Time-dependent flow in a penstock during head-gate closure

ELSEVIER

Time-dependent flow in a penstock during head-gate closure

Sean McKee

University of Strathclyde, Department of Mathematics, Glasgow, Scotland, UK

A. D. Sneyd

University of Waikato, Hamilton, New Zealand

Murilo F. Tom6

ICMSC-Usp de S2o Carlos, Departamento de Computa@o, SLio Carlos, S.P., Brazil

This paper is concerned with the study of the time-dependent water and air flow in a penstock caused by closing the head gate. Two approximate techniques are employed. The first approach uses simple global arguments based on mass and momentum conservation, and the time-dependent Bernoulli equation is used to derive evolution equations for the fluxes of water and air through the various components of the system. To complement this approach, the time-dependent Nacier-Stokes equations were solved with a substantially reduced Reynolds number. Both approaches suggested that the pressure in the penstock remained abor>e 1 / 2 atm, answering a question posed by Electricorp, New Zealand.

Keywords: Bernoulli equation, free surface flow, Navier-Stokes, finite-difference, penstock

1. Introduction

This study was the result of a commission from the Waikato Hydro Group of Electricorp Production, New Zealand, which operates a chain of eight hydrostations along the Waikato River. The aim was to study the time- dependent water and air flow in a penstock caused by closing the head gate.

Normally flow through the turbine is controlled by the wicket gates just in front of the turbine, but in an emer- gency it may be necessary to close the head gate while the system is operating at full power. One worry was that the internal penstock pressure might drop to less than about l/2 atm and cause the penstock to collapse. Another problem was that if the turbine were to be simultaneously

Address reprint requests to Prof. McKee at the Department of Mathemat-

its, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow Gl 1 X H Scotland, UK.

tripped from the grid, its rotation rate would approach “runaway speed” where prolonged operation would cause damage.

Figure 1 gives a diagram of the system. As the gate closes the frictional pressure loss behind it sucks water down the vents until they are drained (usually by the time the gate is about two-thirds closed) and an air pocket forms in the penstock. After the gate is shut, the water level in the penstock continues to fall, sucking air in through the vents until air begins to enter the turbine itself, usually somewhat over a minute after the gate has begun to close.

Two scenarios need to be considered: (a) with the turbine connected to the grid so that it rotates at constant speed, and (b) with the turbine disconnected or rotating freely. In case b the turbine rotation rate may increase until it approaches runaway speed at which point the torque exerted by the water flow falls to zero. Since the speed at which the gate closes may vary, we consider three closure times: 20 set, 30 set, and 40 sec.

Received 10 June 1994; revised 30 November 1994; accepted 2 January A full numerical solution would be a daunting prospect 1996. given the complexity of the flow geometry (including the

Appl. Math. Modelling 1996, Vol. 20, August 0 1996 by Elsevier Science Inc. 655 Avenue of the Americas, New York, NY 10010

0307-904X/96/$15.00 SSDI 0307-904X(96)00009-0

Time-dependent flow in a pencock: S. McKee et al.

Figure 1. Penstock diagram.

turbine) and a Reynolds number of 106-107. Moreover the required accuracy did not warrant such a Herculean task, and we use a simpler two-pronged attack.

Our first method (method I> uses simple global arguments based on mass and momentum conservation. The time-dependent Bernoulli equation is used to derive evolution equations for the fluxes of water and air through the various components of the system. Viscous effects are ignored except for the frictional pressure loss under the gate which is estimated from engineering tables. The turbine characteristics are predicted from a simple (but robust) model, using quoted efficiencies at various power settings and the known runaway speed.

While method I provides global information on mass fluxes, etc., it obviously cannot give a detailed picture of the flow or of local effects. To remedy this defect, we also carry out a full numerical solution of a simplified version of the problem (method II).

The penstock and vents are modelled by two-dimensional channels, and a uniform outflow is assumed at the turbine end of the penstock. The time-dependent laminar Navier-Stokes equations are solved using the GENSMAC code with a Reynolds number of 250. Our second method highlights some important local features of the flow.

