Numerical Modeling of B-Type Hydraulic Jump at an Abrupt Drop
†1Oğuz ŞİMŞEK1 N. Göksu SOYDAN2 Veysel GÜMÜŞ3 M. Sami AKÖZ4 M. Salih KIRKGÖZ5
ABSTRACT
The properties of a B-type hydraulic jump at an abrupt drop are analyzed experimentally and numerically for different flow cases. Using the Standart k-ε, Shear Stress Transport and Reynolds Stress turbulence closure models, the governing equations are solved numerically using ANSYS-Fluent program package which is based on the Finite Volume Method. The Volume of Fluid (VOF) method is used to determine the free surface profile.
Grid independence study is carried out using a Grid Convergence Index (GCI) analysis.
The numerical results for the free surface and velocity profiles of flow from the present turbulence models are compared with experimental data. Mean square errors and mean absolute relative errors of measured and predicted free surface profiles and velocity fields indicate that Reynolds Stress Model is more successful turbulence closure model than the other two for the determination of surface profile and velocity field of the B-type hydraulic jump.
Keywords: B-type hydraulic jump, Velocity field, Free surface profile, Numerical model, Turbulence closure model
1. INTRODUCTION
The most effective way of dissipating the excessive energy of the supercritical open channel flow is to force it go through a process of hydraulic jump to transform into the subcritical regime. Hydraulic jump that is a natural phenomenon in transition from supercritical to subcritical regime observed as the surface discontinuity in which significant loss of energy due to strong turbulence activities occurs during a sudden increase in water depth of the flow. When the tail water depth is greater than the required sequent depth, the location of the hydraulic jump shifts towards upstream then the probability of occurrence of the submerged hydraulic jump increases in that the efficiency of the energy dissipation is
1 Çukurova University, Adana, Turkey [email protected] 2 Çukurova University, Adana, Turkey [email protected] 3 Harran University, Şanlıurfa, Turley - [email protected] 4 Çukurova University, Adana, Turkey - [email protected] 5 Çukurova University, Adana, Turkey - [email protected]
† Published in Teknik Dergi Vol. 26, No. 4 October 2015, pp: 7215-7240
reduced considerably. An effective way of preventing a hydraulic jump from submergence in the channel is to construct an abrupt drop.
Earlier studies of physical model tests show that depending on the Froude number of the incoming flow, drop height and tailwater level, as shown in Fig. 1, two types of hydraulic jump is observed in an abrupt drop: A-type and B-type [1, 2]. As indicated in the figure, while A-type hydraulic jump takes place over the drop, the jump moves towards downstream and forms B-type jump by decreasing the tailwater level.
(a)
(b)
Figure 1. The types of hydraulic jump in abrupt drop: (a) A-type jump, (b) B- type jump
Many experimental studies have been performed to determine the characteristics of hydraulic jump that occurs after an abrupt drop [3,4,5]. In these studies, in general, the types of hydraulic jump for different conditions, its geometrical properties, the velocity distributions in the jump region and the parameters affecting the jump characteristics were investigated. It is a known fact that the some errors due to scale effects influence the results obtained from the laboratory model tests.
On the other hand, developments of Computational Fluid Dynamics (CFD) methods, in recent years, have provided important tools in the numerical analysis of the flow interacting with various types of hydraulic structures. The numerical solutions that can be repeated quickly, provide the opportunity to investigate properties of the flow theoretically under different conditions of flow and structure. Therefore, it seems that the investigation of flow behavior using numerical models, gaining increasing importance against the physical model studies. In this regard, many numerical models complementary to the experimental studies on hydraulic jump studies were performed [6, 7, 8, 9, 10]. Numerical model studies were mostly based on the experimental validations of the numerical predictions for the geometric, dynamic and kinematic properties of hydraulic jump.
