Thermophysics and Aeromechanics, 2013, Vol. 20, No. 5
Numerical simulation of transient processes in hydroturbines* 1
1
1
2
A.Yu. Avdyushenko , S.G. Cherny , D.V. Chirkov , V.A. Skorospelov , 2 and P.A. Turuk 1
Institute of Computational Technologies SB RAS, Novosibirsk, Russia
2
Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia
Е-mail:
[email protected] (Received December 19, 2012; revised February 4, 2013) A method for calculation of unsteady 3D turbulent flows in hydro turbines of power plants developed for simulation of the transient processes is presented herein. The method is based on joint solution to the Reynoldsaveraged Navier—Stokes equations for incompressible fluid flow written for moving mesh, the equation of runner rotation and the system of 1D equations describing propagation of elastic hydraulic shock in the flow domain. The exchange in flow parameters between the penstock and hydro turbine regions is considered in this approach. Results of transient processes simulation are presented for several modes: start-up to the turbine regime, instantaneous load shedding, and output power decrease. Comparison with experimental data is performed. Key words: numerical simulation, modeling, transient processes, hydro turbines.
Introduction Flows in the hydro turbine may be steady or transient. A typical steady flow has a constant water discharge rate Q through the flow domain, a steady frequency of runner rotation, and steady load on the runner shaft. These flows are considered as steady (or periodically unsteady); this class of flows was simulated by CFD solvers by many researchers in the framework of complete 3D statement [1−3]. The transient flow regimes develop during switching of hydro turbine operation from one mode to another through regulation of opening in the wicket gate (WG) or through increasing (decreasing) of runner shaft loading. These flows are significantly unsteady and characterized by drastic oscillations of the water flow rate during time of operation. This leads to a dynamic change of pressure in the flow domain taking a form of hydraulic shock ΔH (ξ , t ) [4], where ξ is a coordinate along the flow domain and t is time. Hydraulic shock ΔH (ξ , t ) can both increase the total head in the turbine H (ξ , t ) = H 0 + ΔH (ξ , t )
*
(1)
The work was financially supported by the Russian Foundation for Basic Research (Grant No. 11-01-00475-а).
© A.Yu. Avdyushenko, S.G. Cherny, D.V. Chirkov, V.A. Skorospelov, and P.A. Turuk, 2013
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A.Yu. Avdyushenko, S.G. Cherny, D.V. Chirkov, V.A. Skorospelov, and P.A. Turuk
if it is positive, or decrease it (when negative). Here H 0 is the difference between total energy fluxes between the upper and lower pools of the hydraulic power plant. Therefore, simulation of 3D flows in transitional regimes is problematic in terms of elaboration of numerical models that are not typical of steady flow simulation. The very first point is description of the value of hydraulic shock ΔH (ξ , t ). The phenomenon of hydraulic shock can be modeled directly if elastic deformation of aerated water and the flow domain walls are taken into account. However, this requires simulation of flow for entire flow domain, including the penstock (PS). This demand leads to too high computational costs which are unattainable for today. If the wall deformation is disregarded and the approximation of incompressible fluid for water flow is accepted, then the pattern of “stiff” hydraulic shock is created. Taking the case of a long flow domain and drastic change in flow mode, a solution very different from reality would appear due to unlimited growth in hydraulic shock magnitude ΔH (ξ , t ). The second problem is using boundary conditions at the inlet and outlet boundaries of flow domain that do not imply fixing of water discharge Q via these boundaries. Typically, flow simulation is carried out not for entire flow domain, but for key elements of hydro turbine: wicket gate (WG), runner, and the draft tube (DT) [1]. One of most widespread boundary conditions statements in the flow domain cross sections for the listed key elements is defining the velocity vectors at the inlet and defining the pressure at the outlet. However, because of inevitable variation in water discharge during our modeling, this approach cannot work for simulation of transient processes. The alternative approach has been proposed in [5] allowing the finding of discharge during the process of flow simulation. For this variant of problem statement, the total energy is assigned for both cross sections; besides, the inlet cross section has information about flow direction, and the pressure profile is given for the outlet cross section. The energy in cross section may vary with time. This statement fits better for transient flow problems and is used in present paper. In simulation of transient flows in WG channels with opening/closing blades, the mesh cell geometry is not fixed: it can accommodate to the position