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Text|Colajun
Editor|Colajun
Supersonic flow is an important research area in fluid mechanics, involving aerodynamics, aerospace engineering, high-speed transportation and many other fields.
In supersonic flow, wedges are one of the common geometric shapes and are of great significance for the study of the flow around them.
In practice, the high temperature effect is often present in supersonic flow, especially in high-speed aerothermodynamic environments. This paper aims to explore the effect of high temperature on supersonic flow around wedges and analyze its mechanism in depth.
●○ Basic characteristics of supersonic flow around wedges ○●
When the shock wave touches the wedge tip, the flow disturbance begins at the wedge tip. Depending on the Mach number and the angle of the wedge, regular reflections (the shock wave connected to the wedge) or irregular reflections (the reflected shock wave has a curved leading edge) are generated.
Depending on the inlet flow conditions (angle of the wedge and number of inlet Mach), the main and reflected waves meet at three points. Between this point and the wedge, the two waves merge into a single wave, forming a Mach configuration.
Downstream of the three stresses, a tangential discontinuous surface is formed, where pressure and normal velocity remain continuous, while density and tangential velocity occur discontinuously.
In solutions with oblique shock waves of lower intensity ("weak" impact), the uniform flow between the shock wave and the wedge is almost always supersonic. After a stronger shock wave ("strong" shock), the flow of complete gas is always subsonic.
For non-viscous compressible supersonic and hypersonic gas flows, there is a possibility of two solutions (strong and weak). Depending on the angle of incidence and the number of input Mach, two typical configurations are formed: two-hop (regular) and three-hop (Mach), and both options may exist within a range of parameters.
When a strong shock wave moves and interacts with an object, the temperature and pressure of the gas behind its front edge increase. A complete gas model does not provide the accuracy of the required numerical solution because molecular weight and heat capacity are not constants.
In high-temperature air, these quantities are a function of pressure and temperature. In this case, the dissociation and ionization processes that occur in the gas need to be considered. In fact, various models are used to account for high-temperature processes in gases, as well as for resolving dependencies and interpolation of tabular values.
From a computational point of view, the model proposed by the experimenters (Kraiko model) for considering reactions between 13 components in air is very interesting and successful. The main advantage of this model is that it takes into account the dissociation and ionization of air at high temperatures.
At temperatures ranging from 20,000 K and pressures from 0.001 to 1000 atm, the error of the model does not exceed 2%, usually in the 1% range.
●○The effect of high temperature on supersonic flow around the wedge ○●
Solve the control equation in the domain shown in Figure 1. The angle of the wedge is β. The length of the domain is 3.2 and the height is 2.2. The distance of the wedge from the origin is 0.2. The shock wave is located at the point x = 0 and appears at time t = 0.
There is no gas movement in front of the shock wave (u = v = 0), and the pressure and temperature are p = 101325 Pa and T = 288.2 K. Flow velocities are fixed at 1000 and 3000 m/s. The amount of flow behind the shock wave was determined by Rankine-Hugoniot conditions. The gas crosses the centerline. The upper and right borders are subject to free outflow conditions.
Figure 1
When a supersonic fluid bypasses the wedge, a oblique shock wave occurs within certain limits of the wedge angle and Mach number. When bypassing the cone, the shock wave front has a tapered surface.
The wedge angle β equal to the steering angle of the flow at the shock wave. The angle between the front of the shock wave and the direction of the undisturbed flow is called the impact slope angle σ. The undisturbed flow velocity v1 is decomposed into components with the impact surface normal and tangential direction, i.e. vn1 and vτ1.
Therefore, the flow before and after the shock wave is interrelated through the law of conservation of mass, momentum and energy, i.e.:
Here, ε represents specific internal energy. The terms in parentheses indicate the sum of specific internal energy and specific kinetic energy before and after the shock wave. This change in quantity is related to the work done by the external force on a certain mass of gas, where only the surface pressure force is taken into account.
Considering the condition of conservation of mass, the tangential velocity components on the shock wave are required to be equal, that is, vτ1=vτ2, ρ1vn1=ρ2vn2, and the relationship has the following form:
where h = ε + p/ρ is the specific enthalpy. The given conservation laws apply to any gas model (ideal gas, real gas, dissociated gas, or ionized gas) when passing through an oblique shock wave, because they express the general relationship of conservation law and do not involve the relationship between correlating thermodynamic variables with each other and determining the form of thermodynamic functions.
In order to meet the conditions of dynamic compatibility at the shock wave, it is necessary to give specific dependencies that determine the characteristics of the thermodynamic state of the gas. Specific enthalpy and molar mass are functions of pressure and temperature, i.e. h = h(p, t) and μ = μ(p, t). The equation of state of an ideal gas is used, the composition of which corresponds to this state:
where R0 is the universal gas constant. The thermodynamic quantities before the shock wave are known (h1 = cp1T1, μ1 = 0.029 kg/mol, for air), and the relationship related to the thermodynamic parameters after the shock wave is introduced from the additional conditions describing the thermodynamic model of high temperature air.
To determine the thermodynamic quantity of high-temperature flow after a shock wave (vτ1 = vτ2), the following equation is applied:
Supersonic flow simulation of ideal and hot gases on wedge-shaped objects with a half-angle of β=30°. The inlet pressure and inlet temperature (flow before the shock wave) are 105 Pa and 290 K, respectively.
The working substance is air (γ=1.4, μ=0.029 kg/mol). The inlet Mach number varies between 2 and 16. For the ideal complete gas, the solution is given in tabular form.
