Based on the continuum hypothesis, problem of compressing wave was studied analytically. By exploring the temperature distribution, the propagation velocity, and the thickness of the transition region, the developing process of compressing wave turning into shock wave was revealed, and the following conclusions were reached:
Compressing wave is an inherent phenomenon for compressible fluid flow. As all actual fluids have compressibility [
In summary, solving compressing problem based on continuum hypothesis is still an important method, and it can also be used to explore the relationship and the transition process between sonic wave and shock wave. Now that the analytical solution of shock wave for
The physical model established in this paper is shown in Figure
Propagation of a one-dimension compressing wave driven by a moving piston.
Between the compressed region and the unaffected region, there exists a transition region, where parameters such as velocity, temperature, and pressure change continuously from the undisturbed state such as pressure
In order to construct the governing equations of the compressing wave, the conservation laws should be considered. First, as the piston moves and the compressing wave propagates, the gas is not leaked, so the mass conservation law must be satisfied; for the one-dimension flow, it can be written as below, which is called continuity equation in fluid dynamics.
Second, as the gas is compressed, the force is exerted between fluid elements, so the conservation law of momentum must be satisfied, which can be written as below, which is the one dimension form of the classical NS equations in fluid dynamics. In the equation, the compressibility of the fluid and the diffusion of the momentum were both considered.
Third, as the temperature of the compressed gas increases, the heat transfer between fluid elements exists, so the conservation law of energy must be considered, and its one dimension form can be written as below, in which the viscous dissipation item was considered and included in the last item.
The governing equations (
In the equations above, velocity
Then a new transformed velocity
Equations (
Thus, the unsteady problem discussed in Figure
Transformed compressing wave model in the new coordinate.
In this model, compressible fluid flowed in a velocity of
To solve (
As for the pressure boundary condition, its initial value in the unaffected region is known, so it can be set as follows:
And so it is with the density boundary condition:
As for the temperature, its initial value in the unaffected region is known, but its value in the compressed region is unknown. In order to construct two boundary conditions for the temperature, it is supposed that its gradient far from the transition region is 0, so the temperature boundary condition can be set as follows:
The nonlinear equations composed of (
Assuming that the dynamic viscosity
For the energy equation (
In order to discuss expediently, the ideal gas state equation is introduced, and
So the closed equations about
The phase diagram for this problem discussed in this article is shown as Figure
Direction field in the phase diagram for Pr = 3/4.
Case for Ma = 2.65
Case for Ma = 1.22
In Figure
The critical points
The solution of compressing wave is composed of two parts: the inner solution and the outer solution: the inner solution corresponds to the transition region; the gradient of it is large and the thickness is thin; the outer solution corresponds to the two steady states, which has gentle gradient and the dimension is larger, and the relationship between the two steady states can be obtained by traditional Rankine-Hugoniot relationship formula and Prandtl formula. The complete solution of a compressing wave must take into account the relationship between the two parts. However, for an arbitrary Pr, the analytical solution of the inner solution is hard to derive. But for the special case with Pr being
As is shown in Figure
This equation is an implicit solution, in which
In the inner solution, the integral constants are still unknown, such as
The velocity of the compressing wave can be obtained by solving the above algebraic equation as below, in which
Based on this velocity, other integral constants can be determined: the total energy
The mass flow rate per unit area
And the total pressure
Figure
Distribution of relative velocity
Distribution of relative velocity
Distribution of temperature
In Figure
For the case of piston velocity being 10 m/s, the thickness
The derived inner analytical solutions (
Direction field in the phase diagram for Pr = 10.
Direction field in the phase diagram for Pr = 0.3.
As the piston velocity increases, the thickness of the transition region
Thickness of transition region and propagation velocity of compressing wave for different piston velocity.
In Figure
The parameters used for calculation in Figure
Calculating parameters for case shown in Figure
Parameter/unit | Value |
---|---|
| 1.4 |
| |
| 1.165 |
| 287 |
| 300 |
| 18.6 × 10−6 |
It can be found that when the piston velocity
It is noticeable that the above results are obtained from the viewpoint of mathematics, while for actual situations the applicability of continuum hypothesis needs to be considered. For a small disturbance velocity
From the analytical solution, the entropy generation process in the compressing wave can be calculated easily as (
The relationship between the entropy generation and the temperature across compressing wave for the case of
Relationship of entropy generation and temperature across compressing wave for different Ma.
Based on the continuum assumptions, distribution of the parameters such as temperature, velocity, and entropy generation across the compressing wave was calculated analytically and the following conclusions were reached.
Sonic velocity
Propagation velocity of the shock wave
Specific heat at constant pressure
Specific heat at constant volume
Enthalpy
Mass flow rate
Integral constant of energy equation
Mach number
Integral constant of momentum equation
Pressure
Prandtl number
Gas constant
Time
Temperature
Absolute velocity relevant to the ground
Disturbance velocity
Specific volume
Relative velocity relevant to the ground defined by (
Absolute coordinate relevant to the ground.
Thickness of the transition region of the shock wave
Entropy generation
Thermal conductivity
Dynamic viscosity
Effective dynamic viscosity defined by (
Density
Relative coordinate relevant to the ground.
Inlet parameters of the shock wave
Outlet parameters of the shock wave.
Dimensionless form of parameters.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This project is supported by the Fundamental Research Funds for the Central Universities, China (Grant no. 2014QNB07), and the Natural Science Foundation of Jiangsu Province (BK20170280).