Nondimensionalization Navier-Stokes equation

It has been stated repeatedly that the boundary-layer and potential-flow equations apply to only the leading term in an asymptotic expansion of the solution for Re F> 1. This is clear from the fact that we derived both in their respective domains of validity by simply taking the limit Re -= oc in the appropriately nondimensionalized Navier-Stokes equations. Frequently, in the analysis of laminar flow at high Reynolds number, we do not proceed beyond these leading-order approximations because they already contain the most important information a prediction of whether or not the flow will separate and, if not, an analytic approximation for the drag. Nevertheless, the reader may be interested in how we would proceed to the next level of approximation, and this is described briefly in the remainder of this section.13... 

Equations (if.4) and (ff.S) are solved, along with the continuity equation (which does not change upon nondimensionalization), in a Cartesian coordinate system using the Fourier-Galerkin (spectral) technique under periodic boundary conditions in all three space dimensions. The scheme is similar to that used by Orszag [8] for direct solution of the incompressible Navier-Stokes equations. More details can be found in [9] and [7], and the scheme may be considered to be pseudospectral. ... 

Nondimensionalization of the species- and energy-conservation equations follows a procedure that is analogous to that for the Navier-Stokes equations. For two-dimensional steady axisymmetric flow of a perfect gas, the full equations are given as... 

In addition to the reference scales and nondimensional variables used for the Navier-Stokes equations, new scaling parameters must be introduced to nondimensionalize the temperature and diffusive mass flux. In a mixture-averaged setting... 

The latter comes from the matching requirement that the magnitude of the pressure in the end regions must be comparable with that in the core. The definitions (6-130) lead to the following nondimensionalized form of the Navier-Stokes equations for the end regions ... 

Thus we begin by considering the full Navier-Stokes equation expressed in terms of the streamfunction characteristic velocity and the sphere radius a as a characteristic length scale. Using spherical coordinates, with ij = cos 9, this equation is... 

It is convenient to nondimensionalize. We must identify a characteristic length scale tc and velocity scale uc. For the former we choose d, though we recognize that this would need to be modified if the fluid is unbounded. For the characteristic velocity scale, it is traditional in this field to choose the maximum value of Uf In addition, we assume that the characteristic time is tc = ic/uc, and the characteristic pressure is pc = pu2c. Then the governing equations, which are the continuity equation and the inviscid Navier-Stokes equations (usually called the Euler equations), can be written in the form 3u... 

Let us nondimensionalize the Navier-Stokes equations by introducing a characteristic velocity u0 and characteristic length L and defining the dimensionless variables... 

For the purpose of understanding pressure filtering, attention may be restricted to the single-component, constant-property, nonreacting equations for a perfect gas. Introducing the nondimensional variables into the vector forms of the mass-continuity, constant-viscosity Navier-Stokes, and perfect-gas thermal-energy equations yields the following nondimensional system ... 

To analyze streaming flow at high Reynolds number past a 2D body, the starting point is the full, steady-state Navier-Stokes and continuity equations, nondimensionalized by use of the streaming velocity Uoo as a characteristic velocity scale and a scalar length of the body in the xy plane, say, a, as the characteristic length scale, namely,... 

The conclusion to be drawn from the preceding discussion is that the potential-flow theory (10-9) [or, equivalently, (10 12) and (10 13)] does not provide a uniformly valid first approximation to the solution of the Navier Stokes and continuity equations (10-1) and (10 2) for Re 1. Furthermore, our experience in Chap. 9 with the thermal boundary-layer structure for large Peclet number would lead us to believe that this is because the velocity field near the body surface is characterized by a length scale 0(aRe n), instead of the body dimension a that was used to nondimensionalize (10-2). As a consequence, the terms V2co and u V >, in (10 6), which are nondimensionalized by use of a, are not 0(1) and independent of Re everywhere in the domain, as was assumed in deriving (10-7), but instead are increasing fimctions of Re in the region very close to the body surface. Thus in... 

With dimensionless variables defined as in Eq. (4), the steady-state Navier-Stokes and continuity equations may be written in the nondimensional forms... 

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