The dimensions in Figure 1 correspond to a typical penstock in the Maraetai I dam on the Waikato River. All lengths are quoted in meters, flow rates in cubic meters per second (cumecs), and turbine rotation rates in radians per second.

In Section 2 we derive the equations necessary for method I, and Section 3 describes our approximate analy- sis of the turbine behavior. Results of the approximate methods are given in Section 4. Section 5 provides a brief description of the GENSMAC calculation, with results, and finally our conclusions are summarized in Section 6.

The following data is provided: 1. 2. 3. 4.

5.

Diameter of the pens&k (D) = 4.88 m Height of the column of water in the vent (h) = 39.5 m Diameter of the vent (d) = 1.08 m Length of the penstock (I,) = 80 m Volumetric flow through the turbine (VT) = 74.4 m3/sec

6. Kinematic viscosity of water (v) = 10e6 m*/sec 7. Density of water ( p> = 1,000 kg/m3 8. Slope of the penstock ( p> = 14” 9. Distance between the vent and the gate (A) = 1.5 m

2. Derivation of equations

2.1. General method

The penstock is essentially a tunnel of length 80 m and diameter 4.88 m through the dam, which brings water from the lake to the turbine. Flow is controlled primarily by the wicket gates (not shown in diagram) just in front of the turbine, but it may also be shut off by lowering the head gate in the dam face. Just behind this gate are two air vents (one only is shown in the diagram) with a combined cross-sectional area of about 1 m*, whose purpose is to equalize air pressure in the penstock when it is drained of water.

We assume that the flow in the penstock is frictionless and ignore viscous energy loss in the interior of the flow and at the walls. This seems a reasonable assumption since an estimate of the frictional head loss is 0.3 m, represent- ing less than 1% of the total head of 59.4 m. Under this assumption the equation of motion for the water can be written in the form,

p(~+~xv)+o(p+;pv’+Pgz)=o (1)

where v is the water velocity, w = VX v is vorticity, p density, p pressure, g gravitational acceleration, and .z a vertical coordinate. Integrating this equation along a streamline between two points X and Y say, we find

+pdh,-h,) =O (2) where ds is a small segment of the streamline, h,, h, is the vertical height at points X and Y respectively, and px denotes the pressure at point X, etc. Equation (2) allows us to formulate the flow problem as a system of ordinary differential equations for the fluxes of water through the various components of the system.

2.2. Phase I: Water in vents

We let ql(t) be the flux of water through the gate and q*(t) the flux of water through the turbine. The flux of water down the vent is therefore q*(t) - q,(t). Consider a streamline C, say, beginning at a point P on the upper water surface, passing through the gate, down the penstock, through the turbine, and finishing at point R on the lower water surface, the so-called tailrace surface (Figure 2Laj). Equation (2) applied to C, gives

/

dV

p--ds= -;pu;+pgH-A~,,-Ap, PR at

(3)

where Ap, and Ap, represent gate and turbine head losses, respectively, and H is the total head. The gate loss Ap, is estimated by cubic spline interpolation of tabulated gate losses in Miller.’ The pressure loss through the turbine, Ap,, depends both on the flow rate through the turbine and on its rotation rate and is discussed in Section 3.

Appl. Math. Modelling, 1996, Vol. 20, August 615


where Ap, is the head loss across the turbine during normal operation. Thus equation (4) can be re-expressed in the form

b

a,i,+a,41+h,~q,=~(AP*,~- AP, - AP,)

(5)

A second equation is derived by integrating equation (2) along the path C, in Figure 2(a), from a point Q on the vent water surface to the point R on the tailrace surface. An argument similar to the above yields

av(42 - 41) + a,&

Figure 2.

