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					ENGR-320 Fluid Mechanics – Chapter 8: Dimensional Analysis & Similitude




                                       Fluid Mechanics
                                        Summer 2005

                                  Prof. Mesut Pervizpour
                                     Office: KH #349B
                                         Ph: x4046




                                                                                                                 1
                                                  Modified from original notes of Dr. Lennon, CE Dept. Lehigh University
       Dimensional Analysis
• Assume quantities are related by
  multiplication and division
• Dimensions and Units
   P Theorem
• Assemblage of Dimensionless Parameters
• Dimensionless Parameters in Fluids
• Model Studies and Similitude

                                           2
              Pendulum Example
    What is the expression for period of the pendulum?
•  Important parameters L, W, g
•  One way  plot Tp-Lp, Tp-W, Tp-g
•  Relationship: Tp = f (L, W, g)                Lp
•  Dimensions: [Tp] = T
                  [Lp] = L
                                                 W
                  [g] = L/T  2

                  [W] = F or ML/T2
Note W has only F dimensions (can’t divide W by anything
   to eliminate F, so Tp is not a function of W)
    Tp = f (L, g)
By observation What is the only way to combine L & g to have same
     dimensions as Tp?     L                      L 
                                 T              
                                          Tp  f    
                             g                    g              3
    Pendulum Example (cont.)
                                       L
•   Run one experiment and plot Tp vs. g
•   Match the slope: Tp = a + mx (where x =   L
                                              g   )
•   a = 0, and m = 2
•   The full relationship:
                                   L
                           Tp  2
                                   g
     Tp

                     L
                     g
                                                  4
       The Buckingham P Theorem
• “in a physical problem including n quantities in
  which there are m dimensions, the quantities
  can be arranged into n-m independent
  dimensionless parameters”
No. Dimensionless Groups = No. Variables Involved – No. Basis Dimensions (3 max)

• We reduce the number of parameters we need
  to vary to characterize the problem!

                                                                           5
     Example – Drag on a falling body
Drag on a model blimp  1530 N
                                                                FD
What is the force on prototype?
                    FD = fnc ( , , V, D)             W
  Dimensions:               [FD] = ML/T2
                            [] = FT/L2 = M/LT          V
                            [] = M/L3 or FT2/L4
                            [V] = L/T
                            [D] = L
Our variables: FD, , , V, D                n=5
Our dimensions: M, L, T                      m=3
Number of  groups = n – m = 2

Choose 3 repeating variables (usually one fluid property, one
  flow property, and one scale): , V, D
                                                            6
              Example – (cont.)
FD = fcn (, , V, D)
Two equations for  groups:
  1 = a Vb Dc FD         Eqn-1
  2 = A VB DC           Eqn-2
Recognize that i = L0 T0 F0
         L0 T0 F0 = (FT2L-4)a (LT-1)b (L)c F
         L0 T0 F0 = L(-4a+b+c) T(2a-b) F(a+1)
         F0 : 0 = a+1                  a = -1
         T0 : 0 = 2a – b               b = 2a = -2
         L0 : 0 = -4a + b + c          c = -2
                                        FD
              1   V D FD 
                     1  2  2
                                                  (Normalized drag   force)
 Similarly for 2:
                                     V D2 2


                                            1
            2   V D  
                     1 1 1
                                          
                                  VD Re                             7
                    Example (cont.)
1 = f (2)
            f Re 
      FD
                          Where f(Re) = CRe : Drag coefficient
    V D
       2 2


      FD
            C Re
    V D
       2 2


   FD  CRe V 2 D2 For sphere projected Area Ap = D2 /4 = const. D2

              1  2
  FD  C D Ap  V           Formula for drag of an object in air
              2   


                                                                 8
                    Dimensional Analysis
 • Want to study pressure drop as function of
                                                       p1
   velocity (V1) and diameter (do )                         p0
 • Carry out numerous experiments with
   different values of V1 and do and plot the         V1     V0
   data                                                     A0
                                                       A1
       V12          V02
p1        p0  
        2            2
p 
       
