Mixing length model
- zero equation model (no transport equations)
- Lm is mixing length, representing the region in which turbulent mixing is expected to act
- main advantage is speed and simplicity of the model
- no transport effects in this model, which are needed to account for CONVECTION and DIFFUSION of patches of turbulence away from site or moment of generation
Eddy viscosity
- turbulence is a viscous phenomena
- analogous to laminar viscosity
Reynolds stresses
- uiuj
- tensor of non-zero correlations that result from the averaging process
RANS +/-
- needs extensive closure modelling
- no information about instantaneous flow, cheap for engineering applications
- generally only models a single scale, the integral scales
LES +/-
- needs closure modelling only at small isotropic scales
- enables instantaneous information to be obtained
- remains expensive and relatively uncertain, not predictive
Hybrid RANS-LES +/-
- RANS used near to a wall, LES used far from a wall
- can provide limited information about instantaneous flow
- a useful compromise which is increasing popularity with industry
- split between GLOBAL and NON-GLOBAL methods
Detached Eddy Simulation
- GLOBAL method, adapts automatically between RANS and LES
- used within an existing RANS framework, with improvements for numerical aspects where the model switches to LES
- the switch is based on the length scale equation, which returns the minimum of the grid scale and the turbulence length scale from which the RANS model its based on
- main problem is grid-induced separation, over refinement of the boundary layers results in undesirable switching to LES in the near wall region, can spontaneously induce unphysical flow separation
Embedded Large Eddy Simulation
- NON-GLOBAL method, requires the user to pre-define regions of the domain where the model will act in RANS and LES
- main challenge is generation of unsteady inlet velocity at the interface from RANS to LES
- generally achieved via introduction of synthetic turbulence at interface
- superimpose unsteady fluctuations on top of a mean flow field from the RANS domain, must be correlated such that they represent coherent turbulent eddies
- white noise is uncorrelated in time and space and therefore doesnt work
Energy Cascade definition
- the transfer of energy, originating from the mean flow with scales of that order, to progressively smaller and smaller eddies my means of the non-linear interaction; dissipated as heat by viscous forces
Characteristics of turbulence
- Strong vorticity
- Irregularity
- chaotic and stochastic
- Three dimensionality
- Unsteadiness
- Broad spectrum of scales
- Dissipative
3 important things about energy spectrum
- midsection of the cascade, ‘inertial subrange’, becomes broader as the Reynolds number increases
- as the turbulent eddies become smaller, directional information is lost
- the gradient of the inertial subrange is a constant value E(k) ~ k^(-5/3)
Newtonian fluid
- viscous stresses vanish when the fluid is at a rest
- viscous stresses are linearly proportional to the strain rate
- there are no preferred directions in the fluid
Turbulence modelling vs simulation
Modelling - reconstruction of bulk effects of turbulent scales without resorting to direct representation of their dynamics
Simulation - more complete resolution of the time and spatial variation of turbulent scales, without resorting to semi-empirical modelling
Length scales
Integral - associated with the bulk of the energy
Taylor - associated with isotropic motion
Kolmogorov - associated with viscous dissipation
Eddy viscosity disadvantages
- modelling is based on local information only (one closure point)
- constants calibrated from simple (equilibrium) flows
- turbulence kinetic energy production is usually over predicted in regions of high strain
- Reynolds stresses are assumed to be isotropic
Bandwidth EQ
integral length scale/kolmogorov length scale
VLES +/-
- lies between URANS and fully resolved LES
- conceptual objective is to resolve flow up to a corresponding wave number
DNS +/-
- best representation of the full Navier-Stokes; no modelling required
- all scales of motion are captured
- extremely expensive, a research tool for low Reynolds numbers only
DES +/-
- apply URANS to attached boundary layers in their entirety
- apply LES to strongly separated flow regions
- particularly relevant to external aerodynamics
Role of Reynolds stresses/turbulence model
- RANS represents time-averaged flow
- Reynolds stresses are the tensor of non-zero correlations that result from the averaging process
- in order to solve the RANS equations, approximations to these terms are required
Low & high Re distinction
- low Re models operate in the viscous sub layer when viscous stress dominate
- they need: terms sensitive to viscous stress, sufficient resolution in this region
- high Re models operate in the log layer where turbulent shear stress dominates
- a coarser mesh is acceptable, though near wall model ‘wall functions’ are required
General benefit
DNS - detailed instantaneous information about turbulence
LES - instantaneous nature retained, with some loss of accuracy and insight
RANS-LES - instantaneous nature retained, significant approximations near to wall
RANS - speed & means for efficient production of mean flow field without need for instantaneous turbulence resolution
General requirement
DNS - extremely high computational resource, knowledge of boundary and initial conditions
LES - retains prohibitively large requirements for practical cases, best practice guidelines exist but are case dependant
RANS-LES - more practical than LES but require time averaging, significantly increases cost
RANS - computational requirements are lower but knowledge of basic modelling limitations is needed
Log layer
- a universal law for turbulent flows which can be derived by applying the assumptions listen in the question to the N-S equations
- U+ is proportional to log y+