The integral on the left-hand side of this equation is somewhat difficult to estimate, since it depends on the detailed geometry of the flow, but in principle can be written in the form,

+ sdt)b,(t) + q*(f)b,(t) where the a,, bi(i = 1, 2) are geometrical coefficients. The a,(i = 1, 2) represent the inertia of the moving water and the b,(i = 1, 2) the effect of the changing flow geometry as the gate closes. The two most important contributions arise from the flow along the penstock and flow from the lake into the gate and can be written in the form,

where the dot implies differentiation with respect to time. Assuming that the flow is approximately uniform across the penstock diameter, the coefficient up = 1,/A,, where I, and A, represent the penstock length and cross-sectional area, respectively. Since the penstock geometry is constant, the b, coefficient is zero. To calculate the lake- flow coefficients uL and b, we assumed that the gate opening is elliptical, with the same cross-sectional area and aspect ratio as the actual rectangle. The details are given in the appendix. In practice a,_ and b, are both much smaller than the penstock coefficient up but nevertheless have an important influence on the flow, particularly in phase II (see below).

Equation (3) can therefore be approximated by

up& + aL4, + b,q,

= -fv;+gH-;(Apo+_(PT) (4)

In the steady state before the gate begins to close, this equation reduces to

0= -;u;+gIkApTo P

(6)

where p0 is the atmospheric pressure. The vent-flow coefficient a, = s/A, where s is the length of the water column in the vent and A, the combined cross-sectional area of the two vents. In some penstocks the two vents are slightly different, but this complication was ignored. Also in equation (6) ha(s) is the height of the vent water surface below the upper water level (Figure 2(aj), and the water speed vp at Q is given by

q2 - 41 =--- “’ A V

The air pressure pa at Q is found by applying the compressible version of Bernoulli’s equation to the airflow in the vents. Assuming an adiabatic gas law, Bernoulli’s equation can be written in the form

ff U$ + c2 = constant ,=’ r-1) 2(

where uv is the air speed, c is local sound speed, and y = 1.401 is the ratio of specific heats. It follows that

pe =po[l - (av&k;)]y’(y-‘) where c0 = 340.6 m/set is the speed of sound at 1 atm at 20°C. This expression for pQ is approximate only, since we have assumed steady flow and neglected the inertia of the air mass in the vents. This neglect is justified by the relatively low density of air. It can be seen from equation (7) that pQ does not drop appreciably below 1 atm until ue (the rate of fall of the water surface in the vents) approaches the speed of sound.

To complete our system of differential equations we also need to describe the evolution of S, the length of the water column in the vents. This is given simply by

91 - q2 s= -“Q=- AV

(8)

Equations (5), (6), and (8) now provide a coupled system of differential equations for the three unknown functions of time: ql(t), q2(t), and s(t).

616 Appl. Math. Modelling, 1996, Vol. 20, August


2.3. Phase II: Air in penstock; gate open

When all water has drained from the vents and air has entered the penstock, our system of equations needs some modification. The fluxes q1 and q2 retain their previous meaning, and s now measures the distance down the penstock of the water surface (Figure 2[6]). We take s = 0 when the water level is at the base of the vents, so that during this phase s is negative. It is necessary to introduce a fourth variable, pQ(t), the air pressure inside the penstock. We assume that this pressure is uniform throughout the air pocket, since the interior velocity must be much less than that of the air down the vents.

The evolution equation for q, is found by considering a path C, from P to Q as shown in Figure 2(b). Applying equation (2) to this path gives

aLg, + b,q, = f(pO -pe) +gh,(s)

-dpo+; (9)

Now considering the path C, from Q to R shown in Figure 2(b), we find

1

1 -ghQ(s) + & (10)

The equation for s is based on the observation that the volume of air in the penstock is increasing at a rate q2 - q,. Thus we can write

41 -q2

‘= A(s) sin /3 (II)

where A(s) is the area of the water/air interface, and p is the inclination of the penstock to the horizontal. Here we assume the water/air interface to be horizontal. In practice the flow may be turbulent in this region, and quantities such as the area A(s) must be interpreted as averages.

