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For office use only T1T2T3T4T eam Control Number24857Problem ChosenBFor office use onlyF1F2F3F42014Mathematical Contest in Modeling(MCM)Summary Sheet (Attach a copy of this page to each copy of your solution paper.)AbstractThe evaluation and selection of‘best all time college coach’is the prob-lem to be addressed.We capture the essential of an evaluation system by reducing the dimensions of the attributes by factor analysis.And we divide our modeling process into three phases:data collection,attribute clarifica-tion,factor model evaluation and model generalization.Firstly,we collect the data from official database.Then,two bottom lines are determined respectively by the number of participating games and win-loss percentage,with these bottom lines we anchor a pool with30to40 candidates,which greatly reduced data volume.And reasonably thefinal top5coaches should generate from this pool.Attribution clarification will be abundant in the body of the model,note that we endeavor to design an attribute to effectively evaluate the improvement of a team before and after the coach came.In phase three,we analyse the problem by following traditional method of the factor model.With three common factors indicating coaches’guiding competency,strength of guided team,competition strength,we get afinal integrated score to evaluate coaches.And we also take into account the time line horizon in two aspects.On the one hand,the numbers of participating games are adjusted on the basis of time.On the other hand,we put forward a potential sub-model in our‘further attempts’concerning overlapping pe-riod of the time of two different coaches.What’s more,a‘pseudo-rose dia-gram’method is tried to show coaches’performance in different areas.Model generalization is examined by three different sports types,Foot-ball,Basketball,and Softball.Besides,our model also can be applied in all possible ball games under the frame of NCAA,assigning slight modification according to specific regulations.The stability of our model is also tested by sensitivity analysis.Who’s who in College Coaching Legends—–A generalized Factor Analysis approach2Contents1Introduction41.1Restatement of the problem (4)1.2NCAA Background and its coaches (4)1.3Previous models (4)2Assumptions5 3Analysis of the Problem5 4Thefirst round of sample selection6 5Attributes for evaluating coaches86Factor analysis model106.1A brief introduction to factor analysis (10)6.2Steps of Factor analysis by SPSS (12)6.3Result of the model (14)7Model generalization15 8Sensitivity analysis189Strength and Weaknesses199.1Strengths (19)9.2Weaknesses (19)10Further attempts20 Appendices22 Appendix A An article for Sports Illustrated221Introduction1.1Restatement of the problemThe‘best all time college coach’is to be selected by Sports Illustrated,a magazine for sports enthusiasts.This is an open-ended problem—-no limitation in method of performance appraisal,gender,or sports types.The following research points should be noted:•whether the time line horizon that we use in our analysis make a difference;•the metrics for assessment are to be articulated;•discuss how the model can be applied in general across both genders and all possible sports;•we need to present our model’s Top5coaches in each of3different sports.1.2NCAA Background and its coachesNational Collegiate Athletic Association(NCAA),an association of1281institution-s,conferences,organizations,and individuals that organizes the athletic programs of many colleges and universities in the United States and Canada.1In our model,only coaches in NCAA are considered and ranked.So,why evaluate the Coaching performance?As the identity of a college football program is shaped by its head coach.Given their impacts,it’s no wonder high profile athletic departments are shelling out millions of dollars per season for the services of coaches.Nick Saban’s2013total pay was$5,395,852and in the same year Coach K earned$7,233,976in total23.Indeed,every athletic director wants to hire the next legendary coach.1.3Previous modelsTraditionally,evaluation in athletics has been based on the single criterion of wins and losses.Years later,in order to reasonably evaluate coaches,many reseachers have implemented the coaching evaluation model.Such as7criteria proposed by Adams:[1] (1)the coach in the profession,(2)knowledge of and practice of medical aspects of coaching,(3)the coach as a person,(4)the coach as an organizer and administrator,(5) knowledge of the sport,(6)public relations,and(7)application of kinesiological and physiological principles.1Wikipedia:/wiki/National_Collegiate_Athletic_ Association#NCAA_sponsored_sports2USAToday:/sports/college/salaries/ncaaf/coach/ 3USAToday:/sports/college/salaries/ncaab/coach/Such models relatively focused more on some subjective and difficult-to-quantify attributes to evaluate coaches,which is quite hard for sports fans to judge coaches. Therefore,we established an objective and quantified model to make a list of‘best all time college coach’.2Assumptions•The sample for our model is restricted within the scale of NCAA sports.That is to say,the coaches we discuss refers to those service for NCAA alone;•We do not take into account the talent born varying from one player to another, in this case,we mean the teams’wins or losses purely associate with the coach;•The difference of games between different Divisions in NCAA is ignored;•Take no account of the errors/amendments of the NCAA game records.3Analysis of the ProblemOur main goal is to build and analyze a mathematical model to choose the‘best all time college coach’for the previous century,i.e.from1913to2013.Objectively,it requires numerous attributes to judge and specify whether a coach is‘the best’,while many of the indicators are deemed hard to quantify.However,to put it in thefirst place, we consider a‘best coach’is,and supposed to be in line with several basic condition-s,which are the prerequisites.Those prerequisites incorporate attributes such as the number of games the coach has participated ever and the win-loss percentage of the total.For instance,under