The problem of supersonic flow past sharp and blunt plates is one of the classic gas dynamics problems that have been studied experimentally, theoretically and with the use of numerical simulation already longer than 70 years. Despite the apparent simplicity of the formulation of such a problem, it turned out that the flow structure in the neighborhood of plate is fairly complex. Especially this is manifested at high supersonic flow velocities when all the transverse dimensions decrease significantly and the nonlinear interactions between the structural flow elements become stronger.
Firstly, it became clear that formulation of the problem of flow past a sharp plate is so idealized that it is not possible to describe the flow structure within the framework of theory of continuum mechanics. In the neighborhood of the leading edge a small part of flow can be described using the Boltzmann equation or, which is implemented significantly more frequently, within the framework of the computational methods of type of imitational simulation based on the Monte Carlo algorithms. In fact, these problems are solved within the framework of the independent scientific direction that investigates strongly rarefied gas flows [1].
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In monograph [2], in discussing the question of what is the plate that can be considered as sharp, an approximate criterion based on theory of continuum mechanics was introduced: the edge can be considered to be sharp if its bluntness has no appreciable effect on the pressure distribution along the plate in inviscid gas flow. The following approximate quantitative estimate of the conditions of flow past the sharp edge is given: the Reynolds number \({{\operatorname{Re} }_{{b,\infty }}} = {{{{\rho }_{\infty }}{{V}_{\infty }}{{R}_{b}}} \mathord{\left/ {\vphantom {{{{\rho }_{\infty }}{{V}_{\infty }}{{R}_{b}}} {{{\mu }_{\infty }}}}} \right. \kern-0em} {{{\mu }_{\infty }}}}\) < 100, where \({{R}_{b}}\) is the rounding radius of the edge and \({{\rho }_{\infty }},\) and μ∞ are the density and the dynamic viscosity coefficient in the free-stream flow traveling at the velocity \({{V}_{\infty }}\).
Obviously, the complexity of set up of the aerodynamic experiments consists in the fact that, firstly, there exists no really absolutely sharp edges and, secondly, measurement organization is not possible in practice at small distances from the edge. In the experimental investigation the plate edges are always blunt. For very small rounding radii, for example, for \({{R}_{b}}\) ~ 0.005 cm [3] we can conservatively assume that flow is formed near the sharp edge. In this case we can note that at least one side of the plate represents a wedge; therefore, the flow structures are although independent but different in the neighborhood of the two surfaces. In [2] the following elements of the structure of compressed layer near the surface of the sharp plate were distinguished: the boundary layer, the shock wave, the region in which the boundary layer closes up with the shock wave, and the flow region in which the boundary layer is separated from the shock wave by an inviscid flow zone. In the neighborhood of the edge, the region of free-molecular motion and the region of using the slip flow models are distinguished. Theory of the strong and weak viscous-inviscid interaction between the boundary layer and the shock wave which obtained the recognition and wide spread occurrence up to now was also outlined in the same monograph. Note that the above structural flow elements can be observed not only in the experiments but also can be identified in the results of computational simulation within the framework of the Navier–Stokes equations in which no artificial flow field separations are performed.
In flow past a blunt plate it is necessary to take into account one more structural element of the flow field, namely, the entropy layer whose appearance is due to formation of the stand-off shock wave in the neighborhood of bluntness. Behind the front of shock wave, not only the boundary layer near the surface, but also the heated gas region between the boundary layer and the front of shock wave are formed. Sometimes, the boundary layer merges with the shock wave, but in the fairly dense gas the inviscid flow region is formed downstream of them. At high Mach numbers, gas can be heated to very high temperatures at which gas dissociation and even ionization is observed. In first studies [4‐7] and in many subsequent papers it was established that the entropy layer strongly affects the entire gas dynamic structure of the compressed layer above the blunt plate, the pressure and surface friction distributions, and the densities of heat fluxes toward the surface.
The important development of studies on hypersonic flow past blunt plates was establishment of the role of plate heating, interaction with external acoustical disturbances, and nonlinear interaction between the acoustical modes of perturbations in the boundary layers and their influence on the development of flow instability, laminar-turbulent transition, and structural features of the turbulent boundary layer [8–10].
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A separate branch of hypersonic aerodynamics of sharp and blunt plates includes the class of the problems of chemically nonequilibrium flows near a surface and, as a separate subclass, – studying the processes of physicochemical kinetics that includes the processes of relaxation of the internal degrees of freedom of the molecular flow components [11–16].
In the present study, we will consider the problem of supersonic rarefied air flow past a blunt plate at the angles of attack α = 10°–45°, nonequilibrium processes of physicochemical kinetics, that include chemical reactions, dissociation, and ionization, proceeding in the air flow. A special attention is focused on studying the regularities of the relaxation processes of thermalization of the vibrational degrees of freedom of N2 and O2 molecules which have an appreciable effect on both the chemical transformation rate in gas and the gas flow structure near the w3indward and leeward surfaces.
The results of solving two problems are given using the computational simulation methods. For the first problem the initial data correspond to the conditions of experiment [3] in which a strongly rarefied gas flow past the plate was formed in the nozzle of a hypersonic wind tube at the Mach number М = 20. The streamwise dimension of the tested plate was equal to L = 10 cm.
In the second problem, flow past the same plate is considered in the case of significantly more dense air stream at М = 10. The major aim of the present investigation carried out within the framework of the full system of the Navier–Stokes equations is to study the configuration of the flow field with regard to chemical reaction right up to dissociation and ionization, as well as thermalization of the internal degrees of freedom with increase in the angle of attack of the blunt plate and the free-stream density right up to appearance of separation flow above the leeward surface.
