19.0 Release Lecture 01: Overview of Engineering Turbulence Models Turbulence Modeling Using ANSYS CFD 1 © 2018 ANSYS, Inc. Outline • Motivation • Characteristics of Turbulent Flow − Energy Cascade − Vortex Stretching − Scales • Overview of Computational Approaches − Direct Numerical Simulation − Eddy Viscosity Models • Boussinesq Approach − Reynolds Stress Models (RSM) − Scale Resolving Simulation Models • Summary 2 © 2018 ANSYS, Inc. Motivation • Turbulence is all around us − Engineering devices • Aerospace applications • Naval applications • Vehicle aerodynamics − • Combustion systems Geophysical applications • Oceanography • Meteorology and weather prediction − • Environmental engineering Biological flows It is critical to predict and analyze the effects of turbulence on mass, momentum and energy transport 3 © 2018 ANSYS, Inc. Observation by O. Reynolds – Pipe Flow with Dye Injection • Flows can be classified as either : − Laminar: • Low Reynolds number • Fluid particles path exhibit no disturbances − Transition: • Increasing Reynolds number • Orderly 2D and 3D structures appear due to flow instability − Turbulent: • Higher Reynolds number • Flow exhibits random 3D unsteady structures 4 © 2018 ANSYS, Inc. Characteristics of Turbulent Flows • Unsteady, irregular (aperiodic) motion in which transported quantities (mass, momentum, scalar species) fluctuate in time and space − The fluctuations are responsible for enhanced mixing of transported quantities Mixing Layer Large Structure • Instantaneous fluctuations are random (unpredictable, irregular) both in space and time − Statistical averaging of fluctuations results in accountable, turbulence related transport mechanisms • Contains a wide range of eddy sizes (scales) − Typical identifiable swirling patterns − Large eddies “carry” small eddies − The behavior of large eddies is different in each flow − 5 • Sensitive to upstream history The behavior of small eddies is more universal in nature © 2018 ANSYS, Inc. Small Structures Turbulence Bowling ball entering still water at 25ft/s Laminar Separation • Turbulence is important in almost all technical flows • Effects of turbulence: − Enhances mixing and entrainment − Dissipates kinetic energy into heat − Increases friction losses − Increases heat transfer − Delays flow separation under pressure gradients (see Figure) − Generates Noise −… Turbulent Separation Has a patch of sand glued onto its leading surface 6 © 2018 ANSYS, Inc. Energy Cascade Turbulence kinetic energy is integral over wave number spectrum: ∞ • Cascade of Turbulence − Turbulence eddies are created at largest − − − − − − 7 scales Large eddies extract energy from mean flow Largest eddies are of size of mixing layer thickness Eddies get stretched and thereby reduced in size. This leads to an energy transfer to smaller and smaller eddies Smallest eddies are then dissipated into heat by molecular viscosity Smallest eddies are of Kolmogorov size Wave number k is invers to eddy size © 2018 ANSYS, Inc. π=ΰΆ± π¬π πΏ πΏ=π Log E Generation of largest eddies Energy transfer Viscous Dissipation π π = π¬π πΏ Log k Vortex Stretching • Existence of eddies implies vorticity • Vorticity is concentrated along vortex lines or bundles • Vortex lines/bundles become distorted from the induced velocities of the larger eddies − As the end points of a vortex line randomly move apart • Vortex line increases in length but decreases in diameter • Vorticity increases because angular momentum is nearly conserved − Most of the turbulence kinetic energy is contained within the largest eddies − Most of the vorticity is contained within the smallest eddies 8 © 2018 ANSYS, Inc. Turbulent Structures in free jet Scales of Turbulence – Smallest Scales • The smallest turbulence scales dissipate turbulence kinetic energy into heat by viscosity • The two relevant quantities are therefore − The dissipation rate, e , (energy dissipated per unit time and volume) − Molecular viscosity, n • The only length-scale which can be formed from these two quantities is: πΌ = (π 3 / πΊ)πΤπ • This scale is called Kolmogorov length scale • In technical fluids (air, water, etc.) the molecular viscosity is very low – therefore the Kolmogorov scales are very small 9 © 2018 ANSYS, Inc. Scales of Turbulence – Largest Scales • The largest turbulence scales are formed by turbulence production Pk • All turbulence which is produced is eventually dissipated. One has therefore on average: π·π ≈ πΊ • Most of the turbulence kinetic energy, π, is stored in the largest scales • The two relevant quantities for estimating the size of the large scales are therefore − The turbulence kinetic energy, k − The dissipation rate, e , (energy dissipated per unit time and volume) • The only length-scale which can be formed from these two quantities is: π³π = (ππΤπ / πΊ) • π³π is