In regard of testing the reliability of the numerical modeling techniques used in the analyses of problems related to fluid motion, the studies need to be enhanced and diversified. In this study, the properties of a B-type hydraulic jump at an abrupt drop are analyzed experimentally and numerically for different flow cases. The governing equations are solved with ANSYS-Fluent software based on the finite volume method using the Standard k-ε, Shear Stress Transport, and Reynolds Stress turbulence closure models. Volume of Fluid method (VOF) is used for the prediction of free surface profile. The numerical results for free surface profiles, the velocity field and the length of the B-type hydraulic jump are compared with experimental measurements. Streamline patterns of the flow in the jump region, velocity profiles along the channel and distribution of the kinetic energy are presented.
2. EXPERIMENTS
Experiments were conducted in a glass-walled, hydraulically smooth horizontal laboratory channel of 0.20 m wide, 0.20 m deep and 1.70 m long (Fig. 2a). For the measurements of discharge, a tank in size of 35x35x60 cm was used at the end of the channel. Free surface profiles were measured using a point gage. The velocity field of the flow was measured using a one-dimensional Laser Doppler Anemometry (LDA) (Dantec® LDA 62N04). The velocity measurements were carried out along the centerline of the channel. LDA measures the velocity component, in the point of measurement where the laser beams focus, with a short time intervals during a specified time. In LDA system, the velocity is measured with the change in frequency of the laser beam and the measured data collected by the photodetector is instantaneously sent to BSA-Flow software by a synchronizer. The point velocity value is determined as the mean value of post-processing the instantaneous velocity measurements recorded at certain time intervals. Some turbulent flow characteristics can be determined by the time series containing the instantaneous velocities.
The instantaneous velocities with the LDA system used in the experiments were measured with uncertainties ±1% within the 95% confidence limit.
As shown in Fig. 2, the drop height was hd = 0.097 m in the measurements. The tailwater depth of flow was controlled by a sharp-crested weir having a fixed height of hk=0.06 m at the end of the channel.
Figure 2. (a) Experimental setup, (b) B-type hydraulic jump at the abrupt drop (a)
(b)
The schematic view of the B-type hydraulic jump and some dimensions of the flow are shown in Fig. 3. The characteristics of the two different flow cases used in the experiments are given in Table 1. In the table, Q is flow discharge, Y0 is water depth at the inlet boundary of the solution domain, Y1 and Y2 are the water depths at the beginning and at the end of the jump region, respectively, Lj is the length of jump, Fr0 (=V0/(gY0)1/2) is Froude number and Re0 (=4V0R0/) is Reynolds number (V0 and R0 are mean velocity and hydraulic radius at the inlet boundary of the solution domain respectively, g is gravitational acceleration and is kinematic viscosity).
Figure 3. The schematic view of the B-type hydraulic jump
Table 1. Experimental conditions for the two flow cases Q
(l/s) Y0
(cm) Y1
(cm) Y2
(cm) Lj
(cm) V0
(m/s) Fr0 Re0
Case 1 3.05 2.30 7.70 8.50 38.50 0.663 1.40 45.500 Case 2 6.05 4.00 9.80 12.10 48.50 0.756 1.21 75.800
3. FORMULATION AND NUMERICAL MODELING 3.1. Governing Equations
2D Reynolds-averaged continuity and Navier-Stokes equations (RANS) were used to theoretically simulate the present open channel flow. For an incompressible, Newtonian fluid flow these equations can be expressed as follows.
Continuity equation:
0
y v x
u (1)
Momentum equation in x direction:
y y x
u x
u x
X p y v u x u u t
u xx xy
22 22 (2)
Y2
Y0
hd Y1
Lj
Momentum equation in y direction:
y y x
v x
v y
Y p y v v x u v t
v xy yy
22 22 (3)
In Eqs. (1-3), u and v are the temporal mean velocity components in xand y direction, respectively, X and Y are the body force components for unit mass, p is temporal mean pressure, μ is dynamic viscosity, ρ is fluid density, t is time and xx , xy and yy are turbulence (Reynolds) stresses.
