Design of structural parameters and study on the energy dissipation characteristics of vertical jet-type energy dissipators

Nappe characteristics

During flow development, rotational flows occur mainly in two regions, namely, the shearing region and the swirling region. In the shearing region, two layers of non-rotational fluid flow in parallel, while in the swirling region, vortex structures cause development of the flow in a third direction. Jeong and Hussain⁵¹ proposed the \(\lambda _2\) criterion to distinguish between the shearing region and the swirling region.

Instantaneous distribution contour maps of \(\lambda _2\) and \(u_x\) on the cross-section (Y/D = 0) of the vertical jet in the span direction: (a) Instantaneous distribution contour map of \(\lambda _2\) when n = 0; (b) Distribution contour map of \(u_x\) when n = 0; (c) Instantaneous distribution contour map of \(\lambda _2\) when n = 2; (d) Distribution contour map of \(u_x\) when n = 2.

In the swirling region, \(\lambda _2<0\) and there are vortex cores, while in the shearing region \(\lambda _2>0\). To investigate the impact of adding fragmentation needles on nappe characteristics, a Reynolds number Re = 32638 needle size m = 0.15, shape s = B, and number of needles n = 0 and n = 2 were selected for the analysis of the jet shearing region and swirling region. Figure 9 shows the distribution of the shearing region and swirling region in the plane of the span section (Y/D = 0) using \(\lambda _2\) and \(u_x\). Figure 9a and c are instantaneous distribution maps of \(\lambda _2\), calculated from the three-dimensional flow field data at t = 207 D/V, when the flow field has fully developed. Figure S1 is the Reynolds shear stress \(\tau _{xz}\) at m = 0.15, Fig. S2 is the turbulence statistical results of velocity field. Figure S3 is the profiles of the normal Reynolds stress intensity components in the central (y = 0) vertical x-z plane. In Fig. 9a and c, the potential core and the surrounding environment are in the shearing region, and two distinct interfaces between shearing and swirling regions are observed. As shown in Fig. 9a , Shearing regions are Region1, and swirling regions are Region2, the envelope of \(u_x/V\in [0.95,1]\) is Region 1, which is located from the nozzle to the point where the flow changes (at Z/D = 4.42), and the end is enclosed by the swirling region on both sides that wraps around the center. The envelope of \(u_x/V\in [0.1,0.95]\) is Region 2, which is located on the outside of the jet and diverges laterally along the flow direction. Downstream at Z/D\(\ge\)6.00, a local swirling region detached from the main flow appears, reflecting the phenomenon of vortex stretching and mixing between the main flow and the surrounding fluid. In Fig. 9a–d, the dashed lines represent the contours of the average velocity \(u_x/V=0.95\) and \(u_x/V=0.1\). It can be seen that the contours of the average velocity can effectively distinguish the average positions of the shearing region and the swirling region. Therefore, in this paper, the contour lines of \(u_x/V=0.95\) and \(u_x/V=0.1\) are used to represent the boundaries of the potential core and the shear layer, respectively.

After the jets are ejected from the nozzles, the flow field exhibits a partitioned structure under the influence of the fragmentation needles, consisting of junction region and multijet region (Fig. 9c). After the two jets are ejected from the nozzles, there is an interaction between them, leading to an adsorption effect (Coanna effect). Due to the jets’ ability to entrain surrounding air, when no fragmentation needles are added, the jet at the nozzle can fully entrain air. However, after the addition of fragmentation needles, the aeration of the jet in the nozzle is limited. In the region between adjacent water jets, the entrained air is not replenished, causing the jets to attract each other and form the junction region.As more air is entrained, the jets begin to separate again, forming the multijet region.

Effect of the number of fragmentation needles on the potential core height: (a) s = A; (b) s = B, where Z is the vertical coordinate.

The effect of the number of fragmentation needles on the height of the nappe crown: (a) s = A; (b) s = B, where \(Z_t\) is the height of the nappe crown.

The effect of the number of fragmentation needles on the maximum diameter of the nappe crown: (a) s = A; (b) s = B, where X is the horizontal coordinate, ranging from 0 to 4.

