Turbulence Models
Turbulence is a physical phenomenon where the inertia of a fluid overcomes the stabilizing effect of viscosity. This causes the various quantities of the flow to vary randomly in time and space over a wide range of scales. These turbulent motions enhance the diffusion of mass, momentum, and other scalars. The irregular nature of turbulent flow requires a statistical description to analyze the quantities of interest. Other than at low Reynolds numbers where viscous effects dominate, turbulence is ubiquitous across all engineering applications.
Various modeling strategies have been developed to account for the effect of turbulence while balancing the computational effort required to include varying levels of physical complexity. One of the most popular modeling approaches is to represent the effect of turbulence as an additional turbulent, or eddy, viscosity that modifies the physical viscosity. The FLOW-3D product family offers a comprehensive turbulence modeling suite covering fully 3D flows, 2D depth-averaged shallow water flows, and hybrid 3D/2D depth-averaged flows. Various turbulence model options are available, ranging from simple mixing length approaches to robust Reynolds-averaged Navier-Stokes models to large-eddy simulation, giving engineers the flexibility to match model complexity to the demands of their application.
Example: Von Kármán Vortex Street
The simulation below illustrates one of LES’s key strengths: the ability to resolve coherent turbulent structures in detail. A bluff body placed in a flow generates a von Kármán vortex street, a classic pattern of alternating vortices shed from either side of the object. The Q-criterion isosurfaces, colored by velocity magnitude, reveal the three-dimensional structure of these vortices as they form, stretch, and dissipate downstream. A RANS model is only capable of capturing the time-dependent mean behavior of the shedding vortices; LES captures their full transient behavior.
3D Turbulence Models
Turbulence models in 3D directly model the effect of the unresolved turbulent fluctuations. For RANS, that is the effect of turbulence on the mean flow. For LES, that is the effect on the resolved transient fluctuating flow.
Prandtl Mixing Length Model
One of the earliest turbulence models, the Prandtl mixing length approach estimates turbulent effects by assuming the turbulent viscosity is proportional to a constant user-specified length scale. It remains useful for simple academic cases and benchmarking but is no longer widely applied to complex industrial flows.
One-Equation Model
The one-equation model solves a transport equation for the turbulent kinetic energy k but requires the specification of a constant turbulent mixing length at all locations to calculate the turbulent viscosity. Because this length scale is rarely known in advance for complex geometries, this model is generally limited to simple flows.
Standard k-epsilon Model
The standard k-epsilon model (Harlow & Nakayama, 1967) solves transport equations for both turbulent kinetic energy k and turbulent dissipation rate epsilon, computing the mixing length and thus the turbulent viscosity dynamically. It is an industry standard with a broad validation base and performs well across a wide range of flows (Rodi, 1980).
RNG k-epsilon Model
The Renormalized Group (RNG) k-epsilon model (Yakhot & Orszag, 1986; Yakhot & Smith, 1992) is a more robust extension of the standard k-epsilon approach. It provides better performance in transitionally turbulent flows, curving flows, wall heat transfer, and mass transfer, and is the recommended starting point for most industrial problems.
k-omega Model
The k-omega model (Wilcox, 1988, 1998, 2008) replaces the dissipation rate with omega (epsilon/k), offering improved accuracy for free shear flows with streamwise pressure gradients such as spreading jets, wakes, and plumes.
Large Eddy Simulation (LES)
Rather than modeling the effect of turbulence statistically, LES resolves most turbulent fluctuations directly and models the unresolved turbulence using an additional subgrid-scale (SGS) viscosity term. Just like with the turbulent viscosity of the RANS models, the SGS viscosity modifies the physical viscosity. It requires significantly finer mesh resolution than RANS models and provides detailed transient statistics of the flow field. Time-averaging LES output produces results very similar to RANS, as shown in the fish passage comparison below.
Example: Turbulence Model Comparison of a Fish Passage
The three simulations below use the same fish passage geometry to compare turbulence models in practice. The first video shows the LES model with instantaneous velocity fluctuations visible. The second shows the time-averaged result from the same LES simulation. The third uses the RNG k-epsilon model. The time-averaged LES and RNG k-epsilon results are closely comparable, demonstrating that RANS models can provide an efficient and accurate alternative to LES for time-averaged quantities in certain types of problems.