often called the “Turbulence Length“ scale or the “Integral Length“ scale 10 © 2018 ANSYS, Inc. Resolution Challenge • Turbulence is a continumm problem and is described by the Navier Stokes equations • Resolving all scales of turbulence in a numerical simulation is called “Direct Numerical Simulation“ (DNS) • DNS is extremely expensive as the ratio of the large to the small scales as: π³π ~ πΌ ππ³π π πΤπ ~ πΉπππ Τπ • These scales have to be resolved in three dimensions making the space resolution ~πΉπππ • In addition, the turbulence scales need to be resolved in time Τ π»π ~ π ππ³π π πΤπ π • CPU cost therefore scales like CPU~πΉπππ - very expensive for high Re number flows π 11 © 2018 ANSYS, Inc. Τπ Example: Aircraft • Turbulence eddies determine aerodynamics of aircraft – without turbulence, wings would stall and aircraft would crash • Dimension of aircraft ~ 100m • Dimension (thickness) of boundary layer 10mm10cm • Dimension of smallest eddies ~10-5-10-6m! • In Direct Numerical Simulation of turbulence (DNS) all turbulence eddies would need to be resolved – Resolution problem (up to 1015-1018 cells) 12 © 2018 ANSYS, Inc. Courtesey Center for Turbulence Research (CTR) Published in: Taraneh Sayadi; Curtis W. Hamman; Parviz Moin; Physics of Fluids 2012, 24, Mixing Layer Overview of Computational Approaches • Different approaches to simulate turbulence − DNS: direct numerical simulation • Full resolution • No modeling required → Too expensive for practical flows − LES: large eddy simulation • Large eddies directly resolved, smaller ones modeled → Less expensive than DNS, but very often still too expensive for practical applications − RANS: Reynolds Averaged Navier-Stokes simulation • Solution of time-averaged equations → Most widely used approach for industrial flows 13 © 2018 ANSYS, Inc. DNS LES RANS Resolved vs Modeled scales Creation of Large Scales Energy transfer – Inertial Range Dissipative Range – conversion to Heat DNS LES RANS 14 © 2018 ANSYS, Inc. βπ«π΅πΊ Resolved Resolved βπ³π¬πΊ Modeled Modeled Direct Numerical Simulation • “DNS” is the solution of the time-dependent Navier-Stokes equations without recourse to modeling π πππ πππ ππ π πππ + ππ =− + π ππ‘ ππ₯π ππ₯π ππ₯π ππ₯π − Numerical time step size required, D t ~ t • For channel example ReH = 30,800 βπ‘2π·πΆβπππππ ≈ – Nuber of cells ~ 107 – Number of time steps ~ 48,000 – This is a very small piece of geometry and a very low Re number! 0.003π» Reπ π’π − DNS is not suitable for practical industrial CFD • DNS is feasible only for simple geometries and low turbulent Reynolds numbers • DNS is a useful research tool 15 © 2018 ANSYS, Inc. RANS Modeling • All turbulence effects are modeled − Reynolds Averaging • • • • Transport equations for mean flow quantities are solved Consider a point in the given flow field: All scales of turbulence are modeled Transient solution D t is set by global unsteadiness Introduces additional terms that must be modeled for closure u u'i Ui ui time ππ π, π = πΌπ π, π + π′π π, π 16 © 2018 ANSYS, Inc. RANS Modeling - Ensemble Averaging π • Ensemble (Phase) average: ππ π₯, Τ¦ π‘ = ππ π₯, Τ¦ π‘ + π’′π π₯, Τ¦ π‘ π€ππ‘β 1 ππ π₯, Τ¦ π‘ = lim ΰ· ππ π→∞ π π=1 • For statistically steady flows one can apply time averaging: π ππ π₯, Τ¦ π‘ = ππ π₯Τ¦ + π’′π π₯, Τ¦ π‘ π€ππ‘β 1 ππ π₯Τ¦ = lim ΰΆ± ππ π₯, Τ¦ π‘ ππ‘ π π→∞ 0 17 © 2018 ANSYS, Inc. π π₯, Τ¦ π‘ Deriving RANS Equations • Substitute mean and fluctuating velocities in instantaneous Navier-Stokes equations and average: π(πα π + π’π′ ) π(πα π + π’π′ ) π πΗ + π′ π π(πα π + π’π′ ) ′ π + (παπ + π’π ) =− + π ππ‘ ππ₯π ππ₯π ππ₯π ππ₯π • Some averaging rules: − Given π = Φ + π′ ; π = Ψ + π′ Φ ≡ π; π′ ≡ 0; ππ = ΦΨ + π′ π ′ ; Φπ ′ = 0; π′ π ′ ≠ 0, ππ‘π. • Mass-weighted (Favre) averaging used for compressible flows • For the momentum equation this means that: −π′π π′π ≠ π 18 © 2018 ANSYS, Inc. RANS Equations • Reynolds Averaged Navier-Stokes equations: π −ππ’π π’π ππα π ππα π ππΗ π ππα π α π + ππ =− + π + ππ‘ ππ₯π ππ₯π ππ₯π ππ₯π ππ₯π (prime notation dropped) • New equations are identical to original except : − The transported variables,πΌα π , r, etc., now represent the mean flow quantities − Additional terms appear: πππ = −ππ’π π’π • tij are called the Reynolds Stresses − Effectively a stress term → π ππα π π − ππ’π π’π ππ₯π ππ₯π • tij represent the influence of turbulence on the mean flow and are the terms to be modeled • tij represents a symmetric tensor, so there are 6 additional unknowns 19 © 2018 ANSYS, Inc. RANS vs LES/DNS Real Flow with turbulent structures resolved • RANS is a very strong simplification • All information on turbulence is lost by averaging • Strong need for modeling • RANS modeling can result in signfificant errors in quantities of interest 20 © 2018 ANSYS, Inc. RANS - L RANS – T RANS - U RANS – nt RANS Modeling : The