Comparison of Zonal, Spectral Solutions for Compressible Boundary Layer and Navier—Stokes Equations

Abstract

In this chapter, the author’s zonal, spectral solutions for the partial differential equations (PDE) of the three-dimensional stationary, compressible boundary layer (CBL) given as in [1]–[3] for the computation of the flow over flattened, flying configurations (FC) are now extended to the Navier-Stokes layer (NSL). If $\eta = \left( {{x_3} - Z\left( {{x_1},{x_2}} \right)} \right)/\delta \left( {{x_1},{x_2}} \right)$ is a new coordinate, the spectral forms of the axial, lateral, and vertical velocity components ${u_{\delta }},{v_{\delta }}, and w\delta$, of the density function $R = \ln \rho$ and of the absolute temperature T (31.1)—(31.5) and their nine boundary conditions (31.6)-(31.14), at the NSL-edge $\left( {\eta = 1} \right)$, are

$$ {u_{\delta }} = {u_e}\sum\limits_{{i = 1}}^N {{u_i}{\eta^i}}, $$

(31.1)

,

$$ {v_{\delta }} = {v_e}\sum\limits_{{i = 1}}^N {{v_i}{\eta^i}} $$

(31.2)

,

$$ {w_{\delta }} = {w_e}\sum\limits_{{i = 1}}^N {{w_i}{\eta^i}} $$

(31.3)

,

$$ R = {R_w} + \left( {{R_e} - {R_w}} \right)\sum\limits_{{i = 1}}^N {{r_i}{\eta^i}} $$

(31.4)

,

$$ T = {T_w} + \left( {{T_e} + {T_w}} \right)\sum\limits_{{i = 1}}^N {{t_i}{\eta^i}} $$

(31.5)

,

$$ {u_{{N - 2}}} = {\alpha_{{0,N - 2}}} + \sum\limits_{{i = 1}}^{{N - 3}} {{\alpha_{{i,N - 2}}}{u_i}} $$

(31.6)

,

$$ {v_{{N - 2}}} = {\alpha_{{0,N - 2}}} + \sum\limits_{{i = 1}}^{{N - 3}} {{\alpha_{{i,N - 2}}}{v_i}} $$

(31.7)

,

$$ {u_{{N - 1}}} = {\alpha_{{0,N - 1}}} + \sum\limits_{{i = 1}}^{{N - 3}} {{\alpha_{{i,N - 1}}}{u_i}} $$

(31.8)

,

$$ {v_{{N - 1}}} = {\alpha_{{0,N - 1}}} + \sum\limits_{{i = 1}}^{{N - 3}} {{\alpha_{{i,N - 1}}}{v_i}} $$

(31.9)

,

$$ {u_N} = {\alpha_{{0,N}}} + \sum\limits_{{i = 1}}^{{N - 3}} {{\alpha_{{i,N}}}{u_i}} $$

(31.10)

,

$$ {v_N} = {\alpha_{{0,N}}} + \sum\limits_{{i = 1}}^{{N - 3}} {{\alpha_{{i,N}}}{v_i}} $$

(31.11)

,

$$ {w_N} = {\gamma_{{0,N}}} + \sum\limits_{{i = 1}}^n {{\gamma_{{i,N}}}{w_i}} $$

(31.12)

,

$$ \sum\limits_{{i = 1}}^N {{r_i} = 1} $$

(31.13)

,

$$ \sum\limits_{{i = 1}}^N {{t_i} = 1} $$

(31.14)

.