FUNDAMENTALS OF FLUID MECHANICSFLUID MECHANICS …

1 FUNDAMENTALS OFFUNDAMENTALS OFFLUID MECHANICSFLUID MECHANICSFLUID MECHANICSFLUID MECHANICSC hapter 7 Dimensional AnalysisChapter 7 Dimensional AnalysisChapter 7 Dimensional AnalysisChapter 7 Dimensional AnalysisModeling, and SimilitudeModeling, and SimilitudeModeling, and SimilitudeModeling, and Similitude1 MAIN TOPICSMAIN TOPICSMAIN TOPICSMAIN TOPICS Diil A l iDiil A l i Dimensional AnalysisDimensional Analysis Buckingham Pi TheoremBuckingham Pi Theorem Determination of Pi TermsDetermination of Pi Terms Comments about Dimensional AnalysisComments about Dimensional Analysis Common Dimensionless Groups in FLUID MechanicsCommon Dimensionless Groups in FLUID MECHANICS Correlation of Experimental DataCorrelation of Experimental Datapp Modeling and SimilitudeModeling and Similitude Typical Model StudiesTypical Model Studies Typical Model StudiesTypical Model Studies Similitude Based on Governing Differential EquationSimilitude Based on Governing Differential

2 Equation2 Dimensional AnalysisDimensional Analysis1/41/4 Dimensional Analysis Dimensional Analysis 1/41/4 A A typical FLUID MECHANICS problemtypical FLUID MECHANICS problemin which in which experimentation is required consider the experimentation is required consider the steady flow of an steady flow of an incompressible Newtonian FLUID through a long, smoothincompressible Newtonian FLUID through a long, smooth--walled, horizontal, circular pipewalled, horizontal, circular pipe.. An important characteristic of this systemAn important characteristic of this system, which would , which would be interest to an engineer designing a pipeline, is the be interest to an engineer designing a pipeline, is the pressure drop per unit lengthpressure drop per unit lengththat develops along the pipe that develops along the pipe as a result of a result of AnalysisDimensional Analysis2/42/4 Dimensional Analysis Dimensional Analysis 2/42/4 The The first step in the planning of an experimentfirst step in the planning of an experimentto study to study this problem would be to this problem would be to decide the factors, or variablesdecide the factors, or variables.

3 That will have an effect on the that will have an effect on the pressure droppressure Pressure drop per unit lengthPressure drop per unit lengthppgppg),,,(VDfp Pressure drop per unit length depends on FOUR variables:Pressure drop per unit length depends on FOUR variables:sphere size (D); speed (V); FLUID density ( ); FLUID viscositysphere size (D); speed (V); FLUID density ( ); FLUID viscositysphere size (D); speed (V); FLUID density ( ); FLUID viscosity sphere size (D); speed (V); FLUID density ( ); FLUID viscosity (m)(m)4 Dimensional AnalysisDimensional Analysis3/43/4 Dimensional Analysis Dimensional Analysis 3/43/4 To perform the experiments in a meaningful and To perform the experiments in a meaningful and systematic manner, it would be necessary to change the systematic manner, it would be necessary to change the variable, such as the velocity, which holding all other variable, such as the velocity, which holding all other constant, and measure the corresponding pressure , and measure the corresponding pressure drop.

4 Difficulty to Difficulty to determine the functional relationship between determine the functional relationship between the pressure drop and the various factsthe pressure drop and the various factsthat influence influence of TestsSeries of TestsSeries of TestsSeries of Tests6 Dimensional AnalysisDimensional Analysis4/44/4 Dimensional Analysis Dimensional Analysis 4/44/4 Fortunately, there is a Fortunately, there is a much simpler approachmuch simpler approachto the to the problem that will eliminate the difficulties described problem that will eliminate the difficulties described Collecting these variables into two nonCollecting these variables into two non--dimensional dimensional ggcombinations of the variablescombinations of the variables(called dimensionless (called dimensionless product or product or dimensionless groupsdimensionless groups)) Only one dependent and one Only one dependent and one independent variableindependent variable VDpD Easy to set up experiments to Easy to set up experiments to determine dependencydetermine dependency Easy to present results (one graph)

5 Easy to present results (one graph) V27 Easy to present results (one graph)Easy to present results (one graph)Plot of Pressure Drop Data UsingPlot of Pressure Drop Data UsingPlot of Pressure Drop Data Using ..Plot of Pressure Drop Data Using ..0003 TLF)L/F(LpD 21242 TLF)FT)(TFL(V 000124)L)(LT)(TFL(VD 0002 TLF)TFL()L)(LT)(TFL(VD dimensionless product or dimensionless product or dimensionless groupsdimensionless groups8 Buckingham Pi TheoremBuckingham Pi Theorem1/51/5 Buckingham Pi Theorem Buckingham Pi Theorem 1/51/5 A fundamental question we must answer is A fundamental question we must answer is how many how many dimensionless products are requireddimensionless products are requiredto replace the original list of to replace the original list of variables ?

