Energy cascade physics & large-eddy simulations
Turbulent flows are vexing for both simulation and analysis because they naturally generate coherent motions with a wide range of length scales and frequencies, even in the most simple flow geometries. The reason this occurs is the energy cascade, an intrinsically nonlinear process by which turbulence inevitably energizes very fine scale motions that are too costly to directly simulate in most applications.
Physics Discovery: Our research is upending the long-held view that the turbulence energy cascade is mostly the result of vortex stretching, a process in which small tornado-like features are elongated under axial stretching. When a vortex is stretched, conservation of angular momentum requires increased rotation rates with smaller radial extent — as is the case when a figure skater pulling in her arms to spin faster. Thus, kinetic energy is progressively moved toward smaller and smaller features. However, our recent work has demonstrated that the energy cascade is actually driven mostly by a different mechanism: the self-amplification of the strain-rate. The tendency of higher velocity fluid to overtake fluid in a nearby slower moving region causes an existing compressive strain-rate to naturally amplify itself as it is squeezed into a thinner layer. Like vortex stretching, this generates larger velocity gradients associated with smaller-scale features. Progress on disentangling these two processes was enabled by our recent mathematical formulation which definitively showed for the first time how the energy cascade can be written exactly in terms of vortex stretching and strain-rate self-amplification. The approach is readily extensible to, e.g., stratified flows, magnetohydrodynamic (MHD) turbulence, and the helicity cascade.
Model Development: Our group is working to leverage these insights about the energy cascade to develop improve engineering models for under-resolved turbulent flow simulations, i.e., so-called large-eddy simulations (LES). It turns out that the same mathematical formulation that precisely relates the energy cascade to vortex stretching and strain-rate self-amplification also provides a basis for LES models. The basic idea is to define a “physics-inspired” coarse-graining procedure that represents a snapshot from coarse grid turbulent simulation as coming from a real turbulent flow that has been artificially subjected to a finite-time diffusion process (more specifically, Stokes flow). The resulting formulation, Stokes Flow Regularization (SFR), has a number of practical and theoretical advantages over the traditional approach based on integral filtering operators. First, SFR theory provides an alternative to the Germano identity for developing dynamic LES models that automatically calculates model coefficients for small-scale unresolved turbulent stresses based on resolved larger scales. Unlike the classical Germano-based dynamic models, SFR-based dynamic models do not require a test filter computation; the dynamic coefficient can be calculated analytically with pen-and-paper. Our group has demonstrated that SFR-based dynamic procedure works quite well for a range of common LES stress model forms. More enticingly, the SFR-based formulation of LES provides governing equations with no commutator errors in the case of non-uniform grid resolution. Going forward, our group will be working to demonstrate that SFR provides an effective way to generalize spatial filtering to attack enduring challenges related to LES such as wall-modeling and multiphase flows.
Further reading:
Johnson, P. L., Wilczek, M., 2024, “Multiscale Velocity Gradients in Turbulence,” Annual Review of Fluid Mechanics, 56, 453-490.
https://doi.org/10.1146/annurev-fluid-121021-031431
Capocci, D., Johnson, P. L., Oughton, S., Biferale, L., Linkmann, M., 2023, “New exact Betchov-like relation for the helicity flux in homogeneous turbulence,” Journal of Fluid Mechanics, 963, R1.
https://doi.org/10.1017/jfm.2023.236
Also available at: https://arxiv.org/abs/2301.04193
Johnson, P. L., 2022, “A physics-inspired alternative to spatial filtering for large-eddy simulations of turbulent flows,” Journal of Fluid Mechanics, 934, A30.
https://doi.org/10.1017/jfm.2021.1150 (PDF)
Johnson, P. L., 2021, “The squeezes, stretches, and whirls of turbulence,” Physics Today, 74 (4), 46-51.
https://doi.org/10.1063/PT.3.4725 (PDF)
Johnson, P. L., 2021, “On the role of vorticity stretching and strain self-amplification in the turbulence energy cascade,” Journal of Fluid Mechanics, 922, A3.
