Modified Ghost Fluid Method with Acceleration Correction (MGFM/AC)

In this work, we show that the modified ghost fluid method might suffer overheating and leads to inaccurate numerical results when directly applied to a moving rigid boundary with acceleration. We discover the insightful reasons and then develop a new technique to take into account the effect of boundary acceleration on the definition of ghost fluid states based on a generalized Piston–Riemann problem. Theoretical analysis and numerical results show that the modified ghost fluid method with acceleration correction can overcome such difficulty effectively.

The research was supported in part by the Science Challenge Project (No. JCKY2016212A502) and National Natural Science Foundation of China (Nos. 11601013 and 91530325).

Authors and Affiliations

1. LMIB and School of Mathematics and Systems Science, Beijing University of Aeronautics and Astronautics, Beijing, 100191, People’s Republic of China

   Tiegang Liu & Chengliang Feng

2. China Academy of Aerospace Aerodynamics, Beijing, 100074, People’s Republic of China

   Liang Xu

Corresponding author

Correspondence to Tiegang Liu.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Basic Differential Relations and the Rankine–Hugoniot Relations for Fluid Flow

To solve the Riemann problem and the generalized Riemann problem, we need to use the concept of Riemann invariants and provide some differential relations among different variables. For more details, we can refer to [35, 39, 40].

For this purpose, we write the system of Euler equations (2.1) in the following form:

where \( D/Dt=\partial /\partial t+u\cdot \partial /\partial x \) is the material derivative, and the entropy S is related to the other variables through the second law of thermodynamics

and T is the temperature. Regarding P as a function of \( \rho \) and S, \( P=P(\rho ,S)\), then the local sound speed c is defined as

Thus, the third equation of (A.1) can be replaced equivalently by

We observe that the entropy S is constant along a streamline. By the hyperbolic quasilinear analysis for equations (A.1), three eigenvalues of the system are obtained.

We recall [35], and here introduce the important Riemann invariants \( \phi \) and \(\psi \) for \( \lambda _{-},\lambda _{+} \)

We carry out the total differential for (A.6)

where

Along the characteristic \( {{C}_{+}}{:}\,dx/dt=u+c\), we have

and along the characteristic \( {{C}_{-}}:dx/dt=u-c\), we have

In particular, when the fluid is polytropic gases, we have

and the sound speed c,

The isentropic relation can be written as

In this case, we can get the Riemann invariants as

Taking the partial derivative of S for (A.13),

Then (A.9) turns to

Bringing (A.18) into (A.7) and (A.8)

and the second thermodynamics law can be written as

Then we have

Taking (A.1) to the Riemann invariants and entropy form

(A.25) still works across the rarefaction wave which is a smooth wave.

If there is a shock (or contact discontinuity) trajectory \( r=r(t) \) with speed \( \sigma =dr/dt\), assuming \( {{U}_{a}},{{U}_{b}} \) are the states on the wave ahead and behind the wave back. Then across the shock, \( {{U}_{a}},{{U}_{b}} \) satisfy the Rankine–Hugoniot relations,

1. (a)

   When \( {{u}_{a}}={{u}_{b}} \) and \( {{\rho }_{a}}\ne {{\rho }_{b}}\), the simple wave is a contact discontinuity, \( {{U}_{a}},{{U}_{b}} \) satisfy \( {{P}_{a}}={{P}_{b}}\), and \( \sigma ={{u}_{a}}\pm {{c}_{a}}={{u}_{b}}\pm {{c}_{a}} \)

2. (b)

   When \( {{u}_{a}}\ne {{u}_{b}}\), the simple wave is a shock wave, from (A.26), we can get

   $$\begin{aligned} \left\{ {\begin{array}{*{20}{l}} {\sigma = \frac{{{u_a}{\rho _a} - {u_b}{\rho _b}}}{{{\rho _a} - {\rho _b}}}}\\ {{{({u_a} - {u_b})}^2} = ({P_a} - {P_b})\left( \frac{1}{{{\rho _b}}} - \frac{1}{{{\rho _a}}}\right) }\\ {{e_a} - {e_b} = \frac{{{\rho _a} - {\rho _b}}}{{2{\rho _a}{\rho _b}}}({P_a} + {P_b})}. \end{array}} \right. \end{aligned}$$

   (A.27)

By the combination of EOS function \( e=e(\rho ,P) \) and (A.27), we can get the following relation form

here, when the shock is related to \( {{\lambda }_{+}}=u+c\), the sign is (+), else should be (−) which is related to \( {{\lambda }_{-}}=u-c \).

