<p>In this paper, we are concerned with the two-dimensional steady supersonic combustion flows with a contact discontinuity moving through a nozzle of finite length. Mathematically, it can be formulated as a free boundary value problem governed by the two-dimensional steady combustion Euler equations with a contact discontinuity as the free boundary. The main mathematical difficulties are that the contact discontinuity is a characteristic free boundary and the equations for all states are coupled with each other due to the combustion process. We first employ the Lagrangian coordinate transformation to fix the free boundary. Then by introducing the flow slope and Bernoulli’s function, we further reduce the fixed boundary value problem into an initial boundary value problem for a first-order inhomogeneous hyperbolic system coupled with several ordinary differential equations. A new iteration scheme is developed near the background states by employing the intrinsic structure of the equation for the mass fraction of unburned gas. We show that there is a fixed point for the iteration by deriving some novel <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2024_2893_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">$C^{1,\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>α</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>-estimates of the solutions and applying the Schaulder fixed point theorem, and then the uniqueness of the fixed point is proved by a contraction argument. Based on the inverse Lagrangian coordinate transformation, we show that the original free boundary problem admits a unique contact discontinuity solution provided that the incoming flows and the nozzle walls are small <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2024_2893_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">$C^{1,\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>α</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>-perturbations of the background states. On the other hand, a quasi-one-dimensional approximate system is often used to simplify the two-dimensional steady supersonic combustion model (cf. Menshov in Fluid Dyn 24:277–284, 1989). The error between these two systems is estimated. Finally, given a piecewise <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2024_2893_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">$C^{1,\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>α</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>-solution containing a contact discontinuity with piecewise constant states on the entrance of the nozzle, we can show that the solution is the piecewise constant states with a straight contact discontinuity.</p>

Steady supersonic combustion flows with a contact discontinuity in two-dimensional finitely long nozzles

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发表日期：

2025年1月6日

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Junlei Gao,
Feimin Huang,
Jie Kuang,
Dehua Wang,
Wei Xiang

摘要

In this paper, we are concerned with the two-dimensional steady supersonic combustion flows with a contact discontinuity moving through a nozzle of finite length. Mathematically, it can be formulated as a free boundary value problem governed by the two-dimensional steady combustion Euler equations with a contact discontinuity as the free boundary. The main mathematical difficulties are that the contact discontinuity is a characteristic free boundary and the equations for all states are coupled with each other due to the combustion process. We first employ the Lagrangian coordinate transformation to fix the free boundary. Then by introducing the flow slope and Bernoulli’s function, we further reduce the fixed boundary value problem into an initial boundary value problem for a first-order inhomogeneous hyperbolic system coupled with several ordinary differential equations. A new iteration scheme is developed near the background states by employing the intrinsic structure of the equation for the mass fraction of unburned gas. We show that there is a fixed point for the iteration by deriving some novel $C^{1,\alpha }$ $C^{1, α}$ -estimates of the solutions and applying the Schaulder fixed point theorem, and then the uniqueness of the fixed point is proved by a contraction argument. Based on the inverse Lagrangian coordinate transformation, we show that the original free boundary problem admits a unique contact discontinuity solution provided that the incoming flows and the nozzle walls are small $C^{1,\alpha }$ $C^{1, α}$ -perturbations of the background states. On the other hand, a quasi-one-dimensional approximate system is often used to simplify the two-dimensional steady supersonic combustion model (cf. Menshov in Fluid Dyn 24:277–284, 1989). The error between these two systems is estimated. Finally, given a piecewise $C^{1,\alpha }$ $C^{1, α}$ -solution containing a contact discontinuity with piecewise constant states on the entrance of the nozzle, we can show that the solution is the piecewise constant states with a straight contact discontinuity.