Closure Problem • The components of the Reynolds-stress tensor, tij, are unknown and have to be determined • The RANS models can be closed in two ways: − Eddy Viscosity Models • The components of tij are modeled using an eddy (turbulent) viscosity µt • Reasonable approach for simple turbulent shear flows: boundary layers, round jets, mixing layers, channel flows, etc. − Reynolds-Stress Models (RSM) • The components of tij are directly solved via transport equations • Advantageous in complex 3D flows with streamline curvature / swirl • Models are complex, computational intensive • The additional complexity does not always result in higher accuracy 21 © 2018 ANSYS, Inc. Eddy Viscosity Models • The key concept of the Eddy Viscosity models is the Boussinesq hypothesis • This hypothesis assumes that the Reynolds Stresses can be expressed analogously to the viscous stresses, but applying a turbulent viscosity mt 2 πππ 2 πππ = −π π’π π’π = 2ππ‘ πππ − ππ‘ πΏ − π ππΏππ ; 3 π π₯π ππ 3 1 πππ πππ πππ = + 2 π π₯π π π₯π − Relation is drawn from analogy with molecular transport of momentum (Brownian velocities u”) velocities πππ π‘ππ = −ππ’π" π’π" = 2ππππ − m t depends on turbulence and needs to be determined from turbulence model equations 22 © 2018 ANSYS, Inc. Reynolds Stress Models (RSM) • Also known as Second Moment Closure Models (SMC) • Based on the solution of a transport equation for each of the independent Reynolds stresses tij in combination with the e- or the w-equation • Some of these models show the proper sensitivity to swirl and system rotation, which have to be modeled explicitly in a two-equation framework • RSM models are also superior for flows in stagnation regions, where no additional modifications are required • RSM models are often much harder to handle numerically − The model can introduce a strong nonlinearity into the CFD method, leading to numerical problems in many applications 23 © 2018 ANSYS, Inc. Scale Resolving Simulation (SRS) Models • DNS – Direct Numerical Simulation − All turbulence scales are resolved in time and space − Extremely expensive as Reynolds number increases • LES – Large Eddy Simulation − Resolves larger eddies; models smaller ones − Inherently unsteady, Dt dictated by smallest resolved eddies • Hybrid SRS Models − Combine features of classical RANS formulation with elements of LES method − SBES • Physically based on blend of the RANS and LES models − WMLES − Zonal 24 © 2018 ANSYS, Inc. SRS refers to methods, which resolved at least a portion of the turbulence spectrum in at least a part of the flow domain Impact of Turbulence Models • The usage of RANS models in CFD reduces the required computing power by many orders of magnitude relative to DNS − E.g. For external airplane simulation the reduction is of order ~1015 or larger! • It is not realistic to expect that such a strong simplification will always result in small errors in the solution • Depending on the application, RANS models can introduce substantial errors into the simulation • Errors can be reduced by: − Optimal selection of turbulence model and sub-models − High quality grids and optimal numerical settings − Investment of more computing power by using Scale-Resolving Simulations (SRS) 25 © 2018 ANSYS, Inc. Turbulence Models in Fluent & CFX • A large number of turbulence models is available, some for very specific applications, others can be applied to a wider class of flows with a reasonable degree of confidence • Many of the models available are there for historical reasons • The large number of turbulence models is often confusing to the user and a optimal selection is difficult • In addition, there are many sub-options which can/should be activated by the user in certain scenarios • ANSYS tries to consolidate the model offering as much as possible and set strong defaults • The user community needs to move along and eventually abandon legacy models 26 © 2018 ANSYS, Inc. Summary • Turbulent flows are inherently unsteady, three-dimensional and irregular • A broad range of time and length scales exist in turbulent flows • Turbulent flows are governed by the Navier-Stokes equations, but the need to resolve all scales from the dissipative (Kolmogorov) scales to the mean flow scales makes direct simulation too expensive to be feasible for industrial applications • Reynolds averaging is one of the approaches used to eliminate the turbulence scales. The application of this approach leads to the Reynolds Averaged Navier-Stokes (RANS) equations • The Reynolds stress terms in the RANS equation require modelling in order to obtain a closed system of equations • Scale resolving simulation opens a path to include at least some resolved scales into the simulation 27 © 2018 ANSYS, Inc. APPENDIX 28 © 2018 ANSYS, Inc. Non Dimensional Numbers in Turbulent flows 29 © 2018 ANSYS, Inc.
