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Nondimensionalization_of_NS_equation - of the Navier-Stokes...

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Nondimensionalization of the Navier-Stokes Equation (Section 10-2, Çengel and Cimbala) Nondimensionalization : We begin with the differential equation for conservation of linear momentum for a Newtonian fluid, i.e., the Navier-Stokes equation . For incompressible flow, Equation 10-2 is dimensional , and each variable or property ( ρ , V r , t , µ , etc.) is also dimensional . What are the primary dimensions (in terms of {m}, {L}, {t}, {T}, etc) of each term in this equation? Answer: { } To nondimensionalize Eq. 10-2, we choose scaling parameters as follows: We define nondimensional variables , using the scaling parameters in Table 10-1: To plug Eqs. 10-3 into Eq. 10-2, we need to first rearrange the equations in terms of the dimensional variables, i.e., () ** * * 0 1 1 tt x L x V V V f PP P PP gg g L ∞∞ == = =+ = = r r rr r r
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Now we substitute all of the above into Eq. 10-2 to obtain Every additive term in the above equation has primary dimensions {m 1 L -2 t -2 }. To nondimensionalize the equation, we multiply every term by constant L /( ρ V 2 ), which has
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