Comparative study of different non-reflecting boundary conditions for compressible flows

Abstract

The numerical simulation of hydrodynamic stability and aeroacoustic problems requires the use of high-order, low-dispersion and low-dissipation numerical methods. It also requires appropriate boundary conditions to avoid reflections of outgoing waves at the boundaries of the computational domain. There are many different methods to avoid wave reflection at the boundaries such as the buffer zone and boundary conditions based on characteristic equations. This paper considers the use of a methodology called perfectly matched layer (PML). The PML is evaluated for the simulation of an acoustic pulse in a uniform flow and the Kelvin–Helmholtz instability in a mixing layer using the linear and nonlinear form of the Euler equation. PML results are compared with other non-reflecting boundary condition methods in terms of effectiveness and computational cost. The other non-reflecting boundary conditions implemented were the buffer zone (BZ), widely used in aeroacoustic and hydrodynamic problems, and the energy transfer and annihilation (ETA), a very simple boundary condition to be implemented. The results show that the PML is an effective boundary condition method, but can be computationally expensive. The PML is also more complex to implement and requires careful stability analysis. The other boundary conditions, the BZ and the ETA, are also effective and may perform better than the PML depending on the flow conditions. These two methods have an advantage in terms of robustness and are much simpler to implement than the PML.

Appendices

Appendix 1: initial conditions for uniform flow

For uniform flow, the following initial conditions were defined for acoustic, vorticity and entropy pulses:

$$\begin{aligned} \rho&= A_0\exp \left[ -{\mathrm{ln}}(2) \left( \frac{(x+x_{a})^2+(y+y_a)^2}{\delta _a}\right) \right] \nonumber \\ &\quad +\, C_0\exp \left[ -{\mathrm{ln}}(2) \left( \frac{(x+x_c)^2+(y+y_c)^2}{\delta _c}\right) \right] , \end{aligned}$$

(19)

$$\begin{aligned} u &= B_0y\exp \left[ -{\mathrm{ln}}(2) \left( \frac{(x+x_{b})^2+(y+y_b)^2}{\delta _b}\right) \right] , \end{aligned}$$

(20)

$$\begin{aligned} v= & {} -B_0(x-x_b)\exp \left[ -{\mathrm{ln}}(2) \left( \frac{(x+x_{b})^2+(y+y_b)^2}{\delta _b}\right) \right] , \end{aligned}$$

(21)

$$\begin{aligned} P= & {} -A_0(x-x_b)\exp \left[ -{\mathrm{ln}}(2) \left( \frac{(x+x_{b})^2+(y+y_b)^2}{\delta _b}\right) \right] . \end{aligned}$$

(22)

The variables $A_0$, $B_0$ and $C_0$ are the amplitudes, and $\delta _a$, $\delta _b$ and $\delta _c$ are the thicknesses for the acoustic, vorticity and entropy pulses, respectively. The positions $x_a$, $x_b$ and $x_c$ are the points of application of the different pulses. All pulses used in this work have unitary amplitude and a thickness of 16 and were located at the origin of the domain.

Appendix 2: initial conditions for a mixing layer

For the mixing layer, the initial conditions are:

$$\begin{aligned} \begin{Bmatrix} \rho _0 \\ u_0 \\ v_0 \\ p_0 \end{Bmatrix} = \begin{Bmatrix} \overline{\rho }(y) \\ \overline{M}_{x}(y) \\ 0 \\ 1/\gamma \end{Bmatrix}, \end{aligned}$$

(23)

where the streamwise non-dimensional velocity $\overline{M}_x$ distribution is

$$\begin{aligned} \overline{M}_{x}(y)= \frac{1}{2}\left[ (M_{x1}+M_{x2})+(M_{x1}-M_{x2}) \text {tanh}\left( \frac{2y}{\delta }\right) \right] , \end{aligned}$$

(24)

and the non-dimensional density is

$$\begin{aligned} \overline{\rho }(y)=\frac{1}{\overline{T}(y)}, \end{aligned}$$

(25)

with the temperature distribution given by the Crocco–Busemann relation,

$$\begin{aligned} \overline{T}(y)= & {} T_1\frac{\overline{M}_{x}-M_{x2}}{M_{x1}-M_{x2}} + T_2\frac{M_{x1}-\overline{M}_{x}}{M_{x1}-M_{x2}}\nonumber \\&+\frac{\gamma -1}{2}(M_{x1}-\overline{M}_{x})(\overline{M}_{x}-M_{x2}). \end{aligned}$$

(26)

The parameter $\delta$ is the thickness of the mixing layer, and $M_x(y)$ represents the streamwise velocity non-dimensionalized by the speed of sound. $M_{x1}$ and $M_{x2}$ are the non-dimensional velocities at the top and bottom of the mixing layer. Table 7 shows the parameters used in the mixing layer definition. A complete study of stability of this mixing layer is found in [20]. 

