On the use of adaptive multiresolution method with time-varying tolerance for compressible fluid flows

Abstract

In this paper, a fully adaptive multiresolution (MR) finite difference scheme with a time-varying tolerance is developed to study compressible fluid flows containing shock waves in interaction with solid obstacles. To ensure adequate resolution near rigid bodies, the MR algorithm is combined with an immersed boundary method based on a direct-forcing approach in which the solid object is represented by a continuous solid-volume fraction. The resulting algorithm forms an efficient tool capable of solving linear and nonlinear waves on arbitrary geometries. Through a one-dimensional scalar wave equation, the accuracy of the MR computation is, as expected, seen to decrease in time when using a constant MR tolerance considering the accumulation of error. To overcome this problem, a variable tolerance formulation is proposed, which is assessed through a new quality criterion, to ensure a time-convergence solution for a suitable quality resolution. The newly developed algorithm coupled with high-resolution spatial and temporal approximations is successfully applied to shock–bluff body and shock-diffraction problems solving Euler and Navier–Stokes equations. Results show excellent agreement with the available numerical and experimental data, thereby demonstrating the efficiency and the performance of the proposed method.

Appendices

Appendix 1: Numerical implementation

The inviscid flux is computed using a classical fifth-order WENO scheme [34, 36] based on Roe flux splitting [42], while a fourth-order central differencing scheme is used to compute the viscous flux. An explicit third-order TVD Runge–Kutta (RK3) scheme is employed for time advancement [14, 15]. A brief description of the numerical solver is given in the context of MR formulation in Algorithm 1.

The node, leaf, and empty points in the algorithm can be referred from Fig. 1 for better clarity.

Appendix 2: Ray tracing

The ray tracing technique [39], commonly used in the field of computer graphics, is used here to identify whether a grid node belongs to a fluid or solid domain. The principle of this method is adapted to define the mask function required in the IB method such that the complex solid obstacles can be carefully defined in the computational domain.

The underlying concept of the ray tracing method makes it very flexible, since it enables the use of any CAD tool, such as AutoCAD, CATIA, to generate complex geometries. The resulting data can be easily plugged into a Cartesian grid, $\mathcal {G}$, comprised of points,

$$\begin{aligned} \mathbf p = \left\{ (x_i,y_j) \in \mathcal {G}\,|\,i = 0,\ldots ,N_x, j = 0,\ldots ,N_y \right\} . \end{aligned}$$

(32)

The rays, from a far away location from the computational domain, are casted to $\mathbf p$. Thus, the equation of lines passing through the source point, $\mathbf S$, and $\mathbf p$ can be expressed as,

$$\begin{aligned} \varLambda = \left\{ \mathbf p + m(\mathbf S -\mathbf p )\,|\,m \in \mathbb {R} \right\} . \end{aligned}$$

(33)

The solid model is converted into triangular cells and is exported in a STereoLithography (STL) format, which includes the vertices and the surface normals of each mesh element. For each triangular element, e, there exists a plane, $\mathcal {P}$, that contains it,

$$\begin{aligned} \mathcal {P}:= \mathbf n \cdot (\mathbf x - \mathbf x _e) = 0, \end{aligned}$$

(34)

where $\mathbf n$ and $\mathbf x _e$ denote the surface normal vector and the vertex of e, respectively, while $\mathbf x$ represents a vector of set of points lying inside $\mathcal {P}$. If the intersection of $\varLambda$ and $\mathcal {P}$ at a point, $p^*$, is enclosed within the element e, the counter, $\sigma$, is incremented by 1 as,

$$\begin{aligned} \exists \,p^*(x,y) \in \mathcal {P}\,\text {s.t.}\,p^* = \varLambda \cap \mathcal {P} \rightarrow \sigma = \sigma + 1. \end{aligned}$$

(35)

Finally, the mask function is obtained by first performing the modulo operation, $\alpha = \sigma \bmod 2$, and then,

$$\begin{aligned} \left( \alpha = 1 \rightarrow \mathrm{mask}_{i,j} = 1 \right) \wedge \left( \alpha \ne 1 \rightarrow \mathrm{mask}_{i,j} = 0 \right) . \end{aligned}$$

(36)

Fig. 14

Concept of the ray tracing method

The ray tracing operation is illustrated schematically in Fig. 14, where $p_n$ and $S_n$ denote the grid and the source points, respectively. It is important to emphasize that a precondition of the watertight (STL) mesh is necessary to achieve reliable results from this technique. Moreover, in certain cases, the rays from the source to the grid point could possibly intersect with either an edge or a corner of the mesh element. In such scenarios, it is necessary to use a secondary source of ray to confirm the intersection (see $p_4, p_5$, and $S_2$ in Fig. 14).