The three basic equations given above contain six different unknown terms: two velocity components, pressure and three Reynolds stresses. To solve the equation system in the numerical solution process, the Reynolds stresses in the Eqs. 2 and 3 are defined using turbulence closure models. Based on the Boussinesq eddy viscosity assumption the turbulence stresses are formulated using the linear constitutive relation for incompressible flow, that is:
x k u x u u
u t
xx
3
2
(4)
y
u x v v
u t
xy
(5)
y k v y v v
v t
yy
3
2
(6)
in which u and v are horizontal and vertical velocity fluctuations, respectively, µt is the turbulent viscosity and k (u iui/2) is the turbulent kinetic energy.
3.2. Turbulence Closure Models
In modeling the turbulent viscosity t in Eqs. (4-6), many different turbulence closure models have been developed. In this study, Standard k-ε (SKE) [11], Shear Stress Transport (SST) [12] and Reynolds Stress Model (RSM) [13] turbulence closure models are used:
3.2.1. SKE Turbulence Model
The SKE model is based on model transport equations for the turbulent kinetic energy (k) and its dissipation rate (). In this model the turbulent viscosity, µt, is expressed as:
t C k2 (7)
Cµ is the model constant. In Eq. (7), the turbulent kinetic energy k and its rate of dissipation ε are obtained from the solution of following two transport equations:
j ij i j k t j
j
j x
u x
k x
x u k t
k) ( )
( (8)
C k x u C k
x x
u x
t j
ij i j
t j
j j
2 2 1
) ( )
(
(9)
The model constants in Eqs. (7)-(9) are proposed asC=0.09, k=1.0, =1.3, C1=1.44, C2=1.92 [11].
3.2.2. SST Turbulence Model
The idea behind the Shear Stress Transport (SST) model is to retain the accurate formulation of the standard k- model in the near wall region [14], and to take advantage of the free stream independence of the k- model in the outer part of the boundary layer and in free flows. To achieve this, the k- model is converted into a k- formulation. The modifications to the original k- model include the addition of a cross-diffusion term in the
-equation and a blending function (F1) to ensure that the model equations behave appropriately with different model constants both in the near-wall and the far-field zones [12]. In SST model, transport equations are expressed as follows for F1 model coefficients:
kx u x
k x
x u k t
k
j ij i j t k j
j
j
( ) ( )
(10)
j j
j ij i t j t j
j j
x x F k
x u x
x u x
t
) 1 1 ( 2 ) ( ) (
2 1
2
(11)
(=/k) is the ratio of specific loss of turbulence kinetic energy. In the model, some different constants are expressed with interpolation of the original k- (1) and converted k-
(2) model coefficients:
2 1 1
1 (1 )
F F , for example:k F1k1(1F1)k2 and F11(1F1)2
1 constants for k-: k10.85, 10.5, 10.075, 0.09,
1 1 1 2 ,
41 .
0
2 constants for k- :k2 1.0, 20.856, 20.0828, 0.09,
2 2 2 2 41
.
0
and the function F1 in the equations is expressed as follows:
4 2 2 1 2
; 4
;500 09 . min 0
tanh
y CD
k y y
mak k F
k
,
2 1 ;1020
2
j j
k x x
mak k
CD
y is the distance of the nearest wall.
In SST model, by taking into account of the transport influence of the turbulent shear stress, the turbulence viscosity is modified as below [12]. It provides an improvement in the prediction of the turbulent boundary layer in an adverse pressure gradient and makes better predictions for the location of boundary layer separation compared to those of SKE model.
Based on the Bradshaw hypothesis, shear stress in the boundary layer is expressed as follow:
k a1
(12)
where a1=0.31. For Eq. (12), the turbulence viscosityis redefined as:
1 2
1
; F a mak
k a
t
,
2
2 500
09 ; . 20
tanh y y
mak k F
where |u/y| is the absolute value of the average vorticity, F2 function is 1 for boundary layer flow, and 0 for free turbulence shear layer.
