G.S. Stelling and L.-X. Wang
Report No. 2-84
Laboratory of Fluid Mechanics
Department of Civil Engineering
Delft University of Technology
8.5. Stelling*> and L.-X. Wang**)
Reoort No. 2 - 84
Laboratorv of Fluid Mechanics Department of Civil Engineering Delft University of Technology Delft, The Netherlands
*) Department of Transport and Public Works~ Data Processing Division~ The Netherlands
**) Water Conservancy and Hydroelectric Power Research Institute, Beijing, China
1. lntroduc:tion
2. Experiments
2.1 Experimental set-up 2.2 Measuring equipment
2.3 Measurements of depth-averaged veloc:ity and water level 2.4 Experimental result.
3. Numeric:al simulation 3.1 The numaric:al model 3.2 Numeric:al experiment.
4. Comparison of c:omputations with ..asurements and c:onc:lusions
Ac:knowledgements Referenc:es
Tables Figures
investigated with a physical model and a numerical model. To simulate tidal flow, the prescribed flow rate at the inflow boundary was a half-period sine function of time. Measurements of velocities and wave heights were conducted by using LDV and a wave-height meter in a straight open channel with a sudden
widening and a non-reflective outlet. Distributions of depth-averaged velocities at different times were obtained. Numerical simulation of this flow was carried out using a computer
programme of second-order accuracy. The development of the recirculation region in course of time was weIl simulated, and the observed splitting of the main eddy in a later phase of the experiment and the secondary eddy in the concave corner were
reproduced. The computational results are markedly influenced by the boundarv conditions at closed boundaries and the eddy
width of the expanding flume Chézy coefficient
focus distance of l.ns in LDV ~ gravitational acceleration
~,H water depth
~ water depth when the velocity is zero K optical constant of LDV
l~
length of th. recirculation regionQ
dischargeQ~
maximum discharge R hydraulic radiusRe Reynolds number
Sdistane. between two la.er beams
T
half period of the sine-function flow ratet time
uJ~ two component. of depth-averaged velocity in
x-
andy
-direction.u:
U-v..
maximum velocity in the recirculation region turbulent fluctuation of velocity
time-mean velocity
depth-averaged velocity
cartesian coordinates in three directions coordinates of the eddy centre
6X,by grid size
E:- eddy viscosity
À laser light wavelength
tP
angleV kinematic viscosity "'C time step
knowledge of these flows is essential for understanding
hvdrodynamic phenomena in many hydraulic structures. While a lot ot work deals with steady flow, only little on unsteady flow can be referred t~. Because of the difficulty of treatment of the governing equations and the complexity of the boundary
conditions, an analytical solution is not feasible. Experiments in physical models and computations using numeri cal models are resorted to to investigate these problems. Dwing to the rapid development of efficient, high-speed computer hardware, the costly, time-consuming physical models have been gradually replaced by sophisticated numerical models in many
investigations. In numerical modelling, the most frequently
applied prediction method is the computation by means of a finite-difference representation of the depth-averaged momentum
equations and continuity equation. The question is, however, to what extent and how weIl a numerical model can reproduce the hydraulic phenomena in real situations. It must be calibrated and verified with either field measurements or a physical model. lhe work presented here is concerned with measurements in an unsteady separating flow in an experimental flume and
computations for this problem using a numerical model.
A series of experimental investigations on this subject has been done recently in the Laboratory of Fluid Mechanics,
Department of Civil Engineering, Delft University of Technology by T. Koppel (Koppel, 1981) and L.-X. Wang (Wang, 1982). The experlments included flow visualization, float observations,
velocity measurements with LDV and water-level masurements with a
wave-height meter. rhe experiments were conducted in a straight,
open channel with a sudden expansion (widening> and a
non-reflective outlet. To simulate tidal flow, the discharge varied
with time according to a half period sine fuJcion. The same flow
was simulated numerically by means of a programme of second-order
accuracy. In the computation, constant eddy viscosities were
assumed and three types of closed-boundary conditions (i.e.
employed.
The experiment.l re.ult. pr•• ented here concern mainly the flow pattern in the expansion region, .s measured by L.-X. Wang
(Wang, 1982) with the discharge ~ equal to
16 sin (~~/150)ix 10-3 m3/s (time in seconds). A comparison is made between the experimental and computational results obtained.
