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7.23.2 General Non-Reflecting Boundary Conditions



Overview


The general non-reflecting boundary conditions in FLUENT are based on characteristic wave relations derived from the Euler equations, and applied only on pressure-outlet boundary conditions. To obtain the primitive flow quantities ( $P, u, v, w, T$) on the pressure-outlet, reformulated Euler equations are solved on the boundary of the domain in an algorithm similar to the flow equations applied to the interior of the domain.

Unlike the turbo-specific NRBC method presented in the previous section, the General NRBC method is not restricted by geometric constraints or mesh type. However, good cell skewness near the boundaries where the NRBC will be applied is advisable for a more stable, converged solution. In addition, the General NRBCs can be applied to steady or transient flows as long as the compressible ideal-gas law is used.



Restrictions and Limitations


Note the following restrictions and limitations on the general NRBCs:



Theory


General NRBCs are derived by first recasting the Euler equations in an orthogonal coordinate system ( $x_1, x_2, x_3$) such that one of the coordinates, $x_1$, is normal to the boundary Figure  7.23.5. The characteristic analysis [ 374, 375] is then used to modify terms corresponding to waves propagating in the $x_1$ normal direction. When doing so, a system of equations can be written to describe the wave propagation as follows:


$\displaystyle \frac{\partial {\rho}}{\partial {t}}$ $\textstyle +$ $\displaystyle d_1 + \frac{\partial {m_2}}{\partial {x_2}} + \frac{\partial {m_3}}{\partial {x_3}} = 0$ (7.23-43)
$\displaystyle \frac{\partial {m_1}}{\partial {t}}$ $\textstyle +$ $\displaystyle U_1 d_1 + \rho d_3 + \frac{\partial {(m_1 U_2)}}{\partial {x_2}} + \frac{\partial {(m_1 U_3)}}{\partial {x_3}} = 0$  
$\displaystyle \frac{\partial {m_2}}{\partial {t}}$ $\textstyle +$ $\displaystyle U_2 d_1 + \rho d_4 + \frac{\partial {(m_2 U_2)}}{\partial {x_2}} ... ...\partial {(m_2 U_3)}}{\partial {x_3}} + \frac{\partial {P}}{\partial {x_2}} = 0$  
$\displaystyle \frac{\partial {m_3}}{\partial {t}}$ $\textstyle +$ $\displaystyle U_3 d_1 + \rho d_5 + \frac{\partial {(m_3 U_2)}}{\partial {x_2}} ... ...\partial {(m_3 U_3)}}{\partial {x_3}} + \frac{\partial {P}}{\partial {x_3}} = 0$  
$\displaystyle \frac{\partial {\rho E}}{\partial {t}}$ $\textstyle +$ $\displaystyle \frac{1}{2} \vert V\vert^2 d_1 + \frac{d_2}{(\gamma - 1)} + m_1 d... ...tial {x_2}} + \frac{\partial {[(\rho E + P)U_3}]}{\partial {x_3}} = 0 \nonumber$  

Where $m_1 = \rho U_1$, $m_2 = \rho U_1$ and $m_3 = \rho U_3$ and $U_1$, $U_2$ and $U_3$ are the velocity components in the coordinate system ( $x_1$, $x_2$, $x_3$). The equations above are solved on pressure-outlet boundaries, along with the interior governing flow equations, using similar time stepping algorithms to obtain the values of the primitive flow variables ( $P, u, v, w, T$).

figure   

Note that a transformation between the local orthogonal coordinate system ( $x_1$, $x_2$, $x_3$) and the global Cartesian system (X, Y, Z) must be defined on each face on the boundary to obtain the velocity components ( $u$, $v$, $w$) in a global Cartesian system.

Figure 7.23.5: The Local Orthogonal Coordinate System onto which Euler Equations are Recasted for the General NRBC Method
figure

The $d_i$ terms in the transformed Euler equations contain the outgoing and incoming characteristic wave amplitudes, $L_i$, and are defined as follows:


$\displaystyle d_1$ $\textstyle =$ $\displaystyle \frac{1}{c^2} [L_2 + \frac{1}{2} (L_5 + L_1)]$ (7.23-44)
$\displaystyle d_2$ $\textstyle =$ $\displaystyle \frac{1}{2} (L_5 + L_1)$  
$\displaystyle d_3$ $\textstyle =$ $\displaystyle \frac{1}{2 \rho c} (L_5 - L_1)$  
$\displaystyle d_4$ $\textstyle =$ $\displaystyle L_3$  
$\displaystyle d_5$ $\textstyle =$ $\displaystyle L_4 \nonumber$  

From characteristic analyses, the wave amplitudes, $L_i$, are given by:


$\displaystyle L_1$ $\textstyle =$ $\displaystyle {\lambda}_1(\frac{\partial{P}}{\partial{x_1}} - \rho c \frac{\partial{U_1}}{\partial{x_1}})$ (7.23-45)
$\displaystyle L_2$ $\textstyle =$ $\displaystyle {\lambda}_2( c^2 \frac{\partial{P}}{\partial{x_1}} - \frac{\partial{P}}{\partial{x_1}})$  
$\displaystyle L_3$ $\textstyle =$ $\displaystyle {\lambda}_3\frac{\partial{U_2}}{\partial{x_1}}$  
$\displaystyle L_4$ $\textstyle =$ $\displaystyle {\lambda}_4\frac{\partial{U_3}}{\partial{x_1}}$  
$\displaystyle L_5$ $\textstyle =$ $\displaystyle {\lambda}_5(\frac{\partial{P}}{\partial{x_1}} + \rho c \frac{\partial{U_1}}{\partial{x_1}}) \nonumber$  