3.2.3. RSM Turbulence Model
In the RSM model, the Reynolds stresses and turbulence energy dissipation rates are computed directly using the differential transport equations. Turbulence stress components are calculated directly by using Eqs. (2) and (3). By taking into consideration of the differentiation depending on the direction of the stress and assuming that more detailed turbulence model than linear and non-linear viscosity turbulence models, the RSM model is known as a second order closure model. The RSM model show superior performance compared to the other models in the flows that include the effects of streamline curvature, the flows with sudden changes in the velocity of fluid particles and secondary flows. The transport equations for the turbulence stresses used in the RSM model are obtained as follows [13]:
ij ij ij
ijv ijt
ij
k j k
i i
j j i
P
k k i j k
j k i
D k k
j i
D
ki j jk i k
j k i
C k ij k ij ij
x u x u x
u x u p x u u x u u u u
x x
u u u
p u p u u x u x
u R t R dt dR
2
1 2
(13)
where Cijis the convection, D is the turbulent diffusion, ijt D is the molecular diffusion, ijv Pij is the stress production,ijis the pressure-strain transport term and ijis the dissipation term.
3.3. Volume of Fluid (VOF) Method for Computation of Free Surface Profile
The VOF method renders the shape and location of constant-pressure free surface boundary. It uses a filling process to determine which cell in the grid is filled and which is emptied [15]. Consider an Eulerian structured fixed grid and a curved liquid surface of a 2D flow field. A volume fraction field (F) is then defined in this grid that can take values between 0 and 1, i.e. F = 0 if the cell is emptied and F = 1 when it is completely filled with liquid. A value of F between 0 and 1 means a fractional fill with the free surface located within the cell.
There are different schemes of VOF application. One of the most popular schemes have so far been used is the “geometric reconstruction” [16]. In the VOF scheme the first step is calculating the position of the linear interface (between air and water) relative to the center of each partially-filled cell, based on information about the volume fraction, F, and its derivatives in the cell. The second step is calculating the advecting amount of fluid through each face using the computed linear interface shape and information about the normal and tangential velocity distribution on the face. The third step is calculating the volume fraction in each cell using the balance of mass fluxes calculated in the previous step.
3.4. Solution Domain, Boundary and Initial Conditions
The geometry of the 2D solution domain and the boundary conditions for open channel flow used for B-type hydraulic jump are shown in Fig. 4. At the air-filled upper boundary of the solution domain, the pressure p =0. At the inflow boundary, the flow velocities were u = 0.663 m/s and 0.756 m/s, for Case 1 and Case 2, respectively, and v =0. The outflow boundary was a free overfall at the outlet boundary where p = 0. The wall boundaries were set as stationary, i.e. the non-slip wall, u=v =0. As seen in Fig. 4, the origin of the coordinate system (x,y) is located at the bottom left corner of the domain. The time- dependent solution procedure was initiated with the initial condition F=0 within the solution domain and F=1 at the inflow boundary for t=0.
To ensure the stability of the time dependent numerical solution, Courant number was fixed as Cn ≤2 and the computational time step Δt was determined as follows:
fluid cell v x Cn t
(14)
in which, Δt is time step, Δxcell is grid size in x direction, vfluid is the velocity of fluid. The Courant number was determined by considering the smallest grid size in thecomputational meshes. From preliminary computations it was found that the time step Δt = 0.001 s for all turbulence closure models satisfies the Courant number criterion.
Figure 4. Geometry and boundary conditions of solution domain with six subdomains
4. COMPUTATIONAL GRID 4.1. Design of Computational Grid
It is well known that the performance of the numerical modeling of the flow field that interacts with hydraulic structures is closely dependent on the design of computational grid.
The computational domain given in Fig. 4 is divided into six local subdomains, as seen in Fig. 5, and the number of elements in each subdomain is increased by about 50% and 75%
to obtain three mesh system with different densities, Mesh 1 (coarse), Mesh 2 (medium) and Mesh 3 (fine). Structured meshes with four-node quadrilateral elements are used in all subdomains. The element numbers of the computational subdomains for the three mesh systems used in the simulations are listed in Table 2. A grid convergence index analysis is carried out to determine the discretization error for the grid-independent solution [17]. The results indicate that the discretization error in the predicted velocities on the fine mesh remains within 2% and it is decided that the Mesh 3 has a sufficient resolution for a grid- independent solution.