2. Experiments
2.1 Experimental set-up
fhe experiments were conductad in a horizontal, straight flume with smooth, transparent side walls and bottom. The 0,4 m wide flume widened suddenly at the cross-section 10 m downstream from the inlet to a width of 0,8 m. Scr •• ns were placed at the entrance, and sand <median diameter 0,7 mm) was glued on the first two metres of the bottom to provide a uniform, fully-developed turbulent flow. The uniformity of flow was confirmed by velocity-profile measurem.nts near the expansion(9 m
downstream from the inlet). In order to diminish reflexion of long waves, a distributed-inflow (discharge Q\) system and a zig-zag weir (total length 2,8 m) were built at the end of the flume. When QI
=
0.,016m3/s and weir heighta.
= 0,07 m.,the water depthho
in the flume was 0.096 m, while the flow velocity in the flume was zero. Wh en the inflow starts, a .mall-amplitude long wave is generated and the related discharge will pass over the zig-zag weir with little or no reflexion. Th. experimental set-up is schematically shown in fig. 2.1.The control system of the water supply consists of a data-track progammer (R.I. Controls U.S.A.), an electromagnetic flow meter <Foxboro mode 696) and a motor-driven valve. The flow was measured automatically by the electromagnetic flow meter and the signal was fed back to the data-track programmer, which
controlled the electromotor driving the valv. according to a prescribed time-function. In this experiment, the maximum water velocity in the flume was about 0,34 mIs and the maximum water depth about 0,12 m. So the maximum Reynolds number was
-
-2.2 MeasurinQ eguipment
wave-here), the velocity measurements with a Laser-Doppler
Velocitymeter (LDV) were part of the experiment. In recent years LDV has been frequantly used in experimental fluid mechanics. Several types of LDV systems have been developed. The choice of LDV type depends on the type of measurements. In this
experiment, a forward scatter orthogonal two-component reference mode arrangement (TPD, type 400) was used. A block-scheme of the LDV measurements is given in fig. 2.2. Measurements were done with the direct configuration, see fig. 2.3. The signals
(frequency) from the measuring point were received by two photodetectors, converted into voltage by a frequency tracker
(TYPE 1077/2M), passed filters (low-pass filter, model 452 Dual Hl/LO, cutoff frequency
=
25 Hz), amplified by a preamplifier built-in in the Data Acquisition System (DAS), and recorded on the paper of a HP recorder (7402A).Parameters of the optical system are as follows: focus distance of the front lens
distance between two laser beams laser light wave length
angle between laser bearns in air optical constant of the system
F = 242.5 mm s
=
29.6 mm~= 632.8 mm
f=
6.9850=
192.5 kHz/ms-1-rhe tracker constant wat» 1 V/200 kHz. Wi th the parameters mentioned above, the following rel_tion.hip between velocity and displacement on recording paper holds:
10.39 cm/s corresponds to 1 cm.
In turbulent flow the valocity can be divided into two parts:
u.(-t)
= ~(.~)
+ ti(-t)where is the mean velocity, a periodic function of time in
this case, and ~ is the turbulent fluctuation, a random
quantity. In this experiment only the mean velocity u..Ct) was read from the recording paper. Fig. 2.4 shows an example of the recorded veloeities.
A small cup with a transparent bottom was fixed at the water surface to avoid deflection of the laser beam by the fluctuating free surf ace. The cup had som influence on the velocity
distribution near the free surface. This imposes some
restrictions on the velocity measurements close to the free surf ace.
2.3 Measurements of depth-averaged velocity and water level
fhe measurements were conducted with the discharge given by
with Q
....
= 0.016 m3/s and T= 75 s. The two velocity-componentsin
x-
and Y-directions were measured at 145 verticals in theseparation region continuously. Fig. 2.5 gives the scheme of the
measuring points, in which the origin of the coordinates is at
the bottom in the expansion section, the ~-axis is in the flow
direction, the y-axis in the transverse direction and the ~-axis
vertically upwards. When an experiment commenced, a pulse was
created to mark (on the recording paper) the start of the flow in
section O( X
=
-32.5 cm). The values of veloeities were read fromthe recording paper at time ~= 5,15,25,35,45,55,65 and 75 s.
The meauring points are identified in the following way.
For example, B2 is the second vertical in section B( ~= 12.5 cm,
y
=
70 cm). At each point, the velocity was measured twice and themean value was taken.