6 Variables ?variables ?variables ? The answer to this question is supplied by the The answer to this question is supplied by the basic theorem of basic theorem of dimensional analysisdimensional analysisthat statesthat statesdimensional analysisdimensional analysisthat statesthat states If an equation involving k variablesis dimensionally homogeneous it can be reduced to a relationship amonghomogeneous, it can be reduced to a relationship among k-r independent dimensionless products, where r is the minimum number of reference dimensionsrequired tominimum number of reference dimensionsrequired to describe the termsPi terms9 Buckingham Pi TheoremBuckingham Pi TheoremPi termsPi termsBuckingham Pi TheoremBuckingham Pi Theorem2/52/5 Buckingham Pi Theorem Buckingham Pi Theorem 2/52/5 Given a physical problem in which the Given a physical problem in which the dependent variabledependent variableis a function of kis a function of k--1 independent variables1 independent )u.

7 ,u,u(fuk321 Mathematically, we can express the functional relationship Mathematically, we can express the functional relationship in the equivalent formin the equivalent formin the equivalent formin the equivalent form0)u,..,u,u,u(gk321 ),,,,(gk321wwhere g is an unspecified function, different from g is an unspecified function, different from Pi TheoremBuckingham Pi Theorem3/53/5 Buckingham Pi Theorem Buckingham Pi Theorem 3/53/5 The Buckingham Pi theorem states that: Given a relation The Buckingham Pi theorem states that: Given a relation among k variables of the formamong k variables of the form ThThkiblkiblbditbditkkidd tidd t0)u,..,u,u,u(gk321 The The k variablesk variablesmay be grouped into may be grouped into kk--r independent r independent dimensionless productsdimensionless products, or terms, expressible in , or terms, expressible in ftilf bftilf bfunctional form byfunctional form by),,,,,(rk321 ?)

8 ??? ?? ??0),,,,,,(orrk321 r ?? r ?? ?? ??11 Buckingham Pi TheoremBuckingham Pi Theorem4/54/5 Buckingham Pi Theorem Buckingham Pi Theorem 4/54/5 ThThbfbfillb llhillb llh The The number of number of rris usually, but not always, equal to the is usually, but not always, equal to the minimum number of independent dimensions required to minimum number of independent dimensions required to specify the dimensions of all the parametersspecify the dimensions of all the parametersUsuallyUsuallythethespecify the dimensions of all the parametersspecify the dimensions of all the parameters.. UsuallyUsuallythe the reference dimensionsreference dimensionsrequired to describe the variables required to describe the variables will be the basic dimensionswill be the basic dimensionsM L and T or F L and TM L and T or F L and Twill be the basic dimensions will be the basic dimensions M, L, and T or F, L, and TM, L, and T or F, L, and ThThisis theorem does theorem does not predict not predict the particular the particular functional functional formformof of or or.

9 The . The functional relationfunctional relation among the among the ggindependent dimensionless products independent dimensionless products must be determined must be determined The kThe k--r dimensionless r dimensionless terms obtained from the terms obtained from the procedure are are Pi TheoremBuckingham Pi Theorem5/55/5 Buckingham Pi Theorem Buckingham Pi Theorem 5/55/5 A term is not independent if it can be obtained from a A term is not independent if it can be obtained from a product or quotient of the other dimensionless products of product or quotient of the other dimensionless products of the problem. For example, ifthe problem. For example, if4/3234/3163215or2 then neither then neither 55nor nor 66is independent of the otheris independent of the other332 then neither then neither 55nor nor 66is independent of the other is independent of the other dimensionless products or dimensionless products or dimensionless groupsdimensionless of Pi TermsDetermination of Pi Terms1/121/12 Determination of Pi Terms Determination of Pi Terms 1/121/12 Several methods can be used to form the dimensionless Several methods can be used to form the dimensionless products, or pi term, that arise in a dimensional , or pi term, that arise in a dimensional analysis.

10 Regardless of the method to be used to determine the Regardless of the method to be used to determine the dimensionless products, dimensionless products, one begins by listingone begins by listingall all ppgy ggy g((dimensionaldimensional)) variables that are known (or believed) to variables that are known (or believed) to affect the given flow phenomenonaffect the given flow Eight steps listedEight steps listedbelow outline a recommended below outline a recommended procedure for determining the for determining the for determining the for determining the of Pi TermsDetermination of Pi Terms2/122/12 Determination of Pi Terms Determination of Pi Terms 2/122/12 Step 1 Step 1 List all the variablesList all the List all the dimensional variables all the dimensional variables involved.