The minimum and maximum values of near-wall mesh size for the three mesh systems used in the numerical modeling are listed in Table 3.
Upper Boundary y (cm) p=0
Solution Domain
II VI
I V
16.5
9.7 IV
III 6.0
x (cm) 0 10 165
Inlet Boundary u=0.663 and 0.756 m/s v=0 F=1
Lower Boundary u=0, v=0
Outlet Boundary p=0
20.0
Figure 5. Subdomains of computational grid, for Mesh 3
Table 2. Subdomain element numbers for the three mesh systems Subdomains Mesh 1 Mesh 2 Mesh 3
I 15x25 24x40 30x50
II 10x25 15x40 20x50
III 15x150 24x225 30x300
IV 10x150 15x225 20x300
V 15x150 24x225 30x300
VI 10x150 15x225 20x300
Table 3. Near-wall mesh size for the three mesh systems Near wall mesh size (mm) Mesh 1 Mesh 2 Mesh 3
Minimum 0.58 0.33 0.16
Maximum 0.69 0.40 0.18
By compressing the mesh toward the solid wall, the mesh resolution was adjusted to have the first mesh point within the viscous sublayer. Fig. 6 shows the variation of the y+(= u∗y/ν
<10) along the near-wall elements of the fine mesh system obtained using RSM model for Case 1 and Case 2 (u∗(= (τo/ρ)1/2) is the shear velocity, y is the height of the first grid element and ν is kinematic viscosity). As may be seen in the figure, the maximum value of y+ occurs in the hydraulic jump region of Case 2 with a value less than 10. Kirkgoz and
Ardiclioglu [18] reported that the experimental data fit the linear velocity distribution in the viscous sublayer if y+≤10. This value is used as a criterion to evaluate the adequacy of the near-wall mesh size for the two-layer model. It can be seen from the figure that the y+ value remains well below 10 and it is reveals that the first mesh points in the present computations take place in the viscous sublayer for both flow cases.
Figure 6. The variation of y+ along the channel for Mesh 3 using RSM for (a) Case 1 and (b) Case 2
In the present numerical simulations, using SKE and RSM turbulence models, enhanced two-layer approach [16] was employed for the near-wall treatment that was originally suggested by Chen and Patel [19] in which the near-wall mesh elements are assumed to remain within the viscous sublayer. In this approach, the whole solution domain is divided into two layers, a fully-turbulent region and a viscosity affected region near the wall. In the viscosity-affected near-wall region, by appropriate descriptions of t and , and by using extremely fine grid topology, the numerical modeling can be achieved down to the solid boundary. For the SST turbulence model no special approach is needed for the near-wall treatment.
5. RESULTS
5.1. Experimental and Numerical Velocity Profiles
A quantitative evaluation of the measured and computed velocity comparisons is made by using mean square error (MSE) and mean absolute relative error (MARE):
2 1
) 1 (
MSE N d h
n
u N u
(15)
1 100 MARE
1 x
u u u N
N
n d
h
d
(16)
where udanduhare measured and computed mean horizontal velocities, respectively; and N is the total number of data on the velocity profile. Tables 4 and 5 give the results for MSE and MARE values using Eqs. (15) and (16) for different channel sections from different turbulence models for two cases.The numbers in parentheses indicate the order of success in regard to compliance with the experimental measurements. Regarding the overall mean values of MSE and MARE in the last row of the tables, the smallest values for both Case 1 and Case 2 were obtained by the RSM turbulence model. In Addition, the SST and SKE are the second and third successful models for Case 1. The ranking of success in predicting the velocity field for Case 2 is RSM, SKE and SST.
In Table 4, it can be seen that the MARE values for x=11.5, 26.5 and 56.5 cm are higher than 100. This is because the experimental and numerical velocities are in opposite directions (positive and negative) in these sections.