Before the depth-averaged velocity measurements, velocity
profiles were taken at several verticals in three sections
(section 0, Band K) with q....
=
0.016 m3/s and T=
90 s. Theseprofiles are depicted in fig. 2.6,2.7 and 2.8. Using these
profiles, two measuring points ('r. = 1.6 cm and 7.6 cm) were
selected for the depth-averaged velocity measurements, since for
the velocity profiles shown,
where
~(t)
=
the depth-averaged velocity calculatad according toafterwards indicate that the two points selected are fairly
representative at verticals where the velocity profile has a more or less logarithmic farm. At a few verticals in the central area of the recirculation region, the velocities near bottom and free surface were in opposite directions, and the values of
3(
U'.b +1.(."1-.1.) are close to zero. In fact, this is the case in the centre
of a recirculation regian.
The water depths were measured using a wave-height meter at
16 verticals in two cross section
(e
and H) under the same flowconditions as the velocity measurements. Before the flow
started, the water depth ho was 0.0096 m. The measured maximum
wave height was about 0.02 m.
2.4 Experimental results
The two components of velocity \:(_(;t;) and
v
Ct) and the anglesbetween the velocity-vectors and the x-axis are listed in Table
1. The flow patterns are plotted according to the results of the
depth-averaged velocity measurements in fig. 2.9. The variations
in water depths at two cross-sections are depicted in fig. 2.12.
It is believed that the characteristics of the unsteady
separating flow (the development of eddies, the splitting of the
main eddy, the meandering of the main stream, etc.) depend on
boundary conditions as weIl as flow conditions, such as the
magnitude of velocity, acceleration and deceleration, the period,
etc. It is insufficient to draw general conclusions from
measurements with one or two sets of flow conditions. The
results presented here, however, can provide a test case for a
numerical investigation as weIl as a few conclusions about
unsteady, separating flow.
(1) A separating eddy occurs immediately behind the protruding
corner at the very beginning of the flow.
In the acceleration phase, the separating eddy
(recirculation region) develops with time. lts size is
increasing while the eddy centre is moving downstream, see
fig. 2.9-a,b,c and d. The eddy still grows in the
experiment> the separating length
l~=
(6 to 8)~ (Abbott and Kline, 1962). In fig. 2.9-h,L~
is at least 2.65 m(= 6.6 B).
During acceleration the main stream is driving the flow in the separation region. In the decel.rating phase the main stream is suppressed by the widening separation region, and a slight meandering of the main stream was observed. A
small reverse velocity was measured near the straight wall in section L, see fig. 2.9-c and h.
(2) In the decelerating phase, the main eddy seems to split into two parts, see fig. 2.9-e,f,g and h. Bath splitted eddies grow simultaneously and move downstream and transversely towards the main flow as weIl. This can also be recognized from Koppel's measurements (Koppel, 1981). Fig. 2.11 is a copy of a flow pattern taken from Koppel's report (flow conditions: Q~= 0.016 m3/s, T= 90 s, t
=
60 s). Fig. 2.10 is a copy of fig. 2.9-f.The splitting of the main eddy in two-dimensional
accelerating, separating flow was observed by other workers, experimentally and numerically (Macagno and Hung, 1970). In their experiments, the first eddy is swept away by the main flow, and the splitted eddy (the third eddy, as it was
named) grows slowly and turns into a captive and steady eddy.
(3) The fluid in the concave corner almost remains
stagnant,except for a secondary circulation existing during a certain time. The valocity in this secondary circulation is at most ten percent of that in the main flow at the same
instant.
(4) In a cross-section the water surface deviates from the horizontal, especially in the accelerating phase. A
relatively steep slope of the water surface appears in the mixing layer. At section C the slope between vertical CS and
0.04 t
=
25(section K), the mixing layer gets wider, a large transverse variation in water level occurs between verticals H3 and H8.
In the decelerating phase the water surface seems smoother than in the accelerating phase.
(5) The prescribed discharge-time relationship (a sine function) holds true at the inlet of the flume. Dwing to the
deformation during propagation of the long wave, neither velocity nor wave height is a simple sine function at the expansion. These variables can be expressed in the form of Fourier series.
According to the measured data, the following expressions were obtained:
for the velocity component in x-direction in section 0 (upstream)
3
\':L(t)
=
u.
0 +2:.