       2
         V02  V12 
          d 4 
p  V12   1   1
                            5 parameters:
    2   d0 
                         p,  , V1, d1, do
                   
                4
 p      d1 
        1
 2  d0               2 dimensionless parameters:
  V1                       p/(V2/2), (d1/do)
2
       d                 Much easier to
Cp  f  1 
       d 
        0                establish functional
                           relations with 2                       9
                           parameters, than 5
        Exponent Method - revisit
1.    List all n variables involved in the problem
     • Typically: all variables required to describe the problem
         geometry (D) or define fluid properties (,  ) and to indicate
         external effects (dp/dx)
2.    Express each variables in terms of MLT dimensions (j)
3.    Determine the required number of dimensionless parameters (n – j)
4.    Select a number of repeating variables = number of dimensions
     • All reference dimensions must be included in this set and each
         must be dimensionalls independent of the others
5.    Form a dimensionless parameter by multiplying one of the
      nonrepeating variables by the product of the repeating variables,
      each raised to an unknown exponent
6.    Repeat for each nonrepeating variable
7.    Express result as a relationship among the dimensionless parameters
                                                                   10
                        Example (8.7)
• Find: Drag force on rough sphere
  is function of D, ,  , V and k.
  Express in form:
               3  f ( 1 ,  2 )

    FD     D                      V      k      1   ( D aV b  c )

   MLT-2    L    ML-3    ML-1T-1   LT-1   L      M 0 L0T 0  ( ML1T 1 )( L) a ( LT 1 ) b ( ML3 ) c
                                                 M:        0  1 c       c  1
                                                 L : 0  1  a  b  3c          a  1
                                                 T : 0  1  b            b  1
n=6       No. of dimensional parameters
j=3       No. of dimensions
                                                                                 VD
k = n - j = 3 No. of dimensionless parameters    1              or  1   
                                                        DV                        

Select “repeating” variables: D, V, and 
Combine these with nonrepeating variables: F,  & k


                                                                                            11
                               Example (8.7)
 FD     D                         V     k     3  FD ( D aV b  c )
MLT-2   L     ML-3       ML-1T-1   LT-1   L    M 0 L0T 0  ( MLT 2 )( L) a ( LT 1 ) b ( ML3 ) c
                                               M:        0  1 c       c  1
Select “repeating” variables: D, V, and 
Combine these with nonrepeating                L : 0  1  a  b  3c  a  2
variables: F,  & k                            T : 0  2  b  b  2

 2  k ( D aV b  c )
                                                        FD
                                               3 
M 0 L0T 0  ( L)( L) a ( LT 1 )b ( ML3 ) c          V 2 D 2
M:        0c         c0
L : 0  1  a  b  3c  a  1
T : 0  b  b  0
                                                   FD     VD k
                                                      f(    , )
2 
     k                                         V D
                                                 2 2        D
     D

                                                                                         12
  Assemblage of Dimensionless
          Parameters
• Several forces potentially act on a fluid
• Sum of the forces = ma (the inertial force)
• Inertial force is always present in fluids
  problems (all fluids have mass)
• Nondimensionalize by creating a ratio with
  the inertial force
• The magnitudes of the force ratios for a
  given problem indicate which forces
  govern
                                            13
              Forces on Fluids
•   Force         parameterdimensionless
•                      
    Mass (inertia) ______
•   Viscosity          
                   ______     ______
                                 R Reynolds#
•   Gravitational ______g     ______ #
                                  F Froude
•   Pressure           p
                   ______     ______Pressure Coeff.
                                 Cp
•                      s
    Surface Tension______     ______ #
                                 W Weber
•   Elastic        ______
                       K      ______ #
                                 M Mach

Dependent variable
                                                14
Inertia as our Reference Force
                           F
• F=ma       F  a          f  a
                           
• Fluids problems always (except for statics)
  include a velocity (V), a dimension of flow (l),
  and a density ()