The outgoing and incoming characteristic waves are associated with the characteristic velocities of the system (i.e eigenvalues), $\lambda_i$, as seen in Figure  7.23.6. These eigenvalues are given by:


$\displaystyle {\lambda}_1$ $\textstyle =$ $\displaystyle U_1 - c$ (7.23-46)
$\displaystyle {\lambda}_2$ $\textstyle =$ $\displaystyle {\lambda}_3 = {\lambda}_4 = U_1$  
$\displaystyle {\lambda}_5$ $\textstyle =$ $\displaystyle U_1 + c \nonumber\nonumber$  

Figure 7.23.6: Waves Leaving and Entering a Boundary Face on a Pressure-Outlet Boundary. The Wave Amplitudes are Shown with the Associated Eigenvalues for a Subsonic Flow Condition
figure

For subsonic flow leaving a pressure-outlet boundary, four waves leave the domain (associated with positive eigenvalues $\lambda_2$, $\lambda_3$, $\lambda_4$, and $\lambda_5$) and one enters the domain (associated with negative eigenvalue $\lambda_1$).

To solve Equations  7.23-44 on a pressure-outlet boundary, the values of $L_2$, $L_3$, $L_4$ and $L_5$ must be first determined from Equations  7.23-46 by using extrapolated values of $\frac{\partial{P}}{\partial{x_1}}$, $\frac{\partial{U_1}}{\partial{x_1}}$, $\frac{\partial{U_2}}{\partial{x_1}}$, and $\frac{\partial{U_3}}{\partial{x_1}}$ from inside the domain. Then, for the lone incoming wave, the Linear Relaxation Method (LRM) of Poinsot [ 287, 288] is used to determine the value of the $L_1$ wave amplitude. The LRM method sets the value of the incoming wave amplitude to be proportional to the differences between the local pressure on a boundary face and the imposed exit pressure. Therefore, $L_1$ is given by


 L_1 = K (P - P_{exit}) (7.23-47)

where $P_{exit}$ is the imposed pressure at the exit boundary, $K$ is the relaxation factor, and $P$ is the local pressure value at the boundary.

In general, the desirable average pressure on a non-reflecting boundary can be either relaxed toward a pressure value at infinity or enforced to be equivalent to some desired pressure at the exit of the boundary.

If you want the average pressure at the boundary to relax toward $P$ at infinity (i.e. $P_{exit} = P_{\infty}$), the suggested $K$ factor is given by:


 K = {\sigma}_1 (1 - {M_{max}}^2 ) \frac{c}{h} (7.23-48)

where $c$ is the acoustic speed , $h$ is the domain size, $M_{max}$ is the maximum Mach number in the domain, and ${\sigma}_1$ is the under-relaxation factor (default value is 0.15) . On the other hand, if the desired average pressure at the boundary is to approach a specific imposed value at the boundary, then the $K$ factor is given by:


 K = {\sigma}_2 c (7.23-49)

where the default value for ${\sigma}_2$ is 5.0



Using General Non-Reflecting Boundary Conditions


The General NRBC is available for use in the Pressure Outlet panel when either the density-based explicit or the density-based implicit solvers are activated to solve for compressible flows using the ideal-gas law.

To activate the general NRBC

1.   Select pressure-outlet from the Boundary Conditions Panel and click the Set... button.

2.   In the Pressure Outlet panel, enable the Non-Reflecting Boundary option.

3.   Select one of the two NRBC Pressure Specification options: Exit pressure is at infinity or Exit pressure is average pressure.

Figure 7.23.7: The Pressure Outlet Panel With the Non-Reflecting Boundary Enabled
figure

(a)   The Exit pressure is at infinity boundary is typically used in unsteady calculations or when the exit pressure value is imposed at infinity. The boundary is designed so that the pressure at the boundary relaxes toward the imposed pressure at infinity. The speed at which this relaxation takes place is controlled by the parameter, sigma, which can be adjusted in the TUI:

define $\rightarrow$ boundary-conditions $\rightarrow$ non-reflecting-bc $\rightarrow$

general-nrbc $\rightarrow$ set

In the set/ submenu, you can set the sigma value. The default value for sigma is 0.15.

(b)   The Exit pressure is average pressure boundary specification is usually used in steady-state calculations when you want to force the average pressure on the boundary to approach the exit pressure value. The matching of average exit pressure to the imposed average pressure is controlled by parameter sigma2 which can be adjusted in the TUI:

define $\rightarrow$ boundary-conditions $\rightarrow$ non-reflecting-bc $\rightarrow$

general-nrbc $\rightarrow$ set

In the set/ submenu, you can set the sigma2 value. The default value for sigma2 is 5.0 .

figure   

There is no guarantee that the sigma2 value of 5.0 will force the average boundary pressure to match the specified exit pressure in all flow situations. In the case where the desired average boundary pressure has not been achieved, the user can intervene to adjust the sigma2 value so that the desired average pressure on the boundary is approached.

Usually, the solver can operate at higher CFL values without the NRBCs being turned on. Therefore, for steady-state solutions the best practice is to first achieve a good stable solution (not necessarily converged) before activating the non-reflecting boundary condition. In many flow situations, the CFL value must be reduced from the normal operation to keep the solution stable. This is particularly true with the density-based implicit solver since the boundary update is done in an explicit manner. A typical CFL value in the density-based implicit solver, with the NRBC activated, is 2.0.


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© Fluent Inc. 2006-09-20