Llk S'VI. kt...>tk~1
for the water level in section K (downstream)
h(.t)~ ho when 0.(. 1:- ~ to
'3
hLt)e, ~o+-
'f;1
~k. S~\I\ Icc..v(t--t"o) when 1:0 '- -t: ~ Twhere 0
=
211'/2T; T=
75 s;U,1
=
0.05 mis,hl.
=
0.001 m,(.(.3
=
0.01 mis;1,10
=
0.0005 m;LLO
=
0,u-,
=
0.375 mis,ho
=
0.1 m, L1.=
0.021 m,to
=
L/(~L.o'/'l..; L=
the distancebetween sections 0 and K. The above expression5are to be
used in the subsequent numerical modelling as the condition5
at the open boundaries.
In this experiment, measurements of turbulence
quantities were not carried out. To meet the need for the
eddy viscosity in the numerical simulations, eddy
viscosities were determined from Koppel's measurements
(Koppel, 1981>, using the following formula,
- (.,(..'v' -vl.: v'
e
::. -::::::."Di.4: )\:1
,u--
+The calculated 6 values in the miMing layer, for Q~= 0.016 m3/s and T= 90 s, are listed in Table 2. E varies in time and space and it is, in this cAse, in the range of
3.1 The numerical model
In this chapter the numerical model that has been used will be described briefly. A detailed description is given by
Stell ing (1983). The numerical model is based upon the well
-known shallow water equations,
(3.1-1a)
(3.1-1b)
(3.1-1c)
where:
7
=
waterlevel with respect to a horizontal reference planeH
=
total waterdepth,H
=
d.+-
5
For the numerical approximation of (3.1_1) first a fully
staggered grid is defined, see fig. 3.1.
n+-I
v
v
v
h
1-11+1
fig. 3.1, Spatial grid
For the inner points of the grid given by fig. 3.1, i.e. for those points that are not influenced by an open or a closed
boundaryof the numerical model, (3.1-1) is approximated by an
ADI type finite-difference scheme. Each derivative of (3.1-1) is
discretized by central differences , except for u.y in (3.1-1a) and
~ in (3.1-1b). These terms are approximated by third-order
upwind differencing. For the numerical integration in time a
non-standard ADI/predictor-corrector approximation has been
Stage 1:
L ,/ I I-~I -Ic 'J( (
=
k+'L k )7k.t-
'
h.
( " ...... 'Io_" "'".. )/-1..-r-...
+
u....
,
\A......"
0,11.. + ..oy v , u,
+-
1
1»)( (3.1-2a) ForF
=
1,2 : --y (VCl>l_v"-)
/
\.
r
+-
s.,
l
~I<., V[t>l,'b(p
tf)1
+V
kV;;l
+-
1
~~y I(1'][(
=")'1..
(
k.)2.1"'i.jCAtHk)_
é:
(
l*]
[I"J)_
+-
'3
Vu..
+-
V j L.. Vox/(+
v oyy - Dat t"tfl
r1+-1
(3.1-2b)(3.1-2c)
Stage 2:
For ~
=
1,2 :kJ.. /
bitf:lx
cl[::::\..t-t
Cr1 ](1.4.'(>]_1.<...
t- ....)
11: +-I.A.-
LtOA +-- Y+y V,u..
,2)(~t-f")+--+-tiJ u.tl'l[(~f,..d.)'1.-+-/l.c..k. ~..\ ..
l
y
\.
·
.,'/"L.
/
(è\tk.,"~\
)_
G(_
u.CP]
OT-X + lA..Clt-,Jf-]
oyy ) :::.0I
at Io\-It·ï.. ,~ (3.1-2d)
at t\1J IH ~ (3.1-2e)
where Hl<.
C
~ ~)~
+- ~
"-
cL
o{_ h- ~) at 1';\+ 1.\ ,''' denotes 1.1...,
....,)
'"
+-
+-~)h rYt,'"
I I(
'
L
k.H
at t\I1 IH-z.. denotesL
S ~,~
+-7
rn, 1'\+/ .j- oL"", +- oL,
VI)
) , \11 1'11-ï.. ' =k I I(l{.k..