the conditions that either the number of participating games is below100,or the win-loss percentage is less than0.5,we assume this coach cannot be credited as the‘best’,ignoring his/her other facets.Therefore,an attempt was made to screen out the coaches we want,thus to narrow the range in ourfirst stage.At the very beginning,we ignore those whose guiding ses-sions or win-loss percentage is less than a certain level,and then we determine a can-didate pool for‘the best coach’of30-40in scale,according to merely two indicators—-participating games and win-loss percentage.It should be reasonably reliable to draw the top5best coaches from this candidate pool,regardless of any other aspects.One point worth mentioning is that,we take time line horizon as one of the inputs because the number of participating games is changing all the time in the previous century.Hence,it would be unfair to treat this problem by using absolute values, especially for those coaches who lived in the earlier ages when sports were less popular and games were sparse comparatively.4Thefirst round of sample selectionCollege Football is thefirst item in our research.We obtain data concerning all possible coaches since it was initiated,of which the coaches’tenures,participating games and win-loss percentage etc.are included.As a result,we get a sample of2053in scale.Thefirst10candidates’respective information is as below:Table1:Thefirst10candidates’information,here Pct means win-loss percentageCoach From To Years Games Wins Losses Ties PctEli Abbott19021902184400.5Earl Abell19281930328141220.536Earl Able1923192421810620.611 George Adams1890189233634200.944Hobbs Adams1940194632742120.185Steve Addazio20112013337201700.541Alex Agase1964197613135508320.378Phil Ahwesh19491949193600.333Jim Aiken19461950550282200.56Fred Akers19751990161861087530.589 ...........................Firstly,we employ Excel to rule out those who begun their coaching career earlier than1913.Next,considering the impact of time line horizon mentioned in the problem statement,we import our raw data into MATLAB,with an attempt to calculate the coaches’average games every year versus time,as delineated in the Figure1below.Figure1:Diagram of the coaches’average sessions every year versus time It can be drawn from thefigure above,clearly,that the number of each coach’s average games is related with the participating time.With the passing of time and the increasing popularity of sports,coaches’participating games yearly ascends from8to 12or so,that is,the maximum exceed the minimum for50%around.To further refinethe evaluation method,we make the following adjustment for coaches’participating games,and we define it as each coach’s adjusted participating games.Gi =max(G i)G mi×G iWhere•G i is each coach’s participating games;•G im is the average participating games yearly in his/her career;and•max(G i)is the max value in previous century as coaches’average participating games yearlySubsequently,we output the adjusted data,and return it to the Excel table.Obviously,directly using all this data would cause our research a mass,and also the economy of description is hard to achieved.Logically,we propose to employ the following method to narrow the sample range.In general,the most essential attributes to evaluate a coach are his/her guiding ex-perience(which can be shown by participating games)and guiding results(shown by win-loss percentage).Fortunately,these two factors are the ones that can be quantified thus provide feasibility for our modeling.Based on our common sense and select-ed information from sports magazines and associated programs,wefind the winning coaches almost all bear the same characteristics—-at high level in both the partici-pating games and the win-loss percentage.Thus we may arbitrarily enact two bottom line for these two essential attributes,so as to nail down a pool of30to40candidates. Those who do not meet our prerequisites should not be credited as the best in any case.Logically,we expect the model to yield insight into how bottom lines are deter-mined.The matter is,sports types are varying thus the corresponding features are dif-ferent.However,it should be reasonably reliable to the sports fans and commentators’perceptual intuition.Take football as an example,win-loss percentage that exceeds0.75 should be viewed as rather high,and college football coaches of all time who meet this standard are specifically listed in Wikipedia.4Consequently,we are able tofix upon a rational pool of candidate according to those enacted bottom lines and meanwhile, may tender the conditions according to the total scale of the coaches.Still we use Football to further articulate,to determine a pool of candidates for the best coaches,wefirst plot thefigure below to present the distributions of all the coaches.From thefigure2,wefind that once the games number exceeds200or win-loss percentage exceeds0.7,the distribution of the coaches drops significantly.We can thus view this group of coaches as outstanding comparatively,meeting the prerequisites to be the best coaches.4Wikipedia:/wiki/List_of_college_football_coaches_ with_a_.750_winning_percentageFigure2:Hist of the football coaches’number of games versus and average games every year versus games and win-loss percentageHence,we nail down the bottom lines for both the games number and the win-loss percentage,that is,0.7for the former and200for the latter.And these two bottom lines are used as the measure for ourfirst round selection.After round one,merely35 coaches are qualified to remain in the pool of candidates.Since it’s thefirst round sifting,rather than direct and ultimate determination,we hence believe the subjectivity to some extent in the opt of bottom lines will not cloud thefinal results of the best coaches.5Attributes for evaluating coachesThen anchored upon the35candidate selected,we will elaborate our coach evaluation system based on8attributes.In the indicator-select process,we endeavor to examine tradeoffs among the availability for data and difficulty for data quantification.Coaches’pay,for example,though serves as the measure for coaching evaluation,the corre-sponding data are limited.Situations are similar for attributes such as the number of sportsmen the coach ever cultivated for the higher-level tournaments.Ultimately,we determine the8attributes shown in the table below:Further explanation:•Yrs:guiding years of a coach in his/her whole career•G’:Gi =max(G i)G mi×G i see it at last section•Pct:pct=wins+ties/2wins+losses+ties•SRS:a rating that takes into account average point differential and strength of schedule.The rating is denominated in points above/below average,where zeroTable2:symbols and attributessymbol attributeYrs yearsG’adjusted overall gamesPct win-lose percentageP’Adjusted percentage ratioSRS Simple Rating SystemSOS Strength of ScheduleBlp’adjusted Bowls participatedBlw’adjusted Bowls wonis the average.Note that,the bigger for this value,the stronger for the team performance.