E ﬀ ect of the Area Contraction Ratio on the Hydraulic Characteristics of the Toothed Internal Energy Dissipaters

: Toothed internal energy dissipaters (TIED) are a new type of internal energy dissipaters, which combines the internal energy dissipaters of sudden reduction and sudden enlargement forms with the open-ﬂow energy dissipation together. In order to provide a design basis for an optimized body type of the TIED, the e ﬀ ect of the area contraction ratio ( ε ) on the hydraulic characteristics, including over-current capability, energy dissipation rate, time-averaged pressure, pulsating pressure, time-averaged velocity, and pulsating velocity, were studied using the methods of a physical model test and theoretical analysis. The main results are as follows. The over-current capability mainly depends on ε , and the larger ε is, the larger the ﬂow coe ﬃ cient is. The energy dissipation rate is proportional to the quadratic of Re and inversely proportional to ε . The changes of the time-averaged pressure coe ﬃ cients under each ﬂow are similar along the test pipe, and the di ﬀ erences of the time-averaged pressure coe ﬃ cient between the inlet of the TIED and the outlet of the TIED decrease with the increase of ε . The peaks of the pulsating pressure coe ﬃ cient appear at 1.3 D after the TIED and are inversely proportional to ε . When the ﬂow is 18 l / s and ε increases from 0.375 to 0.625, the maximum of time-averaged velocity coe ﬃ cient on the line of Z / D = 0.42 reduces from 2.53 to 1.17, and that on the line of Z / D = 0 decreases from 2.99 to 1.74. The maximum values of pulsating velocity on the line of Z / D = 0.42 appear at 1.57 D and those of Z / D = 0 appear at 2.72 D , when the ﬂow is 18 l / s. The maximum values of pulsating velocity decrease with the increase of ε . Finally, two empirical expressions, related to the ﬂow coe ﬃ cient and energy loss coe ﬃ cient, are separately presented.


Introduction
The internal energy dissipater effectively reduces the downstream flow speed, smoothly connects the downstream flow, and avoids the erosion of the river channel by a traditional energy dissipater in a water conservancy project with high water head and high flow. This is because it converts the large area effect of high-speed water flow into a local energy dissipation effect. The most common form of internal energy dissipation is the energy dissipaters with sudden reduction and sudden enlargement. The energy dissipaters with the sudden reduction and sudden enlargement forms are a pressure energy dissipation method that uses the sectional contraction water flow to adjust and dissipate excess energy [1], and its hydraulic characteristics are mainly affected by the geometric size of energy dissipaters.
The common energy dissipaters with sudden reduction and sudden enlargement forms were divided into orifice plates and plugs (thick plates) according to their thickness along the flow direction [2], of the test device.
The TIED is located 143.5 cm away from the inlet of the test pipe, and its length (L) is 13.5 cm. For the TIED, the number of the piers (n) is 4, and the height of the piers (h) is 3.75 cm. The area contraction ratio (ε) can be defined as: . In equation, 0 A is the cross-sectional area of TIED, A is the cross-sectional area of test pipe, and θ is the angle of the piers. In order to study the effect of area contraction ratio (ε) on hydraulic characteristics of the TIED, five types of TIED were experimentally studied via the physical model. The cross-sectional areas of the TIED are displayed in Figure 2.
(a) The schematic diagram of test layout (b) The photo of test pipe During the test, the upstream head (H) was kept constant, the flow was constant in the test pipe, the indoor temperature was 20 °C, and the range of flow was between 18 l/s and 42 l/s. In order to analyze the same flow condition, the flow of each group was about 6 l/s increased sequentially, and the flow measurement error of each experimental flow group was ±0.2 l/s. The TIED is located 143.5 cm away from the inlet of the test pipe, and its length (L) is 13.5 cm. For the TIED, the number of the piers (n) is 4, and the height of the piers (h) is 3.75 cm. The area contraction ratio (ε) can be defined as: . In equation, A 0 is the cross-sectional area of TIED, A is the cross-sectional area of test pipe, and θ is the angle of the piers. In order to study the effect of area contraction ratio (ε) on hydraulic characteristics of the TIED, five types of TIED were experimentally studied via the physical model. The cross-sectional areas of the TIED are displayed in Figure 2.