I:.. L Ic. ) u, at ....,)~Hi. denotes y +- 1--1.. +- \,.t. +k.. It.\ - 11. ' .. Wt+i_, ... )l1+\.,~+1 ~-~I ~+, =1.. V is defined according to at 1\1\)'" denotes-
y
V at is defined according to at denotes Voy• Y ,Y etc.7o~ 7o'f are defined according to ~o~
denotes
~ (CI'J;\ I
v+lt14.// ,i;at I')t) h+t.. denot ..
and ~+y are defined according to ~oy and
'?H'
"
r
is defined according toI
at ~}h+1_ denotes
denotes the sum of ~ over all grid points)
and C~~]
Lloy'l at
KoI+-\
,
I') denotesIn order to represent a unique solution, (3.1-1) has to be completed with boundary condition ••
At closed boundaries the following boundary conditions are prescribed:
(3.1-3a)
if 6
t
0 (3.1-3b)where u.L denotes the vel oc ity normal to the boundary, t(../I
denotes the velocity parallel to the boundary,
o
l
Dn
denotes the derivate normal to the boundary, b a length related to thedistance between the closad boundary and the nearest grid point, and ~ denotes a slip parameter.
At open boundaries a variety of boundary conditions can be prescribed, a detailed description including the numerical
consequences of the open boundary conditions and of (3.1-3a) is given by Stelling (1983).
Since (3.1-3b) plays an important part in the numerical experiments of this work, we will consider this boundary
condition in more detail. (3.1-3b) represents a weighted average of a no-slip and a perfect-slip boundary condition, i.e. if ~
=
0 then (3.1-3b) represents a no-slip boundary condition, whereas if~= 1,then (3.1-3b) represents a perfect-slip boundary condition. In general ~
=
1, i.e. perfect-slip, because for many application the grid size is much too large to represent boundary layers that occur in case of no-slip boundary conditions. Numerically,(3.1-3b) only affects the approximation of 01.u.,///'P":-nearclosed boundaries, i.e. other terms of (3.1-1) are treated irrespective of (3. 1-3b) •
_ ').'- / 1.
~h·1 ~ t'I +-
i.
v
~'7
h- l1. -V h - I ;-tMv
v'.._. point under consideration
0)
~
~
~
---_1/
1/--",-...closed boundary~
5
~
~
" virtual point
Fig.3.2 Boundary treatment for second-order derivative.
The boundary condition (3.1-3b) influences the approximation of
u.yy
at the point under consideration.At inner poi nt s IA.yy is approx imated by
'LlA... +
"'" +
1.
,1'\ (3.1-4)For the situation of fig. 3.2, however, U.1\1+-L 11-1 is located at a
" I
missing gridpoint. Let us assume that this point contains a
"virtual" value for which the following relation holds:
LA. ~ •• L I
..-..,..l.I1'\- ï.
(3.1-5)
Let us assume the slip parameter~, also see figure 3.3 as
follows:
lA. , I
~+1.'\'\-ï. (3.1-6)
From this relation it follows thAt if eX.
=
1 (3.1-6) represents aperfect-slip boundary condition, whereas if
0::::
=
0 (3.1-6) is ano-slip boundary condition.
Substituting (3.1-6) into (3.1-5) yields:
IA. I
1-1.1.+ 1...' 11-\ (3.1-7)
Near closed boundaries (3.1-7) is substituted into (3.1-4). It is
(3.1-3b) with ho= :t
A_f (or
th;). which is indeed the distance of the nearest gridpoint to the closed boundary.y
"
n-.!.1. rl-IFig. 3.3 Values of l.t.
11\1+.1.... ,,.,-1
2.2 Numerical experiments
In this section a few numeri cal .xperiments are described to illustrate the effect of the eddy viscosity
e
and the effect of the sI ip parameter 0(.For the purpose of the numeri cal calculations, the flume as
described is covered by a grid with a gridsize 6X= 4y
=
0.025 m. A timestep,T, has been chosen such that T= 0.125 s.The initial conditions are given by:
u.(o)
=
0,vè)
=
0,?~)
=
0The boundary conditions are given by:
1> at X
=
0:where
=
j(.-t)
=
0, 0 ~t
~ 5 s .3~ Ct)
=
r
Ih' S~iI1 co: [4:-S-)i::.
1d-
J
I=
0.021,hl.
=
0.001 andh ""'
0.0005 m. ")The depth hoiS 0.1 mand the Chézy coefficient is 62.64 m1/2ts.
were tested.
With these boundary conditions sevaral numerical values for t and ~
The results are illustrated by figures 3.4 to 3.8.