of blade surface (which are the boundaries of simulation zone). This creates a necessity of using moving (time-adapting) computation mesh. The allocation of mesh nodes within the computation domain depends on distribution of nodes upon the blade surface. The surface motion is controlled by the law of shift in the hydro turbine control tool. Thus, the third challenge is generalization of previously developed numerical method of 3D solution for incompressible fluid on fixed meshes to the approach with moving meshes. Any numerical method of fluid mechanics on a moving mesh has an important requirement that is a fulfillment of geometric conservation law. The key criterion for this requirement is the following: if the solution is a uniform flow, then numerical method with moving mesh must not produce any disturbance into the flow. Finally, the fourth challenge in modeling of transient flow vs. steady flow is a problem of finding the runner rotation speed: this speed is variable in a transient flow because it follows the solid body rotation law with impact from accelerating or decelerating torque on the runner. The authors are not familiar with any publications where all these challenges in 3D transient flow simulation have found an adequate solution. For today, a typical approach is using 1D hydroacoustic theory for study of transient processes in hydraulic power machinery. This theory is based on a hyperbolic system of conservation equations for mass and momentum of compressible fluid [6, 7]. A serious shortcoming of this approach is that we have to know a priori the hydro turbine’s universal characteristics. Hydroacoustic 1D models take in several input data: the penstock and draft tube length and cross section, runner diameter and its frequency of rotation, concentration of air in the working fluid, cavitation coefficient, water discharge Q, and head H 0 . However, these models do not take into account the geometry of WG and runner as well as velocity profile downstream these elements. All the listed input parameters might be the same for different geometries of the runner and, by theory, will not affect the transient flow characteristics. 578
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The present paper suggests an original combined approach for CFD modeling of real transient 3D flow: the passing of hydraulic shock in a long penstock is described by 1D model of elastic hydraulic shock; the “rigorous” approach is applied for the zone of turbine. Meanwhile, the total approach benefits from accurate three-dimensional geometrical and hydrodynamic statement (Fig. 1). The suggested approach includes a model of transient process: nonstationary 3D Reynolds-averaged Navier—Stokes equations, closed Kim—Chen version of k−ε turbulence model with solution in the time-variable domain [8]; the equations for runner rotation (runner considered as a whole solid object), and 1D equations for elastic hydraulic shock spreading through the penstock. The equations for transient process are completed with new statement of boundary conditions at the inlet and outlet cross sections of the water duct (as developed in [5]), and matching conditions at the penstock-hydroturbine boundary. Here we present the results of simulation of several transient processes within hydraulic turbine: start-up to the turbine regime, instantaneous load shedding, output power decrease. Simulation results are compared with experimental data. 1. Basic equations 1.1. Reynolds equations in the form of integral conservation laws for a moving volume
Reynolds-averaged Navier—Stokes equations are used for simulation of 3D flow. Since the transient processes are related closely to variation in the computation domain boundaries ⎯ turning of the wicket gate blades ⎯ the equations are written in the form of integral laws for a moving volume V(t) in the Cartesian coordinates ( x1 , x2 , x3 ) = ( x, y, z ) [8]: Rt
∂ QdV + ∫ K t dS = ∫ FdV , ∫ ∂t V ( t ) ∂V ( t ) V (t )
(2)
Fig. 1. Diagram of flow section of a hydraulic power plant and turbine configuration. 579
A.Yu. Avdyushenko, S.G. Cherny, D.V. Chirkov, V.A. Skorospelov, and P.A. Turuk
Rt ≡ diag ( 0,1,1,1) ,
n is the external normal to surface ∂V ( t ) ,
where dS = ndS ,
p
2 + k , where p is ρ 3 the static pressure, ρ is the liquid density, k is the kinetic energy of turbulence, ( w1 , w2 , w3 ) are the Cartesian components of velocity vector. The flux matrix is as follows:
Q = ( pˆ , w1 , w2 , w3 )
T
is a vector-column of variables. The value pˆ =
w1 ⎛ ⎜ 2 ⎜ w1 + pˆ − τ11 − w1 xt Kt = ⎜ ⎜ w1w2 − τ 21 − w2 xt ⎜ w w −τ − w x 3 t ⎝ 1 3 31
w2
⎞ ⎟ w1w3 − τ13 − w1 zt ⎟ ⎟, w2 w3 − τ 23 − w2 zt ⎟ w32 + pˆ − τ 33 − w3 zt ⎟⎠ w3
w1w2 − τ12 − w1 yt w22 + pˆ − τ 22 − w2 yt w2 w3 − τ 32 − w3 yt ⎛ ∂wi ∂w j + ⎜ ∂x j ∂xi ⎝
τ ij = ν eff ⎜
(3)
⎞ ⎟. ⎟ ⎠
(4)
In matrix (3) ( xt , yt , zt ) are the components of velocity vector xt of a point moved by ∂V ( t ) . The value ν eff in expression (4) is a sum of molecular ν and turbulent ν t viscosities ν eff = ν +ν t .
