A locally stabilized immersed boundary method for the compressible Navier–Stokes equations

doi:10.1016/j.jcp.2015.04.023

Journal of Computational Physics

Volume 295, 15 August 2015, Pages 475-504

https://doi.org/10.1016/j.jcp.2015.04.023 Get rights and content

Abstract

A higher-order immersed boundary method for solving the compressible Navier–Stokes equations is presented. The distinguishing feature of this new immersed boundary method is that the coefficients of the irregular finite-difference stencils in the vicinity of the immersed boundary are optimized to obtain improved numerical stability. This basic idea was introduced in a previous publication by the authors for the advection step in the projection method used to solve the incompressible Navier–Stokes equations. This paper extends the original approach to the compressible Navier–Stokes equations considering flux vector splitting schemes and viscous wall boundary conditions at the immersed geometry. In addition to the stencil optimization procedure for the convective terms, this paper discusses other key aspects of the method, such as imposing flux boundary conditions at the immersed boundary and the discretization of the viscous flux in the vicinity of the boundary. Extensive linear stability investigations of the immersed scheme confirm that a linearly stable method is obtained. The method of manufactured solutions is used to validate the expected higher-order accuracy and to study the error convergence properties of this new method. Steady and unsteady, 2D and 3D canonical test cases are used for validation of the immersed boundary approach. Finally, the method is employed to simulate the laminar to turbulent transition process of a hypersonic Mach 6 boundary layer flow over a porous wall and subsonic boundary layer flow over a three-dimensional spherical roughness element.

Section snippets

Introduction and motivation of the research

Immersed Boundary Techniques (IBTs) have been developed for many years and have appeared in various forms since they were first introduced by Peskin [1], [2] (see for example Goldstein et al. [3], LeVeque and Li [4], Wiegmann and Bube [5], Linnick and Fasel [6], Johansen and Collela [7], Mittal and Iaccarino [8], Zhong [9], Duan et al. [10] and many others). These methods were first introduced as a nontraditional approach for numerically solving initial/boundary-value problems for complex

Space and time discretization of the interior scheme

The compressible Navier–Stokes equations considering an ideal, Newtonian, non-reactive gas written in vector form are $\frac{\partial W}{\partial t} + \nabla \cdot (\vec{F} - {\vec{F}}_{v}) = 0, with \vec{F} = (F^{1}, F^{2}, F^{3}) and {\vec{F}}_{v} = (F_{v}^{1}, F_{v}^{2}, F_{v}^{3}) .$ The conservative variable vector $W = {(ρ, ρ u, ρ v, ρ w, ρ H - p)}^{T}$ and the inviscid fluxes are $F^{1} = [\begin{matrix} ρ u \\ ρ u^{2} + p \\ ρ u v \\ ρ u w \\ ρ u H \end{matrix}], F^{2} = (\begin{matrix} ρ v \\ ρ v u \\ ρ v^{2} + p \\ ρ v w \\ ρ v H \end{matrix}), and F^{3} = [\begin{matrix} ρ w \\ ρ w u \\ ρ w v \\ ρ w^{2} + p \\ ρ w H \end{matrix}],$ where the total enthalpy is $H = h + u \cdot u / 2$ with $h = γ R T / (γ - 1)$ and the ideal gas law $p = ρ R T$ is used. The viscous fluxes are $F_{v}^{1} = [\begin{matrix} 0 \\ τ_{x x} \\ τ_{x y} \\ τ_{x z} \\ f_{v}^{1} \end{matrix}], F_{v}^{2} = [\begin{matrix} 0 \\ τ_{y x} \\ τ_{y y} \\ τ_{y z} \\ f_{v}^{2} \end{matrix}], and F_{v}^{3} = [\begin{matrix} 0 \\ τ_{z x} \\ τ_{z y} \\ τ_{z z} \\ f_{v}^{3} \end{matrix}$

Truncation error study

The Method of Manufactured Solutions (MMS) is used to verify the formal order-of-accuracy of the convective and viscous terms with and without immersed boundary in 2D and 3D. For the truncation error study of the viscous terms, the viscosity is set to $μ = 1$ to ensure dominance of the viscous terms over the convection terms. Different options for the extrapolation of the pressure and temperature boundary conditions (see Section 2.2.4) and their impact on the accuracy of the overall numerical

Stability analysis

The stability of the immersed boundary method is studied by linearizing the convective terms and conducting a matrix stability analysis. The stability properties of the convective term of the spatial discretization matrix, $\tilde{\underset{̲}{A}}$ , and the spatial and temporal coupled discretization matrix, $\underset{̲}{A}$ , are analyzed. The standard fourth-order accurate Runge–Kutta scheme was used as time-integrator for the stability analysis of $\underset{̲}{A}$ . $\underset{̲}{A} = \underset{̲}{I} + Δ t \tilde{\underset{̲}{A}} + \frac{Δ t^{2}}{2!} {\tilde{\underset{̲}{A}}}^{2} + \frac{Δ t^{3}}{3!} {\tilde{\underset{̲}{A}}}^{3} + \frac{Δ t^{4}}{4!} {\tilde{\underset{̲}{A}}}^{4}$ Matrix $\underset{̲}{A}$ in Eq. (27) is also

Validation

For validation purposes the immersed boundary method is applied to simulate the flow around a cylinder and a sphere at subsonic speeds in a uniform free-stream flow for different Reynolds numbers, ${Re}_{D} = U_{\infty} D / ν$ . These 2D and 3D canonical flows are commonly used to evaluate immersed boundary methods. The flow field around a circular cylinder and a sphere are well-established test cases for IBTs for incompressible flows (Goldstein [3], Saiki [32], Calhoun [33], and Linnick [6]). A large amount of

Applications

In this section, results are presented where the immersed boundary method is applied to various flow problems that are relevant for ongoing laminar to turbulent transition research. To demonstrate that this method can be used for high speed Direct Numerical Simulations (DNS), the propagation of a Tollmien Schlichting wave in a high speed boundary layer was simulated. Additionally, this method was employed to physically resolve a porous wall and to investigate the influence of the porous wall on

Conclusion

A novel higher-order accurate immersed boundary method for solving the compressible Navier–Stokes equations was presented. The key feature of this method is that the irregular FD stencils in the vicinity of the immersed boundary are optimized with respect to numerical stability. This is a very unique feature because commonly only the order-of-accuracy is taken into consideration when discretizing the governing equations in the vicinity of the immersed boundary. In the optimization procedure,

Acknowledgements

The authors gratefully acknowledge the contributions from the LAVA group at Applied Modeling Simulation Branch NASA Ames Research Center (ARC). The work was partially funded by the Applied Modeling Simulation Branch at NASA ARC and the Hypersonic Center for Laminar Turbulent Transition Research.

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