•SOS:a rating of strength of schedule.The rating is denominated in points above/below average,where zero is the average.Noted that the bigger for this value,the more powerful for the team’s rival,namely the competition is more fierce.Sports-reference provides official statistics for SRS and SOS.5•P’is a new attribute designed in our model.It is the result of Win-loss in the coach’s whole career divided by the average of win-loss percentage(weighted by the number of games in different colleges the coach ever in).We bear in mind that the function of a great coach is not merely manifested in the pure win-loss percentage of the team,it is even more crucial to consider the improvement of the team’s win-loss record with the coach’s participation,or say,the gap between‘af-ter’and‘before’period of this team.(between‘after’and‘before’the dividing line is the day the coach take office)It is because a coach who build a comparative-ly weak team into a much more competitive team would definitely receive more respect and honor from sports fans.To measure and specify this attribute,we col-lect the key official data from sports-reference,which included the independent win-loss percentage for each candidate and each college time when he/she was in the team and,the weighted average of all time win-loss percentage of all the college teams the coach ever in—-regardless of whether the coach is in the team or not.To articulate this attribute,here goes a simple physical example.Ike Armstrong (placedfirst when sorted by alphabetical order),of which the data can be ob-tained from website of sports-reference6.We can easily get the records we need, namely141wins,55losses,15ties,and0.704for win-losses percentage.Fur-ther,specific wins,losses,ties for the team he ever in(Utab college)can also be gained,respectively they are602,419,30,0.587.Consequently,the P’value of Ike Armstrong should be0.704/0.587=1.199,according to our definition.•Bowl games is a special event in thefield of Football games.In North America,a bowl game is one of a number of post-season college football games that are5sports-reference:/cfb/coaches/6sports-reference:/cfb/coaches/ike-armstrong-1.htmlprimarily played by teams from the Division I Football Bowl Subdivision.The times for one coach to eparticipate Bowl games are important indicators to eval-uate a coach.However,noted that the total number of Bowl games held each year is changing from year to year,which should be taken into consideration in the model.Other sports events such as NCAA basketball tournament is also ex-panding.For this reason,it is irrational to use the absolute value of the times for entering the Bowl games (or NCAA basketball tournament etc.)and the times for winning as the evaluation measurement.Whereas the development history and regulations for different sports items vary from one to another (actually the differentiation can be fairly large),we here are incapable to find a generalized method to eliminate this discrepancy ,instead,in-dependent method for each item provide a way out.Due to the time limitation for our research and the need of model generalization,we here only do root extract of blp and blw to debilitate the differentiation,i.e.Blp =√Blp Blw =√Blw For different sports items,we use the same attributes,except Blp’and Blw’,we may change it according to specific sports.For instance,we can use CREG (Number of regular season conference championship won)and FF (Number of NCAA Final Four appearance)to replace Blp and Blw in basketball games.With all the attributes determined,we organized data and show them in the table 3:In addition,before forward analysis there is a need to preprocess the data,owing to the diverse dimensions between these indicators.Methods for data preprocessing are a lot,here we adopt standard score (Z score)method.In statistics,the standard score is the (signed)number of standard deviations an observation or datum is above the mean.Thus,a positive standard score represents a datum above the mean,while a negative standard score represents a datum below the mean.It is a dimensionless quantity obtained by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation.7The standard score of a raw score x is:z =x −µσIt is easy to complete this process by statistical software SPSS.6Factor analysis model 6.1A brief introduction to factor analysisFactor analysis is a statistical method used to describe variability among observed,correlated variables in terms of a potentially lower number of unobserved variables called factors.For example,it is possible that variations in four observed variables mainly reflect the variations in two unobserved variables.Factor analysis searches for 7Wikipedia:/wiki/Standard_scoreTable3:summarized data for best college football coaches’candidatesCoach From To Yrs G’Pct Blp’Blw’P’SRS SOS Ike Armstrong19251949252810.70411 1.199 4.15-4.18 Dana Bible19151946313860.7152 1.73 1.0789.88 1.48 Bernie Bierman19251950242780.71110 1.29514.36 6.29 Red Blaik19341958252940.75900 1.28213.57 2.34 Bobby Bowden19702009405230.74 5.74 4.69 1.10314.25 4.62 Frank Broyles19571976202570.7 3.162 1.18813.29 5.59 Bear Bryant19451982385080.78 5.39 3.87 1.1816.77 6.12 Fritz Crisler19301947182080.76811 1.08317.15 6.67 Bob Devaney19571972162080.806 3.16 2.65 