(
The vector of mass forces is defined as F = x1ω 2 + 2 w2ω , x2ω 2 − 2 w1ω , g
)
T
, where g is
the gravity acceleration (axis z is directed downward), ω is the runner rotation speed, ω = 0 for all other elements of the hydraulic turbine. The values ν t and k are found from Kim—Chen k−ε turbulence model [9] with log-law wall function near the solid walls. 1.2 Equations of k−ε turbulence model (Kim—Chen version) in the integral form for a moving volume
Every equation from Kim—Chen k−ε model [9] can be rewritten in the form of integral conservation law for a moving volume V(t)
⎛ ν ⎞ ∂ ϕ dV = − ∫ ϕ ( w − xt ) dS + ∫ ⎜ν + t ⎟ ∇ϕ dS + ∫ Hϕ dV , ∫ ⎜ σ ϕ ⎟⎠ ∂t V ( t ) V (t ) ∂V (t ) ∂V (t ) ⎝
(5)
where ϕ and Hϕ are given in Table 1, w = ( w1 , w2 , w3 ) , ν t = C μ k 2 /ε . Table 1 Values of ϕ and Hϕ in equation (5) Equation
ϕ
Hϕ
For turbulent kinetic energy
k
G−ε
For dissipation rate of turbulent kinetic energy
ε
Cε1ε G/k − Cε2ε /k + Cε3G /k
Remark. G = τ ij
∂wi ∂x j
2
2
; C μ = 0.09, Cε 1 = 1.15, Cε 2 = 1.9, Cε 3 = 0.25, σ k = 0.75, σ ε = 1.15.
1.3. Runner rotation equation
Most of transient processes are accompanied by a change in runner rotation frequency. This relationship is not known in advance, therefore, simultaneously with solving of equations (2), (5) the equation for runner rotation as a solid body has to be solved: Iz
580
dω = M R (t ) − M gen (t ) − sgn (ω ) M fr , dt
(6)
Thermophysics and Aeromechanics, 2013, Vol. 20, No. 5
where I z is the total inertia momentum for the runner and generator, M R is the torque due to impact of flow on the runner, M gen is the torque of useful load applied to the generator shaft, M fr is the entire torque from friction in the electromechanical system of the aggregate.
The law M gen (t ), as a rule, is known, meanwhile M R (t ) is determined by flow mechanics, i.e., found from solving equations (2) and (5). 1.4. Hydraulic shock model
Modeling of hydro-acoustic oscillations in the penstock is carried out with well-proven 1D model for elastic hydraulic shock [4, 6]. If liquid friction on the walls is disregarded, this model can be written in the form ⎧ ∂m c 2 ∂Q = 0, + ⎪ ⎪ ∂t gS ∂ξ ⎨ ⎪ ∂Q + gS ∂m = 0, ⎪⎩ ∂t ∂ξ
ξ ∈ [0, L],
(7)
where m (ξ , t ) = p (ξ , t ) ρ g − z (ξ ) is the potential head, Q (ξ , t ) is the liquid discharge, S (ξ ) is
the penstock cross section area, c (ξ ) is the propagation velocity for elastic shock wave, and L is the penstock length. The velocity c depends on concentration of gas unsolved in water and on elasticity of penstock walls [6]. For real examples of penstocks, velocity is as follows c = 1000 − 1450 m/s. 2. Boundary conditions for joint simulation in the zone penstock-hydroturbine 2.1. Input and outlet boundaries
When a flow in hydro turbines is simulated, a common practice for setting boundary conditions at the inlet and outlet of computation domain is stated like “discharge-pressure”: water discharge and flow entering angle (or velocity vector distribution) is specified at the inlet, and the outlet cross section has a pressure distribution and tangential components of velocity w ⋅ τ i |out , i = 1, 2 ( τ1 , τ 2 are two linearly independent vectors tangential to the outlet cross section) [10]. However, in simulation of transient process, the liquid discharge is a time-variable parameter and is not known a priori. Wherein, the total head, which is the difference between energy at the penstock inlet and draft tube outlet, remains unchanged even for unsteady flow. Therefore, in this paper we propose to use boundary conditions which give us the liquid discharge and the flow field from the given head H 0 . As for the penstock inlet cross section (ξ = 0, see Fig. 1), we have the total energy flux Q 2 ( 0, t ) (8) EPS, in ≡ m(0, t ) + = H0. 2 gS 2 For the outlet cross section of draft tube (Fig. 1), the discharge-averaged total energy can be formulated 2 ⎞ ⎛ p wav 1 EDT, out ≡ − z + (9) ⎜ ⎟ ( wdS ) = 0, Q S ∫ ⎜⎝ ρ g 2 g ⎟⎠ DT, out where wav = Q S DT, out , S DT, out is the draft tube (DT) outlet cross section. One can also write the condition for static pressure profile: p = p0 + ρ g ( z − z0 ).