After the fragmentation needle is arranged in the nozzle of the vertical jet pipe, the motion of the jet tongue can be divided into two stages: The first stage is the movement of the jet from the vertical pipe nozzle to the highest point of jet nappe. During this process, the jet tongue is fragmented by the fragmentation needle into multiple jet tongues, and at the same time, the direction of the motion of the jet nappe is changed, causing it to move to the highest point at a certain angle to the horizontal plane. The second stage is the fall of the jet nappe from the highest point into the water cushion energy dissipation pool. In this stage, the jet nappe performs projectile motion with a certain initial velocity at the highest point, and under the influence of air resistance, it further splits into jet water beams or shatters into water droplets, performing free fall and landing in the water cushion energy dissipation pool. The characteristics of jet nappe are analyzed for Reynolds number Re = 32638 and fragmentation needle numbers n = 0, n = 2, n = 3, n = 4, n = 5, and n = 6. The results of the characteristics of jet nappe are shown in Figs. 10, 11, and 12. From the figures, it can be observed that: Under the influence of no fragmentation needles, the potential core height is 4.42 D, the height of the nappe crown is 1.74 D, and the maximum diameter of the water nappe crown is 1.26 D. When fragmentation needles are present, the potential core height ranges from 0.007 D to 2.80 D, the height of the nappe crown ranges from 2.08 D to 2.50 D, and the maximum diameter of the nappe crown ranges from 2.20 D to 7.40 D. The spatial scale of the nappe crown is greatly influenced by the number, size, shape of the fragmentation needles, and Reynolds number. Compared to cylindrical fragmentation needles, under the same fragmentation needle size and Reynolds number, the potential core height is smaller. As the size and number of fragmentation needles increase, the potential core height gradually decreases, while the height and diameter of the nappe crown gradually increase. This is because the increase in the number of fragmentation needles also reduces the cross-sectional area of the jet nozzle, thereby increasing the initial velocity of the jet, causing the height of the nappe crown to increase with the number of fragmentation needles. The flow near the end of the potential core breaks into numerous small-scale fluctuating vortex structures, located near the interface between the shearing flow and the swirling flow, further indicating that it may dominate the vertical development and horizontal mixing and entrainment phenomena of the nappe crown. For high Reynolds number jets, the trend of vertical development is greater than the trends of horizontal mixing and entrainment, guiding the flow to develop vertically. The increase in fragmentation needle size and number intensifies the horizontal and vertical mixing and entrainment phenomena. Compared to the condition without fragmentation needles, after adding fragmentation needles, the height of the nappe crown increases, and the corresponding size of the swirling zone also increases; the potential core height decreases, and the corresponding size of the shearing region also decreases. Therefore, it can be inferred that, under the same other conditions, fragmentation needles “break” part of the jet’s shearing region into swirling region, and the total height of the jet decreases. This transformation will, to some extent, affect the transport of fluid in that region and dissipate the mechanical energy of the jet.

Multiscale vortices were induced by the velocity gradient that existed for shearing phenomena⁵², these larger-scale coherent structures are normally considered as key factors of turbulence type and turbulence distribution.⁵³ The spatial scale of the coherent structure was represented by the relationship between the distance in the Z direction and the diameter of pipeline D for dimensionless conditions.⁵⁴Hunt et al.⁵⁵proposed to define the region of Q>0 as vortexes.

This means that \(\Vert \Omega \Vert ^{2}>\Vert E\Vert ^{2}\), that is, the irrotational deformation is small compared to the vorticity. For instantaneous flow, the isosurface⁵⁶ of \(Q=0.1[V^2/D^2]\) at time t=120 is shown in Fig. 13. They show large-scale coherent vortices in the cross section. The free shear layer is a typical form of vortex, and its simplified model in inviscid flow is the surface vortex. The formation of a vortex ring is a surface vortex dynamics problem. Its steady state involves the existence problem and its deformation motion involves the stability problem. It plays an important role in the transport and mixing of fluids and vortex interaction.

Coherent structure of three-dimensional flow field: (a) n = 0,Re = 32638; (b) n = 2,Re = 32638, the contours are \(Q=0.1[V^2/D^2]\).

The effect of the number of fragmentation needles on the thickness of the vortex ring: (a) s = A; (b) s = B, where T is the thickness of vortex ring.

