An immersed boundary method on Cartesian adaptive grids for the simulation of compressible flows around arbitrary geometries

Abstract

In this article, we present an immersed boundary method for the simulation of compressible flows of complex geometries encountered in aerodynamics. The immersed boundary methods allow the mesh not to conform to obstacles, whose influence is taken into account by modifying the governing equations locally (either by a source term within the equation or by imposing the flow variables or fluxes locally, similarly to a boundary condition). A main feature of the approach which we propose is that it relies on structured Cartesian grids in combination with a dedicated HPC Cartesian solver, taking advantage of their low memory and CPU time requirements but also the automation of the mesh generation and adaptation. Turbulent flow simulations are performed by solving the Reynolds-averaged Navier–Stokes equations or by a Large-Eddy simulation approach, in combination with a wall function at high Reynolds number, to mitigate the cell count resulting from the isotropic nature of Cartesian cells. The objective of this paper is to demonstrate that this automatic workflow is fast and robust and enables to get quantitative aerodynamics results on geometrically complex configurations. Results obtained are in good agreement with classical body-fitted approaches but with a significant reduction of the time of the whole process, that is a day for RANS simulations, including the mesh generation.

Appendices

Wall functions

Figure 18 shows a typical mean velocity profile in wall units $u^+$ within the inner layer of a turbulent boundary layer. This velocity profile can be split into three portions within this inner layer:

• The viscous sub-layer, for $y^+ \le 5$, where dissipation and viscous diffusion dominate. This yields the linear behavior: $u^+$=$y^+.$

• the log-layer, for $y^+$>30, where there is an equilibrium between turbulence production and dissipation. This region constitutes the junction between the inner and upper layers.

• The buffer layer, for $5 < y^{+} \le 30$, joining the two previously defined layers.

Fig. 18

Asymptotic behaviors of an equilibrium boundary layer

The most common function to describe the evolution of the velocity within an equilibrium turbulent boundary layer (zero-pressure gradient) is the log law of the wall defined as:

$$\begin{aligned} \displaystyle u^{+}=\frac{1}{\kappa }\text {log}(y^{+})+\beta , \end{aligned}$$

(2)

where $\displaystyle u^{+}=\frac{u}{u_{\tau }}$ and $\displaystyle y^{+}=\frac{\rho _w~y~u_{\tau }}{\mu _w}$, with $\kappa$ = 0.41 is the Vón Kármán constant and $\beta =5.2$; $u_{\tau }$ denotes the friction velocity; $\rho _w$ and $\mu _w$ denote the values of density and viscosity at the wall, assumed equal to their values at corresponding image points B.

However, the limitation of the log law is that it is not able to model the inner and buffer layers of the boundary layer, which is critical in our approach, since the dimensionless wall distance $y^{+}$ cannot be controlled at image points. Several algebraic wall functions have been developed to bridge the viscous sub-layer and the log layer: we can cite the law derived by Spalding, by finding a power-series for $y^+$ = $f(u^+)$ or the one proposed by Musker [31], which is very similar (as shown in Fig. 18) but easier to use, since it explicitly provides an expression for the velocity for the point to be addressed. Similarly to the log law, it is based on considerations of the boundary-layer equations. By blending the log layer and the viscous sub-layer asymptotic trends of the turbulent viscosity through an interpolation function, integration of the momentum balance yields the following formula:

$$\begin{aligned} \displaystyle u^{+}&=5.424 \arctan \left[ \frac{2y^{+}-8.15}{16.7}\right] \\&\quad+\log ~\left[ \frac{ \left( y^{+}+10.6 \right) ^{9.6} }{\left( y^{+2}-8.15~y^{+}+86 \right) ^2} \right] -3.52. \end{aligned}$$

(3)

It must be noted that expressions (2) and (3) involve the skin friction velocity $u_{\tau }$, which is unknown. The first step of the process is, therefore, to estimate its value using a Newton–Raphson iterative algorithm.

Determination of the pseudo-viscosity of Spalart–Allmaras at IB target point

In the Spalart–Allmaras model, the turbulent viscosity can be expressed as follows:

$$\begin{aligned} \mu _t = \rho \tilde{\nu } f_{v1}, \end{aligned}$$

(4)

where:

$$\begin{aligned} f_{v1} = \frac{\chi ^3}{\chi ^3 + C_{v1}^3}; \end{aligned}$$

(5)

$C_{v1}$ is a constant and $\displaystyle \chi = \frac{\rho \tilde{\nu }}{\mu }.$ Hence:

$$\begin{aligned} \nu _t = \tilde{\nu } f_{v1}. \end{aligned}$$

(6)

The mixing length assumption can be expressed by:

$$\begin{aligned} \nu _t = \kappa u_{\tau } y D, \end{aligned}$$

(7)

with the Van Driest damping term D, such that $A^{+}$ being a constant, chosen equal to 19:

$$\begin{aligned} D = [1-exp(-\frac{y^{+}}{A^{+}})]^2. \end{aligned}$$

(8)

The pseudo-viscosity $\tilde{\nu }$ must be reconstructed at IB target point A. The friction velocity $u_{\tau }$ is known and has been computed by the algebraic wall function, and y and D are known and $\kappa$ is the Von Kármán constant, equal to 0.4. We have to solve $\tilde{\nu }$ solution of:

$$\begin{aligned} \tilde{\nu } f_{v1} = \kappa u_{\tau } y D; \end{aligned}$$

(9)

that is:

$$\begin{aligned} \tilde{\nu }^4 - \kappa u_{\tau } y D \tilde{\nu }^3 - \kappa u_{\tau } y D \frac{\mu ^3}{\rho ^3}C_{v1}^3 = 0. \end{aligned}$$

(10)

To avoid ill-conditioned problems, the variable that is actually solved is $\displaystyle \frac{\tilde{\nu }}{\nu }$. This leads to solve:

$$\begin{aligned} x^4 - a x^3 - b = 0, \end{aligned}$$

(11)

with:

$$\begin{aligned} x = \frac{\tilde{\nu }}{\nu }; a = \frac{\kappa u_{\tau } y D}{\nu }; b = \frac{\kappa u_{\tau } y D}{\nu } C_{v1}^3 = a C_{v1}^3. \end{aligned}$$

(12)

It is possible to solve this equation explicitly. The following variable change $y=x-\frac{a}{4}$ is performed to remove the monomial of degree 3, leading to an equation of the form $y^4+py^2+qy+r=0$, which is solved by Ferrari’s method. Note that if Eq. (11) was obtained from variable $x=\tilde{\nu }$, q would be very close to zero. Or if it is zero, the equation to be solved would be of the form $y^4+py^2+r=0$, with different solution from the quartic equation above.

Ferrari’s method consists in finding a factorization of two polynomials of degree 2. The main difficulty lies in the fact that four solutions of this equation are possible, and thus, the wrong candidates (especially the complex ones) shall be removed smartly. The monomial of degree 4 is first replaced by the polynomial $(y^2+\lambda ^2)^2-2\lambda ~y^2-\lambda ^2$. This leads to the resolution of a cubic on $\lambda$, and then, the solution $\lambda _0$ is replaced in the quartic on y. This results in a factorization of two polynomials of degree 2. The roots are explicitly obtained, and then, x is derived.