Gaussian Peaks and Clusters of Galaxies

a r X i v :a s t r o -p h /9806074v 2 2 A u g 1998

Draft version February 1,2008

Preprint typeset using L A T E X style emulateapj v.04/03/99

GAUSSIAN PEAKS AND CLUSTERS OF GALAXIES

Renyue Cen 1

Draft version February 1,2008

ABSTRACT

We develop and test a method to compute mass and auto-correlation functions of rich clusters of galaxies from linear density ?uctuations,based on the formalism of Gaussian peaks (Bardeen et al.1986).The essential,new ingredient in the current approach is a simultaneous and unique ?xture of the size of the smoothing window for the density ?eld,r f ,and the critical height for collapse of a density peak,δc ,for a given cluster mass (enclosed within the sphere of a given radius rather than the virial radius,which is hard to measure observationally).Of these two parameters,r f depends on both the mass of the cluster in question and ?,whereas δc is a function of only ?and Λ.These two parameters have formerly been treated as adjustable and approximate parameters.Thus,for the ?rst time,the Gaussian Peak Method (GPM)becomes unambiguous,and more importantly,accurate,as is shown here.

We apply this method to constrain all variants of the Gaussian cold dark matter (CDM)cosmological model using the observed abundance of local rich clusters of galaxies and the microwave background temperature ?uctuations observed by COBE.The combined constraint ?xes the power spectrum of any model to ～10%accuracy in both the shape and overall amplitude.To set the context for analyzing CDM models,we choose six representative models of current interest,including an ?0=1tilted cold dark matter model,a mixed hot and cold dark matter model with 20%mass in neutrinos,two lower density open models with ?0=0.25and ?0=0.40,and two lower density ?at models with (?0=0.25,Λ0=0.75)and (?0=0.40,Λ0=0.60).This suite of CDM models should bracket any CDM model that is currently viable.The parameters of all these models are also consistent with a set of other constraints,including the Hubble constant,the age of the universe and the light-element nucleosynthesis with ?b chosen to maximize the viability of each model with respect to the observed gas fraction in X-ray clusters.Subject headings:Cosmology:large-scale structure of Universe –cosmology:theory –galaxies:

clustering –galaxies:formation –numerical method

1.INTRODUCTION

Clusters of galaxies are cosmologically important be-cause they contain vitally important information on scales from a few megaparsecs to several hundred megaparsecs,and provide fossil evidence for some of the basic cosmolog-ical parameters (Richstone,Loeb,&Turner 1992;Bahcall,Fan,&Cen 1997).The fact that they are among the most luminous objects in the universe renders them an e?ec-tive and economical tracer of the large-scale structure,not only of the local universe (Bahcall 1988)but also of the universe at moderate-to-high redshift.

Perhaps more interesting is the fact that clusters of galaxies are intrinsically rare with typical separations of ～50h ?1Mpc at z ～0and seemingly rarer at high red-shift (Luppino &Gioia 1995;Carlberg et al.1996;Post-man et al.1996).Their rarity is traceable to the fact that they only form at the rare,high peaks in the initial den-sity ?eld.Since the mass in a sphere of radius 10h ?1Mpc roughly corresponds to the mass of a rich cluster like the Coma cluster,the abundance of clusters of galaxies (i.e.,the cluster mass function,Bahcall &Cen 1992)provides a sensitive test of the amplitude of the density ?uctua-tions on that scale and places one of the most stringent constraints on cosmological models to date (Peebles,Daly,&Juszkiewicz 1989;Henry &Arnaud 1991;Bahcall &Cen 1992;Oukbir &Blanchard 1992;Bartlett &Silk 1993;The spatial distribution of clusters of galaxies provides complementary information for cosmological models.A widely used statistic for clusters of galaxies is the two-point auto-correlation function.Earlier pioneering work (Bahcall &Soneira 1983;Klypin &Kopylov 1983)has met with dramatic improvements in recent years thanks to larger and/or new cluster samples that have become available (Postman,Huchra,&Geller 1992;Nichol et al.1992;Dalton et al.1992,1994;Romer et al.1994;Croft et al.1997).Comparing with cosmological models clearly show that the two-point correlation function of clusters of galaxies provides a strong test on cosmological models on scales from several tens to several hundred megaparsecs (Bardeen,Bond,&Efstathiou 1987;Bahcall &Cen 1992;Mann,Heavens,&Peacock 1993;Holtzman &Primack 1993;Croft &Efstathiou 1994;Borgani et al.1995).In addition,the recent studies of superclusters and super-voids by Einasto et al.(1997a,b,c)show a very intriguing property that the correlation function of rich clusters ap-pears to be oscillatory on large scales.If con?rmed,this would challenge most models.

Hence,the combination of cluster mass and correla-tion functions provides a critical constraint on cosmolog-ical models on scales ≥10h ?1Mpc.While uncertainties remain in the current clustering analyses as well as the abundance of observed clusters due chie?y to still limited

2

Sky Survey(SDSS;Knapp,Lupton,&Strauss1996)and 2dF galaxy redshift survey(Colless1998)should provide much more accurate determinations of both.

The groundwork for the gravitational instability picture of cluster formation was laid down more than two decades ago(Gunn&Gott1972).In the context of Gaussian cos-mological models,Kaiser(1984),in a classic paper,put forth the“biased”structure formation mechanism,where clusters of galaxies were predicted(correctly)to form at high peaks of the density?eld to explain the enhanced correlation of Abell clusters over that of galaxies.This idea was subsequently extended to objects on other scales including galaxies and the properties of linear Gaussian density?elds were worked out in exquisite detail(Peacock &Heavens1985;Bardeen et al.1986,BBKS hereafter). While alternatives exist(e.g.,Zel’dovich1980;Vilenkin 1981,1985;Turok1989;Barriola&Vilenkin1989;Bennett &Rhie1990),a Gaussian model is simple and attractive (largely because of it)in that all its properties can be fully speci?ed by one single function,the power spectrum of its density?uctuations.Moreover,it is predicted that ran-dom quantum?uctuations generated in the early universe naturally produced Gaussian density?uctuations,whose scales were then stretched to the scales of cosmological interest by in?ation(Guth&Pi1982;Albrecht&Stein-hardt1982;Linde1982;Bardeen,Steinhardt&Turner 1983).Furthermore,observations of large-scale structure and microwave background?uctuations appear to favor a Gaussian picture(Vogeley et al.1994;Baugh,Gaztanaga, &Efstathiou1995;Kogut et al.1996;Colley,Gott,&Park 1996;Protogeros&Weinberg1997;Colley1997).So mo-tivated,the present study will focus on the family of Gaus-sian CDM models.The reader is referred to Cen(1997c) for a discussion of the cluster correlation function in non-Gaussian models.We will employ the formalism of BBKS of Gaussian density?eld to devise an analytic method that can be used to directly compute the mass and correlation functions of clusters of galaxies.The needed input are:P k (the power spectrum),?0andΛ0.The method developed is calibrated and its accuracy checked by a large set of N-body simulations.

The motivation for having such an analytic method is not only of an economical consideration(fast speed and much larger parameter space coverage possible)but also a necessity,especially for studying very rich clusters.For example,for clusters of mean separation of200h?1Mpc (about richness3and above;Bahcall&Cen1993,BC henceforth),a simulation box of size1170h?1Mpc on a side would contain200such clusters,a number which may be required for reasonably sound statistical calculations.As-suming that the mass of such a cluster is1.0×1015h?1M⊙(approximately the mass of a richness3cluster;BC)and one requires500particles to claim an adequate resolu-tion of the cluster,it demands a requisite particle mass of 2.0×1012h?1M⊙.This particle mass requirement dictates that one discretize the whole simulation box into108.3?0 particles(?0is the density parameter of a model).Mean-time,a minimum nominal spatial resolution of0.5h?1Mpc is needed to properly compute just the cluster masss within Abell radius of1.5h?1Mpc,which translates to a spatial than57GB of RAM to allow for such a large simulation and hence is very expensive,if possible,or an adaptive code such as P3M(Efstathiou et al.1985)or TPM code (Xu1995),where CPU cost will be prohibitively large even if RAM permits.

The paper is organized as follows.Descriptions of GPM for computing cluster mass function are presented in§2.1. Descriptions of GPM for computing the cluster corre-lation function are presented in§2.2.A calibration of Press-Schechter method using the?tted GPM parameters and some comparisons between GPM and Press-Schechter method are presented in§2.3.We discuss the various fac-tors that a?ect the cluster mass function in§3.Detailed constraints by the local rich clusters and the COBE obser-vations(Smoot et al.1992)on all CDM models are pre-sented in§4.A simpleσ8??0relation(with errorbars) for CDM models is presented in§5,derived from?tting to the observed local cluster abundance alone.Conclusions are given in§6.

2.GAUSSIAN PEAK METHOD FOR CLUSTERS OF

GALAXIES

2.1.Gaussian Peak Method for Cluster Mass Function It is convenient to de?ne some frequently used symbols ?rst.Hubble constant is H=100h km/s/Mpc.?0andΛ0 are the density parameters due to non-relativistic mater and cosmological constant,respectively,at redshift z=0.?z andΛz are the same parameters at redshift z.r A is the comoving radius of a sphere in units of h?1Mpc,which in most times represents the Abell radius with value1.5.r v is the virial radius in comoving h?1Mpc.r f is the radius of a smoothing window in comoving h?1Mpc.M A is the mass within a sphere of radius r A in units of h?1M⊙.M v is the mass within a sphere of radius r v(virial mass of a halo)in units of h?1M⊙.For formulae related to Gaussian density ?eld we will follow the notation of BBKS throughout this paper.