The effect of the number of fragmentation needles on the diameter of the vortex ring: (a) s = A; (b) s = B, where \(d_v\) is the diameter of vortex ring.

The effect of the number of fragmentation needles on the equivalent Reynolds number based on local conditions: (a) s = A; (b) s = B, where \(R_\Delta\) is the equivalent Reynolds number based on local conditions.

The thickness of a vortex ring can be defined based on the vorticity distribution within the ring. Q is a measure of the local rotation of the fluid, and the vortex ring’s thickness corresponds to the region where Q is concentrated.

Mathematical Definition: The thickness (\(d_v\)) can be defined as the radial distance between the points where the \(Q=0.1[V^2/D^2]\).

Where,\(d_\textrm{outer}\)is the radial position where the Q drops to the specified fraction on the outer side of the ring. \(d_\textrm{inner}\)is the radial position where the Q drops to the specified fraction on the inner side of the ring.

As shown in Fig. 9d, the position where the flow at the end of the potential flow core changes is Z/D = 2.52. The vortex ring located in the transverse section Z/D = 2.52 is selected for analysis. When Reynolds number Re = 32638, the thickness of the vortex ring T and the diameter of the vortex ring \(d_v\) under different sizes, and numbers of the fragmentation needle are shown in Figs. 14 and 15. Without the influence of the fragmentation needle, the thickness of the vortex ring is 1.00 D, and the diameter of the vortex ring is 2.00 D ; when there are fragmentation needles, the thickness of the vortex ring is between 0.001 D and 0.25 D, and the diameter of the vortex ring is between 0.01 D and 1.13 D. The spatial scale of the vortex ring is greatly affected by the number, size, shape, and Reynolds number of the fragmentation needles. Under the same flow conditions, the characteristic length of the coherent vortex decreases significantly with increasing size and number of fragmentation needles. The diameter of the vortex ring decreases from 2.00 D to 0.01 D horizontally, with a reduction rate of about 99.97%. The thickness of the vortex ring decreases from 1.00 D to 0.001 D vertically, with a reduction rate of about 99.80%. As a geometric description of the vorticity field, the vortex tube should always be a closed vortex ring in three-dimensional space, so that the total integral of the vorticity in space is zero in all directions. When there is a solid surface, the whole space can be treated as a fluid-solid coupling kinematic system. According to the unified fluid-solid system, the vortex tube cannot terminate on the solid wall or the free surface. When the solid wall is stationary, the vortex tube continues to extend along the boundary surface and the cross section of the vortex tube expands in a horn shape along the surface. These vortex structures will produce a damping effect in the flow direction. This damping effect originates from the fluid transport generated by the circumferential rotation of the vortex ring in the vertical direction. It can be inferred that under the same conditions, the relative size of the damping effect will affect the transport of the fluid to a certain extent, and then change the pressure in the region, and dissipate the mechanical energy of the flow-induced vibration when the jet impacts the fragmentation needle.

When Reynolds number Re = 32638, the equivalent Reynolds number based on local conditions \(R_\Delta\) under different sizes, numbers and shapes of the fragmentation needle are shown in Fig. 16. Without the influence of the fragmentation needle, the equivalent Reynolds number based on local conditions \(R_\Delta\) = 772,782; when there are fragmentation needles, the equivalent Reynolds number based on local conditions \(R_\Delta\) is between 1,159,173 and 4,369,992. the equivalent Reynolds number based on local conditions \(R_\Delta\) is greatly affected by the number, size, shape, and Reynolds number of the fragmentation needles. Under the same flow conditions, the equivalent Reynolds number based on local conditions \(R_\Delta\) increases significantly with increasing size and number of fragmentation needles. Compared with the cylindrical fragmentation needle, the equivalent Reynolds number based on local conditions \(R_\Delta\) under the action of the rectangular fragmentation needle are smaller under the same size, number and Reynolds number.

