Modeling turbulence in CFD analyses

Turbulent flows are omnipresent in nature and technology. Pipe flows, flows around buildings, around aircraft, marine vessels and other vehicles are virtually always turbulent. A flow is only laminar in  more specific cases, when a flow is (1)  very viscous, (2) very slow, or (3) happens at microscopic lengthscales. Typical examples of laminar flows are ink in a printhead (microscopic), a glacier flowing (very slowly) or rubber during manufacturing (very viscous). The single important parameter here is the Reynolds number, defined as Re=  ρUL/μ, in which ρ is the density, U the velocity, L a characteristic lengthscale and μ the viscosity. Plainly stated, a higher Reynolds number implies a more turbulent flow. Below some situation-dependent critical Reynolds number, a flow is laminar.

Turbulent flows are characterized by in-stable eddies, which emerge, fall apart, and disappear again, and therefore constantly moving the fluid. Turbulent flows are therefore always unsteady. The turbulent eddies decay into smaller and smaller eddies, until the kinetic energy is dissipated by viscous dissipation. The length scale of these smallest eddies, the so-called Kolmogorov length-scale, is often very small, being on the order of micrometers. Figure 1 An artist impression of a turbulent flow, made by Leonardo da Vinci around 1510. He sketched the flow from a channel in a bath of water. Various eddies of varying lengthscale can be observed clearly.

The set of equations which is solved in a standard CFD analysis are the continuity equation (mass conservation) and the 3 “Navier-Stokes” equations (momentum conservation in 3 dimensions). We thus have 4 equations for 4 unknown variables – the pressure and the velocity in 3 dimensions. To fully solve these equations numerically, including a correct calculation of turbulence is possible. This approach is called DNS: “Direct Numerical Simulation”. The required computational power is extreme, and a DNS approach is not possible for industrial applications. The required computational power scales as Re³, making it impossible to calculate flows with high Reynolds numbers. DNS is used extensively at universities, however, to study the fundamentals of turbulent flows.

In industrial applications, we are constantly making a trade-off between (1) a correct physical approach, and (2) the available budget and computational resources. In most cases, a “steady-state” solution is calculated, thus time-averaging the flow. Here problems arise: we try to solve an unsteady, turbulent flow using a time-averaged method.

For this steady-state solution, the Navier-Stokes equations are time-averaged, which results in the RaNS equations: the Reynolds-averaged Navier Stokes equations. In the mathematical process of Reynolds-averaging, 6 new variables are created: the so-called Reynolds stresses. Here the fundamental problem of solving turbulent flows becomes clear: we have a “closure problem”: more unknown variables than governing equations. Mathematically we cannot solve the RaNS equations as we don’t have sufficient equations. These equations are modeled with “turbulence models”.

With turbulence models, we simulate the characteristics of turbulent flows. A turbulence model therefore always is a simplification of reality – often useful, never fully correct, not seldom completely wrong. The knowledge and experience of the CFD specialist is crucial to apply the turbulence model and asses to the results.

The many different turbulence models are categorized by the amount of partial differential equations (PDE’s) which are solved. The “zero-equation models” are simple, arithmetic correlations. They are computationally inexpensive, and are rarely used to obtain final results. “One-equation models”, such as the Spalart-Allmeras model, are a bit more expensive, and are used in specific fields.

“Two-equation models” are more often used. They solve 2 PDE’s: one for the turbulent kinetic energy κ and the energy dissipation ϵ, or the specific dissipation ω. These κ – ϵ and κ – ω models have proven its use throughout the years, and are able to produce useful results. A mix between these models, the SST model has gained popularity, as it tries to combine the strong points of both.

Numerically more expensive are the Reynolds stress models, which directly solve the 6 components of the Reynolds stresses. This model is used to solve more complex flows, such as flows which larges areas of flow separation, swirl or re-circulation.

A completely different approach is to perform a “Large Eddy Simulation”. Turbulence is not fully calculated as in DNS, but also not completely averaged and modeled as in RaNS. Larger eddies are calculated in a transient simulation, and the smaller (sub-grid length) eddies are modeled. For specific situations, LES is a useful choice. The computational costs are somewhat in between DNS and RaNS.

The computational power continues to increase. Research on turbulence is more active than ever, and much effort is made to make numerical methods faster and more robust. It had been expected that LES would largely replace RaNS methods, but so far a large “transition” did note take place. Who knows: maybe in the far future DNS is the standard, and our current turbulence models are seen as completely outdated. For the coming decades, RaNS and the various turbulence models will remain the golden standard for many practical applications.