                                L=l       l          M=  l3
  f      M                             T=V
         L2T 2                                                   2
                                                            V
                                                     f i= 
                                                             l
                                                         15
                  Viscous Force
 • What do I need to multiply viscosity by to
   obtain dimensions of force/volume?
                        f                  l
  f   C      C              Ll    T     M  l 3
                                           V
      M                                              V2
      L2T 2    C 
                       1                        fi  
C         
                      LT                                l
      M 
       LT 
                        V2
                              f i Vl
        V           fi                        Vl
    C  2              l                R
        l           fμ    V
                         2    fμ             
                          l              Reynolds number
                                                     16
            Gravitational Force
                    M                  l
       fg           L2T 2     Ll   T        M  l     3

Cg           Cg                      V
       g             L                                   2
                     T 2 
                                             fi  
                                                      V
                                                       l
    M            Cg  
Cg  3
    L

         V2
   fi  l
                      fi V 2
                          
                                   V
   fg   g                      F
                       f g gl       gl
                                         Froude number
                                                    17
                  Pressure Force
                    M                   l
       fp           L2T 2    Ll     T       M  l 3
Cp           Cp                       V
       p            M 
                    LT 2                             V2
                                              fi  
                                                        l
     1
Cp            Cp 
                    1
     L              l
        V2
                                      2p
 fi       l      f i V 2        Cp 
                                     V 2
 fp     p        fp   p
        l
                                     Pressure Coefficient
                                                   18
       Dimensionless parameters
                                           Vl
                                        R
• Reynolds Number                           
                                           V
                                        F
• Froude Number                             gl
                                             V 2 l
• Weber Number                          W
                                              s
• Mach Number                           M
                                             V
                                              c
                                                        2Drag
                                           2p C d 
•   Pressure Coefficient             Cp 
                                           V  2         V 2 A
    • (the dependent variable that we measure experimentally)

                                                          19
 Common Dimensionless No’s.
• Reynolds Number (inertial to viscous forces)                  Vd
                                                           
    • Important in all fluid flow problems                       

• Froude Number (inertial to gravitational forces)         F
                                                                  V
    • Important in problems with a free surface                   gh

• Euler Number (pressure to inertial forces)                      p
    • Important in problems with pressure differences      Cp 
                                                                  V 2
• Mach Number (inertial to elastic forces)                        V    V
    • Important in problems with compressibility effects   M        
                                                                  E/ c

• Weber Number (inertial to surface tension forces)           LV 2
    • Important in problems with surface tension effects
                                                           W
                                                               s


                                                                           20
  Application of Dimensionless
           Parameters
• Pipe Flow
• Pump characterization
• Model Studies and Similitude
  •   dams: spillways, turbines, tunnels
  •   harbors
  •   rivers
  •   ships
  •   ...

                                           21
         Example: Pipe Flow
• What are the important forces?
  ______, ______. Therefore _________
  Inertial viscous                Reynolds
  number.
• What are the important geometric
  parameters?
  diameter, length, roughness height
  _________________________
  • Create dimensionless geometric groups
     l/D      e/D
    ______, ______
• Write the functional relationship            l e
                                      C p  f R , , 
                                               D D
                                                22
                                                          l e
Example: Pipe Flow                               C p  f R , , 
                                                          D D
• How will the results of dimensional
  analysis guide our experiments to
  determine the relationships that govern
  pipe flow?
• If we hold the other two dimensionless
  parameters constant and increase the length
  to diameter ratio, how will Cp change?
                               D       e                2p
   Cp proportional to l      Cp     f  ,R        Cp 
                               l       D                V 2
     D      e   
f  Cp   f  , R      f is friction factor
      l     D 
                                                           23
    Frictional Losses in Straight Pipes
                       0.1

                                                                                 0.05
                                                                                 0.04
                                                                                 0.03

                                                                                 0.02

                                                                                            e
                                                                                 0.015
    friction factor