Model Experiment
The test device consists of a flat water tank, a pipeline, an electromagnetic flow meter, a test section, and a valve. The test pipe is made of organic glass to observe the flow regime easily. The inner diameter (D) of the test pipe is 15 cm, and its total length is 370 cm. Figure 1 shows the layout of the test device.
The TIED is located 143.5 cm away from the inlet of the test pipe, and its length (L) is 13.5 cm. For the TIED, the number of the piers (n) is 4, and the height of the piers (h) is 3.75 cm. The area contraction ratio (ε) can be defined as: . In equation, 0 A is the cross-sectional area of TIED, A is the cross-sectional area of test pipe, and θ is the angle of the piers. In order to study the effect of area contraction ratio (ε) on hydraulic characteristics of the TIED, five types of TIED were experimentally studied via the physical model. The cross-sectional areas of the TIED are displayed in Figure 2.
(a) The schematic diagram of test layout (b) The photo of test pipe During the test, the upstream head (H) was kept constant, the flow was constant in the test pipe, the indoor temperature was 20 °C, and the range of flow was between 18 l/s and 42 l/s. In order to analyze the same flow condition, the flow of each group was about 6 l/s increased sequentially, and the flow measurement error of each experimental flow group was ±0.2 l/s. During the test, the upstream head (H) was kept constant, the flow was constant in the test pipe, the indoor temperature was 20 • C, and the range of flow was between 18 l/s and 42 l/s. In order to analyze the same flow condition, the flow of each group was about 6 l/s increased sequentially, and the flow measurement error of each experimental flow group was ±0.2 l/s. The flow (Q) through the test pipe and the transient pressure (p i ) along the bottom of the test pipe were measured for each test group. The center point at the inlet of the TIED is the origin of the coordinate, the direction of water flow is positive for the X axis, and the direction for vertical upwards is the positive Z axis. Q was measured using an intelligent electromagnetic flow meter, and its measured accuracy was ±0.5%. p i was measured with a digital pressure sensor, and its measured accuracy was 0.1%, and its measuring frequency 100 HZ. Figure 3a shows the relative position of the pressure measuring points.
Water 2019, 11, x; doi: FOR PEER REVIEW www.mdpi.com/journal/water measuring frequency 100 HZ. Figure 3a shows the relative position of the pressure measuring points. When Q was 18 l/s, the transient flow velocity (u) was measured by the DOP3010 flow velocity meter for different ε. The sampling frequency of the DOP3010 flow meter was 1MHz and its resolution was 0.01 mm. The angle between the measuring probe of u and the pipe wall was 70°, and their gap was filled with coupling medium to make the measurement of u more precise. Taking the measuring point of 1* as an example, it is possible to obtain the value of u for measuring points at intervals of 1 cm on the line of a. Z/D = 0 is at the central axis of the test pipe and Z/D = 0.42 is 1.2 cm away from the upper side of test pipe. The measured value of u is stable in the position of Z/D = 0.42 and Z/D = 0 for every measuring point. The relative position of the measuring point is shown in Figure 3b.

Over-Current Capability
The flow coefficient ( c μ ) reflects the over-current capability of the test pipe, written as: where H Δ is the head loss between the fifth and fourteenth measuring point. H Δ is calculated by the following equation:  When Q was 18 l/s, the transient flow velocity (u) was measured by the DOP3010 flow velocity meter for different ε. The sampling frequency of the DOP3010 flow meter was 1MHz and its resolution was 0.01 mm. The angle between the measuring probe of u and the pipe wall was 70 • , and their gap was filled with coupling medium to make the measurement of u more precise. Taking the measuring point of 1* as an example, it is possible to obtain the value of u for measuring points at intervals of