The cluster mass function may be derived by relating the initial density peaks to the?nal collapsed clusters, provided that peaks do not merge.Two pieces of observa-tions suggest that merger of initial density peaks of cluster size be infrequent.First,the typical separation between clusters of galaxies is～100h?1Mpc,while the typical size of a cluster is～1h?1Mpc.Second,empirical ev-idence of matter?uctuations,as indicated by observed galaxy number?uctuations(Davis&Peebles1983;Strauss &Willick1995),suggests that the current nonlinear scale is～8h?1Mpc,which is just about the size of?uctuations that collapse to form clusters of galaxies;i.e.,the majority of clusters of galaxies form at low redshift.However,a more quantitative argument,that merger rate should be small,can be made as follows.Suppose that a cluster is moving at velocity v0at z=0,then we can compute the total comoving distance that the cluster has travelled in its entire lifetime as

d cm= t00v0f(t)(1+z)dt,(1) wher

e t0is the current age o

f the universe,f(t)is a func-

3

we will ?rst use ?0=1,which gives the following simple relation:

d cm =v 0H 0.(2)To arriv

e to the above relation we have used the follow-ing simple relations:t =t 0(1+z )?3/2,t 0=2H ?1

0/3,

f (z )=(1+z )?1/2

(linear growth rate of proper peculiar velocities;Peebles 1980).For any reasonable model clus-ters of galaxies do not move at a speed (peculiar velocity)much higher than ～1000km/s at present (Cen,Bahcall &Gramann 1994);it can be obtained approximately in lin-ear theory by integrating a power spectrum,smoothed by an appropriate window,to yield the total kinetic energy (e.g.,Suto,Cen,&Ostriker 1992).Note that some galaxy in a virialized cluster may move at a higher speed,but we are not considering such objects.So,a cluster mov-ing at 1000km /s today moves a total comoving distance of 10h ?1Mpc in an ?0=1universe.The same cluster will move a longer distance in a low ?0universe,but not by a large factor.An upper bound on d cm in such cases may be obtained by setting ?0=0,in which case we have

f (z )=(1+z ),t 0=H ?1

0and t =t 0(1+z )?1.The upper bound is

d cm,ub =v 0H 0(1+z max ),(3)wher

e z max is the maximum redshift to which ?=0ap-plies.Let us make a simple,approximate estimate for a

realistic lower bound by taking ?0=0.2,as follows.For an ?0=0.2model,the redshift at which ?=0.5is 3.0,which we denote as z max .We treat the redshift range z >z max as an ?=1model and treat z

d cm =v 0H 0[(1+z max )+1].

(4)

For z max =3.0,d cm =5v 0H 0.Since velocity decayed

from z max to z =0,a more reasonable upper bound on v 0is 1000/(1+z max )km/s.This gives 12.5h ?1Mpc for v 0=1000km/s and z max =3.

Since one needs to collapse a sphere of 9.5??1/3

0h ?1Mpc in a uniform density ?eld to form a massive cluster of mass 1×1015h ?1M ⊙,i.e.,cluster density peaks have to have a

separation of at least ～20??1/3

0h ?1Mpc and more likely ～50h ?1Mpc (mean separation of rich clusters today),it thus seems quite unlikely that a signi?cant fraction of any massive cluster peaks have merged by z =0.This con-clusion is,however,not in con?ict with observations that seem to show signs of recent and/or ongoing merger ac-tivity.In general,merger is an ever-going processes (at least in the past)in any plausible (i.e.,a plausible range in ?0)hierarchical structure formation model.But,these mergers or substructures seen in some clusters are sub rich cluster scale mergers,i.e.,sub-peaks within a large cluster scale peak are in the process of merging,a result which is in fact expected if clusters have been forming in the recent past in a hierarchical fashion.To our knowledge,there is no major merger event of two massive clusters ob-served.For example,in 55Abell clusters catalogued by Dressler (1980),there is no case of two massive clusters in the process of imminent merging,although there does seem to have signi?cant substructures in a signi?cant fraction clusters with substructures (see Forman &Jones 1994for a review).That being said,one needs to be extra cautious in interpreting such sub cluster scale merger/substructure events,due to unavoidable projection contaminations (see Cen 1997for a thorough discussion of projection e?ects).Having shown that merger should be infrequent,the key link then is to relate a density peak of height

ν=F/σ0

(5)

to the ?nal mass of a cluster de?ned within a ?xed ra-dius,say,the Abell radius r A .Here,F is a density ?eld smoothed by a window of size r f and σ0is the rms ?uctua-tion of F .Gaussian smoothing window (in Fourier space)

W (kr f )=exp(?r 2f k 2

/2),

(6)

will be used throughout this paper,because it guaran-tees convergence of any spectral moment integral with

any plausible power spectrum.Top-hat smoothing does not have this feature.For the sake of de?niteness and convenience in comparing with observations,we de?ne a cluster mass,M A ,as the mass in a sphere of comoving Abell radius,r A =1.5h ?1Mpc,in most cases.Cluster mass de?ned otherwise will be noted in due course.But the formalism developed here should be applicable for any plausible radius.

For a spherical perturbation,the mean density within the virial radius (at redshift z )in units of the global mean density (at redshift z )can be parameterized by

ˉρv (?z ,Λz )=178??0.57

z

C (?z ,Λz ).(7)

For ?z =1and Λz =0,it is well known that ˉρv =178,a result ?rst derived by Gunn &Gott (1972).In equation (7)C (?z ,Λz )(a function of both ?z and Λz )has a value close to unity.It has been shown that C =1is a good approximation for both Λz =0model (e.g.,Lacey &Cole 1993)and Λz +?z =1model (e.g.,ECF)for the range of ?z of interest (0.1

M v (z )

=4πr v

)3?α.

(10)

4

But,in general,the density pro?le of a cluster does not have a power-law form,soα(?z,Λz,M A,P k)should only be considered as a?tting parameter,which should,in prin-ciple,be dependent on both the cluster mass and under-lying cosmology.However,motivated by the insight of Navarro,Frenk,&White(1996)that there seems to be a universal function(as a function of scaled radius in units of the virial radius each individual halo)for density pro-?les of dark matter halos,independent of cosmology and halo mass,it is hoped thatαwill only be a weak func-tion of both the underlying cosmology and cluster mass. As a matter of fact,as we will show below,the best?t to N-body results requires thatαbe a constant equal to 2.3,in harmony with the work of Navarro et al.(1996). Combining equations8,9,10yields

r f=

3.617r(α?3)/α

A M A

2.058×1014 1/α

?0.19(α?3)/αz ??1/α

.(13)

In this equationαis the only adjustable parameter.But as will be shown later,αturns out to be a constant.There-fore,r f is no longer an adjustable parameter,rather it is a unique function of only M A and r A.Another useful ex-pression is to relate M v to M A(obtained by combining equations9and13)in terms of r A:

M v=

2.058×1014r3(α?3)/α

A M A

2.058×1014 α/3

?0.19(3?α) z ?(3?α)/3

.(15)

We note that,in the special case where r A=r v,we have (from equation9)

r f= M A The circular velocity of the(just virialized)halo at redshift z can be expressed as

v c≡

GM v(1+z)

1.0×1014

1/3

??0.095

z

?1/6

(1+z)1/2km/s.(18) Note that the dependences of v c on?z and?0are very weak.But v c depends rather strongly on z and somewhat strongly on M v.The1-d velocity dispersionσ||is just equal to v c/

√

(2π)2R3?

e?ν2/2G(γ,γν),(20) where G(γ,w)is,for the convenience of calculation,an an-

5?1

for w>1,according to BBKS:

w3?3γ2w+[B(γ)w2+C1(γ)]exp[?A(γ)w2]

G(γ,w)=

9?5γ2

432

B=

σ2σ0

R?≡√σ

(23)

2

(equation4.6a of BBKS),andσj are spectral moments:

σj≡ k2dk

6

Table1

List of parameters for32models

Model?0Λ0σ8EP≡3.4σ25/σ8Comment

a the standard CDM model with Hubble constant H o=

50km/s/Mpc,?0=1.0and n=1.0,where n is the power in-

dex at very large scale.BBKS power spectrum transfer function

(equation G3)is used

b an open CDM model with Hubble constant H o=70km/s/Mpc,

?0=0.35and n=1.0;BBKS power spectrum transfer function

(equation G3)is used.

c an open CDM model with Hubble constant H o=60km/s/Mpc,

?0=0.60and n=1.0;BBKS power spectrum transfer function

(equation G3)is used.

d a CDM model with a cosmological constant with Hubbl

e con-

stant H o=65km/s/Mpc,?0=0.40,Λ0=0.60and n=0.95;

the power spectrum transfer function is computed as in Cen et al.

1993.

e a CDM model with a cosmological constant with Hubble con-

7

Fig. 1.—M HR/M LR is plotted against M LR for the three models.The solid line in each plot is the best linear?t:M HR/M LR= a+b log10M LR,where M LR is in h?1M⊙.We?nd(a,b)to be(?0.469,0.103),(?0.0528,0.0662),(?0.952,0.137)for models(1,14,29), respectively.

8

(2b)

Fig.2.—Mass functions of various models.Each curve is labeled by its model number from Table 1.The simulation results are shown as symbols with horizontal errorbars being the uncertainties in the mass determination (15%)and the vertical errorbars being the statistical 1σerrorbars for the number of clusters.The solid curves in Figure 2are the results from GPM.Note that simulation box size limits the density of clusters to >1.56×10?8h 3Mpc ?3,when there is only one cluster in the whole box.At f (>M A )=10?6h 3Mpc ?3,there are 65clusters in the simulation box.