                                                                                 0.01

f                                                                                0.008
                                                                                 0.006
                                                                                 0.004
                                                                                            D
                                 laminar
                                                                                 0.002

                                                                                 0.001
                                                                                 0.0008

                                                                                 0.0004
                                                                                 0.0002
                                                                                 0.0001
                                                                                 0.00005
                      0.01                                                       smooth

                         1E+03       1E+04   1E+05       1E+06   1E+07   1E+08
                                                     R
                                                                                           24
    What did we gain by using
     Dimensional Analysis?
• Any consistent set of units will work
• We don’t have to conduct an experiment
  on every single size and type of pipe at
  every velocity
• Our results will even work for different
  fluids
• Our results are universally applicable
• We understand the influence of
  temperature
                                             25
 Model Studies and Similitude:
    Scaling Requirements
• dynamic similitude (forces)
  • geometric similitude
    • all linear dimensions must be scaled identically (LR,
      AR, ..)
    • roughness must scale
  • kinematic similitude
    • constant ratio of dynamic pressures at corresponding
      points (VR, QR)
    • streamlines must be geometrically similar
    • _______, __________, _________, and _________
        Mach Reynolds Froude                    Weber
      numbers must be the same
            Cp = f (M, R, F, W, geometry)
                                                          26
          Relaxed Similitude
            Requirements
• Impossible to have all force ratios the same
                           same ____
  unless the model is the _____ size as the
  prototype
• Need to determine which forces are
  important and attempt to keep those force
  ratios the same


                                             27
          Similitude Examples
•   Open hydraulic structures
•   Ship’s resistance
•   Closed conduit
•   Hydraulic machinery




                                28
                             Example
• Consider predicting the drag on a thin
  rectangular plate (w*h) placed normal               FD  f ( w, h,  ,  ,V )
  to the flow.
• Drag is a function of: w, h, , , V                  1  f ( 2 ,  3 )
• Dimensional analysis shows:                       FD          w Vw
                                                            f( ,           )
• And this applies BOTH to a model                 w V
                                                    2   2       h 
  and a prototype
• We can design a model to predict the
  drag on a prototype.                                1m  f ( 2m ,  3m )
• Model will have:                                FDm          w  V w
                                                           f ( m , m m m)
                                                wm  mVm
                                                 2      2      hm        m
• And the prototype will have:
                                       1 p  f ( 2 p ,  3 p )
                                   FDp          w p  pV p w p
                                             f( ,             )
                                 w p  pV p
                                  2       2     hp     p
                                                                                  29
                                   Example
•Similarity conditions
       Geometric similarity
                   wm w p               hm
 2m   2 p                     wm     wp        Gives us the size of the model
                   hm h p               hp
       Kinematic similarity
                    mVm wm  pV p w p                 m  p w p
   3m   3 p                                  Vm             Vp
                      m       p                       p m wm
       Then                                     Gives us the velocity in the model


       Dynamic similarity
                                                            2           2
                                   FDp                wp   p Vp 
                                              FDp             F
                      FDm
     1m   1 p                2
                    wm  mVm
                     2     2
                                w p  pV p
                                         2            w   m  V  Dm
                                                      m        m

                                                                            30
                          Example (8.28)
• Given: Submarine moving below surface in sea water
  ( =1015 kg/m3, n   /  = 1.4x10-6 m2/s).
  Model is 1/20th scale in fresh water (20oC).
• Find: Speed of water in the test dynamic similarity and the ratio
  of drag force on model to that on prototype.
• Solution: Reynolds number is significant parameter.
  Re m  Re p                             Fm
                                                
                                                    Fp

 V L     V L                            mVm lm  pV p l 2
                                            2 2       2
                                                         p
             p p
  m m   
  nm        np                             Fm  mVm lm
                                                    2 2
                                              
            Lp n m                         F p  pV p l 2
                                                    2
                                                        p
    Vm              Vp
            Lm n p                                           2
                                                 1000  28.6   1 
                                                                       2
                                                             