Vertical jet energy dissipation rate

In this study, vertical jet energy dissipation refers to the energy loss of the jet in the air after the jet is ejected from the vertical jet pipe. Therefore, the energy dissipation rate \(\eta\) is calculated using the energy difference between section \(1\#\) and section \(2\#\) (Fig. 3) and the energy ratio of section \(1\#\). The formula is as follows:

In the formula, \(\eta _1\) is the vertical jet energy dissipation rate, \(E_1\) is the total energy in the vertical jet pipe of section 1#, \(E_2\) is the total energy of section 2#, \(z_1\) and \(z_2\) are the position heads, that is, the height from the datum plane, \(p_1\) and \(p_2\) are the pressures at the two sections, \(v_1\) and \(v_2\) are the average flow velocities at the two sections, and \(\alpha\) is the flow velocity coefficient, which is generally taken as 1.

In this study, the ratio of the vertical jet energy dissipation rate under the influence of the fragmentation needle to the vertical jet energy dissipation rate without the influence of the fragmentation needle is defined as the vertical jet energy dissipation rate increase ratio r. Figure 17 is the vertical jet energy dissipation rate under different working conditions. Figure 18 is the vertical jet energy dissipation rate increase ratio under different working conditions. It can be seen from the figure:

Under the influence of no fragmentation needle, the vertical jet energy dissipation rate range is between 14.55 and 18.33%. When there are fragmentation needles, the vertical jet energy dissipation rate range is between 15.18 and 89.55%, and the increase ratio of the vertical jet energy dissipation rate ranges from 1.04 to 4.89, and is significantly influenced by the number, size, shape, and Reynolds number of fragmentation needles. When the design of the fragmentation needle is the same, the vertical jet energy dissipation rate is basically positively correlated with the Reynolds number; that is, the vertical jet energy dissipation rate increases with the increase of the Reynolds number. This is because as the Reynolds number increases, the vertical range of the jet increases, the contact surface between the jet and the air increases, and the jet aeration concentration increases, thereby increasing the vertical jet energy dissipation rate.

The dissipation rate under the action of the rectangular fragmentation needle is between 17.94 and 89.55%, the range of the vertical jet energy dissipation rate under the action of the cylindrical fragmentation needle is between 15.18 and 78.54%, the range of the vertical jet energy dissipation rate under the action of the rectangular fragmentation needle is between 1.23 and 4.89, and the range of the vertical jet energy dissipation rate under the action of the cylindrical fragmentation needle is between 1.04 and 4.29. The vertical jet energy dissipation rate under the action of the rectangular fragmentation needle is 1.27–21.80% higher than that under the action of the cylindrical fragmentation needle, and the vertical jet energy dissipation rate is increased by 0.06–1.36. This is because when the jet nappe crown is cut, the area of the jet nappe cut by the rectangular fragmentation needle is larger, which increases the contact friction surface between the jet nappe and the air, so the vertical jet energy dissipation rate is higher. Similarly, when the Reynolds number and the shape and number of the fragmentation needles are the same, the vertical jet energy dissipation rate increases with the increase of the fragmentation needle size.

When the Reynolds number is the same as the shape and size of the fragmentation needle, the vertical jet energy dissipation rate increases with the increase in the number of fragmentation needles. For each additional fragmentation needle of the same shape, the vertical jet energy dissipation rate increases between 1.69 and 21.23%. With an increase in the number of fragmentation needles, the vertical jet energy dissipation rate increases. The larger the ratio, the vertical jet energy dissipation rate increase ratio is between 0.11 and 1.16. This is because as the number of fragmentation needles increases, the number of broken jet nappe increases, the volume ratio of each jet nappe crown decreases, and the contact friction surface with the air increases, thus increasing the vertical jet energy dissipation rate. To clearly highlight the contributions, a table comparing key performance metrics with literature values can be included. The comparison results with the literature are shown in the Table 3

Vertical jet energy dissipation rate of each working condition: (a) s = A,m = 0.1; (b) s = A,m = 0.15; (c) s = A,m = 0.2; (d) s = B,m = 0.1; (e) s = B, m = 0.15; (f) s = B,m = 0.2, where \(\eta\) is the vertical jet energy dissipation rate.

Ratio of the vertical jet energy dissipation rate of each working condition: (a) s = A, m = 0.1; (b) s = A, m = 0.15; (c) s = A,m = 0.2; (d) s = B,m = 0.1; (e) s = B,m = 0.15; (f) s = B,m = 0.2, where r is the ratio of the vertical jet energy dissipation rate.