The partial derivatives are noted as following forms.

In particular, when the EOS function is (A.12), taking \( {{\mu }^{2}}=(\gamma -1)/(\gamma +1)\), we have

Appendix B: Wave Pattern and Interface Solution of Riemann Problem

Considering the following Riemann problem with the EOS of gas (A.12),

As shown in Fig. 15, it is known that the wave pattern is three-waves structure [39], which are a left simple wave \( W_L \) related to \( {{\lambda }_{-}}=u-c\), a contact discontinuity related to \( {{\lambda }_{0}}=u\), and a right simple wave \( W_R \) related to \( {{\lambda }_{+}}=u+c \). The regions between these waves are two constant fluid states \( U_{1*}\), \( U_{2*} \).

Wave pattern of Riemann problem

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Across the three waves, as shown in Table 1, these states satisfy conditions for Riemann invariants to be equal or satisfy the Rankine–Hugoniot relations.

Wave Pattern and Interface Solution of Riemann Problem Obtained by RBC

From the right side state definition (2.6) by RBC, the initial state distributions of Riemann problem (B.1) satisfy

Then, we can get the solution of this problem as following form.

Wave pattern of Riemann problem obtained by RBC when \( u_{I}^{0}<{{u}_{L}} \)

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1. a.

   When \( u_{I}^{0}<{{u}_{L}}\), as shown in Fig. 16, the three-waves structure is composed of two shock wave \( W_L, W_R \) and one contact discontinuity, and the states \( U_{1*},U_{2*} \) satisfy

   $$\begin{aligned}&\begin{array}{*{20}{l}} {{u_{1*}} = {u_L} - H({P_{1*}};{P_L},{\rho _L})}\\ {{\rho _{1*}} = G({P_{1*}};{P_L},{\rho _L})}, \end{array}\end{aligned}$$

   (B.3)
   $$\begin{aligned}&\begin{array}{*{20}{l}} {{u_{2*}} = {u_R} + H({P_{2*}};{P_R},{\rho _R})}\\ {{\rho _{2*}} = G({P_{2*}};{P_R},{\rho _R})}, \end{array}\end{aligned}$$

   (B.4)
   $$\begin{aligned}&\begin{array}{*{20}{l}} {{u_{1*}} = {u_{2*}}}\\ {{P_{1*}} = {P_{2*}}}. \end{array} \end{aligned}$$

   (B.5)

Bringing (B.2) into (B.4)

By the combination of (B.5), (B.3) and (B.6), we can get the following relation form

Here, we note the state on the piston interface as \( (u_{ shock }^{*},P_{ shock }^{*},\rho _{ shock }^{*}):=(u_{1*},P_{1*},\rho _{1*}) \), then they should satisfy

Wave pattern of Riemann problem obtained by RBC when \( u_{I}^{0}\ge {{u}_{L}} \)

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1. b.

   When \( u_{I}^{0}\ge {{u}_{L}}\), as shown in Fig. 17, the three-waves structure is composed of two centered rarefaction wave \( W_L, W_R \) and one contact discontinuity, and the states \( U_{1*},U_{2*} \) satisfy

   $$\begin{aligned}&\begin{array}{*{20}{l}} {\int \limits _{{U_L}}^{{U_{1*}}} {d\psi } = {u_{1*}} - {u_L} + \int \limits _{{P_L}}^{{P_{1*}}} {\frac{1}{{\rho c}}dP} = 0}\\ {S({\rho _{1*}},{P_{1*}}) = S({\rho _L},{P_L})}, \end{array}\end{aligned}$$

   (B.9)
   $$\begin{aligned}&\begin{array}{*{20}{l}} {\int \limits _{{U_R}}^{{U_{2*}}} {d\phi } = {u_{2*}} - {u_R} - \int \limits _{{P_R}}^{{P_{2*}}} {\frac{1}{{\rho c}}dP} = 0}\\ {S({\rho _{2*}},{P_{2*}}) = S({\rho _R},{P_R})}, \end{array}\end{aligned}$$