(i.e.,representing the truth)for the purpose of calibrating our low resolution results.Because the two simulations have identical initial conditions,we are able to identify every rich cluster in one simulation with its counterpart in the other.Having made such a one-to-one correspondence we can compute the ratio M HR /M LR (HR stands for high resolution and LR for low resolution)as a function of clus-ter mass M LR .This allows us to make corrections to M LH in the lower resolution simulation.The above resolution calibration procedure is repeated for an open CDM model (model 14in Table 1)and a CDM model with a cosmo-logical constant (model 23in Table 1).

Figures (1a,b,c)show the results for the three mod-els,where M HR /M LR is plotted against M LR .The solid line in each plot is the best linear ?t:M HR /M LR =a +b log 10M LR ,where M LR is in h ?1M ⊙.We ?nd (a,b )to be (?0.469,0.103),(?0.0528,0.0662),(?0.952,0.137)for models (1,14,23),respectively.Let us call the three ?tting functions (M HR /M LR )as R 1(M ),R 035(M ),R 04(M ),for the three models run:?z =1(model 1in Table 1),?z =0.35and Λz =0(model 14),and ?z =0.40and Λz =0.60(model 23).Then,for the mass of each cluster,M LR ,we correct it by multiplying it by R 1+(R 035?R 1)(1??z )/0.65in an open model,or by R 1+(R 04?R 1)(1??z )/0.60in a Λmodel,where ?z is the density parameter of the model under consideration.From Figure 1we see that the typi-cal correction is about 5-10%in the upward direction with a dispersion of ～5%;i.e.,lower resolution simulations We are now ready to ?nd the best ?tting parameters (α,δc )by comparing results from GPM to the direct N-body results.Before starting the ?tting procedure,we have some rough idea about what the values of αand δc may be.We pick α=2.5and δc =1.5as an initial guess.In the end,the best values are found to be

α=2.3,

(26)

a constant independent of the cluster mass and cosmology,and

δc =

1.40?0.01(1.0??z )for Λz =01.40+0.10(1.0??z )for ?z +Λz =1.(27)

The best overall ?t for all the models is judged by the au-thor by direct visual examination.We ?nd it very di?cult

to design an automated ?tting procedure to be gauged by some objective parameters,because of the enormous range of the number densities of clusters and hard-to-de?ne er-rorbars hence weighting schemes for the densities.But as we will see,the ?nal ?ts are probably as good as one would have hoped,which suggests that our somewhat subjective ?tting procedure works very well.In any case,the ?nal ?t values span very narrow ranges (in fact,αturns out to a constant,and δc varies from 1.40to 1.39from ?z =1to 0in Λz =0models,from 1.40to 1.36from ?z =1to 0in Λz +Λz =1models),which indicates that the ?tting pro-cedure is robust and stable.We note that the ?tted value

9

However,the sensitivity of a?t to the two parameters de-pends on the?tted mass function itself:a low amplitude mass function depends more sensitively on the two param-eters than a high amplitude mass function.This is so,of course,because the abundance of rarer objects depends more sensitively on the parameters.Roughly speaking,αserves more to?x the shape of the mass function in a somewhat less sensitive way,whileδc determines the over-all amplitude and more sensitively the amplitude on the high mass end of the mass function.Our estimates on the uncertainties are?α=0.1and?δc=0.01.

Figures(2a,b)show the simulation results as symbols for32models with horizontal errorbars being the uncer-tainties in the mass determination(15%)and the ver-tical errorbars being the statistical1σerrorbars.Note that simulation box size limits the density of clusters to >1.56×10?8h3Mpc?3,when there is only one cluster in the whole box.At f(>M A)=10?6h3Mpc?3,there are65 clusters.The solid curves in Figure2are the results from GPM.We see that the GPM results?t remarkably well the simulation results for all the thirty two models examined. We note that the actual errorbars should be larger than what are shown for the simulated results because of cosmic variances;i.e.,the simulation boxsize,although quite large being400h?1Mpc,may still not be large enough to have the cosmic variance diminished,especially for models with signi?cant power on several hundred megaparsecs scales. In any case,the GPM results?t the N-body results for all the models within2σin the vertical axis,and within a fac-tor of1.25in the horizontal axis.Since the observed mass function(BC)has uncertainties in mass about a factor2.0 and in number density about a factor≥2.0,the GPM re-sults are practically precise,for the purpose of comparing model results computed using GPM with observations. At this point it seems appropriate to reiterate the virtue of the current method.The essential unique ingredients are the introduction of two adjustable parameters,αand δc,the?rst of which turns out to be a constant and the second of which can be simply expressed as a function of ?z(the density parameter at the redshift in question). Note that the?tted parameterδc(?z)has a slightly dif-ferent form for the case withΛz=0than for the case with?z+Λz=1.The fact thatδc is only a rather weak function of?z in both cases indicates that the method is robust.

2.2.Gaussian Peak Method for Cluster Correlation

Function

Having found that the initial density peaks,appropri-ately de?ned,indeed correspond to the clusters formed at late times,as indicated by the goodness of the?ts of the results from GPM to the N-body results in terms of clus-ter mass function presented in the preceding section,we have some con?dence that we may be able to compute the cluster-cluster two-point correlation function using GPM. We will now proceed along this route.

Even in the linear regime where dynamic contribution to the clustering can be ignored,the primary di?culty in cal-culating the cluster-cluster two-point correlation function using Gaussian peaks is the ambiguity of relating appro-in§2,eliminates this ambiguity by simultaneously?xing both r f andνt.This is achieved by demanding that the appropriate peaks yield the correct cluster mass function, when compared to direct N-body simulations. Following BBKS,we use the following approximate for-mula,which is applicable when the correlation function is smaller than unity[however,BBKS state that it may well be a reasonable approximation even when the statistical correlation function(?rst term at the right hand side of equation28,see below)is not really small],to compute the?nal cluster-cluster correlation function including lin-ear dynamical contributions:

ξpk,pk≈

[3(1?γ2)+0.45+(γν/2)2]1/2+γν/2

(30)

(equation6.14of BBKS).We note that equation28is valid in the linear regime whenξρ,ρis much less than unity and it is not yet clear whether the approximation, coupled with our de?nitive peak identi?cation method, also works in the mild nonlinear regime.Our goal is to ?nd an approximation based on equation28which will give su?ciently accurate results forξpk,pk in the regime whose values are of order unity and below.For this reason we choose to modify equation28in the following man-ner:ξpk,pk≈

x3 x0ξρ,ρ(y)y2dy.The reason for usingˉξρ,ρinstead ofξρ,ρis thatˉξρ,ρis a better indicator of nonlin-earity thanξρ,ρ.The form of D should be constrained at the linear end:D(0)=1.As we will show below,it turns out that equation28?ts results very well;i.e.,?tting to numerical results indicates that D=1is a good approx-imation for the interested range inξpk,pk.To be clear, we use equation28for all the subsequent calculations of cluster correlation functions.

10

(3a)

11

(3c)

12

(3e)

13

(3g)

Fig.3.—The cluster correlation functions for clusters with mean separation of55h?1Mpc,for28models as indicated in the panels.The errorbars are1σstatistical.Three curves are shown in each panel for each model;the middle curve is obtained using equation28and the top and bottom curves are obtained by adding±2h?1Mpc to each point of the middle curve in the x-axis.

14

(4b)

15

(4d)

(4e)

16

from N-body simulations.We compute the two-point cor-relation function from N-body simulations using the fol-lowing estimator:

ξcc(r)=

N CR(r)

2

M v δc

dM v

exp(?

δ2c

dσ0(M A)

17

(5a)

(5b)

18

To calculateσ0(M A)in the above equation we use the Gaussian smoothing window with the radius determined by equation13.The original PS formalism was based on the sharp k-space?lter,but it has been shown sub-sequently by many authors that Gaussian?lter works at least as well.The additional virtue of a Gaussian window is that it guarantees a convergent integral forσ0for any plausible power spectrum.Now,the only parameter left undetermined isδc,which will be?xed by comparing to N-body results.We?nd that the best overall?t of PS results to N-body results is obtained,if

δc= 1.23?0.05(1.0??z)forΛz=0

1.23?0.01(1.0??z)for?z+Λz=1.(34) The results are shown in Figure 5.We see that PS ?ts N-body results quite well except for the P k=k?2 models(models10,11,12,13).The PS results for all the ?0=1models except the P k=k?2models appear to be somewhat above the N-body results at the low mass end(～5×1014h?1M⊙)and somewhat below the N-body results at the high mass end(～

2.5×1015h?1M⊙).On the other hand,the PS results for all the P k=k?2mod-els are signi?cantly above the N-body results.So,there is no room for further adjustments ofδc to achieve better overall?ts,at least for Gaussian smoothing windows. Note thatδc(～1.23)is smaller than1.67,which is in the expected direction because a smoother,Gaussian smoothing window is used here.1.23is also somewhat smaller than that given by Klypin et al.(1995),who give δc=1.40for a Gaussian smoothing window in the context of damped Lyman alpha systems.But Klypin et al.also argue thatδc could be as low as1.3,were waves longer and shorter than those present in the simulation box included. We suspect thatδc also depends on the shape of the power spectrum in a way that is analogous to the di?erence be-tween di?erent smoothing windows:a steeper power spec-trum(i.e.,n being smaller with P k=k n),which conspires to form a sharp k-space?lter like that used in the original derivation of PS,requires a largerδc,while a?atter power spectrum requires a smallerδc.With this conjecture,the trend that applications of PS to smaller cosmic objects tend to require largerδc would have been predicted,since CDM-like spectra have a slowly bending shape which is ?atter at small k(i.e.,for larger systems)and steeper at large k(i.e.,for smaller systems).This conjecture seems to be borne out in the subsequent analyses,as best sum-marized as in equation36,where the dependence ofσ8on Γis consistent with the above hypothesis.This issue will be addressed elsewhere in more detail.