Time-averaged velocity profiles at different times and the mean of these velocity profiles obtained using RSM turbulence closure model at x=26.5 cm in the jump region for Case 1 and Case 2 are given in Figs 7 and 8, respectively. From the figures, the jet flow near the wall region, the geometry of the region where the negative velocities occur and the time dependent change of the jump region can be clearly seen.
Table 4. MSE (m2/s2) and MARE (%) values from different turbulence models for Case 1 x
(cm)
SKE SST RSM MSE MARE MSE MARE MSE MARE
3.5 0.0005 5.0308 0.0032 8.4793 0.0005 3.9057
10.0 0.0012 5.8829 0.0004 1.8417 0.0008 3.3833 11.5 0.0268 186.9051 0.0312 186.8772 0.0408 144.6969 26.5 0.0412 803.2885 0.0364 574.1338 0.0325 224.5140 41.5 0.0222 132.1875 0.0028 23.3451 0.0085 69.9582 56.5 0.0043 266.0103 0.0081 376.9605 0.0023 199.8173 76.5 0.0001 3.3106 0.0049 37.9398 0.0004 10.9807 106.5 0.0002 11.9135 0.0007 45.2350 0.0001 6.7185 126.5 0.0001 10.3930 0.0005 17.6453 0.0001 7.5848 Mean 0.0107(3) 158.3247(3) 0.0098(2) 141.3842(2) 0.0096(1) 74.6177(1)
Table 5. MSE (m2/s2) and MARE (%) values from different turbulence models for Case 2 x
(cm)
SKE SST RSM MSE MARE MSE MARE MSE MARE
3.5 0.0010 2.6419 0.0067 7.2691 0.0053 5.6239
10.0 0.0189 13.6827 0.0040 4.1543 0.0046 4.5117 11.5 0.0082 84.6303 0.0036 17.4581 0.0083 85.9775 26.5 0.0205 51.5785 0.0102 45.4741 0.0245 61.7329 41.5 0.0075 98.0329 0.0135 70.1092 0.0049 29.8989 56.5 0.0016 24.2748 0.0243 115.2046 0.0029 47.5468 76.5 0.0008 11.6528 0.0179 58.1678 0.0045 33.7496 106.5 0.0014 9.8844 0.0054 23.5227 0.0002 3.5334 126.5 0.0005 7.0508 0.0007 7.8241 0.0001 2.0088 Mean 0.0067(2) 33.7143(2) 0.0096(3) 38.7982(3) 0.0062(1) 30.5093(1)
Figure 7. Time averaged and mean velocity profiles from RSM at x=26.5 cm for Case 1
Figure 8. Time averaged and mean velocity profiles from RSM at x=26.5 cm for Case 2
The experimental and computed velocity profiles at different sections and from different turbulence models used in the present study are compared in Figs. 9 and 10 for Case 1 and Case 2, respectively. Considering the time dependent variation of the jump length, the velocity profiles given in Fig. 9 and Fig 10 are obtained from an average of the computed velocity data at seven different instants. A typical velocity profile of a curvilinear flow that is inversely proportional to the radius of curvature takes place in the drop section (x=10 cm), as expected. Just downstream of the drop section (x=11.5 cm), a jet flow appears in the upper part of the velocity profile, and a fluid motion in an horizontal vortex including the reverse flow is located in the lower part of the velocity profile. When the variation of the velocity profile along the channel is considered, it can be clearly seen that the jet flow that is effective in the upper region near the drop section, gradually moves to the lower part of the velocity profile and gradually it transforms into a velocity profile of an open channel flow.
It may be concluded from the Figs. 9 and 10 that, although, the RSM turbulence model generally predicts the velocity field more successfully than the others, at some sections for example x= 11.5 and 76.5 cm for Case 1 and x=3.5, 56.5 and 76.5 cm for Case 2, it can be seen that SKE turbulence model gives better results. Similarly, the SST turbulence model has superiority in predicting the velocity profiles at sections x=10.0 and 41.5 cm for Case 1 and x=10.0, 11.5 and 26.5 cm for Case 2 compared to other models.
