Figure 1: Higher-order CFD: Tandem spheres: Q-criterion
1754 words / 9 minute read
Introduction
Until now the buttress for industrial CFD has been second-order accurate RANS solvers. It is fast, efficient, and robust and has earned the confidence of designers after decades of research and development.
RANS-based CFD is embraced as a reliable design tool complimenting experiments. Though predominantly used, they have acute limitations. For eg: they have difficulty in predicting vortex-dominated flows, flows with large flow separations, computing aeroacoustic problems, etc. Having exhausted the comfort zone within which RANS can provide reliable results, researchers are now focussed on more powerful CFD tools like hybrid RANS-LES, LES, DNS, and high-order methods.
There is no light without darkness, nor do new horizons in CFD. In this article, we will be focussing on higher-order (HO) methods. According to experts, it has the potential to overshadow low order RANS and become the industrial buttress for the coming decades. The article will also introduce you to what higher-order CFD is and how are they different from low-order approaches along with positives, current challenges, etc.
Figure 2: Comparison of Q-criterion and schlieren: 1. hpMusic 3rd order simulation has 9.6M degrees of freedom, 2. Second-order simulation on a commercial solver has 28.7M degrees of freedom. HO simulation costs 1/3 the CPU time of the 2nd order simulation.
Higher-Order (HO) Methods
HO does not have a brick-wall definition. CFD practitioners consent that, numerical schemes which have an order of accuracy equal to 3 and above are termed as high order methods. Since most of the industrial codes are first or second-order accurate, any scheme with order above that is considered HO. Over the years many types of high-order methods have been developed to cater to the needs of various industrial problems, mostly for structured meshes and a few for unstructured.
Though high order schemes have been around for a while, not much attention was paid to them as researchers in the early phases of CFD development were keen to make the 1st and 2nd -order schemes more robust and efficient. Now, having hit the saturation limit in their development, the focus has shifted to high order methods.
Figure 3: a. GridPro higher order mesh, b. Tandem spheres, schlieren
Is Order of Accuracy Important?
Most commercial CFD solvers are first or second-order accurate. First-order schemes are fast, robust, and are good enough for industrial applications, where just knowing the order of magnitude of flow variables is good enough. But, if the accuracy of the solution is a critical need, they fall short.
Theoretically, the accuracy of first-order results can be increased by continuous refinements of the grid. However, this is an inefficient approach. When solution accuracy is the need, high-order approaches fit the bill, as they are more efficient with mesh refinement.
Let’s take a deeper look into this and see what actually happens. If we increase the number of cells in each 3-dimensional space and time by 2 times, the computational cost will increase by a factor of 2^4 = 16. For a 1st -order scheme, the solution error gets reduced by a factor of 2, for a 2nd -order scheme the reduction factor is 4, while for a 4th –order scheme the error is reduced by a factor of 16.
In other words, a 4th –order scheme on a 1-million-element grid will most likely produce a more accurate solution than a 1st –order approach on a 1-billion-element grid !!!
Figure 4: Tandem spheres, Q-criterion, a. P2, b. P3
Misconceptions About HO
Interestingly, though HO methods prove their worth on paper, there are common myths in the CFD communities psyche. The first myth, high-order approaches are expensive and secondly, high-order methods are not needed for engineering accuracy.
The myth arises because of incorrect comparison between high order and low order methods. On a given mesh, the higher the order of accuracy of a scheme, the larger the CPU time taken for computing. In other words, on the same mesh, the first-order scheme is faster than the second-order, which is, in turn, faster than the 3rd-order scheme.
But from the point of accuracy, a first-order scheme will need a finer mesh than second order. In other words, a high order scheme needs a much coarser mesh than that needed for a 2- or 1-order scheme to attain the same accuracy levels and hence is actually more efficient and less time-consuming.
So, the evaluation or comparison should not be mesh-centric to measure the costs involved, but rather on the levels of error sought out. When viewed from this perspective, high-order schemes are not really that expensive.
Video 1: Generic car mirror, Q-criteria colored by stream-wise velocity, P2 – 3rd order, 7. 67M DOFs/equation
Video 2: Generic car mirror, Q-Criteria, colored by stream-wise velocity, P3 – 4th order 18.2M DOFs/equation
Further, there are many fluid flow problems where the accuracy level attained by second-order schemes is just not good enough, especially for flows involving unsteady vortices, aeroacoustic and electromagnetic waves.
In vortex-dominated flows, 1st -order and 2nd -order schemes strongly dissipate unsteady vortices and the mesh resolution required for the flow makes such simulations way too expensive even on present-day supercomputers. Similar issues arise in computational aeroacoustics, where acoustic waves need to propagate for very long distances without significant numerical dissipation or dispersion errors. With superior accuracy, efficiency, and higher resolution, high-order methods are regularly picked for LES and DNS of turbulent flows, especially with unsteady vortices.
