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)

Dp  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): Dp  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 ➢ theThen, number it follows of that original the equation variables can by be r written, where in r is determinedtermsDimensionalDimensional 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

➢ (Osborne Reynolds: 1842-1912)

➢ (William Froude: 1810-1879)

➢ (Leonhard Euler: 1707-1783)

➢ Cauchy Number (Augustin Louis de Cauchy: 1789-1857)

➢ (Ernst Mach: 1838-1916)

➢ (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 V : Flow 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: =pfVD ( ,,, ) ______pVD  This can alternatively be expressed in 2 dimensionless products: 2 =   ______0.5V  −−421 −2 FL TLTL pFL 0 00 VD ( )( )( ) 0 00 ______where, 2 2 FLT and −2 FLT ______0.5V (FL−−421 TLT)( ) ______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 0.5VV220.5( 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 ______15 -0.9996 Cp = 6384.5 Re ______10 ______5 ______PressureCoefficient (Cp) 0 0 500 1000 1500 2000 2500 ______Reynolds Number (Re) ______−0.9996 __ (c) Limitation: correlation equation C p = 6384.5 Re is valid only in the range of 300 Re 2,000 ______ES206 Fluid Mechanics UNIT E: Model and Prototype

ROAD MAP . . . E-1: Dimensional Analysis E-2: Modeling and Similitude

ES206 Fluid Mechanics Unit E-2: List of Subjects

 Model and Prototype  What is Similitude?  Geometric Similitude  Dynamic Similitude

Page 1 of 8 Unit E-2 Model and Prototype (1)

SLIDE 1 UNIT F-2

ModelModel andand PrototypePrototype

➢ Models are widely used in fluid mechanics ➢ An engineering model is a representation of a physical system that may be used to predict the behavior of the system in some desired respect ➢ The physical system for which the predictions are to be made is called the prototype ➢ Model design conditions (also called similarity requirements or modeling laws): ➢ For any given problem: SLIDE 2 UNIT F-2 ➢ For a model (m): ➢ The pi terms of the model must match against the prototype SimilaritySimilarity RequirementsRequirements (1)

➢ Prototype: ➢ Model: ➢ The pi terms can be developed so that contains the variable that is to be predicted from observations made on the model:

➢ Then, with the presumption that the form of  is the same for model and prototype: Page 2 of 8 Unit E-2 Model and Prototype (2)

SLIDE 3 UNIT F-2 → Similarity Requirements (2)

______➢ As an example, consider the problem of determining ______the drag, , on a thin rectangular plate (w  h in size) ______placed normal to a fluid with velocity V ______➢ The dimensional analysis of this problem was ______performed in example 7.1:______➢ Application of pi theorem yielded: ______➢ Now, designing a model that could be used to predict ______SLIDE 4the drag on a prototype with a similar relationship:UNIT F-2 ______Similarity Requirements (3) ______➢ The model design conditions (or similarity ______requirements) are therefore:______➢ The size of the model is obtained from the first ______relationship: ______➢ The similarity requirement provides the ______required velocity for the model: ______➢ With above similarity requirements satisfied, the drag is predicted: ______→ ______Page 3 of 8 Unit E-2 Class Example E-2-1 Related Subjects . . . “Model and Prototype”

______The drag on a small sphere is a function of 3 dimensioned parameters (diameter of the ______sphere, velocity of the sphere, and the fluid density): ______Drag,,= fDV(  ) ______Using Buckingham PI theorem (4 3−= 1 : =1 c onsta nt ), this can be written in terms of ______dimensionless products: ______Drag = constant ______VD22 ______Therefore, Dragconstant= ( ) VD22 ______If V is doubled (2), Drag will be increased by a factor of 4 (4)

______Page 4 of 8 Unit E-2 What is Similitude?

SLIDE 8 UNIT F-2

Ship Interaction Tests VirginiaWhatWhat Tech AOE Water Tunnel isis Similitude?Similitude?Australian Maritime College

➢ Similitude is the theory and art of predicting prototype (actual-scale) performance from model (usually smaller scale) observations

• Since almost all engineering experiments are performed in “scaled model,” it is critically important to maintain similitude

• Although the SIZE IS DIFFERENT, the testing condition of model (testing in smaller scale) SIMILARLY represent the operating condition of prototype (operating in actual-scale) in terms of: o Similarity in Shape (GEOMETRIC SIMILITUDE) o Similarity in Conditions (DYNAMIC SIMILITUDE)

• Although the size is different, the results of modelShip Interaction(testing Tests in smaller scale) canVirginia predict Tech AOE Waterthe Tunnelbehavior of prototype (operatiAustralianng Maritime in actual College -scale): o Results in COEFFICIENTS not in actual values

“NON-DIMENSIONAL” (size-independent) expression of engineering properties is crucially important, in order to conduct engineering experiments!!! Page 5 of 8 Unit E-2 Geometric Similitude

p: Prototype (actual-size)

SLIDE 9 UNIT F-2 m: Model GeometricGeometric SimilitudeSimilitude(small-scale)

p: Prototype (actual-size)

m: Model (small-scale)

 w c m m m Am 2 Vm 3 = = = Lr = L = L  w c r r p p p Ap Vp (L : Length Scale) (Area) (Volume) r Page 6 of 8 Unit E-2 Dynamic Similitude (1)

Fm / Fp = constant

F F Force of gravity: g m g p F F Pressure force: pm p p

Viscous force: Fvm Fv p

SLIDE 11 UNIT F-2

Resultant force: FRm FR p DynamicDynamic SimilitudeSimilitude (2)

Fg M a m = m m (Froude Number) F M a Frm = Frp g p p p F M a vm = m m Rem = Re p (Reynolds Number) Fv p M pa p

Fp M a m = m m C = C (Euler Number) F M a p m p p p p p p (or Pressure Coefficient)

The requirement for similarity of flow between model and prototype is that the significant -groups must be equal for model and prototype

Page 7 of 8 Unit E-2 Dynamic Similitude (2)

Fg M a m = m m (Froude Number) F M a Frm = Frp g p p p F M a vm = m m Re m = Re p (Reynolds Number) Fv p M pa p

Fp M a m = m m C = C (Euler Number) F M a pm p p p p p p SLIDE 12 (or Pressure Coefficient) UNIT F-2

DynamicDynamic SimilitudeSimilitude (3)

Flows without free-surface effects (flow of liquids or gases in closed conduits): C = C Rem = Re p p m p p High Reynolds number flows (high-speed flows): C = C Ma m = Ma p p m p p Free-surface model studies:

Frm = Frp

Page 8 of 8 Unit E-2 Class Example E-2-2 Related Subjects . . . “Dynamic Similitude”

______In order to examine fluid dynamic characteristics of an airplane, the Reynolds number

______must be maintained consistent as a dynamic similarity: Re Repm= ______For prototype (p): V = 2 4 0 m p h ______p ______ =1.75610 slugs/ft−33 (at 10,000 ft) ______p −72 ______ p =3.534  10 lb  s/ft (at 10,000 ft) ______For model (m): _  =2.37710 slugs/ft−33 (at sea-level) ______m  =3.73710 lbs/ft−72 (at sea-level) ______m

______Therefore, ______ pppV mmmV ______=> = ______pm ______−−37  pp 1.756 103.737 1020  VV==m 240 mph = 3,749.67 mph ______mp ( ) −−37 mp m 2.377 103.534 101  ______At sea-level, this is equivalent to Mach 4.93 ______