Figure 9. Experimental and numerical velocity profiles at different channel sections for Case 1
Figure 10. Experimental and numerical velocity profiles at different channel sections for Case 2
5.2. Experimental and Numerical Free Surface Profiles
From the numerical computations of B-type hydraulic jump, the free surface profiles obtained by using Volume of Fluid method are compared with the measurements. MSE and MARE values given by Eqs. 14 and 15 are used for validation purposes of the free surface profiles obtained from SKE, SST and RSM turbulence closure models.
Table 6 gives the MSE and MARE values of the free surface profiles calculated from the turbulence models used in this study. As can be seen from the table, the ranking of success in predicting the free surface profiles for all solution domain, as in the velocity profiles, is RSM, SST and SKE for Case 1 and RSM, SKE and SST for Case 2.
In Figs. 11 and 12, the comparisons between the measured and computed free-surface profiles from the turbulence closure models for the two cases are presented. As may be seen in the figures, the jump zone that is the most complicated region of the flow field is magnified to increase the image sensitivity. The predictions from the RSM turbulence model appear to agree well with experimental measurements, especially in the jump region.
From the figures, it may be said that, in the prediction of free surface profile for the regions before and after the hydraulic jump, SKE and SST turbulence models are as successful as RSM turbulence model.
Table 6. MSE (cm2) and MARE (%) values for free surface profiles from different turbulence models
Turbulence Model
All Solution Domain (x=0-1.65 m) Case 1 Case 2
MSE MARE MSE MARE SKE 0.4788(3) 5.9553(3) 0.8062(2) 7.0153(2)
SST 0.4641(2) 5.1162(2) 0.9689(3) 7.9593(3) RSM 0.3847(1) 4.7566(1) 0.7610(1) 6.8050(1)
5.3. The Geometry of Jump Region
Figs. 13 and 14 give the streamline topologies obtained from SKE, SST and RSM turbulence models. The geometry of the calculated streamline patterns from three turbulence models show similarities near the drop region for both cases. It is seen from the figures that all the turbulence models used in this study are successful in modeling the roller region just downstream of drop and under the water nappe. However, considering the length of the jump, it is clearly seen that the differences appear between the predictions of turbulence models. Table 7 gives the experimental and numerical relative lengths of the jump (Lj/hd), that is nondimensionalised with the drop height.
Figure 11. Experimental and numerical free surface profiles for Case 1
Figure 12. Experimental and numerical free surface profiles for Case 2
Figure 13. Streamline patterns for Case 1
SKE turbulence model is supposedly not successful in modeling the swirling flows, flows with boundary layer separation, and fully developed flows in non-circular open channels [20]. SST turbulence model is a hybrid model which combines the advantages of both the standard k- model and k- model, and it uses the k- model at the wall and k- in the bulk flow. SST model estimates the velocity field better than SKE model in the regions where the Reynolds number is low and flows with boundary layer separations. On the other hand, the RSM turbulence model was reported to give better results than the other models because of direct calculation of the Reynolds stresses for complicated flow problems including water jets, asymmetric channel and non-circular duct flows and curved flows [20]. In the present flow problem, the stream line pattern in the drop region is curvilinear, and in the jump region downstream of the abrupt drop, there exist a mixing of jet flow and roller formations with negative velocities. Accordingly, when considering the hydrodynamic character of the hydraulic jump region, it can be said that RSM model is better in predicting the length of jump than the other two models used in this study.
Actually, these results confirm the information obtained in previous studies dealing with submerged hydraulic jump [6, 21].
Lj RSM
x (m)
y (m)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1
0.0 0.2
SKE
x (m)
y (m)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1
0.0 0.2
SST
x (m)
y (m)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1
0.0 0.2
Figure 14. Streamline patterns for Case 2
Table 7. Experimental and numerical relative jump lengths Relative Length of Jump Region, Lj/hd
Exp. RSM SKE SST
Case 1 3.97 3.97 1.60 4.28
Case 2 5.00 5.21 2.73 5.62
5.4. Computed Velocity Field and Turbulence Kinetic Energy in Jump Region
Figs 15 and 16 give the computed velocity vectors in the jump region obtained from the RSM turbulence model. In the figures the complicated two-dimensional flow structure of B-type hydraulic jump region can clearly be observed in detail. Especially, the typical feature of B-type hydraulic jump that is the occurrence of the flow jet directed towards the base of drop having high velocities and strong power of erosion can clearly be seen from the velocity vector field. In controlling the hydraulic jump, when designing an abrupt drop, knowing the characteristic features of the hydraulic jump is important for the protection of channel bed.
