[Paper Review] Partial Riemann problem, boundary conditions, and gas dynamics

François Dubois|arXiv (Cornell University)|Jan 14, 2011

Computational Fluid Dynamics and AerodynamicsEngineering31 references20 citations

TL;DR

This paper introduces the partial Riemann problem as a novel framework for modeling nonlinear boundary conditions in hyperbolic conservation laws, particularly in gas dynamics. By generalizing the classical Riemann problem to include boundary manifolds, it enables consistent treatment of complex fluid-wall interactions and provides a foundation for high-order finite volume schemes with accurate boundary fluxes.

ABSTRACT

We introduce in this contribution the notion of partial Riemann problem. Recall that the Riemann problem describes a shock tube interaction between two given states ; the partial Riemann problem is a generalization of the previous concept and introduces the notion of boundary manifold. In what follows, we first recall very classical notions concerning gas dynamics and the associated Riemann problem. In a second part, we introduce the partial Riemann problem for general systems of conservation laws and proves that this problem admits a solution in some class of appropriate nonlinear waves. In section 3, we recall the linearized analysis with the method of characteristics, introduce the weak formulation of the Dirichlet boundary condition for nonlinear situations in terms of the partial Riemann problem and show that lot of physically relevant situations are described with this theoretical framework. In the last paragraph, we propose a practical implementation of the previous onsiderations with the finite volume method.

Motivation & Objective

• To bridge the gap between mathematical well-posedness of hyperbolic systems and physical boundary conditions in gas dynamics.
• To develop a unified theoretical framework for handling nonlinear boundary conditions, especially at solid walls and fluid interfaces.
• To extend the classical Riemann problem to include boundary manifolds, enabling solution of complex wave interactions at boundaries.
• To provide a practical implementation of the partial Riemann problem within the Godunov finite volume method for conservation laws.
• To enable second-order accurate schemes with slope limiting at boundaries while preserving consistency with physical constraints like non-penetration.

Proposed method

• Introduces the partial Riemann problem as a generalization of the classical Riemann problem, where one state is replaced by a manifold of boundary conditions.
• Uses the method of characteristics and linearized analysis to derive conditions under which the partial Riemann problem admits a solution in terms of nonlinear waves.
• Defines weak Dirichlet boundary conditions via the partial Riemann problem, ensuring compatibility with nonlinear hyperbolic systems.
• Applies the Godunov finite volume method with reconstructed fluxes at cell faces, using slope limiting to maintain accuracy and stability.
• Implements a face-based reconstruction strategy with boundary-specific extrapolation, where boundary faces use non-penetration conditions to define virtual states.
• Applies a slope limiter that respects boundary constraints by using virtual states derived from the non-penetration condition, ensuring bounded and consistent reconstruction.

Experimental results

Research questions

• RQ1 How can the classical Riemann problem be generalized to handle nonlinear boundary conditions in hyperbolic systems?
• RQ2 What conditions ensure the existence and uniqueness of solutions to the partial Riemann problem for hyperbolic conservation laws?
• RQ3 How can physical boundary conditions such as rigid walls and fluid interfaces be consistently modeled within a Riemann-based framework?
• RQ4 What is the role of the boundary manifold in determining the structure of nonlinear waves near boundaries?
• RQ5 How can high-order accuracy be preserved in finite volume schemes while enforcing complex boundary conditions like non-penetration?

Key findings

• The partial Riemann problem is well-posed for hyperbolic systems under appropriate conditions, providing a solution in terms of nonlinear waves including rarefaction, shock, and contact discontinuities.
• The weak Dirichlet boundary condition is successfully formulated via the partial Riemann problem, enabling consistent treatment of nonlinear boundary data.
• For rigid walls, the non-penetration condition is enforced by defining virtual states at boundary faces using the normal velocity component, ensuring physical consistency.
• The slope limiter is adapted at boundaries by using virtual states derived from the non-penetration condition, preserving boundedness and preventing unphysical oscillations.
• Second-order accuracy is achieved in space through face reconstruction using limited gradients, with the algorithm remaining stable and consistent at boundaries.
• Numerical integration using a second-order Heun scheme applied to the semi-discrete system yields stable and accurate solutions for time-dependent gas dynamics problems with complex boundary interactions.