Nondimensionalization

It is possible to solve a flow problem in either dimensional or dimensionless form. The variables can be assigned values using a consistent set of dimensions, which must be the SI system for turbulent flow. The dimensional formula is convenient since the problem is usually specified in that way, but in some cases the iterations may not converge. Alternatively, the equations can be made dimensionless. The dimensionless formulations are good when you are having trouble getting the iterations to converge, since you have a better sense of the problem when you specify the Reynolds number. This section takes the dimensional Navier-Stokes equation, Eq. (10.40), and derives two different dimensionless versions  

The velocity, pressure, and distance are made dimensionless by dividing them by a constant standard for velocity, pressure, and distance. The nondimensional variable is denoted by a prime  

The variables are substituted into the Navier-Stokes equation, Eq. (10.40) note that the dimensional standards are constants and can be taken through the differentials  

Option 1 This option is most applicable to slow flows. Multiply Eq. (10.43) by x /pM to give px 9u  

Choose the pressure standard so that the coefficient of the pressure gradient term is 1.0 make a similar choice for the time standard to get 

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Reynolds Number. The Reynolds number, Ke, is named after Osborne Reynolds, who studied the flow of fluids, and in particular the transition from laminar to turbulent flow conditions. This transition was found to depend on flow velocity, viscosity, density, tube diameter, and tube length. Using a nondimensional group, defined as p NDJp, the transition from laminar to turbulent flow for any internal flow takes place at a value of approximately 2100. Hence, the dimensionless Reynolds number is commonly used to describe whether a flow is laminar or turbulent. Thus... 

Nusse/t Number. Empidcal correlations can be obtained for a particular size of tube diameter and particular flow conditions. To generalize such results and to apply the correlations to different sizes of equipment and different flow conditions, the heat-transfer coefficient, Z, is traditionally nondimensionalized by the use of the Nusselt number, Nu named after Wilhelm Nusselt,... 

Specific Speed. A review of the dimensionless analysis (qv) as related to pumps can be found in Reference 14. One of these nondimensional quantities is called the specific speed. The universal dimensionless specific speed, Q, is defined as in equation 9 ... 

Spray Correlations. One of the most important aspects of spray characterization is the development of meaningful correlations between spray parameters and atomizer performance. The parameters can be presented as mathematical expressions that involve Hquid properties, physical dimensions of the atomizer, as well as operating and ambient conditions that are likely to affect the nature of the dispersion. Empirical correlations provide useful information for designing and assessing the performance of atomizers. Dimensional analysis has been widely used to determine nondimensional parameters that are useful in describing sprays. The most common variables affecting spray characteristics include a characteristic dimension of atomizer, d Hquid density, Pjj Hquid dynamic viscosity, ]ljj, surface tension. O pressure, AP Hquid velocity, V gas density, p and gas velocity, V. ... 

When such a function is estabUshed or assumed, it will still exist even after the variables are intermultiplied in any manner whatsoever. This means that each variable in the equation can be combined with other variables of the equation to form dimensionless products whose dimensional vectors are the 2ero vector. Equation 4 can then be transformed into the nondimensional form as (eq. 5) ... 

An alternative to the use of generalized charts is an analytical equation of state. Equations of state which are expressed as a function of reduced properties and nondimensional variables are said to be generalized. The term generalization is in reference to the wide appHcabiHty to the estimation of fluid properties for many substances. 

Example The equation dQ/dx = (A/f/)(3 6/3f/ ) with the boundary conditions 0 = OatA.=O, y>0 6 = 0aty = oo,A.>0 6=iaty = 0, A.>0 represents the nondimensional temperature 6 of a fluid moving past an infinitely wide flat plate immersed in the fluid. Turbulent transfer is neglected, as is molecular transport except in the y direction. It is now assumed that the equation and the boundary conditions can be satisfied by a solution of the form 6 =f y/x ) =j[u), where 6 =... 

The hardening index /I is a nondimensional scalar which has the same value... 

While c in (5.112) is a linear function of d, it may be an arbitrary function of s. Truesdell considered cases where c is a polynomial in s, terming (5.112) a hypoelastic equation of grade n, where n is the power of the highest-order term in the polynomial. For a hypoelastic equation of grade zero, the elastic modulus c is independent of s and linear in dand therefore has the representation (A.89). It is convenient to nondimensionalize the stress by defining s = sjljx. Since the stress rate must vanish when d is zero, Cq = 0 and the result is... 