Energy consumption mechanism analysis

The vertical jet energy dissipation is to divide the high-velocity and large-flow water flow into multiple jets along the vertical direction and inject them upward. When the jet is ejected, a special boundary condition (fragmentation needle) is added at its outlet to break the jet nappe, and the jet with higher energy is divided into multiple jet nappe, so as to achieve the purpose of dispersing the water jet and making the jet nappe fully aerated. With the split and break of the jet nappe crown, the contact area between the jet nappe and the air is also increasing. The aeration of the jet nappe is also becoming more and more sufficient. The mixing of a large amount of air into the jet nappe will increase its contact friction surface with the air, so that part of the mechanical energy is converted into heat energy and consumed. The energy of the jet nappe is reduced, and the aeration energy dissipation is realized.

The numerical simulation method is used to simulate the three-dimensional (3 D) aeration characteristics of the vertical jet. The C(x,z) air concentration profile is cut and measured. The vertical spacing between individual profiles is constant at Z = 0.5 D, and the spacing of the sampling points along the coordinate x is about \(0.01 D\sim 0.03 D\). The left (subscript L) and right (subscript R) jet trajectories \(X_L(z)\) and \(X_R(z)\) are defined according to the outer boundary of the shear layer⁵⁷, similar to the isoline of the average velocity \(u_x/V=0.1\). M. Pfister and S. Schwindt used the definition of the average air concentration in the vertical position in the self-mixing airflow to define⁵⁸, and proposed the eigenvalue⁵⁹ that determines the complete distribution of the air concentration of the jet. This eigenvalue is the average air concentration of each jet cross section (subscript a), which is the result of the integral of C(z) divided by the diameter d of the nappe crown⁶⁰. The calculation method of \(C_a\) is as follows:

Effect of fragmentation structure size and Reynolds number on jet air concentration

The influence of fragmentation structure size and Reynolds number on air concentration of the jet: (a) s = A,m = 0.1; (b) s = B,m = 0.1; (c) s = A,m = 0.15; (d) s = B,m = 0.15; (e) s = A,m = 0.2; (f) s = B,m = 0.2, where \(C_a\) is the characteristic value of jet air concentration distribution.

When the size of the fragmentation needle m = 0.1, m = 0.15 and m = 0.2, and the number of fragmentation needles n = 2, the change of the air concentration of the jet nappe is analyzed. The curve of variation of the horizontal average air concentration of the jet along the vertical direction is shown in Fig. 19. It can be seen from the figure that:

Change in jet nappe air concentration when the size of the cylinder fragmentation needle is m = 0.1: the range of characteristic value \(C_a\) under the action of the cylindrical fragmentation needle is between 0.0012 and 0.6795. The Reynolds number Re increases from 24,479 to 32,638, and the eigenvalue \(C_a\) increases from 2.93 to 58.54%. The Reynolds number Re increases from 32,638 to 40,797, and the eigenvalue \(C_a\) increases from 5.53 to 61.34%. The Reynolds number Re increases from 32638 to 40797, and the eigenvalue \(C_a\) increases from 17.69 to 89.92 %. With an increase in Reynolds number, the jet nappe gradually decreases along the vertical unaerated area, and the air concentration in the entire area is increasing. When Reynolds number Re = 48957, the jet nappe is basically doped with air. This is because increasing the jet Reynolds number enhances the fragmentation effect of the jet impinging on the fragmentation needle⁶¹, so that the water flow is doped with more gas, so the jet nappe air concentration is greater.

