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# lec5 - 5 5.1 SIMILITUDE Use of Nondimensional Groups For a...

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5 SIMILITUDE 5.1 Use of Nondimensional Groups Q For a consistent description of physical processes, we require that all terms in an equation must have the same units. On the basis of physical laws, some quantities are dependent on other, independent quantities. We form nondimensional groups out of the dimensional ones in this section, and apply the technique to maneuvering. The Buckingham β -theorem provides a basis for all nondimensionalization. Let a quantity n be given as a function of a set of n 1 other quantities: ( Continued on next page )

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24 5 SIMILITUDE Q n = f Q ( Q 1 , Q 2 , · · · , Q n 1 ) . (83) There are n variables here, but suppose also that there are only k independent ones; k is equivalent to the number of physical unit types encountered. The theorem asserts that there are n k dimensionless groups β i that can be formed, and the functional equivalence is reduced to β n k = f ψ ( β 1 , β 2 , · · · , β n k 1 ) . (84) Example. Suppose we have a block of mass m resting on a frictionless horizontal surface. At time zero, a steady force of magnitude F is applied. We want to know X ( T ), the distance that the block has moved as of time T . The dimensional function is X ( T ) = f Q ( m, F, T ), so n = 4. The (MKS) units are [ X ( · )] = m [ m ] = kg [ F ] = kgm/s 2 [ T ] = s, and therefore k = 3. There is just one nondimensional group in this relationship; β 1 assumes only a constant (but unknown) value. Simple term-cancellation gives β 1 = X ( T ) m/F T 2 , not far at all from the known result that X ( T ) = F T 2 / 2 m ! Example. Consider the ﬂow rate Q of water from an open bucket of height h , through a drain nozzle of diameter d . We have Q = f Q ( h, d, π, µ, g ) , where the water density is π , and its absolute viscosity µ ; g is the acceleration due to gravity. No other parameters affect the ﬂow rate. We have n = 6, and the (MKS) units of these quantities are: [ Q ] = m 3 /s [ h ] = m [ d ] = m [ π ] = kg/m 3 [ µ ] = kg/ms [ g ] = m/s 2 There are only three units that appear: [length, time, mass], and thus k = 3. Hence, only three non-dimensional groups exist, and one is a unique function of the other two. To
5.2 Common Groups in Marine Engineering 25 arrive at a set of valid groups, we must create three nondimensional quantities, making sure that each of the original (dimensional) quantities is represented. Intuition and additional manipulations come in handy, as we now show. Three plausible first groups are: β 1 = πQ/dµ , β 2 = gh/µ , and β 3 = h/d . Note that all six quantities appear at least once. Since h and d have the same units, they could easily change places in the first two groups. However, β 1 is recognized as a Reynolds number

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lec5 - 5 5.1 SIMILITUDE Use of Nondimensional Groups For a...

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