Over-Current Capability
The flow coefficient (µ c ) reflects the over-current capability of the test pipe, written as: where ∆H is the head loss between the fifth and fourteenth measuring point. ∆H is calculated by the following equation: where z 14 and z 5 are the position head for the measuring points of 14 and 5, respectively, and the position head of z 14 is equal to that of z 5 ; p 14 and p 5 are the time-averaged pressure for the measuring points of 14 and 5, respectively; v 14 and v 5 are the average flow velocity for the measuring points of 14 and 5, respectively, and their value are equal to v; v is the averaged velocity of the test pipe; h w is the head loss between the front and back of TIED; ξ is the head loss coefficient, and the sum of the resistance coefficient (λ) along the pipe and the local head loss coefficient (ζ).
Combining Equations (1) and (2) together, we can obtain: The influencing factors of λ are the flow regime and the roughness of the pipe wall, and the roughness of the pipe wall is affected by the geometric parameters of the TIED. When the water flow is laminar, λ is only affected by Re. When the water flow is turbulent and in the transition zone, λ is determined by the roughness of the pipe wall and Re. When the water flow is turbulent in the square zone of resistance, λ is determined by the roughness of the pipe wall and not affected by Re. When Re is between 1.5 × 10 5 and 3.5 × 10 5 , the water flow is located in the square area of turbulent resistance, ξ is determined by the body type of the TIED, the wall roughness of the TIED, and the wall roughness of the testing pipe. Figure 4 shows the change of the flow coefficients (µ c ) with ε for different Re. µ c is little affected by Re and its relative errors are within 2% for the same ε when Re is between 1.5 × 10 5 and 3.5 × 10 5 . For the same Re, µ c increases from 0.4 to 0.9 with the increase of ε, so ε is the main influencing factor of over-current capability.
Water 2019, 11, x; doi: FOR PEER REVIEW www.mdpi.com/journal/water The influencing factors of λ are the flow regime and the roughness of the pipe wall, and the roughness of the pipe wall is affected by the geometric parameters of the TIED. When the water flow is laminar, λ is only affected by Re. When the water flow is turbulent and in the transition zone, λ is determined by the roughness of the pipe wall and Re. When the water flow is turbulent in the square zone of resistance, λ is determined by the roughness of the pipe wall and not affected by Re.
When Re is between 1.5 × 10 5 and 3.5 × 10 5 , the water flow is located in the square area of turbulent resistance, ξ is determined by the body type of the TIED, the wall roughness of the TIED, and the wall roughness of the testing pipe. Figure 4 shows the change of the flow coefficients ( c μ ) with ε for different Re. c μ is little affected by Re and its relative errors are within 2% for the same ε when Re is between 1.5 × 10 5 and 3.5 × 10 5 . For the same Re, c μ increases from 0.4 to 0.9 with the increase of ε, so ε is the main influencing factor of over-current capability.
In order to eliminate the error of c μ caused by the Re in the experiment, the averaged value ( c μ ) of c μ for different ε is acquired in the testing flow range, and the relationship between c μ and ε is presented in Figure 5. It indicates that c μ increases with the addition of ɛ, and the empirical formula of c μ is presented as: 3.1948 0.1213 The calculated values of c μ obtained by Equation (4) and the measured values of c μ are shown in Figure 5. Comparing the calculated values with the measured values for the same ε, their errors are smaller than 5% and within the range of allowable error. Therefore, Equation (4) can calculate the flow coefficient of the TIED in the testing flow range.   In order to eliminate the error of µ c caused by the Re in the experiment, the averaged value (µ c ) of µ c for different ε is acquired in the testing flow range, and the relationship between µ c and ε is presented in Figure 5. It indicates that µ c increases with the addition of ε, and the empirical formula of µ c is presented as:

Energy Dissipation Rate
The energy dissipation rate (η ) can represent the energy dissipation effect of the TIED. The higher the energy dissipation rate, the better the energy dissipation effect. It can be expressed as: The calculated values of µ c obtained by Equation (4) and the measured values of µ c are shown in Figure 5. Comparing the calculated values with the measured values for the same ε, their errors are smaller than 5% and within the range of allowable error. Therefore, Equation (4) can calculate the flow coefficient of the TIED in the testing flow range.

Energy Dissipation Rate
The energy dissipation rate (η) can represent the energy dissipation effect of the TIED. The higher the energy dissipation rate, the better the energy dissipation effect. It can be expressed as: where h w is the head loss between the front and back of TIED, H is the total test head, ξ is the head loss coefficient, and v is the average velocity of test pipe. Figure 6 shows the variation trend of η with ε in the condition of different Re. η increases with Re for the same ε and decreases exponentially with ε for the same Re.  Figure 6. Relationship of the energy dissipation rate (η) and area contraction ratio (ε).