Table 7 Parameters for the mixing layer definition

Appendix 3: basic disturbance equations

The compressible Rayleigh stability equation for pressure disturbance $\pi$ in a two-dimensional, inviscid, parallel flow for a mixing layer can be written as:

$$\begin{aligned} \frac{\partial ^2 \pi }{\partial y^2} -\left( \frac{2}{M-c}\frac{\partial M}{\partial y}-\frac{\bar{T}'}{\bar{T}} \right) \frac{\partial \pi }{\partial y} -\alpha ^2\left[ 1-\frac{1}{\bar{T}}(M-c)^2\right] \pi =0. \end{aligned}$$

(27)

To non-dimensionalize Eq. 27, reference values have been selected. $M=\bar{U}/a_0$ is the Mach number of the uniform flow, where $\bar{U}$ is the base flow velocity in the streamwise direction non-dimensionalized by a reference speed of sound $a_0$, $y=y'/l_0$ where y is the direction normal to the flow non-dimensionalized by a reference scale $l_0$, $T=T'/T_{0}$ where T is the base flow temperature non-dimensionalized by a reference temperature $T_0$, $\pi =p/(a_0^2\rho _0)$ where p is the pressure fluctuation, non-dimensionalized by $a_0^2\rho _0$, where $\rho _0$ is a reference density. $c=\omega /\alpha$ is the disturbance phase velocity, non-dimensionalized by the reference speed of sound $a_0$.

To solve Eq. 27 for spatial stability analyses, it is necessary to find the eigenvalues $\alpha$ for a particular $\omega$ for a nonlinear generalized eigenvalue problem:

$$\begin{aligned}&\alpha ^3 \left( \frac{M^2}{\bar{T}}\bar{U}^3 -\bar{U} \right) \pi \alpha ^2 \left( w -3\frac{M^2}{\bar{T}}\bar{U}^2w \right) \pi \nonumber \\&\qquad +\,\alpha \left( 3\frac{M^2}{\bar{T}}\bar{U}w^2 -2\bar{U}'D + \bar{U}D^2 +\frac{\bar{T}'}{\bar{T}}\bar{U} D \right) \pi \nonumber \\&\quad = \left( w^3\frac{M^2}{\bar{T}} +\frac{\bar{T}'}{\bar{T}}wD +w D^2 \right) \pi , \end{aligned}$$

(28)

where D and $D^2$ represent the derivative operators $\partial / \partial y$ and $\partial ^2 / \partial y^2$, respectively.

To solve the nonlinear eigenvalue problem for $\alpha$, the equations need to be linearized to a linear system of equations using the following transformation:

$$\begin{aligned}&g=\alpha \pi , \end{aligned}$$

(29)

$$\begin{aligned}&\alpha ^2 \left( \frac{M^2}{\bar{T}}\bar{U}^3 -\bar{U} \right) g +\alpha \left( w -3\frac{M^2}{\bar{T}}\bar{U}^2w \right) g\nonumber \\&\qquad + \left( 3\frac{M^2}{\bar{T}}\bar{U}w^2 -2\bar{U}'D + \bar{U}D^2 +\frac{\bar{T}'}{\bar{T}}\bar{U} D \right) g\nonumber \\&\quad = \left( w^3\frac{M^2}{\bar{T}} +\frac{\bar{T}'}{\bar{T}}wD +w D^2 \right) p. \end{aligned}$$

(30)

Using:

$$\begin{aligned}&z=\alpha g, \end{aligned}$$

(31)

$$\begin{aligned}&\alpha \left( \frac{M^2}{\bar{T}}\bar{U}^3 -\bar{U} \right) z + \left( w -3\frac{M^2}{\bar{T}}\bar{U}^2w \right) z \nonumber \\&\qquad + \left( 3\frac{M^2}{\bar{T}}\bar{U}w^2 -2\bar{U}'D + \bar{U}D^2 +\frac{\bar{T}'}{\bar{T}}\bar{U} D \right) g\nonumber \\&\quad = \left( w^3\frac{M^2}{\bar{T}} +\frac{\bar{T}'}{\bar{T}}wD +w D^2 \right) \pi , \end{aligned}$$

(32)

the final linear system of equations can be written as:

$$\begin{aligned} \left[ \begin{array}{ccc} L_{\pi }&{}-L_g&{}-L_z\\ 0&{}1&{}0\\ 0&{}0&{}1\\ \end{array} \right] \left[ \begin{array}{c} \pi \\ g\\ z\\ \end{array} \right] = \alpha \left[ \begin{array}{ccc} 0&{}0&{}R_z\\ 1&{}0&{}0\\ 0&{}1&{}0\\ \end{array} \right] \left[ \begin{array}{c} \pi \\ g\\ z\\ \end{array} \right] , \end{aligned}$$

(33)

where

$$\begin{aligned} L_z= & {} w -3\frac{M^2}{\bar{T}}\bar{U}^2w, \nonumber \\ L_g= & {} 3\frac{M^2}{\bar{T}}\bar{U}w^2 -2\bar{U}'D + \bar{U}D^2 +\frac{\bar{T}'}{\bar{T}}\bar{U} D, \nonumber \\ L_{\pi }= & {} w^3\frac{M^2}{\bar{T}} +\frac{\bar{T}'}{\bar{T}}wD +w D^2,\nonumber \\ R_z= & {} \frac{M^2}{\bar{T}}\bar{U}^3 -\bar{U}. \end{aligned}$$

(34)

The eigenvalue problem 33 is solved using the spectral collocation method, with Chebyshev collocation points, using $N=300$ points. As the collocations points are in the interval [$-\,1,1$], the mapping presented in (35) is used.

$$\begin{aligned} y=\beta \, \tan \left( \frac{\pi }{2N}\right) . \end{aligned}$$

(35)

where $\beta$ controls the grid stretching and $\beta =0.2$ was used.