In fact, as Atluri [17] has pointed out, the hypoelastic equation of grade zero has inadequate latitude to represent realistic nonlinear response of various materials in large deformations, and it is necessary to use a hypoelastic equation of at least grade one to do so. If the grade is one, then, continuing to use Jaumann s stress rate and nondimensionalizing the stress as before, the isotropic representation (A.92) may be used in (5.112) with d = A and s = B to obtain... 

Stresses are usually related to strains through an effective modulus. If the components of stress are nondimensionalized by a suitable scalar modulus c, then they are also of order c. Using (A.94), (A.lOl), and the binomial theorem in (A.39), the relation between the normalized spatial stress s = s/c and the normalized referential stress S = S/c becomes... 

A loss is usually expressed as a loss of heat or enthalpy. A eonvenient way to express them is in a nondimensional manner with referenee to the exit blade speed. The theoretieal total head available (i/ioi) is equal to the head available from the energy equation... 

To determine the deterioration in component performance and efficiency, the values must be corrected to a reference plane. These corrected measurements will be referenced to different reference planes depending upon the point, which is being investigated. Corrected values can further be adjusted to a transposed design value to properly evaluate the deterioration of any given component. Transposed data points are very dependent on the characteristics of the components performance curves. To determine the characteristics of these curves, raw data points must be corrected and then plotted against representative nondimensional parameters. It is for this reason that we must evaluate the turbine train while its characteristics have not been altered due to component deterioration. If component data were available from the manufacturer, the task would be greatly reduced. 

Let us return to the nondimensionalization of the equation. Before we replaced time inconsistently with thetruedimensionofthesecond-order k,wehadthefollowingformof... 

We could also have chosen to nondimensionalize the differential equations before solving theminorderto findageneralsolutioninfewerabsoluteparameters.Wecandivideallby kACAo/Whichwillgiveus ... 

Rectangular Flanged Openings- C i Jhe velocity fields for these openings must be calculated for each direction individually. Here and are the nondimensional velocities parallel to rhe opening plane and v. is the nondimensional velocity perpendicular to the opening plane directed toward the opening plane ... 

Outside the jet and away from the boundaries of the workbench the flow will behave as if it is inviscid and hence potential flow is appropriate. Further, in the central region of the workbench we expect the airflow to be approximately two-dimensional, which has been confirmed by the above experimental investigations. In practice it is expected that the worker will be releasing contaminant in this region and hence the assumption of two-dimensional flow" appears to be sound. Under these assumptions the nondimensional stream function F satisfies Laplace s equation, i.e.. 

Again, as described earlier, we model the exhaust opening as a finite circular opening across which the fluid flows with a constant speed. The axis of the coordinate system is a streamline which we take to be R = 0 and on the surface of the flange of the Aaberg exhaust system is also a streamline on which = 0, due to the nondimensionalization given in Eq. (10,112). Further, at the outer edge of the et, which we assume to be at d> — 0—i.e., it is assumed that the jet is infinitesimally small—the boundary condition (F.q. (10.119)) is appropriate. 

When a coarse grid is used, wall functions are used for imposing boundary conditions near the walls (Section 11.2.3.3). The nondimensional wall distance should be 30 < y < ]Q0, where y = u,y/p. We cannot compute the friction velocity u. before doing the CFD simulation, because the friction velocity is dependent on the flow. However, we would like to have an estimation of y" to be able to locate the first grid node near the wall at 30 < y < 100. If we can estimate the maximum velocity in the boundary layer, the friction velocity can be estimated as n, — 0.04rj, . . After the computation has been carried out, we can verify that 30 nodes adjacent to the walls. 

The next step toward the similarity principles is to develop the governing equations in a nondimensional form. The equations are normalized by first defining the dimensionless, independent variables as... 

The conditions for model experiments can be explained in the following way. The governing equations are made nondimensional in the full scale and in the reduced scale used in the model experiments. For example, the velocity in the room is divided by the diffuser velocity in the room, and the velocity in the model is divided by the supply velocity in the model in order to normalize all velocities. The two sets of equations are identical and they describe the same solution provided that requirements f, 2, and 3 mentioned at the beginning of this section have been met. 

In the method which will be presented in Section 6.3.3., the blast parameters of pressure vessel bursts are read from curves of pentolite, a high explosive, for nondimensional distance R above two. For these ranges, using TNT equivalence makes sense. Pentolite has a specific heat of detonation of 5.11 MJ/kg, versus 4.52 MJ/kg for TNT (Baker et al. 1983). The equivalent mass of TNT can be calculated as follows for a ground burst of a pressure vessel ... 

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