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Viscous Shock Layer Around Slender Bodies with Nonequilibrium Air Chemistry

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Iranian Journal of Science and Technology, Transactions of Mechanical Engineering Aims and scope Submit manuscript

Abstract

A method has been developed which can reliably compute nonequilibrium viscous shock layer over blunt-nosed bodies. This method considerably improves computational efficiency, especially for long slender bodies. The shock shape is generated as a part of the solution in this method, and in the nose region, the shock shape is calculated from an algebraic relation and corrected by global passes through that region. The current algorithm yields a considerable decrease of more than 60% in CPU time relative to other VSL methods. In the present study, both seven and eleven species air models were tested. The seven species analysis includes N, O, N2, O2, NO, NO+ and e−. The eleven species analysis includes the same species plus N+, O+, \(\text{N}_{2}^{ + }\), \(\text{O}_{2}^{ + }\). The chemical reaction models for seven and eleven species are taken from Blottner and Gupta, respectively. The governing equations are solved with a spatial-marching, implicit, finite-difference method. The solution is obtained by solving a couple of normal momentum and continuity equations. The results of the present technique show good agreement compared to the STS_2 flight data and other numerical solutions.

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Abbreviations

C i :

Mass fraction of species i, ρ i /ρ

C p :

Specific heat at constant pressure, \(C_{p}^{\text{*}} \text{/}C_{p\infty }^{\text{*}}\)

\(h\) :

Static enthalpy, \(h^{\text{*}} \text{/}V_{\infty }^{\text{2}}\)

h 1, h 3 :

Metrics

q :

Heating rate

\(J_{i}\) :

Diffusion mass flux of species i, \(J_{i}^{\text{*}} R_{\text{n}}^{\text{*}} \text{/}\mu_{{\text{ref}}}\)

k :

Thermal conductivity, \(k^{\text{*}} \text{/}\mu_{{\text{ref}}}^{\text{*}} C_{p\infty }^{\text{*}}\)

\(k_{{i\text{,}w}}\) :

Surface reaction rate coefficient, \(k_{{i\text{,}w}}^{\text{*}} \text{/}V_{\infty }\)

L e :

Lewis number

M :

Mach number

M i :

Molecular weight of species i

R n :

Body nose radius

\(R_{\text{u}}^{\text{*}}\) :

Universal gas constant

r :

Radius measured from axis of symmetry, \(r^{\text{*}} \text{/}R_{\text{n}}^{\text{*}}\)

s :

Coordinate measured along the shock wave, \(s^{\text{*}} \text{/}R_{\text{n}}^{\text{*}}\)

T :

Temperature, \(T^{\text{*}} C_{p\infty }^{\text{*}} \text{/}V_{\infty }^{\text{2}}\)

u :

Velocity components tangent to the shock, \(u^{\text{*}} \text{/}V_{\infty }^{\text{*}}\)

v :

Velocity component normal to the shock, \(v^{\text{*}} \text{/}V_{\infty }^{\text{*}}\)

ε :

Reynolds number parameter, \(\text{(}\mu_{{\text{ref}}} \text{/}\rho_{\infty }^{\text{*}} u_{\text{n}}^{\text{*}} R_{\text{n}}^{\text{*}} \text{)}^{{\text{1/2}}}\)

Γ, θ b :

Body angles

\(\eta_{n}\) :

Normalized n coordinate, 1 − n/n b

μ :

Viscosity, \(u^{\text{*}} \text{/}\mu_{{\text{ref}}}^{\text{*}}\) (\(\mu_{{\text{ref}}}^{\text{*}}\) is the coefficient of viscosity evaluated at T ref = \(V_{\infty }^{\text{2}} \text{/}C_{p\infty }^{{}}\))

ξ :

Normalized s coordinate, ξ = s

ρ :

Density, \(\rho^{*} /\rho_{\infty }\)

γ :

Catalytic recombination coefficient of species i

\(\dot{\omega }\) :

