[Paper Review] Infinite-thin shock layer solutions for stationary compressible conical flows and numerical results via Fourier spectral method

Aifang Qu, Xueying Su+1 more|arXiv (Cornell University)|Jul 4, 2023

Computational Fluid Dynamics and AerodynamicsEngineering3 citations

TL;DR

This paper proposes a numerical framework based on the Fourier spectral method and Newton's method to solve highly singular, nonlinear ordinary differential equations (ODEs) governing infinite-thin shock layer solutions in stationary compressible conical flows, particularly for Chaplygin gas and hypersonic flows with attack angles. The key contribution is the construction of Radon measure solutions with weighted Dirac measures on the cone's surface, yielding generalized Newton-Busemann pressure laws and accurate numerical solutions that validate physical expectations for pressure distribution and particle trajectories.

ABSTRACT

We consider the problem of uniform steady supersonic Euler flows passing a straight conical body with attack angles, and study Radon measure solutions describing the infinite-thin shock layers, particularly for the Chaplygin gas and limiting hypersonic flows. As a byproduct, we obtain the generalized Newton-Busemann pressure laws. To construct the Radon measure solutions containing weighted Dirac measures supported on the edge of the cone on the 2-sphere, we derive some highly singular and non-linear ordinary differential equations (ODE). A numerical algorithm based on the combination of Fourier spectral method and Newton's method is developed to solve the physically desired nonnegative and periodic solutions of the ODE. The numerical simulations for different attack angles exhibit proper theoretical properties and excellent accuracy, thus would be useful for engineering of hypersonic aerodynamics.

Motivation & Objective

• To model infinite-thin shock layers in steady supersonic conical flows with attack angles using Radon measure solutions.
• To derive and solve highly singular, nonlinear ODEs governing the weights of Dirac measures on the cone's surface.
• To develop a robust numerical algorithm for computing nonnegative, periodic solutions of the ODEs arising from the singular shock layer model.
• To validate the physical consistency of the solutions through numerical simulations of pressure distribution and particle trajectories.
• To generalize the Newton-Busemann pressure law for non-zero attack angles and Chaplygin gas.

Proposed method

• Formulate the problem in spherical coordinates with a conical body and uniform supersonic flow at an attack angle.
• Derive a system of singular, nonlinear ODEs for the weights of Dirac measures representing the infinite-thin shock layer.
• Apply the Fourier spectral method to discretize and solve the periodic, nonnegative ODEs efficiently.
• Combine the Fourier spectral method with Newton's method to enhance convergence and accuracy in solving the nonlinear system.
• Use the solution of the ODEs to reconstruct velocity components $ u^t, w, w_\rho $ and analyze particle trajectories on the cone surface.
• Validate results via numerical simulations showing monotonicity, symmetry, and physical consistency of pressure and flow variables.

Experimental results

Research questions

• RQ1 How can infinite-thin shock layer solutions be constructed for compressible conical flows with non-zero attack angles using Radon measure solutions?
• RQ2 What are the resulting generalized Newton-Busemann pressure laws for Chaplygin gas under arbitrary attack angles?
• RQ3 How can highly singular, nonlinear ODEs governing shock layer weights be numerically solved with high accuracy and nonnegativity constraints?
• RQ4 What physical properties, such as monotonicity and symmetry, are exhibited by the pressure distribution and flow variables in the numerical solutions?
• RQ5 How do particle trajectories on the cone surface evolve, and what do they reveal about flow behavior under varying attack angles?

Key findings

• The numerical method successfully computes nonnegative, periodic solutions of the singular ODEs with high accuracy, confirming the existence of physically consistent infinite-thin shock layer solutions.
• For increasing attack angle $ \alpha_0 $, the maximum pressure $ \max W_C $ increases while the minimum $ \min W_C $ decreases, consistent with physical intuition of higher windward and lower leeward pressure.
• The maximum and minimum of the pressure function $ W_C $ align exactly with the theoretical curves $ \sin^2(\theta_0 + \alpha_0) $ and $ \sin^2(\theta_0 - \alpha_0) $, validating the derived generalized Newton-Busemann law.
• The velocity components $ u^t, w, w_\rho $ exhibit expected singularities at $ \phi = 0, \pm\pi $, attributed to numerical artifacts despite the analytical expectation of $ C^1 $ regularity.
• Particle trajectories on the cone converge toward $ \phi = 0 $, indicating flow accumulation on the leeward side, with behavior symmetric across the $ x^1 $-axis.
• The numerical results demonstrate monotonic trends: $ f(\phi) $ increases with $ \alpha_0 $, and $ w_\rho $ increases near $ \phi = \pm\pi $ while decreasing near $ \phi = 0 $, reflecting changing particle impact density.