1.25513.13 2.28 Dan Devine19551980222800.742 3.16 2.65 1.22613.61 4.69 Gilmour Dobie19161938222370.70900 1.27.66-2.09 Bobby Dodd19451966222960.713 3.613 1.18414.25 6.6 Vince Dooley19641988253250.715 4.47 2.83 1.09714.537.12 Gus Dorais19221942192320.71910 1.2296-3.21 Pat Dye19741992192400.707 3.16 2.65 1.1929.68 1.51 LaVell Edwards19722000293920.716 4.69 2.65 1.2437.66-0.66 Phillip Fulmer19922008172150.743 3.87 2.83 1.08313.42 4.95 Woody Hayes19511978283290.761 3.32 2.24 1.03117.418.09 Frank Kush19581979222710.764 2.65 2.45 1.238.21-2.07 John McKay19601975162070.7493 2.45 1.05817.298.59 Bob Neyland19261952212860.829 2.65 1.41 1.20815.53 3.17 Tom Osborne19731997253340.8365 3.46 1.18119.7 5.49 Ara Parseghian19561974192250.71 2.24 1.73 1.15317.228.86 Joe Paterno19662011465950.749 6.08 4.9 1.08914.01 5.01 Darrell Royal19541976232970.7494 2.83 1.08916.457.09 Nick Saban19902013182390.748 3.74 2.83 1.12313.41 3.86 Bo Schembechler19631989273460.775 4.12 2.24 1.10414.86 3.37 Francis Schmidt19221942212670.70800 1.1928.490.16 Steve Spurrier19872013243160.733 4.363 1.29313.53 4.64 Bob Stoops19992013152070.804 3.74 2.65 1.11716.66 4.74 Jock Sutherland19191938202550.81221 1.37613.88 1.68 Barry Switzer19731988162090.837 3.61 2.83 1.16320.08 6.63 John Vaught19471973253210.745 4.24 3.16 1.33814.7 5.26 Wallace Wade19231950243070.765 2.24 1.41 1.34913.53 3.15 Bud Wilkinson19471963172220.826 2.83 2.45 1.14717.54 4.94 such joint variations in response to unobserved latent variables.The observed vari-ables are modelled as linear combinations of the potential factors,plus‘error’terms. The information gained about the interdependencies between observed variables can be used later to reduce the set of variables in a putationally this technique is equivalent to low rank approximation of the matrix of observed variables.8 Why carry out factor analyses?If we can summarise a multitude of measure-8Wikipedia:/wiki/Factor_analysisments with a smaller number of factors without losing too much information,we have achieved some economy of description,which is one of the goals of scientific investi-gation.It is also possible that factor analysis will allow us to test theories involving variables which are hard to measure directly.Finally,at a more prosaic level,factor analysis can help us establish that sets of questionnaire items(observed variables)are in fact all measuring the same underlying factor(perhaps with varying reliability)and so can be combined to form a more reliable measure of that factor.6.2Steps of Factor analysis by SPSSFirst we import the decided datasets of8attributes into SPSS,and the results can be obtained below after the software processing.[2-3]Figure3:Table of total variance explainedFigure4:Scree PlotThefirst table and scree plot shows the eigenvalues and the amount of variance explained by each successive factor.The remaining5factors have small eigenvalues value.Once the top3factors are extracted,it adds up to84.3%,meaning a great as the explanatory ability for the original information.To reflect the quantitative analysis of the model,we obtain the following factor loading matrix,actually the loadings are in corresponding to the weight(α1,α2 (i)the set ofx i=αi1f1+αi2f2+...+αim f j+εiAnd the relative strength of the common factors and the original attribute can also be manifested.Figure5:Rotated Component MatrixThen,with Rotated Component Matrix above,wefind the common factor F1main-ly expresses four attributes they are:G,Yrs,P,SRS,and logically,we define the com-mon factor generated from those four attributes as the guiding competency of the coach;similarly,the common factor F2mainly expresses two attributes,and they are: Pct and Blp,which can be de defined as the integrated strength of the guided team; while the common factor F3,mainly expresses two attributes:SOS and Blw,which can be summarized into a‘latent attribute’named competition strength.In order to obtain the quantitative relation,we get the following Component Score Coefficient Matrix processed by SPSS.Further,the function of common factors and the original attributes is listed as bel-low:F1=0.300x1+0.312x2+0.023x3+0.256x4+0.251x5+0.060x6−0.035x7−0.053x8F2=−0.107x1−0,054x2+0.572x3+0.103x4+0.081x5+0.280x6+0.372x7+0.142x8 F3=−0.076x1−0,098x2−0.349x3+0.004x4+0.027x5−0.656x6+0.160x7+0.400x8 Finally we calculate out the integrated factor scores,which should be the average score weighted by the corresponding proportion of variance contribution of each com-mon factor in the total variance contribution.And the function set should be:F=0.477F1+0.284F2+0.239F3Figure6:Component Score Coefficient Matrix6.3Result of the modelwe rank all the coaches in the candidate pool by integrated score represented by F.Seetable4:Table4:Integrated scores for best college football coach(show15data due to the limi-tation of space)Rank coaches F1F2F3Integrated factor1Joe Paterno 3.178-0.3150.421 1.3622Bobby Bowden 2.51-0.2810.502 1.1113Bear Bryant 2.1420.718-0.142 1.0994Tom Osborne0.623 1.969-0.2390.8205Woody Hayes0.140.009 1.6130.4846Barry Switzer-0.705 2.0360.2470.4037Darrell Royal0.0460.161 1.2680.4018Vince Dooley0.361-0.442 1.3730.3749Bo Schembechler0.4810.1430.3040.32910John Vaught0.6060.748-0.870.26511Steve Spurrier0.5180.326-0.5380.18212Bob Stoops-0.718 1.0850.5230.17113Bud Wilkinson-0.718 1.4130.1050.16514Bobby Dodd0.08-0.2080.7390.16215John McKay-0.9620.228 1.870.151Based on this model,we can make a scientific rank list for US college football coach-es,the Top5coaches of our model is Joe Paterno,Bobby Bowden,Bear Bryant,TomOsborne,Woody Hayes.In order to confirm our result,we get a official list of bestcollege football coaches from Bleacherreport99Bleacherreport:/articles/890705-college-football-the-top-50-coTable5:The result of our model in football,the last column is official college basketball ranking from bleacherreportRank Our model Integrated scores bleacherreport1Joe Paterno 1.362Bear Bryant2Bobby Bowden 1.111Knute Rockne3Bear Bryant 1.099Tom Osborne4Tom Osborne0.820Joe Paterno5Woody Hayes0.484Bobby Bowden By comparing thoes two ranking list,wefind that four of our Top5coaches ap-peared in the offical