The second myth, high-orders methods are not needed at all and second orders schemes are good enough to meet engineering accuracy for all engineering problems. This is not true, as there are many fluid flow problems that are way out of the scope of second-order schemes or are computationally expensive. As discussed previously, vortex-dominated flows and aero-acoustic problems are two such flow problems where high-order schemes have shown their superiority.
A closer look at the order of accuracy actually achieved by second-order schemes reveals a hidden limitation. For many flow problems, the order of accuracy achieved is not uniform across all flow variables. What could be an acceptable solution error for one variable may not still be acceptable expected solution errors for another. For example, depending on many factors like the flux scheme used, mesh density, Reynolds Number, etc, a 5 percent error in say velocity may translate into a 20 percent error in skin friction coefficient. Another example, for a simulation involving flow past a helicopter, a 5 percent error in CD may require resolving the strength of the tip vortices within 5 percent error over 4 to 8 revolutions. In other words, 2nd -order methods are not the universal mantra for all engineering problems and they fall short in meeting the expectation on many occasions.
Video 3: Q-criterion + Schlieren movie, P3
Limitations
Although HO methods provide more accurate solutions, there are a few roadblocks that need to be ironed out before they can be used as an industrial workhorse.
Building a high-order method is not easy. They are a lot more complicated than low-order methods and it is difficult to build efficient and robust high-order schemes especially for unstructured meshes. And the industry relies heavily on unstructured grids for complex geometries, as they are easy to generate, and are more amenable for efficient load balancing on parallel architectures.
On the other hand, it is relatively easy to develop HO methods for structured meshes. However, care must be taken to make the structured mesh smooth enough to achieve the high-order accuracy, otherwise, the actual order of accuracy may dip down to second or even first order.
In addition, present state-of-the-art high-order methods are slow in converging to study state solutions and they also lack the required robustness to handle complex flows, due to low numerical dissipation.
Further, for non-smooth geometries or solutions, HO fails to achieve high levels of accuracy as expected but is comparable to lower methods in performance. This is seen especially in flow problems with shocks, in regions away from the discontinuity. Although strategies like adding artificial viscosity, solution limiting have been suggested, they are not efficacious approaches as they lead to more user-specified parameters and issues like stalling of iterative convergence.
Some other stumbling blocks include the need for high CPU memory when using implicit time-stepping and the deficiency of robust structured/unstructured high order grid-generators in the market. These narrow the road to wider acceptance and make the reliability of high-order methods in the industry much more difficult.
Figure 5: 1. Regular linear mesh(GridPro), 2. Higher-order cubic mesh( GridPro)
Gridding Challenges
To attain high-order accuracy and exploit the full potential of high-order methods, high-order meshes are required. If regular linear meshes are used in high-order simulations, the solution error caused by the linear mesh may be second-order, thereby negating the high-order accuracy. Therefore, it is critically important to represent the geometry with quadratic or high order elements having curved edges and faces.
Unfortunately, not many grid-generators can generate high-order grids. The difficulty or the challenges appear in generating highly clustered viscous elements near high curvature regions. High order meshes are usually coarser than regular linear meshes and near high curvature regions, coarse cells can overlap/pierce/inter-twine with each other. As a consequence, the lack of a robust high-order grid generator has become a major bottleneck in the development of high-order methods.
You made it to the START, you will make it to the FINISH
Second-order is not the sweet spot the CFD community needs to settle down for with respect to the order of accuracy. It is not the finishing line to be contented with. This was good to hold a couple of decades ago when the computers were less powerful and the first-order solutions were not accurate enough for engineering requirements. Then, attaining 2nd – order accuracy was a sought-after goal. In fact, that drive to attain it had led to the development of robust second-order accurate codes. But now, there is a change in the industry landscape, and engineers demand accuracy far better than what 2nd order can provide. Though high-order methods are still improving and have limitations in meeting the industrial demands, with additional investment in their research and development, it is possible to make them more efficient and robust and ultimately become the buttress of future CFD.
Further Reading
- Nesting your way to mesh Multi-Scale CFD Simulation!
- Efficient Meshing of Turbine Blade Cooling Holes
References
1. “High-order CFD methods: current status and perspective”, Z. J. Wang et al, International journal for numerical methods in fluids, 2013, 72: 811 – 845.
2. “High-Order and High Accurate CFD Methods and Their Applications for Complex Grid Problems”, Xiaogang Deng et al, Commun. Comput. Phys., Vol. 11, No. 4, pp. 1081-1102, April 2012.
3. http://www.hocfd.com/
4. https://how4.cenaero.be/