The numerical values for
e
and~ are given by table 3.1.Table 3.1 Computation number Figure €r (m2/s) 1 3.4 a-d 2.3)(10-4 1 2 3 3.6 a-d 3.5 a-d 2.3Xl0-4
o
o
CommentThis example has a perfect
slip boundary condition. The
growth of a time-dependent
eddy is demonstrated. Only
one eddy develops. The flow
in bac~ward direction follows the rigid walIs.
This example has a no-slip
boundary condition. Here the
emergence of several eddies is shown. Despite the complicated flow patterns numerically
stability was maintained. When compared with
measurements the development of secondary eddies is too fast.
The increased viscosity
4
3.8 a-h 2.3xl0-4
3.7 a~ 2.3xl0-4 0.75
5 0.9
secondary eddies and changes the flow patterns completely. By changing the slip condition at the rigid walls the
emergence of secondary eddies is delayed, which improves the agreement with measurements. By increasing ~ the emergence of secondary eddies is delayed further.
The calculations show the i.portance of eddy viscosity and also the importance of the slip parameter at closed boundaries.
lhe next chapter gives a more detailed study of the numerical results of calculations 4 and 5 through a comparison with measurements.
The results of computations no. 4 and no. 5 (with low eddy viscosity and partial slip condition at the closed boundaries) are compared here with the results of the measurements. The
length of the recirculation region
(ls)'
the eddy centre (Xc.'Yc.) and the maximum velocity (Ve~) in the recirculation region are estimated from measurements and computations, and listed inTables 3, 4 and 5. Comparing the computational results, presented in figs. 3.4 through 3.8, wth the experimental results, presented in fig. 2.9, we can draw the following conclusions:
(1), The development of eddies in an unsteady separating flow with a free surface is fairly weIl simulated by the
numerical model. However, the computational results are much influenced by the conditions specified at the closed boundaries and the applied parameters. As far as the depth-averaged veloeities are concerned, the reproduction of the flow pattern in the numerical model is quite good in the early ph ase of the experiment, but the deviations from the measurements are getting larger in a later ph ase of the experiments, see fig. 4.1-a and b.
(2), From Table 3 it is seen that at the successive instants the computed lengths of the recirculation region is quite close to the measured one. Thi& means that the present numeri cal model could satisfactory predict the development of the separation area.
(3), From Table 5, the computed values of velocities in the recirculation region are seen to be in the same range as those measured. However, th~ listed v~ values may occur at different locations in numerical and physical modeis.
(4), In computation no. 5, the second eddy seems to be the
secondary eddy shedding from the protruding corner, instead of the observed upstream part of the splitted main eddy, see fig. 3.8. In computation no.4 (fig. 3.7), the main eddy tends to split at t
=
45 s. If the no-slip (computation no. 2) or partial-slip (with weak slip) condition is employed atthe closed boundaries9 the splitting of tha main eddy can be
reproduced numerically.
(5)7 The computational results are sensitive to the boundary condition at the closed boundaries (perfect slip versus no-slip). When the amount of slip applied along the side walls is decreased, the main stream becomes more meandering, the recirculation region develops more slowly, the final value of the recirculation length becomes less and the shape of eddy becomes less 'rectangular'.
(6), The results are also sensitive to the eddy viscosity
assumed. Although the specified values of the viscosity é are in
the range of physical values (se. Table 2)9 the differences
between computations with different values of ç are quite
noticeable (see figs. 3.5 and 3.6). lt seem. that the value of é
=
0.001 m2/s is too large9 but that6
=
0.00023 m2/s is toasmalle It is suggested that the eddy viscosity applied in the
computations should be varying with time and location, as in the
physical situation. Prescribing the dependence of eddy viscosity
on location would be cumbersome9 however.
(7), Because of (5) and (6), the computed flow pattern cannot be
expected to agree closely with the measurements. Only very
simple formulations were u.ad in the computation for the
eddy viscosity and for the closed-boundary condition. An
appropriate turbulence model would b. required to improve
Wang~ L.-X. wants to express his gratitude to Prof. M. de Vries for the opportunity to stay in the Laboratory of Fluid Mechanics~ Department of Civil Engineering~ Delft University of Technology. Wang is grateful to Prof. J.P.Th. Kalkwijk and Dr.
c.
Kranenburg for the interesting subject and many instructivedi scussi ons. Many thanks are given to Mr. L.E.A. Calle~ Miss H.
