Effects of the CREST Slip-Wall Model and Higher-Order Continuum Momentum Corrections on Local Non-Equilibrium Flow

As aerospace engineering advances, researchers continue to explore flight at higher speeds and over greater distances than ever before. These developments require new technologies and innovative strategies for designing and testing advanced vehicles. Consequently, high-fidelity numerical simulation has emerged as a central tool in aerospace research. Engineers routinely use computational fluid dynamics (CFD) to evaluate new concepts, identify critical design flaws, and improve performance before costly experiments or flight tests. Yet, in focusing on simulation capability, we sometimes overlook the underlying foundation that makes these analyses possible: the governing equations themselves.

Fluid-flow simulations generally follow one of two approaches: molecular-based modeling or continuum-based modeling.¹ Molecular approaches, such as Direct Simulation Monte Carlo (DSMC) and Boltzmann-based kinetic solvers, are well known in the aerospace community and can accurately represent non-equilibrium effects. However, these methods are computationally expensive and mathematically more difficult to handle than conventional continuum models. For this reason, continuum-based CFD remains the preferred approach whenever the continuum assumption is valid, that is, when the flow remains sufficiently close to local thermodynamic equilibrium. Nevertheless, many challenging flows exhibit strong local non-equilibrium effects, particularly in high-speed and rarefied regimes, where the conventional continuum governing equations may no longer yield accurate predictions.² In such cases, maintaining accuracy within a continuum framework requires modified boundary conditions and, when necessary, adding higher-order continuum corrections to the governing equations, especially in the momentum and energy transport equations.

Slip-Flow Regime (0.001 < Kn < 0.1)

When the Knudsen number, Kn, lies between 0.001 and 0.1, the flow is generally classified as being in the slip-flow regime.¹ In this range, rarefaction effects become important, and the classical no-slip boundary condition is no longer strictly valid. Although the bulk flow may still be described within a continuum framework, the wall boundary conditions must be modified to preserve accuracy of the continuum model.

The most widely used correction is the Maxwell slip boundary condition, which is derived from kinetic theory. This treatment provides a physically motivated wall-slip velocity and improves prediction accuracy in rarefied flows. However, numerical implementation of the Maxwell slip model can introduce strong cell-to-cell velocity variations in the near-wall region. These sharp variations become particularly problematic when higher-order continuum correction terms are added, because the solver then becomes more sensitive to local gradient oscillations and may exhibit unstable behavior.

To address this issue, I proposed a more dissipative and numerically stable slip-wall treatment, CREST (Continuum-Rarefied Explicit Slip Treatment). Compared with the conventional Maxwell slip treatment, CREST produces smoother near-wall velocity variations from cell to cell while still preserving slip effects. This smoothing improves numerical stability when higher-order continuum correction terms are included. At the same time, CREST retains accuracy close to that of the Maxwell slip model.

Figure 2 illustrates this behavior for the double-cone flow problem, a benchmark configuration introduced in 2001 for hypersonic code validation.³ The left panel shows the solution obtained with a no-slip wall boundary condition, the middle panel shows the result with the Maxwell slip treatment, and the right panel shows the result with the CREST slip treatment. The no-slip solution, which assumes continuum behavior at the wall, overpredicts the separation zone generated by shock-shock interactions. In contrast, both the Maxwell and CREST slip treatments predict separation-zone sizes much more accurately.

We implemented the CREST model in CONVERGE CFD software as a user-defined function (UDF). For the double-cone simulations, we employed a hybrid meshing strategy that combines CONVERGE’s automated Cartesian cut-cell approach with inlaid meshes generated in CONVERGE Studio. This strategy is particularly advantageous for compressible-flow simulations because it improves meshing efficiency while allowing greater focus on the underlying flow physics, such as wall momentum transport. In principle, the most accurate mesh for high-speed compressible flow would align shock waves with mesh faces, since shocks represent flow discontinuities. In practice, however, this is difficult to achieve within a structured-grid framework. CONVERGE’s automated Cartesian cut-cell method, together with localized mesh refinement in shock regions, therefore provides a practical and effective alternative that improves solution accuracy without requiring excessive meshing effort.

As shown in the results, the no-slip boundary condition overpredicts the separation region, indicating an overestimation of momentum diffusion in the momentum transport. In contrast, both the Maxwell-slip and CREST boundary conditions provide a more accurate representation of momentum diffusion and yield predictions that better match the experimental behavior.