https://doi.org/10.1017/jfm.2021.490
Also available at: https://arxiv.org/abs/2102.06844
Johnson, P. L., 2020, “Energy Transfer from Large to Small Scales in Turbulence by Multiscale Nonlinear Strain and Vorticity Interactions,” Physical Review Letters, 124, 104501.
https://doi.org/10.1103/PhysRevLett.124.104501
Also available at: https://arxiv.org/abs/1912.00293
Integral equations for turbulent boundary layers
Skin friction and heat transfer are dramatically enhanced when a boundary layer becomes unstable and transitions to turbulence. Designing for, and potentially controlling, this effect requires better understanding of transitional and turbulent boundary layers.
Physics Discovery: Our research has developed a simple formulation to create precise relations that reveal how turbulent fluctuations (and other physical phenomena such as free-stream pressure gradients) alter viscous drag and surface heat transfer. The Angular Momentum Integral (AMI) equation is based on the integral conservation law for the moment of momentum using the boundary layer equations. The AMI equation quantitatively writes the skin friction coefficient as a sum of the skin friction from an equivalent laminar boundary layer plus enhancements and attenuations relative to that baseline laminar case. The other terms include an integral of the Reynolds shear stress across the boundary layer (a “turbulent torque”) and the free stream pressure gradient acting on a moment-of-displacement thickness (a “pressure gradient torque”). As sketched in the above figure, the AMI equation quantifies skin friction enhancement/attenuation in terms of “torques” that alter the angular momentum of the boundary layer and hence the skin friction. For example, turbulence acts like a counter-clockwise torque (red) that rearranges the distribution of momentum in the boundary layer so as to increase the skin friction. An adverse pressure gradient has the opposite effect (green). This basic moment integral approach is readily extensible to study transitional boundary layers, pressure gradient effects, surface heat transfer, compressibility & high enthalpy effects, and the efficacy of flow control schemes for drag reduction or separation delay.
Model Development: Our research on using integral equations to inform turbulence modeling is in its beginning stages, but the history and impact of integral equations on boundary layer predictions stretches back over a century to the work of von Karman and Pohlhaussen on laminar boundary layers. Following clues from accumulated knowledge from many researchers over decades of work on turbulent structures, in combination with our recent quantitative insights of the AMI equation applied to various types of boundary layers, we are pursuing the development of an entirely new framework for modeling turbulent boundary layers, apart from the standard approaches of RANS and LES. More details to come! Stay tuned…
Further reading:
Kianfar, A., Di Renzo, M., Williams, C., Elnahhas, A., Johnson, P. L., 2023, “Angular momentum and moment of total enthalpy integral equations for high-speed boundary layers,” Physical Review Fluids, 8, 054603.
https://doi.org/10.1103/PhysRevFluids.8.054603 (PDF)
Kianfar, A., Elnahhas, A., Johnson, P. L., 2023, “Quantifying How Turbulence Enhances Boundary Layer Skin Friction and Surface Heat Transfer,” AIAA Journal, 61 (9), 3900-3909.
https://doi.org/10.2514/1.J063042 (PDF)
Elnahhas, A., and Johnson, P. L., 2022, “On the enhancement of boundary layer skin friction by turbulence: an angular momentum approach,” Journal of Fluid Mechanics, 940, A36.
https://doi.org/10.1017/jfm.2022.264
Also available at: https://arxiv.org/abs/2107.03043
Johnson, P. L., 2019, “Toward evaluating contributions to skin friction enhancement by transition and turbulence in boundary layer flows,” Center for Turbulence Research Annual Research Briefs, pp. 223-235.
https://web.stanford.edu/group/ctr/ResBriefs/2019/17_Johnson.pdf
Other research topics
Particles in wall-bounded turbulence
Small solid particles or drops often play an important role in applications dealing with turbulent flows. Two out of many examples are: (i) the operation of air-breathing propulsion engines in arid environments faces challenges associated with the ingestion of dust and sand; and (ii) particles may be used to more efficiently absorb radiation for concentrated solar power. Particle deposition, transport, and resuspension can be strongly influenced by turbulent fluctuations near the flow boundary, but key near-wall fluctuations are too small to directly simulate for many applications. Further, algorithms for detecting individual particle-particle collisions can be quite costly, so it is often desirable to simulate flows with artificially reduced number of particles to relax the computational requirements.
