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Introduction to CFX
What is Turbulence?
Unsteady, irregular (non-periodic) motion in which transported quantities (mass, momentum, scalar species) fluctuate in time and space
Identifiable swirling patterns characterize turbulent eddies
Enhanced mixing (matter, momentum, energy, etc.) results
Fluid properties and velocity exhibit random variations
Statistical averaging results in accountable, turbulence related transport mechanisms
This characteristic allows for turbulence modeling
Contains a wide range of turbulent eddy sizes (scales spectrum)
The size/velocity of large eddies is on the order of the mean flow
Large eddies derive energy from the mean flow
Energy is transferred from larger eddies to smaller eddies
In the smallest eddies, turbulent energy is converted to internal energy by viscous dissipation
Is the Flow Turbulent?
External Flows
Internal Flows
Natural Convection
along a surface
around an obstacle
where
where
Other factors such as free-stream turbulence, surface conditions, and disturbances may cause transition to turbulence at lower Reynolds numbers
is the Rayleigh number
is the Prandtl number
Flows can be characterized by the Reynolds Number, Re
Observation by O. Reynolds
Laminar
(Low Reynolds Number)
Transition
(Increasing Reynolds Number)
Turbulent
(Higher Reynolds Number)
Turbulent Flow Structures
Energy Cascade Richardson (1922)
Small
structures
Large
structures
Governing Equations
Conservation Equations
Continuity
Momentum
Energy
where
Note that there is no turbulence equation in the governing Navier-Stokes equations!
Overview of Computational Approaches
Direct Numerical Simulation (DNS)
Theoretically, all turbulent (and laminar / transition) flows can be simulated by numerically solving the full Navier-Stokes equations
Resolves the whole spectrum of scales. No modeling is required
But the cost is too prohibitive! Not practical for industrial flows
Large Eddy Simulation (LES) type models
Solves the spatially averaged N-S equations
Large eddies are directly resolved, but eddies smaller than the mesh are modeled
Less expensive than DNS, but the amount of computational resources and efforts are still too large for most practical applications
Reynolds-Averaged Navier-Stokes (RANS) models
Solve time-averaged Navier-Stokes equations
All turbulent length scales are modeled in RANS
Various different models are available
This is the most widely used approach for calculating industrial flows
There is not yet a single, practical turbulence model that can reliably predict all turbulent flows with sufficient accuracy
RANS Modeling – Time Averaging
Ensemble (time) averaging may be used to extract the mean flow properties from the instantaneous ones
The instantaneous velocity, ui, is split into average and fluctuating components
The Reynolds-averaged momentum equations are as follows
The Reynolds stresses are additional unknowns introduced by the averaging procedure, hence they must be modeled (related to the averaged flow quantities) in order to close the system of governing equations
Fluctuating
component
Time-average
component
Example: Fully-Developed
Turbulent Pipe Flow
Velocity Profile
Instantaneous
component
(Reynolds stress tensor)
RANS Modeling – The Closure Problem
Closure problem: Relate the unknown Reynolds Stresses to the known mean flow variables through new equations
The new equations are the turbulence model
Equations can be:
Algebraic
Transport equations
All turbulence models contain empiricism
Equations cannot be derived from fundamental principles
Some calibrating to observed solutions and “intelligent guessing” is contained in the models
RANS Modeling – The Closure Problem
The RANS models can be closed in one of the following ways
(1) Eddy Viscosity Models (via the Boussinesq hypothesis)
Boussinesq hypothesis – Reynolds stresses are modeled using an eddy (or turbulent) viscosity, μT. The hypothesis is reasonable for simple turbulent shear flows: boundary layers, round jets, mixing layers, channel flows, etc.
(2) Reynolds-Stress Models (via transport equations for Reynolds stresses)
Modeling is still required for many terms in the transport equations
RSM is more advantageous in complex 3D turbulent flows with large streamline curvature and swirl, but the model is more complex, computationally intensive, more difficult to converge than eddy viscosity models