Presentation on theme: "Introduction to Computational Fluid Dynamics"— Presentation transcript:
1Introduction to Computational Fluid Dynamics Course Notes (CFD 4)Karthik DuraisamyDepartment of Aerospace EngineeringUniversity of Glasgow
2Contents Introduction (1.5) Classification of PDE, Model equations (1.5)Finite difference methods: Spatial discretization (3) Temporal discretization (2) Convergence, Consistency, Stability (2)Grids/Boundary conditions (1)RANS Equations and Turbulence modeling (2)DNS/LES (1)Best practices in CFD (1)Case studies/Demonstrations (3)Interaction is very important. I’ll take the steering wheel, but let you guys control the throttle.(.) – Approximate number of lectures
3Numerical Simulations : Estimates of cost of fixed wing calculation Sample fixed wing of AR=10, Re=5e6
5Characteristics of turbulent flow Irregularity (Random and chaotic nature of flow)Increased exchange of momentum (Diffusivity - spreading rate of jets, boundary layers etc.)Large Reynolds numbersDissipation of kinetic energy to internal energyWide range of time and length scalesAlmost all practical flows are turbulent.
6Scales of turbulenceTurbulent flows are characterized by a wide range of length scalesThink of turbulent flow as a collection of eddies of different sizesEddies time/length/velocity scalesThe largest “energy containing” eddies are of the order of the length scale of the object that generated turbulence in the first place (vortex shedding from a cylinder, boundary layer thickness)The smallest eddies are the ones where the energy is dissipatedRoughly speaking, there is a “cascade” of energy from the large scale to the smallest scales This happens because the large eddies interact with each other and breakdown into smaller eddiesThe smallest scales are called the “kolmogorov” scales (η). At these scales, the Reynolds number of the eddies is small enough that viscous effects become dominant and the energy is dissipated.
9Time and length scales Kolmogorov (1941) showed that L/η ~ (Re)3/4 T/τ ~ (Re)1/2The problem with DNS of turbulent flows is that you have to simulate (or resolve) all these scales and in 3 dimensionsTherefore, number of points in each direction ~ (Re)3/4Therefore, total number of points ~ (Re)9/4Therefore, total number of time steps ~ (Re)1/2Therefore, total number of operations ~ (Re)11/4Remember, on top of this, you have to do hundreds of operations per point
10RANS and LESDNS is obviously not feasible for high Reynolds number flows, therefore, LES and RANS to the rescueLES: Simulate (resolve) the large scales of turbulence and model the effect of the smaller scales (smaller scales are universal) by adding a subgrid viscosityRANS: Model all scales of turbulenceProblems: LES still impractical for many flows, RANS is inaccurate in many flowsBut: One can get very useful and sometimes accurate, affordable engineering solutions with a good knowledge of the flow and the pros and cons of RANS and LES.
11In SummaryDNS : All scales of turbulence are resolved. Therefore, smallest grid size is of the order of the Kolmogorov scale η ~ L/(Re)3/4LES: Only the “Large” turbulent scales areresolved. The “smaller” scales are modeledRANS : All the turbulent scales are modeledIn DNS, you just solve the Navier Stokes EquationsIn LES, you solve a filtered version of the Navier-Stokes Equations along with another equation to represent the turbulent small scalesIn RANS, you solve the averaged version of the Navier-Stokes equation along with another equation to represent all the turbulent scales [The extra equations are called turbulence model equations. Typical turbulence models are k-ε, k-ω, Spalart-Allmaras, etc]
12RANS Equations – What are they? DecomposeFinally (Substituting & averaging)Where,
13RANSTherefore, the objective in RANS turbulence modeling is to represent the unknown (Reynolds stress) in terms of the knowns (mean flow)Establishing such a relationship is called the closure modelTypes of closure models:- Algebraic models (Zero equation) [Mixing length model, Baldwin Lomax,Cebecci Smith]- One equation [K-model, Spalart Allmaras]- Two equation [K-e model, k-w model, SST]- Algebraic Reynolds Stress Models (ASBM)- Full Reynolds stress closures
14Eddy viscosity modelsTherefore, we determine eddy viscosity at each point and add it to the laminar viscosityClosure problem is therefore, to determine eddy viscosity
15Zero equation modelsLmix is dependant on the problem. Near a wall, it will be a function of distance to the wall and Reynolds number.Models are reasonably accurate in attached flowsVery easy to code upPoor correlation in off design conditionsNo time history effectsFully local.
162 Equation models 5 free constants Simple, can work well in a variety of flows, history effects, more non-localReasonable results in many flows, but separation is a problemVery diffusive, poor convergence near walls
17Reynolds Stress Models Solve directly for Reynolds stressesTherefore, in theory more accurate than other methods – Production term is very important.But still have to model some terms (other equations required for epsilon)Very expensive, convergence is poor near the wall
18DiscretizationLets now look in little more detail about the mesh spacing required in various types of flowsTurbulent flow Wall bounded flow Free-shear flow
19Wall bounded flowsIn reality, there exist eddies of a wide range of length scales in a boundary layer.In DNS, we simulate (resolve) all these eddiesIn LES we simulate some of the “significant” eddies and model the rest Problem is that some of the “significant” eddies are still smallIn RANS we model all the eddies
20Turbulent Boundary layer Knowledge of boundary layer is essential to understand resolution requirements.
21Turbulent Boundary Layer Scaling u+ = u / U*, y+ = U* y / νU* =Inner layer: u+ = y+Outer layer u+ = 2.5 ln(y+)+5.5Only for zero pressure gradient boundary layer on a flat plateInner layerOuter layer
22LES Problem with LES is that the near wall streaks are of size L+ = 1000, W+=20, H+=30. Therefore to resolve this, we needΔx+ = 100 (streamwise) , Δy+ = 1 (wall normal), Δz+ = 5 (spanwise)When you calculate requirements you find out that it is nearly as expensive as DNS!!Note, this is only for wall bounded flowsRemember, in LES, you look to simulate those eddies that are significant. It so happens that in near-wall flows, eddies of this size are significant.In free shear flows, the “significant” eddies are not as small – loosely speaking.
23RANSIn RANS, you are only concerned about the wall-normal direction. In this direction, you need Δy+ = 1 – In the other two directions, grid sizes can be thousands of times larger and hence the savings.Spacing in other two directions is driven by accuracy considerationsRemember, in DNS, only numerical error.In LES and RANS, both numerical and turbulence modeling errors