One of the most formidable and complex challenges in the field of classical physics has been, and still remains, the complete and accurate description of fluid flow. For centuries, it has attracted the attention and curiosity of great minds, like Leonardo da Vinci, but its complexity has kept it as an unprecedented intellectual challenge. So, why is fluid flow such a difficult subject? The answer to this question is turbulence, which, despite the vast number of experimental and theoretical studies for its understanding, has proved to be a matter of great difficulty.
There is no definition of turbulence despite the clear manifestation of its presence. It's existence has profound importance in engineering applications including fluid flow, and the attempt to mathematically describe turbulence has not produced any satisfactory results. The level of complexity is enormous; turbulent flow settings encountered in practice are an enormously complex subject.
Considering fluid flow over a complex geometry, it is almost impossible to use analytical tools to describe the developing flow field. This is not only for the case of turbulent flows, but even for laminar flows. However, the development, and as of now, the maturity level of Computational Fluid Dynamics (CFD) has established a very reliable and efficient tool, which can provide a description of turbulent fluid flows around general geometries, like the flow around a pitching submarine, the transonic flow over a wing, the simulation of an explosion event and many others.
But still, turbulence is not a completely resolved issue in CFD . At present, setting up an efficient model for a flow simulation is something that could be posed as an art rather as an engineering discipline for the majority of the commercially available CFD simulation tools. The present article is a discussion about the appropriate modeling of turbulence in CFD simulations for several engineering applications. This will hopefully provide some perspective on ideal turbulence modeling approaches, and at the same time will reveal the capacity for high fidelity turbulent flow simulations with the EXN/Aero general purpose CFD solver in short time.
The accumulated experience in "traditional" CFD simulation packages relies mostly on the known solution elements for the mean flow properties. However, as the drive for increasing the fidelity of CFD simulations is always present, the best choice will be one that relies on the implementation of the accumulated knowledge on turbulent flows.
There are nearly 200 models for including the effects of turbulence in CFD simulations. Their number is an indication of the difficulty and intricacy, not only from a theoretical point of view, but for conducting CFD simulations within reasonable time for design purposes. For engineering purposes, time average quantities are appropriate and the large-scale eddies resolution is enough for this. So, how can a CFD engineer feel confident that his choice of turbulence modeling is correct and he will get results within short turnaround?
Mathematical description of fluid flow
The equations describing the flow of fluids are based on three physical principles, which are:
- The conservation of mass
- Newton's second law
- The first law of thermodynamics
There is also the hypothesis of the continuum medium which is correct for flows where the Knudsen number is low. The above principles along with the proper constitutive relations for the shear stresses in the fluid and equation of states for the calculation of thermodynamic properties provide the basic framework for the development of the mathematical description of fluid flow. In principle these equations suffice for the description of laminar and turbulent flows without the need for any additional information.
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Reynolds decomposition - Time average equations
Turbulence manifests its presence as random fluctuations in measurements of flow properties at different locations in space and time. This suggests that the flow properties may be decomposed to a mean, or time-average, and a fluctuating part. This is known as the Reynolds decomposition. Substitution of the decomposition of the flow variables in the governing flow equations leads to the basic equations for the mean motion of turbulent flows, with the new form of the momentum equations known as the Reynolds Averaged Navier Stokes (RANS) equations.
The RANS equations suffer from a closure problem which needs to be addressed before a numerical solution of the equations. The unknowns appearing in the equations of mean motion correspond to the so called Reynolds stresses and other products of the fluctuating components. Turbulence modeling accounts for this issue and all of them rely on a hypothesis for the transfer of momentum caused by the turbulent eddies.
Turbulence models for the RANS equations
Turbulence is a three dimensional unsteady process with multiple time and spatial scales. It arises when the inertia forces of the flow become much higher than viscous forces, so it is a characteristic of high Reynolds number flows. Traditional turbulence models use concepts that relate the mean flow properties to properties of turbulence for including its effect in CFD simulations.
The eddy viscosity models are classified into four categories according to the number of the transport equations solved for the turbulent properties. Two equations models are widely employed in commercial CFD simulation software. They offer a good compromise between accuracy and computational cost with the ability to tune them at specific flow settings. The most widely used are the k-ε and the k-ω eddy viscosity models. Another very popular turbulence model which has proved to be very successful in industrial RANS applications is the k-ω Shear Stress Transport (k-ω SST) model, which combines the best aspects of the k-ε and k-ω models.