Wall Function
Viscous boundary layers that develop along solid surfaces influence the bulk turbulent flow. Since these boundary layers can be extremely thin, it can be computationally expensive to resolve them for general engineering applications. An alternative approach is to model the effect of the unresolved portion of the boundary layer on the rest of the resolved flow using a wall function. Each turbulence model in the FLOW-3D product family employs a wall function based on the logarithmic law of the wall at any smooth or rough solid surface.
Dynamic Maximum Mixing Length
For the two-equation RANS turbulence models, the local bounds of the turbulent length and time scales can be computed dynamically based on limiters from the Kolmogorov scales on the lower bound and integral scales on the upper bound. These dynamic limiters allow for the modeling of a wider range of flows without requiring the specification of a constant maximum turbulent length scale. This is particularly valuable for applications such as contact tanks and water treatment systems where complex flow conditions span a broad range of turbulent states.
Shallow Water Turbulence Models
For 2D depth-averaged and shallow water regions of hybrid 3D/2D flows, the effect of unresolved turbulent fluctuations is accounted for using a scalar model for turbulent viscosity. The effect of dispersion of the velocity in the vertical direction can also be accounted for in some of the models. See the Shallow Water Model capabilities page for more information about the specific shallow water turbulence models available within FLOW-3D HYDRO.
Applications
Turbulence models in the FLOW-3D product family are applied across a wide range of industries and flow types, including:
- Hydraulic infrastructure such as spillways, fish passages, contact tanks, and pump stations
- Industrial mixing and chemical processing
- Metal casting mold filling (HPDC and gravity casting)
- Coastal and estuarine flows
- Internal flows in valves, pipes, and manifolds
- Water and wastewater treatment
- Sediment transport and scour
Example: Spillway Case Study
This side-by-side comparison from a Mott MacDonald project shows the physical model result (left) alongside the FLOW-3D HYDRO simulation (right), colored by velocity magnitude. The close agreement between physical and numerical results demonstrates the accuracy of the turbulence model implementation for complex spillway flows.
References
Driver, D.M. and Seegmiller, H.L., 1985, Features of a reattaching turbulent shear layer in divergent channel flow, AIAA Journal (23), 163-171.
Harlow, F.H. and Nakayama, P.I., 1967, Turbulence transport equations, Physics of Fluids (10), 2323-2332.
Harlow, F.H. and Nakayama, P.I., 1968, Transport of turbulence energy decay rate, Los Alamos Scientific Laboratory report LA-3854.
Kolmogorov, A.N., 1942, Equations of turbulent motion in an incompressible fluid, Izvestia Academy of Sciences, USSR; Physics (6), 56-58.
Pope, S.B., 2000, Turbulent Flows, Cambridge University Press.
Rodi, W., 1980, Turbulence models and their application in hydraulics: a state of the art review, International Association for Hydraulic Research (IAHR), Delft, The Netherlands.
Saffman, P.G., 1970, A Model for Inhomogeneous Turbulent Flow, Proceedings of the Royal Society London A (317), 417-433.
Speziale, C.G., Abid, R., and Anderson, E.C., 1992, Critical evaluation of two-equation models for near-wall turbulence, AIAA Journal (30), 324-331.
Wilcox, D.C., 1988, Reassessment of the scale-determining equation for advanced turbulence models, AIAA Journal (26), 1299-1310.
Wilcox, D.C., 1998, Turbulence Modeling for CFD, DCW Industries, Inc., 2nd edition.
Wilcox, D.C., 2008, Formulation of the k-omega turbulence model revisited, AIAA Journal (46), 2823-2838.
Yakhot, V. and Orszag, S.A., 1986, Renormalization group analysis of turbulence I. Basic theory, Journal of Scientific Computing (1), 3-51.
Yakhot, V. and Smith, L.M., 1992, The renormalization group, the e-expansion and derivation of turbulence models, Journal of Scientific Computing (7), 35-51.