While PS works well for CDM-like models for comput-ing halo mass functions,consistent with earlier works(Efs-tathiou&Rees1988;WEF;ECF),it seems that GPM?ts somewhat better the N-body results for CDM-like mod-els and also works for other models tested.An additional advantage is that GPM allows for a determination of the correlation function as well.Therefore,in subsequent cal-culations we will use GPM,if deemed appropriately appli-cable.

3.VARIOUS F ACTORS THAT AFFECT CLUSTER MASS ing the cluster mass functions for six variants of the stan-dard CDM model(Table2below),as indicated in panels (a,b,e,f,g,h)of Figure6,at di?erent normalization ampli-tudes(σ8).We will return to Table2in§4to discuss the various models in detail.The primordial power spectrum index is assumed to be n=1for the shown models in panels(a,b,e,f,g,h).Also shown as symbols are the obser-vations adopted from BC,and as three dashed curves are the?ts to the symbols.The middle dashed curve is

n fit(>M A)=

2.7×10?5(M A/2.1×1014)?1

exp(?M A/2.1×1014),(35)

where n fit is in h3Mpc?3and M A is the cluster mass within the Abell radius in h?1M⊙.This curve seems to represent the mean value of the observed mass function well(note that equation35is slightly di?erent from the ?tting formula in BC).The top and bottom dashed curves are4n fit(>M A)and0.25n fit(>M A).It is di?cult to estimate the errorbars of the observed mass function.The top and bottom dashed curves are intended to serve as 2σupper and lower bounds(in the vertical axis)of the observed mass function within the indicated mass range, which we deem to be conservative.Subsequent presenta-tions and explanations will follow this assertion.

In all cases,we see that the cluster mass function be-comes progressively steeper at the high mass end as the amplitude of the density perturbations decreases.The physics behind this is simple to understand.As the am-plitude of?uctuations decreases,the required height of the density peaks for clusters with a given mass increases. Since the abundance of the high peaks at the very high end drops exponentially,the mass function steepens as the amplitude of?uctuations decreases.Note that there is only a narrow range inσ8where the model mass func-tion lies within the2σlimits in the mass range from 4.8×1014h?1M⊙to1.2×1015h?1M⊙.

Although it is clear that the amplitude of the power spectrum sensitively determines the abundance of the clus-ters,as seen by comparing di?erent curves within each panel,it is not yet clear how big the e?ect of?0on the mass function is.In Figure6,panel(c)is similar to panel (a)with only one change:?0=0.4instead of?0=1.0. Note that the power spectrum used in panel(c)is identical to that used in panel(a).We see?0has a signi?cant e?ect on the mass function.For example,theσ8=1.0model in panel(c)has comparable mass function to theσ8=0.6 model in panel(a),but about two orders of magnitudes lower than theσ8=1.0model in panel(a).

Next,we examine the e?ect of the shape of the power spectrum on the shape of the mass function.In Figure6 panel(d)is similar to panel(a)but with n=0.75instead of n=https://www.sodocs.net/doc/0b17959141.html,paring panel(d)with panel(a)illustrates the sensitivity of the cluster mass function on the shape of power spectrum.The model shown in panel(d)has sub-stantially more power on large scales(100?300h?1Mpc) than the model shown in panel(a).This di?erence results in?atter mass functions in panel(d)than in panel(a),es-pecially for the cases with lowerσ8.A point that we would

19

(6a,b,c,d)

the mass function.For example,at1.2×1015h?1M⊙,the

model shown in panel(d)has about a factor of four more

clusters than the model shown in panel(a),with the only

di?erence between the two models being a slight tilt of the

power spectrum[n=0.75in(d)vs n=1.0in(a)].

Panels(e,g)and panels(f,h)show two pairs of low den-

sity models,one being open and the other being?at with

a cosmological constant but with a same?0in each pair.

One thing to note is that,other things being?xed,the

cosmological constant does not make much di?erence in

terms of the shape of the mass function.However,for a

?xedσ8,the mass function in the model with a cosmo-

logical constant is systematically lower than that of the

model without a cosmological constant;the di?erence be-

comes larger at lower amplitudes.

Summarizing the above results we see that the factors

that e?ect the cluster mass function in order of decreas-

ing importance areσ8,?0,P k andΛ0.The ordering of

the last two factors is somewhat more complex;the P k

factor is more signi?cant at the high mass end,whereas

theΛ0factor a?ects rather uniformly across-the-board.It

is therefore clear that the tightest constraint may be ob-

tained forσ8for a given model with?0,Λ0and P k being

speci?ed.

4.NORMALIZING ALL CDM MODELS

In§2.1and2.2we have shown that the Gaussian peaks

of cluster size indeed form clusters of galaxies and appro-

priately identi?ed peaks reproduce both cluster mass func-

sentative variants of the standard CDM model,which are

of current interest.The models are listed in Table2.

The baryon densities for Models C,E are computed us-

ing?b h2=0.0125(Walker et al.1991),and for Models

A,B,D,F using?b h2=0.0193(Burles&Tytler1997).

These choices of?b for the models serve to maximize

the viability of each model with respect to the observed

gas fraction in X-ray clusters of galaxies(White et al.

1993;Lubin et al.1996;Danos&Pen1998,ρgas/ρtot=

(0.053±0.004)h?3/2).The power spectrum transfer func-

tions for all the models are computed using the CMBFAST

code developed by Seljak and Zaldarriaga.The choice of

the Hubble constant is made for each model such that

each model is consistent with current measurements of

the Hubble constant.It appears to be a consensus that

H0(obs)=65±10km/s/Mpc can account for the dis-

tribution of the current data from various measurements

(see,e.g.,Trimble1997),except for those from Sunyaev-

Zel’dovich observations(for a discussion of a reconciliation

of this di?erence,see Cen1998).Another consideration is

that the age constraint from latest globular cluster ob-

servations/interpretations(c.f.,Salaris,Degl’Innocenti,&

Weiss1997)is not violated.

The SCDM model(#1in Table2)is the standard CDM

with critical density.The mixed hot and cold dark matter

model(HCDM)has critical density but with20%of the

mass in light massive neutrinos(two species of equal mass

neutrinos are assumed).The next two models,OCDM25

and OCDM40(#3and4),are open models with matter

20

(6e,f,g,h)

Fig.6.—Cluster mass functions for six models at several di?erent normalization amplitudes of the power spectra.Also shown as symbols are the observations adopted from BC,and as three dashed curves are the?ts to the symbols.The solid dots represent the cluster mass function from Abell cluster catalog,the open squares from the Edinburgh-Durham Cluster Catalog(Lumsden et al.1992),and the open circles from the observed temperature function of Henry&Arnaud(1991).The middle dashed curve is computed by equation35,n f it(>M), which represents the mean value of the observed mass function well.The top and bottom dashed curves are4n f it(>M)and0.25n f it(>M), respectively.

Table2

Six variants of CDM models

Model Family?c?hΛ0H0?b t age(Gyrs)

瀑布水景工程计算方式

15米宽，6米高的人工瀑布，泵的流量要多大，怎样计算？ 上水池32方，下水池是一个大湖南 假设瀑布的厚度为A米。那么可以算一下瀑布停止不动是瀑布的体积：15x6xA=90A，那么我们姑且算厚度A=1cm=0.01m，那么此时的体积是0.9立方。 根据瀑布的高度，水从6m处留下来的时间大约是0.6秒，那么此时的流量大概就是0.9x0.6=0.54立方/秒，即1944立方/小时。此时选泵就选流量2000吨/小时，扬程10m左右的泵，此时水泵的功率大概是110Kw左右。 当A=1mm=0.001m的时候，也根据这种算法，那么水泵的流量是194吨/小时。此时选泵就选流量200吨/小时，扬程10m左右的泵，此时水泵的功率大概是11-15Kw左右。 具体选什么泵可根据实况选择潜水泵，或者离心泵（选离心泵是应注意泵不能放在瀑布上方，因为离心泵没有那么高的吸程，放在上方时吸不上水的）。

水景园林给排水：浅谈景观瀑布设计 俗话说“水为庭院灵魂”，由此可见水在园林景观中的重要作用。水与周围景物结合，便会表现出或悠远宁静，或热情昂扬，或天真质朴，或灵动飞扬的意境．艺术地再造自然之魂．从而产生特殊的艺术感染力，使城市景观更添迷人的魅力。因此．景观瀑布作为水景形态之一，在城市景观设计中运用较多。这里，笔者仅就景观瀑布设计谈几点体会。 1 景观瀑布的分类 1．1 自然式瀑布．即模仿河床陡坎的形式，让水从陡坡处滚落下跌形成恢弘的瀑布景观。此类瀑布多用于自然景观与情趣的环境中 1．2 规则式瀑布．即强调落水的规则与秩序性，有着规整的人工构筑落水E1．可形成一级或多级跌落形式的瀑布景观此类瀑布多用于较为规整的建筑环境中。 1．3 斜坡瀑布，即落水由斜面滑落的瀑布景观。它的表面受斜坡表面质地、结构的影响．体现出较为平静、含蓄的意趣，适用于较为安静的场所。 2 景观瀑布的构成 一个完整的景观瀑布一般由背景、上游水源、落水口、瀑身、承瀑潭及溪流构成。其中，瀑身是观赏的主体。 3 景观瀑布的设计要素 3．1 水量 景观瀑布的形式与其上游水源的水量有着密切的关系，瀑布水量应满足景观瀑布的方案设计要求。供水量在lms／s左右时，瀑身可形成重落、离落、布落等形式；供水量在0．1m3／s左右时，瀑身可形成丝落、线落等形式。 3．2 水泵的选择 3．2．1 流量的选择 首先．根据前面提到的瀑布用水量估算表计算流量，再根据《建筑给水排水设计规范》GB50015—2003第3．1 1．9条计算设计循环流量。即：Qs＝1．2Qc 式中：Qc－景观瀑布的设计循环流量，m3/h；