(10) 581
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Note here that the value p0 is not known a priori, but rather found in the process of solving of fluid mechanics equations to fulfill equality (9). This formulation of inlet and outlet conditions is in line with physics of transient process. Actually, conditions (8)−(9) mean that the levels of the upper and lower pools remain constant. 2.2. Interface between penstock and wicket gate
The joint simulation of flow in the zone of penstock-hydroturbine needs correct transfer of parameters from one subzone to another. The difficulty originates from limitation that for the formulated statement, the spiral case (SC) and stator (ST) are not within the computation domain. The exchange of parameters of flow must be performed between the outlet boundary of the penstock ξ = L and WG inlet cross section (Fig. 1). The water discharge for these cross sections is the same, so for the WG inlet we assign the discharge QPS,out obtained during solving of a set of equations (7) in the penstock and the flow inlet angle δ sp = const. The pressure at the WG inlet is extrapolated inside the simulation domain. Let us give more details about transfer of pressure value from the WG zone to the outlet boundary of the penstock. A ratio between the pressure pPS,out at the penstock outlet and pressure pWG,in at the penstock inlet – under condition that the total energy loss is introduced within the SC and ST is estimated by formula ΔhSC + ΔhST = ζ s H 0 ,
(11)
where ζ s is the loss coefficient in the spiral case and the stator. Actually, ζ s Q 2 , but for the simplicity sake we can take ζ s = const 0.01. The total energy at the penstock outlet is 2
| w |PS,out ⎛ p ⎞ − z⎟ + EPS,out = ⎜ , 2g ⎝ ρg ⎠PS,out
(12)
where | w |PS,out = Q S PS,out , since the model assumes that the flow velocity at the penstock outlet is normal to the cross section. For the WG inlet we have 2
| w |WG,in ⎛ p ⎞ . EWG,in = ⎜ − z⎟ + 2g ⎝ ρg ⎠ WG,in
(13)
Within a cylindrical coordinate system, the velocity vector at the WG inlet w WG,in has the components (cr , cz , cu ), meanwhile the axial component of velocity cz = 0. The radial and circumferential components are related through the ratio cr cu = tg δ sp . Therefore, we have | w |WG,in = cr2 + cu2 =
cr Q = . sin δ sp S WG,in sin δ sp
(14)
By these means, with account for equality EPS,out = EWG,in + ζ s H 0
(15)
and equality for levels zPS,out = z WG,in , we have the relation for two static pressures:
pPS,out
ρg 582
=
pWG,in
ρg
+
1 1 Q2 ⎛ ⎜ − 2 2 2 2 g ⎜ SWG,in sin δ sp SWG,out ⎝
⎞ ⎟ + ζ s H0 . ⎟ ⎠
(16)
Thermophysics and Aeromechanics, 2013, Vol. 20, No. 5 2.3. The rest of boundaries
Simulation of 3D flow in wicket gate and runner (WG-R) is performed in cyclic mode ⎯ first for one inter-blade channel of WG, then for one inter-blade channel of runner and whole draft tube. We impose the periodic condition at the interface between two channels, and the solid walls have the no-slip condition. The flow is averaged in circumferential direction when passing the parameters at the exchange boundaries between WG and runner, runner and DT. 3. Numerical method 3.1. Solving Reynolds equations
Numerical code for solving of unsteady equations (2) is based on introducing of artificial compressibility into the model: this is done through adding a derivative by pseudo-time τ for pressure into the continuity equation and derivatives with respect to pseudo-time for corresponding velocity components into the momentum equations. The modified equation (2) takes the form: ∂⎞ ⎛ τ ∂ + Rt ⎟ ∫ QdV + ∫ K βt dS = ∫ FdV , ⎜R τ t ⎠ V (t ) ∂ ∂ ⎝ ∂V (t ) V (t )
(17)
where Rτ = diag(1,1,1,1), K βt = diag( β ,1,1,1) ⋅ K t , β is the coefficient of artificial compressibility, K t is defined in (3). For equation (17), third-order accurate MUSCL discretization scheme is used for convective terms and 2nd-order central difference scheme is used for viscous terms. Second-order upwind difference scheme is used for approximation of t variable, and new derivatives with pseudo-time τ are approximated by 1st-order difference scheme. Meanwhile for every step on physical time t, the algorithm requires iterations by pseudo-time τ. The velocities of shifting for mesh cell sides are approximated to fulfill the geometric conservation law [8]. The linearized system of discrete equations is solved using implicit LUfactorization scheme. For more details of algorithm the reader is referred to [8]. 3.2. Solving equations of elastic hydraulic shock
System (7) is 1D hyperbolic system with constant coefficients. This system can be written in the form: ∂f ∂f +A = 0, (18) ∂t ∂ξ ⎛ 0 c 2 /gS ⎞ ⎛m⎞ f = ⎜ ⎟ , A = ⎜⎜ ⎟ . A uniform mesh with step Δξ : ξ1 = 0, ξ 2 = Δξ , 0 ⎟⎠ ⎝Q ⎠ ⎝ gS ξ3 = 2Δξ , ..., ξ J = L is introduced in the interval [0, L]. Equation (18) is solved using an implicit difference scheme with upwind approximation of 1st-order in space and time:
where
f js +1 − f jn Δt
+ A+
f js +1 − f js−+11 Δξ
+ A−
f js++11 − f js +1 Δξ
= 0,
j = 2,..., J − 1,
(19)
⎛ c / 2 c 2 /2 gS ⎞ ⎛ −c / 2 c 2 /2 gS ⎞ − where A + + A − = A, matrices A+ = ⎜⎜ ⎟⎟ and A = ⎜⎜ ⎟⎟ have only c/2 ⎠ ⎝ gS / 2 ⎝ gS / 2 −c / 2 ⎠ non-negative (λ1+ = c, λ2+ = 0) and non-positive (λ1− = − c, λ2− = 0) eigenvalues, respectively.