Change in jet nappe air concentration when the size of the rectangular fragmentation needle is m = 0.1 : the range of characteristic value \(C_a\) under the action of the rectangular fragmentation needle is between 0.0067 and 0.691. The characteristic value \(C_a\) under the action of the rectangular fragmentation needle is 0.02–93.04% higher than that under the action of the cylindrical fragmentation needle. This is due to the larger area of the jet nappe cut by the rectangular fragmentation needle, so the air concentration involved in the jet nappe increases and the aeration of the jet nappe becomes more and more sufficient. Therefore, under the same conditions, the jet nappe air concentration of the rectangular fragmentation needle is higher. Under the influence of the rectangular fragmentation needle, the whole area of the jet nozzle is basically doped with air. The Reynolds number Re increases from 24,479 to 32,638, and the eigenvalue \(C_a\) increases from 0.88 to 56.77%. The Reynolds number Re increases from 32,638 to 40.797, and the eigenvalue \(C_a\) increases from 0.06 to 69.45%. The Reynolds number Re increases from 32,638 to 40,797, and the eigenvalue \(C_a\) increases from 4.27 to 54.35%. The trend of vertical development of high Reynolds number jet is greater than that of horizontal mixing and entrainment. The higher jet velocity enhances the impact effect of the potential core. With an increase in jet Reynolds number, the conversion efficiency from horizontal velocity to vertical velocity becomes more obvious. The fragmentation caused by the impact of water flow intensifies, resulting in a larger proportion of water converted into a liquid film, line, or droplet that diffuses outward. The flow cluster at the impact point is stretched vertically by a larger vertical velocity, and the air concentration of the two jet nappes is also gradually increasing. When Reynolds number Re = 48957, the aeration of the jet nappe crown is very sufficient.

When m = 0.15, n = 2, the range of the eigenvalue \(C_a\) under the action of the cylindrical fragmentation needle is between 0.04 and 0.71, and the range of the eigenvalue \(C_a\) under the action of the rectangular fragmentation needle is between 0.042 and 0.767. The eigenvalue \(C_a\) under the action of the rectangular fragmentation needle is 6.11–60.12% higher than that under the action of the cylindrical fragmentation needle. The Reynolds number Re increases from 24,479 to 32,638, the eigenvalue \(C_a\) increases from 10 to 75.56%, the Reynolds number Re increases from 32,638 to 40,797, the eigenvalue \(C_a\) increases from 8.92 to 61.82%, the Reynolds number Re increases from 32,638 to 40,797, and the eigenvalue \(C_a\) increases from 5.6 to 28.86%. As the Reynolds number increases, the total height of the jet gradually increases, and the average air concentration of the jet cross section gradually increases. The reason is that, at a lower Reynolds number, the conversion of the mechanical energy of water flow to the internal energy is not sufficient, resulting in the vertical kinetic energy still being interfered with after conversion. Therefore, the height at which the impinging stream is lifted is limited and most of the flow in the shear layer is fragmented and decomposed due to gravity. This leads to the formation of an almost zero velocity region at the top of the water nappe crown. With an increase in Reynolds number, although the volume of water in the central water column increases, the increase of air volume in the surrounding shear layer is more significant. The water flow flows directly to the top of the water nappe crown, resulting in fragmentation and obvious eddy current and horizontal velocity, which aggravates aeration.

When the size of the fragmentation needle is m = 0.2, the variation law of the air concentration of the jet nappe is basically the same as that when m = 0.1 and m = 0.15. When m = 0.2, n = 2, the range of the eigenvalue \(C_a\) under the action of the cylindrical fragmentation needle is between 0.06 and 0.75, and the range of the eigenvalue \(C_a\) under the action of the rectangular fragmentation needle is between 0.08 and 0.80. The eigenvalue \(C_a\) under the action of the rectangular fragmentation needle is 3.33–34.72% higher than that under the action of the cylindrical fragmentation needle. The Reynolds number Re increases from 24,479 to 32,638, the eigenvalue \(C_a\) increases from 2.81 to 66.17%, the Reynolds number Re increases from 32,638 to 40,797, the eigenvalue \(C_a\) increases from 2.22 to 20.75%, the Reynolds number Re increases from 32,638 to 40,797, and the eigenvalue \(C_a\) increases from 4.5 to 31.16%. At the center of the jet impinging on the crushing needle, the increase in the jet Reynolds number enhances the vertical velocity gradient, thereby increasing the shear force, and the collision effect leads to the turbulent energy cascade. Through the non-linear interaction of scale locality and non-locality (large scale), the energy is transferred from large scale to small scale⁶². The scale locality and nonlocality, respectively, mean that the energy cascade is completed by the non-linear effect of adjacent or non-adjacent wavenumber perturbations⁶³, which significantly amplifies the turbulent kinetic energy and its dispersion range in the center, resulting in a significant change in turbulent aeration along the jet path. It can be inferred that under the same conditions, the influence of the Reynolds number on the characteristic value \(C_a\) proves the effectiveness of mixing and mass transfer caused by impact to a certain extent, and emphasizes the key role of the Reynolds number of the jet in the change of turbulent energy.