The Variation of the Time-Averaged Pressure along the Test Pipe
When Q increases in the pipeline, Re becomes larger, the frictional head loss and local head loss also become greater, and the reduced amplitude of the time-averaged pressure ( i p ) decreases sharply along the test pipe. The time-averaged pressure coefficient (α) is introduced to express i p Figure 6. Relationship of the energy dissipation rate (η) and area contraction ratio (ε).
Because of the constant total head in the test, Equation (6) can be transformed as follows, aiming to analyze the change of η: In Equation (6), when ε is constant; ξ, g, A, and H are constant and η is proportional to the square of Q; when Q is constant, η is proportional to ε. Therefore, the value of η can be calculated by ξ in the case of a known Q.
In order to eliminate the influence of Re on ξ, the average value (ξ) is obtained by taking an average of ξ in the testing flow range, and the change of ξ with the increase of ε is shown in Figure 7. ξ decreases from 5.9 to 1.2 when ε is between 0.375 and 0.625. The empirical formula of ξ can be expressed as follows: The calculated values of ξ are obtained by Equation (7), and the measured values of ξ are obtained with the help of the model test. Both of them are presented in Figure 7. Comparing the calculated values of ξ with the measured values, the errors are less than 5%. Thus, this formula can calculate ξ for the TIED in the testing flow range. Moreover, it can obtain η to substitute ξ into Equation (6) Figure 6. Relationship of the energy dissipation rate (η) and area contraction ratio (ε).

The Variation of the Time-Averaged Pressure along the Test Pipe
When Q increases in the pipeline, Re becomes larger, the frictional head loss and local head loss also become greater, and the reduced amplitude of the time-averaged pressure ( i p ) decreases sharply along the test pipe. The time-averaged pressure coefficient (α) is introduced to express i p better, established as: where i p is the time-averaged pressure for each measuring point; min p is the smallest value of the time average pressure among the measuring points along the test pipe; and its position pressure in the test is at 0.2D. Figure 8a shows the change of α along the test pipe. It can be seen that the time-averaged pressure coefficients are less affected by Q and have similar change trends along the test pipe. The reason is that the difference of α between the two measured points is equal to the head loss coefficient, and the effect of Re on the value of α can be neglected, and it was only affected by the energy dissipater. In order to reduce the error caused by the changes of Re, the average value (α ) is obtained to study α better in the range of Q. The relationship between α and ε is shown in Figure   Figure 7. Relationship between the measured or calculated averaged value of the head loss coefficient (ξ) and the area contraction ratio (ε).