Equation (19) is iterated in s , and f n +1 = lim f s . Iteration in s is required because s →∞
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A.Yu. Avdyushenko, S.G. Cherny, D.V. Chirkov, V.A. Skorospelov, and P.A. Turuk
of nonlinear boundary conditions at the penstock inlet (8), and also for exchange of parameters with the WG zone. One iteration in s in the penstock zone corresponds to one step in pseudotime in the hydro turbine. 3.3. Numerical implementation of boundary conditions
Boundary condition (8) for the inlet cross section of the penstock (j = 1) is implemented in the following way. Let us f 2s = ( m2s , Q2s ), then we have f1s +1 = ( H 0 − (Q2s ) 2 /2 gS 2 , Q2s ). By these means, for the inlet cross section, the discharge value Q is extrapolated within the domain from the previous iteration, and the potential head m is calculated to make the total energy (computed for node j = 1) be equal to H 0 . We assign at the interface penstock −WG (j = J): ⎛ mWG ⎞ f Js +1 = ⎜ s ⎟ , ⎜Q ⎟ ⎝ J −1 ⎠
(20)
where mWG = pPS,out /ρ g − z WG , pressure pPS,out is defined from (16), where p WG,in is the cross section averaged static pressure at WG inlet. At the DT outlet, the hydrostatic distribution of pressure p is adjusted to make the total energy at the DT outlet equal zero: (21) EDT,out = 0, meanwhile velocities are extrapolated within the computation domain. One has to notice that in solution of equations (2), the pressure is calculated with accuracy up to a constant, so in calculations, the outlet energy was taken as zero. In reality, for the considered design of turbine, the outlet energy is (22) EDT,out = 13.532 m w.c. Therefore, in order to obtain the absolute values of pressure in the hydro turbine duct one must increase the simulated values with 13.532 m of water column. 4. Computational results Study of unsteady processes and their correct consideration in designing of power unit of hydro power plant is of significant interest. The main task in analysis of transient processes during a change in power is finding of optimal regimes of regulation suitable for change in turbine torque with fast speed and ensuring limitations on developing dynamic impact like a hydraulic shock. The developed method was applied for simulation of main transient processes in a natural-size Francis turbine with the head H 0 = 73.5 m, runner diameter D1 = 3.15 m and at the nominal rotation of frequency nnom = 200 rpm. The friction moment in equation (6) was taken equal M fr = 2 ton-force·meter. Simulations were performed in a cyclic mode for the zone comprising a penstock, single WG channel, one runner’s channel, and the whole DT (Fig. 2). The structured zone in the moving zone of WG is built automatically for every time step. The simulation mesh in the entire domain comprises about 100 000 cells. The penstock zone requires a uniform mesh with the space step Δξ = L/1000. For all kinds of simulations, we used the time step Δt = 0.01 s, which corresponds to the runner rotation by 12°. For every time t step, about 1500 iterations were performed for convergence by pseudo-time τ. The typical time for performing one simulation of nonstationary flow with using three processors is about six days of computation. All simulations were carried out in natural parameters; however, the values of WG clearance a0 (a minimal distance between two neighboring blades of WG) are given here for the model with the runner diameter D1 = 0.46 m. 584
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Fig. 2. Simulation domain for cyclic problem statement: penstock-WG-Runner-DT.