跌水水景流量设计

跌水水景在小区水景中为常见的一种表现形式,水量控制是其中的一个关健点.跌水水景水量过大则能耗大，长期运转费用高；跌水水景水量过小则达不到预期的设计效果。 关健字:水景??跌水跌水水景 在水景设计中，跌水水景是构成溪流、叠流、瀑布等水景的基本单元，具有动态和声响的效果，因而应用较广。 与静态水景不同，动态水景的水是流动的，其流动性一般用循环水泵来维持，水量过大则能耗大，长期运转费用高；水量过小则达不到预期的设计效果。因此，根据水景的规模确定适当的水流量十分重要。 1跌水水景的水力学特征及计算 跌水水景实际上是水力学中的堰流和跌水在实际生活中的应用，跌水水景设计中常用的堰流形式为溢流堰. 根据δ和H的相对尺寸，堰流流态一般分为薄壁堰流、实用堰流、宽顶堰流等三种形式： 当δ/H<0.67，为薄壁堰流；0.67<δ/H<2.5，为实用堰流；2.5<δ/H<10，为宽顶堰流； δ/H>10,为明渠水流，不是堰流。 跌水水景设计中，常用堰流形态为宽顶堰流。 当跌水水景的土建尺寸确定以后，首先要确定跌水水景流量Q，当水流从堰顶以一定的初速度v0落下时，它会产生一个长度为ld的水舌。若ld大于跌水台阶宽度lt，则水景水流会跃过跌水台阶；若ld太小，则有可能出现水景水舌贴着水景跌水墙而形成壁流。这两种情况的出现主要与跌水水景流量Q的大小有关，设计时应尽量选择一个恰当的跌水水景流量以避免上述现象的发生。 水景中的跌水水景设计(二) 跌水水景在小区水景中为常见的一种表现形式,水量控制是其中的一个关健点.跌水水景水量过大则能耗大，长期运转费用高；跌水水景水量过小则达不到预期的设计效果。 关健字:水景??跌水跌水水景 1.1跌水水景流量计算 根据水力学计算公式，一般宽顶堰自由出流的流量计算式为： Q=σc·m·b·(2g)0.5·H1.5=σc·M·b·H1.5 式中b——堰口净宽H——包括行进流速水头的堰前水头， H=H0+υ02/2g 式中υ0——行进流速m——自由溢流的流量系数，与堰型、堰高等边界条件有关σc——侧收缩系数 M=m·(2g)0.5当堰口为矩形时，侧收缩系数σc为1，上述计算式即简化为《给水排水设计手册》中的流量计算式： Q=m·b·(2g)0.5·H1.5=M·b·H1.5

跌水水景中设计中的计算

跌水水景中的计算实例 某宾馆根据其地形条件在大堂内设计一溢流式跌水景，为扇形结构，第一级跌水高度P为2.1 m，堰口为弧线形，长度b=14.65 m，堰顶宽δ=0.15 m，跌水台阶宽度l t =0.7m。 2.1 计算跌水流量Q 根据宾馆大堂环境的要求，跌水流量不须太大，因此，初始选定堰前水头H=0.2 kPa，根据堰流的出口形式，流量系数M=1 417.4，因此试算流量： 2.2 校核跌水水舌 l d 根据试算流量Q可求出跌水景溢流口的单宽流量： q=Q/b=4.007×10-3 m3/(s·m) 由此得 D=q2/(g·p3)=1.767 3×10-7 跌水水舌长度： l d =4.30×D0.27×P=0.136m 0.1

根据δ和H的相对尺寸，堰流流态一般分为薄壁堰流、实用堰流、宽顶堰流等三种形式： 当δ/H<0.67，为薄壁堰流；0.67<δ/H<2.5，为实用堰流；2.5<δ/H<10，为宽顶堰流； δ/H>10,为明渠水流，不是堰流。 跌水水景设计中，常用堰流形态为宽顶堰流。 当跌水水景的土建尺寸确定以后，首先要确定跌水水景流量Q，当水流从堰 顶以一定的初速度v 0落下时，它会产生一个长度为l d 的水舌。若l d 大于跌水台 阶宽度l t ，则水景水流会跃过跌水台阶；若l d 太小，则有可能出现水景水舌贴 着水景跌水墙而形成壁流。这两种情况的出现主要与跌水水景流量Q的大小有关，设计时应尽量选择一个恰当的跌水水景流量以避免上述现象的水景中的跌水水景设计(二) 1.1 跌水水景流量计算 根据水力学计算公式，一般宽顶堰自由出流的流量计算式为： Q=σ c ·m·b·(2g)0.5·H1.5=σ c ·M·b·H1.5 式中b——堰口净宽H——包括行进流速水头的堰前水头，H=H0+υ 2/2g 式中υ ——行进流速m——自由溢流的流量系数，与堰型、堰高等边界条件有关σc——侧收缩系数 M=m·(2g)0.5 当堰口为矩形时，侧收缩系数σc为1，上述计算式即简化为《给水排水设计手册》中的流量计算式： Q=m·b·(2g)0.5·H1.5=M·b·H1.5 上式中，M(或m)为流量系数，与堰的进口边缘形式有关；b为堰口净宽，为已知，因此要求出水景流量Q，关键要确定出堰前水景水头H，堰前水景水头一般先凭经验选定、试算。通常H的初试值可选为0.2～0.4 kPa，当水景堰口为直角时宜取上限，堰口为斜角或圆角时取下限。H初值选定后，根据上述计算式算

跌水水景流量设计

水景中的跌水水景设计(一) 跌水水景在小区水景中为常见的一种表现形式,水量控制是其中的一个关健点.跌水水景水量过大则能耗大，长期运转费用高；跌水水景水量过小则达不到预期的设计效果。 关健字:水景跌水跌水水景 在水景设计中，跌水水景是构成溪流、叠流、瀑布等水景的基本单元，具有动态和声响的效果，因而应用较广。 与静态水景不同，动态水景的水是流动的，其流动性一般用循环水泵来维持，水量过大则能耗大，长期运转费用高；水量过小则达不到预期的设计效果。因此，根据水景的规模确定适当的水流量十分重要。 1 跌水水景的水力学特征及计算 跌水水景实际上是水力学中的堰流和跌水在实际生活中的应用，跌水水景设计中常用的堰流形式为溢流堰. 根据δ和H的相对尺寸，堰流流态一般分为薄壁堰流、实用堰流、宽顶堰流等三种形式：当δ/H<0.67，为薄壁堰流；0.67<δ/H<2.5，为实用堰流；2.5<δ/H<10，为宽顶堰流； δ/H>10,为明渠水流，不是堰流。 跌水水景设计中，常用堰流形态为宽顶堰流。 当跌水水景的土建尺寸确定以后，首先要确定跌水水景流量Q，当水流从堰顶以一定的初速度v0落下时，它会产生一个长度为ld的水舌。若ld大于跌水台阶宽度lt，则水景水流会跃过跌水台阶；若ld太小，则有可能出现水景水舌贴着水景跌水墙而形成壁流。这两种情况的出现主要与跌水水景流量Q的大小有关，设计时应尽量选择一个恰当的跌水水景流量以避免上述现象的发生。 水景中的跌水水景设计(二) 跌水水景在小区水景中为常见的一种表现形式,水量控制是其中的一个关健点.跌水水景水量过大则能耗大，长期运转费用高；跌水水景水量过小则达不到预期的设计效果。 关健字:水景跌水跌水水景 1.1 跌水水景流量计算 根据水力学计算公式，一般宽顶堰自由出流的流量计算式为： Q=σc·m·b·(2g)0.5·H1.5=σc·M·b·H1.5 式中b——堰口净宽H——包括行进流速水头的堰前水头， H=H0+υ02/2g 式中υ0——行进流速m——自由溢流的流量系数，与堰型、堰高等边界条件有关σc——侧收缩系数 M=m·(2g)0.5当堰口为矩形时，侧收缩系数σc为1，上述计算式即简化为《给水排水设计手册》中的流量计算式： Q=m·b·(2g)0.5·H1.5=M·b·H1.5 上式中，M(或m)为流量系数，与堰的进口边缘形式有关；b为堰口净宽，为已知，因此要求出水景流量Q，关键要确定出堰前水景水头H，堰前水景水头一般先凭经验选定、试算。通常H的初试值可选为0.2～0.4 kPa，当水景堰口为直角时宜取上限，堰口为斜角或圆角时取下限。H初值选定后，根据上述计算式算出跌水水景流量Q，由于Q值为试算结果，还须根据跌水水景水舌的长度对Q的大小作进一步的校核和调整。 1.2 校核水景水舌长度 根据水力学的计算公式，溢流堰的跌落水景水舌长度为：

