Turbulence: a necessity!
- Sir Horace Lamb, 1932, in an address to the British Association for the Advancement of Science
"I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic."
The turbulence and boundary layer are two very closely related topics. While overall flow is divided into two zones: boundary layer and free-stream, boundary layers themselves are divided into 4 zones: viscous sub-layer, buffer-layer, log-law region and outer layer.
But turbulence is not an undesirable thing under all circumstances. The image describes the features of "turbulent motions".
The complicated motion of turbulence is the result of the non-linear advection that creates
interactions between different scales of motion. The large scales are governed by the geometrical scales and the smallest scale, known as Kolmogorov scale, is constrained by the viscosity of the fluid.
- Though all fluid flows are governed by Navier-Stokes equations, the wide range of length and time scales poses difficulties in treating turbulence, both analytically and numerically.
- The presence of too may flow scales computational resources too high for most industrial applications. To address this, the governing equations
are either time/spatially filtered to remove all/some of the turbulent scales.
- This approach results in a tractable set of equations for numerical calculations, closure problem arises from the correlated velocity fluctuations. So, a turbulence model is needed to
represent the effect of the small scales on the mean flow.
- There are no turbulence model that fit all the requirements. k-ε model is the workhorse of the industry but needs to be used with caution. k-ω SST generally gives good results in most of the cases where k-ε fall short. Yet one must investigate the effect of different turbulence models for the particular case being investigated.
Last but not the least, an excerpts from Academic Resource Center of Illinois Institute of Technology: "The phenomenon of turbulence, caused by the convective terms, is considered the last unsolved problem of classical mechanics. We know more about quantum particles and supernova than we do about the swirling of creamer in a steaming cup of coffee!"
Key Parameters for Specification of Turbulence
Following plot described the production and dissipation of TKE. Note that k and ε are in almost equalibrium far away from the wall.
- Turbulent Viscosity - μt
- Analogous to molecular viscosity, turbulence viscosity is related to fluctuating component of the velocity.
- μt = Cμ * k2 / ε where Cμ = 0.09, an empirical coefficient.
- Turbulent Kinetic Energy - k
- It is specific (per unit mass) kinetic energy of eddies = 1/2 * v'2
- This is trace of Reynolds stress tensor, computed by summing the diagonals of the matrix of components. Thus, TKE,
- It defines velocity scale of the turbulence (eddies). Remember Ryenolds number is ratio of length scales * velocity scale / kinematic viscosity
- Since u', v' and w' are not modeled in CFD except DNS approach, how do we estimate u' and k? The answer is Boussinesq hypothesis: which relates the Reynolds stresses to the mean velocity gradients. This hypotehsis is used in Spallarat-Allmaras model, k-ε and k-ω models.
- Boussinesq hypothesis:
- Turbulent Intensity - I
- The turbulence intensity I, is defined as the ratio of the root-mean-square of the velocity fluctuations u', to the mean flow velocity, umean.
- A turbulence intensity of 1% or less is generally considered low and turbulence intensities greater than 10% are considered high.
- Turbulent Eddy Dissipation Rate - ε
- It is rate of conversion of k into heat or thermal energy per unit mass.
- Along with k, it defines length scale of the flows (turbulence eddies)
- Turbulent Dissipation Frequency - ω
- It is rate of conversion of k into thermal energy per unit volume.
- It can be defined implicitly by k and ε
- Summary: RANS methods
- Step-1: Mean flow velosity and pressure equations are solved.
- Step-2: TKE (k) and TED (ε) equations are solved.
- Step-3: Turbulent viscosity is solved using equations described above.
- Step-4: Reynolds stress tensor is modeled (estimated) using Boussinesq hypothesis.
Key variable of turbulence modeling is Y-plus (y+). The limitations imposed on this parameter governs the size of mesh near the walls. This refers to two parameters:
- The height of the 1st layer of the mesh near the walls
- Number of layers in the boundary layer (which is not known a priori)!
Further deliberations need to be done along with the wall functions
- "High-Re number wall treatment" refers to case where boundary layers are modeled in terms of empirical (and well-established) correlations.
- "Low-Re number wall treatment" refers to the case when boundary layers are resolved and not modeled.
- Note that it is related to the way boundary layers are resolved and has nothing to do with the "MEAN FLOW REYNOLDS NUMBER".
- A "low Reynolds number" turbulence model doesn't use wall functions, that is does not involve any assumptions about the near-wal variation of velocity.
- Examples include k-ω SST or low Reynolds number k-ε (again, a model that integrates right up to the boundary layer).
- For all "high-Re number wall treatments" such as scalable wall function [exploits a limiter in the (y+) calculations such that y+ = min [y+, 11.36] in CFX, standard wall functions in Fluent and STAR-CCM+.
- the centroid of cell adjacent to walls should lie in logarithmic region: y+ >= 30 in Fluent and STAR CCM+ because they use cell-centred schemes (staggered arrangement)
- the nodes of 1st layer of elements should fall in y+ >= 30 in ANSYS CFX. This software uses vertex-centred scheme (collocated arrangement).
- A scalable wall function is another way of saying wall surface coincides with the edge of the viscous sublayer if the first grid point is too close to the wall and falls in the inner layer
- No noticeable impact on result is likely even if the y+ falls up to the level of 11.36, in any "high-Re number wall treatments"
- It is not possible to maintain y+ value in a very narrow band. It is advised to target a y+ band of 30 ~ 50 for higher accuracy. A range of 30 ~ 100 on y+ yields result which is acceptable for design iterations.
- The y+ value should not be allowed to cross the upper limit of 300.
- It is not possible to maintain this limit (y+ > 11.36) in few areas of the fluid flow such as separation and stagnation (recirculation) zones. This is numerically acceptable and no impact of results should be anticipated because boundary layers are neither orderly nor as steep as normal wall-bounded turbulent flows with separations.
- There are "numerical improvisations" where the wall treatment toggles automatically between "Low-Re number wall treatment" and "high-Re number wall treatments"
- When "Low-Re number wall treatment" is used, following requirements on mesh are imposed:
- The (y+) should be of the order of 1.
- Under no circumstances, the y+ should increase above 5.
- The number of elements in the boundary layer should be ~ 10.