4.1. Start-up to the turbine regime The start-up is a process when the runner from the rest position is accelerated to the idlerun mode with the frequency nnom with the followed-up synchronization and connection of the electric generator to the grid. During this process, the WG opening changes by the law depicted in Fig. 3а. A typical start-up process has two startup-opening stages for the WG. For the simulated turbine case, the first start-up opening of WG is a0,so1 = 0.25a0,max , where a0,max is the maximum opening of WG. After the frequency of runner revolution achieves 90−95 % level
Fig. 3. Startup to operating mode. a ⎯ the law for opening of WG space a0(t) for experiment (1) and adopted for simulation (2), b ⎯ experimental (1) and simulated (2) dependencies of runner rotation frequency as function of time.
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A.Yu. Avdyushenko, S.G. Cherny, D.V. Chirkov, V.A. Skorospelov, and P.A. Turuk
Fig. 4. Calculated dependencies of discharge Q (a), torque MR, and head H (b) vs. time (for the process of start-up into operating mode).
of the nominal
nnom , the WG starts disclosing the second option of start-up opening
a0,so2 = 0.15a0,max . Further, the WG evolves to the opening of the idle run equal to a0,ir = 0.10a0,max . In reality, the whole process of turbine start-up takes about thirty seconds.
Note that the requirement for non-degeneracy of mesh cells does not allow us to close the WG blades completely, therefore, at the initial time moment t = 0 we assign the flow field which was obtained previously for a stationary problem with opening a0 = 1 mm and zero rpm of the runner. Then the blades keep opening for 2.6 second (linear progress up to a0,so1 ), meanwhile the first 2.1 seconds the runner frequency must be n = 0 rpm (as in experiment). Only after this initial interval t = 2.1 s, the frequency n is found from solving equation (6). Figure 3b presents comparison of calculated and experimental rpm for the runner. One can see that the results are in good agreement. It should be noted that for the initial 30 s a slight delay (< 7 %) of the simulated rotation frequency from that in the experiment is observed. Possibly, this takes place because flow parameters averaging (inevitable for cyclic mode of simulation) for the parametric exchange at the WG-runner interface are used. Figure 4а presents the water discharge Q for the WG inlet cross section as a function of time. The shape of this curve almost reproduces the shape of WG opening evolution with time (Fig. 3а). The parameters of points where the pressure on the WG blades is considered, are summarized in Table 2. Figure 4b shows the dependency of the runner torque MR and the hydro turbine head H = EWG,in − EDT,out vs. time. During the initial 5 s, the torque applied to the runner increases rapidly and approaches the level of 55 % from the torque of optimal performance. After the torque reaches the maximum, it starts decreasing gradually, but Table 2 Parameters of points for consideration of pressure on the runner blade during the start-up process 3
−1
No.
time t, s
1
0.5
discharge, m /s 3.02
2 3 4 5
1.5 3 15 20
11.18 24.53 23.35 13.28
0 12.23 162.78 181.36
59.74 142.41 73.19 27.57
6
30
9.58
192.08
8.49
586
frequency n, min 0
torque MR, ton⋅m 13.34
Thermophysics and Aeromechanics, 2013, Vol. 20, No. 5
Fig. 5. Variation of pressure on the runner blades during start-up regime (for every time moment, the image of working side is on left, backside — on the right).
the runner rpm increases. For the idle run mode, the torque value is M R M fr , which corresponds to a constant value of n. One can see from Fig. 4b the effect of hydraulic shock on the head H (hydraulic shock is produced due to fast opening of WG). The pressure distributions on the runner blades at selected moments of time are depicted in Fig. 5. During the initial three seconds, the pressure on the working side near the inlet edge of the blade increases drastically, since the liquid flow speeds up the runner. Later the pressure distribution becomes more uniform: it becomes lower and it is compensated by a growing pressure on the backside of the blade. This decreases the runner torque MR down to the level of friction torque M fr . Figure 6 presents plots for axial Cz (1), radial Cr (2), and circumferential Cu(3) components of velocity as a function Fig. 6. Velocity profiles at the runner outlet for idle mode. 587
A.Yu. Avdyushenko, S.G. Cherny, D.V. Chirkov, V.A. Skorospelov, and P.A. Turuk Fig. 7. Vortex shedding from the inlet edge of runner blade (idle mode operation).
of radius R in the cross section z = 2 m in the draft tube just downstream the runner. As soon as the turbine reaches the idle mode operation, the flow collects and swirls near the DT walls, but the center of this computation zone demonstrates a wide zone of recirculation flow. The vortex shedding from the inlet edge of the runner blade for the idle mode operation (depicted in Fig. 7 through streamlines) helps in understanding the flow structure inside the runner channel. A highly swirled flow which evolves toward the DT walls is being generated on the working side of the blade. The pumping vortex (Fig. 8) is observed near the runner rim; the flow is upward. 4.2. Instantaneous load shedding with trip to the idle mode
The load shedding (switching off in the loaded generator) is a process from emergency operations class. The calculated process of load shedding with final evolution to the idle mode is defined by the law of blades closing (depicted in Fig. 9а). At the initial time moment t = 0,
Fig. 8. Streamlines and contour lines of velocity module for two cross sections of runner. 588
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Fig. 9. Load shedding with trip to idle. a ⎯ law of WG opening a0(t) in experiment (1) and presetted for simulation (2); b ⎯ experimental (1) and simulated (2) dependencies of runner rotation frequency as function of time.