1 COMPUTATIONAL MODEL
We will formulate the two-dimensional problem of laminar flow of viscous, heat-conducting, compressible, chemically and thermally nonequilibrium air past a blunt plate of finite dimension without separation of the shock front on the base of the continuity and Navier–Stokes equations, the energy conservation laws for the translational motion of particles in the form of the Fourier–Kirchhoff heat equation, the conservation laws for the masses of chemical components (diffusion equations) and conservation of the vibrational energy in individual vibrational modes of diatomic molecules and chemical kinetics in each of the elementary volumes of the Euler computational grid:where t is time, x and y are the orthogonal Cartesian coordinates, u and \({v}\) are the projections of the velocity V on the coordinate axes x and y, \(p,\,\;\rho ,\) and T are the pressure, the density, and the temperature of the translational motion of particles, respectively, \(\lambda \) is thermal conductivity coefficient, \({{Y}_{i}}\) is the mass fraction of the ith component of the mixture of gases whose total number is equal to \({{N}_{s}}\), \({{c}_{p}},\) and cp, i are the specific heat capacity at constant pressure of the gas mixture and individual components, \({{c}_{p}} = \sum\limits_i^{{{N}_{s}}} {{{Y}_{i}}{{c}_{{p,i}}}} \), \({{{\mathbf{J}}}_{i}}\) and \({{D}_{i}}\) are the density of diffusive flux and the effective diffusion coefficient of the ith component, respectively, \({{{\mathbf{J}}}_{i}} = - \rho {{D}_{i}}\operatorname{grad} {{Y}_{i}}\).
$$\frac{{\partial \rho }}{{\partial t}} + {\text{div}}\left( {\rho {\mathbf{V}}} \right) = 0,$$
(1.1)
$$\frac{{\partial \rho u}}{{\partial t}} + {\text{div}}\,\left( {\rho u{\mathbf{V}}} \right) = - \frac{{\partial p}}{{\partial x}} + {{S}_{{\mu ,x}}},$$
(1.2)
$$\frac{{\partial \rho {v}}}{{\partial t}} + {\text{div}}\,\left( {\rho {v}{\mathbf{V}}} \right) = - \frac{{\partial p}}{{\partial x}} + {{S}_{{\mu ,y}}},$$
(1.3)
$$\rho {{c}_{p}}\frac{{\partial T}}{{\partial t}} + \rho {{c}_{p}}{\mathbf{V}}\operatorname{grad} T = \operatorname{div} \left( {\lambda \operatorname{grad} T} \right) + {\mathbf{V}}\operatorname{grad} p + {{\Phi }_{\mu }} + {{Q}_{V}} - {{Q}_{h}} + {{Q}_{d}},$$
(1.4)
$$\frac{{\partial {{\rho }_{i}}}}{{\partial t}} + \operatorname{div} {{\rho }_{i}}{\mathbf{V}} = - \operatorname{div} {{{\mathbf{J}}}_{i}} + {{\dot {w}}_{i}},\quad i = 1,2, \ldots ,{{N}_{s}},$$
(1.5)
$$\frac{{\partial {{\rho }_{{i(m)}}}{{e}_{{V{\text{,}}i{\text{(}}m)}}}}}{{\partial t}} + \operatorname{div} \left( {{{\rho }_{{i(m)}}}{{e}_{{V{\text{,}}i{\text{(}}m)}}}{\mathbf{V}}} \right) = - \operatorname{div} \left( {{{e}_{{V{\text{,}}i{\text{(}}m)}}}{{{\mathbf{J}}}_{{i(m)}}}} \right) + {{\dot {e}}_{{V{\text{,}}i{\text{(}}m)}}},\quad m = 1,2, \ldots ,{{N}_{V}},$$
(1.6)
$${{\left( {\frac{{{\text{d}}{{X}_{i}}}}{{{\text{d}}t}}} \right)}_{n}} = \left( {{{b}_{{i,n}}} - {{a}_{{i,n}}}} \right)\left[ {{{k}_{{f,n}}}\prod\limits_j^{{{N}_{s}}} {X_{j}^{{{{a}_{{j,n}}}}}} - {{k}_{{r,n}}}\prod\limits_j^{{{N}_{s}}} {X_{j}^{{{{b}_{{j,n}}}}}} } \right],$$
(1.7)
The right-hand side of Eq. (1.4) contains the terms \({{Q}_{V}},{{Q}_{h}},\) and Qd which represent the volume heat release powers caused by the processes of vibrational relaxation in gas mixture, chemical reactions and diffusion heat transfer processes: \({{Q}_{V}} = - \sum\limits_{m = 1}^{{{N}_{{\text{V}}}}} {{{{\dot {e}}}_{{{\text{V,}}i{\text{(}}m)}}}} \), \({{Q}_{h}} = \sum\limits_{i = 1}^{{{N}_{s}}} {{{h}_{i}}{{{\dot {w}}}_{i}}} \), and \({{Q}_{d}} = \sum\limits_{i = 1}^{{{N}_{s}}} {\rho {{c}_{{p,i}}}{{D}_{i}}} \left( {\operatorname{grad} {{Y}_{i}} \cdot \operatorname{grad} T} \right)\). Here, NV is the number of vibrational modes (in the case considered \({{N}_{{\text{V}}}} = 2\): \(m = 1\) for the vibrational energy of N2, m = 2 for the vibrational energy of O2); \({{h}_{i}},\) and \({{\dot {w}}_{i}}\) are the enthalpy and the mass rate of chemical transformation of the ith component of mixture, \({{\dot {e}}_{{{\text{V,}}m}}}\) is the source of the vibrational energy in the mth mode, \({{R}_{{i(m)}}} = {{{{R}_{0}}} \mathord{\left/ {\vphantom {{{{R}_{0}}} {{{M}_{{i(m)}}}}}} \right. \kern-0em} {{{M}_{{i(m)}}}}}\), \({{R}_{0}} = 8.314 \times {{10}^{7}}\) erg/(K mol) is the universal gas constant, \({{\rho }_{{i(m)}}}\), \({{M}_{{i(m)}}},{{D}_{{i(m)}}},\) and \({{{\mathbf{J}}}_{{i(m)}}} = - {{\rho }_{{i(m)}}}{{D}_{{i(m)}}}\operatorname{grad} {{Y}_{i}}\) are the density, the molecular weight, the effective diffusion coefficient in the multicomponent gas mixture, and the vector of the density of diffusive flux of the ith component of gas mixture, which has the mth mode of vibrational motion, and \({{{{\theta }}}_{m}}\) is the characteristic vibrational temperature (\({{{{\theta }}}_{{m = 1}}}({{{\text{N}}}_{{\text{2}}}})\) = 3396 К and \({{{{\theta }}}_{{m = 2}}}({{{\text{O}}}_{{\text{2}}}})\) = 2275 К).
