Low-Dissipative High-Order Shock-Capturing Methods Using Characteristic-Based Filters

TLDR: In this paper, an approach which closely maintains the non-dissipative nature of classical fourth or higher-order spatial differencing away from shock waves and steep gradient regions while being capable of accurately capturing discontinuities, steep gradient, and fine scale turbulent structures in a stable and efficient manner is described.

About: This article is published in Journal of Computational Physics.The article was published on 1999-03-20 and is currently open access. It has received 626 citations till now. The article focuses on the topics: Shock capturing method & Total variation diminishing.

Figures

FIG. 3.9. Four stages in the vortex pairing, at timest = 40,80,120,160, showing the temperature contours for a 201× 201 grid withκ = 0.7 for the nonlinear fields andκ = 0.35 for the linear fields using ACM44.

FIG. 3.23. Compressible turbulent channel flow: (a) mean velocity profile (solid line) compared to the linear law u+ = y+ and the semi-logarithmic law of the wall, with Karman constant 0.41 and additive constant 5.5, and (b) Favre-averaged (solid line) and total stresses (dashed line) compared to the linear result (chain dotted line).

FIG. 3.11. Effect of order of accuracy on ACM methods (ACM22, ACM44, and ACM66) att = 160 for a 101× 101 grid compared with the reference solution (ACM44, 201× 201 grid) usingκ = 0.7 for the nonlinear fields andκ = 0.35 for the linear fields.

FIG. 3.10. Effect of order of accuracy on TVD methods (TVD22, TVD44, and TVD66), compared with the ACM44 solution att = 160, illustrated by temperature contours att = 160 for a 101× 101 grid.

FIG. 3.3. Convecting vortex: comparison of the ACM44 and ACM66 with exact solution att = 100 for κ = 0.07 andκ = 0.05, illustrated by density contour for a 881× 81 grid.

FIG. 3.2. Convecting vortex: comparison of ACM44 and ACM66 with the exact solution, illustrated by density profiles at the centerliney= 0, att = 50 andt = 100 for a 481× 81 and 881× 81 grid, respectively (κ= 0.05).

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Low-dissipative high-order shock-capturing methods using characteristic-based filters" ?

In this paper, the authors proposed to use the narrow grid stencil of high-order classical spatial differencing as base schemes. 

Q2. Why is the filter used to stabilize the nonlinear governing equations?

Since there are no shock waves or steep gradient regions for this flow, the filter is used to stabilize the nonlinear governing equations. 

Q3. What is the key mechanism for achieving high accuracy of the fine scale flow structure?

The function κθ l j+1/2 is the key mechanism for achieving high accuracy of the fine scale flow structure as well as shock waves in a stable manner. 

Q4. What is the easiest method for obtaining higher than second-order temporal base schemes?

For non-stiff or moderately stiff multidimensional problems, one of the easiest procedures for obtaining higher than second-order temporal base schemes is the Runge–Kutta method. 

Q5. What is the filter numerical flux function for a MUSCL-type approach?

The filter numerical flux function for a MUSCL-type approach using the higher-order Lax–Friedrichs numerical flux (Yee [8, 10]) can be expressed asF̃∗j+1/2,k,l = 1 2 [8◦j+1/2], (2.28a)LOW-DISSIPATIVE 

Q6. What are the disadvantages of modern shock-capturing methods?

Modern shock-capturing methods such as total variation diminishing (TVD) or variants of essentially non-oscillatory (ENO) types of schemes that are higher than third-order accurate are usually CPU intensive, involve large grid stencils, and require special treatment near boundary points. 

Q7. What is the advantage of the small domain calculation as a test case for numerical methods?

The advantage of the small domain calculation as a test case for numerical methods is the small memory requirement, making it feasible to run the calculations on workstations with limited memory. 

Q8. What is the alternative to a more compressive limiter?

To balance the shear and shock capturing, one alternative is to switch to a more compressive limiter (see Yee [31]) for the linear characteristics fields. 

Q9. What are the main factors that affect the performance of the proposed schemes?

Aside from evaluating the vortex preservation property, the performance of these schemes with the presence of shock waves and turbulence are evaluated based on the following factors:(a) effect of the ACM term, (b) effect of the order of the base scheme, (c) effect of the grid size (grid refinement study), (d) effect of employing a compact or non-compact base scheme, (e) effect of the adjustable parameter κ for the particular physics, (f) effect of the flux limiters, (g) shear and fine scale flow structure capturing capability. 

Q10. What is the normalized temperature of the mixing layer?

The normalized temperature, and hence local sound speed squared, is determined from an assumption of constant stagnation enthalpyc2 = c21 + γ − 12( u21 − u2 ) . (3.10)Equal pressure through the mixing layer is assumed. 

Q11. What is the problem arranged with the Mach number at the outflow boundary?

The problem has been arranged with the Mach number at the outflow boundary everywhere supersonic so that no explicit outflow boundary conditions are required. 

Q12. What is the effect of on the resolution of fine scale flow structure?

For viscous flow, in the presence of shocks, shears and turbulence, the effect of κ on the resolution of fine scale flow structure plays a different role than for the inviscid flows with smooth solution. 

Q13. What are the eigenvalues associated with the flux Jacobian matrices?

The eigenvalues associated with the flux Jacobian matrices of F and G are (u, u, u± c) and (v, v, v± c), where c is the sound speed. 

Q14. Why is there no pairing of vortices within the computational box?

In spite of the relatively high amplitude chosen for the subharmonic inflow perturbation there is no pairing of vortices within the computational box. 

Q15. What are the advantages of using higher-order filter operators?

Higher than third-order filter operators are, of course, applicable, but they are more CPU intensive and require special treatment near boundary points for stability and accuracy.