景观水景工程计算书

水景工程

第五章水景工程 导言：园林中最主要的造景法之一是什么？ 水景工程中都包含哪些内容？ 第三章水景工程 第一节水的功能及分类 ，涉及的内容有水体的类型，各种水景的布置，驳岸、护坡、喷泉等。 一、水体的功能 1.造景：水有三态液态：喷泉、瀑布、跌水。 气态：喷雾泉、创造仙境舞台。 固态：滑冰场、冰雕 2.改善小气吸收粉尘，改善环境卫生 3.有利于动植物的生长，特别是水生植物。 4.灌溉与消防 5.水上游乐，划船、游泳、垂钓、漂流 6.组织交通，水上游览 7.水能陶冶人的情操，提高人的修养 二、水系的构成 自然降雨→地表径流（泉水）→涧、溪→瀑布→潭→河→江→海 三、水源种类：

⑴市政给水，自来水（水质好） ⑵地下水 ⑶地表水 四、水体的形式与分类 1.按水体的形式分：水的形式与其所在环境有关。 ⑴自然式水体：边缘不规则，变化自然的水体。例如：河、湖、池、溪、涧等。 ⑵规则式水体：边缘规则，具有明显的轴线的水体，一般是几何形。 ⑶混合式水体：是规则式与不规则式两种交替穿插形成的水环境。 2.按水体的功能分： ⑴观赏性水体：叶饺装饰性水池，面积较小。 ⑵开展活动性水体：游泳馆、游船、垂钓。大规模综合性公园都属此类。 3.按水流状态分： ⑴静态水景：园林中成片汇集的水面，湖、塘、池等。 ⑵动态水景：流动的水，具有动感，溪、涧、瀑布、跌水等。

小结：本次课讲了三个方面 1.水的功能。 2.水的构成。 3.水体的分类。 思考题：1.水体的构成。 2.水体的分类。 引言：上节课我们学习了水景工程的基本知识，也就是水体的分类和功能，下面我们来学习水体中的重要水工措施: 驳岸与护坡。 第二节驳岸与护坡 园林水体要求有稳定、美观的水岸来维持陆地和水面有一定的面积比例，防止陆地被淹或水岸塌陷而扩大水面。因此在水体边缘必须建造驳岸与护坡。同时，作为水景组成的驳岸与护坡直接影响园景，必须从实用、经济、美观几个方面一起考虑。

跌水水景流量设计

跌水水景流量设计 集团企业公司编码：（LL3698-KKI1269-TM2483-LUI12689-ITT289-

跌水水景在小区水景中为常见的一种表现形式,水量控制是其中的一个关健点.跌水水景水量过大则能耗大，长期运转费用高；跌水水景水量过小则达不到预期的设计效果。 关健字:水景??跌水跌水水景 在水景设计中，跌水水景是构成溪流、叠流、瀑布等水景的基本单元，具有动态和声响的效果，因而应用较广。 与静态水景不同，动态水景的水是流动的，其流动性一般用循环水泵来维持，水量过大则能耗大，长期运转费用高；水量过小则达不到预期的设计效果。因此，根据水景的规模确定适当的水流量十分重要。 1跌水水景的水力学特征及计算 跌水水景实际上是水力学中的堰流和跌水在实际生活中的应用，跌水水景设计中常用的堰流形式为溢流堰. 根据δ和H的相对尺寸，堰流流态一般分为薄壁堰流、实用堰流、宽顶堰流等三种形式： 当δ/H<0.67，为薄壁堰流；0.67<δ/H<2.5，为实用堰流； 2.5<δ/H<10，为宽顶堰流； δ/H>10,为明渠水流，不是堰流。 跌水水景设计中，常用堰流形态为宽顶堰流。 当跌水水景的土建尺寸确定以后，首先要确定跌水水景流量Q，当水流从堰顶以一定的初速度v0落下时，它会产生一个长度为ld的水舌。若ld大于跌水台阶宽度lt，则水景水流会跃过跌水台阶；若ld太小，则有可能出现水景水舌贴着水景跌水墙而形成壁流。这两种情况的出现主要与跌水水景流量Q的大小有关，设计时应尽量选择一个恰当的跌水水景流量以避免上述现象的发生。 跌水水景在小区水景中为常见的一种表现形式,水量控制是其中的一个关健点.跌水水景水量过大则能耗大，长期运转费用高；跌水水景水量过小则达不到预期的设计效果。 关健字:水景??跌水跌水水景 1.1跌水水景流量计算 根据水力学计算公式，一般宽顶堰自由出流的流量计算式为：

跌水设计

一、概述 （一）定义 1、跌水：跌落的水，由于地形突然的高差变化而产生的水流现象。 2、瀑布：地形较大的落差变化，使平面的水流呈现直落或斜落的立面水流。 3、叠水：地形呈阶梯状的落差和地貌的凹凸变化，使水流呈现层叠流落而成水流现象 （二）跌水景观的功能 1、跌落的水携带空气中大量的氧进入河流，给水流中的动植物和微生物提供良好的生长条件。 2、飞溅的水花增加了空气湿度，过滤空气中的尘埃。 （三）跌水景观的形式种类 1、水立面形式：线状、点状、帘状、片状、散落状 2、落水方式：直落、飞落、叠落、滑落 3、跌落形式：直接入水式、溅落入水式、可视、可听，具有独特的景观效果。（四）不同形式的形成原因 1、地形的落差决定瀑布形成的高低和水声。 2、地貌的凹凸决定瀑布流落的形状。 3、水流量的多少决定瀑布落水的形式。 4、出水口的大小决定瀑布规模的宽窄。 二、跌水景观的设计要素 1、蓄容 蓄容水流的流量在1m3/s左右的瀑布可行成帘状，片状，和散落状；当仅有0.1m3/s的水流时，则呈现线状、点状。

蓄容分上下两个部位----底池蓄水和堰顶蓄水 2、出水口 （1）隐蔽式：将出水口隐藏在景观环境之中，让水流呈现自然瀑布的形状。（2）外露式：将出水口突显于景观之外，形成明显的人工瀑布造型。 （3）单点式：水流从单一出口跌落，形成单体瀑布。 （4）多点式：出水口以多点或阵列的方式布局，形成规模较大的瀑布景观。

3、瀑布水面 通过控制背景的凹凸肌理加强水面的细节表现，形成造型丰富、形式多样的瀑布景观。 三、叠水景观形式 1、叠水景观以水立面的变化为主要的表现形式。 2、叠水的形式 （1）水帘 水帘是由较大的落差和较宽水流面形成的叠水，控制水流量与出水口的形状将得到不同的水帘形态。 （2）洒落 流量较小的叠水，在较低水压下呈点状或线状跌落。 （3）涌流 涌流是有多层蓄水池不断被注满涌溢而出形成，水流量较大，叠水面呈面状跌落。 （4）管流 由外露式出水管以多种陈列方式形成叠水，水流呈线状。 （5）壁流 流水顺池壁流下，水面可随池壁呈多角度流落。 （6）阶梯式 由多层阶梯造型构成叠水景观。 （7）塔式 多层蓄水池由上至下，由小到大，呈环状倾流而下。 （8）错落式

跌水水景中设计中的计算

跌水水景中设计中的计 算 标准化管理处编码[BBX968T-XBB8968-NNJ668-MM9N]

跌水水景中的计算实例 某宾馆根据其地形条件在大堂内设计一溢流式跌水景，为扇形结构，第一级跌水高度P为2.1 m，堰口为弧线形，长度b=14.65 m，堰顶宽δ=0.15 m，跌水台阶宽度 l t =0.7m。 计算跌水流量Q 根据宾馆大堂环境的要求，跌水流量不须太大，因此，初始选定堰前水头H= kPa，根据堰流的出口形式，流量系数M=1 ，因此试算流量： 校核跌水水舌 l d 根据试算流量Q可求出跌水景溢流口的单宽流量： q=Q/b=×10-3 m3/(s·m) 由此得 D=q2/(g·p3)= 3×10-7 跌水水舌长度： l d =××P=0.136m

在水景设计中，跌水水景是构成溪流、叠流、瀑布等水景的基本单元，具有动态和声响的效果，因而应用较广。 与静态水景不同，动态水景的水是流动的，其流动性一般用循环水泵来维持，水量过大则能耗大，长期运转费用高；水量过小则达不到预期的设计效果。因此，根据水景的规模确定适当的水流量十分重要。 1 跌水水景的水力学特征及计算 跌水水景实际上是水力学中的堰流和跌水在实际生活中的应用，跌水水景设计中常用的堰流形式为溢流堰. 根据δ和H的相对尺寸，堰流流态一般分为薄壁堰流、实用堰流、宽顶堰流等三种形式： 当δ/H<，为薄壁堰流；<δ/H<，为实用堰流；<δ/H<10，为宽顶堰流； δ/H>10,为明渠水流，不是堰流。 跌水水景设计中，常用堰流形态为宽顶堰流。 当跌水水景的土建尺寸确定以后，首先要确定跌水水景流量Q，当水流从堰顶以 一定的初速度v 0落下时，它会产生一个长度为l d 的水舌。若l d 大于跌水台阶宽度l t ，则 水景水流会跃过跌水台阶；若l d 太小，则有可能出现水景水舌贴着水景跌水墙而形成壁流。这两种情况的出现主要与跌水水景流量Q的大小有关，设计时应尽量选择一个恰当的跌水水景流量以避免上述现象的 水景中的跌水水景设计(二)