   (B.10)
   $$\begin{aligned}&\begin{array}{*{20}{l}} {{u_{1*}} = {u_{2*}}}\\ {{P_{1*}} = {P_{2*}}}. \end{array} \end{aligned}$$

   (B.11)

Bringing (B.2) into (B.10)

By the combination of (B.9), (B.11) and (B.12), we can get the following relation form

Here, we note the state on the piston interface as \( (u_{ rare }^{*},P_{ rare }^{*},\rho _{ rare }^{*}):=({{u}_{1*}},{{P}_{1*}},{{\rho }_{1*}})\), then they should satisfy

Appendix C: Wave Pattern and Interface Solution of Generalized Riemann Problem

Considering the following generalized Riemann problem,

As shown in Fig. 18, The solution is piecewise smooth in a small-time range.

Note \( {{U}_{1*}}={\mathop {\lim }\nolimits _{t\rightarrow 0+}}\,{{U}_{1}}({{x}_{I}}-,t) \) and \( {{U}_{2*}}={\mathop {\lim }\nolimits _{t\rightarrow 0+}}\,{{U}_{2}}({{x}_{I}}+,t)\), referring to [35], it is known that \( (U_{1*}^{{}},U_{2*}^{{}}) \) are the solutions on both side of the contact discontinuity for Riemann problem (B.1). With time \( t\rightarrow 0+ \) and \( x\rightarrow {{x}_{I}}-\), the limiting material derivatives \( (Du/Dt)_{1*}\), \( {{(DP/Dt)}_{1*}} \) behind the left wave \( W_L \) satisfy

Wave pattern of generalized Riemann problem

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with

Here, \( H_{i}^{L}={{H}_{i}}({{P}_{*}};{{P}_{L}},{{\rho }_{L}}),i=1,2,3 \) are given in (A.30) and (A.33).

Here, \( {{T}_{L}}{{{S}'}_{L}}\), \( {{{\psi }'}_{L}} \) are given in (A.22) and (A.23).

Particularly, when \( {{u}_{*}}={{u}_{L}}\), (C.7) becomes

The limiting material derivatives \( {{(Du/Dt)}_{2*}}\), \( {{(DP/Dt)}_{2*}} \) behind the right wave \( {{W}_{R}} \) satisfy

with

Here, \( H_{i}^{R}={{H}_{i}}({{P}_{*}};{{P}_{R}},{{\rho }_{R}}),i=1,2,3\), are given in (A.30) and (A.33).

Here, \( {{T}_{R}}{{{S}'}_{R}}\), \( {{{\phi }'}_{R}} \) are given in (A.22) and (A.24).

Particularly, when \( {{u}_{*}}={{u}_{R}}\), (C.14) turns to

Thus, turning the material derivatives to spatial derivatives, we have the limiting spatial derivatives \( {{(\partial u/\partial x)}_{1*}}\), \( {{(\partial \rho /\partial x)}_{1*}}\), \( {{(\partial P/\partial x)}_{1*}} \) behind the left wave \( W_L \) satisfy

Here, \( \frac{\partial \rho _{ shock }^{1*}}{\partial x} \) and \( \frac{\partial \rho _{ rare }^{1*}}{\partial x} \) satisfy

with

where \( {{G}_{i}}\), \( i = 1,2,3 \) are given in (A.35);

The limiting spatial derivatives \( {{(\partial u/\partial x)}_{2*}} \), \( {{(\partial \rho /\partial x)}_{2*}}\), \( {{(\partial P/\partial x)}_{2*}} \) behind the right wave \( {{W}_{R}} \) satisfy

Here, \( \frac{\partial \rho _{ shock }^{2*}}{\partial x} \) and \( \frac{\partial \rho _{ rare }^{2*}}{\partial x} \) satisfy

with

Here \( {{G}_{i}}\), \( i = 1,2,3 \) are given in (A.35).

Wave Pattern and Interface Solution of Generalized Riemann Problem Obtained by RBC

From the right side state definition (2.6) by RBC, the initial state distributions of generalized Riemann problem (C.1) satisfy

Then, as shown in Fig. 19, we can get the solution of this problem as following form.