When the Reynolds number is the same as the shape and number of the fragmentation needle, when the size of the fragmentation needle is different(that is, changing the blockage ratio of the flow around), the variation of the air concentration of the flow field under different blockage ratios is obtained. The size of the fragmentation needle m increases from 0.1 to 0.15, the eigenvalue \(C_a\) increases from 0.34 to 97%, the size of the fragmentation needle m increases from 0.15 to 0.2, and the eigenvalue \(C_a\) increases from 0.05 to 87.35%. The air concentration of the jet nappe crown increases with increasing fragmentation needle size, and the turbulence phenomenon increases with increasing blocking ratio⁶⁴. In the interaction between the jet flow field and the fragmentation needle, the degree of aeration is not only affected by the obstacle itself : the fluid separation occurs at the anchorage of the fragmentation needle and the jet port, so a vortex is formed on the wall of the fragmentation needle. It is also affected by the wake flow generated by the flow field around the obstacle : the vortex formed at the rest of the fragmentation needle is basically fixed, and the change is not large. Only the vortex formed at the anchorage of the jet port will appear similar to the vortex shedding phenomenon around the cylinder, showing an unstable state. The flow around the fragmentation needle will affect the entire downstream area. When flowing through the fragmentation needle, the density and velocity of the compressible fluid in a certain area on both sides of the fragmentation needle are greater than in other areas of the flow field. It can be inferred that under the same conditions, the influence of the size of the fragmentation needle on the characteristic value \(C_a\) proves to some extent that the average air concentration of the jet cross section is related to complex physical factors such as fluid separation, vortex shedding and large eddy structure. The larger the size of the fragmentation needle, the more complex the flow field behind the fragmentation needle.

Analysis of the influence of the number of fragmentation structures on jet air concentration

The Reynolds number Re = 32638, the size of the fragmentation needle m = 0.15, and the number of fragmentation needles n = 0, n = 2, n = 3, n = 4, n = 5 and n = 6 are selected. The change in air jet nappe air concentration is shown in Fig. 20:

Under the influence of no fragmentation needle, the range of characteristic value \(C_a\) is between 0.0044 and 0.3516. When there are fragmentation needles, the range of characteristic value \(C_a\) is between 0.046 and 0.8832. Under the influence of no fragmentation needle, the interior of the jet nappe is basically unaerated. When the Reynolds number and the size of the fragmentation needle are the same, with the increase of the number of fragmentation needles, the unaerated area inside the jet water nappe gradually decreases until the whole jet water nappe is aerated. The volume fraction of water on the surface of the jet nappe gradually decreases, and the air concentration in the whole area is increasing. For each additional fragmentation needle of the same shape, the eigenvalue \(C_a\) increases from 5.09 to 93.81%. Obviously, the more fragmentation needles are arranged, the stronger the entrainment and mixing between the jets, the better the energy dissipation effect. The mixing degree and mixing distance of the jets change with the number of fragmentation needles, and the mixing degree of the intersection of multiple jets gradually increases with the increase of the number of fragmentation needles, until there is no mixing between each other and each maintains the flow characteristics of a single jet, and the corresponding dimensionless vertical distance of mixing will gradually increase. When the number of fragmentation needles is n = 6, the outside and inside of the jet nozzle are basically doped with air. This is because at the nozzle of the vertical pipe, with the increase of the number of fragmentation needles, the number of fragmentation needles breaking the jet nappe increases, and multiple jets undergo deformation, collision, fragmentation and other processes in the air, thereby dispersing the momentum of the jet and enhancing the heat and mass transfer of the jet⁶⁵, the contact area between the jet water and the surrounding air increases greatly, and the entrainment and mixing between each other form multiple strong shear energy dissipation region, using the mutual adsorption between multiple jets and the strong entrainment and mixing with the surrounding ambient air to consume a large amount of residual energy in the pipeline water delivery process, so that the jet energy decays rapidly not far from the jet port. The inner rolling region consumes the rest of the energy. It can be inferred that under the same conditions, the influence of the number of fragmentation needles on the characteristic value \(C_a\) proves to a certain extent that the dispersion degree of the jet momentum and the mutual adsorption effect (Coanna effect) between the jets are the key to determine the energy dissipation effect. When there is no fragmentation needle, the rain curtain falling into the water cushion pool is concentrated, and the impact on the water cushion pool is large ; when there are a large number of fragmentation needles, the rain curtain falling into the water cushion pool is scattered, and the impact on the water cushion pool is small, which is conducive to the safety of the project.