跌水水景流量设计

跌水水景流量设计 Revised by Petrel at 2021

水景中的跌水水景设计(一) 跌水水景在小区水景中为常见的一种表现形式,水量控制是其中的一个关健点.跌水水景水量过大则能耗大，长期运转费用高；跌水水景水量过小则达不到预期的设计效果。 关健字:水景?跌水跌水水景 在水景设计中，跌水水景是构成溪流、叠流、瀑布等水景的基本单元，具有动态和声响的效果，因而应用较广。与静态水景不同，动态水景的水是流动的，其流动性一般用循环水泵来维持，水量过大则能耗大，长期运转费用高；水量过小则达不到预期的设计效果。因此，根据水景的规模确定适当的水流量十分重要。 1跌水水景的水力学特征及计算 跌水水景实际上是水力学中的堰流和跌水在实际生活中的应用，跌水水景设计中常用的堰流形式为溢流堰. 根据δ和H的相对尺寸，堰流流态一般分为薄壁堰流、实用堰流、宽顶堰流等三种形式： 当δ/H<0.67，为薄壁堰流；0.67<δ/H<2.5，为实用堰流；2.5<δ/H<10，为宽顶堰流；δ/H>10,为明渠水流，不是堰流。 跌水水景设计中，常用堰流形态为宽顶堰流。当跌水水景的土建尺寸确定以后，首先要确定跌水水景流量Q，当水流从堰顶以一定的初速度v0落下时，它会产生一个长度为ld的水舌。若ld大于跌水台阶宽度lt，则水景水流会跃过跌水台阶；若ld太小，则有可能出现水景水舌贴着水景跌水墙而形成壁流。这两种情况的出现主要与跌水水景流量Q的大小有关，设计时应尽量选择一个恰当的跌水水景流量以避免上述现象的发生。 水景中的跌水水景设计(二) 跌水水景在小区水景中为常见的一种表现形式,水量控制是其中的一个关健点.跌水水景水量过大则能耗大，长期运转费用高；跌水水景水量过小则达不到预期的设计效果。 关健字:水景?跌水跌水水景 1.1跌水水景流量计算根据水力学计算公式，一般宽顶堰自由出流的流量计算式为： Q=σc·m·b·(2g)0.5·H1.5=σc·M·b·H1.5式中b——堰口净宽H——包括行进流速水头的堰前水头，

水景设计中跌水水景的设计

水景设计中跌水水景的设计及计算 在水景设计中，跌水是构成溪流、叠流、瀑布等水景的基本单元，具有动态和声响的效果，因而应用较广。 与静态水景不同，动态水景的水是流动的，其流动性一般用循环水泵来维持，水量过大则能耗大，长期运转费用高；水量过小则达不到预期的设计效果。因此，根据水景的规模确定适当的水流量十分重要。 1 跌水水景的水力学特征及计算 跌水水景实际上是水力学中的堰流和跌水在实际生活中的应用，跌水水景设计中常用的堰流形式为溢流堰。 根据δ和H的相对尺寸，堰流流态一般分为薄壁堰流、实用堰流、宽顶堰流等三种形式： 当 δ/H<0.67，为薄壁堰流； （ δ：堰顶宽；H：堰前水头） 0.67<δ/H<2.5，为实用堰流； 2.5<δ/H<10，为宽顶堰流； δ/H>10,为明渠水流，不是堰流。 跌水水景设计中，常用堰流形态为宽顶堰流。　 当跌水水景的土建尺寸确定以后，首先要确定跌水流量Q，当水流从堰顶以一定的初速度v0落下时，它会产生一个长度为l d的水舌。 若l d 大于跌水台阶宽度l t ，则水流会跃过跌水台阶；若l d 太小，则有

可能出现水舌贴着跌水墙而形成壁流。这两种情况的出现主要与跌水流量Q的大小有关，设计时应尽量选择一个恰当的流量以避免上述现象的发生。 1.1 跌水流量计算 根据水力学计算公式，一般宽顶堰自由出流的流量计算式为： Q=σc·m·b·(2g)0.5·H1.5=σc·M·b·H1.5 式中 b——堰口净宽 H——包括行进流速水头的堰前水头， 2/2g H=H0+υ ——行进流速 式中 υ m——自由溢流的流量系数，与堰型、堰高等边界条件有关 σc——侧收缩系数 M=m·(2g)0.5 当堰口为矩形时，侧收缩系数σc为1，上述计算式即简化为《给水排水设计手册》中的流量计算式： Q=m·b·(2g)0.5·H1.5=M·b·H1.5 上式中，M(或m)为流量系数，与堰的进口边缘形式有关；b为堰口净宽，为已知，因此要求出流量Q，关键要确定出堰前水头H，堰

跌水水景中设计中的计算定稿版

跌水水景中设计中的计 算 HUA system office room 【HUA16H-TTMS2A-HUAS8Q8-HUAH1688】

跌水水景中的计算实例 某宾馆根据其地形条件在大堂内设计一溢流式跌水景，为扇形结构，第一级跌水高度P为2.1 m，堰口为弧线形，长度b=14.65 m，堰顶宽δ=0.15 m，跌水台阶宽度l t=0.7m。 2.1 计算跌水流量Q 根据宾馆大堂环境的要求，跌水流量不须太大，因此，初始选定堰前水头H=0.2 kPa，根据堰流的出口形式，流量系数M=1 417.4，因此试算流量： 2.2 校核跌水水舌 l d根据试算流量Q可求出跌水景溢流口的单宽流量： q=Q/b=4.007×10-3 m3/(s·m) 由此得 D=q2/(g·p3)=1.767 3×10-7 跌水水舌长度： l d=4.30×D0.27×P=0.136m 0.1

水景中的跌水水景设计(一) 在水景设计中，跌水水景是构成溪流、叠流、瀑布等水景的基本单元，具有动态和声响的效果，因而应用较广。 与静态水景不同，动态水景的水是流动的，其流动性一般用循环水泵来维持，水量过大则能耗大，长期运转费用高；水量过小则达不到预期的设计效果。因此，根据水景的规模确定适当的水流量十分重要。 1 跌水水景的水力学特征及计算 跌水水景实际上是水力学中的堰流和跌水在实际生活中的应用，跌水水景设计中常用的堰流形式为溢流堰. 根据δ和H的相对尺寸，堰流流态一般分为薄壁堰流、实用堰流、宽顶堰流等三种形式： 当δ/H<0.67，为薄壁堰流；0.67<δ/H<2.5，为实用堰流；2.5<δ/H<10，为宽顶堰流； δ/H>10,为明渠水流，不是堰流。 跌水水景设计中，常用堰流形态为宽顶堰流。 当跌水水景的土建尺寸确定以后，首先要确定跌水水景流量Q，当水流从堰顶以一定的初速度v0落下时，它会产生一个长度为l d的水舌。若l d大于跌水台阶宽度l t，则水景水流会跃过跌水台阶；若l d太小，则有可能出现水景水舌贴着水景跌水墙而形成壁流。

跌水水景流量设计

跌水水景在小区水景中为常见的一种表现形式，水量控制是其中的一个关健点?跌水水景水量过大则能耗大，长期运转费用高；跌水水景水量过小则达不到预期的设计效果。 关健字:水景？?跌水跌水水景 在水景设计中，跌水水景是构成溪流、叠流、瀑布等水景的基本单元，具有动态和声响的效果，因而应用较广。 与静态水景不同，动态水景的水是流动的，其流动性一般用循环水泵来维持，水量过大则能耗大，长期运转费用高；水量过小则达不到预期的设计效果。因此，根据水景的规模确定适当的水流量十分重要。 1跌水水景的水力学特征及计算 跌水水景实际上是水力学中的堰流和跌水在实际生活中的应用，跌水水景设计中常用的堰流形式为溢流堰. 根据S和H的相对尺寸，堰流流态一般分为薄壁堰流、实用堰流、宽顶堰流等三种形式： 当S /HvO.67为薄壁堰流；0.67< S /H<2.5为实用堰流；2.5< S /H<10为宽顶堰流； S /H>10为明渠水流，不是堰流。 跌水水景设计中，常用堰流形态为宽顶堰流。 当跌水水景的土建尺寸确定以后，首先要确定跌水水景流量Q，当水流从堰顶以一定的初速度vO落下时，它会产生一个长度为Id的水舌。若Id大于跌水台阶宽度It，则水景水流会跃过跌水台阶；若Id太小，则有可能出现水景水舌贴着水景跌水墙而形成壁流。这两种情况的出现主要与跌水水景流量Q的大小有关， 设计时应尽量选择一个恰当的跌水水景流量以避免上述现象的发生。 水景中的跌水水景设计(二) 跌水水景在小区水景中为常见的一种表现形式，水量控制是其中的一个关健点?跌水水景水量过大则能耗大，长期运转费用高；跌水水景水量过小则达不到预期的设计效果。 关健字:水景？?跌水跌水水景 1.1跌水水景流量计算 根据水力学计算公式，一般宽顶堰自由出流的流量计算式为： Q=c c ? m- b ? (2g)0.5 ? H1.5= c c ? M- b ? H1.5 式中b——堰口净宽H ——包括行进流速水头的堰前水头， H=H0u 02/2g 式中u 0―-亍进流速m――自由溢流的流量系数，与堰型、堰高等边界条件有关 c c艸攵缩系数 M=m (2g)0.5当堰口为矩形时，侧收缩系数cc为1,上述计算式即简化为《给