the It remains to find the evolution equation for pn. From adiabatic equations of state it follows that

pa=c$j&M_pli)

where V(s) is the volume and M the mass of air in the penstock. The airspeed uv in the vents is given by Bernoulli’s theorem as

UV =c,cu-‘/2[1- (py/pu)(y~~~‘y]l’~ (12)

and M = uvpQA,. It follows that

jQ= V(s) Pa(Av+ +a -a) (13)

Equations (91, (lo), (ll), and (13) constitute a system of differential equations for the four variables ql, q2, s, and pQ. The auxiliary functions hQ(s), A(s), and V(s) for the height and area of the air/water interface and the air volume can be determined by simple geometry.

2.4 Phase III: Gate closed

The system of equations just derived describes the flow up until the time at which the gate closes. Thereafter the evolution equations are simply (IO), (ll), and (13) with q1 set equal to zero.

3. Turbine model

3.1. Basic model

If the turbine remains connected to the system, its angular speed of rotation 0 is constant, and no evolution equation is required. If the turbine is disconnected and rotates freely its angular momentum equation can be written

d0 I==7 (14)

where I is the moment of inertia of the turbine, and T the torque due to water flow. The torque depends upon both the water flux q2 and 0, so equation (14) is coupled to the other differential equations. The aim of this section is to derive an approximate expression for r in terms of q2 and fl

A simple model of a turbine is provided by Euler’s turbine equation (see e.g., Prandtl* p. 84). This implies that that the torque 7 exerted by the water on the turbine is given by

r=Aq2-Bfiq (15)

where A and B are constants, q is the flow rate of water through the turbine, and 0 its angular speed of rotation. The two constants can be evaluated from two known operating conditions. First, the runaway speed of the turbine is 31.4 radians/set, and at runaway the flow rate is 0.8 X the normal operating flow = 59.4 (Fazalare’; Figure 2>, under which conditions 7 = 0. Second, under normal operating conditions with ft = 17.4, q = 74.4, the power generated P = L+r= 37.3 MW. These two known operating conditions provide a pair of simultaneous equations in A and B which can be solved to yield,

A = 696.0 B = 1320.0 (16)

3.2. Pressure drop

The pressure drop components:

(i) The pressure and

across the turbine consists of two

drop 6pg due to power generation,

(ii) the pressure drop 6p, due to frictional losses. The first is easily estimated. From equation (1.5) it follows that

P=Q,q=fi(Aq’-Bfiq)

whence

Spg=Af2q-B02 (17)



a

b

80, 1

d

e

Figure 3. Plots of q,, q2, and ~1~ for gates closures of 20, 30, and 40 sec. (a, b, c) Turbine connected; (d, e, f, g) turbine free.



The frictional pressure drop is more difficult to estimate, but it seems reasonable to assume the following form:

6pf= ff,cDL’; (18)

where ~2~ is the entry speed of the water relative to the turbine blades and (Y, the angle of incidence of the entry flow relative to the blades. The drag coefficient cn will be small under normal operating conditions at low angles of incidence, but much larger under abnormal conditions such as runaway.

Suppose the water enters the turbine at an angle p to the circumference. Then,

L’E = :sec p

where q is the flow rate into the turbine and A is the cross-sectional area. The entry velocitity relative to the turbine is therefore

( ; cot P-OR, 3

1

where R is the outer radius, and it follows that equation (18) can be written in the form

6P, = q2f( O/q) (19)

where the drag function f depends only on the ratio O/q and can be tabulated from operational data as follows.