Large-Eddy Simulation and Detached Eddy Simulation techniques
In the Large-Eddy Simulation (LES) technique the time dependent equations of flow motion are solved, but a spatial filtering is applied first. Conceptually the spatial filtering allows the more energetic large-eddies of the turbulent flow to be resolved exactly by the time dependent equations and a so called sub-grid scale model is use to account for the unresolved turbulent fine structures. The sub-grid scale modeling is carried out in a manner similar to the turbulence modeling for the RANS equations. However, the approximation errors are less than in the case of traditional turbulence model since the contribution of the small eddies to the total turbulent kinetic energy and momentum flux is small. The finer the employed mesh the finer the filtering, and thus, the smaller the part of turbulence which has to be modeled. This technique is an expensive approach for the simulation of turbulent flows especially for high Reynolds number, where the resolution of the turbulent boundary layer requires very fine meshes.
An alternative to LES which keeps its benefits but at a lower computational cost is the Detached Eddy Simulation (DES) technique. This is a hybrid of the LES technique and RANS equations and circumvents the difficulties of LES in near-wall regions. This approach treats the near-wall regions using RANS models for turbulence and a LES technique for the rest of the flow field.
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Direct Numerical Simulation
It is possible to numerically solve the general unsteady equations of motion without any time average and filtering. In this case a direct numerical computation is performed which accounts for all the time and spatial scales of turbulence. However, this approach is prohibitively expensive with the current available computer hardware, and only very few computations have been carried out for research purposes. These computations have been for very simple geometries and for low Reynolds numbers and the insight they provide is even better than experiments. Up till now they have been used for understanding turbulence and for the improvement of traditional turbulence models.
Which model or technique to use?
The answer depends on the problem and the available computational resources. Many assumptions are made while making a turbulent model in order to reduce the computational cost of CFD simulations. Different assumptions will be made according to the type of the flow being modeled. It is important for the CFD engineer to understand the developing flow structure and its treatment at near-wall regions.
In the context of turbulence modeling, the accumulated experience has shown that:
- The k-ε model is appropriate for recirculating flows and two dimensional shear flows. This model solves two equations: one for the turbulent kinetic energy (k-equation) and one for the turbulent kinetic energy dissipation rate (ε-equation). It is very popular and it has been the most widely used and validated turbulence model for many industrial applications. It is usually useful for free-shear layer flows with relatively small pressure gradients, and as well as, in confined flows where the Reynolds stresses are important. Wall functions are implemented in the model which lowers the memory requirements for its use, and also, it has demonstrated very good convergence behavior. The k-ε model is not suited for flows containing large adverse pressure gradients and in a variety of important cases such as unconfined flows, jet flows and flows exhibiting curved boundary layers.
- The k-ω turbulence model is similar to the k-ε model but it solves for the specific turbulent kinetic energy dissipation rate (ω-equation). It is suited for internal flow problems, for flows exhibiting strong curvature, separated flows, and jet flows. It uses wall functions as well, so it does not have high memory requirements. It appears to be sensitive to the initial conditions. In practice a solution is used computed using the k-ε model which is used as initial conditions for subsequent simulations with the k-ω model.
- The k-ω SST model has proved to be a very good turbulence model for many industrial applications. However, it does not converges very quickly and requires a good resolution of the near-wall region which is a memory intensive case. Its accuracy though is very good. It exhibits sensitivity to the initial conditions and the usual practice is to employ the k-ε or the k-ω model to compute the flow filed and use it as initial conditions for the k-ω SST.
- When a very accurate resolution of an unsteady flow field is required it is the best choice to use LES or DES. Especially unsteady three dimensional flows with vortex shedding or oscillating shear layers, flows with coherent flow structures, studies for noise generation, small scale processes where turbulence resolution is necessary, and good estimation of loads due to fluid flow and cases where RANS simulation fail are cases where the LES or DES technique is adequate. Both, LES and DES techniques are very expensive computationally and usually require expensive HPC systems and long run times for producing statistically meaningful results.
The final answer
CFD simulation software that offers too few turbulence models could mean that engineers miss the one that is needed, while software that offers too much choice, presents the challenge of not knowing which turbulence model is best suited to answer the question being asked. To counter the above problem, engineers should be fully aware of the strengths and limitations of each model. Here, we take a look at those in common use, before rounding up some of the key factors that come in to play when choosing the most appropriate turbulent models.