各类水景计算

A景区一水力计算 ①跌泉流量计算Q=σc·m·b·(2g)0.5·H1.5 m取0.36，σc取1，初选H=0.2kpa，测量得L=9.0m，L t=1.6m，b=0.4m 则Q1=0.36*0.4*4.43*0.09=0.057 m3/s 单宽流量q=6.3*10-3 m3/(s·m)，则D=40.11*10-6/0.63=6.37*10-5 L d=4.30*D0.27*p=0.127m，0.1< L d <2/3L t,，经校核，跌水景水舌长度lt在合理范围内，因此，选定的流量可作为选用跌水景循环水泵的依据。 ②同理求Q2 m取0.37，σc取1，初选H=0.2kpa，测量得L=8.4m，L t=1.6m，b=0.4m 则Q2=0.37*0.4*4.43*0.09=0.059 m3/s 单宽流量q=0.059/8.4=7.0 *10-3m3/(s·m)，则D=4.9*10-5/0.63=7.78*10-5 水蛇L d=4.30*0.078*0.4=0.134m，可取 故Q=Q1+Q2=0.057+0.059=0.116 m3/s，考虑到二级跌泉接收部分一级的水量，故取Q=0.1 m3/s 即Q=100L/s，H0=21.65-20.84+1=1.81m h=0.065+0.025=0.09m，故扬程为1.9m ③同上求泵二，m取0.36，σc取1，初选H=0.2kpa，测量得L=6.0m，L t=1.6m，b=0.4m Q=0.057 m3/s 单宽流量q=0.057/6=0.0095=9.5*10-3 m3/(s·m)，则D=14.33*10-5 L d=4.30*D0.27*p=0.158m可取，扬程H=1.81+0.045=1.86m A 景区二水力计算 同上L=7.8m，初选Q=0.057 m3/s 单宽流量q=7.3*10-3 m3/(s·m)，则D=5.3*10-5/0.63=8.48*10-5 L d=4.30*D0.27*p=0.137m，符合要求 扬程H=1.29+0.045=1.33m 选泵IS100-80-125，流量60m3/h，扬程4m，功率1.5kw/h 跌水池 m取0.36，σc取1，初选H=0.2kpa，测量得L=5.5m，L t=1.5m，b=0.6m，p=0.85 Q=0.36*0.6*0.43*0.09=8.61*10-2 单宽流量q=1.56*10-2 m3/(s·m) ，则D=2.45*10-4/0.63=3.9*10-4 L d=4.30*D0.27*p=0.44m 0.1< L d <2/3L t故可取Q=86 L/s，Dn=200mm，h=0.09m，扬程H=1.62+0.09=1.71m，选泵为离心清水泵；XA65/13B 流量86.5 m3/s，扬程13.7m，功率7.5kw/h 卵石涌泉 涌泉高度0.4m，喷头选用YQ-201，额定流量6 m3/h，喷头直径DN25，个数15个，管材：钢衬塑复合管，总流量90 m3

跌水水景中设计中的计算

跌水水景中设计中的计 集团文件发布号：（9816-UATWW?MWUB?WUNN?INNUL?DQQTY?

跌水水景中的计算实例 某宾馆根据其地形条件在大堂内设计一溢流式跌水景，为扇形结构，第一级跌水高度 P为2. 1 m,堰口为弧线形，长度b=14. 65 m,堰顶宽6 =0. 15 m,跌水台阶宽度h二0. 7m。2.1计算跌水流量Q 根据宾馆大堂环境的要求，跌水流量不须太大，因此，初始选定堰前水头 H二0.2kPa,根据堰流的出口形式，流量系数M二1417. 4,因此试算流量： 2. 2校核跌水水舌 h根据试算流量Q可求出跌水景溢流口的单宽流量： q=Q/b=4. 007Xl(r ^/(s . m) 由此得 D=q7(g ? p5)=l. 7673 X10'7 跌水水舌长度： la=4. 3OXD O ：7XP=O. 136m 0. Kl d<2/31t 经校核，跌水景水舌长度It在合理范围内，因此，选定的流量可作为选用跌水景循环水泵的依据。 水景中的跌水水景设计(一)

在水景设计中，跌水水景是构成溪流、叠流、瀑布等水景的基本单元，具有动态和声响的效果，因而应用较广。

与静态水景不同，动态水景的水是流动的，其流动性一般用循环水泵来维持，水量过大则能耗大，长期运转费用高；水量过小则达不到预期的设计效果。因此，根据水景的规模确定适当的水流量十分重要。 1跌水水景的水力学特征及计算 跌水水景实际上是水力学中的堰流和跌水在实际生活中的应用，跌水水景设计中常用的堰流形式为溢流堰. 根据S和H的相对尺寸，堰流流态一般分为薄壁堰流、实用堰流、宽顶堰流等三种形式：当6/H<0.67,为薄壁堰流；0. 67〈§/H〈2.5,为实用堰流；2. 5< 6/H<10,为宽顶堰流； 8/H>10,为明渠水流，不是堰流。 跌水水景设计中，常用堰流形态为宽顶堰流。 当跌水水景的土建尺寸确定以后，首先要确定跌水水景流量Q，当水流从堰顶以一定的初速度v。落下时，它会产生一个长度为b的水舌。若b大于跌水台阶宽度X，则水景水流会跃过跌水台阶；若b太小，则有可能出现水景水舌贴着水景跌水墙而形成壁流。这两种情况的出现主要与跌水水景流量Q的大小有关，设计时应尽量选择一个恰当的跌水水景流量以避免上述现象的 水景中的跌水水景设计（二） 1. 1跌水水景流量计算 根据水力学计算公式，一般宽顶堰自由出流的流量计算式为: Q= o e? m ? b ? (2g)0:? H15= o c? M ? b ? H1 °

冯晓东水景计算书

水景计算书编制人：冯晓东2014年5用10日

一、景观水钵专项计算 依据跌水花钵外形可确定水钵为溢流式跌水水景景，上下两级，由甲方购买成品及提供数据可知，第一级跌水高度P 为0.6 m ，堰口为矩形，单个堰宽0.05m ，堰口个数共计38个，第二级跌水高度P 为1.2m ，堰口为弧线形，堰口个数为30个，跌水台阶宽度l t =0.45m 。 根据水力学计算公式，宽顶堰自由出流的流量计算式为： 3c 3c 2H b M H g b m Q ???=????=σσ 式中 b ——堰口净宽 H ——包括行进流速水头的堰前水头， H=H0+υ02/2g 式中 υ0——行进流速

m ——自由溢流的流量系数，与堰型、堰高等边 界条件有关 =H σc ——侧收缩系数 g m M 2?=当堰口为矩形时，侧收缩系数σc 为1，上述计 算式简化为《给水排水设计手册》中的流量计算式： 332H b M H g b m Q ??=???= 校核水景水舌长度 P D l d 27 .030.4= 式中 D=q2/(g·p3) q--堰口单宽流量，q=Q/b ，m3/(s·m) p--跌水高度，1m g--重力加速度，9.81 m/s2 一、计算跌水流量Q 根据方案效果设计要求及现场情况环境的要求，跌水流量不须太大，因此，初始选定堰前水头H=0.15 kPa ，根据堰流的出口形式，流量系数M=1420，因此试算流量： 332H b M H g b m Q ??=???= =17.8m 3 /h 二、校核跌水水舌 l d 根据试算流量Q 可求出跌水景溢流口的单宽流量：

跌水水景中的计算实例

跌水水景中的计算实例 标签：水景跌水跌水水景2007-08-16 23:32 简介:为了更好的理解跌水水景中的水理计算,现以一工程为例. 关健字:水景跌水跌水水景 某宾馆根据其地形条件在大堂内设计一溢流式跌水景，为扇形结构，第一级跌水高度P 为2.1 m，堰口为弧线形，长度b=14.65 m，堰顶宽δ=0.15 m，跌水台阶宽度l t=0.7m。 2.1计算跌水流量Q 根据宾馆大堂环境的要求，跌水流量不须太大，因此，初始选定堰前水头H=0.2 kPa，根据堰流的出口形式，流量系数M=1 417.4，因此试算流量： 2.2校核跌水水舌 l d根据试算流量Q可求出跌水景溢流口的单宽流量： q=Q/b=4.007×10-3 m3/(s·m) 由此得 D=q2/(g·p3)=1.767 3×10-7 跌水水舌长度： l d=4.30×D0.27×P=0.136m 0.1

标签：水景跌水跌水水景2007-08-16 23:30 简介:跌水水景在小区水景中为常见的一种表现形式,水量控制是其中的一个关健点.跌水水景水量过大则能耗大，长期运转费用高；跌水水景水量过小则达不到预期的设计效果。 关健字:水景跌水跌水水景 在水景设计中，跌水水景是构成溪流、叠流、瀑布等水景的基本单元，具有动态和声响的效果，因而应用较广。 与静态水景不同，动态水景的水是流动的，其流动性一般用循环水泵来维持，水量过大则能耗大，长期运转费用高；水量过小则达不到预期的设计效果。因此，根据水景的规模确定适当的水流量十分重要。 1 跌水水景的水力学特征及计算 跌水水景实际上是水力学中的堰流和跌水在实际生活中的应用，跌水水景设计中常用的堰流形式为溢流堰. 根据δ和H的相对尺寸，堰流流态一般分为薄壁堰流、实用堰流、宽顶堰流等三种形式：当δ/H<0.67，为薄壁堰流；0.67<δ/H<2.5，为实用堰流；2.5<δ/H<10，为宽顶堰流； δ/H>10,为明渠水流，不是堰流。 跌水水景设计中，常用堰流形态为宽顶堰流。 当跌水水景的土建尺寸确定以后，首先要确定跌水水景流量Q，当水流从堰顶以一定的初速度v0落下时，它会产生一个长度为l d的水舌。若l d大于跌水台阶宽度l t，则水景水流会跃过跌水台阶；若l d太小，则有可能出现水景水舌贴着水景跌水墙而形成壁流。这两种情况的出现主要与跌水水景流量Q的大小有关，设计时应尽量选择一个恰当的跌水水景流量以避免上述现象的 水景中的跌水水景设计(二)