The Variation of the Time-Averaged Pressure along the Test Pipe
When Q increases in the pipeline, Re becomes larger, the frictional head loss and local head loss also become greater, and the reduced amplitude of the time-averaged pressure (p i ) decreases sharply along the test pipe. The time-averaged pressure coefficient (α) is introduced to express p i better, established as: where p i is the time-averaged pressure for each measuring point; p min is the smallest value of the time average pressure among the measuring points along the test pipe; and its position pressure in the test is at 0.2D. Figure 8a shows the change of α along the test pipe. It can be seen that the time-averaged pressure coefficients are less affected by Q and have similar change trends along the test pipe. The reason is that the difference of α between the two measured points is equal to the head loss coefficient, and the effect of Re on the value of α can be neglected, and it was only affected by the energy dissipater.
In order to reduce the error caused by the changes of Re, the average value (α) is obtained to study α better in the range of Q. The relationship between α and ε is shown in Figure 8b, and the changes of α with the increase of ε for the inlet or outlet of the TIED are presented in Figure 9. 8b, and the changes of α with the increase of ε for the inlet or outlet of the TIED are presented in Figure 9.
As shown in Figures 8 and 9, α drop sharply at the inlet of the TIED, then gradually increases and tends to remain nearly constant in the place of 4D after the inlet of the TIED. The reason is that the sudden changes of the flow velocity, caused by the sudden change of the flow cross-section through the TIED, lead to changes of α. For different types of TIED, α decreases from 9.8 to 2.8 before the inlet of the TIED and falls from 3.9 to 1.6 after the outlet of the TIED with the increase of ε from 0.375 to 0.46, and the minimum of α along the test pipe increases with the increase of ε. For different types of the TIED, α in the outlet and inlet of the TIED both decrease with the increase of ε, and their differences drop from 8.2 to 1.8 when ε grows from 0.375 to 0.625. The change of α within and near the TIED is mainly due to the increase of the head loss coefficient, which is caused by the change of cross-sectional area.
When Q is constant, the smaller ε is, the larger α in the front of the TIED is. Thus, the larger enough value of 6 m i n p p − should be provided. Additionally, the value of 6 p is constant when Q is constant. Therefore, the value of min p will be negative with the decrease of ε. When ε is constant, α is constant. The larger Q is, the smaller i p becomes, so the value of min p will be negative with the increase of Q. In the testing range flow, when the flow is about 42 l/s and ε is equal to 0.375 and 0.46, the value of min p is negative. The air will enter the pipe when the value of time-averaged pressure is negative, and then cavitations are likely where the flow velocity becomes small.    As shown in Figures 8 and 9, α drop sharply at the inlet of the TIED, then gradually increases and tends to remain nearly constant in the place of 4D after the inlet of the TIED. The reason is that the sudden changes of the flow velocity, caused by the sudden change of the flow cross-section through the TIED, lead to changes of α. For different types of TIED, α decreases from 9.8 to 2.8 before the inlet of the TIED and falls from 3.9 to 1.6 after the outlet of the TIED with the increase of ε from 0.375 to 0.46, and the minimum of α along the test pipe increases with the increase of ε. For different types of the TIED, α in the outlet and inlet of the TIED both decrease with the increase of ε, and their differences drop from 8.2 to 1.8 when ε grows from 0.375 to 0.625. The change of α within and near the TIED is mainly due to the increase of the head loss coefficient, which is caused by the change of cross-sectional area.
When Q is constant, the smaller ε is, the larger α in the front of the TIED is. Thus, the larger enough value of p 6 − p min should be provided. Additionally, the value of p 6 is constant when Q is constant. Therefore, the value of p min will be negative with the decrease of ε. When ε is constant, α is constant. The larger Q is, the smaller p i becomes, so the value of p min will be negative with the increase of Q. In the testing range flow, when the flow is about 42 l/s and ε is equal to 0.375 and 0.46, the value of p min is negative. The air will enter the pipe when the value of time-averaged pressure is negative, and then cavitations are likely where the flow velocity becomes small.

Variation of Pulsating Pressure
The size of the pulsating pressure can be represented by the root mean square of the pulsating pressure (σ) in the Equation (9): where N is the measuring times of the pressure and p i is the instant pressure at the measuring points. The pulsation of pressure is caused by the mixing of particles in each layer of turbulence. If air enters the pipe, the stronger the pressure pulsation, the more likely it is for cavitations to occur. The pulsating pressure coefficient (C p ) can be acquired to express σ, written as: Figure 10 presents the change of C p along the test pipe. The variation trend of C p is similar for different ε and Q, and the peaks of C p appear at 1.3D after the outlet of the TIED, because the TIED Water 2019, 11, 1406 9 of 13 increases the turbulence of water flow, resulting in the enhancement of pulsating intensity near the outlet of the TIED.
Water 2019, 11, x; doi: FOR PEER REVIEW www.mdpi.com/journal/water different ε and Q, and the peaks of Cp appear at 1.3D after the outlet of the TIED, because the TIED increases the turbulence of water flow, resulting in the enhancement of pulsating intensity near the outlet of the TIED. Table 1 shows the greatest peak of Cp for each test group. It is clear that the peaks of Cp decrease with the increase of Q and ε, because Cp is inversely proportional to the square of Q from Equation (10) and the changing rate of pressure decreases with the increase of ε. When ε increases from 0.375 to 0.625, the averaged peak of Cp ( max p C ) decreases from 0.747 to 0.306 in the range of the testing flow. When Q is constant, the larger ε is, the smaller Cp is, and the larger σ is, indicating that the smaller ε is, the larger the pressure pulsation is.    Table 1 shows the greatest peak of C p for each test group. It is clear that the peaks of C p decrease with the increase of Q and ε, because C p is inversely proportional to the square of Q from Equation (10) and the changing rate of pressure decreases with the increase of ε. When ε increases from 0.375 to 0.625, the averaged peak of C p (C pmax ) decreases from 0.747 to 0.306 in the range of the testing flow. When Q is constant, the larger ε is, the smaller C p is, and the larger σ is, indicating that the smaller ε is, the larger the pressure pulsation is.

