ES206 Fluid Mechanics UNIT E: Model and Prototype ROAD MAP . E-1: Dimensional Analysis E-2: Modeling and Similitude ES206 Fluid Mechanics Unit E-1: List of Subjects Dimensional Analysis Buckingham Pi Theorem Common Dimensionless Group Experimental Data Page 1 of 9 Unit E-1 Dimensional Analysis (1) p = f (D, , ,V ) SLIDE 1 UNIT F-1 DimensionalDimensional AnalysisAnalysis (1)(1) ➢ To illustrate a typical fluid mechanics problem in which experimentation is required, consider the steady flow of an incompressible Newtonian fluid through a long, smooth-walled, horizontal, circular pipe: ➢ The pressure drop per unit length ( p ) is a function of: ➢ Pipe diameter (D ) SLIDE 2➢ Fluid density ( ) UNIT F-1 ➢ Fluid viscousity ( ) ➢ Mean velocity ( V ) p = f (D, , ,V ) DimensionalDimensional AnalysisAnalysis (2)(2) ➢ To perform experiments, it would be necessaryTextbook (Munson, Young, to and change Okiishi), page 403 one of the variables while holding others constant ??? How can we combine these data to obtain the desired general functional relationship between pressure drop and variables ? Textbook (Munson, Young, and Okiishi), page 347 Page 2 of 9 Unit E-1 Dimensional Analysis (2) Dp VD = 2 V SLIDE 3 UNIT F-1 DimensionalDimensional AnalysisAnalysis (3)(3) ➢ Fortunately, there is a much simpler approach to this problem that will eliminate the difficulties ➢ It is possible to collect variables and combine into two non-dimensional variables (dimensionless products or groups): Dp VD = 2 V SLIDE 4 UNIT F-1 Pressure Reynolds CoefficientDimensionalDimensionalNumber AnalysisAnalysis (4)(4) Textbook (Munson, Young, and Okiishi), page 348 ➢ The basis for this simplification lies in a consideration of the dimensions of the variables involved – recall, three basic dimensions: ➢ Mass (M), Length (L), Time (T) ➢ A quick check of the dimensions of the two or alternatively, Force (F), Length (L), Time (T) groups: ➢ Question: how many dimensionless products are required to replace the original list of variable ? SLIDE 5 UNIT F-1 Page 3 of 9 Unit E-1 BuckinghamBuckinghamBuckingham PiPi Pi TheoremTheorem Theorem(1) ➢ Buckingham Pi Theorem: SLIDE 6 UNIT F-1 ➢ The dimensionless products are frequently BuckinghamBuckinghamreferred to as “pi terms PiPi” TheoremTheorem (2) ➢ For any physically meaningful equation involving k SLIDE➢ 4Thedimensional required variables, number such of pias: terms is fewer thanUNIT F-1 ➢ Then,the number it follows of that original the equation variables can by be r written, where in r is termsdeterminedDimensionalDimensional of dimensionless by the minimum products AnalysisAnalysis number (pi terms) of (4)(4)as: reference dimensions required to describe the ➢ Theoriginal basis list for of this variables simplification lies in a ➢ considerationUsually, the reference of the dimensions dimensions of requiredthe variables to involveddescribe the– recall, variables three will basic be thedimensions: basic dimensions➢ Mass (M), Length (M, L, T(L ),or TimeF, L, (TT)), so r = 3 ➢ AIn quick a few check rare cases, of the thedimensions variables of may the betwo groups:described by some combination of basic dimensions (such as: F/T2 and L), so r would be equal to or less than 2 ➢ Question: how many dimensionless products are required to replace the original list of variable ? Page 4 of 9 Unit E-1 Common Dimensionless Group (1) SLIDE 11 UNIT F-1 CommonCommon DimensionlessDimensionless NumbersNumbers ➢ Reynolds Number (Osborne Reynolds: 1842-1912) ➢ Froude Number (William Froude: 1810-1879) ➢ Euler Number (Leonhard Euler: 1707-1783) ➢ Cauchy Number (Augustin Louis de Cauchy: 1789-1857) ➢ Mach Number (Ernst Mach: 1838-1916) ➢ Strouhal Number (Vincenz Strouhal: 1850-1922) Page 5 of 9 Unit E-1 Common Dimensionless Group (2) : Density of fluid V Re = V : Flow velocity SLIDE 12 : Viscosity of fluid UNIT F-1 ReynoldsReynolds : Characteristic NumberNumber length (Osborne Reynolds: 1842-1912) V : Flow velocity Fr = g : :Gravitational Density of fluid acceleration gV Re = V : :Characteristic Flow velocity length SLIDE 13 : Viscosity of fluid UNIT F-1 SLIDE 13 UNIT F-1 Froude : Characteristic Number length The Reynolds numberFroude is a measure Number of the ratio of the inertia The Reynolds number(William is a measure Froude: of 1810 the ratio-1879) of the inertia force forceon on an anelement element of fluid of fluidto the to viscous the viscous force on force an element on an element If the Reynolds number is very small, it indicates that the viscous forces areV dominantV in: the Flow flow velocity(creeping flow) Fr = g If the Reynoldsg number: Gravitationalis very large, accelerationit indicates that the viscous forces may be neglected : Characteristic (inviscid flow) length TheThe Froude Froude number number is isa measurea