The components of the viscous stress tensor \({{S}_{{\mu ,x}}},\) and \({{S}_{{\mu ,y}}}\) and the dissipation function \({{\Phi }_{\mu }}\) contain the dynamic viscosity coefficients μ that vary in the space and are determined by the temperature of translational degrees of freedom and the chemical composition of gas mixture found in the course of solving the problem
$${{S}_{{\mu ,x}}} = - \frac{2}{3}\frac{\partial }{{\partial x}}\left( {\mu \,{\text{div}}{\mathbf{V}}} \right) + \frac{\partial }{{\partial y}}\left[ {\mu \left( {\frac{{\partial {v}}}{{\partial x}} + \frac{{\partial u}}{{\partial y}}} \right)} \right] + 2\frac{\partial }{{\partial x}}\left( {\mu \frac{{\partial u}}{{\partial x}}} \right),$$
$${{S}_{{\mu ,y}}} = - \frac{2}{3}\frac{\partial }{{\partial x}}\left( {\mu \,{\text{div}}{\mathbf{V}}} \right) + \frac{\partial }{{\partial x}}\left[ {\mu \,\left( {\frac{{\partial {v}}}{{\partial x}} + \frac{{\partial u}}{{\partial y}}} \right)} \right] + \,\,2\frac{\partial }{{\partial y}}\left( {\mu \frac{{\partial {v}}}{{\partial y}}} \right),$$
$${{\Phi }_{\mu }} = \mu \left[ {2{{{\left( {\frac{{\partial u}}{{\partial x}}} \right)}}^{2}}} \right.\left. { + 2{{{\left( {\frac{{\partial {v}}}{{\partial y}}} \right)}}^{2}} + {{{\left( {\frac{{\partial {v}}}{{\partial x}} + \frac{{\partial u}}{{\partial y}}} \right)}}^{2}} - \,\,\frac{2}{3}{{{\left( {\frac{{\partial u}}{{\partial x}} + \frac{{\partial {v}}}{{\partial y}}} \right)}}^{2}}} \right].$$
The system of Eqs. (1.1)–(1.7) is closed by the caloric equation of state, i.e., a thermodynamic relation for determining the internal energy of the gas mixture and each of its components and the thermal equation of state of an ideal gaswhere \({{M}_{\Sigma }} = {{\left( {\sum\limits_i^{{{N}_{s}}} {{{{{Y}_{i}}} \mathord{\left/ {\vphantom {{{{Y}_{i}}} {{{M}_{i}}}}} \right. \kern-0em} {{{M}_{i}}}}} } \right)}^{{ - 1}}}\) is the total molecular weight of gas mixture.
$$\frac{p}{\rho } = \frac{{{{R}_{0}}T}}{{{{M}_{\Sigma }}}},$$
(1.8)
The essential feature of the problem considered is taking into account nonequilibrium excitation of the internal degrees of freedom of diatomic molecules. Equation (1.6) describes the processes of vibrational excitation of the molecular components of N2 and O2 gas mixture in the general form; therefore, in what follows, we will assume that each of the molecules mentioned above, in addition to three translational and two rotational degrees of freedom, possesses also a single vibrational mode of motion (single form of vibrational motion). The rotational molecular excitation is considered to be in equilibrium with the translational degrees of freedom. This was justified in [17‐20]. The measure of vibrational molecular excitation is the so-called temperature of vibrational excitation of the ith molecule in the mth mode \({{T}_{{V,m(i)}}}\). Then the specific internal energy of the ith component of gas mixture can be written in the form:where \({{\delta }_{i}}\) = 1 for diatomic molecules and \({{\delta }_{i}}\) = 0 for atoms. Equation (1.6) makes it possible to determine the specific internal energy of the mth vibrational mode of the ith moleculehence the vibrational excitation temperature can be found. It is assumed to use the internuclear potential of the type of a harmonic oscillator, while the remaining molecules, apart from N2 and O2, are excited in equilibrium with the temperature of translational motion.