The efficiency 77 of the turbine is given by

66 77 = 6p, + 6p, (20)

Using quoted efficiencies for different flow rates, we used equation (19) and (20) to construct a table of the function f. Intermediate values were estimated by linear interpolation (higher order methods gave spurious oscillations since f is a rapidly varying function). To summarize, the total pressure drop Ap, across the turbine is given by

Ap,(q, 0) = a~, + a~, (21)

=AOq-BR2+q2f(12/q) (22)

4. Results

4.1. Results for method I

The calculation of results proceeded in four steps. (8

(ii)

(iiij

Initialization: The fluxes q, and q2 were both set equal to 74.4 m’/sec - the flow rate under normal operating conditions. The water column length s was set to 39.5 m and the turbine rotation rate to 17.4 rad/sec. Phase I: Equations (51, (61, (81, and (14) were integrated forward in time until s became negative, i.e., until the water was fully drained from the tubes. This occurred usually after the about two-thirds closed. Phase II: The system was reinitialized

gate was

using the

(iv)

previously calculated values of q,, q2, and R at the end of phase I, s was set equal to zero, and an initial free surface pressure was calculated. Then equations (9), (101, (111, (13), and (14) were integrated forward in time until the gate was fully closed. Phase III: For the rest of the calculation, q, was set equal to zero. Equations (lo), (111, (13), and (14) were integrated forward in time until air entered the turbine.

For phases I and III, a simple fourth-order Runge-Kutta (RK4) method was used, with a fixed time-step of usually 0.01 sec. Phase II proved more troublesome to integrate, particularly near the point of gate closure when the system of differential equations became “stiff.” During this phase the adaptive routine DO’LBBF was used from the NAG library.

Figure 3 presents results for gate closure when the turbine is operating under normal power - 37.3 MW. Three different gate closure times are considered: (A) 20 set, (B) 30 set, and (C) 40 sec. Figures 3(u), 3(b), and 3(c) give graphs of q,, q2, and uA, the vent air speed, versus time when the turbine is connected to the system. Figures 3(d), 3(e), 3(J) graph the same variables when the turbine is disconnected (or free) and Figure 3(g) plots the turbine rotation rate R versus time. Runaway speed for the turbine is 0 = 31.4 rad/sec, which is quite closely approached in case (Cl.

The main variable of interest, the air pressure inside the penstock, falls only slightly below atmospheric pressure, since the airspeed in the vent is always much less than the speed of sound. By the time the gate is closed, the volumetric water flow has decreased to about 30 m3/sec, and since the combined cross-sectional area of the vents is a little less that 1 m*, the airspeed down the vents is approximately 30 m/set - about one tenth the speed of sound - causing a drop in air pressure of about 1% (equation (7)). This estimate of the air speed is confirmed in the plots of L)*, which show that it never exceeds 47 m/set.

There are two interesting features of the turbine-free results. First q1 and q2 both increase slightly just after the gate begins to close. This is due to spin-up of the turbine which reduces the back pressure on the water column, causing it to accelerate. Eventually the pressure drop under the gate becomes more significant and q, and q2 decrease again. Second, it appears that for the longest gate-closure time, 40 set, the turbine rotates at between 28 and 30 rad/sec over a period of about 20 sec. This is close to the runaway speed of 31.4 rad/sec and could result in damage.

5. Full two-dimensional Navier-Stokes simulation

Here we consider the case when one of the turbines trips out but remains engaged to the system rotating at a constant speed; immediately on tripping out, the gate at the dam end starts to close. We assume that the flow is incompressible and a two-dimensional model is adequate.

Appt. Math. Modelling, 1996, Vol. 20, August 619


i.!!.- IU’UT

Iv.0 u=u(t ) I

I v=!!(t) !V=O

Figure 4. Penstock problem.