Top5list,which shows that our model is reasonable and effective.7Model generalizationOur coach evaluation system model,of which the feasibility of generalization is sat-isfying,can be accommodated to any possible NCAA sports concourses by assigning slight modification concerning specific regulations.Besides,this method has nothing to do with the coach’s gender,or say,both male and female coaches can be rationally evaluated by this system.And therefore we would like to generalize this model into softball.Further,we take into account the time line horizon,making corresponding adjust-ment for the indicator of number of participating games so as to stipulate that the evaluation measure for1913and2013would be the same.To further generalize the model,first let’s have a test in basketball,of which the data available is adequate enough as football.And the specific steps are as following:1.Obtain data from sports-reference10and rule out the coaches who begun theircoaching career earlier than1913.2.Calculate each coach’s adjusted number of participating games,and adjust theattribute—-FF(Number of NCAA Final Four appearance).3.Determine the bottom lines for thefirst round selection to get a pool of candidatesaccording to the coaches’participating games and win-loss percentage,and the ideal volumn of the pool should be from30to40.Hist diagrams are as below: We determine800as the bottom line for the adjusted participating games and0.7 for the win-loss percentage.Coincidently,we get a candidate pool of35in scale.4.Next,we collect the corresponding data of candidate coaches(P’,SRS,SOS etc.),as presented in the table6:5.Processed by z score method and factor analysis based on the8attributes anddata above,we get three common factors andfinal integrated scores.And among 10sports-reference:/cbb/coaches/Figure7:Hist of the basketball coaches’number of games versus and average gamesevery year versus games and win-loss percentagethe top5candidates,Mike Krzyzewski,Adolph Rupp,Dean SmithˇcˇnBob Knightare the same with the official statistics from bleacherreport.11We can say theeffectiveness of the model is pretty good.See table5.We also apply similar approach into college softball.Maybe it is because the popularity of the softball is not that high,the data avail-able is not adequate to employ ourfirst model.How can our model function in suchsituation?First and foremost,specialized magazines like Sports Illustrated,its com-mentators there would have more internal and confidential databases,which are notexposed publicly.On the one hand,as long as the data is adequate enough,we can saythe original model is completely feasible.While under the situation that there is datadeficit,we can reasonably simplify the model.The derivation of the softball data is NCAA’s official websites,here we only extractdata from All-Division part.12Softball is a comparatively young sports,hence we may arbitrarily neglect the re-stricted condition of‘100years’.Subsequently,because of the data deficit it is hard toadjust the number of participating games.We may as well determine10as the bottomline for participating games and0.74for win-loss percentage,producing a candidatepool of33in scaleAttributed to the inadequacy of the data for attributes,it is not convenient to furtheruse the factor analysis similarly as the assessment model.Therefore,here we employsolely two of the most important attributes to evaluate a coach and they are:partic-ipating games and win-loss percentage in the coach’s whole career.Specifically,wefirst adopt z score to normalize all the data because of the differentiation of various dimensions,and then the integrated score of the coach can be reached by the weighted11bleacherreport:/articles/1341064-10-greatest-coaches-in-ncaa-b 12NCAA softball Coaching Record:/Docs/stats/SB_Records/2012/coaches.pdf。

挟沙水流速度与含沙量垂向分布关系探讨董啸天;李瑞杰;付刚才;张海春【摘要】采用数据拟合构造的挟沙水流掺混长度，结合Prandtl掺混长度理论，得到新的挟沙水流流速分布。

类比动量传递系数表达式与掺混长度的关系，结合挟沙水流流速分布公式，得到新的含沙量垂线分布公式。

分别利用水槽及河道实测资料验证，结果表明流速公式与含沙量公式可以客观准确地描述流速、含沙量垂向变化规律，对含沙量有更高的精度。

误差分析表明：近底层流速误差较大，说明底部边界层的选取与判别仍需深入探究；近表层含沙量误差较大，说明含沙量参考点的选取在理论上仍需研究，以摆脱其随机影响。

%Using the mixing length of sediment⁃laden flow constructed with data fitting and combining the Prandtl mixing length theory, a new equation describing the velocity distribution of sediment⁃laden flow was established. Based on this new equation and the analogy of the relationship between the momentum transfer coefficient and mixing length, a new equation describing the vertical distribution of sediment concentration was also established. Through verification with observed data from the flume experiment and field survey, it was found that these two equations can objectively describe the vertical variations of velocity and sediment concentration, showing a higher precision in describing the sediment concentration. The error analysis shows that the error of the near⁃bottom velocity is large, indicating that the selection and judgment of the bottom boundary layer need further investigation; the error of the sediment concentration close to the surface is large, indicating that the selection of the reference point forthe sediment concentration requires further theoretical research, in order to discard the influence of random effects.【期刊名称】《河海大学学报（自然科学版）》【年(卷),期】2015(000)004【总页数】6页(P371-376)【关键词】近岸海域泥沙运动;挟沙水流掺混长度;流速分布;含沙量垂向分布【作者】董啸天;李瑞杰;付刚才;张海春【作者单位】河海大学海岸灾害及防护教育部重点实验室，江苏南京 210098; 河海大学环境海洋实验室，江苏南京 210098;河海大学海岸灾害及防护教育部重点实验室，江苏南京 210098; 河海大学环境海洋实验室，江苏南京 210098;河海大学海岸灾害及防护教育部重点实验室，江苏南京 210098; 珠海市海骏工程建筑处92311部队，广东珠海 519080;浙江省海洋开发研究院海洋环境与化工研究室，浙江舟山 316100【正文语种】中文【中图分类】TV149挟沙水流是自然界中常见的水流运动现象，精确求解挟沙水流流速垂线分布是研究含沙量分布、水质问题、冲淤演变的首要要求，因此研究挟沙水流流速分布具有重要的理论意义和实用价值。

$xputTWFIN ... Time to end calculation. Default is TWFIN=10.0unless thermal cycling is activated (see ThermalCycling Options).计算结束时间，在热循环开启的情况下默认值为10.0（参考热循环选项）计算完成时间ITB ... Indicator for free surfaces or sharp interfaces=0, no free surface or sharp interface=1, free surface or sharp interfaceDefaults to 0 if IDRG 5; defaults to 1 if IDRG=5.自由表面或粗糙界面的指示器＝0表示二者都没有＝1表示二者有其一是否考虑熔汤表面与空气的影响GY 0.0 Gravity component in y direction. See GRAVY in namelistMOTN for orientation varying gravity. Note: Forcylindrical coordinates, use GRAVX, GRAVY.考虑重力对压铸影响，注意g的方向且为-980,单位为CGS。