$${{e}_{i}} = \frac{3}{2}{{R}_{i}}T + {{\delta }_{i}}{{R}_{i}}T + {{\delta }_{i}}{{R}_{i}}\frac{{{{\theta }_{{i(m)}}}}}{{\exp \left( { - \frac{{{{\theta }_{{i(m)}}}}}{{{{T}_{{V,i(m)}}}}}} \right) - 1}},\quad {{R}_{i}} = \frac{{{{R}_{0}}}}{{{{M}_{i}}}},$$
(1.9)
$${{e}_{{V,i(m)}}} = \frac{{{{R}_{{i(m)}}}{{\theta }_{{i(m)}}}}}{{\exp \left( {{{{{\theta }_{{i(m)}}}} \mathord{\left/ {\vphantom {{{{\theta }_{{i(m)}}}} {{{T}_{{{\text{V}},i(m)}}}}}} \right. \kern-0em} {{{T}_{{{\text{V}},i(m)}}}}}} \right) - 1}},$$
Thus, the caloric equation of state of the gas mixture can be formulated as follows:or
$$e = \int\limits_{{{T}_{0}}}^T {{{c}_{V}}dT,} \quad {{c}_{V}} = \sum\limits_i^{{{N}_{s}}} {{{Y}_{i}}{{c}_{{V,i}}}} $$
(1.10)
$$h = e + {p \mathord{\left/ {\vphantom {p \rho }} \right. \kern-0em} \rho }.$$
(1.11)
In determining \({{\dot {e}}_{{{\text{V,}}i{\text{(}}m)}}}\), we took into account the processes of vibrational–translational (VT) relaxation and changes in the specific vibrational energy due to chemical reactions (VC-processes)
$${{\dot {e}}_{{V{\text{,}}m}}} = {{\dot {e}}_{{V{\text{,}}m}}}({\text{VT}}) + {{\dot {e}}_{{V{\text{,}}m}}}({\text{VC}}).$$
In each vibrational mode the relaxation variation in the energy was calculated from the approximate Landau–Teller theory [19]\(e_{{V{\text{,}}m}}^{0} = {{e}_{{V{\text{,}}m}}}\left( {{{T}_{V}} = T} \right)\) is the specific equilibrium energy of the vibrational motion in the mth vibrational mode of the ith component, \({{\tau }_{{i(m)}}}\) is the characteristic relaxation time of the mth vibrational mode determined from the approximate Millikan–White relations [20] with the Park correction [19] which bounds \({{\tau }_{{i(m)}}}\) from below\({{\eta }_{m}}\) is the reduced mass of a diatomic molecule, \({{m}_{{i(m)}}}\) is the molecular mass of the ith component, \({{N}_{t}}\) is the total number density of all molecules at given temperature T and pressure р, and \({{\sigma }_{{V,i(m)}}}\) is the collision cross-section of a vibrational-exited particle whose characteristic value is equal to 3 × 10–17 cm2.
$${{\dot {e}}_{{V{\text{,}}i{\text{(}}m)}}}({\text{VT}}) = {{\rho }_{{i(m)}}}\frac{{e_{{V{\text{,}}i{\text{(}}m)}}^{0} - {{e}_{{V{\text{,}}i{\text{(}}m)}}}}}{{{{\tau }_{{i{\text{(}}m)}}}}},$$
$${{\tau }_{m}} = {{\tau }_{{{\text{VT,}}i{\text{(}}m)}}} + \frac{1}{{{{N}_{t}}{{\sigma }_{{V,i(m)}}}\sqrt {{{8kT} \mathord{\left/ {\vphantom {{8kT} {\left( {\pi {{m}_{{i(m)}}}} \right)}}} \right. \kern-0em} {\left( {\pi {{m}_{{i(m)}}}} \right)}}} }},\quad {{\sigma }_{{V,i(m)}}} = \sigma _{{V,i(m)}}^{'}{{\left( {{{50{\kern 1pt} {\kern 1pt} 000} \mathord{\left/ {\vphantom {{50{\kern 1pt} {\kern 1pt} 000} T}} \right. \kern-0em} T}} \right)}^{2}},$$
$$p{{\tau }_{{{\text{VT,}}i(m)}}} = \exp [{{A}_{{{\text{VT}},i(m)}}}({{T}^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 3}} \right. \kern-0em} 3}}}} - {{B}_{{{\text{VT}},i(m)}}}) - 18.42],$$
$${{A}_{{{\text{VT,}}i(m)}}} = 0.00116\eta _{{i(m)}}^{{0.5}}\theta _{m}^{{1.333}},\quad {{B}_{{{\text{VT,}}i(m)}}} = 0.015\eta _{m}^{{0.25}},$$
Change in the vibrational energy due to chemical reactions was taken into account in accordance with the modelwhere it was assumed that decrease in the vibrational energy in the mth mode in 1 cm3 per second is in proportion to the volume rate of disappearance of the molecules of the same vibrational mode.
$${{e}_{{V,m}}}({\text{VC}}) = {{e}_{{V,m}}}\frac{1}{2}\left( {{{{\dot {w}}}_{{i(m)}}} - \left| {{{{\dot {w}}}_{{i(m)}}}} \right|} \right),$$
Diffuse transfer of the vibrational molecular excitation energy was taken into account due to molecular diffusion, i.e., the density of vibrational energy flux due to diffusion was determined from the formulawhere \({{D}_{{V{\text{, }}i(m)}}}\) is the effective diffusion coefficient of the vibrational excitation energy taken to be equal to the effective diffusion coefficient defined above.
$${{e}_{{V{\text{,}}m}}}{{{\mathbf{J}}}_{{V{\text{,}}i(m)}}} = - {{e}_{{V{\text{,}}m}}}{{\rho }_{{i(m)}}}{{D}_{{V{\text{,}}i(m)}}}{\text{grad}}{{Y}_{i}},$$
In addition to the thermal and caloric equations of state, the method of calculation of the transport properties of multicomponent gas, namely, the viscosity, the thermal conductivity, and the diffusion coefficients, is of importance for the problem under consideration.