Taking the x-axis on the lower penstock wall and the y-axis on the gate wall, the following time-dependent problem depicted in Figure 4 is formulated. The gravitational components are:

g, = 9.81 sin( 14”) = 2.3733 m/set’ and

gY = -9.81 cos(l4”) = -9.5186 m/set*

The outflow velocity, U, is the velocity (assumed constant) through the turbine. It is taken to be 3.96 m/sec2; this is consistent with the volumetric flow of 74.4 m”/sec through the turbine. At the inflow, Ui, is set to UT. We shall denote by T,,,, the time, in seconds, for the gate to shut. Thus, Vgate, the velocity of the closing gate, is taken

to be V,,,, = D/T,,,, m/set. The initial state is obtained by setting u = UT and u = 0 within the penstock and u = u = 0 in the vent. At the penstock and vent walls, free-slip conditions are imposed on the velocity field which often are a far better approximation for narrow pipes.4 Indeed, these last two authors state, “The no-slip condition forces an artificially large drag on the fluid, while the free-slip condition represents no boundary drag, clearly a much better approximation.” By solving the time-dependent Navier-Stokes equations within the penstock and the vent we thereby compute the velocity and pressure fields.

5.1. Method of solution

To solve this problem we employ the GENSMAC” code which solves the two-dimensional time-dependent laminar Navier-Stokes equations for an incompressible viscous fluid. It embodies the SMAC’ (simplified marker-and-cell) methodology which is a numerical solution technique em- ploying pressure and velocity as the primary dependent variables for calculating time-dependent free boundary problems. It is specifically designed to deal with free surface flows within a general domain with free-slip or no-slip rigid boundaries. Multiple inflows and outflows can also be handled. The basic equations are the momentum equations together with the continuity equation which, in the nondimensional form employed, are:

au au z+-=O dY

where Re = UL/v and Fr = U/ fi are the associated Reynolds number and Froude number, respectively. Typi- cal velocity and length scales are denoted by U and L, respectively, and g is the gravitational constant with g = (g,, g,jT the unit gravitational vector; v = (u, u>~ are the nondimensional components of velocity while p is the nondimensional pressure per unit density.

GENSMAC employs a finite difference approach on a staggered grid. An adaptive time stepping technique based on the ideas of Markham and Proctor’ has been used to minimize the number of time steps required to solve explicitly the momentum equations. The correct boundary conditions ensure conservation of vorticity. However, to satisfy the continuity equation, a velocity field update must be made at each time step. This requires the solution of a discrete Poisson equation. GENSMAC has been designed to produce a symmetric positive definite matrix for the discrete Poisson equation for any arbitrarily shaped domain. This allows the use of a robust conjugate gradient solver.#

Particles, created at the inlets, are injected to represent the fluid. These are virtual particles whose coordinates are updated at each time step by solving

dx dy -_=u -_=u dt dt

by Euler’s method. This provides a particle with new coordinates and thus determines whether it moves to a new cell or not or indeed whether it leaves the bulk fluid. GENSMAC also has the facility to cope with splashing. It should be stressed that these particles are virtual and are only for flow visualization. Thus Figure 8 represents a set of “snapshots” of the virtual particles.

A detailed description of the code may be found in Tom& and McKee’ or Tom&’ An extended version to cope with non-Newtonian fluids can be found in Torn& et al.‘O However, none of these papers deal with a closing inlet or gate. A brief description will, therefore, now be given. Consider Figure 5 which displays the gate at time t; time t = 0 is the time when the gate is on the point of starting to

I GATE

PENSTOCK

Figure 5. Position of the gate at time r.


lime-dependent flow in a pencock: S. McKee et al.

Figure 6. Flagging at the penstock entrance a time t.

close. The gate is simulated by moving the y-coordinate of

the upper inflow wall down to zero at a prescribed velocity. In addition to the usual B, F, S, and E-cell? we create G cells (gate cells) (Figure 6). Whether a B cell turns into a G cell depends on the position of Y(r), the lowest point of the gate. If the line y = Y(t) cuts the particular boundary cell above its midpoint then that boundary cell is still deemed to be a boundary cell; otherwise it becomes a gate cell. For inflow boundary cells the boundary conditions are those for prescribed inflow; for the gate cells the boundary conditions imposed are of free slip.

