For coarsely-resolved simulations, especially those for applications that require numerous quick design iterations to asses the plausibility of a design, a simulation that both runs quickly and delivers results in the correct ballpark is needed. The coarse resolution allows for fast computation, but such simulations tend to suffer from gradient underprediction at the walls. This happens in particular for flow regimes which may be considered as turbulent, where the steep gradients at the wall mean that a very fine resolution is required to correctly reproduce them. With the effect of wall-bounded phenomena (e.g. wall shear stress, wall heat flux) being a key result for studies that such simulations are used for, it is clear that there is potential for improvement here.
Gradients at the wall depicted with particles.
As can be seen from the picture above, using a coarse resolution does not resolve the gradient of the velocity profile at the wall properly – the resulting slope at the wall is much less steep than that of the actual profile. We can naturally improve this by resolving the gradient directly, i.e. simulating with a fine resolution to capture the large difference in flow variables near the wall.
However, as an alternative we can avoid the requirement for refinement at the wall to a certain extent by prescribing near-wall behaviour of the fluid a priori based on an empirically determined law.
Plot of dimensionless velocity u+ against dimensionless wall distance y+. Source: Wikipedia
The law of the wall describes, in dimensionless units, how the dimensionless velocity u+ varies logarithmically with the dimensionless wall distance y+ in the range of 30 < y+ < 300 , i.e. when the profile near the wall is no longer linear and has a steeper gradient. This relation is used by PreonLab to correct the behaviour of particles at the wall, such that the steeper gradient is reproduced. Solid objects can be set to utilize this wall function model for fluid particles in contact with them. An analogous process is used to correct temperature gradients at the wall when the thermal wall function is enabled for solids in the scene.
In this article we will examine two benchmark cases that were used to validate PreonLab’s wall function model:
One of the first benchmarks we ran for this new feature was the well-known case of a turbulent flow over a heated flat plate with zero pressure gradient. This is a key benchmark for wall functions as the two main benchmarked quantities, the skin friction coefficient and the heat transfer coefficient, reflect the ability of the wall function to reproduce the correct gradients at the wall. Here the particle spacing is controlled such that the dimensionless wall distance y+ lies within the validity range of the logarithmic wall law.
Time-averaged plots of the skin friction coefficient Cf and heat transfer coefficient along the length of the flat plate (represented by local Reynold’s number Rex).
The results obtained show that the gradients at the wall could be corrected such that the increased gradient resulting from a turbulent flow is reproduced.
The above figure shows a simple setup where a water jet is impinging downwards on a heated flat disk. On the left, a refinement strategy was used for the near-wall region to resolve the near-wall gradients. On the right is a simulation with a relatively coarse resolution. Here the particles are coloured on a spectrum of blue to white according to their y+ values, with deep blue particles not being affected by the wall function model. The only differences between the two simulations are their particle spacings (and by extension, the strategy with which the near-wall area is handled).
Results depicting the Nusselt number (dimensionless heat transfer coefficient) at the wall in relation to the radial distance from the center of the heated disk. Simulation using two-level refinement.
Coarse simulation, with and without wall functions.
The above figure shows the Nusselt number, which is a dimensionless heat transfer coefficient, plotted against the radial distance from the center of the disk. When we compare the results of the coarse simulation and the simulation with refinement, we can see that the simulation with refinement clearly delivers more accurate results especially at the center of the disk. However, further away from the center, the results for the coarse simulation with wall functions activated are quite acceptable compared to the simulation with refinement. We can also see that what was once an unacceptably coarse simulation in the form of the gray line now delivers plausible results when wall functions are used in the simulation.
The runtime for the impinging jet case also shows something remarkable: Due to the much lower number of particles, the runtime for the coarse simulation is minimal compared to the finer simulation which resolves the gradients at the wall correctly. The other advantages of having a lower particle count apply here too, allowing for faster data transfer and much smaller storage requirements. For applications where large numbers of design iterations are needed, having wall functions makes a difference by allowing coarser resolutions to be run with an acceptable level of accuracy, allowing your first estimates to be generated much faster.
As this is currently an experimental feature in 5.0 we are naturally still developing and refining this feature; we look forward to updating you by sharing more results and new developments in the future.