Change of the Time-Averaged Velocity
The water flow in the test pipe is a turbulent flow within the test flow range, and the transient velocity of the water flow (u i ) can be divided into two parts: the time-averaged velocity (u) and the pulsating velocity (u ). The time-averaged velocity coefficient (β) is introduced to describe the characteristic of u, written as: u is logarithmic distribution along the radial direction in the pipe. The value of u at the center is greater than that at the side wall, and u in a different position of the pipe is affected by its position and sectional geometry parameters when Q is constant. According to the comprehensive analysis, the maximum value of u appears on the central axis of the pipe when X is constant. When Z is constant, the changed region of u is located inside and near the TIED, and its maximum value appears inside the TIED. That is caused by the sudden decrease of the cross-sectional area for the TIED. The larger ε is, the smaller the reduced amplitude of the cross-sectional area, and the smaller β is, the smaller the maximum of u .

Change of the Pulsating Velocity
The pulsating strength ( u σ ) is used to represent the fluctuating strength of the velocity for different measuring points and is denoted by the root mean square of the pulsating velocity ( u σ ): Figure 11. Changes of the time-averaged velocity along the test pipe.
According to the comprehensive analysis, the maximum value of u appears on the central axis of the pipe when X is constant. When Z is constant, the changed region of u is located inside and near the TIED, and its maximum value appears inside the TIED. That is caused by the sudden decrease of the cross-sectional area for the TIED. The larger ε is, the smaller the reduced amplitude of the cross-sectional area, and the smaller β is, the smaller the maximum of u.

Change of the Pulsating Velocity
The pulsating strength (σ u ) is used to represent the fluctuating strength of the velocity for different measuring points and is denoted by the root mean square of the pulsating velocity (σ u ): where N 2 is the measured times of the velocity. Introducing the turbulent strength (T u ), and defined as: The change of T u along the test pipe for different ε is illustrated in Figure 12 when Q is 18 l/s. It is shown that the variation of T u is similar for the different ε, and the maximum value of T u on the line of Z/D = 0.42 and Z/D = 0 appears at 1.57D and 2.72D away from the inlet of the TIED, respectively. ε has little effect on T u in the constant section of u and has a great effect on the abrupt section of u. When ε increases from 0.375 to 0.625, the maximum value of T u on the line of Z/D = 0.42 reduces from 0.68 to 0.21 and decreases from 0.56 to 0.13 on the line of Z/D = 0.
As shown in Figure 12, T u in the side wall is larger than that in the central axis when X is constant, which is mainly due to the diffusion of turbulent energy and the interference of the edge wall roughness causing the larger turbulence intensity at this position. The maximum value of T u appears after the outlet of the TIED and decreases with the increase of ε when Z is constant because of the convection of turbulent energy, resulting in downstream movement of the interference wave.
Water 2019, 11, x; doi: FOR PEER REVIEW www.mdpi.com/journal/water line of Z/D = 0.42 and Z/D = 0 appears at 1.57D and 2.72D away from the inlet of the TIED, respectively. ε has little effect on u T in the constant section of u and has a great effect on the abrupt section of u . When ε increases from 0.375 to 0.625, the maximum value of u T on the line of Z/D = 0.42 reduces from 0.68 to 0.21 and decreases from 0.56 to 0.13 on the line of Z/D = 0. As shown in Figure 12, u T in the side wall is larger than that in the central axis when X is constant, which is mainly due to the diffusion of turbulent energy and the interference of the edge wall roughness causing the larger turbulence intensity at this position. The maximum value of u T appears after the outlet of the TIED and decreases with the increase of ε when Z is constant because of the convection of turbulent energy, resulting in downstream movement of the interference wave.