         20 1                                    1015  2   20 
               2m / s
          1 1.4                            Fm
    Vm  28.6 m / s                            0.504                      31
                                           Fp
       Scaling in Open Hydraulic
               Structures
• Examples
  • spillways
  • channel transitions
  • weirs
• Important Forces            NCHRP Request For Proposal on “Effects of Debris on Bridge-Pier Scour “
                                      p://www4.trb.org/trb/crp.nsf/All+Projects/NCHRP+24-26
  • inertial forces
  • gravity: from changes in water surface elevation           V
                                                            F
  • viscous forces (often small relative to gravity forces)     gl
• Minimum similitude requirements                                                       Vl
  • geometric                                                                        R
                                                                                         
  • Froude number
                                                                                          32
    V
 F                                                  Fm  Fp
     gl         Froude similarity
                                                        V  2
                                                                      Vp2
• Froude number the same in model and                      
                                                           m
                                                      g mLm g pLp
  prototype
                                Vm2
                                      Vp2
                                    
• ________________________ L m L p
   difficult to change g
                                                                 Lp
• define length ratio (usually larger than 1)             Lr 
                                                                 Lm
• velocity ratio    Vr  L r
                             Lr
• time ratio            tr        Lr
                             Vr
• discharge ratio   Qr  Vr Ar  L r L r L r  Lr  5/ 2


• force ratio                           3 Lr
                    Fr  M r a r   r L r 2  L3r
                                           tr
                                                                      33
     Example: Spillway Model
• A 50 cm tall scale model of a proposed 50
  m spillway is used to predict prototype
  flow conditions. If the design flood
  discharge over the spillway is 20,000 m3/s,
  what water flow rate should be tested in the
  model?
              Fm  Fp         Lr  100

               Qr  L
                    5/ 2
                    r       100,000
                   20,000 m3 s
              Qm               0.2 m3 s
                    100,000
                                             34
          Ship’s Resistance
• Skin friction ______________
                 Viscosity, roughness
• Wave drag (free surface effect) ________
                                   gravity
• Therefore we need ________ and Froude
                      Reynolds        ______
  similarity

              2Drag           e , R, F 
                     Cd  f           
              V A
                 2
                             l         

                                               35
  Reynolds and Froude Similarity?
    Reynolds                               Froude
       Vl                                    V
    R                                     F
                                              gl
 mVmlm  pV p l p   Water is the only     Vr  L r
       
   m     p         practical fluid
   Vmlm  V pl p
     Vp lm
                             1
    Vm l p                          Lr   Lr = 1
           1                  Lr
    Vr 
           Lr
                                                      36
            Ship’s Resistance
• Can’t have both Reynolds      2D total       e       
                                      C d  f  , R, F 
                                                       
  and Froude similarity         V A
                                  2
                                               D       
• Froude hypothesis: the two
                                 D total  D f  D w
  forms of drag are
  independent
                                           V 2 A  e        
• Measure total drag on Ship     Df                f  ,R 
                                             2         D 
• Use analytical methods to                          analytical
  calculate the skin friction
                                            V 2 A
• Remainder is wave drag           Dw                f F 
                                                 2
                                                     empirical
                                                           37
 Closed Conduit Incompressible
            Flow
• Forces
   • __________
      viscosity
   • __________
       inertia
• If same fluid is used for model and
  prototype
  • VD must be the same
                     velocity
  • Results in high _________ in the model
• High Reynolds number (R)
  • Often results are independent of R for very high
    R
                                                  38
   Example: Valve Coefficient
                                  2p
• The pressure coefficient, Cp         , for a
                                  V 2
  600-mm-diameter valve is to be
  determined for 5 ºC water at a maximum
  velocity of 2.5 m/s. The model is a 60-mm-
  diameter valve operating with water at 5
  ºC. What water velocity is needed?