we assign a steady flow pattern obtained for simulation of maximum power mode: a0 = 35 mm, n = 200 rpm. The torque of useful load (see eq. (6)) becomes zero at t = 2.5 s, and the WG blades start their closure movement. In our simulation, it is impossible to shut down the WG completely to a0 = 0 mm, therefore, for the chosen time interval t ∈ (13.5; 24.5), we assign a minimal level of blades opening a0 = 1 mm. Figure 10 presents the meshes for maximal and minimal opening of WG, which were used for simulation of load shedding. Figures 9b and 11 show the time curves for runner operation frequency n, discharge Q, hyrodynamic torque of runner M R , and for the turbine head H. After the generator was turned off, the runner rotation frequency increased rapidly. A decline in WG opening produces a smaller water discharge. The runner torque drops to zero and even becomes negative for a0 < a0, ir . Meanwhile the rpm passes a maximum value and declines gradually, but until this parameter remains higher than nominal value nnom , the wicket gate keeps closing down to a minimal value (1 mm in simulation, 0 mm in reality). This situation remains unchanged until n approaches to nnom . After that the WG is opened gradually to a0, ir and the hydro turbine starts operating in the idle mode. Note that the calculated increase in the runner rotation frequency during the initial 25 seconds has a slight backlog from the experimental value.
Fig. 10. Meshes for WG for two orientations of blades during load shedding process. a0(2.5 s) = 35 mm (left), a0(12.5 s) = 1 mm (right).
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A.Yu. Avdyushenko, S.G. Cherny, D.V. Chirkov, V.A. Skorospelov, and P.A. Turuk
Fig. 11. Calculated dependencies for water discharge Q (a), torque MR, and head H (b) as function of time; instantaneous load shedding. Table 3 Parameters for points of analysis of pressure on runner blades during the process of load shedding No.
time t, s
1 2 3 4 5 6
3 6 10 14 24 50
3
discharge, m /s 84.42 52.77 5.35 2.05 2.68 10.38
frequency n, min 214.67 278.46 286.00 261.94 216.48 208.56
−1
torque MR, ton⋅m 257.45 110.36 −49.91 −51.74 −28.80 5.29
Pressure distributions on the runner blade for several selected time moments (Table 3) are presented in Fig. 12. Initially, the pressure on the working side of blade decreases, but it increases on the back side. At t = 10 s, the runner torque reaches a minimal value, then it grows gradually to the value M fr . In the same manner as for the start-up process, the pressure distribution over the runner blade depends on current time very much. 4.3. Output power decrease Here we present the results of simulation for a transient process of power decrease. At the initial time moment t = 0, a steady flow field, calculated previously for a flow with maximum efficiency, is applied. Later on, the WG blades opening decreases linearly with time from initial level 27.5 mm (corresponds to the regime with maximum efficiency) down to 18 mm (part load mode) during the other ten seconds, starting from time moment t = 2.5 s. It is a known fact that the flow downstream the runner has low swirling in the maximum efficiency regime, therefore, the flow inside the cone of draft tube becomes almost stationary and axisymmetric. On the contrary, for the part load mode, the circumferential component of velocity in the zone downstream the runner is significant, and this generates a precessing vortex core inside the DT cone. Figure 13 presents the calculated torque for the runner and water discharge as functions of time. Figure 14 shows evolution of pressure upstream the WG and on the DT conical wall. A decrease in discharge while closure of vanes creates a positive hydraulic shock which is seen as an escalation in pressure upstream the WG (Fig. 14a). The nucleated vortex core, observed at the initial time, increases in sizes and precession radius while the WG is closed gradually: this is due to a growing residual swirling of the flow downstream the WG. As this 590
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Fig. 12. Changing the pressure distribution on the runner blades at the instantaneous load shedding (for every time moment the working side on the left, and back side on the right).
motion of WG blades is completed (at t > 12.5 s), the flow possesses a periodic unsteady pattern. Rotation of vortex core induces pulsations in the discharge and the torque on the hydro turbine shaft (Fig. 13). The amplitude of pressure pulsation on the DT cone is about 4 % of the operating head. The vortex core precession period is T = 1.41 s, which is close to the experimental value of Texp = 1.3 s. 4.4. Trajectories of instantaneous modes for different transient processes Whenever we have the curves for change in natural parameters of hydroturbine for different unsteady processes, we can build up a trajectory of consecutive movement of the mode point (in coordinates normalized to H = 1 m,
Fig. 13. Behavior of torque M and discharge Q during the power decrease. 591
A.Yu. Avdyushenko, S.G. Cherny, D.V. Chirkov, V.A. Skorospelov, and P.A. Turuk
Fig. 14. Evolution of pressure in the zone upstream the WG (а) and in the DT cone (b).
