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

Adriana Nastase

Chapter 31 in Integral Methods in Science and Engineering, 2002, pp 197-202 from Springer

Abstract: 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 31.1 $$ {u_{\delta }} = {u_e}\sum\limits_{{i = 1}}^N {{u_i}{\eta^i}}, $$ , 31.2 $$ {v_{\delta }} = {v_e}\sum\limits_{{i = 1}}^N {{v_i}{\eta^i}} $$ , 31.3 $$ {w_{\delta }} = {w_e}\sum\limits_{{i = 1}}^N {{w_i}{\eta^i}} $$ , 31.4 $$ R = {R_w} + \left( {{R_e} - {R_w}} \right)\sum\limits_{{i = 1}}^N {{r_i}{\eta^i}} $$ , 31.5 $$ T = {T_w} + \left( {{T_e} + {T_w}} \right)\sum\limits_{{i = 1}}^N {{t_i}{\eta^i}} $$ , 31.6 $$ {u_{{N - 2}}} = {\alpha_{{0,N - 2}}} + \sum\limits_{{i = 1}}^{{N - 3}} {{\alpha_{{i,N - 2}}}{u_i}} $$ , 31.7 $$ {v_{{N - 2}}} = {\alpha_{{0,N - 2}}} + \sum\limits_{{i = 1}}^{{N - 3}} {{\alpha_{{i,N - 2}}}{v_i}} $$ , 31.8 $$ {u_{{N - 1}}} = {\alpha_{{0,N - 1}}} + \sum\limits_{{i = 1}}^{{N - 3}} {{\alpha_{{i,N - 1}}}{u_i}} $$ , 31.9 $$ {v_{{N - 1}}} = {\alpha_{{0,N - 1}}} + \sum\limits_{{i = 1}}^{{N - 3}} {{\alpha_{{i,N - 1}}}{v_i}} $$ , 31.10 $$ {u_N} = {\alpha_{{0,N}}} + \sum\limits_{{i = 1}}^{{N - 3}} {{\alpha_{{i,N}}}{u_i}} $$ , 31.11 $$ {v_N} = {\alpha_{{0,N}}} + \sum\limits_{{i = 1}}^{{N - 3}} {{\alpha_{{i,N}}}{v_i}} $$ , 31.12 $$ {w_N} = {\gamma_{{0,N}}} + \sum\limits_{{i = 1}}^n {{\gamma_{{i,N}}}{w_i}} $$ , 31.13 $$ \sum\limits_{{i = 1}}^N {{r_i} = 1} $$ , 31.14 $$ \sum\limits_{{i = 1}}^N {{t_i} = 1} $$ .

Keywords: Supersonic Flow; Potential Flow; Spectral Form; Hinge Line; Optimal Shape Design (search for similar items in EconPapers)
Date: 2002
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DOI: 10.1007/978-1-4612-0111-3_31

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