For these purposes, we use the combinatory relations [21]as well as the following relations obtained in the first approximation of the Chapman–Enskog theory for the dynamic viscosity, thermal conductivity, and binary diffusion coefficients [22]where xi is the relative mole concentration, \({{x}_{i}} = {{{{p}_{i}}} \mathord{\left/ {\vphantom {{{{p}_{i}}} {p = {{Y}_{i}}{{{{M}_{\Sigma }}} \mathord{\left/ {\vphantom {{{{M}_{\Sigma }}} {{{M}_{i}}}}} \right. \kern-0em} {{{M}_{i}}}}}}} \right. \kern-0em} {p = {{Y}_{i}}{{{{M}_{\Sigma }}} \mathord{\left/ {\vphantom {{{{M}_{\Sigma }}} {{{M}_{i}}}}} \right. \kern-0em} {{{M}_{i}}}}}}\), σi is the radius of a particle of the ith kind in A; \(\Omega _{i}^{{(2,2)}}{\kern 1pt} \text{*} = f\left( {{{T}_{i}}} \right)\) is the collision integral for the viscosity and the thermal conductivity, \({{T}_{i}} = {{kT} \mathord{\left/ {\vphantom {{kT} {{{\varepsilon }_{i}}}}} \right. \kern-0em} {{{\varepsilon }_{i}}}}\), and εi is the parameter of the Lenard–Jones potential that characterizes the potential well depth. The collision integrals for the viscosity and the diffusion were calculated from the Anfimov approximationwhere \({{T}_{{i,j}}} = \frac{{kT}}{{{{\varepsilon }_{{i,j}}}}},\) \({{\varepsilon }_{{i,j}}} = \sqrt {{{\varepsilon }_{i}}{{\varepsilon }_{j}}} ,\) \({{\sigma }_{{i,j}}} = \frac{1}{2}\left( {{{\sigma }_{i}} + {{\sigma }_{j}}} \right)\).
$$\mu = \frac{1}{{\sum\limits_{i = 1}^{{{N}_{c}}} {\left( {{{{{Y}_{i}}} \mathord{\left/ {\vphantom {{{{Y}_{i}}} {{{\mu }_{i}}}}} \right. \kern-0em} {{{\mu }_{i}}}}} \right)} }},\quad \lambda = 0.5\left[ {\sum\limits_{i = 1}^{{{N}_{c}}} {{{x}_{i}}{{\lambda }_{i}}} + \frac{1}{{\sum\limits_{i = 1}^{{{N}_{c}}} {\left( {{{{{x}_{i}}} \mathord{\left/ {\vphantom {{{{x}_{i}}} {{{\lambda }_{i}}}}} \right. \kern-0em} {{{\lambda }_{i}}}}} \right)} }}} \right],\quad {{D}_{i}} = \frac{{1 - {{x}_{i}}}}{{\sum\limits_{j \ne i}^{{{N}_{c}}} {\left( {{{{{x}_{j}}} \mathord{\left/ {\vphantom {{{{x}_{j}}} {{{D}_{{ij}}}}}} \right. \kern-0em} {{{D}_{{ij}}}}}} \right)} }},$$
$${{\mu }_{i}} = 2.67 \times {{10}^{{ - 5}}}\frac{{\sqrt {{{M}_{i}}T} }}{{\sigma _{i}^{2}\Omega _{i}^{{(2,2){\kern 1pt} }}{\kern 1pt} \text{*}}},\;{\text{g/cm}}\,\,{\text{s,}}$$
$${{\lambda }_{i}} = 8330\sqrt {\frac{T}{{{{M}_{i}}}}} \frac{1}{{\sigma _{i}^{2}\Omega _{i}^{{(2,2)}}{\kern 1pt} \text{*}}},\;{\text{erg/cm}}\,\,{\text{K,}}$$
$${{D}_{{i,j}}} = 1.858 \times {{10}^{{ - 3}}}\sqrt {{{T}^{3}}\frac{{{{M}_{i}} + {{M}_{j}}}}{{{{M}_{i}}{{M}_{j}}}}} \frac{1}{{p\sigma _{{i,j}}^{2}\Omega _{{i,j}}^{{(1,1)}}{\kern 1pt} \text{*}}},\;~{\text{c}}{{{\text{m}}}^{{\text{2}}}}{\text{/s,}}$$
$$\Omega _{i}^{{(2,2)}}{\kern 1pt} \text{*} = 1.157T_{i}^{{ - 0.1472}},\quad \Omega _{{i,j}}^{{(1,1)}}{\kern 1pt} \text{*} = 1.074T_{{i,j}}^{{ - 0.1604}},$$
The mass chemical transformation rate was determined in each elementary computational volume when solving the system of kinetic equations (1.7). Here, \({{X}_{i}}\) is the volume-mole concentration of the ith component, \({{N}_{r}}\) is the number of chemical reactions, \({{S}_{{f,n}}}\), and Sr,n are the forward and inverse reaction rates, respectively, kf,n and kr,n – are the forward and reverse reaction rate constants, and \({{a}_{{i,n}}},\) and bi,n are the stoichiometric coefficients of the nth chemical reaction determined with the use of the canonic form of the chemical kinetics equationwhere [Xj] are the chemical symbols of the reactants and the products of chemical reactions. The mass rate of formation of the ith component in unit volume is determined from the formula
$$\sum\limits_{j = 1}^{{{N}_{s}}} {{{a}_{{j,n}}}\left[ {{{X}_{j}}} \right]} = \sum\limits_{j = 1}^{{{N}_{s}}} {{{b}_{{j,n}}}\left[ {{{X}_{j}}} \right]} ,\quad n = 1,2, \ldots ,{{N}_{r}},$$
$${{\dot {w}}_{i}} = {{M}_{i}}\sum\limits_{n = 1}^{{{N}_{r}}} {\left( {{{b}_{{i,n}}} - {{a}_{{i,n}}}} \right)\left( {{{S}_{{f,n}}} - {{S}_{{r,n}}}} \right)} ,\;{\text{g/(c}}{{{\text{m}}}^{3}}{\text{ s}}).$$
The forward ( f ) and reverse (r) reaction rate constants are determined from the generalized Arrhenius dependencewhere \({{A}_{{f(r),n}}},\;{{n}_{{f(r),n}}},\) and Ef(r),n are the approximating coefficients, k is the Boltzmann constant, and n is the number of chemical reaction in the kinetic model. The forward reaction rate approximation constants were borrowed from [19]. The reverse reaction rate constants were calculated using the equilibrium constants [23] approximated in the form of the generalized Arrhenius law
$${{k}_{{f(r),n}}} = {{A}_{{f(r),n}}}{{T}^{{{{n}_{{f(r),n}}}}}}\exp \left( { - \frac{{{{E}_{{f(r),n}}}}}{{kT}}} \right),$$
(1.12)
$${{K}_{n}} = {{A}_{n}}{{T}^{{{{n}_{n}}}}}\exp \left( { - \frac{{{{E}_{n}}}}{{kT}}} \right).$$
(1.13)
In [24] the approximations of equilibrium constants in form (1.13) were tested by comparing with the Park data [19] based on the JANAF data [25].
The smallness of the charged-particle concentrations made it possible to use the quasi-neutrality condition to find the electron mole concentrations.