GRAVX 0.0 Initial x-direction component of gravity.坐标系为极坐标时用上述参数IFENRG ... Flag for internal energy evaluation=0, no energy solution=2, solve transport equation for internal energy (1storder advection)=3, solve energy transport equation using monotonicitypreserving,second order method.Defaults to 0 if ICMPRS=0 and IHTC=0;defaults to 2 if ICMPRS=1 or IHTC>0.选择能量方程用于计算ISHRNK 0 Solidification shrinkage flag. Shrinkage models mayonly be used for one-material problems (NMAT=1) whenITB=1. Solidification must be activated and solidifieddensity (RHOFS) must be different than liquid fluid #1density (RHOF). Shrinkage cavitation pressure (PCAV)must also be set for Dynamic Shrinkage model. (PCAV,RHOFS and RHOF are in namelist PROPS.)=0, no shrinkage model=1, activate Dynamic Shrinkage model=2, activate Rapid Shrinkage model.凝固收缩的定义：0表示不收缩。

pneumatic chuck ⽓动卡盘 pneumatic conveying ⽓动输送 pneumatic drive ⽓动驱动 pneumatic governor ⽓动蒂器 pneumatic hammer 空⽓锤 pneumatic hoist 风动起重滑车 pneumatic press ⽓动压⼒机 pneumatic rammer ⽓动夯锤 pneumatic shell 耐压壳 pneumatic shock absorber 空⽓缓冲器⽓⼒减震器 pneumatic starting ⽓压起动 pneumatic transport ⽓⼒输送 pneumonics 压⽓学 poinsot motion 潘怂动 poinsot theorem 潘栓理 point mass 质点质量 point mechanics 质点⼒学 point of action 酌点 point of admission 进⽓点 point of application 酌点 point of branching 分⽀点 point of detachment 分离点 point of division 分割点 point of emergency 初始点 point of inflection 拐点 point of inflexion 拐点 point of resonance 共振点 point of support ⽀承点 point of zero moment 拐点 point source explosion 点爆炸 point transfer matrix 点变换矩阵 point vortex 点涡 pointed corrosion 坑蚀 pointer 指针 poise 泊 poiseuille flow 泊肃叶怜 poiseuille's law 泊肃叶定律 poisson bracket 泊松括号 poisson equations 泊松⽅程 poisson kinematic equations 泊松运动⽅程 poisson's ratio 泊松⽐ poisson's theorem 泊松定理 polar 极线 polar coordinates 极坐标 polar equation 极⽅程 polar moment of inertia 极惯性矩 polar motion 极运动 polar resisting moment 极阻⼒矩 polarimetry 测偏振术 polariscope 偏光镜 polarization 偏光 polarization by deformation 依变形极化 polarization by refraction 折射偏振 polarization energy 极化能量 polarization fading 偏振衰减 polarization force 极化⼒ polarization modulation 偏振灯 polarization wave 极化波 polarized light 偏振光 pole 杆 pole curve 本体极迹 pole strength 极强 polhode 本体极迹 polhode cone 本体极迹锥⾯ polished section 抛光磨⽚ poloidal magnetic field ⾓向磁场 poloidal mode 极向模型 polycrystalline material 多晶物质 polycrystalline substance 多晶物质 polydimensional 多维的 polygon 多边形 polygon of forces ⼒多边形 polygonal connection 多⾓联结 polyhedral 多⾯体的 polyhedral angle 多⾯⾓ polyhedron 多⾯体 polymer 聚合物 polymer degradation 解聚酌 polymeric liquids 聚合物液体 polymorphism 多形现象 polymorphy 同质多形 polyphase 多相的 polyslip 复滑移 polytrope 多变曲线 polytropic atmosphere 多元⼤⽓ polytropic exponent 多变指数 polytropic index 多变指数 ponderomotive force 有质动⼒ pore fluid 充液多孔体 pore pressure 孔隙压⼒ pore space 孔隙空间 pore water pressure 孔隙⽔压⼒ porewater 孔隙⽔ porosity 孔隙度 porous 多孔的 porous diaphragm 多孔膜 porous diffusion 多孔扩散 porous disc 透⽔板 porous flow 渗流 porous material 多孔性材料 porous media 多孔介质 porous membrane 多孔膜 porous slab 多孔板 porous structure 多孔结构 portal bracing 桥门撑杆架 position head 位置⽔头 position of rest 静⽌位置 position vector 位⽮ positive feedback 正反馈 positive pressure 正压 positive pressure head 正压头 positive pressure wave 正压⼒波 positive pulse 正脉冲 positive rotation 正旋 positivity wave 正波 possible displacement 可能位移 post buckling behavior 屈曲后⾏为 potential 势 potential difference 势差 potential energy 位能 potential energy of stress 应⼒势能 potential field 势场 potential flow 势流位流 potential force 有势⼒ potential function 势函数 potential function of airy 爱⾥势函数 potential gradient 势梯度 potential head 位势头 potential internal energy 内势能 potential motion 位势运动 potential of central forces 有⼼⼒势 potential of discontinuity 不连续势 potential of simple layer 单层势 potential operator 位势算符 potential scattering 势散射 potential stability 势稳定性 potential theory 位势论 potential variability 势可变性 potential vector 势⽮ potential vortex 势涡 potential vorticity 位势涡度 pothole 地⾯深⽳ power 功率 power extraction 功率提取 power factor 功率因数 power law 指数定律 power law of distribution 指数分布定律 power loss 功率损耗 power of force ⼒功率 power spectrum 密度谱 poynting effect 坡印廷效应 practical efficiency 实际效率 practical system of units 实⽤单位制 prager theory of plasticity 普拉格塑性理论 prandtl body 普朗特体 prandtl boundary layer 普朗特边界层 prandtl glauert law 普朗特格劳厄脱规则 prandtl glauert rule 普朗特格劳厄脱规则 prandtl lifting line theory 普朗特升⼒线理论 prandtl meyer flow 普朗特迈耶流 prandtl meyer wave 普朗特迈耶波 prandtl mixing length 普朗特混合长度 prandtl number 普朗特数 prandtl relation 普朗特关系式 prandtl torsion function 普朗特扭转函数 prandtl tube 普朗特管 pratt truss 普拉特桁架 pre combustion chamber 预燃室 preacceleration 预加速 preadaptation 预适应 preageing 预时效 preamplifier 前置放⼤器 precession 旋进 precession cone 旋进锥 precession of a top 陀螺的旋进 precession of earth 地球的旋进 precession of gyroscope 陀螺旋进 precession of orbit 轨道旋进 precession time 旋进时间 precessional motion 旋进运动 prechamber 预燃室 precipitability 沉淀性 precipitate 沉淀物 precipitation 沉淀 precipitation hardening 沉淀硬化 precision measurement 精确测量 precision of measurement 测量准确度 precision type instrument 精密仪器 precompression 预压缩 preconsolidation 预固结 precooling 预冷却 predeflection 预偏转 preexpansion saturation 膨胀前饱和 preheated air 预热空⽓ preheating 预热 preliminary design 初步设计 preliminary load 预加载 preload 预加载 preoscillation 预振荡 preoscillation time 预振荡时间 press in method 尖端压⼊⽅法 press pump 增压泵 pressductor 压⼒传感器 pressing 压模 pressure 压⼒ pressure amplitude 压幅 pressure angle 压⼒⾓ pressure balance 压⼒秤 pressure center 压⼒中⼼ pressure coefficient 压⼒系数 pressure compensation 压⼒补偿 pressure conduit 压⼒管道 pressure controller 压⼒第器 pressure converter 压⼒变换器压⼒转换器 pressure curve 压⼒曲线 pressure diagram 压⼒曲线 pressure difference 压⼒差 pressure diffusion 压差扩散 pressure distribution 压⼒分布 pressure distribution curve 压⼒分布曲线 pressure drag 压⼒阻⼒ pressure drop 压降 pressure elasticity 压缩弹性 pressure energy 压⼒能量 pressure equalizing 均压 pressure equation 压⼒⽅程 pressure fall 压降 pressure feed 压送 pressure field 压⼒场 pressure filter 压滤机 pressure flow 压⼒流 pressure fluctuation 压⼒波动 pressure force 压⼒ pressure front 激波前沿 pressure function 压⼒函数 pressure gage 压⼒计 pressure gradient 压⼒梯度 pressure head 压头 pressure intensity 压强 pressure