The results presented in the study correspond to two cases, the difference between which lies in the speed in front of the shock wave. The values were 103 m/s (case 1) and 3∙103 m/s (case 2), respectively.
In case 1, the density of the ideal gas is ρ=5.5 kg/m3, the pressure is p=16.78 bar, and the temperature is T=1068 K. The density of high-temperature air is ρ=5.7 kg/m3, the pressure is p=16.59 bar, and the temperature is T=1012 K.
In case 2, the density of the ideal gas is ρ=7.0 kg/m3, the pressure is p=133.5 bar, and the temperature is T=6649 K. The density of high-temperature air is ρ=10.0 kg/m3, the pressure is p=127.1 bar, and the temperature is T=4251 K.
The grid contains 110×160 nodes. Grid nodes cluster near the solid boundary and shock wave front to account for the gradient region of the flow (Figure 2).
The minimum residual level is used as the convergence criterion for difference decomposition to approach the steady-state solution of the problem. It takes about 2200 time steps to reach the specified residual level (R = 10^-10 in the calculation).
The pressure distribution from the ideal gas and the real gas model over time is shown in Figure 3. In this case, the shock wave structure of the two models is similar.
Figure 2
However, the compression region of the real gas is slightly smaller than the compression region in an ideal gas model. In case 1, the temperature does not exceed 1900 K. The temperature is low and no chemical reaction occurs.
The molar mass of the air therefore remains constant. Notably, there are narrow regions with higher temperatures, reaching 2480–3100 K, which have less effect on flow patterns.
The large flow rate results in a significant difference in the flow distribution calculated by the ideal gas model and the real gas model (Figure 5).
Figure 5
The pressure calculated with the real gas model exceeds the pressure calculated with the ideal gas model. The shock wave structure in a real gas has a flat shape compared to an ideal gas model.
Dissociation and ionization processes in high-speed flow result in different temperature distributions. The maximum temperature in a real gas (approximately 11,000 K) is twice as low as the temperature calculated with an ideal gas model (approximately 23,000 K).
Figure 6
The temperature distribution calculated by the two models is compared in Figure 7. The density distribution is similar to the pressure distribution, however, the density calculated with the ideal gas model is twice as small as the density calculated with the real gas model.
Figure 7
Figure 8 shows the distribution of flow characteristics along the x=0 line on the lower wall of the computational domain. When the high temperature effect of air is taken into account, the change in pressure is relatively small. The dashed line corresponds to the ideal gas model, and the solid line corresponds to the real gas model.
The distribution of flow characteristics calculated under the same inlet conditions shows that the shock wave structures calculated by different air models are similar to each other.
Figure 8
However, the distribution of flow characteristics is different (Figure 9). In the impact region, the difference in parameter values is small, similar to what is observed in an ideal gas.
Figure 9
At the same time, the temperature distribution calculated with the ideal gas model and the real gas model is different. Figure 11 compares the temperature distribution calculated using the ideal gas model with the real gas model.
Figure 11
Figure 12 shows the distribution of flow characteristics along the x=0 line. The high temperature effect of air has a significant effect on density and temperature distribution. At the same time, the pressure distribution has a relatively small effect on physical and chemical processes. The dashed line corresponds to the ideal gas model, and the solid line corresponds to the real gas model.
Figure 12
Figure 13 shows the effect of wedge angle and inlet Mach number (β=30°). The pressure distribution is not affected by the effects of high temperatures in the air. At the same time, the temperature distribution calculated by different gas models is different.
Figure 13
The dashed line corresponds to the ideal gas model, and the solid line corresponds to the real gas model. For comparison, Figure 14 shows the distribution of flow characteristics after a positive shock wave versus the inlet Mach number.
The flow velocity after a positive shock wave is subsonic, so the difference between the flow characteristics calculated by the ideal gas model and the real gas model exceeds the mismatch of the flow characteristics observed after the oblique shock wave. Using an ideal gas model under high Mach number inflow conditions results in inaccurate solutions.
Figure 14
Figure 15 shows the relative error of the flow characteristics calculated using the ideal gas model and the real gas model. The increase in wedge angle leads to an increase in the error between solutions calculated using different air models.
Figure 15
In this study, supersonic flow around a wedge with a half-angle of β=30° was simulated, using an ideal gas model and a high-temperature gas model. By calculating the Mach number of different inlets, the distribution and difference of flow characteristics are obtained.
It is found that when considering the air high temperature effect, there is a significant difference in the flow characteristics calculated between the real gas model and the ideal gas model.
In the case of oblique shock waves, the pressure and density distributions calculated by the real gas model are slightly different from the ideal gas model, while the temperature distribution is quite different.
The dissociation and ionization processes in high-speed flow result in differences in temperature distribution, and the maximum temperature calculated by the real gas model is lower than the ideal gas model.
The distribution of flow characteristics along the lower wall of the computational domain was studied, and it was found that the high temperature effect of air had a significant effect on the density and temperature distribution, while the pressure distribution was less affected by physical and chemical processes.
In addition, the effects of wedge angle and inlet Mach number on flow characteristics were explored. The results show that the pressure distribution is not affected by the high temperature effect of air, but the temperature distribution calculated by different gas models is different. In the case of positive shock wave, there are obvious differences in the flow characteristics calculated by different gas models, and the results calculated by the real gas model are more in line with the actual situation.
Overall, using an ideal gas model at high Mach numbers results in inaccurate solutions. The results of this study highlight the importance of considering the effects of high temperatures in supersonic flows and provide a basis for calculations using real gas models.