The dissociation rate constants kf = kD, used in our calculations, were modified with the aim to take into account thermal nonequilibrium of molecules. The investigation of different models of nonequilibrium dissociation demonstrated their appreciable influence on the results of calculations of the intensity of ionization processes. However, in the present study only the heuristic Park model [19] was used. In this model the temperature Ta, used in (1.12), was calculated for each of the vibrational modes from the formulawhere T is the translational temperature and \({{T}_{{V,i(m)}}}\) is the vibrational temperature in the mth vibrational mode.
$${{T}_{{a,i(m)}}} = \sqrt {T{{T}_{{V,i(m)}}}} ,$$
(1.14)
The computational model formulated above, which is based on the system of partial differential equations (1.1)–(1.6), the system of first-order kinetic differential equations (1.7) for each component of gas mixture, the thermal and caloric equations of state (1.8), (1.9)–(1.11), as well as the system of closing relations for determining the thermophysical properties of individual components of gas flow, makes it possible to solve the problem of viscous, heat-conducting and chemically reacting gas flow past a blunt plate of finite dimensions with regard to ionization and nonequilibrium dissociation. We underline that this problem was solved without separation of shock waves and any discontinuities of gas dynamic parameters. The undisturbed free-stream parameters were given on the external boundaries of the computational domain except for the outlet cross-section. Zero derivatives of the velocity components, the density, the pressure, the concentrations of chemical components, and the energy of vibrational modes along the corresponding streamlines were given in the outlet cross-section, in which the flow was always supersonic. It was assumed that charged particle recombination takes place on the surface and the mass fractions of all components recover to the free-stream conditions. The no-slip conditions and a constant surface temperature Tw = 297 K were imposed.
In our calculations we used author’s computer code in which the multigrid multiblock computational technology was implemented. In Fig. 1 we have reproduced the diagram of the problem solved with four computational blocks. In this figure an intermediate computational grid is also given. The final solution was obtained after twice repeated doubling of the number of nodes. The finite-difference method with numerical determination of components of the Jacobian was used within each block. This method was used to transform the real curvilinear grid into a rectangular domain. The first block of the computational grid was aimed to describe the flow in the neighborhood of plate bluntness of radius Rb = 0.1 cm. Block 3 was positioned from the windward side along the plate of length L = 10 and 130 cm and block 2 was positioned from the leeward side. The fourth block was aimed to describe the wake flow. The closing boundary of the plate of thickness h = 0.2 cm was flat. In this region flow was described in fairly detail so that the elements of reverse-vortex flow downstream of the plate edge were distinguished. These elements interacted with the boundary layers developed along the lower and upper surfaces.
Fig. 1.
Schematic of the problem solved and computational four-block grid. The numbers of blocks and characteristic points of the computational grid created analytically are shown.
To obtain the numerical solution, we used a hybrid explicit-implicit algorithm of integration. The continuity and Navier–Stokes equations were integrated using the AUSM-algorithm [26] in variant of approximation of the functions on the second-order contact discontinuity. The system of energy conservation equations for the translational and vibrational degrees of freedom (1.4), (1.6), as well as the diffusion equations (1.5), was integrated using the second-order implicit method. The system of kinetic equations (1.7) was integrated using the generalized Newton’s method [27]. Such a combination of methods made it possible to integrate the equations of motion with the limiting admissible level of the Courant–Friedrichs–Lewy (CFL) criterion. The thermophysical, transfer, and kinetic functions and coefficients that enter into the system of integrated equations were refined in each time step
In the overwhelming majority of cases the steady-state solution was obtained with the discrepancy less than 10–6. In the calculated variants, in which the appearance of separation flow above the leeward surface was observed, a periodic time-dependent solution with the numerical discrepancy of the velocity components and the temperature of the order of 10–3 was also observed. The attempts to increase the accuracy of the time-dependent calculation due to a decrease in the CFL parameter did not lead to apparent change in the solution, but the problem of studying the features of unsteady flow structure was not formulated in the present paper. In [24] the details of the developed methods of computer simulation are given.