RSM
x (m)
y (m)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1
0.0 0.2
SKE
x (m)
y (m)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1
0.0 0.2
SST
x (m)
y (m)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1
0.0 0.2
Figure 15. Computed velocity vectors from RSM for Case 1
Figure 16. Computed velocity vectors from RSM for Case 2
Figs 17 and 18 give the contour lines of computed turbulence kinetic energy kuiui/2 in the jump region obtained from RSM turbulence model. The maximum values for the turbulence kinetic energy is calculated as 0.06 and 0.08 m2/s2 for both cases respectively.
As may be seen from the figures the maximum concentration of the turbulence kinetic energy is experienced in the transition point where the jet flow hits the base and the jump process begins. The peak value of turbulence kinetic energy for the present experimental conditions occurs at a distance of x = 0.85hd and 1.46hd from the drop section for Case1 and Case 2 respectively.
Figure 17. Computed turbulence kinetic energy from RSM for Case 1
Figure 18. Computed turbulence kinetic energy from RSM for Case 2 RSM
x (m)
y (m)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1
0.0 0.2
RSM
x (m)
y (m)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1
0.0 0.2
RSM
x (m)
y (m)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.2
0.0
0.01 0.02
0.04 0.06 0.04 0.02 0.01
RSM
x (m)
y (m)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1
0.0 0.2
0.02 0.04 0.04 0.06 0.08 0.06 0.04
0.02
5.5. Energy Lines
Figs. 19 and 20 show the experimental and computed energy lines obtained from the RSM turbulence model along the solution domain. The total energy head, H, is measured from the channel bed. As can be seen in the figures, the loss of the total energy from the drop section to the end of B-type jump for the two cases is 30% and 35%, respectively, for the present conditions. For both cases the greater part of the energy loss in the flow area between the drop point and the end of the jump region happens during the drop (about 80%), and the rest (about 20%) arises from the fluctuations occurring in the hydraulic jump process.
Figure 19. Experimental and numerical energy lines for Case 1
Figure 20. Experimental and numerical energy lines for Case 2
6. CONCLUSIONS
The experimental and numerical investigation of a B-type hydraulic jump at an abrupt drop is carried out for two different flow cases. The governing equations of the flow are solved by ANSYS-Fluent software based on the finite-volume method using Standard k-, Shear StressTransport, and Reynolds Stress turbulence models. The Volume of Fluid method is used to determine the free surface profile. The discretization error using a grid convergence index for the three-grid comparisons is estimated to obtain a grid independent solution. Free surface profiles and velocity profiles obtained from the numerical analysis are compared with experimental measurements. The results of the mean square error and mean absolute relative error values indicate that the computational results for the free surface and velocity profiles of B-Type hydraulic jump by using Reynolds Stress Model is better than those predicted by other models for both flow cases tested. The total energy head of the flow with respect to the channel bed, is subjected to a loss of 30% and 35% during the drop and jump
0 3 6 9 12 15
0 20 40 60 80 100 120 140 160
H(cm)
x (cm)
RSM Exp.
0 3 6 9 12 15 18
0 20 40 60 80 100 120 140 160
H(cm)
x (cm)
RSM Exp.
processes for Case1 and Case 2, respectively. From the comparisons of the numerical results with the experimental measurements it can be concluded that the computational fluid dynamics methods are successful in modeling the complicated flow cases like B-type hydraulic jump at an abrupt drop; and for the investigation of such flow problems, numerical simulation can provide practical benefits without recourse to laboratory tests.
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