Stream function nondimensional

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.. 

Many boundary-layer problems possess similarity solutions—solutions in which appropriately identified dependent variables may be expressed as functions of a single independent variable, t], the nondimensional similarity variable, through functions defined by ordinary differential equations in f/. To facilitate derivations of similarity solutions, after employing the Moore-Stewartson transformation, we shall further transform the equations to a general similarity variable and introduce a nondimensional stream function F(x, t], t), according to... 

It leads to the use of mass (a conserved quantity) rather than distance as the independent variable. A nondimensional stream function co is next introduced, defined by... 

In Section 5.3.6, activation-energy asymptotics have been applied to the adiabatic version of equation (9) for a particular rate function w burning-rate formulas are given in Section 5.3.6 for this rate function and in Section 5.3.7 for others. Here it is convenient to presume that for L = 0, the burning rate is known on the basis of these results and to employ the known adiabatic mass burning rate for the purpose of nondimensionalization. Thus, in analogy with equation (5-18), we introduce the nondimensional stream wise coordinate = m CpX/X and obtain the equation... 

The three nondimensionalized boundary layer equations (Eqs. 6-64, 6-65, and 6 66) involve three unknown functions /<, i/, and T. two independent variables x and y, and two parameters Rej. and Pr. The pressure P (x ) depends on the geometry involved (it is constant for a flat plate), and it has the same value inside and outside the boundary layer at a specified x. Therefore, it can be,delennined separately from the free stream conditions, and dP ld.x in Eq. 6-65 can he treated as a known function of.v. Note that the boundary condition do not introduce any new parameters. 

To obtain a valid approximate solution for heat transfer from a sphere in a uniform streaming flow at small, but nonzero, Peclet numbers, we must resort to the method of matched (or singular) asymptotic expansions.4 In this method, as we have already seen in Chap. 4, two (or more) asymptotic approximations are proposed for the temperature field at Pe 1, each valid in different portions of the domain but linked in a so-called overlap or matching region where it is required that the two approximations reduce to the same functional form. The approximate forms of (9-1), from which these matched expansions are derived, can be obtained by nondimensionalization by use of characteristic length scales that are appropriate to each subdomain. 

However, in real life situations, wind flow is not deterministic but has random fluctuations about a mean value. Hence, this randomness must be included in the model in order to gain better insights on the behavior of system. Therefore, the nondimensional free stream velocity U is assumed to be a stationary Gaussian random process with the autocorrelation function as given in Eq. 55 with oy= 1 and Cq = 1 x 10 This implies a correlation length of 2.628 x 10 s. It must be... 

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