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2 RESULTS OF NUMERICAL SIMULATION
We have carried out two series of numerical experiments for the initial data given in Table 1. The first series of calculations was carried out for the experimental conditions [3].
Table 1.
Initial data for flow past a plate of length L = 10 cm at various angles of attack
No. var. | \({{p}_{\infty }}\), erg/cm3 | \({{\rho }_{\infty }}\), g/cm3 | \({{T}_{\infty }}\), K |
\({{M}_{\infty }}\)
| \({{T}_{w}}\), K |
|---|---|---|---|---|---|
1 | 12 | 3.25 × 10–7 | 12.4 | 20 | 12.4 |
2 | 12 000 | 1.84 × 10–5 | 227 | 10 | 227 |
In Fig. 2 we have reproduced the Mach numbers and the temperature of the translational degrees of freedom at an angle of attack α = 10° at which the measurements were carried out in the experiments [3]. We can clearly see the above-mentioned basic structural elements of the field, namely, the shock waves departing from the plate bluntness and the boundary layer regions with subsonic velocities and a higher temperature inside the boundary layer. For example, the temperature inside the boundary layer reaches ~380 K at a distance х = 20 cm from the plate edge, whereas it is equal to ~120 K at the outer edge of the dynamic boundary layer and 297 K on the surface. The highest temperature ~1000 K of the translational degrees of freedom is reached in the neighborhood of the plate bluntness. This is close to the theoretical value of the stagnation temperature T0 = 1004 К.
Fig. 2.
Distributions of the Mach numbers (a) and the temperature of translational degrees of freedom (b) for the initial data of the first variant at the angle of attack \(\alpha = 10^\circ \). Discrete dots correspond to the experimental data [3] at \(\alpha = 10^\circ \) (squares) and \(\alpha = - 10^\circ \) (circles).
One more the higher-temperature zone can be observed in the near wake, where the streams shedding from the upper and lower surfaces encounter (Fig. 2b). Here, gas is heated to the temperature Т ~ 350 K. On average, as the angle of attack increases, the temperature decreases in the compressed layer above the leeward surface. On the contrary, above the windward surface and in the near wake the temperature increases with the angle of attack.
In Fig. 3 we have reproduced the distribution of vibrational excitation temperatures of the molecules N2 (in what follows, for brevity, called by vibrational temperatures), as well as the difference between the vibrational excitation temperatures and the temperature of the translational degrees of freedom (in what follows, for brevity, called by translational temperature) \(\Delta {{T}_{V}} = {{T}_{V}} - T\). We underline that positive values of the quantity \(\Delta {{T}_{V}}\) mean that the vibrational temperature is higher than the translational temperature.
Fig. 3.
Vibrational temperature \({{T}_{{V1}}} = {{T}_{V}}\left( {{{{\text{N}}}_{2}}} \right)\) (a) and difference \(\Delta {{T}_{{V1}}} = {{T}_{V}}\left( {{{{\text{N}}}_{2}}} \right) - T\) (b) under the conditions of experiment [3] at \(\alpha = 10^\circ \).
We can clearly see (Fig. 3a) that two regions of the higher temperature of vibrational excitation are formed in the near-wake region (in Fig. 3a these regions are denoted by Roman numerals I and II), while in the boundary layer near the windward side of the plate, on the contrary, the translational temperature is systematically higher than the vibrational temperature (zone III in Fig. 3a). Therefore, we can note the existence of three regions of thermal nonequilibrium in the stream, the first two regions with the higher vibrational temperature being separated by an almost equilibrium region with a slight excess of the vibrational temperature above the translational one. However, we should take into account the fact that formation of these regions of vibrational nonequilibrium depends mainly on the characteristic relaxation times of the vibrational modes. In particular, for oxygen the vibrational temperature distributions testify on the higher rate of the processes of thermalization of О2 molecular vibrations. This is not surprising since the relation between the characteristic times of vibrational relaxation \({{\tau }_{{{{{\text{O}}}_{2}}}}} < {{\tau }_{{{{{\text{N}}}_{2}}}}}\) is well known [19].
Appearance of the zones of nonequilibrium vibrational excitation of diatomic molecules is the consequence of the general regularities of nonequilibrium flow past the plate at an angle of attack. We can observe rapid growth of the translational temperature and significantly slower increase in the vibrational temperatures behind the front of shock formed in the neighborhood of plate bluntness. This relates directly to the basic physical regularities, namely, in order to excite vibrations of diatomic molecule it is necessary that an extremely great number of intermolecular collisions occur than it is required to excite the translational degrees of freedom (the former is greater by ten thousand folds than the latter). Thus, the flow behind the front of shock wave near the bluntness starts from the explicit delay of the vibrational temperature from the translational one. In the neighborhood of the windward surface the translational temperature increases sharply together with the pressure and the density and the vibrational degrees of freedom have no time to be thermalized.
Behind the front of oblique shock the translational temperature is considerably lower than that behind the normal shock; therefore, the vibrational temperature may be even higher than the translational temperature behind the normal shock. The well-studied regularity that the vibrational temperatures are higher than the translational temperature can be observed in the neighborhood of the leeward surface and in the wake. The pattern of the process becomes opposite to that observed behind the shock wave. The translational temperature sharply drops in rarefaction flow, while the intermolecular collision intensity is not sufficient to produce a drop in the vibrational temperature. Sharp increase in the translational temperature is again observed only in the region of collision of streams shedding from the windward and leeward sides of the plate (zone I and II in Fig. 3a).