Transitional-Flow Regime (0.1 < Kn < 10)

When the Knudsen number lies between 0.1 and 10, the flow is generally classified as being in the transitional regime.¹ In this range, rarefaction effects become substantially stronger than in the slip-flow regime, and modifying the wall boundary condition alone is no longer sufficient. To maintain accuracy within a continuum framework, higher-order continuum correction terms must be introduced into the governing equations. As the flow becomes more rarefied, local non-equilibrium effects grow more pronounced. From the perspective of momentum transport, these effects must be represented through the viscous stress tensor, since it is the term that carries the non-equilibrium contribution beyond the isotropic pressure. Consequently, the conventional Navier-Stokes viscous-stress closure must be extended. One systematic way to obtain this extension is through the Chapman-Enskog expansion of the Boltzmann equation, which yields the Burnett viscous-stress closure.²

However, directly incorporating the Burnett viscous-stress closure into the momentum equation introduces well-known stability problems, particularly from the perspective of linear stability theory. To address this issue while retaining Burnett-level accuracy, the augmented Burnett formulation was introduced. Although this formulation improves the short-wavelength stability behavior, it also introduces third-order spatial derivatives into the viscous-stress closure, which produce fourth-order velocity derivatives when substituted into the momentum transport equation. The expanded form of the viscous-stress closure used in the momentum equation is shown below. Here, ω₁ through ω₇ denote coefficients determined by the selected molecular model. In the present study, the Maxwell molecular model is employed.

These higher-order derivatives make the numerical solution much more sensitive to cell-to-cell variations and local oscillations, which can in turn destabilize the solver. To overcome this difficulty, I developed a filtering algorithm with a shock sensor that selectively filters only the augmented Burnett term. Implemented in CONVERGE through a UDF, this approach preserves the accuracy associated with the Burnett viscous-stress closure while improving numerical stability. Figure 3 illustrates the effect of the filtering procedure relative to the raw (unfiltered) augmented Burnett viscous-stress closure. The filter effectively suppresses regions of strong cell-to-cell variation, thereby reducing numerical oscillation errors and preventing distortion of the shock curvature.

Figure 4 demonstrates the influence of the higher-order continuum correction terms on the solution. Velocity is plotted against reference distance to clearly compare the effects of the high-order continuum corrections. Temperature and Mach number contours are also presented using the same reference-distance framework. The top contour represents the Navier-Stokes viscous-closure solution, while the bottom contours represent the filtered augmented Burnett viscous-closure solution.

For this two-dimensional flow over the cylinder, we employed a fully inlaid mesh generated in CONVERGE Studio. As shown in Figures 3 and 4, the inclusion of higher-order continuum correction terms produces noticeable differences in the computed solution, which may be critical in the analysis of hypersonic vehicles.

Conclusion

As discussed throughout this article, numerical methods, particularly continuum-based frameworks, play a central role in the design and analysis of advanced aerospace vehicles. These methods allow engineers to accurately evaluate innovative concepts and to accelerate the development process. However, their successful application depends on recognizing the assumptions that underlie the governing equations. When the flow exhibits stronger non-equilibrium effects, the conventional continuum equations, if used without correction, can produce inaccurate predictions and potentially lead to serious design errors that affect engineering decisions.

For this reason, it is essential to account for non-equilibrium effects when extending continuum CFD to rarefied-flow conditions. In the slip-flow regime, this requires modification of the wall boundary conditions. In more rarefied transitional regimes, boundary-condition corrections alone are not sufficient, and higher-order continuum correction terms must also be incorporated into the governing equations. Therefore, accurate and stable prediction of non-equilibrium flows requires both physically appropriate boundary treatments and carefully implemented higher-order continuum models.

Supported by the CONVERGE Academic Program

This work was performed as part of the CONVERGE Academic Program, which provides free licenses, training, and support for academic research. For this project, Jay collaborated directly with the CONVERGE Development team:

“Our collaboration began through CONVERGE’s code-capability testing for hypersonic flow. From that point, I developed the idea of building our in-house slip-wall treatment, CREST, and I discussed the implementation with Dr. Chai [Solver Development Team Lead]. He helped guide me toward the correct implementation path within CONVERGE, especially in terms of which UDF APIs needed to be used and modified. The same was true for the 2D flow over the cylinder. I presented the new idea, and he helped direct me toward the appropriate implementation approach in CONVERGE.”

The CONVERGE Academic Program enables students, professors, and academic researchers around the world to conduct high-impact CFD research. Learn more about the benefits and how to participate on our webpage!

References

[1] Gad-el-Hak, M., “The Fluid Mechanics of Microdevices—The Freeman Scholar Lecture,” Journal of Fluids Engineering, 121(1), 5-33, 1999. DOI: 10.1115/1.2822013

[2] Cercignani, C. (Author) and Michaelis, C. (Reviewer), “Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations. Cambridge Texts in Applied Mathematics,” Applied Mechanics Reviews, 54(5), B90-B92, 2001. DOI: 10.1115/1.1399679

[3] Holden, M., “Experimental Studies of Laminar Separated Flows Induced by Shock Wave/Boundary Layer and Shock/Shock Interaction in Hypersonic Flows for CFD Validation,” 38th Aerospace Sciences Meeting and Exhibit, Reno, NV, United States, Jan 10-13, 2000. DOI: 10.2514/6.2000-930