For such systems, the fluid flow has to be described with the rarefied flow equations or at least with Knudsen boundary conditions. A fluid is characterized by, among other things, its viscosity. The viscous effects are contained in the viscous stress tensor,. Most commonly encountered fluids, such as water, gases, and ethanol, are Newtonian.

For Newtonian fluids, the viscous stress is proportional to the deviatoric stress tensor:. There are, however, a number of fluids that do not obey the simple relation in Equation 3. Such fluids are known as non-Newtonian and can display a wide range of behaviors. Examples of non-Newtonian fluids include blood; paint; some lubricants; cosmetic products; food products such as honey, ketchup , juice, and yogurt; and many suspensions, for example sand in water or starch suspended in water.

A fluid can be regarded as incompressible if the density variations are very small; that is, if. This is true for liquids unless there are significant temperature variations and gases under moderate pressure and temperature variations. If we can neglect the heating due to viscous dissipation so-called viscous heating and assume that the fluid is Newtonian, Equation 1 can be simplified to:.

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The middle equation in Equation 4 is the famous Navier-Stokes equation , named after the French physicist Navier and the Irish physicist Stokes. Navier was first to derive the equations, but the understanding of the physical mechanism behind the viscous term was first explained by Stokes, hence the name of the equations. In some cases, the first equation, the continuity equation, is also included in the Navier-Stokes equations. As can be seen, the energy equation has been rewritten as a temperature equation, which is much more convenient to work with.

The temperature equation is for incompressible flows completely decoupled from the Navier-Stokes equations, unless the viscosity depends on the temperature. The solution to the Navier-Stokes equations gives the velocity and pressure field for flows of fluids with constant viscosity and density. The temperature can be solved for separately if information about the temperature field is desirable.

Buoyancy is an important physical phenomenon that formally comes from variations in density. For example, buoyancy makes the smoke from a cigar flow upward. In absence of body forces, and if the density and the viscosity are constant, the Navier-Stokes equation middle expression in Equation 4 can be nondimensionalized to read:. As can be seen from Equation 6 , the Reynolds number measures the relative importance of the viscous stresses.

A low Reynolds number means that the flow is completely governed by viscous effects, while the flow is effectively inviscid at very high Reynolds numbers. Observe that there can be several Reynolds numbers associated with a particular flow configuration. A channel flow, for example, can be based on the channel half width or the whole channel width. The velocity can either be the average velocity or the maximum velocity.

Therefore, it is important to know which length scale and velocity scale are associated with a particular Reynolds number, especially when comparing Reynolds numbers between similar flow configurations. Flows at very low Reynolds numbers are known as creeping flows. They can be encountered in, for example, microfluidic systems such as the micromixer shown below or lubrication systems. The limit when is known as Stokes flow.

## Reynolds Number Calculations

Stokes flow can formally support both time dependency and varying material properties, but classical Stokes flow is written for incompressible quasistatic conditions:. The equations are named after the Irish physicist George Gabriel Stokes, who first described viscous momentum transfer through these equations.

Which terms to retain from the energy equation depends on the fluid.

The convective term can often be neglected, as can pressure-work effects. Viscous heating can also be of interest for Stokes flows in bearings and other lubrication applications, for example. The Reynolds number measures the importance of the inertial effects compared to the viscous effects. As long as the Reynolds number is not too large, the viscous effects will damp out perturbations in the flow field.

Such flows are known as laminar flows. It is often feasible to solve, for example, Equation 4 for laminar flows, since the viscosity dissipates any flow structures that are small enough. The higher the Reynolds number, the more inertial effects dominate over viscous effects. When the Reynolds number is high enough, any small perturbation will feed on the mean flow momentum and grow and trigger new flow structures. This phenomenon is called transition. A flow that has undergone transition is denoted turbulent flow.

Turbulent flows are characterized by the seemingly chaotic eddies that have a huge span of length scales, from the large vortices that can be almost as large as the computational domain down to the small dissipative eddies that can be as small as micrometers. This wide span of scales means that not many turbulent flows can be simulated at a reasonable computational cost using the pure Navier-Stokes equations.

It is possible to perform so-called direct numerical simulation DNS for some very simple flow cases, but it requires vast computational resources. In order to make it possible to estimate the flow and pressure fields without having access to a supercomputer, we usually introduce approximate turbulence models. The turbulence models formulate different types of conservation expressions for turbulence in an averaged sense; for example, by looking at the conservation of the kinetic energy that these small eddies may have called turbulent kinetic energy.

The conserved properties, such as the turbulent kinetic energy, are used to generate an additional contribution to the viscosity, called eddy viscosity. The eddy viscosity enlarges the viscous transfer of momentum in order to mimic the momentum that would be transferred by the small-scale eddies that we cannot afford to resolve. The most common turbulence models used in engineering are the Reynolds-averaged Navier-Stokes models RANS models , in which the modeled quantities are time-averaged and the fluctuations are treated in an introduced quantity referred to as the Reynold stresses.

The RANS equations for incompressible flows read:. The unfiltered velocity, for example, can be written. Comparing Equation 8 to the continuity and momentum equations in Equation 4 , it can be seen that the equations are identical except that unfiltered quantities have been replaced by filtered quantities, there is no time derivative since we averaged over time , and there is an extra term in Equation 8. This term is the Reynolds stress tensor and represents the effect of the turbulent fluctuations on the filtered velocity and pressure fields.

It is possible to formulate transport equations for the entries in the Reynolds-stress tensor. With appropriate simplifications and assumptions, these equations can result in so-called Reynolds-stress models. While powerful, these models are often difficult to work with and even if they are computationally much less expensive than DNS, they are still too expensive for most industrial applications. The most common approach is instead to assume that the turbulence acts as an additional viscous effect and write , where is the turbulence viscosity, also known as the eddy viscosity.

There is another class of turbulence models that averages turbulence over a small spatial region instead of over time. This forms a sort of low-pass filter for eddies below a certain length scale. In this way, large turbulent eddies are resolved while the effect of small eddies have to be modeled, hence the name large eddy simulation LES. The continuity and momentum equations for incompressible LES take the same form:.

Equation 9 is identical to Equation 8 , but with a time derivative. Also, the term is the subgrid stress SGS tensor and it represents the effect of the subgrid scales on the resolved scales. LES is often more accurate than RANS, but the simulations must always be 3D, even if the flow is essentially 2D and the simulations are always time dependent. In addition, the required resolution for the SGS models to be valid is often quite high, which means that LES is only used when even the most advanced RANS models fail to capture the essential features of the flow.

The Mach number measures how fast a fluid moves compared to the speed of the pressure waves.

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When the Mach number is small, that is, when , the pressure waves are so fast that they effectively reduce to a mass conservation constraint. The incompressible flow formulation in Equation 4 can formally be reached by letting. As the Mach number approaches 1 — that is, when the velocity approaches the speed of sound — the effects of the pressure waves must also be included.

Viscous heating is also often important in these cases and the result is that we must solve the complete set of continuity, momentum, and energy equations given by Equation 1.

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## Numerical Simulation of Low Reynolds Number Flows Based on CFD

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