Conclusion
In order to gain insight into the optimized body type parameters of the TIED, the effects of the area contraction ratio (ε) on the hydraulic characteristics of the TIED were discussed using the methods of a physical model test and theoretical analysis in this paper. In the testing flow range, the main conclusions are as follows.
During the test, the flow was basically in the square area of turbulent resistance when Re changed from 1.5 × 10 5 to 3.5 × 10 5 , and the Re had little effect on the flow characteristics. The energy dissipation rate (η) was proportional to the head loss coefficient (ξ). The flow characteristics were mainly affected by the body type of the TIED. The over-current capability ( c μ ) and the energy dissipation rate (η) can be characterized by c μ and ξ, respectively. They mainly depended on ε.
With the increase of ε, c μ increased exponentially ( The transient pressure of turbulent flow was composed of time-averaged pressure and pulsating pressure. The change trends of time-averaged pressure coefficients (α) only depended on ε; the differences of the averaged α between the inlet and outlet of the TIED decreased from 8.2 to 1.8 when ε increased from 0.375 to 0.625 in the range of the testing flow; when the flow was about 42 l/s and ε was equal to 0.375 or 0.46, the minimum of time-averaged pressure along the pipe was negative. The pulsating pressure coefficient (Cp) was determined by Re and ε, and its peaks appeared at 1.3D after the outlet of the TIED; the averaged peaks in the range of the testing flow decreased from 0.747 to 0.306 when ε increased from 0.375 to 0.625. Negative pressure and larger peaks of pulsating pressure coefficient were more prone to cavitations behind the outlet of TIED.

Conclusions
In order to gain insight into the optimized body type parameters of the TIED, the effects of the area contraction ratio (ε) on the hydraulic characteristics of the TIED were discussed using the methods of a physical model test and theoretical analysis in this paper. In the testing flow range, the main conclusions are as follows.
During the test, the flow was basically in the square area of turbulent resistance when Re changed from 1.5 × 10 5 to 3.5 × 10 5 , and the Re had little effect on the flow characteristics. The energy dissipation rate (η) was proportional to the head loss coefficient (ξ). The flow characteristics were mainly affected by the body type of the TIED. The over-current capability (µ c ) and the energy dissipation rate (η) can be characterized by µ c and ξ, respectively. They mainly depended on ε. With the increase of ε, µ c increased exponentially (µ c = 0.1213e 3.1948ε (0.375 ≤ ε ≤ 0.625)) and ξ decreased exponentially (ξ = 59.766e −6.198ε (0.375 ≤ ε ≤ 0.625)).
The transient pressure of turbulent flow was composed of time-averaged pressure and pulsating pressure. The change trends of time-averaged pressure coefficients (α) only depended on ε; the differences of the averaged α between the inlet and outlet of the TIED decreased from 8.2 to 1.8 when ε increased from 0.375 to 0.625 in the range of the testing flow; when the flow was about 42 l/s and ε was equal to 0.375 or 0.46, the minimum of time-averaged pressure along the pipe was negative. The pulsating pressure coefficient (Cp) was determined by Re and ε, and its peaks appeared at 1.3D after the outlet of the TIED; the averaged peaks in the range of the testing flow decreased from 0.747 to 0.306 when ε increased from 0.375 to 0.625. Negative pressure and larger peaks of pulsating pressure coefficient were more prone to cavitations behind the outlet of TIED.
The transient velocity of turbulent flow was divided into time-averaged velocity and pulsating velocity, which can be represented by time-averaged flow velocity coefficients (β) and turbulent strength (T u ), respectively. When X was constant, the maximum of β appeared on the line of Z/D =0 in pipe, and β near the side wall was larger than that on the line of Z/D =0 in pipe due to the diffusion of turbulent energy and the interference of the wall roughness. When Z was constant, the maximum value of β appeared inside the TIED for different ε, decreasing with the increase of ε. Moreover, the maximum value of T u appeared after the outlet of the TIED and decreased with the increase of ε because of the convection of turbulent energy. Therefore, the smaller ε is, the more likely it is that cavitations occur near the pipe wall behind the outlet of the TIED.
Comprehensive analysis of over-current capability, energy dissipation rate, and the distribution of flow velocity and pressure showed that the optimized body type parameter (ε) of the TIED in the test was 0.5 when flow in the pipe was about 42 l/s.