5.2. Problem set-up and computation

The length scale L is chosen to be D while U = Ur is chosen for the velocity scale. The input and output veloci- ties, together with the geometrical set-up, are displayed in Figure 7. Since the viscosity of water is small this gives rise to a large Reynolds number of around lo6 which clearly represents turbulent flow. Since GENSMAC has not been designed to cope with highly turbulent flows, it has been necessary to chose a Reynolds number of 250. Whereas this is admittedly a gross approximation, an extremely fine grid has been employed so that large-scale eddies are clearly discernable. Nonetheless, the results displayed in Figure 8 must be regarded as only of qualita- tive value. The nondimensional gravitational field is then set to (1/Fr2)g, = 0.7385 and (l/Fr*)g, = 2.9544. With these parameters the coordinates of the problem is as given in Figure 7. A mesh size of 8x = 6y = 0.05 is employed, giving 20 X 329 cells within the penstock and 5 X 162 in the vent.

“b.l.0

v-00

010s

Figure 7. Problem set-up.

r

t=xo

Figure 8. Particle and pressure contours for gate closure of 30 set at different times.

The code was run for two sets of data, although only the second is reported on in this article for reasons of space; fuller details, in color, may be obtained from Tome.’ For each data set the code was run from t = 0, when the



initial nondimensional pressure is set to zero, so that the velocity and pressure fields may be computed up until the time when the turbine trips out. The calculation then continues, using these values of velocity and pressure as initial data, until the gate is completely closed. The two data sets differ from each other only in the speed at which

not be able to predict. The combination of the two methods seems able to provide the information required for the study, namely that a pressure drop of less than i atm within the penstock would be unlikely.

the gate closes. The penstock problem is run for a gate closure of 30

sec. Figure 8 displays the particle plots and pressure contours at times f = 10.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 19.0, 22.0, and 26.0 sec. Note that the pictures do not display the whole penstock but focus in on the area of most interest at the vent entry point where the pressure is likely to be and can be seen to be a minimum. The problem was also run for a gate closure time of 20 set; similar results were obtained and can be found in Tome.’

Finally, the pressure in atmospheres was recovered from the relation

P=p,+ PU2

101, 325 ’

where pO is a reference level pressure of 1 atm and p is the computed nondimensional pressure.

6. Comparison of results

The most noticeable difference in the results of the two methods, due to differing assumptions, is that Method I predicts flow rates decreasing fairly steadily and more rapidly than Method II, to about half the initial value by the time the air enters the turbine. Thus one would expect everything to happen rather more quickly according to Method Il. One parameter that can be readily compared is the time taken for water to drain from the vent. Method I gives 13 and 19 set for gate-closure times of 20 and 30 set, respectively, while the corresponding figures from Method II are approximately 10 and 12 sec.

Both methods predict that the pressure at the interior free surface decreases only slightly below 1 atm. Excep- tionally, in the 20 set gate-closure time calculation, a region of reduced pressure - about 0.6 atm - occurs just behind the vent.’ However, this appears localized and transitory and may not represent a serious danger.

7. Conclusions

The novel feature of this work is the use of two different approximate methods to study a complicated problem. The first method, based on simple but robust arguments of mass and momentum conservation, could be expected to predict global variables such as flow rates with reasonable accuracy. The second method depends upon the volumetric flow through the turbine being constant and the flow in the penstock not being turbulent. It nevertheless provides information on the detailed structure of both the flow and the pressure. In particular Method II has highlighted a local region of dangerously low pressure which Method I would

Appendix: Expressions for aL and bL

We assume that the flow in the lake is irrotational and incompressible, so that the water velocity v can be expressed in terms of a potential 4 which satisfies Laplace’s equation:

v = 04 v+p=o

We approximate the gate opening by an elliptical hole

x2 2 7+-=l a b=

in an infinite plane wall (the wall of the dam). The major axis and minor axis of the ellipse are chosen so that the area of the hole and its aspect ratio are the same as for the actual rectangle of width 2L, and length 2L, say. Thus we choose

The velocity potential can be found by adapting the formula for electric potential around a charged conducting ellipsoid,” giving

91 - /

ds P=G

5 Js(s+a2)(s+b2)

Here 5 is an ellipsoidal coordinate which is zero on the elliptical hole and tends to infinity at large distances from the hole.

