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Modelling turbulence in CFD analyses

In some industrial applications, wall roughness can easily be avoided. Situations exist in which roughness gradually evolves, for instance biofouling in marine vessels or corrosion. Sometimes the manufacturing process is limiting in avoiding wall roughness, think of welding seams. To understand how roughness affects the flow, we have to understand the physics of the boundary layer.

The relation between roughness and the boundary layer

The thin fluid layer close to the wall is referred to as the boundary layer. For increasing Reynolds numbers the boundary layers continues to decrease in thickness. The Reynolds number, defined asRe = (ρ U L)/μ, is the ratio between inertial forces and viscous forces. In a highly turbulent flow with a high Reynolds number, viscous forces hardly play a role.

For the hypothetical case of Re →∞, the boundary layer is infinitely thin, and consequently every wall is seen as “rough” to some extent by the flow.How much the roughness affects the flow depends on the ratio between the boundary layer thickness and the size of the roughness. A turbulent boundary layer can be divided into a number of layers. The layer directly bordering the wall is called the viscous sublayer. The flow behaviour in the viscous sublayer is identical to a laminar (viscous, layered) flow: in both cases, viscous forces are dominant

The size of the wall roughness kscompared to the thickness of the viscous sublayer is key. We can distinguish 3 different regimes for turbulent flows

ks < viscous sublayer thicknessHydraulically smoothThe roughness is small compared to the viscous sublayer. Consequently, the roughness does not affect the flow.
ks  = 1 – 14 x viscous sublayer thicknessTransitionally roughRoughness affects the flow. Viscous friction and roughness friction both play a role. 
ks > 14 x viscous sublayer thicknessFully roughThe roughness is large compared to the viscous sublayer, and completely alters the boundary layer dynamics. Friction caused by roughness is dominant, and the friction becomes independent of the viscosity.

Earlier we mentioned that the flow behavior of a laminar flow and the viscous sublayer are identical. Consequently, roughness does not affect the flow in a laminar flow.

The friction is made dimensionless using a friction coefficient CF. Together with the Reynolds number and the roughness, CFis related to the pressure drop. In figure 2 we show the well-known Moody diagram[RV1] , in which this relation is made explicit for pipe flow.The roughness does not only affect the friction, but also the velocity profile in the boundary layer. The velocity gradient decreases together with the shear stress on the wall.

For hydraulically smooth flows, the friction coefficient equals CF,smooth: the roughness does not yet affect the flow. For a fully rough flow, CFbecomes a constant value; it no longer depends on the Reynolds number. In fully rough flows, energy is dissipated by pressure forces acting on the roughness rather than by viscous skin friction. Consequently, the viscosity (and thus the Reynolds number) becomes irrelevant. In the transitionally rough regime, both effects (viscous skin friction and pressure forces) are of comparable magnitude. 

Figure 1Moody diagram, in which the relation between the Reynolds number, the roughness and the friction coefficient is shown for pipe flow.

Roughness in itself is difficult to quantify. Let us take figure 2 for instance, in which we show a sawtooth roughness profile. The roughness parameters (such as an averaged height) can be easily calculated. For the fluid friction, however, the direction of the flow matters. A similar behavior is observed for welding seams or turned parts: whether the machined roughness is streamwise or transverse differs.

Figure 2Flow over a flat plate with sawtooth roughness. The direction of the flow (left or right) does not affect the roughness characteristics, but it does affect the friction. 

Roughness in CFD

In many commercial CFD software, the equivalent sandgrain roughness can be used as an input parameter. The roughness must be related to an equivalent sandgrain roughness. How the roughness affects the flowfield and the friction is calculated using available experimental data. Most of these experiments are performed in pipe flow. The classical experiments by Nikuradse in the ’30 of the last century are still heavily used. To which extend the roughness behavior is universal is not yet entirely known. Also for completely different types of flow (e.g. flow over a wing), these pipe flow experiments are used. Active research is ongoing to better understand the universal- and non-universal effects of roughness. 

Consequently the field of roughness in CFD advances quickly. It is expected that many studies on numerical methods and roughness will be published in the coming years. By the implementation of this knowledge in new models, the quality of the roughness models will be improved continuously. 

Johann Nikuradse around 1925
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