In Fig. 4 we have reproduced the results of calculations of flow past the blunt plate under the same conditions but at the angle of attack α = 45°. All the regularities noted above in modification of the flow structure with increase in the angle of attack remain the same but they manifest themselves in the higher degree. The pressure decreases by more than 1000 times from the values in the boundary layer in the neighborhood of the windward surface to the pressure near the leeward surface (Fig. 4d). Two flow regions of the higher translational temperature are observed (Fig. 4c): one region is located in the neighborhood of the windward side, where the free stream creates a compression zone near the surface. The second region is located in the upper part of the wake above the leeward side. In this region the translational temperature reaches approximately 0.5Т0.
Fig. 4.
Distributions of the Mach numbers (a), the y-component of the velocity \({{V}_{y}} = {{v} \mathord{\left/ {\vphantom {{v} {{{V}_{\infty }}}}} \right. \kern-0em} {{{V}_{\infty }}}}\) (b), the temperature of translational degrees of freedom (c) and the pressure (p, in erg/cm3), (d) under the conditions of experiment [3] at the angle of attack \(\alpha = 45^\circ \).
In Figs. 5a and 5b we have reproduced the vibrational temperature distributions and in Figs. 5c and 5d we have shown the differences between the temperatures of the vibrational and translational degrees of freedom. These data illustrate well the effect of the characteristic times of relaxation of molecular oscillations on the vibrational excitation temperature. The vibrational temperatures are higher than the translational temperature by more than 300 K in the regions of maximum rarefaction above the surface.
Fig. 5.
Temperature distributions of the vibrational degrees of freedom of the N2 (a) and O2 (b) molecules and the differences \(\Delta {{T}_{{V1}}} = {{T}_{V}}\left( {{{{\text{N}}}_{2}}} \right) - T\) (c) and \(\Delta {{T}_{{V{\text{2}}}}} = {{T}_{V}}\left( {{{{\text{O}}}_{2}}} \right) - T\) (d) under the conditions of experiment [3] at the angle of attack \(\alpha = 45^\circ \).
In Figs. 6 and 7 we have shown the changes that occur in the flow when the free-stream density increases. These figures correspond to flow conditions at the free-stream static pressure by 103 times higher than that in the previous variant. Under these conditions a small zone of reverse-vortex flow already appears near the windward surface in the neighborhood of the rear plate edge. This zone corresponds to the Mach numbers М = 0.1 (Fig. 6a). In Fig. 6,b we can see the region of negative values of the y-component of the velocity in the neighborhood of the upper rear edge. This confirms the presence of a small vortex zone. Note that an increased oscillation of the numerical solution can be observed in this region; however, this oscillation does not lead to catastrophic consequences in the numerical solution.
Fig. 6.
Distributions of the Mach numbers (a) and the y-component of the flow velocity \({{V}_{y}} = {{v} \mathord{\left/ {\vphantom {{v} {{{V}_{\infty }}}}} \right. \kern-0em} {{{V}_{\infty }}}}\) (b) under the conditions of the second computational variant at the angle of attack \(\alpha = 45^\circ \).
Fig. 7.
Temperature distributions of the translational degrees of freedom (a), the electron density (b) and difference between the temperature of vibrational excitation of molecules \(\Delta {{T}_{{V1}}} = {{T}_{V}}\left( {{{{\text{N}}}_{2}}} \right) - T\) (c) and \(\Delta {{T}_{{V{\text{2}}}}} = {{T}_{V}}\left( {{{{\text{O}}}_{2}}} \right) - T\) (d) for the second computational variant at the angle of attack \(\alpha = 45^\circ \).
The translational temperature reaches the values T ~ 2300 K near the lower (windward) surface, while it drops to the values ~200 K above the upper surface in the rarefaction zone (Fig. 7a). It is noteworthy that the temperature again increases to ~2000 К in the wake in which the streams, that flow past the plate from above and below, converge. Recall that in the previous computational variant (at the lower free-stream density) the temperature scale in the hottest regions was of the order of 500 K. The vibrational degrees of freedom of the N2 and O2 molecules are heated to even higher temperatures (Figs. 7c and 7d). In this case, as before, we can observe the presence of two zones with the higher vibrational temperatures that depart from the leading and trailing plate edges and are separated by a region of relatively high translational temperatures.
One more important result of the increase in the free-stream pressure is appreciable air ionization in the compressed layer near the windward surface and in the near wake (Fig. 7b). No ionization is observed in the reduced-pressure region above the leeward surface.
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3 SUMMARY
To solve the problem of nonequilibrium aerophysics of flow past a blunt plate of finite sizes, the conjugate model of physicochemical gas dynamics is developed and implemented. In this model, together with the Navier–Stokes equations, the chemical kinetics equations, the energy conservation equations for the translational and vibrational degrees of freedom, and the dissociation and ionization kinetics equations are integrated numerically. The multigrid multiblock technology of the hydride explicit-implicit algorithm is used to increase the computational effectiveness of the developed numerical method.
The computational study performed showed that the supersonic rarefied air flow past a blunt plate of finite sizes at high angles of attack is accompanied by formation of several flow regions in which thermalization of the internal degrees of freedom is not reached. This has a strong effect on both the flow structure in the neighborhood of the plate and the chemical reactions in flow which ultimately determine the gas ionization degree at high supersonic velocities.
The developed method was used to interpret experimental data on supersonic flow past a plate at an angle of attack.
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Translated by E.A.Pushkar