production rates on the surface of the cylinder are generally consistent for both the non-cavitation model and the combined SS cavitation model. However, in the region beneath the cylinder, the maximum entropy production rate in the non-cavitation model significantly exceeds that in the combined SS cavitation model. As analyzed earlier, in the trailing-edge region, both operating conditions are significantly influenced by fluctuating entropy production. Consequently, the difference in energy losses mainly depends on the distribution of fluctuating entropy production, which is associated with the dissipation of turbulent kinetic energy in the fluid. Due to the quasi-periodic nature of the wake flow field, the distribution of turbulent kinetic energy dissipation rate on the cylinder surface is also quasi-periodic. At present, the turbulent kinetic energy dissipation on the upper surface concentrates near the leading edge, which is consistent with the flow separation point on the upper surface. This results in minimal generation of fluctuating entropy production in the velocity boundary layer on the upper surface of the cylinder. Therefore, in the velocity boundary layer on the upper surface of the cylinder, the trends of entropy production rates are similar for both the non-cavitation model and the combined SS cavitation model, with the maximum entropy production rate slightly higher when the SS cavitation model is incorporated.

Concurrently, the turbulent kinetic energy dissipation on the lower surface concentrates near the trailing edge, which is consistent with the flow separation point on the lower surface. Therefore, although the trends of entropy production rate distribution in the region near the lower surface are similar for both cases, the region is closer to the trailing-edge region (W1). Under the influence of cavitation bubble compression, the maximum entropy production rate in the velocity boundary layer on the lower surface of the cylinder is significantly smaller when the SS cavitation model is incorporated compared to the non-cavitation model. Meanwhile, in the region near the trailing-edge stagnation point of the cylinder, the effect of cavitation bubble compression in the wake flow field significantly influences the velocity boundary layer’s maximum entropy production rate under non-cavitation model conditions, making it much greater than that under the combined SS cavitation model. This observation indicates that the presence of cavitation bubbles in the wake flow field around the cylinder compresses the upstream fluid, leading to a reduction in the flow velocity and, consequently, a decrease in velocity gradients in the velocity boundary layer on the cylinder surface when the SS cavitation model is incorporated. This further elucidates the sharp decrease in the proportion of fluctuating entropy production in the leading-edge and trailing-edge regions observed in Figure 7 under both non-cavitation and combined SS cavitation model conditions.

According to Figure 9, it is evident that the variation trend of entropy production rate distribution within the velocity boundary layer remains fundamentally consistent under both non-cavitation and combined SS cavitation model conditions. The entropy production rate gradually decreases from the inner to the outer layers, and the location of the maximum entropy production rate also progressively shifts away from the leading-edge stagnation point. This shift is primarily attributed to flow separation occurring on the surface of the cylinder. In summary, the impact of vortex cavitation in the wake flow field around the cylinder on the distribution trend of entropy production rates within the velocity boundary layer on the cylinder surface is relatively minor. However, it significantly affects the magnitude of entropy production rates and the proportion of contributions from different sources to entropy production.