Mass rate of formation of species i, \(\dot{\omega }^{\text{*}} R_{\text{n}}^{\text{*}} \text{/}\rho_{\infty }^{\text{*}} V_{\infty }^{\text{*}}\)

i:

Species index

f:

Chemically frozen value

w:

Wall value

∞:

Free stream condition

*:

Dimensional quantities

b:

Body

ref:

Reference condition

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Authors and Affiliations

  1. Department of Aerospace Engineering, Amirkabir University of Technology, Tehran, Iran

    S. Noori & M. Mani

  2. Space Research Institute, Malek-Ashtar University of Technology, Tehran, Iran

    S. Ghasemloo

Authors
  1. S. Noori
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  2. S. Ghasemloo
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  3. M. Mani
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Corresponding author

Correspondence to S. Noori.

Appendix

Appendix

The coefficients for Eq. (20) are obtained as follows:

$$\begin{aligned} A_{c,j + 1/2} = & - \frac{{(h_{1} h_{3} )_{j + 1/2} }}{{\Delta \eta_{n,j + 1} }} \\ B_{c,j + 1/2} = & \frac{{(h_{1} h_{3} )_{j + 1/2} }}{{\Delta \eta_{n,j + 1} }} \\ C_{c,j + 1/2} = & \frac{{n_{b} h_{3,j + 1/2} u_{i,j + 1/2} }}{{2\Delta \xi_{i} P_{i,j + 1/2} }} - \frac{{h_{3j + 1/2} }}{{\Delta \eta_{n,j + 1} P_{i,j + 1/2} }}\left[ {(\eta_{n,j + 1/2} - 1)u_{i,j + 1/2} \frac{{\text{d}n_{b} }}{{\text{d}\xi }} + h_{1j + 1/2} V_{i,j + 1/2} } \right] \\ D_{c,j + 1/2} = & \frac{{n_{b} h_{3,j + 1/2} u_{i,j + 1/2} }}{{2\Delta \xi_{i} P_{i,j + 1/2} }} + \frac{{h_{3j + 1/2} }}{{\Delta \eta_{n,j + 1} P_{i,j + 1/2} }}\left[ {(\eta_{n,j + 1/2} - 1)u_{i,j + 1/2} \frac{{\text{d}n_{b} }}{{\text{d}\xi }} + h_{1j + 1/2} V_{i,j + 1/2} } \right] \\ E_{c,j + 1/2} = & \frac{{n_{b} h_{3,j + 1/2} u_{i,j + 1/2} }}{{2\Delta \xi_{i} P_{i,j + 1/2} }}(P_{i - 1,j + 1} + P_{i - 1,j} ) \\ & - \frac{{n_{b} h_{3,j + 1/2} }}{{2\Delta \xi_{i} }}\left( {u_{i,j + 1} + u_{i,j} - u_{i - 1,j + 1} - u_{i - 1,j} - \frac{{u_{i,j + 1/2} }}{{T_{i,j + 1/2} }} \times (T_{i,j + 1} + T_{i,j} - T_{i - 1,j + 1} - T_{i - 1,j} )} \right) \\ & - \frac{{n_{b} u_{i,j + 1/2} }}{{2\Delta \xi_{i} }}[(h_{3} )_{i,j + 1} + (h_{3} )_{i,j} - (h_{3} )_{i - 1,j + 1} - (h_{3} )_{i - 1,j} ] \\ & + V_{i,j + 1/2} \left[ {(h_{1} )_{j + 1/2} \left( {\frac{{\partial h_{3} }}{{\partial \eta_{n} }}} \right)_{j + 1/2} + (h_{3} )_{j + 1/2} \left( {\frac{{\partial