measure of the of ratio the ofratio the inertiaof the inertiaforce forceon onan anelement element of fluid of fluidto the to weight the weight of the element of the element The Froude number will be important in problems involving flows with free surfaces since gravity principally affects this type of flow (typical examples include the study of the flow of water around ships or flow through rivers or open conduits) Page 6 of 9 Unit E-1 Common Dimensionless Group (3) p p : Characteristic pressure Eu = V 2 : Density of fluid V : Flow velocity SLIDE 14 UNIT F-1 V 2 EulerEuler: Density NumberNumber Ca = V : Flow velocity (Leonhard Euler: 1707-1783) Ev Ev : Bulk modulus of elasticity V : Flow velocity Ma = p : Characteristic pressure p c c : Speed of sound Eu = 2 : Density of fluid SLIDE 16 V c = E / UNIT F-1 Ma 2 = Ca V : Flow velocityv MachMach NumberNumber (Ernst Mach: 1838-1916) The Euler number is a measure of the ratio of the pressure forces on an element of fluid to the inertial forces of the element The Euler number is often written in2 terms of a pressure c = Ev / Ma = Ca difference, p as: pV Eu = 2 V : Flow“ velocityPressure Coefficient” Ma=V c c : Speed of sound TheThe Cauchy Mach number / Mach is alsonumber a measure are measureof the ratio of of thethe inertialratio of the inertiaforces forces on an on element an element of fluid of to fluidthe compressibility to the compressibility of the fluid If the Cauchy / Mach number is small (i.e., Ma < 0.3), the inertial forces induced by the fluid motion are not large enough to cause a significant change in the fluid density (hence, compressibility of the flow may be ignored – incompressible flow) Page 7 of 9 Unit E-1 Common Dimensionless Group (4) : Flow oscillation frequency SLIDESt 17= UNIT F-1 V StrouhalStrouhal: Characteristic lengthNumberNumber : FlowV velocity (Vincenz Strouhal: 1850-1922) St = : Flow oscillation frequency V : Characteristic length Theodore von Karman V : Flow velocity (1881-1963) The Strouhal number is a measure of the ratio of the inertial forces due to the unsteadiness of the flow (local acceleration) to the inertial forces due to changes in velocity from point to point in the flow field (convective acceleration) The Strouhal number will be quite important in unsteady, oscillating flows • The strouhal number of a stationary circular cylinder is correlated to the Reynolds number Page 8 of 9 Unit E-1 Experimental Data SLIDE 19 UNIT F-1 ExperimentalExperimental DataData (1) ➢ Application of the pi theorem indicates that if the number of variables minus the number of reference dimensions is equal to unity, then only one pi term is required to describe the phenomenon: ➢ If a given phenomenon can be described with two pi terms, such that: the functional relationship among the variables then be determined by varying and measuring the corresponding values of SLIDEThe 20 results can be presented in a UNIT F-1 simple graph within a valid range (extrapolation should not be taken if outsideExperimentalExperimental of valid range) → DataData (2) Textbook (Munson, Young, and Okiishi), page 367 ➢ As the number of required pi terms increases, it becomes more difficult to display the results in a convenient graphical form and to determine a specific empirical equation that describe the phenomenon ➢ For problems involving three pi terms: For problems involving more than two or three pi terms, it is often necessary to use a model to predict specific characteristics Textbook (Munson, Young, and Okiishi), page 367 Page 9 of 9 Unit E-1 Class Example E-1-1 Related Subjects . “Experimental Data” _________________________________________________________(a) The pressure drop is a function of 4 dimensioned parameters: =p f( V,,, D ) _________________________________________________________p VD This can alternatively be expressed in 2 dimensionless products: 2 = 0.5V _________________________________________________________ −−4 2 1 −2 FL T LT L p FL 0 0 0 VD ( )( )( ) 0 0 0 _________________________________________________________where, 2 2 FLT and −2 FLT 0.5V (FL−−4 T 2)( LT 1 ) FL T _________________________________________________________ V p Re Cp _________________________________________________________(b) Using the data given, (ft/s) (lb/ft^2) p p p 3 192 300 21.33 _________________________________________________________C = = = p 0.5VV220.5( 2 slugs/ft32)V 11 704 1100 5.82 _________________________________________________________17 1090 1700 3.77 3 VD (2 slugs/ft)V ( 0.1 ft) 20 1280 2000 3.20 ________________________________________________________=100V _ (2 10−32 lb s/ft ) _________________________________________________________25 _________________________________________________________ 20 _________________________________________________________