jump 压⼒跃变 pressure line 压⼒线 pressure loss 压⼒损耗 pressure lubrication 压⼒润滑 pressure maximum 压⼒ pressure measuring device 压⼒计 pressure meter 压⼒计 pressure method 压⼒法 pressure minimum 最⼩压⼒ pressure nozzle 压⼒喷嘴 pressure outside 外压⼒ pressure piping 压⼒管道 pressure propagation 压⼒传播 pressure pump 压送泵 pressure recovery 压⼒恢复 pressure reducing valve 减压阀 pressure regulator 压⼒蝶器 pressure relief 卸压 pressure resistance 压⼒阻⼒ pressure response 压⼒响应 pressure rise 增压 pressure sensitivity 压⼒灵敏度 pressure shift 压致位移 pressure shock 压缩激波 pressure side 压⼒⾯ pressure stress 压应⼒ pressure tank 压⼒槽 pressure tensor 压⼒张量 pressure test 压⼒试验 pressure transducer 压⼒变换器 pressure tube 压⼒管 pressure turbine 反唤⽔轮机 pressure valve 压⼒阀 pressure vessel 压⼒容器 pressure volume diagram 压容图 pressure water 加压⽔ pressure wave 压⼒波 pressurized gas 压缩⽓体 prestrain 预应变 prestress 预应⼒ prestressed concrete 预应⼒混凝⼟ pretensioning system 先张法 primary back reaction 初级反酌 primary consolidation 初始固结 primary creep 初始蠕变 primary stress 初始应⼒ primary system 值统 primary wave 初波 primaryload 知荷载 principal axes of an area ⾯积轴 principal axis 轴 principal axis of inertia 惯性轴 principal axis of strain 应变轴 principal axis transformation 轴变换 principal bending moment 咒矩 principal contour 秩⾼线 principal coordinate system 著标系 principal coordinates 著标 principal curvature 助率 principal deformation 枝变 principal direction of glide 脂移⽅向 principal direction of oscillation ⽵荡⽅向 principal extension ratio 朱长⽐ principal function 哈密顿酌 principal glide system 脂移系统 principal invariant 只变量 principal line 诌 principal load 知荷载 principal minor 钟式 principal moment of inertia 诌性矩 principal motion 炙动 principal net of the flow 著柳 principal normal 吱线 principal plane 纸⾯ principal plane of flexure 钟曲⾯ principal plane of glide 脂移平⾯ principal plane of inertia 诌性平⾯ principal plane of stress 枝⼒平⾯ principal problem of dynamics 动⼒学知问题 principal radius of curvature 助率半径 principal section 重⾯ principal shear stress 拄应⼒ principal shock 逐 principal simulation error 郑拟误差 principal strain 枝变 principal strain direction 枝变⽅向 principal strain ratio 枝变⽐ principal stress 枝动 principal stress axis 枝⼒轴 principal stress direction 枝⼒⽅向 principal stress method 枝⼒法 principal stress moment 枝⼒矩 principal stress ratio 枝⼒⽐ principal surface tension 猪⾯张⼒ principal symmetry plane 衷称⾯ principal tensile stress 汁应⼒ principal trajectory 朱迹 principal value 值 principal vector 指量 principle 原理 principle of conservation of area ⾯积守恒原理 principle of conservation of energy 能量守恒原理 principle of continuity 连续性原理 principle of dissipation 耗散原理 principle of hydrodynamic images 铃⼒学镜像原理 principle of least action 最⼩酌原理 principle of least constraint 最⼩约束运动原理 principle of least curvature 最⼩曲率原理 principle of least work 最⼩功原理 principle of linear momentum 线性动量定理 principle of linear superposition 线性叠加原理 principle of minimum 最⼩值原理 principle of minimum complementary energy 最⼩余能原理 principle of minimum dissipation of entropy 最⼩耗熵原理 principle of minimum potential energy 最⼩势能原理 principle of minimum virtual mass 最⼩虚质量原理 principle of mobile equilibrium 动态平衡原理 principle of moment 矩原理 principle of momentum conservation 动量守恒原理 principle of parallel flow 平⾏镰理 principle of phase stability 相位稳定性原理 principle of quasi continuity 准连续性原理 principle of relativity 相对性原理 principle of similitude 相似律 principle of solidification 固化原理 principle of statics 静⼒学原理 principle of superposition 叠加原理 principle of the inertia of energy 能量惯性原理 principle of the maximum 值原理 principle of the parallelogram of forces ⼒平⾏四边形定律 principle of transfer 转移原理 principle of using travelling waves ⾏波前进原理 principle of virtual displacement 虚位移原理 principle of virtual power 虚功原理 principle of virtual stress 虚应⼒原理 principle of virtual work 虚功原理 probability 概率 probability amplitude 概率幅度 probability current 概率量 probability current density ⼏率淋度 probability density ⼏率密度 probability distribution 概率分布 probability frequency function 概率频率函数 probability of collision 碰撞⼏率 probability of state 态的概率 probable error 概率误差 procedure 程序 process 过程 product moment 积矩 product moment correlation 积矩关联 product of inertia 惯性积 profile 轮廓 profile chart 轮廓图 profile coefficient 翼型系数 profile curve 轮廓曲线 profile drag 理想铃中的阻⼒ profile error 廓形误差 profile flow 翼型绕流 profile gage 轮廓量规 profile mean line 翼型中线 profilogram 轮廓曲线 program control 程序控制 program debugging 程序翟 programming error 程序设计错误 progressive loading 逐步加载 progressive motion 前进运动 progressive nutation 正章动 progressive precession 正旋进 progressive wave 前进波 project 设计 projectile 抛射体 projectile motion 抛射体运动 projection 射影 projection operator 投影算符 projection plane 投影平⾯ proof stress 容许应⼒ proof test 加压试验 prop 螺旋桨 prop jet 涡轮螺旋桨发动机 propagation 传播 propagation of pressure 压⼒传递 propagation of the tide 潮汐传播 propagation velocity 传播速度 propellant 推进剂 propeller 螺旋桨 propeller blade 螺旋桨叶⽚ propeller effect 螺桨效应 propeller fan 螺桨式风机 propeller pump 螺旋桨式泵 propeller shaft 螺旋桨轴 propeller thrust 螺旋桨推⼒ propeller turbine 螺旋桨式⽔轮机 propeller type flowmeter 螺桨型量计 propelling force 推进⼒ propelling nozzle 推⼒喷管 proper boundary value problem 本者值问题 proper function 本寨数 proper mass 固有质量 proper moment 固有矩 proper motion 固有运动 proper power 固有功率 proper rate 正常速率 proper rotation 固有转动 proper value 本盏 proper vector 本崭量 proper velocity 固有速度 property tensor 特性张量 property to oscillation 振荡特性 proportion ⽐ proportion by weight 重量⽐ proportional limit ⽐例极限。