                                              39
      Example: Valve Coefficient
• Note: roughness height should scale!
• Reynolds similarity
           Vl           VD            ν = 1.52 x 10-6 m2/s
        R          R
                          n
Vm Dm       Vp Dp          Vp Dp
                   Vm 
 nm          np                Dm

       (2.5m / s)0.6m 
  Vm                               Vm = 25 m/s
           0.06m
                                                        40
   Example: Valve Coefficient
        (Reduce Vm?)
• What could we do to reduce the velocity in
  the model and still get the same high    Vl
                                        R
  Reynolds number?                          
                                           VD
 Decrease kinematic viscosity          R
                                          n
     Use a different fluid
     Use water at a higher temperature

                                            41
          Example: Valve Coefficient
 • Change model fluid to water at 80 ºC
           VD
      R                                         νm = 0.367 x 10-6 m2/s
                                                      _____________
           n
                                                 νp = ______________
                                                      1.52 x 10-6 m2/s
 Vm Dm         Vp Dp                n mV p D p
                              Vm 
     nm         np                   n p Dm

     
       0.367 x10 m / s (2.5m / s)0.6m
                     6   2
Vm
            1.52 x10 m / s 0.06m
                          6    2
                                                       Vm = 6 m/s
                                                                    42
Approximate Similitude at High
     Reynolds Numbers
• High Reynolds number means inertial
                                 ______
  forces are much greater than _______
                                viscous
  forces
• Pressure coefficient becomes independent
  of R for high R




                                             43
         Pressure Coefficient for a
              Venturi Meter
             10
         Cp



      2p
Cp 
      V 2

             1
             1E+00 1E+01 1E+02 1E+03 1E+04 1E+05 1E+06
                                R            Vl
Similar to rough pipes in                 R
                                              
Moody diagram!                                      44
  Hydraulic Machinery: Pumps
• Rotational speed of pump or turbine is an
  additional parameter
  • additional dimensionless parameter is the ratio
    of the rotational speed to the velocity of the
    water _________________________________
            streamlines must be geometrically similar
  • homologous units: velocity vectors scale _____Vr  lr
• Now we can’t get same Reynolds Number!
  • Reynolds similarity requires           1
                                      Vr 
                                           lr
  • Scale effects

                                                        45
 Dimensional Analysis Summary
Dimensional analysis:
 • enables us to identify the important
   parameters in a problem
 • simplifies our experimental protocol
 • does not tell us the coefficients or powers
   of the dimensionless groups (need to be
   determined from theory or experiments)
 • guides experimental work using small
   models to study large prototypes
                                                 46
                          Port Model
• A working scale model was used to eliminated danger to boaters
  from the "keeper roller" downstream from the diversion structure




  http://ogee.hydlab.do.usbr.gov/hs/hs.html                  47
              Hoover Dam Spillway
A 1:60 scale
hydraulic model
of the tunnel
spillway at
Hoover Dam for
investigation of
cavitation damage
preventing air
slots.



                    http://ogee.hydlab.do.usbr.gov/hs/hs.html
                                                                48
Irrigation Canal Controls
          http://elib.cs.berkeley.edu/cypress.html




                                           49
                Spillways
Frenchman Dam and spillway (in use).
Lahontan Region (6)




                                       50
                   Dams
Dec 01, 1974
Cedar Springs Dam, spillway & Reservoir
Santa Ana Region (8)




                                          51
                  Spillway
Mar 01, 1971
Cedar Springs Spillway construction.
Santa Ana Region (8)




                                       52
                                            Kinematic Viscosity
                                 1.00E-03
kinematic viscosity 20C (m2/s)




                                 1.00E-04


                                 1.00E-05



                                 1.00E-06


                                 1.00E-07




                                                                  53
Kinematic Viscosity (m /s)
                             Kinematic Viscosity of Water
                             2.0E-06
2




                             1.5E-06

                             1.0E-06

                             5.0E-07

                             0.0E+00
                                       0   20     40      60      80   100
                                                Temperature (C)

                                                                         54

				
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