h_{1} }}{{\partial \eta_{n} }}} \right)_{j + 1/2} } \right] \\ & - \frac{{V_{i,j + 1/2} (h_{1} h_{3} )_{j + 1/2} }}{{\Delta \eta_{n,j + 1} T_{i,j + 1/2} }}\left[ {T_{i,j + 1} - T_{i,j} } \right] \\ & + \frac{{n_{b} h_{3,j + 1/2} (\eta_{n,j + 1/2} - 1)}}{{\Delta \eta_{n,j + 1} }}\left[ {u_{i,j + 1} - u_{i,j} - \frac{{u_{i,j + 1/2} }}{{T_{i,j + 1/2} }}(T_{i,j + 1} - T_{i,j} )} \right] \\ & + (\eta_{n,j + 1/2} - 1)u_{i,j + 1/2} \frac{{\text{d}n_{b} }}{{\text{d}\xi }}\left( {\frac{{\partial h_{3} }}{{\partial \eta_{n} }}} \right)_{j + 1/2} \\ & - \frac{{u_{i,j + 1/2} h_{3,j + 1/2} n_{b} }}{{2\Delta \xi_{i} M_{i,j + 1/2} }}\left[ {M_{i,j + 1} + M_{i,j} - M_{i - 1,j + 1} - M_{i - 1,j} } \right] \\ & + \frac{{(h_{1} h_{3} )_{j + 1/2} V_{i,j + 1/2} }}{{\Delta \eta_{n,j + 1} M_{i,j + 1/2} }}\left[ {M_{i,j + 1} - M_{i,j} } \right] \\ & + \frac{{u_{i,j + 1/2} }}{{M_{i,j + 1/2} \Delta \eta_{n,j + 1} }}\frac{{\text{d}n_{b} }}{{\text{d}\xi }}h_{3j + 1/2} (\eta_{n,j + 1/2} - 1)\left[ {M_{i,j + 1} - M_{i,j} } \right] \\ \end{aligned}$$
$$\begin{aligned} & A_{nm,j + 1/2} = \frac{{n_{b} P_{i,j + 1/2} u_{i,j + 1/2} }}{{2\Delta \xi_{i} h_{1,j + 1/2} }} - \frac{{P_{i,j + 1/2} }}{{\Delta \eta_{n,j + 1} }}\left[ {\frac{{u_{i,j + 1/2} (\eta_{n,j + 1/2} - 1)}}{{h_{1,j + 1/2} }}\frac{{dn_{b} }}{d\xi } + V_{i,j + 1/2} } \right] \\ & B_{nm,i + 1/2} = \frac{{n_{b} P_{i,j + 1/2} u_{i,j + 1/2} }}{{2\Delta \xi_{i} h_{1,j + 1/2} }} + \frac{{P_{i,j + 1/2} }}{{\Delta \eta_{n,j + 1} }}\left[ {\frac{{u_{i,j + 1/2} (\eta_{n,j + 1/2} - 1)}}{{h_{1,j + 1/2} }}\frac{{dn_{b} }}{d\xi } + V_{i,j + 1/2} } \right] \\ & C_{nm,j + 1/2} = \frac{{ - R_{u} T_{i,j + 1/2} }}{{M_{i,j + 1/2} \Delta \eta_{n,j + 1} }} \\ & D_{nm,j + 1/2} = \frac{{ - R_{u} T_{i,j + 1/2} }}{{M_{i,j + 1/2} \Delta \eta_{n,j + 1} }} \\ & E_{nm} = \frac{{n_{b} P_{i,j + 1/2} u_{i,j + 1/2} }}{{2\Delta \xi_{i} h_{1,j + 1/2} }}(V_{i - 1,j + 1} + V_{i - 1,j} ) - \frac{{u^{2}_{i,j + 1/2} P_{i,j + 1/2} }}{{h_{1,j + 1/2} }}\left( {\frac{{\partial h_{1} }}{{\partial \eta_{n} }}} \right)_{j + 1/2} \\ \end{aligned}$$

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Noori, S., Ghasemloo, S. & Mani, M. Viscous Shock Layer Around Slender Bodies with Nonequilibrium Air Chemistry. Iran J Sci Technol Trans Mech Eng 41, 251–264 (2017). https://doi.org/10.1007/s40997-016-0062-0

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