In our previous blog, we described the turbulent energy cascade and the turbulent energy spectrum that characterizes a turbulent flow. To summarize, a turbulent flow consists of eddies (swirling regions of fluid) of different sizes. These eddies range from being a comparable size to the characteristic length of the system and being heavily geometry dependent, to orders of magnitude smaller and more isotropic and universal. The transfer of energy is commonly represented by the turbulent energy cascade, where the turbulent kinetic energy is transferred from the largest eddies which lie in the integral scale (also referred to the energy containing range), to the smallest eddies in the dissipation range (also known as the Kolmogorov scales), where the energy is dissipated as heat. This process is shown in the figure below on the left.

The figure on the right shows the typical turbulent energy spectrum, where turbulent kinetic energy, *E(k)* is plotted against the wave number, *k*, which is the inverse of the eddy size. The spectrum shows a decay of *E(k)* as the eddy size decreases.

To accurately model fluid flow, it is vital to predict this entire turbulent energy transfer process. How is this done in CFD, and what are the different turbulence resolving and modeling techniques that are used?

## Direct Numerical Simulation (DNS)

In DNS, the full turbulent kinetic energy spectrum is resolved by the CFD simulation. This is achieved by employing spatial and temporal discretization (a very fine mesh and a very small timestep size) which is accurately able to capture all length scales from the integral length scale to the Kolmogorov (dissipative) length scale, along with all the time scales. The time dependent Navier-Stokes equations are solved directly, and since no modeling is employed to account for turbulence, this is the most accurate solution that can be obtained using CFD.

However, since DNS requires a very fine spatial and temporal resolution which results in a very fine mesh and a large number of time steps required, these simulations are computationally very expensive, and remain infeasible for complex problems, as well as simpler problems at high Reynolds numbers, even with current supercomputing technology. Therefore, DNS is infeasible and not used for any industrial applications. DNS is primarily used for fundamental research to understand fluid flow physics of turbulent flows and to validate and develop new turbulence models. Some example DNS cases that have been studied in literature are turbulence in boundary layers, jet flows, jet in cross flow, flow around circular/square cylinders, flow over an airfoil at low to moderate Reynolds numbers, isotropic turbulence in a channel flow, etc.

## Large Eddy Simulation (LES)

LES was first proposed by Joseph Smagorinsky in 1963 in his paper on atmospheric turbulence and weather prediction (Smagorinsky, Joseph (March 1963). “General Circulation Experiments with the Primitive Equations”. *Monthly Weather Review*. **91** (3): 99–164). The genius idea was that only the large eddies in a turbulent flow are resolved, whereas the smaller more universal eddies are modelled. Since resolving the smallest eddies requires the most amount of computational cost, modeling these smallest eddies instead provides substantial computational savings. Thus, LES offers a significant reduction in computational costs as compared to DNS, while still maintaining a very high level of accuracy.

In terms of the turbulent energy cascade and the energy spectrum, LES is able to resolve turbulent kinetic energy in the inertial sub-range. A good LES is considered to resolve about 80% of the full turbulent energy spectrum, whereas the other 20% is accounted for using modeling. The turbulent energy spectrum employing an LES simulation is shown in the figure below.

Mathematically, the smallest eddies are filtered using low-pass filtering of the Navier-Stokes equations by applying a low-pass filtering kernel that employs spatial and temporal averaging.

Several LES models have been formulated over the past few decades. For example, Smagorinsky eddy viscosity model, algebraic dynamic model, dynamic Lagrangian Smagorinsky model, etc. A more detailed review of these models will be considered in a future blog post.

LES still remains computationally very expensive, but as computational power and resources are increasing with the advent of supercomputing clusters with GPU acceleration and accessibility via cloud computing, LES is becoming increasingly popular to solve complex industrial problems where other turbulence modeling techniques such as RANS are not able to accurately resolve the flow physics of the system.

LES is widely used in industry and academia to study interactional aerodynamics and even aeroacoustics in turbomachinery flows, flow around rotorcraft blades, wind turbine blades, and highly turbulent wakes behind aircraft, buildings, and wind farms. The automobile and heavy truck industries are utilizing LES to optimize aerodynamics of subsystems and apply active flow control techniques to reduce fuel consumption and reduce noise. LES is also employed to study reactive combustion flows, to study atmospheric turbulence, predict weather patterns, etc.

## Reynolds-Averaged Navier Stokes (RANS)

RANS, which was first proposed by Osborne Reynolds, are the time-averaged Navier-Stokes equations that are obtained after employing Reynolds averaging. The basic principle behind RANS equations is Reynolds decomposition, where a quantity is decomposed into a mean and fluctuating quantity. Here, the mean quantity is resolved using the numerical solution, and any interactions with the fluctuating quantity are modelled.

In terms of the turbulent energy cascade and the energy spectrum, RANS only resolves the energy containing integral scales, and all other scales are modelled. This is represented in the turbulent energy spectrum as shown in the figure below.

As most of the turbulent energy spectrum is modelled in RANS, an appropriate turbulence model is required to accurately represent the transport and decay of turbulent kinetic energy. Several turbulence models have been developed over the past several decades. Some of the widely used models are one equation models such as Spalart-Allmaras (SA) and Prandtl’s one equation model, and two equation models such as standard k-epsilon, realizable k-epsilon, RNG k-epsilon, Wilcox k-omega, Menter k-omega SST, etc. Each of these turbulence models have their own advantages and disadvantages, and careful consideration should be given to choosing a particular turbulence model for CFD simulation. For example, Spalart-Allmaras model is widely used for external aerodynamics and wall bounded attached flows such as flows over airfoils and wings but may not be appropriate and performs poorly for separated flows that are highly turbulent.

As RANS simulations use modeling to account for any turbulent scales smaller than the large energy-containing scales, these simulations require the least computational resources, while still maintaining a good level of accuracy. Hence, they are widely used in all industries, especially for complex configurations, as they provide quick turnaround times for performing design studies and gain valuable insights into how a fluid system will perform without incurring the expenses of actual prototyping.

## Final Thoughts

In this post, we summarized how different turbulent modeling techniques are associated with the turbulence energy cascade and the turbulent kinetic energy spectrum. RANS is widely used in industrial applications due to its capability to provide good approximations at a lower computational cost. LES is growing popular for complex applications and optimization studies of subsystems, where a high level of accuracy and fidelity is required and will find widespread use as computing technology advances and becomes more accessible (supercomputing clusters with GPU acceleration on the cloud). DNS, even with current computing technology, is limited to relatively simple academic and research cases due to its extremely high computational costs. Hybrid turbulence modeling methodologies have also been developed, such as hybrid RANS-LES, Detached Eddy Simulation (DES), Delayed-Detached Eddy Simulation (DDES), etc.

Choosing a turbulence modeling technique and a turbulence model associated with it is not straightforward, and often requires expert consultation. A thorough analysis of the system to be designed must be performed, the level of fidelity of the simulation must be considered, all while maintaining a good trade-off between accuracy, computational costs, and turn-around time.

At Fidelis, we can help you solve your next problem in CFD with the various turbulence models that are available in the 3DEXPERIENCE Platform fluid flow solver or in PowerFLOW. Don’t hesitate to **reach out to our expert team for your CFD needs**!