Computational Fluid Dynamics, or CFD, is the numerical simulation of fluid motion. It means solving the equations describing the fluid flow, heat transfer and material transport over a discrete domain, given boundary conditions.

The first scientist to approach the study of fluid dynamics as we understand it today was Leonardo Da Vinci (1452-1519) in the “Del moto e misura dell'acqua”. Many scientists since then have worked on a rigorous and complete mathematical formulation. However, only in the early 1900 were the final equations developed by the Frenchman, Claude Louis Marie Henry Navier (1785-1836) and the Irishman, George Gabriel Stokes (1819-1903), based on the work of Euler. They introduced the viscous transport into the Euler equations, to become known as the Navier-Stokes equation. The very large discrete domain over which the mathematical transport models are solved constantly challenges the computational power of modern computers. Large computational power is needed to increase the convergence of the equations, the accuracy of the final result and most importantly to resolve the physical time scales of the flow otherwise un-modelled. For this reason, the first real computational simulations were done privately in a few research centres using in-house developed codes. The introduction on the market of newer, faster computers, and the publications of research papers on numerical methods applied to CFD, led a few software houses to start distributing commercially

Conservation Equations

An element of fluid is represented by a mesh element in the discrete domain. Within this element, all the transport equations are solved. Without going into the details of the formulae, it is useful to remember that two changes in the element are most likely to take place, Translation and Rotation. The Navier-Stokes equation can easily be decomposed to show the separate contributions. The translational term can be seen as a change in shape and is usually referred to as convection. The rotational term on the other hand, is characterized by gradients in the velocity fields and referrers to diffusion quantities. Generally these are the main terms governing the evolution of a fluid dynamical system, but in more complex simulations it is not uncommon to find terms describing sources and dissipations related for instance to heat generation, chemical reactions and phase transitions. In a fully and accurately described fluid dynamical system, these equations are coupled, meaning that the changes in one variable can give rise to changes in others.

Continuity

The continuity equation is, in more familiar terms, a mass balance over a rectangular volume, where a mass flow of density ρ possesses the velocity component u1 u2 and u3; done over a Cartesian reference axis of size ?x1 ?x2 and ?x3.

Momentum

The momentum equation is nothing more than a momentum balance over the same volume. It also contains the terms relative to pressure gradients, sources of momentum, gravitational forces (often not considered).

Turbulence

Turbulence is a fluid flow regime where the inertial forces of the fluid are far greater than the frictional forces. In engineering terms, the Reynolds number is greater than approximately 104. But this value changes drastically depending on the geometry of the system, and is often interpreted as an average index of turbulence, while its value changes locally. The main effect that turbulence has on the flow is to create local instabilities observed as time dependent fluctuations of the flow properties (i.e. the velocity).The characteristic length scales of turbulence pose a great obstacle to CFD commercial codes, due to the high resolution needed to model the eddy size and eddy time scales (Kolmogorov microscales). To overcome this difficulty, and include turbulence in the models, a mathematical decomposition is of the instantaneous velocity u used. The time dependent velocity u is divided into time average component U and a time dependent fluctuating u’. Doing so, and grouping all the time dependent terms of the Navier-Stokes equation into the newly created Reynolds stresses, the CFD friendly Reynolds Averaged Navier Stokes (RANS) equations are derived. The following versions of RANS are based on different assumptions and are applicable to different fluid dynamical systems. However, they all share the same base hypothesis introduced by Boussinesq, stating that the Reynolds stresses can be seen as the algebraic sum of a turbulence viscosity and term relative to the turbulence production.

• k-ε Model
• RNG k-ε Model
• Mixing Length Model
• Zero Equation Model

The Reynold Stress Model, RSM, model on the other hand is not based on the Boussinesq hypothesis, but assumes the turbulence to be isotropic, and computing each component of the stress tensor individually. The accuracy of this method is strongly coupled to the isotropicity of the flow. Very useful further information can be found in the book of Computational Fluid Mixing written by Andre’ Bakker and Elizabeth Marden Marshall. It contains an overview of some of the turbulence models just introduced:

Standard k-ε Model: The most widely used model, it is robust, economical, and serves the engineering community. Main advantages include its rapidity, stability and reasonable results for many flows, especially those with high Reynolds number recommended for highly swirling flows, round jets, or flows with strong flow separation.

RNG k-ε Model: A modified version of the k-ε model, this model yields improved results for swirling flows in round jets and flow separation. It is not well suited for round jets, and is not as stable as the standard k-ε.

Realizable k-ε: Another modified version of the k-ε model, the realizable k-ε model correctly predicts the flow in round jets and is also well suited for swirling flows and flows involving separation.

k-Omega: This models turbulent frequency (omega) instead of epsilon. Has advantages in flows with strong wall effects or separation.

Shear Stress Transport (SST): combines the good wall treatment of the k-omega with the good free-stream flow of the k-epsilon using blending functions.

RSM: the full Reynolds stress model provides good prediction for all types of flow, including swirl, separation, and round and planar jets. Because it solves transport equations for the reynold stresses directly, longer calculation times are required.

LES: Large eddy simulation is a transient turbulence model that provides excellent results for all flow systems. It solves the Navier-Stokes equations for large scale turbulent fluctuations and models only the small scale fluctuations. Because it is a transient model, the required computational resources are considerably larger than those required for the RSM and k-ε style models. In addition, a finer grid is needed to gain the maximum benefit from the model, and to accurately capture the turbulence in the smallest, sub-grid scales.

By Andrea Gabriele, David Ryan