D1 = 1 m) for variables of discharge and frequency: Q11 =
Q D12 H
, n11= n
D1 H
,
i. e., to plot the transient process in the field of main hill diagrams. This gives a vivid illustration for the range of operation modes variation and helps to find the operational conditions for different transient processes. Figure 15а presents a typical hill diagram for a radial-axial turbine taken from [6], and line of instantaneous modes corresponding to different types of transient processes. For the starting curve II, the hydrodynamic torque for the runner is M R = 0 and, correspondingly, the efficiency coefficient is η = 0. This line splits the hill diagram into the zone I of turbine operation mode ( M R = 0) and the zone III of retardation operation modes ( M R = 0) .
Fig. 15. Trajectories of performance points for transient processes. а ⎯ typical hill diagram for a radial-axial turbine [6]: operation mode (I), start-up mode (II), retardation mode (III); b ⎯ calculated trajectories of mode points: start-up to the turbine operation mode (1), power decrease (2), load shedding (3).
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The dashed line outlines the zone of working modes for the standard turbine operation. The trajectories calculated in this paper are plotted in Fig. 15b. We observe a good compliance between calculations and experiments. Conclusion This paper describes a new method for simulation of unsteady 3D flow in transient flow of hydro turbines. The developed model takes into account the phenomenon of hydraulic shock, variability in the runner rotation frequency, changes in the water discharge flowing through the penstock and hydro turbine. The results for simulation of different modes are presented: launching to the operating mode, instantaneous load shedding, and output power decrease. We obtained a good compliance between simulation and available experimental data on the frequency of runner rotation and the amount of hydraulic shock. Modeling was carried out in cyclic statement. On the one hand, this model is computationally efficient and suitable for unsteady calculations for transient processes. On the other hand, this model cannot take into account the features of flow in a spiral case, maldistribution of circumferential flow at the runner inlet or rotor-stator interaction. As a further step in development of offered approach, it can be used for complete statement at flow simulation in the entire flow domain. Accounting for water cavitation is another option for improvements in unsteady flow model. References 1. S.G. Cherny, D.V. Chirkov, V.N. Lapin, V.A. Skorospelov, and S.V. Sharov, Numerical Simulation of Flows in Turbomachinery, Nauka, Novosibirsk, 2006. 2. A. Ruprecht, Numerical prediction of vortex instabilities in turbomachinery, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, Springer Verlag, 2006, Vol. 93, Р. 211−224. 3. G.D. Ciocan, M.S. Iliescu, T.C. Vu, B. Nennemann, and F. Avellan, Experimental study and numerical simulation of the FLINDT draft tube rotating vortex, J. Fluids Engng., 2007, Vol. 129, No. 2, Р. 146−158. 4. N.E. Zhukovskii, About Hydraulic Shocks in Water Pipes, Gostekhizdat, Moscow, Leningrad, 1949. 5. D.V. Bannikov, D.V. Yesipov, S.G. Cherny, and D.V. Chirkov, Optimization design of hydroturbine rotors according to the efficiency-strength criteria, Thermophysics and Aeromechanics, 2010, Vol. 17, No. 4, P. 613−620. 6. G.I. Krivcheno, N.N. Arshenevsky, E.E. Kvyatkovsky, and V.M. Klabukov, Hydro-Mechanical Transient Processes in Hydro Power Installations, Ed. G.I. Krivchenko, Energia, Moscow, 1975. 7. C. Nicolet, S. Alligne, B. Kawkabani, J.-J. Simond, and F. Avellan, Unstable operation of Francis pump-turbine at runaway: rigid and elastic water column oscillation modes, J. Fluid Machinery and Systems, 2009, Vol. 2, No. 4, P. 324−333. 8. A.Yu. Avdyushenko, S.G. Cherny, and D.V. Chirkov, Numerical algorithm for modeling three-dimensional flows of an incompressible fluid using moving grids, Computational Technologies, 2012, Vol. 17, No. 6, P. 3−25. 9. Y.S. Chen and S.W. Kim, Computation of turbulent flows using an extended k-ε turbulence closure model, NASA CR-179204, 1987. 10. S.N. Antontsev, A.V. Kazhikhov, and V.N. Monakhov, Boundary-Value Problems in Mechanics of Nonuniform Fluids, Nauka, Novosibirsk, 1983.
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