The influence of the number of fragmentation structures on air concentration of the jet: (a) s = A; (b) s = B, where \(C_a\) is the characteristic value of jet air concentration distribution.

When Re,size and number of fragmentation needles are the same, the range of the eigenvalue \(C_a\) under the action of the cylindrical fragmentation needle is between 0.0044 and 0.873, and the range of the eigenvalue \(C_a\) under the action of the rectangular fragmentation needle is between 0.0044 and 0.8832. The eigenvalue \(C_a\) under the action of the rectangular fragmentation needle is 2.34–60.12% higher than that under the action of the cylindrical fragmentation needle. The total air concentration of the jet nappe formed by the rectangular fragmentation needle is greater than the total air concentration of the jet nappe formed by the cylindrical fragmentation needle. For a fixed point of the same height along the three-dimensional jet nappe, the air concentration of the jet nappe under the rectangular fragmentation needle is higher than that under the cylindrical fragmentation needle. This is because compared to the cylindrical fragmentation needle, the surface of the cuboid crushing needle is not smooth and there are four sharp corners at the anchorage of the jet port. The boundary layer is separated at its two front corners. When the fluid is separated from the front corner of the rectangular fragmentation needle and produces alternately shedding vortices, a vortex structure is generated above and below the rectangular fragmentation needle, which is the recirculation region⁶⁶ under the reverse pressure gradient. At the same time, the aeration of the jet nappe is becoming more and more sufficient. It can be inferred that under the same conditions, the influence of the shape of the fragmentation needle on the characteristic value \(C_a\) proves to some extent that the shedding mode of the vortex behind the rectangular fragmentation needle has different ways. In addition to the recirculation region behind the rectangular fragmentation needle, the vortex structure will be formed on the upper and lower sides of the rectangular fragmentation needle.

The achievements of this study in the design of energy dissipator and the optimization of turbulence model highlight its theoretical and application innovation by comparing with the existing literature. The turbulence model is compared in detail as follows:

Literature comparison: Deng et al.¹ used the standard k-\(\epsilon\) model to simulate the flow of spillway tunnel, and the predicted value of energy dissipation rate was 78 ± 3%, but failed to accurately capture the separation vortex structure (error > 10%). Chen et al.² used the LES model, although it performed well in vortex resolution (error)< 5% ), but the computational cost is about 8 times higher than that of the k-\(\epsilon\) model.The profiles of the averaged streamwise velocity is shown in Fig. 21.The experimental data of Bradbury⁶⁷, Gutmark ⁶⁸and Rampaprian ⁶⁹ are also shown

The averaged velocity profiles.

This study improved: the RNG k-\(\epsilon\) model is introduced and combined with the \(\lambda _2\) criterion threshold screening ( \(\lambda _2<\) 0 ). While ensuring the computational efficiency (the time consumption is reduced by 70% compared with LES), the prediction accuracy of the energy dissipation rate is improved to 92%, and the separation vortex shape error is reduced to 5%, which is significantly better than the literature.

In this study, by introducing the broken needle structure and the advanced turbulence model (RNG k-\(\epsilon\)), combined with the vortex energy dissipation analysis (\(\lambda _2\) criterion threshold \(\lambda _2<0\)),the following quantitative results are achieved:

Compared with the traditional energy dissipater, the arrangement of the fragmentation needle makes the energy dissipation rate increase from 75 to 89.55%, and the local turbulent kinetic energy dissipation rate increased by 40%.