Our recent work has addressed both difficulties. First, we have introduced a mathematical formulation clarifying the important effects dictating near-wall particle-turbulence interactions. Initial simulations results for a simple model are promising in some regimes, but more work is needed to address other regimes. In a related effort, we have developed a systematic procedure for accurate accounting of collisional effects for particles in turbulent flows when only a small fraction of the particles are included in the simulation. This trick has been shown to work across a wide range of regimes and offers substantial computational savings.
Further reading:
Hu, R., Johnson, P. L., Meneveau, C., 2023, “Modeling the resuspension of small inertial particles in turbulent flow over a fractal-like multiscale rough surface,” Physical Review Fluids, 8, 024304.
https://doi.org/10.1103/PhysRevFluids.8.024304
Johnson, P. L., Bassenne, M., and Moin, P., 2020, “Turbophoresis of small inertial particles: theoretical considerations and application to wall-modelled large-eddy simulations,” Journal of Fluid Mechanics, 883, A27.
https://doi.org/10.1017/jfm.2019.865
Johnson, P. L., 2019, “Predicting the impact of particle-particle collisions on turbophoresis with a reduced number of computational particles,” International Journal of Multiphase Flow, 124, 103182.
https://doi.org/10.1016/j.ijmultiphaseflow.2019.103182
Also available at: https://arxiv.org/abs/1908.02869
Drops and bubbles in turbulence
Small bubbles and droplets deform and break when subjected to turbulent flows. Breakage leads to a wide spectrum of drop or bubble sizes in turbulent flows, and is important to a number of applications, such as: (i) fate of oil spills in the ocean, or (ii) characterization of bubbly ship wakes.
When the drops or bubbles are smaller than all turbulent length scales, deformation and breakup is a viscous process driven by velocity gradients along the particle trajectory. We have developed carefully-controlled approximations for stochastic modeling of turbulent velocity gradients and demonstrated effective coupling to under-resolved simulations (large-eddy simulations) to provide a computationally efficient method for predicting such sub-Kolmogorov drops and bubbles.
As is more often the case, the bubble sizes overlap with the sizes of coherent turbulent fluctuations, so that there is more direct turbulence-bubble interactions. In this case, successive bubble breakup events may be a cascade of air mass from large to small bubble. Our recent work has provided a thorough framework for detecting and quantifying the cascade-like nature of turbulent bubbly breakup. Simulation results confirm key underpinnings and results of our theory, which should enable future development of sub-grid breakup modeling the turbulent bubbly flows.
Further reading:
Chan, W. H. R., Johnson, P. L., and Moin, P., 2021, “The turbulent bubble break-up cascade. Part 1. Theoretical developments,” Journal of Fluid Mechanics, 912, A42.
https://doi.org/10.1017/jfm.2020.1083
Also available at: https://arxiv.org/abs/2008.12883
Chan, W. H. R., Johnson, P. L., Moin, P., and Urzay, J., 2021, “The turbulent bubble break-up cascade. Part 2. Numerical simulations of breaking waves,” Journal of Fluid Mechanics, 912, A43.
https://doi.org/10.1017/jfm.2020.1084
Also available at: https://arxiv.org/abs/2009.04804
Chan, W. H. R., Dodd, M. S., Johnson, P. L., and Moin, P., 2021, “Identifying and tracking bubbles and drops in simulations: A toolbox for obtaining sizes, lineages, and breakup and coalescence statistics,” Journal of Computational Physics, 432, 110156.
https://doi.org/10.1016/j.jcp.2021.110156
Also available at: https://arxiv.org/abs/2011.07243
Johnson, P. L., and Meneveau, C., 2018, “Predicting viscous-range velocity gradient dynamics in large-eddy simulations of turbulence,” Journal of Fluid Mechanics, 837, pp. 80-114.
https://doi.org/10.1017/jfm.2017.838