Klaasman and Mrs. T. Capel, for their help concerning
References
Abbott, D.E. and Kline, s.J.,
Experimental inve.tigation of Bubsonic turbulent flow over single and double backNard facing steps.
J. Basic EngineerinQ, Trans. ASME, Sept. 1962, p. 317. Boutier, A.,
Optical systems in Laser Anemo.etry,
Lectures series 1981-3, ven Karman Institute for Fluid Dynamics.
Chow, Ven Te,
Open-channel Hydraulics, 1959, McSraw-Hill. George H. Lean and T. John Weare,
Modelling two-dimensional circulating Flow.
Proc. ASCE, J. Hydr. Div., vol. 105, HY1, Jan. 1979, p. 17. Koppel, T.,
Experiment. on unst •• dy separating flow in an open channel. Internal Report no. 3-81, Lab. of Fluid Mechanics,
Department of Civil Engineering, Delft University of Technol ogy, 1981.
Metha, P.R.,
Flow characteristics in tNo-dimensional expansion.
J. Hydr. Div., Proc. ASCE, vol. 105, HY5, 1979, p. ~01. H-J. Pfei fer,
Introduction to LDA signal analysis and signal Processing. Lectures Series 1981-3, von Karman Institute for Fluid Dynamics.
Stelling, G.S.,
On the construction of computational methods for shallow water flow problems.
Ph.D. Thesis, Delft University of Technology, 1983. TPD, Beschrijving van de Laser Doppier Snelheidsmeter,
Rapport, 1977. Wang, L.-X.,
Experiments on Unsteady Separating Flow with a Free Surface. Internal Report no. 7-82, Lab. of Fluid Mechanic, Department of Civil Engineering, Delft University of Technology.
Macagno, E.O. and Hung, T-K.,
Computational study of accelerated flaN in a two-dimensional conduit expansion,
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0.4 m ~ 0.03 0.2 0.4 0.63 0.9 1.15 1.65 2.65 ) 10 2.0 1.4 20 2.1 2.4 3.4 30 1.4 1.4 5.4 1.3 5.0 40 1.6 1.9 4.1 9.3 6.8 6.0 3.0 2.0 50 1.3 1.9 2.8 4.9 9.1 5.9 2.6 60 1.2 1.4 3.1 4.4 4.4 7.5 5.0 1.7 70 0.8 1.4 2.5 2.7 2.6 2.5 4.3 1.3 80 1.4 2.0 2.0 1.2Table 3. Langth of r.circulation ragian,
Ls)
estimatad from me••ur...nts and ca.putations
case comput.tion L,,<m) no. ti.e (5) 15 25 35 45 55 65 measurement 0.7 1.4 1.9 2.3 2.4 2.6 5 0.67 1.27 1.85 2.22 2.50 2.57 comput.ticn 4 0.71 1.33 1.72 1.92 2.10 2.12
case Computation ~ no. 15 25 35 45 55 65 Co ot. measurement lst Itddy')Cc 0.33 0.58 0.96 1.38 1.88 1.90
'/c.
0.58 0.58 0.58 0.58 0.58 0.56 2nd eddy Xc. 0.53 0.68 0.75 Yc 0.50 0.48 0.45 computation 5 1st eddy X, 0.36 0.72 0.91 1.17 1.17 1.27y,
0.59 0.61 0.59 0.57 0.54 0.49 2nd eddy X, 0.15 0.17y,
0.47 0.47 4 lst Midy Xc 0.36 0.66 0.96 1.15 1.25 1.47'I,
0.59 0.67 0.61 0.58 0.56 0.54 2nd eddyX,
0.50--
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0.45--
0.50computation~ (s) no.
I
V(m~~5 case measurement 45 55 65 25 35v~*)
0.17 0.24 0.22 0.15 0.15 0.16 Vlt\atpoint A8 (X=2.5 cm. 0.29 0.39 0.37 0.33 0.26 0.12 Y-27.5 cm) computation 0.11 0.24 0.25 0.28 0.26 0.24 VtM at{_x
-1. 2~ cmt
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-28.7'5cm 0.3 0.39 0.37 0.35 0.25 0.17 4 0.11 0.23 0.25 0.25 0.2'5 0.24 V"'" at { X -1.25 c. 0.3Y
-28. 75c. 0.4 0.37 0.35 0.2'5 0.17*) rhe measured value. of Vecllisted here ilre not the actual
maximum veloeities in the recirculation region, because the
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