Wave pattern of generalized Riemann problem obtained by RBC

Full size image

1. a.

   When \( u_{I}^{0}<{{u}_{L}}\), the three-waves structure is composed of two shock wave \( W_L, W_R \) and one contact discontinuity, the limiting states \( {{U}_{1*}}\), \( {{U}_{2*}} \) and their material derivatives \( {{(Du/Dt)}_{1*}}\), \( {{(DP/Dt)}_{1*}}\), \( {{(Du/Dt)}_{2*}}\), \( {{(DP/Dt)}_{2*}} \) behind the wave \( W_L, W_R \) satisfy

   $$\begin{aligned}&{U_{1*}} = {U_{2*}},\end{aligned}$$

   (C.25)
   $$\begin{aligned}&A_L^{ shock }\left( {\frac{{Du}}{{Dt}}} \right) _{1*}^ - + B_L^{ shock }\left( {\frac{{DP}}{{Dt}}} \right) _{1*}^ - = D_L^{ shock },\end{aligned}$$

   (C.26)
   $$\begin{aligned}&A_R^{ shock }\left( {\frac{{Du}}{{Dt}}} \right) _{2*}^ + + B_R^{ shock }\left( {\frac{{DP}}{{Dt}}} \right) _{2*}^ + = D_R^{ shock },\end{aligned}$$

   (C.27)
   $$\begin{aligned}&\begin{array}{*{20}{l}} {\left( {\frac{{Du}}{{Dt}}} \right) _{1*}^ - = \left( {\frac{{Du}}{{Dt}}} \right) _{2*}^ + },\\ {\left( {\frac{{DP}}{{Dt}}} \right) _{1*}^ - = \left( {\frac{{DP}}{{Dt}}} \right) _{2*}^ + }. \end{array} \end{aligned}$$

   (C.28)

Here, \( {{U}_{1*}}\), \( {{U}_{2*}} \) are given in (B.8).

Bringing (C.24), (C.25) into (C.4) and (C.11), we can get

Then, (C.27) becomes

From (C.28), (C.26) and (C.30), we have

1. b.

   When \( u_{I}^{0}\ge {{u}_{L}}\), the three-waves structure is composed of two rarefaction wave \( W_L, W_R \) and one contact discontinuity, the limiting states \( {{U}_{1*}}\), \( {{U}_{2*}} \) and their material derivatives \( {{(Du/Dt)}_{1*}}\), \( {{(DP/Dt)}_{1*}}\), \( {{(Du/Dt)}_{2*}}\), \( {{(DP/Dt)}_{2*}} \) behind the wave \( W_L, W_R \) satisfy

   $$\begin{aligned}&{U_{1*}} = {U_{2*}},\end{aligned}$$

   (C.32)
   $$\begin{aligned}&A_L^{ rare }\left( {\frac{{Du}}{{Dt}}} \right) _{1*}^ - + B_L^{ rare }\left( {\frac{{DP}}{{Dt}}} \right) _{1*}^ - = D_L^{ rare },\end{aligned}$$

   (C.33)
   $$\begin{aligned}&A_R^{ rare }\left( {\frac{{Du}}{{Dt}}} \right) _{2*}^ + + B_R^{ rare }\left( {\frac{{DP}}{{Dt}}} \right) _{2*}^ + = D_R^{ rare },\end{aligned}$$

   (C.34)
   $$\begin{aligned}&\begin{array}{*{20}{l}} {\left( {\frac{{Du}}{{Dt}}} \right) _{1*}^ - = \left( {\frac{{Du}}{{Dt}}} \right) _{2*}^ + },\\ {\left( {\frac{{DP}}{{Dt}}} \right) _{1*}^ - = \left( {\frac{{DP}}{{Dt}}} \right) _{2*}^ + }. \end{array} \end{aligned}$$

   (C.35)

Here, \( {{U}_{1*}}\), \( {{U}_{2*}} \) are given in (B.14).

Bringing (C.24), (C.32) into (C.6) and (C.13), we can get

Then, (C.34) becomes

From (C.33), (C.37) and (C.35), we have

Appendix D: Codes for PRP-Solver and GPRP-Solver

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