In the EXN/Aero solver, the accumulated wisdom of decades of turbulence modeling has been taken into account and the k-ω and the k-ω SST have been implemented and validated for CFD simulations in the RANS framework for steady and unsteady flow problems. Furthermore, the revolutionary technology of the EXN/Aero solver has surpassed the limitation of turbulence modeling and the financial burden of buying an expensive HPC system and carrying out an LES or DES simulation in a short time is now possible. The solver can give CFD simulation results on DES and LES techniques for long time integration simulations within a short turnaround as it is demonstrated in the validation case for the NASA ARA M100 validation case. This revolutionary technology permits all CFD organizations around the world to conduct DES or LES simulations within times comparable to, or shorter than, the solution time of RANS and definitely faster than the currently commercially available CFD simulation packages, and at the fraction of the acquisition cost of a HPC system, as it is shown in the Ahmed body benchmark case. Want to get started with CFD? Schedule an EXN/Aero demo today with one of our engineers:
What’s so hard about defining turbulent fluid flow?
The turbulence closure problem is so-called, as the overall aim is to close the Navier-Stokes and Reynolds stress equations that describe turbulent flow. While the solution is yet to be fully satisfied, a number of turbulence models have tried to close the system of equations that describe turbulent flows.
In a paper published in 2009, assessing the industrial application of RANS modeling, David Corson found that many assumptions are made while making a turbulent model, in order to reduce the computational costs of a simulation. Different assumptions will be made according to the type of flow being modeled.
Engineers face a large choice of turbulent models, causing much head-scratching among engineers as they look to choose CFD simulation software.
CFD simulation software that offers too few turbulence models could mean that engineers miss the one that is needed, while software that offers too much choice, presents the challenge of not knowing which turbulence model is best suited to answer the question being asked.
To counter the above problem, engineers should be fully aware of the strengths and limitations of each model. Here, we take a look at those in common use, before rounding up some of the key factors that come in to play when choosing the most appropriate turbulent models.
- Reynolds-averaged Navier Stokes Models
The RANS models are easily the largest in the turbulence field, using viscosity terms to close the turbulence equations.
RANS models are based on the definition of turbulent viscosity, and limitations include unsteady flows like internal combustion engines, swirling flows or flows with rotations, a stagnant region in flows, streamlined curvatures, and a lack of physical description.
- Reynolds Stress Model (RSM)
As the most complete physical representation of turbulent flows, RSM is able to capture complex strains such as swirling flows.
These models are based on the six equations that represent the stresses of turbulence, attempting to model the flow directly in RANS equations. While they have advantages in their representation of flows, they come with high computational costs, so are reserved for complex flows or those being studies for the first time. Further limitations include the requirement of high-quality mesh and their sensitivity to initial conditions.
- Large-Eddy Simulation and Detached Eddy Simulation Models
While RANS models simulate all scales of turbulence and resolve none, large-eddy simulation (LES) and detached-eddy simulation models not only resolve large scales of turbulence, but also use sub-grid turbulence models or blend with a RANS model to model the remainder.
While the LES model predicts large turbulent eddy structures when solving (a CFD model system) with a fine mesh, the model is unable to predict near-wall regions with accuracy, due to the small turbulent scales in this area.
High computational costs restrict the use of both LES and DES models, so are not hugely popular with CFD software vendors.
Key Considerations when choosing a Turbulence Model
When it comes to making the right selection, it’s advisable to have the end goal very much in mind. Having a clear understanding of the question you are asking, will enable the strengths and limitations of each model to be better assessed, increasing the likelihood of more accurate and appropriate answers.
Many specific factors come in to play when choosing the right turbulence model, and the near-wall mesh resolution is a good example. The turbulent flow near the wall is different from that in the bulk area. For mesh that is fine near the wall, it is necessary that a model is chosen that is fully compatible with near-wall turbulent flow, or results will not be accurate or appropriate.
Engineers may also consider a number of other factors before making their choice. How well a model works within applications for which it is not originally intended, could be one factor, and the model’s ability to produce rapid preliminary results (that could rule out inappropriate early design options), could be another.
While many engineering practices often look to reduce computational costs, there may be occasions where utilizing expensive models with limited computational power may still prove necessary. Using boundary layer meshes at the wall along with adaptive mesh requirements within the bulk of the fluid, may prove to be best practice under such circumstances.
A Matter of Experience...
As with most engineering practices, knowledge often comes with experience, and that is particularly applicable to choosing a turbulence model. Many seasoned engineers may be able to select a handful of models simply by looking at an application.
The Role of Experimental Data...
While experience certainly plays a key role, verifying a turbulence model in line with experimental data is good practice, and can prevent errors by identifying issues at an early stage.
In some cases, where an application requires a highly accurate resolution of specific flow features, relying on comparisons to experimental data specific to that application, is the only way to identify the best modeling approach.