If C is a path beginning at a point X on the lake surface far away from the dam, and ending at a point Y in the gate opening, then

ds

s(s+a=)(s+b*) (23)

We take L, to be the fixed width of the gate and L, the decreasing height, so that

where V,,,, is the speed of gate closure. Now equation (23) gives

/

JV

- . dx = &aL + qlb, c at



where

1 =

J ’

ds a!_ = -

877 0 \ils(s+a2)(s+b*)

BV = ld

ds b, = -

4T’ (’ s(.F+a2)(.s+b2)’

These integrals are elliptic integrals, which can be calculated using subroutines from the NAG library.

Acknowledgments

We gratefully acknowledge support from Electricorp Pro- duction, and many useful discussions with Mr. R. Munn. We also would like to acknowledge the financial support provided by the Brazilian funding agencies: CNPq - Conselho National de Desenvolvimento Cientifico and FAPESP - Funda@o de a Amparo a Pesquisa do Estado de S5o Paulo.

Nomenclature

D h d

1, VT v

! V w

P P g z ds

c, 41(t) G(t) PO PT H

diameter of the penstock height of the column of water diameter of the vent length of the penstock volumetric flow through the turbine kinematic viscosity of water density of water slope of the penstock distance between the vent and the gate water velocity vector vorticity density pressure gravitational acceleration vertical coordinate small segment of a streamline a streamline flux of water through the gate flux of water through the turbine gate head loss turbine head loss total head

1, *rJ Re

QL

bL

a,, bv

PO

s

n

Fr 7

4

“pa 6Pf VE ffI CD R

ZPT v, V gate

length of the penstock cross-sectional area of the penstock Reynolds number lake-flow coefficients lake-flow coefficients

atmospheric pressure length of the water column in the vent angular speed of rotation Froude number torque flow rate pressure drop due to power generation pressure drop due to frictional losses entry speed of the water angle of incidence drag coefficient outer radius efficiency total pressure drop the vent air speed velocity of the closing gate

References

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Miller, D. S. Infernal Flow Systems, 2nd edition. BHRA (information Services), 1990 Prandtl, L. Essentials of Fluid Dynamics, Blackie and Son, 1949 Fazalare, R. W. Five technical recommendations for hydro machin- ery. Water Power Dam Const., 1991 Nichols, B. D. and Hirt, C. W. Improved free surface boundary conditions for numerical incompressible flow calculations. /. Com- put. Phys. 1971, 8, 434 Tome, M. F. and McKee, S. GENSMAC: A computational marker and cell method for free surface flows in general domains. .I. Comput. Phys. 1994, 110(l), 171 Amsden, A. A. and Harlow, F. H. The SMAC method: A numerical technique for calculating incompressible fluid flows. Los Alamos Scientific Lab. Rept. LA-4370, Los Alamos, NM, USA, 1970 Markham, G. and Proctor, M. V. Modifications to the two-dimensional incompressible fluid flow code Zuni to provide enhanced performance. C.E.G.B. Rept. TPRD/L/0063/M82, 1983 Ortega, J. M. Introduction to Parallel and Vector Solution of Linear Systems, Plenum Press, New York, NY, USA, 1988 Tome, M. F. GENSMAC: A multiple free surface fluid flow solver. Ph.D. Thesis, University of Strathclyde, Glasgow, 1993 Tome, M. F., Duffy, B. R., and McKee, S. A numerical technique for solving unsteady non-Newtonian free surface flows. J. Non-Newto- nian Fluid Mech., 1996, 62, 9-34 Landau, L. D. and Lifshitz, E. M. Electrodynamics of Continuous Media. Pergamon Press, 1960


Time-dependent flow in a penstock during head-gate closure

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