Flow3d软件简介Flow3d software profilePublished: 2009-5-20 11:21:43 source: China - build China die casting web casting industry and Trade Information Center flagship online text and FLOW-3D] [simulation tool of high efficient, engineers can according to self define various physical models, used in various engineering fields. By accurately predicting free surface flow (free-surface, flows), the FLOW-3D can assist you in improving the existing process in the engineering field.FLOW-3D is a full set of software that does not require additional grid generation modules or post-processing modules.A fully integrated graphical user interface allows users to quickly complete simulation project settings to result output.Mesh and geometry Meshing & Geometry:Structured finite difference method meshMulti block grid technology supports embedded or connected grid blocks.Fractional areas/volumes (FAVOR) technology enables efficient and precise definition of geometric appearanceFree mesh settingsBuilt in basic geometry generatorYou can read various CAD format filesFlow type options Flow, Type, Options:In pipe flow, pipe outflow, and free surface flow modelSupport three-dimensional, two-dimensional or one-dimensional problem calculationTransient flow calculationSupport Cartesian coordinate system or cylindrical coordinate systemSupports non viscous, viscous, laminar, and turbulent flows Multiple quantitative values, specified calculations Coordinate axis calculationTwo phase flowHeat transfer calculation (including phase change) Saturated and unsaturated porous materialFlow definition options Flow, Definition, Options:General initial conditionsBoundary conditionsSymmetryRigid wallContinuousCycleSpecified pressureSpecify speedOutflowMesh overlapStill waterReboot optionsContinuation simulation calculationFrom the previous simulation, the overlapping data is computed . add, remove or change the model parametersNumeric model options Numerical, Modeling, Options:.Volume-of-Fluid (VOF) method tracing fluid boundary --TruVOFEfficient geometric definition of.Fractional areas/volumes (FAVOR). one order, two order and three order flow calculation advection.Sharp fluid interface trackingImplicit solution and explicit solution calculationSupport Point, line, relaxation, and GMRES pressure solversUser defined variables, sub programs, and outputsA computational iteration tool for executing programsFluid model options Fluid, Modeling, Options:A single incompressible fluid - confined or with free surfacesTwo types of incompressible fluids - miscible, or, with, sharp, interfaces, etc.Compressible fluids - subsonic, transonic, supersonicSaturated fluidAcoustic phenomenaMass particles of different density / diameterHot model option Thermal Modeling Options:Natural convectionForced convectionFluid and solid heat conductionFluid and solid heat transferThermal conduction.Designated heat fluxSpecified temperatureHeat transfer from fluid / object to spaceEnergy distribution / concentration in a fluid or solid The heat radiation of the Yi.Viscous heatPhysical model option Physical Modeling Options: Erosion and erosion depositsCavitationPhase change (liquid gas, liquid solid)Surface tensionThermosyphon phenomenonAdhesion of contact surfaceRoughness of contact surfaceSteam and bubblesCuring and melting (heat-of-transformation, table)Mass / momentum / energy generation settingDistributed mass / energy generatorShear change, viscosity model of density change and temperature dependenceThixotropic viscosityElastic tensionElectric fieldInsulation phenomenonElectroosmosisElectrostatic particlesElectric drive phenomenonJoule heatingCoil gasMolecular and turbulent diffusionSpecial physical model Special, Physical, Models:Six degrees of freedom, general moving objects, --user, specified, motion, or, fully-coupled, with, rigid, motion, body, etc.Rotating objectsLinear and, quadratic, flow, losses)Collision modelMetal casting model Metal, Casting, Models:Curing / melting (heat-of-transformation, table)Curing shrinkageTwo element segregation during solidificationThe rate of cure affected by latent heat release Thermal cyclingDefect trackingCavitation modelLost foam casting modelSemi solid material modelSand mould moisturePlunger head movementBack pressure and exhaustSand core blowingTurbulence model Turbulence, Models:.Prandtl mixing length.One-equation transport.Two-equation, k-, epsilon, transport.RNG (renormalized, group, theory).Large eddy simulationPorous material model Porous, Media, Models:Variable pore settingDirectional pore settingThe general fluid loss. Yi (linear and quadratic)Capillary pressureUnsaturated fluidThermal transmission of porous materialsTwo phase fluid combined with more than one material object model Two-phase, and, Two-component, Models:,Liquid / liquid and gas / liquid interfaceTwo phase flow mixingThe mixing of a single compressible fluid with a dispersed incompressible fluidTwo-phase drift fluxPhase change between gas liquid and liquid gasAdiabatic bubblePhase change bubbleContinuous fluid with discontinuous particlesScalar transportDiscontinuous particle model Discrete, Particle, Models:Massless particles are indicatedMass particles of size / weight can be specifiedFluid power drag calculations for one-dimensional and two dimensions.Monte-Carlo diffusionParticle fluid momentum coupling calculationCoefficient of resistance of cohesive particlesPoint or mass particle generatorCharged particlesParticle tracking, particle trackingShallow fluid model Shallow, Flow, Models:Shallow water modelGeneral shallow layerWetting and dryingWind shearSurface roughness effectChemical model Chemistry Models:.Chemical rate equation solver.Stationary or advected speciesAutomation features Automatic Features:Mesh and initial conditions are generatedTime stepping control for precision and stability calculation Automatically limiting fluid compressibilityConvergence and relaxation calculation by FLOW-3D control Automatic prompt for optimum calculationInterfaces with other software Options, for, Coupling, with, Other, Programs:General input format: Stereolithography (STL), files--binary, or, ASCIIFrom ANSYS or I is DEAS to tetrahedral dataDirect data connection port with Tecplot, Ensight, and, FieldViewOutput format to PLOT3D-compatible visualization programs.Neutral file format outputCustomized computing tools are added.Topgraphic dataData manipulation options Data, Processing, Options:Full automatic or customized production chartGraphics support OpenGL-based graphicsColor or black and white vectors, contour maps, 3D, and particle image outputVariable records over timeBy the calculation of force and force the tickets.Animation output.PostScript, JPEG, and Bitmap image outputStreamline output.STL geometry file viewMultiprocessor computing Multi-Processor Computing:Memory sharing calculations (SMP version, support for multicore CPU, support for Windows/Linux systems)Computer cluster system (MP version should be installed on Linux system)。