Tambahan Analisa Dimensional1

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Transcript of Tambahan Analisa Dimensional1


    NazaruddinNazaruddin SinagaSinaga


  • How to find the rank ?

    Consider the pressure drop #p in a pipeline, which is expected to depend on the inside diameter d of the pipe, its length l, the average size e of the wall roughness elements, the average flow velocity U, the fluid density , and the fluid viscosity .

    We can write the functional dependence as


  • The dimensions of the variables can be arranged in the form of the following matrix:

    The rank r of any matrix is defined to be the size of the largest square submatrix that has a nonzero determinant.


  • Testing the determinant of the first three rows and columns, we obtain

    D = K(bfD = K(bf--cece))-- L(L(afaf--cdcd)+ M()+ M(aeae--bdbd) )

    D = 1(1.0D = 1(1.0--1.0))1.0))--0(0(--1.01.0--1.1.--2)+0(2)+0(--1.01.0--1.1.--2)2)

    = 0= 0

    K L MK L M

    a b ca b c

    d e fd e f


  • 5However, there does exist a nonzero third-order determinant, for example, the one formed by the last three columns:

    D = K(bfD = K(bf--cece))-- L(L(afaf--cdcd)+ M()+ M(aeae--bdbd) )

    D = 0(D = 0(--3.3.--1 1 ((--1.0)) 1.0)) 1(1.1(1.--1 1 ((--1.1.--1)) + 1(1.0 1)) + 1(1.0 ((--3.3.--1)) 1))

    = 0 = 0 ((--2) + (2) + (--3) = 2 3) = 2 3 = 3 = --11

  • 6Thus, the rank of the dimensional matrix is r = 3.

    If all possible third-order determinants were zero, we would have concluded that r < 3 and proceeded to test the second-order determinants.

    In most problems in fluid mechanics without thermal effects, r = 3.

  • Example


    A ship 100 m long is expected to sail at 10 m/s. It has a submerged surface of 300 m2. Find the model speed for a 1/25 scale model, neglecting frictional effects. The drag is measured to be 60 N when the model is tested in a towing tank at the model speed.

    Based on this information estimate the prototype drag after making corrections for frictional effects.

  • 8Solution:

    We first estimate the model speed neglecting frictional effects. Then the nondimensional drag force depends only on the Froude number:

    Equating Froude numbers for the model (denoted by subscript m) and prototype (denoted by subscript p), we get


  • 9The total drag on the model was measured to be 60 N at this model speed.

    Of the total measured drag, a part was due to frictional effects.

    The frictional drag can be estimated by treating the surface of the hull as a flat plate, for which the drag coefficient CD is given in Figure 10.12 as a function of the Reynolds number.

    Using a value of = 10 6 m2/s for water, we get

  • 10

    For these values of Reynolds numbers, Figure 10.12 gives the frictional drag coefficients of

    Using a value of = 1000 kg/m3 for water, we estimate

  • 11

    Out of the total model drag of 60 N, the wave drag is therefore 60 2.88 = 57.12 N.

    Now the wave drag still obeys equation (8.20), which means that D/U2L2 for the two flows are identical, where D represents wave drag alone.

    Therefore Wave drag on prototype

  • 12

    Having estimated the wave drag on the prototype, we proceed to determine its frictional drag. We obtain

    If we did not correct for the frictional effects, and assumed that the measured model dragwas all due towave effects, then wewould have found from equation (1.1) a prototype drag of

  • Figure 10.12 Measured drag coefficient for a boundary layer over a flat plate. The continuous line shows the drag coefficient for a plate on which the flow is partly laminar and partly turbulent, with the transition taking place at a position where the local Reynolds number is 5105. The dashed lines show the behavior if the boundary layer was either completely laminar or completely turbulent over the entire length of the plate.


  • ExerciseExercise


    Exercises1. Suppose that the power to drive a propeller of an airplane

    depends on d (diameter of the propeller), U (free-stream velocity), (angular velocity of propeller), c (velocity of sound), (density of fluid), and (viscosity). Find the dimensionless groups. In your opinion, which of these are the most important and should be duplicated in a model testing?

    2. A 1/25 scale model of a submarine is being tested in a wind tunnel in which p = 200 kPa and T = 300 K. If the prototype speed is 30 km/hr, what should be the free-stream velocity in the wind tunnel? What is the drag ratio? Assume that the submarine would not operate near the free surface of the ocean.

  • Example


    In a fuel injction system, small droplets are formed due to the breakup of the liquid jet. Assume the droplet diameter, d, is a function of the liquid density, , viscosity, , and surface tension,

    , and the jet velocity, V, and diameter, D. Form an approriate set of dimensionless parameters using , V and D as repeating variables.

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